Part IV
Group Theory and Its Chemical
Applications
Universe comprises space and matter. These two mutually stipulate their modality of
existence. We often comprehend related various aspects as manifestation of symmetry. In this part we deal with the symmetry from a point of view of group theory.
In this last part of the book, we outline and emphasize chemical applications of the
methods of mathematical physics. This part supplies us with introductory description
of group theory. The group theory forms an important field of both pure and applied
mathematics. Starting with the definition of groups, we cover a variety of topics
related to group theory. Of these, symmetry groups are familiar to chemists, because
they deal with a variety of matter and molecules that are characterized by different
types of symmetry. The symmetry group is a kind of finite group and called a point
group as well. Meanwhile, we have various infinite groups that include rotation
group as a typical example. We also mention an introductory theory of the rotation
group of SO(3)that deals with an important topic of, e.g., Euler angles. We also treat
successive coordinate transformations.
Next, we describe representation theory of groups. Schur’s lemmas and related
grand orthogonality theorem underlie the representation theory of groups. In parallel,
characters and irreducible representations are important concepts that support the
representation theory. We present various representations, e.g., regular representation, direct-product representation, and symmetric and antisymmetric representations. These have wide applications in the field of quantum mechanics and quantum
chemistry, and so forth.
On the basis of the above topics, we highlight quantum chemical applications of
group theory in relation to a method of molecular orbitals. As tangible examples, we
adopt aromatic molecules and methane.
The last part deals with the theory of continuous groups. The relevant theory has
wide applications in many fields of both pure and applied physics and chemistry. We
highlight the topics of SU(2)and SO(3)that very often appear there. Tangible examples help understand the essence.
Group Theory and Its Chemical
Applications
Universe comprises space and matter. These two mutually stipulate their modality of
existence. We often comprehend related various aspects as manifestation of symmetry. In this part we deal with the symmetry from a point of view of group theory.
In this last part of the book, we outline and emphasize chemical applications of the
methods of mathematical physics. This part supplies us with introductory description
of group theory. The group theory forms an important field of both pure and applied
mathematics. Starting with the definition of groups, we cover a variety of topics
related to group theory. Of these, symmetry groups are familiar to chemists, because
they deal with a variety of matter and molecules that are characterized by different
types of symmetry. The symmetry group is a kind of finite group and called a point
group as well. Meanwhile, we have various infinite groups that include rotation
group as a typical example. We also mention an introductory theory of the rotation
group of SO(3)that deals with an important topic of, e.g., Euler angles. We also treat
successive coordinate transformations.
Next, we describe representation theory of groups. Schur’s lemmas and related
grand orthogonality theorem underlie the representation theory of groups. In parallel,
characters and irreducible representations are important concepts that support the
representation theory. We present various representations, e.g., regular representation, direct-product representation, and symmetric and antisymmetric representations. These have wide applications in the field of quantum mechanics and quantum
chemistry, and so forth.
On the basis of the above topics, we highlight quantum chemical applications of
group theory in relation to a method of molecular orbitals. As tangible examples, we
adopt aromatic molecules and methane.
The last part deals with the theory of continuous groups. The relevant theory has
wide applications in many fields of both pure and applied physics and chemistry. We
highlight the topics of SU(2)and SO(3)that very often appear there. Tangible examples help understand the essence.