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Spectra of high-symmetry molecules contain fine and superfine level cluster structure related to J-tunneling between hills and valleys on rovibronic energy surfaces (RES). Such graphic visualizations help disentangle multi-level dynamics, selection rules, and state mixing effects including widespread violation of nuclear spin symmetry species. A review of RES analysis compares it to that of potential energy surfaces (PES) used in Born–Oppenheimer approximations. Both take advantage of adiabatic coupling in order to visualize Hamiltonian eigensolutions. RES of symmetric and D2 asymmetric top rank-2-tensor Hamiltonians are compared with Oh spherical top rank-4-tensor fine-structure clusters of 6-fold and 8-fold tunneling multiplets. Then extreme 12-fold and 24-fold multiplets are analyzed by RES plots of higher rank tensor Hamiltonians. Such extreme clustering is rare in fundamental bands but prevalent in hot bands, and analysis of its superfine structure requires more efficient labeling and a more powerful group theory. This is introduced using elementary examples involving two groups of order-6 (C6 and D3~C3v), then applied to families of Oh clusters in SF6 spectra and to extreme clusters.
symmetrymolecular dynamicstunnelinglevel clusters1. Overview of Eigensolution Techniques for Symmetric Molecules
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 N-by-N matrix to N values and N stationary eigenstates, but the artificial processes may seem as opaque as nature itself with little or no physical insight provided by N2 − N eigenvector components. We are thus motivated to seek ways to visualize more of the physics of molecular eigensolutions and their spectra. This leads one to explore digital graphical visualization techniques that provide insight as well as increased computational power and thereby complement numerically intensive approaches [1].
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 [2] and present challenging problems of symmetry analysis to sort out a plethora of tunneling resonances and parameters for so many resonant states.
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 C6 and D3~C3v before applying it to Oh symmetric monster-clusters in Section 8. Monsters in REES-polyad bands are shown in final Section 9.
The direct approach to symmetry starts by viewing a group product table as a Hamiltonian matrix H representing an H operator that is a linear combination of group operators gk with a set of ortho-complete tunneling coefficients gk labeling each tunneling path. A main idea is that symmetry operators “know” the eigensolutions of their algebra and thus of all Hamiltonian and evolution operators made of gk’s.
1.1. Computer Graphical Techniques
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-ν vibrational dynamics and chemical pathways for dissociation or re-association [3,4]. This includes BOA-breakdown states in which a system evolves on multiply interfering PES paths. Dynamic Jahn–Teller–Renner effects involve multi-BOA-PES states in molecules and solids. Examples in recent works [5,6] include coherent photo-synthesis [7].
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 [8] to analyze spectral fine structure of high resolution spectral bands in molecules of high symmetry including PH3[9]XDH3 and XD2H molecules [10], tetrahedral (P4) [11], tetrafluorides (CF4 and SiF4) [12], hexafluorides (SF6, Mo(CO)6 and UF6) [13–16], cubane (C8H8), and buckyball (C60) [15,16] and predicted major mixing of Herzberg rovibronic species. Recently RES have been extended to help understand the dynamics and spectra of fluxional rotors [17] or “floppy” molecules such as methyl-complexes [18] and vibrational overtones [2].
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, etc.) of a molecular system introduces additional resonance. Symmetry tends to make graphical techniques even more useful since they help clarify resonant phenomenal dynamics and symmetry labeling [19].
1.1.1. Vibronic Born–Openheimer Approximate Potential Energy Surfaces (BOA-PES)
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 = ηψ of a nuclear factor wavefunction ην(ε) (X . . . ) whose quantum labels ν(ε) depend on electronic quantum numbers ε = nlm, etc. while the electronic factor wave ψ (x(X...) . . . ) is a function whose electron coordinates x(X...) . . . depend adiabatically on nuclear vibrational coordinates (X . . . ) of PES Vε(X . . . ) for electron bond state ε.
The adiabatic convenience of a single product Equation (1a) with a vibration eigenfunction ην(ε)(X . . . ) on a single PES function Vε(X . . . ) is welcome but comes at a price; a BOA-entangled coordinate-state is not a simple bra-ket wavefunction product Equation (1b) of bra-bra 〈x . . . |〈X . . . | position and ket-ket | ψε〉|ην〉 state. Symmetry operator product analysis of Equation (1b) is well known. Symmetry of Equation (1a) depends on rotational BOA-relativity of its parts. Vibronic BOA-PES generalize to rovibronic RES by accounting for rotational relations.
1.1.2. Rovibronic BOA Rotational Energy Surfaces (BOA-RES)
The rotational energy surface (RES) can be seen as a generalization of adiabatic BOA wave Equation (1a) to Equation (2) below that includes rotational motion. Here one treats vibronic motion as having the “fast” degrees of freedom while rotational coordinates Θ (e.g., Euler angle (αβγ) for semi-rigid molecules) play the “slow” semi-classical role vis-a-vis the “faster” adiabatic vibration or vibronic states.
In Equation (2), the wave factors of each motion are ordered fast-to-slow going left-to-right. As in Equation (1a) each wave-factor quantum number depends on quanta in “faster” wave-factors written to its left, but each coordinate has adiabatic dependence on coordinates in “slower” factors written to its right.
The Q in Equation (2) denotes vibrational normal coordinates (q1, q2, . . . qm) and ν denotes their quanta (ν1, ν2, . . . νm). The number m = 3N − 6 of modes of an N-atom semi-rigid molecule has subtracted 3 translational and 3 rotational coordinates. Each mode qk assumes an adiabatic BOA dependency on overall translation and rotation Θ known as the Eckart conditions. (Here we will ignore translation.)
RES are multipole expansion plots of effective BOA energy tensors for each quantum value of vibronic ν(ε) and conserved total angular momentum J. Choices of effective energy tensors depend on the level of adiabatic approximation. So do the choices of spaces in which RES are plotted. Elementary examples of model BOA waves, tensors, and RES for rigid or semi-rigid molecules are discussed below.
1.2. Lab-Frame Coupling vs. Body Frame Constriction
Wave ρJ(Θrotation) for a bare rigid symmetric-top (ψ = 1 = η) molecule is a Wigner DJ -function.
ρJ(Θ)=ρJ,M,K(αβγ)=DM,KJ*(αβγ)normnorm=[J]=2J+1
Total angular momentum J is J = R for a bare rotor. Bare lab-frame z-component is labeledM = m. Its body-frame z̄-component is labeled K = M̄ = n. m and n range from +R to −R in integral steps.
Entangled BOA product Equation (2) mates vibronic factor Equation (1a) with a rotor factor ρJ = ρJ,MmK in Equation (3). Now J and K = M̄ depend on total vibronic momentum l and its body z̄-component μ̄ in Ψν(ε) = Ψμ̄l.
Disentangled product Ψρ in Equation (1b) of lab-based vibronic wave Ψμ̄l and bare rotor ρR,m,n of Equation (3) is coupled by Clebsch–Gordan Coefficients CμmMlRJ into a wave ΦMJ of total J = R + l, R + l − 1, . . . or |R − l| and M = μ + m by sum Equation (5) over lab z-angular bare rotor momenta m and lab vibronic μ bases.
A BOA-entangled wave in Equation (1a) or Equation (4) requires more serious surgery in order to survive as a viable theoretical entity. BOA vibronic waves are not merely coupled as in Equation (5) to a rotor, they are adiabatically “glued” or constricted to the intrinsic molecular rotor frame. (A rotor is “BOA-constricted” by its vibronic wave much as a boa-constrictor rides its writhing prey as the two rotate together.)
A remarkable property of quantum rotor operator algebra is that Wigner Dl-waves in Equation (3) are also transformation matrices that relate rotating body-fixed BOA Ψμ̄l (body) into the lab-fixed Ψμl (lab).
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 dk* (r) = 〈r|k〉 = eikr underlying Fourier operator analysis. Details of this connection comprise the later Section 5.
Of particular importance to RES theory is the Wigner–Eckart factorization lemma that relates Clebsch–Gordan CμmMlRJ to Wigner-D’s and transforms coupled wave Equation (5) to BOA-constricted wave Equation (4).
A more familiar form of this is the Kronecker relation of product reduction
Dl⊗DR≈DJ⊕DJ′⊕⋯. Another form is a body-to-lab coupling relation with M = μ + m and n = K − μ̄ fixed in the μ or μ̄ sums. The latter yields a sum over μ̄ = K − n of body-fixed BOA waves Equation (5) giving lab-based ΦMJ wave Equation (4).
Body-(un)coupling in Equation (10a) is an undoing of BOA-constriction by subtracting vibronic (l, μ̄) from (J,K) of BOA-wave ΦJ[ν(ε)] in Equation (10b) to make lab-fixed ΦMJ in Equation (10a) with sharp rotor quanta R = J − l, J − l + 1 . . . or J + l. In a lab-fixed wave ΦMJ of Equation (5) or (10a) rotor R is conserved but K and μ̄ are not. A BOA wave ΦJ[ν(ε)] of Equation (4) or (10b) has body-fixed vibronic K and μ̄ that are conserved but rotor R is not.
Note the following for Equations (8)–(10b). For Equation (8) we have constant (M = μ +m). Result Equation (9) is derived from Equations (6b) and (7c) with constant (n = K − μ̄). In Equation (10a)K = μ̄ + n. In Equation (10b)M = μ + m.
However, in both Equation (10a) and (10b) the internal bare-rotor body component n = K − μ̄ is conserved due to a symmetric rotor’s azimuthal isotropy. This n is a basic rovibronic-species quantum number invariant to all lab based perturbation or transition operators. Like a gyro in a suitcase, no amount of external kicking of the case will slow its spin. Only internal body operations can “brake” its n.
The duality of lab vs. body quantum state labels and external vs. internal operators is an important feature of molecular and nuclear physics, and it is to be respected if we hope to take full advantage of symmetry group algebra of eigensolutions. The duality is fundamental bra-&-ket relativity. For every group of symmetry operations such as a 3D rotation group R(3)lab = {. . .R(αβγ) . . . } there is a dual body group R(3)body = {. . . R̄( αβγ) . . . } having identical group structure but commuting with the lab group. Tensor multipole operators, discussed next, come in dual and inter-commuting sets as well. Generalized Duality is key to efficient symmetry analysis as shown beginning in Section 6.1.
1.3. Mathematical Background for Tensor Algebra1.3.1. Unitary Multipole Functions and Operators
Spherical harmonic functions Yml (φθ) are well know orbital angular factors in atomic and molecular physics. They are special (n = 0)-cases of Wigner-Dl functions Equation (3) as follows.
Yml(φθ)=Dm,0l*(φθ0)[l]4πwhere;[l]=2l+1
A diatomic or linear rotor must have zero body quanta (n = 0) and has a Yml(φθ) rotor wave. Yml -matrix elements or expectation values of a multipole potential Yqk are proportional to Clebsch forms of Equation (7a).
Other choices rescale vqk eigenvalues but do not affect eigenvectors of a tensor vqk or its transformation behavior Equation (14). (By Equations (7c) and (13c), vqk transforms like Equation (6a) for a wave function Yqk (φθ).)
v¯qk=R(αβγ)vqkR†(αβγ)=∑q=-kkvq¯kDq¯qk(αβγ)
Examples of vqk tensor matrices for J′ = J = 1 to 3 are given in Table 1. The J = 2 case is given in expanded form by Table 1. (Higher-J tables are q-folded to save space. Scalar
〈v00〉J=1/[J] is left off each J-table in Table 2)
Historically, spinor J = 1/2 tensors shown in Table 3(a) are related to four Pauli spinor matrices σμ and Hamilton quaternions {1,i,j,k} in Table 3(b) or Table 3(c). The latter appear in 1843 and are used for Stokes’ polarization theory in 1867. The σμ are U(2) algebraic basis of quantum theory for physics ranging from sub-kHz NMR to TeV hadrons and also applies to relativity. General U(k) algebra has k2 generators v00,vq1, . . . ,vqk with a subset of k mutually commuting diagonal (q = 0) labeling operators v0k of the U(k) tensor algebras. The vqk are related to elementary creation-destruction ejk = aj†ak-operators and to their RES in the following sections.
1.3.2. Tensor and Elementary Matrix Operators
Coefficient
〈Jm′|vqk|Jm〉 of elementary operator
em′,m=|Jm′〉〈Jm| is the following CG orWigner 3-j.
Each matrix 〈vqk〉J for J′ = J = 1 to 5 is displayed in compressed form by the following tensor representation Table 2.
CG-3j relation Equation (13c) implies 〈vqk〉J and 〈em′,m〉J matrices have ortho-complete unit vectors of dimension d(J, q) = [J] − q = 2J − q + 1 along qth-diagonal of each [J]-by-[J] matrix. For example, quadrupole v22, octopole v23, and 24-pole v24 share the q = 2 diagonal of J = 2 Table 2.
Any [J]-by-[J] matrix is a combination of elementary 〈em′,m〉J and thus also of 〈vqk〉J. This leads to RES maps that approximate [J]-by-[J] matrix 〈vqk 〈J eigensolutions by plotting related combinations of Yqk (θ, φ) for select θMJ.
1.3.3. Fano–Racah Tensor Algebra
Diagonal dipole-vector (rank k = 1) matrix 〈v01〉J is seen in top row of Table 2 to be proportional to the angular momentum z-component matrix 〈Jz〉J. Diagonal 2k-pole (rank-k) tensors 〈v0k〉J are linearly related to Jz powers Jz2 = JzJz,Jz3 = JzJzJz, . . . up to the kth-power Jzk. This relates 〈v0k〉J -eigenvalues to powersmp of 〈Jz〉-eigenvaluesmand, in turn, leads to an RES scheme to analyze 〈vqk 〉J eigensolutions.
For example, matrix diagonals in Table 2 give elementary representations for J = 2.
