Selfconsistent manybody theory for π electron systems. I. The ethylene moleculeMarcello Baldo and Renato Pucci Citation: The Journal of Chemical Physics 67, 4747 (1977); doi: 10.1063/1.434601 View online: http://dx.doi.org/10.1063/1.434601 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/67/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Selfconsistent manybody theory of πelectron systems. II. Selfenergy effects J. Chem. Phys. 70, 4086 (1979); 10.1063/1.438031 Erratum: Selfconsistent manybody theory for π electron systems. I. The ethylene molecule J. Chem. Phys. 69, 5214 (1978); 10.1063/1.437154 SelfConsistent Approximations in ManyBody Systems. II J. Math. Phys. 10, 2201 (1969); 10.1063/1.1664822 SelfConsistent Approximations in ManyBody Systems J. Math. Phys. 10, 1804 (1969); 10.1063/1.1665030 SelfConsistent Approximations in ManyBody Systems J. Math. Phys. 6, 450 (1965); 10.1063/1.1704293

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Self-consistent many-body theory for 7T electron systems. I. The ethylene moleculea)

Marcello Baldo and Renato Pucci

Istituto di Fisica dell'UniversiUi di Catania, Centro Siciliano di Fisica Nucleare e di Struttura della Materia. Catania. Corso Italia. 57-95129 Catania. Italy (Received 3 May 1977)

We present a general many-body theory of molecular electron systems based on a Green's function formalism. The theory introduces directly the dynamical screened electron-electron interaction and produces an exact expansion for the one-body and two-body (i.e., density--nensity) Green's functions, suitable for self-consistent calculations. In first order we get the usual random-phase approximation (RPA). The successive terms can be interpreted as being due to local field corrections to the self­consistent field. Applications within the Pariser-Parr-Pople model are presented. Explicit calculations of the low-lying excitation energies of ethylene molecule show good agreement with "exact" calculations, and some of the difficulties of other methods, especially for triplet states, are overcome.

I. INTRODUCTION

To understand chemical properties of atoms and mole­cules on the basis of their electronic structures is one of the major aims of chemical physics. This program, despite some relevant successes, has been engaged with several persistent difficulties. The principal ori­gin of these difficulties is the strong correlations among the electrons, as can be clearly seen from the fact that in general the correlation energies are of the same or­der as the diSSOciation energies. Thus many-body ef­fects are surely of decisive relevance and need careful treatment.

Another peculiar characteristic of the interaction among electrons, especially in large atoms and mole­cules~ is the screening effect, which can produce large shifts of spectroscopic levels and corresponding dis­tributions of oscillator strengths. 1,3

The enormous computational complexities of ab initio calculations, which proceed along a separate treatment for the wave functions and for transition probabilities and energies, have suggested the use of techniques which directly produce the desired quantities. An at­tempt along this linel - 8 was the RPA scheme, whose un­doubted success6 stimulated further work and develop­ments on this type of approach. The main failure of the RP A method is the poor values of the triplet energies which, furthermore, can collapse to zero as the inter­action strength increases. The latter feature is due to an instability of the Hartree-Fock ground state. The reason for these difficulties9 can be attributed to some effects which are not included, or only crudely included, in the RPA, like .the screening effect (i. e., polariza­tion of the system), nonlinear contributions (which, for instance, arise from correlations in the ground state), and local field corrections.

Attempts to improve the RP A procedure have fol­lowed essentially two methods: (a) the equation of motion method which produces so called higher RP A schemes10-20 and (b) the propagator or Green's function

alSupported in part by Gruppo Nazionale di Struttura della Materia Unit3. di Catania and Istituto Nazionale di Fisica Nucleare-Sezione di Catania.

method9,21-29 which has been used to derive a variety of approximation schemes. These two methods, as pointed out by some authors17,30,31 are related.

All these approaches improve considerably the RP A results but when it is possible to have a comparison with exact calculations large discrepancies can still be observed. In particular the triplet energies in the ethylene molecule can be too high9 or collapse, 11 even at higher values of the coupling constant (or they can diverge, as in the calculations of J~rgensen et al. 17).

The Green's function method appears to be a prom­ising one for analyzing the physical origin of these difficulties, both because the formulation involves quantities with an easier physical interpretation (e. g., transition amplitudes, screening charge, etc.) and be­cause it is able to indicate successive approximation schemes which in principle should converge to the exact results. The electron many-body theory for molecules through the Green's functions has been formulated by many authors9•21• 30 referring to analogous works in solid state physics.

