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Ionization potentials of CH4 and H2O computed using quasi-degenerate many body perturbation theory (QD-MBPT)
CHEMICAL PHYSICS LETTERS
Volume 47, number 3
IONIZATION POTENTIALS QUASI-DEGENERATE
1 May I977
OF CH4 AND Hz0 COMPUTED USING
MANY BODY PERTURBATION
THEORY (QD-MBPT)
Sally PRIME and Michael A. ROBB Department of Chemistry. Queen Elizabeth College, University of London. London W8 7AH. UK Received 3 November 1976 Revised manuscript received 7 February
1977
The vertical ionization potentials for CH4 and Hz0 are computed using the quasidegencratc many body perturbation method of Brandow in a pti function approximation that neglects nondiagonal hole line diagrams and all the “folded” diagrams. The results obtained for localised orbitals are significantly better than for canonical orbitals for the lowest ionization potentials.
1. Introduction Green’s functions, propagator methods, equations of motion methods and random phase approximations (see for example refs. [l-S]), are now accepted techniques for the calculation of ionization potentials and electron affinities. Cederbaum and von Niessen [6] have been particularly successful in the application of a diagrammatic formulation of the Green’s function method to a large number of atoms and moIecules_ (For their theoretical presentation the reader is referred to ref. [ 11. An example of a recent molecular calculation together with a bibliography of earlier work may be found in ref. [3] .) On the other hand, Brandow [7-91 has developed a form of many body perturbation theory for a multidimensional reference function (quasi-degenerate many body perturbation theory, QD-MBPT) which should also be suitable for the computation of ionization potentials_ (For an example of a successful application of QD-MBPT to quantum chemistry, the reader is referred to the calculation of Ka!dor [lo] on B-H.) It should be pointed out from the outset, however, that Brandow has demonstrated [9] that his perturbation series for a single hole, or particle, is precisely equiuaientto the usual prescription for locating the pole of the causal one-body Green’s function. On the other hand, it would appear that QD-MBPT is conceptually simpler
than Green’s function theories and, consequently, systematic approximation can be more easily formulated in this language.
2. Theory The QD-MBPT method of Brandow is completely documented in refs. [8-lo] _Thus we-will be-content with a brief summary of those aspects of the method that are relevant to the calculation of ionization potentials_ In the QD-MBPT method one calcuiates the projection of tie wavefunction onto a suitable reference space. The solution of a non-hermitean but energy independent eigenvalue problem involving an effective hamiltonian in this reference space yields the energies of those states with non-zero projections on this reference subspace. The matrix elements of this effective hamiltonian are computed by summing diagrams that have many topological simiI?rities to the diagrams of Green’s function theories. However, in QDMBPT the explicit energy dependence of the diagrams is removed using so-called “folded diagrams”. (For a pedagogical example, the reader is referred to the one half page derivation of the simple random phase approximation to be found on page 806 of Brandow’s review [7 ] _) 527
VoIume 47, number 3
CHEMICAL PHYSICS LETI-ERS
In the calculation of ionization potentills, the reference space becomes the space of ail singly ionized states and the fust and second order diagrams for the matrix elements of the corresponding effective hamiitonian can be drawn as in fig. I_ The diagrams are to be evaluated using Brandow’s rules [7-g], and summations in fig. 1 are implied over di the internal lines (greek labels for occupied orbitais and iatin labels for virtua1 orbit&). Diagrams I b and Ic are related to the diagrams of fig. 1b in Cederbaum’s paper [I] . Diagram la is necessary if one is using localized molecular orbitais (LMO’s) [l I] instead of canonical moiecuI= orbit& (CMO’s) and gives the first order correction due to the Hartree-Fock operator. The conceptual significance of diagrams 1b and 1c is related to the concepts first discussed by Sinanoglu [ 141 in his many electron theory for excited states. Thus, diagram 1b corresponds to external coveZation. diagram 1c with y equal to a! or /3 corresponds to a sum of the rearrangement energy of a ASCF caicuiation pIus the spin polarization correlaiion. and diagram Ic with y not equal to CYor 0 corresponds to the dynamic effect of semi-internal correlation. In the QD-MBPT method of Brandow (7-91 the diagrams lb and ic are evaluated unsymmetricalIy (i.e. the diagrams with a! and fl exchanged are not equal since one must discard the label of the bottom external Iine when “cIosing” the diagram in the evaluation of the energy denominators). This procedure leads to an unsymmetric effective hamiltonian which is computationally inconvenient. This problem can be circumvented by introducing a metric matrix. However, Braudow [S] points out, that because of the large amount of canceIlation that takes place when the metric matrix is expanded in a binomial series, the “naively symmetrized” hamiltonian (i.e. average of hamiltonian and its adjoint) may be a good
; j -------
bJ
%?
