Stability conditions and local minima in multicomponent Hartree-Fock and density functional theory

Yang Yang, Tanner Culpitt, Zhen Tao, Sharon Hammes-Schiffer

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9 Scopus citations


Multicomponent quantum chemistry allows the quantum mechanical treatment of electrons and specified protons on the same level. Typically the goal is to identify a self-consistent-field (SCF) solution that is the global minimum associated with the molecular orbital coefficients of the underlying Hartree-Fock (HF) or density functional theory (DFT) calculation. To determine whether the solution is a minimum or a saddle point, herein we derive the stability conditions for multicomponent HF and DFT in the nuclear-electronic orbital (NEO) framework. The gradient is always zero for an SCF solution, whereas the Hessian must be positive semi-definite for the solution to be a minimum rather than a saddle point. The stability matrices for NEO-HF and NEO-DFT have the same matrix structures, which are identical to the working matrices of their corresponding linear response time-dependent theories (NEO-TDHF and NEO-TDDFT) but with a different metric. A negative eigenvalue of the stability matrix is a necessary but not sufficient condition for the corresponding NEO-TDHF or NEO-TDDFT working equation to have an imaginary eigenvalue solution. Electron-proton systems could potentially exhibit three types of instabilities: electronic, protonic, and electron-proton vibronic instabilities. The internal and external stabilities for theories with different constraints on the spin and spatial orbitals can be analyzed. This stability analysis is a useful tool for characterizing SCF solutions and is helpful when searching for lower-energy solutions. Initial applications to HCN, HNC, and 2-cyanomalonaldehyde, in conjunction with NEO Δ SCF calculations, highlight possible connections between stationary points in nuclear coordinate space for conventional electronic structure calculations and stationary points in orbital space for NEO calculations.

Original languageEnglish (US)
Article number084105
JournalJournal of Chemical Physics
Issue number8
StatePublished - Aug 28 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Physics and Astronomy
  • Physical and Theoretical Chemistry


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