Relativistic Pseudopotentials X.Cao and M. Dolg, in book Relativistic Methods for Chemists, edited by Barysz and Ishikawa, Springer UK, in publication

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1 Relatst Pseudopotentals X.Cao and M. Dolg, n book Relatst Methods for Chemsts, edted by Barysz and Ishkawa, Sprnger UK, n publaton 6.1 (Generalzed) Phllps-Klenman Equaton 6. Valene eletron model Hamltonan 6.3 Analytal form of PPs 6.4 Core-Polarzaton potentals 6.5 Core-ore/nuleus repulson orretons 6.6 Energy-onsstent PPs 6.7 Shape-onsstent PPs 6.8 Model potental method 6.9 DFT-Based effete ore potentals

2 Adantage of Valene-Eletrononly Methods Reduton of the omputatonal effort. Relatst effets an be nluded mpltly by means of a sutable parameterzaton. quasrelatst (one- or two- omponent) method All elements n the same group of the perod table an be treated on equal footng. hgher auray n studes of trends wthn a group.

3 The atual defnton of ore and alene orbtals/shells s ether possble on the bass of energet (orbtal energes) or spatal (shape, radal maxma or expetaton alues of orbtals) arguments. Frozen-ore approxmaton s underlyng all alene-only shemes,.e., the ore shells of a system are determned for one physal stuaton and then transferred to other physal stuatons.

4 Phllps-Klenman equaton (1959) wthn an effete one-eletron framework ( ˆ ˆ PK H V ) Vˆ PK + ϕ = ε ϕ a a ' a a p p ( ) = ε ε ϕ ϕ ϕ = ϕ ϕ ϕ a ϕ p p ' { } { } ϕ ϕ = δ a, a ', Hˆ ϕ = ε ϕ a, a a a /5/ Phllps-Klenman equaton 4

5 Adantage of Phllps-Klenman equaton By admxture of ore orbtals ϕ the radal nodes of the alene orbtal ϕ are elmnated and the shape of the ϕ p s as smooth as possble n the ore regon (pseudo-alene orbtal transformaton), then the bass sets used to represent the alene orbtal may be redued.

6 Generalzed Phllps-Klenman equaton (many-eletron system, Weeks and Re, JCP, 49(6):741, 1968) ( ˆ ˆGPK ) H + V Φ = E Φ ˆGPK V = Hˆ Pˆ PH ˆ ˆ + PH ˆ ˆ Pˆ + E Pˆ Pˆ p p ( Pˆ) = ϕ ϕ, Φ = 1 Φ Hˆ Φ = E Φ, Φ ϕ = 0 ϕ : a set of orthonormal funton, not neessarly the egenfunton of Hˆ. p /5/ Phllps-Klenman equaton 6

7 Models suggested by PK and GPK In prnple t s possble to fnd a potental whh, when added to a alene Hamltonan, allows the aratonal soluton of the orrespondng Shrödnger equaton wthout aratonal ollapse by usng a pseudo-alene waefunton wthout explt orthogonalty requrements to the waefunton desrbng the ore system /5/ Phllps-Klenman equaton 7

8 Models suggested by PK and GPK Frozen-ore approxmaton,.e., the ore shells of a system are determned for one physal stuaton and then transferred to other physal stuatons. ore-alene separaton: ore and orealene orrelaton effets are negleted /5/ Phllps-Klenman equaton 8

9 Valene eletron model Hamltonan for an atom ˆ GPK ˆ ˆ GPK, ˆ GPK ˆ GPK H = H + V V = V ( ) () ˆ 1 Q H = + gˆ, j + V ( ) ˆ () ' j > r gˆ, j : eletron-eletron nteraton Q : oreharge, Q = Z - n ˆ ' (): C ore-alene nteratons for the eletron. V 1 H g j ˆ pp ˆ = +, j > ( ) + Vˆ () ˆ Q V V V r () ˆ () ˆ () ˆ ' () ˆ G PK = + V, V = + ( ) () Vˆ : pseudopotental /5/ Valene-eletron model Hamltonan 9

10 Moleular Pseudopotentals ˆ 1 H = + gˆ, j + V + V () = + ˆ () ( ) ˆ ( ) j> ˆ Q V V r QQ µ µ V = + V r < µ rµ ( µ ) /5/ Valene-eletron model Hamltonan 10

11 Analytal form of non-relatst PPs (semloal PP, Kahn and Goddard 197,JCP, 56: ) L 1 ˆ V ( r) V r V r Pˆ ( ) + V r ( ( ) ( )) ( ) ν, aν l L l L l= 0 Pˆ () = lm () lm () l l l l l m = l /5/ Analytal form of PPs 11

