perturbation theories and wave functions for calculation of electronic polarizabilities application...

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. IX, 701-710 (1975)

Perturbation Theories and Wave Functions for Calculation of Electronic Polarizabilities

Application to DNA Bases STEVEN ADAMS, SHLOMO NIR, AND ROBERT REIN

Department of Experimental Pathology, Roswell Park Memorial Institute, 666 Elm Street Buffalo, N . Y . 14203, U S A . and Department of Biophysical Sciences, State University of

New York at Buffalo, 4234 Ridge Lea Road, Amherst, N.Y. 14226, U S A .

Abstracts Three different forms of perturbation theories, variational perturbation, finite pertur-

bation and second-order, are evaluated regarding their value for calculation of electronic polarizabilities of small and intermediate size molecules. It is concluded that with the practical constraint of a small basis set the variational perturbation method is the most promising alternative for calculation of polarizabilities. For several small molecules, our calculated polarizabilities indicate that both IEHT and ab initio wave functions give values in close agreement with each other. Variational perturbation calculations of polarizabilities with IEHT wave functions also include the DNA bases.

Trois formes diff Crentes de la thCorie des perturbations-perturbation variationnelle, finie, et du second ordre-ont ktC dvalukes en ce qui concerne leur valeur pour le calcul des polarisabilitks Clectroniques des molkcules petites ou de grandeur intermtdiaire. I1 en rCsulte qu’avec la contrainte pratique d’un jeu de base limit6 la mkthode des perturbations variationelle forme I’alternative la plus promettante. Pour plusieurs petites molkcules nos polarisabilitks calculkes indiquent que les deux types de fonction d’onde IEHT et ab initio donnent des valeurs cohkrentes. Les calculs IEHT sont faits aussi pour les

Drei verschiedene Formen der Storungstheorie-Variationstorungstheorie, Theorie endlicher und zweiter Ordnung-werden mit Rucksicht auf ihren Wert fur Berechnungen der elektronischen Polarisierbarkeiten von kleinen und intermediaren Molekiilen unter- sucht. Es ergibt sich, dass mit der von den kleinen Basissatzen herruhrenden praktischen Begrenzung die Variations-Storungsmethode die beste Alternative darbietet. Fur mehrere kleine Molekiile deuten unsere berechneten Polarisierbarkeiten an, dass die zwei Wellenfunktionstypen IEHT und ab initio ubereinstimmende Werte geben. Variations-Storungsrechnungen von Polarisierbarkeiten mit IEHT-Funktionen sind auch fur die DNA-Base gemacht worden.

bases DNA.

1. Introduction Since the work of London [l, 21 it was realized that a knowledge of the

polarizability is essential in calculations of intermolecular forces. The value of the electronic contribution to the molecular polarizability seems to be almost independent of the molecular surrounding, e.g., it turns out to be almost the same in the liquid or gaseous states of a substance [3, 41. Therefore, quantum

701 @ 1975 by John Wiley & Sons, Inc.

702 ADAMS, NIR, AND REIN

mechanical calculations of the electronic contribution to the molecular polarizability, which consider the molecules as isolated entities, may be compared with experimental values which are obtained for systems in a condensed state.

However, the results were far from encouraging in our previous attempts to obtain the polarizabilities of several small molecules by an application of second-order perturbation theory and finite perturbation theory [6, 71 using IEHT

orbitals as basis functions [4, 51. A similar failure was observed in finite perturbation calculations in conjunction with CNDO wave functions [8]. The calculated values of the isotropic polarizabilities underestimated the experimental values by a factor of at least two to three. The reason for this failure has been attributed to the limited size of the basis set, which is also illustrated in the case of ab initio calculations [9, 101. For instance, Arrighini et al. [lo] approached up to 80% of the electronic polarizability of water with a basis set of 27 orbitals; on the other hand, with a basis set of seven orbitals the calculated value of the isotropic polarizability was less than 40% of the experimental value.

The calculation of the polarizability by the variational method was proposed by Hylleraas [11] and by Hasst [12] in 1930 and was further elaborated by Hirschfelder, Curtiss and Bird [13] in 1954. The method was applied by Kolker and Karplus [14] who used SCF-LCAO-MO functions to calculate the electronic polarizability tensor for a series of small diatomic molecules, and obtained good agreement with experiment for three molecules for which data were available. The method was generalized by Seprodi, Bicz6 and Ladik [15] who replaced the variational parameter by a three-dimensional vector, and applied their procedure to calculations in the framework of .rr-electron theory to the nucleotide bases.

Recapitulating, there are three different perturbation methods valuable for the calculation of polarizabilities, i.e., second-order, finite and variational perturbation. Also there are two empirical schemes-addition of bond polarizabilities and semiempirical extrapolation of the dispersion of the refractive index-which permit the calculation of polarizabilities from optical data, giving values for a check on the calculations.

