Covariant metric tensor

A covariant metric tensor g p and contravariant inverse metric tensor in the full space are given by... 

The expression given by BCAH for elements of the constrained mobility within the internal subspace is based on inversion of the projection of the modified mobility within the internal subspace, rather than inversion of the projection (at of the mobility within the entire soft subspace. BCAH first define a tensor given by the projection of the modified friction tensor onto the internal subspace, which they denote by the symbol gat and refer to as a modified covariant metric tensor, which is equivalent to our CaT - They then define an inverse of this quantity within the subspace of internal coordinates, which they denote by g and refer to as a modified contravariant metric tensor, which is equivalent to our for afi = 1,..., / — 3. It is this last quantity that appears in their diffusion equation, given in Eq. (16.2-6) of Ref. 4, in place of our constrained mobility Within the space of internal coordinates, the two quantities are completely equivalent. 

The dependence on the curvilinear coordinate system has been made more explicit by replacing the elements by the equivalent covariant metric tensor... 

Dawes and Sceats ° have shown that the reduction to one dimension R of the FPE for two particles on a surface follows identical lines to that of Rodger and Sceats discussed in Section IIB. However, in this case the covariant metric tensor gy gives a determinant of R, which when used in Eq. (2.19), gives the effective potential P (R) = V R) — kTlnR. The first-order steady-state encounter rate is given by Eq. (7.4) with d = 2,... 

The measuring vectors associated with the /th body-frame component of the total angular momentum can be obtained as described in the following sections. As seen from Eq. (110), the contravariant metric tensor, where the rotational part has been expressed in the components of the internal angular momentum, is inverse to the covariant metric tensor, where the rotational part has been given in terms of the components of the rotational velocity. Thus, the vectors are reciprocal to the vectors e ) that is, they obey Eq. (120) ... 

Taking a step thus requires calculating the gradient and covariant metric tensor, and then solving a set of linear equations. 

Removing the polymer chains from explicit consideration directly affects the list of flexible coordinates in the TST formulation. For a monatomic penetrant atom or a pseudo-monatomic united atom, such as methane, the only flexible coordinates are the mass-weighted position (x,y,z) of the penetrant. The fixed polymer coordinates can be considered part of the force field defining the penetrant potential energy. The covariant metric tensor a equals a three-by-three identity matrix. 

The covariant metric tensor is found by matrix multiplication [Eq. (11)],... 

The product of Jacobian matrices can then be replaced with the covariant metric tensor a [Eq. (11)]. If we restrict attention to the transition state, where VqV = 0, we obtain Eq. (8). 

Premultiplying by J, replacing the J J product with the identity matrix, and substituting for the covariant metric tensor using Eq. (11) leads to Eq. (12). 

The covariant metric tensor, a, can be narrowed by subdividing it based on the flexible and stiff coordinates [119, p. 217]... 

The/x/submatrix a° in the upper left is the covariant metric tensor in only the flexible coordinates. Each element... 

From Eq. (72) above, the product Jj equals the covariant metric tensor in the reduced coordinate system, yielding Eq. (15). 

In Monte Carlo simulation, the choice of polymer model is governed by the choice of attempted moves. Typically kinetic energy is integrated analytically over all modes, and the partition function for the flexible model in the limit of infinite stiffness results [105]. In molecular dynamics, constraints (SHAKE, etc.) freeze kinetic energy contributions and the partition function for the rigid model results [105,197]. To achieve sampling from the desired partition function, it is necessary to add a pseudopotential based on the covariant metric tensor a [198]. 

We note that the components of the contravariant metric tensor axe obtained from the components of the covariant metric tensor [gijj by the equation... 

Equation (7.49) determines the covariant metric tensor G (q) in terms of 16 generalized coordinates rZq, dco and we have... 

It can be shown that co- and contravariant tensors may be converted into each other by applying the contravariant and covariant metric tensors (5-1) = 5M-V and as... 

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