DFT in practice : Part II. Ersen Mete

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1 pseudopotentials Department of Physics Balıkesir University, Balıkesir - Turkey August 13, NanoDFT 09, İzmir Institute of Technology, İzmir

2 Outline Pseudopotentials Basic Ideas Norm-conserving pseudopotentials Nonlocal separable Kleinman-Bylander form Ultrasoft pseudopotentials Projector Augmented Waves (PAW) method

3 A one-electron atom For a one-electron atom, the attractive potential is spherically symmetric, V ( r) = V (r) = Z r then, the solutions are separable to radial and angular parts, ψ nlm ( r) = R nl (r)yl m (θ, ϕ) = φ nl(r) Yl m (θ, ϕ) r The radial equation becomes, 1 d 2 [ l(l + 1) 2 dr 2 φ nl + 2r 2 ] + V eff (r) φ nl = εφ nl Kohn-Sham single particle Schrödinger-like equations will be identical if V KS ( r) is spherically symmetric as one-electron Coulomb potential V (r).

4 V KS ( r) = V KS (r)? Pseudopotentials density : n( r) = occ n,l,m occ ψ nlm ( r) 2 = (2l + 1) R n,l (r) 2 = n(r) external potential : υ ext ( r) = Z/r = υ ext (r) n( r ) Hartree potential : r r d r = V H (r) exchange-corr. pot. :V xc ( r) = ɛ xc [n(r)] + n(r) dɛ xc dn [n(r)] =V xc(r) Therefore, total effective potential n,l V KS = υ ext (r) + V H (r) + V xc (r) spherically symmetric. The independent-particle Kohn-Sham equations are analogous to the Schrödinger equation of the one-electron atom, 1 d 2 [ ] l(l + 1) 2 dr 2 φ nl + 2r 2 + V KS (r) φ nl = εφ nl DFT for an atom

5 Why do we need pseudopotentials? DFT calculation with all-electron υ ext is expensive & Core electrons are essentially inert in bonding environments. Pseudopotentials, replace the effect of the core electrons, are smooth in the core region, reproduces all-electron potential behavior out of the core region. Computationally, Sharp oscillations near the core region will be smoothed : Reduction of the number of plane waves The number of electrons will be decreased : Reduction of the number of bands to solve for

6 Pseudopotentials Schematically

7 Pseudopotential terminology transferability : ability to describe the valence electrons in different environments. softness : the need for the number of plane waves. inclusion of semicore states Example : Ti with large core 1s 2 2s 2 2p 6 3s 2 3p 6 4s }{{} 2 3d 2 core Ti with semicore 1s 2 2s 2 2p 6 3s }{{} 2 3p 6 4s 2 3d 2 core locality : all l-channel (s, p, d) electrons feel the same potential efficiency a compromise between accuracy and computational cost

8 Accuracy vs computational load Local PSP V PS = V PS (r) Semilocal PSP V PS = l V PS l (r) χ l χ l Nonlocal separable PSP V PS = Vloc PS (r) + D l β lm β lm lm

9 Total energy in terms of valence electron density Then, N core E[{ψ i }] = ψ i ψ i + i n( r) = n core ( r) + n val ( r) υ ext ( r)n core ( r)d r + 1 ncore ( r)n core ( r ) 2 r r d rd r N val + ψ i ψ i + υ ext ( r)n val ( r)d r + 1 nval ( r)n val ( r ) 2 r r d rd r i ncore ( r)n val ( r ) + r r d rd r + E xc [n core + n val ] N val E val [{ψ i }] = ψ i - 1 ( 2 2 ψ i + - Z r + ncore ( r ) ) r r n val ( r)d r+e xc [n core + n val ] }{{} i }{{} Non linear XC Vion SCR corrections

10 Generic pseudopotential transformation For an atom, assume that the core states, χ n, satisfy H χ n = E n χ n A single valence state, ψ, can be replaced by a smoother pseudofunction, φ, expanding the remaining part in terms of χ n, core ψ = φ + a n χ n Using the orhtogonality of valence and core states, core χ m ψ = χ m φ + a n χ m χ n = 0 = a m = χ m φ n n ψ = φ n χ n φ χ n

11 Then, the eigenvalue equation, ( H φ core ) χ n φ χ n n ( = E φ core H φ + (E E n ) χ n χ n φ = E φ This implies, n (H + V nl ) φ = E φ where l is due to spherical symmetry. E E n > 0 extra potential V nl is repulsive. Cancels the effect of the attractive Coulomb potential. core ) χ n φ χ n Resulting potential is weaker and pseudo eigenstate is smoother. n

