Toward a Microscopic Description of Molecular Electronic Systems*

RENATO PUCCI, MARCELLO BALDO, ANTONIO GRASSI, AND PASQUALE TOMASELLO

Istituto di Fisica dell’Uniuersit2 di Catania, Gruppo Nazionale di Struttura della Materia-Unit2 di Catania, Istituto Nazionale di Fisica Nucleare-Sezione di

Catania, Corso Italia, 57-951 29 Catania, Italy

Abstract

The many-body perturbation theory is reformulated within the dielectric function method pre- sented in the preceding paper [ 121. The self-energy effects are explicitly considered. These corrections turn out to be important and are calculated to the same level of approximation as the corresponding local-field terms. The method is a t the moment applicable to *-electron systems within the zero differential overlap approximation. Explicit calculations of the excitation energies for the benzene molecule, using different parametrizations, are presented. Comparison is made with the results obtained in the random phase approximation (RPA) and other schemes.

1. Introduction

The dielectric function (DF) method [ 1-61 has in the last few years been one of the most useful many-body theory of r-electron systems. Its development is linked to the ability of the method of handling in a simple way the difficult problem of the strong electron correlations typical of conjugated molecules.

Even if different formulations have been presented in the literature, the es- sential feature of the methods are the introduction of a screened interaction [ I ] in the calculation of the density-density correlation function. This screening has its origin in the electron correlation effects which are not included in the usual random phase approximations ( RPA). For instance the local-field corrections ( LFC) [4] modify the free polarization propagator appearing in the RPA and are directly related to nonlinear effects in the response function of the electron system.

One of the main effects of the screening is the renormalization of the two-body Coulomb interaction and of the single particle energies. The relevance of these renormalizations has been analyzed by Cooper and Linderberg [2] and Yam- aguchi [ h ] .

Closely connected to these works is the SPPA method by Oddershede, Jglr- gensen, and co-workers [7,8]. A full dynamical perturbative approach has been followed by Toyoda et al. [3], by 1’Haya et al. [9], and more recently in calcu- lations [ 10,l I ] using the,so-called SOPPA scheme.

* Supported in part by Ccntro Siciliano di Fisica Nucleare e di Struttura della Materia, Catania.

International Journal o fQuan tum Chemistry. Vol. X V I I I , 11-18 (1980) CI I980 by John Wiley & Sons, Inc. 0020-7608/80/0018-001 IS01 .oo

12 PUCCl ET AL.

Recently, Baldo, Pucci, and Tomasello [4,5] have introduced a Green’s function formulation of DF method, based on functional derivative techniques. The starting point of the approach is a set of exact coupled equations for the one- and two-body Green’s functions. From this set of equations an exact expansion in the full dynamical screened interaction is derived. Applications of this method, within the Pariser-Parr-Pople (PPP) model, have been carried out [5,12] for the calculations of the spectra and oscillator strengths of ethylene and benzene molecules.

In this work we study, in the framework of this scheme, the single-particle self-energy effects on the spectrum of benzene molecule. The calculations are performed for three different parametrizations.

2. Method of Calculations

According to Ref. 5 the corrections to the RPA scheme are of two types: (i) the local-field corrections (LFC) and (ii) the self-energy (SE) corrections. Up to second order (in the full dynamical screened interaction), both these correc- tions are depicted by the graphs of Figure 1 . It is apparent from the graphs that the solution of the corresponding equations requires a self-consistent procedure, because the single particle Green’s function G and the density-density correlation function x are nonlinearly coupled. Furthermore, the equation for x can be considered an RPA-type equation with an energy-dependent matrix. The origin of this dependence comes from the introduction of two-particle-two-hole exci- tations, whose effects can be projected into the particle-hole subspace by in- troducing an effective energy-dependent interaction. This point was originally discussed by Shibuya et al. [ 131, and later by many other authors (see, e.g., Ref. 14).

In Ref. 12 the LFC effects has been perturbatively analyzed for the benzene molecule, i.e., inserting for the “intermediate” propagators x (which appear in the second order graphs of Fig. 1) the RPA expressions. This procedure is of course justified only if the RPA approximation can be considered a reasonable starting point. Actually this seems not always to be the case, because in some parametrization schemes, e.g., the Slater assignment, the RPA gives triplet imaginary poles (i.e., it displays an instability of the ground state). The LFC corrections turn out to be relevant for assignment ( A ) and ( B ) (see Table I) and produce a better agreement for triplet states and a poorer one for singlet states, if comparison is made with full configuration interaction (CI) calculations.

