Basis functions Coulomb potential derivatives

In modem quantum chemistry packages, one can obtain moleculai basis set at the optimized geometry, in which the wave functions of the molecular basis are expanded in terms of a set of orthogonal Gaussian basis set. Therefore, we need to derive efficient fomiulas for calculating the above-mentioned matrix elements, between Gaussian functions of the first and second derivatives of the Coulomb potential ternis, especially the second derivative term that is not available in quantum chemistry packages. Section TV is devoted to the evaluation of these matrix elements. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

The choice of Internal coordinates as an object for optimisation Is obvious use of rotational constants maybe less so. They certainly do not give very detailed Information about the conformation of a molecule, but they are the primary structural Information derived from rotational and ro-vlb spectroscopy on small molecules. The Inclusion of dipole moments Is a must when Coulomb terms are present In the potential energy function. Charges are Included, although they are not experimentally observable quantities, because It may be desirable to lock a parameter set to data derived from photoelectron spectroscopy or from ab Initio calculations with a large basis set. Quite naturally we want to optimise on vibrational spectra, and we shall see below that It Is a bit more cumbersome In the consistent force field context than In traditional normal coordinate analysis. 

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