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Inclusion of doubly excited configurations in electron propagators
Volume
36, number
2
INCLUSION
CHEMICAL
OF DOUBLY
EXCITED
PHYSICS
1 November 1975
LETTERS
CONFIGURATIONS
IN ELECTRON
PROPAGATORS
K.D. JORDAN Department of h’ngkeering and Applied New Haven. Connecticut 06520. USA
Received
29 May
Science,
M~:on
Labomtory.
Yale
Utliversity,
L975
Inclcsion of doubly excited configwations in excited states is investigated in a moment conserving decoupling of the two-partic!c Green’s function for molecules with a closed shell ground state. The result obtained is found to correspond to a [2,1] Pad6 approximant decoupling of the Green’s function hierarchy. It is shown that this decoupling is equivalent to the equations of motion method of Shibuya, Rose, and McKay.
1. Introduction The conventional methods for treating the problem of electron correlation such as perturbation theory or configuration interaction (CI) involve considerab!e computational effort especially if one attempts to treat the correlation in the ground and excited states in a consistent manner. Various many-body techniques based on equations of motion [l-9] or Green’s function [lo-IS] techniques provide a means of direct calculation of excitation energies, oscillator strengths, inelastic scattering cross sections and other quantities of interest without explicitly requiring calculation Gf correlated ground and excited state wavefunctions. In addition to computational simplicity, many-body methods have attracted interest due to the possibility of providing a ba!anced treatment of the correlation in various states and also due to the expectation that they will lead to new, powerful semi-empikical approaches. In this paper we study the hierarchy of equations generated from consideration cf the particle-hole (p-h) propagator (a special form of the two-particle Green’s function). The propagator techniques of Linderberg et al. [l l-151 are based on the-assumption that the excited states can be treated in the singly-excited configuration framework. We will show how the effects of the contribution of the double excitations to the excited states enter into the moment expansion of the propagator. The motivation for A&isextension is the growing awareness of the inadequacies of the single excitation description of the excited states of z electron systems [19-211. In order to obtain the correct ordering and relative separation of the !ow-lying electronic states of pi electron systems such as butadiene, hexatriene, and benzene, it is necessary to include double and triple excitations in addition to the single excitations [ 191. The advantages of the EOM or Green’s function methods over the conventional approach of performing separate calculations on each state of interest is not completeiy understood. One of the important aspects of the manybody approaches is th!: cancellation of terms common to both electronic states of interest [23]. This cancellation has been demonstrated by Malrieu et al. [22] by employing a Rayleigh-Schr6dinger perturbation theoretical approach to both states of interest [22].
2. Electron Folldwing 264.. .-
:
propagator Lifiderberg __.
decoup!ing and Ratner
i.1.21, the causal Green’s
function
is defmed
in terms of creation
and annihila-
Volume 36, number 2
CHEMICAL PHYSICS LETTERS
I November 1975
tion operators
where i, i, k, i... refer to orbitals which are occupied ill the Hartree-Fock (I-IF) ground state and the IJL, )I, p, 4, . . . refer to orbit& unoccupied in the HF ground state. 10) corresponds to the correlated ground state. Fourier tnnsformation of the above Green’s function gives the p-h propagator in the energy representation 0
GQTQ, ; Q,'Qjlj~ = J
dt exp (iEf/lfi) (bif(r>~,,, (t); aiUj>) ,:
(2)
--m
For compactness
we make the following
definitions: (3)
si- being the column vector of p-h pair creation operators and s the column operators. In this notation. the equation of motion for the propagator is EG; Iteration
vector of the p-h
S’>j- = COl[S, S’] IO) + (([S, H]; S+NE . of this equation
gives the moment
(4)
expansion
~,=E-‘~01[S,S+]10~tE-~~01[[S,H],S’]10~~E-~~01[[[S,H],H], =E-lM(o)
+E-2&‘)
+E-3M(z)
s+]lo)+...
[S,HJ
=tol[[s,~J,s+]lo,~ol~s,sf]lo,-‘s
(51
+ . .. ,
(6)
where the nth moment, M(“), is the average in which the hamiltonian H appears )I times. Linderberg et a!. [ 11,121 approximate the moment expansion by a geometric sum involving moments. Their basic approximation is the linearization of the commutator of S with H
Substitution
pair annihilation
the first two
-
(7)
of eq. (7) into eq. (4) gives
((S; SfNE
= (Ol[S,
s+]lo>(z~ol[S,Sf]lO>-(OI[[S,H],
s’]ro,)-~
Sf]lO) -
(8)
This geometric sum is equivalent to a [ 1 ,O] Pad6 approximant
decoupling [26]. It corresponds to the p-h equations of motion of Shibuya and McKay [3] and Rowe [&?I*. IftOj[S, St] 10) and tOl[[S, A!], S+] JO) are evaluated using the HF ground state the random phase apprMmation (RPA) is ottained. In order to include the effects of double excitations in the description of the excited states we approximate the commutator as follows:
[S,H] =
(9)
where
in analogy with eq. (3). * In Practice the COmmutatOr [[S, HI, S*] in ccl. (8) is replaced by Its symmetrzed
version, which we have postpoiled
until eq. (37)
Volime
36, number
Substitution
2
CHEMICAL PHYSICS LETTERS
1 November
1975
eq. (9) in to eq. (5) gives
of
ECS; s+)}E = <0l[S,S+I10~~~01[~S,H],S+]10~~01[S,S+]10~-~~~S;S*~~~ +(OI[[S,H],
D+]IOHOl[D,
D+]lO)-‘ND;
S+>,,
,
(12)
.
