Author: Edward Valeev

### General

Program CINTS can evaluate integrals over Gaussian functions of the following one- and two-electron operators which appear in quantum chemical theories:

- overlap

- electron kinetic energy

- nuclear attraction

- electron repulsion (ERI)

- anticoulombic (r12-operator)

- [r12,Ti] operator (i=1,2)

- various one-electron property operators (currently, only dipole moment and electronic nabla operators)

First-order derivatives with respect to nuclear positions may be evaluated for the first four types of integrals. (Derivative) integrals are evaluated over symmetry-adapted Cartesian and Spherical Harmonics contracted Gaussian functions. Symmetry use is restricted to D2h and it's subgroups. The program does not have theoretical limits on the number of basis functions or the maximum angular momentum of Gaussian functions in the basis.

### Tasks

As integrals are evaluated, the program can either write them out to disk (not available for derivative integrals) or contract them with appropriate quantities to form various entities of interest.

#### Disk storage

Integrals are written to disk in the new Integrals With Labels (IWL) format (see PSI 3 Programmer's Manual, section on LIBIWL). LIBIWL functions provide the interface which may be used to access integrals files. Non-zero, unique with respect to index permutations1 two-electron integrals are written in shell-quartet order, i.e. all integrals which belong to the same quartet of (symmetry unique) shells are written together. There are no markers between shell-quartets. However, there's a particular order in which shell-quartets of ERIs get stored to disk. Shell-quartets of ERIs which contribute to the same shell-quartet of supermatrix integrals P and K are written together. The last integral in each PK-block is written with its first index set to the negative of itself so that CSCF knows where it can stop and dump a complete block of P and K elements to disk.

#### Contraction

Depending on the keywords or command-line options specified, the following types of contractions can be performed:

- formation of the two-electron part of Fock matrix (spin-restricted for closed- and high-spin open-shell systems and spin-unrestricted cases) from ERIs (works in conjunction with CSCF)

- restricted integral-direct transformation to compute RHF MP2 energy

- restricted integral-direct transformation to compute and dump MO integrals that appear in the RHF MP2-R12 energy expression (MO integrals are used by program MP2R12)

- evaluate energy gradients at Hartree-Fock and correlated levels

1Two-electron integrals of Hermitian operators have the usual 8-fold permutation symmetry. Operator [r12,Ti] is non-Hermitian, hence a permutation of its integrals's bra and ket does not leave the integral invariant. See Wim Klopper's article in Theor. Chim. Acta..