Energy densities: a systematic approach to correlation in density functional theory

Density functional theory (DFT) has grown to become by far the most widely applied method in the modelling of electronic systems yet, in contrast to wavefunction-based ab initio methods, the reliability of a DFT calculation can be uncertain. This is because the essential ingredient required for a DFT calculation to be meaningful - the exchange & correlation energy functionals, are approximations for a type of electronic interaction with an unknown functional form. Whilst there exist types of system for which DFT does not provide a useful model - those with significant dispersion interactions and those with near-degenerate states, seeking improvements to DFT in these areas can be far from straightforward since the exchange & correlation functionals cannot be systematically improved. In this work, a mathematically rigorous description of DFT is combined with some of the most reliable & accurate ab initio electronic structure methods in a bespoke development code to obtain a detailed picture of how the exact correlation energy functional in DFT behaves. Using these insights, new approaches are investigated for modelling the correlation energy functional in local form, allowing a systematic study of how best to approximate the correlation energy functional in different types of system to be pursued. As an adjunct to this, the present work looks ahead at how to extend & generalise this investigation by considering the behaviour of systems in the presence of a magnetic field. Efficient algorithms are developed and implemented to facilitate this, enabling the advancement of this strand of investigation in subsequent work

Efficient calculation of molecular integrals over London atomic orbitals

The use of London atomic orbitals (LAOs) in a non-perturbative manner enables the determination of gauge-origin invariant energies and properties for molecular species in arbitrarily strong magnetic fields. Central to the efficient implementation of such calculations for molecular systems is the evaluation of molecular integrals, particularly the electron repulsion integrals (ERIs). We present an implementation of several different algorithms for the evaluation of ERIs over Gaussian-type LAOs at arbitrary magnetic field strengths. The efficiency of generalized McMurchie-Davidson (MD), Head-Gordon-Pople (HGP) and Rys quadrature schemes is compared. For the Rys quadrature implementation, we avoid the use of high precision arithmetic and interpolation schemes in the computation of the quadrature roots and weights, enabling the application of this algorithm seamlessly to a wide range of magnetic fields. The efficiency of each generalised algorithm is compared by numerical application, classifying the ERIs according to their total angular momenta and evaluating their performance for primitive and contracted basis sets. In common with zero-field integral evaluation, no single algorithm is optimal for all angular momenta thus a simple mixed scheme is put forward, which selects the most efficient approach to calculate the ERIs for each shell quartet. The mixed approach is significantly more efficient than the exclusive use of any individual algorithm

Efficient calculation of molecular integrals over London atomic orbitals

The use of London atomic orbitals (LAOs) in a non-perturbative manner enables the determination of gauge-origin invariant energies and properties for molecular species in arbitrarily strong magnetic fields. Central to the efficient implementation of such calculations for molecular systems is the evaluation of molecular integrals, particularly the electron repulsion integrals (ERIs). We present an implementation of several different algorithms for the evaluation of ERIs over Gaussian-type LAOs at arbitrary magnetic field strengths. The efficiency of generalized McMurchie-Davidson (MD), Head-Gordon-Pople (HGP) and Rys quadrature schemes is compared. For the Rys quadrature implementation, we avoid the use of high precision arithmetic and interpolation schemes in the computation of the quadrature roots and weights, enabling the application of this algorithm seamlessly to a wide range of magnetic fields. The efficiency of each generalised algorithm is compared by numerical application, classifying the ERIs according to their total angular momenta and evaluating their performance for primitive and contracted basis sets. In common with zero-field integral evaluation, no single algorithm is optimal for all angular momenta thus a simple mixed scheme is put forward, which selects the most efficient approach to calculate the ERIs for each shell quartet. The mixed approach is significantly more efficient than the exclusive use of any individual algorithm

Interpolated energy densities, correlation indicators and lower bounds from approximations to the strong coupling limit of DFT

