Geometry optimization in quantum Monte Carlo with solution mapping: Application to formaldehyde

Abstract

Abstract Since analytic gradients of the energy with respect to nuclear positions are not easily obtained in QMC calculations, determining the locations of critical points, such as energy minima and saddle points, on potential energy surfaces is not straightforward. The statistical uncertainty in QMC energies prevents the direct estimation of these derivatives, and although several techniques for overcoming the problem have been developed their applications have been limited to very small systems. This paper describes a different approach, more readily applicable, in which the number and location of points in the vicinity of a critical point are chosen in such a way as to minimize the computation effort required in predicting the location of the critical point. The method is generally known as “solution mapping” and is often referred to as “factorial design.” The procedure is illustrated with calculations to determine the minimum energy or equilibrium geometry of formaldehyde H2CO, assumed planar and symmetric about the C-0 bond, in terms of the C-0 and C-H bond lengths and the H-C-0 angle. The potential energies calculated at specified points are fit to a polynomial expression of ten terms involving the three variables as l,x1,x2,x3,XI,X ,x ,x1x2,x1x3,x2x3, and ten coeffcients. With 15 points prescribed by the factorial design the determination of the coefficients is simplified by the column orthogonality of the design matrix. Of course, statistical uncertainties in the energies lead to uncertainties in the coefficients, but the method minimizes their effects.