Monte Carlo methods in electronic structures for large systems

Physical binding energies using the electron localization function in 4-hydroxyphenylboronic acid co-crystals with aza donors

Binding energies are traditionally simulated using cluster models by computation of each synthon for each individual co-crystal former. However, our investigation of the binding strengths using the electron localization function (ELF) reveals that these can be determined directly from the crystal supercell computations. We propose a new modeling protocol for the computation of physical binding energies directly from bulk simulations using ELF analysis. In this work, we establish a correlation between ELF values and binding energies calculated for co-crystals of 4-hydroxyphenylboronic acid (4HPBA) with four different aza donors using density functional theory with varying descriptions of dispersion. Boronic acids are gaining significant interest in the field of crystal engineering, but theoretical studies on their use in materials are still very limited. Here, we present a systematic investigation of the non-covalent interactions in experimentally realized co-crystals. Prior diffraction studies on these complexes have shown the competitive nature between the boronic acid functional group and the para-substituted phenolic group forming heteromeric interactions with aza donors. We determine the stability of the co-crystals by simulating their lattice energies, and the different dispersion descriptions show similar trends in lattice energies and lattice parameters. Our study bolsters the experimental observation of the boronic acid group as a competitive co-crystal former in addition to the well-studied phenolic group. Further research on correlating ELF values for physical binding could potentially transform this approach to a viable alternative for the computation of binding energies.

Preliminary Assessment of the First-Order Density Matrix in Quantum Monte Carlo from Density Matrix Theory

The trial wave function commonly used in the quantum Monte Carlo method consists of the product of up-spin and down-spin Slater determinants, allowing accurate calculations of multielectronic properties, although it is not antisymmetric under the exchange of electrons with opposite spins. An alternative description that overcomes these limitations using the Nth-order density matrix was already presented. This study introduces two new strategies based on the Dirac-Fock density matrix for QMC that still fully preserve antisymmetry and electron indistinguishability. Simulations are performed for the ground and excited states of He, Li, and Be showing that the present formulation and the conventional separation of spins are appropriate for a correct description of these systems, except for singlet excited states of the He and Be atoms, and that a part of the antisymmetry (antiparallel spins) can be neglected.

Stochastic evaluation of four-component relativistic second-order many-body perturbation energies: A potentially quadratic-scaling correlation method

A second-order many-body perturbation correction to the relativistic Dirac-Hartree-Fock energy is evaluated stochastically by integrating 13-dimensional products of four-component spinors and Coulomb potentials. The integration in the real space of electron coordinates is carried out by the Monte Carlo (MC) method with the Metropolis sampling, whereas the MC integration in the imaginary-time domain is performed by the inverse-CDF (cumulative distribution function) method. The computational cost to reach a given relative statistical error for spatially compact but heavy molecules is observed to be no worse than cubic and possibly quadratic with the number of electrons or basis functions. This is a vast improvement over the quintic scaling of the conventional, deterministic second-order many-body perturbation method. The algorithm is also easily and efficiently parallelized with demonstrated 92% strong scalability going from 64 to 4096 processors for a fixed job size.

Monte Carlo MP2-F12 for Noncovalent Interactions: The C60 Dimer

A scalable stochastic algorithm is presented that can evaluate explicitly correlated (F12) second-order many-body perturbation (MP2) energies of weak, noncovalent, intermolecular interactions. It first transforms the formulas of the MP2 and F12 energy differences into a short sum of high-dimensional integrals of Green's functions in real space and imaginary time. These integrals are then evaluated by the Monte Carlo method augmented by parallel execution, redundant-walker convergence acceleration, direct-sampling autocorrelation elimination, and control-variate error reduction. By sharing electron-pair walkers across the supermolecule and its subsystems spanned by the joint basis set, the statistical uncertainty is reduced by one to 2 orders of magnitude in the MP2 binding energy corrected for the basis-set incompleteness and superposition errors. The method predicts the MP2-F12/aug-cc-pVDZ binding energy of 19.1 ± 4.0 kcal mol^-1 for the C₆₀ dimer at the center distance of 9.748 Å.

Stochastic evaluation of fourth-order many-body perturbation energies

A scalable, stochastic algorithm evaluating the fourth-order many-body perturbation (MP4) correction to energy is proposed. Three hundred Goldstone diagrams representing the MP4 correction are computer generated and then converted into algebraic formulas expressed in terms of Green's functions in real space and imaginary time. They are evaluated by the direct (i.e., non-Markov, non-Metropolis) Monte Carlo (MC) integration accelerated by the redundant-walker and control-variate algorithms. The resulting MC-MP4 method is efficiently parallelized and is shown to display O(n^5.3) size-dependence of cost, which is nearly two ranks lower than the O(n⁷) dependence of the deterministic MP4 algorithm. It evaluates the MP4/aug-cc-pVDZ energy for benzene, naphthalene, phenanthrene, and corannulene with the statistical uncertainty of 10 mE_h (1.1% of the total basis-set correlation energy), 38 mE_h (2.6%), 110 mE_h (5.5%), and 280 mE_h (9.0%), respectively, after about 10⁹ MC steps.

