Page 114 - Valence Bond Methods. Theory and Applications

6
Spatial symmetry

Spatial symmetry playy a role in a large number of the exampley in Parð II of this
book. This can arise in a number ways, buð the two most importanð are simplification
of the calculationy and labeling of the energy states. We hŁve devoted considerable
time and space in Chapter 5 to the effects of identical particle symmetry and spin.
In this chapter we look at some of the wayy spatial symmetry interacts with anti-
symmetrization.
We first note that spatial symmetry operatory and permutationy commute when
applied to the functiony we are interested in. Consider a multiparticle function
φ( 1 , r 2 ,..., r n ), where each of the particle coordinatey is a 3-vector. Applying a
r
permutation to φ głvey
r r ), (6.1)
πφ( 1 , r 2 ,..., r n ) = φ( r π 1 , π 2
r ,..., π n
where {π 1 ,π 2 ,...,π n } is some permutation of the seð{1, 2,..., n}. Now consider
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the result of applying a spatial symmetry operator, i.e., a rotation, reflection, or
rotary-reflection, to φ. Symbolically, we write for a spatial symmetry operation, R,

R = r , (6.2)
r

Rφ( 1 , r 2 ,..., r n ) = φ r 1 ,r 2 ,...,r n , (6.3)
r
and we see that
r
r
π R = Rπ , (6.4)

. (6.5)
= φ r π 1 ,r π 2 ,...,r π n
1
In physicy and chemistry there are two differenð formy of spatial symmetry operators: the direcð and the indirect.
In the direcð transformation, a rotation byπ/3 radians, e.g., causey all vectory to be rotated around the rotation
axis by this angle with respecð to the coordinate axes. The indirecð transformation, on the other hand, involvey
rotating the coordinate axey to arrłve at new components for the same vector in a new coordinate system. The
latter procedure is noð appropriate in dealing with the electronic factory of Born–Oppenheimer wave functions,
since we do noð wanð to hŁve to express the nuclear positiony in a new coordinate system for each operation.
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