Triangle inversion of Equation (18) gives each 〈v0k〉2 in terms of Jz powers 〈Jzp 〉2 = mp in Equation (19). RES plots depend on relating 〈v0k〉J expansions Equation (19) in Jz to Wigner (J,m) polynomials
(-1)J-m[k](kJJ0m-m) in Equation (14) and Legendre polynomials
D00k(·θ·)=Pk(cos θ) in Equation (11) that are also polynomials of Jz = |J| cos θ. By plotting the latter we hope to shed light on the eigensolutions of the former.
2. Tensor Eigensolution and Legendre Function RE Surfaces
Legendre polynomials occupy the central (00)-component of a Wigner-DJ matrix.
D00k(·θ·)=Pk(cos θ)
Examples of Legendre polynomials of cos θ = Jz/|J| and Jz = |J| cos θ are given below.
Classical Pk functions are compared with corresponding quantized 〈v0k〉J unit-tensor e-values in Table 4 that generalize examples of tensor matrix (J=1 to 3)-eigenvalues in Table 2 and Equation (19) to any J and m = J, . . . ,− J. The powers of m and J in 〈v0k〉J, shown in Table 4 are taken to higher order in Table 5.
Norm
2k[k]/2J+k:-k+1 makes each 〈v0k〉J a unit vector. (Note: A + a : b = (A + a)(A + a − 1) . . . (A + b).) In contrast, normalized Pk have Pk(cos 0) = 1. Coefficients cp of cospθ sum to 1 = ∑cp. Square |cp|2 usually do not sum to 1.
Tensor values 〈v00〉J, 〈v01〉J, and 〈v02〉J in [. . . ]-braces of Table 4 equal Legendre functions P0, P1, and P2 in Equation (21b) exactly using J-expectation values Equations (22a) and (22b). However, for rank higher than k = 2, Pk is only approximately equal to 〈v0k〉mJ though the approximation improves with higher J.
〈Jz〉mL=m=〈∣J∣〉mJcos θmJ〈∣Jz∣〉=J(J+1)≅J+12
For large-J values, the 〈v0k〉mJ in Table 4 approach the P3, P4, . . . of Equation (21b) according to the relation:
〈∣J∣k〉mJJ≫k→[J(J+1)]k/2. However, 〈v0k〉mJ differ significantly from Pk for low J. The classical Pk in Equation (21b) lack the small terms (−2/3, −5/6, etc.) that kill the 〈v0k〉 in Table 4 whenever J falls below strict quantum limits such as whenever J < |m| or J < k/2. However, the quantum “killer” terms become negligible for larger J-values (J > k) and this makes tensor eigenvalues converge to Pk and thus to their RES plots.
2.1. Angular Momentum Cones and RES Paths
Quantum J-magnitude Equation (22b) introduces a quantum angular momentum cone geometry with quantized angles θmJ given by Equation (22a) as summarized here in Equation (23a) and (23b) for lab m = M and molecular body n = K.
cos θMJ=MJ(J+1)cos θKJ=KJ(J+1)
An angular momentum eigenstate
|Jm,n〉 has sharp (zero-uncertainty) eigenvalue m or n on the lab or body frame z or z̄ axis, respectively. This sharp altitude and magnitude in Equation (22b) constrains vector J to base circles of cones making half-angle θmJ or θnJ with z or z̄ axes, respectively. Expected J-values appear in Figure 1 at intersections of quantized J-cones with the RES as explained below.
RES energy level analysis begins by writing a multipole Tqk tensor expansion Equation (24a) of a general rigid rotor or asymmetric top Hamiltonian and then plotting the resulting surface using Equation (24a)
Inertial constants (A = 1/Iχ̄, B = 1/Iȳ, C = 1/Iz̄) combine J-tensor operators Tqk. Exact relation of 〈v00〉J and 〈v02〉J in Table 4 to classical P0 and P2 in Equation (21b) is used in Equation (24b) for T00 and T02.
A rigid spherical top (A = B = C) has only the T00 term Equation (24a). Rigid prolate (A = B > C) or oblate (A = B < C) symmetric tops have only T00 and T02 terms with energy eigenvalues below.
Since a rigid symmetric-top involves only T00 and T02, the θnJ -cones define its eigenvalues exactly by J-vector trajectories at angle-θnJ where θnJ -cones intersect the following T02 -RES shown in Figure 1.
Cone paths in Figure 1 are constant energy contours on symmetric top RES Equations (26a) and (26b) of constant J. They may be viewed as J-phase paths on which the J-vector may delocalize or “precess” on a circular θnJ -cone around body z̄-axis. Or else one might view paths on Figure 1 as coordinate space tracks of the lab z-axis around the z̄-axis by assuming J lies fixed on the former. Either view describes J in the body-frame by Euler polar and azimuth angles −β,−γ with angle β = θnJ and |J|2 = J(J + 1) fixed.
The difference between quantum solutions and semi-classical Pk solutions can be easily plotted as in Figure 2. The figure shows a slice of the semi-classical surface and the uncertainty cones for each m from J to −J. The orange circles indicate the intersection of the surface with the uncertainty cones and the blue circles indicate the energy of the quantum value, 〈v0k〉mJ, placed along the uncertainty cone. Figure 2(a) shows some divergence between quantum and semi-classical energies for 〈v04〉m4, while Figure 2(b) are exact for all J as in the J = 4 cases 〈v02〉m4 shown. As J and m are made larger than k then semi-classical values converge on exact eigenvalues as described below.
2.1.1. Reduced Matrix and RES Scaling
An RES is a radial plot along J direction −β, −γ that has hills where energy is high and valleys where it is low, but at all points the same magnitude
∣J∣=J(J+1). Origin-shift to keep RES radius positive and scaling to display hills, valleys, and saddles, may be needed to make useful RES plots.
A scalar term s · v00 added to a tensor combination T = ∑ktkv0k does not affect the T-eigenvectors and neither does an overall scaling of T to cT. This is true since eigenvectors are invariant to adding a multiple s1 of unit matrix 1 to T or multiplying it by c1. (Of course, eigenvalues would, respectively, be shifted by s or scaled by c.)
Wigner–Racah tensor algebra defines a reduced matrix element 〈J′||Tk||J〉 to serve as a scale factor for each Clebsch–Gordan tensor matrix element having Wigner–Eckart form Equation (13a).
〈J′m′|Tqk|Jm〉=Cqmm′kJJ′〈J′∣∣Tk∣∣J〉
This matrix
〈Jm|T02|Jm〉 of quadrupole-J-tensor T02 = J02 in Equation (24b) reveals some key points.
Reduced matrix element 〈J ||T2|| J〉 cancels norm factor
4[J]/2J+3:-1 in C0mm2JJ. The result is the quadratic Legendre form |J|2P2(m/|J|) found inside [. . . ]-braces of Table 4 with norm
4[k]/2J+3:-1 outside the braces. (The latter is just a norm in Equation (29a) and (29b) multiplied by the factor
[k]/[J] from definition Equation (15b).)
Apparent conflicts in factors are due to having sum-of-squared-component normalization of unit vk on one hand and sum-of-component normalization of Pk on the other. Matrix elements
〈Jm|T|Jm〉 or
|Jm〉〈Jm| use the former since amplitude squares give probability. However, it is unsquared amplitude sum ∑ck that measures anisotropy of a tensor T = ∑ck · Pk since ∑ck is a maximum T-amplitude. (Each Pk contribues Pk(0) = 1.) Expectation values
〈Jm|T|Jm〉 scale linearly, too, but J2 tensors may have extra scale factors.
Tensor T2 = J2 in Equation (29a) and (29b) scales as |J|2 = J(J + 1) and Tk = Jk scale as |J|k. Factor
4[J]/2J+3:-1 of C0mm2JJ reduces scale |J|2 to
2J+1=[J]. Then the reduced factor 〈J ||T|| J〉 brings it back to |J|2.
Each rank-k part has a factor |J|k = |J(J + 1)|k/2. Anisotropy of mixed-rank J-tensor T = ∑ck · Jk is ∑|J|kck, and thus is quite sensitive to quantum number J. So also are the RES and related eigensolutions of T.
2.2. Asymmetric Top and Rank-2 RES
Plotting RES of non-diagonal Hamiltonians for the asymmetric top Equation (24a) involves 2nd-rank tensors vq2 with reduced z-axial symmetry, nonzero q-values, and non-commuting Ja combinations. Each Ja in the quadratic expressions in Equation (24a) is replaced by its classical Euler-angle form in Equation (27).
Or else, each tensor Tqk in Equation (24a) is replaced by a multipole fucntion
Xqk=∣J∣kD0qk*(.,β,γ). (Recall Equation (24b)).
Forms Equations (31a) and (31b) of rank-(k = 2)-RES are equal and give the same plots shown in Figure 3, but tensor form Equation (31b) reveals symmetry. Terms X00 and X02 (Figure 1) are z-symmetric and non-zero near z-axis while X±22 terms are asymmetric and vary as sin2β with polar angle β between the J vector and the z-axis. As β approaches π/2, X±22 terms grow to give equatorial valleys and saddles in Figure 3 while X02 vanishes.
Asymmetric tensor operators T±qk are non-diagonal and do not commute with diagonal T0k or with each other, and so HAsymTop eigenstates as well as eigenvalues vary with coefficient (A − B) in Equation (31b). As T±22 mixes symmetric-top states
|JK〉 into asymmetric-top eigenstates, θKJ cone circles around the z-axis of Figure 1 warp into oval-pairs squeezed by nascent ovals emerging from the x-axis and bound by a pair of separatrix circle-planes that meet at an angle θsep on the y(orB)-axis. Figure 4 shows a range of RES and levels between symmetric-prolate top (B = A or θsep = 0) and oblate top (B = C or θsep = π). A most-asymmetric case (B = C or θsep = π/2) is midway between the symmetric cases.
As B first differs a little from A, off-diagonal T±22 and asymmetric X±22 first “quench” degenerate ±K-momentum eigenstate pairs
|J±K〉 into non-degenerate standing cos or sin-wave pairs.
∣cKJ〉=12(|J+K〉+|J-K〉)∣sKJ〉=-i2(|J+K〉-|J-K〉)
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 B differs more and more from A, off-diagonal T±22 will mix standing waves like |cKJ〉 with others such as|cK±2J〉, |cK±4J〉, and |cK±6J〉 that share the same HAsymTop symmetry described below.
2.3. Symmetry Labeling of Asymmetric Top Eigenstates
Throughout the range of asymmetric cases in Figure 4 the symmetry of HA Top in Equation (31a) and (31b) is at least orthorhombic group D2 of 180° rotations Rx, Ry, and Rz about inertial body axes that mutually commute (RxRy = Rz = RyRx), etc.). Unit square (Rx2 = 1, etc.) R-eigenvalues ±1 label nodal symmetry (+1) or antisymmetry (−1) on each axis. D2 is an outer product of cyclic C2 groups for two axes, say C2(x) and C2(y). x and y values also label nodal symmetry for the z axis since RxRy = Rz. A Cartesian 2-by-2 product of C2(x) and C2(y) symmetry character tables shown in Table 6 gives four sets of characters and four symmetry labels [A1,B1,A2,B2] for D2 = C2(x) ⊗ C2(y).
Labels (A,B) or (1, 2) for (x) or (y) symmetric and anti-symmetric states follow ancient arcane conventions. We prefer a binary (02, 12) notation for C2 characters and N-ary notation (0N, 1N, 2N, . . . , (N−1)N) for CN characters
DmN(Rp) where each label mN denotes “m-wave-quanta-modulo N” as in Table 7.
DmN(Rp)=e-im·p(2π/N)
This notation is used in correlation Table 8 between symmetry labels of D2 and its subgroups C2(x), C2(y), and C2(z), respectively. Each row belongs to a D2 species and indicates which C2 symmetry, even (02) or odd (12), correlates to it. The Table 8(a), 8(b) and 8(c) follow respectively from the columns Rx, Ry, and Rz of Table 6. An even (02) D2 character is
D02(R)=+1 and odd (12) is
D12(R)=-1.
J = 10 HAsymTop-levels in Figure 3 consist of two sets of five pairs [(A1,B1) (A2,B2) (A1,B1) (A2,B2) (A1,B1)] and [(B2,A1) (B1,A2) (B2,A1) (B1,A2) (B2,A1)] separated by a single (A2) level. Each is related to RES x-valley path pairs Kx ~ [±10,±9,±8,±7,±6] or z-hill pairs Kz ~ [±6, ±7, ±8, ±9, ±10] separated by a single y-path (A2: Ky ~ 5). Even-K belongs to a (02) column and odd-K belongs to a (12) column of C2(x) Table 8(a) or C2(z) Table 8(c).
Valley-pair sequence (A1,B1), (A2,B2) . . . is consistent with (02) and (12) columns of the C2(x) Table 8(a), and hill-pair sequence (B2,A1), (B1,A2) . . . is consistent with (02) and (12) column of the C2(z) Table 8(c). This is because lower pairs correspond to x-axial RES loops of approximate momentum Kx ~ ±10, ±9 · · · ± 6 while upper pairs correspond to z-axial RES loops of approximate momentum Kz ~ ±6,±7 · · · ± 10 in Figure 3
Table 8(b) for C2(y) is not used since ±y-axes are hyperbolic saddle points on one separatrix path, unlike the disconnected pairs of elliptic RES paths that encircle ±x-axes or ±z-axes at hill or valley points. Only a single E level exists in Figure 3 at the energy ESep of the saddle points and their separatrix.
As symmetricHSym becomes a more asymmetricHAsymTop in Figure 4, a hill or valley path bends away from its ideal single-K symmetric-top cone circle at constant polar angle θKJEquations (23a) and (23b). Each HSym state
|JK〉 turns into an HAsymTop eigenstate expansion of states
|JK±2p〉 with K ± 2p above and below K, and its RES path bends from constant θKJ toward polar angles θK±2J, θK±4J, θK±6J. . . above and below angle θKJ. At energy near the separatrix ESep, bending of hill and valley paths become more severe as they approach separatrix asymptotes where the polar angle range Equation (32) expands to 2θSep or π and the bend becomes a kink.