(13)
DfllO~-‘y~Ol~[D,H],S+]l0~~0l[S,Si]l~~-1)-1~0l[S,S+]l0~,
(14)
‘E~iD;Sf~~~=~Ol[lD,S’]l0~+~0l[[D,H],Sij(O~!Ol[S,S+]/O~-~~~S;S+~~~ +
D+]lO)(Ol[D,
D+]lOY’(tb:S+)),
GD; S+X from eqs, (12) and (13) yields
Eliminating
<
-K%[S,H], where y is given by ‘Y= (El-(OI[[D,H],
D*]lO)(Ol[D,
D+]IO)-‘)-’
.
(15)
In the derivation ofeq. (15) (OI[D, S+]lO) was taken to be zero. This was done in the spirit of the work of Shibuya, Rcwe, and most other researchers in this area, where one neglects odd p-h configurations in the ground state. The ground state is assumed to consist of the HF single determinant and even p-h excitations from the HF configuration. ‘The work of Sinanoglu [24] indicates that this is a reasonable assumption for systems with a closed shell ground state. in this paper we only consider systems with a closed shell ground state; treatment of the open sheli situation requires modifications [25]. Let us now consider the first few moments. M* =~ol[s,s’]lo~=s,,
,
M3 = Q1,(S:,lQl&Q where the following
MI
= (Ol[[S,H],
S.]lO)=
II+S;,~Q,,Q,-,~QZI)
new symbols
Q 12=KN[[S,Hl,~+l10L S 11 = (01 [S, S*] IO:!,
were introduced
,
+ Q,,(D~~Q,,s;:Q,,~D~~Q,,D~~Q,,)
-
06)
for *he sake of brevity
=i~l~[D,Hl,S+lIO~,
Q,,
QI1
Q,,=~Ol[[D,HJ,D+lIO>,
Dz2 = (01 [D, D+] 10) _
(17)
Unless the system we are treating has special simplifying features, none of the moments are strictly conserved. Even Iwo and WI1 are only approximations to the true values since we have neglected odd p-h pairs in the ground
state. Even with the usual assumption that the ground state can be represented by the HF determinant and even p-h pair excitations, it is found that only MO and M, are “conserved”. Our truncation of [S,H] does not affect WI,-,or MI (with the above stricture
concerning
ground
state correlations).
Some of the terms omitted
in the ap-
proximation to the commutator such as UflJjand &a,, can be shown to contribute to Ma and h&her moments. In the spirit .of Babu and Ratner 1261 we consider M2 and M, as being approximately conserved in our decoupling scheme. “4 and higher moments are given by combinations of products of the lower moments. Eq. (14) is of the form of a [2,1] Pade approximant [26-281. To establish this we reconsider the Green’s function hierarchy. EG;
S$
.g([S,
H]; s’,>,
= M, + (([[S,H].H];
-The Green’s
function
on the right hand side of eq. (19) is approximated
$6
..
:. ... I.
..:
.;..
:.:..
= MO f (<[S,H]; ~‘2~
.. .-.
;
,
(18) S’>>, -
(1% as follows
[26,28]
Volume 36, number 2
CHEMICAL PHYSICS LETTERS
<<[]S,H],N];SiDE lhe
matrices
=a((S;
.
S+>,, + b(([S,H];Sf})E
a and b are chosen
so as to conserve
1 November 1975 (20)
the first four moments.
Combining
eqs. (I Q-(20),
we find
~tS;S+~~,=(E2l-~-a)-1((El-b)~0~[S,S+][0~c~01[[S,H],Sf]10~},
(21)
b = (M,-M,MJ’N1,)(M,-M,M,‘M,)-’
(22)
- bMIMil
a =M2Mi1
,
.
(23)
Eq. (22) corresponds to the exact [2,1] Pad6 decoupling. If we make use of eq. (9), and if the various moments are approximated as in eq. (16), after some simple algebra eq. (14) is obtained. We wi!l now establish the connection between the [2,1] PadC decoupling and the equations of motion technique of Shibuya, Rose and McKay [2]. As mentioned recently by Rowe [25] very little work has been carried out concerning the relationship between the Green’s function and equations of motion approaches. Following Shibuya and co-workers, we consider the excitation operator 0: which operates on the ground state IO) to give the excited state IX) = Ih) )
0;10,
(24)
HIO~=E,IO); The
HIX)=E,lh),
equations
of motion
(01[60,,
q=EA
have the variational
If, o”,] IO> = W,(Ol[6Q~,
where the symmetrized
double
The following 0;
= c
approximation
mi
Adopting
{ Y$#(X)n&[
a notation
(25)
form
011 IO) ,
commutator
M,B,Cl =tcw,~l,Cl
- E,.
