A generalized Romberg differentiation procedure for calculation of hyperpolarizabilities

A generalized Romberg differentiation procedure decreasing truncation errors for numerical evaluation of first-order as well as higher-order derivatives is proposed. In comparison with the original Romberg method the higher-order terms are eliminated more efficiently. Moreover, the generalized scheme is more robust in treating rounding errors, as shown by numerical tests. The proposed method can find a wide range of applications including the evaluation of electronic and vibrational (hyper)polarizabilities.