Legendre transforms and Maxwell relations in density functional theory

摘要

The Hohenbergndash;Kohn principle of the minimum groundhyphen;state energy density functionalElsqb;N,vrsqb;=Evlsqb;rgr;rsqb;equiv;Flsqb;rgr;rsqb;+v(1)rgr;(1)dtgr;1,thinsp;minlcub;Evlsqb;rgr;prime;rsqb;minus;mgr;Nlsqb;rgr;prime;rsqb;rcub;,is reformulated in forms appropriate to the alternative Legendre transformed representations of the system energy. The quantities rgr; andNlsqb;rgr;rsqb; denote the onehyphen;electron density and associated number of electrons, whilevand mgr; stand for the external and chemical potentials, respectively. The following natural Legendre transformed functionals are introduced:Qlsqb;mgr;,vrsqb;=Eminus;(part;E/part;N)cN=Qlsqb;rgr;rsqb;equiv;Flsqb;rgr;rsqb;minus;lsqb;dgr;F/dgr;rgr;(1)rsqb;rgr;(1)dtgr;1,Flsqb;N,rgr;rsqb;=Eminus;lsqb;part;E/part;v(1)rsqb;cv(1)dtgr;1=Flsqb;rgr;rsqb;,R(mgr;,rgr;)=Eminus;lsqb;part;E/part;v(1)rsqb;cv(1)dtgr;1minus;(part;E/part;N)cN=Rlsqb;rgr;rsqb;,wherecimplies all variables kept constant except for the one in the derivative. The Maxwell relations for these representations are derived and their physical implications briefly discussed. Corresponding rsquo;rsquo;thermodynamicrsquo;rsquo; mnemonic square diagrams are introduced to generate the differential expressions and selected Maxwell relations. A typical Maxwell relation islsqb;part;rgr;(1)/part;v(2)rsqb;N,vne;(2)=lsqb;part;rgr;(2)/part;v(1)rsqb;N,vne;(1).It is shown that, as in classical thermodynamics, the extremum principle for each of these basic functionals of stateLlsqb;g,lrsqb;(L=E,Q,F,R;g=N,mgr;;thinsp;l=v,rgr;)can be formulated as follows: The equilibrium value of any unconstructed internal variable of a system specified by its global g and local l parameters, minimizes theLlsqb;g,lrsqb;Legendre transform of the system energy, at constant g andl. It is shown that theFlsqb;N,rgr;rsqb;minimum principle is equivalent to the Levy variational principleFlsqb;rgr;rsqb;=minthinsp;psgr;rgr;Verbar;Tcaret;+Vcaret;eeVerbar;PSgr;rgr;gsim;,which may be interpreted as a process determining the system chemical potential.