Spin-only density at zero temperature

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, p. 726

Section 8.7.4.3.1. Spin-only density at zero temperature

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

8.7.4.3.1. Spin-only density at zero temperature

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Let us consider first an isolated open-shell system, whose orbital momentum is quenched: it is a spin-only magnetism case. Let $[\hat {\bf m}_s]$ be the spin-magnetization-density operator (in units of $[2\mu_B]$ ): $[\hat {\bf m}_s=\textstyle\sum\limits_j\, \hat{\boldsigma}_j\,\delta({\bf r}-{\bf r}_j). \eqno (8.7.4.16)]$ $[{\bf r}_j]$ and $[\hat{\boldsigma}_j]$ are, respectively, the position and the spin operator (in $[\hbar]$ units) of the jth electron. This definition is consistent with (8.7.4.7).

The system is assumed to be at zero temperature, under an applied field, the quantization axis being Oz. The ground state is an eigenstate of $[\hat{\bf S}^2]$ and $[\hat S_z]$ , where $[\hat{\bf S}]$ is the total spin: $[\hat {\bf S}=\textstyle\sum\limits_j\, \hat{\boldsigma}_j. \eqno (8.7.4.17)]$ Let S and M_s be the eigenvalues of $[\hat{\bf S}^2]$ and $[\hat S_z]$ . (M_s will in general be fixed by Hund's rule: M_S = S.) $[2M_S=[n_\uparrow - n_\downarrow], \eqno (8.7.4.18)]$ where $[n_\uparrow]$ and $[n_\downarrow]$ are the numbers of electrons with $[(\uparrow: +{1\over2}\,)]$ and $[(\downarrow:-{1\over2}\,)]$ spin, respectively.

The spin-magnetization density is along z, and is given by $[m_{Sz}({\bf r}) = \left\langle \psi_{SM_S} \left| \,\textstyle\sum\limits_j\, \hat{\boldsigma}_{jz} \delta({\bf r}-{\bf r}_j)\right| \psi_{SM_S} \right\rangle. \eqno (8.7.4.19)]$ $[m_{Sz}({\bf r})]$ is proportional to the normalized spin density that was defined for a pure state in (8.7.2.10). $[m_{Sz}({\bf r})=M_Ss({\bf r}). \eqno (8.7.4.20)]$ If $[\rho_\uparrow({\bf r})]$ and $[\rho_\downarrow({\bf r})]$ are the charge densities of electrons of a given spin, the normalized spin density is defined as $[s({\bf r})=[\rho_\uparrow({\bf r})-\rho_\downarrow({\bf r})]{1\over [n_\uparrow- n_\downarrow]}, \eqno (8.7.4.21)]$ compared with the total charge density ρ(r) given by $[\rho({\bf r})=\rho_\uparrow({\bf r})+\rho_\downarrow({\bf r}). \eqno (8.7.4.22)]$ A strong complementarity is thus expected from joint studies of ρ(r) and s(r).

In the particular case of an independent electron model, $[\rho_\alpha({\bf r})=\textstyle\sum\limits^{N_\alpha}_{i=1}\, |\varphi_{i\alpha}({\bf r})|^2 \quad (\alpha=\,\uparrow,\downarrow), \eqno (8.7.4.23)]$ where $[\varphi_{i\alpha}({\bf r})]$ is an occupied orbital for a given spin state of the electron.

If the ground state is described by a correlated electron model (mixture of different configurations), the one-particle reduced density matrix can still be analysed in terms of its eigenvectors $[\psi_{i\alpha}]$ and eigenvalues $[n_{i\alpha}]$ (natural spin orbitals and natural occupancies), as described by the expression $[\rho_\alpha({\bf r})=\textstyle\sum\limits^\infty_{i=1}\, n_{i\alpha}|\psi_{i\alpha}({\bf r})|^2, \eqno (8.7.4.24)]$ where 〈ψ_iα|ψ_jβ〉 = δ_ijδ_αβ, since the natural spin orbitals form an orthonormal set, and $[n_{i\alpha} \leq 1 \quad \textstyle\sum\limits^\infty_{i=1}\, n_{i\alpha}=n_\alpha. \eqno (8.7.4.25)]$

As the quantization axis is arbitrary, (8.7.4.20) can be generalized to $[{\bf m}_s({\bf r})=\hat{\bf S}s({\bf r}). \eqno (8.7.4.26)]$ Equation (8.7.4.26) expresses the proportionality of the spin-magnetization density to the normalized spin density function.

References

International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 726