Electronic Structure in Periodic Systems. b a

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Marjorie Hawkins
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1 1. Isolted vs. periodic systems How to model thigs tht re ot moleculr, like metl or semicoductor or the surfce of somethig? Could crete very lrge chuk, or cluster, of the mteril, but this c be both expesive d ot lwys cler how to crete d termite the chuk. Altertive is to crete ifiitely repeted, periodic boudry coditio models. Is this esier? Yes, i prticulr i the cotext of DFT. (peroxycetyl itrte) (MgO bulk) (MgO slb) c c b b cluster cluster models models Isolted Isolted molecule molecule i i vcuum vcuum (or (or dielectric dielectric cotiuum) cotiuum) Gs-phse Gs-phse Solutio-phse Solutio-phse Loclized Loclized chemistry chemistry Ameble Ameble to to highly highly ccurte ccurte clcultios clcultios A criticl differece betwee these supercells for bulk mterils like MgO d the previous tom d molecule exmples bove, is tht i the former cse the bodig exteds cross the boudries of the box. I brief, y prticulr o or mo iside the supercell becomes ifiite umber of sttes idexed by the rel vector k. 2. Bloch s theorem d qulittive bd structure Hve to ler some ew lguge, derived from the solid stte physics commuity, to uderstd the bd structure model of electroic structure. Hs trditio t lest s rich s moleculr orbitl/qutum chemistry. Pictures i this sectio borrowed liberlly from Hoffm, Agew. Chemie 1987, 26, 846. Cosider cyclic chi of icresig legth of H toms. Boudry coditio (Bor-v Krm) tht wvefuctios coect t the ed of the chi. supercell supercell models models 3-D 3-D periodic periodic boudry boudry coditios coditios pplied pplied to to tomic tomic cofigurtio cofigurtio Bulk Bulk solids solids Bulk Bulk liquids liquids Surfces Surfces d d Iterfces Iterfces Deloclized Deloclized chemistry chemistry Ameble Ameble to to highly highly efficiet efficiet clcultios clcultios Prof. W. F. Scheider CBE Computtiol Chemistry 1/22

2 How would we solve this? Bsis of sigle s (or p) fuctio o ech ceter, costruct seculr mtrix d digolize (e.g., Hückel model). Get sme umber of output moleculr orbitls s iput tom orbitls. As rig icreses, umber of sttes gets commesurtely lrger, but spredig out is see to sturte. As umber, get cotiuous bd of eergy levels. Note there is some regulrity to the eergy levels, umber of odes icreses i regulr wy. Ech H tom here is ideticl to every other; the electros i the orbitl sttes see the sme ttrctive potetil er ech tom. Defie lttice costt. Atoms: Potetil: H H H H H H H H H H e - e - e - PBC V ( x) = V ( x + ) itegers Bsis: 1s χ0 1s χ 1 1s χ 2 1s χ 3 1s χ 4 1s χ 5 1s χ 6 1s χ 7 1s χ 8 1s χ 9 Wvefuctios: k = 0, λ π k =, λ = 4 2 π k =, λ = 2.. = ψ e χ = χ 0 0 1s 1s ψ = e χ π 2 iπ 2 1s = ψ e χ = ( 1) χ π iπ 1s 1s Bloch s Theorem: k ψ ik = e χ, k π (recll e ikr is like cosie fuctio) Bloch s theorem sys tht the periodic wvefuctios c be writte s the product of cellivrit prt (the H 1s fuctios here) d cell-periodic prt. The periodic prt is idexed by the wvevector k. k tkes s my vlues s there re periodic uits N. If is ifiite, th k is cotiuous d rel vrible, d it exists withi reciprocl spce (ote k hs dimesios 1/legth). The spce of uique vlues of k is clled the first Brilloui zoe. The periodic phses of the bsis fuctios correspod to uderlyig wvelegth ssocited with k: λk = 2π k. By the de Broglie reltioship, the, k reltes to the mometum of electro i tht eergy level. Ech k is degeerte such tht Ek ( ) = E( k), s c be see i the picture bove. The spce of cotiuous eergy levels i k is clled the bd structure. By covetio, structure is plotted gist k. The width, or dispersio, of the eergy bd is determied by the overlp betwee djcet periodic cells. Bottom of bd is bodig, top is ti-bodig. Prof. W. F. Scheider CBE Computtiol Chemistry 2/22

