Frequency-dependent hyperpolarizabilities in the coupled-cluster method: the Kerr effect for molecules

3 March 1995

CHEMICAL PHYSICS LETTERS

ELSEVIER

Chemical Physics Letters 234 (1995) 87-93

Frequency-dependent hyperpolarizabilities in the coupled-cluster method: the Kerr effect for molecules Hideo Sekino, Rodney J. Bartlett Quantum Theory Project, University of Florida, Gainesville, FL 32611-8435, USA

Received 12 September 1994; in final form 13 December1994

Abstract

Introducing frequency dependence in the equation-of-motion coupled-cluster method, we evaluate the optical Kerr effect for butadiene and ammonia. This permits a critical evaluation of dispersion estimates via the uncorrelated time-dependent Hartree-Fock theory. The percentage dispersions are similar for low frequencies, but not for larger values. We also consider other dispersion estimates based upon a power series expansion in the frequency. This leads to a possible resolution of the observed discrepancy between correlated theory and experiment for butadiene. Augmented by vibrational corrections, we offer estimates for the experimentally unknown Kerr values for NH 3 and Call 6.

1. I n t r o d u c t i o n

Recent developments in ab initio quantum chemistry have made the highly accurate evaluation of molecular hyperpolarizabilities and associated nonlinear optical (NLO) properties of molecules possible [1,2]. Many studies [2-7], mostly based upon static molecular hyperpolarizabilities, indicate that electron correlation is a critical factor in the quantitative prediction of hyperpolarizabilities, with the correlation effects amounting to as much as 100% of the measured values [1]. However, dispersion effects at the experimental frequency must also be taken into consideration in order to accurately describe the experimental hyperpolarizabilities. The multitude of different processes all degenerate in the static limit. To evaluate the dispersion effect, we formulated higher-order time-dependent Hartree-Fock theory (TDHF) and applied it for the evaluation of the

dispersion effects in several molecules [8-10]. Since then, the TDHF method has been applied to many different molecular systems [11-12]. Although TDHF, because of the absence of correlation effects, gives a much smaller magnitude for the molecular hyperpolarizabilities than experiment; the TDHF percentage dispersion correction used to scale the static values obtained by advanced correlated methods, such as coupled-cluster (CC) theory, generally reproduces experimental hyperpolarizabilities accurately [1,9,10]. However, the rigorous result would be offered by a correlated frequency-dependent method. There are not many ab initio studies on dynamic hyperpolarizabilities at correlated levels. One such approach is based upon the second-order M~llerPlesset or many-body perturbation theory (MBPT(2)) [13]; and the other upon multi-configurational linear response theory [14]. While the available calculations

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H. Sekino, R.J. Bartlett/Chemical Physics Letters 234 (1995) 87-93

generally tend to support the TDHF percentage dispersion correction [1], there still remains an open question whether the TDHF scaling correction can accurately reproduce the dispersion effects of more sophisticated correlated methods. Recently, the equation-of-motion coupled-cluster methods (EOM-CC), originally developed for the evaluation of excitation energies [15-17], has been formulated for other properties [18-20] including the dynamic polarizability [18]. We present, here, frequency-dependent EOM-CC results for the NLO dispersion effects for experiments based upon the optical Kerr effect (OKE). Augmented by estimates of the vibrational contribution, we are able to offer approximate values for the experimentally unknown optical Kerr effect (OKE) [2] values for NH s and t-butadiene. The latter prototype polyene, uniquely among those we have studied [1,10], significantly overestimates the experimental values for NLO processes when using the TDHF percentage dispersion correction to the correlated static hyperpolarizabilities. Analysis, assisted by a power series estimate of dispersion effects for low frequencies [21], attributes this overestimation to the TDHF dispersion estimate.

