Both kinetically balanced (KB) and kinetically unbalanced (KU) rotational London orbitals (RLO) are proposed to resolve the slow basis set convergence in relativistic calculations of nuclear spin-rotation (NSR) coupling tensors of molecules containing heavy elements [Y. Xiao and W. Liu, J. Chem. Phys. 138, 134104 (2013)]. While they perform rather similarly, the KB-RLO Ansatz is clearly preferred as it ensures the correct nonrelativistic limit even with a finite basis. Moreover, it gives rise to the same "direct relativistic mapping" between nuclear magnetic resonance shielding and NSR coupling tensors as that without using the London orbitals [Y. Xiao, Y. Zhang, and W. Liu, J. Chem. Theory Comput. 10, 600 (2014)].
The idea for separating the algebraic exact two-component (X2C) relativistic Hamiltonians into spin-free (sf) and spin-dependent terms [Z. Li, Y. Xiao, and W. Liu, J. Chem. Phys. 137, 154114 (2012)] is extended to both electric and magnetic molecular properties. Taking the spin-free terms (which are correct to infinite order in ? ? 1/137) as zeroth order, the spin-dependent terms can be treated to any desired order via analytic derivative technique. This is further facilitated by unified Sylvester equations for the response of the decoupling and renormalization matrices to single or multiple perturbations. For practical purposes, explicit expressions of order ?(2) in spin are also given for electric and magnetic properties, as well as two-electron spin-orbit couplings. At this order, the response of the decoupling and renormalization matrices is not required, such that the expressions are very compact and completely parallel to those based on the Breit-Pauli (BP) Hamiltonian. However, the former employ sf-X2C wave functions, whereas the latter can only use nonrelativistic wave functions. As the sf-X2C terms can readily be interfaced with any nonrelativistic program, the implementation of the O(?²) spin-orbit corrections to sf-X2C properties requires only marginal revisions of the routines for evaluating the BP type of corrections.
The relativistic molecular Hamiltonian written in the body-fixed frame of reference is the basis for high-precision calculations of spectroscopic parameters involving nuclear vibrations and/or rotations. Such a Hamiltonian that describes electrons fully relativistically and nuclei quasi-relativistically is just developed for semi-rigid nonlinear molecules [Y. Xiao and W. Liu, J. Chem. Phys. 138, 134104 (2013)]. Yet, the formulation should somewhat be revised for linear molecules thanks to some unusual features arising from the redundancy of the rotation around the molecular axis. Nonetheless, the resulting isomorphic Hamiltonian is rather similar to that for nonlinear molecules. Consequently, the relativistic formulation of nuclear spin-rotation (NSR) tensor for linear molecules is very much the same as that for nonlinear molecules. So is the relativistic mapping between experimental NSR and NMR.
A relativistic molecular Hamiltonian that describes electrons fully relativistically and nuclei quasi-relativistically is proposed and transformed from the laboratory to the body-fixed frame of reference. As a first application of the resulting body-fixed relativistic molecular Hamiltonian, the long anticipated relativistic theory of nuclear spin-rotation (NSR) tensor is formulated rigorously. A "relativistic mapping" between experimental NSR and NMR is further proposed, which is of great value in establishing high-precision absolute NMR shielding scales.
An exact two-component (X2C) relativistic theory for nuclear magnetic resonance parameters is obtained by first a single block-diagonalization of the matrix representation of the Dirac operator in a magnetic-field-dependent basis and then a magnetic perturbation expansion of the resultant two-component Hamiltonian and transformation matrices. Such a matrix formulation is not only simple but also general in the sense that the various ways of incorporating the field dependence can be treated in a unified manner. The X2C dia- and paramagnetic terms agree individually with the corresponding four-component ones up to machine accuracy for any basis.
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.