J&-!@
(b)
approximation. We have adopted this approximation in the present work. The question of which diagrams to include beyond second order must now be considered. The technique of selective summation of diagrams is now well estabIished 1121. C?ne hopes to identify the most inportant diagrams by their intermediate interactions at third order, and then selecrively sum these repeated interactions to infinite order. By analogy with the work of Kelly [ 121 for the ground state correlation problem, one is Ied to suppose that the diagrams with all possible intermediate “ladder” interactions without non-diagonal hole lines in figs. lb and Ic willdominate.(Thisapproximation could aIso be made with respect to diagrams CI and C6 of fig_ 5 in ref. [I] _) The resulting diagrams can then be summed by solving pseudo-linear inhomogeneous “pair function” equations. If we defme “pair functions” for diagrams lb and lc as:
(1) and
(2) where Yp4 and flX_r6 are linear variation and l&l)q(2))
= 2-1’2 rP(l)q(2)
coefficients,
- q(l)p(2)1,
(3)
then we may sum the diagonal hole line interactions to infiiite order by solving the essentially independent pair type equations -
h( 1) -h(2)
+ 1/r12
;/;;-;-:op
(Cl
Fig. 1. QD-MBFT diagrams for ionization potentials. (a) First order Hartree-Fock, giving the Koopmans’ theorem result for LMO. (b) External correlation. (c) Rearrangement plus spin pohization (7 = Q or 0) and semi-internal corrchtion (yfaorfl).
1 May 1977
-_[.!$I)
-KY(l)]
=fi2)W)lY(W3(2))~
-
[J#)
-K,(1)13f’2iX!$2))
iP(2)).
(5)
In eqs. (4) and (5) above, the operators h( 1), J(1) and K( 1) are the usual Hartree-Fock, Coulomb and ex-
Volume 47. number 3
CHEMICALPHYSICSLETTERS
change operators; Qrz is defined as: &
[IP(l)4(2))(P(l)s(2)Il;
= c Pa
(6)
Pz isdefined as p2 = $
Ih?(2))(4(2)11;
(7)
the symbols (76 I7@ denote two electron Coulomb minus exchange integrals; and Ed etc. are diagonal elements of the Hartree-Fock operator_ The resulting ionization potentials are then found as the eigenvalues, Ek, of the equation (H, + VI +
(8)
Vn)Ak =E&
where the operators are defined by their matrix elements in the basis of singIe hole states as:
vn = f c
la) -c
((ax;&g)
+ CX -
p)
where t@e eCrpin eq. (IO) is the offdiagonal element of the Hartree-Fock operator (diagram la). It should be noted that eqs. (4) and (5) are identical to those proposed by Pickup and Goscinski (eqs. (8 1) and (79) of ref. [2] with E set equal to eP)_ (The terms required to go from the Green’s function pair equation (77) of ref. 125 correspond to diagonal hole ladders involving the lines cr and p in diagram lb.) From a computational point of view QD-MBPT offers some advantages over the Green’s function method. Since the QD-MBa method involves energy independent diagrams, only one set of pair function calculations is required for alI ionization potentials. Thus, one avoids the computational problem of the self-consistency of the energy denominators in the usual Green’s function theories_ However, we have neglected the “folded” diagrams which result from the expansion of the explicit energy dependence from the denominators. (Green’s function theories can be regarded as “‘highly summed” [8] versions of the
1 hfay 1977
QD-MBFT.) In the present context the Wded diagrams arise by inserting diagram la into diagrams Ib and lc (giving a third order contribution for the case of L&IO’s only) and by inserting diagrams lb and Ic into themselves (giving a fourth order contribution), Since we are neglecting nondiagonal hole line interactions it is consistent to neglect the third order LMO contribution_ The neglect of the fourth order folded diagrams and all non-diagonal hole line diagrams at third order makes the theory rather crude and these approximations may ultimately only be justified by the quality of the results obtained. Finally, it should be pointed out that if the fokied diagrams are important the self-consistency of the energy denominators in Green’s function theo&~ beCOIPSS a computational advantage because of&e ~10~ convergence [71 of the “folded” diagram expan.sian._ In this case the coupled pair Green’s function theory of Freed [ 13) would provide a better formulation for implementing an “independent” pair approximation of the type discussed in this work.
