Additional Compact Formulas for Vibrational Dynamic Dipole Polarizabilities and Hyperpolarizabilities

Compact expressions, complete through second order in electrical and/or mechanical anharmonicity, are given for the dynamic dipole vibrational polarizability and dynamic first and second vibrational hyperpolarizabilities. Certain contributions not previously formulated are now included. The past decade has seen an increasing number of calculations of vibrational polarizabilities and hyperpolarizabili-ties. At this time, some hundred-odd papers have touched on this subject, and it has recently been reviewed in detail by one of us. 1 For some systems and processes these vibrational properties are even more important than the corresponding electronic ones. Together both contributions govern nonlin-ear optical ͑NLO͒ behavior. A large number of the vibra-tional calculations have been based on one or both of two approximations: ͑a͒ the vibrations are assumed to be harmonic , and the electronic properties are assumed to be linear in the normal coordinates—this is the double-harmonic-oscillator approximation; 2 and ͑b͒ the optical frequencies are taken to be infinite 2 —this is the enhanced, or infinite frequency approximation. 3,4 These approximations have been analyzed by Bishop and Dalskov. 5 The only rigorous treatment of the dynamic vibrational ͑hyper͒polarizabilities is that given by Bishop and Kirtman 6 in the early nineties. The purpose here is to extend their general formulas so that they are complete through second order of perturbation theory in mechanical and/or electrical anharmonicity. All vibrational effects are included except for zero-point vibrational averaging of the electronic ͑hyper͒polarizability. We write the vibrational polarizability (␣ v) and the first and second vibrational hyperpolarizability (␤ v and ␥ v ͒ as a sum of terms derived from the sum-overstates expressions for the total ͑hyper͒polarizability, which are given in terms of vibronic energies and dipole moment matrix elements over the vibronic wave functions,

The past decade has seen an increasing number of calculations of vibrational polarizabilities and hyperpolarizabilities.At this time, some hundred-odd papers have touched on this subject, and it has recently been reviewed in detail by one of us. 1 For some systems and processes these vibrational properties are even more important than the corresponding electronic ones.Together both contributions govern nonlinear optical ͑NLO͒ behavior.A large number of the vibrational calculations have been based on one or both of two approximations: ͑a͒ the vibrations are assumed to be harmonic, and the electronic properties are assumed to be linear in the normal coordinates-this is the double-harmonicoscillator approximation; 2 and ͑b͒ the optical frequencies are taken to be infinite 2 -this is the enhanced, or infinite frequency approximation. 3,4These approximations have been analyzed by Bishop and Dalskov. 5The only rigorous treatment of the dynamic vibrational ͑hyper͒polarizabilities is that given by Bishop and Kirtman 6 in the early nineties.The purpose here is to extend their general formulas so that they are complete through second order of perturbation theory in mechanical and/or electrical anharmonicity.All vibrational effects are included except for zero-point vibrational averaging of the electronic ͑hyper͒polarizability.
We write the vibrational polarizability (␣ v ) and the first and second vibrational hyperpolarizability (␤ v and ␥ v ͒ as a sum of terms derived from the sum-over-states expressions for the total ͑hyper͒polarizability, which are given in terms of vibronic energies and dipole moment matrix elements over the vibronic wave functions, 7 Here, the optical frequencies ( 1 , 2 , 3 ) define a particular NLO process and ϭ ͚ i i .The square-bracket terms 6 are sums over vibrational states of a quotient in which the nu-merator introduces vibrational transition matrix elements of the electronic properties ͑,␣,␤͒ and the denominator involves the vibrational and optical frequencies.For example, where ( ␣ ) 0k ϭ͗0͉ ␣ ͉k͘; ͉k͘ is the kth vibrational wave function with energy ប k ; ␣ is the ␣ component of the dipole moment (␣,␤,␥ϭCartesian coordinates x,y,z͒; ͚ P ␣␤␥ indicates a sum over the simultaneous permutations of frequencies and subscripts (Ϫ ,␣), ( 1 ,␤), and ( 2 ,␥) ͑an alternative equivalent notation is ͚ P Ϫ,1,2 ͒; and the prime on the second summation denotes exclusion of the vibrational ground state.The electronic properties are, next, expanded in the normal coordinates (Q a ,Q b , . . .), e.g.,