A variety of wave functions such as IEHT [16], CNDO and ab initio are now available and easily obtainable for large molecules.

The objective of the present paper is a comprehensive test of the above methods and wave functions for the calculation of polarizabilities.

The conclusions of this study serve to establish the best alternative to calculate polarizabilities of medium size and large molecules. In addition, as a byproduct the study provides a useful insight into the quality of the wave functions employed.

2. Methods

A. The variational perturbation caiculation of electronic polarizabilities We will present an outline of the procedure for calculating the electronic

molecular polarizability following the treatment by Seprodi, Bicz6 and Ladik

CALCULATION OF ELECTRONIC POLARIZABILITIES 703

[15]. The molecular Hamiltonian is written as H=H,,+H'=Ho+e[i r(i)- N ZJ,] - E

i = l 0 = 1

using the perturbed wave function + given by

where +o is the n-electron wave function of the unperturbed system, H, is the Hamiltonian in the absence of a static electric field e ; r(i) is the position vector of the ith electron, r, is the position vector of the a t h nucleus in a fixed molecular coordinate system, Z, is the effective nuclear charge of nucleus a. A is a vector which has as components the variational parameters A,, A,, A, determined to make the molecular energy stationary. N is the number of nuclei. By applying the variational method to the total energy, it was obtained [15] for the k , 1 component of the molecular polarizability tensor,

2 9 i

~ ~ k i = - x QkjQli; ;, k, l = X , y, (3)

in which q is given by [13] q=laoe4n; ao=Bohr radius. The Qi, element of the matrix Q is given as

(4)

(We find it preferable to use the letter Q in place of the letter S which was previously used in References [13] and [15] since S has been widely used to denote the overlap matrix.) In Equation (4) the first moments Iw(i) and the second moments Iw(ij) are given by

Pw are the matrix elements of the charge-bond order matrix P, calculated as occ

Pw = 2 nic$cq

where +i is the molecular orbital obtained in the LCAO treatment, defined according to

(7) +i = C CipXp P

704 ADAMS, NIR, A N D REIN

n, is the occupation number of molecular orbital $,, and xp is an atomic orbital. Because the moment operators 1, in Equation (5) which are employed in the variational treatment are defined with respect to local atomic coordinates, the calculated polarizability tensor referred to the principal axes is invariant with respect to rotation or displacement of the coordinate axes.

We have employed the orbital basis set as 2s- and 2p-orbitals on carbon, nitrogen and oxygen and 1s-orbitals on hydrogen with Slater’s rules for the screening parameters except for a screening parameter of 1.2 on hydrogen. The procedure for obtaining the density matrix P and the local moments I, has been described in detail elsewhere [16-18).

B. Finite perturbation theory The finite perturbation theory is presented in References [6] and [7]. The

components of the polarizability tensor are obtained according to Reference [7] by

where $ is the ground state perturbed wave function obtained from the Hamiltonian operator

(9) H=Ho-p.&

where p is the molecular electric dipole moment. The values of E~ employed should be chosen sufficiently small so that the calculated value of aii becomes independent of magnitude of E , .

C. Second-order perturbation theory Calculation of the polarizability tensor components with second-order

perturbation theory is described in Reference [ 131. We employed one-electron excitations; the number of the excited states contributing to the polarizability varies from 10 to 20, depending on the size of the molecule.

3. Results

We calculated the po!arizability tensor components aii, i, j = x , y , z and the corresponding isotropic polarizability according to the variational perturbation method using IEHT [16] all valence electron wave functions. The components were calculated in the principal axis system. The calculated and experimental values for several molecules, including the DNA bases and other aromatic molecules, are given in Table 1.


TABLE I. Mean and principal molecular polarizabilities (in units of lo-'' cm3)

Variational perturbation Molecule Experiment* treatmentb

Hi N2 CO,

C,H,N' Cytosine' Guanine Adenine Thymine CH:

C6H6C

CzH;

CzH:

C&

HCNd

0.79 1.734 2.594 9.87 9.14

10.33

13.11 11.23 2.60

3.33

4.26

4.47

2.59

-

0.93 2.38 4.10 7.33 5.53 - - - -

2.60

5.12

5.61

5.48

3.92

0.72 1.45 1.93

11.14 10.6 - - - - 2.60

2.43

3.59

3.97

1.92

0.72 1.45 1.93

11.14 11.3 - - - - 2.60

(ab initio) 2.43

(ab initio) 3.59

(ab initio) 3.97

(ab initio) 1.92

(ab initio)

