Energy function, derivatives

The basic idea of NMA is to expand the potential energy function U(x) in a Taylor series expansion around a point Xq where the gradient of the potential vanishes ([Case 1996]). If third and higher-order derivatives are ignored, the dynamics of the system can be described in terms of the normal mode directions and frequencies Qj and Ui which satisfy ... 

In order to solve the classical equations of motion numerically, and, thus, to t)btain the motion of all atoms the forces acting on every atom have to be computed at each integration step. The forces are derived from an energy function which defines the molecular model [1, 2, 3]. Besides other important contributions (which we shall not discuss here) this function contains the Coulomb sum... 

Figure 2-108. Derivation of a syrMbolic potential energy function from the torsion angle distribution of a torsion fragment.

Figure 2-108 shows the correspondence between a histogram and the derived empirical energy function for the torsion angle fragment C-N H)-C(H)(H -C. 

I lagler A T, E Huler and S Lifson 1977. Energy Functions for Peptides and Proteins. I. Derivation of a Consistent Force Field Including the Hydrogen Bond from Amide Crystals. Journal of the American Chemical Society 96 5319-5327. 

Cartesian coordinates, the vector x will have 3N components and x t corresponds to the current configuration of fhe system. SC (xj.) is a 3N x 1 matrix (i.e. a vector), each element of which is the partial derivative of f with respect to the appropriate coordinate, d"Vjdxi. We will also write the gradient at the point k as gj.. Each element (i,j) of fhe matrix " "(xj.) is the partial second derivative of the energy function with respect to the two coordinates r and Xj, JdXidXj. is thus of dimension 3N x 3N and is... 

This section is concerned with the two-dimensional elasticity equations. Our aim is to find the derivative of the energy functional with respect to the crack length. The nonpenetration condition is assumed to hold at the crack faces. We derive the Griffith formula and prove the path independence of the Rice-Cherepanov integral. This section follows the publication (Khludnev, Sokolowski, 1998c). 

The formulae (4.120) will be used in getting the derivative of the energy functional. 

In this section we find the derivative of the energy functional in the three-dimensional linear elasticity model. The derivative characterizes the behaviour of the energy functional provided that the crack length is changed. The crack is modelled by a part of the two-dimensional plane removed from a three-dimensional domain. In particular, we derive the Griffith formula. 

Molecular Dynamics and Monte Carlo Simulations. At the heart of the method of molecular dynamics is a simulation model consisting of potential energy functions, or force fields. Molecular dynamics calculations represent a deterministic method, ie, one based on the assumption that atoms move according to laws of Newtonian mechanics. Molecular dynamics simulations can be performed for short time-periods, eg, 50—100 picoseconds, to examine localized very high frequency motions, such as bond length distortions, or, over much longer periods of time, eg, 500—2000 ps, in order to derive equiUbrium properties. It is worthwhile to summarize what properties researchers can expect to evaluate by performing molecular simulations ... 

Equations (l)-(3) in combination are a potential energy function that is representative of those commonly used in biomolecular simulations. As discussed above, the fonn of this equation is adequate to treat the physical interactions that occur in biological systems. The accuracy of that treatment, however, is dictated by the parameters used in the potential energy function, and it is the combination of the potential energy function and the parameters that comprises a force field. In the remainder of this chapter we describe various aspects of force fields including their derivation (i.e., optimization of the parameters), those widely available, and their applicability. 

When the solid phase 0+ at x = -f oo coexists with the gas phase 0 at X = -oo, the stationary profile of the phase field is determined so as to minimize the free energy functional F (56). The functional derivative gives... 

There are different ways of implementing the cut-off approximation. The simplest is to neglect all contributions if the distance is larger than the cut-off. This is in general not a very good method as the energy function becomes discontinuous. Derivatives of the energy function also become discontinuous, which causes problems in optimization... 

Actually the assumptions can be made even more general. The energy as a function of the reaction coordinate can always be decomposed into an intrinsic term, which is symmetric with respect to jc = 1 /2, and a thermodynamic contribution, which is antisymmetric. Denoting these two energy functions h2 and /zi, it can be shown that the Marcus equation can be derived from the square condition, /z2 = h. The intrinsic and thermodynamic parts do not have to be parabolas and linear functions, as in Figure 15.28 they can be any type of function. As long as the intrinsic part is the square of the thermodynamic part, the Marcus equation is recovered. The idea can be taken one step further. The /i2 function can always be expanded in a power series of even powers of hi, i.e. /z2 = C2h + C4/z. The exact values of the c-coefficients only influence the... 

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