1.2 Operations and Functions
7
Equation (1.12) further reveals an important point. To express the transformation
of a function, one almost automatically encounters the concept of a function space.
To describe the transformation of the cosine function, one really also needs the sine.
The two form a two-dimensional space, which we shall call a vector space. This will
be explained in greater depth in Chap. 2. For now, we may cast the transformation
of the basis components of this space in matrix form. This time we arrange the basis
orbitals in a row-vector notation, so that the transformation matrix is written to the
right of the basis. Thus,
ˆ
R
2p x 2p y
=
2p x 2p y
cos α − sin α
sin α cos α
(1.13)
The matrix that is used here is precisely the same matrix which we used for the
coordinate transformation. How is this possible if functions and points transform in
opposite ways? The reason is of course that we also switched from a column vector
for points to a row vector for functions. Indeed, transposition, T, of the entire matrix
multiplication simultaneously inverts the transformation matrix and interchanges
columns and rows:
D
−1
x
y
T
=
xy
D
−1 T =
xy
D
(1.14)
where we made use of the property that transposition of the rotation matrix changes
α into −α and thus is the same as taking the inverse of D. The final point about
functions is somewhat tricky, so attention is required. Just like the value of a field,
or the amplitude of an orbital, the values of the coordinates themselves are properties
associated with points. As an example, the function that yields the x-coordinate of
a point P 1 , will be denoted as x(P 1 ). The value of this function is x 1 , where we
are using different styles to distinguish the function x, which is a variable, and the
coordinate x 1 , which is a number. We can thus write
x(P 1 ) = x 1
(1.15)
A typical quantum-chemical example of the use of these coordinate functions is
the dipole operator; e.g., the x-component of the electric dipole is simply given by
μ x =−ex, where −e is the electronic charge. We may thus write in analogy with
Eq. (1.5)
ˆ
R
xy
=
xy
cos α − sin α
sin α cos α
(1.16)
In summary, we have learned that when a symmetry operator acts on all the points
of a space, it induces a change of the functions defined in that space. The transformed functions are the result of a direct action of the symmetry operator in a
corresponding function space. Furthermore, there exists a dual relation between the
transformations of coordinate points and of functions. They are mutual inverses. Finally, the active picture also applies to the functions: the symmetry operation sets
the function itself into motion as if we were (physically) grasping the orbitals and
twisting them.
7
Equation (1.12) further reveals an important point. To express the transformation
of a function, one almost automatically encounters the concept of a function space.
To describe the transformation of the cosine function, one really also needs the sine.
The two form a two-dimensional space, which we shall call a vector space. This will
be explained in greater depth in Chap. 2. For now, we may cast the transformation
of the basis components of this space in matrix form. This time we arrange the basis
orbitals in a row-vector notation, so that the transformation matrix is written to the
right of the basis. Thus,
ˆ
R
2p x 2p y
=
2p x 2p y
cos α − sin α
sin α cos α
(1.13)
The matrix that is used here is precisely the same matrix which we used for the
coordinate transformation. How is this possible if functions and points transform in
opposite ways? The reason is of course that we also switched from a column vector
for points to a row vector for functions. Indeed, transposition, T, of the entire matrix
multiplication simultaneously inverts the transformation matrix and interchanges
columns and rows:
D
−1
x
y
T
=
xy
D
−1 T =
xy
D
(1.14)
where we made use of the property that transposition of the rotation matrix changes
α into −α and thus is the same as taking the inverse of D. The final point about
functions is somewhat tricky, so attention is required. Just like the value of a field,
or the amplitude of an orbital, the values of the coordinates themselves are properties
associated with points. As an example, the function that yields the x-coordinate of
a point P 1 , will be denoted as x(P 1 ). The value of this function is x 1 , where we
are using different styles to distinguish the function x, which is a variable, and the
coordinate x 1 , which is a number. We can thus write
x(P 1 ) = x 1
(1.15)
A typical quantum-chemical example of the use of these coordinate functions is
the dipole operator; e.g., the x-component of the electric dipole is simply given by
μ x =−ex, where −e is the electronic charge. We may thus write in analogy with
Eq. (1.5)
ˆ
R
xy
=
xy
cos α − sin α
sin α cos α
(1.16)
In summary, we have learned that when a symmetry operator acts on all the points
of a space, it induces a change of the functions defined in that space. The transformed functions are the result of a direct action of the symmetry operator in a
corresponding function space. Furthermore, there exists a dual relation between the
transformations of coordinate points and of functions. They are mutual inverses. Finally, the active picture also applies to the functions: the symmetry operation sets
the function itself into motion as if we were (physically) grasping the orbitals and
twisting them.