In this work we give first principle derivation of a set of exact equations for the density-density correlation function and one-particle Green's function. Expansion of these equations produces a systematic procedure for calculating the excitation energies and tranSition am­plitudes. The expansion directly introduces the dynam­ical, screened electron-electron interaction and is especially devised for self-consistent calculationso The derivation follows closely the theory first presented by Kadanoff and Baym32 and subsequently refined by Hedin:3

This method has been successfully applied to electron band theory in solids34 and to positron annihilation in metals. 35 The method has been used by Cooper and Linderberg,9 which calculated the excitation energies of some cyclic polyenes.

In this paper we re-derive all equations; firstly be­cause we have to extend the formalism in order to also study the collective states, and secondly because the introduction of the spin orbital basis in Hedin's formal­ism, as we shall see, is not trivial in the sense that it is not equivalent to replace simply the continuous in-

The Journal of Chemical Physics. Vol. 67. No. 10. 15 November 1977 Copyright © 1977 American Institute of Physics 4747

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4748 M. Baldo and R. Pucci: 1r electron systems.

dexes with the discrete ones. Actually our final equa­tions are different from Ref. 9.

The theory is formulated for a general two-body inter­action and it can then be applied, for instance, to the usual schematic interaction adopted in semiempirical calculations.

We have applied our method to model IT-electron sys­tems uSing the Pariser-Parr-Pople (PPP) Hamil­tonian. In particular the low-lying excitation energies and the oscillator strengths of ethylene molecule have been calculated and compared with full Configuration Interaction (Cr) calculations and other approximations. The computation uses a self-consistent procedure, and it is shown that the self-consistency is essential to avoid the difficulties for the triplet state. The results appear in good agreement with complete CI calculations.

In Sec. II the general equations of the theory are de­rived in detail using the atomic orbital representation. Section III is devoted to the description of the expansion procedure and Sec. IV to the use of the PPP approxi­mation in the case of cyclic polyenes. We present the results for the ethylene molecule in Sec. V and then we draw our conclusions in Sec. VI.

II. THEORY

A general Hamiltonian in the atomic spin orbital representation can be written as a sum of a one-body and a two-body terms:

H=L (rlhls)a;as+~ L (pqlvlrs)a;a:asar , (1) rs hrs

where a+ and a's are the creation and annihilation oper­ators for the spin orbitals specified by the indexes r, s, . 0.. If the basis is orthonormal and complete, their relation with the usual electron field operators is:

if!+(x) = L U:(x)a;, s

(2)

where the Us's are the spin orbitals. The Hamiltonian (1) can also be used for calculations with an incomplete basis. 21 We shall specify in the following paragraph the detailed structure of the one-body and two-body terms.

Starting from the Hamiltonian (1) we have to study both the single particle structure and the collective os­cillations of the electron systems. Let us begin by con­sidering the motion of a single electron inside the me­dium of all other electrons. The interactions experi­enced by this electron can be separated into two pieces (a) the mean field produced by all the other electrons which can be calculated within a Hartree-Fock proce­dure, (b) local field corrections deriving from local electron density fluctuations and scattering processes. This separation can be formally performed introduCing a fictitious external field cP through which it is easier to express the response (dielectric) properties of the system. 32 Actually this external field is only a formal device and, in any case, it is put equal to zero at the end of the calculations. Let us then add to the Hamil­tonian (1) an external field:

(3)

rs

and study the propagation of a single electron governed by H' . The equation of motion, of the single particle Green's function

(4)

where T indicates time-ordered product and the brack­ets ground state mean value, can be written

a i at Gnm(tt, t2) = linm li(t1 - (2) + L {en I h! k) + CPnk(t1)

1 k

where, as usual,

x (T[a;(t1) ar (t1) ak(t1)a;"(t2)])

- L.(n/VH(tl)/k)Gkm(tt,t2). k

The matrix M is the single-particle self-energy and

(n / V H(t) / k) = L (nll v I kr) ([a;(t) ar(t)]) lr

(6)

(7)

is the Hartree field. Actually M contains all exchange and scattering processes. A bar over a time variable means integration on that variable. It can be shown33

that the self-energy term can be written by means of a functional derivative:

(8)

We emphasize that all these relations are exact, and they hold in particular for cP =0. In the latter case the above mentioned separation between the mean Hartree field and local field correction for the unperturbed sys­tem is formally achieved [actually we shall see that the exchange term of the Hartree-Fock field arises from the first term in the expansion of (8)].

