Towards a Holomorphic Density Functional Theory

Abstract

Self-consistent-field (SCF) approximations formulated using Hartree−Fock (HF) or Kohn−Sham density-functional theory (KS-DFT) have the potential to yield multiple solutions. However, the formal relationship between multiple solutions identified using HF or KS-DFT remains generally unknown. We investigate the connection between multiple SCF solutions for HF or KS-DFT by introducing a parameterized functional that scales between the two representations. Using the hydrogen molecule and a model of electron transfer, we continuously map multiple solutions from the HF potential to a KS-DFT description. We discover that multiple solutions can coalesce and vanish as the functional changes, forming a direct analogy with the disappearance of real HF solutions along a change in molecular structure. To overcome this disappearance of solutions, we develop a complex-analytic extension of DFTthe "holomorphic DFT" approachthat allows every SCF stationary state to be analytically continued across all molecular structures and exchange−correlation functionals.

Energy Landscapes for Electronic Structure

“…Therefore, although the basis set or SCF potential leaves the general structure and symmetry of the landscape unchanged, it can change the relative energies of stationary points and cause certain solutions to disappear. The disappearance of solutions using different levels of theory is analogous to the case of Coulson–Fischer points in HF theory, where changes in relative components of the energy for different molecular structures can cause HF solutions to coalesce and vanish. ,, …”

“…Furthermore, as the side length decreases, solutions that disappear are seen to coalesce and vanish at two-fold pair-annihilation points and three-fold confluence points along the binding curve. , The disappearance of the four-fold degenerate local minima occurs at three-fold coalescence points corresponding to cusp catastrophes, where one minimum coalesces with two index-1 saddle points to leave a single index-1 saddle point (see Appendix D). When these HF solutions disappear, complex-analytic extensions can be constructed using the recently developed holomorphic HF approach ,, and its KS-DFT counterpart …”

Energy Landscape of State-Specific Electronic Structure Theory

“…Underpinning excited state-specific methods is the fundamental idea that ground-state wave functions can also be used to describe an electronic excited state. This philosophy relies on the existence of additional higher-energy mathematical solutions, which have been found in Hartree–Fock (HF), ,− density functional theory (DFT), − multiconfigurational self-consistent field (MC-SCF), − and coupled cluster (CC) theory. − These multiple solutions correspond to higher-energy stationary points of a parametrized approximate energy function, including local energy minima, saddle points, or maxima. It has long been known that the exact k th excited state forms a saddle point of the energy with k negative Hessian eigenvalues (where k = 0 is the ground state).…”

Strategies for solving the excited-state self-consistent-field problem for highly excited and multiply ionized states

“…Because of the large number of electronic configurations involved, it is critical to employ an efficient and robust electronic structure method. To that end, we seek to calculate excited electronic states by employing a mean-field approximation directly solving the self-consistent-field (SCF) equation for excited electronic configurations [33][34][35][36][37][38][39][40][41][42]. In our previous works [16,25,28,31,43], we employed the excited state orbitals obtained with the Hartree-Fock-Slater (HFS) [44] method as implemented in the XMOLECULE toolkit [25,43] to obtain cross sections and rates.…”