Michael J. G. Peach

Assessment of a Coulomb-attenuated exchange–correlation energy functional

The recently proposed CAM-B3LYP exchange-correlation energy functional, based on a partitioning of the r operator in the exchange interaction into long- and short-range components, is assessed for the determination of molecular thermochemistry, structures, and second order response properties. Rydberg and charge transfer excitation energies and static electronic polarisabilities are notably improved over the standard B3LYP functional; classical reaction barriers also improve. Ionisation potentials, bond lengths, NMR shielding constants and indirect spin-spin coupling constants are comparable with the two functionals. CAM-B3LYP atomisation energies and diatomic harmonic vibrational wavenumbers are less accurate than those of B3LYP. Future research directions are outlined.

Influence of Triplet Instabilities in TDDFT.

Singlet and triplet vertical excitation energies from time-dependent density functional theory (TDDFT) can be affected in different ways by the inclusion of exact exchange in hybrid or Coulomb-attenuated/range-separated exchange-correlation functionals; in particular, triplet excitation energies can become significantly too low. To investigate these issues, the explicit dependence of excitation energies on exact exchange is quantified for four representative molecules, paying attention to the effect of constant, short-range, and long-range contributions. A stability analysis is used to verify that the problematic TDDFT triplet excitations can be understood in terms of the ground state triplet instability problem, and it is proposed that a Hartree-Fock stability analysis should be used to identify triplet excitations for which the presence of exact exchange in the TDDFT functional is undesirable. The use of the Tamm-Dancoff approximation (TDA) significantly improves the problematic triplet excitation energies, recovering the correct state ordering in benzoquinone; it also affects the corresponding singlet states, recovering the correct state ordering in naphthalene. The impressive performance of the TDA is maintained for a wide range of molecules across representative functionals.

Overcoming low orbital overlap and triplet instability problems in TDDFT.

Low orbital overlap and triplet instability problems in time-dependent density functional theory (TDDFT) are investigated for a new benchmark set, encompassing challenging singlet and triplet excitation energies of local, charge-transfer, and Rydberg character. The low orbital overlap problem is largely overcome for both singlet and triplet states by the use of a Coulomb-attenuated functional. For all the categories of functional considered, however, errors associated with triplet instability problems plague high overlap excitations, as exemplified by the excited states of acenes and polyacetylene oligomers. Application of the Tamm-Dancoff approximation reduces these errors for both singlet and triplet states, while leaving low-overlap excitations unaffected. The study illustrates the synergy between overlap and stability and highlights the success of a combined, Coulomb-attenuated Tamm-Dancoff approach.

Influence of Coulomb-attenuation on exchange-correlation functional quality.

The dependence of functional quality on the attenuation parameters--which control the limiting (r12-->0, infinity) values and the rate of attenuation--is investigated for a Coulomb-attenuated exchange-correlation functional. For the attenuation and functional form considered, satisfaction of an exact long-range condition is detrimental for properties such as atomisation energies and bond lengths, but does improve classical reaction barriers and small molecule electronic excitation energies. The functionals considered can provide high quality valence, Rydberg, intramolecular and asymptotic intermolecular charge transfer (CT) excitations, but none are able to provide a simultaneously optimal description of all classes; CT excitations are not necessarily improved compared to those from conventional functionals. The study highlights the need for further development of Coulomb-attenuated functionals.

Modeling the adiabatic connection in H2.

Full configuration interaction (FCI) data are used to quantify the accuracy of approximate adiabatic connection (AC) forms in describing the ground state potential energy curve of H2, within spin-restricted density functional theory (DFT). For each internuclear separation R, accurate properties of the AC are determined from large basis set FCI calculations. The parameters in the approximate AC form are then determined so as to reproduce these FCI values exactly, yielding an exchange-correlation energy expressed entirely in terms of FCI-derived quantities. This is combined with other FCI-derived energy components to give the total electronic energy; comparison with the FCI energy quantifies the accuracy of the AC form. Initial calculations focus on a [1/1]-Padé-based form. The potential energy curve determined using the procedure is a notable improvement over those from existing DFT functionals. The accuracy near equilibrium is quantified by calculating the bond length and vibrational wave numbers; errors in the latter are below 0.5%. The molecule dissociates correctly, which can be traced to the use of virtual orbital eigenvalues in the slope in the noninteracting limit, capturing static correlation. At intermediate R, the potential energy curve exhibits an unphysical barrier, similar to that noted previously using the random phase approximation. Alternative forms of the AC are also considered, paying attention to size extensivity and the behavior in the strong-interaction limit; none provide an accurate potential energy curve for all R, although good accuracy can be achieved near equilibrium. The study demonstrates how data from correlated ab initio calculations can provide valuable information about AC forms and highlight areas where further theoretical progress is required.