2. Computational method In the Kerr experiment, we observe the depolarization of the light and the corresponding Kerr hyperpolarizabilities defined as

~iix),

/3K = 3(3/3~x~-

(1)

with x being the principal axis and ,yx = ~o(3Tij/j _

Yiijj),

(2)

for i and j being coordinates x, y and z. Here, the Einstein summation convention is assumed. T K can be further approximated as

The frequency-dependent polarizability may be expressed, formally, by the sum over states (SOS) representation using the CC ground state laps)= e r 10), and its biorthogonal complement <'Psl (0 I(1 + A)e -r, and the corresponding EOM-CC excited states I ~ > =,-9~er 10), Q/r I = (01 ..~e - r as the intermediate states, where 10) is the Fermi vacuum. T and A are excitation and deexcitation operators, respectively, while ~9~ and ~ are the corresponding excitation and deexcitation operators that define the biorthogonal right and left eigenvectors for EOM excited states, a~g = 1, and .~g = 1 + A. Then, in a sum-over-state form, the polarizability tensor, Olij( -- OJ; to) ~

etc.

<~o I qi I~g> )

," (5)

-

%(

-

to;

~)

-- (01(1 + A ) ( ~ i - (qi>)Ro( + to)qj 10>

A)(Clj- )R(-to)FtilO), (6)

(3)

where we assume

1

Eg -4- to

Here to is the frequency of the applied electric field, Eg and E~ are the corresponding energies of the ground and excited states, and qi represents a component of the dipole vector. The hyperpolarizabilities for the Kerr effect •ijk ( - to; to, 0) and Tijkl(--to; to, 0, 0) are evaluated as the first and second derivatives of the frequency-dependent polarizability a i j ( - t o ; to) with respect to the electric dipole perturbations in the directions k and/or I. The derivatives are obtained by numerical differentiation using a finite value of the field strength of 0.0005 au for each direction. Though Eq. (5) is convenient conceptually and for interpretation, operationally, we evaluate Eq. (5) in the determinantal space I h>, (01 h> = 0 as

+ (0[(1 +

~lxyyx ~- ~yxxy -~- ~ ( ~xxyy -]- "~yyxx)"

E~ -

+

at- "Yyyyy q- "Yzzzz -st- "Yxxyy "~- Yyyxx TK = "5('Yxxxx i

+ Tzzxx + Txxzz + Vrrzz + "Yzzry),

~_.,[ (~/zg [ qi II[re>(~/re !qjlfflg>

(4)

with

Ro= Ih)(h](Eccd=to-H)lh)-l(h[ and qi~-e-Tqe T and (qi> = (atrq Iqlatrq)

(7)

-- <01(1 + A)~ 10> is the generalized expectation value. Us-

H. Sekino, R.J. Bartlett/Chemical Physics Letters 234 (1995) 87-93 ing the resolvent in configuration space permits evaluation without truncation via solving the linear equation for T(~ ) in closed form, (hlEcc _ to-HI

h ) ( h l T ~ ) 10) = ( h i q l 0 ) ,

(8) where H = e - r H e r is the effective Hamiltonian generated by the coupled-cluster similarity transformation, Ecc is the CC ground state energy, and q is the dipole vector. In the E O M - C C S D model, we truncate the excitation manifold to singles and doubles. We restrict our evaluation of Eq. (6) to just linked diagrams, which ensures its extensivity [19,20]. The calculated polarizabilities by the original E O M - C C scheme [18] are identical to that obtained by the extensive version of the EOM-CC [19,20], within the precision we report here. 2.1. N H 3

The experimental geometry is used for the calculation, RN_ H = 1.012 A and 0H_N_ n = 106.7 °. From the molecular symmetry C3~, we have the Kerr hyperpolarizabilities /3 K = 3( /3xxx -- /3yyx -~- 2 /3xyy )

(9)

and 3'K = l ( 3"xxx x + 2( 3"yyyy -t- 3"xxyy + 3"yyxx)

+ 3"z~yy + 3"yyz~)-

(10)

We further assumed 3"zzyy = 3"yyzz" The basis set used is the polarizability consistent basis (POL) [5s3p2d/3s2pld] of Sadlej [22] augmented by a set of diffuse d functions (~d = 0.1) on hydrogen. 2.2. c 4 n 6

