Unconstrained coordinates

As described in the preceding section, the unconstrained coordinates are obtained numerically, using here the basic Verlet recipe ... 

Comparing Eq. [51] with Eq. [52] leads to the identical expression, Eq. [50], for the approximate forces of constraints. Note the factor of in front of the last term in Eq. [52]. A common error in the literature is to extract the constraint forces Gjitg) from the basic Verier expression for the unconstrained coordinates, Eq. [48], and to write the constrained coordinates as follows ... 

Not surprisingly, comparing Eq. [53] with Eq. [51] leads to the forces of constraint in Eq. [50], but with a factor of V2 discrepancy. This discrepancy is due to unwarranted attempts to apply Eq. [48], which should be used only in the computation of the unconstrained coordinates [r (t(j + 8t)J, to the constrained coordinates [r(rQ + 8, (7))). Unfortunately, a factor of Va is often artificially introduced into the equations to mask this inconsistency. For convenience and conformity with the most widely adopted convention, the rest of this chapter redefines the undetermined parameters 7) such that their new values are equal to half their previously defined values. With this new definition, Eq. [46] takes the form... 

Accordingly, the position of the mass is completely determined by specifying the angle i , which may assume any value and is hence not subject to a constraint. This angular coordinate is much better adapted to the symmetry of the system since it automatically respects the constraint. Unconstrained coordinates of this type are called generalized coordinates. This new coordinate i and the Cartesian coordinates x and y are uniquely related to one another according to... 

The free energy differences obtained from our constrained simulations refer to strictly specified states, defined by single points in the 14-dimensional dihedral space. Standard concepts of a molecular conformation include some region, or volume in that space, explored by thermal fluctuations around a transient equilibrium structure. To obtain the free energy differences between conformers of the unconstrained peptide, a correction for the thermodynamic state is needed. The volume of explored conformational space may be estimated from the covariance matrix of the coordinates of interest, = ((Ci [13, lOj. For each of the four selected conform-... 

HyperChem includes only unconstrained optimization. That is, given the coordinates ( X of a set of atoms. A (the inde-... 

Starting from the transition state it was expected the reaction would evolve either forward to the products or backward to the reactants. During the unconstrained CPMD simulations, however, the system was always found to evolve towards the reactants. Because of this it was necessary to apply constrained dynamics to the principal coordinate reaction (the distance between WAT oxygen and GTP y-phosphorus) this enabled investigation of the system evolution towards the products (Fig. 2.7). 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

The corresponding analysis for an unconstrained system in Cartesian coordinates, in which all 3N coordinates are treated as soft, yields a constant Vm = (mi mjv) for the determinant of the mass matrix, which affects only the constant of proportionality, and thus yields a naive Boltzmann distribution /eq(R) cx Analysis of an unconstrained system in generalized... 

We first treat a stiff system as a generic unconstrained system. We consider a joint probability distribution T (g) for all 3N coordinates of a stiff system, soft and hard, given by... 

We next consider the effective force balance for all >N variables, while treating the system as an unconstrained system. For simplicity, we consider the case in which the crossover from ballistic motion to diffusion occurs at a timescale much less than any characteristic relaxation time for vibrations of the hard coordinates, so that the vibrations are overdamped, but in which the vibrational relaxation times are much less than any timescale for the diffusion of the soft coordinates. In this case, we may assume local equilibration of all 3N momenta at timescales of order the vibration time. Repeating the analysis of the Section V.A, while now treating all 3N coordinates as unconstrained, yields an effective force balance... 

Using this definition of we may generalize the diffusion equation for the distribution /( ) on the /-dimensional constraint surface to an equivalent diffusion equation for a distribution V Q) in the 3A-dimensional unconstrained space. We consider a model in which a system of 3N coordinates undergoes Brownian motion in the full unconstrained space under the influence of the mobility, defined above, as described by a diffusion equation... 

The content of diffusion equation (2.175) for such a model is, moreover, independent of our choice of a system of 3,N coordinates for the unconstrained space. Constrained Brownian motion may thus be described by a model with a mobility and an effective potential /eff in any system of 3N coordinates for... 

The connection between a diffusion equation and a corresponding Markov diffusion process may be established through expressions for drift velocities and diffusitivies. The drift velocity for both unconstrained and constrained systems may be expressed in an arbitrary system of coordinates in the generic form... 

Here, is the mobility tensor in the chosen system of coordinates, which is a constrained mobility for a constrained system and an unconstrained mobility for an unconstrained system. As discussed in Section VII, in the case of a constrained system, Eq. (2.344) may be applied either to the drift velocities for the / soft coordinates, for which is a nonsingular / x / matrix, or to the drift velocities for a set of 3N unconstrained generalized or Cartesian coordinates, for a probability distribution (X) that is dynamically constrained to the constraint surface, for which is a singular 3N x 3N matrix. The equilibrium distribution is. (X) oc for unconstrained systems and... 

We first consider derivatives with respect to curvilinear coordinates of the determinant g of the metric tensor in the unconstrained space. Using Eq. (A.14) and definition (2.16) for yields... 

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