333
Progress of Theoretical Physics, Vol. 54, No. 2, August 1975
A Green's Function Method for Calculation of
Electron:ic States of pi-Electron Systems
Kaoru TOYODA, Takeshi IWAI and Osamu TANIMOTO
Department of Applied Physiq, Faculty of Engineering
Osaka City University, Sumiyoshi-ku, Osaka
(Received December 23, 1974)
, This paper presents ~ Green's function appro!!ch to the calculation of the pi-electronic
states of molecules. The Green's functions· are written in terms of the molecular orbital.
Physical quantities are more easily treated in this approach than the previous method using
the AO scheme. Formulae for the excitation and the ground state energies are obtained by
RPA. Using the semi-empirical parameters, several examples are calculated. However, the
well-known instability in RPA appears when the excitation and the ground state energies are
calculated with the non-empirical parameters. From the viewpoint of diagramatic method,
we discuss how this instability can be removed by introducing doubly excited configurations
and renormalization of orbital energies.
§ I. Introduction
Many authorsn,..s> applied the Green's function method which was developed in·
an electron gas and solid state physics to the calculation of electronic states in
molecules. F~r the pi-electron systems, the most part of the investigations along
this line was made within the framework of Pariser-Parr-Pople's (P.P.P,.) appro~
-ximation. 7> On~ of the authors and Shimada1>.a> have proposed a one-body Green's
function method for the calculation of orbital energies. In the method, the orthogonalized atomic orbitals (AO) are adopted as ·basis for second quantization and
a diagrammatic technique is used for the perturbation calculation of the one-body
Green's function. Ecker and Hohln.eicher2> calculated orbital energies of several
molecules with the orie-body Green's function method based on the molecular orbital
(MO) scheme. We also applied the Green's function method to investigate the
excitation energies of pi-electron systems in the framework of random phase approximation (RP A) .5> There, we found the ~xistence of an instability for a triplet
excitation energy, for instance, 3 Bt,. in benzene. Such- an instability in RP A has
already been pointed out by many authors in the theory. of nuclear spectroscopy, 8 >
in electron gas9> and in molecules:10> Several authors have investigated several
methodsm~ta> to remove this instability in molecules.
A representative one is the
12
equation of motion method (HRPA). >
In this paper, the Green's function method is reformulated by the LCAO MO
scheme. The reason why we 'use this scheme is the simplicity for the treatment
of excitation and ground ~tate energies. Using this Green's function, we will
334
K. Toyoda , T. Iwai and 0. Tanimo to
also be able to propose a way to remove the triplet instabil ity m
PRA. In § 2,
we reformu late the pi-electr on hamilto nian in our matrix represen tation.
A derivation of Dyson equation in RPA is discusse d in '§ 3. It is shown that
the equation s
in the present paper have more compac t and simpler forms than
those .. of the
precedi ng paper. 5> Section 4 is devoted to an extensio n of RPA in the
diagram matic
method to include the multipl e scatteri ng diagram s which correspo
nd t~ the inclusion of the doubly excited states. . The scheme discusse d in the
previou s work5>
is refined, insertin g a T matrix to the electron ic interact ion line.
In the viewpoi nt
of this analysis , the two electron transitio n states. give importa nt
contribu tions to
the remova l of the instabili ty. It is also shown that the renorma
lization of resonance integral is necessa ry to retain the consiste ncy of our refined
RPA. Using
Feynma n's theorem / 3> the correlat ion energy of the ground state is
discusse d in § 5.
Applica tions of these procedu res are presente d in the last of each
section.
