Configuration integrals

The integral over the positions is often referred to as the configurational integral,... 

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 ... 

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... 

To obtain thermodynamic perturbation or integration formulas for changing q, one must go back and forth between expressions of the configuration integral in Cartesian coordinates and in suitably chosen generalized coordinates [51]. This introduces Jacobian factors... 

Here, the configuration integral Q has been split into tliree integrals the integration j is over all conformations where 71 is in the ith rotameric state. F is the configurational... 

Configurational integrity of secondary alkylzinc reagents had been demonstrated by Knochel although the preparation of the zinc reagent was not practical (Scheme 8.8) [13],... 

Scheme 8.8 Knochel s precedent for configurational integrity of organozinc reagents.

The path-integral (PI) representation of the quantum canonical partition function Qqm for a quantized particle can be written in terms of the effective centroid potential IT as a classical configuration integral ... 

These are exact expressions for the configuration integrals. Alternatively, we can write the partition function as... 

The angle bracket denotes that the configurational integral is taken over the initial state. The conformational sampling indicated by Equation 4 is generated according to the Boltzmann probability associated with the initial state potential. As discussed in Section 2.1, convergence of conformational... 

Pu depends on the quotient flj, / TT, the calculation of the configurational integral Z(N,V,T) is avoided. The change in potential energy of the system due to the trial move determines if the attempted new configuration is accepted. 

For a classical system of N point particles enclosed in a volume V,at a temperature T, the canonical partition function can be decomposed in two factors. The first one (Qt) comes from the integration over the space of momenta of the kinetic term of the classical Hamiltonian, which represents the free motion of noninteracting particles. The second one, which introduces the interactions between the particles and involves integration over the positions, is the configuration integral. This way, equation (30)... 

The difficulty arises from the fact that the one-step transition probabilities of the Markov chain involve only ratios of probability densities, in which Z(N,V,T) cancels out. This way, the Metropolis Markov chain procedure intentionally avoids the calculation of the configurational integral, the Monte Carlo method not being able to directly apply equation (31). 

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