3. Rest&s and discussion The pair functions corresponding to diagrams lb and lc have been used to compute the vertical ionization potentials of CH, and H,O. The orbital basis set for CH4 has been described previously iI. and gave better than 95% of the ground state valence shell correlation energy. That for Hz0 has been constructed in a similar manner. The computed results for the valence shell ionization potentials of CH4 are collected in table L along with ASCF results in the same basis and the ACEPA (coupled eIectron pair approximation) results of Meyer [ 161. In table 2 we have presented a decomposition of the QD-MBPr results into contributions from diagrams lb and lc with the CEElA results 1161 for comparison. In tables 3 and 4 we present the corresponding results for Hz0 where we bave aIso made a comparison with Green’s function methods. Comparison of the figures in tables I and 3 shows that for all states our result is more accurate than either the ASCF or the Koopmans’ theorem result. This is particutarty true of the lower energy ionization potentiaIs where correlation effects are the greatest and where ASCF and Koopmans’ theorem
52’
Volume 47, number 3 Table 1 Theoretical
1 May 1977
CHEMICAL PHYSICS LETTERS
and experimental
vertical ionization
potentials
Method Koopmans’ ASCF QD-MBPT
theorem
ACEPA d)
experiment
e)
for CH4 (in atomic units)
2ar
t2 a)
0.9431 0.8932 0.9041 b)/o.8821 0.8592 c) 0.840
0.5429 05012 0.5128 b)/o.xmc) 0.5252 =) 0.528
=)
a) Because of neglect of certain third order diagrams and the lack of symmetry and equivalence were not exactly degenerate. The tabulated values are an average with a spread off 0.0003. b) CM0 result. =) LMO result. d) From ref. [ 16]_ d As corrected by ref. [ I6 1.
Table 2 Components
of the vertical ionization
potentials
of CH4
Present work (QD-MBPT) CM0
Koopmans’ (dia. la)
theorem
II external correlation (dia. lb)
Ref. [IS] (ACEPA) CM0
LMO
2al I
the t2 eigenvalues
restrictions,
t2
t2
2a1
2al
t2
0.9431
0.5428
0.943 1
0.5428
0.9443
0.5459
0.0476
0.0773
0.0413
0.0832
0.0437
0.0576
III rearrangement + spin polarization (dia. 1c, y = Q or p)
-0.0627
-0.0893
-0.0871
-0.0980
-0.0575
-0.0541
IV semi-internal corrchrtion (dra. Ic, 7 f a or p)
-0.0239
-0.0182
-0.0153
+0.0012
-0.0746
-0.0200
Table 3 Theoretical
and experimental
Method
Koopmans’ ASCF :;;;;%
theorem
lb2
Sal
Ibt
1.3536
0.7146 0.6476 0.6929 b)/o.7071 0.7000
0.5787 0.4858 0.5395 0.5246
0.5103 0.4093 0.4585 0.4388 b)/o.4677
a) 1.3107 1.1887 b)/l.2934
0
for Ha0 (in atomic units)
2al
hIBGF e) c\perimental
Ionization potentials
-
c)
0.6968 0.688 + 0.008
a) No SCF upper bound is obtainable for the doublet hole state. b) CM0 result. c) LMO result. d) From ref. [17]. e, Many-body Green’s function results from ref. [ 18]_ D As corrected
c)
b)/o.als C)
0.5479 0.545 + 0.004
by ref. [ 171.
