The hyperpolarizability of molecular oxygen

THEO CHEM Journal of Molecular Structure (Theochem) 336 (1995) 61-67

The hyperpolarizability

______

of molecular oxygen

Yi Luoa, Hans Agrena,*, Boris Minaeva, Poul Jm-gensenb aInstitute of Physics and Measurement Technology, University of Linkcping, S-58183 Linkiiping, Sweden bDepartment of Chemistry, Aarhus University, DK-8000 Aarhus C, Denmark

Received 10 October 1994; accepted 14 November 1994

Abstract The static- and frequency-dependent dipole second hyperpolarizabilities y of O2 are calculated by means of quadratic response theory and multi-configuration self-consistent field wavefunctions. It is demonstrated that the O2 molecule presents some unusual and difficult features in the description of its hyperpolarizability, which can be derived from the extreme role of electron correlation.

1. Introduction

Due to its obvious role in atmospheric chemistry an accurate determination of the properties of molecular oxygen is considered an important goal for quantum molecular simulations. Because of its open shell nature, the organization of its lowest excited states, and the unusual role played by electron correlation, with competing valence and dynamic contributions, some of these properties have been notoriously elusive to describe. The hyperpolarizability is a good example of this contention: despite recent developments in computational methods addressing this non-linear property it has to date not been theoretically predicted for molecular oxygen. The general experience from many applications on small molecules is that even if electron correlation is important for the description of the hyperpolarizability, it can often be recovered by fairly modest efforts in terms of perturbation expansions or in * Corresponding author.

terms of the size of the correlating (self-consistent field) wavefunctions. More precisely, small active orbital spaces in which one correlating orbital is added for each occupied orbital give in general hyperpolarizabilities in excellent agreement with experiment [ 11. For molecular oxygen this does not hold; openshell Hartree-Fock gives a hyperpolarizability that is more than two orders of magnitude smaller than that experimentally determined and that even has the wrong sign. From this extremely poor Hartree-Fock description it can be understood that the popular perturbationally oriented approaches have large problems in addressing the hyperpolarizability of OZ. However, approaches based on multi-configurational self-consistent field (MCSCF) wavefunctions that incorporate the important correlation contributions in the reference wavefunctions and then successively add dynamic correlation by increasing the correlating orbital space, would conceivably provide a better strategy. Response theory applied to MCSCF wavefunctions can also address the frequency

0166-1280/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0166-1280(94)04095-8

62

Y. Luo et al./Journal

of Molecular

Structure

dispersion and results can therefore be compared directly with the experiments that are always carried out at finite frequencies. In the present work we explore how far this approach can reach for the difficult second hyperpolarizability of molecular oxygen.

2. Method and computational

details

The equations for MCSCF response theory were derived in Ref. [2] and have been implemented for the linear response (MCLR) [3] and for quadratic response (MCQR) using singlet [4] and triplet operators [5]. For the operators A, B and C the quadratic response function is given as

(Theochem)

336 (1995) 61-67

linear sets of equations N”(w, + w2) = {[E’*’ - (w, + w~)S[*‘]-~A’~‘+}+ x Nb(W,) = (E PI _ wiSP1)-‘B[‘l x NC(WT)= (##?‘*I - w2s’*‘)-‘c’il (2) Ei3] and S13’ in Eq. (1) are generalizations of the Hessian Ei3] and metric SL3]matrices, and C[*] and B[*] are generalizations of the property matrix B[” entering the linear response equations. We refer to the original literature for the exact definitions of the above matrices. If A, B and C are equal to components of the dipole operator, the quadratic response function in Eq. (1) gives the first hyperpolarizability. Using the notation of Eq. (1) the xy.~ component of the hyperpolarizability becomes

((A; B>C)),,,w? &&,;

= Nq(wi + u2)Bjl PIN;(q) PI h + Ny(~i + Wz)Cj, NI (WI) + N;(w,)(A;

+ A;)N;(w*)

