Reduction of derivatives and simple applications of the Legendre transformed density functional theory

摘要

A reduction of derivatives in the Legendre transformed density functional theory (DFT) of the singlehyphen;component (atomic or fixedhyphen;nuclei molecular) systems is described. It is shown that a given derivative can be expressed by an identity in terms of a set of basic derivatives. The basic partial functional derivatives for the general case ofNelectrons in the external fieldv(r) are: agr;(r)=lsqb;part;rgr;(r)/part;Nrsqb;/rgr;(r), bgr;(r,thinsp;rprime;)=minus;lsqb;part;rgr;(r)/part;v(rprime;)rsqb;N/rgr;(r), and ggr;(r,thinsp;rprime;)=minus;lsqb;part;rgr;(r)/part;v(rprime;)rsqb;mgr;/rgr;(r); rgr; is the density and mgr; denotes the chemical potential. For atoms, an additional alternative set of four integral derivatives, involving only global (referring to a system as a whole) parameters, is introduced: agr;tilde;=minus;lsqb;part;vne/part;Nrsqb;Z/vne, bgr;tilde;=minus;lsqb;part;vne/part;Zrsqb;N/vne, ggr;tilde;=lsqb;part;N/part;Zrsqb;mgr;/N, and khgr;tilde;=minus;lsqb;part;vne/part;Zrsqb;mgr;/vne;Zis the nuclear atomic number andvneis the electronhyphen;nuclear attraction energy per unit nuclear charge. The reduction procedure makes free use of the DFT lsquo;lsquo;thermodynamicrsquo;rsquo; diagram equations, Maxwell relations, and mathematical manipulations on functional derivatives. Several representative applications are indicated dealing with the system responses to slight changes in one of the system parameters subject to different constraints.