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Automatic differentiation method for numerical construction of the rotational-vibrational Hamiltonian as a power series in the curvilinear internal coordinates using the Eckart frame.

We present a new numerical method to construct a rotational-vibrational Hamiltonian of a general polyatomic molecule in the Eckart frame as a power series expansion in terms of curvilinear internal coordinates. The expansion of the kinetic energy operator of an arbitrary order is obtained numerically using an automatic differentiation (AD) technique. The method is applicable to molecules of arbitrary size and structure and is flexible for choosing various types of internal coordinates. A new way of solving the Eckart-frame equations for curvilinear coordinates also based on the AD technique is presented. The resulting accuracy of the high-order expansion coefficients for the kinetic energy operator using our numerical technique is comparable to that obtained by symbolic differentiation, with the advantage of being faster and less demanding in memory. Examples for H2CO, NH3, PH3, and CH3Cl molecules demonstrate the advantages of the curvilinear internal coordinates and the Eckart molecular frame for accurate ro-vibrational calculations. Our results show that very high accuracy and quick convergence can be achieved even with moderate expansions if curvilinear coordinates are employed, which is important for applications involving large polyatomic molecules.