§ 2. · The one-bo dy Green' s functio n
In this section, the one-bod y Green's function is discusse d as a
prelimin ary •
for the next section. In the previou s work, 2>-4> it has been shown
that the poles
of the one-bod y Green's function give orbital energies . In rather
large molecul es,
by Koopma nn's theorem,l5) the ionizatio n potentia ls are given in
terms of orbital
energies . Using a perturb ation method, two-bod y Green's function
is represe nted
in terms of the one-bod y one. Then the instabil ity in the triplet
excitatio n closely
relates to the accurac y of calculat ed orbital energies . We assume
~-'lr separab ility
m the followin g sections :
The pi-elect ron hamilto nian is general ly given by
H=Ho +H',
(2·la)
Ho= :En::::; (hrr-tt) a;!A- 9 +-:E' hr,a~a, 9},
'
r
~•
(2·1b)
(2 ·lc)
where
(2·1d)
and at are the creation and destruct ion operato rs of electron at the
i-th atomic
site, respecti vely. It is conveni ent to rewrite this hamilto nian into
the followin g
expressi on:
at+
(2·2)
where
fr =
hrr- ,U- t
:E
,V:!.:rm ,
m
(2·3a)
A Green's Function l'vfethod for Calculation of Electronic States
335
(2. 3b)
In the following calculation, the factors -lL:mv:'.:;rm and -lL:mv:'.:;sm in Eq. (2·3)
could be neglected, because they are cancelled out by the exchange interaction
occurred at the same -time for instantaneous interactions. 16> Now, we define the
destruction (vector) operator of electrons in LCAO MO by
(2·4)
a,=Ca,,
where C is the unitary transformation matrix which diagonalizes Ho by the following manner:
Ho= :E a,+Ea•
•
where the suffix k denotes the k-th molecular orbital.
Eq. (2 · 2), H' becomes
HI
- 2
1 "
"-'' "
"-''
7J7'/ klmn
Substituting Eq. (2 · 4) into
V""'
+
+
km;n!ak,a!,amy'an•'
(2·6)
where
, V""'
""' C*pkC qlC*rmC sn
km;nl = "'
~ Vpr;sq
pqrs
(2·7)
•
The matrix >.. (kl) is the one whose (kl) element is unity and the others are zero,
as shown in the previous articles. 3>. 4> We define the one-body Green's function for
MO as
(2·8)
'in the case of closed shells. Representing the Fourier transform of the unperturbed Green's function as 0G(w), the perturbed Green's function g(w) can be written
as
(2·9)
where A (w) is the irreducible self-energy matrix.
the ha:rniltonian becomes
Introducing a one-body field
x.
H= (Ho+ :E a,+xa.)
•
+ (H'- :E a,+xa.) =3Co+3C'.
•
(2 ·10)
The self-consistent field (SCF) X is deter:rnined by the condition
mA(w) =0.
(2·11)
The first order self-energy part m A (w) involves the fir'st order irreducible diagrams
and X· Equation (2 ·11) can be rewritten as
336
K. Toyoda, T. Iwai and 0. Tanimot o
(2 ·12)
where we define
og (w) = ( {oG (w) }-1+ X}--1
(2 ·13)
and
(2·14)
,Q
--
I
A,{k/)
I
I
a
/
+
I
,{(mn)
/
----
¥
..............
I
I
I
I
I
'\
I
\
I
I
cr J..(kl)
=0
I
,{(mn)
Fig. 1. The equation for the calculation of self-consis tent field
0'
A.(mn)
x is
illustrated diagramati cally.
Equation (2 ·11) is shown diagrama tically in Fig. 1. The rules for diagrams
have
been mentione d in the previous papers, 3).s) X must be __diagonali zed to obtain
SCF
molecula r orbitals. Then, we easily obtain
occ
Xmm= ~ V{kk);mm'
(2·15)
k
where ~occ is. the summatio n over the occupied electroni c states related
to Eq.
(2·12). Substitu ting Eq. (2·15) into Eq. (2·13), we obtain an SCF Green's
function and write it as G (w) hereafter . Once G (w) is given, one ·can calculate
i-th irreducib le self-ener gy part Cil:S(w) (i>2), so that higher order correctio
ns
are able to be taken into account through the followin g equation:
(2·16)
When we make an ·approxim ation,
we have the expressio ns for the orbital energies equivale nt to those in Refs.
2)
and 3).
Now, we discuss the relation of the SCF theory in the Green's function
method with the usual MO scheme. The one-body Green's function in the
atomic
orbital represen tation relates to the MO Green's function by the followip. g equation:
(2·17)
The MO ,Green's
fu~ction
is given by
0 (w) =
G!"'m
'
1
())-
ESCF
!
·:;,
=f zv
(J
!m '
(2·18).
