This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Spectra of high-symmetry molecules contain fine and superfine level cluster structure related to _{2} asymmetric top rank-2-tensor Hamiltonians are compared with _{h}_{6} and _{3}~_{3}_{v}_{h}_{6} spectra and to extreme clusters.

A key mathematical technique for atomic or molecular physics and quantum chemistry is matrix diagonalization for quantum eigensolution. As computers become faster and more available, more problems of chemical physics are framed in terms of choosing bases for eigensolution of time evolution operators or Hamiltonian generator matrices. The resulting eigenvectors and eigenvalues are Fourier amplitudes and frequencies that combine to give all possible dynamics in a given basis choice.

Despite the increasing utility and power of computer diagonalization, it remains a “black box” of processes quite unlike the complex natural selection by wave interference that we imagine nature uses to arrive at its quantum states. Diagonalization uses numerical tricks to reduce each _{2} −

Before describing tensor eigensolution techniques and rovibronic energy surfaces (RES), a brief review is given of potential energy surface (PES) to put the tensor RES in a historical and methodological context. This includes some background on semiclassical approximations of tensor algebra that help explain rotational level clustering and are used to develop the RES graphical tools. Section 2 reviews how RES apply to symmetric and asymmetric top molecules. This serves to motivate the application of RES to more complicated molecules of higher symmetry. Section 3 contains a graphical analysis of octahedral RES and an introductory review of level clusters (fine structure) having 6-fold and 8-fold quasi-degeneracy (superfine structure) due to rank-4 tensor Hamiltonians. Following this is a discussion of mixed-rank tensors that exhibit 12-fold and 24-fold monster-clusters. The latter have only recently been seen in highly excited rovibrational spectra [

Following introductory Section 4, these problems are addressed in Sections 6–8 by redeveloping group algebraic symmetry analysis into a more physically direct and elegantly powerful approach. It uses underlying duality between internal and external symmetry states and their operations. Duality is introduced using the simplest order-6 symmetry groups _{6} and _{3}~_{3}_{v}_{h}

The direct approach to symmetry starts by viewing a group product table as a Hamiltonian matrix _{k}_{k}_{k}

Several graphical techniques and procedures exist for gaining spectral insight. One of the oldest is the Born–Oppenheimer approximate (BOA) potential energy surface (PES) that is a well-established tool for disentangling vibrational-electronic (vibronic) dynamics. While BOA-PES predate the digital age by decades, their calculation and display is made practical by computer. More recent are studies of phase portraits and wavepacket propagation techniques to follow high-

Visualizing eigensolutions and spectra in crystalline solids is helped by bands of dispersion functions in reciprocal frequency-versus-wavevector space. Fermi-sea contours are used to analyze de Haas–van Alphen effects and more recently in understanding quantum Hall effects. Analogy between band theory of solids and molecular rovibronic clusters is made in Section 4 and 7.

Visualization of molecular rotational, rovibrational, and rovibronic eigensolutions and spectra is the subject of this work and involves the rotational energy surface (RES). As described below, an RES is a multipole expansion plot of an effective Hamiltonian in rotational momentum space. Ultra sensitivity of vibronic states to rotation lets the RES expose subtle and unexpected physics. Multi-RES or rovibronic energy eigenvalue surfaces (REES) have conical intersections analogous to Jahn–Teller PES (See Section 9).

The RES was introduced about thirty years ago [_{3}[_{3} and _{2}_{4}) [_{4} and _{4}) [_{6}, _{6} and _{6}) [_{8}_{8}), and buckyball (_{60}) [

Each of the techniques and particularly the RES-based ones described below depend upon the key wave functional properties of stationary phase, adiabatic invariance, and the spacetime symmetry underlying quantum theory. Additional symmetry (point group, space group, exchange, gauge,

A BOA-PES depends on an adiabatic invariance of each electronic wavefunction to nuclear vibration. It is often said that the electrons are so much faster than nuclei that the system “sticks” to a particular PES that electrons provide. Perhaps a better criterion would be that the Fourier spectrum associated with nuclear motion does not overlap that of an electronic transition to another energy level. Nuclei often provide stable configurations that quantize electronic energy into levels separated by gaps much wider than that of low lying vibrational “phonon” states.

A BOA wavefunction is a peculiarly entangled outer product Ψ = _{ν}_{(}_{ε}_{)} (_{(}_{X...}_{)} . . . ) is a function whose electron coordinates _{(}_{X...}_{)} . . . depend adiabatically on nuclear vibrational coordinates (_{ε}

The adiabatic convenience of a single product _{ν}_{(}_{ε}_{)}(_{ε}_{ε}_{ν}

The rotational energy surface (RES) can be seen as a generalization of adiabatic BOA wave

In

The _{1}_{2}_{m}_{1}_{2}_{m}_{k}

RES are multipole expansion plots of effective BOA energy tensors for each quantum value of vibronic

Wave _{J}^{rotation}^{J}

Total angular momentum

Entangled BOA product _{J}_{J,MmK}_{ν}_{(}_{ε}_{)} = Ψ_{μ̄}^{l}

Disentangled product Ψ_{ρ}_{μ̄}^{l}_{R,m,n}_{μmM}^{lRJ}_{M}^{J}

A BOA-entangled wave in

A remarkable property of quantum rotor operator algebra is that Wigner ^{l}_{μ̄}^{l}_{μ}^{l}

This rotational wave relativity is a subset of Lorentz–Einstein–Minkowski space-time-frame relativity that uses symmetry algebra to keep track of the invariant sub-spaces (eigensolutions). D-Matrices underlie all tensor operators, their eigenfunctions and their eigenvalues and are a non-Abelian (non-commutative) generalization of plane waves ^{k*}^{ikr}

Of particular importance to RES theory is the Wigner–Eckart factorization lemma that relates Clebsch–Gordan _{μmM}^{lRJ}

A more familiar form of this is the Kronecker relation of product reduction
_{M}^{J}

Body-(un)coupling in _{J}_{[}_{ν}_{(}_{ε}_{)]} in _{M}^{J}_{M}^{J}_{J}_{[}_{ν}_{(}_{ε}_{)]} of

Note the following for

However, in both

The duality of lab _{lab}_{body}

Spherical harmonic functions _{m}^{l}^{l}

A diatomic or linear rotor must have zero body quanta (_{m}^{l}_{m}^{l}_{q}^{k}

A multipole _{q}^{k}

Factor 〈_{q}^{k}_{q}^{k} and chosen by a somewhat arbitrary convention.

This particular choice simplifies bra-ket coupling and creation-destruction operator expressions for _{q}^{k}

Other choices rescale _{q}^{k}_{q}^{k}_{q}^{k}_{q}^{k}

Examples of _{q}^{k}

Historically, spinor _{μ}_{μ}^{2} generators _{0}^{0}_{q}^{1}_{q}^{k}_{0}^{k}_{q}^{k}_{jk}_{j}^{†}_{k}

Coefficient

Each matrix 〈_{q}^{k}^{J}

CG-3j relation _{q}^{k}^{J}_{m}_{′}_{,m}^{J}^{th}_{2}^{2}, octopole _{2}^{3}, and 2^{4}-pole _{2}^{4} share the

Tensor 〈_{q}^{k}^{J}_{m}_{′}_{,m}^{J}

Any [_{m}_{′}_{,m}^{J}_{q}^{k}^{J}_{q}^{k}^{J}_{q}^{k}_{M}^{J}

Diagonal dipole-vector (rank _{0}^{1}〉^{J}_{z}^{J}^{k}_{0}^{k}^{J}_{z}_{z}^{2} = _{z}_{z}_{z}^{3} = _{z}_{z}_{z}^{th}_{z}^{k}_{0}^{k}^{J}^{p}_{z}_{q}^{k}^{J}

For example, matrix diagonals in

Powers of 〈_{z}^{2} in _{q}^{k}^{2} found by dot products with vectors in

Triangle inversion of _{0}^{k}^{2} in terms of _{z}_{z}^{p}^{2} = ^{p}_{0}^{k}^{J}_{z}_{z}

Legendre polynomials occupy the central (00)-component of a Wigner-^{J}

Examples of Legendre polynomials of cos _{z}_{z}

Classical _{k}_{0}^{k}^{J}_{0}^{k}^{J}

Norm
_{0}^{k}^{J}_{k}_{k}_{p}^{p}_{p}_{p}^{2} usually do not sum to 1.

Tensor values 〈_{0}^{0}〉^{J}_{0}^{1}〉^{J}_{0}^{2}〉^{J}_{0}, _{1}, and _{2} in _{k}_{0}^{k}_{m}^{J}

For large-_{0}^{k}_{m}^{J}_{3}_{4}_{0}^{k}_{m}^{J}_{k}_{k}_{0}^{k}_{k}

Quantum _{m}^{J}

An angular momentum eigenstate
_{m}^{J}_{n}^{J}

RES energy level analysis begins by writing a multipole _{q}^{k}

Inertial constants (_{χ̄}_{ȳ}_{z̄}_{q}^{k}_{0}^{0}〉^{J}_{0}^{2}〉^{J}_{0} and _{2} in _{0}^{0} and _{0}^{2}.

