Hartree-Fock atomic wave function

The only atomic wave-functions that do not have a node at the nucleus are s-functions. The isotropic coupling constant is thus a measure of the s-character of the wave-function of the unpaired electron at the nucleus in question. The coupling constant for an atomic s-electron can be either measured experimentally or calculated from Hartree-Fock atomic wave-functions so that, to a first approximation, the s-electron density may be calculated from the ratio of the experimental and atomic coupling constants. Should the first-order s-character of the wave-function of the unpaired electron be zero, as for example in the planar methyl radical, then a small isotropic coupling usually arises from second-order spin-polarization effects. The ESR spectra of solutions show only isotropic hyperfine coupling. 

E. dementi and C. Roetti, Roothan-Hartree-Fock atomic wave functions. Basic functions and their coefficients for ground and certain excited states and ionized atoms, Z < 54. Atomic Data, Nuclear Data Tables 14, 177-478 (1974). 

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),... 

A Hartree-Fock SCF wave function takes into account the interactions between electrons only in an average way. Actually, we must consider the instantaneous interactions between electrons. Since electrons repel each other, they tend to keep out of each other s way. For example, in helium, if one electron is close to the nucleus at a given instant, it is energetically more favorable for the other electron to be far from the nucleus at that instant. One sometimes speaks of a Coulomb hole surrounding each electron in an atom. This is a region in which the probability of finding another electron is small. The motions of electrons are correlated with each other, and we speak of electron correlation. We must find a way to introduce the instantaneous electron correlation into the wave function. 

SCF MO Wave Functions for Open-Shell States. For SCF MO calculations on closed-shell states of molecules and atoms, electrons paired with each other are almost always given precisely the same spatial orbital function. A Hartree-Fock wave function in which electrons whose spins are paired occupy the same spatial orbital is called a restricted Hartree-Fock (RHF) wave function. (The unmodified term Hartree-Fock wave function is understood to mean the RHF wave function.)... 

Rauhut, G., Puyear, S., Wolinski, K., Pulay, P., 1996, Comparison of NMR Shielding Calculated from Hartree-Fock and Density Functional Wave Functions Using Gauge-Including Atomic Orbitals , J. Phys. Chem., 100,... 

As the wave function is not known analytically for systems larger than a hydrogen-like atom, suitable approximate wave functions have to be found and the accuracy of Eq. (1) depends of course on the level of approximation. A survey of the various quantum chemical methods to generate approximated wave functions can be found in Refs. (22,23). Here, we shall only present the foundations of Hartree-Fock and density functional theory (DFT) needed in later sections. 

Using atomic densities calculated from tabulated atomic wave functions, the summation was found [214] to produce results equivalent to the most elaborate molecular Hartree-Fock calculations for a series of small molecules, at a fraction of the computing expense. Surface areas and volumes computed by the two methods were found virtually identical. The promolecule calculation therefore has an obvious advantage in the exploration of surface electron densities, surface areas and molecular volumes of macromolecules for the analysis of molecular recognition. 

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 situation is quite similar in chemistry. Due to decades of experience with Hartree-Fock and Cl calculations much is known about the construction of basis functions that are suitable for molecules. Almost all of this continues to hold in DFT — a fact that has greatly contributed to the recent popularity of DFT in chemistry. Chemical basis functions are classified with respect to their behaviour as a function of the radial coordinate into Slater type orbitals (STOs), which decay exponentially far from the origin, and Gaussian type orbitals (GTOs), which have a gaussian behaviour. STOs more closely resemble the true behaviour of atomic wave functions [in particular the cusp condition of Eq. (19)], but GTOs are easier to handle numerically because the product of two GTOs located at different atoms is another GTO located in between, whereas the product of two STOs is not an STO. The so-called contracted basis functions , in which STO basis functions are reexpanded in... 

Determinantal MO s may be obtained by a large number of computational methods based on Roothaan s self-consistent field formalism 94> for solving the Hartree-Fock equation for molecules which differ in degree of sophistication as regards the completeness and kind of the set of starting atomic wave functions, as well as the completeness of the Hamiltonian used 9S>. So a chain of various kinds of approximations is available for calculations starting from different ways of non-empirical ab initio" calculations 96>, viasemiempiricalmethods for all-valence electrons with inclusion of electronic interaction 95-97>98)... 

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