{'fraction_non_alphanumeric': 0.01904761904761905, 'fraction_numerical': 0.0, 'mean_word_length': 5.709219858156028, '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}}
Electron-pair densities in position and momentum spaces for multi-determinant wavefunctions

The electronic intracule (relative motion) and extracule (centre-of-mass motion) densities are electron-pair densities which characterize the motion of a pair of electrons in atoms and molecules. A unified method is presented for the evaluation of these electron-pair densities in both position and momentum spaces for wavefunctions expressed as linear combinations of Slater determinants. Detailed expressions are developed for atomic systems where angular integrations can be performed analytically. Interesting relations between atomic intracule and extracule densities and between their moments are discussed. An illustrative application of the results is given for the and states of the helium atom, and the first calculations are reported for the singlet-triplet differences in the extracule densities and in the momentum-space intracule density.