Kinetic energy, Schrodinger

Schrodinger postulated that the form of the hamiltonian in quantum mechanics is obtained by replacing the kinetic energy in Equation (1.20), giving... 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

It turns out that the Klein-Gordon equation cannot describe electron spin in the limit of small kinetic energy, it can be shown to reduce to the familiar Schrodinger equation. 

The one-electron Hamiltonian operator h = h +v, with kinetic energy operator, generates a complete spectrum of orbitals according to the Schrodinger equation... 

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),... 

The Schrodinger equation applied to atoms will thus describe the motion of each electron in the electrostatic field created by the positive nucleus and by the other electrons. When the equation is applied to molecules, due to the much larger mass of nuclei, their relative motion is considered negligible as compared to that of the electrons (Bom-Oppenheimer approximation). Accordingly, the electronic distribution in a molecule depends on the position of the nuclei and not on their motion. The kinetic energy operator for the nuclei is considered to be zero. 

In this picture, the nuclei are moving over a PES provided by the function V(R), driven by the nuclear kinetic energy operator, TN. More details on the derivation of this equation and its validity are given in Appendix A. The potential function is provided by the solutions to the electronic Schrodinger equation,... 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

The Schrodinger equation for nuclear motion contains a Hamiltonian operator Hop,nuc consisting of the nuclear kinetic energy and a potential energy term which is Eeiec(S) of Equation 2.7. Thus... 

To give the reader some further appreciation of the adiabatic correction, we next discuss the so-called Rydberg correction of the hydrogenic atom. The kinetic energy of the electron in the hydrogen atom is expressed by Equation 2.5, where mj is set equal to the electron mass, me. It is well known that the Schrodinger equation for the hydrogen atom separates into two equations, one of which deals with the motion of the center of mass of the system and the other with the motion of the electron... 

In the variational calculations, the expansions of the kinetic energy factors and the pseudo-potential 1/ [1] are taken to fourth order, and, as mentioned above, the potential energy V is expanded through sixth order. In the numerical integration of the inversion Schrodinger equation a grid of 1000 points is used. The basis set [1] is truncated so that... 

The method of many-electron Sturmian basis functions is applied to molecnles. The basis potential is chosen to be the attractive Conlomb potential of the nnclei in the molecnle. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nnclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

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