In order to introduce the dynamical screened electron­electron interaction, we define the following "effective" external potential

Vrs(t) = CPrs(t) + L (Ptu(t» (rt I VA I su) , (9) tu

where Ptu(t) =a;(t)au(t) is the denSity operator and

(rtlvAlsu)=(rtlvlsu)- (rtlvlus) (10)

is the antisymmetrized electrol1-electron interaction. The second term in Eq. (9) describes the effects of the polarization of the electron system (i. e., denSity fluc­tuations). This can be most easily seen if one calcu­lates the functional derivative of V with respect to cp:

[1'iV rs(t1)/IiCPPI(t)] = IirP lis/Ii(tl - t)

+i LXpl.tu(t,tl)(rtlvAI su) , (11) tu •

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M. Baldo and R. Pucci: 1r electron systems. I 4749

where

XPI; mn(t, t') = - (T [P~,(t) p~(t')]) (12)

is just the density-density correlation function; in Eq. (12) P~,(t) =PPI(t) - (PPI(t» is the density fluctuation op­erator. Let us notice that Expression (11) is nothing but the generalized inverse dielectric function of the electron system, E;~; rs(t, t 1).

Using the relationships

oGnm(t lz t2) =_" G (t t) OG;;(4,t5)G ([ t) oeprs(t3) ~ nil lz 4 Oeprs(t3) om 5, 2 ,

_0_ = L:0 Vpo(ljJ _0_ Oeprs(t) Po Oeprs(t) oVpo(t,J'

(13)

one can write the self-energy M in the form

Mn,,(t lz t2)

=i L: Wnm.rs(tl;t4)Gmp(tlz13)Ap".rs(i";t2;4) , (14) ,"prs ' ,

where

(15)

is the dynamical screened effective electron-electron interaction and

(16)

is the irreducible vertex function. Let us notice that this vertex function is dependent on four spin orbital indexes, at variance with Ref. 33 and Ref. 9. Equa­tion (14) is the first result of our formal developments; it separates in the self-energy the polarization effects, expressed by W, and the scattering processes described by A. To neglect the latter, amounts to putting,

(17)

In this case the first term of the effective interaction W, according to Eq. (11), gives just the exchange term of the Hartree-Fock field corresponding to the interac­tion v. The second term describes the local mean field variation around the single electron owing to the denSity fluctuations. In the static limit it takes into account the local electron density depletion (the so called "Coulomb hole" of solid state physics) owing to Coulomb repulsion.

Proceding along the same lines (see Appendix A) one can derive two (exact) integral equations for the vertex function A and density-density correlation function X, respectively, which complete the formal framework of the theory

APk;rs(t2 t3; t 1) = opr Oks 0(t2 - t 1) 0(t3 - t 1)

,,(OMp,,(t2 ,t3) .( I I ) ( ) ( -+ ~ oG IT T) -z pq v zk 15 t2-t3 0 t2-t4) "ofu .0\<4, t 5

X 0(t2 - [S») G .rt{t4,t6) Guq(t7.tS).\fu;rs(t6t7; t 1) , (18)

Xpl. mn(t, t') = - i P PI. mn(t, t') - i L: Xpl' fu(t, t 1) , • rstuv&"

X (rt I VA I su) Gn,,(t', Tz) G .m(t3, t') A",,;rs(i; t 3;t1) , (19)

where P is the so called irreducible polarization prop­agator

P (t t') = 0 (Pm(t'» rs; mn lz - 0 v: (t ) rs 1

=-iL: Gn"(t',tz)G.m(t3t')A,,r,rs(i;t3;tl)' (20) "z

We have seen from Eq. (20) that Expression (17) for A produces the free polarization propagator, i. e., in this case the local field coincides with the mean field (P gives the electron density response to the effective field VI)

Eqs. (14), (18), and (19) forms a set of (exact) coupled equations, which are the initial point for different ap­proximations, particularly for self-consistent proce­dures.

III. EXPANSION PROCEDURE

Starting from Expression (17) for the vertex function and iterating the three coupled Eqs. (14), (18), and (19) one gets an expansion for the self-energy and for the density-density propagator X in terms of the dynamical screened interaction Wand the single particle Green's function G. Of course, because W is expressed again through the propagator X, the expansion implies in gen­eral a self-consistent procedure. It is to be noted that the method of expansion is completely analytic; how­ever, as we shall see, it can be interpreted by means of graphs.

