Rotations of the most asymmetric molecules via 4-step and 1- step ladder operators

The symmetries of the free rotations of asymmetric molecules - at the levels of their respective Hamiltonians, spheroconal harmonic eigenfuntions, and eigenenergies - are reviewed in general. The situation of the most asymmetric molecules is analyzed in particular; their eigenstates with vanishing asymmetry-distribution eigenenergies are identified together with the Valdés and Piña 4-step ladder operators connecting them. Three sets of ladder operators identified by Méndez-Fragoso and Ley-Koo, connecting specific eigenstates of molecules with any symmetry, are also reviewed briefly. The successive applications of the 4- step ladder operators, to the lower eigenstates 1, y, xz, xyz, of the most asymmetric molecules, lead to their respective angular momentum ℓ = 4N, 4N+1, 4N+2, 4N+3, for N = 0,1,2..., companions of the same species with vanishing asymmetry-distribution energy; in turn, the successive application of the cartesian components of the angular momentum Lx, Ly, Lz operators to the above states, for each value of ℓ, lead to the respective companion states with the same angular momentum and companion species [xy, xz, yz,1], [z, xyz, x, y], [yz, 1, xy, xz], [x, y, z, xyz] and complementary excitations, thus completing each set of (2 ℓ+1) eigenstates, with symmetrically distributed energy levels.


Introduction
Kramers and Ittmann pioneered the study of the quantum rotations of asymmetric molecules using Lamé spheroconal harmonic polynomials [1], and Patera and Winternitz identified the corresponding representations of the rotational group [2]. This contribution is focused on the successive use of the 4step ladder operators identified in [3], and of the angular momentum operators as one of the sets identified in [4], on the appropriate eigenstates of the most asymmetric molecules, exhibiting the respective connections among successive pairs of such states.
The article reviews in Sect. 2 the symmetries of the free rotations of asymmetric molecules, including the most asymmetric ones, at the levels of their Hamiltonians, eigenfunctions and eigenenergies, referring the readers to [5][6][7][8] for the details. Sections 3 and 4 describe the actions of the respective 4-step and 1-step ladder operators on the individual pairs of eigenstates. Section 5 provides the overall picture of the successive and combined actions of both types of operators, starting from the lower eigenstates with ℓ = 0, 1, 2, 3 and vanishing asymmetry-distribution energy. Section 6 contains a discussion of the specific results of this investigation and its relevance for developing the theory of angular momentum in bases of Lamé spheroconal harmonics.

Symmetries in the rotations of asymmetric molecules
The section reviews briefly and successively the symmetries of the Hamiltonians, their eigenfunctions and energy spectra for asymmetric molecules, including the most asymmetric ones.

Symmetries in the Hamiltonians
The starting point is the Hamiltonian for the free rotations of asymmetric molecules with moments of inertia I 1 < I 2 < I 3 in the body fixed reference frame with principal axes: the parameters of asymmetry-distribution, The three pairs of Hamiltonians commute with each other and consequently share common eigenfunctions. Their invariance under the individual parity transformations x ® -x, y ® -y, z® -z lead also to eigenfunctions with the respective well-defined parities.
The two conditions on the asymmetry-distribution parameters indicate that only one of them can be chosen independently. An alternative parametrization with a single angle, , can also be used: The particular cases of (s = 0 : e 1 = 1, e 2 = e 3 = -0.5) and correspond to prolate and oblate symmetric molecules with the respective x and z axes of rotational symmetry. The most asymmetric molecules are defined by ( :

