Electron density: Properties of electron density (non-negative): => exchange-correlation functionals should respect these conditions.

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Andrea Parsons
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1 lecton densty: ρ ( =... Ψ(,,..., ds d... d Pobablty of fndng one electon of abtay spn wthn a volume element d (othe electons may be anywhee. s Popetes of electon densty (non-negatve:.. 3. ρ ( d = ρ( = 0 ρ( R A = ma ρ( R... cusp A ρ( ep I lm Z ρ( = => echange-coelaton functonals should espect these condtons

2 Pa densty: ρ (, = (... Ψ(,,..., d... d 3 Pobablty of fndng a pa of two electons wth patcula spns wthn a volume elements d and d (emanng - electons may be anywhee. on-coelated moton: ρ (, = ρ( ρ( on-negatve quantty omalzed to (-, contans all nfomaton about electon coelaton. ymmetc Antsymmetc wavefuncton equement => educed densty mat γ γ (, ;, = (... Ψ(,,..., Ψ (,,..., d... d ' ' * ' ' Vaables n Ψ * whch ae not ncluded n ntegaton ae pmed. γ changes sgn when and (o and ae ntechanged γ = γ (, ;, (, ;, ' ' ' ' Dagonal elements of educed densty mat => pa densty (two-electon densty mat Pobablty of fndng two electons wth the same spn at the same pont s 0!!! ρ (, = ρ (, = ' ' =

3 mall detou: HF pa densty: ρ [ { φ ( σ (s φ ( (s }] HF (, = det σ ρ HF (, = φ( φ( σ( s σ( s + φ( φ( σ( s σ( s φ ( φ ( φ ( φ ( σ ( s σ ( s σ ( s σ ( s σ =σ HF, σ =σ ρ (, ρ, = 0 HF ( ρ HF, σ σ σ σ (, = ρ( ρ( Coelated electon moton ρ = σ = (, 0 HF, σ Completely uncoelated moton (, ( ( HF, σ σ ρ = ρ ρ => Fem coelaton, change coelaton - descbed aleady at the HF level Two electons wth the same spn cannot be at the same pont n space. Ths coelaton does not depend on the electon chage, puely echange effect. Fem coelaton has nothng common wth ( Coulomb coelaton defned fo post HF methods!

4 Pa densty fo completely uncoelated moton: ρ (, = ρ( ρ( Fomulaton of pa densty n tems of electon densty and (whateve s the coelaton: ρ (, = ρ( ρ( + f( ; [ ] Coelaton facto - defnes the dffeence between uncoelated and coelated denstes: f( ; = 0 - completely uncoelated case => => wong nomalzaton of ρ (! (due to self-nteacton Intoducng Condtonal pobablty - pobablty of fndng electon at poston when thee s just one electon at poston. Integates to (-. Ω ( ; = ρ(, ρ( ( ; Ω d =

5 change-coelaton hole: ( ; ( ; h ( =Ω ρ The dffeence between condtonal pobablty Ω and uncoelated (uncondtonal pobablty of fndng electon at. h c accounts fo: echange and coulomb coelaton and self-nteacton ρ(, = ρ( ρ( = ρ( f( ; Coelaton - typcally leads to depleton of electon densty ( ; h d = chödnge equaton n tems of spn-ndependent pa densty (two-electon pat: ρ (, ee = Ψ Ψ = d ddsds ρ j> j (, = ρ( ρ( + ρ( h (; Ingegaton ove 8 vaables only! Pobablty of fndng a pa of electons at, ee = ρ( ρ( d d + ρ( h ( ; d d ds ds J[ρ] change, coelaton, IC

6 ρ( ρ( ρ( h ( ; ee = d d + d d Classcal J[ρ] QM contbuton (coelaton + self-nteacton FORMALLY - echange-coelaton hole can be splt nto the Fem hole and Coulomb hole h = h + h σ, ( ; = σ σ σ ( ; ( ; X C Fem hole - domnates coulomb hole - contans self-nteacton - ntegates to - - equals to mnus densty of electons (same spn at the poston of ths electon (at the same pont - negatve eveywhee - depends also on the densty at - no sphecal symmety h X ( ; = ρ( h X ( ; = ρ( f X ( ; Coulomb hole - ntegates to 0 - negatve at the poston of efeence electon - mos the cusp condton h (; d C = 0

7 ample H molecule: change hole only IC It s just half of the densty! It does not depend on the electon poston ( h X leads to depleton of electon densty HF method consdes only h X and t esults n too dffuse one-el. Functons => undeestmatng V en, low T e, and also J ee s undeestm. α hx ( ; = ρ( = Ψσ g Coulomb hole Changes wth the poston of efeence electon Fo HH h C emoves halves the electon fom one atom and puts t to the othe one Baeends & Gtschenko J. Phys. Chem. A 0 (