In first order one gets

M",,(t h t2) = i L: W" .... paCt r, t2) G mp(t h t 2) , mp •

(21)

Xmn. po(t lz t2) = - iP mn. pit h t2) . . (22)

with

P mn;po(th t2) = - iG om(t2, t 1)G np(th t2) , (23)

Eq. (22) is the usual RPA equation for X. Thus, the RPA corresponds in this scheme to neglect local field correction. We stress that the "full" propagator G ap­pears in Eq. (22), which could be calculated self-con­sistently by means of Eq. (21); the RPA corresponds to take for G the Hartree-Fock propagator.

To next order the integral equation for X becomes

XPI; mn(t, t') = - iF PI; _(t, t')

- i L: XPI,ac(t ,tl)(ra I VA I sc) Gn,,(t' ,72) G am(f3, t') 'rsacv&'

x X"n'; fu(tz,t3)(qt I VA I zu)Gp'r(t2, ~)Gso(t1,t3)

+ L: (vl'\v \ bn')(p'q \ VA \ zu) Gul , (i";, i;) n' qbP','u

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4750 M. Baldo and R. Pucci: 7r electron systems. I

X GbP' (t;, t s) G n'r(fz,tl) G Sq(t;.,ts) + L (vq I v Ibn') gbP' qn'u

X (p'ulv A Izg)Gn'u(i;.t3)G bP' (fz, ts)G rits.tl)GSq(t;.,t2~' (24)

where P has to be calculated to the same order (its ex­plicit expression can be easily calculated, but we do not report it here for simplicity).

Equation (24) is nonlinear, i. e., it introduces an­harmonicities in the density-density propagator. From another point of view looking at Eq. (24) one can say that it takes into account the interactions among the col­lective oscillations. Clearly at this level a full self­consistent procedure for calculating X is entailed.

Analogous expressions can be derived for the self­energy, but because we shall concentrate our attention on the excited states, we report them in Appendix B.

Each term of the expansion can be depicted by means of graphs. For instance in Fig. 1 we !eport the repre­sentation of the integral equations for A and X and the first few expansion terms. The wavy line indicates the density-density propagator, while a dot indicates a bare interaction. The full line is the electron propagator. At any interaction vertex distinction must be made between antisymmetrized and nonantisymmetrized bare interac­tion (see Fig. 1 caption). This distinction, which is not expliCitly present in Ref. 33, is in our case necessary owing to the introduction from the beginning of the anti­symmetrized interaction (10). It is to be emphasized however that these graphs do not follow all the usual rules; for instance, because A is the irredUCible vertex function (i. e., it does not include any polarization pro-

cesses) no bubble has to appear in any graph for X. This is due to the fact that such pieces of diagrams are already included in the wavy lines corresponding to X.

Actually the graphs drawn explicitly in Fig. 1 corre­spond to the terms of Eq. (24).

IV. THE PPP MODEL

Till now the theory has been developed without refer­ence to any particular model. We now speCialize the results of the previous section to the PPP scheme S6

with particular consideration of the cyclic polyenic mol­ecules. 37 The theoretical foundation of this semiem­pirical method has been extensively discussed in the literature,38 and it has been applied by many authors.24,s9

Our theory however can be applied to any other model.

The interaction in this case can be written26

(pal q0"21 v I rO"s S0"4) = Y pq(l - 6 pq 6ala2)6rp \\s 6alas (\2a4' (25)

where, as in the following, we have explicitly written the spin variables, so the indexes p, q, ... denote orbit­als only. As usually we consider the Singlet and triplet density-density correlation function, X(S) and X(I), re­spectively:

(26)

where the + (-) sign refers to singlet (triplet) states. The first spin index refers to p, l orbitals, the second index to m,n ones.

