Atomic self-consistent field calculations

Roothaan, C.C.J. and Bagus, RS. (1963). Atomic self-consistent field calculations by the expansion method, Methods Comput. Phys. 2,47-94. 

ST. Tsuchiya, M. Abe, T. Nakajima, K. Hi-rao. Accurate rdativistic Gaussian basis sets for H through Lr determined by atomic self-consistent field calculations with the third-order Douglas-KroU approximation. /. Chem. Phys., 115(10) (2001) 4463 472. 

The radial parameters are generally obtained from atomic self-consistent field calculations for free atoms, there being essentially no calculations in condensed materials. It is important to recognize that these calculations for the lanthanides and actinides must be done relativistically (i.e. Dirac-Fock), since the neglect of relativity causes serious errors for f electrons. 

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). 

Calculations of vibrational frequencies in a three-center bond as a function of Si—Si separation were performed by Zacher et al. (1986), using linear-combination-of-atomic-orbital/self-consistent field calculations on defect molecules (H3Si—H—SiH3). The value of Van de Walle et al. for H+ at a bond center in crystalline Si agrees well with the value predicted by Zacher et al. for a Si—H distance of 1.59 A. 

By this method one obtains a simple analytic expression for the shielding potential. With the above values of pi and p2 this expression is hardly different from Hartree s self-consistent field calculated via incomparably difficult numerical techniques, and is even perhaps a bit more exact, as it lies between the self-consistent field with and without exchange in the case of the sodium atom. ... 

The results, combined with self-consistent-field calculations by McLean and Yoshimine407 lead to the conclusion that there is nearly no modification of the d orbitals of the silicon atom in monomeric SiO and so the bond system of SiO is very similar to that of carbonmonoxide. 

In principle, this kind of scheme may be carried out for any molecule, with any number of electrons and any number of atomic orbitals y in the LCAO basis set. The practical calculation, however, involves the tedious evaluation of a large number of integrals, a number which increases so rapidly with the number of electrons that, for large molecules, complete self-consistent field calculations are not really feasible on a large scale. 

The bottleneck in all self-consistent field calculations is the difficulty of calculating the integrals [Eq. (16)] over the atomic orbitals. Thus, reducing their number has been an imperative requirement and has led to the fundamental zero-differential overlap approximation (ZDO) which assumes... 

K. Ruud, T. Helgaker, R. Kobayashi, P. Jorgensen, K. Bak, H. Jensen, Multiconfigurational self-consistent field calculations of nuclear shieldings using london atomic orbitals, J. Chem. Phys. 100 (1994) 8178. 

The self-consistent field calculation requires that as a first approximation V j be calculated for each atom j with its associated atomic electron density. The total potential V is then obtained as a superposition of the atomic potentials ... 

Any molecular calculation starts with the evaluation of integrals over the basis functions. This is usually, but not necessarily, followed by a self-consistent field calculation and a transformation from integrals over atomic basis functions to integrals over molecular orbitals. Full details of these particular phases of calculation are well documented elsewhere and we do not consider them further here. 

In order to proceed we first need to know the atomic basis functions from which we can construct the symmetry orbitals and how to evaluate the one-centre and multi-centre integrals of H and S. Finally one has to find an efficient and accurate way to describe the molecular potential in the self-consistent-field calculations. 

The total electron density at the nucleus p(0) depends on the nature of the chemical bonding. Positive contributions to p(0) arise from the atomic 6s populations of the molecular orbitals on the gold ion, while a decrease in p(0) is caused by the atomic bd populations due to their shielding effects on s electrons. The relative magnitudes of the various contributions are known from the results of relativistic free-ion self-consistent field calculations for gold and for other d transition elements. The atomic 6p populations, on the other hand, yield only small contributions both by shielding of s electrons and by their relativistic density at the nucleus. Hence p(0) and consequently the isomer shift will mainly depend on the atomic 6s and 5[Pg.281]

The K(3 IKOi x-ray intensity ratio is an easily measurable quantity with relatively high precision and has been studied extensively for /f-x-ray emission by radioactive decay, photoionization, and charged-particle bombardment (1-3). Except for the case of heavy-ion impact where multiple ionization processes are dominant, it is generally accepted that this ratio is a characteristic quantity for each element. The experimental results are usually compared with the theoretical values for a single isolated atom and good agreement is obtained with the relativistic self-consistent-field calculations by Scofield (4). 

There is generally little ambiguity associated with calculations of molecular orbitals for closed-shell molecules. The Hartree-Fock method (as appoximated by self-consistent field calculations in necessarily finite atomic orbital basis sets11) provides a solution that obeys some of the symmetry properties that must... 

In brief, the CNDO (the acronym stands for complete neglect of differential overlap) approach is an all valence electron, self-consistent field calculation in which multicenter integrals have been neglected and some of the two electron integrals parameterized using atomic data. Slater type atomic orbitals are used as the basis 2s, 2px, 2p, 2p for carbon and oxygen. In these calculations two-electron in egrafs are approximated as... 

At first sight crystal energy calculations should be relatively simple. In the case of neon, the lattice vibrations are so extensive that a rather elaborate procedure was required, which was similar to a self-consistent field calculation. We started first with the static lattice and found the cell potential, assuming a fixed position for neighboring atoms. This cell potential was then used to calculate the vibratory motion of neighboring atoms, which in turn had a large effect on a recalculation of the cell potential. This procedure was repeated until self-consistency was obtained. 

We hope we have demonstrated that the usual delocalized molecular orbitals obtained from self-consistent-field calculations are about as readily understood and interpretable in chemical terms as are the localized orbitals which some people have taken great pains to derive (32) from these delocalized ones. We have demonstrated also through the plots of electron density how these delocalized molecular orbitals are made up from atomic orbitals and how they may be discussed in terms of their atomic-orbital composition. This is particularly interesting for second-row atoms such as sulfur because of the diffuse nature of the parts of the valence-shell atomic orbitals which are involved in bonding. 

Young s modulus was calculated by carrying out a full self-consistent field calculation for each strain imposed along the tube axis, allowing for a full relaxation of atomic positions. The tubes were then congiressed and stretched to about -4% and +4%, respectively. A smaller range of strains [-1%, -i-l%], as previously suggested (31), resulted in incorrect predictions of Poisson s ratio. 

Butadiene is calculated by the simple MO method to have the same average number of it electrons at each atom. Therefore, it is often designated as a molecule with a self-consistent field. The self—consistent field calculated for butadiene is important in lending credence to the validity of of the assumption that Piz = Pzs particularly that i = aj. 

Liberman, D. A., Albritton, J. R., Wilson, B. G., Alley, W. E. (1994). Self-consistent-field calculations of atoms and ions using a modified local-density approximation. Phys. Rev. A 50,171-176. 

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