The geometry is optimized at the Hartree-Fock level with C:h symmetry using a 6-31G basis [23]. The 6-31G * * [23] optimized geometry shows little difference [24] from this. Here, we use the [ 3 s 3 p l d / 2 s ] 6-31G + PD basis set [24,25] which is optimized to well reproduce the averaged hyperpolarizability and the longitudinal component 3,xxxx at the Hartree-Fock level [24]. For checking the validity of Eq. (4), we also calculate all components using TDHF. The extra components ~ t y y x -~- 3343, 3,xzzx =

89

10583 and Yrzzy = 3784 at the frequency 0.0656 are obtained and used for evaluation of 3,n. The calculated Kerr hyperpolarizability 3,K, 17274 by Eq. (3), is almost identical to the value 17277 obtained using a complete set of the components. All the reported numbers in the tables are evaluated using Eq. (3).

3. Results and discussion The Kerr hyperpolarizabilities of the NH 3 molecule at different frequencies are summarized in Table 1 together with the static hyperpolarizabilities. The static hyperpolarizabilities /3 K = _ 14.7 and 3' r = 2400 for Hartree-Fock are close to the values, - 1 5 . 1 and 2400 where a larger basis set was used [1]. EOM-CC hyperpolarizabilities, 13 n = _ 35.9 and y K = 4136, are somewhat larger in amplitude than the MBPT(2) hyperpolarizabilities, /3 K = _ 32.5 and 3, r = 4100; and CCSD hyperpolarizabilities, /3 K = --30.0 and 3,K= 3800 [1]. The prior CCSD(T) hyperpolarizabilities, fl K = _ 34.3 and 3,K = 4200 [1], are quite close to the present EOM-CC results. Rice and Handy [12] reported MBPT(2) dispersion effects on /3 using a larger basis set [5s4p2d/3s3p] + ( l s l p l d / l s l p ) . Their TDHF numbers for/3 K, 14.32, 16.16 and 19.3 t for to = 0, 0.0656 and 0.1, respectively, are very similar to our results; while their MBPT(2) numbers for /3 K, _ 29.7, -- 33.3 (12.0%) and --39.4 (32.7%), are a bit smaller in magnitude than the present EOM-CC values. We ascribe the difference to mostly higher correlation corrections, although other slight differences occur due to basis, and to the fact that the energy derivative and the SOS expression are not precisely equivalent, except when the H e l l m a n - F e y n m a n n is satisfied, which is not the case here [19]. For 3, there is no other reported dispersion estimate. All the components of the dynamic hyperpolarizabilities calculated at the correlated level are much larger than those calculated by TDHF. However, the T D H F percentage corrections between the two methods (shown in parentheses) are not so different for the ruby laser frequency (to--0.0656), further sup-

1 The number is obtained from interpolation of the numbers for to = 0.06 and 0.07.

H. Sekino, R.J. Bartlett/Chemical Physics Letters 234 (1995) 87-93

90

porting the percentage TDHF dispersion estimate for the lower frequency region [1,8,10]. As there is no Kerr experiment for NH 3, we can estimate a value. The vibrational hyperpolarizabilities at the MBPT(2) level are evaluated to be 4.76 and 135 for /3 K and yr: respectively at the frequency 0.07 [26]. Adding by that amount, we would suggest a value of /3 K= - 3 6 and y K= 4800 at to = 0.0656. Once we know the dispersion effect of a certain NLO process, we can estimate the dispersion effects of other NLO processes using the dispersion relation derived for hyperpolarizabilities at low frequencies by Bishop [21]. For the first hyperpolarizability, we can use the simplified dispersion formula

We can easily see that for OKE and for DC-induced second harmonic generation (DC-SHG),

/3(-to,,, ,o 1, to~)

after setting B ' = 0 provides the simple numerical estimates for the various processes. The OKE, intensity-dependent refractive index (IDRI), DC-SHG and third harmonic generation (THG) correspond to 2A'to 2, 4A'to 2, 6A'to 2 and 12A'to 2, respectively. Using this ratio, the DC-SHG CCSD(T) hyperpolarizabilities for NH 3 can be evaluated and are sum-

=/3(0; 0, 0)(1 + A t o Z + B t o 4 +

/ 3 ( - t o ; to, 0) =/3(0; 0, 0)(1 + 2 a t o 2 ) , f l ( - 2 t o ; to, to) =/3(0; 0, 0)(1 + 6 a t o 2 ) .

