Range-Separation Parameter in Tuned Exchange-Correlation Functionals: Successive Ionizations and the Fukui Function.

The range-separation parameter in tuned, range-separated exchange-correlation functionals is investigated in two contexts. First, the system dependence of the parameter is investigated for a series of systems obtained by successively ionizing a single species, paying particular attention to the degree of linearity in the energy versus electron number curve. The parameter exhibits significant system dependence and, therefore, achieving near-linearity in one segment of the curve leads to strong nonlinearity in other segments. This provides a challenging test case for the development of new functionals designed to overcome the known problems of this class of functional. Next, the study considers whether a range-separation parameter tuned to a Koopmans energy condition is also applicable for the analogous density condition. This is tested by comparing two formulations of the Fukui function of conceptual density functional theory, for three representative systems. Both formulations yield the same general features and are not highly sensitive to the range-separation parameter. However, the agreement between the two is near-optimal when the energy-tuned parameter is used, indicating that this parameter is applicable for the analogous density condition.

System-dependent exchange-correlation functional with exact asymptotic potential and εHOMO ≈ - I.

Density scaling considerations are used to derive an exchange-correlation explicit density functional that is appropriate for the electron deficient side of the integer and which recovers the exact r → ∞ asymptotic behaviour of the exchange-correlation potential. The functional has an unconventional mathematical form with parameters that are system-dependent; the parameters for an N-electron system are determined in advance from generalised gradient approximation (GGA) calculations on the N- and (N - 1)-electron systems. Compared to GGA results, the functional yields similar exchange-correlation energies, but HOMO energies that are an order of magnitude closer to the negative of the vertical ionisation potential; for anions, the HOMO energies are negative, as required. Rydberg excitation energies are also notably improved and the exchange-correlation potential is visibly lowered towards the near-exact potential. Further development is required to improve valence excitations, static isotropic polarisabilities, and the shape of the potential in non-asymptotic regions. The functional is fundamentally different to conventional approximations.

Assessment of Tuning Methods for Enforcing Approximate Energy Linearity in Range-Separated Hybrid Functionals.

A range of tuning methods, for enforcing approximate energy linearity through a system-by-system optimization of a range-separated hybrid functional, are assessed. For a series of atoms, the accuracy of the frontier orbital energies, ionization potentials, electron affinities, and orbital energy gaps is quantified, and particular attention is paid to the extent to which approximate energy linearity is actually achieved. The tuning methods can yield significantly improved orbital energies and orbital energy gaps, compared to those from conventional functionals. For systems with integer M electrons, optimal results are obtained using a tuning norm based on the highest occupied orbital energy of the M and M + 1 electron systems, with deviations of just 0.1-0.2 eV in these quantities, compared to exact values. However, detailed examination for the carbon atom illustrates a subtle cancellation between errors arising from nonlinearity and errors in the computed ionization potentials and electron affinities used in the tuning.