Quantum Monte Carlo Methods Sijia Chen University of Chicago

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Abstract Understanding many-body quantum systems usually means solving the Schrödinger equation of a system of many strongly interacting particles. However, there is no analytical solution for most cases. Therefore, numerical approaches are popular in this area. One kind of the famous methods is quantum Monte Carlo. Two sampling schemes based on the Markov chain Monte Carlo introduced in this project are variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). First, we provide a brief introduction to the important Metropolis-Hastings algorithm, which is of great importance during simulation. Then, we review in detail the basic idea and principles of VMC and DMC. Throughout this project, we discuss the applications of these two methods to the first row elements and show the importance of decent trial function choices.

Introduction Quantum effects become to dominant when size of the system of interest is getting smaller and smaller, namely energy is no longer a continuous, but a discrete physical property. It is necessary to use a quantum view to look at microscopic systems which may lead to unusual results, such as the famous blackbody radiation. After a quick development of quantum mechanics in last century, we already have an increasingly deeper understanding of it. In general, we use a language different from classical mechanics to describe the motion and other information of microscopic particles, that is the 'wavefunction' language. The wavefunction itself doesn't have a physical interpretation, however, the square of the wavefunction represents the probability density of the system in a particular state. Thus all information of a system hides behind its wavefunction. It is of priority to solve the wavefunction accurately in quantum mechanics. In order to achieve this goal, we have to write down Schrödinger equation first, where is the wavefunction and is the Hamiltonian operator of a given system, which is an observable and corresponds to the overall energy of the system, then solve the wavefunction. For simple systems, like hydrogen-like atoms with only one electrons in the orbital, particle-in-a-box, and harmonic oscillators, it it possible to solve the exact wavefunctions for them. However, for many-body systems, such as atoms with more than one electrons, we cannot solve analytical exact solutions for them. The Schrödinger equation of a N-electron molecule (or atom) with fixed nuclei (in atomic units) can be written as , where is given by

{ the spatial positions of the N electrons, = the Laplacian operator for electron i of coordinates , and the potential energy function expressed as

where is the interelectronic distance, Z the charge of nuleus, R its position, r = the distance between electron and nucleus, and the internuclear distance. Approximations are needed to give an approximate solution for this system and then we can understand how different parameters impact the system. Nowadays, the most popular and well-established methods are the post-Hartree-Fock approach and Density Functional Theory (DFT). Except for these, numerical approaches become prominent in this area as well. Quantum Monte Carlo (QMC) can be seen as an alternative method which is developed for overcoming the limitations of post-Hartree-Fock and DFT. In recent years, a number of remarkable progresses have been made in QMC, and some of QMC methods have achieve high chemical accuracy. In this project, we shall present the two most popular QMC methods, namely, the variational Monte Carlo (VMC) and the fixed-node diffusion Monte Carlo (FN- DMC) methods. They are both stochastic schemes that can extract important information about the ground state of a many-body system.

Theory Markov Chain Monte Carlo The average of a physical property of the system is given by

, where is the total number of configurations of R , for , sampled according to the distribution function , where R = { . It is a core task in QMC to effectively and accurately simulate the probability density. However, the probability density is always a high dimensional and complicated expression for a many-body system with strong interactions. Here, we would like to discuss a popular sampling scheme called Markov Chain Monte Carlo (MCMC). Markov Chain Consider a sequence of random variables , defined on a finite state space . This sequence is called a Markov chain if it satisfies the Markov property that is, the value of is dependent on its history only through its nearest past, . When is a finite set with dimension , we could use a transition matrix to represent this Markov chain, where represents the 1-step probability of transition from state to state . And the n-step transition function of a Markov chain can be denoted as . Therefore, we could write down the distribution at time according to the initial distribution and transition matrix ,

Metropolis-Hastings Algorithm MCMC mainly uses the property of stationary distribution for some Markov chains. These Markov chain moves have to be both evolutionary oriented (good ones live and bad ones die) and reversible (satisfying the detailed balance). The typical algorithm in MCMC are Metropolis algorithm and Metropolis-Hastings algorithm. Here we introduce the Metropolis-Hasting algorithm. Given target distribution and current state : (i) Propose a random "unbiased perturbation" of current state so to generate a new configuration . Mathematically, can be generated from a unsymmetrical probability transition function (often called the proposal function) (i.e., ) ; if the transition function is symmetric, then it is reduced to Metropolis algorithm. (ii) Draw Uniform[0, 1], and update

where Metropolis and Hastings suggested using . (iii) After sufficient times iteration of (i) and (ii), the Markov chain will converge to the target distribution .

