Operations - Group Theory Applied to Chemistry - page 5

Fig. 1.3 The rotation of the
function f(x,y)
counterclockwise by an angle
α generates a new function,
f (x,y) . The value of the
new function at P 2 is equal to
the value of the old function
at P 1 . Similarly, to find the
value of the new function at
P 1 ,wehavetoretrievethe
value of the old function at a
point P 0 , which is the point
that will be reached by the
clockwise rotation of P 1

in Fig. 1.3 . In order to determine the mathematical form of the new transformed
function, we must be able to compare the value of the new function with the origi-
nal function at the same point , i.e., we must be able to see how the property changes
at a given point. Thus, we would like to know what would be the value of Rf in the
original point P 1 . Equation ( 1.8 ) cannot be used to determine this since the trans-
formed function is as yet unknown and we thus do not know the rules for working
out the brackets in the left-hand side of the equation. However, this relationship
must be true for every point; thus, we may substitute R − 1 P 1 for P 1 on both the left-
and right-hand sides of Eq. ( 1.8 ). The equation thus becomes
Rf R R − 1 P 1 = Rf( EP 1 )

f R − 1 P 1 = Rf(P 1 ) = (1.9)

This result reads as follows: the transformed function attributes to the original point
P 1 the property that the original function attributed to the point R − 1 P 1 .InFig. 1.3
this point from which the function value was retrieved is indicated as P 0 . Thus, the
function and the coordinates transform in opposite ways. 2 This connection transfers
the operation from the function to the coordinates, and, since the original function
is a known function, we can also use the toolbox of corresponding rules to work out
the bracket on the right-hand side of Eq. ( 1.9 ).
As an example, consider the familiar 2 p orbitals in the xy plane: 2 p x , 2 p y . These
orbitals are usually represented by the iconic dumbell structure. 3 We can easily
find out what happens to these upon rotation, simply by inspection of Fig. 1.4 ,in
which we performed the rotation of the 2 p x orbital by an angle α around the z -axis.
Clearly, when the orbital rotates, the overlap with the 2 p x function decreases, and
the 2 p y orbital gradually appears. Now let us apply the formula to determine R 2 p x .
The functional form of the 2 p x orbital for a hydrogen atom, in polar coordinates,
reads: R 2 p (r)Θ 2 p | 1 | (θ)Φ x (φ) , where R 2 p (r) is the radial part, Θ 2 p | 1 | (θ) is the part

2 A more general expression for the transportation of a quantum state may also involve an additional
phase factor, which depends on the path. See, e.g., [ 1 ].
3 The electron distribution corresponding to the square of these orbitals is described by a lemniscate
of Bernoulli. The angular parts of the orbitals themselves are describable by osculating spheres.