We investigate the construction of approximated exchange-correlation functionals by interpolating locally along the adiabatic connection between the weak- and the strong-coupling regimes, focussing on the effect of using approximate functionals for the strong-coupling energy densities. The gauge problem is avoided by dealing with quantities that are all locally defined in the same way. Using exact ingredients at weak coupling we are able to isolate the error coming from the approximations at strong coupling only. We find that the nonlocal radius model, which retains some of the non-locality of the exact strong-coupling regime, yields very satisfactory results. We also use interpolation models and quantities from the weak- and strong- coupling regimes to define a correlation-type indicator and a lower bound to the exact exchange-correlation energy. Open problems, related to the nature of the local and global slope of the adiabatic connection at weak coupling, are also discussed

Self-Consistent Field Methods for Excited States in Strong Magnetic Fields: A Comparison Between Energy- and Variance-based Approaches

Self-consistent field methods for excited states offer an attractive low-cost route to study not only excitation energies but also properties of excited states. Here we present the generalisation of two self-consistent field methods, the Maximum Overlap Method (MOM) and the σ-SCF method, to calculate excited states in strong magnetic fields and investigate their stability and accuracy in this context. These methods use different strategies to overcome the well-known variational collapse of energy-based optimizations to the lowest solution of a given symmetry. MOM tackles this problem in the definition of the orbital occupations to constrain the self-consistent field procedure to converge on excited states, whilst the σ-SCF method is based on the minimisation of the variance instead of the energy. To overcome the high computational cost of the variance minimisation, we present a new implementation of the σ-SCF method with the resolution of identity approximation, allowing the use of large basis sets, which is an important requirement for calculations in strong magnetic fields. The accuracy of these methods is assessed by comparison with benchmark literature data for He, H2 and CH+. The results reveal severe limitations of the variance-based scheme, which become more acute in large basis sets. In particular, many states are not accessible using variance optimisation. Detailed analysis shows that this is a general feature of variance optimization approaches due to the masking of local minima in the optimization. In contrast, MOM shows promising performance for computing excited states under these conditions, yielding results consistent with available benchmark data for a diverse range of electronic states

Connections between variation principles at the interface of wave-function and density-functional theories

A recently proposed variation principle [N. I. Gidopoulos, Phys. Rev. A 83, 040502(R) (2011)] for the determination of Kohn–Sham effective potentials is examined and ex- tended to arbitrary electron-interaction strengths and to mixed states. Comparisons are drawn with Lieb’s convex-conjugate functional, which allows for the determina- tion of a potential associated with a given electron density by maximization, yielding the Kohn–Sham potential for a non-interacting system. The mathematical structure of the two functionals is shown to be intrinsically related; the variation principle put forward by Gidopoulos may be expressed in terms of the Lieb functional. The equivalence between the information obtained from the two approaches is illustrated numerically by their implementation in a common framework

Energy densities: a systematic approach to correlation in density functional theory

Density functional theory (DFT) has grown to become by far the most widely applied method in the modelling of electronic systems yet, in contrast to wavefunction-based ab initio methods, the reliability of a DFT calculation can be uncertain. This is because the essential ingredient required for a DFT calculation to be meaningful - the exchange & correlation energy functionals, are approximations for a type of electronic interaction with an unknown functional form. Whilst there exist types of system for which DFT does not provide a useful model - those with significant dispersion interactions and those with near-degenerate states, seeking improvements to DFT in these areas can be far from straightforward since the exchange & correlation functionals cannot be systematically improved. In this work, a mathematically rigorous description of DFT is combined with some of the most reliable & accurate ab initio electronic structure methods in a bespoke development code to obtain a detailed picture of how the exact correlation energy functional in DFT behaves. Using these insights, new approaches are investigated for modelling the correlation energy functional in local form, allowing a systematic study of how best to approximate the correlation energy functional in different types of system to be pursued. As an adjunct to this, the present work looks ahead at how to extend & generalise this investigation by considering the behaviour of systems in the presence of a magnetic field. Efficient algorithms are developed and implemented to facilitate this, enabling the advancement of this strand of investigation in subsequent work