Variational Monte Carlo

Variational Principle As we mentioned before, it is almost impossible to find out an analytical or exact solution of the Schrödinger equation of a system with more than two particles. Therefore, it is inevitable to introduce approximations into the systems when solving the equation. Considering a trial function with some underdetermined parameters or functions, that are waiting to be optimized, variational principle tells us that, , where E is the ground-state energy of the system. We denote the eigenstates of as , which form a complete basis set. So we can expand arbitrary function as a linear combination of

.

It is easy to verify the variational principle if we substitute the expansion for into the expectation value with the fact that for and ; namely,

.

The Metropolis Step When the system is continuous, we can express the expectation value in the form of an integral

,

where

is the distribution function for the system and can be viewed as a local energy, a physical quantity of the system at configuration . According to above discussion, we could use Metropolis-Hasting algorithm to sample the distribution function and give the expectation value . If the variational wavefunction is relatively simple with only a small number of parameters, we can optimize these parameters by varying them in the simulation. However, if the form of the wavefunction is more complicated, we should use the Euler-Lagrange equation

for , to optimize the wavefunction.

Trial Wavefunction The core task in VMC is to find a decent trial wavefunction. A common choice for trial wavefunctions is of the Slater-Jastrow (SJ) form, , where is a constant for boson systems and a Slater determinant of single-particle orbitals for fermion systems, and is the Jastrow correlation term with the form of = , where

, which is usually truncated at the two-body terms in most Monte Carlo simulations.

Diffusion Monte Carlo Sometimes, we don't have enough information to design a decent trial wavefunction in VMC, which is a main handicap of improving the performance of VMC. Therefore we urgently need a new method which allows us simulate a many-body system with only limited knowledge. Fortunately, a method called fixed-node diffusion Monte Carlo (FNDMC) method has been discovered and developed in the past decades. FNDMC is a stochastic scheme that can extract important information about the ground state of a many-body system. The method maps the Schrödinger equation to a diffusion equation under an imaginary-time transformation and allow the system to evolve toward its ground state over the evolution of time. In principle, it enables us to get the exact solution of the ground state. Basic idea In above sections, we only talked about the time-independent Schrödinger equation. Now, let's consider the time-dependent Schrödinger equation . We can map the time-dependent Schrödinger equation into a diffusion equation with imaginary-time as , where E is a reference energy that and can be viewed as the zero point of the system's energy at the given moment and the imaginary time . This is a simple first-order homogeneous linear differential equation and we can solve the answer as

, if we start from an initial state . Let's consider an arbitrary trial function which can be expanded as a linear combination of the eigenfunctions of the Hamiltonian,

, so that the imaginary time evolution of this function can be expressed as

. Since all the other states' energies are greater than the ground state energy, the exponential part for vanishes quickly. Therefore, after long enough time, only the ground state part leaves, which allows us to get the 'exact' ground state. In practice, to make a more efficient simulation of the original time-dependent equation, we introduce a trial wavefunction (usually optimized by VMC) and define a new time-dependent density

to realize importance sampling. We can easily derive that this time-dependent density satisfies , where is the local energy, is a forward Fokker-Planck operator defined as

, and the drift vector is given by

.

Using imaginary-time propagator K, we could describe the evolution of the density during a time interval

where K is defined as

. For a small enough time-step, we could split the exponential operator into a product of exponentials by using Baker-Campbell-Hausdorff formulas

, and introduce a short-time Gaussian approximation of the propagator

.

By this approximation, we could obtain and sample accurate approximations of K, and it could be written as

. Fixed-Node Approximation In general computation, we assume that the trial functions are always of the same sign, namely, they don't have any nodes. This is the case for ground state of a Bosonic system, and for some simple Fermionic systems. However, for a more common ferminoic system, there are always nodes in the wave function, since we always choose antisymmetric trial function. The positive and negative weights cancel out with statistical noise which would give during calculation. Besides, and are equally good solutions of the Schrödinger equation, the algorithm would sample both with same probability, resulting in the cancellation of positive and negative weights again. These two problems are the famous Fermionic sign problem. To avoid the sign problem in DMC, scientists have introduced fix-node (FN) approximation into DMC. By fixing its nodes to be the same as those of the trial function during convergence, it is possible for us to avoid the sign problem. In order to achieve this goal, we can define the FN Hamiltonian, , by adding to true Hamiltonian infinite potential barriers at the location of the nodes of . And the ground state wavefunction of is called the FN wavefunction and its energy is the FN energy , . If the nodes of is not the exact nodes of true wavefunction, a fixed-node error is introduced. So, the choice of nodes is important for FNDMC.

