## Methane Symmetry Operations

### 4. Relation of Point-Group Operations to Permutation-Inversion Group
Operations

#### 4.1 Algebraic equation relating laboratory-fixed Cartesian coordinates
to rotational angles and vibrational displacement vectors

In this section we wish to relate the point group operations to the
permutation-inversion group operations. This can be accomplished with the least
amount of misunderstanding by using algebraic rather than geometric arguments,
i.e., by considering an equation relating the laboratory-fixed coordinates of
the nuclei to the rotational and vibrational coordinates used in molecular
wave functions [12,13]. In this article we shall take
this equation to have the form

**R**_{i} is a column vector containing the three
laboratory-fixed coordinates *X*_{i}, *Y*_{i},
*Z*_{i} of nucleus *i*. **R** is a column vector
containing the three laboratory-fixed coordinates of the center-of-mass of the
nuclei. *S* is a 3 × 3 rotation matrix relating the orientation
of the molecule-fixed axes *x*,*y*,*z*, to the orientation of
the corresponding laboratory-fixed axes *X*,*Y*,*Z*. We fix
conventions for the Eulerian angles χ θ φ by taking *S* to
have the form

where cχ = cosχ, sχ = sinχ, etc. The
quantity **a**_{i} is a column vector containing the three
molecule-fixed coordinates of nucleus *i* at equilibrium, and
**d**_{i} is a column vector containing the three
molecule-fixed components of the instantaneous displacement vector of nucleus
*i* from its equilibrium position.

Equation (9) can be understood most easily by reading
it from the right-hand side, as illustrated in
Fig. 2. We consider the nuclei to be brought into
their instantaneous positions in the laboratory by four steps. (a) Each
nucleus is placed at its equilibrium position **a**_{i}, as
in Fig. 2a. (b) Each nucleus is displaced from its equilibrium
position by an amount **d**_{i}, as in Fig. 2b.
(c) The entire molecular framework is rotated using the rotation matrix
*S*(χθφ), as in Fig. 2c. (d) The entire molecular
framework is translated by the vector **R**, as in Fig. 2d.

For completeness, though they will not be referred to again in this article,
the center-of-mass and Eckart conditions defining (explicitly) the vector
**R** and (implicitly) the rotational angles (χθφ), are
given in (eq. 11).

The qualitative ideas required to relate the point-group and
permutation-inversion operations are as follows. We seek a set of
transformations for the dynamical variables comprising
vibration-rotation-translation phase space, such that after the
**d**_{i}, χ, θ, φ, and **R** on the
right-hand side of (eq. 9), have been replaced by the
(**d**_{i})_{new}, χ_{new},
θ_{new}, φ_{new}, and **R**_{new},
the new right-hand side will be consistent with a left-hand side in which
**R**_{i} has been replaced by
(**R**_{i})_{new} = ±**R**_{j},
with a sign choice and subscript choice in agreement with the prescription of
the permutation-inversion operation under consideration.

It is sometimes convenient, when a given symmetry operation *P* is to be
applied to the dynamical variables of configuration space, to visualize it as
the product of three separate operators ^{1}*P*,
^{2}*P*, ^{3}*P*, which act on the vibrational,
rotational, and translational variables, respectively. While we shall not use
this notation explicitly in Section 4.2 and
Section 4.3, we shall deal separately and
in turn with vibrational, rotational, and translational transformations. When
considerations involving translation of the molecule as a whole are not
important, the operator ^{3}P can be ignored, as has been done in
Fig. 3 and Fig. 4
illustrating the vibrational and rotational transformations discussed below.