Orbitals single-particle

The approximate molecular wavefunction T is a Slater determinant of the single particle orbitals (j). 

The twin facts that heavy-atom compounds like BaF, T1F, and YbF contain many electrons and that the behavior of these electrons must be treated relati-vistically introduce severe impediments to theoretical treatments, that is, to the inclusion of sufficient electron correlation in this kind of molecule. Due to this computational complexity, calculations of P,T-odd interaction constants have been carried out with relativistic matching of nonrelativistic wavefunctions (approximate relativistic spinors) [42], relativistic effective core potentials (RECP) [43, 34], or at the all-electron Dirac-Fock (DF) level [35, 44]. For example, the first calculation of P,T-odd interactions in T1F was carried out in 1980 by Hinds and Sandars [42] using approximate relativistic wavefunctions generated from nonrelativistic single particle orbitals. 

Equation 7.3 can alternatively be written [5] in terms of single particle orbitals as... 

By construction, the exact TD density of the interacting system can then be calculated from a set of noninteracting, single-particle orbitals fulfilling the TD-KS Equation 8.4 and reads... 

Figure 6.2 Energies of single-particle orbitals in harmonic-oscillator and rounded square-well potentials, the latter with and without spin-orbit coupling. Numbers in parentheses indicate orbital capacities and those in square brackets give cumulative capacity up to the given point. [Reproduced by permission from Gordon and Coryell (1967).]...

This case covers that discussed in the Example 2 of the previous section. While the example 1 can hardly be found in practice, since it rarely occurs in heavy nuclei that protons and neutrons occupy the same single particle orbitals, the example discussed... 

The results obtained here exhibit the connection between the surface delta interaction coupling schemes and dynamical symmetries. It is a challenging problem to see to what extent these symmetries are preserved in nuclei when the relevant single particle orbits are not degenerate. 

Due to the properties of determinants, a Slater determinantal wavefunction ° automatically fulfils the Pauli principle and takes care of the antisymmetric character of fermions. If written explicitly in terms of the single-particle orbitals,... 

The fundamental idea of Hylleraas was that the attractive force between the nuclear charge and each of the electrons is well accounted for in the single-particle orbitals exponential functions with negative exponents in the coordinates rf (see Section 7.1.1) ... 

The FISCI method will be considered in more detail for the 2Se manifold of 3s photoionization in argon, even though finer details of the satellite spectrum can be understood only if further electron correlations and spin-orbit effects are included (see below). In the FISCI approach, uncorrelated functions < >[ with single-particle orbitals provide the starting point. They are abbreviated to their characteristic electron orbital, i.e.,... 

In the limiting case of strong spin-orbit interactions, /s)-coupled single-particle orbitals have to be used. With the help of equs. (7.42)-(7.44) they can be traced back to the common spin orbitals, giving... 

Quantum chemistry is most simply done with single-particle orbitals o,single-particle equations. The exchange-correlation energy Exc is then constructed from the orbitals, or from the spin densities raj and raj. The Hartree-Fock (HF) approximation neglects correlation but treats exchange exactly ... 

At this point we want to emphasize that, by virtue of the Hohenberg-Kohn theorem applied to non-interacting systems, all single-particle orbitals are formally functionals of the densities, i.e. 

These functions can be expressed in terms of single-particle orbitals, similar to the linear response function (157). 

To avoid this problem, we have developed an approximation to the full multielectron Schrodinger equation which treats the evolution of each electron individually, but neglects the dynamic interelectron interactions [8,9], We arrive at this model in the following way. First the single-particle orbitals in the Hartree-Fock wave function are defined in terms of their departure from the initial state. 

The pair product trial wave function is the simplest extension of the Slater determinant of single particle orbitals used in mean field treatment of electronic systems (HF or DFT). This is also the ubiquitous form for trial functions in VMC... 

A simple estimate of the computational difficulties involved with the customary quantum mechanical approach to the many-electron problem illustrates vividly the point [255]. Consider a real-space representation of ( ii 2, , at) on a mesh in which each coordinate is discretized by using 20 mesh points (which is not very much). For N electrons, becomes a variable of 3N coordinates (ignoring spin), and 20 values are required to describe on the mesh. The density n(r) is a function of three coordinates and requires only 20 values on the same mesh. Cl and the Kohn-Sham formulation of DFT (see below) additionally employ sets of single-particle orbitals. N such orbitals, used to build the density, require 20 values on the same mesh. (A Cl calculation employs in addition unoccupied orbitals and requires more values.) For = 10 electrons, the many-body wave function thus requires 20 °/20 10 times more storage space than the density and sets of single-particle orbitals 20 °/10x 20 10 times more. Clever use of symmetries can reduce these ratios, but the full many-body wave function remains inaccessible for real systems with more than a few electrons. 

Ts] ] is not known exactly as a functional of n [and using the LDA to approximate it leads one back to the TF approximation (1.34)], but it is easily expressed in terms of the single-particle orbitals of a noninteracting... 

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