New algorithm for high-order time-dependent hartree–fock theory for nonlinear optical properties

A new formalism and algorithm is developed for solving the general-order time-dependent Hartree–Fock (TDHF) problem. It is shown that for any order a generalization of the TDHF equations can be derived where all lower-order solutions constitute a constant term. This makes it very easy to obtain high-order solutions. As the space required for the mapping of density matrices to Fock matrices in a problem of a giver order is largely reduced, we can perform the most time-consuming steps within the core memory of the machine and easily manipulate vector products via optimum routines. The second hyperpolarizability γ is obtained from the secondorder TDHF solution via a 2n rule. The formalism also allows for expressing all terms in the equation diagrammatically, which provides additional physical insight and a more systematic evaluation of terms. To illustrate the method, TDHF results are presented for trans-butadiene and carbon monoxide for several optical processes, including correlation corrections to their static hyperpolarizabilities obtained via coupled cluster (CCSD) and many-body perturbation theory. The hybrid TDHF/CCSD method provides excellent agreement with the DC–SHG experiments (χ||⁽²⁾ = 11.4 x 10^?32 esu/mol compared to 12.9 ± 11.4 × 10^?32 esu/mol and χ||⁽³⁾ = 149 compared to 144 ± 4 × 10^?39 esu/mol).