Summation over states properties

The three Equations (4.108)—(4.110) are only true for exact wavefunctions and they do indeed provide crude and problematic methods for calculating molecular properties. The advantage of these equations is that they indicate what one is able to obtain from this method but for actual calculations of molecular properties using approximative wavefunctions, it is important to use modern versions of response theory where the summation over states is eliminated [1,10-14,88-90],... 

The sum-over-states expressions that we have presented in Eqs. (69), (70), and (71) are only true for exact wave functions and they are rather cumbersome methods for calculating time-dependent electromagnetic properties of a quantum mechanical subsystem within a structured environment. The advantage of the sum-overstates expressions is that they illustrate the type of information that is obtainable from response functions. We have utilized modem versions of response theory where the summation over states is eliminated when performing actual calculations, that involve approximative wave functions [21,24,45-47,80-83]. 

There are many approaches to compute the polarizabilities and hyperpolarizabilities and also different ways to classify them. One convenient division is between perturbation theory approaches, which express the (hyperjpolarizability using Summation-Over-States (SOS) expressions and those techniques, which are based on the evaluation of derivatives of the energy (or another property). SOS approaches consist in evaluating energies and transition dipoles that appear in the (hyper)polarizability expressions. For instance, in the case of the frequency-dependent electric-dipole electronic first hyperpolarizability, the SOS expression reads ... 

The summation over final states /> has been carried out by the completeness relation, or equivalently, by matrix multiplication. Thus, the zeroth moment is just an equilibrium average of the square of the perturbation amplitude. The first moment can likewise be expressed as an equilibrium property. 

Total derivatives are appropriate here because the properties in the standard state are functions of temperature only. Multiplication of both sides of this equation by Vi and summation over all species gives... 

Equation (2) is an example of a sum-over-states (SOS) expression of a molecular response property. It suggests an easy way of computing / , but in practice the SOS approach is rarely taken because of its very slow convergence, i.e., because of the need to compute many excited states wavefunctions. The summation goes over all excited states and also needs to include, in principle, the continuum of unbound states. As it will be shown below, there are more economic ways of computing [1 within approximate first-principles electronic structure methods. 

In order to facilitate deduction of excited-state properties (Johnson and Peticolas, 1976 Warshel, 1977) considerable activity ensued in the analysis of resonance Raman spectra and excitation profiles. However, progress was limited by the conceptual and computational clumsiness of Eq. (6.1-1). Only for small molecules it has been possible to calculate resonance Raman spectra numerically by means of Eq. (6.1-1) (see for example Strempel and Kiefer, 1991 a-c and references therein). The main problem arises through the summation over many states which for large polyatomic molecules is nearly impossible to perform. 

The popularity of the SOS methods in calculations of non-linear optical properties of molecules is due to the so-called few-states approximations. The sum-over-states formalism defines the response of a system in terms of the spectroscopic parameters, like excitations energies and transition moments between various excited states. Depending on the level of approximation, those states may be electronic or vibronic or electronic-vibrational-rotational ones. Under the assumption that there are few states which contribute more than others, the summation over the whole spectrum of the Hamiltonian can be reduced to those states. In a very special case, one may include only one excited state which is assumed to dominate the molecular response through the given order in perturbation expansion. The first applications of two-level model to calculations of j3 date from late 1970s [93, 94]. The two-states model for first-order hyperpolarizability with only one excited state included can be written as ... 

From the results in the last section it is clear that for particular applied radiative frequencies or frequency multiples, close to resonance with particular molecular states, each molecular tensor will be dominated by certain terms in the summation of states as a result of their diminished denominators—a principle that also applies to all other multiphoton interactions. This invites the possibility of excluding, in the sum over molecular states, certain states that much less significantly contribute. Then it is expedient to replace the infinite sum over all molecular states by a sum over a finite set—this is the technique employed by computational molecular modelers, their results often producing excellent theoretical data. In the pursuit of analytical results for near-resonance behavior, it is often defensible to further limit the sum over states and consider just the ground and one electronically excited state. Indeed, the literature is replete with calculations based on two-level approximations to simplify the optical properties of complex molecular systems. On the other hand, the coherence features that arise through adoption of the celebrated Bloch equations are limited to exact two-level systems and are rarely applicable to the optical response of complex molecular media. 

In order to evaluate this contribution one needs only all excitation energies and corresponding transition dipole moments for molecule A and also for molecule B. Both can be obtained from the poles and residues of a polarization propagator for molecule A and separately for molecule B as described in Section 7.4. However, it is preferable to avoid the simultaneous summation over all states and express the dispersion energy in terms of molecular properties. This can be achieved by using the following integral transform... 

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