Substituting (25) in (24) and using (26) explicit equa­tions for X(S) and XU) can be derived up to second order. The expressions after Fourier transforming in time variable are the following:

x~1;!!.(w) = - iP~~;:;n(w) - iLYra Gnr(W2) [2X~1;~!(w) G rm(w + (2) - X~~;!~(w) G am(w + (2)] ra

- i L YraGnu(Wl)G.m(Wl +w)YUq (GUq(W4)Gqr(Wl- w2){2x~1;~!(w) rau8qu

X[(2 - 0Uq)y6U GII.(W2 +w4)Gru(Wl - W2 +w) - (1- 0Uq)y ... Gvu(W2 +w4)G re(Wl- W2 +w)]

- x~1;!~(w)[(2(s,O) - ova) Y cu Gvz(w2 + ( 4) G au(wl - W2 + w) - (1- 0Vq) Yu• G vu (w2 + ( 4)G a.(wl - W2 + w)]}

- G vr(Wl - (4){2X~I;Oja(w) [(2 - ova) X:a,uu (w4) Y.u G r.(Wl - w4 + w) - (1 - ~ova) X:a,.u(w4) - - 1 t - - - ] (S I) ()[( ") x YueG ru(Wl - w4 + w) + "20Vq Xq",cu (W4) Yu.G ru(Wl - W4 + w) - Xpliar W 2 - V Vq

X X:a,uu(w4) Y cuG ae(wl - W4 + w) - (1- ~ova) X:q;cu(w4) Yu. G au(Wl - w4 + w)

+ ~OUq X!q,cu(w4) rue G au(wl - w4 + w)]} +G ur(W2 - wl){[2X~~;~(w) Grq(W2 - WI + w)

- X~~i!~(w) G aa(w2 - (i\ + w)] [(2(S,O) - 0Vq) Y cuG qu(w2 - w4 + w) Gv .. (W4) - (1- 0Vq)

xYu .. Gqe(W2- w4+w)Gvu(w4)]}) ,

where the superscript (s,O) means zero contribution for the triplet state.

(27)

The main feature of Eq. (27) is that, in contrast to the RPA scheme, the two density-density correlation functions X (s) and X (t) are coupled together. The reason for that lies in the fact that the intermediate propagator can be, in both equations, the singlet as well as the triplet propagator. As a consequence any consistent procedure that is be­yond the RPA must be carried out jointly for X(s) and X(t).

Some simplifications can arise from the symmetry of the molecule. In particular we consider the cyclic poly­enes, whose periodic property allows the following Fourier transformations9 (which are equivalent to going to the molecular orbital representation):

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M. Baldo and R. Pucci: 7r electron systems. I

Ypq= hLY(k)exp[21Tik(P-q)/N] , II

where N = 4n + 2 is the number of carbon atoms and - 2n S kS 2n + 1.

Adopting these transformations Eq. (27) becomes

(s,t) () .p(S,t) () . '""' 1 (S,t) ( )(2 (s,O) ) G(k' - )G(k' k" - ) XIIII'kf' W =-z kll'k" W -z L...J NXII~k" W Y-II"·-Y-k +11' ,Wi +, W+Wi

~ 2

- i L jj'J1 x~:':l, (w) G (k', Wi) G(k' + k", W + Wi) {(2 y~:~?) - Yll -II ) G (k4 - k5' W4) G (k4, Wi - W2) {II} 2 4 2

x [(2Y(~'~~" - y,,_, ) YII - N.! (Y_II -II" - YII '_II ) YII ] G(k' - k5' W2 + W4)G(k /I + kh Wi - W2 + W) -"5 ""4 5 5 4 3

-h (2Y~:~~) - YIlS-~) [(2Yke-ll• - YIl"+lle-1l4)(Yk'-ks - i h Y1l3) X:aIl4k'-lls (w4)

- -21

N.!Yk"+1I -II X: II II'-k (w4)1 G (k6, Wi - W4) G (k" + ke, Wi - W4 + W) 6484 S J

(2 (S,O) )G(k k" - - )G(k - - ) r(2 (B,O) + Y -II" - YIl4 -1l"-1I

2 4 - ,W2 - W1 4, W2 - W 1 + W L YII4 -117-1l"

- Y 11'+11" -II ) Y II -II - N.! (y II -II -II" - Y 1I"+k' -II ) Y II ] G (k7, W2 - W4 + w) G (k' + k4 - k7' W4)} 774 47 73

All the functions appearing in Eq. (29) have to be considered periodic functions of each one of their k arguments with period 4n + 2.

v. THE ETHYLENE MOLECULE

We have applied the theory to the calculation of excited states of ethylene moleculb, which can be considered the prototype for 1T-electron systems. 37 Despite this fact it is the simplest polyene, owing to the strong electron cor­relations, only complex many-body calculations are able to give quantitative results. 6,17,40

The poles41 of the density-density correlation function X give the excitation energies, while the residues at the poles determine the strength distribution. To illustrate in our scheme how the procedure workS, let us derive the equation for X in the RPA case and for the PPP model. So, according to Sec. III, we neglect any non­linear term in Eq. (29). Then it simplifies to

