The Sum-Over-States Formula

The reduced indirect spin-spin coupling tensor is given by 

The operators as well as the unperturbed wavefunction are supposed to be known, thus the only unknown quantities in Eqs. (7a) and (7b) are and 

Of course in practice not all, usually not even one, of the excited-state wave-functions are known and therefore, Eqs. (9a) and (9b) as they stand do not have much practical importance however they greatly facilitate interpretations. For 

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For atoms, the sum-over-states formulas can be simplified considerably and we can write ... 

Oto state m for molecule A andy is the corresponding dipole oscillator strength averaged over degenerate final states. Similarly, the sum-over-states formula for the mean, frequency-dependent, polarizability can be written as... 

Equation (A 1.6.94) is called the KHD expression for the polarizability, a. Inspection of the denominators indicates that the first term is the resonant term and the second term is the non-resonant term. Note the product of Franck-Condon factors in the numerator one corresponding to the amplitude for excitation and the other to the amplitude for emission. The KHD formula is sometimes called the sum-over-states formula, since formally it requires a sum over all intermediate states j, each intermediate state participating according to how far it is from resonance and the size of the matrix elements that coimect it to the states v]/ - and v]/ The KHD formula is fully equivalent to the time domain formula, equation IAl.6.92). and can be derived from the latter in a straightforward way. However, the time domain formula can be much more convenient, particularly as one detunes from resonance, since one can exploit the fact that the effective dynamic becomes shorter and shorter as the detuning is increased. 

In the sum over states formula, excited states for the vibrational modes need to be included up to convergence. A more convenient integral expression is provided by classical or semiclassical theories. At high temperatures and low frequencies, the vibrational motion behaves increasingly classically and the semiclassical Wigner-Kirkwood expression is an excellent approximation to the quantum partition function [78] for low-frequency vibrations at pyrolysis temperatures. The semiclassical... 

The resolvant can be expressed in terms of the eigenfunctions and eigenvalues of the zero-order problem. Substituting equation (38) into the expressions given above for the perturbed wavefunetions and energies gives the sum-over-states formulae most often used in practical applications of perturbation theory. 

For the second-order energy coefficient in Rayleigh-Schrddinger perturbation theory, the sum-over-states formula is... 

One of the drawbacks of Brillouin-Wigner perturbation theory is that the expressions for the energy components in second order and beyond contain the exact energy in the denominator factors. The equations must therefore be solved iteratively until self-consistency is achieved. The generalized Brillouin-Wigner perturbation theory [21] has the advantage that the denominators can be factored from the sum-over-states formulae. 

Nuclear spin-spin coupling constants and were calculated [17] on the basis of the sum-over-states formula given by Nakatsuji [15]. 

A possible computational strategy is to calculate Xs(r O first using the standard sum-over states formula (Equation 24.80). Equation 24.75 can be used next to generate successive Born approximations of the functions f (r). For instance, the first Bom approximation would be... 

Bartkowiak and ZaleSny discuss the sum-over-states (SOS) method which is used for the calculation of NLO properties (electronic contribution) and multi-photon absorption. They comment on the various approximations, including the widely used few-states models, and tlie exact sum-over-states formulas. They show that one of the main advantages of tlie many variants of this approach is the interpretation of the NLO properties in terms of contributions from excited states. They comment on the limited utility of the SOS technique for small molecules, aggregates and clusters, but they point out, that it is still a very attractive tool for large molecules. 

The choice of (Oj, CO2... (static or not) defines the various non-linear optical processes and in Table 1 some of these are listed. Until now there has been no consensus with regards to abbreviations for these processes, so I take the liberty of making a set of recommendations. I have tried to make the abbreviation short, so that, e.g., dc implies the presence of a static electric field (rather than dc-EF), and I have tried to avoid repugnant sounds such as EFISH or OKE. It should be remarked that one can always permute the subscripts a, p, y... along with the frequencies -[Pg.6]

Bishop and Pipin [11] and the H2 (D2) ones by Bishop, Pipin, and Cybulski [12]. The starting off point in all cases is the set of perturbation-theoretic sum-over-states formulas given by Orr and Ward [8] (I include for completeness those for a and p, though only that for y is germane at this point) ... 

