pbasc - calculating radial integrals

After the configuration list has been created, the next step is to calculate the radial integrals using the program pbasc. pbasc calculates one-electron and two-electron radial integrals, which are used by the pconf program to form the Hamiltonian in the CI space. The one-electron radial integrals correspond to the DF potential of the core, and the two-electron radial integrals account for the Coulomb and Breit interactions between the valence electrons. The matrix elements of the Coulomb interaction for the multipolarity \(k\) can be written as

\[\langle c,d|V_q^k|a,b\rangle \equiv G_q^k(ca) G_q^k(bd) R_{abcd}^k,\]

where the angular factors \(G_q^k(fi)\) (known as relativistic Gaunt coefficients) are given by

\[\begin{split}G_q^k(fi)=(-1)^{m_f+1/2}\delta_p\sqrt{(2j_i+1)(2j_f+1)} \begin{pmatrix} j_f & j_i & k \\ -m_f & m_i & q \end{pmatrix} \begin{pmatrix} j_f & j_i & k \\ 1/2 & -1/2 & 0 \end{pmatrix},\end{split}\]

and \(R_{abcd}^k\) are the relativistic Coulomb radial integrals, and \(\delta_p\) accounts for the parity selection rule

\[\begin{split}\delta_p=\xi(l_i+l_f+k), \hspace{0.2in}\xi(n)=\Bigg\{ \begin{matrix} 1 & \text{if \( n \) is even,} \\ 0 & \text{if \( n \) is odd.} \end{matrix}\end{split}\]

The Breit interaction has the same form as the Coulomb interaction, but without the parity selection rule.

The pbasc reads in the files HFD.DAT and CONF.INP to determine which radial integrals are needed. These integrals are calculated and written to the files CONF.INT. The relativistic Gaunt coefficients are written to the file CONF.GNT, and the file CONF.DAT is also formed, storing the basis radial orbitals \(\phi_{nlj}\), as well as functions \(\chi_{nlj} = h_\text{DF}^r\phi_{nlj}\).

Running pbasc

Before running pbasc, make sure the following input files have been appropriately generated by hfd, bass, and add:

  • HFD.DAT (obtained from basis)

  • CONF.INP (obtained from add)

To run parallel pbasc, run the command:

mpirun -n <nprocs> ./basc