Translational eigenstates of He@C60 from four-dimensional ab initio potential energy surfaces interpolated using Gaussian process regression.

We investigate the endofullerene system 3He@C60 with a four-dimensional potential energy surface (PES) to include the three He translational degrees of freedom and C60 cage radius. We compare second order Møller-Plesset perturbation theory (MP2), spin component scaled-MP2, scaled opposite spin-MP2, random phase approximation (RPA)@Perdew, Burke, and Ernzerhof (PBE), and corrected Hartree-Fock-RPA to calibrate and gain confidence in the choice of electronic structure method. Due to the high cost of these calculations, the PES is interpolated using Gaussian Process Regression (GPR), owing to its effectiveness with sparse training data. The PES is split into a two-dimensional radial surface, to which corrections are applied to achieve an overall four-dimensional surface. The nuclear Hamiltonian is diagonalized to generate the in-cage translational/vibrational eigenstates. The degeneracy of the three-dimensional harmonic oscillator energies with principal quantum number n is lifted due to the anharmonicity in the radial potential. The (2l + 1)-fold degeneracy of the angular momentum states is also weakly lifted, due to the angular dependence in the potential. We calculate the fundamental frequency to range between 96 and 110 cm-1 depending on the electronic structure method used. Error bars of the eigenstate energies were calculated from the GPR and are on the order of ∼±1.5 cm-1. Wavefunctions are also compared by considering their overlap and Hellinger distance to the one-dimensional empirical potential. As with the energies, the two ab initio methods MP2 and RPA@PBE show the best agreement. While MP2 has better agreement than RPA@PBE, due to its higher computational efficiency and comparable performance, we recommend RPA as an alternative electronic structure method of choice to MP2 for these systems.

Exploring the parameter space of an endohedral atom in a cylindrical cavity.

Endohedral fullerenes, or endofullerenes, are chemical systems of fullerene cages encapsulating single atoms or small molecules. These species provide an interesting challenge of Potential Energy Surface determination as examples of non-covalently bonded, bound systems. While the majority of studies focus on C60 as the encapsulating cage, introducing some anisotropy by using a different fullerene, e.g., C70 can unveil a double well potential along the unique axis. By approximating the potential as a pairwise Lennard-Jones (LJ) summation over the fixed C cage atoms, the parameter space of the Hamiltonian includes three tunable variables: (M, ɛ, σ) representing the mass of the trapped species, the LJ energy, and length scales respectively. Fixing the mass and allowing the others to vary can imitate the potentials of endohedral species trapped in more elongated fullerenes. We choose to explore the LJ parameter space of an endohedral atom in C70 with ɛ ∈ [20, 150 cm-1], and σ ∈ [2.85, 3.05 Å]. As the barrier height and positions of these wells vary between [1, 264 cm-1] and [0.35, 0.85 Å] respectively, using a 3D direct product basis of 1D harmonic oscillator (HO) wavefunctions centred at the origin where there is a local maximum is unphysical. Instead we propose the use of a non-orthogonal basis set, using 1D HO wavefunctions centred in each minimum and compare this to other choices. The ground state energy of the X@C70 is tracked across the LJ parameter space, along with its corresponding nuclear translational wavefunctions. A classification of the wavefunction characteristics, namely the prolateness and "peanut-likeness" based on its statistical moments is also proposed. Excited states of longer fullerenes are assigned quantum numbers, and the fundamental transitions of Ne@C70 are tracked across the parameter space.

Large scale parallelization in stochastic coupled cluster.

Coupled cluster theory is a vital cornerstone of electronic structure theory and is being applied to ever-larger systems. Stochastic approaches to quantum chemistry have grown in importance and offer compelling advantages over traditional deterministic algorithms in terms of computational demands, theoretical flexibility, or lower scaling with system size. We present a highly parallelizable algorithm of the coupled cluster Monte Carlo method involving sampling of clusters of excitors over multiple time steps. The behavior of the algorithm is investigated on the uniform electron gas and the water dimer at coupled-cluster levels including up to quadruple excitations. We also describe two improvements to the original sampling algorithm, full non-composite, and multi-spawn sampling. A stochastic approach to coupled cluster results in an efficient and scalable implementation at arbitrary truncation levels in the coupled cluster expansion.

Understanding and improving the efficiency of full configuration interaction quantum Monte Carlo.

Within full configuration interaction quantum Monte Carlo, we investigate how the statistical error behaves as a function of the parameters which control the stochastic sampling. We define the inefficiency as a measure of the statistical error per particle sampling the space and per time step and show there is a sizeable parameter regime where this is minimised. We find that this inefficiency increases sublinearly with Hilbert space size and can be reduced by localising the canonical Hartree-Fock molecular orbitals, suggesting that the choice of basis impacts the method beyond that of the sign problem.

Linked coupled cluster Monte Carlo.

We consider a new formulation of the stochastic coupled cluster method in terms of the similarity transformed Hamiltonian. We show that improvement in the granularity with which the wavefunction is represented results in a reduction in the critical population required to correctly sample the wavefunction for a range of systems and excitation levels and hence leads to a substantial reduction in the computational cost. This development has the potential to substantially extend the range of the method, enabling it to be used to treat larger systems with excitation levels not easily accessible with conventional deterministic methods.

Minimising biases in full configuration interaction quantum Monte Carlo.

We show that Full Configuration Interaction Quantum Monte Carlo (FCIQMC) is a Markov chain in its present form. We construct the Markov matrix of FCIQMC for a two determinant system and hence compute the stationary distribution. These solutions are used to quantify the dependence of the population dynamics on the parameters defining the Markov chain. Despite the simplicity of a system with only two determinants, it still reveals a population control bias inherent to the FCIQMC algorithm. We investigate the effect of simulation parameters on the population control bias for the neon atom and suggest simulation setups to, in general, minimise the bias. We show a reweight ing scheme to remove the bias caused by population control commonly used in diffusion Monte Carlo [Umrigar et al., J. Chem. Phys. 99, 2865 (1993)] is effective and recommend its use as a post processing step.