12 Norm-conserving pseudopotentials Hamann, Schlüter and Chiang [Phys.Rev.Lett.43,1494(1979)] criteria : AE & PS wavefunctions correspond to the same energy for the reference level (AE valence level with angular momentum l), H φ AE nl = ε nl φ AE nl (H + V nl ) φ PS nl = ε nl φ PS nl AE & PS wavefunctions match beyond a certain radial cutoff, r c, φ AE nl (r) = φ PS nl (r) r r c

13 AE and PS norm squares integrated upto r r c are equal. r 0 φ AE nl (r ) 2 dr = r 0 φ PS nl (r ) 2 dr Equal amount of charge in the core region. Gauss Law is satisfied for r. Normalization constraint is achieved in the limit r. logarithmic derivatives (their respective potentials) agree for r r c. d dr ln[φps nl (r)] d dr ln[φae nl (r)] V PS must reproduce the same scattering phase shifts as V AE for r. Necessary to improve transferability. PS wavefunction is nodeless. It s twice differentiable and satisfies lim r 0 φ nl (r) r l+1 = continuous.

14 Silicon : wave functions

15 Norm-conserving PSP generation steps 1 Solve the all-electron atomic system. 2 Determine the core and valence states. 3 Apply norm-conservation criteria (e.g. Hamann scheme) and derive a PS wavefunction from the reference AE valence level with angular momentum l. 4 Invert the Schrödinger equation for PS wavefunctions to get screened PSP components. V SCR,PS l = ε PS 2 l(l + 1) 1 d l 2r 2 + 2φ PS nl (r) dr 2 φps l 5 Subtract the Hartree and XC contributions to obtain υ ext (unscreened PSP). V PS l = V SCR,PS l V H (r) V XC (r) Model pseuodopotential replaces the potential of the nucleus. (r)

16 Semilocal pseudopotentials For l-dependent model PSP, treat angular momentum l separately (nonlocal). PSP in the semilocal form : Vnl PS = V loc (r) + V SL where V SL = Y lm V nl (r) Y lm lm It s local in r and nonlocal in θ, ϕ. Drawback : Plane wave representation of the non-local part is expensive.

17 Semilocal PSP matrix elements in plane waves q V SL q = 1 Ω l,m where r = (r, θ, ϕ ) e i q r Y lm(θ, ϕ)v nl (r)y lm (θ, ϕ)e i q r r 2 dωdω dr e i q r = 4π lm i l j l (qr)y lm(ˆq)y lm (ˆr) where j l (qr) are spherical Bessel functions and ˆq and ˆr denote angles associated with the vectors q and r, respectively. m Y lm (ˆq)Y lm(ˆq ) = 2l + 1 4π P l(cos θ q q ) q V SL q = 4π Ω (2l + 1) l j l (qr)j l (q r)p l (cos θ q q )V l (r)r 2 dr N 2 PW such integrations needed!

18 Fully separable Kleinman-Bylander form where V PS nl ( r, r ) = V loc (r)δ( r r ) + V NL V NL = l V NL l = lm V SL l Action of this form on the PS wavefunction, Vl NL φ PS lm = V SL l φ PS lm φps lm V l SL φ PS lm V l φ PS lm φ PS lm φps lm V l SL φ PS lm V l φ PS lm φ SL lm = Vl SL φ PS lm Planewave representation of non-local PSP matrix elements in Kleinman-Bylander form q V NL q = lm q V SL l φ PS lm φ PS lm V SL l V SL l φ PS φ PS lm lm q Number of integrals to be evaluated reduces to N PW. [PRL 48,1425 (1982)]

19 Silicon & Titanium NCPPs

20 Ti log derivatives Pseudopotentials

21 Ultrasoft pseudopotentials (USPP) : formalism Vanderbilt [PRB 41, 7892 (1990)] proposed a new method by relaxing norm-conservation constraint, φ PS i φ PS i ψi AE ψi AE USPPs are norm-conserving in a generalized form φ PS i (1 + ˆN NL ) φ PS i = ψi AE ψi AE where ˆN is nonlocal charge augmentation operator. USPPs require much smaller PW cutoff (less N PW ) ultrasoft. Scattering properties remain to be correct transferable.

22 The aim is to minimize the total energy, E e = φ i ˆV NL φ i + d rv loc ( r)n( r)+ 1 2 i d rd r n( r)n( r ) r r +E xc[n] subject to where φ i (1 + ˆN NL ) φ j = φ i ŜNL φ j = δ ij n( r) = i ( ) φ i r r + ˆK NL ( r) φ i and for consistency ˆN NL = d r ˆK NL ( r) so that n( r)d r = N v Then, the eigenvalue equation, (T + Vloc PS PS + ˆV NL ) φ i = ε n (1 + ˆN NL ) φ i

23 Ultrasoft PSP generation steps Screened V AE is obtained through self-consistent solution of atomic Kohn-Sham system. Cutoff radii are chosen : r cl for the wave functions r loc c for the local PSP R large enough that all PS and AE quantities agree. A smooth local potential, V loc ( r) is generated which approaches V AE ( r) beyond r loc c.

