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Diagrammatic many-body perturbation theory of atomic and molecular electronic structure
DIAGRAMMATIC MANY-BODY PERTURBATION THEORY OF ATOMIC AND MOLECULAR ELECTRONIC STRUCTURE
Stephen WILSON Theoretical Chemistry Depurtmenr,
University
of Oxford, UK
1985
NORTH-HOLLAND
- AMSTERDAM
S. Wilson / Diagrammutic
390
many-body yerturOotion
theory
Contents 1. Introduction ............................................... 2. Diagrammatic many-body perturbation theory ....................... 2.1. Size-consistency of correlation energies ........................ 2.2. Perturbation theory ..................................... 2.3. Rayleigh-Schrodinger perturbation theory ..................... 2.4. Many-body perturbation theory ............................. 2.5. Diagrams. ............................................ 2.6. Second-order and third-order correlation energies ................ ................... 2.7. Diagrammatic expansion for the wavefunction 2.8. Fourth-order and fifth-order correlation energies ................. 2.9. Second-order and third-order terms for closed-shell systems ......... 2.10. Second-order and third-order terms for open shell systems .......... 2.11. Fourth-order terms for closed-shell systems ..................... 2.12. Fourth-order terms for open-shell systems ...................... 2.13. Higher-order terms ...................................... 2.14. Multideterminantal reference functions ........................ 2.15. Valence bond reference functions ............................ 3. Approximate d~agra~lmatic perturbation theory ...................... expansion ............... 3.1. ‘Balanced’ truncation of the perturbation ............. 3.2. Alternative reference wavefunctions and Hamiltonians 3.3. Higher-order terms ...................................... 3.4. Analysis of contemp~)rary approaches to the correlation problem ...... .................... 4. Computational diagrammatic perturbation theory 4.1. Algorithms for perturbation theory ........................... 4.2. Second-order and third-order energy calculations ................. 4.3. The scaling technique .................................... 4.4. Open-shell systems ...................................... 4.5. Parallel computation ..................................... .............................................. 5. Applications. 6. Summary and future directions .................................. Appendix A. Second- and third-order energy expressions for open-shell atoms and molecules .............................................. ........... Appendix B. Fifth-order energy diagrams for closed shell systems Appendix C. Expressions for energy components for a multideterminantal reference function .............................................. Appendix D. Shifted denominator perturbation expansion for the energy through third-order ................................................ References ..................................................
392 392 393 393 395 397 398 403 410 415 417 418 418 431 431 436 448 453 453 454 456 458 458 458 460 467 468 468 469 472 474 476 477 477 478
Computer Physics Reports North-Holland,
391
2 (1985) 389-480
Amsterdam
DIAGRAMMATIC MANY-BODY PERTURBATION MOLECULAR ELECTRONIC STRUCTURE Stephen
WILSON
Theoreticul Chemistr) Received
THEORY
OF ATOMIC
AND
* Department,
Uniuersit~ of Oxford, UK
15 March 1984: in revised form 23 January
1985
The diagrammatic many-body perturbation theory of atomic and molecular electronic structure is reviewed. Attention is centred on the formulation of the method within the algebraic approximation (that is, using finite basis sets) which allows applications to be made to atoms and molecules within a unified framework. The second section is devoted to the basic formalism of the diagrammatic many-body perturbation theory of atoms and molecules. The diagrammatic expansion of both the energy and the wavefunction are considered. the latter being more compact than the former. Section 3 is concerned with aspects of the truncation of the perturbation expansion and the use of many-body perturbation theory in the analysis and comparison of many contemporary approaches to the correlation problem is briefly outlined. The fourth section is devoted to computational aspects of many-body perturbation theory calculations. Some applications are very briefly discussed in section 5.
* S.E.R.C.
Advanced
Fellow.
0167-7977/85/$32.55 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
392
S. Wilson / Diugrummut~c muny-body perturbation
theory
1. Introduction
Diagrammatic many-body perturbation theory forms the basis of a highly systematic and computationally efficient technique for the study of electron correlation effects in atoms and molecules. (See, for example, refs. [l-9] and references therein.) Independent electron models, such as the Hartree-Fock method, account for the major portion, typically 99.5%, of the non-relativistic energy of an atom or molecule. However, the remaining energy, the correlation energy, is, unfortunately, of the same order of magnitude as most energies of interest in chemistry: binding energies, activation barriers, excitation energies, barriers to internal rotation, etc. Over the past twenty years, the diagrammatic perturbation theory has been shown to afford an accurate description of electron correlation effects in atoms and molecules; it provides correlation energies which are as accurate as, and in many cases more accurate than, those obtained by employing other contemporary theories of atomic and molecular structure. Furthermore, the diagrammatic perturbation theory provides a powerful method for the analysis of various contemporary techniques for the study of electron correlation in molecules and yields valuable insight into the relations between them [9]. The pioneering work on the application of the diagrammatic many-body perturbation theory to atomic and molecular systems was performed by Kelly [lo]. Kelly applied the method to atoms using numerical solutions of the Hartree-Fock equations. Many other calculations on atomic systems were subsequently reported [II]. The first molecular applications of diagrammatic perturbation theory were also reported by Kelly [12]. He employed a single-centre expansion and treated the additional nucleus (or nuclei) as a perturbation. This approach is limited to simple hydrides containing one heavy atom which is used as the expansion centre. More recently, the many-body perturbation theory has been applied to arbitrary molecular systems by invoking the algebraic approximation [13], an approximation which is fundamental to almost all molecular calculations and in which the single particle state functions, the orbitals, are parameterized in terms of some finite basis set. A number of reviews and monographs dealing with the application of diagrammatic many-body perturbation theory to molecules have been published in recent years [l-9]. The article is concerned with the application of diagrammatic many-body perturbation theory within the algebraic approximation. It is particularly concerned with the computational aspects of this approach to the calculation of correlation energies. In section 2, the basic theoretical apparatus of the method is given. Various approximations to the perturbation expansion are described in section 3 where the use of perturbation theory in the analysis of different approaches to the correlation problem is discussed and a brief overview of the inter-relation of many contemporary theories is given. In section 4, we turn to the computational aspects of diagrammatic many-body perturbation theory within the algebraic approximation. A very brief review of some applications of the method is given in section 5.
2. Diagrammatic
many-body
perturbation
theory
Diagrammatic many-body perturbation theory provides an expansion for the correlation energy of an atom or molecule the terms in which may be associated with linked diagrams. The
linked diagram expansion is central to the success of the nlany-body perturbation theory in that it ensures that the calculated correlation energies are directly proportional to the number of electrons, IV, being considered. Often the quantum chemist is concerned not with the properties of a single molecule but with the comparison of a number of similar species. Indeed, the chemist is most frequently interested in periodic trends, homologous series and concepts such as fullct~onal group in molecules of different sizes. It is, therefore, important that the theoretical method which we employ to describe such systems yield energies and other expectation values which are directly proportional to the number of electrons in the system. This property, which has been termed size-consistency [14], enables meaningful comparisons of systems of different sizes and also facj~~tates the accurate study of various dissociative and reactive chemical processes.
2.1. Size-consistency of correlation energies A simple illustration of the importance of developing theories of electron correlation yielding energies which are directly proportional to N, is provided by a mode1 system c~nsistjng of n non-interacting atoms, beryllium atoms, say. The linked diagram theorem of many-body perturbation ensures that the correlation energy of the supersystem is equal to n times that of an isolated atom, i.e.
This is illustrated in curve (a) of fig. 1 where the correlation In this figure, curve (b), and estimate of the correlation interaction when limited to single- and double-replacements tal function is also given for comparative purposes. It is well
energy is plotted as a function of n. energy recovered by configuration with respect to a single determinanknown that
where Q is the correlation energy resulting from single- and double-replacement configuration interaction calculations. The curves shown in fig. 1 demonstrate quite clearly that the diagrammatic many-body perturbation theory, in contrast to the limited configuration interaction, is able to handle correlation effects irrespective of the number of electrons in the system under consideration. 2.2. Perturhatim theory In addition to being size-consistent, diagrammatic many-body perturbation theory provides a highly systematic approach to the evaluation of correlation corrections to independent electron models, in that, once the electronic Hamiltonian has been partitioned into a zero-operator and a perturbation, i.e.
394
S. Wilson / Diagrummatic
many-hod,: perturbatron theor?
Fig. 1. Correlation energy for an array of a beryllium atoms obtained theory; (b) single and double replacement configuration interaction.
by (a) diagrammatic
many-body
perturbation
there is a clearly defined order parameter, A, in the expansion for expectation values providing an objective indication of the relative importance of various terms. The total Hamiltonian describing the electronic structure of an atomic or molecular system may be written as a sum of one-electron terms and of two-electron terms $=Ch,
(4)
+ CR,,~ I
’ ‘.J
where h, is the one-electron operator associated with the ith electron and g,, describes the interaction between the ith and the jth electrons. The unperturbed Hamiltonian operator is written as a sum of one-electron operators (5) where /,
is the orbital
operator
defining
the reference
model
/q=h,+u,, and U, is some effective one-electron potential perturbation operator may then be written as 4
= c R,., - c u, ’ 1‘.I
I
such
as the
Hartree-Fock
potential.
The
(7)
S. Wilson / Diagrammatic
many-body perturbation
395
theoq
2.3. Rayleigh-Schriidinger perturbation theory The many-body perturbation theory may be derived from the tion expansion corresponding to the zero-order Hamiltonian (5) For simplicity, we shall restrict our attention to the perturbation a single determinantal reference function. In the non-degenerate tion theory for the total energy of a molecule, the zero-order occupied spin orbital energies, E;
z:
E*=
Rayleigh-Schr~dinger perturbaand the perturbing operator (7). series developed with respect to Rayleigh-Schrodinger perturbaenergy is then the sum of the
(8)
61
i = occupied
and the first-order energy is the expectation determinant, 1G$) that is
value of the perturbing
operator
The sum of the zero-order energy and the first-order energy is the expectation Hamiltonian for the independent electron model wavefunction, that is
for the reference
value of the total
00)
c”,EM=E,+E,. if we use the Hartree-Fock model as an initial approximation to the wavefunction 6 iEM = ‘.4”,, or in a finite basis set b,,, = gsCf. The total energy is given by E = 6i,,
+E,+E,+...
(11)
and, therefore, the correlation energy, with respect to the particular independent employed, is given by the sum of second- and higher-order energy coefficients 6 correlation
=
&
=
0
i--
is the reduced
expansion as
for the correlation
model
energy, with respect to a
resolvent
I@oo)(@ol E,-.&
electron
(14
E* + E3+ . ***
The Rayleigh-Schrodinger perturbation reference determinant QO, may be written
in which 4,
then
(14)
*
The first few orders of the Rayleigh-Schrbdinger
perturbation
theory
expansion
for the correla-
396
S. Wilson / Diagrummuric
tion energy,
therefore,
/‘k,-
muny-ho&
perturbation theog
take the form
PKl~l@K))(Eo-
P’J&I@,>)’
Rayleigh-SchrBdinger perturbation theory leads to an expansion for the correlation energy and other properties which is directly proportional to the number of electrons, N, in the system. To simplify the discussion let us restrict ourselves to closed-shell systems described in zero-order by a Hartree-Fock wavefunction. The matrix element (C+, 12, I tDK) which arises in the numerator in the expansion (15) for the second-order energy is only non-zero, in the closed shell case, if QK is obtained by a double replacement with respect to QO. If QK is obtained by replacing spin-orbitals i and j in QO by spin-orbitals u and h then
where 8 = (1 - (12))r,<’ and (12) is the permutation
operator
which interchanges
the co-ordinates
S. Wilson / Diagrammatic
of electrons
1 and 2. Thus the second-order
marybody
perturbation
theory
397
energy may be written
a, b, c,. . . will be used to denote k,... will be used to denote occupied spin-orbitals unoccupied spin-orbitals. It is clear that if expression (19) were used to calculate the correlation energy of n non-interacting helium atoms, described in zero-order by localized orbitals, an energy would be obtained which would be n times that for a single helium atom; that is E, a: n. In the second term in expression (16) for the third-order energy we observe that E,a n and the summation, which we denote by A and which differs from E, only by an additional denominator factor, also leads to a terms directly proportional to yt. Hence E,A a n2.However, if we compare the diagonal part of the first term in our expression (16) for the third-order energy, i.e. i, j,
with the second
term written
in the form
(21) it can be seen (remembering that GK differs from Cp, by a double replacement) large degree of cancellation between these two terms. Performing this cancellation following expression for the third-order energy is obtained.
that there is a explicitly, the
(22) Each of the terms in expression (22) are directly proportional to hr. Beyond second-order in the Rayleigh-Schrodinger perturbation theory of electron correlation energies, terms arise having a non-linear dependence on the number of electrons in the system. However, such terms can be shown to mutually cancel in each order. This cancellation was first demonstrated in the first few orders by Brueckner [15] and was then shown through infinite order by Goldstone [lci]. By explicitly performing this cancellation of terms in the Rayleigh-Schrodinger perturbation theory one is led to the many-body perturbation theory. 2.4. Many-body ~~r~urb~~ion fheov In diagrammatic many-body perturbation theory, it is recognized that components of the correlation energy which are directly proportional to N may be associated with linked diagrams
whilst terms having a non-linear dependence diagrammatic perturbation theory, therefore, energy.
on N may be described by unlinked diagrams. The leads to the following expression for the correlation
(23) where the subscript L denotes that only terms corresponding to linked diagrams are included in the summation. This is the linked diagram theorem of Brueckner and Goldstone. The linked diagram theorem is of central theoretical importance in the application of quantum mechanical methods to problems of chemical interest. Derivations of the linked diagram theorem of many-body perturbation theory have been given in many text books (see, for example, refs. [17,18]). A time-dependent derivation along the same lines as that first presented by Goldstone 1161 is followed by March, Young and Sampanthar 1171. On the other hand, a time-independent approach, which is perhaps more satisfying since the problem of describing electrons correlation effects does not actually depend on time, is described by Paldus and Cizek in ref. [lg]. In the diagrammatic formation of the many-body perturbation theory the particle-hole formalism is employed. Fig. 2 demonstrates the simple relationship between the particle formalism and the particle-hole formalism. Particles may be created above the Fermi level and holes may be created below the Fermi level. In the diagrammatic notation, particles are represented by upward directed lines and, in the present work, labelled by the indices a, b, c, . . .
whilst holes are represented i
by downward
directed
lines and labelled by the indices i, j, k, . . . (25)
2.5. Diagrams It should be emphasized, at this point, that the use of diagrams in the many-body perturbation theory is by no means obligatory. The whole theoretical apparatus can be set up in entirely algebraic terms. However, the diagrams are both more physical and easier to handle than the corresponding algebraic expressions and it is well worth the effort required to familiarize oneself with the diagrammatic rules and conventions. Furthermore, by using diagram techniques, intricate combinatorial problems which arise in the algebraic approach are replaced by very simple topological considerations. A diagram is said to be closed if it contains no free lines otherwise it is said to be open. Thus, for example, figs. 3(a, c) are closed diagrams; figs. 3(b, d, e) are open diagrams. A diagram containing open lines is said to be connected if it does not contain two or more separate parts. Figs. 3(b, e) provide examples of connected and disconnected diagrams, respectively. A linked diagram is connected and closed; an unlinked diagram contains a disconnected and closed part.
S. Wilson / Diagrammatic
a
b
-__-
0
theory
399
-._--_
----
----------_-----------------. l _ 0 0
C
many-body perturbation
Fermi
level
Fermi
level
_
0
0
~--------------_-_----------~ 0
0 0
0
-_ 0
e
0
0
f
0 0
0
Fermi level ______---_--------------------l l
0
0 __I_-._--... 0 __l_l._...~ _
0
Fig. 2. The particle and the particle-hole formalism. (a) The ground state in the particle formalism; (b) The ground state in the particle-hole formalism; (c) A single excitation in the particle formalism; (d) A single excitation in the particle-hole formalism; (e) A double excitation in the particle formalism; (f) A double excitation in the particle-hole formalism.
Examples of linked and unlinked diagrams are displayed in fig. 3(a) and figs. 3(c, d), respectively. A number of diagrammatic rules and conventions are used in the literature. Here we shall follow the conventions introduced by Brandow [19] and then indicate briefly the relationship between this and other conventions which are in common usage. The basic diagram elements are shown in fig. 4. Downward (upward) directed lines are used to represent holes (particles) created below (above) the Fermi level. Matrix elements for one-electron operators are represented by the diagram elements shown in fig. 4(c). For two-electron matrix elements the diagrammatic representation is given in fig. 4(d). In the Brandow convention the two-electron matrix elements include a permutation of the electrons and are, therefore, sometimes referred to as antisymmetrized vertices. The rules for translating a given Brandow diagram into the corresponding algebraic expression are as follows: (i) Assign a unique ‘hole’ index, i, j, k, . . . to each downward directed line. (ii) Assign a unique ‘particle’ index a, b, c, . . . to each upward directed line. (iii) There is a summation over each hole index covering all occupied spin-orbital indices. (iv) There is a summation over each particle index covering all unoccupied spin-orbital indices. (v) The numerator of the summand consists of a product n integrals in n th order perturbation
400
--..-..---
a
I___
~
-_-__--
~ --L_--
~
-_____
~
Fig. 3. Examples of types of diagrams. Diagrams (a) and (c) are closed and the remaining diagrams are open. The diagrams are described as follows: (a) linked: (h) connected; (c) unlinked; (d) unlinked; (e) disconnected.
theory.
These integrals
may be one-electron
integrals
(26)
S. Wilson / Diagrammatic
many-body perturbation
a hole Zinc
------e
-----b
ow-etactron
line
d
matris alemc2lt Fig. 4. Diagram
two-electron
401
b
particte
C
or antisymmetrized
theory
elements.
integrals
(27) distance. where 4 is spin orbital, ic( r) is some one-electron potential and Y,~ is the interelectronic For each interaction line there is an integral factor. The spin-orbital indices can be read from the labelled diagram. For the two-electron interaction the indices p, q, Y, s should correspond to the hole and particle lines entering or leaving the interaction in the following order: left-in, right-in, left-out, right-out. (vi) The denominator of the summand consists of a product of factors. There are (n - 1) factors corresponding to an n th order diagram. There is a demoninator factor arising between each pair of adjacent lines in the diagram. A denominator factor consists of the terms
where the E’S are orbital energies. The first summation is over all hole lines that extend between adjacent interactions and the second summation is over all particle lines. (vii) There is a multiplicative factor of (i)“‘, where m is the number of pairs of equivalent lines. Equivalent lines begin at the same interaction, they end at the same interaction and the have arrows going in the same direction. (viii) There is a multiplicative factor of (- l)p, where p=h+l,
(29)
h is the number of hole lines in the diagram. 1 is the number of fermion loops. A fermion loop is determined by following the hole and particle lines, in the direction of the arrows, to form a continuous closed loop.
402
S. Wilson / Diagrammatic
many-body perturbation theory
As an example of the application of these rules, consider the following diagram, which involves a triply-substituted intermediate state
fourth-order
energy
(30) The hole and particle
lines in this diagram
may be labelled to give
b
There is a summation over the five hole indices and three particle indices. Working from the bottom to the top of the labelled diagram, the following integrals are obtained in the numerator of the summand:
(34
grjubgklrcgmcklguhm/
and for the denominator
factors,
from the region between
the lower pair of interaction
lines, we
get E, + e,, - E, - Eh from the central
part of the diagram
(33) we have
and from the upper part
k and I label equivalent hole lines and u and b label equivalent particle lines. Thus there is a multiplicative factor of (4)’ = f . There are 3 fermion loops and 5 hole lines giving a factor of 3+5 = + 1. The algebraic expression corresponding to the above diagram is, therefore (-1) (36) Two other diagrammatic conventions are commonly used in many-body perturbation theory. The Gc ldstone diagrams [16] are similar to those of Brandow except that the two-electron interaction lines do not include permutation of the two electrons involved. Thus there is a set of Goldstone diagrams, which are related by electron exchange, corresponding to each Brandow
S. Wilson / Diagrammatic
many-body perturbation
403
theory
og - ----
Q&o @fJ
(yJ
----_----_---
Fig. 5. A set of Goldstone diagrams Brandow diagram the other diagrams
related by electron exchange. If any one of these diagrams are accounted for by using antisymmetrized vertices.
is interpreted
as a
diagram. This correspondence is illustrated in fig. 5. The diagrams of Hugenholtz [20] are in one-to-one correspondence with those of Brandow. The Hugenholtz diagrams can be obtained by replacing the interaction lines in the Brandow diagrams by a single dot. This correspondence is illustrated in fig. 6. It should also be mentioned that some authors rotate the Brandow, Goldstone or Hugenholtz diagram through HIT(see, for example, ref. [18]). 2.6. Second-order
and third-order correlation energies
The complete set of second-order Brandow diagrams for the many-body perturbation theory of the correlation energy developed with respect to an arbitrary single determinant reference function (not necessarily a self-consistent-field function) is given in fig. 7. The one-electron part of the perturbation (U in eqs. (6) and (26)) is represented by a dashed line terminated by a cross.
Fig. 6. Examples
of the one-to-one
correspondence
between
Brandow
and Hugenholtz
diagrams.
S. Wilson / Diagrammatic
404
(B)
(A)
p-g
many-body perturhution
theory
CC)
()__()
----_-
(1))
(E)
Fig. 7. Second-order Brandow diagrams in the many-body with respect to a single determinated wavefunction.
The algebraic
expressions
corresponding
perturbation
to these diagrams
expansion
for the correlation
energy developed
are (37)
E,(B)=
cc=. ‘, ‘I
’
0
(38)
(39)
If real orbitals are used then E,(B) = E,(C). There are 43 Brandow diagrams in the third-order component of the correlation energy with respect to an arbitrary single determinant. The complete set of third Brandow diagrams is given in fig. 8. The algebraic expressions corresponding to these diagrams have the following form:
(42)
(43)
S. Wilson / Diagrammatic E3(C)
=
c
_
405
‘fuUYg/kak
1
(;X_
many-body perturbation theory
u
(Ei-f,)(E;--~)’
(44)
(49) qI)
= _ c
c
U,,g/klkg,~,/
,jkl cd
(50) (51)
E,(K)=
(54
-cc i/k/
a
(53) (54)
(55)
(56) (57)
406
S. Wilson / Diagrammatic
many-body perturbarion theory
(58)
(60) (61) (62) 63)
(64
(65)
w-9
(67)
(68)
(69) (70) (71)
S. Wilson / Diugrummntic
many-body perturbation
theor)
407
(72) (73)
(74)
(75)
Edi) = C C r/k
ub
&/‘,h%kJh (E,
+
E,
-
6,
‘U; -
Eb)G,
-
4
(76)
’
(77) (78)
(82)
o-
(83)
The first 20 terms, corresponding to diagrams (A-T) contain only singly-excited intermediate states. Diagrams (U-j) give rise to expressions which contain both single replacement and double replacements. The last 7 diagrams lead to expressions containing states which are obtained by double replacements.
---_ x -------_ X 0------0 x
40x
S. Wilson / Diagrammatic
(A)
(B)
many-body perturbation
theory
0 -----X ----__
X
CC)
CD)
--_--__ u --_u 0-----\< 0 ------_ u x -----u ---0u a --_.__u 0
(H)
(I)
’
0 -_-.0 -----X 0)
(K)
-0 -__-. 0 -_-_._ 0
-e-m
----
(M)
0
(h’)
(0)
(P)
--_--x 0-___---
0
0 __----
---------_ -------0 0 ---0 0-----u 0----_x CT _---_x (Q)
CR)
(S)
CT)
(W)
03
-e--v
(U)
W)
Fig. 8.
__---_ -_--_ 0 o4 0__---x 0---__-----_ x ---0-----0 -----x 0-----0 --_-x 0----0 _--_ ------__-___ as 00 -----00 __--__ 409
S. Wilson / Diagrammatic many-body perturbation theoq
X
(b)
(2)
(Y)
Cd)
Cc)
K
(9)
(h)
(i)
(j)
(k)
(1)
Cm)
(n)
----_---
(0)
_------
(q)
(P)
Fig. 8. Third-order Brandow diagrams in the many-body with respect to a single determinantal wavefunction.
perturbation
expansion
for the correlation
energy developed
410
S. Wilson / Diagrammatic
man.v-ho& perturbation theoty
v___)( i!LoY--Y (A)
(B)
Fig. 9. First-order Brandow diagrams in the many-body perturbation developed with respect to a single determinantal wavefunction.
2.7. Diagrammatic
(0
expansion
for the correlated
wave function
expansion for the wavefunction
It is well known that the energy through 2n + 1 can be written in terms of the wave function through order n. This is usually referred to as Wigner’s (2n + 1) rule. In diagrammatic many-body perturbation theory Wigner’s rule takes the form [21] J% = N-1
(85)
P+llU_~
E 2n+1= W,l~lG_~
(86)
where the subscript L indicates that only linked energy diagrams are to be included. Wigner’s rule forms the basis of efficient algorithms for the evaluation of the energy coefficients. However, further discussion of this aspect will be postponed until section 4. Here we note that there are a smaller number of wavefunction diagrams in order n than there are energy diagrams in order 2n and 2n + 1. In this sense, the diagrammatic many-body perturbation theory expansion for the wavefunction is more compact than the energy expansion. For example, there are 5 second-order energy diagrams and 43 third-order energy diagrams shown in figs. 7 and 8, respectively, but these diagrams arise from the 3 wavefunction diagrams displayed in fig. 9. Diagrams (A) and (B) in fig. 9 involves states obtained by single replacements with respect to the single determinantal zero-order wavefunction whereas diagram (C) involves states obtained by double replacements. If we represent the state function obtained by replacing spin orbitals i, j, k,. . . by spin orbitals to the diagrams shown in a, b, c,. . . by I @,$T;;- ) then the algebraic expressions corresponding fig. 9 take the following form:
h(A)= q io
h(B)=
w,(*j).
I
(87)
0
+W(~+j)~ IO
I
(88) ‘I
It should be noted that all of the first-order wavefunction diagrams are connected. In order to determine the fourth- and fifth-order energies the second-order wavefunction
is
S. Wilson / Diagrammatic
many-body perturbation
theory
411
required. The 32 second-order wavefunction diagrams are given in fig. 10. Both connected and disconnected wavefunction diagrams arise in second-order. These diagrams involve intermediate states which are obtained by single, double, triple and quadruple replacements with respect to a single determinant. The algebraic expressions corresponding to these wavefunction diagrams have the following form:
(90)
(91)
(92)
(93)
(94)
(95)
(96)
(97)
(98)
(99)
000)
412
S. Wilson / Diagrammatic
(A)
many-hods perturbation theor,
CD)
(B)
c ---x I,: k
--v v v_--vv _Y v x--_-u -----0
I -----
-m-q-
(E)
0
----
(F)
X
0
(11)
o--_ u CL)
V’rJ --
0 -----
(Q)
(R)
\
\p~
L_____
------
X UJ)
(VI
Fig. 10.
(S)
CT)
\
----
-----
/
X
(P)
v ‘I?’ ----_ ----0 0
v _---v ::
I
(0)
i
co
S. Wilson / Diagrammatic
many-body perturbation
theory
413
\/____()\/_LD if__+/ l(J._._y -----_--___- -----(Y)
(2)
(a)
(b)
Cc)
Cd)
(e)
(f)
Fig. 10. Second-order Brandow diagrams in the many-body perturbation developed with respect to a single determinantal wavefunction.
expansion
for the correlated
E,
EC)
wavefunction
giJahgklcl (E;
h(P)
=
+
cj
-
E,
-
Cb)(Ei
+
E, +
Ek -
-
Eb -
’
(103)
005)
c ijkluhc
(106)
(107) 449=
c ijkluh
(108) i
414
S. Wilson / Diagrammatic
&(T) =
c ykatx :
I@$)
(E, + fi _
many-bo&
perturbation theocy
gr~Jocgckhk
009)
i
%&r,kh 011)
013)
(f
i
+ f
(114)
(315)
(E,+f:, 1
_ k
gikhrgb’uk Cb--
C,)(E,
+
f;
-
E,
-
en)
’
0 17)
(118)
gtiohgiklc
i
(fi+‘i-r,-r,)(r,+‘jS.~,--E,--~-f,.)
S. Wilson / Diagrammatic
2.8. Fourth-order
and fifth-order
The fourth-order
many-body perturbation
theory
415
correlation energies
energy may be written
Change in level Interral
-2
type
of
excitation
-1
i +
/
-_ li”u
jb>
--cJ---v 0
+1
-_P J 4 ---‘II’
+2
+
-.__ 4M
ti;iIo^/bc>
i -&
w-_
A
_H
7-
Cij (81 kj>
--0
--of
-0
?4 --a
-r -v--* t
-0
A
_ ._ _ k
-a
0
Fig. 11. Classification
of diagram
components
according
to the change in the level of excitation.
416
S. Wilson / Diagrammatic many-body perturbation theory
and thus using the expression above for $, and I,L~the components can be written down. For example,
of the fourth-order
energy
(123) The matrix element (@,O1J?~ I QJ’) does not involve a change in the degree of replacement with respect to the reference determinant and can, therefore, give rise to terms corresponding to the diagram components displayed in the central column of fig. 11. This can lead, for example, to energy diagrams of the form
Fifth-order energy diagrams can be generated in a similar fashion using the second-order function diagrams displayed in fig. 10 and the corresponding expressions given above formula
,wave in the
For example,
(+,(A) W’IIc/dA)) = c i% ij hd
(125) i
The matrix element (@,! I Xl I @,‘) does not involve a change in the degree of replacement with respect to the reference determinant and again can give rise to the diagram components displayed in the central column of fig. 11. In this way energy diagrams of the form
are obtained.
It should be clear from the above discussion
that the number
of energy diagrams
S. Wilson / Diagrammatic
A__-”
1
__--0
+
+
many-body perturbation
/y(
a
/_____~
theory
417
=
o
x
0
ii!__, +bl--x =o Fig. 12. Diagrammatic representation of the cancellation which occurs if the canonical in the zero-order wave-function for a closed-shell system.
actually
increases
2.9. Second-order
Hartree-Fock
orbital
are used
rapidly with order of perturbation. and third-order terms for closed-shell systems
For closed-shell systems described in zero-order by a single determinant constructed from canonical Hartree-Fock orbitals there are important simplifications. Terms corresponding to diagrams containing “bubbles” are exactly cancelled by terms corresponding to diagrams containing a cross representing the Hartree-Fock potential (multiplied by - 1). This cancellation is illustrated graphically in fig. 12. For example, the following relations are then satisfied E,(A)
+ E,(D)
= 0,
(126)
E,(A)
+ E3(0)
= 0.