0.4664 0.4698
c)
Volume 47, number 3 Table 4 Components
CHEMICAL PHYSICS LETTERS
of the vertical ionization
potentials
2a1 I
Koopmans’ (dia. la)
theorem
II external correlation (dia. Ib) ChIO/LhfO RI redrrangemcnt + spin polarization (dia. Ic, 7 = Q or p) ChfO/LhlO IV semi-internal correlation
of Ha0 (results obtained
1 May I977
in ref. [ 17 ] for CM0 are included in parentheses) 3a1
Wz
lb1
1.3536 (1.3514)
0.7 146 (O-7165)
0.5787 (0.5832)
0.5 103 (0.5092)
0_0418/0.0372 (0.0509)
0.0701/0.0698 (0.0738)
0.0677/0.0610 (0.0743)
(0.07~0)
-0.0754/-0.0778
-0.1127/-0.0854
-0.0667/-0.09 (-0.1046)
13
(-0.1026)
(-0.0802) -0.00931+0_0005 (-0.0165)
give poor results. We also note that except for the lb2 state of H20, the LMO give significantly better results than the CMO. However, the agreement with experimental results is not as good as in the ACEPA and MB-GF methods. This is perhaps not too surprising in view of the approximations made in the present work. On inspection of tables 2 and 4, where the decomposition of our results into contributions from the diagrams of fig. 1 are given, some generaI observations can be made. Firstly, we can see that the semi-internal correlation, although in each case being the smaIIest contribution, is not aIways negIigibIe. Further, there is a partial cancellation of effects between the positive extemaI correIation and the negative rearrangement and spin polarization correction, and both of these effects are of roughly equaI magnitude. We note that also the external correlation term is not very strongly dependent on whether CMO’s or LMO’s are used and that most of the CMO/LMO dependence arises from fig. lc, and further that this effect is again more pronounced for the lower ionization potentials where the LMO’s give the greatest improvement to the overaII result.
4. Conclusions The success of a decoupled pair theory for ionization potentials within QD-MBFT (and also for Green’s
-0.0091/-0.0128 (-0.0139)
0.0666~0.0563
-0_1312/-0.0846
(--O.Ll30) -0.0069/-0.0163 (-0.0105)
functions) depends upon the mutual cancellation of non-diagonal hole line ladders between figs. Ib and lc. Since for each such diagram from fig_ lb there is a corresponding diagram in fig. 1c of opposite sign and roughly equal magnitude, we may expect such a cancellation to occur, at least partially. Our results indicate that for low energy ionization potentials when LMO’s are used, this cancellation is partially effective_ At first sight one might find the fact that the LMO’s give better results than CMO’s rather surprising. CIearIy, there are several factors involved. In our previous calculations [ I5 1 we have demonstrated the partial cancellation of the Iadders involving offdiagonaI elements of the Hartree-Fock operator (which occur onIy for LMO’s) with the non-diagonal hoIe Iine Iadders. In the present work, the influence of the use of LMO’s on the probIem of the non-hermiticity of the effective hamihonian (with LMO’s, the zeroth order hamiItonian comes closer to being degenerate) and the appearance of folded diagrams at third order involving off-diagonal elements of the Hartree-Fock operator when using LMO’s. must aIso be considered_ We hope to examine these factors numerically in future work. FinaIIy, it would appear that the sacrifice in accuracy with the present method over MB-GF or ACEPA methods is not too severe in view cf the obvious computational advantages of the present implementation of QD-MBPT. 531
Volume 47, number 3
CHEMICAL PHYSICS LETTERS
Acknowledgement One of us (S-P.) thanks the Science Research Coun61 for a studentship. The authors are grateful to Professor B.H. Brandow for helpful comments on the unrevised manuscript_
References [I ] L.S. Cedcrbaum. Theoret. Chim. Acta 31 (1973) 239. [Z] B-T. Pickup and 0. Goscinski, hiol. Phys. 26 (1973) 1013. [3] J. Simons, Chem. Phys. Letters 25 (1974) 122. (41 G.D. Purvis and Y. &II, J. Chem. Phys. 60 (1974) 4063. [S] J-D. DolI and W-P. Reinhart, 1 Chem. Phys. 57 (1972) 1169s
532
1 May 1977
[6 ] W. von Niesscn, L.S. Cederbaum, G.H.F. Diercksen and G. Hohineicher. Chem. Phys. ll(1975) 399. [7] B.H. Brandow, Rev. Mod. Phys. 39 (1967) 771. (81 B.H. Brandow, Lectures in Theoretical Physics 1 IB (1969) 55. [9] B.H. Brandow, Ann. Phys. NY 64 (1971) 21. app. C. [IO] P.S. Srem and U. Kaldor, J. Chem. Phys. 64 (1976) 2002. [i 1 J C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 35 (1963) 457. [12] H.P. Kelly, Advan. Chem. Phys. 14 (1969) 129. [I31 F.S.M. Tsui and K. Freed, Chem. Phys. 5 (1974) 337. [I41 0. Sinano@u, Advan. Chem. Phys. 14 (1969) 239. (151 S. Prime and M.A. Robb, Theoret. Chim. Acta 42 (1976) 181. [I61 W. Meyer, J. Cbem- Phys. 58 (1973) 1017. [ 171 W. Meyer, Intern. J. Quantum Chem. 5 (1971) 341_ [ 18) L.S. Cederbaum. G. Hohlneicher and W. von Niessen, Moi. Phys. 26 (1973) 1405.