Equation [86] is sometimes useful, e.g., when a knowledge of the excitation energies and polarized components of the transition moments for a few states could be used to estimate the a tensor from experiment. However, this sum-over-states formula is computationally about the worst approach to the theoretical evaluation of second-order quantities. To use it directly would require computing all the eccited states the finite basis permits and the appropriate transition moments. It can be solved indirealy, however. ... 

Perturbation theory yields a sum-over-states formula for each of the dispersion coefficients. For example, the isotropic C coefficient for the interaction between molecules A and B is given by... 

Dynamic Response Functions. - The perturbation series formula or spectral representation of the response functions can be used only in connection with theories that incorporate experimental information relating to the excited states. Semi-empirical quantum chemical methods adapted for calculations of electronic excitation energies provide the basis for attempts at direct implementation of the sum over states (SOS) approach. There are numerous variants using the PPP,50,51 CNDO(S),52-55 INDO(S)56,57 and ZINDO58 levels of approximation. Extensive lists of publications will be found, for example, in references 5 and 34. The method has been much used in surveying the first hyperpolarizabilities of prospective optoelectronically applicable molecules, but is not a realistic starting point for quantitative calculation in un-parametrized calculations. 

These results show that the presence of the solvent leads to an increase of the (a) values with respect to the vacuum, and it reverses its frequency dependence in aqueous solution, unlike in vacuo, the molecule shows a static (a) value which is larger than that at oj 0. This effect can be related to the shape of the e(w) function. By applying Debye formula it is easy to see that, at the frequency considered in our calculations, the value of e(w) is practically equal to e(oo). which is by far smaller than e(0). K one considers the sum-over-states method for the calculation of polarizabilities with these values in minds, then it is easy to give a qualitative explanation of the behavior indicated above. In fact, when the solvent response function is described by e(oo) the actual stabilization of the excited states will be less than in a situation where the same response depends on c(0), hence smaller it will be the correspondent (a) value. ... 

Chou and Jin have addressed the importance of the vibrational contributions to the polarizability and second hyperpolarizability within the two-level and the two-band models. Their study adopts the sum-over-state (SOS) expressions of the (hyper)polarizabilities expressed in terms of vibronic states and includes two states and a single vibrational normal mode. Moreover, the Herzberg-Teller expansion is applied to these SOS formulas including vibrational energy levels without employing the Plac-zek s approximation. Thus, this method includes not only the vibrational contribution from the lattice relaxation but also the contribution arising... 

In the second, response theory, approach, the response of the Hartree-Fock ground state is calculated by perturbation theory. First-order perturbation theory in the fluctuation potential gives a method known as the random phase approximation (REA). The RPA linear response gives the dynamic polarizability, the quadratic response gives the first hyperpolarizability etc. One can obtain expressions for the response functions as sums over states formulae, but they are not calculated as such, rather they are calculated from coupled linear equations. RPA is equivalent to TDCPHF. 

This approach to calculation of the B term suffers from two main disadvantages. First, the two summations in the expression for B are infinite and must be truncated. Fortunately, correct qualitative and even semiquantitative results are still obtained relatively easily, since only a few terms in the sums make significant contributions in most cases. Second, when approximate wavefunctions are used in the evaluation of the B terms using the truncated sum-over-states formula, the result depends on the choice of origin of the coordinate system. In practice, as long as the origin is chosen somewhere within the molecule, and the molecule is not of inordinate size, the dependence is negligible and does not cause problems. ... 

This is seen to be similar to the photoabsorption formula except that (i) it involves a half Fourier transform, (ii) (t) is projected onto 1- n = representing the final state of interest, and (iii) the integral is squared. Since we have just discussed tractable ways of finding (j)(t), using equation (14) to calculate Raman scattering amplitudes is just as easy as the case of photoabsorption. The time-dependent technique will be far easier than the sum-over-states method of equation (7), and again it focusses our attention on the physically relevant dynamics. For low-lying excited states is still localized to the Franck-Condon vicinity. 

Note that the formula for a t L xx 0), Eq. (1-151), can be rewritten as the following sum-over-states expression,... 

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