- N&

+ ~2)

x ($IJm + Ej31 - wisj.;i - w*s,j9 Jm’ x N:h)N;(w2),

(1)

where N”, Nb and NC are solution vectors to three Table 1 SCF static polarizability

cct cct-ds cct-dsp cct-d2sp cct-pl cct-p2 cct-p3 cct-ds-p 1 cct-ds-p2 cct-ds-p3 POL POL-p 1 POL-pl-f

60 62 68 76 92 124 156 94 126 158 90 122 138

WI, w2)

=

+I

x; Y, z ~w,.w,

where w, = -wl - w2. The second hyperpolarizability measured in the electric-field-induced second harmonic generation (ESHG) experiments is obtained by numerical differentiation of the first hyperpolarizability for wl = w2 = w using the finite field method. Molecular oxygen has an inversion centre and the first hyperpolarizability p is therefore zero. Comparison with experimental ESHG data is performed in terms of the parallel component of the

and hyperpolarizability

4.5994 4.7606 4.7603 4.7676 7.3876 7.5018 7.5041 7.3889 7.5024 7.5050 7.4756 7.4792 7.4873

a See the explanation in the text. b The number of contracted basis functions

19.097 19.144 19.136 19.139 20.669 20.788 20.783 20.675 20.794 20.788 20.718 20.720 20.780

-3779.1 -3819.5 -3806.5 -3807.0 -3863.4 -3598.8 -3613.9 -3871.5 -3605.4 -3620.5 -3548.5 -3535.8 -3643.2

(3)

46.152 50.681 5 1.068 5 1.496 417.96 580.48 582.05 417.71 580.64 582.19 558.78 580.05 584.22

1.1274 3.6701 3.8289 4.0386 224.82 409.96 430.02 224.72 409.91 429.92 393.02 426.42 424.92

Y. Luo et al.lJournal of Molecular Structure (Theochem)

y tensor which may be defined as y(-2w; w, w, 0)

+ 2Y,,zx + ~%xxJ / 15

(4)

In the static limit, Eq. (4) reduces to Y@ o,o, 0) = (3X,,, + 12YXXZZ + 8%X,X)/15

(5)

Eq. (5) is often used also in the non-static limit in which the case so-called Kleinman symmetry is assumed. The present calculations explore the effects of basis sets, correlating active spaces, dispersion and vibrational averaging. The basis set investigation is designed from previous experience on small molecules, especially N2 [6] and Hz0 [7], and from a previous MCLR calculations on the frequency dependent polarizabilities of Oz. The basis sets used are described in the next section and their details are given in footnotes to Table 1. Three types of wavefunctions for the reference ground state of 3C, molecular oxygen have been employed. The first is the open-shell Hartree-Fock wavefunction, the second is a complete active space wavefunction containing all 2p orbitals with eight active electrons (2p-CAS). The third is a restricted active space (RAS) wavefunction consisting of the 2s-shell cT-orbitals in RASl, the 2p-shell cr- and 7r-orbitals in RAS2, and the 3-shell (a, 7r, and S) orbitals with the 30, excluded in RAS3, with a total of 12 active electrons. All excitations with a maximum of two holes in RASl and two electrons in RAS3 are considered. The RAS calculation thus includes part of the dynamic correlation not considered by the 2p-CAS-type wavefunction. Some properties extracted from of these wavefunctions are displayed in Ref. [8]. For instance, the RAS energy was close to the estimated full-C1 limit of Alrichs et al. [9] and to the more recent value of McLean et al. [lo]. Also the quadropole and hexadecapole moments were in good agreement either with experiment or with previous CI-type calculations. Of particular relevance for the present investigation is that the obtained static and dynamic first polarizabilities only deviated by a few % from those experimentally measured, and that 2p-CAS is a sufficient level of theory for

336 (1995) 61-67

63

this property. The 2p-CAS and RAS wavefunctions contain 32 and 53888 determinants, a further enlargement of the RAS wavefunction by the 4th main shell of orbitals, is prohibitive by any computational standard. The results presented in this work have been obtained with the SIRIUS code for MCSCF [11,12] wavefunctions, MCLR [3] and MCQR [4,5] response functions.

