Single-electron wave functions

Because single-electron wave functions are approximate solutions to the Schroe-dinger equation, one would expect that a linear combination of them would be an approximate solution also. For more than a few basis functions, the number of possible lineal combinations can be very large. Fortunately, spin and the Pauli exclusion principle reduce this complexity. 

One of the advantages of this method is that it breaks the many-electron Schrodinger equation into many simpler one-electron equations. Each one-electron equation is solved to yield a single-electron wave function, called an orbital, and an energy, called an orbital energy. The orbital describes the behavior of an electron in the net field of all the other electrons. 

A useful way to write down the functional described by the Hohenberg-Kohn theorem is in terms of the single-electron wave functions, vji (r). Remember from Eq. (1.2) that these functions collectively define the electron density, (r). The energy functional can be written as... 

These equations are superficially similar to Eq. (1.1). The main difference is that the Kohn-Sham equations are missing the summations that appear inside the full Schrodinger equation [Eq. (1.1)]. This is because the solution of the Kohn-Sham equations are single-electron wave functions that depend on only three spatial variables, ij ,(r). On the left-hand side of the Kohn-Sham equations there are three potentials, V, VH, and Vxc- The first... 

If you have a vague sense that there is something circular about our discussion of the Kohn-Sham equations you are exactly right. To solve the Kohn-Sham equations, we need to define the Hartree potential, and to define the Hartree potential we need to know the electron density. But to find the electron density, we must know the single-electron wave functions, and to know these wave functions we must solve the Kohn-Sham equations. To break this circle, the problem is usually treated in an iterative way as outlined in the following algorithm 1 2 3... 

We then applied this formula to various types of single-electron wave functions, for example s,p,d, /, g, and to wave functions for various Russell-Saunders terms characterized by integral values of the quantum number L. 

We consider a metallic phase of volume V, with N free electrons, and hence a free electron density n given by n = NjV. The charge of the core ions is smeared out and leads to a potential energy well keeping the free electrons in the metallic phase (Figure 3). Since in the Sommerfeld model the electrons do not interact with each other, we can describe the electron energy levels by one-electron wave functions. An independent electron can be described by a single-electron wave function ij/ x,y,z) which satisfies... 

The antisymmetric wave function, shown in Equation 28-SI. is a more compact way of writing a Slater determinant (Eq. 28-53). In a Slater determinant, an exchange of any two rows or columns results in the same wave function multiplied by -1. This is another statement of the Pauli exelusion principle. The columns in Equation 28-53 are the single electron wave functions. Equation 28-53. however, is only an approximation, since the electrons are independent of one another and therefore not eorrelated. This eorrelation problem reveals itself when ealculating the energies and is di.s-cussed below. 

Our calculations for multi-electron atoms in magnetic fields are carried out under the assumption of an infinitely heavy nucleus in the (unrestricted) Hartree-Fock approximation. The solution is established in the cylindrical coordinate system (p, p, z) with the 2-axis oriented along the magnetic field. We prescribe to each electron a definite value of the magnetic quantum number mp. Each single-electron wave function depends on the variables p and (p, z)... 

Most of the approaches for the approximate computation of electronic energies are based on the assumption that the electrons are independent of one another. We may then write the electronic wave function cp of the system in terms of separable single-electron wave functions (pf as follows ... 

Consider the amplitude for the creation of secondary electrons upon atom excitation by electron impact. As a result of the Coulomb interaction with the atom, the incident electron loses a part of its energy and goes into an inelastically scattered, state and the atom goes into an excited state characterized by a core hole and a secondary electron. In the context of the single-electron approach, the initial state of the system is characterized by i) = w, a) and the final states are characterized by I/) = Ip, ) where u)) and h) are single-electron wave functions of the incident and inelastically scattered electrons, and p) and a) are singleelectron wave functions of the secondary election and the core level electron, respectively. Then the amplitude for creation of the secondary electron is defined by the matrix element... 

Here E (k) denotes the energy dispersion of a dilute gas of qnasiparticles. In systems with strong correlations it cannot be calcnlated from the overlap of single-electron wave functions. The interactions among the quasiparticles are characterized by the matrix k ). The... 

From the Hartree point, every electron state can be described by a single electron wave function, /,(j ), if the electron-electron interactions are negligible. The wave function of the whole system containing N electrons is written as /(T, X2,. .. Tt,... . 1 ) = /i(x ) /2( ).. - /n( )-... 

Density functional methods provide a convenient framework for treating metallic interfaces [100]. Applications of this methodology to the problem of electron transport through atomic and molecular bridges have been advanced by several workers. In particular, Lang s approach [90, 159-165] is based on the density functional formalism [166,167] in which the single electron wave functions i/ o (r) and the electron density o(r) for two bare metal (jellium) electrodes is computed, then used in the Lippman-Schwinger equation... 

In the case of clusters, a more detailed link with experiment can be established by computing the ionization probabilities for the various charge states. This information can be reliably well extracted from the single-electron wave functions of TDLDA, as has been demonstrated for the case of atoms [101]. First applications for clusters can be found in [102, 103]. 

The development of DFT is based on Kohn and Hohenberg s mathematical theorem, which states that the ground state of the electronic energy can be calculated as a functional of the electron density [18], The task of finding the electron density was solved by Kohn and Sham [19]. They derived a set of equations in which each equation is related to a single electron wave function. From the single electron wave functions one can easily calculate the electron density. In DFT computer codes, the electron density of the core electrons, that is, those electrons that are not important for chemical bonds, is often represented by a pseudopotential that reproduces important physical features, so that the Kohn-Sham equations span only a select number of electrons. For each type of pseudopotential, a cutoff energy or basis set must be specified. 

Refinement of the density function, and consequently of the energy and other related properties, is then performed, using a representation of the electron density through a set of orthonormal single-electron wave functions, by the method of Kohn and Sham (1965), which takes the correlation and exchange into account. 

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