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Methane Symmetry Operations - Quantum Mechanical Operators

10.   Symmetry Properties of Selected Quantum Mechanical Operators

10.1   The Hamiltonian

The Hamiltonian of a molecule in free space is by hypothesis invariant with respect to the interchange of identical particles and to the laboratory-fixed inversion operation. Since all operations of the molecular symmetry group as defined above are equivalent to some permutation-inversion operation, the Hamiltonian is invariant under all symmetry-group operations and is therefore of species A1.

The Hamiltonian is often written as a sum of products of vibrational, rotational and nuclear spin operators. The requirement that it be of species A1 is then used to exclude mathematically possible but physically inadmissible terms. This application of group theory is particularly useful in methane, where interaction terms between vibrational, rotational and nuclear spin motions are relatively complicated to set up correctly. These interaction operators, which are not in general of species A1 with respect to transformations of only the vibrational, only the rotational or only the nuclear spin variables, give rise to many interesting physical effects, e.g., non-zero intensity in the v2 fundamental band [16], pure rotational transitions in the ground state [33-38], and transitions between states of different total nuclear spin [23, 36, 39].

10.2   The electric dipole-moment operator

The laboratory-fixed components µX, µY, µZ of the electric dipole moment operator are by definition sums over all particles i in the molecule of products of the charge qi and laboratory-fixed position coordinates X, Y, Z, i.e., ΣqX, ΣqY, ΣqZ. These sums are clearly invariant to a permutation of the coordinates of identical particles, but change sign under the laboratory-fixed inversion operation. From the characters for Td given in Table 1, and the fact that permutation-inversions correspond to point-group sense-reversing operations, we see that each of the laboratory-fixed components of the electric dipole moment is of species A2.

Symmetry species of the molecule-fixed components µ, µ, µz of the electric dipole moment operator can be determined correctly by intuition. But, as in Sec. 6, it is safer to proceed algebraically, recalling that transformation from laboratory-fixed to molecule-fixed vector components takes place via the direction cosine matrix, and that transformation properties of this matrix under the molecular symmetry group operations were fixed earlier (Sec. 7.2). We have

$$ \left[ \begin{array}{c} \mu_x\\ \mu_y\\ \mu_z \end{array}\right] = \left[ \begin{array}{c} ~\\ S(\chi\theta\phi) \\ ~ \end{array}\right] \left[\begin{array}{c} \mu_X \\ \mu_Y \\ \mu_Z \end{array}\right] ~ .$$

(eq. 35)

We must now apply symmetry operations to the right side of (eq. 35), and then determine transformations for the left side consistent with the changes which occur on the right. It is not difficult to show that the left side of (eq. 35) transforms according to (eq. 1). Thus, as expected intuitively, the molecule-fixed dipole-moment components µ, µ, µz are of species F2, F2, F2, respectively.

10.3   The total angular momentum operator

The laboratory-fixed components J, J, J of the total angular momentum operator can be written as the sums Σi(Yi PZi − Zi PYi), Σi(Zi PXi − Xi PZi), and Σi(Xi PYi − Yi PXi) over all particles. These operators are invariant to permutations of identical particles. They are also clearly invariant to a laboratory-fixed inversion. Consequently, each of the three operators J, J, J is of species A1.

The alert reader will recall that explicit expressions [4] for J, J, J were given in (eq. 25) as functions only of the Eulerian angles and partial derivatives with respect to these angles, and that transformations of the Eulerian angles have already been specified. A somewhat involved mathematical argument shows that indeed J, J, J as defined in (eq. 25) are all invariant with respect to the Eulerian angle transformations of the full Td point group. The demonstration for operations in the D2d subgroup of Table 5, of course, is quite simple.

Arguments using the analog of (eq. 35) indicate that the molecule-fixed components J, J, J of the total angular momentum operator transform according to (eq. 5), i.e., they belong to the symmetry species F1, F1, F1, respectively.

10.4 The proton spin operators

By arguments similar to those above, the laboratory-fixed components I, I, I of the total proton spin operator belong to the species A1, and the molecule-fixed components I, I, I belong to the species F1x , F1y , F1z.

In fact, however, there are twelve proton spin operators, one three-component vector for each of the four protons. The species of twelve linearly independent combinations of both laboratory-fixed [17] and molecule-fixed components of these operators are given in Table 18 and Table 19.


Table 18. Species of linear combinations of the laboratory-fixed components of the individual proton spin operators

Coefficients in the linear combinations are +1, −1, or 0, as indicated in the body of the table by +, −, or blank, respectively.
Species   I1X     I2X     I3X     I4X     I1Y     I2Y     I3Y     I4Y     I1Z     I2Z     I3Z     I4Z  
A1 + + + +    
A1   + + + +  
A1     + + + +
F2x + +    
F2y + +
F2z + +
F2x   + +  
F2y + +
F2z + +
F2x     + +
F2y + +
F2z + +

Table 19.   Species of linear combinations of the molecule-fixed components of the individual proton spin operators

Non-zero coefficients in the linear combinations are given in the body of the table, except for the row pertaining to Ea. These entries must be multiplied by square root of 3.
Species   I1x     I2x     I3x     I4x     I1y     I2y     I3y     I4y     I1z     I2z     I3z     I4z  
A2 +1 −1 +1 −1 −1 +1 +1 −1 +1 +1 −1 −1
Ea +1 −1 +1 −1 +1 −1 −1 +1  
Eb +1 −1 +1 −1 −1 +1 +1 −1 −2 -2 +2 +2
F1x +1 +1 +1 +1    
F1y   +1 +1 +1 +1  
F1z     +1 +1 +1 +1
F1x   +1 +1 −1 −1 −1 +1 +1 −1
F1y +1 +1 −1 −1   +1 −1 +1 −1
F1z −1 +1 +1 −1 +1 −1 +1 −1  
F2x   +1 +1 −1 −1 +1 −1 −1 +1
F2y −1 −1 +1 +1   +1 −1 +1 −1
F2z −1 +1 +1 −1 −1 +1 −1 +1  

10.5 The vibrational angular momentum operator

The molecule-fixed components of the vibrational angular momentum operator Ls associated with the triply-degenerate vibration vs (s = 3, 4) can be taken to be

$$ \left[ \begin{array}{c} L_{sx} \\ L_{sy} \\ L_{sz} \end{array}\right] = \left[ \begin{array}{c} Q_{sy} P_{sz} - Q_{sz} P_{sy}\\ Q_{sz} P_{sx} - Q_{sx} P_{sz}\\ Q_{sx} P_{sy} - Q_{sy} P_{sx}\end{array}\right] ~ , $$

(eq. 36)

if the proportionality constant ζs is suppressed. The quantity [2] ζs, by which (eq. 36) must be multiplied to obtain the true vibrational angular momentum operator, lies between −1 and +1. Its precise value depends on the geometry and force field of the molecule [4041]. Since the vibrational coordinates Qsx , Qsy , Qsz and conjugate linear momenta Psx , Psy , Psz for s = 3, 4 belong to the symmetry species F2, F2, F2, it can be shown fairly easily that the components Lsx , Lsy , Lsz of the vibrational angular momentum belong to the symmetry species F1, F1, F1.

Laboratory-fixed components of the vibrational angular momentum are normally not considered.


Created September 21, 2016, Updated March 1, 2023