Gauge-origin independent formalism of two-component relativistic framework based on unitary transformation in nuclear magnetic shielding constant

Abstract

This article proposes a gauge-origin independent formalism of the nuclear magnetic shielding constant in the two-component relativistic framework based on the unitary transformation. The proposed scheme introduces the gauge factor and the unitary transformation into the atomic orbitals. The two-component relativistic equation is formulated by block-diagonalizing the Dirac Hamiltonian together with gauge factors. This formulation is available for arbitrary relativistic unitary transformations. Then, the infinite-order Douglas-Kroll-Hess (IODKH) transformation is applied to the present formulation. Next, the analytical derivatives of the IODKH Hamiltonian for the evaluation of the nuclear magnetic shielding constant are derived. Results obtained from the numerical assessments demonstrate that the present formulation removes the gauge-origin dependence completely. Furthermore, the formulation with the IODKH transformation gives results that are close to those in four-component and other two-component relativistic schemes.

Electron–Nucleus Hyperfine Coupling Calculated from Restricted Active Space Wavefunctions and an Exact Two-Component Hamiltonian

“…(For properties that only depend on nuclear hyperfine magnetic moments, such as HFC or the indirect nuclear spin–spin coupling, field-dependent basis sets are typically not used because the gauge origin is naturally defined at the position of the nucleus.) Magnetic properties were eventually calculated with DKH to arbitrary order, , offering access to fully relativistic results from 2c calculations. However, in the meantime, “exact two-component” (X2C) methods have emerged, where matrix representations of the 2c (one-electron) Hamiltonian in a basis are constructed directly along with the transformation matrices from the 4c to 2c picture. − The treatment of the electron correlation and quantum electrodynamic effects remains a challenge, , but X2C has opened the way to determine the matrix representations for fully relativistic 2c one-electron operators, that is, capturing all picture-change effects needed for molecular property calculations, including magnetic properties. − …”

“…(For properties that only depend on nuclear hyperfine magnetic moments, such as HFC or the indirect nuclear spin−spin coupling, fielddependent basis sets are typically not used because the gauge origin is naturally defined at the position of the nucleus.) Magnetic properties were eventually calculated with DKH to arbitrary order, 41,42 offering access to fully relativistic results from 2c calculations. However, in the meantime, "exact twocomponent" (X2C) methods have emerged, where matrix representations of the 2c (one-electron) Hamiltonian in a basis are constructed directly along with the transformation matrices from the 4c to 2c picture.…”

RAQET: Large‐scale two‐component relativistic quantum chemistry program package

“…To perform gauge‐origin independent calculations in the NR framework, RAQET employs the gauge‐including/independent atomic orbital (GIAO) . In the 2c relativistic framework, RAQET employs the gauge factorization before unitary transformation (GF‐UT) formulation, which has been derived for gauge‐origin independent calculations by introducing the gauge factor and 2c unitary transformation into the molecular orbitals . The SD relativistic effect, which can include the spin‐dipolar and the Fermi‐contact contributions, is essential for molecules containing heavy elements, particularly for calculation of the nuclear magnetic shielding constant.…”

Relativistic Effects in Magnetic Circular Dichroism: Restricted Magnetic Balance and Temperature Dependence

“…When both spin–orbit coupling and magnetic field are considered in the Dirac equation, cast in a Gaussian basis, the magnetic balance condition between the large and small component bases is preferred over the kinetic balance condition. − The restricted magnetic balance was implemented with four-component methods for the simulation of nuclear magnetic resonance shielding constants . It was later combined with gauge-including atomic orbitals (GIAO) to enforce the gauge-origin independence in both the four- − and two-component − relativistic formalism.…”