2019 5 ' s s j t 41 6t 2 'May, 2019 MATHEMATICA NUMERICA SINICA Vol.41, No.24!p! Hartree-Fock "# *1) 2)

(Y^k/CWVtkA, Ow/CWAyYsh, A/CW, Wn 94720) 3l[wÆldP$M^gkElQ Hartree-Fock KtG8lY"5. QHartree-Fock KQI- Hartree-Fock TyLF*ZHl Kohn-Sham HTyT, dwe2TyTQWwlK. $K-!*lZkL7eAlwe2tT:r. ta~l, Q Hartree-Fock KÆL-U<laeA (FqUgq)we) lt. ZtkÆM, Q Hartree-Fock Kd)Lbj - *Kk, )ta~[wMgjel5, y-d Fock _e. DÆ5LOgfm_e (ACE), l C-DIIS (PC-DIIS) , nGA$ (ISDF), 8^^p *ZHHTylta~. F 1000 )>elA-Y, 8-gkEl*ZHlta~N$2F Fock _el/Htlx. W, 8ÆIQ Hartree-Fock Kltk2y-Q?G6lza80 \lr.|~u: Q Hartree-Fock K; Lb?G6; j - *e; aeZk; we2Ty; HTy

MR (2010) : 65F15, 65R20, 65Z05

1. vd1kkfS hpJ. z(E N (vdk`d, #J v,3N .DS7, W7KPTk (z 1ECT(vd) o'7hpJ. Kohn-Sham GSx (DFT) [1, 2] hpJkzU#~. ~7;kvd15 (N z ,pTfp/), *3~j USYdfX&mU1k:SYi. W Kohn-Sham GSx 0l9kvd1Sx, ($mr:SYi6dm(Vvk>i-Jrk. Kohn-ShamGSxkfS Hohenberg-Kohn S [1] K Levy-Lieb _%qkTY [3, 4]. 1L, 7~7fz(#ekK|Nvdk^6\G. V;(G, C7zfqNko.hpJ, 7~jf`d5fko'. ?;(#ek

* 2019 2 ' 28 Cjg.1) gRÆ: AAyg?kg DMS-1652330, A"5 DE-SC0017867 RÆ.2) o:: hi, Y^k/CWVtkAkOw/CWAyYsht?k5p+ . 2007L 2011 )-&^kL$h^kdk_L._k1; 2011-2013 -Ow/CWAyYshZa._Sp+.o. [wp+l-tG=taeZkt7e?kLL+`tn. 3d DOE Early Career Award(2017-2022)NSF CAREER Award(2017-2022)SIAM Computational

Science and Engineering (CSE) Early Career Prize(2017) L Alfred P. Sloan fellowship(2015-2017) n. JÆ1, - SCI ';MÆ'koy3 70 .

114 t t k 2019 ^6\GC ). f 0,I, 7KCh;(G vzikzfaR, 1Chk4^. W,XwS, Kohn-Sham GSxk'ZvlJ^6\Gk#J. ,C;pT, :SKYik(,#^6\G:jb3k!4. Y.k.#~ (LDA) [5, 6] :\, 9#~ (GGA) [7–9], E|k9#~ (meta-GGA) [10, 11], )YG [12–14], hM0#~ (RPA) [15, 16], o'6\vd#~ (SCE) [17, 18], iW[G [19, 20] mMxP*9fbXwkvd1r.