3 eergy eergy. H H 0 k π/ H H H H H Moleculr orbitl digrm Bd structure digrm The highest occupied eergy level, i this prticulr cse k = π 2, is clled the Fermi eergy. Dispersio of bds depeds o overlp vi spcig betwee cells, d directio depeds o topology of orbitls: If we hd multiple bsis fuctios o ech ceter, would get multiple bds. Cosider p z d d orbitls from chi of metl toms: Prof. W. F. Scheider CBE Computtiol Chemistry 3/22

4 Orbitls tht overlp strogly give big dispersio, d bds ru i directios determied by type of overlp. Extet of bodig depeds o how my levels re filled. Electricl properties relted to bd fillig: prtilly filled bds t Fermi level à metllic behvior; gp t Fermi level à isultor. Alysis is domi of solid stte physics. 3. Bd foldig No reso we hve to choose sigle tom s the fudmetl uit cell. Could e.g double the size of the cell, to crete supercell. Now bsis is doubled i size d Brilloui zoe is hlved. 2 Supercell: Potetil: H H H H H H H H H H e - e - e - PBC V ( x) = V ( x + ) itegers Bsis: Wvefuctios: χ A 1s 1s χa = χ0 + χ1 k χ = χ χ 2 B 1s 1s 0 1 π π k = k = 0 2 χ A Brilloui zoe hlved i size χ B χ B χ bd rus up i eergy, A χ rus dow. Note tht B χ is lwys bodig d A χ lwys B tibodig withi the supercell, but depedig o k, they re either bodig or ti-bodig betwee cells. The two bds re degeerte (ideticl) t right edge of BZ. Plot eergies: Prof. W. F. Scheider CBE Computtiol Chemistry 4/22

5 New bd structure is ideticl to the old, but the bd is folded i hlf. Oe k-poit ow cotis iformtio bout two differet bds. Could cotiue, BZ keeps gettig smller, bds keep multiplyig, evetully retur to MOs of oe big molecule. Notice i this prticulr cse, sice there is oe electro/h tht the Fermi level is right t the poit where the two bds meet. This is ustble situtio: high symmetry degeerte stte. The eergy c be lowered by llowig the hydroges to pir up, cuses sttes er Fermi level to seprte. This is termed Peierl s distortio, d is logous to Jh-Teller distortio i molecules. Of course we kew tht chi of H toms would wt to mke H 2 molecules! 4. Multi-dimesiol periodicity Sme ides c be geerlized to crystls with 2-D d 3-D periodicity. All poits o lttice c be defied i terms of lttice vectors tht defie the uit cell. k becomes wvevector k defied withi 2- or 3-D reciprocl spce. Squre lttice 2- D Squre Brilloui zoe 2 = ŷ 1 = ˆx ij π X M k y Γ X π π k x π Reciprocl lttice vectors b 2π 2π b k b b 1 = xˆ ˆ 2 = y = kx 1 + ky 2 k π π ψ = e χij, kx, ky i ij k i, j Prof. W. F. Scheider CBE Computtiol Chemistry 5/22

6 Some smple 2-D wvefuctios: k = 0 k = ( π,0) k = ( 0, π ) k = ( π, π ) Bd structure E(k) is ow multi-dimesiol, d it is customry to plot eergy log specil directios i reciprocl spce. 5. Desity of sttes Give multi-dimesiol chrcter of k, it is coveiet sometimes to verge out the k cotributios to bds. Desity of sttes (DOS) provides wy: : umber of sttes i itervl E to E+dE Recll prticulr bd (pek i DOS) is bodig or ti-bodig withi the cell, but the width of the DOS pek sps from bodig to tibodig betwee cells. Prof. W. F. Scheider CBE Computtiol Chemistry 6/22

