Abstract
State-specific approximations can provide a more accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere, and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated, and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree–Fock and excited-state mean-field representations of singlet H2 (STO-3G) are then used to explore how the exact energy landscape controls the existence and properties of state-specific approximations. In particular, approximate excited states correspond to constrained stationary points on the exact energy landscape, and their Hessian index also increases for higher energies. Finally, the properties of the exact energy are used to derive the structure of the variance optimization landscape and elucidate the challenges faced by variance optimization algorithms, including the presence of unphysical saddle points or maxima of the variance.
Highlights
Including wave function relaxation in state-specific approximations can provide an accurate representation of excited states where there is significant electron density rearrangement relative to the electronic ground state
Techniques based on linear response theory — including time-dependent Hartree–Fock[14] (TD-HF), time-dependent density functional theory[15–17] (TD-DFT), configuration interaction singles[16,18] (CIS), and equation of motion coupled cluster theory[19,20] (EOM-CC) — are evaluated using the ground-state orbitals, making a balanced treatment of the ground and excited states more difficult.[17]
Counter-intuitively, the restricted HF (RHF) ground state becomes an index-1 saddle point of the excited-state meanfield (ESMF) energy and there is a lower-energy solution at (θ, φ) = (±0.5026, ∓0.3304) that corresponds to the exact ground state
Summary
Including wave function relaxation in state-specific approximations can provide an accurate representation of excited states where there is significant electron density rearrangement relative to the electronic ground state. Recent interest in locating higher-energy SCF solutions has led to several new approaches including: modifying the iterative SCF approach with orbital occupation constraints[1,22] or level-shifting;[10] second-order direct optimisation of higherenergy stationary points;[79,80] minimising an alternative functional such as the variance[4,27,81–83] or the square-magnitude of the energy gradient.[52]. The success of these algorithms depends on the structure of the approximate energy landscape, the stationary properties of excited states, and the quality of the initial guess.
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