Atom-Centered Basis Function Approach

A simple technique that could assess the degree of similarity between atoms in molecules can be based on the technique used by Mezey et al. in their discussion on molecular chiraHty. There the aim was to investigate the degree of similarity between enantiomers, which can naturally be done with the total molecular electron density. Mezey et al. provided an interesting way to calculate the similarity between fragments in both enantiomers. It is based on the well-known fact that the electron density in LCAO-MO theory may be written as 

If the electron density for only one atom is desired, we could reduce the 

An expression for the quantum similarity measure can then be obtained easily. 

This method has been applied to molecular fragments containing three atoms as well as to the case of a single atom by Mezey et A criticism of this approach is that the fragment or atom densities do not sum to the total electron density of an M-atom molecule, as is shown in Eq. [92]  

This inequality is caused by the absence of the off-diagonal blocks 

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A very different approach is the use of non-atom-centered basis functions such as plane waves. Due to their intrinsic periodic nature, they are mostly employed for electronic structure calculations of periodic solids [10]. A more recent development is the usage of real-space wavefunctions either by discretization on real-space grids or in a finite-element fashion [11], In a non-atom-centered basis, the basis set obviously does not depend on the atomic positions, which makes it ideally suited for ab initio molecular dynamics simulations, since the forces acting on the nuclei can be evaluated much more easily than in an atom-centered basis [10]. 

The self-consistent field Hartree-Fock (HF) method is the foundation of AI quantum chemistry. In this simplest of approaches, the /-electron ground state function T fxj,. X/y) is approximated by a single Slater determinant built from antisymmetrized products of one-electron functions i/r (x) (molecular orbitals, MOs, X includes space, r, and spin, a, = 1/2 variables). MOs are orthonormal single electron wavefunctions commonly expressed as linear combinations of atom-centered basis functions ip as i/z (x) = c/ii /J(x). The MO expansion coefficients are... 

It should be first pointed out that the KS equations can be solved numerically in a basis-set-free approach. Such a scheme is used in NUMOL. Assuming the grids employed are sufficiently accurate, this approach has the distinct advantage of being limited only by the quality of the XC energy functional employed and is, therefore, ideally suited for benchmarking their performance on various systems. However, in the vast majority of DFT programs, the KS orbitals, are expressed as a linear combination of atom-centered basis functions. 

The preceding step to both MP2 and coupled-cluster calculations is to solve the Hartree-Fock equations. The standard approach is, of course, to solve the equations in a basis set expansion (Roothaan-Hall method), using atom-centered basis functions. This set of basis functions is used to expand the molecular orbitals and we will call it orbital basis set (OBS). It spans the computational (finite) orbital space. Occupied spin orbitals will be denoted (pi and virtual (unoccupied) spin orbitals pa- In order to address the terms that miss in a finite OBS expansion, the set of virtual spin orbitals in a formally complete space is introduced, pa- If we exclude from this space all those orbitals which can be represented by the OBS, we obtain the complementary space, with orbitals denoted cp i. The subdivision of the orbital space and the index conventions are summarized in the left part of Fig. 2. 

The main purpose of this chapter is to present the basics of ab initio molecular dynamics, focusing on the practical aspects of the simulations, and in particular, on modeling chemical reactions. Although CP-MD is a general molecular dynamics scheme which potentially can be applied in combination with any electronic structure method, the Car-Parinello MD is usually implemented within the framework of density functional theory with plane-waves as the basis set. Such an approach is conceptually quite distant from the commonly applied static approaches of quantum-chemistry with atom-centered basis sets. Therefore, a main... 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

This rescaling reflects the idea that any increase of electronic charge at a center, as a consequence of an enrichment of the basis functions describing it, is unphysical if, lacking equipoise, atoms bonded to it suffer from poorer basis set descriptions. The parameter introduced in Eq. (8.4) is there to correct this imbalance if we follow Mayer s claim [172,173] that Mulliken s half-and-half partitioning of overlap terms between the concerned atoms should not be tampered with. It is felt that the way depends on the basis sets used for describing atoms k and I deserves attention as part of an effort aimed at letting X.rtsi approach Mulliken s limit A = 1 as closely as possible. 

Variational one-center restoration. In the variational technique of one-center restoration (VOCR) [79, 80], the proper behavior of the four-component molecular spinors in the core regions of heavy atoms can be restored as an expansion in spherical harmonics inside the sphere with a restoration radius, Rvoa, that should not be smaller than the matching radius, Rc, used at the RECP generation. The outer parts of spinors are treated as frozen after the RECP calculation of a considered molecule. This method enables one to combine the advantages of two well-developed approaches, molecular RECP calculation in a gaussian basis set and atomic-type one-center calculation in numerical basis functions, in the most optimal way. This technique is considered theoretically in [80] and some results concerning the efficiency of the one-center reexpansion of orbitals on another atom can be found in [75]. 

In Fenske and Hall s parameter-free SCF calculations (80-84), the He1t 1-electron operator is substituted by a model 1-electron operator that has a kinetic energy and potential energy term for each atomic center in the complex. This approach assumes that the electron density may be assigned to appropriate centers. The partitioning of electron density is done through Mulliken population analyses (163) until self-consistency is obtained. The Hamiltonian elements are evaluated numerically, and the energies of the MO s depend only on the choice of basis functions and the intemuclear distance. 

Abstract. Cross sections for electron transfer in collisions of atomic hydrogen with fully stripped carbon ions are studied for impact energies from 0.1 to 500 keV/u. A semi-classical close-coupling approach is used within the impact parameter approximation. To solve the time-dependent Schrodinger equation the electronic wave function is expanded on a two-center atomic state basis set. The projectile states are modified by translational factors to take into account the relative motion of the two centers. For the processes C6++H(1.s) —> C5+ (nlm) + H+, we present shell-selective electron transfer cross sections, based on computations performed with an expansion spanning all states ofC5+( =l-6) shells and the H(ls) state. 

We encountered basis sets in Sections 4.3.4,4.4.1.2, and 5.2.3.6.L A basis set is a set of mathematical functions (basis functions), linear combinations of which yield molecular orbitals, as shown in Eqs. 5.51 and 5.52. The functions are usually, but not invariably, centered on atomic nuclei (Fig. 5.7). Approximating molecular orbitals as linear combinations of basis functions is usually called the LCAO or linear combination of atomic orbitals approach, although the functions are not... 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

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