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Variational Quantum Monte Carlo Abstract Introduction Variational Quantum Monte Carlo Josh Jiang, The University of Chicago > (1.1) Abstract The many electron system is one that quantum mechanics have no exact solution for because of the correlation between each electron with every other electron. Many methods have been developed to account for this correlation. One family of methods known as quantum Monte Carlo methods takes a numerical approach to solving the many-body problem. By employing a stochastic process to estimate the energy of a system, this method can be used to calculate the ground state energy of a system using the variational principle. The core of this method involves choosing an ansatz with a set of variational parameters that can be optimized to minimize the energy of the system. Recent research in this field is focused on developing methods to faster and more efficiently optimize this trial ansatz and also to make the ansatz more flexible by allowing more parameters. These topics along with the theory behind variational Monte Carlo (VMC) will be the subject of this Maple worksheet. Introduction Quantum mechanics is the study of particles at a very small scale. At this scale, particles can be characterized by wave functions and behave much differently than classical systems. By understanding how molecules behave at the quantum level, researchers can apply that knowledge to gain insights into how macroscopic systems work. While the applications are promising, quantum mechanics still struggles to understand the behavior of large atomic systems. For small, simple systems like particle in a box, harmonic oscillator, and the hydrogen atom, the Schrodinger equation: can be solved exactly, revealing the energy levels and exact wave functions for those systems. For an atom with more than one electron, however, the Schrodinger equation becomes impossible to solve and approximation methods must be used. To understand the nature of this problem, consider the Hamiltonian for a single electron system. Using the Born-Oppenheimer approximation to neglect the movement of the nucleus, the Hamiltonian is: This operator contains the translational energy of the electrons and the nucleus-electron attraction. By expressing the Laplacian operator, , in spherical coordinates: and separating the wave function as a product of radial and angular functions: (r) where the radial part, , is given by Laguerre polynomials and the angular part, by spherical harmonics, the energy, E, of the system can be solved analytically. Now, consider a system with a N- electron atom where N>1. The Hamiltonian for such a system now contains a term to account for the electron-electron repulsion: Due to the electron repulsion term, the wave function ansatz above no longer holds as the location of each electron is correlated with the location of every other electron in the system. To overcome this hurdle, many methods have been developed to make accurate approximations of the wave function and energies of a many-body system. This worksheet will focus on one particular method known as variational Quantum Monte Carlo, which uses the variational principle and a stochastic process algorithm to approximate the ground state energy and wave function of any system. Theory/Methodology The Variational Method The variational method is one technique for approximating the ground state energy of a quantum system. The method relies on the variational principle, which states that the true ground state energy, , of a system is less than or equal to the average energy of the trial wave function: Given that is normalized, then it is clear that energy is minimized when = and greater otherwise. This principle can be used in practice by picking a trial wavefunction that has parameter(s) a and then minimize the energy equation: with respect to a a) that best minimizes can be chosen where the minimized E approaches the true wave function. Therefore, this method can only be used to reliably calculate the ground state energy of a system. Monte Carlo Integration Monte Carlo methods are a family of techniques that use random sampling to approximate results that might not be able to be done analytically. One application of this method is in integration. Suppose you want to take the integral of a function, f(x), from a to b: If f is an arbitrary curve or a complex function, then the normal methods of integration might not be sufficient to solve the equation. In Monte Carlo, F can be approximated by first taking N random variables unif(a,b). The approximation, , is given by: where approaches F as N goes to infinity due to the law of large numbers. This method can also be generalized to any arbitrary d dimensional function. Variational Monte Carlo The Monte Carlo method can be applied to calculate the energy of a system, which can be expressed as a multidimensional integral: where the parameters R is the 3N