Potential energy function, partitioned

At this level, the phospholipid and surrounding water are not included explicitly but are described by an empirical potential that characterizes partitioning of hydrophilic and hydrophobic parts of a molecule into the phospholipid. In the simplest way, the phospholipid bilayer is represented by a lipophilic slab oriented in the x, y-plane and the potential energy function of the force field is supplemented with a hydrophobic interaction term (Eq. 6.1) ... 

Heyden A, Lin H et al (2007) Adaptive partitioning in combined quantum mechanical and molecular mechanical calculations of potential energy functions for multiscale simulations. J Phys Chem B 111 2231... 

The central idea of thermodynamic perturbation theory is that the potential energy function can be partitioned in a convenient way i.e., one can write... 

To introduce the formulation, we consider the exact connection between the unperturbed and perturbed systems. We focus on the Helmholtz free energy, A, which is the quantity of interest at constant N, T, and V, where N is the number of particles, T is the temperature, and V is the volume of the system the alternative case (constant N, T, and P), which leads to the Gibbs free energy, can be treated similarly. The Helmholtz free energy for the potential energy function V(r A) can be written in terms of the partition function Zxas... 

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

The volume available for dimers is the total volume V. If we account for the interaction, and if we use relation [6.5] for the corresponding term, which is the potential energy, the partition function of a dimer molecule will be ... 

K. D. Ball and R. S. Berry, Realistic master equation modeling of relaxation on complete potential energy surfaces Partition function and equilibrium results. J. Chem. Phys. 109(19), 8541-8556 (1998). 

Here, the fraction is nothing but the ratio of the partition function for the biased potential energy function and the unbiased one. Taking the natural logarithm on both sides of Eq. (16.37) and multiplying with a constant j, the free energy along the reaction coordinate z can be calculated as ... 

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. 

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... 

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