4
1 Operations
and operate on it (on the left) by means of a transformation matrix D(R):
x 2
y 2
= D(R)
x 1
y 1
=
cos α − sin α
sin α cos α
x 1
y 1
(1.5)
Having obtained the algebraic expressions, it is always prudent to consider whether
the results make sense. Hence, while the point P 1 is rotated as shown in the picture,
its x-coordinate will decrease, while its y-coordinate will increase. This is reflected
by the entries in the first row of the matrix which show how x 1 will change: the
cos α factor is smaller than 1 and thus will reduce the x-value as the acute angle
increases, and this will be reinforced by the second term, −y 1 sin α, which will be
negative for a point with y 1 and sin α both positive. In what follows we also need
the inverse operation, ˆ
R −1 , which will undo the operation itself. In the case of a
rotation this is simply the rotation around the same axis by the same angle but in
the opposite direction, that is, by an angle −α. The combination of clockwise and
counterclockwise rotations by the same angle will leave all points unchanged. The
resulting nil operation is called the unit operation, ˆ
E:
ˆ
R ˆ
R
−1 = ˆ
R
−1 ˆ
R = ˆ
E
(1.6)
1.2 Operations and Functions
Chemistry of course goes beyond the structural characteristics of molecules and
considers functional properties associated with the structures. This is certainly the
case for the quantum-mechanical description of the molecular world. The primary
functions which come to mind are the orbitals, which describe the distribution of the
electrons in atoms and molecules. A function f (x,y,z) associates a certain property
(usually a scalar number) with a particular coordinate position. A displacement of
a point will thus induce a change of the function. This can again be defined in
several ways. Let us agree on the following: when we displace a point, the property
associated with that point will likewise be displaced with it. In this way we create a
new property distribution in space and hence a new function. This new function will
be denoted by ˆ
Rf (or sometimes as f ′ ), i.e., it is viewed as the result of the action
of the operation on the original function. In line with our agreement, a property
associated with the displaced point will have the same value as that property had
when associated with the original point, hence:
ˆ
Rf (P 2 ) = f(P 1 )
(1.7)
or, in general,
ˆ
Rf ( ˆ
RP 1 ) = f(P 1 )
(1.8)
Note that in this expression the same symbol ˆ
R is used in two different meanings,
either as transforming coordinates or a function, as is evident from the entity that follows the operator. This rule is sufficient to plot the transformed function, as shown
1 Operations
and operate on it (on the left) by means of a transformation matrix D(R):
x 2
y 2
= D(R)
x 1
y 1
=
cos α − sin α
sin α cos α
x 1
y 1
(1.5)
Having obtained the algebraic expressions, it is always prudent to consider whether
the results make sense. Hence, while the point P 1 is rotated as shown in the picture,
its x-coordinate will decrease, while its y-coordinate will increase. This is reflected
by the entries in the first row of the matrix which show how x 1 will change: the
cos α factor is smaller than 1 and thus will reduce the x-value as the acute angle
increases, and this will be reinforced by the second term, −y 1 sin α, which will be
negative for a point with y 1 and sin α both positive. In what follows we also need
the inverse operation, ˆ
R −1 , which will undo the operation itself. In the case of a
rotation this is simply the rotation around the same axis by the same angle but in
the opposite direction, that is, by an angle −α. The combination of clockwise and
counterclockwise rotations by the same angle will leave all points unchanged. The
resulting nil operation is called the unit operation, ˆ
E:
ˆ
R ˆ
R
−1 = ˆ
R
−1 ˆ
R = ˆ
E
(1.6)
1.2 Operations and Functions
Chemistry of course goes beyond the structural characteristics of molecules and
considers functional properties associated with the structures. This is certainly the
case for the quantum-mechanical description of the molecular world. The primary
functions which come to mind are the orbitals, which describe the distribution of the
electrons in atoms and molecules. A function f (x,y,z) associates a certain property
(usually a scalar number) with a particular coordinate position. A displacement of
a point will thus induce a change of the function. This can again be defined in
several ways. Let us agree on the following: when we displace a point, the property
associated with that point will likewise be displaced with it. In this way we create a
new property distribution in space and hence a new function. This new function will
be denoted by ˆ
Rf (or sometimes as f ′ ), i.e., it is viewed as the result of the action
of the operation on the original function. In line with our agreement, a property
associated with the displaced point will have the same value as that property had
when associated with the original point, hence:
ˆ
Rf (P 2 ) = f(P 1 )
(1.7)
or, in general,
ˆ
Rf ( ˆ
RP 1 ) = f(P 1 )
(1.8)
Note that in this expression the same symbol ˆ
R is used in two different meanings,
either as transforming coordinates or a function, as is evident from the entity that follows the operator. This rule is sufficient to plot the transformed function, as shown