[w - ~~ - h(B.tl]X<:otl(w)

T 1 (s.tl{) .p(s.tl()[ A] T 2" Y! Xllll W = - Z k01 W W - ... ~ ,

1 ( •• t l {) .p(s.tl()[ A] ± 2" Y! XIlO1 W = - Z 1111 W W + ... ~ , (30)

(a) I\. =

(b) x =

4751

(28)

(29)

where y(BI =2Y1- Yo, y(tl =- Yo and ~E =E1- Eo, Ell being the single particle Hartree-Fock energies. It has to be noted that in this case the matrix of the system for X is independent of the first k index, while the third index of X is fixed to be one. The latter is a consequence of the simplicity of the ethylene molecule {for which k =0,1~ while the former is characteristic of the RPA in the PPP model.

FIG. 1. Graph representation of integral Eqs. (18) and (19). The first few terms of the expansion are also depicted. A full circle indicates a bare nonantisymmetrized interaction, while a full square an antisymmetrized one. A full line represents a Single-particle propagator. The dashed line appearing in (a) indicates the effective external potential.

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4752 M. Baldo and R. Pucci: 1T electron systems.

The poles of X are the zeroes of the determinant of the system. Solving this secular equation one gets the usual result4

:

(31)

where {3 is the resonance integral and g = Y d21 (31 is the interaction strength.

From Eq. (30) it is clear that the residues are deter­mined essentially by the polarization propagator P. The final explicit expressions for the X components are:

(S,t)( ) 1 [2'( A 1 (a,t»] X001 W = 2 2 - Z W + ... E +"2 Y ,

w - W(s,t)

(S,tl( ) (1/ 2 2 )[2'( A ~ (s,t»] X1ll W = W -W(s,t) Z W- ... E-2Y . (32)

As is well known the RPA solution breaks down at g= 1 (see Eq. 31), where the triplet energy collapses. To overcome this difficulty one is forced to go beyond RPA using Eq. (29). To solve this nonlinear equation one could, to first order, linearize it using as inter­mediate X propagator (see also Fig. 1) The RPA ex­pression (32), so reduCing the problem to a conventional secular problem. The result of such a substitution (after explicit integration over energy variables) is

[W2 _ (Ad2 - t(w + AE) i O, t) ± Yl CJxk~rt) (w) + [ - t(w + AE) (± Yl) + y(" t) CJxHit) (w) = - iPk~i 0 (w)[ w2 - (Ad2] ,

[t( W - AE) (± Yl) + y( .. t) C] Xk~ n (w) + [w2 - (AE)2 +t( W - AE) y(" tl± Yt CJxkfitl (w) = - iPkto( w)[ w 2 - (AE)2], (33)

where

However this procedure faces necessarily with troubles. In fact the appearance of the RPA energies in the inter­mediate propagator gives rise to divergencies at g = 1, and the procedure itself is meaningless for g> 1. Of course the difficulty remains to any order, and a non­perturbative overcoming of the RPA scheme is needed. As a consequence the full self-consistent structure of Eq. (29) has to be preserved in order to get any suc­cessful result.

The self-consistent calculation can be performed as follows. One knows, on general groundS, that the den­sity-denSity propagator must have the analytic form

A (0,0

(s,n (w) :\"2"3 Xklk2 k3 2E (W2_E2 )' (s,n (s,t)

(35)

where the A's are the residues at the poles and E(a,t)

are the (unknown) singlet and triplet energies. Then one inserts Eq. (35) into Eq. (29) and perform analytical­ly the w integrations. The results are a system of alge­braic equations for X, whose coefficients are explicit functions of the E' s and A's. Thus, by comparing the solutions of these equations with (35), one gets the de­sired (coupled) equations for the energies E(S,t) and residues A(·,t). It is easy to verify that the number of equations just equals the number of unknowns.

