Gradient of energy

The positions on the potential energy surface of reactants or products (or reaction intermediates in a stepwise process) and the transition stmcmre are intrinsically different. In both cases, the gradient of energy is zero, that is, the derivatives of E iR) with respect to all the reaction coordinates are zero ... 

Following the Hellmann-Feynman theorem [49, 50] the gradient of energy is simply determined by the electron density p(f) ... 

All the reviewed programs carry out fundamental tasks of a computational chemist or a computational molecular physicist calculation of energy for various hamUtonians evaluation of gradients of energy (needed to locate stationary points on the potential energy surface) evaluation of the energy hessian (required to analyze the character of the located stationary point, identify local minima and saddle points, and perform vibrational frequency calculation) and evaluation of basic properties (population analysis, dipole moments). The components of the programs include basis set libraries and pseudopotentials. 

One of the most usefiil applications of the mean free path concept occurs in the theory of transport processes in systems where there exist gradients of average but local density, local temperature, and/or local velocity. The existence of such gradients causes a transfer of particles, energy or momentum, respectively, from one region of the system to another. 

P is the critical exponent and t denotes the reduced distance from the critical temperature. In the vicinity of the critical point, the free energy can be expanded in tenns of powers and gradients of the local order parameter m (r) = AW - I bW ... 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

One of the discussion points is how the quantum system reacts back on the classical d.o.f., i.e., how the forces on the classical system should be derived from the quantum system. One can use the gradient of the effective energy, i.e., of the expectation value of the total energy... 

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... 

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,... 

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