An algorithm to sample the exact (within the nodal error) ground state distribution to find nondifferential properties of multielectron systems is developed and applied to first-row atoms. Calculated properties are the distribution moments and the electronic density at the nucleus (expected value of the δ operator). For this purpose compact trial functions are developed and optimized, and a new estimator for the δ is formulated. A comparison is made with results of highly accurate post-Hartree-Fock calculations, thereby illuminating the nodal error in our estimates. In general, we obtain more accurate estimates for the distribution moments than those obtained previously using Monte Carlo methods, despite the relative crudeness of our trial functions. We confirm the literature values for the electron density at the nucleus for the lighter atoms (Li-C), but disagree with previous (Monte Carlo) estimates for the heavier ones (N-Ne).
By introducing perturbations of O(τ) type (where τ is the time step used in the simulation) to our diffusion quantum Monte Carlo algorithm we obtain a simulated energy which is reasonably constant over a wide range of τ values. Reliable estimation of the fixed-node energy (τ=0 intercept) results, as extrapolation becomes more robust and a radical change in small τ behavior becomes less likely. We apply our techniques to the problem of estimating the ground-state energy of LiH and H2O.
We show how to estimate, for a given molecule, the first and higher derivatives of the expected value of an operator with respect to one or more physical parameters. This is done with high accuracy achieved by sampling to within a certain approximation from the exact electron distribution, compatible with the Hellmann–Feynman theorem. Finite difference approximations are avoided. The required derivatives of the unknown exact wave function are determined by averaging expressions involving only the total serial correlation of known quantities. The operator is not restricted to the case of the molecular Hamiltonian. This allows for computation of virtually all ground-state properties of a molecule by a single, relatively trivial computer program. Our formulas are presented and applied in the context of a diatomic molecule (LiH), but they can be readily extended to polyatomics.
We show how to extend the formalism of infinitesimal differential diffusion quantum Monte Carlo to the case of higher derivatives of the ground-state energy of a molecule with respect to the molecular geometry. We use LiH as an example, but the technique can be extended to more complicated, nonliner molecules as well. We obtain good agreement with experimental values for the energy derivatives and for the harmonic and anharmonic frequencies of LiH and LiD, despite using a compact single-determinant wave function.
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