(26)
of three operators
A, B, and C is defined
as
+ [A>P,CllI to the excitation
(27)
operator
- Z#+fG,}
#I. 1 m,njc )Qi+jQ;l
{Y(2) nlrnJ(x )47QiQn+q-
+ c
“1in;
similar to that considered
is employed
in the propagator
derivation
enables
.
us to rewrite
(28)
0;
o+ = y(‘&+ _ z(‘& f Y(“)d+ _ #)d ?. We now substitute
I 1
eq. (29) into (26) and obtain
All
A12
HI1
HI2
A21
A22
B21
822
*;1
Hi2
*;I
*;a
%I
“I2
AZ1
AS2
where the definitions A,,
(29)
of the matrices
=(Oi[s,H,s’]i0),
B,, = (01 [s,H, s] 10) , U ,,,=(0i[s,s+]lO),
r
u11
=C+,
0
“22
0
0
1
0
entering
= (O([s, H, d] IO) ,
U,,=(Ol[d~d+]lO~.
0
0
0
a
0
(30)
0
-UT,
0
1
-“Z2
J
into (30) are
A12=(Ol[~,H,d’-]IOL Bit
r
O
A2,=
A,,=(Ol[d,H,d+]lO),
B21 = (Ol[d, H, s] IO),
B22 = (Ol[d, H, d] IO)
(31) ,
(32) 267
Volume 36, number WE also
2
CHEMICAL PHYSICS LJXTERS
1 November
1975
define the following matrices i,i= 1,2 ;
Q;;
= Q&
,
(33)
Rewriting eq. (30) one obtains
which
“;I X1 + Q;&
= wSll X1 ,
QilX1
= uD22X2
may be
-i Qi2X2
,
(35)
combined to give (36)
(37) If in the propagator approach WCmake the replacement Qii + Qb, that is we replace the unsyrnmetrized commutator by the symmetrized commutators, then the poles of (14) correspond to the zeros of (36). We could have defied the moments in the Green’s function hierarchy in this manner from the start [Xl. The philosophy behind the use of the symmetrized commutators is discussed in ref. [S] . Thus we have succeeded in demonstrating the equivalence of the equations of motion approach of Shibuya, Rose and McKay to a [3,1] Padi approximant, with the first four moments only approximately given.
3. Comments
on choice of IO) and 01
Eqs. (15) dr (36) are rather difficult to solve without additional approximations 12,301. If one makes the crude assutiption that IO) can be replaced by the HF ground state, then the equations reduced to the second RPA of da [29]. In the second RPA, AI1 describes the mixing of the sin$y excited configurations, AI2 the mixbetween the singly and doubly excited configurations, and AZ2 the mixirlg between doubly excited states. S12 and Bs2 vanish when evaluated over the HF ground state and B,, contains the HF-double excited configuration r-nixing and the.refore includes certain types of ground state correlations. Further if the singly excited states are not strongly perturbed by the doubly excited states. and if our interest is orJy in the lowest lying states, it may be possible to neglect Qs2. In general it is necessary to employ a correlated ground state. The choice 1301 Providencia ing
which corresponds to the first order Rayleigh-Schrod-hger perturbation theory ground state, is expected to prove adequafe for closed-shell systems. hi impoitant, apparently previously not considered, problem became apparent in the course of this work and is currently under investigation. The excitation operators employed in the second RP.4 of Shibuya, Rose and McKay [2] contain the combinations a+ ,,, a-,5 ai0 , m, LT~~ CI~CI~Q~~ and ara,?a, a,, . The difficulty appears when one considers that,in addition to eq. (24) the.equation of motion is based upon the equation [S] ,o~lo~=o,
‘-
where 0, k,thc adjoint of the&citation
(39)
operattir 0:.
