Potential energy partitioning

The following derivation is modified from that of Fowler and Guggenheim [10,11]. The adsorbed molecules are considered to differ from gaseous ones in that their potential energy and local partition function (see Section XVI-4A) have been modified and that, instead of possessing normal translational motion, they are confined to localized sites without any interactions between adjacent molecules but with an adsorption energy Q. 

McMillan-Mayer theory of solutions [1,2], which essentially seeks to partition the interaction potential into tln-ee parts that due to the interaction between the solvent molecules themselves, that due to die interaction between the solvent and the solute and that due to the interaction between the solute molecules dispersed within the solvent. The main difference from the dilute fluid results presented above is that the potential energy u(r.p is replaced by the potential of mean force W(rp for two particles and, for particles of solute in the solvent, by the expression... 

The potential energy of the electrons, V, which is a negative quantity that can be partitioned into bulk and surface contributions, as shown. 

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 implicit-midpoint (IM) scheme differs from IE above in that it is symmetric and symplectic. It is also special in the sense that the transformation matrix for the model linear problem is unitary, partitioning kinetic and potential-energy components identically. Like IE, IM is also A-stable. IM is (herefore a more reasonable candidate for integration of conservative systems, and several researchers have explored such applications [58, 59, 60, 61]. 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

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... 

While thermodynamics does not describe the nature of this internal energy, it is helpful to consider the insights gained from kinetic molecular theory. According to this theory, the internal energy can be partitioned into kinetic and potential energy terms associated with various motions and positions of the nuclei of the atoms or molecules that make up the gas, and with energies associated with their electrons. 

Quite similar equations can be formulated for AG and AH by use of the partition function f of the activated complex. It follows from equations (6) and (7) that AEp can only be evaluated if the partition functions and AEz are available from spectroscopic data or heat capacity measurements. However, if AG = AH, the entropy change AS equals zero, and if AEz also equal to zero, either AG or AH can then be identified with the potential energy change. If... 

One of the first studies to predict log P by using potential energy fields calculated using the GRID and CoMFA approaches was done by Kim [60]. The author investigated H, CH3 and H2O probes, and calculated the best models using the hydro-phobic probe H2O for relatively small series (20 or less compounds each) of furans, carbamates, pyridines and pyrazines. A similar study was performed by Waller [61] who predicted a small series of 24 polyhalogenated compounds. Recently, Caron and Ermondi [62] used a new version of Cruciani s software, VolSurf [63], to predict the octanol-water and alkane-water partition coefficients for 152 compounds with r = 0.77, q = 0.72, SDEP = 0.60 for octanol-water and r = 0.76, q = 0.71, SDEP = 0.85 for alkane-water. 

Another approach to calculating AA relies on estimating the appropriate probability density functions. The connection between the probabilities of different states and the partition function is natural in statistical mechanics. Equation (1.19) is a reflection of this connection. Similarly, the probability of observing the potential energy of the system being equal to U is ... 

To understand better the difficulties connected with the partitioning of free energy into contributions, let us break down the potential energy into a sum of two terms,... 

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