Reduced free-particle Green’s functions in quantum-mechanical perturbation calculations

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Abstract
Several new methods are given by which contributions to terms involving single excitations in many‐body perturbation series expansions of electronic properties of atoms and molecules can be calculated accurately. All are based upon solution of a Fredholm identity connecting the reduced free‐particle Green’s function with a reduced Green’s function appearing in the Goldstone diagram that is to be evaluated. The most generally applicable approach appears to be one based upon the method of moments and the expansion of an appropriately defined first‐order wavefunction in a set of polynomials that are orthogonal with respect to a weight function appearing naturally in the configuration‐space representation of the diagram. This ’’polynomial moment’’ method is compared with a ’’power moment’’ method and also with iterative solutions of the Fredholm identity. Accuracy of the numerical values of the diagram so obtained varies from exact (for the dipole polarizability for the hydrogen atom), through 10 significant figures (for the single‐particle excitation term in the second‐order energy for a two‐electron atom in a 1/Z expansion) to 5 significant figures (for the first‐order term in a 1/Z expansion of the electron density at the nucleus of a two‐electron atom). These methods obviate explicit use of hydrogenic Green’s functions and require at most the inversion of a matrix of order ∼15.