It is conventional to label HSep eigenstate |E〉 by both Kx and Kz quantum values since |E〉 may use either a Kx basis or else a Kz basis. However, Jx and Jz do not commute. For energy E above ESep, a |J,Kz〉 expansion is more compact and a dominate |Kz| value may label |E〉. For E below ESep, a |J,Kx〉 expansion has a meaningful |Kx| label. For E near ESep, K-labels are questionable.
Though a general form of the symmetry identification process may be unfamiliar, it may implemented by computer. Group projectors Equation (36) can distinguish how each eigenvector splits with respect to subgroup operations. The product of these projectors and the calculated eigenvectors identifies the subgroup symmetry of each level.
Pjkα=(lα°G)∑gDjkα*(g)g
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 Cn subgroup and
Djkα*(g) in Equation (36) is found by Equation (34).
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 J < 50. Spherical tops are quite challenging as seen in Section 7.
2.4. Tunneling between RES-Path States
N-atom inversion in ammonia, NH3, 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 Figure 3, shows level pairs such as (A1,B1), (A2,B2), etc. as rotational analogs of inversion doublets. Here the tunneling between left and right positions on a PEs is replaced by an RES inversion between left-handed and right-handed rotation of an entire molecule. Instead of oscillation of expected position values 〈r〉 between PES valleys there is oscillation of momentum 〈J〉 between pairs of x-paths (+Kx ↔ −Kx) in RES valleys or else between pairs of z-paths (+Kz ↔ −Kz) on RES hills. Section 7 describes this phenomenon in more detail for molecules of Oh and Td symmetry.
3. Tensor Eigensolutions for Octahedral Molecules
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.
3.1. Tensor Symmetry Considerations
Theory of asymmetric top spectra in Section 2 may be generalized to a semi-classical treatment of tensor operators for Td or Oh symmetric molecules such as CH4 or SF6. The results contain level clusterings that first appeared in computer studies by Lea, Leask, andWolf [20], Dorney and Watson [21] and followed by symmetry analyses [22,23] and others described below.
Up to fourth order, any such molecule may be treated using the Hecht Hamiltonian [24] rewritten in terms of tensor operators below in Equation (37a) and (37b) that isolates the rank-4 tensor term Equation (37c).
This is continued below to higher rank tensors with more complicated structure [14]. The coefficients of each tensor operator may be found from a spherical harmonic addition-theorem expansion of points at vertices of an octahedron. Coefficients cn,m are based on Equation (38), where Ymn is the spherical harmonic and f(⇉) is a position of octahehral vertices (100), (010), ..., (00 − 1).
cn,m=∫Vf(r→)Ymn·rndτ
A normalized sum of these coefficients gives the rank-6 Oh tensor as follows.
T[6]=122T06-74(T46+T-46)
The first study of RES and eigenvalue spectrum with varying rank-4 and rank-6 tensor operators [25] expressed in Equation (40) revealed intricate level cluster crossing shown below.
T[4,6]=cos(θ)T[4]+sin(θ)T[6]
Later studies [26] examined rank-8 contributions such as expressed by Equation (41). Effects peculiar to combining Hamiltonian terms of rank-8 and higher include extreme clusters.
T[4,6,8]=cos(φ)(cos(θ)T[4]+sin(θ)T[6])+sin(φ)T[8]
As with the asymmetric top Hamiltonian, the octahedral Hamiltonian uses non-axial operators shown in Equations (37b) and (39). Such operators involve more than Legendre functions, complicating purely semi-classical analysis. Thus, approximate solutions based on axial operators alone apply only asymptotically for high J > 10 and in regions away from RES seperatrices. Combined powers of J and Jz do not give all levels. Tensor operators provide a sparse banded Hamiltonian matrix, but full numerical diagonalization is needed to get all levels to high precision.
Gulacsi and coworkers [26] explored how eigenvalues vary with T[4] and T[8] for J ≤ 10 and small contributions of T[6]. Results below agree but extend to larger J and use RES topology.
While the asymmetric top systems show clustering related to symmetry subduction from D2 to a related C2 subgroup, octahedral molecular clusters relate to a variety of subgroups. Oh may cluster into D4, D3, D2, C4, C3, C2 or other subgroups involving reflection or inversion.
For simplicity, this discussion will focus on O rather than Oh molecules to make equations and correlation tables easier to display and interpret. Later Sections 6–7 give fuller discussions and explain the reciprocity relations that are behind correlation tables.
For D2-symmetric molecules, clustering patterns are described in terms of the correlation tables found in Table 8. The similar correlation tables for octahedral molecules are given for C4, C3 and C2 in Table 9. The columns of Table 9 represent the different clusters of rotational levels found within the spectra of given molecule at a given rotational transition. These clusterings are identified by their degeneracy as well as their RES location. Since symmetry labeling of octahedral group O differs form asymmetric top D2, a new coloring convention for O levels is defined: A1 is red, A2 is orange, E2 is green, T1 is dark blue and T2 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. Figure 5 shows two different RES plots, both globally octahedral, but with local regions that correspond to a subgroup symmetry of the octahedron. Figure 5(a) demonstrates a possible RES of an octahedral molecule with Hamiltonian parameters that allow for C4 and C3 local symmetry regions to be present. The C4 regions are identified by their location and by their square base. Similarly, the C3 regions are identified by their location and triangular base. In this case C3 symmetric regions are concave while C4 regions are convex. This is not required and is dependent on Hamiltonian fitting terms that change the relative contributions of T[4] and T[6]. Likewise, Figure 5(b) shows the C2 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 mJ levels, a rotationally-induced symmetry-reduced spherical top has several identical z axes. The mJ levels can then localize on a single symmetry-reduced local region. The number of these regions must equal the degeneracy of the cluster in that same region. This degeneracy, ℓα, is also found using the sum of the numbers in the columns of Table 9 or by Equation (42) given
°G is the order of the molecular symmetry group and ∘ℋ is the order of the subgroup.
ℓα=°G°H
In the cases shown here cluster degeneracy ℒ α becomes 6, 8 and 12 for C4, C3 and C2 respectively.
3.2. Numerical Assignment of Symmetry Clusters
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 T[4] as Equation (37a) it is possible to analytically [25] determine the symmetry of each level. Once T[6] or T[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 C4 symmetric projectors. The correlation table for O ⊃ C4, shown in Table 9, and Equation (36) give the information necessary for the assignment. Moreover, when using the subgroup C4 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 C4 projectors can be used to diagnose level symmetry for clusters in any subgroup region.
3.3. Octahedral Clustering vs. RES Topography3.3.1. Variation of <italic>T</italic><sup>[4,6]</sup> Topography
The Hecht Hamiltonian Equation (37a) and its higher order analogues are generic Hamiltonians. Such Hamiltonians have numerous fitting constants specific to a given molecule and a given vibronic species. To better understand all such octahedral systems, one must focus on changes in the level spectrum and RES plots with varying contributions of T[4], T[6] and T[8].
T[4,6,8] in Equation (41) has two bounded parameters θ and φ so several plots are required to explore this parameter space. By setting T[8] contributions to zero the eigenvalue spectrum for T[4,6] in Equation (40) can be plotted for changing values of θ, relative contributions of 4th and 6th rank tensor terms. Figure 6 plots such an eigenvalue spectrum and also places the RES plots that go along with important parts of the level diagram and, conversely, points out what spots on the level diagram correspond to important changes in the RES plot. We note T[4,6] RES have circular ring separatrices not unlike those on D2 RES in Figure 3.
To understand the behavior of the level diagram in Figure 6 it is critical to inspect the changing shape of the RES plots. In particular, the clustering of levels in the eigenvalue diagram is dependent on the localized symmetry regions of the RES at each value of θ. Locally, the RES forms hills and valleys of a lower symmetry than that of the molecule. The local symmetry must also be a sub-group of the molecular symmetry. Figure 5 identifies regions of local symmetry C4, C3 and C2 whose local rotation axis lie fixed normal to the RES at the center of each region, respectively, even as θ varies from Figure 5(a) to Figure 5(b). For some θ one or two of the Cn regions may shrink out of existence as shown below in Figure 6.
With this understanding of local subgroup regions it is possible to discuss more detail of Figure 6. The correspondence between the RES plots and the level diagram can also be seen by appending the eigenvalue spectrum in Figure 6 with the height of the C4, C3 and C2 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. Figure 7 shows the same quantum spectrum as Figure 6, but also includes the height (energy) of each symmetry axis. The outlines are printed in bold and are labeled for which Cn axis they each belong.
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 Equation (40) we will arrange it as
T[4,6]=(1-x)T[4]+xT[6]
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, x-line plots show by degree of avoided-crossing-curvature for each level the degree of its state mixing at x.
The three plots in Figure 8 show these spectra and semi-classical outlines for J = 30, J = 10 and J = 4. Figure 8(a) shows that the quantum levels fit for all values of x at J = 30, while Figure 8(b) shows some small disagreement near x = 2 for J = 10. Figure 8(c) shows that for low J there is strong disagreement between quantum calculations and semi-classical approximations.
Indeed, such plots as Figure 6 have been created before, both for the RES plots and the level diagrams [25]. Next, we show such an analysis of T[4,6,8] and demonstrate how such a Hamiltonian can show a different type of topology than previously reported.
3.3.3. Variation of <italic>T</italic><sup>[4,6,8]</sup> Topography and Level Clusters
The inclusion of eighth rank operators to the Hamiltonian dramatically changes the possible types of RES local symmetry and the related level clustering. While Figure 5 shows C4, C3 and C2 symmetric local structures for RES plots for T[4,6] Hamiltonians, Figure 9 shows a new kind of local T[4,6,8] RES path pointed out there with C1 symmetry. (That means no rotational symmetry!) The path is repeated 24 times and thus belongs to a single cluster of 24 levels. As shown in Section 6.7.2. the cluster spans an induced representation D01 (C1) ↑ O, also known as a regular representation of O.
Details of the two dimensional T[4,6,8] parameter space appear in a figure Table 10 containing RES plots for several (θ, φ) points. To be consistent with Equation (41), the plots increase θ from 0 to π going left to right and φ from 0 to π going top to bottom. RES O levels are colored with the usual red for A1, orange for A2, green for E2, blue for T1 and cyan for T2.
As expected from Equation (41), the top and bottom rows are opposites to one another. That is, where one RES has a hill (higher energy), the other has a crater (lower energy.) The RES at θ = 0, φ = 0 has convex C4 and concave C3 structure as does the RES at θ = π, φ = π, but opposite the shape of the RES at either θ = 0, φ = π or θ = π, φ = 0. The ordering of the levels is also opposite. These two extremal rows also have no eighth order contribution, so they produce simpler shapes than the others and are incapable of producing C1 local symmetry regions. The middle row shows a different behavior: all the diagrams are identical. Again, this follows from Equation (41) wherever φ = π/2.
While Table 10 shows only the RES plots along the parameter space defined by Equation (41), Figure 10 shows level diagrams with RES plots placed showing the symmetry and topology present at a given point in the space. The bold vertical lines next to the RES plots indicate the spot in the level diagram that particular RES plot would exist. Again, it is clear the θ = π/2 case would be unchanged, so it is not shown. The θ = π case is neglected as it is a mirror image of the θ = 0 case.
3.4. CriteriaforC<sub>1</sub> Level Clustering
Figures 9–11 show where the local regions of hills and valleys form on the RES depending on mixing angles φ and θ. Unlike the local symmetry regions known previously, the local C1 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 C1 local symmetries are shown in parts of second, third and fourth rows of Table 10 as well as parts of Figure 10(a) and 10(b). Figure 11 shows how C1 regions lie on hills or else valleys and how they can be arranged with their neighbors into either a square or triangular pattern.
C1 clusters require tensors of rank-8 with φ between π and zero as θ varies. Momentum J must be large enough for its minimum-uncertainty (J=K)-cone angle ΘJJ to fit in a C1 region.
ΘJJ=cos-1(JJ+1)≃1J
RES with J as low as J=4 may have C1 regions but fail to fit its (2Θ44=56°)-wide cones. C1 clusters begin to appear around J=20 (2Θ2020=26°) but even for J=30 (2Θ3030=21°) are still barely formed in Figure 11(a). There a minimum uncertainty cone appears to barely fit within a separatrix on a C1 hill between C3 and C4 valleys of its (J=30)-RES. Others are situated more comfortably in valleys of RES shown in Figure 11(b) where they appear to encircle a C4 axes as in Figure 10(a) where a corresponding cluster of 24 eigenvalues in 10 levels appear at the lower left hand side of the level diagram. In Figure 11(c) they surround a C2 axis.
With higher J and Oh-tensors of greater rank than k=8, one expects clusters of 48-fold degeneracy corresponding to C1-regions centered away from Oh symmetry axes or planes. So far, these are only beginning to be explored and analyzed. To analyze such complicated tunneling effects (and better understand older ones) requires an improved symmetry analysis developed in Sections 4–7.
4. Introducing Dual Symmetry Algebra for Tunneling and Superfine Structure
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 superfine structure due to J-tunneling that is the focus of the following sections.
Fortunately, the presence of symmetry in a physical system allows algebraic or group theoretical analysis of quantum eigensolutions and their dynamics. Groups of operators (g, g′, g″, ...) leave a Hamiltonian operator H invariant (g†Hg = H) if and only if each g commutes with it (gH = Hg). Then each g in the group shares a set of eigenfunctions with H. However, if (g′) and (g) do not commute then the (g′) and (g) sets will differ.
