Some Remarks on Scaling Relations in Density Functional Theory

Abstract

In wavefunction theory both kinetic and potential energy as functionals of wavefunction scale homogeneously; on the contrary in density functional theory, as previously shown [M. Levy and J. P. Perdew, Phys. Rev. A 32, 2010 (1985)], based on the constrained search definition the kinetic and potential energies dg not exhibit naively expected scaling properties, that is, $T\left[ {{\rho _\lambda }} \right] \ne {\lambda ^2}T\left[ \rho \right]$ and Vee $\left[ {\rho \lambda } \right] \ne \lambda Vee\left[ \rho \right]$, where ${\rho _\lambda } = {\lambda ^3}\rho \left( {\lambda \overrightarrow r } \right)$, the scalecl density. To preserve the naive scaling property, a new functional of $\rho \left( {\overrightarrow r } \right)$ and X, $\lambda ,F\left[ {\rho ,\lambda } \right]$ is defined as $F\left[ {\rho \left( {\vec r} \right),\lambda } \right] = \langle U\left( \lambda \right)\phi _\rho ^{\min }\left( {{{\vec r}^N}} \right)|\,\hat T\left( {{{\vec r}^N}} \right)\, + \,\hat Vee\,\left( {{{\vec r}^N}} \right)|U\left( \lambda \right)\phi _\rho ^{\min }\left( {{{\vec r}^N}} \right)\rangle$, where $U\left( \lambda \right)\phi _\rho ^{\min }\left( {{{\overrightarrow r }^N}} \right)$ is ghat anti symmetric N-particle wavefungtion which yieldi ${\rho _\lambda }\left( {\overrightarrow r } \right) = {\lambda ^3}\rho \left( {\lambda \overrightarrow r } \right)$ and minimizes $< \hat T\left( {{{\vec r}^N}} \right) + \lambda Vee\left( {{{\vec r}^N}} \right) >$, and where $U\left[ \lambda \right]$ is the unitary transformation that performs the scaling of wavefunction. Then the new variational principle for ground state energy $E_{G.S}^V$ for potential $V\left( {\overrightarrow r } \right)$ is proved to be $E_{G.S}^v = _{\lambda ,\rho \left( {\overrightarrow r } \right)}^{\min }\{ \int {d\overrightarrow r } v{\left( {\overrightarrow r } \right)_{\rho \lambda }}\lambda \left( {\overrightarrow r } \right) + F\left[ {\rho \left( {\overrightarrow r } \right),\lambda } \right]\}$ This allows one to satisfy virial theorem and lower the energy by optimum scaling. Further it is demonstrated that $F\left[ {{p_\lambda }} \right] = {\lambda ^2}{F_{1/\lambda }}\left[ \rho \right]$ where ${F_{1/\lambda }}\left[ \rho \right]$ is the Levy functional $F\left[ \rho \right]$, except that the electron-electron interaction potential is multiplied by a factor of $_{1/\lambda }$ in the definition. As results of such a relation, the correlation component of kinetic energy ${T_C}\left[ \rho \right]$ is related to a property of $F\left[ {{\rho _\lambda }} \right]$ and a 1/Z expansion in density functional theory for atoms exists. This paper is based on a previous work [M. Levy, W. Yang and R. G. Parr, J. Chem. Phys. 83, 2334 (1985)].