1 2 LDA Local Density Approximation 2 LDA 1 LDA LDA N N N H = N [ 2 j + V ion (r j ) ] + 1 e 2 2 r j r k j j k (3) V ion V ion (r) = I Z I e 2 r

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1 11 March [ { } ] 3 1/3 2 + V ion (r) + V H (r) 3α 4π ρ σ(r) ϕ iσ (r) = ε iσ ϕ iσ (r) (1) KS Kohn-Sham [ 2 + V ion (r) + V H (r) + V σ xc(r) ] ϕ iσ (r) = ε iσ ϕ iσ (r) (2)

2 1 2 LDA Local Density Approximation 2 LDA 1 LDA LDA N N N H = N [ 2 j + V ion (r j ) ] + 1 e 2 2 r j r k j j k (3) V ion V ion (r) = I Z I e 2 r j R I (4) Z I N Ψ HΨ = EΨ (5) e Ψ N r j Ψ = Ψ (r 1,, r N ) (6) 2

3 Ψ (r 1... r N ) = ϕ 1 (r 1 ),, ϕ N (r N ) (7) 7 N 3 [ 2 + V ion (r) + V H (r) ] σ =σ ϕ iσ (r) j ˆF jσ,iσ(r)ϕ jσ (r) = ε iσ ϕ iσ (r) (8) ˆF jσ,iσ(r) r 8 i It is now perfectly clear that a single configuration (Slater determinant) wave function must inevitably lead to a poor energy Coulson (1960) N N Ψ N K = {ϕ 1 ϕ N } Ψ N = K c K Ψ(K) (9) 3 3

4 CI Configuration Interaction CI CI α 2/3 α Xα Hohenberg-Kohn N Hohenberg-Kohn HK [1] N v 1 ( ) E G ρ(r) ρ(r) v 2 ( ) E G [ρ] N ρ (r) ρ (r) ρ(r) E G [ρ(r)] < E G [ρ (r)] (10) Hohenberg-Kohn 4

5 ψ ~ e ikr E = h2 k 2 2m ρ ~ const 1: E v [Ψ] Ψ Hohenberg-Kohn HK Ψ Ψ N 7 N N N 1 Ψ ρ HK ρ Ψ 1 Kohn 5

6 1 ψ k = e ikr k k k 2 ρ(r) = ψ(r) 2 k HK ψ ρ v ρ v ψ (11) HK ρ ψ v Hohenberg-Kohn ρ v v ρ v ψ ρ (12) 12 v ψ ρ 12 v ψ HK Hohenberg-Kohn 12 ρ v (13) 6

7 ρ ρ ψ v v 2: ρ v ρ v v ρ Hohenberg-Kohn v ρ E G [ρ] E G [ρ] = F HK [ρ] + ρ(r)v(r)dr (14) Hohenberg- Kohn F HK [ρ] F HK [ρ] N Ψ F HK [ρ] = Ψ ˆT + ˆV ee Ψ F HK [ρ] ρ v ρ v Löwdin A. J. Coleman, (15)

8 Q ρ ψ 3: Levy ρ v n Hohenberg-Kohn 3.2 ρ n ρ HK Hohenberg Kohn Sham ρ v ρ v Levy n [3] 3 v ρ ρ v F HK [ρ] v ρ ψ ρ 8

9 HK ψ v ψ ρ Levy Q[ρ] = min Ψ ρ ˆT + ˆVee Ψρ Ψ ρ Q[ρ] HK 16 ρ ρ Ψ ρ ˆT + ˆV ee Ψ ρ Q[ρ] Q[ρ] ρvdr E[ρ] ρ v Q[ρ] 15 F HK [ρ] v 16 ρ v HK 3 ρ (16) 3.3 Thoms-Fermi HK N 15 N E[ρ] = T [ρ] + ρ(r)v(r)dr + U ee [ρ] (17) U ee [ρ] U ee [ρ] = 1 ρ(r)v H (r)dr + E xc [ρ] (18) 2 V H ρ(r ) V H (r) = r r dr (19) U ee Thoms-Fermi T [ρ] = C TF dr [ρ(r)] 5/3 (20) 9

