@dc1394

Pseudopotential of Vanderbilt

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,

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.

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)

Property of pseudo-wave-function

Agreement of value and derivative at cutoff radius

(agreement of logarithmic derivative)

Agreement of energy derivative of logarithmic

derivative

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.

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.

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.

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.

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).

Property of the nonlocal

pseudopotential operator VNL

The nonlocal pseudopotential operator VNL becomes the

following,

where,

Property of the nonlocal

pseudopotential operator VNL

VNL satisfies Schrodinger equation as shown below.

Property of the nonlocal pseudopotential

operator VNL (continued)

Property of the nonlocal pseudopotential

operator VNL (conclusion)

where, because it is

in fact we can confirm the relation.

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.

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.

The norm-conserving pseudopotential

of Vanderbilt

The nonlocal pseudopotential operator of Vanderbilt is

Thus, Bij is diagonalized, and it is assumed,

where,

The norm-conserving pseudopotential

of Vanderbilt

Therefore, it becomes

(∵U=U*)

The norm-conserving pseudopotential

of Vanderbilt

Here,

Therefore, it becomes

The norm-conserving pseudopotential of Vanderbilt is

obtained.