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Variational Monte Carlo
In computational physics, variational Monte Carlo (VMC) is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system. The basic building block is a generic wave function , \Psi(a) \rangle depending on some parameters a . The optimal values of the parameters a is then found upon minimizing the total energy of the system. In particular, given the Hamiltonian \mathcal , and denoting with X a many-body configuration, the expectation value of the energy can be written as: E(a) = \frac = \frac . Following the Monte Carlo method for evaluating integrals, we can interpret \frac as a probability distribution function, sample it, and evaluate the energy expectation value E(a) as the average of the so-called local energy E_(X) = \frac . Once E(a) is known for a given set of variational parameters a , then optimization is performed in order to minimize the energy and obtain the best possible representation of the g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Physics
Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline (or offshoot) of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics - an area of study which supplements both theory and experiment. Overview In physics, different theories based on mathematical models provide very precise predictions on how systems behave. Unfortunately, it is often the case that solving the mathematical model for a particular system in order to produce a useful prediction is not feasible. This can occur, for instance, when the solution does not have a closed-form expression, or is too complicated. In such cases, numerical approximations are required. Computational phys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization (mathematics)
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of Molecule, molecules, Material, materials, and solutions at the atomic level. These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed Wave function, wave functions as well as to observable properties such as structures, spectra, and thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics. Chemists rely heavily on spectroscopy through which information regarding the Quantization (physics), quantization of energy on a molecular scale can be obtained. Common metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time-dependent Variational Monte Carlo
The time-dependent variational Monte Carlo (t-VMC) method is a quantum Monte Carlo approach to study the dynamics of closed, non-relativistic quantum systems in the context of the quantum many-body problem. It is an extension of the variational Monte Carlo method, in which a time-dependent pure quantum state is encoded by some variational wave function, generally parametrized as : \Psi(X,t) = \exp \left ( \sum_k a_k(t) O_k(X) \right ) where the complex-valued a_k(t) are time-dependent variational parameters, X denotes a many-body configuration and O_k(X) are time-independent operators that define the specific ansatz. The time evolution of the parameters a_k(t) can be found upon imposing a variational principle to the wave function. In particular one can show that the optimal parameters for the evolution satisfy at each time the equation of motion : i \sum_\langle O_k O_\rangle_t^c \dot_=\langle O_k \mathcal\rangle_t^c, where \mathcal is the Hamiltonian of the syst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rayleigh–Ritz Method
The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz. The name Rayleigh–Ritz is being debated vs. the Ritz method after Walther Ritz, since the numerical procedure has been published by Walther Ritz in 1908-1909. According to, Lord Rayleigh wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places in his book and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the Rayleigh quotient make the arguable misnomer persist. According to, citing Richard Courant, both Lord Rayleigh and Walther Ritz independently conceived the idea of utilizing the equivalence between boundary value problems of partial differential equations on the one hand and problems of the calculus of variations on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metropolis–Hastings Algorithm
In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This sequence can be used to approximate the distribution (e.g. to generate a histogram) or to compute an integral (e.g. an expected value). Metropolis–Hastings and other MCMC algorithms are generally used for sampling from multi-dimensional distributions, especially when the number of dimensions is high. For single-dimensional distributions, there are usually other methods (e.g. adaptive rejection sampling) that can directly return independent samples from the distribution, and these are free from the problem of autocorrelated samples that is inherent in MCMC methods. History The algorithm was named after Nicholas Metropolis and W.K. Hastings. Metropolis was the first author to appear on the list of authors of the 1953 article ''Equation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronic Structure
