Single particle wave functions

Writing the Euler-Lagrange equations in terms of the single-particle wave functions (tpi) the variation principle finally leads to the effective singleelectron equation, well-known as the Kohn-Sham (KS) equation ... 

The next step is the decomposition of the total density into single particle densities which are related to single particle wave functions by... 

In the next step, which is numerically the most demanding, the differential equations (3) are solved. Two possible strategies using a variational expansion of the single particle wave functions, /., are described below. After the eigenvalues and eigenfunctions have been found, a new ("output") charge density can be... 

We may express the single-particle wave function tpniqd fhe product of a spatial wave function 0n(r,) and a spin function % i). For a fermion with spin such as an electron, there are just two spin states, which we designate by a(i) for m = and f i) for Therefore, for two particles there are three... 

Now, if the many-body (electron) problem can be arranged in such a way that the many-body, nonseparable wave function is expressed in terms of a separable wave function, which depends on N single-particle wave functions (Hartree approximation), i.e.,... 

In Section 8.4.2, we considered the problem of the reduced dynamics from a standard DFT approach, i.e., in terms of single-particle wave functions from which the (single-particle) probability density is obtained. However, one could also use an alternative description which arises from the field of decoherence. Here, in order to extract useful information about the system of interest, one usually computes its associated reduced density matrix by tracing the total density matrix p, (the subscript t here indicates time-dependence), over the environment degrees of freedom. In the configuration representation and for an environment constituted by N particles, the system reduced density matrix is obtained after integrating pt = T) (( over the 3N environment degrees of freedom, rk Nk, ... 

The basic variable in density functional theory (DFT)22 is the electron density n(r). In the usual implementation of DFT, the density is calculated from the occupied single-particle wave functions (r) of an auxiliary system of noninteracting electrons... 

Solve the Kohn-Sham equations defined using the trial electron density to find the single-particle wave functions, i i,(r). 

As an example of the preceding analysis, we examine the rotation of the single particle wave function of a trapped charged particle around the z-axis. We consider a particle with charge q and mass m in a Penning trap... 

The single-particle wave function for the free photoelectron may be expressed as an expansion in angular momentum partial waves characterized by an orbital angular momentum quantum number l and and associated quantum number X for the projection of l on the molecular frame (MF) z axis [22, 23, 63-66],... 

In usual practice, all single-particle wave functions and energies are typically obtained by solving the single-particle Kohn-Sham equation of density-functional theory in the so-called local-density approximation (LDA) (see, e.g.. Ref. [48]). 

The exchange density in equations (8.52) and (8.53) is important only when the single-particle wave functions V (q) and overlap substantially. 

Density Functional Theory and the Local Density Approximation Even in light of the insights afforded by the Born-Oppenheimer approximation, our problem remains hopelessly complex. The true wave function of the system may be written as i/f(ri, T2, T3,. .., Vf ), where we must bear in mind, N can be a number of Avogadrian proportions. Furthermore, if we attempt the separation of variables ansatz, what is found is that the equation for the i electron depends in a nonlinear way upon the single particle wave functions of all of the other electrons. Though there is a colorful history of attempts to cope with these difficulties, we skip forth to the major conceptual breakthrough that made possible a systematic approach to these problems. 

Non-relativistic quantum theory of atoms and molecules is built upon wave-functions constructed from antisymmetrized products of single particle wave-functions. The same scheme has been adopted for relativistic theories, the main difference now being that the single particle functions are 4-component spinors (bispinors). The finite matrix method approximates such 4-spinors by writing... 

In order to apply the perturbation formalism, we now need to calculate the matrix elements of v and v with respect to the unperturbed single-particle wave functions. These will be taken as the eigenfunctions of the momentum operator, normalized in a cubic box of volume Q with periodic boundary conditions ... 

In this figure we have used both a harmonic-oscillator (HO) and a Brueckner-Hartree-Fock (BHF) basis for the single-particle wave functions in order to study the behavior of the RS perturbation theory at low orders. What can be seen from this figure is that the BHF basis yields a smaller overlap between states in the excluded space and the model space, reflected in the small change when going from second order to third order in the perturbation expansion. However, the BHF spectra are too compressed and in poor agreement with experiment. This is probably related to the fact that the radii obtained for the self-consistent single-particle wave functions are much smaller than the empirical ones [53]. 

As shown below, this finding is almost independent of the order of the perturbative expansion, rather it seems to be a property akin to the potential itself, if we apply the harmonic oscillator choice for the single-particle wave functions. A similar observation was made by us in Ref. [8]. Its implications will be discussed below. 

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