0.66 0.71 0.64 0.64 1.10 1.14 1.08 1.08 1.36 1.58 1.25 1.25 7.19 6.51 7.53 7.53 6.40 5.86 6.70 6.63 6.30 6.48 6.72 5.70 8.17 8.59 8.63 7.28 8.12 8.62 8.47 7.27 7.15 7.31 7.63 6.52 1.95 1.95 1.95 1.95 1.68 1.68 1.68 1.68 2.44 2.74 2.29 2.29 1.88 2.17 1.74 1.74 2.90 3.04 2.88 2.78 2.36 2.49 2.33 2.27 3.40 3.50 3.36 3.36 2.85 2.92 2.82 2.82 1.68 1.78 1.63 1.63 1.31 1.39 1.28 1.28

Experimental values from Reference [ 191. The variational perturbation method following the expressions of Reference [ 151.

See also in Table 11. Experimental value from Reference [23].

IEHT wave functions have been employed.

For all molecules treated, the relative magnitudes of the polarizability component parallel to the bond axis (all) and the polarizability components perpendicular to the bond axis (al) are in agreement with the experimental values. The calculated polarizabilities yield an underestimate of about 15-50% with a value of 30-40% as an average. We also present in Table I results of our calculations on several small molecules with a b initio wave functions taken from the literature [24]. It is seen that the values from both IEHT and ab initio wave functions are in close agreement with each other.

We also undertook a comparison of the results of several methods for the cases of benzene, pyridine and the DNA bases. This comparison is given in Table 11.

In the cases of benzene and pyridine, we employ IEHT wave functions and compare the results of three methods: second-order perturbation theory, finite perturbation theory and variational perturbation theory with experimental values of the principal axis and isotropic polarizabilities. The worst results are


TABLE 11. Comparison between different methods in the calculation of the principal and isotropic polarizabilities of benzene, pyridine, and cytosine.

Method %x %Y a z z a.

benzene experiment" second-order perturbationb finite perturbationb variational perturbationb.'

pyridined

cytosine'

guanine

adenine

thymine

experiment* second-order perturbationb finite perturbationb variational perturbationb.'

experiment ZDO .sr-electron variational

IEHT all-valence electron perturbations

variational perturbationb

experiment ZDO r-electron variational

IEHT all valence electron perturbations


experiment ZDO r-electron variational

IEHT all valence electron perturbation8


experiment ZDO n-electron variational

IEHT all valence electron perturbations


11.14 0.55 4.93 7.53

10.6 0.50 4.29 6.70

11.14 7.33 0.55 0 4.93 0 7.53 6.51

11.3 5.53 0.55 0.05 4.39 0 6.63 5.86

9.87 0.37 3.29 7.19

9.14 0.37 2.89 6.40

10.33'

25.43

6.30

16.03

8.17

13.11'

17.47

8.12

11.23f

25.42

7.15

a From Reference [20]. Benzene molecule in x-y plane. Using IEHT wave functions. According to Equation (3). Pyridine in x- y plane with nitrogen atom in the x-direction. Cytosine in x- y plane.

' The static polarizability as obtained from the refractive index. sFrom Table I of Reference [15].


those with the second-order perturbation theory, which, because of the limited size of the basis set, underestimate the polarizabilities by an order of magnitude. This has also been the conclusion of previous studies [4, 7, lo]. The finite perturbation method represents an improvement over the use of second-order perturbation theory, but it totally neglects the polarizability component perpendicular to the molecular plane. This is discussed in the next section. The corresponding polarizability component from finite perturbation theory using C~D0/2(8) wave functions also appears to underestimate the experimental value. It is apparent from Table I1 that the variational perturbation method gives more superior results than both the other methods.

In the case of the DNA bases, we compare the isotropic polarizabilities calculated in the 7r-electron approximation [15] and with the IEHT all valence electron method, in both cases within the framework of the variational perturbation approach. For this purpose we used the same geometry as that in Reference [15]. The principal axis components of the polarizability tensor have not been measured. However the isotropic polarizability is directly obtainable [3, 41 from the experimental value of the molar refraction [21]. The results of this comparison (see Table 11) indicate that the m-electron approximation overesti- mates by a factor of more than two the value of the isotropic polarizability whereas the IEHT all valence electron wave function gives an average underestimate of about 3 0 4 0 % . The overestimate obtained in Reference [15] is attributed to the exclusive we of T electrons (vide infru).

It is of interest to point out that for the isotropic polarizability, the variational perturbation method of Seprodi et al. [15] reduces to that of Hirschfelder et al. [13]. In \general, the isotropic polarizability obtained with Equations (1)-(5) was only slightly larger (less than 1%) than that obtained with the simpler method of Hirschfelder et al. [13] which does not give the off -diagonal tensor components.