10 5 3 C TFρ(r) 2/3 + v(r) + ρ(r ) r r dr = µ (21) µ N 21 ρ(r) v(r) Thoms-Fermi Thoms-Fermi Kohn-Sham [2] ρ(r) N {ϕ i (r)} ρ(r) = N ϕ i (r) 2 (22) i 22 2 T T s [ρ] = N ϕi 2 ϕ i i (23) 23 N E xc T s [ρ] T E xc 2 v xc (r) = δe xc[ρ] δρ 2 ρ(r) Ψ 22 ρ(r) (24) 10

11 ρ N N 2 ε i E = N ε i 1 2 i=1 ρ(r)v H (r)dr + E xc [ρ] ρ(r)v xc (r)dr (25) N 2 22 ϕ i (r) 22 {ϕ i (r)} ρ(r) {ϕ i (r)} 4.2 E xc [ρ] E xc [ρ] E xc [ρ] = Exc HOM (ρ) E xc [ρ] LDA ɛ xc (ρ(r)) [] () 11

12 LDA LDA LDA [9] LDA E xc [ρ] E xc [ρ] LDA LDA E xc [ρ] It is advisable to stop at the simple LDA W. Kohn (1984) LDA Exc HOM (ρ) Exc HOM (ρ) LDA LDA LDA 4.3 KS KS KS 12

13 i i ε i I i i E(n 1,, n i,, n N ) E(n 1,, n i 1,, n N ) I i = E(, n i, ) E(, n i 1, ) (26) I (1) = I N I (2) N 0 E(N) = N I (i) (27) N 27 E = i ε i I i = ε i 27 I (i) = ɛ N+1 i 25 I i = ε i I (i) = ɛ N+1 i N i=1 E(, n i, ) E(, n i 1, ) = ε i (28) 28 E n i = ε i (29) 13

14 Janak [4] E i E(, n i, ) n i E(, n i,, n N ) E(n 1,, n i 1, ) ε i (, n i 0.5, ) (30) KS ε N 0 < n N 1 n N [5] E(N) E(N 1) = ε N (31) E(N) = N ε i (i) (32) i=1 N ɛ i = ɛ i (N)

15 1: HF DFT Ψ(x 1,, x N ) ρ(r) Ψ ρ ρ Ψ Ψ Ψ H Ψ E[ρ] LDA HF(S) KS [ 2 + V ion (r) + V H (r) + V Xα (r) ] [ 2 + V ion (r) + V H (r) + V xc (r) ] ϕ i (r) = ε i ϕ i (r) ϕ i (r) = ε i ϕ i (r) CI CI N LDA N ρ(r) ρ (r) ρ (r) ρ (r) ρ (r) 2 2 ρ αβ (r) LDA LSD 15

16 [1] P. Hohenberg and W. Kohn, Phys. Rev. 136 B864 (1964). [2] W. Kohn and L. Sham, Phys. Rev. 140 A1133 (1965). [3] M. Levy, Proc. Natl. Acad. Sci. USA (1979). [4] J. F. Janak, Phys. Rev. B (1978). [5] J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Jr., Phys. Rev. Lett (1982). [6] O. Gunnarsson, M. Jonson, and B. I. Lundqvist, Phys. Rev. B 20, 3136 (1979). [7] S. Lundqvist and N. H. March, eds., Theory of the Inhomogeneous Electron Gas (Plenum, New York, 1983). [8] J. Callaway and N. H. March, Solid State Physics 38 (Academic, New York, 1984) p [9] R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989) [10] S. B. Trickey ed., Adv. in Quantum Chemistry 21, (Academic, San Diego, 1989) [11] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford, New York, 1989). [12] C. Fiolhais, F. Nogueira, M. Marques, eds., A Primer in Density Functional Theory (Springer, Berlin, 2003). 16

1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr ψ σ + (r)ψ +

1 1.1 21 11 22 10 33 cm 10 29 cm 60 6 8 10 12 cm 1cm 1 1.2 2 1 1 1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr