Negative gradient

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,... 

Molecular clynainics sim illations calculate future position s and velocities of atoms, based on their current positions and velocities. A sim Illation first determ in es the force on each atom (lY) as a function of time, ct ual to the negative gradient of the polen tial en ergy (ct]uation 2 I ),... 

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. 

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

If we put Inp = x, hiKi = /, the curve InK = f(lnp) will be a straight line having a positive, zero, or negative gradient according... 

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

The negative gradient of QV(r) equals the electrostatic force that is exerted by the molecule s unperturbed charge distribution upon the point charge Q. 

Series 28 contains a diverse range of electronic effects on the benzoyloxy ring. A plot of log TA100 against Hammett a constants for all but the 4-methoxy 28b (toxic) and 4-phenyl 28c (significantly more hydrophobic) was linear with negative gradient (Fig. 29).48... 

Suppose that the negative gradient of/(x), — V/(x), is selected as the search direction starting at the point xr = [1 2]. The negative gradient is the direction that maximizes the rate of change of /(x) in moving toward the minimum. To move in this direction we want to calculate a new x... 

The preceding results may be stated in algebraic terms. For V/ to lie within the cone described earlier, it must be a nonnegative linear combination of the negative gradients of the binding constraints that is, there must exist Lagrange multipliers u such that... 

In step 3 of the descent algorithm (because the gradient is clearly not small enough to stop), the first search direction is chosen as the negative gradient direction ... 

Since force is the negative gradient of the corresponding potential, one can also define an electrostatic potential (ESP) at a point r in the space around a molecule or solid, in terms of p(r), as... 

From classical physics, the negative gradient of a potential energy is a force - AVTr) = F(r). It follows that a force can also be associated with a process described by a potential energy VTRC) ... 

From elementary classical mechanics (43), the force acting on an atom in a molecular mechanics system is the negative gradient, or derivative, of the potential energy function with respect to atomic position. 

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