7 6. Brvis lttices I geerl, the trsltiol symmetry c be chrcterized i terms of Brvis lttice, which describes ifiite periodic rry of uits. The thig tht is periodiclly replicted, the periodic uit, is clled the bsis. The bsis could be tom, molecule, frgmet of solid, chuk of liquid, b, whtever. I oe dimesio, there is oly oe type of Brvis lttice, defied by sigle trsltio vector,. Positio of y lttice poit c be give s some iteger multiple of, R =. Showed picture of this bove. I two dimesios, there re five distict types of Brvis lttices, oblique, rectgulr, cetered rectgulr, hexgol, d squre (see below). The lttice vectors defie the uit cell. The uit cell is ot uique; primitive cell cotis exctly oe Brvis lttice poit. Lrger uit cells c be costructed; show exmple of covetiol cell for cetered lttice. Wiger-Seitz cell is most compct cell possible, formed by regio creted by bisectors to ll eighbors. Show exmple bove. I three dimesios, defied by three lttice vectors 1, 2, d 3, which defies ifiite periodic rry of uits. Positio of y lttice poit c be give by three itegers, R = Prof. W. F. Scheider CBE Computtiol Chemistry 7/22

There re 14 Brvis lttices i three dimesios, combitio of 7 crystl

Cetered (F), d Bse Cetered (C) (ot ll 28 combitios re uique).

Our POSCARs will lwys fll ito oe of these 14 types.

90 Moocliic 2 Primitive b c α = β = 90 γ 90 3 Bse Cetered Orthorhombic

8 There re 14 Brvis lttices i three dimesios, combitio of 7 crystl systems d four types of ceterigs, Primitive (P), Body Cetered (I), Fce Cetered (F), d Bse Cetered (C) (ot ll 28 combitios re uique). (Imges courtesy Wikipedi). Our POSCARs will lwys fll ito oe of these 14 types. I order of roughly icresig symmetry: Tricliic 1 Primitive b c α, β, γ 90 Moocliic 2 Primitive b c α = β = 90 γ 90 3 Bse Cetered Orthorhombic 4 Primitive b c α = β = γ = 90 (I thik) 5 Bse Cetered 6 Body Cetered 7 Fce Cetered Tetrgol Prof. W. F. Scheider CBE Computtiol Chemistry 8/22

9 8 Primitive = b c α = β = γ = 90 9 Body Cetered Rhombohedrl/Trigol 10 Primitive = b = c α = β = γ 90 Tret s hexgol with bsis of d Hexgol 11 Primitive = b c α = β = 90 γ = 120 Cubic 12 Primitive = b = c α = β = γ = Body Cetered 14 Fce Cetered Ay of the cetered oes c be defied s covetiol with bsis lrger th 1. NEED TO MAKE POSCARS OF ALL OF THESE! 7. Qutittive supercell clcultios Wt to solve Koh-Shm equtio qutittively: where the potetil terms hve the periodicity of Brvis lttice, υ(r) = υ(r + R) for ll R = i idexes the vrious eergy levels. Prof. W. F. Scheider CBE Computtiol Chemistry 9/22

10 Coveiet to defie the reciprocl lttice, b, defied i geerl by b T = 2π1 (where colums of mtrix correspod to lttice vectors). I 1 dimesio, b = 2π/. I 3 dimesios, c be show to be b = 2π b = 2π b = 2π k = kxb + kyb + kzb ( 2 3) 1 ( 2 3) 1 ( 2 3) The Wiger-Seitz cell of the reciprocl lttice is the first Brilloui zoe. Reciprocl lttice importt for two resos: (1) Fourier trsforms: Ay fuctio (like the potetil, υ(r)) tht hs the periodicity of the Brvis lttice c be writte s Fourier series whose oly o-zero terms re elemets of the reciprocl lttice, G = m 1 b 1 + m 2 b 2 + m 3 b 3 : C tke dvtge of this fct to efficietly evlute mtrix elemets. (2) Bloch s theorem, rigorously: The eigesttes ψ i c be chose to hve the form of ple wves times fuctio with the periodicity of the Brvis lttice: k ( ) i k r k k k ψ r = e χ ( r), where χ ( r) = χ ( r+ R ) i i i i where the vectors k re withi the first Brilloui zoe. Koh-Shm equtio c the be writte: Hve to solve this for ll the orbitls i d for ll possible (ifiity!) vlues of k, get the bd structure d wvefuctios iside the periodic cell, d, of course, the totl eergy E. k Now we re bck to where we were i the previous lecture. C i priciple expd χ i i whtever bsis we wt, but bsed o (1) bove, prticulrly coveiet to describe usig plewves, tht is, to do Fourier expsio: 2π The c re our vritiol prmeters to solve for. I oe dimesio, b = : k 1 k ig 2 mx π m χ i ( x) = cime, Gm = mb= m Prof. W. F. Scheider CBE Computtiol Chemistry 10/22

