Density-Functional-Theory Lecture I
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1 8/0/009 Desty-Fuctoal-Theory Lecture I Dr. Bors Kefer Physcs Departmet New Mexco State Uversty Overvew: Moday Buldg the perodc table: Classcal mechacs. Electrcty. Quatum mechacs: wave fucto ad probablty. Hydroge atom/uts. Mult-electro atoms. Perodc table. Covalet bodg. Adabatc-decouplg/tme scales. Rtz varatoal prcple. p Hartree-Fock: Exchage Iteracto. Desty-Fuctoal-Theory. Exchage-correlato eergy. Examples of DFT applcatos. DFT lectures, 08/08+08/09/009, UNM Materal Propertes Crystal Chemstry. Groud State Eergy. Phase Dagrams. Vbratoal Propertes. Electroc Desty of States. Magetsm. Sold Solutos. Alloyg. Elastcty. Mmal Requremets Perodc table. Thermodyamcs. Sgle theoretcal framework for sold, lqud, gas. Bulk, surfaces, ad molecules. Requremet Trasport Propertes: Thermal coductvty. Electrcal coductvty. Vscosty. DFT lectures, 08/08+08/09/009, UNM 3 Descrpto of atoms ad teractos. Forces. Esembles. DFT lectures, 08/08+08/09/009, UNM 4
2 8/0/009 Scetfc Goal Classcal Mechacs, 8 th Cetury m a F Hˆ E Atoms Iteracto World Sr Isaac Newto (643 77) m x m a F V r F j DFT lectures, 08/08+08/09/009, UNM 5 DFT lectures, 08/08+08/09/009, UNM 6 Emprcal Potetals Assessmet Bor-Meyer potetal: " " Z Z V r C r r Aexp B r 6 Coulomb-Potetal No-Coulomb Newto s d law: m a F V ( r j ) j Dsperso Hemoglob Actve ste: Fe + Trasferablty? Electroc propertes? Predctve Power? Suggest that electros should be treated explctly Advatage: Fast. May atoms (> Mllos). Explct dsperso forces. Dsadvatage: Charges (Z); Costats: A, B, C. Multple valace states: 3d, 4d, Fe +, Fe 3+, DFT lectures, 08/08+08/09/009, UNM 7 DFT lectures, 08/08+08/09/009, UNM 8
3 8/0/009 Electrcty, 8 th +9 th Cetury Costtuets of Atoms Electro (Thomso, 897): egatvely charged partcle. eed postvely charged partcle. Proto (Thomso + Rutherford, 907). Charles Coulomb ( ) q q F 4 R Charles_Coulomb.jpg DFT lectures, 08/08+08/09/009, UNM James Clerk Maxwell (83 879) Electrodyamcs 9 Proto Electro DFT lectures, 08/08+08/09/009, UNM Electrodyamcs predcts that t the electro sprals to the ucleus wth ~0 8 s. 0 Classcal Physcs Predcts that Matter s Ustable?!? Soluto: Quatum Mechacs Max Plack (899). Albert Este (905). Nels Bohr (95). Werer Heseberg (95). Erw Schroedger (95). E Hˆ t E Hˆ Auguste Rod; The Thker DFT lectures, 08/08+08/09/009, UNM DFT lectures, 08/08+08/09/009, UNM 3
4 8/0/009 Iterpretato (Tme depedet) Schroedger equato: Eergy egevalue E ˆ H Total eergy (operator) New quatty: Wave fucto Hˆ Kˆ Uˆ Total eergy = Ketc eergy + Potetal eergy What s a Wave Fucto? Iterpretato (Max Bor, 97): probablty dv dv Example: Hydroge Atom Kˆ m The square of a wave fucto descrbes a probablty (desty). e ; Uˆ r DFT lectures, 08/08+08/09/009, UNM 3 DFT lectures, 08/08+08/09/009, UNM 4 Hydroge Atom Thermodyamcs p e Hˆ ˆ rˆ ev = J = kj/mol. kcal/mol = 4.84 kj/mol 3.6eV E a l 0,,..., m l,..., l l Degeeracy : No sp : Sp : m How to dstrbute two or more electros over avalable states? Lowest eergy that s compatble wth costrats == Groud State DFT lectures, 08/08+08/09/009, UNM 5 DFT lectures, 08/08+08/09/009, UNM 6 4
