Lecture VIII : The pseudopotential

Transcription

1 Lecture VIII : The pseudopotentia I. KOHN-SHAM PROBLEM FOR AN ISOLATED ATOM For a one-eectron atom, the Couombic potentia, V ( r) = V (r) = Z/r is sphericay symmetric. The soutions may then be spit into a radia and an anguar part ψ nm ( r) = ψ n (r)y m (θ, φ) = r 1 φ n (r)y m (θ, φ) (1) The ast equaity in Eq. 1 is given in order to simpify the sphericay-symmetric Schrödinger equation and Y m (θ, φ) are normaized spherica harmonics. The sphericay-symmetric Schrödinger equation for principe quantum number n can be reduced to the radia equation 1 d 2 [ ] ( + 1) 2 dr 2 φ n + 2r 2 + V ext (r) ǫ n, φ n, = 0. (2) In the Kohn-Sham approach to the many-partice system, the form of the singe-partice equations are identica to the above radia Schrödinger equation, with an effective potentia, V eff repacing the pure Couomb potentia. The substitution can be done directy provided V eff is aso sphericay symmetric. Let s see if that is the case : Density : n( r) = occ n,,m Externa potentia : Ionic Couomb potentia occ ψ n,,m (r) 2 = (2 + 1) ψ n, (r) 2 = n(r) (3) n, V ext ( r) = Z/r = V ext (r) (4) The Hartree potentia V H ( r) = = n( r ) r r d r n(r ) [(r r cosθ sin φ) 2 + (r cosθ sinθ) 2 + (r sin θ) 2 ] 1/2 r 2 dr d(cos θ)dφ = V H (r) (5) The exchange-correation potentia V xc ( r) = ǫ xc [n(r)] + n(r) dǫ xc dn [n(r)] = V xc(r) (6) Tota effective potentia V eff (r) = V ext (r) + V H (r) + V xc (r) (7) Thus the independent-partice Kohn-Sham states may be cassified by the anguar quantum numbers L =, m. The one-partice equations are then anaogous to the Schrödinger equation for the one-eectron atom, 1 d 2 [ ] ( + 1) 2 dr 2 φ n + 2r 2 + V eff (r) ǫ n, φ n, = 0. (8)

2 2 II. THE PSEUDOPOTENTIAL IDEA The concept of a pseudopotentia is reated to repacing the effects of the core eectrons with an effective potentia. The pseudopotentia generation procedure starts with the soution of the atomic probem using the Kohn-Sham approach. Once the KS orbitas are obtained, we make an arbitrary distinction between vaence and core states. The core states are assumed to change very itte due to changes in the environment so their effect is repaced by a mode potentia derived in the atomic configuration and it is assumed to be transferabe. The vaence states are seen to osciate rapidy cose to the core regions. With the introduction of the new potentia, the vaence states are made smoother. Let s now work out the pseudopotentia transformation in its most genera framework : Assume that we have a Hamitonian Ĥ, core states { χ n } and core eigenvaues {E n }. We are ooking at a singe vaence state ψ Let s repace the vaence state by the smoother φ and expand the remaining portion in terms of core states : core ψ = φ + a n χ n (9) Next, we take the inner product of Eq. 9 with one of the core states. Because the vaence state has to be orthogona to the core states, we have n core χ n ψ = χ n φ + a n χ m χ n = 0 (10) n } {{ } a m We now write the right-hand side of Eq. 10 in terms of the pseudofunction, φ ψ = φ n χ n φ χ n (11) Appying the Hamitonian onto the expression in Eq. 11, we get core Ĥ φ + (E E n ) χ n χ n φ = E φ (12) n As a resut, the smooth, pseudowavefunction satisfies an effective equation with the same eigenenergy of the rea vaence wavefunction. In the case of isoated atoms, the indices n corresponds to the combined index nm incuding the principa quantum number n, the anguar momentum quantum number and the magnetic quantum number m. However, since the potentia does not incude any terms (such as L S couping) that ift the m degenaracy, we incude in what foows ony n and. The reason why the -degenaracy is broken instead wi become cear ater on. Eq. 12 may then be written as an eigenvaue equation for the smooth pseudo wavefunction as (Ĥ + ˆVn ) φ = E φ (13) where the extra potentia ˆV n depends on due to spherica symmetry and its effect is ocaized to the core. Since E > E n by definition, it is a repusive potentia and it partiay cances the effect of the attractive Couombic potentia. The resuting potentia is then a much weaker one than the origina potentia. This shows that the eigenstates of this new potentia are smoother. III. NORM-CONSERVING PSEUDOPOTENTIALS In order to ensure the optimum smoothness and transferabiity Hamann, Schüter and Chiang (Phys Rev Lett (1979)) came up with four criteria that the pseudopotentias are expected to obey in the reference configuration (i.e. the atomic configuration for which the pseudopotentia is generated.) : 1. A-eectron and pseudo eigenvaues agree for the reference configuration. Ĥ ψn AE = ǫ n ψn AE (14) ) (Ĥ + ˆVn ψn PS = ǫ n ψn PS (15) (16)

