H.Eschrig Here,I means a general (possibly mixed)quantum state, T=∑VMta)PMa(Mal,PMa≥0，∑PMa=1 (2.4) Mo Ma where yMe is the many-body wave function of M particles in the quantum state a.In (2.3),HT]and NI]are the expectation values of the Hamil- tonian with external potential i and of the particle number operator,resp., in the state T.In the (admitted)case of non-integer N,non-pure (mixed) quantum states are unavoidable. The variational principle by Hohenberg and Kohn states that there exists a density functional Hn]so that E[西，N]=min{H问例+()|(i|)=N}, (2.5) ∑rewm,=∫rm-B:m.0=∑∫ rngs· (2.6) Given an external potential i and a(possibly non-integer)particle number N,the variational solution yields E,N]and the minimizing spin-matrix density n(r),the ground state density. There is a unique solution for energy since His convex by construction. The solution for n is in general non-unique since Hn need not be strictly convex.The ground state (minimum of Hn+(n))may be degenerate with respect to n for some i and N (cf.Fig.2.1). In what follows,only the much more relevant spin dependent case is con- sidered and the checks above v and n are dropped. H[例 H[+(l) 抗 () Fig.2.1.The functionals H[n,()and H[n+()for a certain direction in the functional n-space and a certain potential8 H. Eschrig Here, Γ means a general (possibly mixed) quantum state, Γ =  Mα |ΨMα
pMαΨMα| , pMα ≥ 0 ,  Mα pMα = 1 (2.4) where ΨMα is the many-body wave function of M particles in the quantum state α. In (2.3), Hvˇ[Γ] and N[Γ] are the expectation values of the Hamil￾tonian with external potential ˇv and of the particle number operator, resp., in the state Γ. In the (admitted) case of non-integer N, non-pure (mixed) quantum states are unavoidable. The variational principle by Hohenberg and Kohn states that there exists a density functional H[ˇn] so that E[ˇv,N] = minnˇ  H[ˇn] + (ˇv | nˇ)   (ˇ1 | nˇ) = N  , (2.5) (ˇv | nˇ) =  ss
 d3rvss
ns
s =  d3r(vn − B · m), (ˇ1 | nˇ) =  s  d3rnss . (2.6) Given an external potential ˇv and a (possibly non-integer) particle number N, the variational solution yields E[ˇv,N] and the minimizing spin-matrix density ˇn(r), the ground state density. There is a unique solution for energy since H[ˇn] is convex by construction. The solution for ˇn is in general non-unique since H[ˇn] need not be strictly convex. The ground state (minimum of H[ˇn] + (ˇv | nˇ)) may be degenerate with respect to ˇn for some ˇv and N (cf. Fig. 2.1). In what follows, only the much more relevant spin dependent case is con￾sidered and the checks above v and n are dropped. nˇ





















(ˇv | nˇ) ✘✘✘✘✘✘ H[ˇn] H[ˇn] + (ˇv | nˇ) Fig. 2.1. The functionals H[ˇn], (ˇv | nˇ) and H[ˇn] + (ˇv | nˇ) for a certain direction in the functional ˇn-space and a certain potential ˇv.