24 Then, for each angular momentum channel, a few reference energy values, ε i, are chosen where i = {τlm} and τ is the number of reference energies. (T + V AE ε i ) ψ i = 0 ψ i not determined self-consistently. So, ψ i ψ j R = R ψ i ( r)ψ j( r)d r New orbitals defined, χ i = (ε i T V loc ) φ i which vanish at and beyond R where V loc = V AE and φ i = ψ i. To define a nonlocal PSP, χ i are used as projectors to define new wave functions, β i = j χ i χ j φ i = j (B 1 ) ji χ j

25 To compensate valence charge deficit, generalized augmentation charges needed, q ij = ψ i ψ j R φ i φ j R The nonlocal overlap operator can be defined as, S = 1 + i,j q ij β i β j Then the nonlocal potential operator is V NL = ij D ij β i β j where D ij = B ij + ε j q ij PS wave functions satisfy generalized orthonormality condition, φ i S φ j R = φ i φ j R + q ij = ψ i ψ j R = δ ij Then, PS wave functions satisfy generalized eigenvalue problem, (H ε i S) φ i = 0 where H = T + V loc + V NL

26 Verification of generalized eigenvalue problem ( T + V loc + ) ( D nm β n β m φ i = ε i 1 + ) q nm β n β m φ i nm nm where D nm = B nm + ε m q nm Since, ( ) B nm β n β m φ i nm = φ n χ m nm k χ k χ k φ n l φ m χ l χ l φ i = km χ k n χ k φ n φ n χ m δ im = km χ k δ km δ im = χ i Substitution yields, H φ i = ε i S φ i

27 Valence electron density The electron density is augmented, n v ( r) = [ φ n ( r) 2 + n i,j ] Q ij ( r) φ n β i β j φ n where Q ij ( r) = ψ i ( r)ψ j ( r) φ i ( r)φ j ( r) so that it must integrate to the correct number of valence electrons, n v ( r)d r = φ n ( r) 2 d r + Q ij ( r) φ n β i β j φ n d r n ij = nm δ nm [ φ n φ m + ij ] q ij φ n β i β j φ m = nm δ nm φ n S φ m = nm δ nm δ nm = n δ nn = N valence

28 Minimization of total energy δn( r ) δφ n( r) = φ n( r)δ( r r ) + ij Q ij ( r )β i ( r) β j φ n Then, the modified Kohn-Sham equations, δe e δφ n = = d r δe e δn( r ) δn( r ) δφ n( r) [ V eff + ij ( D (0) ij + V eff ( r )Q ij ( r )d r ) ] β i β j φ n where V eff = V H + V loc + V xc The coefficients in the non-local part of the PSP gets updated self-consistently.

29 Ionic USPP Pseudopotentials Ionic potential are obtained by unscreening, V ion loc = V loc V H V xc D (0) ij = D ij d r V loc ( r )n( r )

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31 Projector Augmented Waves (PAW) method : basic idea For a particular reference energy, P.E. Blöchl, PRB (1994) the behavior of an arbitrary PS wavefunction ψ PS at the atomic site can be calculated by projection at that site in terms of partial waves (spherical), c lm =< P lm ψ PS In each sphere, ψ PS = lm c lm φ PS lm and ψ AE = c lm φ AE lm lm where φ lm are partial waves. Projectors must be dual to partial waves, P lm φ PS l m = δ ll δ mm ψps = i φ i p m ψ PS = ψ PS

32 Then, the PS transformation is given by, ψn AE = ψn PS φ PS lmε p lmε ψn PS lmε + lmε φ AE lmε p lmε ψ PS n Transformation involves AE wavefunction One can derive AE results.

33 References Pseudopotentials E. Kaxiras, Atomic and electronic structure of solids, Cambridge University Press, Cambridge, R.M. Martin, Electronic Structure : Basic Theory and Methods, Cambridge University Press, Cambridge, D.R.Hamann, M. Schlüter, C. Chiang, Phys. Rev. Lett. 43, 1494 (1990). G.B. Bachelet and M. Schlüter, Phys. Rev. B 25, 2103 (1982). L. Kleinman and D.M. Bylander, Phys. Rev. Lett. 48, 1425 (1982). G.B. Bachelet, D.R.Hamann, M. Schlüter, Phys. Rev. B 26, 4199 (1982). A.M. Rappe, K.M. Rabe, E. Kaxiras, and J.D. Joannopoulos, Phys. Rev. B 41, 1227 (1990). N. Troullier and J.L. Martins, Phys. Rev. B 43, 1993 (1991). D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). P.E. Blöchl, Phys. Rev. B 50, (1994).