DFT in practice : Part II - pseudopotentials

Pseudopotentials

DFT in practice : Part II pseudopotentials

Ersen Mete Department of Physics Balıkesir University, Balıkesir - Turkey

October 2009 - ITAP, Turun¸c

Ersen Mete

DFT in practice : Part II

Pseudopotentials

Outline

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

Ersen Mete

DFT in practice : Part II

Pseudopotentials

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, ψn`m (~r ) = Rn` (r )Y`m (θ, ϕ) =

φn` (r ) m Y` (θ, ϕ) r

The radial equation becomes,   `(` + 1) 1 d2 − φn` + + Veff (r ) φn` = εφn` 2 dr 2 2r 2 Kohn-Sham single particle Schr¨ odinger-like equations will be identical if VKS (~r ) is spherically symmetric as one-electron Coulomb potential V (r ).

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DFT in practice : Part II

Pseudopotentials

VKS (~r ) = VKS (r ) ? density : n(~r ) =

occ X

|ψn`m (~r )|2 =

n,`,m

occ X (2` + 1)|Rn,` (r )|2 = n(r ) X n,`

external potential : υext (~r ) = −Z /r = υext (r ) X Z n(~r 0 ) Hartree potential : d~r 0 = VH (r ) X |~r − ~r 0 | exchange-corr. pot. :Vxc (~r ) = xc [n(r )] + n(r )

dxc [n(r )] = Vxc (r ) X dn

Therefore, total effective potential VKS = υext (r ) + VH (r ) + Vxc (r ) spherically symmetric. The independent-particle Kohn-Sham equations are analogous to the Schr¨ odinger equation of the one-electron atom,   1 d2 `(` + 1) φn` + + VKS (r ) φn` = εφn` DFT for an atom − 2 dr 2 2r 2 Ersen Mete

DFT in practice : Part II

Pseudopotentials

Why do we need pseudopotentials? DFT calculation with all-electron υext 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 X The number of electrons will be decreased : Reduction of the number of bands to solve for X Ersen Mete

DFT in practice : Part II

Pseudopotentials

Schematically

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DFT in practice : Part II

Pseudopotentials

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 | {z } core

Ti with semicore

2

2

1s 2s 2p 6 3s 2 3p 6 4s 2 3d 2 | {z } core

locality : all `-channel (s, p, d) electrons feel the same potential

efficiency → a compromise between accuracy and computational cost

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DFT in practice : Part II

Pseudopotentials

Accuracy vs computational load

Local PSP V PS = V PS (r ) Semilocal PSP V PS =

X

V`PS (r )|χ` ihχ` |

`

Nonlocal separable PSP PS V PS = Vloc (r ) +

X

D` |β`m ihβ`m |

`m Ersen Mete

DFT in practice : Part II

Pseudopotentials

Total energy in terms of valence electron density n(~r ) = ncore (~r ) + nval (~r ) Then, E [{ψi }] =

N core X i

1 hψi |- ∇2 |ψi i + 2

Z

1 υext (~r )ncore (~r )d~r + 2

Z

ncore (~r )ncore (~r 0 ) d~r d~r 0 |~r − ~r 0 |

Z Z Nval X 1 2 1 nval (~r )nval (~r 0 ) + hψi |- ∇ |ψi i + υext (~r )nval (~r )d~r + d~r d~r 0 2 2 |~r − ~r 0 | i Z ncore (~r )nval (~r 0 ) + d~r d~r 0 + Exc [ncore + nval ] |~r − ~r 0 |

Eval [{ψi }] =

Nval X i

Z Z 1 Z ncore (~r 0 )  hψi |- ∇2 |ψi i+ - + nval (~r )d~r +Exc [ncore + nval ] | {z } 2 r |~r − ~r 0 | | {z } Non−linear XC SCR Vion

