Atomic structure Schrodinger equation

In the discussion of the electronic structure of atoms, the Schrodinger equation could be reduced to one involving only the electrons. This was achieved by separating the electronic energy of the atom from the nuclear kinetic energy, which is essentially determined by the translational motion of the atom. 

Much of quantum chemistry attempts to make more quantitative these aspects of chemists view of the periodic table and of atomic valence and structure. By starting from first principles and treating atomic and molecular states as solutions of a so-called Schrodinger equation, quantum chemistry seeks to determine what underlies the empirical quantum numbers, orbitals, the aufbau principle and the concept of valence used by spectroscopists and chemists, in some cases, even prior to the advent of quantum mechanics. 

The Hydrogenic atom problem forms the basis of much of our thinking about atomic structure. To solve the corresponding Schrodinger equation requires separation of the r, 0, and (j) variables... 

There are two types of basis functions (also called Atomic Orbitals, AO, although in general they are not solutions to an atomic Schrodinger equation) commonly used in electronic structure calculations Slater Type Orbitals (STO) and Gaussian Type Orbitals (GTO). Slater type orbitals have die functional form... 

And yet in spite of these remarkable successes such an ab initio approach may still be considered to be semi-empirical in a rather specific sense. In order to obtain calculated points shown in the diagram the Schrodinger equation must be solved separately for each of the 53 atoms concerned in this study. The approach therefore represents a form of "empirical mathematics" where one calculates 53 individual Schrodinger equations in order to reproduce the well known pattern in the periodicities of ionization energies. It is as if one had performed 53 individual experiments, although the experiments in this case are all iterative mathematical computations. This is still therefore not a general solution to the problem of the electronic structure of atoms. 

The initial purpose of pioneer quantum mechanics was to provide the theoretical framework to account for the structure of hydrogen and the nuclear model of atoms in general. The final result, a quantum theory of atomic structure can be discussed in terms of the time-independent Schrodinger equation, in its most general form... 

Solutions to the Schrodinger equation Hcj) = E(f> are the molecular wave functions 0, that describe the entangled motion of the three particles such that (j) 4> represents the density of protons and electron as a joint probability without any suggestion of structure. Any other molecular problem, irrespective of complexity can also be developed to this point. No further progress is possible unless electronic and nuclear variables are separated via the adiabatic simplification. In the case of Hj that means clamping the nuclei at a distance R apart to generate a Schrodinger equation for electronic motion only, in atomic units,... 

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 ... 

Our model of positive atomic cores arranged in a periodic array with valence electrons is shown schematically in Fig. 14.1. The objective is to solve the Schrodinger equation to obtain the electronic wave function ( ) and the electronic energy band structure En( k ) where n labels the energy band and k the crystal wave vector which labels the electronic state. To explore the bonding properties discussed above, a calculation of the electronic charge density... 

Consider an electron of mass m and charge e moving on a two-dimensional surface that defines the x,y plane (perhaps the electron is constrained to the surface of a solid by a potential that binds it tightly to a narrow region in the z-direction), and assume that the electron experiences a constant potential Vo at all points in this plane (on any real atomic or molecular surface, the electron would experience a potential that varies with position in a manner that reflects the periodic structure of the surface). The pertinent time independent Schrodinger equation is ... 

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