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.

Adiabatic connection forms in density functional theory: H2 and the He isoelectronic series

Full configuration interaction (FCI) data are used to quantify the accuracy of approximate adiabatic connection (AC) forms in describing two challenging problems in density functional theory--the singlet ground state potential energy curve of H(2) in a restricted formalism and the energies of the helium isoelectronic series, H(-) to Ne(8+). For H(2), an exponential-based form yields a potential energy curve that is virtually indistinguishable from the FCI curve, eliminating the unphysical barrier to dissociation observed previously with a [1,1]-Pade-based form and with the random phase approximation. For the helium isoelectronic series, the Pade-based form gives the best overall description, followed by the exponential form, with errors that are orders of magnitude smaller than those from a standard hybrid functional. Particular attention is paid to the limiting behavior of the AC forms with increasing bond distance in H(2) and increasing atomic number in the isoelectronic series; several forms describe both limits correctly. The study illustrates the very high quality results that can be obtained using exchange-correlation functionals based on simple AC forms, when near-exact data are used to determine the parameters in the forms.

Fractional electron loss in approximate DFT and Hartree–Fock theory

Plots of electronic energy vs electron number, determined using approximate density functional theory (DFT) and Hartree-Fock theory, are typically piecewise convex and piecewise concave, respectively. The curves also commonly exhibit a minimum and maximum, respectively, in the neutral → anion segment, which lead to positive DFT anion HOMO energies and positive Hartree-Fock neutral LUMO energies. These minima/maxima are a consequence of using basis sets that are local to the system, preventing fractional electron loss. Ground-state curves are presented that illustrate the idealized behavior that would occur if the basis set were to be modified to enable fractional electron loss without changing the description in the vicinity of the system. The key feature is that the energy cannot increase when the electron number increases, so the slope cannot be anywhere positive, meaning frontier orbital energies cannot be positive. For the convex (DFT) case, the idealized curve is flat beyond a critical electron number such that any additional fraction of an electron added to the system is unbound. The anion HOMO energy is zero. For the concave (Hartree-Fock) case, the idealized curve is flat up to some critical electron number, beyond which it curves down to the anion energy. A minimum fraction of an electron is required before any binding occurs, but beyond that, the full fraction abruptly binds. The neutral LUMO energy is zero. Approximate DFT and Hartree-Fock results are presented for the F → F(-) segment, and results approaching the idealized behavior are recovered for highly diffuse basis sets. It is noted that if a DFT calculation using a highly diffuse basis set yields a negative LUMO energy then a fraction of an electron must bind and the electron affinity must be positive, irrespective of whether an electron binds experimentally. This is illustrated by calculations on Ne → Ne(-).

Shielding Constants and Chemical Shifts in DFT: Influence of Optimized Effective Potential and Coulomb-Attenuation

The influence of the optimized effective potential (OEP) and Coulomb-attenuation on shielding constants and chemical shifts is investigated for three disparate categories of molecule: main group, hydrogen bonded, and transition metal systems. Expanding the OEP in the orbital basis leads to physically sensible exchange-correlation potentials; OEP generalized gradient approximation results provide some indication of the accuracy of the expansion. OEP uncoupled magnetic parameters from representative hybrid and Coulomb-attenuated functionals can be a dramatic improvement over conventional results; both categories yield similar accuracy. Additional flexibility is introduced by expanding the OEP in an extensive even-tempered basis set, but this leads to the well-known problem of unphysical, oscillatory potentials. Smooth potentials are recovered through the use of a smoothing norm, but deficiencies in the procedure are highlighted for transition metal complexes. The study reiterates the importance of the OEP procedure in magnetic response calculations using orbital-dependent functionals, together with the need for careful attention to ensure physically sensible potentials. It also illustrates the utility of Coulomb-attenuated functionals for computing short-range molecular properties.

On the triplet instability in TDDFT

The utility of a Hartree–Fock triplet stability threshold, for identifying time-dependent density functional theory triplet excitations for which the inclusion of exact orbital exchange is detrimental, is illustrated for a benchmark set of 63 excitations in 20 organic molecules. Following earlier spin–spin coupling observations that suggest a relatively small triplet instability problem, the accuracy of triplet excitation energies from the B97-2 hybrid functional is also quantified. As anticipated, the results are relatively accurate and this is readily understood in terms of the stabilities. Application of the Tamm–Dancoff approximation dramatically improves all the triplet excitation energies corresponding to low stabilities, whilst also providing a modest improvement in the corresponding singlet states.