Transformation matrices for representations of the spin permutation group. A graphical approach

✍ Scribed by Stephen Wilson

• Publisher: Elsevier Science
• Year: 1977
• Tongue: English
• Weight: 383 KB
• Volume: 49
• Category: Article
• ISSN: 0009-2614
• DOI: 10.1016/0009-2614(77)80467-3

✦ Synopsis

Transformation matrices for the representations of the spin permutation group may be easily derived using graphical methods of angular momentum theory. The transformation from the genealogical spin basis to the Jahn-Serber basis is considered as an illustrative example.

The spin functions for an N-electron system form a basis for an irreducible representation of the symmetric group, II,; that is i=f.

VPE I-l,-.

(1) S is the total spin quantum number, M is the quantum number corresponding to the projection of the spin on the z-axis, and k numbers the f[ spin functions for a given N and S. A detailed knowledge of the representation matrices is required in the study of the electronic structure of atoms and molecules [l]. The representation matrices depend on the mode of coupling employed in the construction of the spin functions. The most commonly used coupling is the genealogical scheme in which the one-electron spin functions are coupled according to the branching diagram shown in fig. la. Yamanouchi [2] derived a recursive method for constructing the representation matrices corresponding to the genealogical spin functions which does not require the explicit construction of the spin functions * _ In this work, we investigate the transformation from the genealogical basis to some non-standard spin basis**.

It is demonstrated that the graphical techniques'of angular momentum theory* are particularly useful in this respect. We base our discussion on an important example; the transformation from the genealogical basis to the Jahn-Serberbasis*:.The Jahn-Serber basis [13] is constructed according to the branching diagram shown in fig. lb'" _ In the following discussion we shall use si to denote the spin of the ith electron, and Si to denote the resultant spin obtained when spins si. s~+~, . . _ , s~_~, sj are coupled.