Four-component relativistic theory for nuclear magnetic shielding: Magnetically balanced gauge-including atomic orbitals

Abstract

It is recognized only recently that the incorporation of the magnetic balance condition is absolutely essential for four-component relativistic theories of magnetic properties. Another important issue to be handled is the so-called gauge problem in calculations of, e.g., molecular magnetic shielding tensors with finite bases. It is shown here that the magnetic balance can be adapted to distributed gauge origins, leading to, e.g., magnetically balanced gauge-including atomic orbitals (MB-GIAOs) in which each magnetically balanced atomic orbital has its own local gauge origin placed on its center. Such a MB-GIAO scheme can be combined with any level of theory for electron correlation. The first implementation is done here at the coupled-perturbed Dirac-Kohn-Sham level. The calculated molecular magnetic shielding tensors are not only independent of the choice of gauge origin but also converge rapidly to the basis set limit. Close inspections reveal that (zeroth order) negative energy states are only important for the expansion of first order electronic core orbitals. Their contributions to the paramagnetism are therefore transferable from atoms to molecule and are essentially canceled out for chemical shifts. This allows for simplifications of the coupled-perturbed equations.

Perspective: Relativistic effects

“…Implementations of 4c NMR shielding tensor calculations with restricted magnetic balance have been reported not long ago, 86,91,92 including extensions to using gauge-including (magnetic-field dependent) basis functions. 86,92 Olejniczak et al recently reported a simplified magnetic balance scheme for 4c NMR calculations whereby the flexibility of unrestricted magnetic balance can be combined with the advantages of a restricted magnetic balance formalism. 93 Quasi-relativistic two-component Hamiltonians are approximate 2c operators where the decoupling of Eq.…”

“…24 Further, these approaches may be combined with 4c matrix formulations for magnetic properties. [84][85][86] For spin-free (scalar relativistic) X2C and NESC, analytic methods for calculations of nuclear gradients and second derivatives have been reported. [87][88][89] Regarding magnetic properties such as NMR shielding, the concept of "magnetic balance" 51 becomes important: The lower-component basis set should not only contain {σ · pχ } functions, but also {σ · Aχ } (see Eq.…”

Relativistic calculations of magnetic resonance parameters: background and some recent developments

“…As in the case of kinetic balance, the term 'restricted' indicates a lower-component basis that consists of just this set of functions, whereas 'unrestricted' indicates that additional functions may be present in the basis. Implementations of four-component NMR shielding tensor calculations with restricted magnetic balance (RMB) were reported not long ago [134][135][136][137], including extensions to using gauge-including (magnetic-field-dependent) basis functions [135,136]. Olejniczak et al [138] have come up with a simplified magnetic balance scheme for four-component magnetic property calculations, in which the flexibility of unrestricted magnetic balance can be combined with the advantages of a formalism that assumes restricted magnetic balance.…”

Relativistic four-component calculations of Buckingham birefringence using London atomic orbitals

“…The use of RKB can therefore lead to significant errors for magnetic properties. This problem can be solved by invoking magnetic balance [58,59] or the unrestrictedkinetic-balance (UKB) scheme. In the latter approach, the Cartesian components of the linear momentum p are treated separately, providing a more flexible basis set to ensure the correct nonrelativistic limit.…”