As it was found in Ref. 5 the self-energy effects, are essential in the ethylene molecule to give the right trend, as a function of the coupling constant g , for the singlet and triplet energies. In view of these results we have investigated the self-energy effects in the benzene molecule on the excitation spectrum, following the above mentioned perturbative approach.

The general expression for the self-energy matrix can be written

DESCRIPTION OF MOLECULAR S Y S T E M S 13

RPA+SE

+ . * . . . . . .

- LFC+ SE

+ .... *... b Figure I . Graphical representation for the expansion of the density-density correlation function x and of the single-particle propagator G. Wavy line represents a propagator x; il full thick line is Ihe renormalized Green’s function G, and a full thin line the HF propagator Go. Full squares represent antisymmetrized two-body interactions and full circles, nonan-

tisymmetrized ones.

where a bar means integration on the corresponding variable; m,n,l, . . . , are spin-orbital indexes, G is the single-particle Green’s function, W is the full dy- nainical screened interaction

Wnn?;rs(tlrt4) = 6( t1 - td (n+lms)

and is the irreducible vertex function. For the interaction u, we use the PPP approximation [ 151. The symbol u A means antisymmetrized matrix elements. To the order of approximation displayed in Figure 1 , the vertex function ap- pearing in the self-energy expansion can be replaced by

- i i p l ; r s ( f 3 t 2 ; r 4 ) = 6 ( t 3 - f 2 ) d ( t 3 - t 4 ) 6 p r 6 / s . (3)

The equation for the propagator x consistent with this approximation is Eq. (24) of Ref. 4.

For a 7r-electron system it is useful to adopt the molecular orbital represen-

14 PUCCl ET AL.

TABLE I . Parameter sets used in the calculations ( in eV). Parameters 0 and y ( k ) are the resonance integral and the electron repulsion integrals (in the molecular orbital

representation), respectively.

AssigMlent(A) Assignment (B) Assignment (C) (PPP) (SLATER) (YAMAGUCH I )

2.68

8 . 6 1

2.69

1.35

1

2.68

8 . 6 2 5

0 . 8 4 2

0 . 4 9 5

0 . 2 1 2

a Reference 15, p. 68. Reference 6.

tation, which is equivalent to perform a Fourier transformation of Eqs. (1) and (2). Then the Lehman representation for x can be written

where the w, are the excitation energies and the A’s are the corresponding residues; the index k labels the molecular orbitals.

Equation ( l ) , when Fourier transformed with respect to time variables, gives the following correction to the Hartree-Fock ( HF) energies:

[ 2 y ( k ” ) - 373 [ 2 y ( k ” ) - Y ( k - k’ - k ” ) ] 2 N 3 k’k”

>.k/k” (u:) [ 1 - f ( k - k ” ) ] w - E(k - k”) - W ;

- y y ( k - k’ - k”)

where the symbols s and t refer to singlet or triplet states, the y ( k ) ’ s are the Fourier transform of the yrs, 7 = (l/N)Zk y ( k ) , N is the number of carbon atoms,f(k) is the usual step function [ 2 ] , and finally, the E(k) are the single- particle H F energies.

We have calculated the shifts of the single-particle and single-hole energies for the benzene molecule, using for the propagators x the R P A expression and taking, in Eq. ( 5 ) , M ( ’ ) at w = E(k). This procedure is in agreement with the perturbative approach used for the LFC calculations [ 121. The results for the particle-hole energies are reported in Table 11. For the assignments ( A ) and ( B ) the shifts are sizable in some case even up to 30%. There is a tendency to lower the particle-hole energy values. For assignment (C), the corrections are

DESCRIPTION OF MOLECULAR SYSTEMS 15

T A B L E 11. Particle-hole energies (in eV) for the three different parametrizations of Table I in the RPA with and without the self-energy (SE) corrections.

Differences of Assignment (A) Assignment (B) Assignment (C) single-particle PPP SLATER energies in eV. without SE with SE without SE with SE without SE with SE

YAMAGUCH I

F ( ~ ) - E ( Z ) - 1 1 . 1 7 - 1 0 . 4 6 - 1 2 . 9 7 - 1 4 . 6 9 - 1 3 . 7 1 - 1 3 . 9 4

s ( O ) - e ( 3 1 - 1 7 . 3 9 - 1 4 . 1 4 - 2 1 . 0 1 - 1 3 . 3 7 - 1 9 . 8 3 -18.31

E [ O ) - t ( Z ) - 1 4 . 2 8 - 1 2 . 3 0 - 1 6 . 9 9 - 1 4 . 0 3 - 1 6 . 8 0 - 1 6 . 1 2

only of the order of few percent. This is hardly surprising because the Yamaguchi parameters are extracted through a self-consistent procedure, so that the 7’s should effectively already take into account some of the self-energies.

