Kleinman-Bylander Scheme

The other method is based on the resolution of the identity in a local basis set

$$\sum_{\alpha} \mid \chi_{\alpha} \rangle \langle \chi_{\alpha} \mid = 1 \enspace ,$$ (187)

where $\{ \chi_{\alpha} \}$ are orthonormal functions. This identity can now be introduced in the integrals for the non-local part

$$V^{\rm nl}({\bf G},{\bf G}^\prime) = \sum_{L} \int_0^{\infty} dr \langle {\bf G} \mid Y_{\rm L} \rangl... ...ega} r^2 \Delta V^{L}(r) \langle Y_{\rm L} \mid {\bf G}^\prime \rangle_{\omega}$$

$$= \sum_{\alpha, \beta} \sum_{L} \int_0^{\infty} dr \langle {\bf G} ... ...a} \rangle_{\omega} \langle \chi_{\beta} \mid {\bf G}^\prime \rangle \enspace ,$$ (188)

and the angular integrations are easily performed using the decomposition of the basis in spherical harmonics

$$\chi_{\alpha}({\bf r}) = \chi^{lm}_{\alpha}(r) Y_{lm}(\omega) \enspace .$$ (189)

This leads to

$$V^{\rm nl}({\bf G},{\bf G}^\prime) = \sum_{\alpha, \beta} \sum_{L} \langle {\bf G} \mid \chi_{\alpha} ... ... V^{L}(r) \chi^{lm}_{\beta}(r) \langle \chi_{\beta} \mid {\bf G}^\prime \rangle$$ (190)

$$= \sum_{\alpha, \beta} \sum_{L} \langle {\bf G} \mid \chi_{\alpha} ... ...le \Delta V^{l}_{\alpha \beta} \langle \chi_{\beta} \mid {\bf G}^\prime \rangle$$ (191)

which is the non-local pseudopotential in fully separable form. The coupling elements of the pseudopotential

$$\Delta V^{l}_{\alpha \beta} = \int_0^{\infty} dr \chi^{lm}_{\alpha}(r) r^2 \Delta V^{L}(r) \chi^{lm}_{\beta}(r)$$ (192)

are independent of the plane wave basis and can be calculated for each type of pseudopotential once the expansion functions $\chi$ are known.

The final question is now what is an optimal set of basis function $\chi$ . Kleinman and Bylander[98] proposed to use the eigenfunctions of the pseudo atom, i.e. the solutions to the calculations of the atomic reference state using the pseudopotential Hamiltonian. This choice of a single reference function per angular momenta guarantees nevertheless the correct result for the reference state. Now assuming that in the molecular environment only small perturbations of the wavefunctions close to the atoms occur, this minimal basis should still be adequate. The Kleinman-Bylander form of the projectors is

$$\sum_{L} {\mid \chi_{L} \rangle \langle \Delta V^L \chi_L \mid \over \langle \chi_{L} \Delta V^L \chi_L \rangle } = 1 \enspace ,$$ (193)

where $\chi_{L}$ are the atomic pseudo wavefunctions. The plane wave matrix elements of the non-local pseudopotential in Kleinman-Bylander form is

$$V^{\rm KB}({\bf G},{\bf G}^\prime) = { \langle {\bf G} \mid \Del... ...^\prime \rangle \over \langle \chi_{L} \Delta V^L \chi_L \rangle } \enspace .$$ (194)

Generalizations of the Kleinman-Bylander scheme to more than one reference function were introduced by Blöchl [99] and Vanderbilt [100]. They make use of several reference functions, calculated at a set of reference energies.

In transforming a semilocal to the corresponding Kleinman-Bylander(KB) pseudopotential one needs to make sure that the KB-form does not lead to unphysical ``ghost'' states at energies below or near those of the physical valence states as these would undermine its transferability. Such spurious states can occur for specific (unfavorable) choices of the underlying semilocal and local pseudopotentials. They are an artefact of the KB-form nonlocality by which the nodeless reference pseudo wavefunctions need to be the lowest eigenstate, unlike for the semilocal form [101]. Ghost states can be avoided by using more than one reference state or by a proper choice of the local component and the cutoff radii in the basic semilocal pseudopotentials. The appearance of ghost states can be analyzed by investigating the following properties:

• Deviations of the logarithmic derivatives of the energy of the KB-pseudopotential from those of the respective semilocal pseudopotential or all-electron potential.
• Comparison of the atomic bound state spectra for the semilocal and
  KB-pseudopotentials.
• Ghost states below the valence states are identified by a rigorous criteria by Gonze et al. [101].