Perdew m, 2001 CkG!_b?PS [21]. 8;(?P, a^6\G )k2z, GSxk#'xak%K. ,;U, , )kG1K, )k&^, 3~EP Kohn-Sham Jkri1. XE, LDAGGAmeta-GGA GvUSk )d, (kKaJxK|Nvd. ; 0lCk Kohn-Sham J, ;PGG,.G. )YGv7USi - )d, (kKaJ|N(vd0=. Æ)YGFb Fock ^n, k Kohn-Sham JK`dYiSk Hartree-Fock JP~. W)YGk Kohn-ShamJxG,P Hartree-FockJ.ECntY:;P )k, M.JG, )YG~G\& Kohn-Sham GSxk'. )YGSl\kXd Becke m, 1990 `Pk B3LYP )YG, *H,b`dYiSk%`G (f 0,I, +2 ISI Web of Science, B3LYP G(EC 6 %X). RPA m, )kG1I|N0=, x v62vdhk>, (o' vf'gGrSx.)YGK RPA mGxG,.G.2Zv xk P Hartree-Fock JkrF. `dYi5krFkg0''C|NrFRJ', Gfj. &zfj [22]dhfj [23, 24] mC~~~d 10 M 100 (fGrk`L&^vdh, kRJ Kohn-Sham 0= (x Fock0=, (D2fk Fock ^0=) k. (un Nb) z ,pfp%D, WG,Tfj. Tfjku AMbvd:S`, ERJ Kohn-Sham 0=0=m, ~ErK[T. ,P Hartree-Fock Jk5ID, rk )''Vrvvdi1/RJ Kohn-Sham0=. Æ;z3kr ) O(N4), o[ [bTfj, P Hartree-Fock Jz xK7Tk`d (N z ,pTfp). Tfjkm#z(=u fjg0|N(r, Waq%K'. TfjZv,`dYik9[. MTfjq, `d:S6dk''[fj, E, [25]J [26–29]T, [30] m. ;Pk u Ak, ('~CEPz(ep(:r%K. fjk=u ~dkfÆ'',pfp%SM,, k Nb '' p%fp%. WRJ Kohn-Sham 0=K0=1~N0=k[T,[S. ;(aR[jfjkrFKTfjRk1. fjk|qKVrriUS)Jk,,#. m#arhUS[k&Kk%!, fjx&L&9q. W2aV xfjDkP Hartree-Fock Jk.

2. Hartree-Fock yt, Born-Oppenheimer #~D, Kohn-Sham GSx % 7~DkE%qkk

2 ' hi: Q Hartree-Fock KltG 115TY5EKS = inf

ψiNi=1

,〈ψi|ψj〉=δijFKS(ψi) + EII (2.1)(S ψi

Ni=1 (62k)Kohn-Sham h*ziA%q. d - dMVn EII K|Nd0O, W z(vd_96kCr. FKS vd4k`G

FKS(ψiNi=1) =

1

2

N∑

i=1

∫

Ω

|∇rψi(r)|2 dr+

∫

Ω

ρ(r)Vext(r) dr

+1

2

∫

Ω×Ω

ρ(r)ρ(r′)

|r− r′|

drdr′ + Exc[P ]. (2.2)(SszQ vdk|, sQ d - vdMVn, sIQ vd - vdFvMVn (x Hartree ), x, 9BDkvdMVn. lRzQ ^6\. ,.GK)YGS, (|N0= P (r, r′) =∑N

i=1 ψi(r)ψ∗i (r

′). vd~Æ0=,XD&^k Ejfρ(r) = P (r, r) =

N∑

i=1

|ψi(r)|2.r9 Ω ~ <D R3, x~ Jk9. ,bY+, 7UubvdkfffÆ, ~m Ω 4k!k x. u,bYuI,7,~Dk xS1+GqZPi9 Ω. )YGk^6\CEDk&^

Exc[P ] = Ec[ρ] + (1− α)Ex[ρ] + αEEXx [P ]. (2.3),;U 0 ≤ α ≤ 1 z(x Fock ^, '>^ (EX) k:r. Ex K Ec Lf

LDA e GGA k^K6\#~, K|Nvd ρ. EEXx [P ] |N0=k

Fock ^EEX

x [P ] = −1

2

∫∫|P (r, r′)|2K(r, r′) drdr′, (2.4)(S K(r, r′) vd - vdnkiJ. ,"k Fock ^nSxS [14] K(r, r′) =

erfc(αs|r− r′|)/|r− r

′| k "kFvMVn, αs z(x"Dk:r. dαs = 0 V, K(r, r′) = 1/|r− r

′| H,%`kFvMVn.