Applications The choice of trial functions has great impact on the performance of VMC and FNDMC. Brown et al. compared four different forms of trial functions, which are single-determinant SJ wavefunction, a single- determinant SJ wavefunction with backflow (BF), multiple determinants (MD), and both together (MDBF). The multi-configuration state functions (multi-CSF) SJ wavefunction with BF formations can be written in the form

, where each results from the backflow transformation of the electronic coordinates , each is a multideterminant representation of a CSF, and are the optimize wavefunction parameters: the CSF coefficients a, the Jastrow parameters b, and the backflow parameters c. In this work, the authors used Jastrow factors containing up to three-body terms, namely electron- electron (e-e), electron-nucleus (e-n), and electron-electron-nucleus (e-e-n) terms, and the same for backflow transformations. The backflow displacement for electron-electron terms can be described as

, where N is the number of electrons. Similarly, the e-n contributions is of the form , where N is the number of nuclei. And e-e-n terms looks like

, where and . The total backflow displacement is the sum of these three terms .

Here are the results for the ground state energies (in Hartrees) of the first row atoms (The DMC2-MDBF results are for a second DMC run with half the time step of DMC-MDBF):

VMC

VMC-BF

VMC-MD

VMC- MDBF DMC

DMC-BF

DMC-MD

DMC- MDBF DMC2- MDBF

DMC2 corr% Table 1. The VMC and DMC energies for each atom. Also shown are the Hartree-Fock single- determinant energies E E , the correlation energies , and the percentage of the correlation energy recovered by the DMC2-MDBF calculations (DMC2 corr%). Reprinted from M. D. Brown, J. R. Trail, P. López Ríos, and R. J. Needs, "Energies of the first row atoms from quantum Monte Carlo," J. Chem. Phys. 126, 224110 (2007), with the permission of AIP Publishing.

Figure 1. The percentage of the correlation energy recovered for each atom within VMC, using a single- determinant SJ wave function (VMC) with the addition of backflow (VMC-BF), multiple determinants (VMC-MD), and both together (VMC-MDBF). The statistical error bars (not shown) are smaller than the symbols. Reprinted from M. D. Brown, J. R. Trail, P. López Ríos, and R. J. Needs, "Energies of the first row atoms from quantum Monte Carlo," J. Chem. Phys. 126, 224110 (2007), with the permission of AIP Publishing.

As shown in the Table 1 and Figure 1, the DMC2-MDBF method can recover about 99% of the correlation energy for these atoms, which means it is an much improved method than HF. Compare these results with the energy obtained from DFT: >

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DFT DFT corr% Table 2. The DFT energies for each atom using basis ccpvtz, and the percentage of the correlation energy recovered by the DFT calculations (DFT corr%).

Here, in order to compare the results of DMC2 and DFT intuitively, we draw the correlation energy of the two methods in one picture, which can clearly see the excellent performance of DMC2.

Figure 2. The correlation energy recovered for each atom within DFT and DMC2.

Conclusion In this project, we discuss the algorithm technique, MCMC, the principles of VMC and DMC along with their applications to simple first row atoms. It shows that VMC and DMC with decent trial wavefunctions can recover 99% of the correlation energy. Compared to post-HF methods and DFT methods, one advantage of quantum Monte Carlo methods is that no analytical integration is needed, which accelerate the calculation. The key task for VMC and DMC is to find decent trial functions and optimize the parameters more efficiently.

References 1, T. Pang, An Introduction to Quantum Monte Carlo Methods (IOP Publishing, Bristol, 2016) 2, J. Toulouse, R. Assaraf, and C. J. Umrigar, "Introduction to the variational and diffusion Monte Carlo methods," Advances in Quantum Chemistry 73, 285-314 (2016). 3, J. Liu, Monte Carlo Strategies In Scientific Computing (Springer, New York, 2001) 4, M. D. Brown, J. R. Trail, P. López Ríos, and R. J. Needs, "Energies of the first row atoms from quantum Monte Carlo," J. Chem. Phys. 126, 224110 (2007).