Atomic orbital basis methods

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. 

In an ab initio method, all the integrals over atomic orbital basis functions are computed and the Fock matrix of the SCF computation is formed (equation (61) on page 225) from the integrals. The Fock matrix divides into two parts the one-electron Hamiltonian matrix, H, and the two-electron matrix, G, with the matrix elements... 

The basis of constructing the MOs is the linear combination of atomic orbitals (LCAO) method. This takes account of the fact that, in the region close to a nucleus, the MO wave function resembles an AO wave function for the atom of which the nucleus is a part. It is reasonable, then, to express an MO wave function 1/ as a linear combination of AO wave functions Xi on both nuclei ... 

All of the above methods require the evaluation of one- and two-electron integrals over the N atomic orbital basis <%a and - Eventually, all of these methods provide their working equations and energy expressions in terms of one- and two-electron integrals over the N final molecular orbitals <(f>ilfk )j> and <( )j( )jlgl( )k( ) >. 

All of the measurements employed the technique described above that involves the analysis of the isotope composition of 02 released from the carrier complexes in preequilibrated solutions. In addition, an established DFT method (mPWPW91)34 with the atomic orbital basis functions, Co, Fe, and Cl (the compact relativistic effective core potential basis CEP-31G),35 N and O (6-311G ), P (6-311G ), C(6-31G), and H (STO-3G),36 were used to calculate the 180 EIE in terms of actual and model structures. The latter approach has also been employed for hypothetical intermediates in enzymes as described below. 

NMR shielding tensors were calculated using the Gauge-Independent Atomic Orbital (GIAO) method as implemented in Gaussian 98 (44-46). The basis sets and level of theory are the same as used in the geometry optimizations and frequency calculations mentioned above. 

The original VB method has been difficult to use in practice because of the nonorthogonality of the atomic orbital basis, but there has been a revival of interest in it recently. Gallup and co-workers116-118 have described a new technique for carrying out such calculations, and the results of applications to the first-row hydrides. Calculations were carried out using a minimal basis set constructed from gaussian lobe orbitals. The orbitals were scaled to their best atom value and also optimally scaled in the molecule. Atomic populations were also computed.118... 

The one-electron wave function in an extended solid can be represented with different basis sets. Discussed here are only two types, representing opposite extremes the plane-wave basis set (free-electron and nearly-free-electron models) and the Bloch sum of atomic orbitals basis set (LCAO method). A periodic solid may be considered constmcted by the coalescence of these isolated atoms into extended Bloch-wave functions. On the other hand, within the free-electron framework, in the limit of an infinitesimal periodic potential (V = 0), a Bloch-wave function becomes a simple... 

A CNDO/2 method extending the atomic orbitals basis to sulfur d-orbitals has been applied to the study of the cathodic reduction of 1,2-dithiolium cations. ... 

The method that is used in most of the work described in this chapter is the distributed multipole analysis (DMA) of Stone,which is implemented in the CADPAC ab initio suite. DMA is based on the density matrix p,y of the ab initio wavefunction of the molecule, expressed in terms of the Gaussian primitives q that comprise the atomic orbital basis set ... 

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... 

Normally these molecular orbitals are obtained as expansions in a set of atom-centred basis functions (the linear combination of atomic orbitals (LCAO) method), m being the number of such functions. Recently, two-dimensional numerical integration methods have been developed to solve the MCSCF equations for linear molecules. The dimension m is then, in principle, infinite (practice, it is determined by the size of the grid used in the numerical integration). The molecular-orbital space is further divided into three subspaces the inactive, the active and the external orbitals. The inactive and active subspaces constitute the internal (occupied) orbital subspace, while the external orbitals are unoccupied. The CASSCF wavefunction is formed as a linear combination of configuration state functions (CSFs) generated from these orbitals in the following way. 

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