Kohn-ShamJ kTY5 (2.1)k Euler-LagrangeJ,x, P Hartree-FockJH [P ]ψi =

(−1

2∆ + Vext + VHxc[ρ] + αV EX

x [P ]

)ψi = εiψi,

∫ψ∗i (r)ψj(r) dr = δij , P (r, r′) =

N∑

i=1

ψi(r)ψ∗i (r

′).

(2.5);U VHxc K Hartreeb ∫ρ(r′)/|r−r

′| dr′,~mK|Nvd ρk^6\b. V EXx [P ]Lf Fock ^ EEX

x [P ]V EXx [P ](r, r′) = −P (r, r′)K(r, r′). (2.6)

116 t t k 2019 P Hartree-Fock J z(i - )KaJj, (SRJ Kohn-Sham 0= H [P ] `F*k^, KCf.By` (SCF) jf.

Fock ^d V EXx [P ] z("kfd. Æ P K K k Hadamard Iijf. ?0= P kQK N (N ,vdr), V EX

x [P ] kQvk. ,fjk5ID,

Kohn-ShamJ dCy`7 [31]. ~3y`v7RJ Kohn-Sham0=n,62h ψiNi=1 L. WP Hartree-Fock J, 79x1 v V EX

x [P ] n,N0=[,[S, K v V EXx [P ] n62h

(V EXx [P ]ψj

)(r) = −

N∑

i=1

ϕi(r)

∫K(r, r′)ϕ∗

i (r′)ψj(r

′) dr′, (2.7),;U P (r, r′) =∑Ni=1 ϕi(r)ϕ

∗i (r

′). ,f.By`i^D0, &^ P khKv7k62h ψiNi=1 1zM. W7[1kuI (ϕi) K (ψj) L9(;_U62h.~( ψ, ÆiJ K k5i^, i ∫

K(r, r′)ϕ∗i (r

′)ψ(r′) dr′ ~C7 N (P0Jjf. ~,,X, EBXDRJk'ur, Ng, [H Fourier #^(FFT), ~(P0J7k`, O(Ng logNg). ,vd1SxkrS, &^~(d vkfÆ~<nz(Cr,W Ng ∼ O(N), ~ V EX

x [P ]n62hk`, O(Ng logNgN2) ∼ O(N3). ,s 1 S, 7fC[Tfj7P Hartree-FockJkr` O(N4). |Ry`Xr N 96, |d N i3V, [fjSM[Tfj,Hq7P Hartree-Fock J. ?, ,C;krS, V EX

x [P ] n62h krV''6hrVk 95%~L.;,/auk`d,)YG DFT k7V .G DFT 7Vk 20 ~L. ;JPb)YG, )Sk.,bTP Hartree-Fock Jkr`, zU [;k*_r. ; ,,~3y` v7 N2 (~ZkP0J.WH1r`,7~[ O(N2)(US,*_7 [32, 33]. x, y, d N = 1000 V, 1L[% 106 (US,7P Hartree-Fock J. ?, Rb v`krb!#, ;*_rk=u RbnV EXx [P ]k4, Kohn-ShamJ7k(GMW[p/(US,. W;uk;*_r*1$<Wk[rb!.mzPTr`k [.YS [34, 35]. V62vdh!_KajNjf Wannier Gr, V EX

x [P ] n Wannier Grk`~, O(N), x, Ka%.Ka%ZvBk8# [36, 37]. eKn, Wannier Gr,XDku''C, WKa%x1+e.2ZvYI( xEMoP Hartree-Fock Jr`: fel^d (ACE), k C-DIIS (PC-DIIS) , ~m?F#~ (ISDF). ;Xv~,8#, x~,eKnS. Pk=s, 7K,;U*Pz(rFXLykWa. E 1000(=dk ( 1),[ PWDFTG (DGDFT G [38] Skz(dG) K 2000 ( CPU *_r, YhOFP ,

GGA Gi^ vkrV 360 . Y GGA Gi^kBP , [Vk!_ HSE )YG [14] r kV 12425 . ;(, EB[ ACE z,i^ vkV 1659. EBV[ ACE/ISDF/PC-DIIS, vkVM 280. W;(, 7()YGkr`Mb GGA Grkw.