coordinates of the N electrons, a are the parameters that are to be optimized, / (R,a)= / is the probability distribution of such a configuration. The integral can be evaluated using . An added easily integrated. Sampling with the Metropolis-Hastings Algorithm therefore, cannot be sampled from directly. The Metropolis-Hastings algorithm is a technique that can chain is a stochastic process where the transition probability for a final state depends only on the state equation is imposed on the process. The detailed balance equation states that the probability flux between two states and be the same in both directions: P( | ) = P( | ) With this constraint and after a sufficiently large number M of steps, the random walk converge to the stationary distribution ). How this algorithm works in practice is as follows: 1. The Markov chain is initialized by any arbitrary state . 2. For step t: A proposed point, is drawn from the distribution P( | ), which is a distribution that can be sampled directly (typically a Gaussian distribution centered at ). This is accompanied with ). If ), then . A = , otherwise, = . 3. Step 2 is repeated for a sufficiently large number of steps such that the Markov chain converges to 3. Figure 1: Demonstration of MH algorithm. Available via license: Creative Commons Attribution 4.0 International After a large number M of steps, the energy can be approximated as: In practical applications of this algorithm, M is in the range of (2). Common practice is also to throw out the first k points when the random walk is correlated to the choice of starting point . The Linear Method for Ansatz Optimization The Metropolis-Hastings Algorithm provides a useful method for calculating the energy of system as a function of parameter set a. The next step for variational Monte Carlo is to optimize the trial wave function. This is a central problem in developing quantum Monte Carlo methods and a wide variety of methods have been and are being developed to improve optimization performance. One such standard way to do so is known as the linear method (LM). For a set of variational parameters a, LM begins by first taking the first order Taylor approximation of the wavefunction: where and is the wave function with current parameter values. The solution for the new set of parameters is obtained by solving the eigenvalue problem: Hc = ESc where H is the hamiltonian and S is the overlap matrix. The matrix diagonalization to solve this for c=(1, then yields the new set of parameters . The linear method is one of the most widely used ansatz optimization methods but it also has certain limitations. For one, memory cost becomes a limiting factor for the method at high numbers of parameters, as memory cost scaled with the square of the number of variational parameters. Furthermore, this method does not perform well with optimizating wave functions that have a nonlinear relation with its parameters. Current on this field is working towards reducing memory cost, therby increasing the number of variational parameters allowed in the wave function, and developing methods that perform well for non-linear wave functions. Applications/Results 3. Many trial wave functions have been developed for variational Monte Carlo. Typically the trial function takes the form accuracy of VMC depends on the form of the trial wave function. Prasad et al., compared the ground state energy for second period atoms found using HF and VMC with different kinds of trial functions with only two correlation parameters (3): Trial Li Be B C N O F Ne function HF-Limit -7.43273 -14.57302 -24.52906 -37.68862 -54.40093 -74.80940 -99.40935 -128.54710 HF- -7.4605 -14.60820 -24.5683 -37.7285 -54.441 -74.8487 -99.4437 -128.5641 Jastrow HF-Sun -7.4616 -14.6070 -24.56686 -37.73018 -54.4480 -74.8660 -99.4730 -128.6123 HF- -7.4722 -14.6267 -24.5973 -37.7743 -54.5093 -74.9580 -99.6036 -128.7900 SMBH Exact -7.4781 -14.6674 -24.6539 -37.8450 -54.5892 -75.0673 -99.7339 -128.9376 value VMC is an improvement over HF however, even the best results here shows that VMC only recovers about 60% of the correlation energy. Using diffusion Monte Carlo has been found to recover 90-100% of the correlation energy for these atoms (3). Compare these results with the energy obtained from coupled cluster: > (5.1) > (5.2) > (5.3) > (5.4) This coupled cluster calculation outperforms the VMC methods, however, with more complex wave functions and a larger set of parameters, the estimates can get more accurate. Current research is done to develop faster ways to optimize trial wave functions. Sabzevari, Mahajan, and Sharma (2020) developed an accelerated linear method for optimizing non-linear wave functions. Using a wavefunction with 364 parameters, they estimated the ground state energy of Be to be -14.667 in 23.52s and used a wavefunction with 492 parameters to estimate the ground state of Ne to be -128.907 in about one hour.
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