2 The Essentials of Density Functional Theory and the Full-Potential Local-Orbital Approach H.Eschrig IFW Dresden,P.O.Box 27 00 16,01171 Dresden,Germany Abstract.Density functional theory for the ground state energy in its modern un- derstanding which is free of representability problems or other logical uncertainties is reported.Emphasis is on the logical structure,while the problem of modeling the unknown universal density functional is only very briefly mentioned.Then,a very accurate and numerical effective solver for the self-consistent Kohn-Sham equations is presented and its power is illustrated.Comparison is made to results obtained with the WIEN code. 2.1 Density Functional Theory in a Nutshell Density functional theory deals with inhomogeneous systems of identical par- ticles.Its general aim is to eliminate the monstrous many-particle wave func- tion from the formulation of the theory and instead to express chosen quan- tities of the system directly in terms of the particle density or the particle current density.There are basically three tasks:(i)to prove that chosen quan- tities are unique functionals of the density and to indicate how in principle they can be obtained,(ii)to find constructive expressions of model density functionals which are practically tractable and approximate the unique func- tionals in a way to provide predictive power,and (iii)to develop tools for an effective solution of the resulting problems. As regards task (i),final answers have been given for the ground state energy 2.2,and citations therein.In the following these results are summa- rized.Task (iii)is dealt with in the next section as well as in Chap.3. Central quantities of the density functional theory for the ground state energy are: the external potential v(r)or its spin-dependent version i=vgs(r）=v(r)6gs-BB(r）·oss, (2.1) the ground state density n(r)or spin-matrix density i=nsrr)=厂n(r)=∑ns(r), 1m(r)=B∑sgns(r)ags了 (2.2) the ground state energy E[,N]=min {H[]N[r]=N}. (2.3) H.Eschrig,The Essentials of Density Functional Theory and the Full-Potential Local-Orbital Approach,Lect.Notes Phys.642,7-21(2004) http://www.springerlink.com/ C Springer-Verlag Berlin Heidelberg 2004
2 The Essentials of Density Functional Theory and the Full-Potential Local-Orbital Approach H. Eschrig IFW Dresden, P.O.Box 27 00 16, 01171 Dresden, Germany Abstract. Density functional theory for the ground state energy in its modern understanding which is free of representability problems or other logical uncertainties is reported. Emphasis is on the logical structure, while the problem of modeling the unknown universal density functional is only very briefly mentioned. Then, a very accurate and numerical effective solver for the self-consistent Kohn-Sham equations is presented and its power is illustrated. Comparison is made to results obtained with the WIEN code. 2.1 Density Functional Theory in a Nutshell Density functional theory deals with inhomogeneous systems of identical particles. Its general aim is to eliminate the monstrous many-particle wave function from the formulation of the theory and instead to express chosen quantities of the system directly in terms of the particle density or the particle current density. There are basically three tasks: (i) to prove that chosen quantities are unique functionals of the density and to indicate how in principle they can be obtained, (ii) to find constructive expressions of model density functionals which are practically tractable and approximate the unique functionals in a way to provide predictive power, and (iii) to develop tools for an effective solution of the resulting problems. As regards task (i), final answers have been given for the ground state energy [2.2, and citations therein]. In the following these results are summarized. Task (iii) is dealt with in the next section as well as in Chap. 3. Central quantities of the density functional theory for the ground state energy are: – the external potential v(r) or its spin-dependent version vˇ = vss (r) = v(r)δss − µBB(r) · σss , (2.1) – the ground state density n(r) or spin-matrix density nˇ = nss (r) ˆ= n(r) = s nss(r), m(r) = µB ss nss (r)σss , (2.2) – the ground state energy E[ˇv,N] = minΓ {Hvˇ[Γ] | N[Γ] = N } . (2.3) H. Eschrig, The Essentials of Density Functional Theory and the Full-Potential Local-Orbital Approach, Lect. Notes Phys. 642, 7–21 (2004) http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2004
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 potential
8 H. Eschrig Here, Γ means a general (possibly mixed) quantum state, Γ = Mα |ΨMαpMαΨMα| , 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 Hamiltonian 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 d3rvssnss = 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 considered 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.