CHEMICAL
Volume 36, number 2
For the F-h
(Y(l)%-2
PHYSICS
1 November 1975
LETTERS
R?A where only the s and st operators are retained in eq. (2% eq. (39) becomes
(l)*,‘)(IHF)cKlde))=O,,
(40)
where Ide) refers to the vector of doubly excited configurations is of order one since eq. (40) requires [3,25]
out of the HF ground state. This implies that Z(l)
-7(l)* = y(l)*K
(41)
and ~(1) is of order zero and K is of order one. If the full 0: of eq. (28) is considered, and if eq. (39) is to be obeyed it is necessary to include higher order ground state correlations in (0). After the first order double excitations the next important contribution to 101 consists of quadruple excitations Iqe) [24,3 1,321. In this case eq. (30) becomes (Y(‘)‘s-Z(1)*s++Y(2)~d-Z(2)*di)
(IHF)+Klde)+Llqe>)=
0,
(42)
where the L vector consists of the coefficients of the quadruple excitations. From eqs. (42) ar!d (30) it can be determined that Y(*) is of order one while 2 c2) is of order three and that the presence of these terms in 0; leads to a third order contribution to the excitation energ. It is not at all a ~4s that terms such as+ a+aTa $ I 1114, which have been included in 0: are more important than f an (or for tl- -t rnatteraiaj f ,a 3 a-a etc;). One popular justification of the inclusion of S as well a~ urn ; 4’ I r’=p=q=r=v Sf in the RPA “1 is that singly excited states can be reached from destroying a p-h pair in a doubly excited con-
figuration of the correlated ground state in addition to the direct p-h excitation from the ground state HF component. Extension of this line of reasoning would lead to the following justification of the c7pfaGaiajand aT~;a~a~ terms: Doubly excited states can be reached from creatin, 0 two p-h pairs in the HF ground statz or by destroying two p-h pairs in the quadruply excited configurations in the ccrrelated ground state. But what of the possibility of reaching the doubly excited state from the doubly excited conflgurations present in the ground state and whose contribution to the ground state is first order, while that of the quadruple excitations is second order? This is precisely the effect of terms like the CT:,,&, p roduct which were omitted in both eqs. (12) and (29) of this paper. It is rather difficult to assess the importance of such terms since they cause a complicated metric (overlap) matris as opposed to the relatively simple block diagonal metric mntrk o feq. (30). In our work on this problem we have found that although the right and left hand sides of the analog to eq. (30) becomes algebraicauy very com-
plicated, when these new terms are included in Oi, many of these terms cancel out.
4. Conclusions
We have investigated the [2,1] PadC approximant decoupling to the. p-h Green’s function hierarchy and discussed the additional assumptions that were necessary to arrive at the result recently developed in ref. [2]. Such an approach will be necessary in many mo!ecules due to stron, 0 single-double excited configurational mixing. Furthsr
consideration
of the method,
and computations
are planned
in order to answer
the important
question
of
how well this approach compares with various levels of CI. In a paper which appeared since the completion of this work, Jjrgensen, Oddershede and Ratner [33] describe a method for treating 2p-2h excitations within a self consistent time-dependent Hartree-Fock method. Their decoupling approach is very similar but not equivalent to that described in this paper. Further they advocate evaluation of all matrix elements over the Hartree-Fack ground state. We feel that, in general, it will be necessary to include at least double excitations in the grcund state. It has often been noted that for systems for which correlation is important it is essential to employ a correlated ground state [3,6,7,22,29,30]. In fact simple RPA based on the HF ground state may give rise to imaginary excitation energies [6,22]. The inclusion of double excitations accentuates the tendency of instability of the RPA equation [29], making the explicit inclusion of ground state correlations even more essential than in the simple RPA case. .. 269
Volume 36, number 2
CHEMICAL PHYSICS LETTERS
1 November
1975
Acknowledgement The author No. P33277Xl
has enjoyed
profitable discussions is also gratcfuily acknowledged.
with
T-T. Chen,
R. Silbey
and
J. Simons. Support
of NSF
Grant
References [ 11 MA. Ball snd A.D. McLachlaa, hlol. Phys. 7 (1964) 501. [Z] T. Shibuya, J. Rose and V. McKay, J. Chem. Phys 58 (1973) 500. 131 T. Shibuya and V. McKay, I. Chem. Phys. 53 (1970) 3306; 5’i (1971) 1738. [4] J. Rose. T. Shibuya and V. McKay, J. Chem. Phys 61 (1974) 7.55. [S] D.L. Yea&r and V. hkKcy, J. Chem. Phys. 60 (1974) 2714. [6] K.D. Jordan, Intern. J. @ranturn Chem. 7 (19733 491. [7] N. OsUund and hi. Karpius Chem. Phys Letters 11 (1970) 450. [8] D. Rowe, Nuclear collective motion (Methuen, London, 1970). [9] A.M. Lane, Nuclear theory (1964). [lo] J. Simons, J. Chem. Phys 57 (1972) 3787. [ll] J. Linderberg, P. Jdrgensen, J. Oddershede and M. Ratner, J. Chem. Phys. 56 (1972) 6213. [ 121 J. Linderberg and M. Rstner. Chem. Phys. Letters 6 (1970) 37. [13] I. Cooper and J. Linderberg, Mol. Phys. 25 (1973) 265. [14] P. Jdrgensen, J. Oddershede and M. Ratner, J. Chem. Phys. 61 (1974) 710. !15] J. Linderberg and Y. ahrn, Propagators in quantum chemistry (Academic Press, New York, 1973). 1161 J. Pddus and J. ciiek, J. Chem. Phys. 60 (1974) 149. [ 171 K. Kitaura and K. Nishimoto, Theoret. Chim. Acti 28 (i973) 111. [18] 0. Tanimoto end K. Toyad& hlol. Phys. 24 (1972) 1257. [19] K. Schutten end M. Kaxplu~ Chem. Phys. Letters 14 (1972) 305. [ZO] R.P. Hosteny, T-H. Dunning Jr., I. Shnvitt, R.R. Gilman and A. Pipano, J. Chem. PhyL (1975) to be published. 1211 P.J. Hay and I. Shavitt, J. Chem. Phys 60 (1974) 2865. [22] J.P. Malrieu, P. Qaverie and S. Diner, Theoret. Chim. Acta 8 (1967) 606; J.P. Xfalrieu, J. Chem. Phys 47 (1967) 4555. [23] K.D. Jordan, Fh.D. Thesis, MIT (1974), unpublished. (241 0. Sinano~lu, J. Chem. Phys. 36 (1962) 706. [251 G. Rowe, Dynamic structure of nuclex states (1972). [26] S. Babu and M. Ratner, I. Chem. Phys. 57 (1972) 3156. 1271 0. Goscinski and B. Lukman, Chem. Phys. Letters 7 (1970) 573. [28] R. Tahir-fieli, Phys.. Rev. Bl (1970) 3163. [29] J. de. Providencia, Nucl. Phys 61 (1965) 87. [30] J. Simons and W.D. Smith, I Cham. Phys 58 (lP73) 4899. [3 l] 0. Sinanogiu, in: Three app::oaches to electron correlation in atoms, eds 0. Sinano~lu and K.A. Brueckner (Yale Univ. Press, New Hover., 1970). !32] J. Pald~s, J. &ek and 1. Shxviti Phys. Rev. AS (1972) 50. 1331 P. Jgrgensen, J. Oddershede and M.A. Ratner, Chem. Phys. Letters 32 (1975) 111.
36, number
2
INCLUSION
CHEMICAL
OF DOUBLY
EXCITED
PHYSICS
1 November 1975
LETTERS
CONFIGURATIONS
IN ELECTRON
PROPAGATORS
K.D. JORDAN Department of h’ngkeering and Applied New Haven. Connecticut 06520. USA
Received
29 May
Science,
M~:on
Labomtory.
Yale
Utliversity,
L975
Inclcsion of doubly excited configwations in excited states is investigated in a moment conserving decoupling of the two-partic!c Green’s function for molecules with a closed shell ground state. The result obtained is found to correspond to a [2,1] Pad6 approximant decoupling of the Green’s function hierarchy. It is shown that this decoupling is equivalent to the equations of motion method of Shibuya, Rose, and McKay.
1. Introduction The conventional methods for treating the problem of electron correlation such as perturbation theory or configuration interaction (CI) involve considerab!e computational effort especially if one attempts to treat the correlation in the ground and excited states in a consistent manner. Various many-body techniques based on equations of motion [l-9] or Green’s function [lo-IS] techniques provide a means of direct calculation of excitation energies, oscillator strengths, inelastic scattering cross sections and other quantities of interest without explicitly requiring calculation Gf correlated ground and excited state wavefunctions. In addition to computational simplicity, many-body methods have attracted interest due to the possibility of providing a ba!anced treatment of the correlation in various states and also due to the expectation that they will lead to new, powerful semi-empikical approaches. In this paper we study the hierarchy of equations generated from consideration cf the particle-hole (p-h) propagator (a special form of the two-particle Green’s function). The propagator techniques of Linderberg et al. [l l-151 are based on the-assumption that the excited states can be treated in the singly-excited configuration framework. We will show how the effects of the contribution of the double excitations to the excited states enter into the moment expansion of the propagator. The motivation for A&isextension is the growing awareness of the inadequacies of the single excitation description of the excited states of z electron systems [19-211. In order to obtain the correct ordering and relative separation of the !ow-lying electronic states of pi electron systems such as butadiene, hexatriene, and benzene, it is necessary to include double and triple excitations in addition to the single excitations [ 191. The advantages of the EOM or Green’s function methods over the conventional approach of performing separate calculations on each state of interest is not completeiy understood. One of the important aspects of the manybody approaches is th!: cancellation of terms common to both electronic states of interest [23]. This cancellation has been demonstrated by Malrieu et al. [22] by employing a Rayleigh-Schr6dinger perturbation theoretical approach to both states of interest [22].
2. Electron Folldwing 264.. .-
:
propagator Lifiderberg __.
decoup!ing and Ratner
i.1.21, the causal Green’s
function
is defmed
in terms of creation
and annihila-
Volume 36, number 2
CHEMICAL PHYSICS LETTERS
I November 1975
tion operators
where i, i, k, i... refer to orbitals which are occupied ill the Hartree-Fock (I-IF) ground state and the IJL, )I, p, 4, . . . refer to orbit& unoccupied in the HF ground state. 10) corresponds to the correlated ground state. Fourier tnnsformation of the above Green’s function gives the p-h propagator in the energy representation 0
GQTQ, ; Q,'Qjlj~ = J
dt exp (iEf/lfi) (bif(r>~,,, (t); aiUj>) ,:
(2)
--m
For compactness
we make the following
definitions: (3)
si- being the column vector of p-h pair creation operators and s the column operators. In this notation. the equation of motion for the propagator is EG; Iteration
vector of the p-h
S’>j- = COl[S, S’] IO) + (([S, H]; S+NE . of this equation
gives the moment
(4)
expansion
+E-2&‘)
+E-3M(z)
s+]lo)+...
[S,HJ
=tol[[s,~J,s+]lo,~ol~s,sf]lo,-‘s
(51
+ . .. ,
(6)
where the nth moment, M(“), is the average in which the hamiltonian H appears )I times. Linderberg et a!. [ 11,121 approximate the moment expansion by a geometric sum involving moments. Their basic approximation is the linearization of the commutator of S with H
Substitution
pair annihilation
the first two
-
(7)
of eq. (7) into eq. (4) gives
((S; SfNE
= (Ol[S,
s+]lo>(z~ol[S,Sf]lO>-(OI[[S,H],
s’]ro,)-~
Sf]lO) -
(8)
This geometric sum is equivalent to a [ 1 ,O] Pad6 approximant
decoupling [26]. It corresponds to the p-h equations of motion of Shibuya and McKay [3] and Rowe [&?I*. IftOj[S, St] 10) and tOl[[S, A!], S+] JO) are evaluated using the HF ground state the random phase apprMmation (RPA) is ottained. In order to include the effects of double excitations in the description of the excited states we approximate the commutator as follows:
[S,H] =
(9)
where
in analogy with eq. (3). * In Practice the COmmutatOr [[S, HI, S*] in ccl. (8) is replaced by Its symmetrzed
version, which we have postpoiled
until eq. (37)
Volime
36, number
Substitution
2
CHEMICAL PHYSICS LETTERS
1 November
1975
eq. (9) in to eq. (5) gives
of
ECS; s+)}E = <0l[S,S+I10~~~01[~S,H],S+]10~~01[S,S+]10~-~~~S;S*~~~ +(OI[[S,H],
D+]IOHOl[D,
D+]lO)-‘ND;
S+>,,
,
(12)
.
(13)
DfllO~-‘y~Ol~[D,H],S+]l0~~0l[S,Si]l~~-1)-1~0l[S,S+]l0~,
(14)
‘E~iD;Sf~~~=~Ol[lD,S’]l0~+~0l[[D,H],Sij(O~!Ol[S,S+]/O~-~~~S;S+~~~ +
D+]lO)(Ol[D,
D+]lOY’(tb:S+)),
GD; S+X from eqs, (12) and (13) yields
Eliminating
<
-K%[S,H], where y is given by ‘Y= (El-(OI[[D,H],
D*]lO)(Ol[D,
D+]IO)-‘)-’
.
(15)
In the derivation ofeq. (15) (OI[D, S+]lO) was taken to be zero. This was done in the spirit of the work of Shibuya, Rcwe, and most other researchers in this area, where one neglects odd p-h configurations in the ground state. The ground state is assumed to consist of the HF single determinant and even p-h excitations from the HF configuration. ‘The work of Sinanoglu [24] indicates that this is a reasonable assumption for systems with a closed shell ground state. in this paper we only consider systems with a closed shell ground state; treatment of the open sheli situation requires modifications [25]. Let us now consider the first few moments. M* =~ol[s,s’]lo~=s,,
,
M3 = Q1,(S:,lQl&Q where the following
MI
= (Ol[[S,H],
S.]lO)=
II+S;,~Q,,Q,-,~QZI)
new symbols
Q 12=KN[[S,Hl,~+l10L S 11 = (01 [S, S*] IO:!,
were introduced
,
+ Q,,(D~~Q,,s;:Q,,~D~~Q,,D~~Q,,)
-
06)
for *he sake of brevity
=i~l~[D,Hl,S+lIO~,
Q,,
QI1
Q,,=~Ol[[D,HJ,D+lIO>,
Dz2 = (01 [D, D+] 10) _
(17)
Unless the system we are treating has special simplifying features, none of the moments are strictly conserved. Even Iwo and WI1 are only approximations to the true values since we have neglected odd p-h pairs in the ground
state. Even with the usual assumption that the ground state can be represented by the HF determinant and even p-h pair excitations, it is found that only MO and M, are “conserved”. Our truncation of [S,H] does not affect WI,-,or MI (with the above stricture
concerning
ground
state correlations).
Some of the terms omitted
in the ap-
proximation to the commutator such as UflJjand &a,, can be shown to contribute to Ma and h&her moments. In the spirit .of Babu and Ratner 1261 we consider M2 and M, as being approximately conserved in our decoupling scheme. “4 and higher moments are given by combinations of products of the lower moments. Eq. (14) is of the form of a [2,1] Pade approximant [26-281. To establish this we reconsider the Green’s function hierarchy. EG;
S$
.g([S,
H]; s’,>,
= M, + (([[S,H].H];
-The Green’s
function
on the right hand side of eq. (19) is approximated
$6
..
