10 H.Eschrig Model functionals consist of an orbital variation part K and a local density expression L: H[n]=K[n]+L[n], K问=n{o,n∑enoi=n}, (2.12) Lin] 形rn(rl(ns(r),7n,), which cast the variational principle(2.5)into the KS form Ee,W=m,{oe,nl+no]+(②omol (2.13) (klpk〉=dk,0≤nk≤1，∑knk=N oi and ni must be varied independently.The uniqueness of solution now depends on the convexity of k[ok,n]and L[n]. Variation of oi yields the(generalized)KS equations: 16k nk6中1 （迈+U)=k· (2.14) Since nk and o enter in the combination no only,the relation 16 nk onk =〈 (2.15) is valid which yields Janak's theorem: （k+L+(n)=k Onk (2.16) Variation of nk,in view of the side conditions,yields the Aufbau principle: Let nknk,then (cf.Fig.2.2) ≥0 for nk'=0ornk=1, òn Onk (2.17) Onk （k+L+olm)0r0<w,胜<1. Hence, nk =1 for Ek <EN, 0≤nk≤1 for Ek=eN, (2.18) nk=0 for Ek EN. Finding suitable expressions for k and L mainly by physical intuition is the way task (ii)is treated.The standard L(S)DA or GGA is obtained by putting10 H. Eschrig Model functionals consist of an orbital variation part K and a local density expression L: H[n] = K[n] + L[n] , K[n] = min {φk,nk} k[φk, nk]    kφknkφ∗ k = n  , L[n] =  d3rn(r)l nss
(r), ∇n, . . .  , (2.12) which cast the variational principle (2.5) into the KS form E[v,N] = min {φk,nk} k[φk, nk] + L[Σφnφ∗]+(Σφnφ∗ | v)      φk|φk

= δkk
, 0 ≤ nk ≤ 1, knk = N  . (2.13) φ∗ i , φi and ni must be varied independently. The uniqueness of solution now depends on the convexity of k[φk, nk] and L[n]. Variation of φ∗ k yields the (generalized) KS equations: 1 nk δk δφ∗ k + δL δn + v  φk = φkk . (2.14) Since nk and φ∗ k enter in the combination nkφ∗ k only, the relation nk ∂ ∂nk =  φk    δ δφ∗ k (2.15) is valid which yields Janak’s theorem: ∂ ∂nk  k + L + (v | n)  = k . (2.16) Variation of nk, in view of the side conditions, yields the Aufbau principle: Let nk
< nk, then (cf. Fig. 2.2) δn ∂ ∂nk
− ∂ ∂nk k + L + (v | n)   ≥ 0 for nk
= 0 or nk = 1 , = 0 for 0 < nk
, nk < 1 . (2.17) Hence, nk = 1 for k < N , 0 ≤ nk ≤ 1 for k = N , nk = 0 for k > N . (2.18) Finding suitable expressions for k and L mainly by physical intuition is the way task (ii) is treated. The standard L(S)DA or GGA is obtained by putting