Several four-component relativistic approaches for nuclear magnetic shielding constant have recently been proposed and their formal relationships have also been established [Xiao et al., J. Chem. Phys. 126, 214101 (2007)]. It is shown here that the approaches can be recast into a unified form via the generic ansatz of orbital decomposition. The extension of the formalisms to magnetizability (and nuclear spin-spin coupling) is straightforward. Exact analytical expressions are also derived for both the shielding constant and magnetizability of the hydrogenlike atom in the ground state. A series of calculations on Rn(85+) and Rn is then carried out to reveal the performance of the various methods with respect to the basis set requirement, leading to the conclusion that it is absolutely essential to explicitly account for the magnetic balance condition. However, different ways of doing so lead to quite similar results. It is also demonstrated that only extremely compact negative energy states are important for the total shieldings and their effects are hence essentially canceled out for chemical shifts. This has important implications for further theoretical developments.
The previously proposed exact two-component (X2C) relativistic theory of nuclear magnetic resonance (NMR) parameters [Q. Sun, W. Liu, Y. Xiao, and L. Cheng, J. Chem. Phys. 131, 081101 (2009)] is reformulated to accommodate two schemes for kinetic balance, five schemes for magnetic balance, and three schemes for decoupling in a unified manner, at both matrix and operator levels. In addition, three definitions of spin magnetization are considered in the coupled-perturbed Kohn-Sham equation. Apart from its simplicity, the most salient feature of X2C-NMR lies in that its diamagnetic and paramagnetic terms agree individually with the corresponding four-component counterparts for any finite basis. For practical applications, five approximate schemes for the first order coupling matrix X(10) and four approximate schemes for the treatment of two-electron integrals are introduced, which render the computations of X2C-NMR very much the same as those of approximate two-component approaches.
The separation of the spin-free and spin-dependent terms of a given relativistic Hamiltonian is usually facilitated by the Dirac identity. However, this is no longer possible for the recently developed exact two-component relativistic Hamiltonians derived from the matrix representation of the Dirac equation in a kinetically balanced basis. This stems from the fact that the decoupling matrix does not have an explicit form. To resolve this formal difficulty, we first define the spin-dependent term as the difference between a two-component Hamiltonian corresponding to the full Dirac equation and its one-component counterpart corresponding to the spin-free Dirac equation. The series expansion of the spin-dependent term is then developed in two different ways. One is in the spirit of the Douglas-Kroll-Hess (DKH) transformation and the other is based on the perturbative expansion of a two-component Hamiltonian of fixed structure, either the two-step Barysz-Sadlej-Snijders (BSS) or the one-step exact two-component (X2C) form. The algorithms for constructing arbitrary order terms are proposed for both schemes and their convergence patterns are assessed numerically. Truncating the expansions to finite orders leads naturally to a sequence of novel spin-dependent Hamiltonians. In particular, the order-by-order distinctions among the DKH, BSS, and X2C approaches can nicely be revealed. The well-known Pauli, zeroth-order regular approximation, and DKH1 spin-dependent Hamiltonians can also be recovered naturally by appropriately approximating the decoupling and renormalization matrices. On the practical side, the sf-X2C+so-DKH3 Hamiltonian, together with appropriately constructed generally contracted basis sets, is most promising for accounting for relativistic effects in two steps, first spin-free and then spin-dependent, with the latter applied either perturbatively or variationally.
Related JoVE Video
Journal of Visualized Experiments
What is Visualize?
JoVE Visualize is a tool created to match the last 5 years of PubMed publications to methods in JoVE's video library.
How does it work?
We use abstracts found on PubMed and match them to JoVE videos to create a list of 10 to 30 related methods videos.
Video X seems to be unrelated to Abstract Y...
In developing our video relationships, we compare around 5 million PubMed articles to our library of over 4,500 methods videos. In some cases the language used in the PubMed abstracts makes matching that content to a JoVE video difficult. In other cases, there happens not to be any content in our video library that is relevant to the topic of a given abstract. In these cases, our algorithms are trying their best to display videos with relevant content, which can sometimes result in matched videos with only a slight relation.