Potential energy function, equivalent representations

Each time step thus involves a calculation of the effect of the Hamilton operator acting on the wave function. In fully quantum methods the wave function is often represented on a grid of points, these being the equivalent of basis functions for an electronic wave function. The effect of the potential energy operator is easy to evaluate, as it just involves a multiplication of the potential at each point with the value of the wave function. The kinetic energy operator, however, involves the derivative of the wave function, and a direct evaluation would require a very dense set of grid points for an accurate representation. 

The calculation of a point on a potential-energy hypersurface is equivalent to calculating the energy of a diatomic or polyatomic system for a specified nuclear configuration and thus presents considerable practical computational difficulty. For certain problems or nuclear configurations, the maximum possible accuracy is needed, and under these conditions relatively elaborate ab initio methods are indicated. For other problems, the description to a uniform accuracy of many electronically excited states of a given system is required. Such is the situation for the atmospheric systems described here, and thus most of our final potential curves are based on the analysis of VCI wave functions constructed to uniform quality for representation of the excited states. 

Equations 12.14 and 12.16 are solved for the surface function S and electrostatic potential 0, respectively. These coupled "Laplace-Beltrami and Poisson-Boltzmann" equations are the governing equation for the DG-based solvation model in the Eulerian representation. The Lagrangian representation of the DG-based solvation model has also been derived [72]. Both the Eulerian and Lagrangian solvation models have been shown [71, 72] to be essentially equivalent and provide very good predictions of solvation energies for a diverse range of compounds. 

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