8 Hamltonan only contans pats dependng on ( one electon o ( on two electons => chödnge equaton can be ewtten n tems of one- and two-patcle densty matces Knowledge of ( ρ h ( ; ρ( ρ( ρ( h ( ; ee = d d + d d asy soluton of chödnge equaton n tems of spn-denstes (8 vaables Is t possble? Does the densty contans all the nfomaton? Answe: Hohenbeg-Kohn Theoem (964 Popetes of electon densty (non-negatve:.. 3. ρ ( d = ρ( = 0 ρ( R A = ma ρ( R... cusp A ρ( ep I lm Z ρ( 0 + = Fst attempts: Thomas-Fem Model (97

9 mall detou: Unfom lecton Gas Hypothetcal system, Homogeneous lecton Gas lectons move on the postve backgound chage Oveall system chage s 0 Volume umbe of electons el lecton densty (constant / V = ρ V o so model fo smple metals; constant densty s fa fom ealty fo molecules! Only system fo whch we know echange-coelaton functonal eactly.

10 Thomas-Fem Model Patally classcal eglects echange and coelaton contbuton Cude appomaton fo knetc enegy (fa fom eal molecules Poo pefomance! VRTHL negy s gven as a functonal of electon densty! T TF TF ρ [ (] = [ ρ(] = (3π (3π / 3 / 3 ρ 5 / 3 5 / 3 (d (d Z ρ( d + ρ( ρ( oluton vaatonal pncple unde the constant of numbe of electons. ρ d d umeous etentons and mpovments: chemcal accuacy neve eached (by a dstance! ven when V ee descpton mpoved poblems stay due to knetc enegy descpton. It was goously poofed that wthng T-F model all molecules wll dssocate nto the fagments!

late s appomaton fo electon echange Used befoe by Dac: Thomas-Fem-Dac model 95 to fnd an appomate way to calculated echange n HF ( h X ( ; X = d d ρ One needs a good appomaton to h X.

11 late s appomaton fo electon echange Used befoe by Dac: Thomas-Fem-Dac model 95 to fnd an appomate way to calculated echange n HF ( h X ( ; X = d d ρ One needs a good appomaton to h X. Assumng sphecally symmetc hole centeed aound the efeence electon.. Assumng that densty s constant wthn the hole and that t ntegates to -. 3 / 3 phee adus = ρ( (Wgne-etz adus 4π mple ntepetaton aveage dstance between electons / 3 Appomate soluton: X [ ρ ] C X ρ ( 4 / 3 d Densty functonal fo echange enegy! Ognal wok Hatee-Fock-late (HF method known also as X α method: change ntegals eplaces by (α s a paamete between /3 and Xα [ ρ] = π / 3 α ρ ( 4 / 3 d

12 Densty functonal theoy Tadtonal ab nto: fndng the -electon wavefunton Ψ(,,, dependng on 4 cood. DFT: fndng the total electon spn-denstes dependng on 8 coodnates Hohenbeg & Kohn: Theoem I: negy of the system s unque functonal of electon densty ρ ( =>, R, Z => V H, Ψ { } ˆ el J J et Theoem II: Vaatonal pncple [ ρ] [ ρ] 0 0[ ρ0] = ρ0( Ved + T[ ρ0] + ee[ ρ0] system dependent unvesally vald [ ρ ] = ρ ( V d + F [ ρ ] e HK 0 Hohenbeg-Kohn functonal: Knetc enegy of electon Coulomb epulson on-classcal nteacton (self-nteacton, echange, and coelaton All popetes (defned by V et ae detemned by the gound state densty H&K only poofed that F HK est, howeve, we do not know t H&K do not gve a decton how can we fnd densty H&K theoems allow us to constuct the goous many-body theoy usng densty as a fundamental popetes F[ ρ( ] = T[ ρ( ] + J[ ρ( ] + [ ρ( ] ncl

13 The econd Hohenbeg-Kohn Theoem: Densty functonal F HK [ρ] wll gve the lowest enegy of the system only f the ρ s a tue gound state densty. ~ VARIATIOAL PRICIPL [ ρ ~ ] = T[ ρ ~ ] + [ ρ ~ ] + [ ~ ] 0 e ee ρ Poof lteally tval. Tal densty defnes ts own Hamltonan, thus, wave functon:. Applyng vaatonal theoem fo ths tal wava functon: ~ ~ Ψ Ĥ Ψ = T[ ρ ~ ] + V [ ρ ~ ] + ~ ρ(v d = [ ρ ~ ] ee et 0 [ ρ ρ( Hˆ Ψ 0 ] = Ψ 0 Ĥ Ψ 0 OT: tctly VP holds only fo eact functonal. Appomate functonals can easly gve eneges below a tue mnmum (dffeent fom HF. Mathematcal vs. Physcal meanng of VP.

