an important position in algebra. These include a triangle matrix, diagonalizable
matrix as well as a nilpotent matrix and idempotent matrix. The most general form
will be Jordan canonical form. We present its essential parts in detail taking a
tangible example. Next to the general discussion, we deal with an inner product
space. Once an inner product is defined between any couple of vectors, the vector
space is given a fruitful structure. An example is a norm (i.e., “length”) of a vector.
Also, we gain a clear relationship between Part III and Part I. We define various
operators or matrices that are important in physical applications. Examples include
normal operators (or matrices) such as Hermitian operators, projection operators, and
unitary operators. Once again, we emphasize the importance of Hermitian operators.
In particular, two commutable Hermitian matrices share simultaneous eigenvectors
(or eigenstates) and, in this respect, such two matrices occupy a special position in
quantum mechanics.
Finally, Part IV describes the essence of group theory and its chemical applications. Group theory has a broad range of applications in solid-state physics, solidstate chemistry, molecular science, etc. Nonetheless, the knowledge of group theory
does not seem to have fully prevailed among chemists. We can discover an adequate
reason for this in a preface to the first edition of Chemical Applications of Group
Theory written by F. A. Cotton. It might well be natural that definition and statement
of abstract algebra, especially group theory, sound somewhat pretentious for chemists, even though the definition of group is quite simple. Therefore, we present
various examples for readers to get used to notions of group theory. The notion of
mapping is important as in the case of the linear vector spaces. Aside from being
additive with calculation for a vector space and multiplicative for a group, the
fundamentals of calculation regulations are pretty much the same regarding the
vector space and group. We describe the characteristics of symmetry groups in detail
partly because related knowledge is useful for molecular orbital (MO) calculations
that are presented in the last section of the book. Representation theory is probably
one of the most daunting notions for chemists. Practically, however, the representation is just homomorphism that corresponds to a linear transformation in a vector
space. In this context, the representation is merely denoted by a number or a matrix.
Basis functions of representation correspond to basis vectors in a vector space.
Grand orthogonality theorem (GOT) is a “nursery bed” of the representation theory.
Therefore, readers are encouraged to understand its essence apart from the rigorous
proof of the theorem. In conjunction with Part III, we present a variety of projection
operators. These are very useful to practical applications in, e.g., quantum mechanics
and molecular science. The final parts of the book are devoted to applications of
group theory to problems of physical chemistry, especially those of quantum
chemistry, more specifically molecular orbital calculations. We see how symmetry
consideration, particularly the use of projection operators, saves us a lot of labor.
Examples include aromatic hydrocarbons and methane.
The previous sections sum up the contents of this book. Readers may start with
any part and go freely back and forth. This is because contents of many parts are
interrelated. For example, we emphasize the importance of Hermiticity of differential operators and matrices. Also projection operators and nilpotent matrices appear
Preface to the First Edition
xi
matrix as well as a nilpotent matrix and idempotent matrix. The most general form
will be Jordan canonical form. We present its essential parts in detail taking a
tangible example. Next to the general discussion, we deal with an inner product
space. Once an inner product is defined between any couple of vectors, the vector
space is given a fruitful structure. An example is a norm (i.e., “length”) of a vector.
Also, we gain a clear relationship between Part III and Part I. We define various
operators or matrices that are important in physical applications. Examples include
normal operators (or matrices) such as Hermitian operators, projection operators, and
unitary operators. Once again, we emphasize the importance of Hermitian operators.
In particular, two commutable Hermitian matrices share simultaneous eigenvectors
(or eigenstates) and, in this respect, such two matrices occupy a special position in
quantum mechanics.
Finally, Part IV describes the essence of group theory and its chemical applications. Group theory has a broad range of applications in solid-state physics, solidstate chemistry, molecular science, etc. Nonetheless, the knowledge of group theory
does not seem to have fully prevailed among chemists. We can discover an adequate
reason for this in a preface to the first edition of Chemical Applications of Group
Theory written by F. A. Cotton. It might well be natural that definition and statement
of abstract algebra, especially group theory, sound somewhat pretentious for chemists, even though the definition of group is quite simple. Therefore, we present
various examples for readers to get used to notions of group theory. The notion of
mapping is important as in the case of the linear vector spaces. Aside from being
additive with calculation for a vector space and multiplicative for a group, the
fundamentals of calculation regulations are pretty much the same regarding the
vector space and group. We describe the characteristics of symmetry groups in detail
partly because related knowledge is useful for molecular orbital (MO) calculations
that are presented in the last section of the book. Representation theory is probably
one of the most daunting notions for chemists. Practically, however, the representation is just homomorphism that corresponds to a linear transformation in a vector
space. In this context, the representation is merely denoted by a number or a matrix.
Basis functions of representation correspond to basis vectors in a vector space.
Grand orthogonality theorem (GOT) is a “nursery bed” of the representation theory.
Therefore, readers are encouraged to understand its essence apart from the rigorous
proof of the theorem. In conjunction with Part III, we present a variety of projection
operators. These are very useful to practical applications in, e.g., quantum mechanics
and molecular science. The final parts of the book are devoted to applications of
group theory to problems of physical chemistry, especially those of quantum
chemistry, more specifically molecular orbital calculations. We see how symmetry
consideration, particularly the use of projection operators, saves us a lot of labor.
Examples include aromatic hydrocarbons and methane.
The previous sections sum up the contents of this book. Readers may start with
any part and go freely back and forth. This is because contents of many parts are
interrelated. For example, we emphasize the importance of Hermiticity of differential operators and matrices. Also projection operators and nilpotent matrices appear
Preface to the First Edition
xi