3 3 2. AE and PS wavefunctions agree beyond a certain cutoff, r c. 3. Rea and pseudo norm squares integrated from 0 to R for a R < r c agree. ψ AE n (r) = ψ PS n (r) for r r c (17) R 0 φ AE n 2 r 2 dr = R 0 φ PS n 2 r 2 dr (18) 4. The ogarithmic derivatives and the energy derivative of the ogarithmic derivative agree for a R < r c. [ (rφ AE (r)) 2 d ] [ d de dr nφae (r) = (rφ PS (r)) 2 d ] d de dr nφps (r) R R (19) Items 1 and 2 are obvious. Item 3 is necessary so that by Gauss theorem, outside the core region where r < r c, the eectrostatic potentia is identica for the rea and pseudo wavefunctions. Finay, 4 is necessary in order to improve transferabiity, reproducing phase shifts of the rea potentia in the scattering sense. When we go from an atom to a different environment such as a soid or a moecue, the eigenvaues change. A pseudopotentia satisfying this item reproduces these changes to inear order. Using the Friede sum rue, one can show that item 4 is impied by item 3. In order to generate a norm-conserving pseudopotentia, we must foow the steps beow : 1. Sove the a-eectron atomic system. 2. Decide which of the states are to be taken as core and which are to be taken as vaence. 3. Generate the pseudopotentias, V n (r), for the vaence states. There are severa methods for doing that. We describe here the method due to Kerker. In this method, we assume a parametrized anaytica function for the core region. The anaytic form chosen by Kerker is φ PS n (r) = ep(r), where p(r) is a fourth-order poynomia matching the true soution and its various derivatives at r c. 4. Once we have a form for the pseudowavefunction, invert the Schro dinger equation, yieding [ d ( + 1) 2 dr φ PS V n,tota (r) = ǫ 2r 2 2 n (r) ] φ PS n (r) (20) Keep in mind that we are deaing with φ(r), which is defined by ψ(r) = rφ(r). 5. The potentia V,tota (r) found by inversion of the Schrödinger equation is not ony the ionic potentia that we want, but incudes aso contributions from the Hartree and exchange-correation terms. We must therefore subtract these two contributions. V n (r) = V,tota (r) VHxc PS (r) (21) where the Hartree and xc contributions are computed for the pseudowavefunctions. This operation is caed unscreening. There is no unique way to determine the V n s. There are two opposing considerations : 1. Good transferabiity sma r c. 2. Large r c smoother pseudopotentias. A good pseudopotentia is one that strikes a baance between these two contraints. Athough the generation procedure is simiar, the form of the pseudopotentia may be assumed differenty depending on considerations such as ease of computation, accuracy etc. In the foowing subsections, we sha see two different ways of forming norm-conserving pseudopotentias. Keep in mind that these pseudopotentias are formed within the atomic configuration and then exported to the soid with the hope of decent transferabiity. An actua exampe of the construction of the pseudopotentia can be seen in Fig. 1 for the radia parts of 2S and 2P orbitas of the eement B. The agreement of the wavefunctions after the cutoff radius r c is evident.