12 Construton of V m (r ) (m=l, L) Kahn, Baybutt, and Truhlar 1976, JCP,65: a pseudo-alene orbtal ϕ p for eah l s wrtten as lnear ombnaton of the AE alene and ore orbtals (fntedfferene HF atom orbtals). Plug ϕ p n the radal Fok equaton together wth the orbtal energy ε and to sole the resultng expresson for V m a funtonal should be mnmzed wth respet to the expanson oeffents a. After V m were generated on a grd, they were ftted to the expanson n a least-square sense. n ( ) ( km ) V r = A r exp a r wth m = l, L n m km km k km =, 1,0 /5/ Analytal form of PPs 1

13 Three ondtons for ϕ p Kahn, Baybutt, and Truhlar 1976 JCP,65: (1) ϕ p should hae no radal node () ϕ p should be as lose as to ϕ (3) ϕ p should hae a mnmal number of spatal undulatons /5/ Analytal form of PPs 13

14 Relatst PPs ˆ ( ) ˆ V r V ( r ) Vˆ = + ( r ) ν, aν ν, so S alar-relatst P P : L 1 ˆ V ( r ) V r V r P ( ) + V r ( ( ) ( )) ˆ ( ) ν, aν l L l L l = 0 Pˆ ( ) = lm ( ) lm ( ) S pn-orbt P P : l l l l l m = l ( r ) ˆ V V r = lpˆ l + Pˆ L 1 l ν, so ( ) l, l + 1/ ( 1) l, l 1/ l = 1 l + 1 ( r ) ˆ V ˆ V r = P l sˆ P L 1 l ν, so ( ) l l l = 1 l + 1 () () () ˆ () WC Ermler, YS Lee, PA Chrstansen, KS Ptzer, CPL, 1981, 81:70-74 RM Ptzer, NW Wnter, JPC, 1988, 9: /5/ Analytal form of relatst PPs 14

15 CPP: Why? Core-alene orrelaton (dynam orepolarzaton) negleted Leadng ontrbuton n an AE CI treatment would be sngle extatons from the ore orbtals oupled to sngle and hgher extatons from the ouped alene orbtals to the rtual ones. Frozen ore approxmaton (stat orepolarzaton mssng) the ndued error may beome large for LPP and only a few alene eletrons. /5/ CPP 15

16 V α f pp 1 = α f denotes the dpole polarzablty of the ore eletr feld at ore generated by all other ores, nule and alene eletrons. Problem: Only apply to large dstane of the polarzng harge(s) from the polarzed ore(s) and een derges n the lmt anshng dstanes. Meyer et. al (J.C.P 80,397(1984)) hae suggested a ut-off funton F remong the sngulartes, the feld at ore then reads as: r r µ f = F 3 ( r, δe ) + Qµ F 3 ( rµ, δ ) r r F : Cutoff funton µ µ n e ( ) ( 1 exp δ ) (, δe ) = 1 exp( δe ) F r r ( µ, δ ) = ( µ ) F r r n /5/ CPP 16

17 Dffultes exstng for CPP: One- and two-partle ontrbutons arsng from the alene eletrons as well as the ores/nule omplex of ntegral ealuaton oer Cartesan Gaussan funtons, energy gradents for geometry optmzaton are stll mssng. /5/ CPP 17

18 To orret pont harge repulson model (Born-Mayer-typ e ansatz): V ( r ) = B exp( b r B µ and b an be obtaned dretly by fttng to µ µ µ µ µ µ the dfferene between the eletrostat potental of the atom ore eletron system modelled by the ECP and the Coulomb potental due to the ECP ore harge, multpled wth the harge of the approahng nuleus. ) G. Igel, U. Wedg, M. Dolg, P. Fuentealba, H. Preuss, H. Stoll, R. Frey, JCP, 1984, 81: /5/ Core-ore/nuleus repulson orreton 18

19 Gaussan expanson of radal parts: V V ( ) n ( exp ) ljk r = Aljkr aljkr lj k ( ) n ( exp ) lk r = Alkr alkr l k ( ) n ( exp ) lk r Alkr alkr V = l k 'Stuttgart pseudopotentals' : typally V = 0 and n =n = 0 L ljk lk are hosen. /5/ Energy-onsstent PPs 19