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DFT in practice : Part II

corrections

Pseudopotentials

Generic pseudopotential transformation For an atom, assume that the core states, |χn i, satisfy H|χn i = En |χn i A single valence state, |ψi, can be replaced by a smoother pseudofunction, |φi, expanding the remaining part in terms of |χn i, |ψi = |φi +

core X

an |χn i

n

Using the orhtogonality of valence and core states, hχm |ψi = hχm |φi +

core X

an hχm |χn i = 0

=⇒

n

|ψi = |φi −

X

hχn |φi|χn i

n

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DFT in practice : Part II

am = −hχm |φi

Pseudopotentials

Then, the eigenvalue equation, core   X H |φi − hχn |φi|χn i =

core   X E |φi − hχn |φi|χn i

n

H|φi +

core X

n

(E − En )|χn ihχn |φi =

E |φi

n

This implies, (H + Vn` )|φi = E |φi where ` is due to spherical symmetry. E − En > 0 → extra potential Vn` is repulsive. Cancels the effect of the attractive Coulomb potential. Resulting potential is weaker and pseudo eigenstate is smoother.

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DFT in practice : Part II

Pseudopotentials

Norm-conserving pseudopotentials

Hamann, Schl¨ uter 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 `), AE H|φAE n` i = εn` |φn` i

⇒

PS (H + Vn` )|φPS n` i = εn` |φn` i

AE & PS wavefunctions match beyond a certain radial cutoff, rc , PS φAE n` (r ) = φn` (r )

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r ≥ rc

DFT in practice : Part II

Pseudopotentials

AE and PS norm squares integrated upto r ≥ rc are equal. Z r Z r AE 0 2 0 0 2 0 |φn` (r )| dr = |φPS n` (r )| dr 0

0

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 ≥ rc . d d ln[φPS ln[φAE n` (r )] −→ n` (r )] dr dr VPS must reproduce the same scattering phase shifts as VAE for r . Necessary to improve transferability.

PS wavefunction is nodeless. It’s twice differentiable and satisfies limr →0 φn` (r ) ∝ r `+1 =⇒ continuous. Ersen Mete

DFT in practice : Part II

Pseudopotentials

Silicon : wave functions

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DFT in practice : Part II

Pseudopotentials

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 `.

4

Invert the Schr¨ odinger equation for PS wavefunctions to get screened PSP components. V`SCR,PS = εPS ` −

5

1 d 2 PS `(` + 1) + PS φ (r ) 2 2r 2φn` (r ) dr 2 `

Subtract the Hartree and XC contributions to obtain υext (unscreened PSP). V`PS = V`SCR,PS − VH (r ) − VXC (r ) Model pseuodopotential replaces the potential of the nucleus. Ersen Mete

DFT in practice : Part II

Pseudopotentials

Semilocal pseudopotentials

For `-dependent model PSP, treat angular momentum ` separately (nonlocal). PSP in the semilocal form : PS Vn` = Vloc (r ) + VSL

where

VSL =

X

|Y`m iVn` (r )hY`m |

`m

It’s local in r and nonlocal in θ, ϕ. Drawback : Plane wave representation of the non-local part is expensive.

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DFT in practice : Part II

Pseudopotentials

Semilocal PSP matrix elements in plane waves h~q |VSL |~q 0 i =

1X Ω

ZZ

0

0

∗ (θ, ϕ)Vn` (r )Y`m (θ, ϕ)e i~q ·~r r 2 dΩdΩ0 dr e −i~q·~r Y`m

`,m

where ~r 0 = (r , θ0 , ϕ0 ) e i~q·~r = 4π

X

∗ i ` j` (qr )Y`m (ˆ q )Y`m (ˆr )

`m

where j` (qr ) are spherical Bessel functions and qˆ and ˆr denote angles associated with the vectors ~q and ~r , respectively. X