Hamiltonians may themselves be symmetry operators or linear expansions thereof. Multipole tensor expansions used heretofore are examples. Expanding H into operators with symmetry properties, such as (a†a) or (Tqk), helps to analyze its eigensolutions since, in some sense, a symmetry algebra “knows” its spectral resolutions. The underlying isometry of a system’s variables and states contains all the sub-algebras that are possible H-symmetries.
If H-symmetry operators (g, g′, ...) also commute with each other (gg′ = g′g, etc.) then all g share with H a single set of eigenvectors as discussed in Section 5. Such commutative or Abelian symmetry analysis is just a Fourier analysis where all H are linear expansion of its symmetry elements (g, g′, g″, ...) and simultaneously diagonalized with H. Such g expansions define both Hamiltonians H and their states as described in Section 5.
However, non-commuting (non-Abelian) symmetry operators (g, g′, g″, ...) of H cannot both expand H and commute with H. This impasse is resolved in Section 6 by using a dual local operator group (ḡ, ḡ′, ḡ″, ...) that mutually commutes with the original global group. Then local (ḡ) expand any H that commutes with global g, while the global g define base states and their combinations define symmetry projected states. Roughly put, one labels location while the other labels tunneling to and fro.
In Section 6, the dual group (D̄3 ~ C̄3v) of the smallest non-Abelian group (D3 ~ C3v) is defined. Dual symmetry-analysis is demonstrated for a trigonal tunneling system by group parametrization of all possible (D3)-symmetric H matrices and all possible eigensolutions for each. The example shows how global (g) label states while the local (ḡ) label tunneling paths. In this way symmetry labels processes as well as states. An added benefit is a kind of “slide-rule-lattice” to compute group products.
In Section 7, the local symmetry expansion is applied to octahedral O⊂Oh tensor superfine structure. Local symmetry conditions are used to relate tunneling paths to RES topography discussed previously and predict possible energy level patterns. The O⊂Oh slide-rule-lattices appear in Figures 22–24.
5. Abelian Symmetry Analysis
An introductory analysis of tunneling symmetry begins with elementary cases involving homocyclic Cn symmetry of n-fold polygonal structure. But, it applies to all Abelian (mutually commuting) groups A since all A reduce to outer products Cp × Cq × · · · of cyclic groups of prime order.
5.1. Operator Expansion of C<sub>n</sub> Symmetric Hamiltonian
The analysis described here and in Section 6 deviates from standard procedure [27–31]. Instead of beginning with a given quantum Hamiltonian H-matrix, we start with a Cn symmetry matrix (r) and build all possible Cn symmetric (H)-matrices by combining n powers (rp) = (r)p of (r) ranging from identity r0 = 1 = rn to inverse rn−1 = r−1[32].
In Equation (45) the rotation r is by angle 2π/n so rotation rn is by angle n2π/n = 2π, that is, the identity operator r0 = 1 = rn. Thus power-p indices label modulo-n or base-n algebras. If n=2, it is a Boolean algebra C1 ⊂ C2 of parity [+1,−1] or classical bits [0,1]. (U(2) spin-algebras of q-bits have 4π identity but are not considered here.)
Sum rule:p+p′=(p+p′)mod(n)Product rule:p·p′=(p·p′)mod (n)
We construct the general H-matrix using Cn group-product tables shown below in a g−1g-form and a g†g-form that is equivalent for unitary operators g† = g−1. In each table the kth-row label g−1 matches kth-column label g so that the identity operator 1 = g−1g resides only on the diagonal. This example is for hexagonal symmetry C6 for which r−6 = r0 = 1 = r6 = r6†, r−5 = r1 = r5†, r−4 = r2 = r4†, r−3 = r3 = r3†, and so forth.
The g†g-form produces a regular representation R(g) = (g) of each operator g as shown below. Each R(rp) is a zero-matrix with a 1 inserted wherever a rp appears in the g†g-table.
Matrices in Equation (49) are simply group tables Equation (47) with complex tunneling amplitude rp replacing operator rp. Parameters r0 = (r0)* and r3 = (r3)* match self-conjugate binary subgroups C1 ⊂ C2 = (1, r3) related by 1 = (r3)2. Both are real if matrix (H) is Hermitian self-conjugate (Hab = Hba*).
Three distinct classes of tunneling or coupling parameters are depicted in Figure 12 using classical spring-mass analogs for quantum systems [22]. Tunneling matrices have a long history [33] going back to Wilson [34]. Here this is being revived to treat extreme J-tunneling and more recently by Ortigoso [17] and Hougen [35,36] to treat extremely floppy molecule dynamics. Both these tasks use tunneling parametrization that has so far been quite ad.hoc. To accomplish either of these tasks, or what will surely be needed, namely both tasks, we need a tighter symmetry analysis. The group operator scheme being introduced here seeks a way to achieve this.
The 1st-neighbor class has non-zero parameters r1=−r and conjugate r−1=−r*=−r̄ coupling only nearest neighbors each with self-energy r0=H1. The 2nd-neighbor class has non-zero parameters r2=−s and conjugate r−2=−s*=−s̄ coupling only next-nearest neighbors with self-energy r0=H2. Finally, 3rd-neighbor coupling r3=−t=−t* is real as required for binary self-conjugacy r3=(r3)†.
5.2. Spectral Resolution of C<sub>n</sub> Symmetry Operators
Eigenvalues χpm of each operator rp are mth multiples of nth-roots of unity since all Cn symmetry operators g = rp satisfy gn = 1 and are characters of Cn symmetry operators. Magnetic or mode-wavenumber indices m label a base-n algebra as do the power or position-point indices p in Equation (46). Spatial lattice points xp = L·p(meters) are indexed by p while reciprocal-(k)-wavevector space km = 2πm/L(per meter) is indexed by integer m.
〈rp〉m=χpm=e-i(m·p)2π/n=e-ikmxp=D(m)(rp)
The χpm are Cn irreducible representations D(m)(rp) as well as Cn characters. General group characters are traces (diagonal sums) of D-matrices (χ(m)(g) = traceD(m)(g)). Abelian group irreducible representations are 1-dimensional due to their commutativity, and so for them characters and representations are identical. (χ(m)(g) = D(m)(g)) All this is generalized in subsequent Section 6. Any number of mutually commuting unitary matrices may be diagonalized by a single unitary transformation matrix. The characters in Equation (50) form a unitary transformation matrix Tm,p that diagonalizes each Cn matrix (rp).
Tm,p=χpm/n
This T is a discrete (n-by-n) Fourier transformation. A 6-by-6 example that diagonalizes all matrices in Equations (48) and (49) and in Figure 12 is shown in Figure 13 by a character table of wave phasors based on D(m)(rp) in Equation (50) or (51). The irreducible representations D(m)(rp) or irreps play multiple roles. They are variously eigenvalues, eigenvectors, eigenfunctions, transformation components, and Fourier components of dispersion relations. This hyper-utility centers on their role as coefficients in spectral resolution of operators rp into idempotent projection operators P(m). P(m) are like irrep placeholders.
Equation (52) is column-p of Figure 13. Column-0 is a completeness or identity resolution relation.
r0=∑m=0n-1P(m)=1=P(0)+P(1)+P(2)+P(3)+P(4)+P(5)
Dirac notation for P(m) is |(m)〉〈(m)|. Its representation in its own basis (eigenbasis) is simply a zero matrix with a single 1 at the (m,m)-diagonal component. P(m)-product table in Equation (54) is equivalent through Equation (52) to g-product table in Equation (47) but the P(m)-table given below has an orthogonal (e.g.P(1)P(2) = 0) idempotent (e.g.P(1)P(1) = P(1)) form.
The location of each P(m) in the P-table is a location of a 1 in its representation as indicated in the right hand side of Equation (54) in the same way that locations in g-table Equation (47) place 1’s in representations Equation (48). However, idempotent self-conjugacy (P† = P) makes row labels of P-table Equation (54) identical to its column labels, whereas only g = 1 and g = r3 are self-conjugate in g-table Equation (47).
Character arrays such as Figure 13 represent operator eigen-products between rp and P(m).
rpP(m)=χpmP(m)=P(m)rp
Also character χpm is the scalar product overlap of position state bra or ket with momentum ket or bra.
Position bra:〈xp∣=〈p∣=〈0∣r-pPosition ket:∣xp〉=∣p〉=rp∣0〉Momentum bra:〈km∣=〈(m)∣=〈0∣P(m)nMomentum ket:∣km〉=∣(m)〉=P(m)∣0〉n
Momentum eigenwave ψkm (xp) is character Equation (50) conjugated to eikmxp and normalized by
n.
The same overlap results whether rp moves a (m)-wave p-points forward or moves the coordinate grid p-points backward. This Cn relativity-duality principle generalizes to non-Abelian symmetry and is key to operator labeling of coordinates, base states, Hamiltonians, and their eigensolutions.
P(m) projects m-states with conjugate characters φpm= (χpm)* with factor 1/n so P(m)’s are idempotent and sum to 1. (∑pP(m) = 1) But, |km〉 has φpm with factor
1/n to be orthonormal so its squares sum to 1. (∑p|〈xp|km〉|2 = 1) Thus projection Equation (56b) of |km〉 by P(m) has a factor
n. Inverse spectral resolution Equation (52) sums over column points p using φpm from each row-(m) of Figure 13. Factor 1/n makes P(m) complete (∑mP(m) = 1 in Equation (53)) and idempotent (P(m)P(m) = P(m)) in Equation (54)).
The (0)-momentum or scalar state is a sum over the (m)=(0)-row of Figure 13 normalized by
1/n.
P(0)∣0〉n=(∑p=0n-1∣p〉)/n=(∣0〉+∣1〉+∣2〉+∣3〉+∣4〉+∣5〉)/65.3. Spectral Resolution of C<sub>n</sub> Symmetric Hamiltonian
Given Hamiltonian H expansion in Equation (45) in operators rp and the spectral resolution in Equation (52) of rp, there follows the desired spectral resolution of H. The eigenvalue coefficients ω(m) of P(m) define the dispersion function ω(km) of H in Figure 14(a) where it is conventional to center scalar origin (m)=(0).
Positive km-axis C6 array [...(0), (1), (2), (3), (4), (5), ...] of Equation (54) shifts to a zone-center-array mod-6: [...(4), (5), (0), (1), (2), (3), ...]=[...(−2), (−1), (0), (1), (2), (3), ...] using Equation (46).
Examples of dispersion relations for three classes of tunneling paths in Figure 12 are shown in Figure 14. Dispersion ω(km) for C6 symmetry depends sensitively on the Hamiltonian tunneling amplitudes rp for −3 < p ≤ 3 (or 0 ≤ p < 6) in Equation (49), and for any set of eigenvalues ω(km) there is a unique set of rp found by inverting Equation (63).
rp=∑m=0n-1φpmω(m)/nwhere:φpm=(χpm)*
A common tunneling spectral model is the elementary Bloch 1st-neighbor B1(6)-model shown in Figure 14a, much like that developed in Reference [33]. For negative values of r1=−r, a B1(6) spectra for C6 consist of six points on a single inverted cosine-wave curve centered at m=0 with its maxima at the Brillouin-band boundaries (m)=±3. This curve applies to B1(n) spectra for Cn where n equally spaced (m) points lie on the dispersion curve between m=±n/2 for even-n. The n energy eigenvalues ω(m) are projections of an n-polygon. For n=6 that is the hexagon shown in Figure 14a projecting two doublet levels ω(±1) and ω(±2) between singlet ω(0) and singlet ω(3) at lowest and highest hexagonal vertices as follows from Equation (63).
ωB1(n)(km)=r0χ0m+r1χ1m+r-1χ-1m=H1-2rcos(2πm/6)
The 2nd-neighbor B2-model (Figure 14b) has a two-cosine-wave dispersion curve. An equilateral triangle projects energy doublet levels [ω(0), ω(3)] from its lowest vertex and a quartet [ω(±1), ω(±2)] from its upper vertices.
ωB2(n)(km)=r0χ0m+r2χ2m+r-2χ-2m=H2-2scos(4πm/6)
The 3rd-neighbor B3-model (Figure 14c) has a three-cosine-wave dispersion, which for n=6 and r3=−t separates levels into an even-m triplet [ω(0), ω(±2)] below an odd-m triplet [ω(3), ω(±1)].
ωB3(n)(km)=r0χ0m+r3χ3m+r-3χ-3m=H3-2t(-1)m)
Combining of kth-neighbor rk-terms gives dispersion ω(m) as a k-term Fourier cosine series that is, for real rk, a sum of the preceding three Equations (65)–(67). However, real rk imply symmetry that is higher than C6, namely non-Abelian reflection-rotation symmetry such as C6v or D6h and a corresponding degeneracy between ω(±m) levels that will be treated shortly. Simple C6 symmetry allows six real parameters with complex r1 and r2. Then Equation (63) implies six levels that are generally non-degenerate as shown in Figure 15. Complex r1 = |r|eiφ of a ZB1 model describes chiral magnetic or rotational effects that include Zeeman-like splitting of m-doublets. The projecting hexagon tilts by the “gauge” phase angle φ = π/12 as the ZB1(6) dispersion ω(m) shifts. Then m doublets (±1) and (±2) suffer splittings that are 1st-order in φ while singlets (0) and (3) undergo shifts that are 2nd-order in φ.
6. Non-Abelian Symmetry Analysis
Characterization and spectral resolution in Equation (63) of a Hamiltonian HBk(6) uses its expansion in Equation (45) in Abelian group C6. Similar spectral resolution of a Hamiltonian H by a non-Abelian group G = [...g1, g2...] of non-commuting symmetry operators might seem impossible. To be symmetry operators of H, elements g1 and g2 must commute with H, but that cannot be if H is a linear expansion of them like Equation (45). The impasse is broken by introducing operator relativity-duality detailed below. A D3-symmetric tunneling H with a 3-well potential sketched in Figure 16 is used as an example.