Symmetries in the eigenfunctions in spheroconal coordinates
The spheroconal coordinates are expressed in terms of their transformation equations to cartersian the Jacobi elliptical integrals. Their parameters 2 i k are subject to the condition that their sum is one, ensuring that the sum of the squares of the cartesian coordinates is equal to the square of the spherical radial coordinate. Fixed values of the angular coordinates c 1 = c 10 and c 2 = c 20 correspond to elliptical cones with axis along the z-axis and the x-axis, respectively. Notice that the change of the indices 1 and 2 in the coordinates and parameters is equivalent to the exchange of x and z, while y remains the same.
The scale factors for these coordinates are readily obtained [3][4][5][6][7][8]: In fact, the separated equations for Substitution in the Lamé differential equation leads to three term recurrence relations for the expansion coefficients, equivalent to a tridiagonal matrix eigenvalue problem; diagonalization of the latter yields the B is guided by their species and parities, and also by the condition that their sum is ℓ(ℓ+ 1). Additionally, the number of nodes for the product eigenstates with a given value of ℓ, using the notation ℓ[AB]n 1 n 2 , is such that: . The readers may examine them in order to recognize some of their properties as described in the previous paragraphs. Additionally, notice the same shapes in the first and fifth columns, second and fourth, as well as in the upper and lower parts of the third one, differing only in orientations. The latter follow from the symmetries in the last paragraph in Sect. 2.1 and in the first paragraph of this section.

Angular Momentum 4-step ladder operators for the most asymmetric molecules
Valdés and Piña investigated the rotational spectra of the most asymmetric molecules, identifying their eigenstates with vanishing energies as well as the 4-step ladder operators connecting them [3]. Here we simply borrow their equations describing the pertinent states and ladder operators.
In their notation, the Lamé polynomials L ℓE * become the products of the singularity removing factors and Jacobi polynomials of degree n with labels a,g in the variable w = cn 2 (c i | k i 2 = 0.5) ) f n (7 / 4, 5 / 4, w 2 ) The 4-step ladder operators, connecting eigenstates of the same species, do it as follows: in their up and down versions, respectively.

Three sets of ladder operators for molecules of any asymmetry
The ladder operators for the Lamé spheroconal harmonic polynomials, identified in [4], come in three sets: 1) For eigenstates with the same angular momentum ℓ, same species AB, and different and complementary angular excitations n 1 + n 2 = n' 1 + n' 2 = ℓn AB , h n        above and below E * = 0 come in pairs with almost the same energy, including the connections of some of those pairs represented only by the tip of the respective arrows. The readers may also ascertain, for each value of ℓ, the numbers of states for the respective species, as described in Sect.
2.2, as well as the matching and the types of their nodal surfaces when going from the lower to the higher energy levels, following the changes in the c 1 and c 2 excitations.

Discussion
The symmetries of the rotations of asymmetric molecules reviewed in Sect. 2 allow the identification of the commuting set of operators H, H Q , H * and the individual x, y and z parity operators; with the corresponding labeling of their common eigenstates ℓ[AB]n 1 n 2 characterizing their angular momentum, species and excitations, and also of their respective eigenenergies, especially E * . The 4step angular momentum ladder operators in Sect. 3 provide the connections between the eigenstates of the most asymmetric molecules with vanishing asymmetry-distribution energy E * = 0, ℓ = 0, 1, 2, 3 modulo 4 as graphically illustrated in Sect. 5. The cartesian components of the angular momentum operators in Sect. 4 applied successively to each of the above eigenstates lead to their sets of 2ℓ companion eigenstates with the common value of ℓ, companion species, and complementary excitations and, with energies ±E * of the same magnitude above and below the level E * = 0 , as illustrated also in Sect 5.
In addition to this specific investigation for the most asymmetric molecules, a few remarks about molecules of any asymmetry can be added to the discussion. First, we recall the connection between the Hamiltonians H * (s ) and H * (60º -s ) involving the change of sign in their respective asymmetrydistribution parameters as well as the exchange of  figure  3, for molecules of any asymmetry; for the most asymmetric molecules, four successive applications should take us from ℓ to ℓ+ 4 making contact with Sect. 3. Our concluding remark is that these investigations contribute to the development of the theory of angular momentum in bases of Lamé spheroconal harmonic polynomials.