4 rea 2S for B pseudo 2S for B rc= rea 2P for B pseudo 2P for B rc= FIG. 1: Semiog pot of the rea and pseudo wavefunctions for 2S (a) and 2P (b) orbitas of B. A. Semioca pseudopotentias Since a mode pseudopotentia repaces the potentia of a nuceus, it is sphericay symmetric and each anguar momentum may be treated separatey. This eads to the nonoca, i.e. -dependent mode pseudopotentias V n (r). Thus, the genera form of a pseudopotentia may be written in the semioca form as ˆV SL = m Y m V n (r) Y m. (22) Here the anguar and the radia parts are separated due to spherica symmetry and the Y m s are there because any function that depends upon anges ony may be written as an expansion. This form is caed semioca because it is nonoca ony in the anguar coordinates and it s oca in the radia coordinate. Athough conceptuay simpe, semioca pseudopotentias create a severe probem in the panewave basis representation due to the form of the nonoca part. This can be seen by considering a particuar matrix eement of the pseudopotentia and using a ot of straighforward but tedious agebra q ˆV SL q = 1 e i q r Y Ω m(θ, φ)v n (r)y m (θ, φ)e i q r r 2 dω dω dr (23) m where r = (r, θ, φ ). Notice the fact that we cacuate a singe integra over the radia component r but two integras over the anges because the nonoca part of the pseudopotentia is in the form of projectors instead of a reguar function as for the radia coordinate. In order to make use of their orthonormaity, we expand the panewaves in the integra in Eq. 23 in terms of spherica harmonics e i q r = 4π m i j (qr)y m (ˆq)Y m(ˆr) (24) where j (qr) are spherica Besse functions and ˆq and ˆr denote the anges associated with the vectors q and r. Substituting Eq. 24 into Eq. 23 we have q ˆV NL q = (4π)2 ( i) i j (qr)j (q r)y m Ω m (ˆq )V n (r)r 2 dr m m m Y m (ˆr )Y m (ˆr )dω Ym m (ˆr)dΩ } {{ }} {{ } δ,mm δ,mm (25) where we have used the orthonormaity of spherica harmonics as mentioned before. When the deta functions are imposed the two sums over m and m are gone and the ony indices that remain are and m. Next in order to dea with the remaining spherica harmonics, we utiize the summation theorem for spherica harmonics. m Y m (ˆq)Y m (ˆq ) = π P (cos θ q q ) (26)

