Journal of Structural Chemistry, VoZ 36, No. 4, 1995

EFFECTIVE ACCOUNT OF 2p-2h EXCITATIONS

IN MOLECULAR CALCULATIONS BY THE

EQUATION-OF-MOTION METHOD

A. V. Glushkov, N. G. Serbov, S. V. Ambrosov, V. 1~. Orlova, G. M. Dormostuchenko, and O. V. Troitskaya

UDC 539.184 + 539.186

The excitation energies and oscillator strengths of transitions in the helium atom (as a test system) and carbon dioxide and ethylene molecules are calculated by a modified equation-of-motion method with effective account of the most important polarization effects, including the 2p-2h (two particles-two holes) interactions, in the quasiparticle approximation of the-density functional Two different types of the effective potential of the particle-hole polarization interaction are used. The accuracy of the calculations is sensitive to the type of the potential. It is shown that the account of the polarization corrections for the 2p-2h

effects is of fundamental importance since it contributes up to 30% to the values of the excitation energies and oscillator strengths.

Information about the excitation energies and the probabilities and oscillator strengths of electronic transitions is important in solution of physicochemical problems [1, 2]. To determine these characteristics, one should first calculate the total energies and wave functions of the corresponding states with the greatest possible accuracy, which presents a real challenge. The adequate calculations of these parameters by traditional quantum chemical methods, in particular, ab initio multicortfiguratiou approximations, which take into account the correlation effects at different orders of perturbation theory with the Hartree-Fock zero approximation (the MNler-Plesset theory), and others [3-6] involve certain difficulties. In this case, a more suitable technique is the equation-of-motion (EOM) method, which takes into account the correlation effects in the random phase approximation (RPA) [%9]. This approach allows one to directly calculate the amplitudes of various quantum processes, including absorption and emission of photons, etc., and to avoid the problems in calculating the wave functions and total energies of molecules. Although tbi~ method is not absolutely accurate, unlike the limiting variational solutions, it is very efficient in calculating the excitation energies and oscillator strengths of electron transitions in molecules. The relativistic version of the method has been successfully employed to calculate the effect of nonconservafion of parity in heavy atoms and ions [10]. In this work, to calculate the excitation energies and oscillator strengths of some transitions in the helium atom and in carbon dioxide and ethylene molecules, we used the modified EOM method, which differs from the standard version in the effective cal~klation of the 2p-2h (two particles-two holes) polarization interactions (PI) [11, 12]. As shown, e.g., in [7], a satisfactory accuracy of calculations can be gained by using small basis sets. However, in this case, proper account must be taken of the most important PI effects, such as the effects associated with the 2p-2h interactions, the "pressure" of continuum, the energy dependence of the serf-consistent field potential, etc. The accurate account of most of these effects by standard methods (e.g., in terms of perturbation theory) leads to considerably complicated calculations [4-11]. Our procedure is based on the use of the corresponding exchange-correlation density-funcfionals (DF) [11, 12]. For example, in terms of the EOM method, these DF allow one to take into account the 2p-2h PI without complicating the calculations and without loss in accuracy. The contributions of the effects associated with the-'~ove interactions to transition energies are up to 3 eV. In the calculations of the spectroscopic characteristics of benzene, formaldehyde, and other molecules [12], the 2p-2h interaction effects were taken into account using a DF based on the correlation one-quasiparticle

Odessa Institute of Hydrometeorology. Translated from Zhurnal Struktumoi Khimii, Vol. 36, No. 4, pp. 615-622, July-August, 1995. Original article submitted December 20, 1993.