2 ' hi: Q Hartree-Fock KltG 117

! 1 F 1000 )>elF3. yx (ACE),P Hartree-Fock Jk7CJS, ~3 SCF y` v+[E Davidson [39]my`7KaYRkJ (2.5) (,;UKaYHk5 P jfkKa>F5). ~3y` vn Fock ^d, x, v7 N2 (P0J. ?, ~3y`vdhk,[ Tk, ~~Xn Fock ^djfk>#~,Wq[. ,s 2 S7f, Fock ^d1 z(oQd. fel^d

(ACE) [40] kO 1/z( Fock^dkQ N # V EXx . Æ V EX

x kQ$o V EXxkQ, 7Kv7;_(dn,62hLjfkF#M.,bO ACE , 7F=sEDkKa>F5

(A+B)vi = λivi, i = 1, . . . , N. (3.1)(S A,B ∈ CNb×Nb 0=. ,;U Nb RJ Kohn-Sham 0=k.. ,02fk,fjS, Nb %mXDRJk'ur Ng. 7k % rlok N (>(λi, vi), *|RB λg := λN+1 − λN > 0. ,;U B ~<H Fock ^d, A RJ Kohn-Sham 0=R; B k4. 7|R B "0= (un B ≺ 0), *30= A k%* ‖A‖2 $ ‖B‖2, 0=S`( Bv k`v$ Av k`. ; Æ A RJfz(9k)d. B RJf Fock ^d, z(kid.,[y`7>F5 (3.1) kCJS, V = [v1, . . . , vN ], 7Fr0=W = [w1, . . . , wN ], (S wi := Bvi. felk|O 1/EDk0=

B[V ] =W (W ∗V )−1W ∗. (3.2)

B[V ] Q N k0=, *ziB[V ]V =W (W ∗V )−1W ∗V =W = BV. (3.3)W*k V , B[V ] n, spanV Lm B, WG, B kfel. (k,

B[V ] zi (3.3) k+zkQ N k0=.|Rs k 3y`k#~>S` V (k), (kfeld B[V (k)]. ACE 7EDk1|u5(A+ B[V (k)])v

(k+1)i = λ

(k+1)i v

(k+1)i , i = 1, . . . , N. (3.4)|R1|u5i^, (>S`, V , |+2 B[V ]V = BV , ACE ~jf'>k>FK>S`. *k k, J (3.4) vCy`7. ?, Mz k

118 t t k 2019 >F, ACE 0=S`( Av K Bv k<n9:L, YTb Bv (kXrLTr`.`fP Hartree-Fock J, 7F V EXx n,d0k62h ϕi

Ni=1 L

Wi(r) = (V EXx [ϕ]ϕi)(r) i = 1, . . . , N (3.5)|

V EXx (r, r′) =

N∑

i,j=1

Wi(r)(M−1)ijW

∗j (r

′), (3.6)(SMij =

∫ϕ∗i (r)Wj(r) dr. (3.7)

ACE ~N Fock ^dknXr. #(E Quantum ESPRE-

SSO [41, 42], PWMat [43] mvd1G9.YrF<k , z(f?k5 1|u5 (3.4) i^f5 (3.1) k.EBi^k[, i^ a&. 7;(5jfbAk\ [44]n P (k) =

V (k)(V (k))∗, P = V V ∗, |1|uy`5 (3.4) EDk.4i^aR‖P − P (k)‖2 . γk‖P − P (0)‖2, ,;U γ ≤

‖B‖2‖B‖2 + λg

. (3.8)FjHPk ,.4i^K|N ‖B‖2, λg, Ak%*96.f.3i^k<,7jfb,z k<.i^BpTk A,B KOF V (0), ACE <.i^a [44].