:. ... I.
..:
.;..
:.:..
= MO f (<[S,H]; ~‘2~
.. .-.
;
,
(18) S’>>, -
(1% as follows
[26,28]
Volume 36, number 2
CHEMICAL PHYSICS LETTERS
<<[]S,H],N];SiDE lhe
matrices
=a((S;
.
S+>,, + b(([S,H];Sf})E
a and b are chosen
so as to conserve
1 November 1975 (20)
the first four moments.
Combining
eqs. (I Q-(20),
we find
~tS;S+~~,=(E2l-~-a)-1((El-b)~0~[S,S+][0~c~01[[S,H],Sf]10~},
(21)
b = (M,-M,MJ’N1,)(M,-M,M,‘M,)-’
(22)
- bMIMil
a =M2Mi1
,
.
(23)
Eq. (22) corresponds to the exact [2,1] Pad6 decoupling. If we make use of eq. (9), and if the various moments are approximated as in eq. (16), after some simple algebra eq. (14) is obtained. We wi!l now establish the connection between the [2,1] PadC decoupling and the equations of motion technique of Shibuya, Rose and McKay [2]. As mentioned recently by Rowe [25] very little work has been carried out concerning the relationship between the Green’s function and equations of motion approaches. Following Shibuya and co-workers, we consider the excitation operator 0: which operates on the ground state IO) to give the excited state IX) = Ih) )
0;10,
(24)
HIO~=E,IO); The
HIX)=E,lh),
equations
of motion
(01[60,,
q=EA
have the variational
If, o”,] IO> = W,(Ol[6Q~,
where the symmetrized
double
The following 0;
= c
approximation
mi
Adopting
{ Y$#(X)n&[
a notation
(25)
form
011 IO) ,
commutator
M,B,Cl =tcw,~l,Cl
- E,.
(26)
of three operators
A, B, and C is defined
as
+ [A>P,CllI to the excitation
(27)
operator
- Z#+fG,}
#I. 1 m,njc )Qi+jQ;l
{Y(2) nlrnJ(x )47QiQn+q-
+ c
“1in;
similar to that considered
is employed
in the propagator
derivation
enables
.
us to rewrite
(28)
0;
o+ = y(‘&+ _ z(‘& f Y(“)d+ _ #)d ?. We now substitute
I 1
eq. (29) into (26) and obtain
All
A12
HI1
HI2
A21
A22
B21
822
*;1
Hi2
*;I
*;a
%I
“I2
AZ1
AS2
where the definitions A,,
(29)
of the matrices
=(Oi[s,H,s’]i0),
B,, = (01 [s,H, s] 10) , U ,,,=(0i[s,s+]lO),
r
u11
=C+,
0
“22
0
0
1
0
entering
= (O([s, H, d] IO) ,
U,,=(Ol[d~d+]lO~.
0
0
0
a
0
(30)
0
-UT,
0
1
-“Z2
J
into (30) are
A12=(Ol[~,H,d’-]IOL Bit
r
O
A2,=
A,,=(Ol[d,H,d+]lO),
B21 = (Ol[d, H, s] IO),
B22 = (Ol[d, H, d] IO)
(31) ,
(32) 267
Volume 36, number WE also
2
CHEMICAL PHYSICS LJXTERS
1 November
1975
define the following matrices i,i= 1,2 ;
Q;;
= Q&
,
(33)
Rewriting eq. (30) one obtains
which
“;I X1 + Q;&
= wSll X1 ,
QilX1
= uD22X2
may be
-i Qi2X2
,
(35)
combined to give (36)
(37) If in the propagator approach WCmake the replacement Qii + Qb, that is we replace the unsyrnmetrized commutator by the symmetrized commutators, then the poles of (14) correspond to the zeros of (36). We could have defied the moments in the Green’s function hierarchy in this manner from the start [Xl. The philosophy behind the use of the symmetrized commutators is discussed in ref. [S] . Thus we have succeeded in demonstrating the equivalence of the equations of motion approach of Shibuya, Rose and McKay to a [3,1] Padi approximant, with the first four moments only approximately given.
