The Energy Surface

Each atom consists of a central, positively charged nucleus which contains practically all of the mass of the atom, and which is surrounded by a cloud of electrons. From the standpoint of chemistry, the internal energy of the atom may be measured in terms of the energy of the electrons. 

The energy of the electrons may be broken into component parts. Those which raise the total energy of the atomic molecular system, and hence raise the energy content of one gram atom or one mol of the substance, will be termed positive, whereas those which lower the energy 

If the atom is not at rest, the kinetic energy of motion of the atom as a whole must be included in the total energy. For the moment we shall assume that it is at rest. 

k is Boltzmann s constant, namely 1.39 10-16 ergs/deg. and N (E) is the number of independent ways in which the system of atoms may have the total energy E. The subject of quantum statistics is concerned with the detailed determination of the quantity N(E). 

The general condition for the stability of a cluster of atoms is again that the free energy (1) be a minimum. At very low temperatures this 

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These electronic energies dependence on the positions of the atomic centres cause them to be referred to as electronic energy surfaces such as that depicted below in figure B3.T1 for a diatomic molecule. For nonlinear polyatomic molecules having atoms, the energy surfaces depend on 3N - 6 internal coordinates and thus can be very difficult to visualize. In figure B3.T2, a slice tln-oiigh such a surface is shown as a fimction of two of the 3N - 6 internal coordinates. 

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods. 

A number of issues need to be addressed before this method will become a routine tool applicable to problems as the conformational equilibrium of protein kinase. E.g. the accuracy of the force field, especially the combination of Poisson-Boltzmann forces and molecular mechanics force field, remains to be assessed. The energy surface for the opening of the two kinase domains in Pig. 2 indicates that intramolecular noncovalent energies are overestimated compared to the interaction with solvent. 

This section deals with the question of how to approximate the essential features of the flow for given energy E. Recall that the flow conserves energy, i.e., it maps the energy surface Pq E) = x e P H x) = E onto itself. In the language of statistical physics, we want to approximate the microcanonical ensemble. However, even for a symplectic discretization, the discrete flow / = (i/i ) does not conserve energy exactly, but only on... 

Thus, this multi-level process produces a finer and finer covering of rsE -r) B)-Up to now, the parameters 6Ek are adapted to the size of the boxes according to some heuristics. Recall that an approximation of the energy surface Eo(E) would only be possible in the limit r —+ 0 which implies 5E t) —+ 0. 

Inadequate availability of experimental data can considerably inhibit the development of improved energy functions for more accurate simulations of energetic, structural, and spectroscopic properties. This has led to the development of class II force fields such as CFF and the Merck Molecular Force Field (MMFF), which are both based primarily on quantum mechanical calculations of the energy surface. The purpose of MMFF, which has been developed by Thomas Halgren at Merck and Co., is to be able to handle all functional groups of interest in pharmaceutical design. 

Energy Minimisation and Related Methods for Exploring the Energy Surface... 

A random search is, in many ways, the antithesis of a systematic search. A systematic searc explores the energy surface of the molecule in a predictable fashion, whereas it is n(... 

The second energy derivatives with respect to the x, y, and z directions of centers a and b (for example, the x, y component for centers a and b is Hax,by = (3 E/dxa3yb)o) form the Hessian matrix H. The elements of H give the local curvatures of the energy surface along the 3N cartesian directions. 

Order 0 minimization methods do not take the slope or the curvature properties of the energy surface into account. As a result, such methods are crude and can be used only with very simple energy surfaces, i.e., surfaces with a small number of local minima and monotonic behavior away from the minima. These methods are rarely used for macro-molecular systems. 

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