4. Discussion and Conclusions

We see from the results that for the employment of semiempirical wave functions which use a limited basis set, the variational perturbation gives the best results. At this level of approximation, the two other perturbation methods are not of much practical value. In examining the reasons for the failure of second-order perturbation theory, the excited state wave function for the perturbation expansion seems to be too limited and inaccurate at the present level of molecular orbital approximation.

The finite perturbation method underestimates the experimental values of the polarizability, even with the employment of ab initio wave functions [9]. As noted previously, the results can be improved by employing a more extended basis set [22]. With the use of semiempirical wave functions, one must be careful to obtain the correct 2s-2p hybridization on a single center [8]. The C N D O / ~


method gives a component of the moment perpendicular to the bond axis through induced 2s-2p hybridization. In our application of the finite perturbation method, we do not obtain any contribution to the polarization perpendicular to the bond. This is because the off-diagonal matrix elements of the Hamiltonian are made proportional to the overlap integral in the IEHT definition of the Hamiltonian [16]. Note that at the same time the matrix elements ( ~ 2 ~ ~ 1 y Ixz~,A) (bond axis in the Z direction) do not vanish. Attempts to employ the one-center moment over the 2s- and 2p-orbitals in the perturbation without making the off -diagonal elements in the perturbation proportional to the overlap did reproduce the perpendicular component of the polarizability, but still gave poor results for the isotropic polarizability.

In addition to the problem of treating one-center hybridization, one must properly match the magnitude of the perturbed Hamiltonian to that of the effective Hamiltonian in the IEHT. This limitation could be perhaps removed by introducing the proper scaling parameter to match both terms. In the application of finite perturbation theory using C N D O / ~ wave functions, the degree of anisotropy of the polarizability is reasonably well predicted [8], but the isotropic polarizability underestimates the experimental value in the same way as the use of IEHT wave functions.

The reason why the variational perturbation method using IEHT wave functions gives a fair agreement with experiment can be understood from inspection of Equations (3)-(5). Since it is necessary to know only the first and second moments and the density matrix in the variational perturbation method, a wave function which gives reasonable dipole and quadrupole moments may also give reasonable values of the polarizability. Since it has been previously demonstrated that the IEHT gives dipole and quadrupole moments in good agreement with experiment [18], it may have been also expected that the method would yield more acceptable polarizability values.

As mentioned in the previous section, the application of the variational perturbation method with .sr-electron wave functions [15] gave an overestimate of the polarizability by a factor of about two, whereas the application of all valence electron IEHT or ab initio ground state wave functions gave an underestimate by approximately 30-40% or 50%, respectively. From an examination of the matrix elements Iw(ij) and I,(i) in Equation ( 5 ) for P and S atomic orbitals, we have arrived at a conclusion that an overestimate by a factor of two or more can be obtained by an exclusive use of P orbitals. An increase in the percent of P orbitals in the wave function can enhance the magnitude of the polarizability. In order to check and exemplify the above conclusions we performed variational perturbation calculations of the polarizability on Hz using various atomic orbitals with IEHT wave functions. When 2s-Slater orbitals were used, the calculated polarizability was several times larger than with 1s-electrons, and even a larger overestimate was observed with the use of 2p-orbitals. This result explains why the calculated values for the polarizability from IEHT wave functions are slightly


larger than the results from the employment of ab initio wave functions. The ab initio wave functions take into account the 1s-electrons which contribute relatively little to the integrals determining the matrix Q in Equation (3), which contains the total number of electrons in the denominator. (See definition of q in Equation (3).) In fact, a multiplication of the a b initio values for the polarizabilities (see Table I) by n/(n-nl,), (where nl, is the number of l s - electrons) gives values which are close to those obtained with IEHT wave functions.

It should be restated that with a significant increase in the size of the basis set, both the finite and second-order perturbation methods give results which approach the experimental value of the polarizability. However, the variational perturbation method yields values which are generally from within 60-70% of that of the experimental values even with a limited basis. On the basis of our calculations we would speculate that a slight increase in the percentage of orbitals with higher principal quantum numbers in the basis set would result in good predictions for the values of electronic molecular polarizabilities.

Acknowledgements

The authors wish to express their thanks to Professor J. Ladik for his sugges- tion to use the variational perturbation method in calculations of polarizabilities and for reading the manuscript and commenting on it.

One of the authors (S. A.) wishes to acknowledge the financial assistance of NIH training grant. They would also like to acknowledge funds from NASA grant NGR 33-015-002. The generous allotment of computer time from the SUNY at Buffalo Computing Center is also appreciated.

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Received July 29, 1974.

perturbation theories and wave functions for calculation of electronic polarizabilities application...

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