3. Comments
on choice of IO) and 01
Eqs. (15) dr (36) are rather difficult to solve without additional approximations 12,301. If one makes the crude assutiption that IO) can be replaced by the HF ground state, then the equations reduced to the second RPA of da [29]. In the second RPA, AI1 describes the mixing of the sin$y excited configurations, AI2 the mixbetween the singly and doubly excited configurations, and AZ2 the mixirlg between doubly excited states. S12 and Bs2 vanish when evaluated over the HF ground state and B,, contains the HF-double excited configuration r-nixing and the.refore includes certain types of ground state correlations. Further if the singly excited states are not strongly perturbed by the doubly excited states. and if our interest is orJy in the lowest lying states, it may be possible to neglect Qs2. In general it is necessary to employ a correlated ground state. The choice 1301 Providencia ing
which corresponds to the first order Rayleigh-Schrod-hger perturbation theory ground state, is expected to prove adequafe for closed-shell systems. hi impoitant, apparently previously not considered, problem became apparent in the course of this work and is currently under investigation. The excitation operators employed in the second RP.4 of Shibuya, Rose and McKay [2] contain the combinations a+ ,,, a-,5 ai0 , m, LT~~ CI~CI~Q~~ and ara,?a, a,, . The difficulty appears when one considers that,in addition to eq. (24) the.equation of motion is based upon the equation [S] ,o~lo~=o,
‘-
where 0, k,thc adjoint of the&citation
(39)
operattir 0:.
CHEMICAL
Volume 36, number 2
For the F-h
(Y(l)%-2
PHYSICS
1 November 1975
LETTERS
R?A where only the s and st operators are retained in eq. (2% eq. (39) becomes
(l)*,‘)(IHF)cKlde))=O,,
(40)
where Ide) refers to the vector of doubly excited configurations is of order one since eq. (40) requires [3,25]
out of the HF ground state. This implies that Z(l)
-7(l)* = y(l)*K
(41)
and ~(1) is of order zero and K is of order one. If the full 0: of eq. (28) is considered, and if eq. (39) is to be obeyed it is necessary to include higher order ground state correlations in (0). After the first order double excitations the next important contribution to 101 consists of quadruple excitations Iqe) [24,3 1,321. In this case eq. (30) becomes (Y(‘)‘s-Z(1)*s++Y(2)~d-Z(2)*di)
(IHF)+Klde)+Llqe>)=
0,
(42)
where the L vector consists of the coefficients of the quadruple excitations. From eqs. (42) ar!d (30) it can be determined that Y(*) is of order one while 2 c2) is of order three and that the presence of these terms in 0; leads to a third order contribution to the excitation energ. It is not at all a ~4s that terms such as+ a+aTa $ I 1114, which have been included in 0: are more important than f an (or for tl- -t rnatteraiaj f ,a 3 a-a etc;). One popular justification of the inclusion of S as well a~ urn ; 4’ I r’=p=q=r=v Sf in the RPA “1 is that singly excited states can be reached from destroying a p-h pair in a doubly excited con-
figuration of the correlated ground state in addition to the direct p-h excitation from the ground state HF component. Extension of this line of reasoning would lead to the following justification of the c7pfaGaiajand aT~;a~a~ terms: Doubly excited states can be reached from creatin, 0 two p-h pairs in the HF ground statz or by destroying two p-h pairs in the quadruply excited configurations in the ccrrelated ground state. But what of the possibility of reaching the doubly excited state from the doubly excited conflgurations present in the ground state and whose contribution to the ground state is first order, while that of the quadruple excitations is second order? This is precisely the effect of terms like the CT:,,&, p roduct which were omitted in both eqs. (12) and (29) of this paper. It is rather difficult to assess the importance of such terms since they cause a complicated metric (overlap) matris as opposed to the relatively simple block diagonal metric mntrk o feq. (30). In our work on this problem we have found that although the right and left hand sides of the analog to eq. (30) becomes algebraicauy very com-
plicated, when these new terms are included in Oi, many of these terms cancel out.
4. Conclusions
We have investigated the [2,1] PadC approximant decoupling to the. p-h Green’s function hierarchy and discussed the additional assumptions that were necessary to arrive at the result recently developed in ref. [2]. Such an approach will be necessary in many mo!ecules due to stron, 0 single-double excited configurational mixing. Furthsr
consideration
of the method,
and computations
are planned
in order to answer
the important
question
of
how well this approach compares with various levels of CI. In a paper which appeared since the completion of this work, Jjrgensen, Oddershede and Ratner [33] describe a method for treating 2p-2h excitations within a self consistent time-dependent Hartree-Fock method. Their decoupling approach is very similar but not equivalent to that described in this paper. Further they advocate evaluation of all matrix elements over the Hartree-Fack ground state. We feel that, in general, it will be necessary to include at least double excitations in the grcund state. It has often been noted that for systems for which correlation is important it is essential to employ a correlated ground state [3,6,7,22,29,30]. In fact simple RPA based on the HF ground state may give rise to imaginary excitation energies [6,22]. The inclusion of double excitations accentuates the tendency of instability of the RPA equation [29], making the explicit inclusion of ground state correlations even more essential than in the simple RPA case. .. 269
Volume 36, number 2
CHEMICAL PHYSICS LETTERS
1 November
1975
Acknowledgement The author No. P33277Xl
has enjoyed
profitable discussions is also gratcfuily acknowledged.
with
T-T. Chen,
R. Silbey
and
J. Simons. Support
of NSF
Grant
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