Unitary group approach to the many-electron correlation problem via graphical methods of spin algebras

Physica Scripta

Volumn 21, Issue 3-4, 1980, Pages 295-311

  Paldus, J ^a,b   Boyle, M J ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84953347532     PISSN: 00318949     EISSN: 14024896     Source Type: Journal    
DOI: 10.1088/0031-8949/21/3-4/012     Document Type: Article

References (56)

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30. Paldus, J., Unitary Group Approach to the Many-Electron Correlation Problem, in Electrons in Finite and Infinite Structures (ed. P. Phariseau and L. Scheire), p. 411, Plenum. New York, 1977.
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38. The reader should beware of a number of misprints in [8], particularly in eqs. (3.2) and (3.6). Also, the graphical representation used [eq. (4.1)] is different from ours, being more appropriate for Young-Yamanouchi rather than Yamanouchi-Kotani states (cf. [28, 30]).
39. Kotani, M., Amemiya, A., Ishiguro, E. and Kimura, T., Tables of Molecular Integrals. 2nd edition. Maruzen, Tokyo, 1963.
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43. These phase and normalization factors, arising from a transition to 3 — jm vertices, will be cancelled out by similar factors which result from diagram factorization. It would thus be more convenient to work directly with the CG representation, as developed by Jucys and Bandzaitis [14], However, since this text is not available in English translation, as well as for other reasons (higher symmetry of 3 — jm graphs and their more general usage) we refrain from using the formalism of [14]. We would also like to mention that a very convenient unification of both CG and 3 — jm representations is formulated in the forthcoming book by Lindgren and Morrison [37].
44. Lindgren, I. and Morrison, J., Atomic Many-Body Theory, Springer Verlag, New York (to be published).
45. Paldus, J. and Boyle, M. J. (to be published).
46. r. Additional arguments or indices X and/or Y are then introduced for two-particle segments.
47. k relate the b-values of the same rows of the ket and bra states involved in the matrix element under consideration.
48. The single prime reflects the fact that only one generator line is involved in this type of internal segment.
49. Since Shavitt [6] uses the symmetric scheme, our lowering generator segment values are different from his, which are simply obtained by “inversions” [cf., eqs. (47)] of raising generator values.
50. Assuming, of course, that the lines associated with doubly occupied orbitals are equivalent.
51. The normal ordering and normal product are defined relative to the true (physical) vacuum state| 0〉, since in this paper we work solely within the particle formalism.
52. These expressions can be very useful, for example, in calculating the matrix elements of U(n) Casimir operators, or of similar operators arising from spatial symmetry considerations (cf., e.g., [3], p. 269).
53. These can be easily remembered either as the number of resulting orbital diagrams or as the number of terms in eq. (50) for the operator which has the same indices.
54. ij (k ≠ i, j)].
55. ji – yields the same matrix element above and below the diagonal, only the lower one is considered). In other words, every term in Table III of [3] must give a contribution to the lower diagonal part of the Hamiltonian matrix; if this contribution vanishes, the upper diagonal contribution is taken instead and is transposed into the lower diagonal part.
56. b.