4. v C-DIIS yx (PC-DIIS),`dYikS, l9k SCF y` Pulay , 1980 `Pk^d- y`dDSE7 (C-DIIS) [45]. Ka5, DIIS rF`rSmCkGMRESEk\ [46, 47]. C-DIISk u (rLi^H (?1Bi^), (=u vE[TRJ Kohn-Sham 0=0=, ~mkdk>

R[P ] = H [P ]P − PH [P ]. (4.1),s 2 S7f, ,fjk5ID, ;X0= 1~Nk^[Tk. W%7Zpk`dYiGXHb C-DIIS , #,`d:SK6dkPN[. ,fjS,Ck 2Æ SCF y`k [42], (i^M.JGkb> SCF y`vzX. , [48] S, 7Pbk C-DIIS (PC-DIIS), [jP C-DIIS ~7fjDkP Hartree-Fock J.P~ C-DIIS , PC-DIIS kfO [p3y`Sd R[P ] k>jfl kbNr, YbN0p3 H [P ] e P k>jf[k0=. Æ P z(Q N 0=, z(f?kO [ Kohn-Sham h Ψ = [ψ1, . . . , ψN ] n,bN#`!_y`. ?Kohn-ShamhLf Kohn-Sham>F5k. W~z(>S` ψi I~Akb0 r eıθ, θ ∈ R |? >S`, ;( rx1%#0=. ,z k, EB=s0=U ∈ CN×N , | Ψ ΨU uk0=. ,`d:SS, ;(fÆ U G,;

2 ' hi: Q Hartree-Fock KltG 119. Æ;~ Ak, EBbN#`|N;, |Yz3y`fDz3y`SkbN#`,1z,ea, x,1Wq?F.,b7;(5, PC-DIIS kO g0Cg0z(:=0= Φref ∈ CNb×N Ljf;1#kbN#`:

Φ = PΦref = Ψ(Ψ∗Φref). (4.2),;U Φ ∈ CNb×N K Ψ Mk[T`. Æ P ;1#k, Φref z(5k0=,~ Φ x ;1#k. :=0= Φref 1L~Ag:, ( v ,~3y`S Φ# z(fzQk0=. P Hartree-FockJkr''[.Gkn,O\F,~7,~"qg:k Kohn-Sham hn, Φref. ,;(VQ, P x~Æ ΦKV1PL:

P = Φ(Φ∗Φ)−1Φ∗. (4.3),;(L, P K Φ zz, *Euk>., PC-DIIS S, s k + 1 3y`Sk Φ ~Æ0 ℓ + 1 3k> Φk−ℓ, . . . ,Φk nKajNjfΦk+1 =

k∑

j=k−ℓ

αjΦj . (4.4),bjfbNr αj, C-DIIS |Rdx~Æ0 ℓ+ 1 3kdnukKajNjfR :=∑k

j=k−ℓ αjRΦj. ,vd1rS, ℓ k:Fz , 5 ∼ 20. ,bTr`,

PC-DIIS !z3|dxf:=0= Φref LRΦ = H [P ]PΦref − PH [P ]Φref = (H [P ]Ψ) (Ψ∗Φref)−Ψ((H [P ]Ψ)∗Φref) . (4.5)?R7EDklTI5jf αj

minαj

∥∥∥∥∥∥

k∑

j=k−ℓ

αjRΦj

∥∥∥∥∥∥

2

F

, s.t.

k∑

j=k−ℓ

αj = 1 (4.6),(rCJS,71 v P,R,H n,N0=[T,K v H n, Ψ L. Wk<nKQmf Nb ×N K N ×N Tk0=, zifjrkv7.%7 C-DIIS ,`dYik:jbPkH/, (i^aKi^k< 0BPJ. W7x9*P PC-DIIS ki^a<. ?, d:=0= Φref g:,f.y`i^Rkh Ψ kVQ, EB 0= Λ &^(k>F, |kd~&^,RΦ = H [P ]Ψ−ΨΛ. (4.7)x, y, kd~<n "9Lk>F5k;`. 7Ch,Ka>F5ky`S, ;`kTY z(Vvk-4. ;,z(=Lgb PC-DIIS ,XwrFrSkWa.

