Correcting deficiencies in approximate density functionals

Abstract

In the last fifty years, approximate density functional theory (DFT) has become firmly established as the de facto standard for electronic structure calculations in chemistry. Although the theory itself is formally exact, approximations must be made for the unknown exchange–correlation (XC) functional, and whilst many successful approximate functionals exist, a number of deficiencies still persist, leading to many cases where the approximation breaks down completely. This thesis addresses two prevalent deficiencies, and examines some novel approaches to reducing and eliminating them.

Chapter 1 provides a background to electronic structure theory, with particular reference to the approximate solution of the electronic Schrödinger equation through ab initio wavefunction methods. Chapter 2 then provides the formal justification for DFT as an alternative to wavefunction-based methods, and outlines common approximations to the XC functional. Two prominent deficiencies of approximate DFT are discussed: delocalisation error due to non-linearity in the energy variation with number of electrons, and incorrect long-range behaviour of the XC potential.

Chapter 3 examines a system-dependent tuning technique for the range-separated hybrid class of XC functionals, whereby the range-separation parameter is non-empirically tuned to self-consistent energy-linearity conditions, which has been successfully used to improve the calculation of quantities affected by the delocalisation error. A full, systematic assessment of this tuning technique is provided, and it is demonstrated that the success of the technique is aided by a convenient cancellation of errors.

In Chapter 4, the tuned functionals are applied to quantities relevant to conceptual DFT. It is shown that functionals tuned to the energy conditions of Chapter 3 remain appropriate for calculation of the electronegativity from orbital energies, however the density variation with number of electrons — described by the Fukui function — is better modelled by conventional non-tuned functionals.

Finally, an entirely new approach to functional development is provided in Chapter 5. The behaviour of a functional under density scaling is used to impose homogeneity constraints on a simple functional form, culminating in an electron-deficient functional that satisfies the appropriate energy-linearity condition and exhibits the correct asymptotic XC potential.