Modern approaches to the exchange-correlation problem

Abstract

Kohn-Sham density functional theory (DFT) is the most prevalent electronic structure method in chemistry. Whilst formally exact, in practice it affords reasonable accuracy with reasonable computational cost and is the method of choice when considering molecules of non-trivial size. The key quantity is the exchange-correlation energy functional, the exact form of which is unknown. Approximate exchange-correlation functionals, particularly B3LYP and PBE, are routinely applied to chemical problems. However, it is not possible to guarantee a given accuracy in advance, nor is there a systematic means of obtaining a more accurate answer. Existing functionals are applied to ever more challenging problems and the accuracy required of them is continually increasing the need for more accurate functionals is one of the major challenges in electronic structure theory. This thesis focuses on several approaches that attempt to address this issue. In chapter 1 the electronic structure problem is outlined and discussed in terms of the Schrödinger equation and solutions involving wavefunctions. In chapter 2, the formal foundations of DFT are presented and methods of approximating the exchange-correlation functional are introduced. A promising new direction for developing exchange-correlation functionals, through attenuation of the exchange term, is introduced and discussed in detail in chapter 3. The accuracy of such functionals is investigated and compared to that obtained from conventional approaches, with a particular emphasis on the dependence on the attenuation parameters. It is then demonstrated that attenuated functionals offer the prospect of significantly improved descriptions of excitation energies, particularly for those of charge-transfer character. Apphcation of attenuated functionals to excitation energies that are problematic for conventional functionals is undertaken in chapter 4. Insight into the conflicting performance of conventional methods for different charge-transfer excitations is provided through a consideration of the orbital overlap between the orbitals involved in an excitation. Through this overlap quantity, a diagnostic test is proposed that enables a user to judge in advance the reliability of excitation energies from conventional functionals. Attenuated functionals are then applied to other difficult properties in chapter 5. Firstly they are used to study the bond length alternation and band gap in poly acetylene and polyyne oligomers and infinite chains. Then they are used to calculate nuclear magnetic resonance parameters in both main-group and first-row transition metal systems, through the theoretically rigorous optimised effective potential method. An entirely different approach to functional development is considered in chapter 6, where the adiabatic connection formalism is introduced as an alternative method of obtaining the exchange-correlation functional. For a series of two-electron systems, exact input data is used to determine the applicability of a number of simple mathematical forms in modelling the exact adiabatic connection. The conclusions from these simple systems are then used to provide insight into the possibility of using this approach in functional development.