20 Energy adjustment: ( ) PP AE EI E I + : = mn I I : A multtude of eletron onfguratons/states/leels of the neutral atom and the low-harged ons. E PP I : Total alene energy obtaned from fnte-dfferene alene-only alulatons. AE E I : All-eletron referene data from Wood-Borng quasrelatst HF approah, or n the most reent erson fnte-dfferene AE MCDHF alulatons based on the DC o r DCB Hamltonan. /5/ Energy-onsstent PPs 0

21

22 Methods of adjustment of ab nto pseudopotentals: Orbtal adjustment: shape-onsstent Referene data: all-eletron alene orbtals and orbtal energes (ndependent partle model) Ptzer, Chrstansen, ; Durand, Barthelat, ; Hay, Wadt; Steens, ) Energy adjustment: energy-onsstent Referene data: all-eletron total alene energes (quantum mehanal obserables; ndependent partle model and beyond) Stoll, Dolg, Shwerdtfeger, Adantage: ndependent of the qualty of the waefunton (SCF, MCSCF, CI, CC), e.g., adjustment n an ntermedate ouplng sheme possble! Dsadantage: relately hgh omputatonal effort; problems wth neutral or negate harged ores. /5/ Shape-onsstent PPs

23 Requrement for ϕ p n the shapeonsstent PP ϕ ϕ f plj, lj, lj ( r) ( r) ( r) ( ) ( ) ϕlj, r for r r = flj r for r < r s AE alene orbtal s radally nodeless and smooth n the ore regon. The pseudopotental ˆ 1 d dr V PP lj fufll the below radal Fok equaton: l( l + 1) PP ˆ + V lj + W plj, ϕ p', l' j' ϕplj, r = εlj, ϕplj, r r + { } ( ) ( ) ( ) Q = + PP nlj, k αlj, kr V r A r e P lj, k lj r j k ˆ /5/ Shape-onsstent PPs 3

24 Hay and Wadt: aalable for man group and transton elements based on salar-relatst Cowan-Grffn AE alulatons JCP,1985(8):99-310, 70-8, ϕ remans normalzed. F l (r) and ts frst 3 derates math ϕ and ts frst 3 derates at r. ϕ p, l C r l α e r b 3 4 ( ) ( ) f r = r a + a r + a r + a r + a r for r < r lj = b = l + 3 n the non-relatst ase b= + for relatst ase ( 0, l ) + l l l,0 + ( Z ) = 1 δ ( 1) (1 δ ) α 4 For relatst s orbtals the hoe b = + a s 6 th degree polynomal. 3 and f /5/ Shape-onsstent PPs 4

25 A useful rteron for more ompat bass sets JC Barthelat, P Durand, A Serafn, Mol. Phys. 33, (1977) The mnmzaton of the followng operator norm: 1/ ˆ ϕ ˆ plj, Ο ϕplj, wth Ο = Ο= ˆ ε ϕ ϕ ε ϕ ϕ ϕ lj, plj, plj, lj, plj, plj,, ε obtaned wth the exat V tabulated on a grd from pp p, lj, lj lj the radal Fok equaton. pp,, p lj, lj obtaned wth the analytal potental Vlj ϕ ε Aalable for almost all elements, as well as for heaer atoms based on DHF AE alulatons applyng the DC Hamltonan. WJ Steens et al, Can. J. Chem. 199, 70:61-630, JCP, 98: , JCP, 81, (1984) /5/ Shape-onsstent PPs 5

26 Generalzed relatst ECP Tto, Mosyagn Possble problem exstng n shape-onsstent PPs: 1 d l( l + 1) PP ˆ + + V lj + W plj, p', l' j' plj, r = lj, plj, r dr r { ϕ } ϕ ( ) ε ϕ ( ) PP 1 1 d l( l + 1), ( ) ˆ Vlj = ϕ plj r ε lj, + W plj, { ϕ p', l' j' } ϕ plj, ( r) dr r In the ase of pseudo-alene orbtals wth nodes, sngularty appear n ( r ) 1 the PPs due to the term ϕ plj,. Soluton: (1) Most of shape-onsstent PPs are dered for poste ons (FC errors may be large). ( ) Interpolatng the potentals n the nty of the nodes (GRECP) addtonal nonloal terms added besdes the standard sem-loal form. Physs of atom nule, 003, 66(6): more parameters, not supported by most of the standard quantum hemstry odes. /5/ Shape-onsstent PPs 6

