The AO bases

The calculations described in this section of the book have, for the most part, been carried out using three of the basis sets developed by the Pople school. 

ST03G A minimal basis. This contains exactly the number of orbitals that might be occupied in each atomic shell. 

6-31G A valence double-f basis. This basis has been constmcted for atoms up through Ar. 6-31G A valence double-f basis with polarization functions added. Polarization functions are functions of one larger /-value than normally occurs in an atomic shell in the ground state. 

Any departures from these will be spelled out at the place they are used. 

Our general procedure is to represent the atoms in a molecule using the Hartree-Fock orbitals of the individual atoms occurring in the molecule. (We will also consider the interaction of molecular fragments where the Hartree-Fock orbitals of the fragments are used.) These are obtained with the above bases in the conventional way using Roothaan s RHF or ROHF procedure[45], extended where necessary. 

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The mo-based integrals ean only be evaluated by transforming the AO-based integrals as follows ... 

The energy terms in the third row are then summed up yielding the energy of the AO-based determinant ... 

By means of Equation 4.26, these six values suffice to calculate the 45 overlaps between the nine possible AO-based determinants. To save space, we will use a simplified notation for the latter for example, the determinant IX2X2XiX3l> which is made of the half-determinants IX2X1I and IX2X3I will be noted 2123. One then expresses 2x2 as a function of the AO-based determinants ... 

A difference between the qualitative VB theory, discussed in Chapter 3, and the spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO-based determinant, without any a priori bias for a given electronic coupling into bond pairs like those used in the Rumer basis set of VB structures. The bond coupling results from the diagonalization of the Hamiltonian matrix in the space of the determinant basis set. The theory is restricted to determinants having one electron per AO. This restriction does not mean, however, that the ionic structures are neglected since their effect is effectively included in the parameters of the theory. Nevertheless, since ionicity is introduced only in an effective manner, the treatment does not yield electronic states that are ionic in nature, and excludes molecules bearing lone pairs. Another simplification is the zero-differential overlap approximation, between the AOs. 

The most useful method for background studies is based on the MNDO model [18], which is a valence-electron self-consistent-field (SCF) MO treatment. It takes up a minimal basis of atomic orbitals (AOs) and the NDDO integral estimation. The molecular orbitals, ([), , and the corresponding orbital energies, s , are obtained from the linear combination of the AO base functions, cjaM, and the solution of the secular equations with Suv ... 

Bartlett looks a lot like the usual random phase approximation (discussed later). The same authors also expressed the THDF approach in terms of the sum-over-states procedure and discussed the effects of truncation with this context. Recently, Kama extended the AO-based THDF procedure to open-shell systems using a UHF-like approach. 

In the AO-based implementation of the CCSD(F12) model we need the where the xy are the geminal indices (efectively they belong to the occupied space within this implementation, see Subsection 3.2) and yu belong to the covariant AO-basis. In the subsequent sections we will use the term covariant for the description of the quantities in the ordinary AO basis (denoted by upper-case AO left superscript) whereas the contravariant basis is used for the back-transformed quantities (vide infra) (denoted with lower-case ao left superscript). The final equation for the Al-noCABS V intermediate can be written in the following way... 

The contraction over the cd virtual indices in the above formula is the most expensive term within the whole conventional CCSD scheme. It scales like /iocc vir- Moreover, it requires the T similarity-transformed Coulomb integrals with four virtual MO indices available. The storage of them on disk could very easily limit applicability of the code. Therefore, the AO-based integral-direct scheme was implemented here. The alternative approach to this transformation scales like without carrying out expensive AO-MO... 

Primes with the AO basis of the product are used to denote the fact that the corresponding atomic orbitals x can differ from the AO basis of the reactant, for example because of different spatial orientation. This distinction between the AO bases of the reactant and the product is very important since it is just precisely from here that the possibility arises to exploit the formalism for the discrimination between the forbidden and the allowed mechanisms, i.e., in our case, between the conrotation and the disrotation. The basis of this discrimination are the so-called assigning tables, the physical meaning of which is just in providing the detailed specification of the mutual transformation between bases % and x. which is the necessary prerequisite for the calculation of the overlap integral S p. The same problem was encountered also by Trindle [33], and his mapping analysis failed to find broader use only because of considerable numerical complexity. On the other hand, the overlap determinant method solves this problem much more simply and its use is really a matter of seconds using only pen and paper. 

Similarly as with the overlap determinant method the primes are used to denote the fact that the AO bases x and % serving to describe the molecular orbitals of the reactant and the product are generally different. Using the known expressions for the first order spinless density matrices (136), the original definition expression for the similarity index (137) can be rewritten in the form (138). 

To implement CCM into the HF LCAO calculations [100] the periodic crystalline-orbital program CRYSTAL95 [322] was used with the modifications [97] allow the DM idempotency (6.76) to be checked. In rutile Ti02 calculations the AO bases, lattice parameters and effective core potentials were taken to be the same as those used in [323] to calculate the optimized lattice parameters. In the CRYSTAL code... 

Half-determinants have no physical meaning but are defined here as a convenient mathematical intermediary. Each of these MO-based half-determinants can be expanded into AO-based half-determinants, just as has been done for the determinants in Eq. [A4]. After orbital permutations the AO-based halfdeterminants that are equivalent are regrouped, and we are left with some AO-based half-determinants each having a unique collection of AOs... 

In earlier chapters we classified the symmetry of atomic orbitals (AOs) in a number of example molecules. It is now time to develop the ideas of molecular orbital (MO) theory and use it to describe chemical bonding. Symmetry classifications help in the MO description of chemical bonding because symmetry controls how the AOs on neighbouring atoms mix together. MOs are the wavefunctions for electrons in the complex field of the many nuclei and other eleetrons that make up a molecule. The complexity of MOs can be dealt with by constructing them from the AOs of the isolated atoms. The MOs are formed by mixing the AOs based on the idea of interference described by the superposition of waves when waves come together in the same phase they reinforce one another, whereas waves of opposite phase will tend to cancel each other out. 

The basis functions used in constructing MOs are the AOs based on the hydrogen atom solutions of the Schrddinger equation discussed in Appendix 9, with the proviso that accurate energies will require flexibility in the radial decay constants. Before moving on to molecules more complex than H2, it is worth looking at the shapes of the AOs relevant for the first row of the periodic table. We have already used the shapes of s-, p- and d-functions to discuss the symmetry of particular AOs (e.g. the d-orbitals of the central metal atom in transition metal complexes were covered in Section 5.8). These shapes are based on the... 

In contrast to the conventional MO-based formulation, the AO-based Laplace formalism allows one to reduce the conventional O(N ) scaling of the computational cost for large molecules. However, for small molecules, the overhead consists of the need to compute x = 5-8 exponentials and the larger prefactor for the transformations scaling formally as compared with the occ , and N scaling for the different MO-based... 

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