Summation over states

Ab Initio Summation Over States/SCI for Static and Dynamic Hyperpolarizabilities of Small Molecules... 

Ab initio summation over states/SCI for static and dynamic first hyperpolarizabilities of small molecules... 

For molecules with very small moments of inertia, for example, LiH, the rotational energy levels are widely spaced with respect to kgT. In this case, converting the summation over states into an integral is not appropriate. However, because very few excited rotational levels are likely to be occupied, one can evaluate qrot by performing the summation explicitly,... 

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],... 

To apply this result to thiophosgene, we consider the vibronic interaction induced by the activity of the antisymmetric out-of-plane vibrational mode Q4. The summation over states in Eq. (3) in the simplest case can be reduced to a single term, the state S2. 

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]. 

The electrostatic effects associated with the relative orientation of the chromophores show a specific connection with the corresponding electronic excited states and dipole transition moments that can, in turn, be related to changes in the hyperpolarizabilities via summation-over-states (SOS) expressions [1 3]. When the molecules interact, the electronic excited states split. In the case of a collinear arrangement, the intensities (oscillator strengths) are shifted to the red, that is, to the states of lower energies, whereas for a side by side and parallel configuration, the intensities are shifted to the blue. Furthermore, since the excited state polarizabilities are usually larger than the... 

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