A Green's Function Method for Calculation of Electronic States
337
where ErcF is the SCF orbital energy of J;he 1-th molecular orbital. -Equation
(2 ·18) implies
(2 ·19)
Substituting this relation into Eq. {17) in the previous paper, 3l we have the selfconsistent field in terms of LCAO scheme.
(2·20)
where
Using this self-consistent field, the poles of the one-body SCF Green's function
is given by the following se'cular equation:
(2·21a)
where
(2·21b)
This result is identical with the qsual SCF equation.
§ 3.
Two-body Green's function and excitation energy
In the previous article, 5l we have already formulated a method for calculation
of electronic excitation energies and applied it to benzene and ethylene. It is not
easy: however, to treat excitation energies starting from the AO scheme. Therefore
we now reformulate it into the MO scheme. They can be converted to ea-ch
other by unitary transformation, but the latter method is more useful than the
former in the calculations of excitation energies and correlation energy in the
ground state. The validity of RP A framework is discussed in the next section.
The two-body Green's function (the polarization propagator) Dtt, (t', t) is de,fined as
(3·1)
The Fourier transform of it is given by
J
Glz (w' + w) c-::,. (w') iJ~~, .
nz-:: (w) = dw~
2nt
(3·2)'
The Dyson equation for the polarization propagator in the RP A is shown in Fig.
2. Usi1,1g the SCF one-body Green's function G(w), Dyson equation which corresponds to Fig. 2 is written as
338
K. Toyoda , T. Iwai and 0. Tanim oto
.~~n+ Q;-:n-w: ~~~Fig. 2.
The Dyson equation for the polariza tion propaga tor in RPA.
J~:;
P~'(a>)m,.G(a>l)}
sda>~ Tr{>..(lk)G(a>1 +a>)>..{mn)G(a>1)}ff~,~
2:: 2:: [vp;;.,;"'" Jda>~ Tr{>..(lk)G(~l+w)>..(k'l')G(a>l)}
Tr{>..(Zk)G(a>l +a>)
=
2nz
+
-2nz
,. k'l'm'n'
- v;;;,,; ..'v
sda>~ sdw~
2m
2nz
Tr{>..(lk)G(a>1 + a>)>..(k'l')G(a>z+ a>)
(3·3)
where T"' (w)mn is the vertex part. Since the SCF Green'
s functio n G(a>) is a
diagon al matrix, the above equatio n is rewritt en as follows :
Dk,r~~'(a>)k!;mn = Dk! [d'mkd' ,.dJ ~~,
+ 2:: D,.,m,{:E v::;,,;,.,,r·~'(a>),.,m';mn- V%!:-,;,,.,p'~'(a>),.,m';m ..} l
m'n'
'YI"
(3·4)
The excitat ion energie s are determ ined as the peles of D,kP''
(a>) 1k;mn· From Eq.
(3·4), we can show that the poles of the vertex parts
(a>)tk;mn justly give ,the
excitat ion energie s. Theref ore, Dyson equatio n for the vertex
parts,
r··
r'J~'(a>)kZ;mn = d'mkd'ntd'~~,
(3·5)
can be used, in order tu calcula te excitat ion energie s.
By employ ing the followi ng transfo rmation for the
ver~ex par~,
r[s]_=r•["':._1r-•
(3·6)
Dyson equatio n for the triplet and singlet states is obtaine d,
(3·7)
where x = 1, 3 corresp ond to singlet and triplet excitati ons,
respect ively. Dyson
equatio n for the two-bo dy Green' s functio n which is derived
by multipl ying Dkz
A Green's Function Method for Calculation of Electronic States
Fig. 3. The transformed Dyson equation in RP A.
with the effective interaction V(.,J;mn·
339
In this case, the wavy line is associated
to the both sides of Eq. (3 · 7) is illustrated m Fig. 3. In Fig. 3, the wavy line
represents the effective interaction, -iV(~!);m'n'· Thus, we see that the diagrams
in Fig. 2 are reduced to only the bubble diagrams as shown in Fig. 3.