Volume 47, number 3
IONIZATION POTENTIALS QUASI-DEGENERATE
1 May I977
OF CH4 AND Hz0 COMPUTED USING
MANY BODY PERTURBATION
THEORY (QD-MBPT)
Sally PRIME and Michael A. ROBB Department of Chemistry. Queen Elizabeth College, University of London. London W8 7AH. UK Received 3 November 1976 Revised manuscript received 7 February
1977
The vertical ionization potentials for CH4 and Hz0 are computed using the quasidegencratc many body perturbation method of Brandow in a pti function approximation that neglects nondiagonal hole line diagrams and all the “folded” diagrams. The results obtained for localised orbitals are significantly better than for canonical orbitals for the lowest ionization potentials.
1. Introduction Green’s functions, propagator methods, equations of motion methods and random phase approximations (see for example refs. [l-S]), are now accepted techniques for the calculation of ionization potentials and electron affinities. Cederbaum and von Niessen [6] have been particularly successful in the application of a diagrammatic formulation of the Green’s function method to a large number of atoms and moIecules_ (For their theoretical presentation the reader is referred to ref. [ 11. An example of a recent molecular calculation together with a bibliography of earlier work may be found in ref. [3] .) On the other hand, Brandow [7-91 has developed a form of many body perturbation theory for a multidimensional reference function (quasi-degenerate many body perturbation theory, QD-MBPT) which should also be suitable for the computation of ionization potentials_ (For an example of a successful application of QD-MBPT to quantum chemistry, the reader is referred to the calculation of Ka!dor [lo] on B-H.) It should be pointed out from the outset, however, that Brandow has demonstrated [9] that his perturbation series for a single hole, or particle, is precisely equiuaientto the usual prescription for locating the pole of the causal one-body Green’s function. On the other hand, it would appear that QD-MBPT is conceptually simpler
than Green’s function theories and, consequently, systematic approximation can be more easily formulated in this language.
2. Theory The QD-MBPT method of Brandow is completely documented in refs. [8-lo] _Thus we-will be-content with a brief summary of those aspects of the method that are relevant to the calculation of ionization potentials_ In the QD-MBPT method one calcuiates the projection of tie wavefunction onto a suitable reference space. The solution of a non-hermitean but energy independent eigenvalue problem involving an effective hamiltonian in this reference space yields the energies of those states with non-zero projections on this reference subspace. The matrix elements of this effective hamiltonian are computed by summing diagrams that have many topological simiI?rities to the diagrams of Green’s function theories. However, in QDMBPT the explicit energy dependence of the diagrams is removed using so-called “folded diagrams”. (For a pedagogical example, the reader is referred to the one half page derivation of the simple random phase approximation to be found on page 806 of Brandow’s review [7 ] _) 527
VoIume 47, number 3
CHEMICAL PHYSICS LETI-ERS
In the calculation of ionization potentills, the reference space becomes the space of ail singly ionized states and the fust and second order diagrams for the matrix elements of the corresponding effective hamiitonian can be drawn as in fig. I_ The diagrams are to be evaluated using Brandow’s rules [7-g], and summations in fig. 1 are implied over di the internal lines (greek labels for occupied orbitais and iatin labels for virtua1 orbit&). Diagrams I b and Ic are related to the diagrams of fig. 1b in Cederbaum’s paper [I] . Diagram la is necessary if one is using localized molecular orbitais (LMO’s) [l I] instead of canonical moiecuI= orbit& (CMO’s) and gives the first order correction due to the Hartree-Fock operator. The conceptual significance of diagrams 1b and 1c is related to the concepts first discussed by Sinanoglu [ 141 in his many electron theory for excited states. Thus, diagram 1b corresponds to external coveZation. diagram 1c with y equal to a! or /3 corresponds to a sum of the rearrangement energy of a ASCF caicuiation pIus the spin polarization correlaiion. and diagram Ic with y not equal to CYor 0 corresponds to the dynamic effect of semi-internal correlation. In the QD-MBPT method of Brandow (7-91 the diagrams lb and ic are evaluated unsymmetricalIy (i.e. the diagrams with a! and fl exchanged are not equal since one must discard the label of the bottom external Iine when “cIosing” the diagram in the evaluation of the energy denominators). This procedure leads to an unsymmetric effective hamiltonian which is computationally inconvenient. This problem can be circumvented by introducing a metric matrix. However, Braudow [S] points out, that because of the large amount of canceIlation that takes place when the metric matrix is expanded in a binomial series, the “naively symmetrized” hamiltonian (i.e. average of hamiltonian and its adjoint) may be a good
; j -------
bJ
%?
