The following codes are available:

• SAPT0.py: A simple Psi 4 input script to compute SAPT interaction energies.
• SAPT0_ROHF.py: A SAPT0(ROHF) script for the oxygen dimer (two triplets making a quintet).
• SAPT0_no_S2.py: A script to compute the SAPT0 interaction energy without the the Single-Exchange Approximation.
• SAPT0ao.py: A Psi 4 input script to compute SAPT interaction energies in atomic orbitals.

Helper programs:

• helper_SAPT.py: A collection of helper classes and functions for SAPT.

The most important operator in SAPT is the intermolecular interaction operator \tilde{V}, that is defined as follows:

As all SAPT quantities will make use of this expression, a SAPT helper object that can automatically build this value greatly simplifies many routine SAPT tasks. In psi4numpy this helper object can be initialized as follow:

sapt = helper_SAPT(dimer, memory=8)

Where the dimer object is a Psi4 Molecule with exactly two fragments.

Indices used:

• a - occupied indices of monomer A
• b - occupied indices of monomer B
• r - virtual indices of monomer A
• s - virtual indices of monomer B

The sapt object now contains all of the information that we need to compute the following:

• v: Two electron repulsion integrals in the MO basis (e.g. sapt.v('arar'))
• s: Overlap integrals in the MO basis (e.g. sapt.s('ab'))
• eps: SCF eigenvalues (e.g. sapt.eps('a'))
• potential: Electrostatic potential of a monomer (e.g. sapt.potential('bb', 'A'))
• vt: Intermolecular interaction operator \tilde{V} (e.g. sapt.vt('arar'))
• chf: Computes CPHF orbitals for each monomer (e.g. chf('A'))

Where for all 4-index quantities the order of the indices is as follows:

\tilde{V}_{0, 1}^{2, 3}

For example, the expression for first order electrostatics is as follows:

4 * \tilde{V}_{a, b}^{a, b}

Computation of the \tilde{V} operator can be accomplished as:

vt_abab = sapt.vt('abab')

vt_abab is a 4-index tensor that needs to be summed over following einsum convention:

Elst10 = 4 * np.einsum('abab', vt_abab)

By first building a tool to compute arbitrary \tilde{V} tensors we can, in principle, compute up to third order SAPT easily.

1. General SAPT Theory & SAPT0 Equations
   • [Szalewicz:2005:43] K. Szalewicz, K. Patkowski, and B. Jeziorski, in Intermolecular Forces and Clusters II (Springer Berlin Heidelberg, Berlin/Heidelberg, 2005), pp. 43–117.
   • [Jeziorski:1994:1887] B. Jeziorski, R. Moszynski, and K. Szalewicz, Chem. Rev. 94, 1887 (1994).
   • [Szalewicz:2012:254] K. Szalewicz, WIREs: Comput. Mol. Sci. 2, 254 (2012).
   • [Hohenstein:2012:304] E. G. Hohenstein and C. D. Sherrill, WIREs: Comput. Mol. Sci. 2, 304 (2012).
2. SAPT0 without the Single-Exchange approximation:
   • [Jeziorski:1976:281] B. Jeziorski et al. Int. J. Quantum Chem. 10, 281 (1976)
   • [Schaffer:2012:1235] R. Schaffer et al. Theor. Chem. Acc. 131, 1235 (2012)
   • [Schaffer:2013:2570] R. Schaffer et al. Mol. Phys. 111, 2570 (2013)
3. SAPT(ROHF) for multiplet splittings:
   • [Patkowski:2018:164110] K. Patkowski, P. S. Zuchowski, and D. G. A. Smith, J. Chem. Phys. 148, 164110 (2018)