Molecular Photochemistry: Recent Developments in Theory

Photochemistry is a fascinating branch of chemistry that is concerned with molecules and light. However, the importance of simulating light-induced processes is reflected also in fields as diverse as biology, material science, and medicine. This Minireview highlights recent progress achieved in theoretical chemistry to calculate electronically excited states of molecules and simulate their photoinduced dynamics, with the aim of reaching experimental accuracy. We focus on emergent methods and give selected examples that illustrate the progress in recent years towards predicting complex electronic structures with strong correlation, calculations on large molecules, describing multichromophoric systems, and simulating non-adiabatic molecular dynamics over long time scales, for molecules in the gas phase or in complex biological environments.

Nanoscale boron carbonitride semiconductors for photoredox catalysis

The conversion of solar energy to chemical energy achieved by photocatalysts comprising homogeneous transition-metal based systems, organic dyes, or semiconductors has received significant attention in recent years. Among these photocatalysts, boron carbon nitride (BCN) materials, as an emerging class of metal-free heterogeneous semiconductors, have extended the scope of photocatalysts due to their good performance and Earth abundance. The combination of boron (B), carbon (C), and nitrogen (N) constitutes a ternary system with large surface area and abundant activity sites, which together contribute to the good performance for reduction reactions, oxidation reactions and orchestrated both reduction and oxidation reactions. This Minireview reports the methods for the synthesis of nanoscale hexagonal boron carbonitride (h-BCN) and describes the latest advances in the application of h-BCN materials as semiconductor photocatalysts for sustainable photosynthesis, such as water splitting, reduction of CO₂, acceptorless dehydrogenation, oxidation of sp³ C-H bonds, and sp² C-H functionalization. h-BCN materials may have potential for applications in other organic transformations and industrial manufacture in the future.

Complementary first and second derivative methods for ansatz optimization in variational Monte Carlo

This Perspective contrasts first and second derivative methods in variational Monte Carlo and presents a hybrid optimization approach that combines their advantages.

Bias cancellation in one-determinant fixed-node diffusion Monte Carlo: Insights from fermionic occupation numbers

The accuracy of the fixed-node diffusion Monte Carlo (FNDMC) depends on the node location of the supplied trial state Ψ_{T}. The practical FNDMC approaches available for large systems rely on compact yet effective Ψ_{T}, most often containing an explicitly correlated single Slater determinant (SD). However, SD nodes may be better suited to one system than to another, which may possibly lead to inaccurate FNDMC energy differences. It remains a challenge how to estimate nonequivalence or appropriateness of SDs. Here we use the differences of a measure based on the Euclidean distance between the natural orbital occupation number (NOON) vector of the SD and the exact solution in the NOON vector space, which can be viewed as a measure of SD nonequivalence and as a qualitative measure of the expected degree of nondynamic-correlation-related bias in FNDMC energy differences. This is explored on a set of small noncovalent complexes and covalent bond breaking of Si_{2} vs N_{2}. It turns out that NOON-based measures well reflect the magnitude and sign of the bias present in the data available, thus providing insights into the nature of bias cancellation in SD FNDMC energy differences.

Generic expansion of the Jastrow correlation factor in polynomials satisfying symmetry and cusp conditions

Jastrow correlation factors play an important role in quantum Monte Carlo calculations. Together with an orbital based antisymmetric function, they allow the construction of highly accurate correlation wave functions. In this paper, a generic expansion of the Jastrow correlation function in terms of polynomials that satisfy both the electron exchange symmetry constraint and the cusp conditions is presented. In particular, an expansion of the three-body electron-electron-nucleus contribution in terms of cuspless homogeneous symmetric polynomials is proposed. The polynomials can be expressed in fairly arbitrary scaling function allowing a generic implementation of the Jastrow factor. It is demonstrated with a few examples that the new Jastrow factor achieves 85%-90% of the total correlation energy in a variational quantum Monte Carlo calculation and more than 90% of the diffusion Monte Carlo correlation energy.

Electron Structure Quantum Monte Carlo

The diffusion quantum Monte Carlo method (DMC) is capable of calculating accurately the electronic energy for molecules and molecular aggregates. An overview is given on recent developments for the optimization of the guide function that determines the accuracy of the method. Furthermore, the versatility of DMC is shown with applications to Rydberg states, transition metal compounds, and weakly interacting systems.

Periodic orbits of the ensemble of Sinai-Arnold cat maps and pseudorandom number generation

We propose methods for constructing high-quality pseudorandom number generators (RNGs) based on an ensemble of hyperbolic automorphisms of the unit two-dimensional torus (Sinai-Arnold map or cat map) while keeping a part of the information hidden. The single cat map provides the random properties expected from a good RNG and is hence an appropriate building block for an RNG, although unnecessary correlations are always present in practice. We show that introducing hidden variables and introducing rotation in the RNG output, accompanied with the proper initialization, dramatically suppress these correlations. We analyze the mechanisms of the single-cat-map correlations analytically and show how to diminish them. We generalize the Percival-Vivaldi theory in the case of the ensemble of maps, find the period of the proposed RNG analytically, and also analyze its properties. We present efficient practical realizations for the RNGs and check our predictions numerically. We also test our RNGs using the known stringent batteries of statistical tests and find that the statistical properties of our best generators are not worse than those of other best modern generators.

