Application to Internal Coordinate Constraints

The Taylor expansion, Eq. [57], is linearized, and no further iterations are carried out using the lowest nonlinear term, as in the matrix method. 

The current time step unconstrained molecular configuration is assumed to be (approximately) identical to the preceding time step molecular configuration, as described in connection with Eq. [93]. 

Clearly, approximation 1 leads to an Eq. [85] that is linear in the Lagrangian multipliers. Not surprisingly, its solution by the TB method is found to be inaccurate.- The reason for the inaccuracy is obvious in light of the steps of the matrix method the solution in Eq. [89] is just a linearization first estimate of the true solution, and no further iterations, using at least the lowest nonlinear term in the expansion, are carried out to refine this first estimate, unlike the procedure followed in the matrix method. To deal with this problem, Tobias and Brooks decouple the constraints and iterate over them until convergence is reached to within a certain tolerance.  

Therefore, in the final analysis the TB method is equivalent to SHAKE with the additional approximation of Eq. [93]. Because the equations being decoupled and iterated over are Eq. [92] with the approximation of Eq. [93], or equivalently Eq. [85], the SHAKE parameters are given in the TB method by 

In contrast to the correct expression for of the true SHAKE algorithm, given by Eq. [69], the denominator in the expression for [7jJ ] of Eq. [94] is constant during successive iterations, and its value is determined using only the preceding time step particle positions. Hence the only feedback in this iterative process is through the numerator -I- 8t) ). One side effect of 

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Lipkowitz and D. B. Boyd, Eds., VCH, Weinheim, Germany, 1998, pp. 75-136. Molecular Dynamics with General Holonomic Constraints and Application to Internal Coordinate Constraints. 

Kutteh, R., Straatsma, T.P. Molecular dynamics with general holonomic constraints and application to internal coordinate constraints. In Reviews in Computational Chemistry (eds... 

R. Kutteh and T. P. Straatsma, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., Wiley-VCH, New York, 1998, Vol. 13, pp. 75-136. Molecular Dynamics with General Holonomic Constraints and Applications to Internal Coordinate Constraints. J. C. Shelley and D. R. Berard, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., Wiley-VCH, New York, 1998, Vol. 12, pp. 137-205. Computer Simulation of Water Physisorption at Metal-Water Interfaces. 

Molecular Dynamics with General Holonomic Constraints and Application to Internal Coordinate Constraints... 

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