(127)
418
S. Wilson
(A)
/
~~u~r~rnrn~ti~
(B)
mango-body perturbutjo~
(Cl
theory
CD)
Fig. 13. Non-zero second-order and third-order Brandow diagrams in the many-body perturbation expansion correlation energy of a closed-shell system developed with respect to a single determinantal wavefunction.
for the
If canonical Hartree-Fock orbitals are employed for a closed-shell system then the only terms which have to be considered correspond to the four diagrams shown in fig. 13 (see, for example, ref. [13]) since all other diagrams shown in figs. 9 and 10 mutually cancel. If the occupied orbitals are subjected to an arbitrary unitary transformation, for example, if some localization procedure were used, then diagrams such as those shown in figs. 8(k, m) would have to be considered. On the other hand, if the virtual orbitals were subjected to a unitary transformation then one has to include the diagrams shown in fig. 8(1, n). 2. IO. Second-order
and third-order terms for open-shell systems
For an open-shell system described in zero-order by a restricted Hartree-Fock determinant, the second- and third-order energy diagrams which have to be considered are shown in fig, 14 (see, for example, ref. [22]). In this figure the solid dot represents a potential whose precise form is determined by the particular electronic state under consideration. A full set of second- and third-order energy expressions for open-shell systems is given in Appendix A. 2. I I. Fourth-order
terms for closed-shell systems
The perturbation expansion for the correlation energy of a closed-shell system through third-order only involves doubly substituted states as can readily be seen from fig. 13. In fourth-order terms, which are obtained by single replacements. double replacements, triple replacements and quadruple replacements, can arise in the energy diagrams. In fig. 1.5 the complete set of fourth-order Brandow diagrams for a closed-shell system, described in zero-order by a determinant constructed from Hartree-Fock canonical orbitals (see, for example, refs. [2,23]), is given. There are 39 diagrams in total; four contain single replacements, twelve contain only double replacements, sixteen contain triple replacements and seven contain quadruple replacements. The algebraic expressions corresponding to each of these diagrams are as follows:
(129)
---* ---0--_-_ 0----0 _--+ --_ 6-----_ 0 (F)
S. Wilson / Diagrammatic many-body perturbation theory
419
0
(D)
(B)
(A)
l
(El
(1)
(G)
(H)
(J)
(K)
CL)
(N)
(0)
(P)
Fig. 14. Non-zero second-order and third-order Brandow diagrams in the many-body perturbation expansion for the correlation energy of an open-shell system developed with respect to a restricted Hartree-Fock single determinantal wavefunction. grJabgkbiJgalcd!?cdkl
(130) E,(D,)
= +c
(E + ~ _ (
gijubgkbrJglmkcgac/m
031) I
Et(&)
=
&c
0 -
J
(E
+ I
~
_ J
E
cb)(ck
~ u-b
-
%h+
E,
-
cu
-
)4i:y:p;bc~~;y~;;(< I
J
cc)
(132)
+ ‘
d
1
’
‘J-‘e-
grJuhgabcdgk/iJgcdkl E4(BD)=
Kc
(
(133)
420
--------- --_------_ - ------ --0 0 0 0 --------------------_---------_---00 ---- c.H? ----_00 --_--00 ------------_----_. ----------_ ------_---_ --_-a0 ----- 00 ----- CJ @O 0 --------------_-_ --_ -__ --00 _-__-0----0_ c!J S. Wilson / Diagrammatic
many-body perturbation
theory
0 --___ __--_ 0 -_---_
0 -----
---_--0
0 ---------0
0 _---_---0
PSI
tBsl
PSI
PSI
b)
ICnl
kD1
kD1
(GDI
PDI
b)
0 _-_ ------.0
(ID)
b)
(BT)
kD1
Fig. 15.
PTI
-----_ -_ _ the --_ -__ _ _ _c)” ---_ _-_ _ @Q9 -----__-_ -tbo S. Wilson / Diagrammatic
many-body perturbation
tETl
(GTI
71-j-
LKTl
theory
(“4
--& ----__ 3 -----
--_-0-b ------_-_ --_ -Qeo 421
(IiT)
--------@a -(iTi
----@Ii0 1’3
(“d
@O
lECl Fig. 15. Fourth-order Brandow diagrams in the many-body perturbation expansion closed-shell system developed with respect to a single determinantal wavefunction.
for the correlation
energy
of a
S. Wilson / Diagrammatic
422
many-hodJ perturbation
theory
(134)
g,.,ohgkl,,g,,,nklguhm,,
E4(DD)= iE (c,+c
-6
,
c
u--h
(135)
)(q+c,-6
G)(%?,
u-
+
-
cu
-
%)
’
(136)
(137)
(138)
(139)
(140)
E4(JD)
=
_
c
(~
+
L
_
E
(141) I
E4wD)= -c
u
I
Eb)(
-
6,
+
6k
-
E,
-
c,.)(
6k
+
6,
-
fu
-
cd)
’
gr,~bgkbc/gulXdg~~drl (c
+E
(142)
_~
I
‘I
J
-
Eb)(
6,
+
6k
-
E,
-
cc)(
c,
+
c,
-
cu
-
cd)
’
(143)
E4(b)= -2
g,,ub~ukcd&wk&dr/ (c
I
(144)
_-E
+c
I
u-
(146)
E4(W= -E
gr.,crh~kl,~gm~~k/guhml +~
(E
1
(147)
_~
I
u-
c,
+
6k
+
c,
-
co
-
-
cc)(
c,
+
cm
-
<,
-
’
S. Wilson / Diagrammatic
+E I
E4(%)
=
+C
-
J
(c
+
E
’
E,
_
-
Eh)(Ci
~
E
J
+
)(c
EJ
many-body perturbation
+
Ek
-
Eh
-
C‘
-
Ed)(
+g~+““:“c’;‘kf;hnlJ
u-h
E,
+
~ ,)(c
f
J
u
h-
423
theory
CJ -
+
<
-
E,,)
049)
’
(150)
E Eb)
n,-co-
,
’
g~/uhgklr~gn,bkJg,,.~,l E4(W
=
c
( cr
+
c., -
E‘,
Cb)(
-
C/ +
Ek
+
6,
-
E,
-
Cb -
E,.)(
E,
+
en1 -
E,
-
E,
)
-
E,
-
Ed)
051)
’
~~Jub~uk~d~lb,,~~~dIk E4(IT)
=
-
aC
(
6,
+
c,
-
cu
-
Cb)(
E,
+
C/ +
Ck
-
Eb
-
cc, -
Cd)(
fk
+
E,
&(JT) = - c (E + c I
053) b-
E )(E+g;;;":"":"dy:bk' (E + ~ _~ ‘<& I LI b' J u-b J
E4(KT)= -c
054
’
054) +
‘h
-
cb -
Ed)
’
giJubgkircguhdjgdckl Eh)(
u-
E,,
+
Ek
+
c,
-
cu
-
Eh
-
c,.)(
Ck
+
E,
-
6‘.
-
Ed)
’
gIJ”h&k
Eh)(
E,
+
E/
+
Ck
-
Eb
-
E,.
-
Ed)(
E,
+
c,
-
E,
-
Cd)
’
-
6,.
-
Ed)
’
grjuhgklrcgrrhdlgdck] E4(W=
tC
(
E,
+
CJ -
E,
-
Eh)(
C/ +
Ek
+
c,
-
ca
-
Cb -
cc)(
C/ +
Ek
055)
056) (157)
(159)
grJuhgklcdgcbilgudkJ
(160)
gi,~rrbgklcdgcdrJgabki E4(BA+CQ)=&z
(E
+~__~
I
J
u-
Cb)(E,
+
cj
-
E,
-
Cd)(Ek
+
6,
-
cu
-
Eh)
’
(161)
424
---------------_ -* -__ -_ -00 -----00 0-i) /so -.__._ ------ -_ 00 0 0 _----_ -_-_. _--__ c!sJ” ---_--------_ _ __ ----_-__ b-O ---fn c!nlb0 ----_ -_ -----. k4J S. Wilson / Diagrammatic many-body perturbation theory
-e --
-----
0
-----
----
(2)
(I)
0
0
(3)
(4)
(7)
(8)
0
--
----
- --
-
(5)
-----
(6)
0
(12)
--_-
-a -_ -
-----_-(13)
-0 _- -
-----
-----
-a
-0
(14)
(15)
(16)
(20)
(22)
Fig. 16.
(23)
-------_ _ __ . -___ --o o _-----_ ----_ ---_o u ----* ao
---- .----0 D 0 a0 0 _---_0D 00 __-_----- l-h----* _--_--__-----_ ---_0 0 u 0 0:0 Q----*I3 S. Wilson / Diagrammatic
-0
many-body perturbation
theory
425
---__L_
0
__-__
-_---
----.
--_-
-_---
----__
----
-__.-.
.
-a
(26)
(25)
0
----
_----
*
(27)
(28)
_--w-e
-we--
*
----_
_-_-
s-e
_--
--_--
-m-v-
--
--__-_
-.
_--
--
*
(29)
(30)
(31)
(32)
0
----
v--e
__-__
_-
-
__-_
---__-
---_-
v-m--
*
----_ a0 (33)
(34)
(35)
(35)
--_ _-_-_ A 0 $3 D D. -----_ --_---_ ----__ __-__ 0-_____@0 0 ----___^__ ___ _-__ -_ __ ---0 c0 0 0 0 0 0 -0
._ - ---
-___-
-_
t
-0
-em_---
_---
-____ j / ___--
__
-___
-_-__
__---
-_-_-_
(37)
0
(38)
(39)
l
V___-__. (40)
_.a-_
0
__-
_-----_
(42)
(43)
-w--D
a ---_
0 -__
_-___ (45)
_
-em--
__
_--___(46)
l
_---__
Fig. 16.
(47)
---0 _______
(48)
426
-_--_ --_--Q a --_--!J 0---_-_-___ 0 ._ -_--_ @---_--0 B o-----_-_-_---
S. Wilson / Diagrammatic
*_--__-
0 --_-_-. ---_----__ _--a
0
*-
_
many-ho&
perturbation
theoq
_ _ ._
V_______ *.-_-
-- 4
_ _ _ _ -----_ -_-----_ -_ *__-*-__.._---0 0 o--0 ______-. --_ -__ -__ __ ___ Q--0 --_ 0 iz __._._~ 0 0--0---_--0 0--_ 0 - - -- _ -.-____-0 (‘..
t-
-
0 ----__. __-_-__ 0
1
(52)
(51)
(50)
(49)
-3 l_ ___ 1 -___---
(54)
(55)
(58)
(59)
0 t---_---0 (56)
*_--
4
(57)
-____-_ -o0 __ -_ --. -----_-_-_-0 0 0 0--0 o(62)
(64)
0 *---__(68)
(G6)
0
_-
D_-_---
0
___--__
(70)
l
/
Fig. 16.
(71)
0 -_--_-_. l
(72)
---_ _---- -___ * _--__-_Q _---. 0 0 Q_ 0 0 t? ______ -_-__ .__ -_ ----__ ___ ^-_ * ---____-_ 0 Q0 ______ 0__--_000o _---_ -_ ----_ --*0 a___ -_ 3 0 a0 o0 ___--____ --_-_ ---_ a3 0 a----_ 0 -. --_ ----0 0 0 0 0 o-S. Wilson / Diagrammatic
e----.-.-.-e _---_ -__--- -
many-hod,:
theor)
421
.._-----
t------
l
-e
.______-V
@
t-----
(73)
\
t-------
-e _-___ __-_--
l
-e *___-_
(79)
(80)
l
(78)
(77)
l --_____,
(76)
(75)
(74)
*_____
perturbation
l -_-___
--se-
_-__ -
-e _.___-_ l _--_-
a
----l ---__-
---a -__----
(81)
--
(82)
(84)
(83)
*-_-_-_
---e ___-@ @__._-___
-e _---. l _-_---_ _
l
(85)
(86)
(88)
(87)
“_____
_-___
--e __-----
_-
-e t--_-_---
e
a---
a___
_----_
----------• --_-_ __----_ -----_ --__-_ 0 _---0 ---_-_ Q o0--_-__0..___ _.__* a _____ 0 (89)
(90)
l _--_----
*-____-
(92)
(91)
e
---_- _ _- ----
e________
l
(93)
(94)
Fig. 16.
(95)
l
(96)
428
--_---*a -----__ _ ----._ ----0 a 0 Q... o:-. ___. _._. ___-* :. __ ----_--* * -__ ._ ___. _ 6-G 0 0 0 0 O----O -------__ -_ __-_-_ -_--\ --_ _-_ 0 0 0 0 0 o-_._ -.-. ----__---_____ --___-_ ---u --_ a o--acl____ _ _ ._ . _--__ --_-* -_ -__-__-_-* _---__ o 0 0 0 u ‘s o0_. ____-_--- -_ _-3 0 w b S. Wilson / Diagrammatic
.I .._._ -e
-0
-_-
many-body perturbation
-3
--- -* - _. __--_ -
-0
-_____-
0
0
--_---
a
-0
(loo!
(99)
(98)
(97)
theor)
0
_-----
-e ______--
0
------
(102)
(101)
0
a
(104)
(103)
-0
0
0
0
-_ 0 ---_-__
_-- l __-__
(105)
-0
e-e_
_-_--
b :------* -_-
(106)
0
--
___ - - . -
---.-
_---_
0
-__---
0
(108)
0
A _- -_---0
e
_--.--
(110)
(109)
-3 ,-_.-e
(107)
-0
_-
--
-e -4
-----
(112)
(111)
0
-_
--
0
-_--
-0
---_---
(114)
(113)
-0
__- ___--
0 --____ 0 (117)
-0
______-_
--_---
--___
----
-
A----
___ ._-_
__.
0 __---_ 0
(118)
(116)
(115)
-0 .- _ ___ . - _ .
Fig. 16.
l
0
--
-1) _ _0
-0
-0
_--
_-_-~-.
(119)
0
_--
_--
-____--
(120)
0
S. Wilson / Diagrammatic
many-body perturbation
theory
429
(123)
(125)
(126)
(127)
__--__ a____-_ -0 l
_---0
__ --_--_--_--_ __ D-*0___ a t-.6 a-----_ Q 0-----o0 (132)
l
0
-a
-_0
------_-------* ------a_-_ *___ * __ 0 0 o0____ 0 ii 0 ----- -_---____ -_-_ * 0 0 a 0----_ t-# (133)
__ 8 0-_
_0
-0
----_
(137)
-0
_--_--
--.
#
--__
b
-------o (141)
-0 --_----
(139)
-0
__ _ --_-0
a_.._
--o
_t
(138)
_---4f
(136)
0
___
-0
_---
(135)
(134)
(142)
-a ---_--_ 0 ._-_
---
---_
Fig. 16.
(140)
---
(143)
l
0
-
(144)
430
__--__*____ 0 S. Wilson / Diagrammatic
many-body perturbation
theory
-0
0 ____
o---_- -_0
-I)
-_-__
--
-e
-_--_--0
----_0 0 Q0 l --_---_
@.j-j)
l ------
---l
-a -_ -0
0
---__-
__-_
(148)
(145)
(,ijj_FD
--___ 00 -__--_
--_-_--
-e
__ __--_-___-__ ____ a Q-_-___ _ ______ --_---* ___-__ --_-_______. Q; _-__0 0 a 0 I) -----_-__ _----_ 0 0 _-___--0__.___ 0----_-_I:---0 ___------_____-_ * --Q_-__ 0 0 o(149)
(150)
(151)
(152)
-0
_-_-___ _-___--
*___--_
D
0
--_---a
--_--__ l (157)
t-------
4 __-__-a
---_____ 0
0 ___ - _ _ -
_-__ *
l
-0 ____--
---_-_ *-__
0
-0
(165)
-0
-0
(162)
-0
(163)
-0 _- _- -. *____
-0
0
-_ _--
-0
_----
8
----_
Fig. 16.
(167)
l --- -__-- ..-
0 -----0
(164)
-0
- __. _--a (166)
(160)
o------
-Q
(161)
*__-_____
-0
_ _ _. - -.-
l
0
(159)
.---_-___
t-------
0
l
(156)
@_________
(158)
l
l
0
(155)
e-------
___-___
-0
-___ ___
-0
___- __0 __--__ l (168)
S. Wilson / Diagrammatic
many-body perturbation
431
(172)
(171)
(170)
(169)
theory
Fig. 16. Additional fourth-order Brandow diagrams which have to be considered developed with respect to a single determinantal wavefunction.
when treating
a-
&(Fq+Gq)= -iC the last three expressions the relation In
to simplify
an open-shell
E
(162)
)’
d
gfJahgktcdgcdilgahkJ Cc
+~
’
_~
J
a-
Q)(
C/ +
Ek
-
we have added the formula
E,
-
Eh)(
6,
+
corresponding
E,
-
cc -
system
Cd)
(163)
.
to two diagrams
and used
the sum.
2.12. Fourth-order
terms for open-shell systems
The additional fourth-order diagrams which arise in the diagrammatic perturbation theory expansion for an open-shell system described in zero-order by a restricted Hartree-Fock wavefunction are shown in fig. 16 (see, for example, ref. [24]). These energy diagrams were function diagrams shown in fig. 10. The algebraic expressions corresponding to the diagrams shown in fig. 16 have the general form
matrix element UP4 = (p, 1u 1q,) or a two-electron ‘factor of the form and D, is a denominator
where X, may be either a one-electron element
gp,q,r,$ = (P,q,l4r,sJ
D, = Dp ,4, . . . ‘,,~,. . . = cp, + c4, + . . . -E,., - es a is a constant
corresponding
...,
determined by the number of pairs of equivalent lines. The algebraic to the energy diagrams displayed in fig. 16 are presented in table 1.
2. I3. Higher-order
matrix
W)
expressions
terms
The fifth-order energy diagrams can also be written down from the second-order wavefunction diagrams given in fig. 10. Some examples of fifth-order energy diagrams are presented in fig. 17.
432
S. Wilson / Diagrammatic
many-body perturbation
theory
Table 1 Fourth-order energy expressions for open-shell atoms and molecules. The diagrams which have to be considered in addition to those which arise in the closed-shell case are given in fig. 16. The first column gives the number of the diagram in fig. 16. The remaining columns give the quantities occurring in expression (165). Values of p, and q, are given for one-electron matrix elements; values of p,, q,, r, and s, are given for the two-electron matrix elements. The final three columns give the subscripts occurring in the denominator factors defined in (166) No. 1
2 3 4 5 6
a
XI
X*
x,
X.4
D,
D,
D,
-1 +a -1
ijab
kbic
dckj
jkcd klbc jkcd
+: -a
ijab
klij ac ac
ijab vab ijab
jkac
ijab ijab
ad ac kbid
gab ijab
klij
cbkl cdkj cbkl
ijab
ijhc
klbc
ki Ii
cdkj cblk
ijab ijab
ijcd ikbc
abed alcj cdG cbil
cdkj cbkl ak ak adkj acjl ci
ijab jab ijab ijab
ci bdik
ijab ia
jkab jkab ijkacd iklabc ijkabc ijkabc ijkacd iklabc ijkabc
jkcd klbc jkcd klbc ka ka jkad jlac ic ic ikbd
bcil adkj
ia ijab
ijkabc ijkacd
abkl adkj abkl
ijab ijab
iklabc ijkabc
-1 -1
ijab ijab ijab ijab ia ia ijab
ahcd akcj ki ki kbcd klcj kjbc jkbc kbcd
+$ -1
ijab ia
klcj jkbc
16 17
+: -1
ia ijab
jkbc kbcd
18 19 20
+$ -1
ijab ijab ijab
klcj kc
arjk ci ci chid
kc
clij
7 8 9 10 11 12 13 14 15
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
+1 -f +1 -a -1 -f
+$ -a +1 -: +1 +1
I 4 -1
I 4 +: -1 +f -1 -4 +1 -f +1 -1 +: -1 +$ +1 -f
bcid blik adkj abkl acjd
Gab ijab ijab ijab ijab ijab ia ia
kbcd klcj kc kc kbic klij jkib jkib
ak ak
ijab ijab
abed akcj
cdej cddk
ia ia
ajbc ajbc
bcde bkdj
ijab ijab ia ia ijab ijab ia ia ijab ijab
abed akcj jkib jkib kbic klij ajbc ajbc kbic
kdij Ibik abed alck acdj abcl kcid klij lk mk
klij
abkd alkj lckj nbkl lbjc lnjk
cdij cbil cdij abil al am aclk ablm ei di deij dcik ck Cl
cdjk cbjk dk ck bdkj bckl aclj abml
ia ia ijab
klab ijbc
ijab
ijkabc
ijab ijab ijab gab ijab ijab ia ia ijab ijab ia in ijab ljab ijab ia ijab qab ia ia ijab qab
ijkacd iklabc
ilbc jkad klab jkad klab
ijkabc ijkabc jkac klab jkab jkab
ijcd ilbc ijcd ilbc la na klac lmab
ijcd ikbc
ie id
ijbc ijbc
ijde ikcd
ijcd ikbc jkab jkab jkac klab ijbc ijbc jkac klab
kc Ic jkcd jlbc kd kc jkbd klbc jlac lmab
433
S. Wilson / Diagrammatic many-body perturbation theory Table 1 (continued) No.
a
4
X*
4
X4
D,
D,
D,
43 44
+1
ijab
ki
aclj
ijab
ki
ablm
ijab ijab ijab ijab
abed akcj
Cl
ac ac
cd cbde ckdj
edij dbik deij dbik
ijab ijab ijab ijab ijab
jkab jkab
jlac
ijab
lbkc lmkj
ijab ijab
kbij abcj
alcd ckid
cl dk
ijab ijab
ka
ia ia
jabi jabi
kijc bkcd
bckl cdjk
ia .ia
ijab
kbij
Ic
ijab
ijab
abcj
kd
ackl cdik
jb jb ka
ijab
ic
klac ikcd
ijab
bj
klic
ackl
ijab
ia
klac
ijab
akcd
cdik
ijab
ia
ikcd
ijab
bj klic
abkj
cl
ijab
jklabc
Ic
ijab
akcd
cbij
kd
jkbc
laji
ijab ia
ijkbcd
ia
dk bclk
ijkabc
klbc
++ -5 -f I
ia
jkbc
badi
dcjk
ia
ijkabc
jkcd
qab
klic
ijab
jklabc
klac
akcd kc
bj nk
ackl
ijab ijab
cdik
ijab
ijkbcd
ikcd
++ I
ijab ijab
kc klic
lbij abdj
aclk dcik
ijab ijab
ijkabc ijkabc
klac ikcd
ackl
bj
ijab
kacd
cdik
jklabc ijkbcd
ia
jkbc
ia
ijkabc
jb ilab
ia
jkbc
lcjk bcdk
bj bali
ijab ijab
jb
++ --i ++ -+ ++ I
gab
klic
cl
daji abkj
ia ijab
ijkabc jklabc
ijad jkab
ijab
kacd
cl
cbij
ijab
ijkbcd
ijbc
ijab
kc
lcik
abrj
ijab
ijkabc
jlab
72 13 14 75 76
+f
kc kbic abed
acdk ak
dbij cj
ijab ijab
ijkabc jkac
jc
4 bcki cdji
ijab ia ia
ijcd ijbc ijab
jd ijcd ijcd
77 78 79 80 81
++ -1
ijab ijab
jb jb klij akcj
ci kajc
+:
ijab ijab ijab ia ia
bl bk
uab ijab
klab ikbc
+f -1 +1 -f +1 +1 +1 -1 +1 -1 -1
jb jb ki ac jkbi jabc ki ac jkbi jkbi ki
bakl cajk bl bk klak dcji 61 bk balk cajk ackj
ia ia ijab
82 83 84 85 86 87 88 89
ia 9a ijab ijab ia ia ijab ijab ia ia ijab
ijab ijab jkab ijbc jkab ijbc jkab ijbc jkab jkab jkab
lb kb klab jkac lb
45 46 47 48
I 4 +a -1 +$ -1
49
I
50 51
++ I
52
+4 -+ ++ ++ +f -f +j I
53 54 55 56 51 58 59 60 61 62 63 64 65 66 67 68 69 70 71
-1 +f -1
bacd ak ci krji bkci alkj ckij !i bd alkj ckij G bc bc
ijab ia ia ijab ijab ia ia ijab
ijcd ikbc ijbc ijbc ic
lmab ijde ikbd ijde ikbd Ic kd klbc ikcd
ijbd
kb klab ijcd lb kb klab jkac jkac
434
S. Wilson / Diugrammatic
many-body perturbation
theory
Table 1 (continued) No. 90 91 92 93 94 95
cl
t-4 -1 +i -1
-kj
Xl
X2
x3
x4
DI
D2
D3
ijab ijab
ac bc
bd ki
cdij ackj
ijab ijub
ijbc ijac
ijcd jkac
ijab ia
kj jkib
li bc
ablk ck
ijub ia
klub kc
ia ia
ajbc jucb
cd
dj ck
iu iu
ikab jkab ijhc ijbc
jd kc lb
96
-1 +t
kjbi
bl
ia
Jkab
97
4-4
ia ijub
kj Ik
kbij
NC
ul
ijnb
ka
la
98
+4
ijab
abcj
cd
di
ijab
ic
id
99
kitrb Qcd la id k/ah ijcd j/ah
+f
ia
ji
kljh
abkl
ia
100 101 102 103 104 10.5
++ .+ 1 +1 -i-l +1 +i
iu ijub @b ia ia ijah
ah ki ac jkih ujbc ki
bjcd lbkj cbdj
cdij al di ul dcij
IL1 ijub rJub ia ia
ah/j
ijab
ju ib jkab ijbc jkab ijbc jkab
106
+A
ijrrb
107 108
+i -t+
1U
ac jkib
ijab ia
tjbc jkub
r;ihd la
ia
ujbc
lbjk bcdj
dbij al
109
+i
ijub
kbij
ac
di ck
ia gub
ijbc ka
id kc
110
-2 _ i
ijab
ubcj
ki
ck
ijub
ic
kc
ia
ab
jkic
ra
ib
_I
jb
ia
ji
ukbc
bcjk bcjk
iu
- ._;
ijab
ki
abcj
ck
ijab
ja ,jkab
jb kc
114
-4
ljuh
UC
kbij
ck
ijub
ijbc
kc
115
-f
ia
ajbc
ki
bck;
ia
ijbc
kb
116 117 118 119 120 121 122 123 124
- 1 -1 -1 -1 -1 -1 -1 -1 - 4
iu ia iu ijab ijab ia ia ijab ijub
jkih jkib ajbc kbcj kbcj
ac ac ki ci ak bjic akjc cbij abkj
cbjk
ia ia ic rJub ijab ia ia ijab
jkub jkab ijbc ikac ikac ijab ijab ijkabc ijkubc
jc jc kb ka I(’ jkuc ikbc ka ic
jkbc jkhc
hi
acjk
ijkabc
jkac
- .;.
ia ia
bcik
-L -$ - t
ijab gab
kc kc kc jkbc
aj ci ak uk bcik
111 112 113
125 126 127 127 128 129 130 131 132 133 134 135
-$.
- .; -+ +1 +1 il +1 +1
(jab ra ia ijab ia ijab ia ijab
jb jb kc kc
jkbc ai ajib bkjc jb kc
ci bd Ik cd
acjk bkjc kc ai akic cbkj
:k ak ci acjk bcik ak ci
abkj ebij rbij aj hi ck bcjk ck bcjk ai
ijab ia ia ijab ijab ijab ia ia ijub ia ijab ia ijab
ijkabc
ikbc
ijkabc ijkabc ijkahc ijkabc
jkab (jbc ijbc
ijkabc jb .ib ikac ijab ijkabc
ja ib kc jkbc kc jkbc ia
S. Wilson / Diagrammatic
many-body perturbation
theory
Table 1 (continued) No. 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172
+1 +1 +1 +1 +1 +1 +1 -1 -1 -1 +1 -1 +1 +1 +1 -1 +1 -1 +1 -1 +1 -1 -1 -1 +1 -1 +1 -1 +1 -1 +1 +1 -1 -1 -1 -1 -1
ia ia ijab gab ijab ia ia ijab ijub ia ia ijab ijab ia ia ijab ijab ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia
jkbc
bj
jb bkjc kc ai ajib kjcb
bkjc ck ai kc bkjc abij ki
bj bj ji ab ki ac jb jb ki ac jb jb jabi ja ji ab jc jb ji ab ji ab jb jb ji ab ji ab jb jb
2 jc bj ti ki ac ak ci kj bc kj bc akjb bjic kaji baci aj bi bk cj kbij abcj kj bc ah ji bi ah
acik acik ai bcjk bcjk ck ck ak ci abjk bcij ak ci abkj cbij hj hj baki caji bk :k cj ck :k cj aj bi ak ci ak ci bj bj aj bi
ia ia ijab ijab ijab ia ia ijab ijab ia ia ijab ijab ia ca ijab ijab ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia
ijkabc ijab ikac ijkabc jb jb ijkabc ia ia ja ib jkab ijbc ijab ijab jkab ijbc ijab ijab jb jb ja ib ijrrc ijab jkab ijhc jkbc ijbc ijab ijab
ikac ikac ia jkbc jkbc kc kc ka ic jkab ijbc ka ic jkab ijbc jb jb ikab ijac kb jb kb kc kc jc kb jc
ja ia
ja ib ka ic ka ic
ja ib ijab ijab
jb jb ja ib
A full set of fifth-order energy diagrams for a closed-shell system are given in appendix B. It should be emphasized that the number of diagrams increases rapidly with the order of the perturbation expansion. Sixth- and seventh-order energy can be obtained according the Wigner’s (2~ + 1) rule from a knowledge of the third-order wavefunction diagrams. As an example of third-order wavefunction diagrams the complete set of such diagrams which have to be considered for a closed-shell system, described in zero-order by a single determinant, is displayed in fig. 18.
S. Wilson / Diagrammatic
436
Fig. 17. Examples
2.14. Multideterminantal
many-body perturbation
of higher-order
Brandow
theory
diagrams.
reference functions
The diagrammatic perturbation theory which we have considered above can only be applied when the perturbed wave function can be described by a single determinant. Let us now consider the use of a reference function which consists of a linear combination of determinants
IV =
c IqJCp7
(167)
PGS
where S denotes the “model” space spanned by the set of reference determinants 1Qp). The zero-order space is multidimensional and quasi-degenerate diagrammatic perturbation theory must be used. The problem of obtaining a many-body perturbation theory with respect to a multideterminantal reference functions has been addressed by a number of authors. Brandow [19] gave the first complete derivation of quasi-degenerate diagrammatic perturbation theory [25,26]. The orbital basis set is partitioned into core orbitals, valence or active orbitals and virtual orbitals as illustrated in fig. 19. The core orbitals are doubly occupied in all of the determinants, 1BP), included in the model space, S; the valence orbitals are sometimes occupied and sometimes unoccupied in S and the virtial orbitals are always unoccupied in S. The total Hamiltonian is
S. Wilson / Diagranvnatic
_-_
--_
- ---.
~
many-body perturbation
-----I_
0 D -_------_~
theory
S, Wilson / Diagrammatic
----_
many-body perturbation
theory
~
__--_-
_-__--__
b
0 ----_---~ -_---_ -----_ ~ ~-----0 ~------_--_ ------_- ---_-_ ~-----~ ~----~ _---. __ -__-~ ~ ~-___-~
~
-A_---
a-_-
------_
--
---
--
~
-------
_----
__---
_--_--
~
~
w-s_
-----
~
-_
_
-----
~
-_
m--e--
~
~
-_----
--em---
__-_-
-----
~
~
__-----_
I_---
---_-_
~
------
-._
---_---
~
~ ~ -----
Fig. 18.
~
-__---
-
~
S. Wilson / Diagrammatic
many-bode perturbation
Fig. 18.
theory
439
S. Wilson / Dia~ra~~~atic many-body perturbation
440
theory
---
~~_---
~~ -..-Fig. 18.
