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The diffusion of passing over the saddle point of a three-dimensional quadric potential energy surface was studied by analytically solving a set of coupled generalized Langevin equations. An accurate expression of the passing probability was obtained. The effect of the coupling between different degrees of freedom which is represented by the off-diagonal elements of the inertia, friction and potential-curvature tensors was analyzed in detail. It is found that some of the coupling have great influence on the diffusion process, while others not. The combination of them results in an optimal injecting direction of the diffusing particles, revealing an optimal three-dimensional diffusion path. • Off-diagonal elements of the tensors in the generalized Langevin equation were all recovered. • Optimal path of diffusion on the three-dimensional potential energy surface was determined. • Two angles were defined to determine the optimal path of diffusion of Brownian particles. [ABSTRACT FROM AUTHOR]