Poisson-Schrodinger equations

Therefore it is necessary to solve simultaneously the quantum mechanical problem (the Schrodinger equation) and the classical electrostatic problem (the Poisson and Laplace equations) using the boundary conditions as defined by the physical problem. 

The mutual polarization process between the solute and the polarizable medium is obtained by solving a system of two coupled equations, i.e., the QM Schrodinger equation for the solute in presence of the polarized dielectric, and the electrostatic Poisson equation for the dielectric medium in presence of the charge distribution (electrons and nuclei) of the solute. The solute occupies a molecular shaped cavity within the dielectric continuum, whose polarization is represented by an apparent surface charge (ASC) density spread on the cavity surface. The solute-solvent interaction is then represented by a QM operator, the solvent reaction potential operator, Va, corresponding to the electrostatic interaction of the solute electrons and nuclei with the ASC density of the solvent. 

The Polarizable Continuum Model (PCM)[18] describes the solvent as a structureless continuum, characterized by its dielectric permittivity e, in which a molecular-shaped empty cavity hosts the solute fully described by its QM charge distribution. The dielectric medium polarized by the solute charge distribution acts as source of a reaction field which in turn polarizes back the solute. The effects of the mutual polarization is evaluated by solving, in a self-consistent way, an electrostatic Poisson equation, with the proper boundary conditions at the cavity surface, coupled to a QM Schrodinger equation for the solute. 

In general terms, we may say that the atomic or molecular calculations in quantum chemistry have the aim of finding self-consistent solutions of the Schrodinger equation and Poisson equation for the distributions of nuclei and electrons making up the system of our interest in the chosen state. 

In a natural way the problems of 5.3 combine some of the above elements of confinement and the binding of an electron by the hydrogen atom and by the hydrogen molecular ion, respectively. The superintegrability of both the Poisson and Schrodinger equations in common coordinate systems provide the tools for the analysis of alternative models for the binding of an electron in pure and complete dipole fields in 5.4. 

Scalar and vector fields that depend on more than one independent variable, which we write in the notation T fx, y), TCx, t), TCr), TCr, t), etc., are very often obtained as solutions to PDEs. Some classic equations of mathematical physics that we will consider are the wave equation, the heat equation, Laplace s equation, Poisson s equation, and the SchrOdinger equation for some exactly solvable quantum-mechanical problems. 

Before we discuss in detail the numerical discretization scheme used for the Poisson equation, which by the way is very similar to the discretization of the radial Schrodinger equation given in Eq. (9.120) [491], we sum up some of its general features. We consider the general form of the radial Poisson equation,... 

We consider here two of the most basic equations in computational chemistry the Poisson equation and the Schrodinger equation. The Poisson equation yields the electrostatic potential due to a fixed distribution of... 

Solution of the Schrodinger equation can be viewed similarly. The quantum energy functional (analog of the Poisson action functional above) is... 

W. D. Wilson, C. M. Schaldach, and W. L. Bourcier, Chem. Phys. Lett., 267, 431 (1997). Single- and Double-Layer Coupling of Schrodinger and Poisson-Boltzmann Equations. 

The one-particle equations for the wave functions p z, p) are solved by means of the fully numerical mesh method described in refs. [3,18,20]. In our first works on the helium atom in magnetic fields [18,20] we calculated the Coulomb and exchange integrals by means of a direct summation over the mesh nodes. But this direct method is very expensive with respect to the computer time and due to this reason we obtained in the following works [28-31] these potentials as solutions of the corresponding Poisson equation. The problem of the boundary conditions for the Poisson equation as well as the problem of simultaneously solving Poisson equation on the same meshes with Schrodinger-like equations for the wave functions p z, p) have been discussed in ref. [20],... 

The LSDA approach requires simultaneous self-consistent solutions of the Schrodinger and Poisson equations. This was accomplished using the Layer Korringa-Kohn-Rostoker technique which has many useful features for calculations of properties of layered systems. It is, for example, one of only a few electronic structure techniques that can treat non-periodic infinite systems. It also has the virtue that the computational time required for a calculation scales linearly with the number of different layers, not as the third power as most other techniques. 

We should not finish this discussion without mentioning the basis-set-fi-ee method of Becke. The grids used in this approach are the same as those described in Section 4.1. The grids were designed to accurately describe orbitals and densities in the neighborhood of each nucleus. A finite-difference approximation (in spherical polar coordinates) is used to solve Poisson s and Schrodinger s equations. The accuracy obtained with this basis-set-fi-ee approach for all-electron calculations is impressive, and, with the techniques described in Sections 3, 4.1 and 4,2, an 0(N) implementation is feasible. Delley s DMol program also uses a related approach. 

---