  The density-functional (DF) theory provides a simple method for calculating
the properties of an interacting system under an external potential by
associating it with a corresponding non-interacting system. Here, we find some
relations in this non-interacting system, which can enable us to set up the
thermodynamical relations for a neutral electron-nucleus mixture in terms of
quantities of the non-interacting system and the exchange-correlation effect.
In this way, Andersen's force theorems are clearly described and easily proved
in conjunction with other types of force theorems, Janak's theorem extended to
a finite-temperature system, pressure formulas and some thermodynamical
relations. For this purpose, the thermodynamics in the DF theory is presented
in a systematic and explicit way.
