Hartree-Fock-Slater wave function

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

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

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... 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

An interesting improvement of the SRC model has been discussed by Wang and Bulou (1995). They considered simplified expansion factors used in the Hartree-Fock radial wave-functions for the f-electrons. With these factors it was possible to introduce a k dependence for the pressure-induced change of different Slater parameters. This procedure would remove the weak point of the model which could not account for the observed -dependence of the parameters under pressure. 

We have used the terms SCF wave function and Hartree-Fock wave function interchangeably. In practice, the term SCF wave function is applied to any wave function obtained by iterative solution of the Roothaan equations, whether or not the basis set is large enough to give a really accurate approximation to the Hartree-Fock SCF wave function. There is only one true Hartree-Fock SCF wave function, which is the best possible wave function that can be written as a Slater determinant of spin-orbitals. Some of the extended-basis-set calculations approach the true Hartree-Fock wave... 

M. Towler, An introductory guide to Gaussian basis sets in solid state electronic structure calculations, http //www.orystal.unito.it/tutojan2004/tutorials/index.html A.D. McLean, R.S. McLean, Roothaan-Hartree-Fock atomic wave functions (Slater basis-set expansions for Z=55-92),... 

Density functional theory, 21, 31, 245-246 B3LYP functional, 246 Hartree-Fock-Slater exchange, 246 Kohn-Sham equations, 245 local density approximation, 246 nonlocal corrections, 246 Density matrix, 232 Determinantal wave function, 23 Dewar benzene, 290 from acetylene + cyclobutadiene, 290 interaction diagram, 297 rearrangement to benzene, 290, 296-297 DFT, see Density functional theory... 

LCAO expansion of the MOs [15]. In the DV-Xa MO method based on the Hartree-Fock-Slater approach, the exchange-correlation potential is approximated by the simple Slater form [16] Vxc(r) = —3a 3p(r)/47i 1/3, where the coefficient a is a scaling parameter (fixed at 0.7 in the present study) and p(r) is the local electron density at a position r. The basis functions for the MO calculation consisted of atomic orbital wave eigenfunctions obtained in numerical form, which included the ls-6s, ls-5s, ls-6p, ls-4p, and ls-2p orbitals for Ba, Sr, Pb, Ti, and O ions, respectively... 

The calculations of the photoionization cross section of the atomic subshell have previously been performed using Hartree-Fock-Slater one-electron model by several workers. Table 1 compares the photoionization cross sections of the atomic orbital electrons obtained in the present work with those previously reported by Scofield for some atoms. Scofield has used the relativistic wave functions. The... 

The MO calculations are made with the DV-Xa method, which has been described in detail elsewhere [17]. First the Hartree-Fock-Slater (HFS or Xo ) calculations are performed for each constituent atom in the molecule and the atomic wave functions obtained in the numerical form are used as the basis functions for the MO calculation. The molecular wave function for the A-th MO is expressed... 

The potential surrouding each atom in a molecule is not the same as that for the free atom, because electron transfer occurs between atoms in the molecule. This means that atomic orbitals in the molecule are distinct from those in the free atom. Accordingly, it is necessary to use atomic orbitals optimized for each atomic potential in the molecule, as basis functions. In the present methods, the molecular wave functions were expressed as linear combinations of atomic orbitals obtained by numerically solving the Dirac-Slater or Hartree-Fock-Slater equations in the atomic-like potential derived from the spherical average of the molecular charge density around the nuclei [15]. Thus the atomic orbitals used as basis functions were automatically optimized for the molecule and thus the minimum size of the present basis set has enough flexibility to form accurate molecular orbitals. 

Accurate distance determinations depend critically on the accurate determination of phase shifts. There are two general approaches to this problem theoretical and empirical determination. The main approaches to the theoretical calculation of phase shifts are based on the Hartree-Fock (HF) and Hartree-Fock-Slater (HFS) methods. However, both of these are too involved for general use. Teo and Lee used the theoretical approach of Lee and Beni to calculate and tabulate theoretical phase shifts for the majority of elements. Use of these theoretical phase shifts requires the use of an adjustable q in the data analysis (vide supra). Most recently, McKale and co-workers performed ab initio calculations of amplitude and phase functions using a curved wave formalism for the range of k values 2 < k < 20. 

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