Total electronic energy of the system

The total electronic energy of the system in Kohn–Sham (KS) approach has a form (see formula (7.2.10) from [1])

  1. $$E = \sum\limits_i {{\varepsilon _i} - {1 \over 2}\int {{V^{ee}}\left( {\bf{r}} \right)\rho \left( {\bf{r}} \right)d{\bf{r}}} }  + {E^{XC}}\left[ \rho  \right] - \int {{V^{XC}}\left( {\bf{r}} \right)\rho \left( {\bf{r}} \right)d{\bf{r}}} $$

where $\rho \left( {\bf{r}} \right)$ is the electron density, $V^{ee}\left(\bf{r}\right)$ and $V^{XC}\left(\bf{r}\right)$ are electron-electron repulsion and exchange-correlation potentails.

The first term in Eq. (1) $\sum\limits_i {\varepsilon _i}$ means sum of all KS orbital energies, the second term means correction for double-counting of electron–electron Coulomb (Hartree) interaction, the term $E^{XC}\left[ \rho  \right]$ is the exchange–correlation energy functional, and the last term $-\int {V^{XC}}\left( {\bf{r}} \right)\rho \left( {\bf{r}} \right)d{\bf{r}}$ removes the contribution of exchange–correlation potential already included in orbital energies.

The exchange-correlation energy in local-density approximation (LDA) is the integral of energy density (see formula (7.4.1) from [1])

  1. $${E^{XC}}\left[ \rho  \right] = \int {{\varepsilon^{XC}}\left[ \rho  \right]\rho \left( {\bf{r}} \right)d{\bf{r}}} $$

where 

  1. $${\varepsilon^{XC}} = {\varepsilon^X} + {\varepsilon^C}$$

The total energy (1) in LDA can be rewritten as

  1. $$E = \sum\limits_i {{\varepsilon _i} + \int {\left\{ {{\varepsilon^{XC}}\left[ \rho  \right] - {1 \over 2}{V^{ee}}\left( {\bf{r}} \right) - {V^{XC}}\left( {\bf{r}} \right)} \right\}\rho \left( {\bf{r}} \right)d{\bf{r}}} } $$

The exchange energy density in LDA has a form (see formula (7.4.5) from [1])

  1. $${\varepsilon^X}\left[ \rho  \right] =  -{3\over 4} {\left( {\frac{3}{\pi }\rho } \right)^{\frac{1}{3}}}$$

The correlation energy density in LDA in the form of Ceperley-Alder with Perdew-Zunger parameterization for unpolarized case has a form (see formulas (C3) and (C5) in [2])

  1. $${\varepsilon^C}\left[ \rho  \right] = \left\{ \begin{array}{l}A\ln {r_s} + B + C{r_s}\ln {r_s} + D{r_s},\;\;\;\;{\rm{if}}\;{r_s} < 1;\\ \frac{{\gamma}}{{{{\left( {1 + {\beta _1}\sqrt {{r_s}}  + {\beta _2}{r_s}} \right)}}}},\;\;\;\;\;\;\;{\rm{if}}\;\;\;{r_s} \ge 1;\end{array} \right.$$

here ${r_s} = {\left( {\frac{3}{{4\pi \rho }}} \right)^{\frac{1}{3}}}$, $A=0.0311$, $B=-0.048$, $C=0.002$, $D=-0.0116$, ${\beta _1} = 1.0529$, ${\beta _2} = 0.3334$, $\gamma  =  - 0.1423$. 

References:

[1] Parr, R.G.; Yang, W. Density functional theory of atoms and molecules. Oxford University. Press, New York, 1989.

[2] Perdew, J.P.; Zunger, A. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23, 5048, 1981.

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