0022-4766/95/3604-0557512.50 �9 Plenum Publishing Corporation 557

G,mnarson-Lundqvist pseudopotential [13]. In this work, we employ an alternative effective two-quasiparticle polarization operator, which was obtained [11, 14] by calculating the Rayleigh--Schr6dinger polarization diagrams of the second-order perturbation theory in the Thomas-Fermi approximation. The calculations of the above-mentioned systems show that the accuracy of the results is sensitive to the pseudopotential used in the calculation scheme. In addition, it is shown that applying polarization corrections for the 2p-2h effects is of basic importance since they contribute up to 30% to the excitation energies and oscillator strengths. It should be noted that although the systems under consideration have been extensively studied (see, e.g., [1-3, 7, 9, 15-18]), some spectroscopic parameters of ethylene and carbon dioxide molecules have not been reliably establishedi some theoretical values of these parameters have not received experimental support (see below). Moreover, in some cases, different theoretical estimations give different values of the same parameters (see below).

CALCULATION PROCEDURE

According to [7], the operator Q~" that generates the excited state [2) of the molecule from its ground state

10), i.e., 12) =a~-10), is an exact solution of the equation of motion

(01 [6Q~, H, a~-] I 0) --- oJx(0 [[6Q2 , a~'] I 0), (1)

where o92 is the transition frequency, the amplitudes of Q+ are the matrix elements for the 10) ~ [2) transition, and

the double commutator is

2H, H ,B ] = [.4, [H,B]] + [H,B] ,B] . (2)

With the l p - l h (one particle-one hole) excitations incorporated in Eq. (1), the latter may be represented as the matrix equation for the {Ymr} and {Ziny} amplitudes

-B" z(~) I o~ z(,D ' (3)

where the matrix elements A, B, and D are defined as

A , ~ 6 = (01 [Cr,,~,H, C+,A 10),

Bmen6 = -(01 [Cm-/, H, Cn6] IO), (4)

Dmy,,a = (01[C., r, C~a] 10);

C + and C are the particle-hole creation and annihilation operators, respectively;, the subscripts m and n are for particles and 6 and y are for holes; H is the Hamiltonian of the molecule in the secondary quantization representation. Matrix elements (4) are estimated using the approximate wave function of the ground state [9]

10) ---- No(1 + U) IHF), (5)

where U = 1/2- E Cm;,n,3C,~eCr~ , IH~ is the Hartree-Fock function. Thus, the matrix elements .4, B, and D have the

form

Bm~6 = BOmTn6 + ( - 1 ) z " Smrn 3, (6)

Here matrices A ~ and B ~ are the standard RPA matrices [7-9]. The other terms of Eq. (6) are determined as follows:

558

plz

= {Vpq . p q + V pq. p C , pqv

1 = + (7> guy

V/jkz = (i(1)j(2) I r~-211k(1)l(2)>. (8)

In Eqs. (6)-(8), the subscripts p and q, as well as m and n, refer to the states of particles, while the subscripts/.t and

v denote the states of holes, as do y and 3. In (6), e defines the Hartree-Fock orbital energies; p(m~ 2) , (2) ann p~,~ are the

second-order corrections to the density matrix, which are quadratic functions of the correlation coefficients. With the correlation coefficients ignored, matrix elements (6) are reduced to the corresponding matrix elements of RPA [7]. In this approximation, the equations of motion for the {II} and {Z} amplitudes of l p - l h and for the corresponding excitation energies ~o are solved by standard methods of matrix algebra. However, in Eq. (1), the l p - l h approximation is not adequate to always provide a sufficient accuracy of spectroscopic calculations. In this case, a satisfactory accuracy is gained even with a small orbital basis set, due to correct consideration of the most important PI effects that are associated with the 2p-2h excitations. Although the most important low-lying excited states correspond to the l p - l h pairs and have the largest amplitudes in the complete expansion of Q~-, the 2p-2h components are also very

important since they can give shifts of ~o~ of several electron-volts. Physically the account of these components

corresponds to the consideration of the serf-consistent relaxation of hole orbitals during virtual excitations in the core of the ground-state configuration of a molecule. Account of the 2p-2h components in Q f is equivalent to

