Hamann-Schlüter-Chiang Conditions

Norm-conserving pseudopotentials are derived from atomic reference states, calculated from the atomic Schrödinger equation

$$\left( T + V_{\rm AE} \right) \vert \Psi_l \rangle = \epsilon_l \vert \Psi_l \rangle \enspace ,$$

where $T$ is the kinetic energy operator and $V_{\rm AE}$ the all-electron potential derived from Kohn-Sham theory. This equation is replaced by a valence electron only equation of the same form

$$\left( T + V_{\rm val} \right) \vert \Phi_l \rangle = \hat \epsilon_l \vert \Phi_l \rangle \enspace .$$

Hamann, Schlüter, and Chiang [93] proposed a set of requirements for the pseudo wavefunction and pseudopotential.

The pseudopotential should have the following properties

1. Real and pseudo valence eigenvalues agree for a chosen prototype atomic configuration. $\epsilon_l = \hat \epsilon_l$
2. Real and pseudo atomic wave functions agree beyond a chosen core radius r$_c$ .
   $$\Psi_l(r) = \Phi_l(r)$$   for$$r \geq r_c$$

3. The integrals from 0 to R of the real and pseudo charge densities agree for R $\geq$ r$_c$ for each valence state (norm conservation).
   $$\langle \Phi_l \vert \Phi_l \rangle_R = \langle \Psi_l \vert \Psi_l \rangle_R$$   for$$R \geq r_c$$
   where
   $$\langle \Phi \vert \Phi \rangle_R = \int_0^R r^2 \vert \phi(r) \vert^2 dr$$

4. The logarithmic derivatives of the real and pseudo wave function and their first energy derivatives agree for r $\geq$ r$_c$ .

Property 3) and 4) are related through the identity

$$-{1 \over 2} \left[ (r \Phi)^2 {d \over d\epsilon} {d \over dr} ln \Phi \right]_R = \int_0^R r^2 \vert \Phi \vert^2 dr$$

They also gave a recipe that allows to generate pseudopotentials with the above properties.

1. $$V_l^{(1)}(r) = V_{AE}(r) [ 1 - f_1 \left( {r \over r_{cl}} \right) ]$$
   r$_{cl}$ : core radius $\approx$ 0.4 - 0.6 R$_{max}$ , where R$_{max}$ is the outermost maximum of the real wave function.

2. $$V_l^{(2)}(r) = V_l^{(1)}(r) + c_l f_2 \left( {r \over r_{cl}} \right)$$
   determine c$_l$ so that $\hat \epsilon_l = \epsilon_l$ in
   $$( T + V_l^{(2)}(r) ) w_l^{(2)}(r) = \hat \epsilon_l w_l^{(2)}(r)$$

3. $$\Phi_l(r) = \gamma_l \left[ w_l^{(2)}(r) + \delta_l r^{l+1} f_3 \left( {r \over r_{cl}} \right) \right]$$
   where $\gamma_l$ and $\delta_l$ are chosen such that
   $$\Phi_l(r) \rightarrow \Psi_l(r)$$   for$$r \geq r_{cl}$$
   and
   $$\gamma_l^2 \int \vert w_l^{(2)}(r) + \delta_l r^{l+1} f_3 \left( {r \over r_{cl}}\right) \vert^2 dr = 1$$

4. Invert the Schrödinger equation for $\hat \epsilon_l$ and $\Phi_l(r)$ to get V$^l_{val}$ (r).
5. Unscreen V$^l_{val}$ (r) to get V$^l_{ps}$ (r).
   $$V^l_{ps}(r) = V^l_{val}(r) - V_H(n_v) - V_{xc}(n_v)$$
   where V $_H(\rho_v)$ and V $_{xc}(\rho_v)$ are the Hartree and exchange and correlation potentials of the pseudo valence density.

Hamann, Schlüter and Chiang chose the following cutoff functions $f_1(x) = f_2(x) = f_3(x) = \exp(-x^4)$ .

These pseudopotentials are angular momentum dependent. Each angular momentum state has its own potential that can be determined independently from the other potentials. It is therefore possible to have a different reference configuration for each angular momentum. This allows it for example to use excited or ionic states to construct the pseudopotential for $l$ states that are not occupied in the atomic ground state.

The total pseudopotential in a solid state calculation then takes the form

$$V_{ps}(r) = \sum_L V^L_{ps}(r) {\bf P}_L$$

where $L$ is a combined index {l,m} and ${\bf P}_L$ is the projector on the angular momentum state {l,m}.