3. Collective Oscillations in Benzene The fact that the particle-hole energies tend to be depressed by the SE effects

increases the instability of some excited states in the RPA, when self-energy corrections (RPASE) are included.

For this reason the ’BTU state in the PPP parametrization collapses (Table 111). In the Slater parametrization (Table IV), besides the collapse of the 3B:u state, there is a strong depression of the

The RPASE calculations show the large effect of the self-energy shifts. However, RPASE; is not a consistent approximation, according to the theory of Refs. 5 and 16, which gives a well-defined prescription for the inclusion of the L F C terms consistent with the corresponding S E corrections.

The calculation of the LFC terms including the self-energy shifts (LFCSE) can be performed using the same method of Ref. 12, where we used a program that computes all the integrals applying directly the residues theorem.

state.

TABLE 111. Energy levels (in eV) of the benzene obtained with different approxi- mation schemes (for the meaning of the symbols see the text), in the case of the

assignment ( A ) of Table I , compared with CI results. Assignment (A) - PPP

k” K A T E RPA LFC RPASE LFCSE CI

3 2 . 2 1 3 . 8 6 2 . 6 5 3 . 8 0 [ 2 0 ]

1 ’1 : + 4 . 1 1 3 . 8 7 3 . 3 9 3 . 2 - 4 . 5 0

3 ’I - 4 . 6 6 4 . 5 8 3 . 9 4 5 . 0 4 5. 50

2 3 , ; + 5.85 6 . 2 0 3 . 8 1 5 . 1 5 6 . 3 4

2 IE - 8 . 5 6 8 . 4 6 6 . 5 8 6 . 4 6 8 . 7 3

3 1 0 . 8 2 1 1 . 2 - 7 . 8 0 9 . 2 3 9 . 3 4

”3”+ 1“

2”

2K

2Y ’ R l ” +

3 ‘11 - 4 . 6 6 3 . 9 4 3 . ’ ) 4 3 . 4 0 4 . 2 [ 2 2 ]

1 1 L l ” + 6 . 4 4 5 . 8 8 5.70 5 . 6 9 6 . 8

3 ‘ I l l”+ 1 1 . 5 0 1 0 . 8 8 8 . 2 4 8 . 9 2

2u

1“ 3 ‘11 + 5 . 8 8 5 . 9 1 5 . 1 4 5 . 5 0 5 . 7

2 8 . 1 3 7 . 9 9 6 . 1 2 6 . 0 1 6 . 9 1 1

2 ‘E - 8 . 5 6 8 . 2 9 6.SR 6 . 4 3 8 . 4 2g+

2 g

16 PUCCl ET AL.

TABLE IV. Energy levels (in eV) of the benzene obtained with different approximation schemes in the case of the assignment ( B ) of Table 1.

Assignment (B) - SLATER k" STATE RPA LFC RPASE LFCSE

3 PBlu+ 2 . 2 2 5 . 1 3

1 3 E l u + 4 . 2 4 3 . 8 0 6.00 5 . 6 7

5 . 4 5 6 . 1 6 7 . 2 4 6 . 4 7 3 %l-

2 'E + 5 . 1 8 5 . 8 1 1 . 3 9 3 . 3 8

2 ' E - 1 1 . 0 6 1 0 . 9 4 8 . 1 0 8 . 1 2 2g

2g 3 Y u + 1 3 . 7 2 9 . 1 1 8 . 3 9 1 3 . 9 3

3 lB2u- 5 . 4 6 3 . 8 2 6 . 1 6 4 . 7 0

3 'B l "+ 6 . 8 2 3 . 8 6 7 . 2 4 6 . 1 6

8 . 7 0 4 . 1 2 1 0 . 5 9 1 1 . 9 7

2 'E + 1 0 . 6 6 8 . 5 0 7 . 5 4 6 . 8 9

2 ' E - 11.06 9 . 3 6 8 . 1 0 8 . 2 4

3 'BlU+ 1 4 . 4 6 1 3 . 5 7 9 . 1 2 1 3 . 8 8

1 l E l U +

2g

2g

Generally speaking, the LFC and SE terms, as can be seen in Tables 111, IV, and V, tend to correct the RPA collective oscillation energies in opposite direc-

TABLE V . Energy levels (in eV) of the benzene obtained with different approximation schemes in the case of the assignment ( C ) of Table 1, compared with experimental values.