renormalization of matrices in Eq. (3), w~ch leads to their dependence on co and, according to [11, 12], is reduced to the appearance of the weight factor

a(r) = [1 - E 2(r)]-I (9)

in the matrix elements. With the quasiparticle DF approximation, E is defined using the Gunnarson-Lundqvist

exchange-correlation DF [13] as

~] (r) = -0.0834p1/3(r) - 0.0518pl/3(r) / [1 + 18.377p~(r)], (10)

where p is the electron density. For simplicity, a(r) may be replaced with a(0) according to the familiar procedure based on the RPA with exchange [13], which is well justified, e.g., in atomic photoeffect theory. In this case, there is no loss in the accuracy of calculations. Another method of taking into account the 2p-2h polarization interaction employs the effective two-particle polarization potential [11, 14], which is obtained by calculating the Rayleigh--SchrSdinger /1=~ (direct) and ~--J~J. (exchange) polarization diagrams of perturbation' theory in" the Thomas-Fermi approximation. The corresponding second-order correction to energy (for the direct polarization interaction) is of the form [11, 14]:

E(A) = f f arldr2pl(rl)V o, (rlr2)P2(r2), (11)

with the effective two-quasiparticle interaction

Vdol(rlr2) = X ( f dr'p(O)Y3(r') / Irl - r2l Ir' - rzl -

( f dr'p(O)l/3(r') / I rl - r' [ f dr' 'p(~ ') / ] r " - r 2 [ ) / (p(~

<p(0)l/3 > = f drp(0)l/3(r), (12)

where X is the numerical coefficient [14]. A similar approximate potential distribution was found for the exchange

559

polarization interaction E(B). However, as follows from atomic calculations [14], the corresponding correction E(B) is usually two orders of magnitude lower than the direct polarization correction and is thus negligible. Strictly speaking, this correction should be considered together with the first-order correction for electron gas irdaomogeneity [13]. In practice, potential (12) is added to the corresponding matrix elements of the pure Coulomb potential. The procedure for numerical calculations of the matrix elements of operators (12) was described elsewhere (see, e.g., [12, 14]). The {Ym-/} and {Zm~,} amplitudes define the transition moment M0; "

and the oscillator strength

MO; t = (2)1/2 E {I~m~,(2 ) "Mrrrr + Zm~,(2)"Mm-;,} mr

fox = (2/3)'G'w,t" IM0Xl 2.

(13)

(14)

Here G is the degeneracy factor and M 0,1 is the particle-hole matrix element of M. With the exception of the account

of the 2p-2h effects, the present calculation scheme is completely similar to the standard EOM method [7]. This work is concerned with comparison of the results of taking into account the 2p-2h effects in terms of RPA and quasiparticle DF [11, 12] to demonstrate the efficiency of the latter approach. It is shown that the calculation scheme for taking into account the particle-hole polarization effects is sensitive to the type of polarization potential.