The projector @=
onto the model space is (170)
c I@JC@J P=s
and
(171) For as exactly degenerate AE=b--
may be written (172)
E,,
where d is an eigenvalue p=
model space with energy E,, the energy correction
c t$)@i !JEs
of 2.
For a quasi-degenerate
model space L%$ and 2,
are shifted by 073)
- EJPJ
that is 074) The modified
problem
%/fpIL)
=KI@p)
is exactly
degenerate.
0751 A E is an eigenvalue
I P}
is the exact wavefunction
sif = &9$@+ @ is the Green’s 9=
.&W,@2,9. function
59()+ 9&%$,
operator,
b’, 076)
I??+1 9) = A E&l 9)) where
of the reaction
and 077)
of .Y? (178)
S. Wilson / Diagrammatic
many-body perturbation
theory
v v v-vHI v v ---vv‘i__vvv _--_ v ----vl/v V___Y \ / v----v ----_ -----------_ 0 ----_-_ D
\1 ----------
v-_ ----_ -----
v-_--_-----Y
-----
v-- - ----
v-- -_ --_------v -----_ -----
441
442
S. Wilson / Diugranmutic
many-body perrurhotion
rlzeorj
S. Wilson / Diagrammatic
many-body perturbation
443
theory
Fig. 18.
where @‘Ois the Green’s
function
j2=&+fi
of &.
In the case of quasi degeneracy,
the operator
[25] (179)
can be defined,
satisfying
li2@]\k) =(AE+
ti)B]\k).
(180)
Almost all formulations of quasi-degenerate many-body perturbation theory require the use of a model space consisting of determinants corresponding to all possible occupations of the valence orbitals - termed a complete model space. This can lead to the situation in which determinants which are not in the model space have an energy significantly below the energies corresponding to some of the highest states included in S. These @;‘s are called intruder states. Their presence can impair or even completely destroy the convergence of the quasi-degenerate perturbation expansion. There are three possible solutions to this problem: (i) Formation of matrix PadC approximants to the expansion for the reaction operator or effective interaction operator [9]. For example, in third order the expansion P [ 3/o]
= Q2, + Q, + Q* + 9,
(181)
Fig. 18.
S. Wiison / Lkgrammaric
many-bogv perturbation rheory
Fig. 18.
446
S. Wilson / Diagrammatic
many-body perturbation
theory
\I_-v
I:i ---_
-
v -11;‘v -_-1 w -__ ul
---
v -_ v-----__ v --
--
Fig. 18. Third-order wavefunction diagrams for a closed-shell system described constructed from canonical Hartree-Fock orbitals. Only those arrows which diagram are shown.
v--_--Y
in zero-order by a single determinant are necessary to uniquely define the
S. Wilson / Diagrammatic
many-body perturbation
theory
virtual
orbitals
valence
orbitals
441
-___-__-__------
--__-------------
core
Fig. 19. Partitioning
is replaced Q[2/1]
of the orbitals
in quasi-degenerate
orbitals
diagrammatic
perturbation
theory.
by = 9, + !& + &(I - w,‘fiJ’.
(182)
Such PadC approximations have been found to be useful in studies of effective interaction in nuclei in which the problem of intruder states also arises [27]. (ii) Including th e intruder states in the model space. This, of course, leads to an incomplete model space which can also be much larger than the initial complete model space. It can also destroy the size-consistency of the method. (iii) Excluding the higher energy configurations from the model space. Again this may lead to a loss of size consistency. Let us consider the diagrams which arise in the calculation of the matrix elements of fi, (@, 1h 1a]). The diagrammatic rules and conventions of Hose and Kaldor [25,28,29] will be followed. In the work of these authors the orbitals are divided into holes, which are occupied in @J,, and particles which are unoccupied. The diagonal elements ( Qi 1b I@;) lead to the usual Goldstone diagrams and to folded diagrams (discussed below) which may be derived from them. Off-diagonal matrix elements (@, I h I @,) i #j lead to open diagrams which can also be folded. Open diagrams contain incoming and outgoing lines describing the transition from @, to a,. Open diagrams may be derived by “cutting” Goldstone diagrams in all possible ways. Fig. 20 gives all of the second-order diagrams to be evaluated in Qi and @, differ by one orbital. The algebraic expressions corresponding to these diagrams are given in Appendix C. In figs. 21, 22 and 23, the second order diagrams in which Qi, and @, differ by two, three and four orbitals, respectively, are displayed. Internal lines are summed over hole and particle orbitals, defined with respect to @,. No model state, or linear combination of model states, may appear as an intermediate state. If a complete model space is used or if the model space is exactly degenerate, there are no unlinked diagrams in the expansion. When an incomplete quasi-degenerate model space is employed then unlinked diagrams can occur [25]. The third diagram in fig. 21, for example, is unlinked, but none of the two parts of this diagram can represent a transition connecting Qi and @].
448
Fig. 20. Second-order diagrams in quakkgenerate one-or&taI.
diagrammatic perturbation theory far configurations differing by
Johnston and Baranger [30] have given a detailed analysis of folded diagrams which arise in quasi-degenerate diagrammatic perturbation theory. In the scheme of Hose and Kaidor, the folded diagrams are generated from the unfolded ones as follows: ‘“A group of lines (particles, holes or bath) is designated ‘“folded” if a simultaneous cut through them produces two vertically overlapping parts, each of which is a legitimate open diagram in the model space, This process may be repeated”. The type of folded lines, holes or particles, is determined by the requirement that particle lines enter the portion of the diagram containing the highest interaction line and hole lines enter the other part, Thus there are equal numbers of hole and particle lines in a fold and the line directions are as in the parent diagram, A double arrow notation is employed with the upper arrow giving the line type and the tower arrow the line direction. In fig. 24, an example of the fording of a closed diagram is presented. In fig. 25, the folding of an open diagram is illustrated. When evaluati~l~ folded diagrams, the usual summation rules apply for the unfolded lines. For the folded lines hole and particle ‘fines are summed over transition hole and particle lines corresponding to aXI possible transitions from the vacuum state of the relevant matrix element to all other model states. In the evaluation of the denominator factors the rules given previously again apply; folded lines are classified accarding to their upper arrow and external lines coals into the diagram are always downgoing. The overall sign factor is
where h is the number of hok lines, i~~I~ding folded lines,I is the number of loops after any open incoming lines are connecting to outgoing lines in ascending order or orbital labels and f is the number of folds. The rules outhned above, enable the matrix of b to be evaluated in the model space. It should be remembered that the di~gona~izatio~ of this matrix may introduce size inconsistency. 2.15. VaImx bond refermcefu~c~~~~~ A correct description of many dissociative processes can be obtained by using a valence bond reference function in diagrammatic perturbation theory. Furthermore, this could be achieved
-_-
--
~~_-__
~
3-__
\JJ
~
--_ ~ ~ - ._ _ _ .- ~ Fig. 21. Second-order diagram in quasi-degenerate ~gr~mmatic perturbation theory for c~nf~guratjo~~ fering by two orbitals.
~~~ Fig. 23. Second-order four orbitals.
diagrams
didif-
Fig. 22. Second-order diagrams in quasi-degenerate diagrammatie perturbation theory for c~~figur~t~~~~ diffeting by three orbit&.
-...--_ v ____
in quasi-degenerate
diagrammatic
perturbation
theory
for configurations
differing
by
450
----- ---__----&jfo =+ ()(yo @@?o __ -_ -S. Wilson / Diagrummutic
Fig. 24. Folding
many-body perturbation
theory
of a closed diagram.
without having to resort to multideterminantal reference wavefunctions and thereby suffering the attendant lost of physical interpretability of the correlated wavefunction and energy expansion. The work of Gerratt [31-351, Gallup [36] and Goddard [37] suggests a possible reference function. Newman [38,39] has examined a many-body perturbation theory formalism in which biorthogonal sets of orbitals are employed and the use of reference functions constructed from non-orthogonal orbitals in perturbation theory has also been discussed by Kvasnicka [40], Moshinsky and Seligman [41], Gouyet [42], Cantu et al. [43] and Kirtman and Cole [44]. A brief outline of the generalization of the formalism of Brueckner and Goldstone, to the case of non-orthogonal basis states is now given. The total Hamiltonian is, of course, Hermitian, but
_-
-_
-_
-_-
3 -_
-_ ~~~
~~~_--_
I__--
Y
--
_~~-~--Fig. 25. Folding of an open diagram
S. Wilson / Diagrammatic many-body perturbation theory
may be split into components 3P=&+.&
451
which are not Hermitian
3&z*,
(184)
%ld%n) = Gh#%,)~
(185a)
(we=~,(~“I~
(185b)
with WV%,>
(186)
=L?. state wavefunction,
If the true ground
-wel)
\k,, satisfies
=~oI%)
and the intermediate
(187) normalization
condition
Pol+o,) = 1 is assumed, &I=
(188)
then
P%I%I+o,)
(189)
and Ecl= WoPwo)
= Pol‘%lGoo)
+
ew%l+o,)~
(190)
and the level shift is given by AE,=IE,-E,=(~~I~~I~~). Defining
the projection
@= r+ox+”
(191) operators
I
(192)
and (193) and following a treatment analogous to that given for the case where the zero-order is Hermitian leads to an expression for the wavefunction
Hamiltonian
094)
S. Wilson / Diagrammatic
452
many-body perturbation
theory
and for the level shift L AEo=(+‘I Further
“+
f ,I = 0
iteration
)n*,+o).
can then be performed
AE 0 = E”’ + Ec2) + Ec3’ + 0 0 0 which then leads to “linked
095)
0
0
for the level shift
. . .
diagram”
096) expressions
which are denoted
by the subscript
L
J%“=
(+“l&l+o)~
097)
Et’=
(+“l%l+o)~~
(198) 099)
(200) If a biorthogonal basis sets is introduced, creation and annihilation operators can be introduced which satisfy the usual anticommutation relations but are no longer Hermitian conjugates. Given a basis set x its biorthogonal complement is
(201)
@=A-‘x9 A,,= (x,Ix,>. The covariant
creation
and annihilation
operators
[a,, a,]+= 0, a; [a:, 1+=o, [a,?,a,1+= A,j. The corresponding ai+=
CA’Jq!,
obey the anticommutation
relations
(202) (203) (204)
contravariant d” = (A,,)-‘,
operators
are (205)
.i a’ = -&YJa,
(206)
.I
which obey [
a’+, a,] += [ ai, a;]
[a’+,
a']+=alJ.
= sj,
(207) (208)
S. Wilson / Diagrammatic
In terms of these operators
the total molecular
many-body perturbation
Hamiltonian
453
theory
may be written
The diagrammatic expansion of the Rayleigh-Schrodinger theory with a non-Hermitian partition of the total Hamiltonian operators is topologically equivalent to the usual one. In the algebraic expressions corresponding to these diagrams each summation is over one corresponding to these diagrams each summation is over one covariant and one contravariant index. For example, in second order, the diagram ___(210) 00 _-__ leads to the algebraic
expression
3. Approximate diagrammatic perturbation theory 3.1. ‘Balanced’ truncation of the perturbation
expansion
Diagrammatic many-body perturbation theory forms the basis of a most systematic technique for the evaluation of correlation corrections to independent electron models of atomic and molecular structure in that, once the total Hamiltonian has been partitioned, according to eq. (3) into a zero-order operator and a perturbation, there is a clearly defined order parameter, A, in the expansion which affords an objective indication of the relative importance of the various
Table 2 Second- and third-order energy coefficients for some small atoms and molecules obtained using the Hartree-Fock model perturbation series; The self-consistent-field energies are in hartrees and the correlation energies are in mhartree. Valence correlation energies are given. The basis set and nuclear geometries used are given by Guest and Wilson [45] System Ne FH H,G NH, N, co SiS
E scf
4
- 128.54045 - 100.06007 + 76.05558 + 56.21635 - 108.97684 - 112.77551 + 686.48488
-
210.784 223.633 220.304 198.600 326.887 300.509 206.813
6
E[2/11
+ 0.753 + 1.212 - 3.108 - 11.003 + 9.201 + 5.083 + 16.878
-
210.034 222.427 223.457 210.345 317.938 295.510 225.190
S. Wilson / Diagrummutic many-ho& perturbation theory
454
components of the correfation energy expansion. Second-order energies often yield rather accurate estimates of the correlation energy. Through third-order the correlation energy of a closed shell system is represented by just four diagrams, as shown in fig. 8, if antisymmetrized vertices and a Hartree-Fock reference function are employed. Some typical second- and third-order correlation energies are displayed in table 2 [45]. It has been shown that, provided a ‘balanced’ procedure is followed and all terms through third-order are evaluated, third-order calculations are accurate and very competitive with other approaches to the correlation problem. The dominant error in such third-order calculations is almost invariably attributable to the truncation of the basis sets rather than the neglect of higher-order terms in the perturbation series. There are two possible ways of improving calculations based on third-order perturbation theory developed with respect to a Hartree-Fock wavefunction and Hamiltonian: (i) by employing an alternative reference wavefunction and/or Hamiltonian in order to obtain an improved approximation to the correlation energy in third order; (ii) by extending the perturbation series to fourth and higher orders. We consider these two options in turn. 3.2. Afternative reference ~~~~ef~n~tions and ~amiit~~ians 3.2. I. Shifted denominators In the expressions for the components of the correlation energy of ^ a closed-shell system given in section 2, the N-electron Hartree-Fock model Hamiltonian, yi”*, was used as a zero-order operator. This leads to the perturbation series of the type first discussed through second-order by Moller and Plesset [49,50]. Flowever, it is clear that any operator 2 satisfying the commutation relation
[J& q =o may be used to develop a perturbation operator which satisfies (22) is %hifted
=nwww(kl~
(212) series using
the same reference
wave function.
One
(213)
in which the single determinant ]k) is an eigenfunction of 203 and this gives rise to the shifted denominator, or Epstein-Nesbet 151-531, perturbation series. The resulting perturbation expansion has the same diagrammatic representation as that given in section 2. The corresponding algebraic expressions are also as given in section 2 except that [2,9]: (i) the denominator factors are each shifted by an amount
(ii) diagonal scattering terms are omitted in third (and higher) order. The use of shifted demoninators may also be interpreted as the summation to all orders of certain diagonal terms in the perturbation series based on the Hartree-Fock model Hamiltonian. A full set of expressions for the energy coefficients in the shifted denominator perturbation series are given in appendix D.
S. Wilson / Diagrammatic
man.v-hodv perturbation
455
theor)
3.2.2. Pad& approximants and scaling of the reference Hamiltonian Following Feenberg [54], let us consider two modifications of a given zero-order operator, $O; namely, a uniform displacement of the reference spectrum and a uniform change of scale. This defines the new zero-order Hamiltonian as follows:
The perturbation
is then
2iy.y =.2P,+ (1 - p).2&- Yi, So that the full molecular
2= (3ey-t
Hamiltonian
(216) is recovered
E, +(l
that is
h.Py)x=l.
The energy coefficients, El”,“, where i denotes those of the original series by the expressions
El.“=
on adding (215) and (216)
-/A)E~-Y,
(217) the order of perturbation,
may be obtained
from
(219)
(220) It can be shown [55,57] that the [M + l/M] Pade approximants, alone among all Pade approximants of order 2M + 1 (including the [2M + l/O] PadC approximant), are invariant to arbitrary choices of p and V, when A is set to unity. The sum of the perturbation expansion, with respect to given reference wavefunction, through infinite order is, of course, independent of the choice of zero-order operator. The degree of agreement between two perturbation expansions based on different %‘“‘s is, therefore, a qualitative measure of the convergence of the series. In table 3, third-order results are presented for a number of small atoms and molecules obtained using both the Hartree-Fock model perturbation series and the shifted denominator series. For each of these expansions, [2/l] PadC approximants are formed. It can be seen, from this table, that for systems which are well described by a single determinant, such as Ne, FH or NH,; the third-order model perturbation series E[3/0] is in close agreement with the [2/l] PadC approximant formed from it, whereas the shifted denominator series leads to a significant change on forming the [2/l] Pade approximant; although the [2/l] PadC approximant is in close agreement with that constructed for the model series. For systems such as the BH, BeH, and BH, molecules, which involve some degree of quasi-degeneracy, the results obtained from the model perturbation series change significantly on forming the [2/l] PadC approximant and, furthermore, are thereby brought into closer agreement with the shifted denominator results E[3/0] and E[2/1] which for these cases are in close agreement with each other.
456
S. Wilson / Diagrammatic
many-body perturbation theory
Table 3 Comparison of third-order diagrammatic perturbation theory calculations for some small atoms and molecules obtained using the Hartree-Fock model perturbation series and the shifted denominator perturbation series; All correlation energies are in mhartrees. The basis sets are nuclear geometries used are given by Guest and Wilson [45]
A, is the percentage difference between E[3/0] and E[2/1] for the shifted denominator expansion, A,s,al and A,2,,,I are the difference between E[3/0] and E[2/1] respectively in the model and shifted denominator perturbation expansion System
Ne
BH FH BeH, H,G BH, NH, HZ co SiS
Model perturbation
series
Shifted denominator
&3/O]
E12/‘11
A,
- 210.031 - 80.615 - 222.421 - 64.596 -223.413 - 116.931 - 209.600 - 317.686 - 295.679 - 223.691
- 210.034 - 86.731 - 222427 - 68.623 - 223.457 - 121.471 - 210.245 - 317.938 - 295.510 - 225.190
0.0 7.3 0.0 6.0 0.0 3.8 0.3 0.1 0.1 0.7
(%I
perturbation
series
Ef3PJl
EP/fl
A, (%)
- 201.253 - 89.893 - 201.262 -69.181 -201.657 - 122.542 - 202.576 - 257.443 - 269.302 - 205.403
- 209.472 - 89.893 - 221.208 - 69.187 - 221.856 - 122.593 - 209.543 - 310.457 - 293.641 - 222.582
4.0 0.0 5.1 0.0 5.2 0.0 3.4 18.7 8.6 8.0
A[,,,, (%)
AIVll (%I
4.3 10.9 5.6 6.9 5.9 4.7 3.4 20.9 9.3 8.5
0.3 3.6 0.5 0.8 0.7 0.9 0.3 2.4 0.6 1.2
3.2.3. Alternative reference wavefunctions The modification of the perturbation expansion considered above have not departed from the use of the canonical molecular orbitals, both in the reference function and in the excited configuration functions. Let us briefly consider the use of orbitals other than the Hartree-Fock canonical orbitals. If electron-electron interactions are completely neglected, i.e. the bare nucleus model, in zero order, we then have to consider in the corresponding diagrammatic perturbation diagrams which then arise are series diagrams which contain ‘bubbles’. The second-order displayed in figs. 7(D) and (E) and the third-order diagrams are shown in figs. 8 (0, P, T, W, X, b, e, f, j, m, n, o, p, cl). If instead of using the canonical Hartree-Fock virtual orbitals, we use the orbitals corresponding to the so-called F/“‘-’ potential [58,59], then the diagrams shown in figs. 8(1, n) have to be considered the ‘cross’ representing the VN-’ potential. It has been demonstrated [59] that results obtained by using the VN-’ potential are in fairly close agreement with those corresponding to the Hartree-Fock, VN, potential if all terms through third order are included. Similarly, if localized Hartree-Fock orbitals are employed in the reference configurations then figs. 8(k, m) have to be considered. 3.3. Higher-order
terms
As an alternative to modification of the reference wavefunction and/or zero-order Hamiltonian in order to obtain an improved approximation for the correlation energy in third order, the perturbation series can be extended to fourth order. The fourth-order terms, which correspond to
457
S. Wilson / Diagrammatic many-body perturbation theory Table 4 Fourth-order
calculations
for some small molecules
Molecule
W/11
E[3/01
FH H,D NH, N, co
+ -
- 0.2207 - 0.2221 - 0.2107 -0.3193 - 0.3028
0.2206 0.2176 0.1989 0.3291 0.3076
- 0.2276 - 0.2300 -0.2133 - 0.3486 - 0.3312
-
0.0039 0.0050 0.0050 0.0180 0.0161
+0.0011 + 0.0020 + 0.0027 + 0.0065 - 0.0049
the diagrams shown in fig. 15, can involve singly, doubly, triply and quadruply excited intermediate states [23]. Diagrammatic perturbation theory provides a systematic scheme for including triply and quadruply, and indeed higher-order excited intermediate states. The results
Table 5 Perturbation molecules
theoretic
analysis
of some approaches
to the electron
correlation
problem
in closed-shell
atoms
Method
Order of perturbation expansion
Comments
Refs.
single and double excitation configuration interaction (SD CI)
3
includes unlinked diagrams components of the correlation energy in fourth and higher order
[9.62]
single, double, triple and quadruple excitation configuration interaction (SDTQ
5
includes unlinked diagram components of the correlation energy in sixth and higher orders.
[9X2]
coupled electron pair approximation (CEPA)
3
CEPA(0) includes linked diagrams double excitation through all orders. Other versions of the CEPA include approximations to fourth and higher-order linked diagram quadruple excitation components of the correlation CEPA(0) = L - CPMET ‘) = DE MBPT h’
W,651
coupled pair many electron theory (CPMET)
3
neglects triple excitations in fourth order: there we include in the extended CPMET
WI
CI)
a) Linear CPMET. b, Double excitation
many-body
perturbation
theory.
and
458
S. Wilson / Diagrammatic
shown in table 4 illustrate the importance for a number of small molecules [60,61]. order in the perturbation expansion, it mhartree) can be achieved within the basis 3.4. Anulysis of contemporary
many-body perturbation
theog
of fourth-order components of the correlation energy Provided that all terms are included through fourth has been demonstrated that chemical accuracy (1 set employed.
approaches to the correlation problem
Perturbation theory not only provides a very efficient scheme for the computation of correlation effects in atoms and molecules but also affords a highly systematic technique for the analysis of various approaches employed in many contemporary calculations. In table 5, an overview of the perturbation theoretic analysis of some of the techniques currently being employed in correlation energy calculations is presented. It should be noted that many of these techniques are third-order theories, in that they neglect or approximate fourth- and higher order terms. Perturbation theory can indicate the dominant corrections to these theories. An analysis of some contemporary theories of electron correlation effects in atoms and molecules in terms of the diagrammatic perturbation theory has been presented elsewhere [9]. We wish to emphasise at this point that techniques which have been described for the summation of certain classes of diagrams through infinite order are, in fact, exactly equivalent to well-established, non-perturbative approaches to the correlation problem. For example, the so-called double-excitation many-body perturbation theory, in which double-excitation linked diagrams are summed through all orders, is equivalent to the linear coupled pair many-electron theory. Since these infinite order summations of certain classes of diagrams (classes of diagrams which are often chosen because that can be easily evaluated rather than because they are physically important!) have abandoned the ‘balanced’ truncation of the perturbation series advocated in section 3.1 and because they are merely reformulations of other methods, we shall not consider them in this article.
4. Computational 4. I. Algorithms
diagrammatic perturbation theory
for perturbation
theory
The theoretical properties of diagrammatic many-body perturbation theory, which were described in section 2, ensure that it leads to a computational scheme of high efficiency. The particular strengths of the diagrammatic perturbation theory from a computational point of view are: (a) it leads to a non-iterative algorithm (unless one abandons the spirit of the perturbation approach and sums classes of terms); (b) all possible excited states are included through a given order of the perturbation series of configuration selection schemes, such as those widely used in practical implementations of the method of configuration interaction are avoided; (c) the diagrammatic representation of the correlation energy components can be very easily translated into computer code; (d) it leads to algorithms which are extremely well suited to implementation on parallel computers, vector processing and array processing computers.
S. Wilson / Diagrammatic
f/=1
[ABICDI
Fig. 26. Classification
[AIIJKI
of integrals
many-body perturbation
[IAIBCI
according
theory
[IJ(ABl
to diagram
459
[IAl JBl
components.
Furthermore, the scaling technique, which is described further below, enables the diagrammatic perturbation theory to be used to calculate correlation energies and reduce the basis set truncation error in molecular calculations simultaneously. After integrating over the spin coordinates, the various expressions in the perturbation series can be written in terms of integrals over spatial orbitals. In the present work, lower case indices are employed to represent spin orbitals and upper case indices are used to represent spatial orbitals. The indices I, J, K,. . . are used for occupied orbitals and the indices A, B, C,. . . for unoccupied orbitals. In order to facilitate the evaluation of the algebraic expressions corresponding each of the diagrams in the expansion for the energy, it is convenient to separate the list of two-electron integrals over molecular orbitals into six lists [67]. Each of these lists corresponds to the classes of diagrammatic components shown in fig. 26. Of course, if all of the two-electron integrals can be held in store simultaneously this separation into six lists is not necessary, although it may be convenient. However, this separation is certainly necessary if, as is usually the case, we cannot avoid storing the integrals on backing store. Efficient algorithms for the evaluation of the correlation energy components corresponding to the various terms in the diagrammatic perturbation expansion may be devised by: (i) Taking account of spin orthogonalities. For example, the second-order energy for a closed shell system may be written
E2= aC Cg,jobguhij/(Ei IJ
=EC IJ
+
'j -
‘a -
‘h)
oh
c c gIo,.JoJ,Ao.,.B~HgA~,,B~,,.I~,.J~,/(~I + 'J - ‘A -‘B)-
AB o,o,, o,o,,
(221)
S. Wilson / Diagrammatic
460
where p = Pa, and up is a one-electron
many-body perturbation
spin function.
theory
Now performing
the spin integrals
(223)
the second-order
energy takes the form
E,=~~~{([ZAIJB] IJ
+[zsIJA])([AzpJ]
-[AJIBZ])
AB
(&v?QS = [PQIRSI) which for real orbitals
leads to
E,=fCC{[z~IJB]2+[z~IJA]2 IJ
AB
-[zAIJB][zBIJ/4]}(~,+~,-~,-r,)-‘.
(225)
(ii) Exploiting any spatial symmetry properties of the system being studied. If the Pth spatial orbital transforms according to the r(P) irreducible representation of the point symmetry group then the integral [ PQ I RS] is zero unless the direct product r(P)
@ r(Q)
@ r(R)
@ T(S)
(226)
contains the totally symmetric (A,) irreducible representation. This restriction will often lead to a reduction in the number of integrals which have to be considered in evaluating the summation involved in the various energy expansions. (iii) Recognizing certain permutational symmetry properties of various intermediates which arise in the calculation. (iv) Using the diagrammatic form of Wigner’s (2n + 1) rule discussed in section 2. Algorithms based on the (2n + 1) rule are clearly more efficient than the recursive schemes described in refs. [70,71]. If the shifted denominator perturbation schemes is employed the integrals of the type [ZZ I JJ] and [ZJ/Z.Z] can be retained in the computer’s fast store. 4.2. Second-order
and third-order energy calculations
An efficient algorithm for evaluating the energy of closed-shell systems through third order has been described by Silver and Wilson [67,69]. In order to evaluate the second-order energy Silver
S. Wilson / Diagrammatic
many-body perturbation theory
461
[67] arranges the required list of integrals of the type [IA 1JB] in blocks, each block corresponding to a unique I, J pair (I >, J) and containing all non-zero integrals for all possible values of A and B. Furthermore, if I # J and A # B, then the two integrals [IA 1JB] and [IB 1JA] occupy consecutive positions in the list whenever both are non-zero. With this file handling scheme the evaluation of the second-order energy is reduced to a “near triviality”. The energy components corresponding to the diagrams labelled (B) and (D) in fig. 13 can be evaluated using similar algorithms [68]. The component arising from (B) requires integrals of the type [IA 1JB] and [IJ 1KL] whilst that arising from (D) requires integral types [IA I JB] and [ AB ICD]. For the (D) component the [IA I JB] integrals are blocked and ordered in exactly the fashion described above for the second-order energy whereas for the (B) component another list of these integrals is generated in which each block of integrals corresponds to a particular A, B pair (A 2 B) and contains all non-zero integrals for all possible values of I/J. The number of [ ZJI KL] integrals is not excessively large and all of these integrals can usually be held in the computers fast core. In contrast, the list of [ AB I CD] integrals can be enormous; the number of these integrals increases as the fourth power of the number of basis functions. Thus the problem logic is constructed so that the [AB I CD] (or [ IJ I KL]) integral list is not reordered and these lists are read only once. Note that diagrams (B) and (D) are related by “time reversal” (i.e. inverting the diagram) and thus the same procedure can be used to evaluate both energy components. The algorithm for evaluating the (B) component is as follows: Read a block of [ AB I CD] integrals into fast storage. Read a block of [IA ) JB] integrals for a given A. B pair. Form all integral products required for the expression corresponding to (D) Return to (b) until all [IA I JB] integrals have been read. Return to (a) until all [ AB I CD] integrals have been read. The possible combinations of spin functions which can arise in the evaluation of diagrams (B) and (D) of fig. 13 are shown in fig. 27. In fig. 28, the possible combinations of spin functions which can arise in the case of the diagram shown in fig. 13(c) are summarised [69]. Two lists of two-electron integrals are required to evaluate this energy component. The [IA I JB] = (IJ ) g I AB) (g = l/r,,) are ordered in the manner described above for the second-order energy calculations.
Fig. 27.
S. Wilson / Diagrammatic
cl
theor):
------_-----_-_-_0 00 Q-O0 0 ttkh -__-_ _ -__ ----0 bo w P 0 P 0 a
P__
CL
a
a
-_
CY
a -----
_----
s-v
Lx
a --
a
a
cf.
0.
cl.
--__-
0.
CY
c1
__
-_-_ w t33 ------
----
CY
__
0.
cx
CL
6
G
6
a
c1
es-mm_
a
c1
__
6
a
a
many-body perturbation
c1
cl.
CY
----_
6
-_-_--
-e-B-
a
a
a
a
CY
-_
CL
-
_--
-_ a -
cf
cf. --_-_
Fig. 28
For an atom or molecule which is described by N doubly occupied orbitals, N secondary lists are created. The Ith of these secondary lists contains the integrals (ZJ 1g ) AB) for all J, A a B. This secondary list is divided into blocks according to the index J. The integrals (IB 1g 1AJ) are divided into blocks corresponding to I, J (I > J) pairs and each block is ordered with A > B. These integral lists are then processes in the following manner: (i) Integrals of the type (IC I g I AK) are read into fast core for all A > C for a given I > K. (ii) The integrals (ZJ I g I AB) and (JK ) g I BC) are read into core from the Ith secondary list, for all A > B, and from the Kth secondary list, for all B > C, respectively. (iii) All integrals depending on a certain I, J, K are thus in fast core simultaneously. The contribution to the energy components corresponding to the diagram (C) can be evaluated for fixed 1, J, K; summation over A, B, C can be performed. (iv) If the end of the Ith and Kth secondary lists has not been reached, J is increased and the algorithm returns to (ii).