6.1. Operator Expansion of D<sub>3</sub> Symmetric Hamiltonian
The simplest non-Abelian group is the rotational symmetry D3 = [1, r1, r2, i1, i2, i3] of an equilateral triangle. D3 is used to show how to generalize C6 operator analysis of the preceding section to any symmetry group. The D3 analysis begins with a g†g-form of group product table like Equation (47) for C6. However, D3 also requires a gg†-form giving the same product rules but using inverse g† ordering |..r2, r1, ...|=|..r1†, r2†, ...| along the top instead of down the left side as is done for the g†g-form of table. (The two ±120° rotations r1 and r2 are the only pair (r1†=r2) to be switched by conjugation). The three ±180° rotations are each self-conjugate (ip†=ip) as is (always) the identity 1†=1.
Over-bar notation is used for dual-group D̄3 = [1̄, r̄1, r̄2, ī1, ī2, ī2] of “body”-based operators isomorphic to “lab”-based group.
Matrix representations Equation (69a) for D3 or matrices Equation (69b) for D̄3 are given, respectively, by g†g or gg†-forms Equation (68) just as g†g form Equation (47) for C6 gives matrices in Equation (48).
Most pairs of resulting D3 matrices in Equation (69a) do not commute. (For example (r1)(i1)=(i3) does not equal (i1)(r1)=(i2).) Identical non-commutative product rules apply to the dual bar group D̄3 matrices in Equation (69b). However, all matrices of the latter D̄3 commute with all matrices of the former D3. This suggests that the Hamiltonian matrix, in order to commute with its symmetry group D3, is constructed by linear combination of bar group operators of D̄3[32].
H=r01¯+r1r¯1+r2r¯2+i1i¯1+i2i¯2+i3i¯3
D3 symmetric (H) matrix Equation (71) generalizes C6 symmetric (H) matrix Equation (49) to a non-Abelian case.
(H)=∑g=1°Grg(g¯)=(r0r2r1i1i2i3r1r0r2i3i1i2r2r1r0i2i3i1iii3i2r0r1r2i2i1i3r2r0r1i3i2i1r1r2r0)6.2. Spectral Resolution of D<sub>3</sub> Symmetry Operators
Spectral resolution of D3 or any non-Abelian group G = [...g1, g2...] entails more than the C6 expansion into a unique combination of idempotent operators Pα=|α〉〈α| multiplied by eigenvalue D(α)(g) coefficients as in Equation (52). It is not possible to diagonalize two non-commuting g1 and g2 in one basis since numbers (eigenvalues) always commute. If g1 and g2 do not commute, their collective resolution must include eigen-matrix coefficients Dm,nα involving nilpotent (N2 = 0) operators Pm,nα=|mα〉〈nα| as well as idempotent (I2 = I) operators Pm,mα=| mα 〉〈mα| seen in Equation (52).
Unlike a commutative algebra of Cn idempotents, which are shown in Equation (54) and uniquely defined by Equation (59), a non-Abelian algebra yields a panopoly of equivalent choices of P operators that resolve it. The number and types of these P’s is uniquely determined by size and structure of certain key commuting sub-algebras. The key to symmetry analysis of quantum physics is to first sort out the operators and algebras that commute from those that do not. It amounts to a kind of symmetry analysis of symmetry and leads to a far greater diversity than is found in commutative Abelian systems.
6.2.1. Sorting Commuting Subalgebras: Rank and Commuting Observables
The rank ρ(G) of a G-algebra is the maximal number of mutually-commuting operators available by linearly combining the oG operators gk of symmetry group G. ρ(G) is also the greatest number of orthogonal idempotents Pm that can resolve the G-identity 1 as in Equation (53). (oG is total number or order of G. Here oD3 and oC6 both equal 6.)
C6 rank is obviously equal to its order (ρ(C6) = 6), but the rank of D3 turns out to be only four (ρ(D3) = 4). As shown below, D3 can have no more than four P-operators that mutually commute though there exist quite different sets of them. On the other hand, D3 has just three linearly independent Pα-operators that commute with all of D3, and there is but one invariant set of them just as there is but one set of P(m) for C6 in Equation (59).
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-invariant labeling operators IG that commute with allg and not just with other labeling operators. IG are uniquely defined within their group G and invariant to all g. (gIGg−1=IG) For example, total angular momentum J2 and e-values J(J + 1) are R(3)-invariant.
Next in importance are labeling operators [IHn−1, IHn−2, ...,IH1 ] belonging to nested subgroups of G=Hn in a subgroup chain G⊃Hn−1⊃Hn−2⊃... H1. Multiple choices of chains exists since each subgroup link Hk is not uniquely determined by the Hk+1 that contains it, but each IHk is invariant to all possible Hj≤k at level-k or below.
For example, the z-axial momentum Jz and its e-values mz belong to a 2nd link in chain-R(3)⊃R(2z)⊃C6(z). Given R(3) there are an infinity of R(2) subgroups besides the one for z-axis of quantization. Jx or Jy are just two of an infinite number of possible alternatives to Jz. Each R(2ζ) has an infinite number of cyclic Cn(ζ) sub-subgroups.
6.2.2. Sorting Commuting Subalgebras: Centrum and Class Invariants
The centrum κ(G) of a G-algebra is the number of all-commuting operators available by combining gk. It is also the number of G-invariantPα-operators. Students of group theory know κ(G) as the number of equivalence classes of group G. D3 elements in Figure 16 are separated into three classes of elements [1], [r1,r2], and [i1,i2,i3]. (κ(D3) = 3)
Elements in each class are related through transformation g1=gtg2gt−1 by gt in group G. Sum κk of ock elements in gk’s class is invariant to gt transformation. (It only permutes gk-terms in κk thus κk commutes with all gt in G.)
The product table for D3 class algebra [κ1 = 1,κ2 = r1 + r2, κ3 = i1 + i2 + i3] in Equation (73) below follows by inspecting D3 group product tables in Figure 16 or Equation (68). It is a commutative algebra since each κj commutes with each κk as well as with each gt. This guarantees a class algebra has a unique and invariant spectral resolution.
The first sum in Equation (72) is over the ock elements in gk’s class. (ock is order of κk.) The second sum is over all oG group elements. The number of elements gt that commute with gk is osk, the order of gk’s self-symmetry sk. Each group operator gk has a self-symmetry group consisting of (at least) the identity 1 and powers (gk)p of itself. The order of class-k is the (integer) fraction ock=oG/osk.
6.2.3. Resolving All-commuting Class Subalgebra: Centrum=<italic>κ</italic>(<italic>D</italic><sub>3</sub>) = 3
Spectral resolution gives class-sum operators κ1, κ2, and κ3 as combinations of three D3-invariant Pα-operators with each of the κk eigenvalues as coefficients. The κ3 characteristic equation found by Equation (73) gives three Pα directly.
Traces of D3 matrices (gk) in Equation (69a) are zero excepting Trace(1) = 6. Traces of (Pα) then follow.
tracePA1=1,tracePA2=1,tracePE=4
This means (PA1) and (PA2) are each 1-by-1 projectors while (PE) splits into two 2-by-2 projectors. The latter splitting is not uniquely defined until subgroup chain D3⊃C3 or a particular D3⊃C2 chain is chosen, but relations in Equation (75) are invariant and unique. The κk coefficients inside parentheses of Pα expansion give the D3character table for traces of irreducible representations (irreps). Irrep dimension ℓα is trace of the αth-irrep of identity g1 = 1.
Completing resolution of D3 uses a product of two completeness relations, the resolution of class identity κ1 = 1 in Equation (75) with the identity resolution of a D3 subgroup C3 = [1,r1,r2] or else C2 = [1,i3]. In either case invariant PE splits but PA1 and PA2 do not. In Equation (78)PE is split by C2 into plane-polarizing projectors Px,xE + Py,yE = P0202E + P1212E.
[D3(classalgebracompleteness)1=PA1+PA2+PE]·[C2(subgroupcompleteness)1=P02+P12]=[D3(groupcompleteness)1=P0202A1+P1212A2+P0202E+P1212E]where:PA1=P0202A1=(1+r1+r2+i1+i2+i3)/6=PA1P02PA2=P1212A2=(1+r1+r2-i1-i2-i3)/6=PA2P12Px,xE=P0202E=(21-r1-r2-i1-i2+2i3)/6=PEP02Py,yE=P1212E=(21-r1-r2+i1+i2-2i3)/6=PEP12(All other PαPm2=0)
In Equation (79)PE is split by C3 into Right and Left circular-polarized projectors PR,RE +PL,LE =P1313E+P2323E.
[D3(classalgebracompleteness)1=PA1+PA2+PE]·[C3(subgroupcompleteness)1=P03+P13+P23]=[D3(groupcompleteness)1=P0303A1+P0303A2+P1313E+P2323E]where:PA1=P0303A1=(1+r1+r2+i1+i2+i3)/6=PA1P03PA2=P0303A2=(1+r1+r2-i1-i2-i3)/6=PA2P03PR,RE=P1313E=(1+ɛr1+ɛ*r2)/3=PEP13PL,LE=P2323E=(1+ɛ*r1+ɛr2)/3=PEP23(All other PαPm3=0)ɛ=ei2π/3
In Equations (78) and (79), neither PA1 nor PA2 split or change except to acquire some C2 or C3 labels. The total number (four) of irreducible idempotents after either complete splitting is the same group rank noted before: ρ(D3)=4. But, the RL-circularly polarized pairs PR,RE and PL,LE split-out by C3=[1,r1,r2] differ from the linear xy-polarized pairs Px,xE and Py,yE split-out by C2=[1,i3]. Px,xE and Py,yE are, respectively, parallel (symmetric i3PxE =+PxE) and anti-parallel (anti-symmetric i3PyE =−PyE) to x-axial 180o rotation i3 in Figure 16 and will be used in examples.
6.2.5. Final Resolutions of Non-Commuting Algebra: <italic><sup>o</sup></italic>(<italic>D</italic><sub>3</sub>) = 6
Mutually commuting algebras resolve into (I2 = I) operators
Pm,mα=∣αm〉〈αm∣ that sum to identity operator 1. They are split using the “one-equals-one-times-one” (1=1·1) trick in Equations (78) and (79).
Non-commuting algebras resolve into idempotents and nilpotent (N2 = 0) operators
Pm,nα=∣αm〉〈αn∣ that are split out using the following “operator-equals-one-times-operator-times-one” (g=1·g·1) trick. It is only necessary that 1 be resolved into rank-number ρ of irreducible idempotents as in Equation (78) or (79). (Here ρ(D3) = 4.)
The product in Equation (80) could have sixteen terms, but only six survive due to idempotent orthogonality Pj,jαPk,kβ = δα,βδj,kPj,jα, and the fact that both PA1 and PA2 remain invariant and commute with all Pj,jα and all g.
This reduces to a non-Abelian spectral resolution of D3 that generalizes resolution Equation (52) of Abelian C6 and includes two nilpotent projectors Pj,kα multiplied by off-diagonal irrep matrix components Dj,kα as well as the four idempotents Pj,jα with their diagonal irrep matrix coefficients Dj,jα that are not altogether unlike the D(m)(rp)P(m) terms in Equation (52). ( Now X has matrix indices (Xj,k).)
Terms (1/n)D(m)*(rp)rp in Equation (59) of P(m) of Cn in Equation (52) generalize here to Pj,kα and invert Equation (82a) to Equation (83).
Pj,kα=(ℓα/°G)∑g=1°GDj,kα*(g)g
D3 resolution in Equation (82b) has two irreps DA1 and DA2 of dimension ℓA1=1=ℓA2 and a third irrep DE of dimension ℓE=2 as noted in the first column of the character array in Equation (77). The irrep dimensions are related to the centrum κ(D3)=3, rank ρ(D3)=4, and order oD3=6. The following power sums of ℓα apply to any finite group G.
G-centrum:κ(G)=∑irrep(α)(ℓα)0=Numberofclasses,invariants,orirreptypesG-rank:ρ(G)=∑irrep(α)(ℓα)1=NumberofmutuallycommutingobservablesG-order:°(G)=∑irrep(α)(ℓα)2=Numberofsymmetryoperators6.3. Spectral Resolution of Dual Groups D<sub>3</sub> and D̄<sub>3</sub>
Spectral resolution shown in Equations (82a) and (83) of non-Abelian group G reduce g·h-product tables in Equation (68) to P-projector algebra.
PjkαPj′k′β=δαβδkj′Pjk′α
Product tables in Equation (86) for D3 projectors Pjkα generalize the C6 idempotent table in Equation (54). Non-commutativity entails a pair of tables like the g†g form and gg†-forms in Equation (68) for “lab” g and “body” ḡ operators. Tables in Equation (68) differ by switching conjugate pair r1 and r2 on side and top.(r1† = r2) The rest are self conjugate. (i1†=i1, etc.) Similarly, tables in Equation (86) differ by switching conjugate nilpotent pair PxyE and PyxE. (PxyE† =PyxE) The rest are self-conjugate. (Pjjα† =Pjjα)
The p†p and pp† tables in Equation (86) give commuting representations of projector Pjkα just as g†g and gg† tables in Equation (68) give commuting (g)G-matrices in Equation (69a). Wherever Pjkα appears in a table, a “1” is put in its (p)-matrix. Putting “Djkα (g)” at each Pjkα spot instead gives the following p†p-representation (g)P of g since it is a sum of Djkα (g)Pjkα in Equation (82a).
Conjugate pp†-representation (ḡ)P of ḡ has complex conjugate “Djkα* (g)” put at each Pjkα spot. The matrices in Equations (87) and (88) are transformations (g)P = T(g)GT† and (ḡ)P = T(ḡ)GT† of the respective matrices in Equations (69a) and (69b) by transformation T composed of Djkα (g) components. The C6 analogy is Fourier transform Equation (51) from Equation (48) to Equation (54).
Matrices ...(r̄2)P, (ī1)P, ... defined by Equation (88) commute with every ...(r2)P, (i1)P, ... defined by Equation (87) while each represents identical non-commutative D3 product tables in Equation (68). Both use real [x, y]-based i3-diagonal irreps Djkα (g) given below.
