Page 93 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

78 P. CASSAM-CHENAÏ

The fundamental representation of the generators as n x n matrices is easily obtained; the
matrix elements have the following expressions :

is that it decomposes on with real numbers, even when the integrals
are complex (case of an electromagnetic field, of a molecule whose symmetry group has
irreducible representations which are not realizable over real numbers...) :

with w (respectively v ) one-electron (respectively two-electrons) Hermitian operator
and Re(x) (respectively Im(x)) real part (respectively imaginary part) of the complex
number x.

So the genuine generators of the unitary group have original properties and do not deserve
to be forgotten. It would seem weird to build the theory of angular momentum using only
with no mention of and . It is equally surprising that only the appear
in the theory of the unitary group. In short, in the traditional approach, one builds the Lie
algebra of the linear group but uses only the Lie subalgebra corresponding to the unitary
group. A more satisfactory approach would consist in generating the Lie algebra of the
unitary group only, using its real generators, then to define in this algebra with Eq.(3) the
rising and lowering operators

References

1. J. Paldus, J. Chem. Phys. 61, 5321 (1974).