Variational Density Functional Calculations of Excited States via Direct Optimization

Abstract

The development of variational density functional theory approaches to excited electronic states is impeded by limitations of the commonly used self-consistent field (SCF) procedure. A method based on a direct optimization approach as well as the maximum overlap method is presented, and the performance is compared with previously proposed SCF strategies. Excited-state solutions correspond to saddle points of the energy as a function of the electronic degrees of freedom. The approach presented here makes use of a preconditioner determined with the help of the maximum overlap method to guide the convergence on a target nth-order saddle point. This method is found to be more robust and converge faster than previously proposed SCF approaches for a set of 89 excited states of molecules. A limited-memory formulation of the symmetric rank-one method for updating the inverse Hessian is found to give the best performance. A conical intersection for the carbon monoxide molecule is calculated without resorting to fractional occupation numbers. Calculations on the excited states of the hydrogen atom and a doubly excited state of the dihydrogen molecule using a self-interaction corrected functional are presented. For these systems, the self-interaction correction is found to improve the accuracy of density functional calculations of excited states.

The photodissociation of solvated cyclopropanone and its hydrate explored via non-adiabatic molecular dynamics using ΔSCF

“…The DFT-based ΔSCF method is a variational method that independently optimizes excited state electronic densities ( ρ i ( r⃑ ) corresponding to electronic state i ). 38–45 Excited electronic states of systems with discrete eigenstates are obtained by a Kohn–Sham (KS) DFT optimisation with non-Aufbau fixed KS molecular orbital (MO) occupations ( n i j corresponding to KS MO φ i j ( r⃑ )) using the ground-state algorithms, so that the total final electron density resembles that of a certain excited electronic state i .The sum of the orbital occupation numbers equals the total number of electrons N e and each KS MO φ i j ( r⃑ ) is an eigenfunction of the KS equation corresponding to eigenenergy ε i j …”

Modeling of the spectroscopy of core electrons with density functional theory

“…The ionization energy can then be simply evaluated as the difference between the ground state and core‐ionized state energies in what is usually referred to as a ΔSCF method. This approach originated in the work of Bagus and Schaefer, 31 and more recently it has been implemented and developed further with an emphasis on preventing variational collapse in difficult cases 32‐37 . This approach is used widely, and many applications have been reported 38‐45 .…”

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

“…To overcome this issue, several techniques have been suggested [35,41,42,[52][53][54][55]. In our calculations we employ the maximum-overlap method (MOM) [35,53], where the above ambiguity is resolved by assigning an occupation number to each orbital based on orbital shape rather than orbital energy.…”