Coulomb Schrodinger equation

The absorption rate can be calculated by treating —iVabs as a small perturbation and by adopting the Coulomb Schrodinger equation... 

The absorption potential — z(1,3Vabs) is quite weak compared with the Coulomb potential, as is noted below Eq. (109). Therefore,1,3in Eq. (115) can normally be replaced by the solution of the spin-independent Coulomb Schrodinger equation... 

The problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

This, at first glance innocuous-looking functional FHK[p] is the holy grail of density functional theory. If it were known exactly we would have solved the Schrodinger equation, not approximately, but exactly. And, since it is a universal functional completely independent of the system at hand, it applies equally well to the hydrogen atom as to gigantic molecules such as, say, DNA FHK[p] contains the functional for the kinetic energy T[p] and that for the electron-electron interaction, Eee[p], The explicit form of both these functionals lies unfortunately completely in the dark. However, from the latter we can extract at least the classical Coulomb part J[p], since that is already well known (recall Section 2.3),... 

Calculations of the electronic structure of molecules, crystals and surfaces are often performed in atomic units. They are defined by setting the most important constants equal to unity h — eo — me — 1, where me is the electronic mass. The Coulomb law is written in electrostatic units V(r) = q/r, so that the time-independent Schrodinger equation for the hydrogen atom takes on the simple form ... 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

Technically, the time-independent Schrodinger equation (2) is solved for clamped nuclei. The Hamiltonian is broken into its electronic part, He, including the nuclear Coulomb repulsion energy, and the nuclear Hamiltonian HN. At this level, mass polarization effects are usually neglected. The wave function is therefore factorized as usual (r,X)= vP(r X)g(X). Formally, the electronic wave function d lnX) and total electronic energy, E(X), are obtained after solving the equation for each value of X ... 

We consider an /V-electron system where the electrons experience the mutual Coulomb interaction along with an external potential vv( r) due to the nuclei. The system is then subjected to an additional TD scalar potential (r, t) and a TD vector potential A(r, t). The many-electron wavefunction [Pg.74]

Consequently, from the density the Hamiltonian can be readily obtained, and then every property of the system can be determined by solving the Schrodinger equation to obtain the wave function. One has to emphasize, however, that this argument holds only for Coulomb systems. By contrast, the density functional theory formulated by Hohenberg and Kohn is valid for any external potential. Kato s theorem is valid not only for the ground state but also for the excited states. Consequently, if the density n, of the f-th excited state is known, the Hamiltonian H is also known in principle and its eigenvalue problem ... 

The singularity in the interelectron Coulomb potential r 1 creates a cusp in the exact solution of the Schrodinger equation,... 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

It is interesting to note that the Coulomb matrix and the matrix of the nuclear potential present in Vc are opposite in sign. This means that an underestimation, or complete neglect, of the Coulomb matrix will lead to a larger Vc and thus to an overestimation of the relativistic effect. If Vc is negligable compared to 2c the ZORA equation reduces to the non relativistic Schrodinger equation. 

No assumptions concerning potentials u r,r ) and v(r) (like Coulombic character), unless they lead to bounded solutions of the Schrodinger equation [see Eqs. (10)-(12)]... 

Momentum-space methods, pioneered by McWeeny, Fock, Shibuya, Wulfman, Judd, Koga, Aquilanti and others [4,17-26] provide us with an easy and accurate method for constructing solutions to the Schrodinger equation of a single electron moving in a many-center Coulomb potential... 

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