3. Results 3.1. Basis set dependence It is well known that the basis set plays an important role for determining reliable hyperpolarizabilities. In the present study, two series of basis sets are used to investigate the basis set dependence of the polarizability and the hyperpolarizability. Firstly, we have chosen the correlation consistent VTZ basis set of Dunning (cct) [13] generally contracted as 0(4s3p2dlf), as the starting point to evaluate different basis sets. By decontraction of one set of diffuse s functions, one set of diffuse s and p functions, and two sets of diffuse s and p functions, we have obtained the basis sets cct-ds, cct-dsp and cct-d2sp, respectively. The basis sets cct-pl, cct-p2 and cct-p3 are obtained by adding one, two and three sets of diffuse s, p, d and f functions (with a scale factor of 3.0), respectively. Furthermore, the combination of adding and decontraction procedures is also employed here to construct the cct-ds-pl, cct-ds-p2 and cct-ds-p3 basis sets. Our previous calculations for the hyperpolarizability of Hz0 [7], used the polarizability consistent (POL) basis set of Diercksen et al. [ 141in which the functions contracted as O(8s5p3dlf) gave a very good description for this quantity. It is important that selected basis sets will give results that are uniformly accurate from one molecule to the next; therefore, the ability of this basis set is examined here. Based on the POL basis set, we have also constructed two more basis sets, firstly by adding one set of diffuse s, p, d and f functions (as we did for the cct basis set) giving the POLpl basis set, and, secondly, by adding one diffuse f function to POL-pl giving the POLpl -f basis set.

Y. Luo et aLlJournal of Molecular Structure (Theochem) 336 (1995) 61-67

64 Table 2 Correlated BAS

static polarizability

and hyperpolarizability

orx

a::

4.620 4.773 4.778 4.773 7.244 7.336 7.338 7.245 7.337 7.339 7.311 7.314 7.320

13.157 13.171 13.172 13.174 13.438 14.438 14.430 14.336 14.440 14.432 14.372 14.384 14.396

7.832

14.978

cy

Y;:;r

?xrr

^ixw

Y

7.466 7.572 7.576 7.573 9.309 9.703 9.702 9.609 9.705 9.703 9.665 9.670 9.679

145.69 149.67 149.98 147.23 534.67 719.43 717.12 534.89 719.59 717.39 704.11 722.44 712.44

12.526 14.406 14.578 14.191 188.06 263.74 265.97 187.84 263.86 266.08 254.78 264.45 266.01

0.2853 3.0605 3.2735 2.9020 207.91 367.22 384.38 207.8 1 367.17 384.29 356.29 381.74 380.67

39.309 43.091 43.404 42.347 368.29 550.73 561.20 368.08 550.83 561.30 534.67 559.64 558.32

10.214

720.80

340.45

483.63

674.46

CAS-2p cct cct-dsp cct-d2sp cct-ds cct-pl cct-p2 cct-p3 cct-ds-pl cct-ds-p2 cct-ds-p3 POL POL-p 1 POL-pl-fl

RAS POL

The SCF static polarizabilities and hyperpolarizabilities obtained from different basis sets are shown in Table 1. For the cct series the basis set dependence is quite evident. The cct basis set gives too small values for the xx component of the polarizability and yxxx and zzxx components of the hyperpolarizability. The decontraction procedure does not improve the quality of the basis set. By adding one set of diffuse s, p, d and f functions, significant changes have been found for the polarizability and hyperpolarizability, especially for the components perpendicular to the molecular axis. The differences between cct-pl and cct-p2 basis sets are quite large; however, very small changes were found between the ycct-p2 and cct-p3 basis sets. The combination of decontraction and addition of diffuse functions has a negligible effect. For the cct basis set, one has to add at least two sets of diffuse s, p, d and f functions in order to describe the polarizability and the hyperpolarizability properly. It was found for N2 [6] that the f functions are important to obtain an accurate xxxx component of the hyperpolarizability y. The strong effect off functions can be traced to the fact that the @ orbitals can be reached in two successive dipole allowed transitions where lI orbitals are occupied in the reference state. The g functions were found to be unimportant for both the polarizability and hyperpolarizability [6].

