Wavefunction stability analysis without analytical electronic Hessians: application to orbital-optimised second-order Møller–Plesset theory and VV10-containing density functionals

Abstract

Wavefunction stability analysis is commonly applied to converged self-consistent field (SCF) solutions to verify whether the electronic energy is a local minimum with respect to second-order variations in the orbitals. By iterative diagonalisation, the procedure calculates the lowest eigenvalue of the stability matrix or electronic Hessian. However, analytical expressions for the electronic Hessian are unavailable for most advanced post-Hartree–Fock (HF) wave function methods and even some Kohn–Sham (KS) density functionals. To address such cases, we formulate the Hessian-vector product within the iterative diagonalisation procedure as a finite difference of the electronic gradient with respect to orbital perturbations in the direction of the vector. As a model application, following the lowest eigenvalue of the orbital-optimised second-order Møller–Plesset perturbation theory (OOMP2) Hessian during H₂ dissociation reveals the surprising stability of the spin-restricted solution at all separations, with a second independent unrestricted solution. We show that a single stable solution can be recovered by using the regularised OOMP2 method (δ-OOMP2), which contains a level shift. Internal and external stability analyses are also performed for SCF solutions of a recently developed range-separated hybrid density functional, ωB97X-V, for which the analytical Hessian is not yet available due to the complexity of its long-range non-local VV10 correlation functional.

Keywords:

• quantum chemistry
• stability analysis
• density functional theory
• Brueckner orbitals
• orbital optimisation

Acknowledgements

The authors would like to thank Narbe Mardirossian for providing instructive examples for stability analysis with the ωB97X-V functional and Dr Julien Panetier for electrocatalyst geometries. The authors acknowledge computational resources obtained under National Science Foundation (NSF) award numbers CHE-1048789 and CHE-0840505.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by a grant from Chevron Energy Technology Co. and the Department of Energy Office of Science Graduate Fellowship Program (DOE SCGF). This study was made possible, in part, by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU [contract number DE-AC05- 06OR23100]; the Office of Science, Office of Basic Energy Sciences, the US Department of Energy [contract number DE-AC02-05CH11231].