S. Wilson / Diagrammatic
many-body perturbation
theory
463
If the Zth and Kth secondary lists have been completely read, they are rewound. If the (IC 1gj AK) list has not been completely read, the algorithm returns to (i). The list of (IC 1g ( AK) integrals is, therefore, read into core once. To illustrate algorithms devised for evaluating fourth-order components of the correlation energy, the fourth-order closed-shell triple-excitation diagrams will be considered [72]. These are the most demanding of the fourth-order diagrams for a closed shell system leading to an algorithm which depends on the seventh power of the number of electrons. The energy components corresponding to the sixteen diagrams labelled X, in fig. 15 may all be written in terms of the intermediate quantities
(v)
(227) and
(228) Each diagram K~
c
i/k ohc
leads to an expression +rJk;ahc{
of the form
p~jkpuhcytjk;ohc)
(229)
( 6, + E, + Ek - E, - Lb - cc) ’
where 4 and y denote either an _f or a R intermediate, permutation operators. The following expressions arise:
K is a constant
and
P,,k Pa,,<,are
(230)
(231)
EL&,) = J%(D,)= E‘iW = c
(232)
4 c c gijk:ohcgrkJ;ohc/~rJkohc, iJk ah<
(233)
(234)
ijk
(235)
(236)
S. Wilson / Diagrammatic
464
many-body perturbation
theory
Table 6 Spin case for the intermediate FhKEABC P
RDKAC
glJDB
E&T) = E4(JT)
=
-
ac Cfr,k;ahrgik,~orb/~r,kahrr
(238)
.(239)
c Cfrjk:ohcgijk;ubc/~i,kabc, ijk ohc
(240)
E4(oT)
=
(241)
$ c Cf;jk;ohrgikj;ahr/~,jkabc’ !jk ulx
where 9
rJkahL
=
6, +
Ej +
Ck -
ca -
Ch -
E,.
(244
It should be noted that diagrams L,, K,, N,, P, lead to expressions which are numerically equal to those for diagrams I,, J,, M,, Or, respectively, for real orbitals. The nine different spin cases are summarized in table 6. Definition of which arise in the evaluation of the frjkiahc intermediates these intermediates enables a spin-free formulation of the problem to be achieved. Defining the secondary intermediate quantities.
(243)
(244)
Similar spin-Free ~nterme~i~t~s may be obtained for the ~~~~~~~~~ Then, fr>r example, the energy corresponding to the diagram A, in fig. I5 may be written as &fA,)
= - c IJK
( F;:K: ABC-FI:K,ACB
c ABC
+ F~K;Ad%;Aa+ ~?K:ABcF&:AcB +GKT;Ald%L;ACB
+
~~K;AeCGwB
-+GK:ABCF~K;ACB
-t- Fr%;Afd%;ACLi
+ ~~K;A~C~~K~*C~)~~~,
it
+ “J f- CK -
=
hK;CBA
=
-%K;ABC
=
and thus only nine unique intermediates %%.4BC intermediates.
QB -
4.
F&ABC obey the permutational
should be noted that the intermediates I:IJK:ABC
EA -
-
F,,;CBA
cw
symmetry properties w-0
have to be considered. Similar relations exist for the
S. Wilson / Diagrummotic
466
many-body perturbution
theory
Similar algorithms to that described above for the fourth-order triple-excitation energy can be devised for other components of the diagrammatic perturbation expansion of the correlation energy (see, for example, ref. [73]). For example, in fourth order for closed-shell systems the single-excitation component, corresponding to the diagrams labelled As-D, in fig. 15, may be written in terms of the intermediates
Cg,hbg,b.,k/(~, + c/,-
g,, =
60
-Ebb
(25%
jkb
Thus, for example
&(A,) = f ~Mw’k
- E,.).
(260)
id
The double-excitation components, corresponding may be written in terms of the intermediates
g ,,oh
=
&kloh&k,/(‘k
+
‘/
+
‘k
-
‘u
-
to the diagrams
labelled
An-L,,
in fig. 15,
(262)
‘b)?
XI
h r/oh
=
&ikocg,db/(‘,
-
‘,
-
‘<
(263)
>.
Thus, for example,
EdA,) = & ~f,,,,f,,,,/(~;+ 6, - cu- d
(264)
l/“h
The quadruple-excitation components, written in terms of the intermediates f rkhd
=c
gr,ub&,ud/(~,
+
6,
-
cu
-
corresponding
to diagrams
A,-Go
in fig. 15 may be
(265)
Cb),
P grkhd
=
&?,,ubgk,o&,
+
‘,
-
E,
-
‘b)h,
+
-
cu
-
%/)3
(266)
I”
fi,kl
=
~&obgklob/(E,
+
‘,
-
“J
-
(267)
‘b>3
oh g,,k/
=
~~,,ob&,ob/‘+,
+
f,
-
E,
-
‘b)hk
+
E/ -
cu
-
d
(268)
S. Wilson / Diagrammatic
many-body perturbation
theory
467
(269)
(270)
guc=Cgrjuhg,,~h/(Ei+tl-E,--~)(C,+~,-Ch-E,), i/h
f,k = Cgr,uhgk,oh/k + 6, - ca- 4
(271)
.Ph
g,k = Cg,,uhgk,u(,/(‘r .l”h
+ ‘j - ‘u - ‘h)(‘,
+ ‘k - ‘o - ‘h)*
(272)
Thus E,(AQ)
=
f
c
frkhdgrkhd~
(2;3)
lkhd
&tBQ
+ &)
= ii Cf;,k,gi,ki’ r/k1
E,(Do+ E,) = - acf,cs,o
(274)
(275)
4
C&+GQ)=
(276)
-ichkg;k. rk
4.3. The scaling technique
In view of the difficulty of evaluating all terms through fourth order in the perturbation expansion various approximation schemes have been devised. The scaling technique, suggested by the present author [74], enables second-order perturbation calculations, using a large basis set, to be used to approximate calculations taken to fourth-order. The correlation energy is thus determined by an algorithm which depends on the fourth power of the number of basis functions. A scaling parameter, p, is introduced into the zero-order Hamiltonian. This leaves the self-consistent-field reference energy unchanged but the second-order energy is scaled according to /PE,.
(277)
In the scaling technique, I_Lis chosen so that the perturbation expansion converges rapidly and so that the scaled second-order energy provides an accurate correlation energy [74]. It has been demonstrated that a useful value of p can be determined from calculation taken beyond second order using basis sets of moderate size [74]. An example, of the application of this technique is given in section 5. The use of constant denominator perturbation theory has been advocated [75]. This leads to an algorithm for the fourth-order energy component depending on the sixth power of the number of basis functions.
468
S. Wilson / Diagrammatic
many-body perturbation
theory
4.4. Open-shell systems In the above discussion, attention has been restricted to the evaluation of the perturbation energy components for closed-shell systems; however, the same approach can be adopted for calculations on open-shell systems using either single determinant or multi-determinantal reference functions. The computational problem is one of efficiently forming the products of one- and two-electron integrals which occur in the numerator of the perturbation expressions; the data required to form the denominator can always be held in the computer’s fast store. 4.5. Parallel computation The linked diagram theorem of many-body perturbation is of considerable theoretical importance as the discussion of section 2 demonstrates. It is also of computational importance. The linked diagram theorem allows the energy of any system to be written as a sum of energies of its component parts no matter how these component parts are refined. It provides an additive decomposition of the energy which aids the development of algorithms for parallel computation by allowing calculations on arbitrarily defined subsystems to be performed concurrently. It is clear that a central problem of atomic and molecular physics over the next few years is the development of efficient techniques for the ab initio treatment of electron correlation effects which allow the organization of concurrent computation on a very large scale. Since the linked diagram theorem ensures that the diagrammatic perturbation theory satisfies certain additive separability conditions of the type first discussed by Primas [76], it forms the basis of algorithms affording the maximum degree of parallelism. The paracomputer model, introduced by Schwartz [77], plays a useful theoretical role in the determination of the limits of parallel computation possible for a given method. The paracomputer consists of a very large number of identical processors, each with a conventional order code set, that share a common memory which they can read simultaneously in a single cycle. Such machines cannot be physically realized, since any computing element can have no more than some fixed number of external connections. However, the model is theoretically valuable. Consider, for example, the evaluation of the second-order energy for a closed-shell system on a paracomputer. This involves a summation over the indices I, J, A, B with I > J and A > B. The number of terms in the summation is M=~N,N,(N,+l)(N,+l),
(278)
where N, and NV are the numbers of occupied and of virtual orbitals, respectively. computer the evaluation of the second-order energy will take a time Tqeria,given by qeriai = M7i + ( M - 1) 5 7
On a serial
(279)
where 7, is the time required to compute the contribution to the sum by a given set of spatial orbitals ZJAB and r2 is the time required to perform an addition. On a paracomputer with P processors, the computation of the second-order energy will take a time Tparalle,given by M Tparall. =pr,+
i
SP-2 p
i
72.
469
S. Wilson / Diagrammatic many-body perturbation theory
If the number
of processors
is less than the number
of terms in the summation,
i.e. M B P, then
(281) and the paracomputer performs the calculation P time faster than a serial machine. The above considerations can be generalized to other components of the correlation energy in the perturbation expansion. Any contribution can be evaluated approximately P times faster on a P-processors paracomputer than on a serial machine. Considerations, using the paracomputer model, such as those outlined above, clearly demonstrate the suitability of diagrammatic perturbation theory for concurrent computation [78].
5. Applications
It is not intended to give a detailed review of applications of diagrammatic perturbation theory and molecules here but to provide a few examples to illustrate its potential. In a series of papers by Eggarter and Eggarter [79] and by Jankowski and his co-workers [80], accurate calculations of correlation energies using second-order and, in the recent work of Jankowski et al., third-order Rayleigh-Schrodinger perturbation theory have been reported for a variety of atomic systems. Such calculations have clearly demonstrated the need to include basis functions corresponding to higher angular quantum numbers if the error due to basis set truncation error is to be significantly reduced. Jankowski and Malinowski, for example, showed that the second-order energy for the ground state of the neon atom provides a very accurate approximation if functions corresponding to higher I quantum number are included in the basis set. Jankowski et al., have recently extended their calculations for neon to third-order and the to atoms
Table 7 Calculated
correlation
energies
for the neon atom ground
Radially
Calculated & -
‘) In units of lop4
1918.0 3219.3 3596.1 3742.8 3799.2 3825.1 3838.3 3845.5 3849.8
hartree.
state ( Escr = - 128.54705 h) a)
E,
E,
42.3 11.1 2.0 10.0 20.5 27.1 31.1 33.4 34.9
-
extrapolated E,
1920.4 3223.8 3602.7 3751.2 3809.2 3836.3 3850.7 3858.8 3863.8 3879
42.5 11.7 3.0 11.6 22.7 29.8 34.3 37.0 38.7 44
470
S. Wilson / Diagrummatic
Table 8 Calculated correlation Al9 (1979) 1375
energies
for the beryllium
Correlation E acf E, E, + EX
many-body perturbation
atom ground
state; taken from Silver, Wilson and Runge, Phys. Rev.
energy
Total energy
- 0.07195 - 0.08380 - 0.08613 - 0.09434
W/11
E non.rral
theory
-
14.57302 14.64497 14.65682 14.65915 14.66736
results which they obtained are summarized in table 7. For the Be atom ground state there is a low lying 2p2 excited state and the accuracy of the perturbation expansion through third-order is not as high as for the Ne atom as the results of Silver et al. [81] summarized in table 8 clearly demonstrates. For molecular calculations it is usually necessary to employ basis sets of a more modest size than it is possible to use in atomic studies. Some indication of the usefulness of perturbation theory can be obtained by comparisons with full configuration interaction calculations, although for the latter calculations to be tractable the basis set truncation error is enormous. In table 9 a comparison of perturbation theory and full configuration interaction is made for two different
Table 9 Comparison interaction
of calculated
molecular
correlation
energies
obtained
by perturbation
theory
with full configuration
System
Hz dimer case (a)
H 2 dimer case (b)
E2
- 0.031053 - 0.012711 - 0.000136 - 0.006023 - 0.000887 - 0.003051 - 0.010097 - 0.043764 - 0.052572 - 0.053861
-
0.043648 0.018811 0.000002 0.010743 0.~003 0.006036 0.016783 0.062459 0.076706 0.079242
- 0.139478 - 0.001391 - 0.000908 - 0.003083 - 0.001364 - 0.~0815 - 0.006170 - 0.140869 -0.140883 - 0.147039
- 0.052515
- 0.105552
- 0.139340
- 0.053690
- 0.109196
-0.148028
~53
E 4s E 4D E 4T E 4Q E4
E[3/01 E[2/11 EI4/01 double excitation configuration mixing full configuration mixing Footnotes: Jankowski
based on the work of Jankowski and Paldus (1982).
and Paldus (1980); Saxe et al. (1981); Wilson and Guest (1980); Wilson,
S. Wilson / Diagrammatic
many-body perturbation
theary
471
Table 10 Calculations on diatomic molecules obtained using diagrammatic perturbation theory and a universal basis set; all energies are in atomic units. The following abbreviations are used: Ci: configuration interaction; PNO-CI: pair natural orbital-configuration interaction; CEPA: coupled electron pair approximation; DPT: diagrammatic perturbation theory; MCSCF: multiconfiguration self-consistent field; E[2/1] denotes the [2/l] PadC approximant to the perturbation series based on the Hartree-Fock model zero-order Hamiltonian and i[2/1] denotes the [2/l] PadC approximant to the shifted denominator expansion Method LiH (empirical
correlation
energy,
m/11
Diagrammatic perturbation E]2/11 theory/Universal basis sets (d) &‘[2/1] correlation
energy,
correlation
energy,
correlation
energy,
8.0606 8.0647 8.0660 80643 - 8.0652 - 8.0653 - 8.0661
88.3 93.3 94.8 92.8 93.9 94.0 94.9
- 14.9649 - 14.9842 -14.9845
74.3 89.6 89.9
-
100.3564 100.3274 100.3392 100.3727 100.3707 100.3837 100.3770
75.1 67.5 70.6 79.4 78.9 82.3 80.6
- 109.2832 - 109.4180
58.4 79.5
- 109.443
83.7
e,,r = -0.381
Bender and Davidson (a) Cl Meyer and Rosmus (b) PNO-CI CEPA Wilson and Silver(c) E[2/1] Ef2/11 E12/1] Diagrammatic perturbation theory/Universal basis set (d) E[2/1] N, (empirical
-
erxp = - 0.126)
Werner and Reinsch (e) MCSCF-CI Diagrammatic perturbation E]2/11 theory/Universal basis set (f) E[2/1] FH (empirical
eexp @)
ecxp = - 0.083)
Bender and Davidson (a) CI Meyer and Rosmus (A), PNO CI CEPA Wilson and Silver(b) DPT, E[2/1)
Liz (empirical
-&,,I
e,xp = -0.538)
Langhoff and Davidson (g) CI Wilson and Silver DPT (h) E[2/1] Diagrammatic perturbation theory/Universal basis set (5) E[2/1]
conformations of the H, dimer, the first involving no near degeneraties in the reference spectrum and the second demonstrating the effects of quas~degeneracy. A comparison is also made in table 9 of calculations performed for the water molecule using a double zeta basis set. The results presented in table 9 demonstrate that the non-degenerate perturbation theory is in good agreement with full configuration interaction if quasi-degeneracy is absent. To illustrate the accuracy of molecular calculations using diagrammatic perturbation theory, results are presented in table 10 which were obtained by evaluating the expansion through third order in the energy and by using a universal basis set of functions with s, p and d symmetry. The
472
S. Wilson / Diagrammatic
many-body perturbation
theory
Table 10 (continued) Method CO (empirical
E,,,,, correlation
energy,
ecxp = - 0.525)
Siu and Davidson (j) CI Bartlett et al. DPT (k) E[2/1] Diagrammatic perturbation theory/Universal basis set (i) E[2/1] BF (empirical
correlation
energy,
(f) (g) (h) (i) (j) (k) (1)
69.4 77.4
- 113.2286
82.7
- 124.235 - 124.5028
65.2
- 124.5782
77.1
ecxp = - 0.531)
Bender and Davidson (a) MCSCF Wilson et al. DPT (1) E[2/1] Diagrammatic perturbation theory/Universal basis set (i) E[2/1] (a) (b) (c) (d) (e)
- 113.1456 - 113.1952
C.F. Bender and E.R. Davidson, J. Chem. Phys. 70 (1966) 2675; Phys. Rev. 183 (1969) 23. W. Meyer and P. Rosmus, J. Chem. Phys. 63 (1975) 2356. S. Wilson and D.M. Silver, J. Chem. Phys. 66 (1977) 5400. S. Wilson and D.M. Silver, J. Chem. Phys. 77 (1982) 3674. H.-J. Werner and E.-A. Reinsch, Proc. 5th Seminar on Computational Problems in Quantum van Duijnen and W.C. Nieuwpoort, Groningen (1982). S. Wilson, Specialist Periodical Reports: Theoretical Chemistry 4 (1981) 1. S. Langhoff and E.R. Davidson, Intern. J. Quantum Chem. 8 (1974) 61. S. Wilson and D.M. Silver, J. Chem. Phys. 67 (1977) 1689. S. Wilson and D.M. Silver, J. Chem. Phys. 72 (1980) 2159. A.K.Q. Siu and E.R. Davidson, Intern. J. Quantum Chem. 4 (1970) 223. R.J. Bartlett, S. Wilson and D.M. Silver, Intern. J. Quantum Chem. 13 (1977) 737. S. Wilson, D.M. Silver and R.J. Bartlett, Mol. Phys. 33 (1977) 1177.
Chemistry,
eds. P.Th
results are compared with the previous calculations and in every case it can be seen that perturbation theory affords an accuracy equal to or better than previous work. Diagrammatic perturbation theory has been used to calculate potential energy curves and surfaces and as an example we show a potential energy curve for the ground state of the fluorine molecule, obtained by Urban and coworkers [92], in fig. 29. Spectroscopic constants obtained from these results are shown in table 11 where they are also compared with experiment. Also in fig. 29 a potential energy curve obtained by means of the scaling technique, described in section 3, is shown and can be seen to be in good agreement with the full fourth-order calculation [93].
6. Summary and future directions We have emphasized in this article the highly systematic approach to the electron correlation problem in atoms and molecules afforded by the diagrammatic many-body perturbation theory. Not only does this method provide a powerful and tractable approach for performing calculations of high accuracy but also it provides a theoretical formalism in terms of which many contemporary theories of molecular electronic structure can be analysed.
S. Wilson / Diagrammatic
many-body perturbation
473
theory
-199.27
-199.28
-199.29
internuclear distances, bohr -199.31
-199.32
-199.33
1
Fig. 29. Potential
energy curves for the ground
state of the fluorine molecule.
It is clear from the many calculations which have been performed to date that the dominant source of error in molecular calculations utilizing diagrammatic perturbation theory (and indeed in most contemporary calculations) is attributable to basis set truncation [94]. Future work will have to be directed towards the reduction of the magnitude of this error. Indeed, some progress has already been made in this direction by using universal even-tempered basis sets and by using a systematic sequence of universal even-tempered basis sets [95]. The latter approach enables extrapolation procedures to be employed to determine the basis set limit and when used in conjunction with th,e scaling technique described in section 5 shows promise [96]. Table 11 Comparison values
of calculated
spectroscopic
Energy
R, (lo-”
E rcr
1.329
W/u1
1.400
E[3/01 E [4/O] experiment
1.386 1.420 1.412
m)
constants
for the ground
state of the fluorine
molecule
with experimental
0, (cm-‘)
w,x, (cm-‘)
a, (cm-‘)
1265 1007 1039 913 917
6.41 8.63 9.16 11.7 11.2
0.008 0.010 0.010 0.0013 0.014
474
S. Wilson / Diagrammatic
many-body perturbation
theory
Finally, it should be mentioned that relativistic effects will become important when the techniques described in this report are applied to heavy atoms and molecules containing heavy atoms. Work is in progress to develop a relativistic many-body perturbation theory suitable for applications to atoms and molecules including the most important relativistic effects in the reference model [97].
Appendix A. Second- and third-order energy expressions for open-shell atoms and molecules The secondare as follows:
E&J=
and third-order
energy expressions
_ c (il~l~>(~l~li>(~lal~> k-&--A ?iU
E,(E)= c
corresponding
’
3
+lh
(il~lu)(jl~lb)(ublQI~) E3(G)=z
(E,-cE,)(c,+<,-cE,-cZh)
’
(il~lu)(ujl~lbc)(bcIBlij) E3(J)=+vz<
(Ei-E,)(c,+cJ-cb-cr)
’
to the diagrams
given in fig. 14
S. Wilson / Diagrammatic
many-body perturbation
475
theory
. (ijlolab)(kIicIj)(abl~lik) (
E&j
ijkoh
+
%bk
where the e are the usual orbital
and the one-electron matrix non-degenerate doublet state
energies,
elements
CPlW) =%,F(P; P> 4L
+
(PK
6k -
E,
ca -
-
%)
6h)
’
’
the two-electron
depend
integrals
on the electronic
are given by
state
being
studied.
For
a
l4),
where the value of the spin factor F( p; p, q) depends on the nature of the spin orbitals p and q. The nine possible cases for cx spin are given in table 12. The factors for j3 spin are given by
m? P, Table 12 Spin factors
F(a;
q)=F(a;
p,
q)-1.
p, q) for the matrix elements Occupation
of u number
for p in reference
function
Occupation number for q in reference function
2
1
0
2 1 0
312 1 l/2
1 l/2 0
l/2 0 -l/2
416
S. Wilson / Diagrammatic
For triplet states the one-electron (AGlq) K denotes
Pl d((PIK,,
=&J%-C the exchange
operator
operator
many-hodJ perturhution theory
is
14) + (PVC,
14))
for the singly occupied
orbitals
IM and n.
Appendix B. Fifth-order energy diagrams for closed shell systems In this appendix we give the fifth-order energy diagrams for a closed-shell system which have been given previously in refs. [4,6]. Following Paldus and Wong [98,99] we give the essentially distinct Hugenholtz diagrams from which all other non-equivalent Hugenholtz or Brandow diagrams can be obtained either (i) as a different time versions of the given diagram resulting from a permutation of the vertices; (ii) as the particle-hole inversion of the given diagram, resulting from the reversal of the orientation of all the arrows; or (iii) as the time reversal, which is a result of reflecting the diagram in the plane perpendicular to the time axis.
Fig. 30. Essentially
distinct
Hugenholtz
diagrams
which describe
the correlation
energy through
fifth-order.
S. Wilson / Diagrammatic
many-body perturbation
theory
417
In fig. 30 we give the full set of essentially distinct diagrams which describe the correlation energy of a closed-shell system through fifth order. It should be emphasized that the number of distinct Brandow diagrams increases rapidly with the order of perturbation.
Appendix C. Expressions for energy components for a multideterminantal
reference function
As an example of the algebraic expressions for the energy components which arise when a multideterminantal reference wave function is employed. We give the expressions corresponding to the diagrams given in fig. 20: + c .ik
GwbwwQ~~) E, -co
Appendix D. Shifted-denominator
3
perturbation expansion for the energy through third-order
The algebraic expressions for the energy denominators are employed are as follows:
-
components
dijoh)(c; +
c/ -
through
E,. -
E(/ -
third
drJ_/)’
order
when
shifted
478
The denominator
S. Wilson / Diagrammatic
many-bed)) perturbation
theory
shift diiuh is defined in eq. (214).
References
111H.P. Kelly, Adv. Chem. Phys. 14 (1969) 129.
PI
S. Wilson, in: Correlated Wavefunctions, Proc. of Daresbury Study Weekend (December 1977) ed. V.R. Saunders, Science Research Council, London (1978). Topics in Current Chemistry 75 (1978) 97. I31 I. HubaE and P. &sky, 141 S. Wilson, in: Electron Correlation, Proc. of Daresbury Study Weekend (November 1979) eds, M.F. Guest and S. Wilson, Science Research Council, London (1980). and M. Urban, Ab Initio Calculations: Methods and Applications in Chemistry, Lecture Notes in PI P. &sky Chemistry, vol. 16 (Springer, Berlin, 1980). PI S. Wilson, Specialist Periodical Report: Theoretical Chemistry 4 (1981) 1. 171 R.J. Bartlett, Ann. Rev. Phys. Chem. 32 (1981) 359. See also Nobel Symposium issue of Physica Scripta 21 (1981) Molecular Physics, Proc. of Nato Advanced Study Institute, Bad PI S. Wilson, in: Methods in Computational Windsheim (August 1982) eds. G.H.F. Diercksen and S. Wilson (D. Reidel, Dordrecht, 1983). [91 S. Wilson, Electron Correlation in Molecules (Clarendon Press, Oxford, 1984). PO1 H.P. Kelly, Phys. Rev. 131 (1963) 684, 136 (1964) 896, 144 (1966) 39; Adv. Theoret. Phys. 2 (1968) 75; Intern. J. Quantum Chem. Symp. 3 (1969) 349. t111 ES. Chang, R.T. Pu and T.P. Das, Phys. Rev. 174 (1968) 1. NC. Dutta, C. Matsubara, R.T. Pu and T.P. Das, Phys. Rev. 177 (1969) 33. R.T. Pu and ES. Chang, Phys. Rev. 151 (1966) 31. T. Lee, N.C. Dutta and T.P. Das, Phys. Rev. Al (1970) 995, A4 (1971) 1410. P21 H.P. Kelly, Phys. Rev. Lett. 23 (1969) 455. J.H. Miller and H.P. Kelly, Phys. Rev. Lett. 26 (1971) 679. 1131 S. Wilson and D.M. Silver, Phys. Rev. Al4 (1976) 1949. See also J.M. Schulman and D.N. Kaufman, J. Chem. Phys. 53 (1970) 477, 57 (1972) 2328. U. Kaldor, Phys. Rev. A7 (1973) 427; Phys. Rev. Lett. 31 (1973) 1338. M.A. Robb, Chem. Phys. Lett. 20 (1973) 274. 1141 J.A. Pople, J.S. Binkley and R. Seeger, Intern, J. Quantum Chem. Symp. 10 (1976) 1. E.R. Davidson and D.W. Silver, Chem. Phys. Lett. 52 (1977) 403. R.J. Bartlett and I. Shavitt, Intern. J. Quantum Chem. Symp. 11 (1977) 165. S. Wilson and D.M. Silver, Theoret. Chim. Acta. 54 (1979) 83. 1151 K.A. Brueckner, Phys. Rev. 100 (1955) 36. 1161 J. Goldstone, Proc. Roy. Sot. A239 (1951) 267. v71 N.H. March, W.H. Young and S. Sampanthar, The Many-Body Problem in Quantum Mechanics (1967) CUP. P81 J. Paldus and J. Ciiek, Adv. Quantum. Chem. 9 (1975) 105. [I91 B.H. Brandow, Rev. Mod. Phys. 39 (1967) 771. bw N. Hugenholtz, Physica 23 (1957) 481. WI V. Kvasnicka, V. Laurinc and S. Biskupic, Mol. Phys. 39 (1980) 143. PI I. HubaE and P. Carsky, Phys. Rev. A22 (1980) 2392.
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theory
S. Wilson and D.M. Silver, Intern. Quantum Chem. 15 (1979) 683. S. Wilson, Theoret. Chim. Acta 61 (1982) 343. G. Hose and U. Kaldor, J. Phys. B 12 (1979) 3827. See also; D. Hegarty and M.A. Robb, Mol. Phys. 37 (1979) 1455, 38 (1979) 1795. H. Baker, D. Hegarty and M.A. Robb, Mol. Phys. 41 (1980) 653. M.G. Sheppard and K.F. Freed, J. Chem. Phys. 75 (1981) 4507 and references therein. I. Shavitt and L.T. Redman, J. Chem. Phys. 73 (1980) 5711. L.T. Redman and R.J. Bartlett, J. Chem. Phys. 76 (1982) 1938. [27] E.M. Krenciglowa and T.T.S. Kno, Nucl. Phys. A235 (1974) 171. H.M. Hoffmann, S.Y. Lee, J. Richert and H.A. Weidenmuller, Ann. Phys. 85 (1974) 410. J.M. Leinaas and T.T.S. Kno, Ann. Phys. 111 (1978) 19. T.H. Schucan and H.A. Weidenmuller, Ann. Phys. 73 (1972) 108. J. Richert, T.H. Schucan, M.H. Scrubel and H.A. Weidenmuller, Ann. Phys. 96 (1976) 139. T.H. Schucan and H.A. Weidenmuller, Ann. Phys. 76 (1973) 483. [28] G. Hose and U. Kaldor, Phys. Scripta 21 (1980) 357. [29] G. Hose and U. Kaldor, J. Phys. Chem. 86 (1982) 2133. [30] M.B. Johnston and M. Baranger, Ann. Phys. 62 (1971) 172. [31] W.N. Lipscomb and J. Gerratt, Proc. Nat. Acad. Sci. 59 (1968) 332. (321 J. Gerratt, Advan. Atom. Mol. Phys. 7 (1971) 141. [33] J. Gerratt, Specialist Periodic Reports: Theoret. Chem. 1 (1974) 60. [34] S. Wilson and J. Gerratt, Mol. Phys. 30 (1975) 777. [35] N.C. Pyper and J. Gerratt, Proc. Roy. Sot. A355 (1976) 407. [36] R.C. Morrison and G.A. Gallup, J. Chem. Phys. 50 (1969) 1214. [37] W.A. Goddard Jr., Phys. Rev. 157 (1967) 81. [38] D.J. Newman, J. Phys. Chem. Solids 30 (1969) 1709. [39] D.J. Newman, J. Phys. Chem. Solids 31 (1970) 1143. [40] V. Kvasnicka, Chem. Phys. Lett. 51 (1977) 165. (411 M. Moshinsky and Seligman, Ann. Phys. 66 (1971) 311. [42] J.F. Gouyet, J. Chem. Phys. 59 (1973) 4637. [43] A.A. Cantu, D.J. Klein, F.A. Matsen and T.H. Seligman, Theoret. Chim. Acta 38 (1975) 341. [44] B. Kirman and Cole, J. Chem. Phys. 69 (1978) 5055. [45] M.F. Guest and S. Wilson, Chem. Phys. Lett. 72 (1980) 49. [46] S. Wilson and D.M. Silver, J. Chem. Phys. 66 (1977) 5400. [47] S. Wilson, D.M. Silver and R.J. Bartlett, Mol. Phys. 33 (1977) 1177. [48] R.J. Bartlett, S. Wilson and D.M. Silver, Intern. J. Quantum Chem. 12 (1977) 737. S. Wilson, ibid. 12 (1977) 609. [49] Chr. Moller and M.S. Plesset, Phys. Rev. 46 (1934) 618. [SO] J.S. Binkley and J.A. Pople, Intern. J. Quantum Chem. 9 (1975) 229. [51] P.S. Epstein, Phys. Rev. 28 (1926) 695. 1521 R.K. Nesbet, Proc. Roy. Sot. A230 (1955) 312, 322. [53] P. Claverie, S. Diner and J. Malrieu, Intern. J. Quantum Chem. 1 (1967) 751. [54] E. Feenberg, Ann. Phys. 3 (1958) 292. [55] A.T. Amos, Intern. J. Quantum Chem. 6 (1972) 125. [56] S. Wilson, D.M. Silver and R.A. Farrell, Proc. Roy. Sot. A356 (1977) 363. [57] P.M. Silver and S. Wilson, Proc. Roy, Sot. A383 (1982) 477. [58] D.M. Silver and R.J. Bartlett, Phys. Rev. Al3 (1976) 1. [59] D.M. Silver, S. Wilson and R.J. Bartlett, Phys. Rev. Al6 (1976) 477. [60] R. Krishnan, J.S. Binkley, R. Seeger and J.A. Pople, J. Chem. Phys. 72 (1980) 650. [61] S. Wilson and M.F. Guest, J. Phys. B (1981). [62] I. Shavitt, in: Modern Theoretical Chemistry, vol. 3, Methods of Electronic Structure Theory, III (Plenum, New York, 1977). [63] R. Ahlrichs, Comput. Phys. Commun. 17 (1979) 31.