A rigid spherical top (_{0}^{0} term _{0}^{0} and _{0}^{2} terms with energy eigenvalues below.

Since a rigid symmetric-top involves only _{0}^{0} and _{0}^{2}, the _{n}^{J}_{n}^{J}_{n}^{J}_{0}^{2} -RES shown in

Inserting quantized-body cone relation

Cone paths in _{n}^{J}_{n}^{J}^{2} =

The difference between quantum solutions and semi-classical _{k}_{0}^{k}_{m}^{J}_{0}^{4}〉_{m}^{4}, while _{0}^{2}〉_{m}^{4} shown. As

An RES is a radial plot along

A scalar term _{0}^{0} added to a tensor combination _{k}_{k}_{0}^{k}

Wigner–Racah tensor algebra defines a reduced matrix element 〈^{k}

This matrix
_{0}^{2} = _{0}^{2} in

Reduced matrix element 〈^{2}|| _{0}_{mm}^{2}^{JJ}^{2}_{2}(

Apparent conflicts in factors are due to having sum-of-^{k}_{k}_{k}_{k}_{k}_{k}_{k}_{k}^{2} tensors may have extra scale factors.

Tensor ^{2} = ^{2} in ^{2} = ^{k}^{k}^{k}_{0}_{mm}^{2}^{JJ}^{2} to
^{2}.

Each rank-^{k}^{k/}^{2}. Anisotropy of mixed-rank _{k}^{k}^{k}_{k}

Plotting RES of non-diagonal Hamiltonians for the asymmetric top ^{nd}_{q}^{2} with reduced z-axial symmetry, nonzero _{a}_{a}

Or else, each tensor _{q}^{k}

Forms _{0}^{0} and _{0}^{2} (_{±2}^{2} terms are asymmetric and vary as sin^{2}_{±2}^{2} terms grow to give equatorial valleys and saddles in _{0}^{2} vanishes.

Asymmetric tensor operators _{±}_{q}^{k}_{0}^{k}^{AsymTop} eigenstates as well as eigenvalues vary with coefficient (_{±2}^{2} mixes symmetric-top states
_{K}^{J}^{sep} on the ^{sep} = 0) and oblate top (^{sep} = ^{sep} =

As B first differs a little from A, off-diagonal _{±2}^{2} and asymmetric _{±2}^{2} first “quench” degenerate ±

These states have nodes or anti-nodes standing on hills, saddles, or valleys of the RES topography at the principal body axes. Whether a wave is cos-like or sin-like at an axial point depends on whether it is symmetric or antisymmetric at the point and thus whether that point is an anti-node or node. Nodal location can determine whether a cos-like or sin-like wave has higher energy.

As _{±2}^{2} will mix standing waves like _{K}^{J}_{K}_{±2}^{J}_{K}_{±4}^{J}_{K}_{±6}^{J}^{AsymTop} symmetry described below.

Throughout the range of asymmetric cases in ^{A Top} in _{2} of 180° rotations _{x}_{y}_{z}_{x}_{y}_{z}_{y}_{x}_{x}^{2} = _{2} is an outer product of cyclic _{2} groups for two axes, say _{2}(_{2}(_{x}_{y}_{z}_{2}(_{2}(_{1}_{1}_{2}_{2}] for _{2} = _{2}(_{2}(

Labels (_{2}_{2}) notation for _{2} characters and N-ary notation (0_{N}_{N}_{N}_{N}_{N}_{N}

This notation is used in correlation _{2} and its subgroups _{2}(_{2}(_{2}(_{2} species and indicates which _{2} symmetry, _{2}) or _{2}), correlates to it. The _{x}_{y}_{z}_{2}) _{2} character is
_{2}) is

^{AsymTop}-levels in _{1}_{1}) (_{2}_{2}) (_{1}_{1}) (_{2}_{2}) (_{1}_{1})] and [(_{2}_{1}) (_{1}_{2}) (_{2}_{1}) (_{1}_{2}) (_{2}_{1})] separated by a single (_{2}) level. Each is related to RES _{x}_{z}_{2}: _{y}_{2}) column and odd-_{2}) column of _{2}(_{2}(

Valley-pair sequence (_{1},_{1}), (_{2},_{2}) . . . is consistent with (0_{2}) and (1_{2}) columns of the _{2}(_{2},_{1}), (_{1},_{2}) . . . is consistent with (0_{2}) and (1_{2}) column of the _{2}(_{x}_{z}

_{2}(^{Sep} of the saddle points and their separatrix.

As symmetric^{Sym} becomes a more asymmetric^{AsymTop} in _{K}^{J}^{Sym} state
^{AsymTop} eigenstate expansion of states
_{K}^{J}_{K}_{±2}^{J}_{K}_{±4}^{J}_{K}_{±6}^{J}_{K}^{J}^{Sep}, bending of hill and valley paths become more severe as they approach separatrix asymptotes where the polar angle range ^{Sep} or

It is conventional to label ^{Sep} eigenstate |_{x}_{z}_{x}_{z}_{x}_{z}^{Sep}, a |_{z}_{z}^{Sep}, a |_{x}_{x}^{Sep},

Though a general form of the symmetry identification process may be unfamiliar, it may implemented by computer. Group projectors

Only projectors in lower symmetry subgroups are used because they are easy to calculate and there are fewer in number. With the eigenvector projection lengths and knowledge of the correlation table between the molecular group itself and the subgroup one can start to deduce the eigenvector symmetry. As mentioned earlier, one correlation table is not enough to fully identify an eigenvector’s symmetry, but using several subgroups one can assign symmetry. This process is simpler than calculating projectors of the full group, particularly if one can use a _{n}

This method can be significantly simpler than a traditional block diagonalization. Block diagonlalizing the Hamiltonian requires projectors of the entire molecular symmetry group rather than of the smaller subgroups.

The disadvantage of this method is that it becomes unstable when clusters are tightest. As eigenvectors become more mixed with tighter clustering the algorithm may be unable to distinguish. Some RES paths and level curves indistinguishable to numeric projector then appear black. Symmetry definitions hold for asymmetric tops where

N-atom inversion in ammonia, _{3}, is an example of molecular tunneling modeled by a particle whose closely paired levels (inversion doublets) lie below the barrier of a double-well PES. An RES generalization, sketched in _{1},_{1}), (_{2},_{2}), _{x}_{x}_{z}_{z}_{h}_{d}

Section 2 has shown that asymmetric top molecules may be treated semi-classically, using only tensor operators and RES plots with a seperatrix between regions of local symmetry. Spherical top molecules experience such symmetry locality, but with greater variety of local symmetry. This section focuses on the added complication and convenience of higher symmetry as well as showing novel rotational level clustering patterns related to RES paths and tunneling.

Theory of asymmetric top spectra in Section 2 may be generalized to a semi-classical treatment of tensor operators for _{d}_{h}_{4} or _{6}. The results contain level clusterings that first appeared in computer studies by Lea, Leask, andWolf [

Up to fourth order, any such molecule may be treated using the Hecht Hamiltonian [

This is continued below to higher rank tensors with more complicated structure [_{n}_{m}_{m}^{n}

A normalized sum of these coefficients gives the rank-6 _{h}

The first study of RES and eigenvalue spectrum with varying rank-4 and rank-6 tensor operators [

Later studies [

As with the asymmetric top Hamiltonian, the octahedral Hamiltonian uses non-axial operators shown in _{z}

Gulacsi and coworkers [^{[4]} and ^{[8]} for ^{[6]}. Results below agree but extend to larger

While the asymmetric top systems show clustering related to symmetry subduction from _{2} to a related _{2} subgroup, octahedral molecular clusters relate to a variety of subgroups. _{h}_{4}, _{3}, _{2}, _{4}, _{3}, _{2} or other subgroups involving reflection or inversion.

For simplicity, this discussion will focus on _{h}

For _{2}-symmetric molecules, clustering patterns are described in terms of the correlation tables found in _{4}, _{3} and _{2} in _{2}, a new coloring convention for _{1} is red, _{2} is orange, _{2} is green, _{1} is dark blue and _{2} is light blue.

In the RES, rotationally induced deformation or symmetry breaking is seen from the shape of local regions of the RES involving a specific contour. _{4} and _{3} local symmetry regions to be present. The _{4} regions are identified by their location and by their square base. Similarly, the _{3} regions are identified by their location and triangular base. In this case _{3} symmetric regions are concave while _{4} regions are convex. This is not required and is dependent on Hamiltonian fitting terms that change the relative contributions of ^{[4]} and ^{[6]}. Likewise, _{2} regions that are determined by their location and rectangular base.

