Single-particle wavefunction

Kohn and Sham provided a further contribution to make the DFT approach useful for practical calculations, by introducing the concept of fictitious non-interacting electrons with the same density as the true interacting electrons [8]. Non-interacting electrons are described by orthonormal single-particle wavefunctions y/i (r) and their density is given by ... 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

Renaming the electron numbers shows that the first and fourth terms in the matrix element M i(M1,ms) and also the second and third terms are identical, and they can be combined. As a next step one calculates the action of the photon operator on the single-particle wavefunctions. Omitting for simplicity the wavefunction... 

All three forms of the dipole matrix element are equivalent because they can be transformed into each other. However, this equivalence is valid only for exact initial- and final-state wavefunctions. Since the Coulomb interaction between the electrons is responsible for many-body effects (except in the hydrogen atom), and the many-body problem can only be solved approximately, the three different forms of the matrix element will, in general, yield different results. The reason for this can be seen by comparing for the individual matrix elements how the transition operator weights the radial parts R r) and Rf(r) of the single-particle wavefunction differently ... 

When discussing the electronic spectra and optical properties of Ceo, it is important to include the effect of screening of the external field. Using the ground state single-particle wavefunctions and energy levels, the dipole transition matrix elements and transition frequencies have been evaluated in order... 

The energy functional of any many-electron system in the one-particle approximation with the single-particle wavefunction can be written as... 

Except in the ad-atom s vicinity, the metal screens out the effects of the ad-atom on the total charge density and on the Kohn-Sham potential, even though the disturbance in the individual Kohn-Sham single-particle wavefunctions is not short-ranged [38, p. 618]. That result is reproduced in Figure 2, together with the... 

We have resorted to an approximate technique which attempts to include the above mentioned main quantum effects via the construction of effective potentials V. Basically, each pmticle is represented by a single particle wavefunction tmd the Ehrenfest theorem is applied. Similar ideas have been used with good success ev( n for quantum solids like hydrogmi [38]. Effective quantum potentials ajx also among the results of the Feynman-Hibbs treatment [12] which have been apjjlied to pure neon clusters in the past [34]. 

The flux is defined up to the constant A. Taking A = (a single particle wavefunction normalized in the volume Q) implies that the relevant observable is 2VJ(r), that is, is the particle flux for a system with a total of N particles with N Sometimes it is convenient to normalize the wavefunction to unit flux, J = 1 by choosing A =... 

In order to begin to understand the ideas behind, and information available from experiments probing MQCOH in polymers, let us remind ourselves of the meaning of phase coherence in quantum mechanics. We start with the simplest case, a noninteracting ensemble of spin j systems, and with spin basis functions that are eigen functions of the largest interaction present, the Zeeman Hamiltonian These are I, m) =, ) = a), and, - ) = )3). A spin 2 system will have single particle wavefunction... 

A A th single-particle wavefunction in which the global phase, 6, on the two basis functions is the same. 

Also shown in Fig. 4.4 is the single-particle wavefunction, of the mid-gap state, which is localized at the soliton. In the continuum limit, a,... 

Consistent with time-independent Hartree-Fock theory the main approximation in time-dependent Hartree-Fock theory is, that the system is represented by a single Slater determinant, which now is composed of time-dependent single-particle wavefunctions. The time-dependent Schrodinger equation that has to be solved is given in eqn (1). The time-dependent Hamiltonian consists of a static Hamiltonian and an additional time-dependent operator describing the time-dependent perturbation, e.g. an electric field, which is a sum of time-dependent single-particle potentials ... 

In (2.3) the first three terms on the right-hand side (Ek, Eh, Exc) describe all the electron-electron interactions, the fourth term (Eei) refers to the electron-nucleus interaction and the fifth one ( //) corresponds to the nucleus-nucleus interaction. More explicitly, Ek is the Schrddinger-like kinetic energy expressed in terms of the single-particle wavefunctions, (x) as... 

MOs first appear in the framework of the Hartree-Fock (HF) method, which is a mean-field treatment [17,22]. The basic idea is to start from an A-particle wave-function that is appropriate for a system of non-interacting electrons. Having fixed the Ansatz for the A-particle wavefunetion in this way, the variational principle is used in order to obtain the best possible approximation for the fully interacting system. Such independent particle wavefunctions are Slater-determinants, which consist of antisymmetrized products of single-particle wavefunctions (x)J (the antisymmetry brought about by the determinantal form is essential in order to satisfy die Pauh principle). Thus, the Slater-determinant is written as... 

We wish to derive the Lindhard dielectric response function for the free-electron gas, using perturbation theory. The charge density is defined in terms of the single-particle wavefunctions as... 

A different formulation of Bloch s theorem is that the single-particle wavefunctions must have the form... 

The formulation of the TEA up to this point has assumed that the TEA hamiltonian matrix elements depend only on two single-particle wavefunctions centered at two different atomic sites. Eor example, we assumed that the hamiltonian matrix elements depend only on the relative distance and orientation of the two atomic-like orbitals between which we calculate the expectation value of the single-particle hamiltonian. This is referred to as the two-center approximation, but it is obviously another implicit approximation, on top of restricting the basis to the atomic-like wavefunctions. In fact, it is plausible that, in the environment of the solid, the presence of other electrons nearby will affect the interaction of any two given... 

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