The percentage dispersion corrections of the processes is given by the ratio. Similarly, the power series expression [21] for the second hyperpolarizability

~/ll(-to~, toa, to2, '°3) = y(O; 000) (1 + a'to~ + B'to 4a + ... ), to 2

...)

--~/3(0; O, 0)(1 +Ato2),

(11)

where

= to: + ,o12 + tog

(12)

=

too,2 +

to~ + o)22+ to2,

Table 1 K e r r effect tensors for N H 3 calculated at different frequencies (in au) ~ Frequency (au/nm)

0

0.0656/694.3

0.1/455.6

flxxx(- to; to, 0)

-41.0 - 10.1 - 8.6 - 7.2 - 9.1 - 7.2 7.2 8.3 -- 35.9 -- 14.7 8100 4500 1800 1200 1900 1100 1900 1100 600 400 4136.63 2404.99

-47.6(16.3%) - 11.8(16.8%) - 9.4(9.6%) - 7.8(7.6%) - 10.3(12.8%) - 7.9(9.1%) 7.6(5.3%) 8.7(4.5%) -- 4 1 . 5 ( 1 5 . 7 % ) -- 16.6(12.7%) 9300(15.3%) 5000(11.3%) 2000(7.2%) 1300(6.0%) 2100(9.0%) 1200(11.1%) 2400(23.4%) 1300(16.3%) 600(6.1%) 400(5.4%) 4703.00(13.7%) 2646.36(10.0%)

-59.5(45.2%) - 14.7(45.1%) - 10.7(24.5%) - 8.6(19.1%) - 12.3(35.0%) - 8.9(23.6%) 8,2(13.3%) 9.3(11.0%) -- 5 1 . 5 ( 4 3 . 5 % ) -- 19.7(33.8%) 11500(42.3%) 5800(29.7%) 2100(18.1%) 1400(15.0%) 2400(22.9%) 1400(29.0%) 3300(70.2%) 1600(45.3%) 700(15.3%) 500(13.4%) 5711.56(38.1%) 3039.95(26.4%)

/3yyx( - to; to, 0) /3xyy( - to; to, 0) / 3 r y y ( - to; to, 0) /3 K

Yxxxx(--to; to, 0, 0)

~/yyyy(-- to;

to, 0, 0)

Y y y x x ( - t o ; to, 0, 0) Y x x y r ( - to; to, 0, 0)

~yyzz( yK

to; to, 0, 0)

(13)

a The n u m b e r s in u p p e r and l o w e r r o w s are evaluated by E O M - C C and T D H F respectively, x is the C 3 m o l e c u l a r axis.

(14)

H. Sekino, RJ. Bartlett/Chemical Physics Letters 234 (1995) 87-93

marized in Table 2. The similarity between the two results suggests that for this example the two dispersion estimates are consistent. The vibrational hyperpolarizabilities for the processes are almost ignorable (0.% and - 1 % f o r / 3 and T, respectively [26]). Table 3 shows the Kerr hyperpolarizabilities of trans-butadiene at different frequencies. Again, the correlated hyperpolarizabilities are much larger than the T D H F hyperpolarizabilities, but, here, the EOM-CCSD percentage correction compared to that for T D H F is different. The overenhancement of the percentage T D H F dispersion is especially apparent in the longitudinal component Txxxx rather than in y K. If real, this overenhancement partially explains our earlier observation that an estimate from the static correlated values using MBPT(2) together with the T D H F percentage correction significantly overestimates the hyperpolarizability of the molecule, which is unusual among the molecules we have studied [1,10]. The lowest dipole-allowed excited state 1B~ is 5.93 eV (0.22 au) above the ground state. The transition to this state enhances the xx component of the polarizability and therefore the xxxx component of the Kerr tensor. Although the frequency of the dynamic field is much lower than the excitation energy, T D H F theory, which underestimates the excitation energy to the lowest states for such conjugated systems, tends to overestimate the dispersion even at the frequency of interest. The typical overestimate of the dispersion in the T D H F theory must be particularly enhanced in the xxxx component rather than in the average values. Table 2 DC-SHG of NH3 by different dispersionestimates (in au) a TDHF CCSD(T)+ % TDHF dispersionb CCSD(T) + % EOM-CCSD dispersion ¢ experiment a