The POL basis set is of higher quality compared with the cct type basis set with the same size. The effect of adding one set of s, p, d and f diffuse functions is noticeable, but not significant. The POL-pl basis set has the same quality as the cct-p2 basis set. Again, very small changes are found between the POL-pl and POLpl-f basis sets. The cct-p3 and POL-p2 basis sets seem almost to have reproduced the Hartree-Fock limit values. 3.2. Correlation dependence The correlation effects on static polarizability and hyperpolarizability obtained at the CAS-2p level with different basis sets are shown in Table 2. The correlation contributes significantly to the zz component of the polarizability. It decreases the SCF value by about 70%. Very small contributions are found for the xx component of y polarizability, while the value of CY, is hardly changed. There is a very significant contribution to the hyperpolarizability from electron correlation, it even changes the sign of the zzzz component. The values of the two components zzxx and xxxx are decreased by correlation. The basis set dependence has been found to be quite similar to that at the SCF level. It thus turns out that the calculations at the Hartree-Fock level give a completely erroneous description of the hyperpolarizability

Y. Luo et al./Journal Table 3 Calculated

0.06563 0.07200 0.07700 0.09300

values of y( -2~;

835.16 867.71 898.76 1027.86

of Molecular Structure

w,w, 0) for O2 as a function

302.78 314.56 325.75 371.79

292.51 301.36 309.63 342.19

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336 (1995)

of frequency

w for basis set POL at the CAS-2p

393.89 402.33 410.10 439.54

293.43 302.33 310.58 342.42

of 02; it is two orders of magnitude too small and has the wrong direction. The correlation effect has been further studied using a RAS space with the POL basis set. As we can see from the SCF and CAS-2p calculations, the POL basis set is of quite high quality, despite its moderate size. It reproduces 96% of the hyperpolarizability value obtained from the basis sets cct-p2 and pol-pl which is very close to the Hartree-Fock limit. The static polarizabilities and hyperpolarizabilities from the RAS calculation are also shown in Table 2. It turns out that the zzzz component of y changes only minimally on going from CAS-2p to RAS. However, the results from RAS for the zzxx and xxxx components are increased by about 35% compared with CAS-2p. The averaged hyperpolarizability, “i,-,increases by about 26% from CAS-2p to RAS calculations. The polarizability of O2 has been calculated by Hettema et al. [8] who used the corresponding linear response method and applied the same wavefunctions as in the present work, although with different basis sets. They obtained a very good agreement with the experimental measurement already at the 2p-CAS level. Our best values for the polarizabilities, obtained in the RAS calculation, are a: = 10.21 and Ao = 7.146, which are in good agreement with the best values o = 10.22 and Ao = 7.157 of Hettema et al. [S], and with the experimental values cx = 10.78 and Ao = 7.43 at 632.8 nm by Bridge and Buckingham [15]. 3.3. Frequency dependence The dispersion of the hyperpolarizability y has been investigated in some detail at the equilibrium internuclear distance using the CAS-2p wavefunction. The selected frequency values correspond to the wavelengths 694.3, 632.8 and 590.0 nm at

309.81 323.66 336.94 392.83

65

61-67

level (in Da)

616.40 635.96 654.37 728.45

which the experimental measurements were conducted [16]. The frequency dependent hyperpolarizabilities are listed in Table 3. They show a normal dispersion feature. The dispersion percentages goes from 15 to 36% for the frequencies covered. It is found that the dispersion for the various components is quite different. For instance, at the wavelength 590.0 nm (w = 0.093 Da), the dispersion percentages range between 46% and 23% for the zzzz and xxxx components, respectively. It confirms that Kleinman symmetry can only be applied at frequencies far below the resonance. The difference between the zzxx and zxxz components becomes appreciable as w increases. 3.4. Vibrational averaging Vibrational averaging for the hyperpolarizability in the w = 0 state was carried out for the frequencies 0.0 and 0.0656 Da. Calculations were performed at five internuclear distances using the 2p-CAS potential function. The vibrationally averaged hyperpolarizabilities were found to be about 3% smaller than the results at the equilibrium internuclear distance. The effect of vibrational averaging is thus negligible in the present context. The purely vibrational contributions to the hyperpolarizability of O2 have been calculated by Shelton [17] both in the static limit and at optical frequencies. The vibrational contribution was estimated to be about 14% of the electronic contribution in the static limit. However, it was found to be very small for ESHG at 633 nm.

