Molecular states, bound, Schrodinger equation

Bound-state photoabsorption, direct molecular dynamics, nuclear motion Schrodinger equation, 365-373... 

Schrodinger equation for each molecular electronic state, we seek an approximation that will represent U reasonably well for most diatomics. For a bound electronic state, we know that U has the general appearance of the solid curve in Fig. 4.2. We expect the nuclei to vibrate about the position of minimum potential energy therefore, we expand U in a Taylor series (Section 1.2) about Re, the equilibrium internuclear separation ... 

The rationale behind this approach is the variational principle. This principle states that for an arbitrary, well-behaved function of the coordinates of the system (e.g., the coordinates of all electrons in case of the electronic Schrodinger equation) that is in accord with its boundary conditions (e.g., molecular dimension, time-independent state, etc.), the expectation value of its energy is an upper bound to the respective energy of the true (but possibly unkown) wavefunction. As such, the variational principle provides a simple and powerful criterion for evaluating the quality of trial wavefunctions the lower the energetic expectation value, the better the associated wavefunction. 

There have been a number of instances where EAs have been used to obtain approximate solutions to the Schrodinger equation. Zeiri et al. use a real-value encoding to aid in the calculation of bound states in a double well potential and in the non-linear density functional calculation. Rossi and Truhlar have devised a GA to fit a set of energy differences obtained by NDDO semiempirical molecular orbital theory to reference ab initio data in order to yield specific reaction parameters. The technique was applied to the reaction Cl -I- (THa. In a third example, Rodriguez et al. apply a GA to diagonalization of the... 

In view of this discussion, it is tempting to conceptualize the weakly-bound anions of polar molecules as dipole-bound anions. Solution of the Schrodinger equation for an electron interacting with a point dipole reveals that bound states are obtained for dipole moments g > 1.625 debye, ° ° with no further molecular structure required. (In practice this threshold should be modified to something like g > 2.4 debye, owing to the possibility of rotational-to-electronic energy transfer, but the point remains that... 

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