Methane Symmetry Operations - Vibration-rotation operators

16.4 Vibration-rotation operators in the transformed Hamiltonian

It is common to treat the vibration-rotation problem in methane using the Nielsen transformed-Hamiltonian approach [10, 14, 60, 65]. Some of the terms in such a transformed Hamiltonian are obtained by considering explicitly various orders of the appropriate contact transformations. Other terms are introduced phenomenologically. Because of the variety of notation used and the different methods of introducing new interaction terms, relationships between the parameters determined in various recent vibration-rotation spectral analyses [43-45] of CH₄ are not all known with certainty. But in all cases, the form of the interaction terms in the Hamiltonian operator is constrained by the group theoretical requirement that they be of species A₁ in the point group T_d.

Consider first the form of the centrifugal distortion operator, which consists of terms quartic in the molecule-fixed components of the total angular momentum operator. Since J_x , J_y , J_z transform as F₁, we are interested in the species of F₁⁴. However, we need actually only consider species belonging to the symmetrized fourth power [F₁⁴]. Operators belonging to other species contained in F₁ × F₁ × F₁ × F₁ can be written in terms of products cubic and lower in the angular momentum components, by making use of the commutation relations

Table 9 indicates that [F₁⁴] contains the representation A₁ only twice. Thus, there are for methane only two linearly independent fourth power expressions in the total angular momentum components, associated with two independent quartic centrifugal distortion constants. The quartic centrifugal distortion operator is expressed in many different notations in the literature. One which parallels symmetric top notation somewhat is the following (suppressing ℏ's)

The construction of this operator from first principles is beyond the scope of this work, but it can be verified using (eq. 5) and Table 3, that it is of species A₁ for any values of D_J and D_t . We note also that the second term contains the operator J²J_z², associated in a symmetric top with the constant D_JK, and the operator J_z⁴, associated in a symmetric top with the constant D_K.

J_x , J_y , J_z can also be considered to transform according to the irreducible representation D_g⁽¹⁾ of the continuous three-dimensional rotation-reflection group. The symmetrized fourth power [(D_g⁽¹⁾)⁴] contains the species D_g⁽⁰⁾ + D_g⁽²⁾ + D_g⁽⁴⁾. It can be shown [14] that the first term in (eq. 63) transforms as D_g⁽⁰⁾ under the three-dimensional rotation-reflection group, while the second term corresponds to a linear combination of terms transforming according to D_g⁽⁴⁾. If we represent functions transforming according to D_g⁽⁴⁾ by f_m⁽⁴⁾, where -4 ≤ m ≤ +4, the second term in (eq. 63) is proportional to [14]

It is common [10, 14, 65] to handle higher-order vibration-rotation operators for υ_s = 1 states, s = 3 or 4, rather formally as follows. Consider a set of (2k₁ + 1) vibrational operators f_m1^(k₁) which transform according to the irreducible representation D^(k₁) of the continuous three-dimensional rotation group. These vibrational operators are taken to be functions of the vibrational coordinates Q_sx , Q_sy , Q_sz , their conjugate linear momenta, and the components L_sx , L_sy , L_sz of the vibrational angular momentum. Consider also a set of (2k₂ + 1) rotational operators g_m2^(k₂) which transform according to D^(k₂) and are taken to be functions of J_x , J_y , J_z . It is then possible to couple these operators using the vector coupling techniques to obtain vibration-rotation operators

which transform according to D^(k). Matrix elements of these coupled vibration-rotation operators can be obtained formally using the results of irreducible tensor theory.

16.5 Operators involving the nuclear spin angular momentum

It is possible to construct all operators involving components of the nuclear spin angular momentum raised to a given power by considering only molecule-fixed components of these spin operators, since the molecule-fixed components are automatically invariant with respect to overall rotations in the laboratory-fixed axis system. However, the spin basis set functions are characterized by projections quantized along the laboratory-fixed Z axis, so that it is necessary in calculating matrix elements to use the analog of (eq. 35) to write the spin operators in terms of laboratory-fixed components and elements of the direction cosine matrix.

The application of symmetry considerations to the resulting expressions, which involve tensors in the rotational operators and tensors in the nuclear spin operators, is particularly complicated because care must be taken to insure that final coupled tensor operators transform as D_g⁽⁰⁾ under overall rotations in the laboratory, and according to A₁ under the point-group symmetry operations. Loosely speaking, the laboratory-fixed "projections" of the rotational tensors must be correctly coupled to the laboratory-fixed "projections" of the spin tensors, and the molecule-fixed "projections" of the rotational tensors must be correctly coupled to the molecule-fixed "projections" of the spin tensors.

16.6 Reduced matrix elements: the Wigner-Eckart theorem

The formalism of irreducible tensorial sets displays its greatest power in systematizing the evaluation of matrix elements. If we consider two wave functions Ψ_m1^(j₁) and Ψ_m2^(j₂), and an operator O_m3^(j₃), which transform, respectively, according to the irreducible representation D^(j₁), D^(j₂), and D^(j₃) of the continuous three-dimensional rotation group, then matrix elements can be written [22, 62, 64] in the form

The first quantity on the right is a so-called reduced matrix element; its value does not depend on m₁, m₂, or m₃ and must in general be calculated for the specific molecular problem under consideration. The second factor is a vector coupling coefficient; its value is determined by symmetry and can be tabulated once and for all, or conveniently manipulated algebraically. Because of (eq. 66), (2j₁ + 1) (2j₂ + 1) (2j₃ + 1) matrix elements can be related to a single unknown parameter, achieving thereby an enormous simplification and reduction of labor.

Equation (66) gives an example of a matrix element in an uncoupled basis set. It is also written in terms of 3 - j symbols in the literature. Extensive formalism has been developed to yield matrix elements of coupled operators in coupled basis sets, which requires the introduction of 6 - j and 9 - j symbols, etc. There are numerous different phase conventions used in the literature in the definitions of 3 - j symbols and related quantities. Users of irreducible tensor formalism must be absolutely certain that their own applications are internally self-consistent.