5. szwyx (ISDF)L2Sk ACE K PC-DIIS k k C%Cy`rk, 9q&rWt. +2J (2.7), P Hartree-Fock JkZvkr` h~(h ϕ∗i (r)ψj

120 t t k 2019 (r)Ni,j=1 7P0J. q, ; N2 (hEk> Dk. |R~(h~&^,Æ Ng ('ujHk&'L, ~m Ng ∼ N . |d N MV, hkr ,a'ukr (N2 > Ng). ~z(h &^,uk&'L, Wh ?EDk>. Lk x fl\5Ik4r. EB ϕi, ψj 8XkGr, |hk0=k*FkuaHk [49]. ,`dYiS, z(Ckl #~ [50, 51], ~<H h!_*F, ?RoVvk*FK*S`. #~kr ) O(N4), (`&7P Hartree-Fock Jkr ).?F#~ (ISDF) [52, 53] k|O [EDk'h!_#

ϕ∗i (r)ψj(r) ≈

Nµ∑

µ=1

ϕ∗i (rµ)ψj(rµ)ζµ(r). (5.1),;U?Fu rµ

Nµ

µ=1 XD&' riNg

i=1 kz(dl, ζµ(r)Nµ

µ=1 ,XD&'Lk?FS`. 8Xkh, ?FS`kr Nµ ∼ O(N) *3~$T Ng. (k, ISDF krF&H*1|NBkT [54], ; #.YmK8#WkkVvk1u. ?FuK?FS`U1kg:, EhkgZk QR , lTI, K K- 91Pm. ;Xkr`K O(N3), W~f;kS. ÆJP, 6;XkNCO~m#;AkP,769:= [52, 54, 55].[ ISDF , 7K v?FS` ζµ(r)Nµ

µ=1 7P0J,?RkmKajNjfhkP0J(V EXx [P ]ψj

)(r) ≈ −

N∑

i=1

Nµ∑

µ=1

ϕi(r)

(∫K(r, r′)ζµ(r

′) dr′)ϕ∗i (rµ)ψj(rµ)

:= −

Nµ∑

µ=1

P (r, rµ)Vζµ (r)ψj(rµ),

(5.2)(S V ζµ (r) =∫K(r, r′)ζµ(r

′) dr′.

ISDF 'x~f?q ACE K PC-DIIS N[. , ACE N[V,7^f,J (3.7) S, 0= M # z("k0=. Æ ISDF l, EBE[ (3.7) r, jfk0= M 1B k. ;aPkrF4a. ÆM [hS K(r, r′) _!mijf, 7~_!khVm ISDF LB;(5

Mij ≈−

N∑

l=1

Nµ∑

µ,ν=1

(∫ζ∗µ(r)K(r, r′)ζν(r

′) drdr′)ϕl(rµ)ϕ

∗l (rν)ψ

∗i (rµ)ψj(rν)

=−

Nµ∑

µ,ν=1

(∫ζ∗µ(r)K(r, r′)ζν(r

′) drdr′)P (rµ, rν)ψ

∗i (rµ)ψj(rν)

:=

Nµ∑

µ,ν=1

ψ∗i (rµ)Mµνψj(rν),

(5.3)

Æ M "k=, J (5.3) k M x "k=, Wzi ACE vk.

