The total energy of a crystal as a function of the electron 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, p. 721

Section 8.7.3.4.4. The total energy of a crystal as a function of the electron 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

8.7.3.4.4. The total energy of a crystal as a function of the electron density

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One can write the total energy of a system as $[ E=e_c\left [\rho \right] +T+E_{xc},\eqno (8.7.3.70)]$ where T is the kinetic energy, $[E_{xc}]$ represents the exchange and electron correlation contributions, and $[E_c[\rho] ]$ , the Coulomb energy, discussed in the previous section, is given by $[ E_c= {\textstyle{1\over2}}\sum_{{i}\neq j} \displaystyle{Z_{i}Z_j \over R_{{i}j}}-\sum_{i}Z_{i}\Phi({\bf R}_{i}) + {\textstyle {1\over2}}\int\int \displaystyle {\rho ({\bf r}) \rho ({\bf r}^{\prime }) \over | {\bf r}-{\bf r}^{\prime }| }{\,{\rm d}}{\bf r}{\,{\rm d}}{\bf r}^{\prime },\eqno (8.7.3.71)]$ where $[Z_{i}]$ is the nuclear charge for an atom at position R_i, and R_ij = R_j − R_i.

Because of the theorem of Hohenberg & Kohn (1964), E is a unique functional of the electron density ρ, so that $[T+E_{xc}]$ must be a functional of ρ. Approximate density functionals are discussed extensively in the literature (Dahl & Avery, 1984) and are at the centre of active research in the study of electronic structure of various materials. Given an approximate functional, one can estimate non-Coulombic contributions to the energy from the charge density ρ(r).

In the simplest example, the functionals are those applicable to an electronic gas with slow spatial variations (the `nearly free electron gas'). In this approximation, the kinetic energy T is given by $[ T=c_k\textstyle\int \rho t[\rho] {\,{\rm d}}^3{\bf r},\eqno (8.7.3.72)]$ with $[c_k=(3/{10})(2\pi {^2}){^{2/3}}]$ ; and the function $[t[\rho] =\rho ^{2/3} ]$ . The exchange-correlation energy is also a functional of ρ, $[ E_{xc}=-c_x\textstyle\int \rho e_{xc}\left [\rho \right] {\,{\rm d}}^3{\bf r},]$ with $[c_x=(3/4)(3/\pi)^{1/3}]$ and $[\ e_{xc}[\rho] =\rho ^{1/3}]$ .

Any attempt to minimize the energy with respect to ρ in this framework leads to very poor results. However, cohesive energies can be described quite well, assuming that the change in electron density due to cohesive forces is slowly varying in space.

An example is the system AB, with closed-shell subsystems A and B. Let ρ_A and ρ_B be the densities for individual A and B subsystems. The interaction energy is written as $[\eqalignno{ \Delta E &=E_c [\rho] -E_c [\rho _A] -E_c [\rho _B] \cr &\quad +c_k\textstyle\int {\,{\rm d}}{\bf r}\left \{ \rho t [\rho] -\rho _At [\rho _A] -\rho _Bt [\rho _B] \right \} \cr &\quad-c_x\textstyle\int {\,{\rm d}}{\bf r}\left \{ \rho e_{xc} [\rho] -\rho _Ae_{xc} [\rho _A] -\rho _Be_{xc} [\rho _B] \right \}. & (8.7.3.73)}]$ This model is known as the Gordon–Kim (1972) model and leads to a qualitatively valid description of potential energy surfaces between closed-shell subsystems. Unlike pure Coulombic models, this density functional model can lead to an equilibrium geometry. It has the advantage of depending only on the charge density ρ.

References

Dahl, J. P. & Avery, J. (1984). Local density approximations in quantum chemistry and solid state physics. New York/London: Plenum.Google Scholar

Gordon, R. G. & Kim, Y. S. (1972). Theory for the forces between closed-shell atoms and molecules. J. Chem. Phys. 56, 3122–3133.Google Scholar

Hohenberg, P. & Kohn, W. (1964). Inhomogeneous electron gas. Phys. Rev. B, 136, 864–867.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, p. 721