5 where P is the generaized Legendre poynomia and θ q q is the ange between the two wavevectors q and q. Putting everything together the expectation vaue in Eq. 25 becomes q ˆV NL q = 4π (2 + 1) j (qr)j (q r)p (cos θ q q )V (r)r 2 dr (27) Ω As seen in Eq. 27, the semioca pseudopotentia mixes panewaves with different q vectors. This means that if we have N pw panewaves in the basis that we choose we have to evauate Npw 2 such integras as in Eq. 27, which brings a arge computationa oad. In their semina work Keinman and Byander (Physica Review Letters, 48, 1425) proposed a different scheme for composing the anguar momentum-dependent part of the pseudopotentia such that the number of integras to be cacuated coud be reduced from Npw 2 to just N pw. This form is caed the fuy nonoca form and we sha see how it is formed in the next section. 5 B. The Keinman-Byander Transformation The difficuty with the semioca pseudopotentias above is that due to the ocaity in the radia coordinate there is a singe radia variabe for both panewaves whie the anguar part is nonoca and the integras reated to each anguar coordinate may be evauated separatey. The fuy nonoca formuation incudes the radia part in the projectors thereby separating the integra reated to each projector. The starting point for the transformation is the usua semioca form expained above where the oca radia part is given by V. We then find the pseudowavefunctions in the atomic configuration, φ PS corresponding to this pseudopotentia. Using these wavefunctions, we construct a new pseudopotentia who is composed of a oca part, V L, which is a competey arbitrary function that has the ong-range behavior of the rea potentia and a nonoca part V NL, which is the remaining part of the pseudopotentia. After the subtraction of the oca part, the nonoca part is eft behind ocaized in the core region. The nonoca part is then given by ˆV NL = ˆV NL = m ˆV SL φ PS m φps SL m ˆV φ PS m ˆV φ PS m (28) This particuar form is chosen such that its action on a pseudowavefunction yieds the same resut as the action of the semioca form ˆV NL φ PS m In order to see this, et s appy this operator on the pseudowavefunction ˆV NL φ PS m = SL ˆV φ PS SL = ˆV φ PS m (29) m φps m φ PS m ˆV SL SL ˆV φ PS φ PS m m = SL ˆV φ PS m (30) In the panewave representation, an arbitrary matrix eement of the Keinman-Byanderized pseudopotentia is q ˆV NL q = q ˆV φ PS m ˆV φ PS m q φ PS m ˆV φ PS m. (31) The two integras in Eq. 31 do not share any common integra variabes and may thus be performed separatey. One coud then compute N PW such integras, i.e. q ˆV φ PS m in advance and cacuate the appropriate product q ˆV φ PS m ˆV φ PS m q for each q, q pair. IV. ULTRASOFT PSEUDOPOTENTIALS In 1990, Vanderbit (see Physica Review B, 41, 7892) proposed a new and radica method for generating pseudopotentias by reaxing the norm-conservation constraint. Pseudopotentias generated in this way (due to their softness) require a much smaer panewave cutoff and thus a much smaer number of panewaves. For this reason they are usuay caed utrasoft pseudopotentias. First et s see how utrasoft pseudopotentia is integrated into the Kohn-Sham equations and then see how it is generated. The tota energy in the presence of utrasoft pseudopotentias is written as

6 6 E e = i φ i V NL φ i i d rd r n( r)n( r ) r r + E xc [n] + d rv oc ( r)n( r) (32) where the Hartree, exchange-correation, oca and kinetic terms have the usua form. The nonoca part of the pseudopotentia instead is given in terms of a new set of projection operators as V NL = D (0) nm βi n βi m (33) where the superscript I indicates that the projector is centered on atom I. The norm-conserving condition is reaxed by introducing a generaized orthonormaity condition through the overap matrix S where φ i S φ j = δ ij (34) S = 1 + nm,i q nm β I n βi m (35) with q nm = d rq nm ( r). Because the norm-conserving condition is now reaxed, the eectronic density needs to be augmented in order to preserve the correct number of eectrons. The new density is then n( r) = i φ i ( r) 2 + Q I nm ( r) φ i βn I βi m φ i (36) nm,i In order to verify that the density as given in Eq. 36 integrates to the correct number of eectrons, consider n( r)d r = φ i ( r) 2 + d rq I nm( r) φ i βn β I m φ I i i nm,i = ij ij δ ij φ i φ j + nm,i,j δ ij q nm φ i βn β I m φ I j δ ij φ i S φ j = δ ij δ ij = δ ii = 1 = N v (37) ij i i where N v stands for the number of vaence eectrons in the system. The new form of the density, athough it gives us the fexibiity to create softer pseudopotentias, inevitaby makes the Kohn-Sham equations more compex. The functiona derivative of the density with respect to the orbitas is δn( r ) δφ i ( r) = φ i( r)δ( r r ) + Q I nm ( r )βn I ( r) βi m φ i. (38) nm,i Utiizing Eq. 38 and Eq. 32 we arrive at the modified Kohn-Sham equations δe e δφ = d r δe e δn( r [ ) i δn( r ) δφ i ( r) = 1 ( ) ] V eff + D nm (0) + d r V eff ( r )Q nm ( r ) βn β I m I φ i. (39) where V eff is the reguar potentia V eff = V H + V oc + V xc. (40) As easiy seen in Eq. 39, unike norm-conserving pseudopotentias, the coefficients of the projectors in the nonoca part of the pseudopotentia gets updated in a sef-consistent manner at every iteration. This introduces a sight overhead to the cacuation but the gain in the panewave expansion surpasses the oss thus introduced. Now, we take a step back and see the justification and generation of such a radica impementation. The generation of utrasoft potentias start out in the same way as norm-conserving pseudopotentias. For a given species, the atomic

