Diffusion Monte Carlo approaches for studying nuclear quantum effects in fluxional molecules

Abstract

Diffusion quantum Monte Carlo (DMC) provides a powerful approach for obtaining the ground state energy and wave function of molecules, ions, and molecular clusters. The approach is uniquely well suited for studies of fluxional molecules, which undergo large amplitude vibrational motions even in their ground state. In contrast to the electronic structure problem, where the wave function must be antisymmetric with respect to exchange of any pair of electrons, the wave function for the ground vibrational state is nodeless. This greatly simplifies the application of DMC for vibrational problems. Because there is not a single potential function that can be used to describe the intramolecular and intermolecular interactions in all molecular systems, most methods that are used to describe nuclear quantum effects rely on a carefully chosen zero-order description of the molecular vibrations. In contrast, DMC calculations can be performed in Cartesian coordinates, making the DMC algorithm easily transferable between different chemical systems. In this contribution, the theory that underlies DMC will be discussed along with important considerations for performing DMC calculations. Extensions for evaluating vibrationally excited states and molecular properties are also discussed. Insights that can be obtained from DMC calculations are illustrated in the context of the protonated water clusters.

Optimal Mode Combination in the Multiconfiguration Time-Dependent Hartree Method through Multivariate Statistics: Factor Analysis and Hierarchical Clustering

“…In the simplest case, when a ground state is sought, this density is given by the absolute square of the ground state wave function. Here, we apply the diffusion Monte Carlo (DMC) method, although one might think of a number of alternative strategies (e.g., quasi-classical normal mode sampling, metropolis sampling, or snapshots of molecular dynamics trajectories). From this discrete distribution in Cartesian coordinates, we compute the corresponding correlation matrix for the variables in the desired system of coordinates. Again, a number of approaches might be applied (e.g., Pearson correlation, distance correlation, mutual information, etc.…”

“…The probability density thus can be correspondingly estimated as ∥ ϕ 0 ( r , τ ) ∥ 2 = ∑ j = 1 n w j δ ( r − r j false( τ false) ) where the index j runs over the number of walkers n , and the amplitudes w j are obtained through the descendent weighting approach. We refer the reader to refs , , and for more details about the procedure. From these, the expectation value and variance for any multiplicative operator of interest (e.g., potential energy, dipole moment, coordinate, etc.)…”

Isotope Effects on Ground and Excited States of Ethyl Cation, H+(C2H4)

“…Likewise, application of the propagator in the potential energy leads the weight of the i th walker to become w i ( τ + Δ τ ) = exp [ − false( V ( x i false( τ + normalΔ τ false) ) − E ref ( τ ) false) Δ τ ] w i ( τ ) In order to retain the form for the wave function that is provided by eq where all of the weights are equal to one, the value of w i (τ + Δτ ) is used to determine the number of walkers at x i (τ) at the start of the next iteration of the propagation. Specifically, the integer part of w i (τ + Δτ ) provides the number of walkers at this geometry, while the fractional part provides the probability that one additional walker is added at that configuration. , …”

“…Once this is done for the parts of the wave function with positive and negative amplitude, the location of the node is found by identifying the value of the scanned coordinate at which the energies obtained from the two simulations are equal. We refer to this approach as adiabatic DMC or ADMC, and more complete descriptions of this approach and our implementation can be found elsewhere. ,, …”

“…We refer to this approach as adiabatic DMC or ADMC, and more complete descriptions of this approach and our implementation can be found elsewhere. 23,34,35 Once the position of the node is known, DMC simulations are performed that focus on the configurations of the molecule in which the amplitude of the wave function is either positive or negative. Within this region, the part of the excited state wave function that is being considered resembles a ground state, where the wave function goes to zero at the node with finite slope.…”