~ S ~ T S

H e l i u m atom. The helium atom has been well studied and as a rule is used as a reference system in testing the validity of theoretical approaches. The energies of the excited states, transition oscillator strengths, and other spectroscopic characteristics of helium were calculated by many authors; in the literature, there are numerous theoretical and very accurate experimental data for this atom (see, e.g., [1, 17]). In this work, we restrict ourselves to considering only the information that is important for our calculations. We calculated the He atom in order to estimate the correctness of the DF approximation of the 2p-2h PI effects and to compare the results obtained with this approximation with the data obtained by other EOM techniques (in particular, RPA). Available experimental data and the corresponding calculation results are presented below. Our EOM calculations were performed in the [12s/8p] basis set of Gaussian orbitals (the basis set was constructed using the Huzinaga exponents; for details see [9]) supplemented by diffuse s functions and a polarization p function. Table 1 li~ts the experimental values of transition energies and oscillator strengths together with the corresponding values obtained in this work and in [9], where the 2p-2h effects were taken into account using RPA. The oscillator strengths calculated in terms of the multiconfiguration approximation (MC) are also given in Table 1. A comparison of our transition energies and oscillator strengths with the corresponding parameters obtained by other techniques shows that the effective method of calculating the 2p-2h PI effects in the DF approximation, which is proposed in this work, yields sufficiently accurate values. It should be noted that a similar accuracy was obtained by McKoy with coworkers [9], but the RPA employed by these authors is much more computationally expensive than the DF approximation. As expected, the account of the 2p-2h effects is very important, since it allows one to decrease the error in the EOM calculations of the energy of transitions to low-lying excited states from 1.5-2 eV in the l p - l h approximation to tenth fractions of an electron volt in the 2p-2h approximation. Correlation of the calculated and experimental values shows that the calculation error is below 1%. However, for the 21'3S and 21"~P terms, the PI effects seem to be underestimated, while those for the 31"3S and 3t'3P terms are overestimated. A comparison between the techniques of taking into account the 2p=2h effects using DF (10) and DF (12) shows that DF-(12) gives more accurate results and is theoretically more consistent and correct. For transitions to the 2,31P states, our values of f are similar in accuracy to those obtained in [9] and are consistent with the experimental values and with the result of the MC calculation, which is very accurate [1, 17].

1 ~2 11.2 o~2 o~,2 11.2 a~2 lh2 "lh2 Ethylene. The electronic configuration of ethylene in the ground state is ~.,,,3,,.,,~,,.o3u~z~_,,,lS.,,qS~lu. The

geometry of the ethylene molecule taken for the calculation corresponds to the conventional experimental geometry of ethylene in the ground state. Table 2 gives the energies and oscillator strengths of the N-T, N-V, and N-R'" transitions in the C2H 4 molecule obtained in this work and the corresponding values calculated in the l p - l h

560

TABLE 1. Energies AE (eV) and Oscillator Strengths f of Transitions to Low-Lying Excited States for the Helium Atom

State I b c(10) c(12). ] d

23S 2 1S 22.22

23p 2 1p 22.78

33s 3 1S 24.47

33? 3 1p 24.63

20.00

20.88

21.24

21.51 22.91 23.14

23.22

23.32

zXE

19.98

20.74

21.04

21.26 22.64

22.82

22.85

22.96

19.90

20.68

21.01

21.24

22.66

22.84

22.90

23.02

19.82

20.61

20.96

21.22 22.72

22.92

23.01

23.09

b I c(10) I

0.264

0.0~5

f

0.26

0.73

0.276

0.0734

Note: a and b - EOM calculations in the l p - l h and 2p-2h approximations, respectively, according to [9]; c(10) and c(12) - results obtained in this work using DF (10) and DF (12), respectively; d - experimental values; e - values obtained in the configuration interaction approximation.

TABLE 2. Vertical Excitation Energies 1327 (eV) and Oscillator Strengths f for the Ethylene Molecule

ZLE zkE z3,E zkE ZkEex p f f f Transition a b c(10) c(12) d b c(10) d

N - - V N - . R ' "

9.0 10.4

7.9 8.4

7.56 "8.94

7.49 8.36

7.6 0.4 8.05 0.02

0.33 0.016

0.34

Note: a and b - EOM calculations [7] in the l p - I h and 2p-2h approximations, respectively; c(10) and c(12) - this work; d - experimental values.