4. Discussion The frequency dependent ESHG hyperpolarizabilities y( -2~; w, w, 0) of O2 have been determined

Y. Luo ef al./Journal qf Molecular Structure (Theochem)

66

Table 4 Comparison of theoretical and experimental frequencydependent values of y(-2w;w,w,O) for Oz. The calculated results are from the basis set POL (in Da) w

CAS-2p

RASd

Experimentb

0.0 0.06563 0.07200 0.07700 0.09300

534.67 616.40 635.96 654.37 728.45

674.46 775.63 795.86 822.84 917.27

840.50 1164.75 1214.18 1271.04 1510.49

a See discussion b Ref. [16].

in the text.

experimentally by Mizrahi and Shelton [16] for wavelengths from 700.0 to 476.5 nm. The ratios of y for O2 and He were measured and the ab initio results of Sitz and Yaris [18] for y of He were used to predict y for 02. The static value of y was estimated to be 840.50 Da by expanding the of w*. In our YN~(W)IYH~( w 1 in a linear function previous study for the hyperpolarizability of N2 we showed that in order to determine the dispersion of the hyperpolarizability with high accuracy, the term in w4 should be included in the fitting function [6]. In Table 4, the frequency dependent hyperpolarizabilities y(-2~; w, w, 0) calculated at the 2p-CAS and RAS levels are shown together with the corresponding experimental data. The RAS results are obtained by adding the dispersion percentages from the 2p-CAS calculations to the static value from the RAS wavefunction. This is based on the assumption that the dispersion feature of the hyperpolarizability at the CAS-2p level is the same as that at the RAS level. Compared with experimental data at the static limit, it turns out that the 2p-CAS function just covers 50% of the correlation. The RAS space used here is very large, and a reasonable agreement with the experimentally estimated polarizability has been obtained, but the calculated value of the hyperpolarizability is still 20% off from the experimental data. It indicates that the correlation effect is extremely strong for the hyperpolarizability. For molecular oxygen there is a delicate balance between intra-valence and dynamic correlation. In MCSCF response theory we start out from the former (2p-CAS) and add parts of the latter (RAS). From Table 2 one can anticipate

336 (1995) 61-67

that the xxxx component could be further raised by adding even more correlation. Evidently a prohibitively large space would be required to obtain a converged value of this component. The error increases when the frequency gets larger: the dispersion obtained from y2p-CAS function is significantly underestimated. The spin coupling between the ground triplet state ‘C, and the excited singlet state ‘Cl of O2 is relatively large M 150 cm-’ [19], and has been suggested to contribute to the large hyperpolarizability of 02. The contributions of the ‘Cp’ singlet state were considered here at the 2p-CAS level and using first-order perturbation theory. It was found to be small and increased the ground state hyperpolarizability only by approximately 1%.

Acknowledgements This work was supported by a grant from CRAY Research Inc and by the Danish Natural Science Research Council (Grant No. 11-6844).

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Y. Luo et al./Journal of Molecular Structure (Theochem) [14] G.H.F. Diercksen, V. Kelb and A.J. Sadlej, J. Chem. Phys., 79 (1983) 2918. [I 51 N.J. Bridge and A.D. Buckingham, Proc. R. Sot. London, Ser. A, 295 (1966) 334. 1161 V. Mizrahi and D.P. Shelton, Phys. Rev. Lett., 55 (1985) 696.

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[17] D.P. Shelton, Mol. Phys., 60 (1987) 65. [18] P. Sitz and R. Yaris, Chem. Phys., 49 (1968) 3546. [19] 0. Vahtras, H. Agren, P. Jorgensen, H.J.Aa. Jensen, T. Helgaker and J. Olsen, J. Chem. Phys., 96 (1992) 2118.