A diffusion quantum Monte Carlo study on the lowest singlet and triplet electronic states of BN molecule

Ab initio calculation of both the lowest singlet and triplet electronic states of BN has been performed by the fixed-node Ornstein-Uhlenbeck diffusion quantum Monte Carlo method with the floating spherical Gaussian orbitals and spherical Gaussian geminals. The Monte Carlo calculation gives equilibrium bond lengths and equilibrium harmonic frequencies of 1.3317(7) A and 1529(7) cm(-1), respectively, for the lowest triplet state and 1.2751(7) A and 1709(8) cm(-1), respectively, for the lowest singlet state. Also, the Monte Carlo calculation reports an energy separation of 178(83) cm(-1) between the two electronic states and recommends the ground state is the lowest triplet state.

Accuracy of a random-walk-based approach in the determination of equilibrium bond lengths and harmonic frequencies for some doublet first-row diatomic radicals

The accuracy of equilibrium bond lengths and harmonic frequencies for 12 doublet first-row diatomic radicals is presented as predicted by the fixed-node diffusion quantum Monte Carlo method based on the Ornstein-Uhlenbeck random walk guided by the floating spherical Gaussian orbital and spherical Gaussian geminal-type trial wave function. Compared to the experimental determined values, the random-walk-based approach gives the absolute mean deviations of 0.0019 A and 18 cm-1 for the equilibrium bond length and harmonic frequency, respectively. We also compare the random-walk-based results with some coupled-cluster-based values.

A version of diffusion Monte Carlo method based on random grids of coherent states. II. Six-dimensional simulation of electronic states of H2

We report a new version of the diffusion Monte Carlo (DMC) method, based on coherent-state quantum mechanics. Randomly selected grids of coherent states in phase space are used to obtain numerical imaginary time solutions of the Schrodinger equation, with an iterative refinement technique to improve the quality of the Monte Carlo grid. Accurate results were obtained, for the appropriately symmetrized two lowest states of the hydrogen molecule, by Monte Carlo sampling and six-dimensional propagation in the full phase space.

Theoretical study of transition state structure and reaction enthalpy of the F+H2→HF+H reaction by a diffusion quantum Monte Carlo approach

Ab initio calculations of transition state structure and reaction enthalpy of the F + H2-->HF + H reaction has been carried out by the fixed-node diffusion quantum Monte Carlo method in this study. The Monte Carlo sampling is based on the Ornstein-Uhlenbeck random walks guided by a trial wave function constructed from the floating spherical Gaussian orbitals and spherical Gaussian geminals. The Monte Carlo calculated barrier height of 1.09(16) kcal/mol is consistent with the experimental values, 0.86(10)/1.18(10) kcal/mol, and the calculated value from the multireference-type coupled-cluster (MRCC) calculation with the aug-cc-pVQZ(F)/cc-pVQZ(H) basis set, 1.11 kcal/mol. The Monte Carlo-based calculation also gives a similar value of the reaction enthalpy, -32.00(4) kcal/mol, compared with the experimental value, -32.06(17) kcal/mol, and the calculated value from a MRCC/aug-cc-pVQZ(F)/cc-pVQZ(H) calculation, -31.94 kcal/mol. This study clearly indicates a further application of the random-walk-based approach in the field of quantum chemical calculation.

Local-MP2 electron correlation method for nonconducting crystals

Rigorous methods for the post-HF (HF-Hartree-Fock) determination of correlation corrections for crystalline solids are currently being developed following different strategies. The CRYSTAL program developed in Torino and Daresbury provides accurate HF solutions for periodic systems in a basis set of Gaussian type functions; for insulators, the occupied HF manifold can be represented as an antisymmetrized product of well localized Wannier functions. This makes possible the extension to nonconducting crystals of local correlation linear scaling On techniques as successfully and efficiently implemented in Stuttgart's MOLPRO program. These methods exploit the fact that dynamic electron correlation effects between remote parts of a molecule (manifesting as dispersive interactions in intermolecular perturbation theory) decay as an inverse sixth power of the distance R between these fragments, that is, much more quickly than the Coulomb interactions that are treated already at the HF level. Translational symmetry then permits the crystalline problem to be reduced to one concerning a cluster around the reference zero cell. A periodic local correlation program (CRYSCOR) has been prepared along these lines, limited for the moment to the solution of second-order Moller-Plesset equations. Exploitation of point group symmetry is shown to be more important and useful than in the molecular case. The computational strategy adopted and preliminary results concerning five semiconductors with tetrahedral structure (C, Si, SiC, BN, and BeS) are presented and discussed.