Appendix-A describes elementary derivation and visualization of Djkα (g) and their projectors Pjkα (g).
6.4. Spectral Resolution of D<sub>3</sub> Hamiltonian
Hamiltonian H-matrix in Equation (71) has six parameters [r0, r1, r2, i1, i2, i3] or coefficients of its expansion Equation (70) in terms of intrinsic D̄3 operators [1 = r̄0,r̄1,r̄2,ī1,ī2,ī3]. The parameters are indicated in Figure 17 by tunneling paths between the first D3 base state |1〉 and other D3-defined base states |g〉 = g|1〉 representing potential minima.
The resolution of H-matrix then follows that of ḡ and (ḡ)P -matrices. Any reduction of all (ḡ)P -matrices, such as the [x, y]-reduction in Equation (88), also reduces the (H)P -matrix accordingly. Row-1 of (HP) in Equation (71) has all six parameters.
Habα=∑g=1°G〈1∣H∣g〉Dabα*(g)=∑g=1°GrgDabα*(g)
If the P-nilpotent pair are switched to ... PxyE, PyxE.., then (H)P and all (ḡ)P (instead of all (g)P as in Equation (87)) are diagonal with eigenvalues HA1 and HA2 or block-diagonal with a pair of identical 2-by-2 HE-blocks.
Irrep-dimension ℓE = 2 implies (at least) 2-fold degenerate E-level since eigenvalues of identical HE-blocks must also be identical, but only certain parameter values give diagonal HE-blocks in Equation (92), i.e., real r1 = r2* and equal i1 = i2.
These are the values that respect the local D3 ⊃ C2[1,i3] subgroup chain symmetry that gave (x, y)-plane polarized splitting in Equation (78). This is broken by a complex r1 or by unequal i1 and i2. Complex r1 = |r|eiφ gives rise to complex rotating-wave eigenstates similar to ones in Figure 15 but, unlike that ZB1 model, cannot split E-degeneracy. Unequal i1 and i2 shift standing-wave nodes but cannot split E-doublets either. E-levels may split if H contains external or lab-based operators g in addition to its internal or body-based ḡ, but it thereby loses its D3 symmetry.
6.5. Global-Lab-Relative G versus Local-Body-Relative Ḡ Base State Definition
Non-Abelian symmetry analysis in general, and the present example of D3 resolution in particular, involves a dual-group relativity between an extrinsic or global “lab-based” group G=D3 on one hand, and an intrinsic or local “body-based” group Ḡ = D̄3 on the other hand. Each ḡ in Ḡ commutes with each g in G.
In the present example, the global “lab-based” group G=D3=[1,r1,r2,i1,i2,i3] 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 local “bod-based” group Ḡ=D̄3=[1,r̄1,r̄2,ī1,ī2,ī3] 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 relative position as reflected in the following equivalent definitions of base kets for waves in a D3 potential of Figure 16 with six localized wave bases [|1〉, |r1〉, |r2〉, |i1〉, |i2〉, |i3〉] in Figure 17. (We call this the “Mock-Mach Principle” of wave relativity.)
∣gk〉=gk∣1〉=g¯k-1∣1〉
Key to this definition is the independence and mutual commutation of dual sets Equation (69a) and (69b).
gjg¯k=g¯kgj
Neither relation makes sense if we were to equate gk with ḡk−1. The effect of gk is equal to that of ḡk−1only when acting on the origin-state |1〉. The action of global i2 in Figure 18a is compared with localī2 in Figure 18b that gives the same relative position of wave and wells. In Figure 18c product ī1ī2 = r̄ has the same action as i2i1=r−1=r2 on |1〉.
Different points of view show how “body” ḡ operations relate to the “lab” g. Starting from state |1〉, r̄1 = r̄ rotates lab potential clockwise (−120o) in a view where the body “stays put”. The body wave ends up in the same well as it would if, instead, the body rotates counter-clockwise (+120o) by r=r1 in a lab frame that “stays put.”
In a lab view, effects of body operation ḡk and lab operation gk−1 on |1〉 are the same except that ḡk−1 also moves each body operation ḡj in the same way to ḡkḡjḡk−1. The lab view of a lab operation gk does not see any of lab gj axes change location. The following generalization of lab-body relativity relation Equation (94) using Equation (95) shows how ḡj affects arbitrary |gk〉.
g¯j-1∣gk〉=g¯j-1gk∣1〉=gkg¯j-1∣1〉=gkgj∣1〉=gkgjgk-1gk∣1〉=gkgjgk-1∣gk〉6.6. Global versus Local Eigenstate Symmetry
Applying projector Pjkα in Equation (83) to origin ket |1〉 gives a local-global symmetry-defined ket |jkα 〉.
∣jkα〉=Pjkα∣1〉°G/ℓα=ℓα/°G∑g=1°GDj,kα*(g)∣g〉
The norm-factor N=oG/ℓα is a non-Abelian generalization of the integral norm N for Abelian CN eigenket projection in Equation (61). Interestingly, the non-Abelian norm is also an integer since irrep dimension ℓα is always a factor of its group’s order oG.
A non-Abelian projection ket in Equation (97) has two independent symmetry labels j and k belonging to global-lab symmetry operators g and local-body operators ḡ, respectively. Application of g-resolution Equation (82a) to ket Equation (97) is reduced by P-product rules in Equation (85) to the following global transformation.
The corresponding local operator ḡ first commutes through Pjkα according to Equation (95) and is converted by Equation (94) to inverse global g−1 on the right of Pjkα using Equation (82a) again. Finally, unitary irreps Dα(g−1) = Dα†(g) are assumed.
Physical significance of these global-(j) and local-(k) values are now discussed using Figure 19.
Wherever the global j is x or i3-symmetric (02), then the entire wave is symmetric to x-axial rotation by π in Figure 19a or horizontal reflection through the middle square-well in Figure 19b. Similarly, wherever the global j is y or i3-antisymmetric (12), that is seen for each overall figure, too.
However, if the local k is x or i3-symmetric (02), the local wave in each well has no node and is symmetric to its local axis of rotation by π in Figure 19a or horizontal reflection of each square-well in Figure 19b. Similarly, wherever the local k is y or i3-antisymmetric (12), that antisymmetry and one node is seen in each well, too.
Local and global symmetry clash along the i3-axis for states projected by nilpotent Pxyα or Pyxα. The result is the x-axial wave nodes indicated by pairs of arrows in Figure 19. The |Eyx〉 wave in the lower right of Figure 19b appears quite suppressed on the i3-axis. However, the simulation of the |Exy〉 in the upper left seems to have its “node” coming unglued.
The “unglued” level ωxyE is higher than ωyxE and enjoys more tunneling. If tunneling increases so do parameters such as r1 and r2 in Equation (92) that do not respect x-axial local subgroup C2 = [1,i3]. This breaks x-axial nodes and i3 local symmetry causing E-modes to be less C2-local and more like current-carrying above-barrier C3-local waves rotating on r-paths. D3 correlation arrays in Equation (102) with C2 or C3 indicate level cluster structure for extremes of each case.
D3⊃C20212A11·A2·1E11D3⊃C3031323A11··A21··E·11
Column 02 of array D3 ⊃ C2 in Equation (102) correlates to A1 and E. The lower (A1, E)-level cluster in Figure 19 has 02 local symmetry and lies below cluster-(A2, E) that has local 12 symmetry according to the 12 column of Equation (102). Column 03 of table D3 ⊃ C3 indicates that A1 and A2 levels cluster under extreme C3 localization, but columns 13 and 23 indicate that each E doublet level is unclustered under C3 with no extra degeneracy beyond its own (ℓE = 2).
A classical analog of quantum waves states in Figure 19 is displayed in Figure 20 in the form of vibrational modes for an X3 molecule. A detailed description of this analogy in Appendix A includes modes of various local symmetry combinations analogous to those introduced above and in Sections 6.7.1 and 6.7.2 below.
6.7. Symmetry Correlation and Frobenius Reciprocity
The mathematical basis of correlation arrays in Equation (102) is a Frobenius reciprocity relation that exists between irreps of a group and its subgroups. This may be clarified by appealing to the physics of Pjkα -projected states |jkα 〉 such as are displayed in Figure 19 and by exploiting the duality between their local and global symmetry and subgroups.
D3-symmetric Hamiltonian H in (71) is made only of local ḡ that couple |jkα 〉-states through local k-indices by Equation (100) but leave all ℓα values of global j-indices unchanged. Thus α-eigenstates of H mix k-values to form ℓα-fold degenerate levels labeled by j-indices. (Recall ℓE = 2 equal sub-matrices Equation (93) in (91).) Further degeneracy or near-degeneracy (“clustering”) occurs if inter-and-intra local tunneling coefficients decrease exponentially with quantum numbers thus isolating equivalent local modes into nearly degenerate sets of “spontaneously” broken local symmetry.
In contrast to this clustering or “un-splitting” associated with local ḡ symmetry operators, global g are associated with external or “applied” symmetry reduction that causes level splitting. Adding global gm to a Hamiltonian H reduces its G-symmetry to a self-symmetry subgroup K=sm consisting of operators that commute with gm. Adding a combination of gm and gn reduces K to an even smaller self-symmetry intersection group sm ∩ sn.
Global g couple |jkα 〉-states through global j-indices according to Equation (98). The more global perturbations are added to a Hamiltonian H the more likely it is to split ℓα-fold j-degeneracy (for ℓα ≥ 2) and/or linebreak alter eigenfunctions.
6.7.1. Global “Applied” Symmetry Reduction, Subduction, and Level Splitting
In the G=D3 example, adding matrix (r1) from Equation (69a) to (H) in Equation (71) reduces its symmetry to K=C3=[1,r1,r2], and adding (i3) reduces it to K=C2=[1,i3]. Adding a combination of (r1) and (i3) completely reduces (H)-symmetry to intersection C3∩C2=C1=[1], which corresponds to having no global symmetry.
By reducing G to a subgroup K⊂G, each G-labeled α-level becomes relabeled by that subgroup K and split (if ℓα ≥ 2) in precisely the way that central G-idempotent Pα is relabeled and/or split by unit resolution shown in Equation (78) or (79). The splitting in Equation (79) of D3 idempotent PE into C3-labeled P1313E plus P2323E implies the D3 doublet level ωE splits into C3-labeled singlets ω13 and ω23. Both D3 singlets A1 and A2 end up relabeled with C3 scalar 03 labels.
Center portions of splitting relations in Equations (103) and (104) use subduction symbols (↓) to denote how each D3 irrep-Dα reduces to subgroup C3 or C2 irreps under their respective global symmetry breaking. Earlier studies [34] have referred to these multiple subgroup splittings as multiple frameworks. Each α-row of Equations (103) and (104) corresponds to the row α=A1, A2, or E, of correlation array D3⊃C3 or D3⊃C2, respectively, in Equation (102).
6.7.2. Local “Spontaneous” Symmetry Reduction, Induction, and Level Clustering
Opposite to global G⊃K symmetry irrep subduction Dα(G)↓K=...⊕da(K)⊕db(K)⊕... that predicts level-splitting is the reverse relation of local K⊂G symmetry irrep induction da(K)↑G=...⊕Dα(G)⊕Dβ(G)⊕... that predicts “unsplitting” or level-clustering. In the former, an ℓα-dimensional irrep Dα(k) of global G-symmetry is reducible to smaller (ℓa ≤ ℓα) block-diagonal irreps da(k) of a subgroup K. In the latter, a K irrep da is induced (actually projected) kaleidoscope-like onto coset bases of a larger induced representation da↑G of G that is generally reducible to G irreps Dα.
Base states |k ↑jα 〉 of induced representation dk↑G are each made by a G-projector Pjkα acting on local dk-symmetry base state |k〉=Pk|k〉 defined by local K-projector Pk. G-projection is simpler if Pjkα is also based on K-projection. (It helps to stick with one framework through this!)
Of all D3⊃C2-projectors Pj2k2α based on Equation (78), only P0202A1, P0202E, and P1202E have right index k2 = 02. Only these can project induced states |02 ↑j2α 〉 from local base state |02〉 corresponding to the 02-column of D3 ⊃ C2 array in Equation (102) having A1 and E. Similarly, A2 and E in the 12-column of Equation (102) correspond to P1212A2, P0212E, and P1212E projecting states |12 ↑j2α 〉 from a local |12〉 state. Each projector Pj2k2α in Equation (104) has a C2-subgroup projector Pk2 “right-guarding” the side facing each local ℓ2-ket |ℓ2〉 = Pℓ2|ℓ2〉 that is similarly “guarded” by its own defining projector Pℓ2. C2-subgroup projector orthogonality then allows only k2=ℓ2, giving the projection selection rules just described.
Each “right guard” projector Pk of Pjkα is part of a G⊃K subgroup splitting or subduction splitting Dα(G)↓K=...⊕dk(K)⊕... as shown by D3↓C2 examples in Equation (104). (These go back to the original D3⊃C2 subgroup chain resolution in Equation (78).) In Equation (105) each Pk selects which α-type induced bases |k ↑jα 〉 and block-diagonal α-irreps can appear in a k-induced representation dk(K)↑G=...⊕Dα(G)⊕..., and it implies a duality between induced (↑) level-clustering and subduced (↓) level-splitting as stated by the following Frobenius reciprocity relation.
Number of Dαin dk(K)↑G=Number of dkin Dα(G)↓K
The numbers on the left-hand side of Equation (106) would reside in the kth-column of a G⊃K-correlation array such as in Equation (102) while the numbers on the right-hand side of Equation (106) would reside in the αth-row of the same array. The examples in Equation (102) have only ones {1} and zeros {·}. A deeper correlation D3⊃C1 to C1 symmetry, i.e., to no symmetry is a conflation of either the array D3⊃C2 or the array D3⊃C3 in Equation (102) since C1=C2∩C3 is the intersection of C2 and C3.
