Modelling the spin density

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

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International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 729-730

Section 8.7.4.5. Modelling the spin density

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

8.7.4.5. Modelling the spin density

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In this subsection, the case of spin-only magnetization is considered. The modelling of $[{\bf m}_s({\bf r})]$ is very similar to that of the charge density.

8.7.4.5.1. Atom-centred expansion

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We first consider the case where spins are localized on atoms or ions, as it is to a first approximation for compounds involving transition-metal atoms. The magnetization density is expanded as $[{\bf m}_S({\bf r}) = \textstyle\sum\limits_j\langle {\bf S}_j\rangle\,\langle s_j({\bf r}-{\bf R}_j)\rangle, \eqno (8.7.4.54)]$ where $[\langle{\bf S}_j\rangle]$ is the spin at site j, and $[\langle s_j\rangle]$ the thermally averaged normalized spin density $[f_j({\bf h})]$ , the Fourier transform of $[s_j({\bf r})]$ , is known as the `magnetic form factor'. Thus, $[{\bf M}({\bf h})=\textstyle\sum\limits_j\langle{\bf S}_j\rangle\;f_j({\bf h}) T_j \exp \,(2\pi{\bf h}\cdot {\bf R}_j), \eqno (8.7.4.55)]$ where [T_j] and $[{\bf R}_j]$ are the Debye–Waller factor and the equilibrium position of the jth site, respectively.

Most measurements are performed at temperatures low enough to ensure a fair description of [T_j] at the harmonic level (Coppens, 2001). [T_j] represents the vibrational relaxation of the open-shell electrons and may, in some situations, be different from the Debye–Waller factor of the total charge density, though at present no experimental evidence to this effect is available.

8.7.4.5.1.1. Spherical-atom model

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In the crudest model, $[s_j({\bf r})]$ is approximated by its spherical average. If the magnetic electrons have a wavefunction radial dependence represented by the radial function U(r), the magnetic form factor is given by $[f(h)=\textstyle\int\limits^\infty_0\,U^2(r)4 \pi r^2\,{\rm d} r \,j_0(2\pi hr) = \langle\, j_0\rangle, \eqno (8.7.4.56)]$ where [j_0] is the zero-order spherical Bessel function. For free atoms and ions, these form factors can be found in IT IV (1974).

One of the important features of magnetic neutron scattering is the fact that, to a first approximation, closed shells do not contribute to the form factor. Thus, it is a unique probe of the electronic structure of heavy elements, for which theoretical calculations even at the atomic level are questionable. Relativistic effects are important. Theoretical relativistic form factors can be used (Freeman & Desclaux, 1972; Desclaux & Freeman, 1978). It is also possible to parametrize the radial behaviour of U. A single contraction-expansion model [κ refinement, expression (8.7.3.6)] is easy to incorporate.

8.7.4.5.1.2. Crystal-field approximation

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Crystal-field effects are generally of major importance in spin magnetism and are responsible for the spin state of the ions, and thus for the ground-state configuration of the system. Thus, they have to be incorporated in the model.

Taking the case of a transition-metal compound, and neglecting small contributions that may arise from spin polarization in the closed shells (see Subsections 8.7.4.9 and 8.7.4.10), the normalized spin density can be written by analogy with (8.7.3.76) as $[s({\bf r}) = \textstyle\sum\limits^5_{i=1}\, \textstyle\sum\limits^5_{j\ge i}\,D_{ij}\, d_i ({\bf r}) d_j ({\bf r}), \eqno (8.7.4.57)]$ where $[D_{ij}]$ is the normalized spin population matrix. If $[\rho_{d\uparrow}]$ and $[\rho_{d\downarrow}]$ are the densities of a given spin, $[s({\bf r}) = {\rho_{d\uparrow}-\rho_{d\downarrow} \over n_\uparrow - n_\downarrow}, \eqno (8.7.4.58)]$ the d-type charge density is $[\rho_d({\bf r})=\rho_{d\uparrow}+\rho_{d\downarrow} \eqno (8.7.4.59)]$ and is expanded in a similar way to s(r) [see (8.7.3.76)], $[\rho_d({\bf r})=\textstyle\sum\limits_i\, \textstyle\sum\limits_{j\ge i}\,P_{ij} d_i d_j\semi \eqno (8.7.4.60)]$ writing $[\rho_{d \sigma} = \textstyle\sum\, P^\sigma_{ij}\, d_i d_j \eqno (8.7.4.61)]$ with σ = $[ \uparrow]$ and $[\downarrow]$ , one obtains $[\matrix{ P_{ij} = P^\uparrow_{ij} +P^\downarrow_{ij} \cr \vphantom{}\cr D_{ij} = (P^\uparrow_{ij} - P^\downarrow_{ij})/(n_\uparrow - n_\downarrow).} \eqno (8.7.4.62)]$ Similarly to $[\rho_d({\bf r})]$ , [s(r)] can be expanded as $[s({\bf r})= U^2 (r)\, \textstyle\sum\limits^4_{l=0}\; \textstyle\sum\limits^l_{m=0}\; \textstyle\sum\limits_p\, D_{lmp}\, y_{lmp}(\theta, \varphi), \eqno (8.7.4.63)]$ where U(r) describes the radial dependence. Spin polarization leads to a further modification of this expression. Since the numbers of electrons of a given spin are different, the exchange interaction is different for the two spin states, and a spin-dependent effective screening occurs. This leads to $[\rho_{d\sigma}({\bf r}) = \kappa^3_\sigma U^2(\kappa_\sigma r)\textstyle \sum\limits_{lmp}\, P^\sigma_{lmp}\, y_{lmp}, \eqno (8.7.4.64)]$ with σ = $[\ \uparrow]$ or $[\downarrow]$ , where two κ parameters are needed.