479
[23] [24] [25] [26]
ed. H.F. Schaefer
480 [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80]
[81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99]
S. Wilson / Diagrammatic
many-body perturbation theory
A.C. Hurley, Electron Correlation in Small Molecules (Academic Press, New York, 1976). M.R.A. Blomberg and P.E.M. Siegbahn, Intern. J. Quantum. Chem. 14 (1978) 583. J. Ciiek and J. PaIdus, Phys. Scripta 21 (1980) 251 and references therein. D.M. Silver, Comput. Phys. Commun. 14 (1978) 71. D.M. Silver, Comput. Phys. Commun. 14 (1978) 81. S. Wilson, Comput. Phys. Commun. 14 (1978) 91. R.J. Bartlett and I. Shavitt, Chem. Phys. Lett. 50 (1977) 190. H.J. Monkhorst, B. Jeziorski and F.E. Harris, Phys. Rev. A23 (1981) 1639. S. Wilson and V.R. Saunders, Comput. Phys. Commun. 19 (1980) 293. S. Wilson and D.M. Silver, Comput. Phys. Commun. 17 (1979) 47. S. Wilson, J. Phys. B 12 (1979) L135, L599, 14 (1981) L31. S. Wilson, Theoret. Chim. Acta 59 (1981) 71. J.M. Cullen and M.C. Zerner. Theoret. Chim. Acta 61 (1982) 203. H. Primas, in: Modern Quantum Chemistry II, ed. 0. Sinanoglu (Academic Press, New York, 1965). J.T. Schwartz, ACM Trans. Prog. Lang & Systems 2 (1980) 484. S. Wilson, in: Proc. 5th Seminar in Computational Methods in Quantum Chemistry, eds. P.Th. van Duijnan and W.C. Nieuwpoort, Groningen (1981). E. Eggarter and T.P. Eggarter, J. Phys. B 11 (1978) 1157, 2069, 2969. T.P. Eggarter and E. Eggarter, ibid. K. Jankowski and P. Malinowski, Chem. Phys. Lett. 54 (1978) 68. K. Jankowski, P. Malinowski and M. Polasik, J. Phys. B 12 (1979) 345, 3157, ibid. 13 (1980) 3909; Phys. Rev. A22 (1980) 51. K. Jankowski, D. Rutkowska and A. Rutkowski, J. Phys. B 15 (1982) 1137. D.M. Silver, S. Wilson and C.F. Bunge. Phys. Rev. A 19 (1979) 1375. S. Wilson, K. Jankowski and J. Paldus, Intern. J. Quantum Chem. 23 (1983) 1781. P. Saxe, H.F. Schaefer 111, N.C. Handy, Chem. Phys. Lett. 79 (1981) 202. S. Wilson and M.F. Guest, Chem. Phys. Lett. 73 (1980) 607. C.F. Bender and E.R. Davidson, J. Phys. Chem. 70 (1966) 2675; Phys. Rev. 183 (1969) 23. W. Meyer and P. Rosmus, J. Chem. Phys. 63 (1975) 2356. S. Wilson and D.M. Silver, J. Chem. Phys. 67 (1977) 5400. H.J. Werner and E.A. Reinsch, Proc. 5th Seminar on Computational Problems in Quantum Chemistry, eds. P.Th. van Duijknen and W.C. Nieuwpoort, Groningen (1981). S. Langhoff and E.R. Davidson, Intern. J. Quantum Chem. 8 (1974) 61. S. Wilson and D.M. Silver, J. Chem. Phys. 67 (1977) 1689. A.K.Q. Sui and E.R. Davidson, Intern. J. Quantum Chem. 4 (1970) 223. M. Urban, J. Noga and V. Kello, Theoret Chim. Acta (1983). M. Urban, J. Noga, V. Kello and S. Wilson, Mol. Phys. (1983). S. Wilson, Advan. Chem. Phys. (in preparation). S. Wilson, Theoret. Chim. Acta 58 (1980) 31. S. Wilson, Theoret. Chim. Acta 59 (1981) 71. H.M. Quiney, I.P. Grant and S. Wilson, J. Phys. B (1985). J. Paldus and H.C. Wong, Comput. Phys. Commun. 6 (1973) 1. H.C. Wong and J. RaIdus, 1973, Comput. Phys. Commun. 6 (1973) 9.
Stephen WILSON Theoretical Chemistry Depurtmenr,
University
of Oxford, UK
1985
NORTH-HOLLAND
- AMSTERDAM
S. Wilson / Diagrammutic
390
many-body yerturOotion
theory
Contents 1. Introduction ............................................... 2. Diagrammatic many-body perturbation theory ....................... 2.1. Size-consistency of correlation energies ........................ 2.2. Perturbation theory ..................................... 2.3. Rayleigh-Schrodinger perturbation theory ..................... 2.4. Many-body perturbation theory ............................. 2.5. Diagrams. ............................................ 2.6. Second-order and third-order correlation energies ................ ................... 2.7. Diagrammatic expansion for the wavefunction 2.8. Fourth-order and fifth-order correlation energies ................. 2.9. Second-order and third-order terms for closed-shell systems ......... 2.10. Second-order and third-order terms for open shell systems .......... 2.11. Fourth-order terms for closed-shell systems ..................... 2.12. Fourth-order terms for open-shell systems ...................... 2.13. Higher-order terms ...................................... 2.14. Multideterminantal reference functions ........................ 2.15. Valence bond reference functions ............................ 3. Approximate d~agra~lmatic perturbation theory ...................... expansion ............... 3.1. ‘Balanced’ truncation of the perturbation ............. 3.2. Alternative reference wavefunctions and Hamiltonians 3.3. Higher-order terms ...................................... 3.4. Analysis of contemp~)rary approaches to the correlation problem ...... .................... 4. Computational diagrammatic perturbation theory 4.1. Algorithms for perturbation theory ........................... 4.2. Second-order and third-order energy calculations ................. 4.3. The scaling technique .................................... 4.4. Open-shell systems ...................................... 4.5. Parallel computation ..................................... .............................................. 5. Applications. 6. Summary and future directions .................................. Appendix A. Second- and third-order energy expressions for open-shell atoms and molecules .............................................. ........... Appendix B. Fifth-order energy diagrams for closed shell systems Appendix C. Expressions for energy components for a multideterminantal reference function .............................................. Appendix D. Shifted denominator perturbation expansion for the energy through third-order ................................................ References ..................................................
392 392 393 393 395 397 398 403 410 415 417 418 418 431 431 436 448 453 453 454 456 458 458 458 460 467 468 468 469 472 474 476 477 477 478
Computer Physics Reports North-Holland,
391
2 (1985) 389-480
Amsterdam
DIAGRAMMATIC MANY-BODY PERTURBATION MOLECULAR ELECTRONIC STRUCTURE Stephen
WILSON
Theoreticul Chemistr) Received
THEORY
OF ATOMIC
AND
* Department,
Uniuersit~ of Oxford, UK
15 March 1984: in revised form 23 January
1985
The diagrammatic many-body perturbation theory of atomic and molecular electronic structure is reviewed. Attention is centred on the formulation of the method within the algebraic approximation (that is, using finite basis sets) which allows applications to be made to atoms and molecules within a unified framework. The second section is devoted to the basic formalism of the diagrammatic many-body perturbation theory of atoms and molecules. The diagrammatic expansion of both the energy and the wavefunction are considered. the latter being more compact than the former. Section 3 is concerned with aspects of the truncation of the perturbation expansion and the use of many-body perturbation theory in the analysis and comparison of many contemporary approaches to the correlation problem is briefly outlined. The fourth section is devoted to computational aspects of many-body perturbation theory calculations. Some applications are very briefly discussed in section 5.
* S.E.R.C.
Advanced
Fellow.
0167-7977/85/$32.55 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
392
S. Wilson / Diugrummut~c muny-body perturbation
theory
1. Introduction
Diagrammatic many-body perturbation theory forms the basis of a highly systematic and computationally efficient technique for the study of electron correlation effects in atoms and molecules. (See, for example, refs. [l-9] and references therein.) Independent electron models, such as the Hartree-Fock method, account for the major portion, typically 99.5%, of the non-relativistic energy of an atom or molecule. However, the remaining energy, the correlation energy, is, unfortunately, of the same order of magnitude as most energies of interest in chemistry: binding energies, activation barriers, excitation energies, barriers to internal rotation, etc. Over the past twenty years, the diagrammatic perturbation theory has been shown to afford an accurate description of electron correlation effects in atoms and molecules; it provides correlation energies which are as accurate as, and in many cases more accurate than, those obtained by employing other contemporary theories of atomic and molecular structure. Furthermore, the diagrammatic perturbation theory provides a powerful method for the analysis of various contemporary techniques for the study of electron correlation in molecules and yields valuable insight into the relations between them [9]. The pioneering work on the application of the diagrammatic many-body perturbation theory to atomic and molecular systems was performed by Kelly [lo]. Kelly applied the method to atoms using numerical solutions of the Hartree-Fock equations. Many other calculations on atomic systems were subsequently reported [II]. The first molecular applications of diagrammatic perturbation theory were also reported by Kelly [12]. He employed a single-centre expansion and treated the additional nucleus (or nuclei) as a perturbation. This approach is limited to simple hydrides containing one heavy atom which is used as the expansion centre. More recently, the many-body perturbation theory has been applied to arbitrary molecular systems by invoking the algebraic approximation [13], an approximation which is fundamental to almost all molecular calculations and in which the single particle state functions, the orbitals, are parameterized in terms of some finite basis set. A number of reviews and monographs dealing with the application of diagrammatic many-body perturbation theory to molecules have been published in recent years [l-9]. The article is concerned with the application of diagrammatic many-body perturbation theory within the algebraic approximation. It is particularly concerned with the computational aspects of this approach to the calculation of correlation energies. In section 2, the basic theoretical apparatus of the method is given. Various approximations to the perturbation expansion are described in section 3 where the use of perturbation theory in the analysis of different approaches to the correlation problem is discussed and a brief overview of the inter-relation of many contemporary theories is given. In section 4, we turn to the computational aspects of diagrammatic many-body perturbation theory within the algebraic approximation. A very brief review of some applications of the method is given in section 5.
2. Diagrammatic
many-body
perturbation
theory
Diagrammatic many-body perturbation theory provides an expansion for the correlation energy of an atom or molecule the terms in which may be associated with linked diagrams. The
linked diagram expansion is central to the success of the nlany-body perturbation theory in that it ensures that the calculated correlation energies are directly proportional to the number of electrons, IV, being considered. Often the quantum chemist is concerned not with the properties of a single molecule but with the comparison of a number of similar species. Indeed, the chemist is most frequently interested in periodic trends, homologous series and concepts such as fullct~onal group in molecules of different sizes. It is, therefore, important that the theoretical method which we employ to describe such systems yield energies and other expectation values which are directly proportional to the number of electrons in the system. This property, which has been termed size-consistency [14], enables meaningful comparisons of systems of different sizes and also facj~~tates the accurate study of various dissociative and reactive chemical processes.
2.1. Size-consistency of correlation energies A simple illustration of the importance of developing theories of electron correlation yielding energies which are directly proportional to N, is provided by a mode1 system c~nsistjng of n non-interacting atoms, beryllium atoms, say. The linked diagram theorem of many-body perturbation ensures that the correlation energy of the supersystem is equal to n times that of an isolated atom, i.e.
This is illustrated in curve (a) of fig. 1 where the correlation In this figure, curve (b), and estimate of the correlation interaction when limited to single- and double-replacements tal function is also given for comparative purposes. It is well
energy is plotted as a function of n. energy recovered by configuration with respect to a single determinanknown that
where Q is the correlation energy resulting from single- and double-replacement configuration interaction calculations. The curves shown in fig. 1 demonstrate quite clearly that the diagrammatic many-body perturbation theory, in contrast to the limited configuration interaction, is able to handle correlation effects irrespective of the number of electrons in the system under consideration. 2.2. Perturhatim theory In addition to being size-consistent, diagrammatic many-body perturbation theory provides a highly systematic approach to the evaluation of correlation corrections to independent electron models, in that, once the electronic Hamiltonian has been partitioned into a zero-operator and a perturbation, i.e.
394
S. Wilson / Diagrummatic
many-hod,: perturbatron theor?
Fig. 1. Correlation energy for an array of a beryllium atoms obtained theory; (b) single and double replacement configuration interaction.
by (a) diagrammatic
many-body
perturbation
there is a clearly defined order parameter, A, in the expansion for expectation values providing an objective indication of the relative importance of various terms. The total Hamiltonian describing the electronic structure of an atomic or molecular system may be written as a sum of one-electron terms and of two-electron terms $=Ch,
(4)
+ CR,,~ I
’ ‘.J
where h, is the one-electron operator associated with the ith electron and g,, describes the interaction between the ith and the jth electrons. The unperturbed Hamiltonian operator is written as a sum of one-electron operators (5) where /,
is the orbital
operator
defining
the reference
model
/q=h,+u,, and U, is some effective one-electron potential perturbation operator may then be written as 4
= c R,., - c u, ’ 1‘.I
I
such
as the
Hartree-Fock
potential.
The
(7)
S. Wilson / Diagrammatic
many-body perturbation
395
theoq
2.3. Rayleigh-Schriidinger perturbation theory The many-body perturbation theory may be derived from the tion expansion corresponding to the zero-order Hamiltonian (5) For simplicity, we shall restrict our attention to the perturbation a single determinantal reference function. In the non-degenerate tion theory for the total energy of a molecule, the zero-order occupied spin orbital energies, E;
z:
E*=
Rayleigh-Schr~dinger perturbaand the perturbing operator (7). series developed with respect to Rayleigh-Schrodinger perturbaenergy is then the sum of the
(8)
61
i = occupied
and the first-order energy is the expectation determinant, 1G$) that is
value of the perturbing
operator
The sum of the zero-order energy and the first-order energy is the expectation Hamiltonian for the independent electron model wavefunction, that is
for the reference
value of the total
00)
c”,EM=E,+E,. if we use the Hartree-Fock model as an initial approximation to the wavefunction 6 iEM = ‘.4”,, or in a finite basis set b,,, = gsCf. The total energy is given by E = 6i,,
+E,+E,+...
(11)
and, therefore, the correlation energy, with respect to the particular independent employed, is given by the sum of second- and higher-order energy coefficients 6 correlation
=
&
=
0
i--
is the reduced
expansion as
for the correlation
model
energy, with respect to a
resolvent
I@oo)(@ol E,-.&
electron
(14
E* + E3+ . ***
The Rayleigh-Schrodinger perturbation reference determinant QO, may be written
in which 4,
then
(14)
*
The first few orders of the Rayleigh-Schrbdinger
perturbation
theory
expansion
for the correla-
396
S. Wilson / Diagrummuric
tion energy,
therefore,
/‘k,-
muny-ho&
perturbation theog
take the form
PKl~l@K))(Eo-
P’J&I@,>)’
Rayleigh-SchrBdinger perturbation theory leads to an expansion for the correlation energy and other properties which is directly proportional to the number of electrons, N, in the system. To simplify the discussion let us restrict ourselves to closed-shell systems described in zero-order by a Hartree-Fock wavefunction. The matrix element (C+, 12, I tDK) which arises in the numerator in the expansion (15) for the second-order energy is only non-zero, in the closed shell case, if QK is obtained by a double replacement with respect to QO. If QK is obtained by replacing spin-orbitals i and j in QO by spin-orbitals u and h then
where 8 = (1 - (12))r,<’ and (12) is the permutation
operator
which interchanges
the co-ordinates
S. Wilson / Diagrammatic
of electrons
1 and 2. Thus the second-order
marybody
perturbation
theory
397
energy may be written
a, b, c,. . . will be used to denote k,... will be used to denote occupied spin-orbitals unoccupied spin-orbitals. It is clear that if expression (19) were used to calculate the correlation energy of n non-interacting helium atoms, described in zero-order by localized orbitals, an energy would be obtained which would be n times that for a single helium atom; that is E, a: n. In the second term in expression (16) for the third-order energy we observe that E,a n and the summation, which we denote by A and which differs from E, only by an additional denominator factor, also leads to a terms directly proportional to yt. Hence E,A a n2.However, if we compare the diagonal part of the first term in our expression (16) for the third-order energy, i.e. i, j,
with the second
term written
in the form
(21) it can be seen (remembering that GK differs from Cp, by a double replacement) large degree of cancellation between these two terms. Performing this cancellation following expression for the third-order energy is obtained.
that there is a explicitly, the
(22) Each of the terms in expression (22) are directly proportional to hr. Beyond second-order in the Rayleigh-Schrodinger perturbation theory of electron correlation energies, terms arise having a non-linear dependence on the number of electrons in the system. However, such terms can be shown to mutually cancel in each order. This cancellation was first demonstrated in the first few orders by Brueckner [15] and was then shown through infinite order by Goldstone [lci]. By explicitly performing this cancellation of terms in the Rayleigh-Schrodinger perturbation theory one is led to the many-body perturbation theory. 2.4. Many-body ~~r~urb~~ion fheov In diagrammatic many-body perturbation theory, it is recognized that components of the correlation energy which are directly proportional to N may be associated with linked diagrams
whilst terms having a non-linear dependence diagrammatic perturbation theory, therefore, energy.
on N may be described by unlinked diagrams. The leads to the following expression for the correlation
(23) where the subscript L denotes that only terms corresponding to linked diagrams are included in the summation. This is the linked diagram theorem of Brueckner and Goldstone. The linked diagram theorem is of central theoretical importance in the application of quantum mechanical methods to problems of chemical interest. Derivations of the linked diagram theorem of many-body perturbation theory have been given in many text books (see, for example, refs. [17,18]). A time-dependent derivation along the same lines as that first presented by Goldstone 1161 is followed by March, Young and Sampanthar 1171. On the other hand, a time-independent approach, which is perhaps more satisfying since the problem of describing electrons correlation effects does not actually depend on time, is described by Paldus and Cizek in ref. [lg]. In the diagrammatic formation of the many-body perturbation theory the particle-hole formalism is employed. Fig. 2 demonstrates the simple relationship between the particle formalism and the particle-hole formalism. Particles may be created above the Fermi level and holes may be created below the Fermi level. In the diagrammatic notation, particles are represented by upward directed lines and, in the present work, labelled by the indices a, b, c, . . .
whilst holes are represented i
by downward
directed
lines and labelled by the indices i, j, k, . . . (25)
2.5. Diagrams It should be emphasized, at this point, that the use of diagrams in the many-body perturbation theory is by no means obligatory. The whole theoretical apparatus can be set up in entirely algebraic terms. However, the diagrams are both more physical and easier to handle than the corresponding algebraic expressions and it is well worth the effort required to familiarize oneself with the diagrammatic rules and conventions. Furthermore, by using diagram techniques, intricate combinatorial problems which arise in the algebraic approach are replaced by very simple topological considerations. A diagram is said to be closed if it contains no free lines otherwise it is said to be open. Thus, for example, figs. 3(a, c) are closed diagrams; figs. 3(b, d, e) are open diagrams. A diagram containing open lines is said to be connected if it does not contain two or more separate parts. Figs. 3(b, e) provide examples of connected and disconnected diagrams, respectively. A linked diagram is connected and closed; an unlinked diagram contains a disconnected and closed part.
S. Wilson / Diagrammatic
a
b
-__-
0
theory
399
-._--_
----
----------_-----------------. l _ 0 0
C
many-body perturbation
Fermi
level
Fermi
level
_
0
0
~--------------_-_----------~ 0
0 0
0
-_ 0
e
0
0
f
0 0
0
Fermi level ______---_--------------------l l
0
0 __I_-._--... 0 __l_l._...~ _
0
Fig. 2. The particle and the particle-hole formalism. (a) The ground state in the particle formalism; (b) The ground state in the particle-hole formalism; (c) A single excitation in the particle formalism; (d) A single excitation in the particle-hole formalism; (e) A double excitation in the particle formalism; (f) A double excitation in the particle-hole formalism.
Examples of linked and unlinked diagrams are displayed in fig. 3(a) and figs. 3(c, d), respectively. A number of diagrammatic rules and conventions are used in the literature. Here we shall follow the conventions introduced by Brandow [19] and then indicate briefly the relationship between this and other conventions which are in common usage. The basic diagram elements are shown in fig. 4. Downward (upward) directed lines are used to represent holes (particles) created below (above) the Fermi level. Matrix elements for one-electron operators are represented by the diagram elements shown in fig. 4(c). For two-electron matrix elements the diagrammatic representation is given in fig. 4(d). In the Brandow convention the two-electron matrix elements include a permutation of the electrons and are, therefore, sometimes referred to as antisymmetrized vertices. The rules for translating a given Brandow diagram into the corresponding algebraic expression are as follows: (i) Assign a unique ‘hole’ index, i, j, k, . . . to each downward directed line. (ii) Assign a unique ‘particle’ index a, b, c, . . . to each upward directed line. (iii) There is a summation over each hole index covering all occupied spin-orbital indices. (iv) There is a summation over each particle index covering all unoccupied spin-orbital indices. (v) The numerator of the summand consists of a product n integrals in n th order perturbation
400
--..-..---
a
I___
~
-_-__--
~ --L_--
~
-_____
~
Fig. 3. Examples of types of diagrams. Diagrams (a) and (c) are closed and the remaining diagrams are open. The diagrams are described as follows: (a) linked: (h) connected; (c) unlinked; (d) unlinked; (e) disconnected.
theory.
These integrals
may be one-electron
integrals
(26)
S. Wilson / Diagrammatic
many-body perturbation
a hole Zinc
------e
-----b
ow-etactron
line
d
matris alemc2lt Fig. 4. Diagram
two-electron
401
b
particte
C
or antisymmetrized
theory
elements.
integrals
(27) distance. where 4 is spin orbital, ic( r) is some one-electron potential and Y,~ is the interelectronic For each interaction line there is an integral factor. The spin-orbital indices can be read from the labelled diagram. For the two-electron interaction the indices p, q, Y, s should correspond to the hole and particle lines entering or leaving the interaction in the following order: left-in, right-in, left-out, right-out. (vi) The denominator of the summand consists of a product of factors. There are (n - 1) factors corresponding to an n th order diagram. There is a demoninator factor arising between each pair of adjacent lines in the diagram. A denominator factor consists of the terms
where the E’S are orbital energies. The first summation is over all hole lines that extend between adjacent interactions and the second summation is over all particle lines. (vii) There is a multiplicative factor of (i)“‘, where m is the number of pairs of equivalent lines. Equivalent lines begin at the same interaction, they end at the same interaction and the have arrows going in the same direction. (viii) There is a multiplicative factor of (- l)p, where p=h+l,
(29)
h is the number of hole lines in the diagram. 1 is the number of fermion loops. A fermion loop is determined by following the hole and particle lines, in the direction of the arrows, to form a continuous closed loop.
402
S. Wilson / Diagrammatic
many-body perturbation theory
As an example of the application of these rules, consider the following diagram, which involves a triply-substituted intermediate state
fourth-order
energy
(30) The hole and particle
lines in this diagram
may be labelled to give
b
There is a summation over the five hole indices and three particle indices. Working from the bottom to the top of the labelled diagram, the following integrals are obtained in the numerator of the summand:
(34
grjubgklrcgmcklguhm/
and for the denominator
factors,
from the region between
the lower pair of interaction
lines, we
get E, + e,, - E, - Eh from the central
part of the diagram
(33) we have
and from the upper part
k and I label equivalent hole lines and u and b label equivalent particle lines. Thus there is a multiplicative factor of (4)’ = f . There are 3 fermion loops and 5 hole lines giving a factor of 3+5 = + 1. The algebraic expression corresponding to the above diagram is, therefore (-1) (36) Two other diagrammatic conventions are commonly used in many-body perturbation theory. The Gc ldstone diagrams [16] are similar to those of Brandow except that the two-electron interaction lines do not include permutation of the two electrons involved. Thus there is a set of Goldstone diagrams, which are related by electron exchange, corresponding to each Brandow
S. Wilson / Diagrammatic
many-body perturbation
403
theory
og - ----
Q&o @fJ
(yJ
----_----_---
Fig. 5. A set of Goldstone diagrams Brandow diagram the other diagrams
related by electron exchange. If any one of these diagrams are accounted for by using antisymmetrized vertices.
is interpreted
as a
diagram. This correspondence is illustrated in fig. 5. The diagrams of Hugenholtz [20] are in one-to-one correspondence with those of Brandow. The Hugenholtz diagrams can be obtained by replacing the interaction lines in the Brandow diagrams by a single dot. This correspondence is illustrated in fig. 6. It should also be mentioned that some authors rotate the Brandow, Goldstone or Hugenholtz diagram through HIT(see, for example, ref. [18]). 2.6. Second-order
and third-order correlation energies
The complete set of second-order Brandow diagrams for the many-body perturbation theory of the correlation energy developed with respect to an arbitrary single determinant reference function (not necessarily a self-consistent-field function) is given in fig. 7. The one-electron part of the perturbation (U in eqs. (6) and (26)) is represented by a dashed line terminated by a cross.
Fig. 6. Examples
of the one-to-one
correspondence
between
Brandow
and Hugenholtz
diagrams.
S. Wilson / Diagrammatic
404
(B)
(A)
p-g
many-body perturhution
theory
CC)
()__()
----_-
(1))
(E)
Fig. 7. Second-order Brandow diagrams in the many-body with respect to a single determinated wavefunction.
The algebraic
expressions
corresponding
perturbation
to these diagrams
expansion
for the correlation
energy developed
are (37)
E,(B)=
cc=. ‘, ‘I
’
0
(38)
(39)
If real orbitals are used then E,(B) = E,(C). There are 43 Brandow diagrams in the third-order component of the correlation energy with respect to an arbitrary single determinant. The complete set of third Brandow diagrams is given in fig. 8. The algebraic expressions corresponding to these diagrams have the following form:
(42)
(43)
S. Wilson / Diagrammatic E3(C)
=
c
_
405
‘fuUYg/kak
1
(;X_
many-body perturbation theory
u
(Ei-f,)(E;--~)’
(44)
(49) qI)
= _ c
c
U,,g/klkg,~,/
,jkl cd
(50) (51)
E,(K)=
(54
-cc i/k/
a
(53) (54)
(55)
(56) (57)
406
S. Wilson / Diagrammatic
many-body perturbarion theory
(58)
(60) (61) (62) 63)
(64
(65)
w-9
(67)
(68)
(69) (70) (71)
S. Wilson / Diugrummntic
many-body perturbation
theor)
407
(72) (73)
(74)
(75)
Edi) = C C r/k
ub
&/‘,h%kJh (E,
+
E,
-
6,
‘U; -
Eb)G,
-
4
(76)
’
(77) (78)
(82)
o-
(83)
The first 20 terms, corresponding to diagrams (A-T) contain only singly-excited intermediate states. Diagrams (U-j) give rise to expressions which contain both single replacement and double replacements. The last 7 diagrams lead to expressions containing states which are obtained by double replacements.
---_ x -------_ X 0------0 x
40x
S. Wilson / Diagrammatic
(A)
(B)
many-body perturbation
theory
0 -----X ----__
X
CC)
CD)
--_--__ u --_u 0-----\< 0 ------_ u x -----u ---0u a --_.__u 0
(H)
(I)
’
0 -_-.0 -----X 0)
(K)
-0 -__-. 0 -_-_._ 0
-e-m
----
(M)
0
(h’)
(0)
(P)
--_--x 0-___---
0
0 __----
---------_ -------0 0 ---0 0-----u 0----_x CT _---_x (Q)
CR)
(S)
CT)
(W)
03
-e--v
(U)
W)
Fig. 8.
__---_ -_--_ 0 o4 0__---x 0---__-----_ x ---0-----0 -----x 0-----0 --_-x 0----0 _--_ ------__-___ as 00 -----00 __--__ 409
S. Wilson / Diagrammatic many-body perturbation theoq
X
(b)
(2)
(Y)
Cd)
Cc)
K
(9)
(h)
(i)
(j)
(k)
(1)
Cm)
(n)
----_---
(0)
_------
(q)
(P)
Fig. 8. Third-order Brandow diagrams in the many-body with respect to a single determinantal wavefunction.
perturbation
expansion
for the correlation
energy developed
410
S. Wilson / Diagrammatic
man.v-ho& perturbation theoty
v___)( i!LoY--Y (A)
(B)
Fig. 9. First-order Brandow diagrams in the many-body perturbation developed with respect to a single determinantal wavefunction.
2.7. Diagrammatic
(0
expansion
for the correlated
wave function
expansion for the wavefunction
It is well known that the energy through 2n + 1 can be written in terms of the wave function through order n. This is usually referred to as Wigner’s (2n + 1) rule. In diagrammatic many-body perturbation theory Wigner’s rule takes the form [21] J% = N-1
(85)
P+llU_~
E 2n+1= W,l~lG_~
(86)
where the subscript L indicates that only linked energy diagrams are to be included. Wigner’s rule forms the basis of efficient algorithms for the evaluation of the energy coefficients. However, further discussion of this aspect will be postponed until section 4. Here we note that there are a smaller number of wavefunction diagrams in order n than there are energy diagrams in order 2n and 2n + 1. In this sense, the diagrammatic many-body perturbation theory expansion for the wavefunction is more compact than the energy expansion. For example, there are 5 second-order energy diagrams and 43 third-order energy diagrams shown in figs. 7 and 8, respectively, but these diagrams arise from the 3 wavefunction diagrams displayed in fig. 9. Diagrams (A) and (B) in fig. 9 involves states obtained by single replacements with respect to the single determinantal zero-order wavefunction whereas diagram (C) involves states obtained by double replacements. If we represent the state function obtained by replacing spin orbitals i, j, k,. . . by spin orbitals to the diagrams shown in a, b, c,. . . by I @,$T;;- ) then the algebraic expressions corresponding fig. 9 take the following form:
h(A)= q io
h(B)=
w,(*j).
I
(87)
0
+W(~+j)~ IO
I
(88) ‘I
It should be noted that all of the first-order wavefunction diagrams are connected. In order to determine the fourth- and fifth-order energies the second-order wavefunction
is
S. Wilson / Diagrammatic
many-body perturbation
theory
411
required. The 32 second-order wavefunction diagrams are given in fig. 10. Both connected and disconnected wavefunction diagrams arise in second-order. These diagrams involve intermediate states which are obtained by single, double, triple and quadruple replacements with respect to a single determinant. The algebraic expressions corresponding to these wavefunction diagrams have the following form:
(90)
(91)
(92)
(93)
(94)
(95)
(96)
(97)
(98)
(99)
000)
412
S. Wilson / Diagrammatic
(A)
many-hods perturbation theor,
CD)
(B)
c ---x I,: k
--v v v_--vv _Y v x--_-u -----0
I -----
-m-q-
(E)
0
----
(F)
X
0
(11)
o--_ u CL)
V’rJ --
0 -----
(Q)
(R)
\
\p~
L_____
------
X UJ)
(VI
Fig. 10.