As the simplest example, Eq. (3 · 7) is applied to ethylene. The excitation
energies are given by
(3 ·Sa)
and
(3 ·Sb)
in P.P.P. approximation for N~T .and N~ V excitation, respectively,*) where (3
is the resonance integral and rli-JI = 7iJ is the electron repulsion integral associated
with the i-th and j-th atomic sites. In Eqs. (3 ·Sa) and (3 · Sb), the orbital energies
are shown to be ± (3. For the actual calculation, we use not only SCF orbitals
but also the correlated ones with higher order diagrams. From these result, we
see that the instability of excitation can be removed by the choice of the set of
paramete~s.
When we use the theoretical parameters (Lowdih's, Slater's, etc.),
the instability arises in the triplet excitation, but does not appear in the calculation
based on the semi~empirical parameters. As it is well known, the Pauli exclusion
principle is violated in RP A. Thus in order to refine RP A method, we must
consider some kind of diagrams which compensates this violation. We discuss this
problem in the next section. For the case of allyl-cation, the results are presented
m Table I.
Table I. The excitation energies of allyl-cation in RPA. In the cases 2 and 3, the parameters in Salem's text"J are used. The resonance integral fiu adopted is zero or -_0.451.
The latter is the value in the semi-empirical case.
~
n
'B•
'B.
·•A,
'A,
Assignment 1
Assignment 2
Assignment 3
In the case of semiempirical parameters
(fiu=O)
(fiu= -0.415)
3.510
5.723
6.857
8.390
0.888
5.284
2.815
2.827
7.245
.
In the unit of eV.
*l
The result for the excitation energies of ethylene in Ref. 5) is not correct and revised here.
340
K. Toyoda, T. Iwai and 0. Tanimo to.
§ 4.
Refinem ent of RPA by means of multipl e scatterin g effect
In order to removt) the (triplet) instabilit ies in RPA, we consider a number
of diagrams , whose typical ones, chosen by us, are shown in . Fig. 4.
In the
word of configura tion inter~ction, these diagrams represen t the contribut ion
of two
electron excitation s. When we replace the. interactio n line V<%i);mn by the effective
a -·-D a----D
-------
--·--
'
...
etc.
Fig. 4. Some diagrams which involves the doubly excited
configurati ons.
interactio n line W(%!);mn, then
we can take these diagrams
into the RPA calculatio n.
The effective interactio n, in
general, depends on the
internal- variables W1 and W2
in Eq. (3 · 3), so that we
represen t it more accuratel y
b.,v W<%!); mn ( w1, w2; w) .
If the effective interaction is obtained, the Dyson
equation of vertex parts. is
given by
(4·1)
Now, our problem is to get the actual form of W(%!);mn(Wh W2 ; w).
V(%!);mn is defined by Eq. (2 ·14), W(%!);mn is given. as follows:
Since
Wt£Rmn = W{~;n!- W{~;!n[~J W{,;;~~.
(4·2)
Thus, the evaluatio n of Wc'~:!);mn is reduced to the calculati on of W~;n!·
For
convenie nce, we introduc e a coupling constant g and define W%~;n! (g) as ·
~~Wkm;nl
=
(g) -"'
- .4:....1 (}) w~~km;nl g' ,
J=O
(4·3)
where CJ>W%~;n! is the j-th effective interactio n and
(4·4)
Substitu ting Eq. (4·3) into Eq. (4·2), W(%!);mn(g) is also obtained,
(4 ·5a)
where
(4·5b)
It is not easy to obtain the j-th effective potential <J> W%~;n! accuratel y. We
take first the ladder approxim ation for W%~;n! (g). Since the ladder diagrams
.re-
·A Green's Function Method for Calculation of Electronic States
w,
w •w,
I
I
"
I
w •w,
I
.I
·tw
I
'ri
w.w, w.w,-x
k:
+
t1w
I
I
I
I
w,
I
I
I
I
k
~
m
w•w2
I
I
I1
k'
I
I
I
x t1
I
I
I
w.
w•w, W+w,-x w•W,-x-x' w,
: •
•k :
I
I
1
+
t1w-x
I
I
I
x
l
I
I
I
1
I
I
rr}' I, n•
m. I1
w•w. W+W.-X
~
w1
341
mI
•
cv2
W+cut
1
k' : k• : •
I
I
I
x'f
I
I
I
I
m' I
I
1
1
~1 w-x-x' +
1
I
.
