Configurations, canonical

Foster JM, Boys SF (1960) Canonical configurational interaction procedure. Rev Mod Phys 32 300... 

The stationary state of the molecule is, therefore, a combination of various configurations for which we preferably but not necessarily choose the canonical configurations. The square of the coefficient represents the contribution of the configuration in question to (the charge distribution of) the stationary state. For benzene we have ... 

In benzene it is found by superposition of the five canonical configurations mentioned that each bond has a double bond character of 0.389 + 0.073 = 0.462. In view of the single o-bond always present one also speaks of total bond character equal to 1 -j- the double bond character. Though this bond character ( indice de liaison double , Daudel and Pullman) from the calculations based on the Valence Bond method is different from the bond order (Penney, Coulson), e.g. in the Molecular Orbital Method, both terms are frequently used without discrimination. In Pauling s earlier very simple concept (p. 210) the bond character of benzene is just 0.5 since there only the Kekule configurations are taken into account. 

Both the bond character and the free valence index (but not the energy) depend in principle on the choice of the canonical configurations. In general, however, it is easy to make a proper choice on the basis of chemical intuition. In naphthalene the results for both quantities are as given below (see also Table 24 and p. 210). o.I0... 

The agreement of these results according to the S method with those of the M.O. method is very satisfactory, which gives confidence in their reliability. The ready illustration of the usual V.B. method by the structural formulae of the canonical configurations is, however, no longer present. 

Foster, J. M., Boys, S. F. (1960). Canonical configurational interaction procedure. Reviews of Modem Physics, 32,300-302. 

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

Since H=K. + V, the canonical ensemble partition fiinction factorizes into ideal gas and excess parts, and as a consequence most averages of interest may be split into corresponding ideal and excess components, which sum to give the total. In MC simulations, we frequently calculate just the excess or configurational parts in this case, y consists just of the atomic coordinates, not the momenta, and the appropriate expressions are obtained from equation b3.3.2 by replacing fby the potential energy V. The ideal gas contributions are usually easily calculated from exact... 

The grand canonical ensemble corresponds to a system whose number of particles and energy can fluctuate, in exchange with its surroundings at specified p VT. The relevant themiodynamic quantity is the grand potential n = A - p A. The configurational distribution is conveniently written... 

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

In an ideal gas there are no interactions between the particles and so the potential ener function, 1 ), equals zero. exp(- f (r )/fcBT) is therefore equal to 1 for every gas partic in the system. The integral of 1 over the coordinates of each atom is equal to the volume, ai so for N ideal gas particles the configurational integral is given by (V = volume). T1 leads to the following result for the canonical partition function of an ideal gas ... 

The criterion used to accept or reject a new configuration is slightly different for the is thermal-isobaric simulation than for a simulation in the canonical ensemble. The followi] quantity is used ... 

Eor many systems the ensemble that is used in an MC simulation refers to the canonical ensemble, (N, F/ T). This ensemble permits a rise and fall in the pressure of the system, P, because the temperature and volume are held constant. Thus, the probabiUty that any system of N particles, in a volume H at temperature Tis found in a configuration x is proportional to the Boltzmann weighted energy at that state, E, and it is given by... 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

FIG. 21 Dependence of the average density on the configurational chemical potential. The solid line denotes the grand canonical Monte Carlo data, the long dashed fine corresponds to the osmotic Monte Carlo results for ZL = 40, and the dotted line for ZL = 80. (From Ref. 172.)... 

Let us consider a simple model of a quenched-annealed system which consists of particles belonging to two species species 0 is quenched (matrix) and species 1 is annealed, i.e., the particles are allowed to equlibrate between themselves in the presence of 0 particles. We assume that the subsystem composed of 0 particles has been a usual fluid before quenching. One can characterize it either by the density or by the value of the chemical potential The interparticle interaction Woo(r) does not need to be specified for the moment. It is just assumed that the fluid with interaction woo(r) has reached an equlibrium at certain temperature Tq, and then the fluid has been quenched at this temperature without structural relaxation. Thus, the distribution of species 0 is any one from a set of equihbrium configurations corresponding to canonical or grand canonical ensemble. We denote the interactions between annealed particles by Un r), and the cross fluid-matrix interactions by Wio(r). 

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

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