Basis set in the 𝘎𝘞 calculation

Representation of one-particle wave functions

The wave functions are expands by two types of basis set as follows:

where is atomic LMTO basis and is interstitial plane wave to comprised interstitial area.

Detail of

The is generated from bloch sum of atomic orbital to satisfy the bloch symmetry on basis functions, i.e.:

The radial part of atomic orbital are constructed from , , and based on argumentation of Smooth Henkel functions.

The index of atom is omitted for simplify. Each has 2 or 3 reference energy of . is the channel orbital of solution on radial Scrhödinger equation at energy , is energy derivative of , is local orbital which is the also solution on radial Scrhödinger equation but the given energy is far from . The energy of is set as the center of gravity for occupied states. This means that this basis set is not fixed in the QSGW cycle.

?? About radial equation??

• Is LDA used?

• In the in radial equation obtained from self-consistent calculation?
• What is the boundary conditions of radial equation.
• is not the solution of radial equation, the does not satisfy the some boundary conditions, for example
• how to set the reference energy in the case of ?

Detail of

The interstitial plane wave is expressed as follows:

Since the hollow out the MT region, these basis have overlap matrix between them, i.e.:

where,

Formula of inner product

This is for checking the orthonormality of wave function, but it is good for better understanding.

The cross term are vanished.