I
m• 1'
• 1 n.
w.•w-x ~w-x-x' Wt
Fig. 5. The ladder summation of the effective interactions is illustrated diagramatically.
present the {:!£feet of multiple scattering of electrons, we can expect that they
would contribute to the compensation for the violation of Pauli principle. The
ladder part of W%~; .. z is shown in Fig. 5 and represented as Wg;,z· In terms of
the power series of. g, wg;nz (g) is written as
W. '1J11'
km;nz ( g )
co
" " <J> W'IJ1/'
=...:...,
· km;nzg J
J=O
'
(4·6)
where
<o> wn~'
km.; nt =
V'IJ1/'
km~ nt ( (j) ) '
(4·6a)
X G,.,(£0 + £0 2 -x) V%1;,.,;,.z(l0 -x)},
<2>Wt",;.;,.z=L;·
L:; s dx.
27rt
lc'm' k•m•
.X
(4·6b)
sdx~ {V%~;m'a:'(x)Gk,(w+w~-x)
27rt
G,., (w + £02 -x) yzy;,_,;m•k•(x') Gk.(lO + £01-x-x')
x· G,.. (w + £02 -x -x') v~;,..; .. z(w -x-x')}, (4 ·6c)
etc. In Fig. 5, the diagram equation which corresponds to Eq. (4·6), when we put
g = 1, is given by removing the four solid lines labelled as uh, lO +lOr, £02 and lO + w2
from every term iP. both sides.. The thick dotted line in the left-hand side, therefore, corresponds to -iW%~;nz and the parts composed of solid lines and the thin
dotted ones of every term 'in the right-hand side to -i<f>Wg;,.z respectively. We
represent the thin dotted line of the first term in the right-hand side as - i
<o>wz~;nz· Putting Eq. (4·6) into. Eq. (4·1) and performing the integration>over
Q)l and (.1)2, we get the simultaneous equations for r:m:nz (w)
which involves an
infinite series of functions with w-dependence. In contrast to the electron gas problem, the ladder sum is not easily carried out in our case. Therefore an approximation is introduced here. The most plausible approximation would be Pade approximant method.m After carrying out the [1, 1] Pade approximant, we obtain
342
K. Toyoda, T. Iwai and 0. Tanimoto
c.J +
w,
k
w•w,-x
\
k'
\
\
\
x \
I
\
excitation energies including the multiple
scattering effect· of electrons,
Except for the ladder diagrams, the most
important contribution is given from the diagram in Fig. 6. This is one of the lowest
order diagrams to be included in the extension of RPA. As a correction for the ladder
approximation, we adopt the above di_agram.
Its contribution to W%;',;;-n! is given by
I
I
I
I
w-x
I
\
I
\ I
I
I
I
+
I
\
I
\
\
\
\
I
m
w
I
/\
m'
w2
\
n
W2 + X
Fig. 6. The lowest order diagram which
is not contained in the ladder sum.
In a quite similar way, as the case of ladder approximation, the excitation energies
involving this correction could be calculated. We notice that the contributions
from the diagrams in Figs. 5 and 6' are only included in the diagonal elements of
the secular equation. As an example, the present method is applied to ethylene.
The results are given in Table II. The Coulomb integrals were calculated, using
Slater and Lowdin orbitals. In this Table, /3+ denotes the correlated resonance
integral, which is given by
(4·8)
It can be derived by considering the second order irreducible diagrams. 3> In the
calculations of the columns 1, 2 and 3, fiscF, renormalized by SCF calculation, is
used as the resonance integral. It is shown from these results that the inclusion
of ladder diagrams in RP A framework rempves the triplet instability as expected.
To obtain a good agreement with experimental results, the correctiqn diagram in
Fig. 6 must also be taken into consideration. In the case 3, the singlet excited
state becomes lower than the triplet state on the contrary to the other cases.
This fact indicates that the use of the renormalized resonance integral /3+ instead
of the SCF one is necessary for retaining the consistency of the refined RP A.
Table II. The excitation energies of ethylene, in the unit of eV.
In cases of
The 0 7pes 1 fJscF
3 fJscF 4 {J.