14 Kohn-ham Appoach A Basc Idea HK theoems = 0 mn ρ ( ρ + ρ F[ ] (V d e F[ ρ(] = T[ ρ(] + J[ ρ(] + Kohn-ham: Most poblems of Thomas-Fem type appoaches come fom knetc enegy Kohn-ham establshng a smla stategy as used n Hatee-Fock method Hatee-Fock method fom a dffeent pont of vew: late detemnant appomaton to the tue -electon wave functon It can be vewed as the eact wave functon of fcttous system of non-nteactng electons movng n the effectve potental V HF ( electons vewed as unchaged femons not eplctly nteactng va Coulomb epulson Knetc enegy of such system s then eactly [ ρ(] One electon functons, spn-obtals, obtaned fom vaatonal pncple ncl T HF = χ χ HF = Φ mn D Φ D Tˆ + Vˆ e + Vˆ ee Φ D In analogy wth above non-nteactng electons K& ntoduced a non-nteactng efeence system fo patcles nteactng va effectve local potental V, that n some way ncludes desed nteactons between patcles.

15 Hamltonan wth effectve local potental V : late detemnant s then eact wave functon: One electon functons obtaned (n analogy wth Fock equatons by solvng Kohn-ham equatons, usng a one-electon Kohn-ham opeato f K : Resultng obtals ~ Kohn-ham (K obtals Ĥ = + V ( ϕ ϕ ϕ ϕ Θ =! ϕ ϕ K fˆ fˆ K ϕ = = ε ϕ ( ( ϕ( ( ( ϕ ( ( ( ϕ ( + V ( ffectve potental V s such that the densty consttuted fom K obtals eactly equals the gound state densty of eal system wth nteactng electons ρ ( = ϕ (,s = ρ0 ( s

16 Kohn-ham Appoach Adoptng a bette epesson fo knetc enegy: Usng eact knetc enegy of the non-nteactng efeence system that has the same densty as a eal one. uch knetc enegy cannot be the same as a tue one; t s epected to be close. Resdual pat of knetc enegy (T C s shfted to the functonal. Kohn-ham functonal s then: [ ρ] T = T T ( T[ ρ] T [ ρ] + ( [ ρ] J[ ρ] = T [ ρ] + [ ρ] ee C ϕ F[ ρ(] = T [ ρ(] + J[ ρ(] + ϕ ncl [ ρ(] Knetc enegy of non-nteactng efeence system Coulomb epulson of uncoelated denstes change-coelaton functonal Includes: lecton echange lecton coelaton Resdual pat of knetc en.

17 Puttng thngs togethe: [ ρ(] = T [ ρ] + J[ ρ] + [ ρ] + e [ ρ] ρ( ρ( = T[ ρ] + d d + [ ρ] + Veρ(d = ϕ ϕ + ϕ ( ϕ j ( dd j + [ ρ(] M ZA ϕ ( A A d Applyng vatatonal pncple Contans all poblematc tems ρ( + d + V ( = + Veff ( ϕ = εϕ M A Z A A ϕ = ε ϕ atsfyng the condtons stated fo non-nteactng efeence system V ( V eff ( = ρ( d + V ( M A Z A A Iteatve soluton. What s V? V δ δρ

18 Kohn-ham method n a nut-shell: on-nteactng efeence system wth effectve local potental V s ntoduced: H V ˆ = + ( Θ =! ϕ ϕ ϕ ( ( ( ( ( ( ϕ ϕ ϕ ϕ ϕ ϕ ( ( ( fˆ K ϕ = εϕ Kohn-ham equatons ˆ K f = + V ( Poblematc knetc enegy tem s dvded between non-nteactng system and echangecoelaton functonal: F[ ρ( ] = T [ ρ( ] + J[ ρ( ] + [ ρ( ] ( ( [ ρ] T[ ρ] T [ ρ] + [ ρ] J[ ρ] = T [ ρ] + [ ρ] ee C ncl T C =T-T... the esdual pat of knetc enegy

19 How does t wok: ρ( ρ( [ ρ( ] = T [ ρ] + J[ ρ] + [ ρ] + e[ ρ] = T [ ρ] + dd + [ ρ] + Veρ( d = + ϕ ϕ ϕ( ϕ j( dd j M Z A [ ( ] ( ρ ϕ d + A A Applyng vaatonal pncple we solve Kohn-ham equatons n teatve way veythng unknown s n If we know we have an XACT method s not known => we have to ely of appomate echange coelaton functonals change-coelaton functonals: I. local densty appomaton - [ρ] II. Genealzed Gadent appo. - [ρ, ρ] III. Hybd densty functonals - [ρ, ρ] + combnes wth eact =HF echange Local Densty Appomaton: (LDA ~ LD ~ VW deved fo the model of unfom electon gas LDA 3 3 ρ( [ ρ] = ρ( ε ( ρ( d ε 3 X = used by late ε ( ρ( = ε ( ρ( + ε ( ρ( 4 π X C ε C (ft of QMC data VW

Density Functional Theory I

Density Functional Theory I Densty Functonal Theoy I cholas M. Hason Depatment of Chemsty Impeal College Lonon & Computatonal Mateals Scence Daesbuy Laboatoy ncholas.hason@c.ac.uk Densty Functonal Theoy I The Many Electon Schönge