Exchange-correlation energy functional and virial theorem in the extended constrained-search theory

Recently we have proposed the extended constrainedsearch sECSd theory in which arbitrary physical quantities can be chosen as basic variables in addition to the electron density.1,2 The validity of this theory has been confirmed by revisiting on the basis of it the spin-density functional theory sSDFTd, the current-density functional theory sCDFTd, the LDA+U method, and the Hartree–Fock–Kohn–Sham scheme.1,2 The ECS theory has two advantages over the conventional density functional theory sDFTd, as stated in preceding papers.1,2 One is that the ground-state values of quantities which are characteristic of the system can be reproduced directly by means of the Kohn-Sham orbitals if they are chosen as basic variables. Another is that the approximate form of the exchange-correlation energy functional may be given in a more effective form than the conventional DFT.3–6 However, in order to perform the actual energy-band calculation within the ECS theory, a tractable form of the exchange-correlation energy functional is indispensable, i.e., some approximation has to be made on the exchangecorrelation energy functional. Generally, there exist two strategies for developing such an approximate functional. One is to devise the approximate form on the basis of the coupling-constant expression for the exchange-correlation energy functional. The examples are the average-density approximation7–10 and weighted-density approximation.8–12 Another is to utilize some exact relations which should be satisfied for the exchange-correlation energy functional. Virial theorem and scaling relations have been first discussed by Löwdin.13 In the DFT, virial and scaling relations are utilized in devising the density-moment expansion of density functionals such as the exchange and correlation energy functionals.14–18 In Sec. II, we shall give the exact expression for the exchange-correlation energy functional by means of the coupling-constant integration technique. In Sec. III, an exact relation for the exchange-correlation energy functional is derived with the aid of the virial theorem of the ECS theory. Both expressions are expected to be a good starting point toward the development of the approximate form of the ECS exchange-correlation energy functional.