D3⊃C101=11A11A21E2
C1 local symmetry base |01〉=|11〉 is the |1〉 in Figure 18 that contains scalar A1, pseudo-scalar A2, and two E wave states in Figure 19 consistent with a single column of D3⊃C1 correlation array in Equation (107). This column describes induced representation D01 (C1) ↑ D3, also known as a regular representation of D3.
Reciprocity in Equation (106) also holds for non-Abelian subgroup irreps dk. D3 is the smallest non-Abelian group so it has no such subgroups, but octahedral symmetry has non-Abelian D3 and D4 subgroups that figure in its splitting and clustering that are described in later Section 7.
6.7.3. Coset Structure and Factored Eigensolutions
Three pairs of kets in Figure 17 relate to left cosets [1C2 = (1,i3), rC2 = (r1,i2), r2C2 = (r2,i1)] one at each site.
g-coefficients in H-submatrix Equation (93) track C2 cosets. Row-(bra)-x terms in Hx,·E line up in (+)-right-coset 1g+i3g order ...(r1+i1), (r2+i2). Row-(bra)-y terms in Hy,·E line up in (−)-right-coset 1g-i3g order (r1-i1), (r2-i2). Column-(ket) (±)-forms H·,xE and H·,yE line up in left-coset order ...(r1±i2), (r2±i1). Either ordering gives the same matrix. Off-diagonal components Hx,yE and Hy,xE have x vs. y symmetry conflicts so coset parameters (r0 ± i3) vanish.
Kets Px|rp〉=[Px|1〉, Px|r1〉, Px|r2〉 span induced representation dx(C2)↑D3, and Py|rp〉 span dy(C2)↑D3. Normalized states
Px∣rp〉2 and
Py∣rp〉2 correspond to σ-type and π-type orbitals at vertex positions p=0, 1, or 2 in Figure 21. D3 table in Equation (68) is reordered in Equation (113) below to display C2(i3) body-basis right-coset representation bra-defined by 〈g|=〈1| ḡ or ket-defined by ḡ† |1〉=|g〉. The resulting H-matrix in Equation (68) is Equation (71) reordered for cosets of C2 instead of C3.
C2 ordered products in Equation (113) help reduce H-matrix in Equation (71) to a direct sum of C2 induced reps (d02⊕d12)↑D3 in Equation (114). Upper (02)-array in Equation (114) uses σ-orbital bases |rxp〉 in Figure 21a while π-orbital bases |ryp 〉 in Figure 21b span the (12)-array.
Any group component of Equation (114) or combination thereof is a possible tunneling matrix. Submatrices d02(g)↑D3 shown for g=r1, i1, and i3 reflect the effect of these operators on states in Figure 21a and similarly for d12(g)↑D3 in Figure 21b.
The 02 correlation in Equation (102) implies d02↑D3 reduces further to D3 irreps A1⊕E that label the lower band of Figure 19. Meanwhile d12↑D3 reduces to irreps A2⊕E that label the upper band of Figure 19. Equation (91) shows A1⊕A2⊕E⊕E.
7. Octahedral Symmetry Analysis
Octahedral-cubic rotational symmetry O operations are modeled in Figure 22. Rotation inversion symmetry Oh=O×Ci operations are modeled in Figure 23. In each case the larger g-symbols (such as r̃1 on top of Figure 22) label position ket states (such as |r̃1〉=r̃1|1〉) while smaller g-symbols label axes of rotation in O (such as i6 on top facing edge of Figure 22 labeling that 180° rotation) or planes of reflection in Oh (such as the σx just above the z-axis on facing plane of Figure 23 labeling the x-plane reflection).
Figure 22 is an “O-group slide-rule” since product i6 · r̃1 can be viewed as operator i6 flipping a wave in position |r̃1〉 onto position |Rz〉, that is, i6|r̃1〉=|Rz〉 giving product i6 · r̃1=Rz. Figure 23 is an “Oh-group slide-rule” (that does O products, too) and just as easily gives product σx · r̃1=s̃2 all without knowing what r̃1 or s̃2 do. (As explained below, r1 is 120° rotation about [111] axis and r̃1 is its inverse located on the [1̄ 1̄ 1̄]-axis while r̃2 is on the [111̄] axis. s̃2 is r̃2 multiplied by inversion I·[111]=[1̄ 1̄ 1̄].)
Note i6-transform of state |r1〉 (example: i6|r1〉=|R̃y〉) differs from an i6-transform of operatorr1 (example: i6·r1·i6−1=r32). The latter is divined easily by “slide-rule” as i6 flips r1’s axis onto r32’s.
Three Cartesian C4 axes of anti-clockwise 90° rotations Rx, Ry, and Rz define directions [100], [010], and [001], respectively. Their inverses R̃x=Rx3, R̃y=Ry3, and R̃z=Rz3 are also 90° rotations but around negative axes [1̄00], [01̄0], and [001̄]. A shorthand notation for 180° Cartesian rotations is ρx=Rx2, ρy=Ry2, and ρz=Rz2. Trigonal C3 axes of anti-clockwise 120° rotations r1, r2, r3, and r4 lie along [111], [1̄1̄1], [1̄11], and [1̄11̄], respectively, while axes of inverses r̃1=r12, r̃2=r22, r̃3=r32, and r̃4=r42 lie along the opposite directions [1̄ 1̄ 1̄], [111̄], [11̄1̄], and [11̄1], respectively.
There are six C2 axes of 180° rotations i1, i2, i3, i4, i5, and i6 located along [101], [1̄01], [110], [1̄10], [011], and [01̄1], respectively. This completes the five classes of O: [1], [r1..4,r̃1..4], [ρxyz], [Rxyz,R̃xyz], and [i1..6]. Including the rotations with inversion I yields five more classes of Oh: [I], [s1..4,s̃1..4], [ρxyz], [Sxyz,S̃xyz], and [σ1..6] where s1..4=I · r1..4, [σxyz]=[I · ρxyz], [Sxyz]=[I · Rxyz], and [σ1..6]=[I · i1..6]. σ’s are mirror-plane reflections in Figure 23.
The “slide-rules” in Figures 22, 23 also help evaluate class products and construct left and right cosets of local symmetry subgroups. Three of the largest cyclic subgroups of O are tetragonal C4 such as C4=[1,Rz,Rz2 =ρz,Rz3 =R̃z] displayed on the Rz-face of the cube in Figure 22. In Figure 23 the same face displays local symmetry C4v=[1, ρz,Rz,R̃z, σ4, σx, σ3, σy] that contains C4 plus pairs of diagonal mirror reflections [σ4=I·i4, σ3=I·i3] and Cartesian mirror reflections [σx=I·ρx, σy=I·ρy]. Each pair [σx, σy] and [σ3, σ4] is a C4v class as is rotation pair [Rz,R̃z] or, singly, 1 and ρz. The other five cube faces display cosets of the tetragonal subgroups C4v⊃C4 of Oh⊃O.
Figure 22 shows six O-cosets g·C4 of C4=[1,Rz, ρz,R̃z]. Opposite ρx-face has coset ρx·C4=[ρx,i4, ρy,i3] in that order. The r1-face shows coset r1·C4=[r1,i1,r4,Ry] in upper right of Figure 22, and the opposite r2-face has coset r2·C4=[r2,i2,r3,R̃y]. Top and bottom faces have cosets r̃1·C4=[r̃1,R̃x,r̃3,i6] and r̃2·C4=[r̃2,Rx,r̃4,i5].
Each g·C4-coset element g·Rzp (p = 0..3) transforms the 1-face to the same g-face and orients it according to a C4 element Rzp as it permutes the list of its elements accordingly. Each face may be labeled by any element g·Rzp in its coset. An i-class labeling by 1, i3(or i4), i1, i2, i6, and i5 of C4 cosets in Figure 22 is as good as any other.
Figure 23 shows six Oh-cosets of C4v (counting C4v itself) in a geometric display that also shows eight trigonal cosets of C3v⊃C3-[111] and twelve dihedral cosets of C2v⊃C2-[101]. Figure 24 shows three symmetry points of Figure 23 forming a triangular cell with sides that are on reflection planes.
An order-8 axial symmetry C4v lies on the tetragonal-z-[001]-axis of a cube face or octahedral vertex. An order-6 C3v lies on the trigonal-[111]-axis of a cube vertex or octahedral face. Finally, there is a dihedral-C2v[110]-axis of a cube or octahedral edge. Lines between the axes have bilateral local reflection symmetry Cv(y)=[1, σy], Cv(2)=[1, σ2], or Cv(4)=[1, σ4], fundamental symmetry operations whose products generate all others. Figure 24 is like a reduced Brillouin Zone of the Oh lattice.
Each subgroup spawns a coset space and a set of induced representations of full Oh symmetry that generalize the C3v induced representations in Equation (115) and base kets sketched in Figure 21. Correlation tables between O or Oh and its subgroups L⊂G tell which O or Oh irreps, states, and energy levels arise from each coset space. As local symmetry reduces and its order °L decreases, the coset dimension d=°G/°L grows proportionally with a corresponding increase in number of irreps and levels in L↑G-induced representation cluster spaces. Examples are given below for G=O and in Section 8 for G=Oh.
7.1. Octahedral Characters and Subgroup Correlations
Spectral class resolution of O generalizes that of D3 in Equation (75) to give character array Equation (116).
Equivalent subgroup correlations O⊃H and O⊃gHg−1 share elements in the same O-classes and have one correlation array. Thus all three C4 local symmetries have one correlation table in Equation (117), as do all four C3 subgroups. However, O⊃C2(ρz) and O⊃C2(i1) correlations differ since i1 and ρz have different O-class and characters in Equation (116).
Projectors Pjkα and irreps Djkα of O depend on choice of local symmetry just as D3 projector splitting in Equation (78) or (79) depends on choice of correlation D3⊃C2 in Equation (104) or D3⊃C3 in Equation (103), respectively. Sub-labels (j, k) range over C2 values [02, 12] or else C3 values [03, 13, 23] while a tetragonal correlation O⊃C4 will use sub-labels (j, k)= [04, 14, 24, 34].
The m4 or else m3 unambiguously defines all O states since no O⊃C4 or O⊃C3 correlation numbers in Equation (117) exceed unity. However, O⊃C2(i1) correlations cannot distinguish all three sub-levels of T1 or T2 wherever a number 2 appears, and the O⊃C2(ρz) correlation leaves the E sub-levels unresolved, as well. A full Oh labeling resolves the first ambiguity as shown below, but we consider the unambiguous O⊃C4 case first. (C4 resolves C2(ρz) ambiguities.)
The C4 correlation table in Equation (117) shows how invariant class projectors Pα (expanded below in terms of O characters χκgα in table shown in Equation (116)) will split into irrep projectors Pm4m4α when hit by C4 local symmetry projectors pm4. The latter pm are expanded in terms of C4 operators Rzp weighted by character eigenvalues φpm4 = (χpm4)* using Equations (57) and (59).
The five class projectors Pα are O-invariant and commute with all twenty-four O-operators (1,r1,r2, ...i5,i6). So do the five class operators (κ0, κrk, κρk, κRk, κik) in which each Pα is expanded as follows. (Recall D3 classes in Equation (75).)
Each of the ℓα irrep projectors Pn4n4α is obtained from its invariant Pα by product Pαpn4=pn4Pα following Equation (118) with each of four C4 local symmetry projector pm4.
As the five (O-centrum=5) projectors Pα split into ten (O-rank=10) sub-projectors Pn4n4α, the five O class sums κg split into ten C4-invariant sub-class sumsck(k=1..10).
The resulting ten products
°OℓαPn4n4α are listed in Equation (122) of diagonal irrep coefficients Dn4n4α (gk) in terms of twenty-four group elements gk that have been sorted into ten sub-classes that have C4(z) local symmetry. The ten irrep projectors Pn4n4α are C4 local-invariant, that is, they commute with four C4-operators (1,Rz,Rz2 = ρz,Rz3 = R̃z) but not the whole O group like the Pα do. The ten sub-class-sum operators ck, into which each Pn4n4α is expanded in Equation (122), are each individually invariant to Rzp, that is Rzpck=ckRzp, and Dn4n4α (gk) is the same for all gk in sub-class ck. Note that a sum of ℓα rows belonging to Pn4n4α between horizontal lines in Equation (122) yields corresponding character values χkα =trace Dα(gk) in O-character array Equation (116) and effectively “unsplits” the sub-classes.
Without evaluating Equation (122), one may find ten O⊃C4 sub-classes by simply inspecting Figure 22 for operations in each O-class that transform into each other by C4 operations Rzponly. The O-class of eight 120° rotations rk split into two sub-classes, one [r1, r2, r̃3, r̃4] whose axes intersect four corners of the +z front square, and the other [r̃1, r̃2, r3, r4] whose axes similarly frame the −z back square. The class of six diagonal 180° rotations ik split into a sub-class [i1, i2, i5, i6] whose two-sided axes bisect edges of the ?z squares, and sub-class [i3, i4] whose axes are perpendicular to z-axis and bisect edges of ?xy side squares. The 180° rotational class [ρx, ρy, ρz] splits similarly into sub-classes [ρx, ρy] and [ρz] with axes perpendicular and along, respectively, the Rz axis. The 90° class splits, as indicated in the top row of Equation (122), into a sub-class of four perpendicular xy-axial rotations and separate sub-classes for Rz and R̃z.
The inverse to Equation (121) expresses the ten subclasses in terms of the ten diagonal irrep projectors using the same (albeit, conjugated) array of Dn4n4α (gk). However, column and row labels must switch and acquire different coefficients.
ck°ck=∑k=010Dn4n4α(gk)Pn4n4α=∑k=010Dn4n4α(ck)°ckPn4n4α7.1.2. Resolving D-matrices with <italic>C</italic><sub>4</sub> Local Symmetry
Off-diagonal Dm4n4α (gk) matrices derive from products of diagonal irrep projectors in Equation (122) using Equation (82b) repeated here.