In quantum chemistry, electronic structure is the state of motion of electrons in an electrostatic field created by stationary nuclei. The term encompasses both the wave functions of the electrons and the energies associated with them. Electronic structure is obtained by solving quantum mechanical equations for the aforementioned clamped-nuclei problem. Electronic structure problems arise from the Born–Oppenheimer approximation. Along with nuclear dynamics, the electronic structure problem is one of the two steps in studying the quantum mechanical motion of a molecular system. Except for a small number of simple problems such as hydrogen-like atoms, the solution of electronic structure problems require modern computers. Electronic structure problem is routinely solved with quantum chemistry computer programs. Electronic structure calculations rank among the most computationally intensive tasks in all scientific calculations. For this reason, quantum chemistry calculatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics. Some fermions are elementary particles (such as electrons), and some are composite particles (such as protons). For example, according to the spin-statistics theorem in relativistic quantum field theory, particles with integer spin are bosons. In contrast, particles with half-integer spin are fermions. In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin-statistics relation is, in fact, a spin statistics-quantum number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neural Network Quantum States
Neural Network Quantum States (NQS or NNQS) is a general class of variational quantum states parameterized in terms of an artificial neural network. It was first introduced in 2017 by the physicists Giuseppe Carleo and Matthias Troyer to approximate wave functions of many-body quantum systems. Given a many-body quantum state , \Psi\rangle comprising N degrees of freedom and a choice of associated quantum numbers s_1 \ldots s_N , then an NQS parameterizes the wave-function amplitudes \langle s_1 \ldots s_N , \Psi; W \rangle = F(s_1 \ldots s_N; W), where F(s_1 \ldots s_N; W) is an artificial neural network of parameters (weights) W , N input variables ( s_1 \ldots s_N ) and one complex-valued output corresponding to the wave-function amplitude. This variational form is used in conjunction with specific stochastic learning approaches to approximate quantum states of interest. Learning the Ground-State Wave Function One common application of NQS is to find an approxi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matthias Troyer
Matthias is a name derived from the Greek Ματθαίος, in origin similar to Matthew. People Notable people named Matthias include the following: In religion: * Saint Matthias, chosen as an apostle in Acts 1:21–26 to replace Judas Iscariot * Matthias of Trakai (–1453), Lithuanian clergyman, bishop of Samogitia and of Vilnius * Matthias Flacius, Lutheran reformer * Matthias the Prophet, see Robert Matthews (religious impostor) Claimed to be the reincarnation of the original Matthias during the Second Great Awakening * Matthias F. Cowley, Latter-day Saint apostle In the arts: * Matthias Grünewald, highly regarded painter from the German Renaissance * Matthías Jochumsson, Icelandic poet * Matthias Lechner, German film art director * Matthias Paul (actor), German actor * Matthias Schoenaerts, Belgian actor In nobility: * Matthias Corvinus of Hungary, King of Hungary * Matthias, Holy Roman Emperor, Emperor of the Holy Roman Empire (Habsburg dynasty) In music: * Matthias Ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giuseppe Carleo
Giuseppe Carleo (born 1984) is an Italian physicist. He is a professor of computational physics at EPFL (École Polytechnique Fédérale de Lausanne) and the head of the Laboratory of Computational Quantum Science. Career Carleo studied physics at the Sapienza University of Rome and in 2011 earned his PhD in theoretical physics at the International School for Advanced Studies under the supervision of Stefano Baroni. His thesis on "Spectral and dynamical properties of strongly correlated systems" was dedicated to novel numerical simulation techniques to study condensed-matter systems, such as the time-dependent variational Monte Carlo. As a Marie Curie Fellow he joined the École supérieure d'optique to work in the Lab directed by Alain Aspect on theoretically model and simulate ultra-cold atoms systems. In 2015, he went to work with the group of Matthias Troyer at the ETH Zurich where he later became a lecturer of computational quantum physics. Here he investigated the idea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronic Correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons. Atomic and molecular systems Within the Hartree–Fock method of quantum chemistry, the antisymmetric wave function is approximated by a single Slater determinant. Exact wave functions, however, cannot generally be expressed as single determinants. The single-determinant approximation does not take into account Coulomb correlation, leading to a total electronic energy different from the exact solution of the non-relativistic Schrödinger equation within the Born–Oppenheimer approximation. Therefore, the Hartree–Fock limit is always above this exact energy. The difference is called the ''correlation energy'', a term coined by Löwdin. The concept of the correlation energy was studied earlier by Wigner. A certain amount of electron c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