- 22.0 - 49.1

3276 5600

- 50.5

5900

- 48.9 -I-1.2

6147 + 110

a All values are at frequency to = 0.0656. These TDHF and CCSIXT) results use basis 3 in Ref. [1]. b Static CCSD(T) value scaled by % TDHF dispersion. c Value evaluated by EOM-CCSD Kerr effect number assuming the dispersionratio among the differentNLO processes mentioned in the text. d Refs. [2,27]

91

Table 3 Kerr effect tensors of C4H 6 calculatedat differentfrequencies(in au) a Frequency (au/nm) 3'xx~,x(- to; to, 0, 0)

0

41200 23514 yyyyy(-- to; to, 0, 0) 10000 6860 Yzz~z(-- to; to, 0, 0) 14900 13292 ")lxxyy(-- to; to, 0, 0) 4 9 0 0 2965 7yyxx(-- to; to, 0, 0) 5100 2965 "Y~z~x(-- to; to' 0, 0) 9 8 0 0 8837 %,x~z(- to; to, O, O) 9 6 0 0 8837 Yyyzz( - to; to, O, O) 4 1 0 0 3396 7zzyy(- to; to, O, O) 4 1 0 0 3396 7K 20700 14812

0.043/1060

0.0656/694.3

44000(6.9%) 25571(8.7%) 10400(3.6%) 7088(3.3%) 15600(4.5%) 13916(4.7%) 5200(5.4%) 3172(7.0%) 5300(3.2%) 3045(2.7%) 10400(5.9%) 9394(6.3%) 10400(7.9%) 9704(9.8%) 4300(3.5%) 3518(3.6%) 4300(4.6%) 3560(4.8%) 21900(5.7%) 15794(6.6%)

48100(16.8%) 28733(22.2%) 10900(8.5%) 7408(8.0%) 16500(11.1%) 14821(11.5%) 5500(12.6%) 3489(17.7%) 5500(7.9%) 3157(6.5%) 11200(14.7%) 10234(15.8%) 11600(20.0%) 11058(25.1%) 4500(8.4%) 3693(8.8%) 4600(11.4%) 3803(12.0%) 23700(14.1%) 17277(16.7%)

a The numbersin upper and lower rows are evaluatedby EOM-CC and TDHF, respectively. The x component corresponds to the longitudinalmolecularaxis and the z componentis perpendicular to the molecularplane.

The Kerr effect vibrational hyperpolarizabilities at the Hartree-Fock level are estimated to be 1 6 5 6 / 1285 and 1 7 6 2 / 1 3 9 5 at frequencies to = 0.043 and 0.0656 [28]. Therefore we conclude that the EOM-CC Kerr electronic contribution to the hyperpolarizability of 23700 at to -- 0.0656 would become = 25 000 if the experiment were performed. To further analyze this result, consider that the experimentally observed DC-SHG values are 20 180 -4-110 and 27700 _+ 1600 at the frequencies shown [2,29]. Using the EOM-CC evaluation of the Kerr effect dispersion to relate the other processes, the DC-SHG hyperpolarizabilities would be 24 300 and 29 700 at the frequencies of interest. The DC-SHG vibrational hyperpolarizabilities at the Hartree-Fock are estimated to be (scaled/unscaled) - 7 3 4 / 630 and - 6 1 5 / - 504, respectively [26]. While the DCSHG hyperpolarizability including the vibrational correction, 23 600 at to = 0.043, is a little bit larger than the experiment, the DC-SHG value plus the vibrational correction of 29100 at to = 0.0656 falls within the error bars of the experimental value.

