Electronic orbital response of infinite periodic systems to uniform
external magnetic fields
Bernard Kirtman,a Mohammad Molayem,b and Michael Springborgb
a Department of Chemistry and Biochemistry, University of California, Santa Barbara,
California 93106, U.S.A.
b Physical and Theoretical Chemistry, University of Saarland, 66123 Saarbrucken, Germany
The electronic orbital (as opposed to spin) response to a uniform external magnetic field –
particularly when coupled with the effect of either simultaneous electric fields, mechanical
forces, or structural distortions, as well as internal magnetic fields – is a key ingredient in a
wide variety of important physical properties. Such properties include optical rotation, NMR
shielding, ESR g-tensor, vibrational circular dichroism, Faraday rotation and others. For
ordinary molecules the theoretical treatment of this response is well established, but that is not
the case for infinite periodic systems. Here, we present a treatment of the latter based on an
extension of our previous approach to the corresponding electric field problem. Important
differences arise because the magnetic field term depends upon the gauge origin and also
upon the electronic linear momenta, in addition to the electronic positions. As a consequence,
the obvious choice for the many-electron Hamiltonian operator becomes non-hermitian and
must be modified accordingly. The hermitian character is lost, however, for the subsequently
derived Hartree-Fock (HF) or Kohn-Sham (KS) single-particle operator that determines the
electronic orbitals in the general case. It can be shown, however, that this does not occur for
either the canonical orbitals or the non-canonical orbitals that occur in coupled HF/KS
perturbation theory. As far as the gauge origin is concerned, one can readily convert the
common origin to an individual one for each unit cell by modifying the usual k-dependent
phase of the Bloch orbitals and, then, magnetic field-dependent phases can be included for
each atom of the unit cell as in the molecular GIAO method. On the other hand, in contrast
with electric fields, the field-free phase of the crystal orbitals has no effect.