(S)
CT)
\
----
-----
/
X
(P)
v ‘I?’ ----_ ----0 0
v _---v ::
I
(0)
i
co
S. Wilson / Diagrammatic
many-body perturbation
theory
413
\/____()\/_LD if__+/ l(J._._y -----_--___- -----(Y)
(2)
(a)
(b)
Cc)
Cd)
(e)
(f)
Fig. 10. Second-order Brandow diagrams in the many-body perturbation developed with respect to a single determinantal wavefunction.
expansion
for the correlated
E,
EC)
wavefunction
giJahgklcl (E;
h(P)
=
+
cj
-
E,
-
Cb)(Ei
+
E, +
Ek -
-
Eb -
’
(103)
005)
c ijkluhc
(106)
(107) 449=
c ijkluh
(108) i
414
S. Wilson / Diagrammatic
&(T) =
c ykatx :
I@$)
(E, + fi _
many-bo&
perturbation theocy
gr~Jocgckhk
009)
i
%&r,kh 011)
013)
(f
i
+ f
(114)
(315)
(E,+f:, 1
_ k
gikhrgb’uk Cb--
C,)(E,
+
f;
-
E,
-
en)
’
0 17)
(118)
gtiohgiklc
i
(fi+‘i-r,-r,)(r,+‘jS.~,--E,--~-f,.)
S. Wilson / Diagrammatic
2.8. Fourth-order
and fifth-order
The fourth-order
many-body perturbation
theory
415
correlation energies
energy may be written
Change in level Interral
-2
type
of
excitation
-1
i +
/
-_ li”u
jb>
--cJ---v 0
+1
-_P J 4 ---‘II’
+2
+
-.__ 4M
ti;iIo^/bc>
i -&
w-_
A
_H
7-
Cij (81 kj>
--0
--of
-0
?4 --a
-r -v--* t
-0
A
_ ._ _ k
-a
0
Fig. 11. Classification
of diagram
components
according
to the change in the level of excitation.
416
S. Wilson / Diagrammatic many-body perturbation theory
and thus using the expression above for $, and I,L~the components can be written down. For example,
of the fourth-order
energy
(123) The matrix element (@,O1J?~ I QJ’) does not involve a change in the degree of replacement with respect to the reference determinant and can, therefore, give rise to terms corresponding to the diagram components displayed in the central column of fig. 11. This can lead, for example, to energy diagrams of the form
Fifth-order energy diagrams can be generated in a similar fashion using the second-order function diagrams displayed in fig. 10 and the corresponding expressions given above formula
,wave in the
For example,
(+,(A) W’IIc/dA)) = c i% ij hd
(125) i
The matrix element (@,! I Xl I @,‘) does not involve a change in the degree of replacement with respect to the reference determinant and again can give rise to the diagram components displayed in the central column of fig. 11. In this way energy diagrams of the form
are obtained.
It should be clear from the above discussion
that the number
of energy diagrams
S. Wilson / Diagrammatic
A__-”
1
__--0
+
+
many-body perturbation
/y(
a
/_____~
theory
417
=
o
x
0
ii!__, +bl--x =o Fig. 12. Diagrammatic representation of the cancellation which occurs if the canonical in the zero-order wave-function for a closed-shell system.
actually
increases
2.9. Second-order
Hartree-Fock
orbital
are used
rapidly with order of perturbation. and third-order terms for closed-shell systems
For closed-shell systems described in zero-order by a single determinant constructed from canonical Hartree-Fock orbitals there are important simplifications. Terms corresponding to diagrams containing “bubbles” are exactly cancelled by terms corresponding to diagrams containing a cross representing the Hartree-Fock potential (multiplied by - 1). This cancellation is illustrated graphically in fig. 12. For example, the following relations are then satisfied E,(A)
+ E,(D)
= 0,
(126)
E,(A)
+ E3(0)
= 0.
(127)
418
S. Wilson
(A)
/
~~u~r~rnrn~ti~
(B)
mango-body perturbutjo~
(Cl
theory
CD)
Fig. 13. Non-zero second-order and third-order Brandow diagrams in the many-body perturbation expansion correlation energy of a closed-shell system developed with respect to a single determinantal wavefunction.
for the
If canonical Hartree-Fock orbitals are employed for a closed-shell system then the only terms which have to be considered correspond to the four diagrams shown in fig. 13 (see, for example, ref. [13]) since all other diagrams shown in figs. 9 and 10 mutually cancel. If the occupied orbitals are subjected to an arbitrary unitary transformation, for example, if some localization procedure were used, then diagrams such as those shown in figs. 8(k, m) would have to be considered. On the other hand, if the virtual orbitals were subjected to a unitary transformation then one has to include the diagrams shown in fig. 8(1, n). 2. IO. Second-order
and third-order terms for open-shell systems
For an open-shell system described in zero-order by a restricted Hartree-Fock determinant, the second- and third-order energy diagrams which have to be considered are shown in fig, 14 (see, for example, ref. [22]). In this figure the solid dot represents a potential whose precise form is determined by the particular electronic state under consideration. A full set of second- and third-order energy expressions for open-shell systems is given in Appendix A. 2. I I. Fourth-order
terms for closed-shell systems
The perturbation expansion for the correlation energy of a closed-shell system through third-order only involves doubly substituted states as can readily be seen from fig. 13. In fourth-order terms, which are obtained by single replacements. double replacements, triple replacements and quadruple replacements, can arise in the energy diagrams. In fig. 1.5 the complete set of fourth-order Brandow diagrams for a closed-shell system, described in zero-order by a determinant constructed from Hartree-Fock canonical orbitals (see, for example, refs. [2,23]), is given. There are 39 diagrams in total; four contain single replacements, twelve contain only double replacements, sixteen contain triple replacements and seven contain quadruple replacements. The algebraic expressions corresponding to each of these diagrams are as follows:
(129)
---* ---0--_-_ 0----0 _--+ --_ 6-----_ 0 (F)
S. Wilson / Diagrammatic many-body perturbation theory
419
0
(D)
(B)
(A)
l
(El
(1)
(G)
(H)
(J)
(K)
CL)
(N)
(0)
(P)
Fig. 14. Non-zero second-order and third-order Brandow diagrams in the many-body perturbation expansion for the correlation energy of an open-shell system developed with respect to a restricted Hartree-Fock single determinantal wavefunction. grJabgkbiJgalcd!?cdkl
(130) E,(D,)
= +c
(E + ~ _ (
gijubgkbrJglmkcgac/m
031) I
Et(&)
=
&c
0 -
J
(E
+ I
~
_ J
E
cb)(ck
~ u-b
-
%h+
E,
-
cu
-
)4i:y:p;bc~~;y~;;(< I
J
cc)
(132)
+ ‘
d
1
’
‘J-‘e-
grJuhgabcdgk/iJgcdkl E4(BD)=
Kc
(
(133)
420
--------- --_------_ - ------ --0 0 0 0 --------------------_---------_---00 ---- c.H? ----_00 --_--00 ------------_----_. ----------_ ------_---_ --_-a0 ----- 00 ----- CJ @O 0 --------------_-_ --_ -__ --00 _-__-0----0_ c!J S. Wilson / Diagrammatic
many-body perturbation
theory
0 --___ __--_ 0 -_---_
0 -----
---_--0
0 ---------0
0 _---_---0
PSI
tBsl
PSI
PSI
b)
ICnl
kD1
kD1
(GDI
PDI
b)
0 _-_ ------.0
(ID)
b)
(BT)
kD1
Fig. 15.
PTI
-----_ -_ _ the --_ -__ _ _ _c)” ---_ _-_ _ @Q9 -----__-_ -tbo S. Wilson / Diagrammatic
many-body perturbation
tETl
(GTI
71-j-
LKTl
theory
(“4
--& ----__ 3 -----
--_-0-b ------_-_ --_ -Qeo 421
(IiT)
--------@a -(iTi
----@Ii0 1’3
(“d
@O
lECl Fig. 15. Fourth-order Brandow diagrams in the many-body perturbation expansion closed-shell system developed with respect to a single determinantal wavefunction.
for the correlation
energy
of a
S. Wilson / Diagrammatic
422
many-hodJ perturbation
theory
(134)
g,.,ohgkl,,g,,,nklguhm,,
E4(DD)= iE (c,+c
-6
,
c
u--h
(135)
)(q+c,-6
G)(%?,
u-
+
-
cu
-
%)
’
(136)
(137)
(138)
(139)
(140)
E4(JD)
=
_
c
(~
+
L
_
E
(141) I
E4wD)= -c
u
I
Eb)(
-
6,
+
6k
-
E,
-
c,.)(
6k
+
6,
-
fu
-
cd)
’
gr,~bgkbc/gulXdg~~drl (c
+E
(142)
_~
I
‘I
J
-
Eb)(
6,
+
6k
-
E,
-
cc)(
c,
+
c,
-
cu
-
cd)
’
(143)
E4(b)= -2
g,,ub~ukcd&wk&dr/ (c
I
(144)
_-E
+c
I
u-
(146)
E4(W= -E
gr.,crh~kl,~gm~~k/guhml +~
(E
1
(147)
_~
I
u-
c,
+
6k
+
c,
-
co
-
-
cc)(
c,
+
cm
-
<,
-
’
S. Wilson / Diagrammatic
+E I
E4(%)
=
+C
-
J
(c
+
E
’
E,
_
-
Eh)(Ci
~
E
J
+
)(c
EJ
many-body perturbation
+
Ek
-
Eh
-
C‘
-
Ed)(
+g~+““:“c’;‘kf;hnlJ
u-h
E,
+
~ ,)(c
f
J
u
h-
423
theory
CJ -
+
<
-
E,,)
049)
’
(150)
E Eb)
n,-co-
,
’
g~/uhgklr~gn,bkJg,,.~,l E4(W
=
c
( cr
+
c., -
E‘,
Cb)(
-
C/ +
Ek
+
6,
-
E,
-
Cb -
E,.)(
E,
+
en1 -
E,
-
E,
)
-
E,
-
Ed)
051)
’
~~Jub~uk~d~lb,,~~~dIk E4(IT)
=
-
aC
(
6,
+
c,
-
cu
-
Cb)(
E,
+
C/ +
Ck
-
Eb
-
cc, -
Cd)(
fk
+
E,
&(JT) = - c (E + c I
053) b-
E )(E+g;;;":"":"dy:bk' (E + ~ _~ ‘<& I LI b' J u-b J
E4(KT)= -c
054
’
054) +
‘h
-
cb -
Ed)
’
giJubgkircguhdjgdckl Eh)(
u-
E,,
+
Ek
+
c,
-
cu
-
Eh
-
c,.)(
Ck
+
E,
-
6‘.
-
Ed)
’
gIJ”h&k
Eh)(
E,
+
E/
+
Ck
-
Eb
-
E,.
-
Ed)(
E,
+
c,
-
E,
-
Cd)
’
-
6,.
-
Ed)
’
grjuhgklrcgrrhdlgdck] E4(W=
tC
(
E,
+
CJ -
E,
-
Eh)(
C/ +
Ek
+
c,
-
ca
-
Cb -
cc)(
C/ +
Ek
055)
056) (157)
(159)
grJuhgklcdgcbilgudkJ
(160)
gi,~rrbgklcdgcdrJgabki E4(BA+CQ)=&z
(E
+~__~
I
J
u-
Cb)(E,
+
cj
-
E,
-
Cd)(Ek
+
6,
-
cu
-
Eh)
’
(161)
424
---------------_ -* -__ -_ -00 -----00 0-i) /so -.__._ ------ -_ 00 0 0 _----_ -_-_. _--__ c!sJ” ---_--------_ _ __ ----_-__ b-O ---fn c!nlb0 ----_ -_ -----. k4J S. Wilson / Diagrammatic many-body perturbation theory
-e --
-----
0
-----
----
(2)
(I)
0
0
(3)
(4)
(7)
(8)
0
--
----
- --
-
(5)
-----
(6)
0
(12)
--_-
-a -_ -
-----_-(13)
-0 _- -
-----
-----
-a
-0
(14)
(15)
(16)
(20)
(22)
Fig. 16.
(23)
-------_ _ __ . -___ --o o _-----_ ----_ ---_o u ----* ao
---- .----0 D 0 a0 0 _---_0D 00 __-_----- l-h----* _--_--__-----_ ---_0 0 u 0 0:0 Q----*I3 S. Wilson / Diagrammatic
-0
many-body perturbation
theory
425
---__L_
0
__-__
-_---
----.
--_-
-_---
----__
----
-__.-.
.
-a
(26)
(25)
0
----
_----
*
(27)
(28)
_--w-e
-we--
*
----_
_-_-
s-e
_--
--_--
-m-v-
--
--__-_
-.
_--
--
*
(29)
(30)
(31)
(32)
0
----
v--e
__-__
_-
-
__-_
---__-
---_-
v-m--
*
----_ a0 (33)
(34)
(35)
(35)
--_ _-_-_ A 0 $3 D D. -----_ --_---_ ----__ __-__ 0-_____@0 0 ----___^__ ___ _-__ -_ __ ---0 c0 0 0 0 0 0 -0
._ - ---
-___-
-_
t
-0
-em_---
_---
-____ j / ___--
__
-___
-_-__
__---
-_-_-_
(37)
0
(38)
(39)
l
V___-__. (40)
_.a-_
0
__-
_-----_
(42)
(43)
-w--D
a ---_
0 -__
_-___ (45)
_
-em--
__
_--___(46)
l
_---__
Fig. 16.
(47)
---0 _______
(48)
426
-_--_ --_--Q a --_--!J 0---_-_-___ 0 ._ -_--_ @---_--0 B o-----_-_-_---
S. Wilson / Diagrammatic
*_--__-
0 --_-_-. ---_----__ _--a
0
*-
_
many-ho&
perturbation
theoq
_ _ ._
V_______ *.-_-
-- 4
_ _ _ _ -----_ -_-----_ -_ *__-*-__.._---0 0 o--0 ______-. --_ -__ -__ __ ___ Q--0 --_ 0 iz __._._~ 0 0--0---_--0 0--_ 0 - - -- _ -.-____-0 (‘..
t-
-
0 ----__. __-_-__ 0
1
(52)
(51)
(50)
(49)
-3 l_ ___ 1 -___---
(54)
(55)
(58)
(59)
0 t---_---0 (56)
*_--
4
(57)
-____-_ -o0 __ -_ --. -----_-_-_-0 0 0 0--0 o(62)
(64)
0 *---__(68)
(G6)
0
_-
D_-_---
0
___--__
(70)
l
/
Fig. 16.
(71)
0 -_--_-_. l
(72)
---_ _---- -___ * _--__-_Q _---. 0 0 Q_ 0 0 t? ______ -_-__ .__ -_ ----__ ___ ^-_ * ---____-_ 0 Q0 ______ 0__--_000o _---_ -_ ----_ --*0 a___ -_ 3 0 a0 o0 ___--____ --_-_ ---_ a3 0 a----_ 0 -. --_ ----0 0 0 0 0 o-S. Wilson / Diagrammatic
e----.-.-.-e _---_ -__--- -
many-hod,:
theor)
421
.._-----
t------
l
-e
.______-V
@
t-----
(73)
\
t-------
-e _-___ __-_--
l
-e *___-_
(79)
(80)
l
(78)
(77)
l --_____,
(76)
(75)
(74)
*_____
perturbation
l -_-___
--se-
_-__ -
-e _.___-_ l _--_-
a
----l ---__-
---a -__----
(81)
--
(82)
(84)
(83)
*-_-_-_
---e ___-@ @__._-___
-e _---. l _-_---_ _
l
(85)
(86)
(88)
(87)
“_____
_-___
--e __-----
_-
-e t--_-_---
e
a---
a___
_----_
----------• --_-_ __----_ -----_ --__-_ 0 _---0 ---_-_ Q o0--_-__0..___ _.__* a _____ 0 (89)
(90)
l _--_----
*-____-
(92)
(91)
e
---_- _ _- ----
e________
l
(93)
(94)
Fig. 16.
(95)
l
(96)
428
--_---*a -----__ _ ----._ ----0 a 0 Q... o:-. ___. _._. ___-* :. __ ----_--* * -__ ._ ___. _ 6-G 0 0 0 0 O----O -------__ -_ __-_-_ -_--\ --_ _-_ 0 0 0 0 0 o-_._ -.-. ----__---_____ --___-_ ---u --_ a o--acl____ _ _ ._ . _--__ --_-* -_ -__-__-_-* _---__ o 0 0 0 u ‘s o0_. ____-_--- -_ _-3 0 w b S. Wilson / Diagrammatic
.I .._._ -e
-0
-_-
many-body perturbation
-3
--- -* - _. __--_ -
-0
-_____-
0
0
--_---
a
-0
(loo!
(99)
(98)
(97)
theor)
0
_-----
-e ______--
0
------
(102)
(101)
0
a
(104)
(103)
-0
0
0
0
-_ 0 ---_-__
_-- l __-__
(105)
-0
e-e_
_-_--
b :------* -_-
(106)
0
--
___ - - . -
---.-
_---_
0
-__---
0
(108)
0
A _- -_---0
e
_--.--
(110)
(109)
-3 ,-_.-e
(107)
-0
_-
--
-e -4
-----
(112)
(111)
0
-_
--
0
-_--
-0
---_---
(114)
(113)
-0
__- ___--
0 --____ 0 (117)
-0
______-_
--_---
--___
----
-
A----
___ ._-_
__.
0 __---_ 0
(118)
(116)
(115)
-0 .- _ ___ . - _ .
Fig. 16.
l
0
--
-1) _ _0
-0
-0
_--
_-_-~-.
(119)
0
_--
_--
-____--
(120)
0
S. Wilson / Diagrammatic
many-body perturbation
theory
429
(123)
(125)
(126)
(127)
__--__ a____-_ -0 l
_---0
__ --_--_--_--_ __ D-*0___ a t-.6 a-----_ Q 0-----o0 (132)
l
0
-a
-_0
------_-------* ------a_-_ *___ * __ 0 0 o0____ 0 ii 0 ----- -_---____ -_-_ * 0 0 a 0----_ t-# (133)
__ 8 0-_
_0
-0
----_
(137)
-0
_--_--
--.
#
--__
b
-------o (141)
-0 --_----
(139)
-0
__ _ --_-0
a_.._
--o
_t
(138)
_---4f
(136)
0
___
-0
_---
(135)
(134)
(142)
-a ---_--_ 0 ._-_
---
---_
Fig. 16.
(140)
---
(143)
l
0
-
(144)
430
__--__*____ 0 S. Wilson / Diagrammatic
many-body perturbation
theory
-0
0 ____
o---_- -_0
-I)
-_-__
--
-e
-_--_--0
----_0 0 Q0 l --_---_
@.j-j)
l ------
---l
-a -_ -0
0
---__-
__-_
(148)
(145)
(,ijj_FD
--___ 00 -__--_
--_-_--
-e
__ __--_-___-__ ____ a Q-_-___ _ ______ --_---* ___-__ --_-_______. Q; _-__0 0 a 0 I) -----_-__ _----_ 0 0 _-___--0__.___ 0----_-_I:---0 ___------_____-_ * --Q_-__ 0 0 o(149)
(150)
(151)
(152)
-0
_-_-___ _-___--
*___--_
D
0
--_---a
--_--__ l (157)
t-------
4 __-__-a
---_____ 0
0 ___ - _ _ -
_-__ *
l
-0 ____--
---_-_ *-__
0
-0
(165)
-0
-0
(162)
-0
(163)
-0 _- _- -. *____
-0
0
-_ _--
-0
_----
8
----_
Fig. 16.
(167)
l --- -__-- ..-
0 -----0
(164)
-0
- __. _--a (166)
(160)
o------
-Q
(161)
*__-_____
-0
_ _ _. - -.-
l
0
(159)
.---_-___
t-------
0
l
(156)
@_________
(158)
l
l
0
(155)
e-------
___-___
-0
-___ ___
-0
___- __0 __--__ l (168)
S. Wilson / Diagrammatic
many-body perturbation
431
(172)
(171)
(170)
(169)
theory
Fig. 16. Additional fourth-order Brandow diagrams which have to be considered developed with respect to a single determinantal wavefunction.
when treating
a-
&(Fq+Gq)= -iC the last three expressions the relation In
to simplify
an open-shell
E
(162)
)’
d
gfJahgktcdgcdilgahkJ Cc
+~
’
_~
J
a-
Q)(
C/ +
Ek
-
we have added the formula
E,
-
Eh)(
6,
+
corresponding
E,
-
cc -
system
Cd)
(163)
.
to two diagrams
and used
the sum.
2.12. Fourth-order
terms for open-shell systems
The additional fourth-order diagrams which arise in the diagrammatic perturbation theory expansion for an open-shell system described in zero-order by a restricted Hartree-Fock wavefunction are shown in fig. 16 (see, for example, ref. [24]). These energy diagrams were function diagrams shown in fig. 10. The algebraic expressions corresponding to the diagrams shown in fig. 16 have the general form
matrix element UP4 = (p, 1u 1q,) or a two-electron ‘factor of the form and D, is a denominator
where X, may be either a one-electron element
gp,q,r,$ = (P,q,l4r,sJ
D, = Dp ,4, . . . ‘,,~,. . . = cp, + c4, + . . . -E,., - es a is a constant
corresponding
...,
determined by the number of pairs of equivalent lines. The algebraic to the energy diagrams displayed in fig. 16 are presented in table 1.
2. I3. Higher-order
matrix
W)
expressions
terms
The fifth-order energy diagrams can also be written down from the second-order wavefunction diagrams given in fig. 10. Some examples of fifth-order energy diagrams are presented in fig. 17.
432
S. Wilson / Diagrammatic
many-body perturbation
theory
Table 1 Fourth-order energy expressions for open-shell atoms and molecules. The diagrams which have to be considered in addition to those which arise in the closed-shell case are given in fig. 16. The first column gives the number of the diagram in fig. 16. The remaining columns give the quantities occurring in expression (165). Values of p, and q, are given for one-electron matrix elements; values of p,, q,, r, and s, are given for the two-electron matrix elements. The final three columns give the subscripts occurring in the denominator factors defined in (166) No. 1
2 3 4 5 6
a
XI
X*
x,
X.4
D,
D,
D,
-1 +a -1
ijab
kbic
dckj
jkcd klbc jkcd
+: -a
ijab
klij ac ac
ijab vab ijab
jkac
ijab ijab
ad ac kbid
gab ijab
klij
cbkl cdkj cbkl
ijab
ijhc
klbc
ki Ii
cdkj cblk
ijab ijab
ijcd ikbc
abed alcj cdG cbil
cdkj cbkl ak ak adkj acjl ci
ijab jab ijab ijab
ci bdik
ijab ia
jkab jkab ijkacd iklabc ijkabc ijkabc ijkacd iklabc ijkabc
jkcd klbc jkcd klbc ka ka jkad jlac ic ic ikbd
bcil adkj
ia ijab
ijkabc ijkacd
abkl adkj abkl
ijab ijab
iklabc ijkabc
-1 -1
ijab ijab ijab ijab ia ia ijab
ahcd akcj ki ki kbcd klcj kjbc jkbc kbcd
+$ -1
ijab ia
klcj jkbc
16 17
+: -1
ia ijab
jkbc kbcd
18 19 20
+$ -1
ijab ijab ijab
klcj kc
arjk ci ci chid
kc
clij
7 8 9 10 11 12 13 14 15
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
+1 -f +1 -a -1 -f
+$ -a +1 -: +1 +1
I 4 -1
I 4 +: -1 +f -1 -4 +1 -f +1 -1 +: -1 +$ +1 -f
bcid blik adkj abkl acjd
Gab ijab ijab ijab ijab ijab ia ia
kbcd klcj kc kc kbic klij jkib jkib
ak ak
ijab ijab
abed akcj
cdej cddk
ia ia
ajbc ajbc
bcde bkdj
ijab ijab ia ia ijab ijab ia ia ijab ijab
abed akcj jkib jkib kbic klij ajbc ajbc kbic
kdij Ibik abed alck acdj abcl kcid klij lk mk
klij
abkd alkj lckj nbkl lbjc lnjk
cdij cbil cdij abil al am aclk ablm ei di deij dcik ck Cl
cdjk cbjk dk ck bdkj bckl aclj abml
ia ia ijab
klab ijbc
ijab
ijkabc
ijab ijab ijab gab ijab ijab ia ia ijab ijab ia in ijab ljab ijab ia ijab qab ia ia ijab qab
ijkacd iklabc
ilbc jkad klab jkad klab
ijkabc ijkabc jkac klab jkab jkab
ijcd ilbc ijcd ilbc la na klac lmab
ijcd ikbc
ie id
ijbc ijbc
ijde ikcd
ijcd ikbc jkab jkab jkac klab ijbc ijbc jkac klab
kc Ic jkcd jlbc kd kc jkbd klbc jlac lmab
433
S. Wilson / Diagrammatic many-body perturbation theory Table 1 (continued) No.
a
4
X*
4
X4
D,
D,
D,
43 44
+1
ijab
ki
aclj
ijab
ki
ablm
ijab ijab ijab ijab
abed akcj
Cl
ac ac
cd cbde ckdj
edij dbik deij dbik
ijab ijab ijab ijab ijab
jkab jkab
jlac
ijab
lbkc lmkj
ijab ijab
kbij abcj
alcd ckid
cl dk
ijab ijab
ka
ia ia
jabi jabi
kijc bkcd
bckl cdjk
ia .ia
ijab
kbij
Ic
ijab
ijab
abcj
kd
ackl cdik
jb jb ka
ijab
ic
klac ikcd
ijab
bj
klic
ackl
ijab
ia
klac
ijab
akcd
cdik
ijab
ia
ikcd
ijab
bj klic
abkj
cl
ijab
jklabc
Ic
ijab
akcd
cbij
kd
jkbc
laji
ijab ia
ijkbcd
ia
dk bclk
ijkabc
klbc
++ -5 -f I
ia
jkbc
badi
dcjk
ia
ijkabc
jkcd
qab
klic
ijab
jklabc
klac
akcd kc
bj nk
ackl
ijab ijab
cdik
ijab
ijkbcd
ikcd
++ I
ijab ijab
kc klic
lbij abdj
aclk dcik
ijab ijab
ijkabc ijkabc
klac ikcd
ackl
bj
ijab
kacd
cdik
jklabc ijkbcd
ia
jkbc
ia
ijkabc
jb ilab
ia
jkbc
lcjk bcdk
bj bali
ijab ijab
jb
++ --i ++ -+ ++ I
gab
klic
cl
daji abkj
ia ijab
ijkabc jklabc
ijad jkab
ijab
kacd
cl
cbij
ijab
ijkbcd
ijbc
ijab
kc
lcik
abrj
ijab
ijkabc
jlab
72 13 14 75 76
+f
kc kbic abed
acdk ak
dbij cj
ijab ijab
ijkabc jkac
jc
4 bcki cdji
ijab ia ia
ijcd ijbc ijab
jd ijcd ijcd
77 78 79 80 81
++ -1
ijab ijab
jb jb klij akcj
ci kajc
+:
ijab ijab ijab ia ia
bl bk
uab ijab
klab ikbc
+f -1 +1 -f +1 +1 +1 -1 +1 -1 -1
jb jb ki ac jkbi jabc ki ac jkbi jkbi ki
bakl cajk bl bk klak dcji 61 bk balk cajk ackj
ia ia ijab
82 83 84 85 86 87 88 89
ia 9a ijab ijab ia ia ijab ijab ia ia ijab
ijab ijab jkab ijbc jkab ijbc jkab ijbc jkab jkab jkab
lb kb klab jkac lb
45 46 47 48
I 4 +a -1 +$ -1
49
I
50 51
++ I
52
+4 -+ ++ ++ +f -f +j I
53 54 55 56 51 58 59 60 61 62 63 64 65 66 67 68 69 70 71
-1 +f -1
bacd ak ci krji bkci alkj ckij !i bd alkj ckij G bc bc
ijab ia ia ijab ijab ia ia ijab
ijcd ikbc ijbc ijbc ic
lmab ijde ikbd ijde ikbd Ic kd klbc ikcd
ijbd
kb klab ijcd lb kb klab jkac jkac
434
S. Wilson / Diugrammatic
many-body perturbation
theory
Table 1 (continued) No. 90 91 92 93 94 95
cl
t-4 -1 +i -1
-kj
Xl
X2
x3
x4
DI
D2
D3
ijab ijab
ac bc
bd ki
cdij ackj
ijab ijub
ijbc ijac
ijcd jkac
ijab ia
kj jkib
li bc
ablk ck
ijub ia
klub kc
ia ia
ajbc jucb
cd
dj ck
iu iu
ikab jkab ijhc ijbc
jd kc lb
96
-1 +t
kjbi
bl
ia
Jkab
97
4-4
ia ijub
kj Ik
kbij
NC
ul
ijnb
ka
la
98
+4
ijab
abcj
cd
di
ijab
ic
id
99
kitrb Qcd la id k/ah ijcd j/ah
+f
ia
ji
kljh
abkl
ia
100 101 102 103 104 10.5
++ .+ 1 +1 -i-l +1 +i
iu ijub @b ia ia ijah
ah ki ac jkih ujbc ki
bjcd lbkj cbdj
cdij al di ul dcij
IL1 ijub rJub ia ia
ah/j
ijab
ju ib jkab ijbc jkab ijbc jkab
106
+A
ijrrb
107 108
+i -t+
1U
ac jkib
ijab ia
tjbc jkub
r;ihd la
ia
ujbc
lbjk bcdj
dbij al
109
+i
ijub
kbij
ac
di ck
ia gub
ijbc ka
id kc
110
-2 _ i
ijab
ubcj
ki
ck
ijub
ic
kc
ia
ab
jkic
ra
ib
_I
jb
ia
ji
ukbc
bcjk bcjk
iu
- ._;
ijab
ki
abcj
ck
ijab
ja ,jkab
jb kc
114
-4
ljuh
UC
kbij
ck
ijub
ijbc
kc
115
-f
ia
ajbc
ki
bck;
ia
ijbc
kb
116 117 118 119 120 121 122 123 124
- 1 -1 -1 -1 -1 -1 -1 -1 - 4
iu ia iu ijab ijab ia ia ijab ijub
jkih jkib ajbc kbcj kbcj
ac ac ki ci ak bjic akjc cbij abkj
cbjk
ia ia ic rJub ijab ia ia ijab
jkub jkab ijbc ikac ikac ijab ijab ijkabc ijkubc
jc jc kb ka I(’ jkuc ikbc ka ic
jkbc jkhc
hi
acjk
ijkabc
jkac
- .;.
ia ia
bcik
-L -$ - t
ijab gab
kc kc kc jkbc
aj ci ak uk bcik
111 112 113
125 126 127 127 128 129 130 131 132 133 134 135
-$.
