Improved convergence of Hartree–Fock style excited-state wavefunctions using second-order optimisation with an exact Hessian

Abstract

An augmented optimisation of a previously described (Richings and Karadakov in Mol Phys 105:2363, 2007), variationally stable, Hartree–Fock style excited-state wavefunction is presented. The matrix of second derivatives (Hessian) of the electronic energy with respect to the molecular orbital coefficients is derived, and the matrix elements, necessary for the evaluation of the derivatives, are explicitly laid out. The Hessian is then used in a second-order optimisation procedure to demonstrate the significant improvement, in comparison with the simple steepest descent method used previously, in the rate of convergence of the energies of the selection of small molecules from the earlier work. The improvement is both in terms of the computational time required and in the tighter convergence of the gradient norms. The former factor is particularly significant when using an unrestricted reference wavefunction. A brief discussion of the merits and disadvantages of the use of the Hessian, as well as ideas for future work to improve to further improve the method, is also included.