Schrodinger equation periodic

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. 

W.Kohn and Rostoker, Solution of the Schrodinger equation in periodic lattices with an apphcation to metaUic hthium , Phys.Rev.94 1111 (1954). 

Suppose we get a little more sophisticated about our question. The more advanced student might respond that the periodic table can be explained in terms of the relationship between the quantum numbers which themselves emerge from the solutions to the Schrodinger equation for the hydrogen atom.5... 

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. 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

For a periodic lattice, it can be shown (Bloch theorem) that the solutions to the one-electron Schrodinger equation are of the... 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrodinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

Ab initio quantum mechanics is based on a rigorous treatment of the Schrodinger equation (or equivalent matrix methods)4-7 which is intellectually satisfying. While there are a number of approximations made, it relies on a set of equations and a few physical constants.8 The use of ab initio methods on large systems is limited if not impossible, even with the fastest computers available. Since the size of an ab initio calculation is defined by the number of basis functions in the system, ab initio calculations are extremely costly for anything past the second row in the periodic table, and for all systems with more than 20 or 30 total atoms. 

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... 

Because spatially localized functions are the natural choice for isolated molecules, the quantum chemistry methods developed within the chemistry community are dominated by methods based on these functions. Conversely, because physicists have historically been more interested in bulk materials than in individual molecules, numerical methods for solving the Schrodinger equation developed in the physics community are dominated by spatially periodic functions. You should not view one of these approaches as right and the other as wrong as they both have advantages and disadvantages. 

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 ... 

We know that not all solids conduct electricity, and the simple free electron model discussed previously does not explain this. To understand semiconductors and insulators, we turn to another description of solids, molecular orbital theory. In the molecular orbital approach to bonding in solids, we regard solids as a very large collection of atoms bonded together and try to solve the Schrodinger equation for a periodically repeating system. For chemists, this has the advantage that solids are not treated as very different species from small molecules. 

The explanation of the periodic system by quantum mechanics, for example, is only partial. The possible lengths of the various periods in the table follow deductively from the solution of the SchrOdinger equation for the hydrogen atom and the relationship between the four quantum numbers, which is also obtained deductively. However, the repetition of all but the first period length remains a source of debate (/). The repetition of all the other period lengths has not been deduced from first principles however (2). Stated more precisely, the empirical order in which the atomic orbitals are filled has not been deduced. If this were possible the explanation for the lengths of successive periods, including the repetitions, would follow trivially. 

Quantum chemistry is a diverse discipline which uses many different methods to correlate a wide variety of phenomena. In the earliest period of the subject the Schrodinger equation was solved exactly for a few simple model situations. These model solutions were then used to interpret the spectra, kinetics, and thermodynamics of molecules and solids. 

During this period, accurate solutions for the electronic structure of helium (1) and the hydrogen molecule (2) were obtained in order to verify that the Schrodinger equation was useful. Most of the effort, however, was devoted to developing a simple quantum model of electronic structure. Hartree (3) and others developed the self-consistent-field model for the structure of light atoms. For heavier atoms, the Thomas-Fermi model (4) based on total charge density rather than individual orbitals was used. 

According to this model, the solution of the Schrodinger equation for the electron in periodical potential is the sum of two flat waves... 

J.H. Shirley, Solution of the Schrodinger equation with a Hamiltonian periodic in time, Phys. Rev. 138 (4B) (1965) B979. 

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