Cluster degeneracy is a hallmark of a specific symmetry breaking. While a symmetric top spectra may be resolved into _{J}_{J}^{α}

In the cases shown here cluster degeneracy ^{α}_{4}, _{3} and _{2} respectively.

As mentioned previously, it is possible to diagonalize the Hamiltonian and organize species by the order of each block, yet this alone will not distinguish all levels. For Hamiltonians defined by ^{[4]} as ^{[6]} or ^{[8]} terms are present, a numerical examination of eigenvectors is required to assign the symmetry of each level. Subgroup projectors are used here where the cluster degeneracy increases and the symmetry becomes challenging to distinguish. These projectors represent a simplification of the symmetry analysis of an octahedral molecule into projections onto _{4} symmetric projectors. The correlation table for _{4}, shown in _{4} there are only four projectors to create and a clever choice of axis can force several of these projectors to be entirely real or entirely imaginary. Conveniently, the _{4} projectors can be used to diagnose level symmetry for clusters in any subgroup region.

The Hecht Hamiltonian ^{[4]}, ^{[6]} and ^{[8]}.

^{[4,6,8]} in ^{[8]} contributions to zero the eigenvalue spectrum for ^{[4,6]} in ^{th}^{th}^{[4,6]} RES have circular ring separatrices not unlike those on _{2} RES in

To understand the behavior of the level diagram in _{4}, _{3} and _{2} whose local rotation axis lie fixed normal to the RES at the center of each region, respectively, even as _{n}

With this understanding of local subgroup regions it is possible to discuss more detail of _{4}, _{3} and _{2} axes. This serves two purposes: To confirm that the quantum spectrum sits inside the semi-classical boundaries and to see that there is a change in the eigenvalue spectrum corresponding to changes in RES topology. _{n}

Section 2 described how to predict the error between a fully quantum mechanical calculation and a semi-classical approximation of the symmetric rotor rotational spectra. For the symmetric rotor this was done analytically. It is difficult to be as exact in calculating error for an octahedrally symmetric Hamiltonian, but a line plot can show when an RES plot fails to describe quantum mechanical behavior.

Rather than plotting the Hamiltonian as

This changes semi-classical outlines from cosines to lines and shows where quantum levels exceed semi-classical bounds and where an RES approximation fails. Also,

The three plots in

Indeed, such plots as ^{[4,6,8]} and demonstrate how such a Hamiltonian can show a different type of topology than previously reported.

The inclusion of eighth rank operators to the Hamiltonian dramatically changes the possible types of RES local symmetry and the related level clustering. While _{4}, _{3} and _{2} symmetric local structures for RES plots for ^{[4,6]} Hamiltonians, ^{[4,6,8]} RES path pointed out there with _{1} symmetry. (That means ^{01} (_{1}) ↑

Details of the two dimensional ^{[4,6,8]} parameter space appear in a figure _{1}, orange for _{2}, green for _{2}, blue for _{1} and cyan for _{2}.

As expected from _{4} and concave _{3} structure as does the RES at _{1} local symmetry regions. The middle row shows a different behavior: all the diagrams are identical. Again, this follows from

While

_{1} structures associated with 24-fold level clusters have no rotation axis to locate their central maxima or minima on the RES. However, they do have bisecting reflection planes that must contain surface gradient vectors and an extreme point for which the gradient points radially. RES plots with _{1} local symmetries are shown in parts of second, third and fourth rows of _{1} regions lie on hills or else valleys and how they can be arranged with their neighbors into either a square or triangular pattern.

_{1} clusters require tensors of rank-8 with _{J}^{J}_{1} region.

RES with _{1} regions but fail to fit its (2Θ_{4}^{4}=56°)-wide cones. _{1} clusters begin to appear around _{20}^{20}=26°) but even for _{30}^{30}=21°) are still barely formed in _{1} hill between _{3} and _{4} valleys of its (_{4} axes as in _{2} axis.

With higher _{h}_{1}-regions centered away from _{h}

For a system to have symmetry means two or more of its parts are the same or similar and therefore subject to resonance. This can make a system particularly sensitive to internal parameters and external perturbations and give rise to interesting and useful effects. However, resonances can make it more difficult to analyze and understand a system’s eigensolutions. The tensor level cluster states give rise to spectral fine structure discussed in the preceding sections and that splits further into complex

Fortunately, the presence of symmetry in a physical system allows algebraic or group theoretical analysis of quantum eigensolutions and their dynamics. Groups of operators (^{†}

Hamiltonians may themselves be symmetry operators or linear expansions thereof. Multipole tensor expansions used heretofore are examples. Expanding ^{†}_{q}^{k}

If

However, non-commuting (non-Abelian) symmetry operators (

In Section 6, the dual group (_{3} ~ _{3}_{v}_{3} ~ _{3}_{v}_{3})-symmetric

In Section 7, the local symmetry expansion is applied to octahedral _{h}_{h}

An introductory analysis of tunneling symmetry begins with elementary cases involving homocyclic _{n}_{p}_{q}

The analysis described here and in Section 6 deviates from standard procedure [_{n}_{n}^{p}^{p}^{0} = ^{n}^{n}^{−1} = ^{−1}[

In ^{n}^{0} = ^{n}_{1} ⊂ _{2} of

We construct the general _{n}^{−1}^{†}^{†} = ^{−1}. In each table the ^{th}^{−1} matches ^{th}^{−}^{1}_{6} for which ^{−6} = ^{0} = ^{6} = ^{6†}, ^{−5} = ^{1} = ^{5†}, ^{−4} = ^{2} = ^{4†}, ^{−3} = ^{3} = ^{3†}, and so forth.

The ^{†}^{p}^{p}^{†}

The _{n}

Matrices in _{p}^{p}_{0} = (_{0})^{*} and _{3} = (_{3})^{*} match self-conjugate binary subgroups _{1} ⊂ _{2} = (^{3}) related by ^{3})^{2}. Both are real if matrix (_{ab}_{ba}^{*}).

Three distinct classes of tunneling or coupling parameters are depicted in

The 1^{st}_{1}=−_{−1}=−^{*}=−_{0}=_{1}. The 2^{nd}_{2}=−_{−2}=−^{*}=−_{0}=_{2}. Finally, 3^{rd}_{3}=−^{*} is real as required for binary self-conjugacy ^{3}=(^{3})^{†}.

Eigenvalues _{p}^{m}^{p}^{th}^{th}_{n}^{p}^{n}_{n}_{p}_{m}

The _{p}^{m}_{n}^{(}^{m}^{)}(^{p}_{n}^{(}^{m}^{)}(^{(}^{m}^{)}(^{(}^{m}^{)}(^{(}^{m}^{)}(_{m}_{p}_{n}^{p}

This ^{(}^{m}^{)}(^{p}^{(}^{m}^{)}(^{p}^{p}^{(}^{m}^{)}. ^{(}^{m}^{)} are like irrep

Dirac notation for ^{(}^{m}^{)} is |(^{(}^{m}^{)}-product table in ^{(}^{m}^{)}-table given below has an orthogonal (^{(1)}^{(2)} = ^{(1)}^{(1)} = ^{(1)}) form.

The location of each ^{(}^{m}^{)} in the ^{†} = ^{3}

Character arrays such as ^{p}^{(}^{m}^{)}.

Also character _{p}^{m}

Momentum eigenwave _{k}_{m} (_{p}^{ik}^{m}^{x}^{p} and normalized by

Action of ^{p}_{m}_{q}

The same overlap results whether ^{p}_{n}

^{(}^{m}^{)} projects _{p}^{m}_{p}^{m}^{*} with factor 1^{(}^{m}^{)}’s are idempotent and sum to _{p}^{(}^{m}^{)} = _{m}_{p}^{m}_{p}_{p}_{m}^{2} = 1) Thus projection _{m}^{(}^{m}^{)} has a factor
_{p}^{m}^{(}^{m}^{)} complete (∑_{m}^{(}^{m}^{)} = ^{(}^{m}^{)}^{(}^{m}^{)} = ^{(}^{m}^{)}) in

First row ((

Thus factors

The (0)-momentum or

Given Hamiltonian ^{p}^{p}^{(}^{m}^{)} of ^{(}^{m}^{)} define the dispersion function _{m}

Positive _{m}_{6} array [...(0), (1), (2), (3), (4), (5), ...] of

Examples of dispersion relations for three classes of tunneling paths in _{m}_{6} symmetry depends sensitively on the Hamiltonian tunneling amplitudes _{p}_{m}_{p}

A common tunneling spectral model is the elementary Bloch 1^{st}_{1}=−_{6} consist of six points on a single inverted cosine-wave curve centered at _{n}^{(}^{m}^{)} are projections of an ^{(±1)} and ^{(±2)} between singlet ^{(0)} and singlet ^{(3)} at lowest and highest hexagonal vertices as follows from

The 2^{nd}^{(0)}, ^{(3)}] from its lowest vertex and a quartet [^{(±1)}, ^{(±2)}] from its upper vertices.

