Calculations of Excited Electronic States by Converging on Saddle Points Using Generalized Mode Following

Abstract

Calculations of excited electronic states are carried out by finding saddle points on the surface describing the variation of the energy of the system as a function of the electronic degrees of freedom. This approach has several advantages over commonly used methods especially in the context of density functional calculations, as collapse to the ground state is avoided, and yet, the orbitals are variationally optimized for the excited state. Such a state-specific optimization makes it possible to describe excitations with large charge transfer, where calculations based on ground state orbitals, such as linear response time-dependent density functional theory, can be problematic. A generalized mode following method is presented where an n th-order saddle point is found by inverting the components of the gradient in the direction of the eigenvectors of the n lowest eigenvalues of the electronic Hessian matrix. This approach has the distinct advantage of following a chosen excited state by its saddle point order through molecular configurations where the symmetry of the single determinant wave function is broken, thereby making it possible to calculate potential energy curves even at avoided crossings, as demonstrated here in calculations of the ethylene and dihydrogen molecules. Results of calculations are, furthermore, presented for charge transfer excitations in nitrobenzene and N-phenylpyrrole, corresponding to fourth- and sixth-order saddle points, respectively, where an approximate initial estimate of the saddle point order could be found by energy minimization with excited electron and hole orbitals frozen. Finally, calculations of a diplatinum–silver complex are presented, illustrating the applicability of the method to larger molecules.

Generalization of One-Center Nonorthogonal Configuration Interaction Singles to Open-Shell Singlet Reference States: Theory and Application to Valence-Core Pump-Probe States in Acetylacetone

“…89−92 Even still, converging the required orbitals remains one of the challenges to 1C-NOCIS 2eOS but progress on this subject is active and we expect advances to further streamline this step. 93 As a concluding remark, we mention that localization of the core orbitals (when the canonical ground-state orbitals delocalize over several atoms) prior to the SCF re-optimization in the presence of a core hole is crucial for improvements in accuracy. 94 2.2.…”

“…While converging high-energy core excited-state solutions is often difficult, there has been tremendous progress in the past decades. The techniques we employ for that purpose include the maximum-overlap methods, the state-targeted energy projection scheme, and square-gradient minimization, which provides a reliable suite of techniques to stabilize most core excited states. − Even still, converging the required orbitals remains one of the challenges to 1C-NOCIS 2eOS but progress on this subject is active and we expect advances to further streamline this step . As a concluding remark, we mention that localization of the core orbitals (when the canonical ground-state orbitals delocalize over several atoms) prior to the SCF re-optimization in the presence of a core hole is crucial for improvements in accuracy …”

Orbital-Optimized Versus Time-Dependent Density Functional Calculations of Intramolecular Charge Transfer Excited States

“…49,53 Preliminary studies show that the calculations can collapse to lower-energy solutions where the charge is too delocalized, giving an inadequate description of the excited state. 17,49 Moreover, even when MOM manages to prevent variational collapse, the convergence can still be problematic when using SCF algorithms based on the eigendecomposition of the Hamiltonian matrix, such as the direct inversion in the iterative subspace (DIIS). 51,54−58 Recently, direct orbital optimization (DO) methods have been developed for time-independent, variational excited state calculations.…”

Seniority and Hierarchy Configuration Interaction for Radicals and Excited States

“…Instead, one may optimize the orbitals for the excited state of interest (described with an appropriate reference), followed by a separate CI calculation, in a so-called state-specific approach (ΔCI). There has been a recent surge in the development of state-specific methods, covering single-reference and multiconfigurational self-consistent field, − density functional theory, − perturbation theory, − quantum Monte Carlo, − and coupled-cluster (CC) − methods. In particular, by employing a minimal configuration state function (CSF) reference, we have recently shown that excitation-based ΔCI models deliver far more accurate excitation energies than their ground-state-based analogs …”