Ladd Ladd
+Corr RPA
-
N~T
N~V
I
9.122
parameters
2 fJscF
RPA
transition
Sla~er's
I
8.088
8.899
I
7.1181
6.309
10~391
5
Lowdin's parameters
6 {J. 7 {J• . 8 {J. 9 {J.
Ladd Ladd
Ladd Ladd
+Corr RPA
+Corr
{J.
7.790
9.915
I
4.963
7.454
-
10.552
8.402
9.692
I
5.6051
6.715
Obs
4.6
7.6
A Green's Function A1ethod for Calculation of Electronic States
343
§ 5. ·correlation energy of pi-electrons in the ground state
According to Feynman's famous theorem,w the ground state energy is given
by
E=Eo+
I
l
0
dg
-<Wo(g)jgH' IP'o(g)),
g
(5·1)
where H 1 is the perturbed hamiltonian and g is the coupling constant. Pi-hamiltonian consists of two parts; the unperturbed part and the perturbation. In order
to identify the SCF ground state energy with E 0, we divide the hamiltonian as
follows:
(5·2)
where
Jlo=Ho+t ~ a~+xa~,
(5·3)
~
Jl'=H' -t ~ a/xaq.
(5·4)
~
Then the SCF ground state energy EscF and the correlation energy Ecorr are
given by
(5·5)
Ecorr=E' -Ex,
(5·6)
where
E'= _ _!_ [1 _<![_<Wo(g)jgH'IP'o(g)),
2 Jo g
(5·7)
Ex= 21 ~
~
(5·8)
+xa~IP'o(g)).
Jof _<![_<Wo(g)jga,
g
1
Expressing <W (g) IH 1 1 W (g)) (O<g<1) in terms of the density correlation
function II~~(; km (g, w), we can represent E' as
1
E, = -2
Il -
dg
o
g
S--.
dw ""
"" v~~
"'-' "-'.g
2rct
~q· k!mn
km;n! Jlqq'
n!;km
c· g, (J) )
•
(5·9)
The diagram corresponding to this equation is illustrated in Fig. 7.. One of the
simplest diagrams in Fig. 7 is shown in Fig. 8, whose contribution to E 1 , e, is
In electron gas, e does not contribute to E 1 , but in pi-electron system, we cannot
neglect it. If, we adopt RPA for the polarization part, the correlation energy in
RPA is easily obtained. In this approximation, the two-body Green's function could
be solved exactly as shown in the preceding section. The correlation energy in
344
K. Toyoda, T. Iwai and 0. Tanimoto
0----~~~o·.
A.(kl)
A.(mn)
Fig. 7. The diagram corresponding to the energy E' is illustrated. The shadowed part
corresponds to iiH;?.,.(g, ro).
RPA is written as
ERPA = -
rl
_!_
Jo
2
dg
g
.
Fig. 8. The simplest one of the diagrams illustrated in Fig. 7.
s
dw_ .I; .I; _gVZ't.;nzDzkrn''l(g, w)zk;mn.
2rct nn' klmn
(5·11)
When we replace V%'t.;nz by
(5·12)
Eq. (5 ·11) can be rewritten as
1
E RPA= - 2
ll J
0
-dg
g
-2dw. "'
£..J
7Ct
:c=l, 3
"'
£..J g U"'
km;n! n··Zk T"' Cg,. (f).)Zk;mn •
k!mn
(5 ·13).
When the relation
U{m;nl = U/"m;nk = U{n;ml
(5·14)
is satisfied, it is convenient to use new vertex parts:
(5 ·15)
Then, we have· the following Dyson equation:
The new vertex part fx(ro) has the _same poles as rx(w).
(5 ·15) and (5 ·13), we obtain another expression for ERPA
Using Eqs. (5·14),
The above procedure is applied to pi-electron system; ethylene, benzene and
allyl-cation. Results o{ correlation energy, Ecorr=e+ERPA-Ex,where Ex is given.
by the following equation:
(5 ·18)
A Green's Function l'vfethod for Calculation of Electronic States
Table III.' The values of
Ecorr
Ecorr·
345
In cases of using semi·empirical parameters.