2 DFT and the Full-Potential Local-Orbital Approach 9 The mathematical basis of the variational principle is (for a finite total volume,for instance provided by periodic boundary conditions,to avoid for- mal difficulties with a continuous energy spectrum)that Elv,N]is convex in N for fired v and concave in v for fired N,and E[v const.,N]E[v,N]+const..N. (2.7) Because of these simple properties of the ground state energy(which are not even mentioned by Hohenberg and Kohn in their seminal paper [2.5)it can be represented as a double Legendre transform, E[v,N]=inf sup]+(vln)+[N-(n)]u, (2.8) which is equivalent to (2.5)because the u-supremum is +oo unless (1n)= N.The inverse double Legendre transformation yields the universal density functional: H[n]inf sup{E[v,N]-(nlv). (2.9) Universality means that given a particle-particle interaction (Coulomb inter- action between electrons say)a single functional Hn]yields the ground state energies and densities for all (admissible)external potentials. The expression(2.9)need not be the only density functional which pro- vides(2.5).Generally two functions which have the same convex hull have the same Legendre transform.(Here the situation is more involved because of the intertwined double transformation.)Nevertheless,the outlined analysis can be put to full mathematical rigor,and the domain of admissible potentials is very broad and contains for instance the Coulomb potentials of arbitrary arrangements of nuclei.There are no representability problems.For details see[2.2. This solves task (i)for the ground state energy as chosen quantity,which, for given v and N,is uniquely obtained via (2.5)from the functional Hn+ (vn).There are attempts to consider other quantities as excitation spectra or time-dependent quantities which are so far on a much lower level of rigor.Of course,Hn]is unknown and of the same complexity as Elv,N].It can only be modeled by guesses.This turns out to be uncomparably more effective than a direct modeling of Elv,N]. Modeling of Hn]starts with the Kohn-Sham(KS)parameterization [2.7] of the density by KS orbitals ok(rs)and orbital occupation numbers nk: n(r）=ns(r)=∑(rs)nk(rs), (2.10) 0≤nk≤l,(〉=dk,(1|n)=nk=N, (2.11)
2 DFT and the Full-Potential Local-Orbital Approach 9 The mathematical basis of the variational principle is (for a finite total volume, for instance provided by periodic boundary conditions, to avoid formal difficulties with a continuous energy spectrum) that E[v,N] is convex in N for fixed v and concave in v for fixed N, and E[v + const., N] = E[v,N] + const. · N. (2.7) Because of these simple properties of the ground state energy (which are not even mentioned by Hohenberg and Kohn in their seminal paper [2.5]) it can be represented as a double Legendre transform, E[v,N] = inf n sup µ H[n]+(v|n) + N − (1|n) µ , (2.8) which is equivalent to (2.5) because the µ-supremum is +∞ unless (1|n) = N. The inverse double Legendre transformation yields the universal density functional: H[n] = inf N sup v E[v,N] − (n|v) . (2.9) Universality means that given a particle-particle interaction (Coulomb interaction between electrons say) a single functional H[n] yields the ground state energies and densities for all (admissible) external potentials. The expression (2.9) need not be the only density functional which provides (2.5). Generally two functions which have the same convex hull have the same Legendre transform. (Here the situation is more involved because of the intertwined double transformation.) Nevertheless, the outlined analysis can be put to full mathematical rigor, and the domain of admissible potentials is very broad and contains for instance the Coulomb potentials of arbitrary arrangements of nuclei. There are no representability problems. For details see [2.2]. This solves task (i) for the ground state energy as chosen quantity, which, for given v and N, is uniquely obtained via (2.5) from the functional H[n] + (v|n). There are attempts to consider other quantities as excitation spectra or time-dependent quantities which are so far on a much lower level of rigor. Of course, H[n] is unknown and of the same complexity as E[v,N]. It can only be modeled by guesses. This turns out to be uncomparably more effective than a direct modeling of E[v,N]. Modeling of H[n] starts with the Kohn-Sham (KS) parameterization [2.7] of the density by KS orbitals φk(rs) and orbital occupation numbers nk: n(r) = nss (r) = k φk(rs) nk φ∗ k(rs ), (2.10) 0 ≤ nk ≤ 1, φk|φk = δkk , (1 | n) = k nk = N . (2.11)
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 putting
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[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
2 DFT and the Full-Potential Local-Orbital Approach 11 on Fig.2.2.A free minimum of a function of on and a minimum under the constraint 6m>0. 72 ,n=∑nk《o-2l〉+ +"】 rd(rs)o(r's) (2.19) T-T This completes the brief introduction to the state of the art of density func- tional theory of the ground state energy. Just to mention one other realm of possible density functionals,quasipar- ticle excitations are obtained from the coherent part(pole term)of the single particle Green's function (2.3-2.5) Gr,ro)=∑aC+cr,r, (2.20) 1 w-Ek /r[,-(-+uo)+m) +(r,r';ex)x.(r)=x.(r)ek.(2.21) Here,in the inhomogeneous situation of a solid,the self-energy is among other dependencies a functional of the density.This forms the shaky ground (with rather solid boulders placed here and there on it,see for instance also 2.4)for interpreting a KS band structure as a quasi-particle spectrum. In principle from the full ss(r,r;w)the total energy might also be ob- tained. 2.2 Full-Potential Local-Orbital Band Structure Scheme (FPLO) This chapter deals with task (iii)mentioned in the introduction to Chap.1. A highly accurate and very effective tool to solve the KS equations self consistently is sketched.The basic ideas are described in 2.6,see http://www.ifw-dresden.de/agtheo/FPLO/for actual details of the imple- mentation
2 DFT and the Full-Potential Local-Orbital Approach 11 ✏✏ δn ✏ Fig. 2.2. A free minimum of a function of δn and a minimum under the constraint δn > 0. k[φk, nk] = k nkφk| − ∇2 2 |φk + + kk nknk 2 ss d3rd3r |φk(rs)| 2|φk (r s )| 2 |r − r | . (2.19) This completes the brief introduction to the state of the art of density functional theory of the ground state energy. Just to mention one other realm of possible density functionals, quasiparticle excitations are obtained from the coherent part (pole term) of the single particle Green’s function ([2.3–2.5]) Gss (r, r ; ω) = k χs(r)η∗ s (r ) ω − εk + Gincoh ss (r, r ; ω) , (2.20) s d3r δ(r − r ) −∇2 2 + u(r) + uH(r) + Σss (r, r ; k) χs (r ) = χs(r)εk . (2.21) Here, in the inhomogeneous situation of a solid, the self-energy Σ is among other dependencies a functional of the density. This forms the shaky ground (with rather solid boulders placed here and there on it, see for instance also [2.4]) for interpreting a KS band structure as a quasi-particle spectrum. In principle from the full Σss (r, r ; ω) the total energy might also be obtained. 2.2 Full-Potential Local-Orbital Band Structure Scheme (FPLO) This chapter deals with task (iii) mentioned in the introduction to Chap. 1. A highly accurate and very effective tool to solve the KS equations selfconsistently is sketched. The basic ideas are described in [2.6, see http://www.ifw-dresden.de/agtheo/FPLO/ for actual details of the implementation].