approximation of the EOM method [7]. Our calculations were performed using the valence basis set [4s2p/2s] of Gaussian orbitals (for details see [7]) complemented with diffuse functions. The N - R ' " transition is the first term of the series of N - n R " ' Rydberg transitions. The R ' " state is the Rydberg state of the same symmetry as V and appears during the n-nd.Tr z transition. Our value of the N - R ' " transition energy shows good agreement with the corresponding

experimental value. The N - R ' " transition energy calculated using the l p - l h approximation differs from the experimental value by 1.3 eV. For the N - V transition, calculation of the 2p-2h PI is of fundamental importance since it decreases the error for w from 18 to 1%. Our calculations have shown that it is important to involve a sufficient number of virtual valence orbitals and to take into account the a-z~ correlations in addition to the effect calculated by adding diffuse orbitals to the basis set. The N - V transition energy calculated in the MC approximation, 7.8-8.0 eV, is closer to the experimental value than the result of the l p - l h EOM calculation, but is less accurate than the values obtained in the 2p-2h approximation. For the N - T transition, the EOM calculation gives a transition energy of 4.32 eV, which is in good agreement with the experimental value, 4.6 eV. Our calculation of this energy in the 2p-2h approximation gives a value that is somewhat lower than the experim_ental one, which seems to be due to a certain overestimation of the 2p-2h effects for the given transition. The value o f f for the N - V transition obtained in this work is consistent with the experimental valuer = 034. The calculation shows that here the account of the 2p-2h polarization effects is very important. The most interesting result of the present work is the calculation of the oscillator strength f of the N - R ' " Rydberg transition, for which there is no reliable experimental value (possible values are estimated at 0.002-0.01). For the oscillator strength of thi~ transition, we recommend the value f = 0.016. Here, as for helium, more accurate results are obtained when the 2p-2h effects are taken into account through DF (12) rather than through DF (10).

Carbon dioxide. In this work, we calculated the energies of the singlet and triplet excited states of the CO 2

561

molecule and the oscillator strengths for some transitions. The molecular geometry was the conventional experimental geometry of CO 2. We used a contracted Ganssian basis set {[3s / 2p] + (Is / lp)} (for details see [5]). The zim.~ of the calculations were to investigate certain peculiarities of the spectrum of CO 2 as well as to estimate the correctness of taking into account the 2p-2h effects by the techniques based on the quasiparticle approximation of DF. The results of our calculations are compared with the corresponding data obtained with the RPA. The results of ab initio calculations that take into account the configuration interaction are not more accurate than those reported in [7] (except for the results of complete calculations, e.g., by Mtller-Plesset perturbation theory including the fourth-order effects [14]).

The energies of CO 2 states and the oscillator strengths for the transitions to the Ix+ and 1H u states, obtained in this

work and calculated by the standard EOM method with allowance made for the 2p-2h effects in the RPAI are given in Tables 3 and 4 together with the corresponding experimental values. For comparison, the tables give the corresponding state energies calculated in the l p - l h approximation disregarding the 2p-2h polarization effects. An analysis of these results suggests that taking into account the polarization effects is essential. Thus, for example, the excited state energies calculated in the l p - l h approximation are in error by 2 or more electronvolts (up to 40%). Taking into account the 2p-2h effects decreases the error to 2-3%; on the average, the results obtained using the DF approximation and RPA are of the same accuracy. The difference between the state energies calculated in the l p " lh approximation and the corresponding experimental energies is associated with the limited basis set used for this molecule. EOM calculations with consideration for the 2p-2h effects for other complex molecules, e.g., benzene and formaldehyde [12], show that if the energies calculated in the l p - l h approximation differ from the experimental values by 5-10%, subsequent correct account of the next-order effects, in particular of the 2p-2h effects, allows one to obtain the highly accurate state

energies despite the limited basis set. As seen from Table 3, our energies of the transitions to the lyg, liig ' 1Hu ' states

and 21~u are in good agreement with the experimental data. At the same time, the energies of the 1yg-llAu transition

differ significantly from the corresponding experimental values (in [9], the situation is opposite: for the former transitions, the agreement between calculated and experimental values is rather poor, while for the latter transition the calculated energy is close to the experimental one). It is seen that the approach under discussion describes the singlet states

better than the triplet ones. For the transition to the 1 + Yu state, the calculated oscillator strength shows good fit to the

experimental value, which appears to be due to the compensation of the errors that are associated, on the one hand, with the limited basis set and, on the other hand, with the overestimated contribution from the 2p-2h effects owing

to the simplified procedure for taking into account the a factor. For the transition to the lIIu state, the experimental

value of the oscillator strength is not available, and the theoretical value o f f obtained in this work may be considered

