Optimization of quantum Monte Carlo wave functions by energy minimization

We study three wave function optimization methods based on energyminimization in a variational Monte Carlo framework: the Newton, linear andperturbative methods. In the Newton method, the parameter variations arecalculated from the energy gradient and Hessian, using a reduced variancestatistical estimator for the latter. In the linear method, the parametervariations are found by diagonalizing a non-symmetric estimator of theHamiltonian matrix in the space spanned by the wave function and itsderivatives with respect to the parameters, making use of a strongzero-variance principle. In the less computationally expensive perturbativemethod, the parameter variations are calculated by approximately solving thegeneralized eigenvalue equation of the linear method by a nonorthogonalperturbation theory. These general methods are illustrated here by theoptimization of wave functions consisting of a Jastrow factor multiplied by anexpansion in configuration state functions (CSFs) for the C$_2$ molecule,including both valence and core electrons in the calculation. The Newton andlinear methods are very efficient for the optimization of the Jastrow, CSF andorbital parameters. The perturbative method is a good alternative for theoptimization of just the CSF and orbital parameters. Although the optimizationis performed at the variational Monte Carlo level, we observe for the C$_2$molecule studied here, and for other systems we have studied, that as moreparameters in the trial wave functions are optimized, the diffusion Monte Carlototal energy improves monotonically, implying that the nodal hypersurface alsoimproves monotonically.Comment: 18 pages, 8 figures, final versio