92

H. Sekino, R~I. Bartlett / Chemical Physics Letters 234 (1995) 87-93

Table 4 DC-SHG of C4H 6 calculated at different frequencies (in au) Frequency (au/nm) 0.043/1060 0.0656/694.3 TDHF CCSD + % TDHF dispersion a EOM-CCSD b experiment c

18121 25300 24300 20180 + 110

24932 34800 29700 27700+ 1600

a Static CCSD value scaled by % TDHF dispersion correction. b Value evaluated by EOM-CCSD Kerr effect number with the dispersion ratio among the different NLO processes mentioned in the text. c Refs. [2,29].

In Table 4, we present the D C - S H G hyperpolarizabilities evaluated by different methods. The T D H F numbers are unusually close to the experimental values, yet the static T D H F (CPHF) is quite different from the correlated calculations. The correlated static hyperpolarizabilities scaled by the T D H F dispersion effect would seem to overestimate for this example. However, dispersion based upon the E O M - C C S D O K E values evaluated as above, tend to be closer to the experimental values. This, of course, might be purely fortuitous due to mutual cancellation of errors. A better fit would be offered by determining B by O K E evaluation for a molecule at several values o f to, and then assume the same A' and B' values for the other N L O processes [30]. A s s u m i n g the reliability of the single parameter ratio, the above observation offers an explanation for the failure o f the T D H F percentage dispersion correction for t-butadiene [1,8-10]. It will clearly fail when T D H F ( = RPA) must fail for excitation energies. In particular, it can be expected that T D H F dispersion estimates will be less appropriate for the longer polyenes where the dipole allowed states lie at even lower energies. In this work, we calculate the hyperpolarizabilities via a finite-field derivative o f the frequency-dependent polarizabilities. This is a similar idea to that used to evaluate hyperpolarizabilities from finite field derivatives of the induced dipole evaluated via the relaxed density [10]. Since E O M - C C as used here does not provide the complete second-derivative energy response of the system, neglecting the orbital and a part o f the CC amplitude relaxation [19], the static hyperpolarizabilities will be slightly different from that obtained from the finite-field method based

on the total energy (or quasi-energy [13] in the frequency-dependent case). However, this has so far been found to have only a small numerical effect. Also, E O M - C C S D accounts for most of the orbital relaxation effects through T 1 in the exponential Ansatz [31]. For other N L O processes which involve two photon absorptions, there are other states in addition to the lower dipole-allowed states which can enhance the nonlinearity. Therefore, it is critical to use a theory which can describe dipole forbidden states as well. The E O M - C C S D model does so, but in the absence of substantial comparison to experiment, its reliability is not yet established.

Acknowledgement The authors are greatly indebted to Professor B. Kirtman for providing his vibrational hyperpolarizability numbers before publication. We also acknowledge helpful discussions with Professor Kirtman and Professor H. Kurtz. This work has been supported by the US A i r Force Office of Scientific Research (Grant No. AFOSR-F49620-93-1-0118).

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[24] G.J.B. Hurst, M. Dupuis and E. Clementi, J. Chem. Phys. 89, 385 (1988). [25] H.O. Villar, M. Dupuis, J.D. Watts, G.J.B. Hurst and E. Clementi, IBM Research Report, Jan. 28, 1987. [26] D.M. Bishop, B. Kirtman, H.A. Kurtz and J.E. Rice, J. Chem. Phys. 98, 8024 (1993). [27] J.F. Ward and C.K. Miller, Phys. Rev. A19, 826 (1979). [28] B. Kirtman, private communication: A / B ; A: scaled vibrational frequencies, B: unscaled vibrational frequencies. [29] J.F. Ward and D.S. Elliott, J. Chem. Phys. 69, 5438 (1978). [30] H. Kurtz, private communication. [31] E.A. Salter, H. Sekino and R.J. Bartlett, J. Chem. Phys. 87, 502 (1987),