  We study several approaches to orbital optimization in selected configuration
interaction plus perturbation theory (SCI+PT) methods, and test them on the
ground and excited states of three molecules using the semistochastic heatbath
configuration interaction (SHCI) method. We discuss the ways in which the
orbital optimization problem in SCI resembles and differs from that in complete
active space self-consistent field (CASSCF). Starting from natural orbitals,
these approaches divide into three classes of optimization methods according to
how they treat coupling between configuration interaction (CI) coefficients and
orbital parameters, namely uncoupled, fully coupled, and quasi-fully coupled
methods. We demonstrate that taking the coupling into account is crucial for
fast convergence and recommend two quasi-fully coupled methods for such
applications: accelerated diagonal Newton and Broyden-Fletcher-Goldfarb-Shanno
(BFGS).
