12 H.Eschrig The KS(2.14)represents a highly non-linear set of functional-differential equations of the form 72 io:=-2+to:=o6 (2.22) since the effective potential parts contained in ok/oo;and in 6L/on depend on the solutions oi.The general iterative solving procedure is as follows: Guess a densityn(r）. Determine the potentials vH(r)(part of ok/6o from the second line of(2.l9)and vxe,ss(r）=iL/6ns's· Solve the KS equation for i(rs),Ei. -Determine the density n(r)(s)(u(rs) withμ=(N)from∑，（μ-e)=N. Determine a new input densityn(r）=f(ngou(r),ng》(r)》from n(out)of the previous step and n(in)of a number of previous cycles;f has to be chosen by demands of convergence. Iterate until n(out)=n(in)=n(SCF) SCF density:nss(r)(n(r),m(r)), Total energy:Elv,N=Hn+(v n). In the following the most demanding second step is sketched 2.2.1 The Local Orbital Representation The KS orbitals on of a crystalline solid,indexed by a wave number k and a band index n,are expanded into a nonorthogonal local orbital minimum basis (one basis orbital per band or per core state): Okn(r)=>L(r-R-5)CL..kneik(R+). (2.23) RsL This leads to a secular equation of the form HC=SCe, (2.24) Hr,aL=∑《0s'RsL)ek(R+s-), (2.25) SL=>(0s'L/RsL)eik(R+-). (2.26) R By definition,core states are local eigenstates of the effective crystal po- tential which have no overlap to neighboring core states and are mutually12 H. Eschrig The KS (2.14) represents a highly non-linear set of functional-differential equations of the form Hφˆ i =  −∇2 2 + veff φi = φii (2.22) since the effective potential parts contained in δk/δφ∗ i and in δL/δn depend on the solutions φi. The general iterative solving procedure is as follows: Guess a density n(in) ss
(r) . – Determine the potentials vH(r) (part of δk/δφ∗ i from the second line of (2.19)) and vxc,ss
(r) = δL/δns
s . – Solve the KS equation for φi(rs), i . – Determine the density n(out) ss
(r) = i φi(rs)θ(µ − i)φ∗ i (rs ) with µ = µ(N) from i θ(µ − i) = N . – Determine a new input density n(in) ss
(r) = f n(out) ss
(r), n(in,j) ss
(r)  from n(out) of the previous step and n(in,j) of a number of previous cycles; f has to be chosen by demands of convergence. Iterate until n(out) = n(in) = n(SCF) . SCF density: nss
(r) ˆ=(n(r), m(r)) , Total energy: E[v,N] = H[n]+(v | n) . In the following the most demanding second step is sketched. 2.2.1 The Local Orbital Representation The KS orbitals φkn of a crystalline solid, indexed by a wave number k and a band index n, are expanded into a nonorthogonal local orbital minimum basis (one basis orbital per band or per core state): φkn(r) =  RsL ϕsL(r − R − s)CLs,kneik(R+s) . (2.23) This leads to a secular equation of the form HC = SC , (2.24) Hs
L
,sL =  R 0s L |Hˆ |RsL
eik(R+s−s
) , (2.25) Ss
L
,sL =  R 0s L |RsL
eik(R+s−s
) . (2.26) By definition, core states are local eigenstates of the effective crystal po￾tential which have no overlap to neighboring core states and are mutually