5 8/0/009 Iteractos Mult-Electro Atoms Electros are dstgushable t 0 Physcs/Chemstry must be depedet of umberg. Wolfgag Paul: Paul excluso Prcple (95) t 0 Thermodyamcs Fte T : Mmze F=F(V,T,N) or G=G(P,T,N) T=0 K : Mmze E=E(V,N) or H=H(P,N) I quatum mechacs we have two possbltes: r, r r r, r, r r r, Fermos: electros, protos, eutros, Bosos: photos, phoos, sp-waves, DFT lectures, 08/08+08/09/009, UNM 7 DFT lectures, 08/08+08/09/009, UNM 8 Paul-Excluso-Prcple Probablty r, r dv dv Two Fermos the same locato: r, r r, r r, r 0 Probablt y 0 Slater determats: r,..., r r r N! r... r Fermos the exact same quatum state are mpossble DFT lectures, 08/08+08/09/009, UNM 9 Hud s Rule # For a gve electro cofgurato, the term wth maxmum multplcty has the lowest eergy. Sce multplcty s equal to S+, ths s also the term wth maxmum S., * + == symmetrc; - == at-symmetrc. Fredrch Hud Cosder two electros: S=0 at-symmetrc sp state symmetrc spatal state. S= symmetrc sp state at-symmetrc spatal state zero probablty to fd electros at same locato. reduced Coulomb repulso. S= (trplet) state has a lower eergy. DFT lectures, 08/08+08/09/009, UNM 0 5
6 8/0/009 Hud s Rule #, cot d Spatal States Electrostatc repulso reduced for at-symmetrc spatal state lower eergy. DFT lectures, 08/08+08/09/009, UNM Hud s Rule # For a gve multplcty, the term wth the largest value of L has the lowest eergy. Hgh L electros orbt the same drecto. For low L some electros rotate oppostely must pass crease Coulomb repulso hgher eergy. Classcally speakg: Larger L More electros orbt ucleus the same sese. Reduced Coulomb repulso. Lower eergy. DFT lectures, 08/08+08/09/009, UNM Hud s Rule #3 Perodc Table For a gve term, a atom wth outermost subshell halfflled or less, the level wth the lowest value of J=L+S les lowest eergy. If the outermost shell s more tha halfflled, the level wth hghest value of J s lowest eergy. Reaso: Sp-orbt couplg ~ L * S J=L+S eergy lower f L ad S are opposte drectos lower J. DFT lectures, 08/08+08/09/009, UNM 3 N l- l z = -l +l = * l +; s=±/ N=: l=0, s = ±/ states N=: l=0, s = ±/ states l=, s = ±/ 6states DFT lectures, 08/08+08/09/009, UNM 4 6
7 8/0/009 Summary Falure of classcal mechacs/electrodyamcs. Quatum mechacs to the rescue. New quatty: wave fucto Probablty. Buldg blocks of the perodc table: Paul excluso prcple. Hud s rules. DFT lectures, 08/08+08/09/009, UNM 5 DFT lectures, 08/08+08/09/009, UNM 6 Bor-Oppehemer or Adabatc Decouplg Bor-Oppehemer or Adabatc Decouplg Do we eed to treat ucle quatum mechacally as well? Hˆ Kˆ o Uˆ o Kˆ el Uˆ el Uˆ elo m p /m e ~ 000 moto of electro much faster tha ucle. Tme scales: Dffuso : D= 0 - cm /s; d= A t=d /D ~ 0-5 s Vbratos : H ~ 4400 cm - t=/(c k) ~ s. Electros : t= h/c m e ~ 0-0 s. Electroc tme-scale shortest Electros follow uclear moto stataeously. (except may be for the lghtest elemet: hydroge) DFT lectures, 08/08+08/09/009, UNM 7 DFT lectures, 08/08+08/09/009, UNM 8 7