J&-!@
(b)
approximation. We have adopted this approximation in the present work. The question of which diagrams to include beyond second order must now be considered. The technique of selective summation of diagrams is now well estabIished 1121. C?ne hopes to identify the most inportant diagrams by their intermediate interactions at third order, and then selecrively sum these repeated interactions to infinite order. By analogy with the work of Kelly [ 121 for the ground state correlation problem, one is Ied to suppose that the diagrams with all possible intermediate “ladder” interactions without non-diagonal hole lines in figs. lb and Ic willdominate.(Thisapproximation could aIso be made with respect to diagrams CI and C6 of fig_ 5 in ref. [I] _) The resulting diagrams can then be summed by solving pseudo-linear inhomogeneous “pair function” equations. If we defme “pair functions” for diagrams lb and lc as:
(1) and
(2) where Yp4 and flX_r6 are linear variation and l&l)q(2))
= 2-1’2 rP(l)q(2)
coefficients,
- q(l)p(2)1,
(3)
then we may sum the diagonal hole line interactions to infiiite order by solving the essentially independent pair type equations -
h( 1) -h(2)
+ 1/r12
;/;;-;-:op
(Cl
Fig. 1. QD-MBFT diagrams for ionization potentials. (a) First order Hartree-Fock, giving the Koopmans’ theorem result for LMO. (b) External correlation. (c) Rearrangement plus spin pohization (7 = Q or 0) and semi-internal corrchtion (yfaorfl).
1 May 1977
-_[.!$I)
-KY(l)]
=fi2)W)lY(W3(2))~
-
[J#)
-K,(1)13f’2iX!$2))
iP(2)).
(5)
In eqs. (4) and (5) above, the operators h( 1), J(1) and K( 1) are the usual Hartree-Fock, Coulomb and ex-
Volume 47. number 3
CHEMICALPHYSICSLETTERS
change operators; Qrz is defined as: &
[IP(l)4(2))(P(l)s(2)Il;
= c Pa
(6)
Pz isdefined as p2 = $
Ih?(2))(4(2)11;
(7)
the symbols (76 I7@ denote two electron Coulomb minus exchange integrals; and Ed etc. are diagonal elements of the Hartree-Fock operator_ The resulting ionization potentials are then found as the eigenvalues, Ek, of the equation (H, + VI +
(8)
Vn)Ak =E&
where the operators are defined by their matrix elements in the basis of singIe hole states as:
vn = f c
la) -c
((ax;&g)
+ CX -
p)
where t@e eCrpin eq. (IO) is the offdiagonal element of the Hartree-Fock operator (diagram la). It should be noted that eqs. (4) and (5) are identical to those proposed by Pickup and Goscinski (eqs. (8 1) and (79) of ref. [2] with E set equal to eP)_ (The terms required to go from the Green’s function pair equation (77) of ref. 125 correspond to diagonal hole ladders involving the lines cr and p in diagram lb.) From a computational point of view QD-MBPT offers some advantages over the Green’s function method. Since the QD-MBa method involves energy independent diagrams, only one set of pair function calculations is required for alI ionization potentials. Thus, one avoids the computational problem of the self-consistency of the energy denominators in the usual Green’s function theories_ However, we have neglected the “folded” diagrams which result from the expansion of the explicit energy dependence from the denominators. (Green’s function theories can be regarded as “‘highly summed” [8] versions of the
1 hfay 1977
QD-MBFT.) In the present context the Wded diagrams arise by inserting diagram la into diagrams Ib and lc (giving a third order contribution for the case of L&IO’s only) and by inserting diagrams lb and Ic into themselves (giving a fourth order contribution), Since we are neglecting nondiagonal hole line interactions it is consistent to neglect the third order LMO contribution_ The neglect of the fourth order folded diagrams and all non-diagonal hole line diagrams at third order makes the theory rather crude and these approximations may ultimately only be justified by the quality of the results obtained. Finally, it should be pointed out that if the fokied diagrams are important the self-consistency of the energy denominators in Green’s function theo&~ beCOIPSS a computational advantage because of&e ~10~ convergence [71 of the “folded” diagram expan.sian._ In this case the coupled pair Green’s function theory of Freed [ 13) would provide a better formulation for implementing an “independent” pair approximation of the type discussed in this work.
