Effective homogeneity of Fermi–Amaldi-containing exchange–correlation functionals

Parr and Ghosh [Phys. Rev. A. 51 3564 (1995)] demonstrated that when near-exact electron densities and potentials are used, the exchange–correlation energies of first- and second-row atoms are well-described by a combination of the Fermi–Amaldi functional with a functional that is homogeneous of degree one under density scaling. Insight into this observation is provided by considering their work from the perspective of the effective homogeneity of the overall exchange–correlation functional. By considering a general form that combines the Fermi–Amaldi functional with a functional that is homogeneous of degree k, it is shown that for these atoms, the functional of Parr and Ghosh (k = 1) exhibits essentially optimal effective homogeneities on the electron-deficient side of the integer. Percentage errors in effective homogeneities are close to percentage errors in exchange–correlation energies

Incorporation of the Fermi–Amaldi Term into Direct Energy Kohn–Sham Calculations

In direct energy Kohn–Sham (DEKS) theory, the density functional theory electronic energy equals the sum of occupied orbital energies, obtained from Kohn–Sham-like orbital equations involving a shifted Hartree exchange–correlation potential, which must be approximated. In the present study, the Fermi–Amaldi term is incorporated into approximate DEKS calculations, introducing the required −1/r contribution to the exchange–correlation component of the shifted potential in asymptotic regions. It also provides a mechanism for eliminating one-electron self-interaction error, and it introduces a nonzero exchange–correlation component of the shift in the potential that is of appropriate magnitude. The resulting electronic energies are very sensitive to the methodologies considered, whereas the highest occupied molecular orbital energies and exchange–correlation potentials are much less sensitive and are similar to those obtained from DEKS calculations using a conventional exchange–correlation functional

Hybrid exchange-correlation functional determined from thermochemical data and ab initio potentials

Multiplicative potentials, appropriate for adding to the non- multiplicative fractional orbital exchange term in the Kohn- Sham equations, are determined from correlated ab initio electron densities. The potentials are examined graphically and are used in conjunction with conventional thermochemical data to determine a new hybrid exchange-correlation functional, denoted B97-2. Calculations using B97-2 are compared with those from (a) the B97-1 functional [J. Chem. Phys. 109, 6264 (1998)], which has the same functional form and fraction of orbital exchange, but was fitted to just thermochemical data; and (b) the widely used B3LYP functional [J. Chem. Phys. 98, 5648 (1993)]. B97-2 atomization energies are close to those from B97-1; total electronic energies and ionization potentials are less accurate, but remain an improvement over B3LYP. Molecular structures from all three functionals are comparable. Static isotropic polarizabilities improve from B3LYP to B97-1 to B97-2; the B97-2 functional underestimates experimental values, which is consistent with the neglect of zero-point vibrational corrections. NMR shielding constants-determined as the conventional second derivative of the electronic energy- improve from B3LYP to B97-1 to B97-2. Shieldings determined directly from these DFT electron densities using the recently proposed MKS approach [Chem. Phys. Lett. 337, 341 (2001)] are two to three times more accurate than the conventional shieldings, and exhibit an analogous improvement across the three functionals. Classical reaction barriers for sixteen chemical reactions improve significantly from B3LYP to B97-1 to B97-2. The introduction of multiplicative potentials into semi- empirical hybrid functional development therefore appears beneficial

Correlation potentials and functionals in Hartree-Fock-Kohn-Sham theory

We compute molecular Hartree-Fock-Kohn-Sham correlation potentials from ab initiocoupled-cluster densities via a modified Zhao, Morrison and Parr [Phys. Rev. A, 50, (1994) 2138] scheme involving exact exchange. We examine the potential for several small systems, and observe complex structure. By fitting a functional expansion to our potentials we obtain a closed-shell functional which is an improvement over other pure correlationfunctionals in Hartree-Fock-Kohn-Sham calculations. The leading term in our functional is dependent on the number of electrons. Our results lead us to question the utility of correlation defined within the Hartree-Fock-Kohn-Sham scheme, and to consider alternative partitionings of the exchange-correlation energy

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

On the description of conical intersections between excited electronic states with LR-TDDFT and ADC(2).

Conical intersections constitute the conceptual bedrock of our working understanding of ultrafast, nonadiabatic processes within photochemistry (and photophysics). Accurate calculation of potential energy surfaces within the vicinity of conical intersections, however, still poses a serious challenge to many popular electronic structure methods. Multiple works have reported on the deficiency of methods like linear-response time-dependent density functional theory within the adiabatic approximation (AA LR-TDDFT) or algebraic diagrammatic construction to second-order [ADC(2)]-approaches often used in excited-state molecular dynamics simulations-to describe conical intersections between the ground and excited electronic states. In the present study, we focus our attention on conical intersections between excited electronic states and probe the ability of AA LR-TDDFT and ADC(2) to describe their topology and topography, using protonated formaldimine and pyrazine as two exemplar molecules. We also take the opportunity to revisit the performance of these methods in describing conical intersections involving the ground electronic state in protonated formaldimine-highlighting in particular how the intersection ring exhibited by AA LR-TDDFT can be perceived either as a (near-to-linear) seam of intersection or two interpenetrating cones, depending on the magnitude of molecular distortions within the branching space. [Abstract copyright: © 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Correlation potentials and functionals in Hartree-Fock-Kohn-Sham theory

We compute molecular Hartree-Fock-Kohn-Sham correlation potentials from ab initiocoupled-cluster densities via a modified Zhao, Morrison and Parr [Phys. Rev. A, 50, (1994) 2138] scheme involving exact exchange. We examine the potential for several small systems, and observe complex structure. By fitting a functional expansion to our potentials we obtain a closed-shell functional which is an improvement over other pure correlationfunctionals in Hartree-Fock-Kohn-Sham calculations. The leading term in our functional is dependent on the number of electrons. Our results lead us to question the utility of correlation defined within the Hartree-Fock-Kohn-Sham scheme, and to consider alternative partitionings of the exchange-correlation energy