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Title: Relativistic theory of nuclear magnetic resonance parameters in a Gaussian basis representation

Abstract

The calculation of NMR parameters from relativistic quantum theory in a Gaussian basis expansion requires some care. While in the absence of a magnetic field the expansion in a kinetically balanced basis converges for the wave function in the mean and for the energy with any desired accuracy, this is not necessarily the case for magnetic properties. The results for the magnetizability or the nuclear magnetic shielding are not even correct in the nonrelativistic limit (nrl) if one expands the original Dirac equation in a kinetically balanced Gaussian basis. This defect disappears if one starts from the unitary transformed Dirac equation as suggested by Kutzelnigg [Phys. Rev. A 67, 032109 (2003)]. However, a new difficulty can arise instead if one applies the transformation in the presence of the magnetic field of a point nucleus. If one decomposes certain contributions, the individual terms may diverge, although their sum is regular. A controlled cancellation may become difficult and numerical instabilities can arise. Various ways exist to avoid these singularities and at the same time get the correct nrl. There are essentially three approaches intermediate between the transformed and the untransformed formulation, namely, the bispinor decomposition, the decomposition of the lower component, andmore » the hybrid unitary transformation partially at operator and partially at matrix level. All three possibilities were first considered by Xiao et al. [J. Chem. Phys. 126, 214101 (2007)] in a different context and in a different nomenclature. Their analysis and classification in a more general context are given here for the first time. Use of an extended balanced basis has no advantages and has other drawbacks and is not competitive, while the use of a restricted magnetic balance basis can be justified.« less

Authors:
 [1];  [2]
  1. Lehrstuhl fuer Theoretische Chemie, Ruhr-Universitaet Bochum, D-44780 Bochum (Germany)
  2. Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, State Key Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry and Molecular Engineering, and Center for Computational Science and Engineering, Peking University, Beijing 100871 (China)
Publication Date:
OSTI Identifier:
21559745
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Chemical Physics; Journal Volume: 131; Journal Issue: 4; Other Information: DOI: 10.1063/1.3185400; (c) 2009 American Institute of Physics
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 36 MATERIALS SCIENCE; ACCURACY; DECOMPOSITION; DEFECTS; DIRAC EQUATION; EXPANSION; GAUSSIAN PROCESSES; MAGNETIC BALANCES; MAGNETIC FIELDS; MAGNETIC PROPERTIES; MAGNETIC SHIELDING; MAGNETIZATION; NUCLEAR MAGNETIC RESONANCE; NUCLEAR SCREENING; NUMERICAL ANALYSIS; QUANTUM MECHANICS; RELATIVISTIC RANGE; SINGULARITY; TRANSFORMATIONS; WAVE FUNCTIONS; CHEMICAL REACTIONS; DIFFERENTIAL EQUATIONS; ENERGY RANGE; EQUATIONS; FIELD EQUATIONS; FUNCTIONS; MAGNETIC RESONANCE; MATHEMATICS; MEASURING INSTRUMENTS; MECHANICS; PARTIAL DIFFERENTIAL EQUATIONS; PHYSICAL PROPERTIES; RESONANCE; SHIELDING; WAVE EQUATIONS

Citation Formats

Kutzelnigg, Werner, and Liu Wenjian. Relativistic theory of nuclear magnetic resonance parameters in a Gaussian basis representation. United States: N. p., 2009. Web. doi:10.1063/1.3185400.
Kutzelnigg, Werner, & Liu Wenjian. Relativistic theory of nuclear magnetic resonance parameters in a Gaussian basis representation. United States. doi:10.1063/1.3185400.
Kutzelnigg, Werner, and Liu Wenjian. 2009. "Relativistic theory of nuclear magnetic resonance parameters in a Gaussian basis representation". United States. doi:10.1063/1.3185400.
@article{osti_21559745,
title = {Relativistic theory of nuclear magnetic resonance parameters in a Gaussian basis representation},
author = {Kutzelnigg, Werner and Liu Wenjian},
abstractNote = {The calculation of NMR parameters from relativistic quantum theory in a Gaussian basis expansion requires some care. While in the absence of a magnetic field the expansion in a kinetically balanced basis converges for the wave function in the mean and for the energy with any desired accuracy, this is not necessarily the case for magnetic properties. The results for the magnetizability or the nuclear magnetic shielding are not even correct in the nonrelativistic limit (nrl) if one expands the original Dirac equation in a kinetically balanced Gaussian basis. This defect disappears if one starts from the unitary transformed Dirac equation as suggested by Kutzelnigg [Phys. Rev. A 67, 032109 (2003)]. However, a new difficulty can arise instead if one applies the transformation in the presence of the magnetic field of a point nucleus. If one decomposes certain contributions, the individual terms may diverge, although their sum is regular. A controlled cancellation may become difficult and numerical instabilities can arise. Various ways exist to avoid these singularities and at the same time get the correct nrl. There are essentially three approaches intermediate between the transformed and the untransformed formulation, namely, the bispinor decomposition, the decomposition of the lower component, and the hybrid unitary transformation partially at operator and partially at matrix level. All three possibilities were first considered by Xiao et al. [J. Chem. Phys. 126, 214101 (2007)] in a different context and in a different nomenclature. Their analysis and classification in a more general context are given here for the first time. Use of an extended balanced basis has no advantages and has other drawbacks and is not competitive, while the use of a restricted magnetic balance basis can be justified.},
doi = {10.1063/1.3185400},
journal = {Journal of Chemical Physics},
number = 4,
volume = 131,
place = {United States},
year = 2009,
month = 7
}
  • A theory is developed for nuclear magnetic resonance spectra of A/sub 2/B/sub 2/ systems with nuclei of higher spin. It is assumed that all nuclei have the same spin value. Otherwise no arbitrary limit is set on the spin. Although the development is made for NMR it also has application to the magnetic properties of clusters of transition-metal ions.
  • This article is concerned with the time dependent quantum theory of collision of heavy particles that result in the excitation of internal degrees of freedom, such as electronic excitations or charge transfer. Attempts to treat the motion of the heavy particle classically encounter difficulties typical to all problems in which the classical degrees of freedom are strongly coupled to quantum degrees of freedom: They lack a feedback mechanism that will force the classical degree of freedom to respond to the excitation of the quantum companion. To avoid such difficulties we propose a method in which the nuclear wave function associatedmore » with each electronic state is represented by a Gaussian wave packet. Each packet is propagated by the time dependent Schroedinger equation on a different electronic energy surface in a manner that resembles classical mechanics but perserves many quantum properties such as Heisenberg uncertainty principle, quantum interference, zero point motion, and the quantum mechanical rules for computing observables. Various limiting cases are discussed in detail and compared numerically.« less
  • An analytical form of expansion coefficients of a diffracted field for an arbitrary Hermite-Gaussian beam in an alien Hermite-Gaussian basis is obtained. A possible physical interpretation of the well-known Young phenomenological diffraction principle and experiments on diffraction of Hermite-Gaussian beams of the lowest types (n = 0 - 5) from half-plane are discussed. The case of nearly homogenous expansion corresponding to misalignment and mismatch of optical systems is also analyzed. 7 refs., 2 figs.
  • Gaussian basis functions, routinely employed in molecular electronic structure calculations, can be combined with numerical grid-based functions in a discrete variable representation to provide an efficient method for computing molecular continuum wave functions. This approach, combined with exterior complex scaling, obviates the need for slowly convergent single-center expansions, and allows one to study a variety of electron-molecule collision problems. The method is illustrated by computation of various bound and continuum properties of H{sub 2}{sup +}.