On derivatives of the energy with respect to total electron number and orbital occupation numbers. A critique of Janak's theorem

The relation between the derivative of the energy with respect to occupation number and the orbital energy, $\partial E/\partial n_i = \epsilon_i$, was first introduced by Slater for approximate total energy expressions such as Hartree-Fock and exchange-only LDA, and his derivation holds for hybrid functionals as well. We argue that Janak's extension of this relation to (exact) Kohn-Sham density functional theory is not valid. The reason is the nonexistence of systems with noninteger electron number, and therefore of the derivative of the total energy with respect to electron number, $\partial E/\partial N$. How to handle the lack of a defined derivative $\partial E/\partial N$ at the integer point, is demonstrated using the Lagrange multiplier technique to enforce constraints. The well-known straight-line behavior of the energy as derived from statistical physical considerations [J.P. Perdew, R. G. Parr, M. Levy and J.J. Balduz, Phys. Rev. Lett. 49, 1691 (1982)] for the average energy of a molecule in a macroscopic sample ("dilute gas") as a function of average electron number is not a property of a single molecule at $T=0$. One may choose to represent the energy of a molecule in the nonphysical domain of noninteger densities by a straight-line functional, but the arbitrariness of this choice precludes the drawing of physical conclusions from it.Comment: 20 pages, 2 figure

Asymptotic nodal planes in the electron density and the potential in the effective equation for the square root of the density

It is known that the asymptotic decay of the electron density outside a molecule is informative about its first ionization potential. It has recently become clear that the special circumstance that the Kohn-Sham (KS) highest-occupied molecular orbital (HOMO) has a nodal plane that extends to infinity may give rise to different cases for the asymptotic behavior of the exact density and of the exact KS potential [Mol. Phys. 114 (2016) 1086]. Here we investigate the consequences of such a HOMO nodal plane for the effective potential in the Schr\"odinger-like equation for the square root of the density, showing that for atoms and molecules it will usually diverge asymptotically on the plane, either exponentially or polynomially, depending on the coupling between Dyson orbitals. We also analyze the issue in the etxernal harmonic potential, reporting an example of an exact analytic density for a fully interacting system that exhibits a different asymptotic behavior on the nodal plane.Comment: Submitted for EPJB (Hardy Gross special issue

Response calculations based on an independent particle system with the exact one-particle density matrix: polarizabilities

Recently, we have demonstrated that the problems finding a suitable adiabatic approximation in time-dependent one-body reduced density matrix functional theory can be remedied by introducing an additional degree of freedom to describe the system: the phase of the natural orbitals [Phys. Rev. Lett. 105, 013002 (2010), J. Chem. Phys. 133, 174119 (2010)]. In this article we will show in detail how the frequency-dependent response equations give the proper static limit ($\omega\to0$), including the perturbation in the chemical potential, which is required in static response theory to ensure the correct number of particles. Additionally we show results for the polarizability for H$_2$ and compare the performance of two different two-electron functionals: the phase-including L\"owdin-Shull functional and the density matrix form of the L\"owdin-Shull functional.Comment: 10 pages, 6 figure

Hydration of methanol in water. A DFT-based molecular dynamics study

We studied the hydration of a single methanol molecule in aqueous solution by first-principle DFT-based molecular dynamics simulation. The calculations show that the local structural and short-time dynamical properties of the water molecules remain almost unchanged by the presence of the methanol, confirming the observation from recent experimental structural data for dilute solutions. We also see, in accordance with this experimental work, a distinct shell of water molecules that consists of about 15 molecules. We found no evidence for a strong tangential ordering of the water molecules in the first hydration shell.Comment: 5 pages, 3 figures, submitted to Chemical Physics Letter

An evaluation of the density functional approach in the zero order regular approximation for relativistic effects: Magnetic interactions in small metal compounds

The density functional approach was evaluated for electron spin resonance (ESR) parameters in the relativistic zero order regular approximation in small metal compounds. The effects of spin-orbit coupling and spin polarization on the magnetic hyperfine interaction were investigated. It was shown that relativistic effects in the calculation of the hyperfine parameters were large not only for the heavy metals but also for ligands bound to heavy elements due to secondary effects

A vibrational circular dichroism implementation within a Slater-type-orbital based density functional framework and its application to hexa- and hepta-helicenes

We describe the implementation of the rotational strengths for vibrational circular dichroism (VCD) in the Slater-type orbital based Amsterdam Density Functional (ADF) package. We show that our implementation, which makes use of analytical derivative techniques and London atomic orbitals, yields origin independent rotational strengths. The basis set dependence in the particular case of Slater-type basis functions is also discussed. It turns out that the triple zeta STO basis sets with one set of polarization functions (TZP) are adequate for VCD calculations. The origin- dependence of the atomic axial tensors is checked by a distributed origin gauge implementation. The distributed and common origin gauge implementations yield virtually identical atomic axial tensors with the Slater-type basis sets employed here, proving that our implementation yields origin independent rotational strengths. We verify the implementation for a set of benchmark molecules, for which the dependence of the VCD spectra on the particular choice of the exchange–correlation functional is studied. The pure functionals BP86 and OLYP show a particularly good performance. Then, we apply this approach to study the VCD spectra of hexa- and hepta- helicenes. In particular we focus on relationships between the sign of the rotational strengths of the two helicenes

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022