3. Rest&s and discussion The pair functions corresponding to diagrams lb and lc have been used to compute the vertical ionization potentials of CH, and H,O. The orbital basis set for CH4 has been described previously iI. and gave better than 95% of the ground state valence shell correlation energy. That for Hz0 has been constructed in a similar manner. The computed results for the valence shell ionization potentials of CH4 are collected in table L along with ASCF results in the same basis and the ACEPA (coupled eIectron pair approximation) results of Meyer [ 161. In table 2 we have presented a decomposition of the QD-MBPr results into contributions from diagrams lb and lc with the CEElA results 1161 for comparison. In tables 3 and 4 we present the corresponding results for Hz0 where we bave aIso made a comparison with Green’s function methods. Comparison of the figures in tables I and 3 shows that for all states our result is more accurate than either the ASCF or the Koopmans’ theorem result. This is particutarty true of the lower energy ionization potentiaIs where correlation effects are the greatest and where ASCF and Koopmans’ theorem
52’
Volume 47, number 3 Table 1 Theoretical
1 May 1977
CHEMICAL PHYSICS LETTERS
and experimental
vertical ionization
potentials
Method Koopmans’ ASCF QD-MBPT
theorem
ACEPA d)
experiment
e)
for CH4 (in atomic units)
2ar
t2 a)
0.9431 0.8932 0.9041 b)/o.8821 0.8592 c) 0.840
0.5429 05012 0.5128 b)/o.xmc) 0.5252 =) 0.528
=)
a) Because of neglect of certain third order diagrams and the lack of symmetry and equivalence were not exactly degenerate. The tabulated values are an average with a spread off 0.0003. b) CM0 result. =) LMO result. d) From ref. [ 16]_ d As corrected by ref. [ I6 1.
Table 2 Components
of the vertical ionization
potentials
of CH4
Present work (QD-MBPT) CM0
Koopmans’ (dia. la)
theorem
II external correlation (dia. lb)
Ref. [IS] (ACEPA) CM0
LMO
2al I
the t2 eigenvalues
restrictions,
t2
t2
2a1
2al
t2
0.9431
0.5428
0.943 1
0.5428
0.9443
0.5459
0.0476
0.0773
0.0413
0.0832
0.0437
0.0576
III rearrangement + spin polarization (dia. 1c, y = Q or p)
-0.0627
-0.0893
-0.0871
-0.0980
-0.0575
-0.0541
IV semi-internal corrchrtion (dra. Ic, 7 f a or p)
-0.0239
-0.0182
-0.0153
+0.0012
-0.0746
-0.0200
Table 3 Theoretical
and experimental
Method
Koopmans’ ASCF :;;;;%
theorem
lb2
Sal
Ibt
1.3536
0.7146 0.6476 0.6929 b)/o.7071 0.7000
0.5787 0.4858 0.5395 0.5246
0.5103 0.4093 0.4585 0.4388 b)/o.4677
a) 1.3107 1.1887 b)/l.2934
0
for Ha0 (in atomic units)
2al
hIBGF e) c\perimental
Ionization potentials
-
c)
0.6968 0.688 + 0.008
a) No SCF upper bound is obtainable for the doublet hole state. b) CM0 result. c) LMO result. d) From ref. [17]. e, Many-body Green’s function results from ref. [ 18]_ D As corrected
c)
b)/o.als C)
0.5479 0.545 + 0.004
by ref. [ 171.
0.4664 0.4698
c)
Volume 47, number 3 Table 4 Components
CHEMICAL PHYSICS LETTERS
of the vertical ionization
potentials
2a1 I
Koopmans’ (dia. la)
theorem
II external correlation (dia. Ib) ChIO/LhfO RI redrrangemcnt + spin polarization (dia. Ic, 7 = Q or p) ChfO/LhlO IV semi-internal correlation
of Ha0 (results obtained
1 May I977
in ref. [ 17 ] for CM0 are included in parentheses) 3a1
Wz
lb1
1.3536 (1.3514)
0.7 146 (O-7165)
0.5787 (0.5832)
0.5 103 (0.5092)
0_0418/0.0372 (0.0509)
0.0701/0.0698 (0.0738)
0.0677/0.0610 (0.0743)
(0.07~0)
-0.0754/-0.0778
-0.1127/-0.0854
-0.0667/-0.09 (-0.1046)
13
(-0.1026)
(-0.0802) -0.00931+0_0005 (-0.0165)
give poor results. We also note that except for the lb2 state of H20, the LMO give significantly better results than the CMO. However, the agreement with experimental results is not as good as in the ACEPA and MB-GF methods. This is perhaps not too surprising in view of the approximations made in the present work. On inspection of tables 2 and 4, where the decomposition of our results into contributions from the diagrams of fig. 1 are given, some generaI observations can be made. Firstly, we can see that the semi-internal correlation, although in each case being the smaIIest contribution, is not aIways negIigibIe. Further, there is a partial cancellation of effects between the positive extemaI correIation and the negative rearrangement and spin polarization correction, and both of these effects are of roughly equaI magnitude. We note that also the external correlation term is not very strongly dependent on whether CMO’s or LMO’s are used and that most of the CMO/LMO dependence arises from fig. lc, and further that this effect is again more pronounced for the lower ionization potentials where the LMO’s give the greatest improvement to the overaII result.
