Product basis set in the 𝘎𝘞 calculation

Product basis

The basis set to represent product of one-particle wave functions. This is the way to reduce the dimension of product.

,where . Some of pairs of (,) or (,) which has the same or are got rid of in the set of .

WARNING

In this notation, DOES NOT indicate the direct product between and .

Product basis

The MPB introduced above does not satisfy orthogonality. Therefore, we introduce an orthogonal basis. This basis also diagonalizes the Coulomb matrix. By doing so, the calculation of the exchange self-energy becomes easier. In equations, new product basis is represent by the liner combination of , i.e.:

Then, is satisfy the following releations.

where is Coulomb matrix and is eigen value. The coefficient and are obtained by soliving following generalized eigenvalue equation:

where is Coulomb matrix represented by , namely, . By using , Coulomb interaction operator is represented as follows:

about

Since is a non-local function, the calculation of this matrix element includes cross terms of and .