The 3^{rd}_{3}=−^{(0)}, ^{(±2)}] below an odd-^{(3)}, ^{(±1)}].

Combining of ^{th}_{k}^{(}^{m}^{)} as a _{k}_{k}_{6}, namely non-Abelian reflection-rotation symmetry such as _{6}_{v}_{6}_{h}^{(±}^{m}^{)} levels that will be treated shortly. Simple _{6} symmetry allows six real parameters with complex _{1} and _{2}. Then _{1} = |^{iφ}^{st}^{nd}

Characterization and spectral resolution in ^{Bk}^{(6)} uses its expansion in _{6}. Similar spectral resolution of a Hamiltonian _{1}, _{2}...] of non-commuting symmetry operators might seem impossible. To be symmetry operators of _{1} and _{2} must commute with _{3}-symmetric tunneling

The simplest non-Abelian group is the rotational symmetry _{3} = [^{1}, ^{2}, _{1}, _{2}, _{3}] of an equilateral triangle. _{3} is used to show how to generalize _{6} operator analysis of the preceding section to any symmetry group. The _{3} analysis begins with a ^{†}_{6}. However, _{3} also requires a ^{†}-form giving the same product rules but using inverse ^{†} ordering |..^{2}, ^{1}, ...|=|..^{1†}, ^{2†}, ...| along the top instead of down the left side as is done for the ^{†}^{1} and ^{2} are the only pair (^{1†}=^{2}) to be switched by conjugation). The three ±180° rotations are each self-conjugate (_{p}^{†}=_{p}^{†}=

Over-bar notation is used for _{3} = [^{1}, ^{2}, _{1}, _{2}, _{2}] of “body”-based operators isomorphic to “lab”-based group.

Matrix representations _{3} or matrices _{3} are given, respectively, by ^{†}^{†}-forms ^{†}_{6} gives matrices in

Most pairs of resulting _{3} matrices in ^{1}_{1}_{3}_{1}^{1}_{2}_{3} matrices in _{3} commute with all matrices of the former _{3}. This suggests that the Hamiltonian matrix, in order to commute with its symmetry group _{3}, is constructed by linear combination of bar group operators of _{3}[

_{3} symmetric (_{6} symmetric (

Spectral resolution of _{3} or any non-Abelian group _{1}, _{2}...] entails more than the _{6} expansion into a unique combination of idempotent operators ^{α}^{(}^{α}^{)}(_{1} and _{2} in one basis since numbers (eigenvalues) always commute. If _{1} and _{2} do not commute, their collective resolution must include eigen-_{m}_{n}^{α}^{2} = _{m}_{n}^{α}_{m}^{α}_{n}^{α}^{2} = _{m}_{m}^{α}_{m}^{α}_{m}^{α}

Unlike a commutative algebra of _{n}

The ^{o}G_{k}^{m}^{o}G^{o}D_{3} and ^{o}C_{6} both equal 6.)

_{6} rank is obviously equal to its order (_{6}) = 6), but the rank of _{3} turns out to be only four (_{3}) = 4). As shown below, _{3} can have no more than four _{3} has just ^{α}_{3}, and there is but ^{(}^{m}^{)} for _{6} in

Rank is a key quantum concept since it is the total number of commuting observables, the operators that label and define eigenstates. Of primary importance are _{G}_{G}_{G}^{−1}=_{G}^{2} and e-values

Next in importance are labeling operators [_{H}_{n}_{−1}, _{H}_{n}_{−2}, _{H}_{1} ] belonging to nested subgroups of _{n}_{n}_{−1}⊃_{n}_{−2}⊃_{1}. Multiple choices of chains exists since each subgroup link _{k}_{k}_{+1} that contains it, but each _{H}_{k} is invariant to all possible _{j}_{≤}_{k}

For example, the _{z}_{z}^{nd}_{z}_{6}(_{x}_{y}_{z}_{ζ}_{n}

The _{k}^{α}_{3} elements in ^{1}^{2}], and [_{1}_{2}_{3}]. (_{3}) = 3)

Elements in each class are related through transformation _{1}=_{t}_{2}_{t}^{−1} by _{t}_{k}^{o}c_{k}_{k}_{t}_{k}_{k}_{k}_{t}

The product table for _{3} class algebra [_{1} = _{2} = ^{1} + ^{2}, _{3} = _{1} + _{2} + _{3}] in _{3} group product tables in _{j}_{k}_{t}

The first sum in ^{o}c_{k}_{k}^{o}c_{k}_{k}^{o}G_{t}_{k}^{o}s_{k}_{k}_{k}_{k}_{k}^{p}^{o}c_{k}^{o}G^{o}s_{k}

Spectral resolution gives class-sum operators _{1}_{2}_{3}_{3}-invariant ^{α}_{k}_{3}^{α}

Standard notation _{1}, _{2}, and ^{α}

Traces of _{3} matrices (_{k}^{α}

This means (^{A}^{1}) and (^{A}^{2}) are each 1-^{E}_{3}⊃_{3} or a particular _{3}⊃_{2} chain is chosen, but relations in _{k}^{α}_{3}^{α}^{th}_{1} =

Completing resolution of _{3} uses a product of two completeness relations, the resolution of class identity _{1} = _{3} subgroup _{3} = [^{1}^{2}] or else _{2} = [_{3}]. In either case invariant _{E}^{A}^{1} and ^{A}^{2} do not. In ^{E}_{2} into plane-polarizing projectors _{x,x}^{E}_{y,y}^{E}_{0202}^{E}_{1212}^{E}

In ^{E}_{3} into Right and Left circular-polarized projectors _{R,R}^{E}_{L,L}^{E}_{1313}^{E}_{2323}^{E}

In ^{A}^{1} nor ^{A}^{2} split or change except to acquire some _{2} or _{3} labels. The total number (four) of irreducible idempotents after either complete splitting is the same group rank noted before: _{3})=4. But, the _{R,R}^{E}_{L,L}^{E}_{3}=[^{1}^{2}] differ from the linear _{x,x}^{E}_{y,y}^{E}_{2}=[_{3}]. _{x,x}^{E}_{y,y}^{E}_{3}_{x}^{E}_{x}^{E})_{3}_{y}^{E}_{y}^{E})^{o}_{3} in

Mutually commuting algebras resolve into (_{2} =

Non-commuting algebras resolve into idempotents and nilpotent (^{2} = _{3}) = 4.)

The product in _{j,j}^{α}_{k,k}^{β}^{α,β}δ_{j,k}_{j,j}^{α},^{A}^{1} and ^{A}^{2} remain invariant and commute with all _{j,j}^{α}

This reduces to a non-Abelian spectral resolution of _{3} that generalizes resolution _{6} and includes two nilpotent projectors _{j,k}^{α}_{j,k}^{α}_{j,j}^{α}_{j,j}^{α}^{(}^{m}^{)}(^{p}^{(}^{m}^{)} terms in _{j,k}

Terms (1^{(}^{m}^{)*}(^{p}^{p}^{(}^{m}^{)} of _{n}_{j,k}^{α}

_{3} resolution in ^{A}^{1} and ^{A}^{2} of dimension ^{A}^{1}=1=^{A}^{2} and a third irrep ^{E}^{E}_{3})=3, rank _{3})=4, and order ^{o}D_{3}=6. The following power sums of ^{α}

Spectral resolution shown in

Product tables in _{3} projectors _{jk}^{α}_{6} idempotent table in ^{†}^{†}-forms in ^{1} and ^{2} on side and top.(^{1†} = ^{2}) The rest are self conjugate. (_{1}^{†}=_{1}, _{xy}^{E}_{yx}^{E}_{xy}^{E}^{†} =_{yx}^{E}_{jj}^{α}^{†} =_{jj}^{α}

The ^{†}^{†} tables in _{jk}^{α}^{†}^{†} tables in _{G}_{jk}^{α}_{jk}^{α}_{jk}^{α}^{†}_{P}_{jk}^{α}_{jk}^{α}

Conjugate ^{†}-representation (_{P}_{jk}^{α}^{*} (_{jk}^{α}_{P}_{G}T^{†} and (_{P}_{G}T^{†} of the respective matrices in _{jk}^{α}_{6} analogy is Fourier transform

Matrices ...(^{2})_{P},_{1})_{P}, ...^{2})_{P},_{1})_{P}, ..._{3} product tables in _{3}-diagonal irreps _{jk}^{α}

_{jk}^{α}_{jk}^{α}

Hamiltonian _{0}_{1}_{2}_{1}_{2}_{3}] or coefficients of its expansion _{3} operators [^{0}^{1}^{2}_{1}_{2}_{3}]. The parameters are indicated in _{3} base state |1〉 and other _{3}-defined base states |

The resolution of _{P}_{P}_{P}_{P})

If the _{xy}^{E}_{yx}^{E}_{P}_{P}_{P}^{A}^{1} and ^{A}^{2} or block-diagonal with a pair of identical 2-^{E}

The

Irrep-dimension ^{E}^{E}^{E}_{1} = _{2}^{*} and equal _{1} = _{2}.