We have followed the above described method using Hubbard's prescription for the interaction, i. e., Ypq

= Yppflpq. Actually, because we will use the coupling con­stant g as the only parameter of the calculations, the Hubbard approximation is completely equivalent to the PPP approximation, 43,44 in the sense that they are re­lated simply by a relative shift of the nand n + 1 elec­tron levels.45 The main simplification introduced by this parametrization is that now X has only one indepen­dent component, which for convenience we take XkOl(W) +X"u(w)=X(w); for this component, following the same lines as in gettting (33), we obtain the equation

(34)

[ g3( A' At)] 1 ( .. 0 + ±g'f"2 1+E. +1+E

t E2_1X (E),

(36) where all the energies are in units of 2 1{31. The self­consistent procedure is greatly simplified if one intro­duces the quantity 0:

[ g2 ( AS At )]

O=g 1- - --+--2 I+Es I+Et

(37)

for which the self consistency condition gives the follow­ing equation:

[03 _g02 _ (1 +g2/2)0+g] + (g_ O)~

- [02 + (1- g+g2/4) 0 -g].f1-1'f

+[02_(I+g+g 2/4)0+g]v'1+0=0. (38)

In terms of 0 the singlet and triplet energies can be ex­pressed as:

Es=v'1+0,

E t =v'1-0, (39)

so 0 can be viewed as an effective coupling constant.

The numerical results are displayed in Figs. 2 and 3 (label LFC: Local Field Corrections) where com­parison is made with a number of other methods. All the upper curves refer to singlet energies, the lower curves to the triplet ones.

First of all we notice that the triplet energies do not collapse (actually there is no collapse until much higher values of g than reported in the figures) and more re­markably they are rather close to the "exact" CI cal­culations. The former feature is intrinSically bound to the self-consistent procedure, which prevents any en­ergy to vanish. This means that the nonlinear terms become large when g approaches one, and are essen-

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M. Baldo and R. Pucci: 1f electron systems. I 4753

E FULL CI

SPPA

HRPA

__ -----------------LFC

SPPA

------~~--~~LFC FULL CI

HRPA

0.5 10. 1.5 9 FIG. 2. Triplet and Ejinglet excitation energies for ethylene are plotted as a function of the interaction strength. CI means full configuration interaction; HRPA, higher random phase approximation11 ; SPPA self-consistent polarization propagator approximation. 17 Finally the results of this paper are called LFC (local field corrections).

tial in the collapse region. The fact that on the con­trary the higher RPA schemesll (HRPA) do not avoid the collapse indicates that the ground state correlations are not the only mechanism that plays a role in effecting appreciably the excitation energies as the coupling con­stant increases.

E FULL CI

COHSEX

RPA

~ ____ ------------LFC

~-------------------------COHSEX

--____ -==_=LFC FULL CI

RPA

0.5 1.0. 1.5 9 FIG. 3. The LFC results of Fig. 2 are compared with RPA, CI, and COHSEX9 (coulomb hole and screened exchange).

~~-----------------------RPA

FULL CI

LFC

2 3 9 FIG. 4. Oscillator strength as a function of the strength pa­rameter g. The SPPA 38 collapses at a value slightly larger than 1. All the values are in units ~ R2.

Local field corrections thus appear to become near g ""1 the dominant physical origin of the deviation from RPA result, i. e., the polarization propagator is strong­ly modified by local density distortions. The results for the triplet energies are considerably improved with respect to any other preceding calculations. Actually we have checked that the agreement with complete CI calculations perSists up to g - 3, even if the trend of the curves indicates a different asymptotical behavior. Finally the fact that the COHSEX approximation9 over­estimates the triplet energies could be ascribed to a too large weight of the screening effects.

The Singlet energies are rather underestimated. Thus the. higher order terms in the equation for X appear to be more relevant for the Singlet state, i. e., the con­vergence of the expansion seems to be slower for the Singlet than for the triplet states. The other reason for the discrepancy can be, of course, the self-energy ef­fects that could playa different role in the singlet and triplet case.

In Fig. 4 we report the oscillator strength f as a func­tion of the parameter g. It is extracted directly from the residue ~. of X through

f=iR2~sE =iR2 E.Et {Et +1)(Es +1) n B g2[E.(E" + 1) +Et{Et + 1)] ,

(40)

where R is the C-C bond distance. Expression (40) holds in the dipole length approximation43 assuming a diagonal form of the dipole operator in the atomic orbit­al representation {which is consistent with the PPP model). 46

From the figure it is again apparent of the crucial role of the self-consistency and local field corrections. The displayed discrepancy is due essentially to the in­accurate values of the singlet energies. In any case the

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4754 M. Baldo and R. Pucci: 7r electron systems. I

overall trend, until very large values of g, can be con­sidered satisfactory, and relevant improvement is ob­tained with respect to RPA and SPPA.

