Self-Consistent Field Methods for Excited States in Strong Magnetic Fields: a Comparison between Energy- and Variance-Based Approaches

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 selfconsistent 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 energybased 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, H 2 , 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.

On the origin of the inverted singlet–triplet gap of the 5th generation light-emitting molecules

“…73,[77][78][79] The DSCF solutions may be used as reference wave functions for ab initio correlation approaches. 77 However, the DSCF calculations at the HF level on the present molecules led to significant spin contamination hampering its use as reference wave function in ab initio correlation calculations.…”

“…The DE ST values were also estimated at the B3LYP and oB97X-D levels using the DSCF approach. [73][74][75][76][77] We do not discuss any details of the DSCF method since its history has recently been reviewed in ref. 77.…”

“…[73][74][75][76][77] We do not discuss any details of the DSCF method since its history has recently been reviewed in ref. 77. The orthogonality of the non-aufbau SCF solutions can be easily introduced for five of the six studied molecules because the HOMO and the LUMO belong to different irreducible representations of the C 2v point group.…”

“…Molecule 4 belongs to the C s point group with the HOMO and LUMO in the same irreducible representation (a 00 ) implying that more advanced optimization approaches must be employed in the DSCF calculations. 73,[77][78][79] The DSCF solutions may be used as reference wave functions for ab initio correlation approaches. 77 However, the DSCF calculations at the HF level on the present molecules led to significant spin contamination hampering its use as reference wave function in ab initio correlation calculations.…”

Supporting Notebook for "Energy Landscape of State-Specific Electronic Structure Theory"

“…Approximate state-specific variance minimisation can also be considered as an optimisation of the exact variance constrained to the approximate wave function submanifold. Variance minimisation is increasingly being applied to target excited states because it turns the optimisation of higher-energy stationary points into a minimisation problem 27,81,82,86,87,123 and is easily applied for correlated wave functions using VMC. 23,83 In practice, these algorithms often use the foldedspectrum objective function 89…”