27 Huznaga-Cantu equaton S. Huznaga,AA Cantu, JCP, 1971,55: ; S. Huznaga,D MWllams, AA Cantu, Ad. Quantum. Chem. 7, 187-0, 1973 ˆ F + ( ε ) ϕ ϕ ϕ = ε ϕ Fˆ ϕ = ε ϕ a, a a a a a ' aa ' { } { } ϕ ϕ = δ a, a ', Comparng to AE HF equaton: bass sets do not need to represent ore orbtals ==> smaller bass sets than AE! Comparng to Phllps-Klenman equaton: ϕ ϕ p here keeps ts orret nodal struture n Phllps-Klenman does not neessarly hae radal nodes ==> larger bass sets than PPs! /5/ Model potental 7

28 Atom alene-eletron model Hamltonan 1 Q F J K V V () () ˆ = + ˆ ˆ ˆ ˆ + C + X r ˆ n V ˆ ˆ ˆ C = + J VX = K r () (), () () () ˆ () Jˆ and K stand for the usual Coulomb and exhange operators related to the ore orbtal ϕ, Insertng nto the HC equaton 1 ˆ ˆ Q ˆ + J K + VMP ϕ = ε ϕ r Vˆ = Vˆ + Vˆ + Pˆ MP C X ( ) Pˆ = ε ϕ ϕ 1 ˆ ˆ Q ˆ MP J K r /5/ Model potental 8 () () () ˆ MP H = + + V

29 Moleular alene-eletron model Hamltonan A moleular MP s onsdered to ontan an assembly of non-oerlappng ore leels: ˆ 1 QQ MP Q H ˆ = + g, j VMP, j> < µ rµ r Vˆ Vˆ Vˆ Pˆ () = () + () + () MP, C, X, AIMP (Huznaga, Sejo, Barandaran): 1 V r = C e C = n ( ) ˆ α k r, C, k k r k k Vˆ r A ( ) χ ( ) χ () X, p pq q pq, SO operator (ft to the WB SO term) : ˆ B lk β ˆ ˆ lk ˆ V, so = e P l l spl l k r DKH type AIMP also aalable. µ ( ) ˆ () () ˆ () /5/ Model potental 9

30 DFT-based MP ombned wth LSD-VWN, Andzelm, Radzo, Salahub, JCP, 83, Startng from the Kohn-Sham(KS) equatons for a spn-polarzed system of n alene eletrons, and assumng orthogonalty between σ σ ϕ and ϕ of spn σ ( σ =+, ), Huznaga-Cantu equaton may be rewrtten as: ˆ σ σ σ σ σ σ σ F + ( ε ) ϕ ϕ ϕ = ε ϕ ˆ σ ˆ σ ˆ σ, MP F = F + V ˆ 1 Q ρ ( r ') dr ' F = + + ρ ρ ρ σ x +,,, r r r ' ρ +,,, orbtals, ρ = ρ + ρ +,, denote the spn-up ans spn-down denstes for the alene /5/ DFT-based ECPs 30

31 n ( ), s an oupaton number ( ) σ σ σ σ, r = f ϕ r f ˆ σ,m P ˆ σ, MP V Pˆ Pˆ σ σ σ = + = ( ε ) ϕ ϕ ρ V ˆ σ,m P The moleular V s wrtten as a sum oer the atom MPs: ( MP V P ) ˆ σ,m P ˆ σ,mp ˆ σ, = V ˆ = + V ˆ σ, MP n V = + r ρ ( r ') Vˆ dr ' + x +,,, σ, MP The analytal form of may be wrtten as: e V = A A = n α k r ˆ σ, MP k, k k r k r r ' The ore orbtals for the projeton operator are approxmated by a least square ft proedure usng an expanson of Gaussan funtons. The referene atom orbtals were obtaned form CG/WB-type /5/ DFT-based ECPs 31 ρ LSD-VWN fnte-dfferene atom alulatons. ρ

32 Norm-onserng DFT-based shape-onsstent PPs H through Pu, Hamann, Shlueter, and Chang, Phys. Re Lett. 1979, 43: Comparng to ab nto shape-onsstent PPs: ϕ p,lj s radal KS orbtals at the orgnal AE orbtal energes ε,lj Addtonal norm-onserng propertes of ϕ p,lj : 1. The ntegral from 0 to r of the real and pseudo harge denstes agree for r>r for eah alene state.. The logarthm derates of the real and pseudo waefunton and ther frst energy derates agree for r>r /5/ DFT-based ECPs 3

33 Densty funtonal sem-ore PPs from H to Am DSPP, Delley, Phys. Re. B 66, ,00 Suggested for use wth loal orbtal methods. Based on a mnmzaton of errors wth the norm onseraton ondtons for two to three releent on onfguratons of the atom. AE referene were defned usng PBE funtonal. /5/ DFT-based ECPs 33