TABLE 3. Energies E (eV) of the Excited Singlet States of the CO 2 Molecule and the Corresponding Oscillator Strengths f

E E E E E f f State a b c(10) c(12) d b c(10)

Iz 7 -. %- I Z ; --~ llAu

1 + 1 + Z~'-" Yu

Ix;-~ 21~u ly.; _~ li-iu lz[-, %

11.63

11.98

11.87

13.87

13.91

14.20

15.56

16.09

15.42

8.53

8 .56

8.62

i0.29

10.42

10.97

12.60

12.63

12.78

9.22

9.50

9.41

10.99

11.03

11.26

12.34

12.76

12.22

8.74

8.82

9.26

i0.73

10.78

11.04

12.12

1238

12.09

8.41

9.31 11.08 0.116

(0.12)

11.4 0.168

0.124

0.173

Note: a and b -- EOM calculations [5] in the l p - l h and 2p-2h approximations, respectively; c(10) and c(12) -- thi.~ work; d - experimental values; the experimental values o f f are given in parentheses.

562

State

1S~; ~ 13Au

1Z; ---> 23Au 1Z; ~ 3IIu

1~; " 3Ag

TABLE 4. Energies (eV) of the Excited Triplet States of the CO 2 Molecule

b c(10) c(12)

7.94

8.95

9.16

8.98

11.03

11.28

11.10

12.53

12.12

7.72

7.80

8.84

7.82

7.35

8.06

8.39

8.40

10.43

10.99

11.33

12.42

12.67

10.85

10.96

10.94

12.37

12.00

d

7.5

8.0

Note: b -- results borrowed from [5]; c(10) and c(12) - this work; d - experimental values.

sufficiently accurate. The better agreement between the EOM calculations [7, 8, 12] of the excited state energies and the experimental values for the H2CO , C2H4, N2, and CO molecules is accounted for by the fact that the basis sets used for these molecules form a more adequate one-quasiparticle representation, which allows most correlation effects to be taken into account even in the l p - l h approximation. The present calculation has shown that the basis set used in [9] and in this work is not an optimal basis set for the CO 2 molecule. Desimaing an optimi7ed one-quasiparticle representation has been one of the key theoretical problems over many years (see, e.g., [2-4, 20]) and still remain~ to be solved. The calculation technique proposed in the present work can be efficiently employed in the design of optlmzl basis sets. An optimal basis set is the one that takes into account most p - h effects even in the lp-- lh approximation; the contribution of the 2p--2h effects should be insis A procedure for minimiTation of this contribution may be employed to optimize the corresponding basis set. In fact, such a procedure, developed in terms of quantum electrody~amlcs, was employed in [20] in order to construct optimal atomic basis sets for precise calculations of the spectroscopic characteristics of atoms and ions. It seems reasonable to extend this procedure to molecular systems.

.

2.

.

4. 5. 6. 7. 8. 9.

10.

11.

12. 13. 14.

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15. 16. 17. 18. 19. 20.

M. J. Hubin and J. E. Collins, Bull. Soc. R. Sci. Liege, 40, 361-370 (1971). V. A. Kondaurov and N. I. Filippov, Opt. Spektrosk., 74, No. 5, 846-850 (1993). L. A. Vainstein and V. I. Safronova, Phys. Scr., 31, 519 (1985). A. V. Glushkov, Opt. Spelarosk., 64, No. 2, 446-448 (1988). M. Ya. Amus'ya, Atomic Photoeffect [in Russian], Nauka, Moscow (1987). A. V. Glushkov~and L. N. Ivanov, Phys. Lett..4, 170, 34-37 (1992).

Translated by-I. Izvekova

564