11 H H H H H H H Atom- cetered: χ= χ 1s Ple wves: m=1, G=2π/. m=2, G=4π/ If we could let m rge from - to, we d hve exct expsio. C t do tht, of course, d i prctice the sum is tructed such tht 2 2 h 2 h 2 k + Gm = Gcut < E 2m 2m e e cutoff Agi, the cutoff cotrols the size of the bsis set. Bigger the bsolute vlue, higher the mximum frequecy d higher the kietic eergy of the bsis fuctio. Substitutig ito Bloch s theorem, k ikx ψ ( x) = e k c e m = k c e + m ig x i( k G ) x i im im m m i ( k Gm ) x For give vlue of k, hve to evlute mtrix elemets of Kietic eergy: Digol i bsis fuctios esy! e 2 i( k+ G ) 1 ( ) 1 m x d i k+ Gmʹ x e 2 2 dx e +. ( ) = k + Gm δ 2 Potetil eergy terms: Sice potetil terms hve the periodicity of the system, it is possible to Fourier trsform them: 1 ( ) ( ) ig m x igx υ x = υ Gm e υ( Gm) = υ( x) e dx m Trsforms require rel spce/reciprocl spce grids. Oce trsformed, though, evlutio is rpid: e υ( x) e = υ( G ) e e e = υ( G ) i ( k + G m) x i ( k + G m ʹ ) x i ( k + G m) x ig m x i ( k G m ) x m ʹ ʹ + ʹ ʹ ʹ mʹ m mʹ ʹ These mtrix elemets re compoets of FT of potetil, d thus c be evluted quickly with ple wves. Efficiet code depeds o beig ble to evlute some of the potetil terms i rel spce d some prts i reciprocl spce. Just eed to be ble to do fst discrete Fourier trsforms of the potetil. Trde-off is tht idividul ple wves re t very good bsis fuctios, so we ll eed lot of them. Ed up with lrge (# PW) (# PW) mtrix to digolize for the coefficiets. 2 mmʹ Prof. W. F. Scheider CBE Computtiol Chemistry 11/22

12 Hve to do this for ech k poit. So how do the k poits couple to ech other? Through the chrge desity, which ultimtely determies the potetils: k ρ(r) = occ i ψ k i (r) 2 k dk = f i ψ k i (r) 2 dk, i Efficiet evlutio requires Fourier trsforms here s well. C show tht the exct Fourier expsio hs to iclude terms up to 2G cut. Note itegrtio over ll k i the first Brilloui zoe. This itegrtio cot be doe lyticlly; hve to use umericl qudrture over k poits. Ad sice the coefficiets re ot kow hed of time, equtios hve to be solved itertively, SCF-like. i Would be ice to work this 1-D cse out more explicitly. itegrtio over it. Show step fuctio, error i Multiple dimesios follow the sme bsic ide. k is vector, ow, d hve to do everythig o 3-D grids. The bsis fuctios hve to fill the whole supercell, so there re bsis fuctios eve where you do t wt them! k 1 k igm r χ ( r) =, 1/2 cg e G= m1b1+ m2b2+ m3b3, m I m Ω m k 1 k i( G+ k) r ψ () r = ρ(r) c 1/2 = G occ e Ω k ψ k (r) 2 dk = f k ψ k (r) 2 dk 8. k-poit smplig The totl eergy of system give by G 1 f k =, Fermi fuctio 1+ e (ε k ε fermi )/k BT The sttes i re discrete (like orbitls), the ple-wves G re determied by the ple wve cutoff, but there re ifiite k poits to sum over. C t clculte every k vlue, so hve to replce itegrtio of first BZ with discrete smplig of k spce. Qulity of this smplig hs lot to do with qulity of the clcultio. Too few poits, poor results, too my, clcultio is too expesive. Hdled i Vsp i KPOINTS file. If bds re very flt (thik H chi well spced out) the sigle poit would be dequte. If dispersio is lrge the more re eeded. Geerlly eed to test. How to choose k poits to smple over? Most widely used pproch is tht of Mokhorst d Pck, which provides formul for choosig uiform sets of poits withi first BZ: Prof. W. F. Scheider CBE Computtiol Chemistry 12/22