∗ Y`m (ˆ q )Y`m (ˆ q0) =

m

h~q |VSL |~q 0 i =

2` + 1 P` (cos θ~q~q0 ) 4π

Z 4π X (2` + 1) j` (qr )j` (q 0 r )P` (cos θ~q~q0 )V` (r )r 2 dr Ω `

2 NPW such integrations needed! Ersen Mete

DFT in practice : Part II

Pseudopotentials

Fully separable Kleinman-Bylander form PS Vn` (~r ,~r 0 ) = Vloc (r )δ(~r − ~r 0 ) + VNL

where VNL =

X

V`NL =

X |V SL φPS ihφPS V SL |

`

`

`m

`m

`

PS hφPS `m |V` |φ`m i

`m

Action of this form on the PS wavefunction, V`NL |φPS `m i =

PS SL SL |V`SL φPS `m ihφ`m V` |φ`m i = V`SL |φPS `m i PS hφ`m |V` |φPS i `m

Planewave representation of non-local PSP matrix elements in Kleinman-Bylander form h~q |VNL |~q 0 i =

X h~q |V SL φPS ihV SL φPS |~q 0 i `

`m

`m

`

`m

SL PS hφPS `m |V` |φ`m i

Number of integrals to be evaluated reduces to NPW . [PRL 48,1425 (1982)] Ersen Mete

DFT in practice : Part II

Pseudopotentials

Silicon & Titanium NCPPs

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DFT in practice : Part II

Pseudopotentials

Ti log derivatives

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DFT in practice : Part II

Pseudopotentials

Ultrasoft pseudopotentials (USPP) : formalism Vanderbilt [PRB 41, 7892 (1990)] proposed a new method by relaxing norm-conservation constraint, PS hφPS 6 hψiAE |ψiAE i i |φi i =

USPPs are norm-conserving in a generalized form PS AE AE ˆ hφPS i |(1 + NNL )|φi i = hψi |ψi i

ˆ is nonlocal charge augmentation operator. where N USPPs require much smaller PW cutoff (less NPW ) ⇒ ultrasoft. Scattering properties remain to be correct ⇒ transferable.

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DFT in practice : Part II

Pseudopotentials

The aim is to minimize the total energy, Z ZZ X n(~r )n(~r 0 ) 1 ˆNL |φi i+ d~r Vloc (~r )n(~r )+ 1 d~r d~r 0 +Exc [n] Ee = hφi |- ∇2 +V 2 2 |~r − ~r 0 | i

subject to ˆ NL )|φj i = hφi |SˆNL |φj i = δij hφi |(1 + N where n(~r ) =

  X ˆ NL (~r ) |φi i hφi | |~r ih~r | + K i

and for consistency ˆ NL = N

Z

ˆ NL (~r ) d~r K

Z so that

n(~r )d~r = Nv

Then, the eigenvalue equation, PS PS ˆNL ˆ NL )|φi i (T + Vloc +V )|φi i = εn (1 + N

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DFT in practice : Part II

Pseudopotentials

Ultrasoft PSP generation steps

Screened V AE is obtained through self-consistent solution of atomic Kohn-Sham system. Cutoff radii are chosen : rcl for the wave functions rcloc for the local PSP R large enough that all PS and AE quantities agree.

A smooth local potential, Vloc (~r ) is generated which approaches V AE (~r ) beyond rcloc .

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DFT in practice : Part II

Pseudopotentials

Then, for each angular momentum channel, a few reference energy values, εi , are chosen where i = {τ `m} and τ is the number of reference energies. (T + V AE − εi )|ψi i = 0 ψi not determined self-consistently. So, hψi |ψj iR =

R R

ψi∗ (~r )ψj (~r )d~r

New orbitals defined, |χi i = (εi − T − Vloc )|φi i which vanish at and beyond R where Vloc = VAE and φi = ψi . To define a nonlocal PSP, |χi i are used as projectors to define new wave functions, X X |βi i = |χi ihχj |φi i = (B −1 )ji |χj i j

j

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DFT in practice : Part II

Pseudopotentials

To compensate valence charge deficit, generalized augmentation charges needed, qij = hψi |ψj iR − hφi |φj iR The nonlocal overlap operator can be defined as, X qij |βi ihβj | S =1+ i,j

Then the nonlocal potential operator is X VNL = Dij |βi ihβj | where

Dij = Bij + εj qij

ij

PS wave functions satisfy generalized orthonormality condition, hφi |S|φj iR = hφi |φj iR + qij = hψi |ψj iR = δij Then, PS wave functions satisfy generalized eigenvalue problem, (H − εi S)|φi i = 0 Ersen Mete

where

H = T + Vloc + VNL

DFT in practice : Part II

Pseudopotentials

Verification of generalized eigenvalue problem ! T + Vloc +

X

!

