Mario Motta

Imaginary time correlations and the phaseless auxiliary field quantum Monte Carlo

The phaseless Auxiliary Field Quantum Monte Carlo (AFQMC) method provides a well established approximation scheme for accurate calculations of ground state energies of many-fermions systems. Here we address the possibility of calculating imaginary time correlation functions with the phaseless AFQMC. We give a detailed description of the technique and test the quality of the results for static properties and imaginary time correlation functions against exact values for small systems.

Imaginary time density-density correlations for two-dimensional electron gases at high density

We evaluate imaginary time density-density correlation functions for two-dimensional homogeneous electron gases of up to 42 particles in the continuum using the phaseless auxiliary field quantum Monte Carlo method. We use periodic boundary conditions and up to 300 plane waves as basis set elements. We show that such methodology, once equipped with suitable numerical stabilization techniques necessary to deal with exponentials, products, and inversions of large matrices, gives access to the calculation of imaginary time correlation functions for medium-sized systems. We discuss the numerical stabilization techniques and the computational complexity of the methodology and we present the limitations related to the size of the systems on a quantitative basis. We perform the inverse Laplace transform of the obtained density-density correlation functions, assessing the ability of the phaseless auxiliary field quantum Monte Carlo method to evaluate dynamical properties of medium-sized homogeneous fermion systems.

Ab initio computations of molecular systems by the auxiliary-field quantum Monte Carlo method

The auxiliary‐field quantum Monte Carlo (AFQMC) method provides a computational framework for solving the time‐independent Schrödinger equation in atoms, molecules, solids, and a variety of model systems. AFQMC has recently witnessed remarkable growth, especially as a tool for electronic structure computations in real materials. The method has demonstrated excellent accuracy across a variety of correlated electron systems. Taking the form of stochastic evolution in a manifold of nonorthogonal Slater determinants, the method resembles an ensemble of density‐functional theory (DFT) calculations in the presence of fluctuating external potentials. Its computational cost scales as a low‐power of system size, similar to the corresponding independent‐electron calculations. Highly efficient and intrinsically parallel, AFQMC is able to take full advantage of contemporary high‐performance computing platforms and numerical libraries. In this review, we provide a self‐contained introduction to the exact and constrained variants of AFQMC, with emphasis on its applications to the electronic structure of molecular systems. Representative results are presented, and theoretical foundations and implementation details of the method are discussed.

Communication: Calculation of interatomic forces and optimization of molecular geometry with auxiliary-field quantum Monte Carlo

We propose an algorithm for accurate, systematic, and scalable computation of interatomic forces within the auxiliary-field quantum Monte Carlo (AFQMC) method. The algorithm relies on the Hellmann-Feynman theorem and incorporates Pulay corrections in the presence of atomic orbital basis sets. We benchmark the method for small molecules by comparing the computed forces with the derivatives of the AFQMC potential energy surface and by direct comparison with other quantum chemistry methods. We then perform geometry optimizations using the steepest descent algorithm in larger molecules. With realistic basis sets, we obtain equilibrium geometries in agreement, within statistical error bars, with experimental values. The increase in computational cost for computing forces in this approach is only a small prefactor over that of calculating the total energy. This paves the way for a general and efficient approach for geometry optimization and molecular dynamics within AFQMC.

Review of Probability Theory

This chapter provides a self-contained review of the foundations of probability theory, in order to fix notations and introduce mathematical objects employed in the remaining chapters. In particular we stress the notions of measurability, related to what it is actually possible to observe when performing an experiment, and statistical independence. We present several tools to deal with random variables; in particular we focus on the normal random variables, a cornerstone in the theory of random phenomena. We conclude presenting the law of large numbers and the central limit theorem.

Applications to Mathematical Statistics

In this chapter we present applications of probability theory within the science of extracting information from data: mathematical statistics. We present, on a rigorous basis, the theory of statistical estimators and some of the most widely employed hypothesis tests. Finally, we briefly discuss linear regression, a mandatory topic for physicists.

Stochastic Calculus and Introduction to Stochastic Differential Equations

In this chapter we introduce the basic notions of stochastic calculus, starting from the Brownian motion. Stochastic processes describe time-dependent random phenomena, generalizing the usual deterministic evolution. The description of the latter requires the notions of differential and integral, which need to be properly extended to stochastic properties. Stochastic calculus is the branch of mathematics dealing with this important topic. The reason why traditional calculus is not suitable for stochastic processes is revealed by the Brownian motion. Since \(Var(B_t) = t\), implying that \(B_t\) “scales” as \(\sqrt{t}\), its trajectories are not differentiable in the usual sense. The stochastic calculus allows us to introduce generalized notions of differential and integral, notwithstanding this difficulty. Moreover, it allows to write differential equations involving stochastic processes, providing thus a powerful generalization of ordinary differential equations to study phenomena evolving in time in a non deterministic way.

Sampling of Random Variables and Simulation

In this chapter we introduce the art of sampling of random variables. Sampling a random variable X, for example real valued, means using a random number generator to generate n real numbers \((x_1,\ldots ,x_n)\), realizations of a sample \((X_1,\ldots ,X_n)\) of independent and identically distributed random variables sharing the same law as X. The ability of sampling is crucial to deal with integrals in high dimensions, appearing in quantum mechanics and in statistical physics, and gives the possibility to simulate physical systems. We first introduce simple tools to sample random variables, that can be used only in quite special situations. In the last part of the chapter, we will introduce and discuss a very general sampling technique, relying on the Metropolis theorem.

Conditional Probability and Conditional Expectation

In this chapter we deal with conditional probability. After having sketched the elementary definitions, we introduce the advanced notion of conditional expectation of a random variable with respect to a given \(\sigma \)-field. The intuitive meaning of the conditional expectation is the best prediction we can do about the values taken by the random variable, once we have observed the family of events inside the \(\sigma \)-field. The conditional expectation is widely used in the theory of stochastic processes we will present in the following chapters.