- .; -+ +1 +1 il +1 +1
(jab ra ia ijab ia ijab ia ijab
jb jb kc kc
jkbc ai ajib bkjc jb kc
ci bd Ik cd
acjk bkjc kc ai akic cbkj
:k ak ci acjk bcik ak ci
abkj ebij rbij aj hi ck bcjk ck bcjk ai
ijab ia ia ijab ijab ijab ia ia ijub ia ijab ia ijab
ijkabc
ikbc
ijkabc ijkabc ijkahc ijkabc
jkab (jbc ijbc
ijkabc jb .ib ikac ijab ijkabc
ja ib kc jkbc kc jkbc ia
S. Wilson / Diagrammatic
many-body perturbation
theory
Table 1 (continued) No. 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172
+1 +1 +1 +1 +1 +1 +1 -1 -1 -1 +1 -1 +1 +1 +1 -1 +1 -1 +1 -1 +1 -1 -1 -1 +1 -1 +1 -1 +1 -1 +1 +1 -1 -1 -1 -1 -1
ia ia ijab gab ijab ia ia ijab ijub ia ia ijab ijab ia ia ijab ijab ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia
jkbc
bj
jb bkjc kc ai ajib kjcb
bkjc ck ai kc bkjc abij ki
bj bj ji ab ki ac jb jb ki ac jb jb jabi ja ji ab jc jb ji ab ji ab jb jb ji ab ji ab jb jb
2 jc bj ti ki ac ak ci kj bc kj bc akjb bjic kaji baci aj bi bk cj kbij abcj kj bc ah ji bi ah
acik acik ai bcjk bcjk ck ck ak ci abjk bcij ak ci abkj cbij hj hj baki caji bk :k cj ck :k cj aj bi ak ci ak ci bj bj aj bi
ia ia ijab ijab ijab ia ia ijab ijab ia ia ijab ijab ia ca ijab ijab ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia ia
ijkabc ijab ikac ijkabc jb jb ijkabc ia ia ja ib jkab ijbc ijab ijab jkab ijbc ijab ijab jb jb ja ib ijrrc ijab jkab ijhc jkbc ijbc ijab ijab
ikac ikac ia jkbc jkbc kc kc ka ic jkab ijbc ka ic jkab ijbc jb jb ikab ijac kb jb kb kc kc jc kb jc
ja ia
ja ib ka ic ka ic
ja ib ijab ijab
jb jb ja ib
A full set of fifth-order energy diagrams for a closed-shell system are given in appendix B. It should be emphasized that the number of diagrams increases rapidly with the order of the perturbation expansion. Sixth- and seventh-order energy can be obtained according the Wigner’s (2~ + 1) rule from a knowledge of the third-order wavefunction diagrams. As an example of third-order wavefunction diagrams the complete set of such diagrams which have to be considered for a closed-shell system, described in zero-order by a single determinant, is displayed in fig. 18.
S. Wilson / Diagrammatic
436
Fig. 17. Examples
2.14. Multideterminantal
many-body perturbation
of higher-order
Brandow
theory
diagrams.
reference functions
The diagrammatic perturbation theory which we have considered above can only be applied when the perturbed wave function can be described by a single determinant. Let us now consider the use of a reference function which consists of a linear combination of determinants
IV =
c IqJCp7
(167)
PGS
where S denotes the “model” space spanned by the set of reference determinants 1Qp). The zero-order space is multidimensional and quasi-degenerate diagrammatic perturbation theory must be used. The problem of obtaining a many-body perturbation theory with respect to a multideterminantal reference functions has been addressed by a number of authors. Brandow [19] gave the first complete derivation of quasi-degenerate diagrammatic perturbation theory [25,26]. The orbital basis set is partitioned into core orbitals, valence or active orbitals and virtual orbitals as illustrated in fig. 19. The core orbitals are doubly occupied in all of the determinants, 1BP), included in the model space, S; the valence orbitals are sometimes occupied and sometimes unoccupied in S and the virtial orbitals are always unoccupied in S. The total Hamiltonian is
S. Wilson / Diagranvnatic
_-_
--_
- ---.
~
many-body perturbation
-----I_
0 D -_------_~
theory
S, Wilson / Diagrammatic
----_
many-body perturbation
theory
~
__--_-
_-__--__
b
0 ----_---~ -_---_ -----_ ~ ~-----0 ~------_--_ ------_- ---_-_ ~-----~ ~----~ _---. __ -__-~ ~ ~-___-~
~
-A_---
a-_-
------_
--
---
--
~
-------
_----
__---
_--_--
~
~
w-s_
-----
~
-_
_
-----
~
-_
m--e--
~
~
-_----
--em---
__-_-
-----
~
~
__-----_
I_---
---_-_
~
------
-._
---_---
~
~ ~ -----
Fig. 18.
~
-__---
-
~
S. Wilson / Diagrammatic
many-bode perturbation
Fig. 18.
theory
439
S. Wilson / Dia~ra~~~atic many-body perturbation
440
theory
---
~~_---
~~ -..-Fig. 18.
The projector @=
onto the model space is (170)
c I@JC@J P=s
and
(171) For as exactly degenerate AE=b--
may be written (172)
E,,
where d is an eigenvalue p=
model space with energy E,, the energy correction
c t$)@i !JEs
of 2.
For a quasi-degenerate
model space L%$ and 2,
are shifted by 073)
- EJPJ
that is 074) The modified
problem
%/fpIL)
=KI@p)
is exactly
degenerate.
0751 A E is an eigenvalue
I P}
is the exact wavefunction
sif = &9$@+ @ is the Green’s 9=
.&W,@2,9. function
59()+ 9&%$,
operator,
b’, 076)
I??+1 9) = A E&l 9)) where
of the reaction
and 077)
of .Y? (178)
S. Wilson / Diagrammatic
many-body perturbation
theory
v v v-vHI v v ---vv‘i__vvv _--_ v ----vl/v V___Y \ / v----v ----_ -----------_ 0 ----_-_ D
\1 ----------
v-_ ----_ -----
v-_--_-----Y
-----
v-- - ----
v-- -_ --_------v -----_ -----
441
442
S. Wilson / Diugranmutic
many-body perrurhotion
rlzeorj
S. Wilson / Diagrammatic
many-body perturbation
443
theory
Fig. 18.
where @‘Ois the Green’s
function
j2=&+fi
of &.
In the case of quasi degeneracy,
the operator
[25] (179)
can be defined,
satisfying
li2@]\k) =(AE+
ti)B]\k).
(180)
Almost all formulations of quasi-degenerate many-body perturbation theory require the use of a model space consisting of determinants corresponding to all possible occupations of the valence orbitals - termed a complete model space. This can lead to the situation in which determinants which are not in the model space have an energy significantly below the energies corresponding to some of the highest states included in S. These @;‘s are called intruder states. Their presence can impair or even completely destroy the convergence of the quasi-degenerate perturbation expansion. There are three possible solutions to this problem: (i) Formation of matrix PadC approximants to the expansion for the reaction operator or effective interaction operator [9]. For example, in third order the expansion P [ 3/o]
= Q2, + Q, + Q* + 9,
(181)
Fig. 18.
S. Wiison / Lkgrammaric
many-bogv perturbation rheory
Fig. 18.
446
S. Wilson / Diagrammatic
many-body perturbation
theory
\I_-v
I:i ---_
-
v -11;‘v -_-1 w -__ ul
---
v -_ v-----__ v --
--
Fig. 18. Third-order wavefunction diagrams for a closed-shell system described constructed from canonical Hartree-Fock orbitals. Only those arrows which diagram are shown.
v--_--Y
in zero-order by a single determinant are necessary to uniquely define the
S. Wilson / Diagrammatic
many-body perturbation
theory
virtual
orbitals
valence
orbitals
441
-___-__-__------
--__-------------
core
Fig. 19. Partitioning
is replaced Q[2/1]
of the orbitals
in quasi-degenerate
orbitals
diagrammatic
perturbation
theory.
by = 9, + !& + &(I - w,‘fiJ’.
(182)
Such PadC approximations have been found to be useful in studies of effective interaction in nuclei in which the problem of intruder states also arises [27]. (ii) Including th e intruder states in the model space. This, of course, leads to an incomplete model space which can also be much larger than the initial complete model space. It can also destroy the size-consistency of the method. (iii) Excluding the higher energy configurations from the model space. Again this may lead to a loss of size consistency. Let us consider the diagrams which arise in the calculation of the matrix elements of fi, (@, 1h 1a]). The diagrammatic rules and conventions of Hose and Kaldor [25,28,29] will be followed. In the work of these authors the orbitals are divided into holes, which are occupied in @J,, and particles which are unoccupied. The diagonal elements ( Qi 1b I@;) lead to the usual Goldstone diagrams and to folded diagrams (discussed below) which may be derived from them. Off-diagonal matrix elements (@, I h I @,) i #j lead to open diagrams which can also be folded. Open diagrams contain incoming and outgoing lines describing the transition from @, to a,. Open diagrams may be derived by “cutting” Goldstone diagrams in all possible ways. Fig. 20 gives all of the second-order diagrams to be evaluated in Qi and @, differ by one orbital. The algebraic expressions corresponding to these diagrams are given in Appendix C. In figs. 21, 22 and 23, the second order diagrams in which Qi, and @, differ by two, three and four orbitals, respectively, are displayed. Internal lines are summed over hole and particle orbitals, defined with respect to @,. No model state, or linear combination of model states, may appear as an intermediate state. If a complete model space is used or if the model space is exactly degenerate, there are no unlinked diagrams in the expansion. When an incomplete quasi-degenerate model space is employed then unlinked diagrams can occur [25]. The third diagram in fig. 21, for example, is unlinked, but none of the two parts of this diagram can represent a transition connecting Qi and @].
448
Fig. 20. Second-order diagrams in quakkgenerate one-or&taI.
diagrammatic perturbation theory far configurations differing by
Johnston and Baranger [30] have given a detailed analysis of folded diagrams which arise in quasi-degenerate diagrammatic perturbation theory. In the scheme of Hose and Kaidor, the folded diagrams are generated from the unfolded ones as follows: ‘“A group of lines (particles, holes or bath) is designated ‘“folded” if a simultaneous cut through them produces two vertically overlapping parts, each of which is a legitimate open diagram in the model space, This process may be repeated”. The type of folded lines, holes or particles, is determined by the requirement that particle lines enter the portion of the diagram containing the highest interaction line and hole lines enter the other part, Thus there are equal numbers of hole and particle lines in a fold and the line directions are as in the parent diagram, A double arrow notation is employed with the upper arrow giving the line type and the tower arrow the line direction. In fig. 24, an example of the fording of a closed diagram is presented. In fig. 25, the folding of an open diagram is illustrated. When evaluati~l~ folded diagrams, the usual summation rules apply for the unfolded lines. For the folded lines hole and particle ‘fines are summed over transition hole and particle lines corresponding to aXI possible transitions from the vacuum state of the relevant matrix element to all other model states. In the evaluation of the denominator factors the rules given previously again apply; folded lines are classified accarding to their upper arrow and external lines coals into the diagram are always downgoing. The overall sign factor is
where h is the number of hok lines, i~~I~ding folded lines,I is the number of loops after any open incoming lines are connecting to outgoing lines in ascending order or orbital labels and f is the number of folds. The rules outhned above, enable the matrix of b to be evaluated in the model space. It should be remembered that the di~gona~izatio~ of this matrix may introduce size inconsistency. 2.15. VaImx bond refermcefu~c~~~~~ A correct description of many dissociative processes can be obtained by using a valence bond reference function in diagrammatic perturbation theory. Furthermore, this could be achieved
-_-
--
~~_-__
~
3-__
\JJ
~
--_ ~ ~ - ._ _ _ .- ~ Fig. 21. Second-order diagram in quasi-degenerate ~gr~mmatic perturbation theory for c~nf~guratjo~~ fering by two orbitals.
~~~ Fig. 23. Second-order four orbitals.
diagrams
didif-
Fig. 22. Second-order diagrams in quasi-degenerate diagrammatie perturbation theory for c~~figur~t~~~~ diffeting by three orbit&.
-...--_ v ____
in quasi-degenerate
diagrammatic
perturbation
theory
for configurations
differing
by
450
----- ---__----&jfo =+ ()(yo @@?o __ -_ -S. Wilson / Diagrummutic
Fig. 24. Folding
many-body perturbation
theory
of a closed diagram.
without having to resort to multideterminantal reference wavefunctions and thereby suffering the attendant lost of physical interpretability of the correlated wavefunction and energy expansion. The work of Gerratt [31-351, Gallup [36] and Goddard [37] suggests a possible reference function. Newman [38,39] has examined a many-body perturbation theory formalism in which biorthogonal sets of orbitals are employed and the use of reference functions constructed from non-orthogonal orbitals in perturbation theory has also been discussed by Kvasnicka [40], Moshinsky and Seligman [41], Gouyet [42], Cantu et al. [43] and Kirtman and Cole [44]. A brief outline of the generalization of the formalism of Brueckner and Goldstone, to the case of non-orthogonal basis states is now given. The total Hamiltonian is, of course, Hermitian, but
_-
-_
-_
-_-
3 -_
-_ ~~~
~~~_--_
I__--
Y
--
_~~-~--Fig. 25. Folding of an open diagram
S. Wilson / Diagrammatic many-body perturbation theory
may be split into components 3P=&+.&
451
which are not Hermitian
3&z*,
(184)
%ld%n) = Gh#%,)~
(185a)
(we=~,(~“I~
(185b)
with WV%,>
(186)
=L?. state wavefunction,
If the true ground
-wel)
\k,, satisfies
=~oI%)
and the intermediate
(187) normalization
condition
Pol+o,) = 1 is assumed, &I=
(188)
then
P%I%I+o,)
(189)
and Ecl= WoPwo)
= Pol‘%lGoo)
+
ew%l+o,)~
(190)
and the level shift is given by AE,=IE,-E,=(~~I~~I~~). Defining
the projection
@= r+ox+”
(191) operators
I
(192)
and (193) and following a treatment analogous to that given for the case where the zero-order is Hermitian leads to an expression for the wavefunction
Hamiltonian
094)
S. Wilson / Diagrammatic
452
many-body perturbation
theory
and for the level shift L AEo=(+‘I Further
“+
f ,I = 0
iteration
)n*,+o).
can then be performed
AE 0 = E”’ + Ec2) + Ec3’ + 0 0 0 which then leads to “linked
095)
0
0
for the level shift
. . .
diagram”
096) expressions
which are denoted
by the subscript
L
J%“=
(+“l&l+o)~
097)
Et’=
(+“l%l+o)~~
(198) 099)
(200) If a biorthogonal basis sets is introduced, creation and annihilation operators can be introduced which satisfy the usual anticommutation relations but are no longer Hermitian conjugates. Given a basis set x its biorthogonal complement is
(201)
@=A-‘x9 A,,= (x,Ix,>. The covariant
creation
and annihilation
operators
[a,, a,]+= 0, a; [a:, 1+=o, [a,?,a,1+= A,j. The corresponding ai+=
CA’Jq!,
obey the anticommutation
relations
(202) (203) (204)
contravariant d” = (A,,)-‘,
operators
are (205)
.i a’ = -&YJa,
(206)
.I
which obey [
a’+, a,] += [ ai, a;]
[a’+,
a']+=alJ.
= sj,
(207) (208)
S. Wilson / Diagrammatic
In terms of these operators
the total molecular
many-body perturbation
Hamiltonian
453
theory
may be written
The diagrammatic expansion of the Rayleigh-Schrodinger theory with a non-Hermitian partition of the total Hamiltonian operators is topologically equivalent to the usual one. In the algebraic expressions corresponding to these diagrams each summation is over one corresponding to these diagrams each summation is over one covariant and one contravariant index. For example, in second order, the diagram ___(210) 00 _-__ leads to the algebraic
expression
3. Approximate diagrammatic perturbation theory 3.1. ‘Balanced’ truncation of the perturbation
expansion
Diagrammatic many-body perturbation theory forms the basis of a most systematic technique for the evaluation of correlation corrections to independent electron models of atomic and molecular structure in that, once the total Hamiltonian has been partitioned, according to eq. (3) into a zero-order operator and a perturbation, there is a clearly defined order parameter, A, in the expansion which affords an objective indication of the relative importance of the various
Table 2 Second- and third-order energy coefficients for some small atoms and molecules obtained using the Hartree-Fock model perturbation series; The self-consistent-field energies are in hartrees and the correlation energies are in mhartree. Valence correlation energies are given. The basis set and nuclear geometries used are given by Guest and Wilson [45] System Ne FH H,G NH, N, co SiS
E scf
4
- 128.54045 - 100.06007 + 76.05558 + 56.21635 - 108.97684 - 112.77551 + 686.48488
-
210.784 223.633 220.304 198.600 326.887 300.509 206.813
6
E[2/11
+ 0.753 + 1.212 - 3.108 - 11.003 + 9.201 + 5.083 + 16.878
-
210.034 222.427 223.457 210.345 317.938 295.510 225.190
S. Wilson / Diagrummutic many-ho& perturbation theory
454
components of the correfation energy expansion. Second-order energies often yield rather accurate estimates of the correlation energy. Through third-order the correlation energy of a closed shell system is represented by just four diagrams, as shown in fig. 8, if antisymmetrized vertices and a Hartree-Fock reference function are employed. Some typical second- and third-order correlation energies are displayed in table 2 [45]. It has been shown that, provided a ‘balanced’ procedure is followed and all terms through third-order are evaluated, third-order calculations are accurate and very competitive with other approaches to the correlation problem. The dominant error in such third-order calculations is almost invariably attributable to the truncation of the basis sets rather than the neglect of higher-order terms in the perturbation series. There are two possible ways of improving calculations based on third-order perturbation theory developed with respect to a Hartree-Fock wavefunction and Hamiltonian: (i) by employing an alternative reference wavefunction and/or Hamiltonian in order to obtain an improved approximation to the correlation energy in third order; (ii) by extending the perturbation series to fourth and higher orders. We consider these two options in turn. 3.2. Afternative reference ~~~~ef~n~tions and ~amiit~~ians 3.2. I. Shifted denominators In the expressions for the components of the correlation energy of ^ a closed-shell system given in section 2, the N-electron Hartree-Fock model Hamiltonian, yi”*, was used as a zero-order operator. This leads to the perturbation series of the type first discussed through second-order by Moller and Plesset [49,50]. Flowever, it is clear that any operator 2 satisfying the commutation relation
[J& q =o may be used to develop a perturbation operator which satisfies (22) is %hifted
=nwww(kl~
(212) series using
the same reference
wave function.
One
(213)
in which the single determinant ]k) is an eigenfunction of 203 and this gives rise to the shifted denominator, or Epstein-Nesbet 151-531, perturbation series. The resulting perturbation expansion has the same diagrammatic representation as that given in section 2. The corresponding algebraic expressions are also as given in section 2 except that [2,9]: (i) the denominator factors are each shifted by an amount
(ii) diagonal scattering terms are omitted in third (and higher) order. The use of shifted demoninators may also be interpreted as the summation to all orders of certain diagonal terms in the perturbation series based on the Hartree-Fock model Hamiltonian. A full set of expressions for the energy coefficients in the shifted denominator perturbation series are given in appendix D.
S. Wilson / Diagrammatic
man.v-hodv perturbation
455
theor)
3.2.2. Pad& approximants and scaling of the reference Hamiltonian Following Feenberg [54], let us consider two modifications of a given zero-order operator, $O; namely, a uniform displacement of the reference spectrum and a uniform change of scale. This defines the new zero-order Hamiltonian as follows:
The perturbation
is then
2iy.y =.2P,+ (1 - p).2&- Yi, So that the full molecular
2= (3ey-t
Hamiltonian
(216) is recovered
E, +(l
that is
h.Py)x=l.
The energy coefficients, El”,“, where i denotes those of the original series by the expressions
El.“=
on adding (215) and (216)
-/A)E~-Y,
(217) the order of perturbation,
may be obtained
from
(219)
(220) It can be shown [55,57] that the [M + l/M] Pade approximants, alone among all Pade approximants of order 2M + 1 (including the [2M + l/O] PadC approximant), are invariant to arbitrary choices of p and V, when A is set to unity. The sum of the perturbation expansion, with respect to given reference wavefunction, through infinite order is, of course, independent of the choice of zero-order operator. The degree of agreement between two perturbation expansions based on different %‘“‘s is, therefore, a qualitative measure of the convergence of the series. In table 3, third-order results are presented for a number of small atoms and molecules obtained using both the Hartree-Fock model perturbation series and the shifted denominator series. For each of these expansions, [2/l] PadC approximants are formed. It can be seen, from this table, that for systems which are well described by a single determinant, such as Ne, FH or NH,; the third-order model perturbation series E[3/0] is in close agreement with the [2/l] PadC approximant formed from it, whereas the shifted denominator series leads to a significant change on forming the [2/l] Pade approximant; although the [2/l] PadC approximant is in close agreement with that constructed for the model series. For systems such as the BH, BeH, and BH, molecules, which involve some degree of quasi-degeneracy, the results obtained from the model perturbation series change significantly on forming the [2/l] PadC approximant and, furthermore, are thereby brought into closer agreement with the shifted denominator results E[3/0] and E[2/1] which for these cases are in close agreement with each other.
456
S. Wilson / Diagrammatic
many-body perturbation theory
Table 3 Comparison of third-order diagrammatic perturbation theory calculations for some small atoms and molecules obtained using the Hartree-Fock model perturbation series and the shifted denominator perturbation series; All correlation energies are in mhartrees. The basis sets are nuclear geometries used are given by Guest and Wilson [45]
A, is the percentage difference between E[3/0] and E[2/1] for the shifted denominator expansion, A,s,al and A,2,,,I are the difference between E[3/0] and E[2/1] respectively in the model and shifted denominator perturbation expansion System
Ne
BH FH BeH, H,G BH, NH, HZ co SiS
Model perturbation
series
Shifted denominator
&3/O]
E12/‘11
A,
- 210.031 - 80.615 - 222.421 - 64.596 -223.413 - 116.931 - 209.600 - 317.686 - 295.679 - 223.691
- 210.034 - 86.731 - 222427 - 68.623 - 223.457 - 121.471 - 210.245 - 317.938 - 295.510 - 225.190
0.0 7.3 0.0 6.0 0.0 3.8 0.3 0.1 0.1 0.7
(%I
perturbation
series
Ef3PJl
EP/fl
A, (%)
- 201.253 - 89.893 - 201.262 -69.181 -201.657 - 122.542 - 202.576 - 257.443 - 269.302 - 205.403
- 209.472 - 89.893 - 221.208 - 69.187 - 221.856 - 122.593 - 209.543 - 310.457 - 293.641 - 222.582
4.0 0.0 5.1 0.0 5.2 0.0 3.4 18.7 8.6 8.0
A[,,,, (%)
AIVll (%I
4.3 10.9 5.6 6.9 5.9 4.7 3.4 20.9 9.3 8.5
0.3 3.6 0.5 0.8 0.7 0.9 0.3 2.4 0.6 1.2
3.2.3. Alternative reference wavefunctions The modification of the perturbation expansion considered above have not departed from the use of the canonical molecular orbitals, both in the reference function and in the excited configuration functions. Let us briefly consider the use of orbitals other than the Hartree-Fock canonical orbitals. If electron-electron interactions are completely neglected, i.e. the bare nucleus model, in zero order, we then have to consider in the corresponding diagrammatic perturbation diagrams which then arise are series diagrams which contain ‘bubbles’. The second-order displayed in figs. 7(D) and (E) and the third-order diagrams are shown in figs. 8 (0, P, T, W, X, b, e, f, j, m, n, o, p, cl). If instead of using the canonical Hartree-Fock virtual orbitals, we use the orbitals corresponding to the so-called F/“‘-’ potential [58,59], then the diagrams shown in figs. 8(1, n) have to be considered the ‘cross’ representing the VN-’ potential. It has been demonstrated [59] that results obtained by using the VN-’ potential are in fairly close agreement with those corresponding to the Hartree-Fock, VN, potential if all terms through third order are included. Similarly, if localized Hartree-Fock orbitals are employed in the reference configurations then figs. 8(k, m) have to be considered. 3.3. Higher-order
terms
As an alternative to modification of the reference wavefunction and/or zero-order Hamiltonian in order to obtain an improved approximation for the correlation energy in third order, the perturbation series can be extended to fourth order. The fourth-order terms, which correspond to
457
S. Wilson / Diagrammatic many-body perturbation theory Table 4 Fourth-order
calculations
for some small molecules
Molecule
W/11
E[3/01
FH H,D NH, N, co
+ -
- 0.2207 - 0.2221 - 0.2107 -0.3193 - 0.3028
0.2206 0.2176 0.1989 0.3291 0.3076
- 0.2276 - 0.2300 -0.2133 - 0.3486 - 0.3312
-
0.0039 0.0050 0.0050 0.0180 0.0161
+0.0011 + 0.0020 + 0.0027 + 0.0065 - 0.0049
the diagrams shown in fig. 15, can involve singly, doubly, triply and quadruply excited intermediate states [23]. Diagrammatic perturbation theory provides a systematic scheme for including triply and quadruply, and indeed higher-order excited intermediate states. The results
Table 5 Perturbation molecules
theoretic
analysis
of some approaches
to the electron
correlation
problem
in closed-shell
atoms
Method
Order of perturbation expansion
Comments
Refs.
single and double excitation configuration interaction (SD CI)
3
includes unlinked diagrams components of the correlation energy in fourth and higher order
[9.62]
single, double, triple and quadruple excitation configuration interaction (SDTQ
5
includes unlinked diagram components of the correlation energy in sixth and higher orders.
[9X2]
coupled electron pair approximation (CEPA)
3
CEPA(0) includes linked diagrams double excitation through all orders. Other versions of the CEPA include approximations to fourth and higher-order linked diagram quadruple excitation components of the correlation CEPA(0) = L - CPMET ‘) = DE MBPT h’
W,651
coupled pair many electron theory (CPMET)
3
neglects triple excitations in fourth order: there we include in the extended CPMET
WI
CI)
a) Linear CPMET. b, Double excitation
many-body
perturbation
theory.
and
458
S. Wilson / Diagrammatic
shown in table 4 illustrate the importance for a number of small molecules [60,61]. order in the perturbation expansion, it mhartree) can be achieved within the basis 3.4. Anulysis of contemporary
many-body perturbation
theog
of fourth-order components of the correlation energy Provided that all terms are included through fourth has been demonstrated that chemical accuracy (1 set employed.
approaches to the correlation problem
Perturbation theory not only provides a very efficient scheme for the computation of correlation effects in atoms and molecules but also affords a highly systematic technique for the analysis of various approaches employed in many contemporary calculations. In table 5, an overview of the perturbation theoretic analysis of some of the techniques currently being employed in correlation energy calculations is presented. It should be noted that many of these techniques are third-order theories, in that they neglect or approximate fourth- and higher order terms. Perturbation theory can indicate the dominant corrections to these theories. An analysis of some contemporary theories of electron correlation effects in atoms and molecules in terms of the diagrammatic perturbation theory has been presented elsewhere [9]. We wish to emphasise at this point that techniques which have been described for the summation of certain classes of diagrams through infinite order are, in fact, exactly equivalent to well-established, non-perturbative approaches to the correlation problem. For example, the so-called double-excitation many-body perturbation theory, in which double-excitation linked diagrams are summed through all orders, is equivalent to the linear coupled pair many-electron theory. Since these infinite order summations of certain classes of diagrams (classes of diagrams which are often chosen because that can be easily evaluated rather than because they are physically important!) have abandoned the ‘balanced’ truncation of the perturbation series advocated in section 3.1 and because they are merely reformulations of other methods, we shall not consider them in this article.
4. Computational 4. I. Algorithms
diagrammatic perturbation theory
for perturbation
theory
The theoretical properties of diagrammatic many-body perturbation theory, which were described in section 2, ensure that it leads to a computational scheme of high efficiency. The particular strengths of the diagrammatic perturbation theory from a computational point of view are: (a) it leads to a non-iterative algorithm (unless one abandons the spirit of the perturbation approach and sums classes of terms); (b) all possible excited states are included through a given order of the perturbation series of configuration selection schemes, such as those widely used in practical implementations of the method of configuration interaction are avoided; (c) the diagrammatic representation of the correlation energy components can be very easily translated into computer code; (d) it leads to algorithms which are extremely well suited to implementation on parallel computers, vector processing and array processing computers.
S. Wilson / Diagrammatic
f/=1
[ABICDI
Fig. 26. Classification
[AIIJKI
of integrals
many-body perturbation
[IAIBCI
according
theory
[IJ(ABl
to diagram
459
[IAl JBl
components.
Furthermore, the scaling technique, which is described further below, enables the diagrammatic perturbation theory to be used to calculate correlation energies and reduce the basis set truncation error in molecular calculations simultaneously. After integrating over the spin coordinates, the various expressions in the perturbation series can be written in terms of integrals over spatial orbitals. In the present work, lower case indices are employed to represent spin orbitals and upper case indices are used to represent spatial orbitals. The indices I, J, K,. . . are used for occupied orbitals and the indices A, B, C,. . . for unoccupied orbitals. In order to facilitate the evaluation of the algebraic expressions corresponding each of the diagrams in the expansion for the energy, it is convenient to separate the list of two-electron integrals over molecular orbitals into six lists [67]. Each of these lists corresponds to the classes of diagrammatic components shown in fig. 26. Of course, if all of the two-electron integrals can be held in store simultaneously this separation into six lists is not necessary, although it may be convenient. However, this separation is certainly necessary if, as is usually the case, we cannot avoid storing the integrals on backing store. Efficient algorithms for the evaluation of the correlation energy components corresponding to the various terms in the diagrammatic perturbation expansion may be devised by: (i) Taking account of spin orthogonalities. For example, the second-order energy for a closed shell system may be written
E2= aC Cg,jobguhij/(Ei IJ
=EC IJ
+
'j -
‘a -
‘h)
oh
c c gIo,.JoJ,Ao.,.B~HgA~,,B~,,.I~,.J~,/(~I + 'J - ‘A -‘B)-
AB o,o,, o,o,,
(221)
S. Wilson / Diagrammatic
460
where p = Pa, and up is a one-electron
many-body perturbation
spin function.
theory
Now performing
the spin integrals
(223)
the second-order
energy takes the form
E,=~~~{([ZAIJB] IJ
+[zsIJA])([AzpJ]
-[AJIBZ])
AB
(&v?QS = [PQIRSI) which for real orbitals
leads to
E,=fCC{[z~IJB]2+[z~IJA]2 IJ
AB
-[zAIJB][zBIJ/4]}(~,+~,-~,-r,)-‘.