4. Conclusions The success of a decoupled pair theory for ionization potentials within QD-MBFT (and also for Green’s
-0.0091/-0.0128 (-0.0139)
0.0666~0.0563
-0_1312/-0.0846
(--O.Ll30) -0.0069/-0.0163 (-0.0105)
functions) depends upon the mutual cancellation of non-diagonal hole line ladders between figs. Ib and lc. Since for each such diagram from fig_ lb there is a corresponding diagram in fig. 1c of opposite sign and roughly equal magnitude, we may expect such a cancellation to occur, at least partially. Our results indicate that for low energy ionization potentials when LMO’s are used, this cancellation is partially effective_ At first sight one might find the fact that the LMO’s give better results than CMO’s rather surprising. CIearIy, there are several factors involved. In our previous calculations [ I5 1 we have demonstrated the partial cancellation of the Iadders involving offdiagonaI elements of the Hartree-Fock operator (which occur onIy for LMO’s) with the non-diagonal hoIe Iine Iadders. In the present work, the influence of the use of LMO’s on the probIem of the non-hermiticity of the effective hamihonian (with LMO’s, the zeroth order hamiItonian comes closer to being degenerate) and the appearance of folded diagrams at third order involving off-diagonal elements of the Hartree-Fock operator when using LMO’s. must aIso be considered_ We hope to examine these factors numerically in future work. FinaIIy, it would appear that the sacrifice in accuracy with the present method over MB-GF or ACEPA methods is not too severe in view cf the obvious computational advantages of the present implementation of QD-MBPT. 531
Volume 47, number 3
CHEMICAL PHYSICS LETTERS
Acknowledgement One of us (S-P.) thanks the Science Research Coun61 for a studentship. The authors are grateful to Professor B.H. Brandow for helpful comments on the unrevised manuscript_
References [I ] L.S. Cedcrbaum. Theoret. Chim. Acta 31 (1973) 239. [Z] B-T. Pickup and 0. Goscinski, hiol. Phys. 26 (1973) 1013. [3] J. Simons, Chem. Phys. Letters 25 (1974) 122. (41 G.D. Purvis and Y. &II, J. Chem. Phys. 60 (1974) 4063. [S] J-D. DolI and W-P. Reinhart, 1 Chem. Phys. 57 (1972) 1169s
532
1 May 1977
[6 ] W. von Niesscn, L.S. Cederbaum, G.H.F. Diercksen and G. Hohineicher. Chem. Phys. ll(1975) 399. [7] B.H. Brandow, Rev. Mod. Phys. 39 (1967) 771. (81 B.H. Brandow, Lectures in Theoretical Physics 1 IB (1969) 55. [9] B.H. Brandow, Ann. Phys. NY 64 (1971) 21. app. C. [IO] P.S. Srem and U. Kaldor, J. Chem. Phys. 64 (1976) 2002. [i 1 J C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 35 (1963) 457. [12] H.P. Kelly, Advan. Chem. Phys. 14 (1969) 129. [I31 F.S.M. Tsui and K. Freed, Chem. Phys. 5 (1974) 337. [I41 0. Sinano@u, Advan. Chem. Phys. 14 (1969) 239. (151 S. Prime and M.A. Robb, Theoret. Chim. Acta 42 (1976) 181. [I61 W. Meyer, J. Cbem- Phys. 58 (1973) 1017. [ 171 W. Meyer, Intern. J. Quantum Chem. 5 (1971) 341_ [ 18) L.S. Cederbaum. G. Hohlneicher and W. von Niessen, Moi. Phys. 26 (1973) 1405.