͑5͒
Here the quadratic term is considered to be first order in electrical anharmonicity and the cubic term is considered to be second order.Thus, it is sufficient for our purposes to truncate the expansion as above after the third derivatives.The vibrational potential, V, is expanded in a similar manner, a͒ Permanent address: Institute of Computational Chemistry and Department of Chemistry, University of Girona, 17017 Girona, Catalonia, Spain.
but, in this case, the cubic term is taken to be first order in mechanical anharmonicity and the series is terminated after the ͑second-order͒ term involving the quartic force constants, F abcd , which are fourth derivatives of V. Treating the electrical and mechanical anharmonicity by double perturbation theory and substituting into Eqs.͑1͒-͑3͒ leads to where we have used the abbreviations ͓ ͔ 0 ϭ͓ ͔ 0,0 , ͓ ͔ I ϭ͓ ͔ 1,0 ϩ͓ ͔ 0,1 , and ͓ ͔ II ϭ͓ ͔ 2,0 ϩ͓ ͔ 1,1 ϩ͓ ͔ 0,2 .The notation ͓ ͔ n,m indicates the order of electrical anharmonicity (n) and the order of mechanical anharmonicity (m).
As stated in our earlier work, 6 we previously neglected the cubic term in Eq. ͑5͒ and all of the ͓ ͔ 0,2 contributions in ͓ ͔ II .We now include these and, in order that there be a single source of information, we give the complete formulas in Tables I and II.
expression for ͓ 4 ͔ 0,2 .To be more precise, with the abbreviations in the footnote to Table I and changing indices where appropriate, the essential form of ͓ 4 ͔ 2,0 is whereas for ͓ 4 ͔ 1,1 it is It is apparent that increasing the mechanical anharmonicity and decreasing the electrical anharmonicity simultaneously by one order introduces b Ϯ1 ; changes the second derivative of ␤ ͑associated with 1 ͒ with respect to Q a and Q c to a single derivative with respect to Q b ͑note the recurrence of b͒; and introduces the cubic force constant F abc .Repeating this step in a consistent way we have ‫ץ‬Q c ‫ץ‬Q d →‫ץ‬Q e and, then, introducing e Ϯ2 and F cde gives the essential form of ͓ 4 ͔ 0,2 as With the known numerical factor for the static quantity and with some rearrangement of the indices, the formula for the ''F 2 '' part of ͓ 4 ͔ 0,2 given in Table I was obtained.
][10] This led us to discover that the diatomic formulas of Ref. 9 are only approximate because the anharmonic terms in Eq. ͑5͒ of that reference were not carried through in deriving Eqs.͑11b͒ and ͑11c͒.These terms were included subsequently in the corresponding treatment of polyatomics. 10However, in the latter case the fact that the field-dependent harmonic force constant matrix contains off-diagonal elements, when expressed in terms of field-free normal coordinates, was ignored.One can determine the effect of these off-diagonal matrix elements on the field-dependent harmonic vibrational frequencies, which are the quantities of interest, by carrying out an appropriate perturbation treatment.Thus, we write where K͑F͒ is the field-dependent harmonic force constant matrix, ͑F͒ is the diagonal matrix of field-dependent harmonic frequencies, and T͑F͒ is a unitary transformation, All quantities are, then, expanded as a power series in F with K͑0͒ being diagonal and T(0)ϭ1.After some simplification it can be shown that the resulting formulas agree exactly with those given in Tables I and II when the static limit is taken ͑see the Appendix for an example͒.Thus, the equiva- P ␣␤ →P ␣␤␥␦ Each term is replaced by two: in the first ␤ →␤ ␤␥␦ and in the second lence between the perturbation theory 6 and finite field 10 approaches to static vibrational hyperpolarizabilities demonstrated in Ref. 8 is now extended to polyatomic molecules.A similar extension can be made for the infinite frequency limit.
To conclude, we have presented general ͑dynamic͒ expressions, complete through second order in perturbation theory, for the vibrational polarizability (␣ v ), and the first and second vibrational hyperpolarizability (␤ v and ␥ v ͒.The only additional contributions from the effects of vibration to this order arise from zero-point vibrational averaging.

APPENDIX
If the vibrational potential is expanded in terms of fieldfree normal coordinates, Q, then the off-diagonal quadratic force constants will no longer vanish when an external field is present.The effect of these force constants on the harmonic vibrational frequencies and, hence, on the linear and nonlinear vibrational polarizabilities, can be evaluated through a perturbation treatment of Eqs.͑10͒ and ͑11͒.The purpose of this Appendix is to illustrate how this can be accomplished using the linear polarizability as an example.
The term linear in F is considered to be first order; the term quadratic in F is second order; and so forth.Through second order the harmonic vibrational frequencies from Eq. ͑13͒ are given by, In deriving Eqs.͑A4͒-͑A6͒ we have used the fact that K is Hermitian; K͑0͒ is diagonal; and T(F)ϭ1ϩ͚T ␣ F ␣ ϩ ͚T ␣␤ F ␣ F ␤ ϩ¯is unitary.Thus, From Eqs. ͑A6͒-͑A8͒ one obtains

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.M.B. acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada.J.M.L. was supported by the Generalitat de Catalunya through the CIRIT Project No. FI/95-5101.