Shells of basis functions

It is appropriate here to use the symmetry of the shells of basis functions as these groupings are called and this development has been pioneered by M. Dupuis and H. F. King using the methods of Dacre and Elder. 

Outline of the parallel algorithm (c) for two-electron integral computation using dynamic manager-worker distribution of shell pairs MN. Indices M, N, R, and S represent shells of basis functions, Wsheii is the number of shells, p is the process count, and this proc (0 < this proc < p) is the process identifier. 

Outline of a parallel algorithm for Fock matrix formation using replicated Fock and density matrices. A, B, C, and D represent atoms M, N, R, and S denote shells of basis functions. The full integral permutational symmetry is utilized. Each process computes the integrals and the associated Fock matrix elements for a subset of the atom quartets, and processes request work (in the form of atom quartets) by caUing the function get quartet. Communication is required only for the final summation of the contributions to F, or, when dynamic task distribution is used, in get quartet. 

Outline of a parallel algorithm for Fock matrix formation using distributed Fock and density matrices. A, B, C, and D represent atoms, M, N, R, and S denote shells of basis functions, and only unique integrals are computed. 

The relativistic effects (Rl) and (R2) can be simulated by adjusting the sizes of basis functions used in a standard variational treatment. This adjustment is usually combined with an effective-core-potential [ECP] approximation in which inner-shell electrons are replaced by an effective [pseudo] potential of chosen radius. The calculations of this chapter were carried out with such ECP basis sets in order to achieve approximate incorporation of the leading relativistic effects.)... 

The simplest approximation corresponds to a single-determinant wave function. The best possible approximation of this type is the Hartree-Fock (HF) molecular-orbital determinant. The HF wavefunction is constructed from the minimal number of occupied MOs (i.e., NI2 for an V-eleclron closed-shell system), each approximated as a variational linear combination of the chosen set of basis functions (vide infra). 

With the topological analysis of the total charge density, the distinction between a covalent and a closed-shell ionic interaction can be based on the value of the Laplacian and its components at the bond critical point. Such an analysis will be most conclusive when done on a series of related compounds, analyzed with identical basis sets, as the topological values of the model density from experimental data have been found to be quite dependent on the choice of basis functions. 

The total electron density, or more simply, the electron density, p(r), is a function of the coordinates r, defined such that p(r)dr is the number of electrons inside a small volume dr. This is what is measured in an X-ray diffraction experiment. For a (closed-shell) molecule, p(r) is written in terms of a sum of products of basis functions, ( ). 

We need at least enough spatial MO s i// to accommodate all the electrons in the molecule, i.e. we need at least n ij/ s for the 2n electrons (recall that we are dealing with closed-shell molecules). This is ensured because even the smallest basis sets used in ab initio calculations have for each atom at least one basis function corresponding to each orbital conventionally used to describe the chemistry of the atom, and the number of basis functions

initio calculation on CH4, the smallest basis set would specify for C ... 

First consider what we could denote as the simple 3-21G basis set. This splits each valence orbital into two parts, an inner shell and an outer shell. The basis function of the inner shell is represented by two Gaussians, and that of the outer shell by one Gaussian (hence the 21 ) the core orbitals are each represented by one basis function, each composed of three Gaussians (hence the 3 ). Thus H and He have a Is orbital (the only valence orbital for these atoms) split into Is (Is inner) and Is" (Is outer), for a total of two basis functions. Carbon has a Is function represented by three Gaussians, an inner 2s, 2px, 2py and 2pz (2s, 2px, 2py, 2pj)... 

Molecular orbitals (MOs) were constructed using linear combinations of basis functions of atomic orbitals. The MO eigenfunctions were obtained by solving the Schrodinger equations in numerical form, including Is— (n+l)p, that is to say, Is, 2s, 2p, -ns, np, nd, (n+l)s, (n+l)p orbitals for elements from n-th row in the periodic table and ls-2p orbitals for O, where n—1 corresponded to the principal quantum number of the valence shell. 

We finally observe that the effective orbital probabilities of Eqs. (52-54) and the associated condensed probabilities of bonded atoms (Eq. 55) do not reflect the actual AO participation in all chemical bonds in AB, giving rise to comparable values for the bonding and nonbonding (lone-pair) AO in the valence and inner shells. The relative importance of basis functions of one atom in forming the chemical bonds with the other atom of the specified diatomic fragment is reflected by the (nonnormalized) joint bond probabilities of the two atoms, defined by the diatomic components of the simultaneous probabilities of Eqs. (52 and 53) ... 

As the atom becomes larger, the number of basis functions needed to describe it increases as well. However, since one is most interested in the valence shell where most of the action occurs, the increasingly larger number of inactive or core functions become more and more of a nuisance. One cannot simply omit them as the valence orbitals would then collapse into smaller core orbitals (which are of much lower energy). One solution is development of core pseudopotentials or effective core potentials (ECP) which eliminate the need to include core functions explicitly, yet keep the valence functions from optimizing themselves into core orbitals ° . Such pseudopotentials are commonly used in elements of the lower rows of the periodic table, like Br or I. 

Method RB3LYP basis set 6-31 IG number of shells 26 number of basis functions 78. 

Notation and Matrix Storage Requirements The number of basis functions which we have called m will appear in the code as m and the number of electrons (2n in the closed-shell case) will be nelec. 

The implementation of the GUHP method is surprisingly similar to that of the closed-shell MO method. Both of them are concerned with the calculation of a set of orbitals which optimise the energy of a single-determinant wavefunction. The only real difference, apart from the doubling of the number of basis functions, is in the processing of the spatial basis functions to form the electron-repulsion matrices. 

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