13 You specify totl umber of k poits i ech directio, N i, d formul spreds them out. For exmple, i 1-dimesiol cse, sy N = 3, the k 1 = 1 (π/), k 2 = 0 (π/), k 3 = 1 (π/). This is gmm-cetered (cotis 0), d hs 2 symmetry-distict k. Prof. W. F. Scheider CBE Computtiol Chemistry 13/22

14 Prof. W. F. Scheider CBE Computtiol Chemistry 14/22

15 Rules of thumb: Should keep desity of k poits ~ costt i ech of three reciprocl lttice directios,. More totl k poits geerlly more precise results. Metllic systems require more th semicoductors or isultors. Number of k poits does ot ecessrily scle with N. Eve d odd grids, i prticulr, c scle differetly becuse of symmetry. Odd grids will lwys coti the Γ poit. Eve grids c be Γ-cetered, or shifted, to coti the Γ poit. Tured o usig Gmm rther th Mokhorst i KPOINTS file. Symmetry c be used to reduce the umber tht ctully eed to be evluted; get weighted by degeercy. Geerlly eed to test! Do t forget, k poit smplig is ecessry to ccout for bodig tht exteds cross cell boudries; o k poit smplig ecessry for toms or molecules! I Vsp, c either list k poits explicitly i KPOINTS file (Crtesi optio), or c specify usig Mokhorst-Pck descriptio. Prof. W. F. Scheider CBE Computtiol Chemistry 15/22

16 Prof. W. F. Scheider CBE Computtiol Chemistry 16/22

17 9. Fermi level smerig A chrcteristic of isultors (like MgO) is tht there is gp t the Fermi level: bds below the gp re filled for ll vlues of k, bds bove re empty. Itegrtig over k works esily. Metl re more tricky. By defiitio the Fermi level cuts through some bds (s i the 1-D H chi bove), d so occupcies chge discotiuously with k, ccordig to shrp Fermi fuctio. k Itegrtios over k hve to hdle these discotiuities. For istce, thik bout ϵ i cotributio to totl eergy, chges discotiuously t Fermi level for y bd tht crosses Fermi level. Possible to trsform this itegrl ito oe tht icludes delt fuctio (s proposed by Methfessel d Pxto, PRB 40, 3616): How to itegrte delt fuctio? Smer out the discotiuity. ISMEAR d SIGMA prmeters do this i INCAR. ISMEAR = 1: Fermi smerig, very simple, itroduces fictitious T (>> rel T) tht you c t get rid of ISMEAR = 0: Gussi smerig, chges delt ito Gussi of width SIGMA. Similr to itroducig fictitious T, but this oe you c extrpolte wy. ISMEAR > 0: Methfessel d Pxto itegrtio, like Gussi but expds delt fuctio i fuctios. ISMEAR = 4 or 5: Tetrhedro method, either without or with Blöchl correctios. Iterpoltes o k poits, either lierly or qudrticlly. Most ccurte, hve to hve eough k poits to do iterpoltio o. Sigm = smerig prmeter, i ev. Typiclly o order of 0.1 ev. Prof. W. F. Scheider CBE Computtiol Chemistry 17/22

18 Prof. W. F. Scheider CBE Computtiol Chemistry 18/22

19 Prof. W. F. Scheider CBE Computtiol Chemistry 19/22

20 Prof. W. F. Scheider CBE Computtiol Chemistry 20/22

21 Prof. W. F. Scheider CBE Computtiol Chemistry 21/22

22 10. Resources R. Hoffm, Solids d Surfces: A Chemist s View of Bodig i Exteded Structures, VCH: New York, Ashcroft d Mermi, Solid Stte Physics R. M. Mrti, Electroic Structure: Bsic Theory d Prcticl Methods Prof. W. F. Scheider CBE Computtiol Chemistry 22/22

Particle in a Box. and the state function is. In this case, the Hermitian operator. The b.c. restrict us to 0 x a. x A sin for 0 x a, and 0 otherwise

Particle in a Box. and the state function is. In this case, the Hermitian operator. The b.c. restrict us to 0 x a. x A sin for 0 x a, and 0 otherwise Prticle i Box We must hve me where = 1,,3 Solvig for E, π h E = = where = 1,,3, m 8m d the stte fuctio is x A si for 0 x, d 0 otherwise x ˆ d KE V. m dx I this cse, the Hermiti opertor 0iside the box The