Nodal Pulay terms for accurate diffusion quantum Monte Carlo forces

The diffusion quantum Monte Carlo DMC method is the most accurate approach available for calculating the total energies of solids and large molecules.1 Energy gradients are also of great significance in quantum mechanical calculations, because they give the forces on atoms which may be used to relax structures and perform molecular dynamics simulations. Progress in such calculations using DMC methods has, however, been held up by difficulties encountered in evaluating forces accurately and efficiently. The DMC method is based on imaginary time evolution, which projects out the lowest energy many-body wave function.1 The fermionic symmetry is maintained by fixing the nodal surface the surface on which the wave function is zero and across which it changes sign to be that of an antisymmetric trial wave function, T. The nodal surface divides the wave function into nodal pockets, and the DMC algorithm gives the lowest energy solution within each pocket. The Hellmann-Feynman theorem HFT implies that the force is given by the expectation value of the gradient of the Hamiltonian with respect to the relevant parameter, , when the wave function is an exact eigenstate.2,3 Standard fixednode DMC samples the “mixed” probability distribution T . It is straightforward to evaluate the HFT expression for the energy gradient within DMC, but it does not give the exact gradient of the DMC energy unless T is exact. If T is not exact, the correct energy gradient is obtained only when the Pulay correction terms4 are included, which contain the gradient of the wave function with respect to . Mixed DMC calculations of forces including approximate Pulay terms have been reported in Refs. 5 and 6. One approach to reducing the size of the Pulay terms is to evaluate the “pure” estimate of the HFT operator7 by sampling the probability distribution . This approach still does not produce the exact gradient of the DMC energy unless the trial nodal surface is exact, because it neglects a Pulay term which can be written as an integral over the nodal surface. The existence of this nodal term in the pure estimate was pointed out in Ref. 8, and an explicit expression for it was given in Ref. 9. However, a practical scheme for evaluating this nodal term was not developed and it has been neglected in force calculations.7,10 Here, we show that the gradients of both the mixed and pure estimates of the energy contain nodal Pulay terms, and we describe and test a practical scheme for estimating them. This paper is organized as follows: In Sec. II, we introduce the DMC energy when integrating over a single nodal pocket. In Sec. III, we derive exact expressions for the first derivative of the DMC energy, and in Sec. IV, we give practical expressions for estimating them. In Sec. V, we present and discuss the results obtained for a test system, and we draw our conclusions in Sec. VI.