Ethylene
I
-0.236
I
Allyl Cation
Benzene
-0.175
-1.019
In the unit of eV.
Table IV. The values of
fiscF
11 Ladd
I
in the case of ethylene. In unit of eV.
Low?Jn' s parameters
In cases ··of Slater's parameters
I
Ecorr
Ecorr,
-0.077
1· 2
I
fiscF
Ladd+Corr
+2.363
fi+
13 Ladd
I
-0.226
fi+
14
. Ladd + Corr
I
-0.320
5
fi+
Ladd
-0.128
16
I
fi+
Ladd+Corr
-0.139
are shown m Table III. Semi-empirical parameters are used for each system,
because they give no instability for ,the excitation energies. However, when we
use theoretical parameters, we obtain some instabilities in the excitation and the
ground state energies in RPA calcuhition. In order to improve RPA, we consider
. two electron transition configurations. We. define the correlation energy as
(5 ·19)
where JfRPA Is the. quantity calculation by a similar way to Eq. (5 ·17), with the
modification that the resonance intergral and the electronics interactions are replaced
by the effective ones. It is difficult to calculate ERPA exactly, for the effective interactions have a w-dependence. To estimate ERPA, the effective potentials are detc;!rmined approximately under the condition that the excitation energies calculated by refined
RP A are expressed by the same formula with those of standard RP A. At the
same time, we can use the SCF orbitals or the refined ones mentioned in § 4.
By this procedure we obtain the w-independent effective potentials. Results are
tabulated in Table IV. for ethylene. In the case 2, Ecorr has a positive value.
It means that the value for !ERPAI is smaller than (s-Ex), on the contrary to
the other cases. Such circumstance comes from the inversion of triplet and singlet
excited states as shown in Table II. Results of the approximate calculation, where
g-dependence of the effective interactions are simplified, show similar behavior.
§ 6.
Concluding remark
We have formulated a refinement of RP A, starting from the standard RP A
and. including doubly excited states into the polarization propagator.
As an advantage of the diagramatic method, one is able to see what types
must be included to remove the instability. One type of such diamechanism
of
grams is the self-energy part which contributes to the improvement of the one-body
Green's function. This fact means that the energy gap between the refined orbitals
is larger than that of the SCF orbitals. The multiple scattering has another
346
K. Toyod a, T. Iwai and 0. Tanim oto
effect which stabili zes the excitat ion and the groun d
state. It replac es the el~c­
tronic interac tion by the effecti ve one and contrib utes
to remov al of the instabi lity.
In the case of the semi-e mpiric al param eters, the
instab ility does not occur in
RPA. This might be interp reted as the higher order
correc tions includ ed effecti vely
in these param eters. One of our purpos es is to find
the relatio n betwe en the
semi-e mpiric al param eters and the theore tical ones.
As the multip le scatter ing
effect is introd uced into RPA, the secula r equati on,
which gives the excitat ion
energi es, has a much compl icated w-depe ndence in_ the
diagon al elemen ts. Thus,
a simple relatio n betwe en the semi-e mpiric al and the non-em
pirical param eters seems
not to exist. Howe ver, the higher order calcul ation
includ ing the multip le scatter ing effect is much laborio us and it is desira ble to devise
' the reno~malized RPA
in which the effecti ve electro nic interac tions are contain
ed. The attemp t discus sed
in § 5 shows that the study along this line would
be promis ing. As alread y
mentio ned, the higher order effect in orbita l energi es
closely related to the disappearan ce of the instab ility in the refined theory . This
indica tes the necess ity of
.the renorm alizati on of resona nce integr als in remov al
of instabi lity. The contrib ution of the second order irredu cible diagra ms in the
previo us paper3> to the orbital
energi es is studied . A more accura te calcul ation is
now in progre ss.
Pade approx imant metho d has been used to calcul ate
cJ> Wg;nt in § 4, for it
is difficu lt to carry out the summa tion of the infinit
e series appear ed in the ladder
diagra ms. When we can make an accura te calcul ation
of aJw:;;:;nL> the more reliabl e
estima tes of the excitat ion energi es are expect ed.
Ackn owled geme nts
The author s would like to thank Dr. K. Nishim oto
for advice .
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