The document summarizes the Vanderbilt pseudopotential method. It starts from all-electron calculations and defines a pseudo-wavefunction. The pseudopotential V'loc is constructed as the sum of the local potential Vloc, Hartree potential VH, and exchange-correlation potential VXC. Unlike other pseudopotentials which subtract VH and VXC, Vanderbilt's adds them. The method can produce both norm-conserving and ultra-soft pseudopotentials depending on whether the generalized norm-conserving condition is satisfied.

1. @dc1394
Pseudopotential of Vanderbilt

2. It starts from the all electron calculation
 The Schrodinger equation in the all electron
calculation can be written as follows,
 where, T is the kinetic-energy operator, VAE is the all
electron potential, εi is the eigenenergy, |ψi> is the
all electron wave function.
 or, when you use Hamiltonian H, we have the
following relation,

3. The pseudo-wave-function is used
 where, it is assumed that pseudo-wave-function φ(r) is
defined as follows,
 where, rc is a suitable cutoff radius, ψ(r) is the all
electron wave function, and concrete shape of pseudo-
wave-function φ(r) is still undecided.

4. Property of pseudo-wave-function
 Pseudo-wave-function φ(r) requires the following
conditions.
 Norm-conserving condition
 Generalized norm-conserving condition.(Feature of
Vanderbilt pseudopotential)

5. Property of pseudo-wave-function
 Agreement of value and derivative at cutoff radius
(agreement of logarithmic derivative)
 Agreement of energy derivative of logarithmic
derivative

6. Feature of pseudized Hamiltonian H
 Hamiltonian H can be written,
 by making use of the pseudo-wave-function,
 where, T is the kinetic-energy operator, V’loc is the
screened pseudopotential, VNL is the nonlocal
pseudopotential operator.

7. Property of pseudopotential V’loc
 V’loc satisfies the following property,
 where, Vloc is the unscreened pseudopotential, ρv is the
electron density of valence electron in applied system,
ρc: is the electron density of core electron.
 It is necessary to give the electron density ρc of the core
electron as data.
 Unlike the TM pseudopotential, V’loc is constructed by
the sum of Vloc, VH, and VXC.

8. Particularity of Vanderbilt
pseudopotential
 In the pseudopotential of TM, the pseudopotential is
obtained by subtracting VH and VXC from V(L)
PS.(It is
said the unscreening)
 However, V’loc is constructed by the sum of Vloc, VH,
and VXC for the pseudopotential of Vanderbilt.

9. Hartree potential VH
 VH is a functional of the electron density of the valence
electron in the applied system, the shape of classic
Coulomb potential is given by
 and it is called the Hartree potential.
 In general, VH is linear with respect to ρ.

10. Exchange-correlation potential VXC
 VXC is a functional of the electron density of the valence
electron in the applied system and the electron density
of the core electron, and it is called exchange-
correlation potential.
 This becomes shape different depending on the method
of the selected density functional theory.
 The problem caused by this nonlinearity can be solved
to some degree by the method of Louie(Partial Core
Correction (PCC) method).

11. Property of the nonlocal
pseudopotential operator VNL
 The nonlocal pseudopotential operator VNL becomes the
following,
 where,

12. Property of the nonlocal
pseudopotential operator VNL
 VNL satisfies Schrodinger equation as shown below.

13. Property of the nonlocal pseudopotential
operator VNL (continued)

14. Property of the nonlocal pseudopotential
operator VNL (conclusion)
 where, because it is
 in fact we can confirm the relation.

15. Ultra-soft pseudopotential of
Vanderbilt
 If generalized norm-conserving condition Qij = 0 is
satisfied, this pseudopotential is the norm-conserving.
 However, Vanderbilt showed that it is not necessary to
satisfy generalized norm-conserving condition Qij = 0
when the following conditions
 are satisfied.

16. Ultra-soft pseudopotential of
Vanderbilt
 Where, S is
 and it is called the overlap operator.
 Because of unnecesarly of the generalized norm-
conserving condition, the number of plane waves for
sufficient convergence can be reduced.
 This is an ultra-soft pseudopotential of Vanderbilt.

17. The norm-conserving pseudopotential
of Vanderbilt
 The nonlocal pseudopotential operator of Vanderbilt is
 Thus, Bij is diagonalized, and it is assumed,
 where,

18. The norm-conserving pseudopotential
of Vanderbilt
 Therefore, it becomes
 (∵U=U*)

19. The norm-conserving pseudopotential
of Vanderbilt
 Here,
 Therefore, it becomes
 The norm-conserving pseudopotential of Vanderbilt is
obtained.