Analytical derivatives in quantum chemistry.

by Pulay, Peter

Anal. energy derivs., particularly those with respect to the coordinates of the nuclei, made, together with d. functional theory, modern computational chem. a widely used and indispensable tool in chem. research. Anal. derivs. do not offer anything that cannot be calcd. in principle from energies only. However, their advantages in efficiency and numerical accuracy makes their use almost mandatory for larger systems. The first step in quantum mech. modeling, the optimization of mol. geometries, is virtually impossible for large mols. without anal. derivs. Ab initio mol. dynamics, the detn. of transition states, IR and Raman frequencies and intensities, non-Born-Oppenheimer couplings, NMR chem. shifts and other magnetic properties, vibrational CD, among others require anal. derivs. to be computationally feasible. Although closely related to perturbation theory, deriv. theory differs (and is more demanding) by using perturbation-dependent basis sets. The key original insight was that energy derivs. (gradients) can be evaluated straightforwardly with a computational effort that is comparable (and often less) than that of the energy evaluation, even though the quantum mech. energy is the result of a complicated iterative procedure and cannot be written as a simple function of the perturbational variables (say nuclear coordinates). Efficient first derivs. were at first restricted to energy expressions obtained as (constrained) min., e.g., the Hartree-Fock energy. A major further advance was the realization that this can be generalized to all energy expressions, variational or not, at a reasonable expense. A mol. with N nuclei has 3N nuclear forces (neg. gradients), giving potentially a speed-up approaching 3N. Unfortunately, this is not true for second derivs. which scale the same way as the numerical derivs. of the gradients. However, second derivs. still offer significant savings and increased numerical accuracy, at the cost of greatly increased demand for computer resources. Higher derivs. can be formulated but have not been widely used, although third derivs. offer significant advantages. The talk will focus on concepts and computational efficiency, in an historical context encompassing the past 60 years.