VI. DISCUSSION AND CONCLUSION

Starting from the equations of motion of the Green's functions, and using functional derivative techniques, we have derived an expansion for the one body self­energy and density-density correlation function in terms of the dynamical screened electron-electron interac­tion. Such an expansion is obtained from a s~t of exact equations for the irreducible vertex function A and den­sity-density propagator X [Eqs. (18), (19)].

This fact assures that any double counting of terms is avoided and that the Pauli principle is restored as the expansion proceeds even if in the starting approximation (the RP A) it is violated. For instance the graph which appears in Fig. 1 for the expansion of the X including a particle-hole propagation inside a bubble is just a graph correcting for these effects. To each order the ex­pansion entails a complete self-consistent procedure for both the single particle and two particle Green's func­tions. Of course in practical calculations the self-con­sistency can be fulfilled only for one of the two Green's functions, and one has to take a suitable approximation for the other. In particular if one is interested in the excited states, the Hartree-Fock approximation for the single particle propagator can be chosen. In this case use is made of the Lehmann representation for the X,

(w) = _ i" [<01 a;asl n) <nl a;aqIO) Xrs,pq L...J w _ W +iE

n n

(0 I ap aq I n) <n I a; as I 0) ] + .

W +wn - tE (41)

where {I n)} are the excited states, wn the corresponding excitation energies, and 10) is the ground state; after substitution in the equations one gets a set of coupled relations for the amplitudes (i. e., residues of X) and the energies wn by imposing the self-consistency condi­tion. The latter condition requires some constraints on the amplitudes in order that the same structure (41) is preserved along the whole procedure. In the c'ase of ethylene molecule such constraints are automatically satisfied.

The expansion should be connected in some way with the theory by Cooper and Linderberg9

; however our derivation is from first principles and the final equa­tions for the density-density propagator appear rather

APPENDIX B: SELF·ENERGY EXPANSION

different. The main physical difference, in any case, can be identified in the use of a dynamical screened in­teraction instead of the static one of Ref. 9. According to the results for the ethylene molecule, the dynamical character of the effective interaction seems to be essen­tial to avoid an overestimation of the screening effects. In this respect, by comparing our results with other methods, it is possible to draw the conclusion that local field corrections are the main mechanism that is active for large values of the coupling constant (g~ 1). Finally we have to emphasize that the calculations use the ex­pansion up to second order. This approximation seems to be a posteriori justified for the triplet states, while the successive terms could playa role in explaining the observed discrepancies for the Singlet energies.

The firm theoretical basis of our method and the promising results for the ethylene molecule encourage further work along these lines.

ACKNOWLEDGMENTS

One of us (R. P. ) wishes to express his acknowledg­ment to J. Linderberg, P. JrOrgensen, and J. Odder­shede for several stimulating and illuminating discus­sions.

APPENDIX A: DERIVATION OF THE EQUATIONS FOR X AND A

Starting from Eq. (16) and looking at the definition of the self-energy (6), one gets

AI>/l; r.(t3ta; t4 ) = I5pr 151<.15 (ta - t 1) l5(t3 - (1 ) + I5Mpl«t3, fa)/ I5V rs(t4 )

= 151>r0I<.o(fa _ t 1)0(t3 - t 1) + L ~~I>It~' :a» 15~lr(t(t' ~6). 1m 1m 5, 6 rs 4

(AI)

USing now Eq. (13) (with cp repl~ced by V), one reob­tains on the right hand side the A function and then the integral equation (18).

Analogously from Eqs. (9), (11), and (13) one has

(t f ') . ° <Pmn{t '» XPI;mn, = - l 0CPPI(t)

I5Vh (t1)

I5CPPI(t)

(A2) Using agin the first of Relations (13) one immediately gets Eqs. (19) and (20).

The self-energy expansion is obtained from Eq. (14) by expanding A according the integral Eq. (18). To first order M is given by Eq. (21), to second order one has to add

M ~~)(fl> fa) =i L Wnm;rs(th t4 )Gmp(tl> t3) [- L (Pl/ v /ae)x IC;h(t3, ta)(qt / VA /ku) mPrs aqlctu

x Gar(t;,t' 4) Gsq(T4J fa) + L (pl / v / be) (aq / VA / ku) Gy/(ta, ~) Gba(t3,t a) Ccr(t3,t4) Cs.(t4, f 2) aQbclu

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M. Baldo and R. Pucci: 1f electron systems. I 4755

To get Eq. (Bl) one has to follow completely analogous manipulations explained in Appendix A.

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