Exchange representations in Kohn-Sham theory
Kohn-Sham density functional theory (DFT) is the most widely used method in quantum chemistry. It has the potential to provide accurate results at low computational cost. The quality of a DFT calculation is determined by the exchange-correlation energy functional. Hybrid functionals, which contain a fraction of exact orbital exchange, are extensively used due to their accuracy in a variety of applications. However, as commonly implemented, these functionals are outside the Kohn-Sham scheme, since the exchange operator is not a local multiplicative potential. In order to handle orbital dependent functionals correctly, schemes which determine a local multiplicative potential must be employed. The implementation and application of several such methods is the focus of this thesis. In Chapter 1 we outline the Hartree-Fock scheme, which defines the exchange energy, and overview wavefunction based procedures that recover correlation energy. Alternative theories based on the electron density are then considered and the foundations of modern DFT are reviewed. The formalism of the optimized effective potential (OEP) method is introduced, which is the rigorous way to handle orbital dependent functionals. A number of approximations to the exchange only OEP method are outlined in Chapter 2 and their implementation is described. The methods are applied to the calculation of NMR shielding constants, highlighting differences between the approximations; their use in the construction of multiplicative hybrid functionals is also considered. In Chapter 3 these approximations are further investigated in the calculation of excited states and structural perturbations. In Chapter 4, the theory and implementation of a direct optimization procedure to determine OEPs is outlined, along with an implementation of the constrained search procedure, which allows the determination of the Kohn-Sham exchange-correlation potential from any input density. Chapter 5 compares the performance of the approximate exchange potentials with those of OEP, highlighting the presence of correlated character in some of the approximate methods. The OEP implementation is extended to include hybrid exchange-correlation functionals in Chapter 6. The performance of these methods for the calculation of NMR shielding constants, rotational g tensors and transition metal NMR chemical shifts is investigated. In all cases, substantial improvements over conventional results are obtained. In Chapter 7 DFT is used to investigate an interaction of relevance in organic chemistry. Concluding remarks are given in Chapter 8.