Wave function Coulomb potential derivatives

Numerical calculations of the Coulomb interaction O Eq. 23.68 are not trivial. They were simplified in Allan and Delerue (2008) by using a screened Coulomb potential involving one-electron wave functions which were derived for bulk semiconductor materials (Landsberg 1991). To calculate the impact ionization interaction from an initial state of Ret ri)Ye mi 0u(l>i)Re r2)Ye mi(d2,2) to a final state Re, ri)Ye mA i>h)Re,(r2)Ye, (02. 2) (the first term one in O Eq. 23.69), we notice... 

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. 

To improve on the wave function one has to accept that the standard multideterminantal expansion [Eq. (13.3)] is unsuitable for near-exact but practical approximations to the electronic wavefunction. The problem is dear from a simple analysis of the electronic Hamiltonian in Eq. (13.2) singularities in the Coulomb potential at the electron coalescence points necessarily lead to irregularities in first and higher derivatives of the exact wave function with respect to the interpartide coordinate, rj 2. The mathematical consequences of Coulomb singularities are known as electron-electron (correlation) and electron-nuclear cusp conditions and were derived by... 

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 Hamiltonian operator in Eq. 1 contains sums of different types of quantum mechanical operators. One type of operator in Ti gives the kinetic energy of each electron in by computing the second derivative of the electron s wave function with respect to all three Cartesian coordinates axes. There are also terms in H that use Coulomb s law to compute the potential energy due to (a) the attraction between each nucleus and each electron, (b) the repulsion between each parr of electrons, and (c) the repulsion between each pair of nuclei. 

For the ground electronic states, there is a one-to-one relation beween the wave function and the stationary Coulomb potential. This derives from eq.(5) following a line of analysis Gilbert used to study the density functional theory approach[13],... 

We have derived the total Hamiltonian expressed in a space-fixed (i.e. non-rotating) coordinate system in (2.36), (2.37) and (2.75). We can now simplify the electronic Hamiltonian 3Q,i by transforming the electronic coordinates to the molecule-fixed axis system defined by (2.40) because the Coulombic potential term, when expressed as a function of these new coordinates, is independent of 0, ip and x From a physical standpoint it is obviously sensible to transform the electronic coordinates in this way because under the influence of the electrostatic interactions, the electrons rotate in space with the nuclei. We shall take the opportunity to refer the electron spins to the molecule-fixed axis system in this section also, and leave discussion of the alternative scheme of space quantisation to a later section. Since we assume the electron spin wave function to be completely separable from the spatial (i.e. orbital) wave function,... 

To date, analytic first and second derivatives of the electronic energy functional (eq.(7)) with respect to geometric parameters defining position of the external sources of Coulomb potential have been used to actually obtain models for both the electronic wave function and the geometry [2]. The art of obtaining saddle points [29] has been now fully developed. 

In the Z — 1 limit, the kinetic terms involving derivatives remain, the centrifugal terms drop out, and the scaled Coulombic potentials metamorphize into delta functions. This hyperquantum limit is tantamount to — oo in the unsealed wave equation. For electronic structure, the low-D limit is generally less useful than the large-D limit, because only the ground state of a delta-function potential [2,5,9] is bound and that can accommodate only two electrons. However, for... 

Inclusion of the spin-orbit and other relativistic terms in Eq. (5), as we have done, is, strictly speaking, the most correct approach. This yields, as we have seen, a set of nuclear wave functions Xa(R) whose uncoupled motion is governed by the potentials (7a (R) and which are coupled only by the nuclear-derivative terms Fa (R) and Ga (R). In practice, though, Hrei(R) is difficult to treat on an equal footing with the coulombic terms in the Hamiltonian. Therefore one sometimes works with... 

Here, emphasis is given to the application of few-state models in the description of the near-resonant vacancy exchange between inner shells. It is well known that the quantities relevant for inner-shell electrons may readily be scaled. Therefore, the attempt is made to apply as much as possible analytic functional forms to describe the characteristic quantities of the collision system. In particular, analytic model matrix elements derived from calculations with screened hydrogenic wave functions are applied. Hydrogenic wave functions are suitable for inner shells, since the electrons feel primarily the nuclear Coulomb field of the collision particles. Input for the analytic expressions is the standard information about atomic ionization potentials available in tabulated form. This procedure avoids a fresh numerical calculation for each new collision system. 

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