Assignment (C) - YAMAGUCHI 'r

k" STATE RPA LFC RPASE LFCSE EXP

3

1

3

2

2

3

3

3

1

2

2

3

4 . 2 4

5 . 1 4

5 . 6 1

'E + 7 . 3 0

'E - 9 . 0 1

'Blu

3Elu+

'BZu-

2g

2g 1 1 . 4 0

5 . 4 4

5 . 6 1

6 . 6 7

' E + 9 . 0 1

'E - 9 . 2 3

1 1 . 6 8

3Blu+

'Bzu-

I B l U +

l E I U +

2g

2g

B l u +

4 . 6 2 4 . 3 3

5 . 1 3 5 . 3 1

5 . 6 1 5 . 7 8

7 . 4 4 6 . 6 2

8 . 9 8 8 . 3 3

1 1 . 3 7 9 . 9 5

5 . 3 2 5 . 5 9

5 . 5 9 5 . 7 8

6 . 5 3 6 . 8 4

8 . 7 9 8 . 3 3

9 . 2 7 8 . 5 4

1 1 . 6 6 1 0 . 1 8

4 . 7 3

5 . 3 0

5 . 7 8

6 . 7 6

8 . 3 0

9 . 8 9

5 . 4 8

5 . 7 4

6 . 7 1

8 . 1 2

8 . 5 7

1 0 . 1 4

3 . 9 5 [23]

4 . 7 5

5 . 6 0

6 . 7 5

8 . 9 0

4 . 8 9 [ 2 4 ]

6 . 1 7

6 . 9 8

DESCRIPTION OF MOLECULAR SYSTEMS 17

tions. Thus one could expect some compensation when both of them are included. This is confirmed by the explicit calculations reported in the LFCSE column of Tables 111, IV, and V. Similar cancellation effects between individual diagrams of the same order have been discussed in Ref. 11, where a second-order calcu- lation in the framework of the polarization propagator method is performed. The fact that w e sum some diagram subseries up to infinite order seems to in- dicate that these cancellation effects are independent of the order.

For assignment ( A ) , the 3B:u state, which is unstable including only the SE corrections, is again present in the LFCSE calculations.

Analogously in assignment ( B ) the 3B:u state again appears, while ' E & is raised by more than a factor of 2 with respect to the RPASE calculations. For few states this compensationdoes not occur, but in these cases the total correc- tions are much smaller. These results seem to indicate that the SE corrections a re rather important, and it is very difficult to have a good estimate of them in perturbative-type scheme as the one we have proposed.

The main reason for that appears to be the extremely large correlations present in 7r-electron system, as can be seen from the fact that the 3B:u state energy, according to CI calculations with assignment ( A ) , is about three times smaller then the lowest unperturbed particle-hole energy. Thus a self-consistent pro- cedure, including both SE and LFC corrections, appear to be necessary in order to describe these large correlations.

This is confirmed by the calculations of Ref. 5 for the ethylene molecule and by comparing the LFCSE results with CI calculations in assignment ( A ) (see Table 111). Work along this line is in progress.

A completely different trend is shown by parametrization (C) reported in Table V. Both SE and LFC corrections turn out to be much smaller than for parametrizations ( A ) and ( B ) . As already mentioned this can be ascribed to the fact that the Yamaguchi interaction parameters are already extracted through a self-consistent procedure, even if the static approximation is made in the equations, i.e., they are taken at zero frequency.

Finally, we have found that the modification on the residue values produced by the self-energy terms is much larger for parametrization ( B ) than for par- ametrization ( A ) and especially for (C), where the changes are of the order of few percent.

These results can be considered a further confirmation that in our scheme assignment (C) is the most reliable, in the sense that the other two parametri- Lation ( A ) and ( B ) clearly indicate a divergence of the perturbative expan- sion.

The (C) assignment gives better agreement of our results with the experi- mental values (see Table V) and the ordering of the singlet and triplet states seems in agreement with recent experimental findings [ 17-20].

18 PUCCl ET AL

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Received April 10, 1980