{'fraction_non_alphanumeric': 0.030303030303030304, 'fraction_numerical': 0.002635046113306983, 'mean_word_length': 4.891472868217054, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Thermodynamics of Strongly Anharmonic Vibrations Using Bogoliubov's Inequality

The self-consistent method, based on Bogoliubov's inequality, is a scheme which provides optimised vibrational frequencies Ω. We have derived an explicit equation for Ω when a single particle is bound to a potential which is specified by its series expansion in the particle displacement from the potential minimum. This equation is applied to the potentials V tan2 (πx/a) and V[(x/a) - (a/x)]2. It is discussed to what extent the true vibrational entropy can be approximated by a proper choice of a vibrational frequency in the harmonic entropy formula. We find that Ω obtained by the self-consistent scheme may be less adequate than the result of low order perturbation theory.