7 Kohn-Sham system is soved sef-consistenty, resuting in the screened a-eectron potentia, V AE. Then for each anguar momentum channe, a few energy vaues, ǫ τ are chosen (τ=number of such energies, usuay max 3). For the V AE aready obtained, the Schrödinger equation is found [T + V AE ] ψ n (r) = ǫ n ψ n (r) (41) where n is a composite index hoding for the anguar momentum quantum number, m for the projection and τ. In the genera case, ψ n is not going to be normaizabe but we use the bra-ket notation anyway. Next, we choose three cutoff radii : 1. A smooth oca (-independent) potentia is created which matches V AE (r) after the radius r oc c. 2. A second cutoff radius, r c, is chosen for each anguar momentum channe and a pseudowavefunction, φ n is constructed which matches the rea wavefunction ψ n at r c. 3. Finay a diagnostic radius R is chosen which is sighty arger than the maximum of the maximum of a the r c and r oc C. Then new orbitas are formed that satisfy which are zero beyond R. Next, an auxiiary matrix of inner products and the projectors necessary for the definition of the nonoca part of the potentia χ n = (ǫ n T V oc ) φ n (42) B nm = φ n χ m (43) β n = m (B 1 ) mn χ m (44) 7 are defined. Whie determining the pseudowavefunctions mentioned above, norm-conservation has not been imposed. We thus define a set of generaized augmentation functions and the reated augmentation charges Q nm ( r) = ψ n( r)ψ m ( r) φ n( r)φ m ( r) (45) q nm = ψ n ψ m R φ n φ m R. (46) where the subscript R denotes an integration within the cutoff radius, R. With a these descriptions, we can verify that the pseudowavefunctions φ n obey the equation ( ˆT + V oc + ) ( D nm β n β m φ i = ǫ i 1 + ) q nm β n β m φ i (47) nm nm where D nm = B nm + ǫ m q nm. In order to prove the centra equation Eq. 47, et s ook at ( ) B nm β n β m φ i = B nm (B 1 ) n χ (B 1 km ) χ k phi i }{{} nm nm k Bik = m χ n (B 1 ) n B nm (Bik (B 1 ) km ) = m δ m δ im χ = χ i = (ǫ i ˆT V oc ) φ i (48) Sustituting Eq. 48 into Eq. 47 competes the proof.

8 Thus the fundamenta equation of the pseudopotentia theory has been satisfied, that is pseudowavefunctions that satisfy the same equation as the rea wavefunctions have been found. These pseudopotentias, however, may be proven to satisfy, instead of being orthonorma in the usua sense, satisfy a generaized orthonormaity equation through an overap matrix S = 1 + nm q nm β n β m (49) 8 Once the D nm are obtained, the oca part and the nonoca coefficients of the bare pseudopotentia are obtained through a descreening procedure V ion oc = V oc V H V xc (50) D nm (0) = D nm d r V oc ( r )n( r ) (51) Whie generating pseudopotentias, we don t enforce norm conservation. This aows us to increase the cutoff radii to arger vaues. This convenient fact however brings about the necessity of defining a separate panewave cutoff for the density because as seen in the expression in Eq. 36, the density now has a smooth part and a sharp part. This sharp part needs to be expressed using panewaves that have high frequencies, i.e. a pane wave cutoff that is typicay severa times the panewave cutoff used to express the wavefunctions.