The complementarity between charge and spin density in the crystal field approximation is obvious. At this particular level of approximation, expansions are exact and it is possible to estimate d-orbital populations for each spin state.

8.7.4.5.1.3. Scaling of the spin density

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The magnetic structure factor is scaled to $[F_N({\bf h})]$ . Whether the nuclear structure factors are calculated from refined structural parameters or obtained directly from a measurement, their scale factor is not rigorously fixed.

As a result, it is not possible to obtain absolute values of the effective spins 〈S_n〉 from a magnetic neutron scattering experiment. It is necessary to scale them through the sum rule $[\textstyle\sum\limits _n\, \langle{\bf S}_n\rangle = {\boldmu}, \eqno (8.7.4.65)]$ where $[\boldmu]$ is the macroscopic magnetization of the sample.

The practical consequences of this constraint for a refinement are similar to the electroneutrality constraint in charge-density analysis (§8.7.3.3.1).

8.7.4.5.2. General multipolar expansion

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In this subsection, the localized magnetism picture is assumed to be valid. However, each subunit can now be a complex ion or a radical. Covalent interactions must be incorporated.

If $[\chi_\mu({\bf r})]$ are atomic basis functions, the spin density s(r) can always be written as $[s({\bf r})=\textstyle\sum\limits_\mu\, \textstyle\sum\limits_\nu\, D_{\mu\nu}\chi_\mu ({\bf r}-{\bf R}_\mu)\chi _\nu({\bf r}-{\bf R}_\nu), \eqno (8.7.4.66)]$ an expansion that is similar to (8.7.3.9) for the charge density.

To a first approximation, only the basis functions that are required to describe the open shell have to be incorporated, which makes the expansion simple and more flexible than for the charge density.

As for the charge density, it is possible to project (8.7.4.66) onto the various sites of the ion or molecule by a multipolar expansion: $[\matrix{ s({\bf r}) \sim \sum\limits_j\, s_j ({\bf r}-{\bf R}_j) * P_j({\bf R}_j), \cr \vphantom{}\cr s_j({\bf r}) = \sum\limits_j\, \kappa^3_j\, R_{lj} (\kappa_j\,r)\sum\limits_m\, \sum\limits_p\, D_{jlmp}\, y_{lmp}.} \eqno (8.7.4.67)]$ P(R_j) is the vibrational p.d.f. of the jth atom, which implies use of the convolution approximation .

The monopolar terms D_j00 give an estimate of the amount of spin transferred from a central metal ion to the ligands, or of the way spins are shared among the atoms in a radical.

The constraint (8.7.4.65) becomes $[\textstyle\sum\limits_n\langle{\bf S}_n\rangle\,\textstyle\sum\limits_j\, D^{(n)}_{j00}={\boldmu}, \eqno (8.7.4.68)]$ where (n) refers to the various subunits in the unit cell. Expansion (8.7.4.67) is the key for solving noncentrosymmetric magnetic structure (Boucherle, Gillon, Maruani & Schweizer, 1982).

8.7.4.5.3. Other types of model

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One may wish to take advantage of the fact that, to a good approximation, only a few molecular orbitals are involved in s(r). In an independent particle model, one expands the relevant orbitals in terms of atomic basis functions (LCAO ): $[\varphi_{i\sigma}=\textstyle\sum\limits_\mu\, c^\sigma_{i\mu}\chi^\sigma_\mu({\bf r}-{\bf R}_\mu) \eqno (8.7.4.69)]$ with σ = $[\uparrow]$ or $[\downarrow]$ , and the spin density is expanded according to (8.7.4.21) and (8.7.4.23). Fourier transform of two centre-term products is required. Details can be found in Forsyth (1980) and Tofield (1975).

In the case of extended solids, expansion (8.7.4.69) must refer to the total crystal, and therefore incorporate translational symmetry (Brown, 1986).

Finally, in the simple systems such as transition metals, like Ni, there is a d–s type of interaction, leading to some contribution to the spin density from delocalized electrons (Mook, 1966). If [s_l] and [s_d] are the localized and delocalized parts of the density, respectively, $[s({\bf r})=as_l({\bf r})+[1-a]s_d({\bf r}), \eqno (8.7.4.70)]$ where a is the fraction of localized spins. s_d(r) can be modelled as being either constant or a function with a very small number of Fourier adjustable coefficients.

References

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International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 729-730