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Thermodynamic Integration
Thermodynamic integration is a method used to compare the difference in Thermodynamic free energy, free energy between two given states (e.g., A and B) whose potential energies U_A and U_B have different dependences on the spatial coordinates. Because the free energy of a system is not simply a function of the phase space coordinates of the system, but is instead a function of the Boltzmann distribution, Boltzmann-weighted integral over phase space (i.e. Partition function (statistical mechanics), partition function), the free energy difference between two states cannot be calculated directly from the potential energy of just two coordinate sets (for state A and B respectively). In thermodynamic integration, the free energy difference is calculated by defining a thermodynamic path between the states and integrating over ensemble-averaged enthalpy changes along the path. Such paths can either be real chemical processes or alchemical processes. An example alchemical process is the Ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermodynamic Free Energy
In thermodynamics, the thermodynamic free energy is one of the state functions of a thermodynamic system. The change in the free energy is the maximum amount of work that the system can perform in a process at constant temperature, and its sign indicates whether the process is thermodynamically favorable or forbidden. Since free energy usually contains potential energy, it is not absolute but depends on the choice of a zero point. Therefore, only relative free energy values, or changes in free energy, are physically meaningful. The free energy is the portion of any first-law energy that is available to perform thermodynamic work at constant temperature, ''i.e.'', work mediated by thermal energy. Free energy is subject to irreversible loss in the course of such work. Since first-law energy is always conserved, it is evident that free energy is an expendable, second-law kind of energy. Several free energy functions may be formulated based on system criteria. Free energy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form: :p_i \propto \exp\left(- \frac \right) where is the probability of the system being in state , is the exponential function, is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . The symbol \propto denotes proportionality (see for the proportionality constant). The term ''system'' here has a wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore, the Boltzmann distribution can be used to sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (statistical Mechanics)
In physics, a partition function describes the statistics, statistical properties of a system in thermodynamic equilibrium. Partition functions are function (mathematics), functions of the thermodynamic state function, state variables, such as the temperature and volume. Most of the aggregate thermodynamics, thermodynamic variables of the system, such as the energy, total energy, Thermodynamic free energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless. Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular Thermodynamic free energy, free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the Environment (systems), environment at fixed temperature, volume, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kirkwood's Coupling Parameter
Kirkwood (formerly Kirk, Kirkwood's, and Roundtop) is an unincorporated community in Alpine and Amador counties, California, United States. Kirkwood's main attraction is the Kirkwood Mountain Resort. The town is accessible by State Route 88. Kirkwood is within the Eldorado National Forest. The population was 190 at the 2020 census. For statistical purposes, the United States Census Bureau has defined Kirkwood as a census-designated place (CDP). History Zack Kirkwood, a cattle rancher who had settled in the area, opened an inn, named Kirkwood's, in 1863 with the opening of the Amador/Nevada Wagon Toll Road, the primary route through the Sierra Nevada Mountains. The following year, Alpine County was created and the redrawing of the county borders placed the inn at the convergence of Alpine, El Dorado, and Amador counties (the borders were later changed so that the inn is no longer in El Dorado County, but the original signpost marking the intersection of the three counties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Ensemble
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat bath, so that the states of the system will differ in total energy. The principal thermodynamic variable of the canonical ensemble, determining the probability distribution of states, is the absolute temperature (symbol: ). The ensemble typically also depends on mechanical variables such as the number of particles in the system (symbol: ) and the system's volume (symbol: ), each of which influence the nature of the system's internal states. An ensemble with these three parameters, which are assumed constant for the ensemble to be considered canonical, is sometimes called the ensemble. The canonical ensemble assigns a probability to each distinct microstate given by the following exponential: :P = e^, where is the total energy of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Umbrella Sampling
Umbrella sampling is a technique in computational physics and chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a physical science within the natural sciences that studies the chemical elements that make up matter and chemical compound, compounds made of atoms, molecules a ..., used to improve sampling of a system (or different systems) where ergodicity is hindered by the form of the system's energy landscape. It was first suggested by Torrie and Valleau in 1977. It is a particular physical application of the more general importance sampling in statistics. Systems in which an energy barrier separates two regions of configuration space may suffer from poor sampling. In Metropolis Monte Carlo runs, the low probability of overcoming the potential barrier can leave inaccessible configurations poorly sampled—or even entirely unsampled—by the simulation. An easily visualised example occurs with a solid at its melting point: cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Energy Perturbation
Free-energy perturbation (FEP) is a method based on statistical mechanics that is used in computational chemistry for computing free-energy differences from molecular dynamics or Metropolis Monte Carlo simulations. The FEP method was introduced by Robert W. Zwanzig in 1954. According to the free-energy perturbation method, the free-energy difference for going from state A to state B is obtained from the following equation, known as the ''Zwanzig equation'': : \Delta F(\mathbf \to \mathbf) = F_\mathbf - F_\mathbf = -k_\text T \ln \left\langle \exp\left(-\frac \right) \right\rangle_\mathbf, where ''T'' is the temperature, ''k''B is the Boltzmann constant, and the angular brackets denote an average over a simulation run for state A. In practice, one runs a normal simulation for state A, but each time a new configuration is accepted, the energy for state B is also computed. The difference between states A and B may be in the atom types involved, in which case the Δ''F'' obtained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bennett Acceptance Ratio
The Bennett acceptance ratio method (BAR) is an algorithm for estimating the difference in free energy between two systems (usually the systems will be simulated on the computer). It was suggested by Charles H. Bennett in 1976. Preliminaries Take a system in a certain super (i.e. Gibbs) state. By performing a Metropolis Monte Carlo walk it is possible to sample the landscape of states that the system moves between, using the equation : p(\text_x \rightarrow \text_y) = \min \left(e ^ , 1 \right) = M(\beta \, \Delta U) where Δ''U'' = ''U''(State''y'') − ''U''(State''x'') is the difference in potential energy, β = 1/''kT'' (''T'' is the temperature in kelvins, while ''k'' is the Boltzmann constant), and M(x) \equiv \min(e^ , 1) is the Metropolis function. The resulting states are then sampled according to the Boltzmann distribution of the super state at temperature ''T''. Alternatively, if the system is dynamically simulated in the canonical ensemble (al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Chemistry
Computational chemistry is a branch of chemistry that uses computer simulations to assist in solving chemical problems. It uses methods of theoretical chemistry incorporated into computer programs to calculate the structures and properties of molecules, groups of molecules, and solids. The importance of this subject stems from the fact that, with the exception of some relatively recent findings related to the hydrogen molecular ion (dihydrogen cation), achieving an accurate quantum mechanical depiction of chemical systems analytically, or in a closed form, is not feasible. The complexity inherent in the many-body problem exacerbates the challenge of providing detailed descriptions of quantum mechanical systems. While computational results normally complement information obtained by chemical experiments, it can occasionally predict unobserved chemical phenomena. Overview Computational chemistry differs from theoretical chemistry, which involves a mathematical description of chem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