8 8/0/009 Cosequece If T=0 K Ketc eergy of the ucle s zero. Note: Nucle are ot elmated from the problem, U uc-uc ad U uc-el. Statc t problem. If T>0 K Nucle are o loger fxed. Lattce vbratos; thermodyamcs. Electros: always treated quatum mechacally. Covalet Bods- I p Hˆ ˆ m e e rˆ R / Electro boud to ucleus : I> Electro boud to ucleus : I> e e rˆ R / R ˆ ˆ H j Hˆ Hˆ Hˆ Hˆ H j DFT lectures, 08/08+08/09/009, UNM 9 DFT lectures, 08/08+08/09/009, UNM 30 Covalet Bods- II Rtz s Varatoal Prcple E H H E 0 * r,..., rn Hˆ r,..., r N * r r r,...,,..., N r N 0 H r H r dv 0 dv Overlap of electroc wave fuctos s ecessary for covalet bodg. DFT lectures, 08/08+08/09/009, UNM 3 For a gve H the groud state eergy ca be estmated as the lowest eergy foud for ay possble wave fucto. Also remscet of Paul excluso prcple, Hud s rules, ad thermodyamcs whch all requred the mmzato of eergy. DFT lectures, 08/08+08/09/009, UNM 3 8
9 8/0/009 Geeralzato: Hartee-Fock May-electro systems: Keep: Adabatc decouplg. Paul-prcple. r r,..., r... N! r r... r Ths theory s called Hartree-Fock theory. Note: Hartee-Fock: Electrostatcs Notably the theory cotas expressos such as: r j r dvdv r Electrostatc eergy Electros at the same locato: r =0 j Icrease of electrostatc eergy. Ths s mpossble accordg to the Paul-excluso prcple. DFT lectures, 08/08+08/09/009, UNM 33 DFT lectures, 08/08+08/09/009, UNM 34 Hartee-Fock: Exchage But aother term appears as well: r r r r r dvdv j r Exchage eergy: Org: Paul-excluso prcple. No classcal aalogue. j Probablty Exchage: Aother Look j Exact compesato of electrostatc term: Paul excluso prcple s fulflled. Paul excluso prcple Exchage eergy Reduces Coulomb repulso Lowers eergy of system. DFT lectures, 08/08+08/09/009, UNM 35 DFT lectures, 08/08+08/09/009, UNM 36 9
10 8/0/009 Chagg the Pot of vew Hartree-Fock: cetral quatty: wave fuctos. Alteratvely: Choose to focus o the charge desty. Desty-Fuctoal-theory. y Desty-Fuctoal-Theory (DFT) Hoheberg-Koh (964) Theorem For ay system of teractg partcles a exteral potetal V ext (r), the potetal s determed uquely by the groud state partcle desty 0 (r): E ext 3, V r V r d r F F[] s uque ad depeds oly o the desty,. Completely geeral. DFT lectures, 08/08+08/09/009, UNM 37 DFT lectures, 08/08+08/09/009, UNM 38 Theorem A uversal fuctoal for the eergy E[,V ext ] terms of the desty (r) exsts wth a global mmum for the exact groud state desty 0 (r). Vald for ay system (uversal): gas, lqud, d sold Iteratve Soluto HK V ext r r r r r All States r Groud State Koh ad Sham Asatz (965) Replace correlated may-electro problem by a equvalet sgle electro depedet partcle problem a effectve potetal. t e r r ' 3 3 F d r d r' r r ' Ketc eergy T 0 E XC Coulomb teracto Exchage- Correlatoeergy DFT lectures, 08/08+08/09/009, UNM 39 DFT lectures, 08/08+08/09/009, UNM 40 0
11 8/0/009 Koh ad Sham Equatos Koh ad Sham (965) r Veff, mel V eff r, r V r r r r r r e dr r r E XC r r Exteral potetal. Ca be geeralzed to clude magetsm: Coulomb teracto. Exchage-correlato. DFT lectures, 08/08+08/09/009, UNM 4 What s the Exchage-Correlato Potetal, E XC? Thermodyamcs: groud state == lowest eergy state Paul excluso prcple Exchage eergy. DFT lectures, 08/08+08/09/009, UNM Repulso of opposte sps Correlato eergy. Both effects lower the eergy Actve research feld