These are the values that respect the local _{3} ⊃ _{2}[_{3}] subgroup chain symmetry that gave (_{1} or by unequal _{1} and _{2}. Complex _{1} = |^{iφ}_{1} and _{2} shift standing-wave nodes but cannot split _{3} symmetry.

Non-Abelian symmetry analysis in general, and the present example of _{3} resolution in particular, involves a dual-group relativity between an _{3} on one hand, and an _{3} on the other hand. Each

In the present example, the _{3}=[^{1}^{2}_{1}_{2}_{3}] labels equivalent locations in a potential or lab-based field and is a reference frame for an excitation wave or “body” occupying lab locations.

On the other hand, the _{3}=[^{1}^{2}_{1}_{2}_{3}] regards the excitation wave as a reference frame to define relative location of the potential or laboratory field.

Quantum waves provide the most precise space-time reference frames that are possible in any situation due to the ultra-sensitive nature of wave interferometry. This is the case for optical coherent waves or electronic and nuclear matter waves. The latter derive their space-time symmetry properties from the former, and these are deep classical and quantum mechanical rules of engagement for currently accepted Hamiltonian quantum theory.

Interference of two waves depends only on _{3} potential of ^{1}〉^{2}〉_{1}〉_{2}〉_{3}〉] in

Key to this definition is the independence and

Neither relation makes sense if we were to equate _{k}_{k}^{−1}. The effect of _{k}_{k}^{−1}_{2} in _{2} in _{1}_{2} = _{2}_{1}=^{−1}=^{2} on |

Different points of view show how “body” ^{1} = ^{o}^{o}^{1} in a lab frame that “stays put.”

In a lab view, effects of body operation _{k}_{k}^{−1} on |_{k}^{−1} also moves each body operation _{j}_{k}_{j}_{k}^{−1}. The lab view of a lab operation _{k}_{j}_{j}_{k}

Applying projector _{jk}^{α}_{jk}^{α}

The norm-factor ^{o}G^{α}_{N}^{α}^{o}G

A non-Abelian projection ket in

The corresponding local operator _{jk}^{α}_{−}_{1}_{jk}^{α}_{α}^{−1}) = ^{α}^{†}(

A summary of the results is consistent with the block matrix forms in

Choice of subgroup _{2} = [_{3}_{2} labeled by their _{3} eigenvalues (−1)^{m}

Physical significance of these global-(

Wherever the global _{3}-symmetric (0_{2}), then the entire wave is symmetric to _{3}-_{2}), that is seen for each overall figure, too.

However, if the local _{3}-symmetric (0_{2}), the local wave in _{3}-_{2}), that antisymmetry and one node is seen in

Local and global symmetry clash along the _{3}-axis for states projected by nilpotent _{xy}^{α}_{yx}^{α}_{yx}_{3}-axis. However, the simulation of the |_{xy}

The “unglued” level _{xy}^{E}_{yx}^{E}_{1} and _{2} in _{2} = [_{3}_{3} local symmetry causing _{2}-local and more like current-carrying above-barrier _{3}-local waves rotating on _{3} correlation arrays in _{2} or _{3} indicate level cluster structure for extremes of each case.

Column 0_{2} of array _{3} ⊃ _{2} in _{1} and _{1}_{2} local symmetry and lies below cluster-(_{2}_{2} symmetry according to the 1_{2} column of _{3} of table _{3} ⊃ _{3} indicates that _{1} and _{2} levels cluster under extreme _{3} localization, but columns 1_{3} and 2_{3} indicate that each _{3} with no extra degeneracy beyond its own (^{E}

A classical analog of quantum waves states in _{3} molecule. A detailed description of this analogy in

The mathematical basis of correlation arrays in _{jk}^{α}_{jk}^{α}

_{3}-symmetric Hamiltonian _{jk}^{α}^{α}^{α}^{E}

In contrast to this clustering or “un-splitting” associated with local _{m}_{m}_{m}_{m}_{n}_{m}_{n}

Global _{jk}^{α}^{α}^{α}

In the _{3} example, adding matrix (^{1}) from _{3}=[^{1}^{2}], and adding (_{3}) reduces it to _{2}=[_{3}]. Adding a combination of (^{1}) and (_{3}) completely reduces (_{3}∩_{2}=_{1}=[

By reducing ^{α}^{α}_{3} idempotent ^{E}_{3}-labeled _{1313}^{E}_{2323}^{E}_{3} doublet level ^{E}_{3}-labeled singlets ^{13} and ^{23}. Both _{3} singlets _{1} and _{2} end up relabeled with _{3} scalar 0_{3} labels.

Global _{3}⊃_{2} relabeling and/or splitting is by _{3} singlets have different labels 0_{2} and 1_{2}.

Center portions of splitting relations in _{3} irrep-^{α}_{3} or _{2} irreps under their respective global symmetry breaking. Earlier studies [_{1}, _{2}, or _{3}⊃_{3} or _{3}⊃_{2}, respectively, in

Opposite to global ^{α}^{a}^{b}^{a}^{α}^{β}^{α}^{α}^{a}^{α}^{a}^{a}^{a}^{α}

Base states |_{j}^{α}^{k}_{jk}^{α}^{k}^{k}^{k}_{jk}^{α}

Of all _{3}⊃_{2}-projectors _{j}_{2}_{k}_{2}^{α}_{0202}^{A}^{1}, _{0202}^{E}_{1202}^{E}_{2} = 0_{2}. Only these can project induced states |0_{2} ↑_{j}_{2}^{α}_{2}〉 corresponding to the 0_{2}-column of _{3} ⊃ _{2} array in _{1} and _{2} and _{2}-column of _{1212}^{A}^{2}, _{0212}^{E}_{1212}^{E}_{2} ↑_{j}_{2}^{α}_{2}〉 state. Each projector _{j}_{2}_{k}_{2}^{α}_{2}-subgroup projector ^{k}^{2} “right-guarding” the side facing each local _{2}-ket |_{2}〉 = ^{ℓ}^{2}|_{2}〉 that is similarly “guarded” by its own defining projector ^{ℓ}^{2}. _{2}-subgroup projector orthogonality then allows only _{2}=_{2}, giving the projection selection rules just described.

Each “right guard” projector ^{k}_{jk}^{α}^{α}_{k}_{3}↓_{2} examples in _{3}⊃_{2} subgroup chain resolution in ^{k}_{j}^{α}^{k}^{α}

The numbers on the left-hand side of ^{th}^{th}_{3}⊃_{1} to _{1} symmetry, _{3}⊃_{2} or the array _{3}⊃_{3} in _{1}=_{2}∩_{3} is the intersection of _{2} and _{3}.

_{1} local symmetry base |0_{1}〉=|1_{1}〉 is the |_{1}, pseudo-scalar _{2}, and two _{3}⊃_{1} correlation array in ^{01} (_{1}) ↑ _{3}, also known as a _{3}.

Reciprocity in ^{k}_{3} is the smallest non-Abelian group so it has no such subgroups, but octahedral symmetry has non-Abelian _{3} and _{4} subgroups that figure in its splitting and clustering that are described in later Section 7.

Three pairs of kets in _{2} = (_{3}), _{2} = (^{1}_{2}), ^{2}_{2} = (^{2}_{1})] one at each site.

Conjugate bras 〈^{†}relate to _{2}=(_{3}), _{2}^{2}=(^{2}_{2}), _{2}_{1})], again, one per _{2}-well site.

_{2} projectors ^{02}=_{2}^{1} (_{3})=^{x}^{12}=_{2}^{1} (_{3})=^{y}^{†}_{2}.

The same projectors split ket |^{m}^{2} |_{2}

_{2} cosets. Row-(bra)-_{x,}_{·}^{E}_{3}_{1}+_{1})_{2}+_{2}). Row-(bra)-_{y,}_{·}^{E}_{3}_{1}-_{1})_{2}-_{2}). Column-(ket) (±)-forms _{·}_{,x}^{E}_{·}_{,y}^{E}_{1}±_{2})_{2}±_{1}). Either ordering gives the same matrix. Off-diagonal components _{x,y}^{E}_{y,x}^{E}^{0} ± _{3}) vanish.

