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New methods of determining the steepest descent reaction path in dynamical calculations based on the reaction path Hamiltonian are investigated. These methods use first and second energy derivatives and are correct to third order in the Taylor series expansion of the path in the arc length. The reaction H+H2O→H2+OH using small basis set three electron-three active orbital CASSCF wave functions with first and second analytically calculated energy derivatives is used as a test platform. Reaction paths using a number of different step sizes are computed and their accuracy determined by evaluating the deviations of the computed path curvature from the exact curvature and the RMS deviation of the computed transverse frequencies from the exact values, criteria more sensitive and more relevant to dynamical studies than are geometrical parameters. The new methods are compared to one another in computational efficiency and accuracy. For this example reaction and level of theory it is found that, in addition to being more accurate at any step size, the new methods are less expensive at any accuracy than those that utilize only first energy derivatives. [ABSTRACT FROM AUTHOR]