Abstract: A fast and generally-applicable method to calculate an internal coordinate Jacobian at quadratic (order N2 where N is the number of internal coordinates) cost, which can be used to dramatically speed up molecular modeling methods. In one embodiment, the present invention provides methods and algorithms useful for converting a Cartesian Hessian into a Torsion Jacobian without limitation to pair-potential energy terms, and for performing such calculation with highly-efficient usage of computer memory. Such methods and algorithms can do the conversion in computation time quadratic in the number of internal coordinates used to model any multibody system, e.g., a molecular system. In a related embodiment, the present invention provides methods and algorithms useful for computing the stiffness matrix without limitation to pair-potential energy terms, and for performing such calculation with highly-efficient usage of computer memory.

Abstract: In the molecular modeling of a molecular system represented by constituent charged rigid bodies, the electrostatic interactions are computed between the rigid bodies, rather than the individual atoms which constitute the bodies. Multipole representations of the charge distribution of the bodies are used, and the total translational force and moment acting on each rigid body are computed using approximations of the form: 1 F ~ 2 , M ~ 2 = u 3 × 1 ⁢ q 2 + M 3 × 3 · p 2 + N 3 × 5 · Q ^ 2 + T 3 × 10 · O ^ 2 .

Abstract: A method for obtaining analytic Jacobians used in implicit integration methods in the computations for the dynamics of a physical system. With this method, the Jacobian with at least twice the number of digits of accuracy as a numerical Jacobian can be computed. This also results in the implicit integration method being more efficient because a smaller number of iterations are required to solve the nonlinear stage equations of the equations of motion, as well as the ability to take larger timesteps. This speedup in computation is very useful in molecular modeling.

Abstract: For the computer modeling of molecules, a model with reduced coordinates is used with sufficiently stable implicit integration methods integrating the model's equations of motion. The timesteps in the integration method can vary in a range over 100 to greatly increase the computer's efficiency and to hasten the computational results. Both static analysis and molecular dynamics simulations are some ready applications.

Abstract: The Residual Form of the equations of motion of a molecular model is used to reduce the computational load by a factor of approximately 7 as compared to the conventional Direct Form (not including the force computations). Implicit integrators are used with the Residual Form, especially L-stable integrators, such as implicit Euler and Radau5. A preferable molecular model is an Order (N), torsion angle, rigid multibody system.