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20
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84950781443
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Ph.D. thesis, University of Waterloo,
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(1972)
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Keating, B.A.1
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43
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0004276833
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(Institute of Physics and Mathematics of the Academy of Sciences of the Lithuanian S. S. R., Mintis, Vilnius)(in Russian). English translations: Israel Program for Scientific Translations, Jerusalem, 1962, and Gordon and Breach, New York, 1964;
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(1960)
Mathematical Apparatus of the Theory of Angular Momentum
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Jucys, A.P.1
Levinson, I.B.2
Vanagas, V.V.3
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49
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85034680613
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To avoid misunderstanding let us recall that in the terminology of the many‐body perturbation theory the connected part is represented by a sum of connected diagrams. A diagram is connected if and only if there exists at least one path of oriented and/or nonoriented lines between any two vertices. In contrast, a linked diagram is a diagram which does not contain any disconnected vacuum part(s). Thus, for a diagram having only internal lines (i.e., lines leaving and entering some vertex or vertices) the concepts of linked and connected diagrams are synonymous. However, for the diagram containing also external lines, the linked diagram may be either connected or disconnected. Of course, an unlinked diagram is always disconnected.
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52
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84950758891
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In order to achieve a compact representation of Hartree‐Fock‐type diagrams it is convenient to use the Schrödinger equation with the Hamiltonian in the normal product form with respect to [formula omitted] (i.e., [formula omitted]) as explained in Appendix D of Ref. 19.
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53
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85034678938
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Strictly speaking this assumes that we use for [formula omitted] the restricted Hartree‐Fock ground state. Otherwise, [formula omitted] simply represents the difference between the exact and reference state energies.
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54
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85034679346
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If only the spin‐orbital formalism is desired it is most convenient to use Brandow’s form even for the interaction vertices (cf. Ref. 5).
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55
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85034674519
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In fact, Fig. 3 shows the corresponding orbital diagrams pertinent to a spin‐adapted theory. The diagrams pertinent to the spin‐orbital form are simply obtained from the given diagrams by replacing the lower case orbital labels by corresponding capitals labeling the spin orbitals.33 Further, the labels [formula omitted] in Brandow’s vertices should be replaced by A [cf., Fig. 11(a)] indicating that antisymmetrized matrix elements are associated with them.
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56
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84950853415
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It should be noted that the singlet spin‐adapted states as defined by (17)23 are the same as those35 used in Ref. 28, except for an opposite phase of the [formula omitted] singlet. Correspondingly, the [formula omitted] matrix elements have opposite signs to the corresponding doubly excited state coefficients in relations (23) of Ref. 28.
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71
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84950898103
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There is clearly no essential difference between various forms for one‐particle (i.e., [formula omitted] [formula omitted] or [formula omitted]) operators.
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