Dnm |βn ihβm | |φi i = εi

1+

nm

X

qnm |βn ihβm | |φi i

nm

where Dnm = Bnm + εm qnm Since, X  X X X Bnm |βn ihβm | |φi i = hφn |χm i |χk ihχk |φn i hφm |χ` ihχ` |φi i nm

nm

=

X

=

X

k

`

X |χk i hχk |φn ihφn |χm iδim n

km

|χk iδkm δim = |χi i

km

Substitution yields, H|φi i = εi S|φi i Ersen Mete

DFT in practice : Part II

Pseudopotentials

Valence electron density The electron density is augmented, i Xh X nv (~r ) = |φn (~r )|2 + Qij (~r )hφn |βi ihβj |φn i n

i,j

where Qij (~r ) = ψi∗ (~r )ψj (~r ) − φ∗i (~r )φj (~r ) so that it must integrate to the correct number of valence electrons,   Z X Z XZ  |φn (~r )|2 d~r + nv (~r )d~r = Qij (~r )hφn |βi ihβj |φn id~r  n

=

X

=

X

ij

h

δnm hφn |φm i +

X

nm

qij hφn |βi ihβj |φm i

i

ij

δnm hφn |S|φm i =

nm

X

δnm δnm =

nm

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DFT in practice : Part II

X n

δnn = Nvalence X

Pseudopotentials

Minimization of total energy X δn(~r 0 ) 0 ~ = φ (~ r )δ(~ r − r ) + Qij (~r 0 )βi (~r )hβj |φn i n δφ∗n (~r ) ij

Then, the modified Kohn-Sham equations, δEe δφ∗n

Z =

=

d~r 0

δEe δn(~r 0 ) δn(~r 0 ) δφ∗n (~r )

h 1  i X  (0) Z - ∇2 + Veff + Dij + Veff (~r 0 )Qij (~r 0 )d~r 0 |βi ihβj | |φn i 2 ij

where Veff = VH + Vloc + Vxc The coefficients in the non-local part of the PSP gets updated self-consistently.

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DFT in practice : Part II

Pseudopotentials

Ionic USPP

Ionic potential are obtained by unscreening, ion Vloc (0)

Dij

= Vloc − VH − Vxc Z = Dij − d~r 0 Vloc (~r 0 )n(~r 0 )

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DFT in practice : Part II

Pseudopotentials

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DFT in practice : Part II

Pseudopotentials

Projector Augmented Waves (PAW) method : basic idea P.E. Bl¨ ochl, PRB 50 17953 (1994)

For a particular reference energy, 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`m = hP`m |ψ PS i In each sphere, |ψ PS i =

X

c`m |φPS `m i

|ψ AE i =

and

`m

X

c`m |φAE `m i

`m

where φ`m are partial waves. Projectors must be dual to partial waves, hP`m |φPS `0 m0 i = δ``0 δmm0

⇒

|ψ PS i =

X

PS PS |φPS `m ihP`m |ψ i = |ψ i

`m Ersen Mete

DFT in practice : Part II

Pseudopotentials

Then, the PS transformation is given by,

|ψnAE i

=

|ψnPS i

−

X

PS |φPS `mε ihP`mε |ψn i+

`mε

X

PS |φAE `mε ihP`mε |ψn i

`mε

Transformation involves AE wavefunction ⇒ One can derive AE results. X

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DFT in practice : Part II

Pseudopotentials

References E. Kaxiras, Atomic and electronic structure of solids, Cambridge University Press, Cambridge, 2003. R.M. Martin, Electronic Structure : Basic Theory and Methods, Cambridge University Press, Cambridge, 2004. D.R.Hamann, M. Schl¨ uter, C. Chiang, Phys. Rev. Lett. 43, 1494 (1990). G.B. Bachelet and M. Schl¨ uter, 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. Schl¨ uter, 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. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994).

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DFT in practice : Part II