{'fraction_non_alphanumeric': 0.021526418786692758, 'fraction_numerical': 0.0, 'mean_word_length': 5.089285714285714, '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}}
Missing particles and modes.

We investigate the validity of the Fourier transformations that have long been used to transform a given Hamiltonian written in configuration space into momentum space. We show that although these transformations are valid for ideal (perfect) lattices, their use for lattices with vacancies poses mathematical problems and renders inaccuracies in the form of the Hamiltonian by not conserving the number of degrees of freedom in the Hilbert space. Consequently, transformations that avoid this problem are found. We also investigate another solution to this problem that was proposed in the literature. We study the difference between these three methods by investigating the ground-state energy of an electron system in the context of the [ital s]-band tight-binding model due to removing (a) one site and (b) two sites. The three methods give similar expressions for the ground-state energy for both of these cases, but they predict different binding energies of electrons at the vacancies.