1

Pseudopotential

plane wave:exp(ik · r) =P

ll (2l + 1)i jl (kr)Pl (k̂ · r̂)

Figure 1: scattering of plane wave
the scattering amplitude
f (θ) = (2ik)−1X(2l + 1)[exp(2iηl ) − 1]Pl (cos θ)(1)ηl = nl π + δl

(2)l

the phase shift ηlnl is an integer so |δl | < π/2. We put
nl = n − l − 1(3)where n and l is the principal and angular quantum number of the wavefunction ψnl .
(n − l − 1) is the number of the nodes in the wavefunction ψnl , and corresponds tothe number of the core states with angular momentum l. [eg. 3-2-1=0 for 3d(no core dstate), 4-0-1=3 for 4s(core 1s, 2s, 3s state), 4-1-1=2 for 4p(core 2p, 3p state)]

Figure 2: All electron and pseudowave functions of Mo atom1

Pesudopotential:n−l−1⇒0The pseudo-wave function has no radical nodes, i.e. no rapid oscillations so that we
expect a rapidly convergence series of plane wave.
For atom and ion system(−h̄2 2
∇ + Vion (r) + VH (r) + Vxc (r) − ²nlm )ψnlm (r) = 02m(4)ψnlm (r) = Rnl (r)Ylm (θ, φ)(5)Ll ≡ (Rl0 /Rl )|r=rc(6)

HereLogarithmic derivative
The phase shift ηl can be obtained from the logarithmic derivative Lltan ηl =jl (krc )Ll − kjl0 (krc )nl (krc )Ll − kn0l (krc )(7)

In periodic potential if we replace the real potential inside the core sphereby a pseudopotential with the same scattering properties, then it also gives
the same energy band structure.

Figure 3: Pseudopotential and logrithmic derivative of all-electron and psudo wave functions of Mo atom2v̂ps =Xvps,l (r)P̂l(8)lP̂l is projection operater that picks out l-th angular momentum component of the wavefunction so that vps,l (r) acts only on that. (reference: M. L. Cohen and V. Heine,
Ed. by H. Ehrenreich, F. Seitz and D. Turnbull, in: “The fitting of pseudopotentials
to experimental data and their subsequent application”, (Academic, New York, 1970)
p. 37-248.)PP
psP̂l = 1, then v̂ps = vloc + l (vps,l − vloc ). If we put vloc = vion,l=0
psv̂ionψ(r) =psvion(r, r0 ) = v0local (r)δ(r − r0 ) +lXmaxZps
dr0 vion(r, r0 )ψ(r0 )(vlps (r) − v0local (r))2δ(r − r0 )/r02

l=1

(9)

X

|Ylm (r̂)ihYlm (r̂0 )|m(10)(11)δvlnonlocal = vlps − v0local

Norm-Conserving Pseudopotentials (Hamman, Schlüter and Chiang, PRL, 43,1494, 1979)
For real(all electron atom) and pseudoatom(I) eigenvalues of the valence states are the same(II) wavefunctions
agree beyound a chosen core radius rc .
R rc2(III) 0 drr 4πρ(r) agree ≡ normconservation. (The electrostatic potential produced outside rc is identical for real and pseudo charge distributions.)(IV) logarithmic derivatives and their energy derivatives agree. (scattering propertiesof the real ion cores are reproduced with minimum error as bonding and banding shifts
eigenvalues away from the atomic levels.)
The properties (III) and (IV) are crucial for the pseudopotential transferability, andare related by a simple identity.
Z rcut1∂ ∂1l−ln Rl (r, ²)|²=²l ,r=rlcut = cut
drr2 Rl2 (r, ²l )cut
22 ∂² ∂r(rl Rl (rl , ²l )) 0(12)Derivation of logarithmic derivative(Harrison, Solid State Theory)²Rl(² + δ²)Rl0h̄2 l(l + 1)h̄2 1 ∂ 2 ∂= −r
Rl + V (r)Rl +Rl2m r2 ∂r ∂r
2m r2h̄2 l(l + 1) 0h̄2 1 ∂ 2 ∂ 00+= −+V(r)RrRRll
2m r2 ∂r ∂r l2m r2(13)(14)

If we apply Rl0 to lhs of Eq.(13) and Rl to lhs of Eq.(14) and integral from 0 to R−δ²Z R02dr4πr Rl0 RlÃ!
Z R
4πh̄2 Z R 0 ∂ 2 ∂∂ 2∂ 0= −
Rl r
Rl −
Rl r
R2m
∂r ∂r
∂r ∂r l00(15)Here we assume V (r) had no ² dependence. Usi ng partial integral−δ²Z R0"

dr4πr2 Rl0 Rl#RZ R
4πh̄2  0 2 ∂Rl∂R0 ∂Rl= −Rl r−dr l r22m
∂r 0∂r∂r0(16)

Z R0∂Rl 2 ∂Rl0 2 ∂Rl−drr− Rl r
∂r 0∂r∂r0(17)#R"3ÃRl0−δ²−δ²Z R0Z R0dr4πr2 (Rl +∂Rlδ²)Rl∂²∂Rldr4πr (Rl2 + Rlδ²)
2∂²!

4πh̄2
∂Rl
∂R0= −(18)Rl0 r2− Rl r 2 l2m∂r∂r R∂Rl= Rl +δ²(19)∂²Ã!
∂Rl4πh̄2∂Rl2 ∂Rl2 ∂(Rl + ∂² δ²)= −(Rl +δ²)r− Rl r2m∂²∂r∂rRÃ4πh̄2∂Rl ∂Rl∂ 2 Rl= −δ² r2− r 2 Rl2m
∂² ∂r
∂²∂r!

(20)
R

If we pick up the first-order of δ²Z R0Ã

drr2 Rl2h̄2 2 ∂Rl ∂Rl∂ 2 Rl=R− Rl
2m
∂² ∂r
∂²∂r
Ã!

∂Rl ∂Rlh̄2 2 2∂ 2 Rl=R Rl Rl−2− Rl−1
2m
∂² ∂r
∂²∂r
Ãh̄2 2 2 ∂∂RlR Rl
Rl−1= −2m
∂²∂r

(21)
!

(22)

!

h̄2 2 2 ∂ ∂= −R Rl
ln Rl
2m
∂² ∂r

(23)
(24)
(25)

Troullier and Martins(Kerker) Pseudopotential
psvion,l(r) = ²l −l(l + 1)
1 d2
ps
ps
− vxc
+(rRlps ) − vH
2r2
2rRlps dr2

Rlps (r) = rl exp[p(r)],r ≤ rlcut

(26)
(27)

where p(r) is a polynominal of order six in r2 .

For nonbound states Hamman’s generalized normconserving pseudopotential isused.

4