2 ' hi: Q Hartree-Fock KltG 121

6. n,- ,IlVvkvd1Sx, GSxv zMHmki>, ,1k 4S. ?, 1990 `O`dYik(Zf)YGkVva, Ær`kJP,)YGK,l#T8:\,6dkS,9q.2 xk )YGkP Hartree-Fock JkrF, *Zv.J7,C;pS,;(Sk-n.fjk5^,P Hartree-FockJkZvJLf vq Fock^dn,k62hL. ;(J~YI( LSFock ^dn,62hk>MV2Æ SCF y`i^Fock ^dknEDkr. <;I( , 7Pk7 fel^d (ACE) 1/oQ#N Fock ^dknXrk C-DIIS (PC-DIIS) 2Æ SCF y`#Hb> SCF y`zi^[?F#~ (ISDF) NDr. ~9[k,fj,X, rB&IUMN, 7~ (Ep/(vdk) )YGrk`M.Grkw. Rb~Lk-n, l#C GPU zk [56], ~m1[f.By`k Y [57] mx*B Wko)YGr`k. 7?(;X!4~z)YG, )k`dSk.r Æ [1] Hohenberg P, Kohn W. Inhomogeneous electron gas [J]. Phys. Rev., 1964, 136: B864–B871.

[2] Kohn W, Sham L. Self-consistent equations including exchange and correlation effects [J]. Phys.

Rev., 1965, 140: A1133–A1138.

[3] Levy M. Universal variational functionals of electron densities, first-order density matrices, and

natural spin-orbitals and solution of the v-representability problem [J]. Proc. Natl. Acad. Sci.,

1979, 76: 6062–6065.

[4] Lieb E H. Density functional for Coulomb systems [J]. Int J. Quantum Chem., 1983, 24: 243.

[5] Ceperley D M, Alder B J. Ground state of the electron gas by a stochastic method [J]. Phys.

Rev. Lett., 1980, 45: 566–569.

[6] Perdew J P, Zunger A. Self-interaction correction to density-functional approximations for many-

electron systems [J]. Phys. Rev. B, 1981, 23: 5048–5079.

[7] Lee C, Yang W, Parr R G. Development of the Colle-Salvetti correlation-energy formula into a

functional of the electron density [J]. Phys. Rev. B, 1988, 37: 785–789.

[8] Becke A D. Density-functional exchange-energy approximation with correct asymptotic behav-

ior [J]. Phys. Rev. A, 1988, 38: 3098–3100.

[9] Perdew J P, Burke K, Ernzerhof M. Generalized gradient approximation made simple [J]. Phys.

Rev. Lett., 1996, 77: 3865–3868.

[10] Staroverov V N, Scuseria G E, Tao J, Perdew J P. Comparative assessment of a new nonem-

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NUMERICAL METHODS FOR HARTREE-FOCK-LIKE

EQUATIONS

Lin Lin

(Department of Mathematics, University of California, Berkeley, and Computational Research

Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA)

Abstract

The main goal of this paper is to introduce some recent developments of numerical

methods for solving Hartree-Fock-like equations in the context of large basis sets. Hartree-

Fock-like equations are an important type of equations in electronic structure theory. They

appear in the Hartree-Fock theory, as well as the Kohn-Sham density functional theory

with hybrid exchange-correlation functionals, and are widely used in electronic structure

calculations of complex chemical and materials systems. Because of its high computational

cost, Hartree-Fock-like equations are typically only used in systems consisting of tens to

hundreds of electrons. From a mathematical perspective, Hartree-Fock-like equations are a

system of nonlinear integro-differential equations. The computational cost is mainly due to

the integral operator part, namely the Fock exchange operator. Through the development of

the adaptive compression method (ACE), the projected commutator-direct inversion in the

iterative subspace (PC-DIIS) method, and the interpolative separable density fitting (ISDF)

method, we demonstrate that the cost of Kohn-Sham density functional theory calculations

with hybrid functionals can be significantly reduced. Using a silicon system with 1000 atoms

for example, we have reduced the cost of hybrid functional calculations with a planewave

2 ' hi: Q Hartree-Fock KltG 125

basis set to a level that is close to the cost of semi-local functional calculations, which do

not involve the Fock exchange operator. Meanwhile, we find that the structure of Hartree-

Fock-like equations provides new insights for the iterative solution of one type of eigenvalue

problems.

Keywords: Hartree-Fock-like equation; nonlinear eigenvalue problem; integro-differen-

tial operator; quantum chemistry; electronic structure theory; Density

functional theory

2010 Mathematics Subject Classification: 65F15, 65R20, 65Z05