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Abstract
Self-consistent field methods for excited states offer an attractive low-cost route to study not only excitation energies but also properties of excited states. Here, we present the generalization of two self-consistent field methods, the maximum overlap method (MOM) and the σ-SCF method, to calculate excited states in strong magnetic fields and investigate their stability and accuracy in this context. These methods use different strategies to overcome the well-known variational collapse of energy-based optimizations to the lowest solution of a given symmetry. The MOM tackles this problem in the definition of the orbital occupations to constrain the self-consistent field procedure to converge on excited states, while the σ-SCF method is based on the minimization of the variance instead of the energy. To overcome the high computational cost of the variance minimization, we present a new implementation of the σ-SCF method with the resolution of identity approximation, allowing the use of large basis sets, which is an important requirement for calculations in strong magnetic fields. The accuracy of these methods is assessed by comparison with the benchmark literature data for He, H2, and CH+. The results reveal severe limitations of the variance-based scheme, which become more acute in large basis sets. In particular, many states are not accessible using variance optimization. Detailed analysis shows that this is a general feature of variance optimization approaches due to the masking of local minima in the optimization. In contrast, the MOM shows promising performance for computing excited states under these conditions, yielding results consistent with available benchmark data for a diverse range of electronic states.
Highlights
Atoms and molecules may exhibit exotic chemistry in strong magnetic fields. Such chemistry is of particular interest in astrophysics since ultrastrong magnetic fields may occur in, for example, the vicinity of white dwarf stars.[1−4] Since this range of magnetic fields cannot be investigated on Earth, quantum chemistry is an essential tool for understanding chemistry under these conditions
The maximum overlap method (MOM), initial MOM (IMOM), and σ-self-consistent field (SCF) have been generalized for calculations in strong magnetic fields and using London atomic orbitals (LAOs) in our development platform QUEST,[85,86] where all three methods have been implemented using standard two electron integrals (TEIs) and with the RI approximation
The MOM/ IMOM may be used with DFT, in the present work, they are only applied to Hartree−Fock to enable a direct comparison between the MOM/IMOM and σ-SCF, since the latter is not directly applicable to DFT
Summary
Atoms and molecules may exhibit exotic chemistry in strong magnetic fields. Such chemistry is of particular interest in astrophysics since ultrastrong magnetic fields may occur in, for example, the vicinity of white dwarf stars.[1−4] Since this range of magnetic fields cannot be investigated on Earth, quantum chemistry is an essential tool for understanding chemistry under these conditions. The computation of excited states through mean-field approaches has seen a great renewal of interest, and different strategies have been explored.[66−74] These include the possibility to overcome the variational collapse of the energy optimization with direct optimization techniques In this context, Hait and Head-Gordon proposed an extension of direct orbital optimization to optimize excited states by minimizing the square of the orbital gradient to avoid variational collapse; this is termed the squared-gradient minimization (SGM) approach.[71] the use of quasi-Newton optimization approaches in this context has been exploited by Levi et al.[73,74] In order to improve further on the Δ-SCF method, as the MOM does by tracking the orbital occupations, the statetargeted energy projection method enforces the computation of non-auf bau configurations by restricting the occupiedvirtual orbital rotations by means of a level shift in the definition of the Fock matrix.[72] following a different approach, Neuscamman et al developed excited-state meanfield theory (ESMFT), in which a generalized variation principle is introduced to target excited states using the method of Lagrange multipliers.[66−68].
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