to develop better E XC fuctoals: LDA : Ceperley ad Alder. GGA : PW9, PBE, rpbe, revpbe, 4 Perodc Structures: Bloch s Theorem Cosequece? g of Na ~3 x 0 3 electros. Need to solve determe for 0 3 electros?!? Recogze that crystals are perodc structures. Suffcet to descrbe oe ut cell. (Bloch s theorem). Crystals are 3-d perodc structures: That s they ca be created by perodcally repeatg a smaller ut. bcc-na: electros suffcet. Our problem has become tractable. Crystal == Lattce + Motf DFT lectures, 08/08+08/09/009, UNM 43 DFT lectures, 08/08+08/09/009, UNM 44
12 8/0/009 Sutable Electroc Wave Fuctos Costat potetal Pseudopotetals Problem : Steep potetal close to ucle plae waves are ot well suted. Steep potetal Solutos to Schrödger equato wth costat potetal: Plaewaves: ψ~exp( k r) Idea: Most materal/chemcal propertes rely o valece electros. Pseudopotetal. Elmate core electros DFT lectures, 08/08+08/09/009, UNM 45 DFT lectures, 08/08+08/09/009, UNM 46 Optmzato: Forces Hellma-Feyma (force) theorem: F E Hˆ electroc U oo Summary: Perodc Table. Pseudopotetal fewer electros. Uversal E XC : sold, lqud, gas. Bulk, surface, molecule. Optmzato. Predctve power. DFT lectures, 08/08+08/09/009, UNM 47 Summary Covalet bodg. Adabatc decouplg. T=0 K ad fte temperatures. Hartree-Fock Exchage. DFT Exchage ad correlato. Perodc structures, Bloch s theorem. Pseudopotetal. Forces. Predctve power. DFT lectures, 08/08+08/09/009, UNM 48
13 8/0/009 Some Applcatos of DFT Fudametals of Codesed Matter Physcs/Chemstry: Sodum at hgh pressure. Octet-rule at hgh pressure. Studet presetatos Adrew: Pt -x Re x sold solutos. Erc: Pd o γ-al O 3. Lev: Pd o α-al O 3. Sam: o-pt based catalysts. DFT lectures, 08/08+08/09/009, UNM 49 DFT lectures, 08/08+08/09/009, UNM 50 Sodum at Hgh Pressures (Ma et al., 009) Backgroud: Sodum at ambet codtos s a metal. Expected: structures adopt dese packed structures at suffcetly hgh pressure. Expermet GPa ~ 0 4 atm Theory trasparet Phases: bcc fcc (close-packed) o-closed-packed Trasparet (o-metallc) hgh pressure phase. DFT lectures, 08/08+08/09/009, UNM 5 DFT lectures, 08/08+08/09/009, UNM 5 3
14 8/0/009 Octet rule at Hgh Pressures (Kefer ad Tschauer) Octet rule: ma group elemets ted to acheve a complete outermost shell: eght electros. CH 4 : C: 4e - ; each hydroge doates e -. H S: S: 6e - ; each hydroge doates e -. CHOONa: Na-formate. DFT lectures, 08/08+08/09/009, UNM 53 thalpy (ev/0 atoms) E Expermetal Observatos Sample becomes opaque at ~0 GPa durg laser heatg to T ~ 500 K, reverses at T ~ 300 K. H lbro observed C 5 Na-formate C C 5 C 3 C C 6 Na-formate stable -45 C 5 Na C O 4 H :C4-50 Sold squares: Na C O 4 + H (molecule) Hydrous: H Ope squares: Na C O 4 + H (sold) Isulators, -55 E gap >~.3 ev Pressure (GPa) Hydrous: C-H Isulators, E gap >~ 4.eV Ahydrous: Metallc DFT lectures, 08/08+08/09/009, UNM 54 Summary DFT versatle tool. Exchage ad correlato. Pseudopotetals. Perodc table. Applcable to sold, lqud, ad gas. Bulk, surface, molecule. T=0 K, fte temperature. Revew. DFT: s ad out s. Outlook DFT lectures, 08/08+08/09/009, UNM 55 4
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