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 file 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 \(\chi_{nlj} = h_\text{DF}^r\phi_{nlj}\).

Input Files:

  • HFD.DAT - basis set radial orbitals from hfd

  • CONF.INP - list of relativistic configurations and system parameters from add

Output Files:

  • CONF.DAT - basis radial orbitals \(\phi_{nlj}\) and functions \(\chi_{nlj} = h_\text{DF}^r\phi_{nlj}\)

  • CONF.INT - one- and two-electron radial integrals (stored in single precision by default; see note below)

  • CONF.GNT - relativistic Gaunt coefficients

  • BASC.RES - program log

Note

If Ncpt > Nc in CONF.INP (i.e. a PT block is defined), pbasc will interactively ask whether to compute integrals over Ncpt or Nc configurations. Enter 1 to use Ncpt (recommended if conf_pt will be run subsequently) or 2 to use Nc.

Note

By default, the two-electron radial integrals in CONF.INT are written in single precision to reduce file size and memory usage. To write them in double precision, compile with the CMake option -DUSE_DP_INTEGRALS=ON. Both pbasc and pconf must be compiled with the same setting, as pconf reads CONF.INT using the same precision. Double-precision integrals are only necessary for isotope shift calculations, where the higher numerical precision is required.

Running pbasc

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

  • HFD.DAT (obtained from hfd and bass)

  • CONF.INP (obtained from add)

To run parallel pbasc, run the command:

mpirun -n <nprocs> pbasc