{'fraction_non_alphanumeric': 0.026103464641670623, 'fraction_numerical': 0.008068343616516375, 'mean_word_length': 4.888268156424581, '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}}
Many-Body Perturbation Theory of the Effective Electron-Electron Interaction for Open-Shell Atoms

The effective-operator form of many-body theory is reviewed and applied to the calculation of the effective interaction of electrons in an open-shell atom. Numerical results are given for the 1s22s22p2 configuration of carbon. The effect of correlation upon the interaction of the 2p electrons of this configuration is represented by effective two-body operators of the form ΣakTk(1) · Tk(2). These operators are evaluated using angular-momentum diagrams and solving numerically a two-particle equation for the linear combination of excited states which contribute to the Goldstone diagrams. The effect of the operators of even rank is to depress the values of the two-electron Slater integrals Fk(2p, 2p) below their Hartree-Fock values. The two-body operator of odd rank does not appear in the Hartree-Fock theory. Our second-order values of the Slater integrals agree quite well with experiment but the value which we obtain of the coefficient of odd rank is much too small. This is partly due to a large cancellation which occurs for the contribution of the outer 2s2, 2s2p, 2p2 pair excitations. In order to study the convergence properties of the theory and to obtain more accurate values of the interaction integrals, we consider the higher-order terms in the perturbation expansion. An important family of two-particle effects is included to all orders by solving the pair equations iteratively until self-consistency is achieved. A more accurate description of the electron-electron interaction is obtained in this way. There are three additional families of wave-operator diagrams which can have an important effect. One family has an additional open-shell line which polarizes a closed-, open-, or excited orbital. There are also the coupled-cluster diagrams and a family of diagrams involving two polarizing open-shell lines, which appears first in fourth order. All of these diagrams can be included in our iterative scheme and they include all possible two-particle effects to self-consistency.