Chapter 19
Applications of Group Theory to Physical
Chemistry
On the basis of studies of group theory, now in this last chapter we apply the
knowledge to the molecular orbital (MO) calculations (or quantum chemical calculations). As tangible examples, we adopt aromatic hydrocarbons (ethylene,
cyclopropenyl radical, benzene, and ally radical) and methane. The approach is
based upon a method of linear combination of atomic orbitals (LCAO). To seek an
appropriate LCAO MO, we make the most of a method based on a symmetryadapted linear combination (SALC). To use projection operators is a powerful tool
for this purpose. For the sake of correct understanding, it is desired to consider
transformation of functions. To this end, we first show several examples.
In the process of carrying out MO calculations, we encounter a secular equation
as an eigenvalue equation. Using a SALC eases the calculations of the secular
equation. Molecular science relies largely on spectroscopic measurements and
researchers need to assign individual spectral lines to a specific transition between
the relevant molecular states. Representation theory works well in this situation.
Thus, the group theory finds a perfect fit with its applications in the molecular
science.
19.1 Transformation of Functions
Before showing individual examples, let us consider the transformation of functions
(or vectors) by the symmetry operation. Here we consider scalar functions.
For example, let us suppose an arbitrary function f(x, y) on a Cartesian xycoordinate. Figure 19.1 shows a contour map of f(x, y) ¼ constant. Then suppose
that the map is rotated around the origin. More specifically, the position vector r 0
fixed on a “summit” [i.e., a point that gives a maximal value of f(x, y)] undergoes a
symmetry operation, namely, rotation around the origin. As a result, r 0 is
transformed to r
0
0 . Here, we assume that a “mountain” represented by f(x, y) is a
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_19
729
Applications of Group Theory to Physical
Chemistry
On the basis of studies of group theory, now in this last chapter we apply the
knowledge to the molecular orbital (MO) calculations (or quantum chemical calculations). As tangible examples, we adopt aromatic hydrocarbons (ethylene,
cyclopropenyl radical, benzene, and ally radical) and methane. The approach is
based upon a method of linear combination of atomic orbitals (LCAO). To seek an
appropriate LCAO MO, we make the most of a method based on a symmetryadapted linear combination (SALC). To use projection operators is a powerful tool
for this purpose. For the sake of correct understanding, it is desired to consider
transformation of functions. To this end, we first show several examples.
In the process of carrying out MO calculations, we encounter a secular equation
as an eigenvalue equation. Using a SALC eases the calculations of the secular
equation. Molecular science relies largely on spectroscopic measurements and
researchers need to assign individual spectral lines to a specific transition between
the relevant molecular states. Representation theory works well in this situation.
Thus, the group theory finds a perfect fit with its applications in the molecular
science.
19.1 Transformation of Functions
Before showing individual examples, let us consider the transformation of functions
(or vectors) by the symmetry operation. Here we consider scalar functions.
For example, let us suppose an arbitrary function f(x, y) on a Cartesian xycoordinate. Figure 19.1 shows a contour map of f(x, y) ¼ constant. Then suppose
that the map is rotated around the origin. More specifically, the position vector r 0
fixed on a “summit” [i.e., a point that gives a maximal value of f(x, y)] undergoes a
symmetry operation, namely, rotation around the origin. As a result, r 0 is
transformed to r
0
0 . Here, we assume that a “mountain” represented by f(x, y) is a
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_19
729