Project title: Calculating total energy derivatives using quantum Monte Carlo
Quantum Monte Carlo (QMC) methods are powerful techniques for calculating the electronic structure of atoms, molecules and solids with an accuracy comparable to the standard ab initio methods of quantum chemistry. The two most commonly used methods, variational Monte Carlo and the higher accuracy fixed-node diffusion Monte Carlo (DMC), rely on the stochastic sampling of an accurate many-body wave function with a cost that scales favourably when compared to post-Hartree-Fock methods such as configuration interaction and coupled cluster calculations. This makes the QMC approach the ideal candidate for computing the total energies of large collections of interacting quantum particles where conventional density functional theory methods fail to capture the role of dispersion interactions or electronic correlation.
Despite great success in computing total energies, and a number of recent advances in the area, the calculation of total energy derivatives via DMC remains problematic. The principal difficulty is that the fully interacting many-body wave function is represented solely by a distribution of sampling points which lack an algebraic formulation. Whilst this allows the calculation of total energies that are free from basis set restrictions, the wave function derivatives required for analytic total energy derivatives are undefined. Calculating derivatives using a finite difference approach also encounters difficulties: as the finite difference step size approaches zero, the stochastic error in the energy difference from two independent energy evaluations remains constant, leading to a divergence in the error of the derivative.
An alternative route to total energy derivatives is to use algorithmic differentiation (AD), a programming technique for the efficient evaluation of the derivatives of a computed function through the repeated application of the chain rule to the lines of computer code defining the function's operation. This technique allows access to the derivatives of functions that are too complex to permit an algebraic solution; any computed function can be decomposed into a (potentially very long) sequential list of basic arithmetic operations, each of which is differentiable. The primary attraction of the AD method is that the `reverse mode' of operation allows access to the derivatives of a program output with respect to many inputs, simultaneously, in a small multiple (the value of which is independent of the number of inputs) of the computational cost of evaluating the underlying function. Cheap access to accurate DMC energy gradients (the derivative with respect to multiple variables) motivates two main applications: wave function optimisation and molecular dynamics (MD) simulations. Energy gradients with respect to the wave function parameters enable the application of gradient-based wave function optimisation schemes, such as the conjugate gradients and BFGS methods; gradients with respect to the nuclear positions enable the calculation of the many-body force calculations required by MD simulations in feasible time scales.
Finite difference total energy derivatives can be calculated in DMC using a correlated sampling method. Correlated sampling reduces the error in the relative energies of two related systems, with the error vanishing in the limit of the two systems becoming identical, by coupling the sampling distribution in each. More details of the approach employed can be found athttp://www.cmth.ph.ic.ac.uk/people/t.poole/research.html