Kets ^{x}^{p}^{x}^{x}^{1}〉, ^{x}^{2}〉 span induced representation ^{x}_{2})↑_{3}, and ^{y}^{p}^{y}_{2})↑_{3}. Normalized states
_{3} table in _{2}(_{3}) body-basis right-coset representation bra-defined by 〈^{†} |_{2} instead of _{3}.

_{2} ordered products in _{2} induced reps (^{02}⊕^{12})↑_{3} in _{2})-array in _{x}^{p}_{y}^{p}_{2})-array.

Any group component of ^{02}(_{3} shown for ^{1}_{1}_{3}^{12}(_{3} in

The 0_{2} correlation in ^{02}↑_{3} reduces further to _{3} irreps _{1}⊕^{12}↑_{3} reduces to irreps _{2}⊕_{1}⊕_{2}⊕

Octahedral-cubic rotational symmetry _{h}_{i}_{1}_{1}_{1}_{6} on top facing edge of _{h}_{x}

_{6}_{1}_{6} flipping a wave in position |_{1}_{z}_{6}|_{1}_{z}_{6}_{1}_{z}_{h}_{x}_{1}_{2}_{1}_{2}_{1}_{1}_{2}_{2}_{2}

Note _{6}-transform of _{1}_{6}|_{1}_{y}_{6}-transform of _{1}_{6}_{1}_{6}^{−1}=_{3}^{2}). The latter is divined easily by “slide-rule” as _{6} flips _{1}_{3}^{2}’s.

Three Cartesian _{4} axes of anti-clockwise 90° rotations _{x}_{y}_{z}_{x}_{x}^{3}, _{y}_{y}^{3}_{z}_{z}^{3} are also 90° rotations but around negative axes [1̄00], [01̄0], and [001̄]. A shorthand notation for 180° Cartesian rotations is _{x}_{x}^{2}, _{y}_{y}^{2}_{z}_{z}^{2}. Trigonal _{3} axes of anti-clockwise 120° rotations _{1}, _{2}, _{3}, and _{4} lie along [111], [1̄1̄1], [1̄11], and [1̄11̄], respectively, while axes of inverses _{1}=_{1}^{2}, _{2}=_{2}^{2}, _{3}=_{3}^{2}, and _{4}=_{4}^{2} lie along the opposite directions [1̄ 1̄ 1̄], [111̄], [11̄1̄], and [11̄1], respectively.

There are six _{2} axes of 180° rotations _{1}, _{2}, _{3}, _{4}, _{5}, and _{6} located along [101], [1̄01], [110], [1̄10], [011], and [01̄1], respectively. This completes the five classes of _{1..4}_{1..4}], [_{xyz}_{xyz},_{xyz}_{1..6}]. Including the rotations with inversion _{h}_{1..4}_{1..4}], [_{xyz}_{xyz},_{xyz}_{1..6}] where _{1..4}=_{1..4}, [_{xyz}_{xyz}_{xyz}_{xyz}_{1..6}]=[_{1..6}].

The “slide-rules” in _{4} such as _{4}=[_{z},_{z}^{2} =_{z},_{z}^{3} =_{z}_{z}_{4}_{v}_{z},_{z},_{z}, σ_{4}_{x}, σ_{3}_{y}_{4} plus pairs of diagonal mirror reflections [_{4}=_{4}, _{3}=_{3}] and Cartesian mirror reflections [_{x}_{x}_{y}_{y}_{x}, σ_{y}_{3}_{4}] is a _{4}_{v}_{z},_{z}_{z}_{4}_{v}_{4} of _{h}

_{4} of _{4}=[_{z}, ρ_{z},_{z}_{x}_{x}_{4}=[_{x},_{4}_{y},_{3}] in that order. The _{1}-face shows coset _{1}·_{4}=[_{1}_{1}_{4}_{y}_{2}-face has coset _{2}·_{4}=[_{2}_{2}_{3}_{y}_{1}·_{4}=[_{1}_{x},_{3}_{6}] and _{2}·_{4}=[_{2}_{x},_{4}_{5}].

Each _{4}-coset element _{z}^{p}_{4} element _{z}^{p}_{z}^{p}_{3}(or _{4}), _{1}, _{2}, _{6}, and _{5} of _{4} cosets in

_{h}_{4}_{v}_{4}_{v}_{3}_{v}_{3}-[111] and twelve dihedral cosets of _{2}_{v}_{2}-[101].

An order-8 axial symmetry _{4}_{v}_{3}_{v}_{2}_{v}_{v}_{y}_{v}_{2}], or _{v}_{4}], fundamental symmetry operations whose products generate all others. _{h}

Each subgroup spawns a coset space and a set of induced representations of full _{h}_{3}_{v}_{h}_{h}_{h}

Spectral class resolution of _{3} in

Cyclic subgroup _{4}(_{z}^{p})_{3}(_{1}^{p}_{2} characters correlate to

Equivalent subgroup correlations ^{−1} share elements in the same _{4} local symmetries have one correlation table in _{3} subgroups. However, _{2}(_{z}_{2}(_{1}) correlations differ since _{1} and _{z}

Projectors _{jk}^{α}_{jk}^{α}_{3} projector splitting in _{3}⊃_{2} in _{3}⊃_{3} in _{2} values [0_{2}_{2}] or else _{3} values [0_{3}_{3}_{3}] while a tetragonal correlation _{4} will use sub-labels (_{4}_{4}_{4}_{4}].

The _{4} or else _{3} unambiguously defines all _{4} or _{3} correlation numbers in _{2}(_{1}) correlations cannot distinguish all three sub-levels of _{1} or _{2} wherever a number 2 appears, and the _{2}(_{z}_{h}_{4} case first. (_{4} resolves _{2}(_{z}

The _{4} correlation table in ^{α}_{κ}_{g}^{α} in table shown in _{m}_{4}_{m}_{4}^{α}_{4} local symmetry projectors _{m}_{4}. The latter _{m}_{4} operators _{z}^{p}_{p}^{m}^{4} = (_{p}^{m}^{4}^{*} using

The five class projectors ^{α}_{1}_{2}_{5}_{6}). So do the five class operators (_{0}_{r}_{k}, κ_{ρ}_{k}, κ_{R}_{k}, κ_{i}_{k}) in which each ^{α}_{3} classes in

Each of the ^{α}_{n}_{4}_{n}_{4}^{α}^{α}^{α}_{n}_{4}=_{n}_{4}^{α}_{4} local symmetry projector _{m}_{4}.

As the five (^{α}_{n}_{4}_{n}_{4}^{α}_{g}_{4}-_{k}

The resulting ten products
_{n}_{4}_{n}_{4}^{α}_{k}_{k}_{4}(_{n}_{4}_{n}_{4}^{α}_{4} local-invariant, that is, they commute with four _{4}-operators (_{z},_{z}^{2} = _{z},_{z}^{3} = _{z}^{α}_{k}_{n}_{4}_{n}_{4}^{α}_{z}^{p},_{z}^{p}_{k}_{k}_{z}^{p},_{n}_{4}_{n}_{4}^{α}_{k}_{k}_{k}^{α}_{n}_{4}_{n}_{4}^{α}_{k}^{α}^{α}_{k}

Without evaluating _{4} sub-classes by simply inspecting _{4} operations _{z}^{p}only_{k}_{1}_{2}_{3}_{4}] whose axes intersect four corners of the +_{1}_{2}_{3}_{4}] whose axes similarly frame the −_{k}_{1}_{2}_{5}_{6}] whose two-sided axes bisect edges of the ?_{3}_{4}] whose axes are perpendicular to _{x}, ρ_{y}, ρ_{z}_{x}, ρ_{y}_{z}_{z}_{z}_{z}

The inverse to _{n}_{4}_{n}_{4}^{α}_{k}

Off-diagonal _{m}_{4}_{n}_{4}^{α}_{k}

Scalar _{1} and pseudo-scalar _{2} are given first then _{1}, and _{2} irrep matrices for the fundamental _{k}

Symmetry of _{4}⊂_{1}_{2}_{5}_{6}] and [_{3}_{4}] would demand equality of parameters for each.

Setting each parameter to the inverse of its sub-class order (_{k}_{i}_{k})) reduces each matrix to diagonal form and gives the diagonal _{n}_{4}_{n}_{4}^{α}_{k}

An octahedral Hamiltonian _{k}_{=1}^{24}_{k}_{k}_{4}(_{k}_{4}(_{k}_{n}_{4}_{n}_{4}^{α}_{n}_{4}^{α}_{m}_{4}_{n}_{4}^{α}_{4}-_{1256}=_{I} from +^{st}_{34} = _{II} to 2^{nd}_{I} and _{II} contributions to eigenvalues _{n}_{4}^{α}_{n}_{04}^{A}^{1}, _{04}^{T}^{1}, _{04}^{E}_{34}^{T}^{2}, _{34}^{T}^{1}) are plotted in _{I} = _{1256} and _{II} = _{34}.

