Properties derivable from spin densities

Coppens, P.; Su, Z.; Becker, P. J.

doi:10.1107/97809553602060000615

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

pdf | chapter contents | chapter index | related articles

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. Su^b and P. J. Becker^c

^a 732 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

| top | pdf |

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

8.7.4.7.1. Vector fields

| top | pdf |

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 m_s, 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): $[{\bf A}({\bf r}') = \int\ \displaystyle{ [{\boldnabla}\times {\bf m}_S({\bf r})] \over |{\bf r} - {\bf r}'|}\, {\rm d} {\bf r}. \eqno (8.7.4.91)]$

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 m_s(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

| top | pdf |

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 R_n is a nuclear position and r_n = r − R_n. 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).

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).

References

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

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