International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 8.7, pp. 731-732

Section 8.7.4.7. Properties derivable from spin densities

P. Coppens,a Z. Sub and P. J. Beckerc

a732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France

8.7.4.7. Properties derivable from spin densities

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The derivation of electrostatic properties from the charge density was treated in Subsection 8.7.3.4[link]. Magnetostatic properties can be derived from the spin-magnetization density ms(r) using parallel expressions.

8.7.4.7.1. Vector fields

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The vector potential field is defined as [{\bf A}({\bf r}) = \int\, \displaystyle{\bf m_s({\bf r}')\times ({\bf r}-{\bf r}') \over |{\bf r}-{\bf r}'|^3} \,{\rm d}{\bf r}'. \eqno (8.7.4.87)]In the case of a crystal, it can be expanded in Fourier series: [{\bf A}({\bf r})={2i\over V}\, \sum_{\bf h}\, \left\{ \displaystyle{{\bf M}_S({\bf h})\times {\bf h} \over h^2}\right\} \exp\,(-2\pi i{\bf h}\cdot{\bf r})\semi \eqno (8.7.4.88)]the magnetic field is simply [\eqalignno{ {\bf B}({\bf r}) &={\boldnabla}\times{\bf A}({\bf r}) \cr &=-{4\pi \over V}\, \sum_{\bf h}\, \left[\displaystyle{{\bf h}\times{\bf M}_S\,({\bf h})\times{\bf h} \over h^2}\right] \exp\,(-2\pi i{\bf h}\cdot{\bf r}). &(8.7.4.89)}]One notices that there is no convergence problem for the h = 0 term in the B(r) expansion.

The magnetostatic energy, i.e. the amount of energy that is required to obtain the magnetization ms, is [\eqalignno{ E_{ms} &= -\textstyle{1\over2}\, \int\limits_{\rm cell} {\bf m}_s({\bf r})\cdot {\bf B}({\bf r}) \,{\rm d} r \cr &={2\pi\over V}\, \sum_{\bf h}\, \displaystyle{ [{\bf M}_s(-{\bf h})\times{\bf h}]\cdot [{\bf M}_S({\bf h})\times{\bf h}] \over h^2}. & (8.7.4.90)}]

It is often interesting to look at the magnetostatics of a given subunit: for instance, in the case of paramagnetic species.

For example, the vector potential outside the magnetized system can be obtained in a similar way to the electrostatic potential (8.7.3.30)[link]: [{\bf A}({\bf r}') = \int\ \displaystyle{ [{\boldnabla}\times {\bf m}_S({\bf r})] \over |{\bf r} - {\bf r}'|}\, {\rm d} {\bf r}. \eqno (8.7.4.91)]

If [r'\gg r, 1/|{\bf r}-{\bf r}'|] can be easily expanded in powers of 1/r′, and A(r′) can thus be obtained in powers of 1/r′. If ms(r) = <S>s(r), [{\bf A}({\bf r}') = \langle{\bf S}\rangle \times \int \displaystyle{ {\boldnabla}s ({\bf r}) \over |{\bf r}-{\bf r}'|}\, {\rm d}{\bf r}. \eqno (8.7.4.92)]

8.7.4.7.2. Moments of the magnetization density

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Among the various properties that are derivable from the delocalized spin density function, the dipole coupling tensor is of particular importance: [D_{nij}({\bf R}_n) = \int\, s({\bf r})\, \displaystyle{ [3\, r_{ni}r_{nj} - r^2_n\, \delta_{ij}] \over r^5_n}\, {\rm d} {\bf r}, \eqno (8.7.4.93)]where Rn is a nuclear position and rn = rRn. This dipolar tensor is involved directly in the hyperfine interaction between a nucleus with spin [{\bf I}_n] and an electronic system with spin s, through the interaction energy [\textstyle\sum\limits_{i, j}\, I_{ni}\, D_{nij}\, S_j. \eqno (8.7.4.94)]This tensor is measurable by electron spin resonance for either crystals or paramagnetic species trapped in matrices. The complementarity with scattering is thus of strong importance (Gillon, Becker & Ellinger, 1983[link]).

Computational aspects are the same as in the electric field gradient calculation, ρ(r) being simply replaced by s(r) (see Subsection 8.7.3.4[link]).

References

Gillon, B., Becker, P. & Ellinger, Y. (1983). Theoretical spin density in nitroxides. The effect of alkyl substitutions. Mol. Phys. 48, 763–774.








































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