Abstract
An efficient implementation of geometrical derivatives at the Hartree–Fock (HF) and current-density functional theory (CDFT) levels is presented for the study of molecular structure in strong magnetic fields. The required integral derivatives are constructed using a hybrid McMurchie–Davidson and Rys quadrature approach, which combines the amenability of the former to the evaluation of derivative integrals with the efficiency of the latter for basis sets with high angular momentum. In addition to its application to evaluating derivatives of four-center integrals, this approach is also applied to gradients using the resolution-of-the-identity approximation, enabling efficient optimization of molecular structure for many-electron systems under a strong magnetic field. The CDFT contributions have been implemented for a wide range of density functionals up to and including the meta-GGA level with current-density dependent contributions and (range-separated) hybrids for the first time. Illustrative applications are presented to the OH and benzene molecules, revealing the rich and complex chemistry induced by the presence of an external magnetic field. Challenges for geometry optimization in strong fields are highlighted, along with the requirement for careful analysis of the resulting electronic structure at each stationary point. The importance of correlation effects is examined by comparison of results at the HF and CDFT levels. The present implementation of molecular gradients at the CDFT level provides a cost-effective approach to the study of molecular structure under strong magnetic fields, opening up many new possibilities for the study of chemistry in this regime.
Highlights
Interest has grown over recent years in the nonperturbative calculation of electronic structure in strong magnetic fields.[1−17] Such calculations are of interest since they are one means by which static response properties with respect to an applied magnetic field may be evaluated and since they are essential for modeling the behavior of molecular systems in strong magnetic fields that cannot be treated perturbatively and of the kind found on stellar objects such as magnetic white dwarf stars.[18−20] Molecular hydrogen has been observed in spectra from nonmagnetic white dwarf stars,[21] suggesting that molecules and even chemistry may be possible in such extreme environments
The methods for computing one- and two-electron derivative integrals described in Section 4, along with xc analytical derivatives for current-density functional theory (CDFT) described in Section 6.2, have been implemented in the QUEST electronic structure code,[28] which is used in the present work for the geometry optimization of two small molecules in strong magnetic fields
The principal focus here is on the generalized McMurchie−Davidson algorithm[1,33,34,37−40,115,116] due to its amenability to the evaluation of derivative integrals; due to the superior efficiency of the Rys quadrature[30,51−53,117] for integrals of high angular momentum, a method using a hybrid of Rys and McMurchie−Davidson approaches was proposed for the evaluation of derivative integrals over London atomic orbitals (LAOs)
Summary
Interest has grown over recent years in the nonperturbative calculation of electronic structure in strong magnetic fields.[1−17] Such calculations are of interest since they are one means by which static response properties with respect to an applied magnetic field may be evaluated and since they are essential for modeling the behavior of molecular systems in strong magnetic fields that cannot be treated perturbatively and of the kind found on stellar objects such as magnetic white dwarf stars.[18−20] Molecular hydrogen has been observed in spectra from nonmagnetic white dwarf stars,[21] suggesting that molecules and even chemistry may be possible in such extreme environments. Initial developments and investigations concerned the application of Hartree−Fock theory to systems in strong magnetic fields,[1,2] with subsequent work employing configuration interaction,[24] coupled-cluster theory,[9] equation of motion coupled-cluster theory,[10,11] current−density functional theory,[12,13] and most recently the calculation of spectra using real-time time-dependent self-consistent field methods.[6−8,17] Several electronic structure packages have been developed or generalized to treat systems in strong magnetic fields, starting with the LONDON quantum chemistry program[24] and followed by the BAGEL program,[25] the CHRONUSQ package,[26] TURBOMOLE,[27] and our own development code QUEST.[28] Central to these developments has been the use of Gaussiantype London atomic orbitals (LAOs),[29] which comprise a standard Gaussian basis function multiplied with a complex plane-wave phase factor dependent on the external magnetic field and the gauge origin.
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