Full implementation linear combinations

We determine the ground-state electronic structure of solids within Density Functional Theory (DFT) and the usual KS variational procedure, all implemented in the computational package gtoff [14]. The results of the all-electron, full-potential calculations are Bloch eigenfunctions p,k(r), expressed as linear combinations of Gaussian Type Orbitals (GTOs), and KS eigenvalues p k-... 

Having seen how the implementation goes for the simplest possible case, we can proceed to the full case when symmetry operations induce linear combinations of basis functions on symmetry-equivalent centres. 

All explicitly correlated calculations were performed at the CCSD(F12) level of theory, as implemented in the TurbomOLE program [58, 69]. The Slater-type correlation factor was used with the exponent 7 = 1.0 aQ. It was approximated by a linear combination of six Gaussian functions with linear and nonlinear coefficients taken from Ref. [44]. The CCSD(F12) electronic energies were computed in an all-electron calculation with the d-aug-cc-pwCV5Z basis set [97]. For all cases we used full CCSD(F12) model (see Subsection 4.9 for the discussion about models implemented in Turbomole), the open-shell species were computed with a UHF reference wave function. The explicitly correlated contributions to the relative quantities are collected in Tables 10 and 11 under the label F12 . 

We then present ab initio molecular orbital theory. This is a well-defined approximation to the full quantum mechanical analysis of a molecular system, and also the basis of an array of powerful and popular computational approaches. Molecular orbital theory relies upon the linear combination of atomic orbitals, and we introduce the mathematics and results of such an approach. Then we discuss the implementation of ab initio molecular orbital theory in modern computational chemistry. We also describe a number of more approximate approaches, which derive from ab initio theory, but make numerous simplifications that allow larger systems to be addressed. Next, we provide an overview of the theory of organic TT systems, primarily at the level of Hiickel theory. Despite its dramatic approximations, Hiickel theory provides many useful insights. It lies at the core of our intuition about the electronic structure of organic ir systems, and it will be key to the analysis of pericyclic reactions given in Chapter 15. 

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