Sum-over-states approach

Although both the ab-initio derivative method and the semi-empirical sum-over-states approach have been used with some success to predict qualitative trends, they are not sufficiently developed to have predictive capabilities for structure-property relationship. Clearly, there is a need to develop semi-empirical theoretical methods which can reliably be used to predict, with cost-effectiveness and with reasonable computational time, molecular and polymeric structures with enhanced optical nonlinearity. 

In resonant infrared multidimensional spectroscopies the excitation pulses couple directly to the transition dipoles. The lowest order possible technique in noncentrosymmetrical media involves three-pulses, and is, in general, three dimensional (Fig. 1A). Simulating the signal requires calculation of the third-order response function. In a small molecule this can be done by applying the sum-over-states expressions (see Appendix A), taking into account all possible Liouville space pathways described by the Feynman diagrams shown in Fig. IB. The third-order response of coupled anharmonic vibrations depends on the complete set of one- and two-exciton states coupled to thermal bath (18), and the sum-over-states approach rapidly becomes computationally more expensive as the molecule size is increased. 

The components of the signal calculated according to these diagrams, using the sum-over-state approach presented in Appendix A, coincide with Equations (32) and (33) in the narrow line limit r (A, J)... 

When the external electric field is time-dependent, there is no well-defined energy of the molecular system in accordance with Eq. (100), and the wave function response can thus not be retrieved from a variational condition on the energy as in the analytic derivative approach described above. Instead the response parameters have to be determined from the time-dependent Schrodinger equation, a procedure which was illustrated in Section 3 for the exact state case. In approximate state theories, however, our wave function space only partially spans the 7V-electron Hilbert space, and the response functions that correspond to an approximate state wave function will clearly be separate from those of the exact state wave function. This fact is disregarded in the sum-over-states approach, and, apart from the computational aspect of slowly converging SOS expressions, it is of little concern when highly accurate wave function models are used. But for less flexible wave function models, the correct response functions should be used in the calculation of nonlinear optical properties. 

The effect of a-n coupling in a series of donor/acceptor organic molecules including zwitterions and betaines has been investigated within an INDO/S sum over state approximation by Rao and Bhanuprakash.177 Yuan et al.m have investigated structural effects on ft using the CNDO/S Cl sum over states approach. 

In the following it will be outlined, how the parity violating potentials are computed within a sum-over-states approach, namely on the uncoupled Hartree-Fock (UCHF) level, and within the configuration interaction singles approach (CIS) which is equivalent to the Tamm-Dancoff approximation (TDA), that avoids, however, the sum over intermediate states. Then a further extension is discussed, namely the random phase approximation (RPA) and an implementation along similar lines within a density functional theory (DFT) ansatz, and finally a multi-configuration linear response approach is described, which represents a systematic procedure that... 

Hartree-Fock and sum-over-states approaches. Although at present many ab initio coupling calculations take into account electron correlation effects, several calculations at the CHF or at a lower level of approximation were reported during the review period. Some coupling trends calculated at the CHF level compare favourably with those calculated with approaches that include electron correlation effects, as shown, for example, by Carmichael et al. ... 

A virtual Vibrational Self-Consistent Field (v-VSCF) method has then been developed and implemented for calculating the thermal elfects on molecular properties and has been apphed, among other properties, to the static polarizability. This method is more effieient than expKcit sum-over-states approaches and can be applied to larger systems. In an illustration of this scheme, when going from 0 to 298.15 K, the amplitude of the thermal effects on the diagonal polarizability tensor eomponents has been shown to amount generally to 2-10% but it can also attain 35% like in the case of SOCI2. 

The sum-over-states approach provides the following expressions for the MCD terms ... 

BEYOND THE BORN-OPPENHEIMER APPROXIMATION A SUM OVER STATES APPROACH... 

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