One expects the parameter _{II} for 2^{nd}_{I} for adjacent tunneling so the (_{II} = 0)-cases are drawn first in _{k}^{st}_{xy}_{I}(120°), and _{I}(180°) label 1^{st}_{4} axial tunneling paths that have the same _{I}-level patterns and splitting ratios as (_{II}=0)-cases in _{4}_{z}_{I}(180°) patterns (with _{I}_{1}_{1}_{2}_{1})(_{2}_{2})(_{2}_{1}) on the left hand side of _{1}_{2})(_{1}_{2})(_{2}_{2}_{1}_{1}) of _{3[111]} local symmetry across the separatrix break on the right-hand side of _{1})_{1})_{2})_{1}_{2}_{2})_{1}_{2}_{1})

For an isolated three-level (_{4} or else 2_{4} the splitting pattern requires only two parameters. This could be either the 180°(_{I}, _{II}) or the 90°(_{xy}_{z}_{I}. Parameters _{I}, _{xy}_{I} each split (

Local symmetry 1_{4} and 3_{4} each have two-level (_{I}, or else _{xy}_{z}_{z}_{n}_{n}_{4} clusters but are necessary in order that the whole set be orthonormal and complete.

_{nn}^{(}^{α}^{)} by _{1}_{2}_{6}). It acts on |_{n}_{4}_{n}_{4}^{(}^{α}^{)} 〉 eigenkets in

An _{1}_{2}_{6}) with coefficients _{a}_{0}_{1}_{2}_{6}. Local symmetry _{4} or _{3} reduces the sum to _{G}_{a}_{a}_{a}_{a}_{a}

From these arise expansions like _{n}_{4}^{α} in terms of its coefficients _{a}_{j}_{k}_{k}_{j}_{nn}^{(}^{α}^{)} and ^{α*}_{a}_{a}

Each _{4} sub-class of order °_{a}_{a}_{a}D_{nn}^{(}^{α}^{)*} (_{a}_{a}D_{nn}^{(}^{α}^{)*} (_{a}_{n}_{4}^{α}_{G}_{a}g_{a}D_{nn}^{(}^{α}^{)*} (_{a}_{n}_{4}^{α}^{A}^{1}^{A}^{2}_{34}^{T}^{2} ) expand to ten _{4}-local tunneling parameters _{a}_{0}_{I}_{II}_{II}) and

One might count twelve real parameters in _{I},_{I}) and (_{z}_{z}_{I} = _{I}, which are real. If ^{†}) it should only require ten, the rank of

With no conjugation symmetry, such as for a _{4}-symmetric matrix, the

The previous two sections have detailed of symmetry-based level clustering and cluster splitting for _{4}. In _{6} for _{4}_{3} sub-group quickly. Starting with _{n}_{3}_{n}_{3}^{α}_{3} is shown in _{3} clustering fits patterns of (_{1}_{2}_{2}_{2}) and two of (_{1}_{2}), each with a total degeneracy of 8. As before in _{3} make different patterns depending on which tunneling parameters are active. This is demonstrated in

As the order °^{λ}^{λ}^{λ↑G}^{λ}^{A}^{1}+^{A}^{2}+^{E}^{T}^{1}+^{T}^{2} where ^{α}

The number of _{4}⊂_{3}⊂

This happens for the _{2}(_{1}) ⊂_{2} label to two bases of _{1} and of _{2}. (Two _{2} symmetries 0_{2} and 1_{2} cannot distinctly label three bases.) _{2}(_{1}) splits _{1}_{4}), (_{2}_{2}), (_{3}_{3}), (_{4}_{1}), (_{x}ρ_{z}_{y}_{x}_{z}_{x}_{z}_{y}_{y}_{1}), (_{2}), (_{3}_{5}), (_{4}_{6}). The _{2}⊂_{3}⊂_{4}⊂

Spectral resolution of fourteen _{2}(_{1})⊂_{nn}^{α}_{2} basis, two off-diagonal pairs _{ab}^{T}^{1}=_{ba}^{T}^{1†} and _{ab}^{T}^{2}=_{ba}^{T}^{2†} of non-commuting nilpotent projectors are needed to finish _{2}-labeling of _{ℓ}_{2}(_{1})⊂_{a,b}_{b,a}^{*}.)

The other class of _{2} symmetry has similar problems. Local _{2}(_{z}_{ab}^{T}^{1}=_{ba}^{T}^{1†} and _{ab}^{T}^{2}=_{ba}^{T}^{2†}, too, but then another nilpotent pair _{ab}^{E}_{ba}^{E}^{†} must be added to label repeated _{2}(_{z}^{†} matrices for _{2}(_{z}

For the lowest local symmetry _{1}=[_{1} provides no distinguishing labeling, and all twenty-four _{α}^{α}^{2}=24) are active in its resolution. The 24-parameter_{a}_{j,k}^{α}^{*}(_{p}_{p}_{p}_{α}ℓ^{α}^{α}

For _{α}^{α}^{0}=5 invariant idempotents ^{α}^{α}^{2}. This 5-parameter resolution is sketched in

Any non-Abelian local symmetry such as _{4} also fails to split ^{α}^{α}_{nn}^{α}_{nn}^{α}_{4}⊂_{3}⊂_{2}⊂

Each matrix display lists ^{α}^{λ}^{↑}^{G}^{λ}^{α}_{a,λ}^{α}^{λ}^{↑}^{G}^{λ}

Including inversion _{n}_{2} subgroups lie in _{2}-correlations in _{h}_{2}_{v}_{2}_{v}^{i}_{y},_{1}_{2}] is the one of the three local symmetries shown in _{2}_{v}^{z}_{z}, σ_{y}, σ_{x}_{4}_{v}_{2}_{v}^{34}=[_{z}, σ_{3}_{4}].

The local symmetry _{2}_{v}^{i}_{h}_{2}_{v}_{g}_{u}

As the order of the local sub-group symmetry goes down, the degeneracy and complexity of the rotational cluster must increase. _{h}_{2}_{v}_{2} will show many of the same effects. To actually resolve the doubled _{1} or _{2} triplets of _{2} requires distinguishing the _{2} clusters are 12 fold degenerate, but they are also easily displayed.

As noted earlier, _{3}⊃_{2} and _{4}⊃_{2} local symmetries give identical cluster degeneracies and groupings, but with cluster splittings and structure dependent on the sub-group chain. Though it neglects inversion, _{2} (and, thus _{h}_{2}_{v}_{4}⊃_{2}(_{4}) sub-group chain.

Compared with _{4} and _{3}, the splittings of _{2} are relatively simple to calculate since the terms in _{2} is done in the same way as for

_{2} local-symmetry structures. What will result is a transformation between cluster-splitting energy and tunneling parameters. The inverse of this transformation is also easily defined.

_{m}_{2} cluster may not interact with a 0_{2} cluster. Second we recognize that only half of the subclasses are needed to fully define the possible splittings, the others simply repeat the same information. _{2} cluster. Looking at the _{1} level in the 0_{2} cluster, one can see that the subclasses _{n}, ρ_{n}_{n}, i_{n}_{2} level in the 1_{2} cluster shows the same similarity, but the _{n}, i_{n}

By using only the minimum number of splitting parameters and including only a single cluster gives a condensed version of

There are multiple ways to use

An example demonstrates this process for a model (4^{[4]} and ^{[6]} model rotational distortions written in an octahedral basis of fourth and sixth order respectively in ^{[4]} and ^{[6]} while keeping them normalized. Because ^{[4]} and ^{[6]} each have octahedral symmetry,

As noted in Section 3 cluster structure location and the RES shape will change significantly as the Hamiltonian parameters change in

RES diagrams in _{2} clusters exist. The lowest 0_{2}(_{2})↑_{2} local symmetry. Its superfine levels are magnified about 100 times in the central inside plot of _{2}(_{2})↑_{2} replacing _{1} and _{1} switched with _{2}.

At certain ^{α}_{1}_{1}_{2}_{2}) are given by vectors _{dash}_{dot}_{solid}_{2} cluster region at ^{α}_{r}^{[4}^{,}^{6]} Hamiltonian _{r}

Dotted-line and solid-line curve patterns appear alternately flipped in sign. Dotted-line patterns have a crossing (_{1},_{2}) pair while the solid-line patterns have a crossing (_{2},

Solid-line patterns appear to be centered on quasi-hyperbolic avoided-level-crossing episodes involving the pair of repeated _{2} tensor species of _{1}_{1}_{2}_{2}) of solid-line superfine level patterns reflects Bohr-like orbital ordering (_{34} or equivalent _{xy}_{2} valleys.

