Gradient of the potential

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 ... 

The forces in a protein molecule are modeled by the gradient of the potential energy V(s, x) in dependence on a vector s encoding the amino acid sequence of the molecule and a vector x containing the Cartesian coordinates of all essential atoms of a molecule. In an equilibrium state x, the forces (s, x) vanish, so x is stationary and for stability reasons we must have a local minimizer. The most stable equilibrium state of a molecule is usually the... 

Our work is targeted to biomolecular simulation applications, where the objective is to illuminate the structure and function of biological molecules (proteins, enzymes, etc) ranging in size from dozens of atoms to tens of thousands of atoms today, with the desire to increase this limit to millions of atoms in the near future. Such molecular dynamics (MD) simulations simply apply Newton s law to each atom in the system, with the force on each atom being determined by evaluating the gradient of the potential field at each atom s position. The potential includes contributions from bonding forces. 

Molecular dynamics conceptually involves two phases, namely, the force calculations and the numerical integration of the equations of motion. In the first phase, force interactions among particles based on the negative gradient of the potential energy function U,... 

The steepest descent method is a first order minimizer. It uses the first derivative of the potential energy with respect to the Cartesian coordinates. The method moves down the steepest slope of the interatomic forces on the potential energy surface. The descent is accomplished by adding an increment to the coordinates in the direction of the negative gradient of the potential energy, or the force. 

Molecular dynamics simulations calculate future positions and velocities of atoms, based on their current positions and velocities. A simulation first determines the force on each atom (Fj) as a function of time, equal to the negative gradient of the potential energy (equation 21). 

Evaluate the force F as the negative gradient of the potential energy function ... 

The limitation of the gradient of the potential is particularly important for calculations with ADRs and for data sets that potentially contain noise peaks, since it facilitates the appearance of violations due to incorrect restraints. A standard hannonic potential would put a high penalty on large violations and would introduce larger distortions into the structure. 

In the discussion of the physical meaning of the calculations of the gradient of the potential energy in the S1 (or T1) state at the initial geom-... 

Since the gradient of the potential is the field given by Equation 7.33, its substitution in Equation 7.35 gives... 

Usually, the terms attraction and repulsion refer to ihe forces operating between the particles. The force is the gradient of the potential. Therefore, it is not always true that a negative potential is attractive, or that a positive potential is repulsive. The important quantity that determines the cooperativity is the potential and, in more general cases, the work W(l, 1), not the force. 

Because, in a conservative field, the force is linked to the gradient of the potential by... 

An important equation of electrostatics, which follows directly from Maxwell s equations (Jackson 1975) is Poisson s equation. It relates the divergence of the gradient of the potential charge density at that point ... 

Like the potential, other electrostatic functions can be expressed as Fourier summations over the structure factors (Stewart 1979). The electric field, being the (negative) gradient of the potential, is a Fourier series in which the power of the magnitude of H increases from —2 to —1, as expected from the reciprocal relationship between direct space and Fourier space. Starting with... 

Similar statements can be made about holes. They, too, have to be transported to the interface to be available for the receipt of electrons there. These matters all come under the influence of the Nernst-Planck equation, which is dealt with in (Section 4.4.15). There it is shown that a charged particle can move under two influences. The one is the concentration gradient, so here one is back with Fick s law (Section 4.2.2). On the other hand, as the particles are changed, they will be influenced by the electric field, the gradient of the potential-distance relation inside the semiconductor. Electrons that feel a concentration gradient near the interface, encouraging them to move from the interior of the semiconductor to the surface, get seized by the electric field inside the semiconductor and accelerated further to the interface. 

The interpretation of p, then, in terms of the gradients of the potential energy surfaces in electrode kinetics seems a reasonable one, and it does lead to values of P that are in fairly good accord with those observed. They are always near one-half but seldom exactly one-half, and that is just what potential energy surfaces indicate when calculations are made. 

This is precisely the same as the force that a unit positive charge would experience at the same location. Since force is the negative gradient of the potential, Equation (7) also supplies a second definition of field ... 

The first derivatives of a potential energy function define the gradient of the potential and the second derivatives describe the curvature of the energy surface (Fig. 3.4). In most molecular mechanics programs the potential functions used are relatively simple and the derivatives are usually determined analytically. The second derivatives of harmonic oscillators correspond to the force constants. Thus, methods using the whole set of second derivatives result in some direct information on vibrational frequencies. 

The rate constant can be expressed in terms of the potential of mean force at the activated complex. This potential may, e.g., be defined such that the gradient of the potential gives the average force on an atom in the activated complex due to the solvent molecules, Boltzmann averaged over all configurations. 

If the London force is expressed as the gradient of the potential energy of interaction 4>> the total flux may be written as the sum of the London and diffusive fluxes ... 

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