(225)
(ii) Exploiting any spatial symmetry properties of the system being studied. If the Pth spatial orbital transforms according to the r(P) irreducible representation of the point symmetry group then the integral [ PQ I RS] is zero unless the direct product r(P)
@ r(Q)
@ r(R)
@ T(S)
(226)
contains the totally symmetric (A,) irreducible representation. This restriction will often lead to a reduction in the number of integrals which have to be considered in evaluating the summation involved in the various energy expansions. (iii) Recognizing certain permutational symmetry properties of various intermediates which arise in the calculation. (iv) Using the diagrammatic form of Wigner’s (2n + 1) rule discussed in section 2. Algorithms based on the (2n + 1) rule are clearly more efficient than the recursive schemes described in refs. [70,71]. If the shifted denominator perturbation schemes is employed the integrals of the type [ZZ I JJ] and [ZJ/Z.Z] can be retained in the computer’s fast store. 4.2. Second-order
and third-order energy calculations
An efficient algorithm for evaluating the energy of closed-shell systems through third order has been described by Silver and Wilson [67,69]. In order to evaluate the second-order energy Silver
S. Wilson / Diagrammatic
many-body perturbation theory
461
[67] arranges the required list of integrals of the type [IA 1JB] in blocks, each block corresponding to a unique I, J pair (I >, J) and containing all non-zero integrals for all possible values of A and B. Furthermore, if I # J and A # B, then the two integrals [IA 1JB] and [IB 1JA] occupy consecutive positions in the list whenever both are non-zero. With this file handling scheme the evaluation of the second-order energy is reduced to a “near triviality”. The energy components corresponding to the diagrams labelled (B) and (D) in fig. 13 can be evaluated using similar algorithms [68]. The component arising from (B) requires integrals of the type [IA 1JB] and [IJ 1KL] whilst that arising from (D) requires integral types [IA I JB] and [ AB ICD]. For the (D) component the [IA I JB] integrals are blocked and ordered in exactly the fashion described above for the second-order energy whereas for the (B) component another list of these integrals is generated in which each block of integrals corresponds to a particular A, B pair (A 2 B) and contains all non-zero integrals for all possible values of I/J. The number of [ ZJI KL] integrals is not excessively large and all of these integrals can usually be held in the computers fast core. In contrast, the list of [ AB I CD] integrals can be enormous; the number of these integrals increases as the fourth power of the number of basis functions. Thus the problem logic is constructed so that the [AB I CD] (or [ IJ I KL]) integral list is not reordered and these lists are read only once. Note that diagrams (B) and (D) are related by “time reversal” (i.e. inverting the diagram) and thus the same procedure can be used to evaluate both energy components. The algorithm for evaluating the (B) component is as follows: Read a block of [ AB I CD] integrals into fast storage. Read a block of [IA ) JB] integrals for a given A. B pair. Form all integral products required for the expression corresponding to (D) Return to (b) until all [IA I JB] integrals have been read. Return to (a) until all [ AB I CD] integrals have been read. The possible combinations of spin functions which can arise in the evaluation of diagrams (B) and (D) of fig. 13 are shown in fig. 27. In fig. 28, the possible combinations of spin functions which can arise in the case of the diagram shown in fig. 13(c) are summarised [69]. Two lists of two-electron integrals are required to evaluate this energy component. The [IA I JB] = (IJ ) g I AB) (g = l/r,,) are ordered in the manner described above for the second-order energy calculations.
Fig. 27.
S. Wilson / Diagrammatic
cl
theor):
------_-----_-_-_0 00 Q-O0 0 ttkh -__-_ _ -__ ----0 bo w P 0 P 0 a
P__
CL
a
a
-_
CY
a -----
_----
s-v
Lx
a --
a
a
cf.
0.
cl.
--__-
0.
CY
c1
__
-_-_ w t33 ------
----
CY
__
0.
cx
CL
6
G
6
a
c1
es-mm_
a
c1
__
6
a
a
many-body perturbation
c1
cl.
CY
----_
6
-_-_--
-e-B-
a
a
a
a
CY
-_
CL
-
_--
-_ a -
cf
cf. --_-_
Fig. 28
For an atom or molecule which is described by N doubly occupied orbitals, N secondary lists are created. The Ith of these secondary lists contains the integrals (ZJ 1g ) AB) for all J, A a B. This secondary list is divided into blocks according to the index J. The integrals (IB 1g 1AJ) are divided into blocks corresponding to I, J (I > J) pairs and each block is ordered with A > B. These integral lists are then processes in the following manner: (i) Integrals of the type (IC I g I AK) are read into fast core for all A > C for a given I > K. (ii) The integrals (ZJ I g I AB) and (JK ) g I BC) are read into core from the Ith secondary list, for all A > B, and from the Kth secondary list, for all B > C, respectively. (iii) All integrals depending on a certain I, J, K are thus in fast core simultaneously. The contribution to the energy components corresponding to the diagram (C) can be evaluated for fixed 1, J, K; summation over A, B, C can be performed. (iv) If the end of the Ith and Kth secondary lists has not been reached, J is increased and the algorithm returns to (ii).
S. Wilson / Diagrammatic
many-body perturbation
theory
463
If the Zth and Kth secondary lists have been completely read, they are rewound. If the (IC 1gj AK) list has not been completely read, the algorithm returns to (i). The list of (IC 1g ( AK) integrals is, therefore, read into core once. To illustrate algorithms devised for evaluating fourth-order components of the correlation energy, the fourth-order closed-shell triple-excitation diagrams will be considered [72]. These are the most demanding of the fourth-order diagrams for a closed shell system leading to an algorithm which depends on the seventh power of the number of electrons. The energy components corresponding to the sixteen diagrams labelled X, in fig. 15 may all be written in terms of the intermediate quantities
(v)
(227) and
(228) Each diagram K~
c
i/k ohc
leads to an expression +rJk;ahc{
of the form
p~jkpuhcytjk;ohc)
(229)
( 6, + E, + Ek - E, - Lb - cc) ’
where 4 and y denote either an _f or a R intermediate, permutation operators. The following expressions arise:
K is a constant
and
P,,k Pa,,<,are
(230)
(231)
EL&,) = J%(D,)= E‘iW = c
(232)
4 c c gijk:ohcgrkJ;ohc/~rJkohc, iJk ah<
(233)
(234)
ijk
(235)
(236)
S. Wilson / Diagrammatic
464
many-body perturbation
theory
Table 6 Spin case for the intermediate FhKEABC P
RDKAC
glJDB
E&T) = E4(JT)
=
-
ac Cfr,k;ahrgik,~orb/~r,kahrr
(238)
.(239)
c Cfrjk:ohcgijk;ubc/~i,kabc, ijk ohc
(240)
E4(oT)
=
(241)
$ c Cf;jk;ohrgikj;ahr/~,jkabc’ !jk ulx
where 9
rJkahL
=
6, +
Ej +
Ck -
ca -
Ch -
E,.
(244
It should be noted that diagrams L,, K,, N,, P, lead to expressions which are numerically equal to those for diagrams I,, J,, M,, Or, respectively, for real orbitals. The nine different spin cases are summarized in table 6. Definition of which arise in the evaluation of the frjkiahc intermediates these intermediates enables a spin-free formulation of the problem to be achieved. Defining the secondary intermediate quantities.
(243)
(244)
Similar spin-Free ~nterme~i~t~s may be obtained for the ~~~~~~~~~ Then, fr>r example, the energy corresponding to the diagram A, in fig. I5 may be written as &fA,)
= - c IJK
( F;:K: ABC-FI:K,ACB
c ABC
+ F~K;Ad%;Aa+ ~?K:ABcF&:AcB +GKT;Ald%L;ACB
+
~~K;AeCGwB
-+GK:ABCF~K;ACB
-t- Fr%;Afd%;ACLi
+ ~~K;A~C~~K~*C~)~~~,
it
+ “J f- CK -
=
hK;CBA
=
-%K;ABC
=
and thus only nine unique intermediates %%.4BC intermediates.
QB -
4.
F&ABC obey the permutational
should be noted that the intermediates I:IJK:ABC
EA -
-
F,,;CBA
cw
symmetry properties w-0
have to be considered. Similar relations exist for the
S. Wilson / Diagrummotic
466
many-body perturbution
theory
Similar algorithms to that described above for the fourth-order triple-excitation energy can be devised for other components of the diagrammatic perturbation expansion of the correlation energy (see, for example, ref. [73]). For example, in fourth order for closed-shell systems the single-excitation component, corresponding to the diagrams labelled As-D, in fig. 15, may be written in terms of the intermediates
Cg,hbg,b.,k/(~, + c/,-
g,, =
60
-Ebb
(25%
jkb
Thus, for example
&(A,) = f ~Mw’k
- E,.).
(260)
id
The double-excitation components, corresponding may be written in terms of the intermediates
g ,,oh
=
&kloh&k,/(‘k
+
‘/
+
‘k
-
‘u
-
to the diagrams
labelled
An-L,,
in fig. 15,
(262)
‘b)?
XI
h r/oh
=
&ikocg,db/(‘,
-
‘,
-
‘<
(263)
>.
Thus, for example,
EdA,) = & ~f,,,,f,,,,/(~;+ 6, - cu- d
(264)
l/“h
The quadruple-excitation components, written in terms of the intermediates f rkhd
=c
gr,ub&,ud/(~,
+
6,
-
cu
-
corresponding
to diagrams
A,-Go
in fig. 15 may be
(265)
Cb),
P grkhd
=
&?,,ubgk,o&,
+
‘,
-
E,
-
‘b)h,
+
-
cu
-
%/)3
(266)
I”
fi,kl
=
~&obgklob/(E,
+
‘,
-
“J
-
(267)
‘b>3
oh g,,k/
=
~~,,ob&,ob/‘+,
+
f,
-
E,
-
‘b)hk
+
E/ -
cu
-
d
(268)
S. Wilson / Diagrammatic
many-body perturbation
theory
467
(269)
(270)
guc=Cgrjuhg,,~h/(Ei+tl-E,--~)(C,+~,-Ch-E,), i/h
f,k = Cgr,uhgk,oh/k + 6, - ca- 4
(271)
.Ph
g,k = Cg,,uhgk,u(,/(‘r .l”h
+ ‘j - ‘u - ‘h)(‘,
+ ‘k - ‘o - ‘h)*
(272)
Thus E,(AQ)
=
f
c
frkhdgrkhd~
(2;3)
lkhd
&tBQ
+ &)
= ii Cf;,k,gi,ki’ r/k1
E,(Do+ E,) = - acf,cs,o
(274)
(275)
4
C&+GQ)=
(276)
-ichkg;k. rk
4.3. The scaling technique
In view of the difficulty of evaluating all terms through fourth order in the perturbation expansion various approximation schemes have been devised. The scaling technique, suggested by the present author [74], enables second-order perturbation calculations, using a large basis set, to be used to approximate calculations taken to fourth-order. The correlation energy is thus determined by an algorithm which depends on the fourth power of the number of basis functions. A scaling parameter, p, is introduced into the zero-order Hamiltonian. This leaves the self-consistent-field reference energy unchanged but the second-order energy is scaled according to /PE,.
(277)
In the scaling technique, I_Lis chosen so that the perturbation expansion converges rapidly and so that the scaled second-order energy provides an accurate correlation energy [74]. It has been demonstrated that a useful value of p can be determined from calculation taken beyond second order using basis sets of moderate size [74]. An example, of the application of this technique is given in section 5. The use of constant denominator perturbation theory has been advocated [75]. This leads to an algorithm for the fourth-order energy component depending on the sixth power of the number of basis functions.
468
S. Wilson / Diagrammatic
many-body perturbation
theory
4.4. Open-shell systems In the above discussion, attention has been restricted to the evaluation of the perturbation energy components for closed-shell systems; however, the same approach can be adopted for calculations on open-shell systems using either single determinant or multi-determinantal reference functions. The computational problem is one of efficiently forming the products of one- and two-electron integrals which occur in the numerator of the perturbation expressions; the data required to form the denominator can always be held in the computer’s fast store. 4.5. Parallel computation The linked diagram theorem of many-body perturbation is of considerable theoretical importance as the discussion of section 2 demonstrates. It is also of computational importance. The linked diagram theorem allows the energy of any system to be written as a sum of energies of its component parts no matter how these component parts are refined. It provides an additive decomposition of the energy which aids the development of algorithms for parallel computation by allowing calculations on arbitrarily defined subsystems to be performed concurrently. It is clear that a central problem of atomic and molecular physics over the next few years is the development of efficient techniques for the ab initio treatment of electron correlation effects which allow the organization of concurrent computation on a very large scale. Since the linked diagram theorem ensures that the diagrammatic perturbation theory satisfies certain additive separability conditions of the type first discussed by Primas [76], it forms the basis of algorithms affording the maximum degree of parallelism. The paracomputer model, introduced by Schwartz [77], plays a useful theoretical role in the determination of the limits of parallel computation possible for a given method. The paracomputer consists of a very large number of identical processors, each with a conventional order code set, that share a common memory which they can read simultaneously in a single cycle. Such machines cannot be physically realized, since any computing element can have no more than some fixed number of external connections. However, the model is theoretically valuable. Consider, for example, the evaluation of the second-order energy for a closed-shell system on a paracomputer. This involves a summation over the indices I, J, A, B with I > J and A > B. The number of terms in the summation is M=~N,N,(N,+l)(N,+l),
(278)
where N, and NV are the numbers of occupied and of virtual orbitals, respectively. computer the evaluation of the second-order energy will take a time Tqeria,given by qeriai = M7i + ( M - 1) 5 7
On a serial
(279)
where 7, is the time required to compute the contribution to the sum by a given set of spatial orbitals ZJAB and r2 is the time required to perform an addition. On a paracomputer with P processors, the computation of the second-order energy will take a time Tparalle,given by M Tparall. =pr,+
i
SP-2 p
i
72.
469
S. Wilson / Diagrammatic many-body perturbation theory
If the number
of processors
is less than the number
of terms in the summation,
i.e. M B P, then
(281) and the paracomputer performs the calculation P time faster than a serial machine. The above considerations can be generalized to other components of the correlation energy in the perturbation expansion. Any contribution can be evaluated approximately P times faster on a P-processors paracomputer than on a serial machine. Considerations, using the paracomputer model, such as those outlined above, clearly demonstrate the suitability of diagrammatic perturbation theory for concurrent computation [78].
5. Applications
It is not intended to give a detailed review of applications of diagrammatic perturbation theory and molecules here but to provide a few examples to illustrate its potential. In a series of papers by Eggarter and Eggarter [79] and by Jankowski and his co-workers [80], accurate calculations of correlation energies using second-order and, in the recent work of Jankowski et al., third-order Rayleigh-Schrodinger perturbation theory have been reported for a variety of atomic systems. Such calculations have clearly demonstrated the need to include basis functions corresponding to higher angular quantum numbers if the error due to basis set truncation error is to be significantly reduced. Jankowski and Malinowski, for example, showed that the second-order energy for the ground state of the neon atom provides a very accurate approximation if functions corresponding to higher I quantum number are included in the basis set. Jankowski et al., have recently extended their calculations for neon to third-order and the to atoms
Table 7 Calculated
correlation
energies
for the neon atom ground
Radially
Calculated & -
‘) In units of lop4
1918.0 3219.3 3596.1 3742.8 3799.2 3825.1 3838.3 3845.5 3849.8
hartree.
state ( Escr = - 128.54705 h) a)
E,
E,
42.3 11.1 2.0 10.0 20.5 27.1 31.1 33.4 34.9
-
extrapolated E,
1920.4 3223.8 3602.7 3751.2 3809.2 3836.3 3850.7 3858.8 3863.8 3879
42.5 11.7 3.0 11.6 22.7 29.8 34.3 37.0 38.7 44
470
S. Wilson / Diagrummatic
Table 8 Calculated correlation Al9 (1979) 1375
energies
for the beryllium
Correlation E acf E, E, + EX
many-body perturbation
atom ground
state; taken from Silver, Wilson and Runge, Phys. Rev.
energy
Total energy
- 0.07195 - 0.08380 - 0.08613 - 0.09434
W/11
E non.rral
theory
-
14.57302 14.64497 14.65682 14.65915 14.66736
results which they obtained are summarized in table 7. For the Be atom ground state there is a low lying 2p2 excited state and the accuracy of the perturbation expansion through third-order is not as high as for the Ne atom as the results of Silver et al. [81] summarized in table 8 clearly demonstrates. For molecular calculations it is usually necessary to employ basis sets of a more modest size than it is possible to use in atomic studies. Some indication of the usefulness of perturbation theory can be obtained by comparisons with full configuration interaction calculations, although for the latter calculations to be tractable the basis set truncation error is enormous. In table 9 a comparison of perturbation theory and full configuration interaction is made for two different
Table 9 Comparison interaction
of calculated
molecular
correlation
energies
obtained
by perturbation
theory
with full configuration
System
Hz dimer case (a)
H 2 dimer case (b)
E2
- 0.031053 - 0.012711 - 0.000136 - 0.006023 - 0.000887 - 0.003051 - 0.010097 - 0.043764 - 0.052572 - 0.053861
-
0.043648 0.018811 0.000002 0.010743 0.~003 0.006036 0.016783 0.062459 0.076706 0.079242
- 0.139478 - 0.001391 - 0.000908 - 0.003083 - 0.001364 - 0.~0815 - 0.006170 - 0.140869 -0.140883 - 0.147039
- 0.052515
- 0.105552
- 0.139340
- 0.053690
- 0.109196
-0.148028
~53
E 4s E 4D E 4T E 4Q E4
E[3/01 E[2/11 EI4/01 double excitation configuration mixing full configuration mixing Footnotes: Jankowski
based on the work of Jankowski and Paldus (1982).
and Paldus (1980); Saxe et al. (1981); Wilson and Guest (1980); Wilson,
S. Wilson / Diagrammatic
many-body perturbation
theary
471
Table 10 Calculations on diatomic molecules obtained using diagrammatic perturbation theory and a universal basis set; all energies are in atomic units. The following abbreviations are used: Ci: configuration interaction; PNO-CI: pair natural orbital-configuration interaction; CEPA: coupled electron pair approximation; DPT: diagrammatic perturbation theory; MCSCF: multiconfiguration self-consistent field; E[2/1] denotes the [2/l] PadC approximant to the perturbation series based on the Hartree-Fock model zero-order Hamiltonian and i[2/1] denotes the [2/l] PadC approximant to the shifted denominator expansion Method LiH (empirical
correlation
energy,
m/11
Diagrammatic perturbation E]2/11 theory/Universal basis sets (d) &‘[2/1] correlation
energy,
correlation
energy,
correlation
energy,
8.0606 8.0647 8.0660 80643 - 8.0652 - 8.0653 - 8.0661
88.3 93.3 94.8 92.8 93.9 94.0 94.9
- 14.9649 - 14.9842 -14.9845
74.3 89.6 89.9
-
100.3564 100.3274 100.3392 100.3727 100.3707 100.3837 100.3770
75.1 67.5 70.6 79.4 78.9 82.3 80.6
- 109.2832 - 109.4180
58.4 79.5
- 109.443
83.7
e,,r = -0.381
Bender and Davidson (a) Cl Meyer and Rosmus (b) PNO-CI CEPA Wilson and Silver(c) E[2/1] Ef2/11 E12/1] Diagrammatic perturbation theory/Universal basis set (d) E[2/1] N, (empirical
-
erxp = - 0.126)
Werner and Reinsch (e) MCSCF-CI Diagrammatic perturbation E]2/11 theory/Universal basis set (f) E[2/1] FH (empirical
eexp @)
ecxp = - 0.083)
Bender and Davidson (a) CI Meyer and Rosmus (A), PNO CI CEPA Wilson and Silver(b) DPT, E[2/1)
Liz (empirical
-&,,I
e,xp = -0.538)
Langhoff and Davidson (g) CI Wilson and Silver DPT (h) E[2/1] Diagrammatic perturbation theory/Universal basis set (5) E[2/1]
conformations of the H, dimer, the first involving no near degeneraties in the reference spectrum and the second demonstrating the effects of quas~degeneracy. A comparison is also made in table 9 of calculations performed for the water molecule using a double zeta basis set. The results presented in table 9 demonstrate that the non-degenerate perturbation theory is in good agreement with full configuration interaction if quasi-degeneracy is absent. To illustrate the accuracy of molecular calculations using diagrammatic perturbation theory, results are presented in table 10 which were obtained by evaluating the expansion through third order in the energy and by using a universal basis set of functions with s, p and d symmetry. The
472
S. Wilson / Diagrammatic
many-body perturbation
theory
Table 10 (continued) Method CO (empirical
E,,,,, correlation
energy,
ecxp = - 0.525)
Siu and Davidson (j) CI Bartlett et al. DPT (k) E[2/1] Diagrammatic perturbation theory/Universal basis set (i) E[2/1] BF (empirical
correlation
energy,
(f) (g) (h) (i) (j) (k) (1)
69.4 77.4
- 113.2286
82.7
- 124.235 - 124.5028
65.2
- 124.5782
77.1
ecxp = - 0.531)
Bender and Davidson (a) MCSCF Wilson et al. DPT (1) E[2/1] Diagrammatic perturbation theory/Universal basis set (i) E[2/1] (a) (b) (c) (d) (e)
- 113.1456 - 113.1952
C.F. Bender and E.R. Davidson, J. Chem. Phys. 70 (1966) 2675; Phys. Rev. 183 (1969) 23. W. Meyer and P. Rosmus, J. Chem. Phys. 63 (1975) 2356. S. Wilson and D.M. Silver, J. Chem. Phys. 66 (1977) 5400. S. Wilson and D.M. Silver, J. Chem. Phys. 77 (1982) 3674. H.-J. Werner and E.-A. Reinsch, Proc. 5th Seminar on Computational Problems in Quantum van Duijnen and W.C. Nieuwpoort, Groningen (1982). S. Wilson, Specialist Periodical Reports: Theoretical Chemistry 4 (1981) 1. S. Langhoff and E.R. Davidson, Intern. J. Quantum Chem. 8 (1974) 61. S. Wilson and D.M. Silver, J. Chem. Phys. 67 (1977) 1689. S. Wilson and D.M. Silver, J. Chem. Phys. 72 (1980) 2159. A.K.Q. Siu and E.R. Davidson, Intern. J. Quantum Chem. 4 (1970) 223. R.J. Bartlett, S. Wilson and D.M. Silver, Intern. J. Quantum Chem. 13 (1977) 737. S. Wilson, D.M. Silver and R.J. Bartlett, Mol. Phys. 33 (1977) 1177.
Chemistry,
eds. P.Th
results are compared with the previous calculations and in every case it can be seen that perturbation theory affords an accuracy equal to or better than previous work. Diagrammatic perturbation theory has been used to calculate potential energy curves and surfaces and as an example we show a potential energy curve for the ground state of the fluorine molecule, obtained by Urban and coworkers [92], in fig. 29. Spectroscopic constants obtained from these results are shown in table 11 where they are also compared with experiment. Also in fig. 29 a potential energy curve obtained by means of the scaling technique, described in section 3, is shown and can be seen to be in good agreement with the full fourth-order calculation [93].
6. Summary and future directions We have emphasized in this article the highly systematic approach to the electron correlation problem in atoms and molecules afforded by the diagrammatic many-body perturbation theory. Not only does this method provide a powerful and tractable approach for performing calculations of high accuracy but also it provides a theoretical formalism in terms of which many contemporary theories of molecular electronic structure can be analysed.
S. Wilson / Diagrammatic
many-body perturbation
473
theory
-199.27
-199.28
-199.29
internuclear distances, bohr -199.31
-199.32
-199.33
1
Fig. 29. Potential
energy curves for the ground
state of the fluorine molecule.
It is clear from the many calculations which have been performed to date that the dominant source of error in molecular calculations utilizing diagrammatic perturbation theory (and indeed in most contemporary calculations) is attributable to basis set truncation [94]. Future work will have to be directed towards the reduction of the magnitude of this error. Indeed, some progress has already been made in this direction by using universal even-tempered basis sets and by using a systematic sequence of universal even-tempered basis sets [95]. The latter approach enables extrapolation procedures to be employed to determine the basis set limit and when used in conjunction with th,e scaling technique described in section 5 shows promise [96]. Table 11 Comparison values
of calculated
spectroscopic
Energy
R, (lo-”
E rcr
1.329
W/u1
1.400
E[3/01 E [4/O] experiment
1.386 1.420 1.412
m)
constants
for the ground
state of the fluorine
molecule
with experimental
0, (cm-‘)
w,x, (cm-‘)
a, (cm-‘)
1265 1007 1039 913 917
6.41 8.63 9.16 11.7 11.2
0.008 0.010 0.010 0.0013 0.014
474
S. Wilson / Diagrammatic
many-body perturbation
theory
Finally, it should be mentioned that relativistic effects will become important when the techniques described in this report are applied to heavy atoms and molecules containing heavy atoms. Work is in progress to develop a relativistic many-body perturbation theory suitable for applications to atoms and molecules including the most important relativistic effects in the reference model [97].
Appendix A. Second- and third-order energy expressions for open-shell atoms and molecules The secondare as follows:
E&J=
and third-order
energy expressions
_ c (il~l~>(~l~li>(~lal~> k-&--A ?iU
E,(E)= c
corresponding
’
3
+lh
(il~lu)(jl~lb)(ublQI~) E3(G)=z
(E,-cE,)(c,+<,-cE,-cZh)
’
(il~lu)(ujl~lbc)(bcIBlij) E3(J)=+vz<
(Ei-E,)(c,+cJ-cb-cr)
’
to the diagrams
given in fig. 14
S. Wilson / Diagrammatic
many-body perturbation
475
theory
. (ijlolab)(kIicIj)(abl~lik) (
E&j
ijkoh
+
%bk
where the e are the usual orbital
and the one-electron matrix non-degenerate doublet state
energies,
elements
CPlW) =%,F(P; P> 4L
+
(PK
6k -
E,
ca -
-
%)
6h)
’
’
the two-electron
depend
integrals
on the electronic
are given by
state
being
studied.
For
a
l4),
where the value of the spin factor F( p; p, q) depends on the nature of the spin orbitals p and q. The nine possible cases for cx spin are given in table 12. The factors for j3 spin are given by
m? P, Table 12 Spin factors
F(a;
q)=F(a;
p,
q)-1.
p, q) for the matrix elements Occupation
of u number
for p in reference
function
Occupation number for q in reference function
2
1
0
2 1 0
312 1 l/2
1 l/2 0
l/2 0 -l/2
416
S. Wilson / Diagrammatic
For triplet states the one-electron (AGlq) K denotes
Pl d((PIK,,
=&J%-C the exchange
operator
operator
many-hodJ perturhution theory
is
14) + (PVC,
14))
for the singly occupied
orbitals
IM and n.
Appendix B. Fifth-order energy diagrams for closed shell systems In this appendix we give the fifth-order energy diagrams for a closed-shell system which have been given previously in refs. [4,6]. Following Paldus and Wong [98,99] we give the essentially distinct Hugenholtz diagrams from which all other non-equivalent Hugenholtz or Brandow diagrams can be obtained either (i) as a different time versions of the given diagram resulting from a permutation of the vertices; (ii) as the particle-hole inversion of the given diagram, resulting from the reversal of the orientation of all the arrows; or (iii) as the time reversal, which is a result of reflecting the diagram in the plane perpendicular to the time axis.
Fig. 30. Essentially
distinct
Hugenholtz
diagrams
which describe
the correlation
energy through
fifth-order.
S. Wilson / Diagrammatic
many-body perturbation
theory
417
In fig. 30 we give the full set of essentially distinct diagrams which describe the correlation energy of a closed-shell system through fifth order. It should be emphasized that the number of distinct Brandow diagrams increases rapidly with the order of perturbation.
Appendix C. Expressions for energy components for a multideterminantal
reference function
As an example of the algebraic expressions for the energy components which arise when a multideterminantal reference wave function is employed. We give the expressions corresponding to the diagrams given in fig. 20: + c .ik
GwbwwQ~~) E, -co
Appendix D. Shifted-denominator
3
perturbation expansion for the energy through third-order
The algebraic expressions for the energy denominators are employed are as follows:
-
components
dijoh)(c; +
c/ -
through
E,. -
E(/ -
third
drJ_/)’
order
when
shifted
478
The denominator
S. Wilson / Diagrammatic
many-bed)) perturbation
theory
shift diiuh is defined in eq. (214).
References
111H.P. Kelly, Adv. Chem. Phys. 14 (1969) 129.
PI
S. Wilson, in: Correlated Wavefunctions, Proc. of Daresbury Study Weekend (December 1977) ed. V.R. Saunders, Science Research Council, London (1978). Topics in Current Chemistry 75 (1978) 97. I31 I. HubaE and P. &sky, 141 S. Wilson, in: Electron Correlation, Proc. of Daresbury Study Weekend (November 1979) eds, M.F. Guest and S. Wilson, Science Research Council, London (1980). and M. Urban, Ab Initio Calculations: Methods and Applications in Chemistry, Lecture Notes in PI P. &sky Chemistry, vol. 16 (Springer, Berlin, 1980). PI S. Wilson, Specialist Periodical Report: Theoretical Chemistry 4 (1981) 1. 171 R.J. Bartlett, Ann. Rev. Phys. Chem. 32 (1981) 359. See also Nobel Symposium issue of Physica Scripta 21 (1981) Molecular Physics, Proc. of Nato Advanced Study Institute, Bad PI S. Wilson, in: Methods in Computational Windsheim (August 1982) eds. G.H.F. Diercksen and S. Wilson (D. Reidel, Dordrecht, 1983). [91 S. Wilson, Electron Correlation in Molecules (Clarendon Press, Oxford, 1984). PO1 H.P. Kelly, Phys. Rev. 131 (1963) 684, 136 (1964) 896, 144 (1966) 39; Adv. Theoret. Phys. 2 (1968) 75; Intern. J. Quantum Chem. Symp. 3 (1969) 349. t111 ES. Chang, R.T. Pu and T.P. Das, Phys. Rev. 174 (1968) 1. NC. Dutta, C. Matsubara, R.T. Pu and T.P. Das, Phys. Rev. 177 (1969) 33. R.T. Pu and ES. Chang, Phys. Rev. 151 (1966) 31. T. Lee, N.C. Dutta and T.P. Das, Phys. Rev. Al (1970) 995, A4 (1971) 1410. P21 H.P. Kelly, Phys. Rev. Lett. 23 (1969) 455. J.H. Miller and H.P. Kelly, Phys. Rev. Lett. 26 (1971) 679. 1131 S. Wilson and D.M. Silver, Phys. Rev. Al4 (1976) 1949. See also J.M. Schulman and D.N. Kaufman, J. Chem. Phys. 53 (1970) 477, 57 (1972) 2328. U. Kaldor, Phys. Rev. A7 (1973) 427; Phys. Rev. Lett. 31 (1973) 1338. M.A. Robb, Chem. Phys. Lett. 20 (1973) 274. 1141 J.A. Pople, J.S. Binkley and R. Seeger, Intern, J. Quantum Chem. Symp. 10 (1976) 1. E.R. Davidson and D.W. Silver, Chem. Phys. Lett. 52 (1977) 403. R.J. Bartlett and I. Shavitt, Intern. J. Quantum Chem. Symp. 11 (1977) 165. S. Wilson and D.M. Silver, Theoret. Chim. Acta. 54 (1979) 83. 1151 K.A. Brueckner, Phys. Rev. 100 (1955) 36. 1161 J. Goldstone, Proc. Roy. Sot. A239 (1951) 267. v71 N.H. March, W.H. Young and S. Sampanthar, The Many-Body Problem in Quantum Mechanics (1967) CUP. P81 J. Paldus and J. Ciiek, Adv. Quantum. Chem. 9 (1975) 105. [I91 B.H. Brandow, Rev. Mod. Phys. 39 (1967) 771. bw N. Hugenholtz, Physica 23 (1957) 481. WI V. Kvasnicka, V. Laurinc and S. Biskupic, Mol. Phys. 39 (1980) 143. PI I. HubaE and P. Carsky, Phys. Rev. A22 (1980) 2392.
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[23] [24] [25] [26]
ed. H.F. Schaefer
480 [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80]
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many-body perturbation theory
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