Dashed-line pattern level curve _{2}-species surrounding a degenerate (_{1}_{1}_{2}(_{2})↑_{2}_{2}

Such crossings are quite ironic if we recall that it was (_{1}_{1}_{2}_{2}_{1}_{2}) clusters noted by Lea, Leask, and Wolf [_{4}(_{4})↑_{4}(_{4})↑_{1}⊕_{1}⊕_{4}(_{4})↑_{2}⊕_{2}⊕_{4}(_{4})↑_{1}⊕_{2}. (Recall _{4} columns of _{4}-axial tunneling model [_{4}-axial tunneling cluster splitting dies exponentially as body momentum-_{4}(

However, the _{2}(

It would be easy to write off such (_{1}_{2}) or (

Clearly there is still much to learn about multi-path tunneling models in general and the octahedral ones in particular. Here we can only offer a potentially elegant way to treat these kinds of high-symmetry cases.

Semiclassical treatment of rovibronic or rovibrational states provides some insight into the transition between lab-coupled and body-coupled vibronic momentum that are related in _{2}_{3}_{1}[_{2} + _{3} “hot-bands” in 1979 [

These methods are similar in philosophy to those described in Section 2 that approximate tensor eigenvalues with Legendre formulas and thereby construct rotational energy based on a semiclassical _{2}_{3}_{1} system, and a 5-by-5 matrix for the _{2} + _{3} system.

The resulting _{3}_{1} bands by 3-sheet REES was done using the high-resolution 3D-graphics at Los Alamos in 1987 and reported in 1988 [_{3}_{1} REES include conical intersections that occur for zero scalar Coriolis coupling. These are analogous to well known conical intersections of Jahn–Teller PES that lend insight into BOA breakdown of single adiabatic surfaces. The following contains two examples of REES models. The first is a simplified internal rotation model involving a 2-sheet REES, and the second is an excerpt of a recent study of the _{3}_{4} dyad of _{4} that involves a 9-sheet REES.

A first application by Ortigosa and Hougen [

Total momentum _{RS}

Replacing bare-rotor momentum _{RS}

The simplest classical theory of the rotor-

A quantum theory of multiple RES involves mixing extreme cases |

Eigensolutions of matrix form

The first practical REES application includes 9-sheet displays of the _{3}_{4} dyad of _{4} recently shown by Boudon _{4} axes (shaded red), _{3} axes (shaded blue), or _{2} axes (shaded green).

Each of the symmetry axes may take turns as central loci for clusters of their type of local symmetry _{2}, _{3}, or _{4}, or else, they may sit on a REES separatrix or saddle point between two or more different types of clusters. A third option involves _{1} clusters that have no rotation axis point but are likely to belong to vertical _{v}_{z}_{d}_{3}]. These label clusters of 24 levels associated with 24 equivalent REES hills or valleys.

A final option involves true-_{1} clusters with no local symmetry whatsoever and 48 REES hills or valleys. So far this extreme type has not been identified, but one may speculate that it may actually become most common at extremely high

A common ordering noted before on the left hand side of ^{[4]}) and in _{6}) is (_{3}-valley→_{2}-saddle→_{4}-hill). It is present in the lowest REES band of ^{nd}

A cutaway view at ^{[4]} RES on the right hand side of ^{[4]}). However, the first and lowest REES for

After the first two REES sheets the cluster topography become more complicated with multiple conical intersections and avoided crossing points.

On the 5^{th}_{3}_{4} REES are found examples of _{1}-local symmetry valleys as shown in _{1} loop occupies an area that is comparable to the minimum uncertainty (_{4} axis of the figure and a nascent 24-level cluster of type 1_{2}(_{2})↑

The symmetry details in this rovibrational spectra and the potential richness of quantum dynamics it represents should be quite evident from the few examples glimpsed here. We seem to be just scratching the surface of quantum systems of a great but potentially comprehensible complexity.

Semiclassical methods for visualizing and analyzing rovibrational dynamics of symmetric polyatomic molecules have been reviewed. This includes improved understanding of RES and REES phase spaces and development of more powerful symmetry methods to calculate tunneling dynamics of symmetric molecules that are highly resonant. A group-table-matrix analysis of intrinsic

Local symmetry theory applies to classical vibrational modes as well as to quantum tunneling. Examples of classical _{3} modes given below help clarify global-_{3} modes defined by local _{2}(_{3}) in _{jk}^{α}_{jk}^{α*}

In particular, global _{xx}^{E}^{1}〉 and |_{xy}^{E}^{1}〉 (left _{1} column in figure) “point” along global _{yx}^{E}^{1}〉 and |_{yy}^{E}^{1}〉 (right _{1} column) “point” along global _{xℓ}^{E}^{1}〉_{yℓ}^{E}^{1}〉](_{3}-symmetric-antisymmetric pair like lab unit vectors [|

This exposes easy derivations of _{jk}^{E}^{1} (_{jk}^{E}^{1} (_{j}_{r}_{·}_{k}

This also fixes local transformations. Local _{xx}^{E}^{1}〉 and |_{yx}^{E}^{1}〉 (lower _{1} row in figure) “point” along local _{xy}^{E}^{1}〉 and |_{yy}^{E}^{1}〉 (upper _{1} row) “point” along local _{xy}^{E}^{1}〉 (or _{yx}^{E}^{1}〉), a zero appears on the _{3}-axis in _{xx}^{A}^{1}〉 and |_{yy}^{A}^{2}〉 avoid such conflicts by being all one or the other.

For group-defined cases like

Classical modes are eigenvectors of force-field matrix _{0}, _{a}_{3}-labeled axes _{a}_{ab}^{a}^{b}_{â}_{a}_{b̂}_{b}_{a}_{b}_{a}_{b}_{a}_{b}_{ab}

This sum of harmonic Hooke (^{2}_{aa}_{ab}_{ba}

The classical equation of coupled harmonic motion is a Newtonian _{3}-symmetric case treated here. The driving force

It is instructive to compare this classical equation of motion to that of Schrodinger’s equation for quantum motion.

Squaring quantum time generator _{t}^{2} and ^{2}.

The (_{m}/ħ_{m}_{m}/M_{m}^{2}. The former is Planck’s oscillator frequency relation ^{2}. Apart from normalization, eigenvectors of quantum

The preceding relates eigensolutions _{3}⊃_{2}(_{3})-locally-symmetric _{a}_{0}, _{1}, _{12}, and _{3} allowed for _{3}⊃_{2}(_{3})-symmetric _{h}

Only 1^{st}_{b}_{b}_{1}_{b}_{ab}_{r}_{i}_{3}(angular) and _{0}(radial) constants represent external connections between each mass and an outside lab frame.

Generic group parameters _{b}_{1}_{b}_{0}_{1}_{2}_{1}_{2}_{3}] for the _{b}_{1}_{b}_{b}_{j}_{1} ● _{1})( _{b}_{b}_{b}_{1} or _{b}^{st}_{a}_{ab}

One may visualize each −_{1}_{b}_{1} due to setting a (tiny) unit _{b}_{11} must be negative or else _{1} blows up. Higher order ^{b}^{b}

Substitution of generic _{a}_{3}⊃_{2}(_{3})-symmetric _{ℓℓ}^{α}_{i}_{r}_{3}, or _{0} in _{i}_{r}_{3}, and _{0} since none can mix local

Any set of four ^{A}^{1}, ^{A}^{2}, _{x}^{E}_{y}^{E}_{1}-stretching modes among the highest frequencies. In contrast, tunneling amplitudes are often negative so their _{1} states lie low. As a rule, fewer quantum nodes imply lower energy.

In some ways the direct-_{1}-connection spring model of _{3}⊃_{2}(_{3}) model just treated since it involves maximal (50-50) mixing of _{b}_{b}_{1} using (

Again, a substitution of generic _{a}

Eigenvectors of the

_{1}. It distinguishes genuine vector modes (|_{x,}_{(−)}^{E}_{y,}_{(−)}^{E}_{x,x}^{A}^{1}〉) from non-genuine (low or zero-frequency) vector modes of pure _{x,}_{(+)}^{E}_{y,}_{(+)}^{E}_{y,y}^{A}^{2}〉). The _{3}-local symmetry is wiped out by direct connection-_{1}.

In order to reestablish approximate _{3}⊃_{2}(_{3})-local-symmetry there needs to be a _{2}(_{3})-“_{3} and _{0} in _{3} + 2_{r}_{0}) between eigenvalues _{1}_{3}⊃_{2}(_{3}) locality seen in

Meanwhile the constant _{3} that determines eigenvalue _{y,y}^{A}^{2} does not affect locality for either of the singlet _{1} or _{2} modes. Singlet eigenvectors are non-negotiable as long as master symmetry _{3} holds.

Another choice for _{3} local symmetry is the _{3} subgroup of _{1}=_{2}=_{3}=0) ma