Slater wave functions

We have just explained that the wave equation for the helium atom cannot be solved exacdy because of the term involving l/r12. If the repulsion between two electrons prevents a wave equation from being solved, it should be clear that when there are more than two electrons the situation is worse. If there are three electrons present (as in the lithium atom) there will be repulsion terms involving l/r12, l/r13, and l/r23. Although there are a number of types of calculations that can be performed (particularly the self-consistent field calculations), they will not be described here. Fortunately, for some situations, it is not necessary to have an exact wave function that is obtained from the exact solution of a wave equation. In many cases, an approximate wave function is sufficient. The most commonly used approximate wave functions for one electron are those given by J. C. Slater, and they are known as Slater wave functions or Slater-type orbitals (usually referred to as STO orbitals). 

Slater wave functions have the mathematical form 

When the radial function Rnj(r) is approximated, the wave functions can be written as 

The calculation of the screening constant for a specific electron is as follows. 

Electrons residing outside the shell in which the electron being described resides do not contribute to the screening constant. 

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One of the simplest approaches to comprehensive molecular orbital calculations is the extended Hiickel method. This method was developed by Roald Hoffman in the 1960s, and it was applied to hydrocarbon molecules. From the discussion presented in Chapters 2 and 3, we know that one of the first things that has to be done is to choose the atomic wave functions that will be used in the calculations. One of the most widely used types of wave functions is that known as the Slater wave functions (see Section 2.4). In the extended Hiickel method, the molecular wave functions are approximated as... 

Table 3. Core-valence electron coulomb interaction terms, evaluated from relativistic Hartree-Fock-Slater wave functions...

Cromer, D. T., J. T. Waber Scattering Factors Computed from Relativistic Dirac-Slater Wave Functions. Acta Cryst. 18, 104 (1965). 

The Hartree-Fock-Slater wave function for the singlet state of helium is the single determinant... 

The QMC method is thought to be promising for the treatment of dynamical and static eiectron correiation effects with the compact functional form of wave functions. One standard form of the wave function is the Jastrow-Slater wave function. The Jastrow-Slater wave function is defined by... 

Let us consider the cusp correction scheme for the Jastrow-Slater wave function. This type of wave function allows us to use a MO correction scheme. The s type component of the Is MO inside a given radius is replaced with the correction function,... 

Wave function Single or Multi-determinant Jastrow-Slater wave function and their linear combinations are available. Jastrow factors of Schmidt-Moskowitz type are available. 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

The QMC calculations are done with Jastrow-Slater wave functions using a single determinant (JSD), or a state-average (SA) or state-specific (SS) complete-active-space multideterminant expansion (JCAS). The lists of parameters optimized by energy minimization in VMC are indicated within square brackets Jastrow (J), CSF coefficients (c), and orbitals (o). For comparison, the DMC results of Ref. [13] obtained with state-average CAS(6,5) wave functions and a Gaussian basis set are also shown... 

K Schwarz, P Blaha. Electron densities in solid compounds. In JP Dahl, J Avery, eds. Local Density Approximations in Quantum Chemistry and SoUd State Physics. New York Plenum, 1984, p 605. JC Slater. Wave functions in a periodic potential. Phys Rev 51 846, 1937. 

Desclaux [11] performed true relativistic Dirac-Fock calculations with exchange to obtain orbital binding energies for every atom. Relativistic Hartree-Fock-Slater calculations were made by Huang et al. [12] and later improved by ab initio Dirac-Hartree-Slater wave functions for elements with Z = 70 to 106 in [13]. 

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