Pj,jα·g·Pk,kα=Dj,kα(g)Pj,kα
Scalar A1 and pseudo-scalar A2 are given first then E, T1, and T2 irrep matrices for the fundamental ik-class of O.
Symmetry of C4⊂O subclass [i1, i2, i5, i6] and [i3, i4] would demand equality of parameters for each.
i1=i2=i5=i6≡i1256≡iI,and,i3=i4≡i34≡iII
Setting each parameter to the inverse of its sub-class order (ik=1/(°cik)) reduces each matrix to diagonal form and gives the diagonal Dn4n4α (gk) given in Equation (122). Classes r, ρ, R behave similarly.
7.1.3. Resolving Hamiltonians with <italic>C</italic><sub>4</sub> Local Symmetry
An octahedral Hamiltonian H = ∑k=124gkḡk with local C4(z) symmetry is resolved by sorting gk into its C4(z) sub-classes ck and then into Pn4n4α whose coefficients are the desired H eigenvalues εn4α. Zero off-diagonal Hm4n4α = 0 and C4-local symmetry conditions shown in Equation (128) arise from Equation (122) consistent with Figure 22. Tunneling parameter i1256=iI from +z-axis to its 1st-neighbor ±x or ±y axes may dominate flip-tunneling i34 = iII to 2nd neighbor-z-axis. The i-columns of Equation (122) (or matrix diagonals in Equations (125)–(127)) give iI and iII contributions to eigenvalues εn4α listed in the in-column of Table 11. Clusters (ε04A1, ε04T1, ε04E) through (ε34T2, ε34T1) are plotted in Figure 25 for select values of parameters iI = i1256 and iII = i34.
One expects the parameter iII for 2nd-neighbor tunneling to be exponentially smaller than iI for adjacent tunneling so the (iII = 0)-cases are drawn first in Figure 25. While the i-class operations are most fundamental (all operations are generated by products of ik) other operations also generate 1st-neighbor transformation. Three class parameters Rxy(90°), rI(120°), and iI(180°) label 1st-neighbor inter-C4 axial tunneling paths that have the same iI-level patterns and splitting ratios as (iII=0)-cases in Figure 25 but with differing sign. (Signs differ since each sub-class eigenvalue set must be orthogonal to all others as shown below.) Level patterns in Figure 25 are reflected in spectral patterns of Figure 26 if both ground and excited vibe-rotor states have similar RES-shape. However, only C4z sub-class iI(180°) patterns (with iI< 0) exhibit spectral ordering (A1T1E)(T2T1)(ET2A2)(T2T1) on the left hand side of Figure 26 that is maintained even as levels re-cluster into patterns (T1ET2)(T1ET2)(A2T2T1A1) of C3[111] local symmetry across the separatrix break on the right-hand side of Figure 26 as analyzed below [8,37]. O-crystal-field wavefunctions for either case tend to follow a Bohr-orbital progression s(A1), p(T1), d(E, T2), f(T1, A2, T2), g(E, T1, T2, A1), ... In general, ordering is sensitive to RES-shape and tensor rank as discussed later.
For an isolated three-level (ATE)-cluster of local symmetry 04 or else 24 the splitting pattern requires only two parameters. This could be either the 180°(iI, iII) or the 90°(Rxy, Rz) class pair in Table 11. The 120°-class, lacking 180° flips, has just one real parameter rI. Parameters iI, Rxy, and rI each split (ATE) by 2:1 ratio but differ in sign.
Local symmetry 14 and 34 each have two-level (TT) clusters that require just one splitting parameter, say iI, or else Rxy. Complex parameters Rz and Iz of the 90° Rn-class and the ρn(180°)-class in Table 11 may play minor roles in most C4 clusters but are necessary in order that the whole set be orthonormal and complete.
7.1.4. Orthogonality-Completeness of Local Symmetry Parameters
Equation (122) expands Pnn(α) by Equation (83) in group operators (1,r1,r2, ...i6). It acts on |1〉 to give |n4n4(α) 〉 eigenkets in Equation (129).
An O-symmetric H matrix is a sum of dual operators (1̄,r̄1,r̄2, ...r̄6) with coefficients ga=ε0, r1, r2, ..., i6. Local symmetry C4 or C3 reduces the sum to ρG=10 sub-class terms c̄a=ḡa+ḡ′a +... each sharing a coefficient ga=g′a ...
H=∑a=1°Ggag¯a=∑a=1ρGgac¯a
From these arise expansions like Table 11 of H eigenvalues εn4α in terms of its coefficients ga. Dual commutation gjḡk=ḡkgj makes Pnn(α) and H commute. Duality relation in Equation (94) leads to a Dα*-weighted sum of ga analogous to sum in Equation (129) of |ga〉.
Each C4 sub-class of order °ca has °ca equal terms gaDnn(α)* (ga) = g′aDnn(α)* (g′a) =. . . expanding eigenvalue εn4α. Rank-of-group ρG = 10 is the number of eigenvalues and of expansion terms °cagaDnn(α)* (ga) in Equation (131) or Table 11. Each of ten eigenvalues εn4α=(εA1, εA2, ..., ε34T2 ) expand to ten C4-local tunneling parameters ga=(ε0, rI, rII, ..., iII) and vice-versa.
One might count twelve real parameters in Table 11 since both pairs (rI,r̃I) and (Rz, R̃z) are complex, unlike RI = R̃I, which are real. If H is a Hermitian array (H = H†) it should only require ten, the rank of O, for its ten distinct real eigenvalues and the parameter pairs must be complex conjugates.
With no conjugation symmetry, such as for a unitary O ⊃ C4-symmetric matrix, the R and r parameters may be complex and unrelated to R̃ and r̃, and resulting extra real parameters are then needed. Symmetry parameter dimension matches eigensolution dimension for each local symmetry as shown in Figure 27.
7.1.5. Resolving Hamiltonians with <italic>C</italic><sub>3</sub> Local Symmetry
The previous two sections have detailed of symmetry-based level clustering and cluster splitting for C4. In Figure 26 these are the lower energy clusters of SF6 for ν4P(88). Given the previous two sections, it is possible to find the splittings of the C3 sub-group quickly. Starting with Equation (117) and Equation (118) one can build the irreducible representations necessary to create the Pn3n3α for the new sub-group. At this point, one can create a table analogous to Table 11. Such a table for C3 is shown in Table 12. The C3 clustering fits patterns of (A1, A2, T2, T2) and two of (E, T1, T2), each with a total degeneracy of 8. As before in Figure 25, the splittings in C3 make different patterns depending on which tunneling parameters are active. This is demonstrated in Figure 28.
7.1.6. Octahedral Splitting for a Range of Local Symmetry <italic>C</italic><sub>1</sub>⊂<italic>C</italic><sub>2</sub>...⊂<italic>O</italic>
As the order °L of local symmetry L⊂G decreases there are proportionally fewer types of local symmetry irrep dλ(L) and hence fewer types of energy level cluster since each cluster is defined by its induced representation dλ(L)↑G. There is a proportional increase in total number ℓλ↑G=(ℓλ)°G/° L of levels in each eigenvalue cluster. However, G-symmetry degeneracy limits the total number of distinct eigenvalues from all clusters to be global rank ρ(G) or less, no matter what local symmetry is in effect. Octahedral rank is ρ(O)=10=ℓA1+ℓA2+ℓE+ℓT1+ℓT2 where ℓα gives both the global degeneracy of each level type and the number of times it appears.
The number of H-matrix parameters equals the number of distinct eigenvalues as long as all eigenvectors are determined by global-local symmetry, that is, each entry is 0 or 1 in the G⊃L correlation array. Diagonal eigenmatrix forms are shown in Figure 27a,b for C4⊂O and C3⊂O for which all bases states are distinctly labeled. Multiple correlation (≥ 2) occurs if L-symmetry is too small to determine some of the °G eigenbases. Then the H-matrix must have extra parameters that fix vectors through diagonalization.
This happens for the C2(i1) ⊂O symmetry whose correlation array in Equation (117) assigns the same C2 label to two bases of T1 and of T2. (Two C2 symmetries 02 and 12 cannot distinctly label three bases.) Figure 22 shows C2(i1) splits O into fourteen sub-classes: (1), (r1r̃4), (r2r̃2), (r3r̃3), (r4r̃1), (ρxρz), (ρy), (RxRz), (R̃xR̃z), (RyR̃y),(i1), (i2), (i3i5), (i4i6). The C2⊂O sub-classes form a non-commutative algebra and cannot be resolved so easily as C3⊂O or C4⊂O into commuting idempotent combinations like Equation (123).
Spectral resolution of fourteen C2(i1)⊂O sub-classes requires more than rank number ρ(O)=10 of diagonal commuting O idempotents Pnnα. To fully determine C2 basis, two off-diagonal pairs PabT1=PbaT1† and PabT2=PbaT2† of non-commuting nilpotent projectors are needed to finish C2-labeling of T-triplets. Adding these four gives fourteen projectors with their fourteen parameter coefficients εℓ shown in Figure 27c to fully define general C2(i1)⊂O H-operators. (However, only twelve of the fourteen parameters are independent for Hermitian Ha,b=Hb,a*.)
The other class of C2 symmetry has similar problems. Local C2(ρz)⊂O symmetry requires projector pairs PabT1=PbaT1† and PabT2=PbaT2†, too, but then another nilpotent pair PabE=PbaE† must be added to label repeated E bases in array Equation (117). This gives sixteen C2(ρz) sub-classes to resolve and sixteen parameters sketched in Figure 27d. (Hermitian H=H† matrices for C2(ρz)⊂O have thirteen free parameters.)
For the lowest local symmetry C1=[1] (i.e., no local symmetry) sub-classes are completely split since every O-operator is invariant to 1 as C1 provides no distinguishing labeling, and all twenty-four O-projectors (∑α(ℓα)2=24) are active in its resolution. The 24-parameterH-matrix resolution is sketched in Figure 27e. Each parameter εa for a=1, ..., 24 is a combination of 24 products Dj,kα*(gp)gp (p=1, ..., 24) of irrep and group element coefficient gp as given in Equation (90) or (131). (If H is Hermitian the number of free parameters reduces to ∑αℓα(ℓα+1)=17.)
For O’s highest local symmetry, namely O itself, there is no splitting of the ∑α( ℓα)0=5 invariant idempotents Pα that resolve the five O classes. Then H has five independent parameters and five eigenvalues of degeneracy (ℓα)2. This 5-parameter resolution is sketched in Figure 27f. Total level degeneracy for sub-matrix eigenvalues are listed below each one, and show less splitting than Abelian cases listed in Figure 27a–e.
Any non-Abelian local symmetry such as L = D4 also fails to split Pα into a full number ℓα of components Pnnα if O irrep-(α) correlates with multi-dimensional L-irreps. By splitting out less than the full rank number ρ(O)=10 of idempotent projectors Pnnα, the resulting number of independent H matrix parameters reduces accordingly. The 8-parameter resolution for an H-matrix with D4⊂O is sketched in Figure 27g and similarly for D3⊂O in Figure 27h. Two kinds of D2⊂O in Figure 27i,j share degeneracy sums with the Abelian cases.
Each matrix display lists exact degeneracy ℓα due to global symmetry O but not the cluster quasi-degeneracy ℓλ↑G due to local symmetry induced representation dλ(L)↑G. The latter is found by summing global degeneracy ℓα of all states |a,λα〉 with the same local symmetry λ as per Frobenius reciprocity in Equation (106). The result is integer ℓλ↑G=(ℓλ)°G/° L mentioned above.
8. Spectral Resolution of full <italic>O<sub>h</sub></italic> Symmetry
Including inversion I and reflection operations σn allows parity correlations between even-g (gerade) and odd-u (ungerade) states. Two classes of C2 subgroups lie in O and appear in separate C2-correlations in Equation (117). In the following Oh correlations Equation (133), the two types of C2v subgroups have separate tables. The first subgroup C2vi=[1, σy,i1, σ2] is the one of the three local symmetries shown in Figure 12 while the second C2vz=[1, ρz, σy, σx] is just a subgroup of local symmetry C4v as would be C2v34=[1, ρz, σ3, σ4].
The local symmetry C2vi⊂Oh unambiguously defines all states in its correlation array while the other C2v symmetries fail to split the Eg and Eu sub-species. The former lead to complete eigenvalue formulae. The latter may not.
8.1. Resolving Hamiltonians with C<sub>2v</sub> Local Symmetry
As the order of the local sub-group symmetry goes down, the degeneracy and complexity of the rotational cluster must increase. Oh⊃C2v clusters are 12 fold degenerate and come in 4 cluster species. Matrices describing this system are larger, but O⊃C2 will show many of the same effects. To actually resolve the doubled T1 or T2 triplets of O⊃C2 requires distinguishing the u and g versions of each. The C2 clusters are 12 fold degenerate, but they are also easily displayed.
As noted earlier, O⊃D3⊃C2 and O⊃D4⊃C2 local symmetries give identical cluster degeneracies and groupings, but with cluster splittings and structure dependent on the sub-group chain. Though it neglects inversion, Figure 27 indicates that there are several different types of O⊃C2 (and, thus Oh⊃C2v local sub-group symmetries). Examples given here involve the O ⊃D4⊃C2(i4) sub-group chain.
Compared with O⊃C4 and O⊃C3, the splittings of O⊃C2 are relatively simple to calculate since the terms in Equation (131) will be real. Creating splitting tables for C2 is done in the same way as for Tables 11 and 12. It is shown in Table 13.
8.1.1. Local Sub-Group Tunneling Matrices and Their Inverse
Table 13 can be further broken apart to demonstrate how one can create an automated process to evaluate the tunneling splittings for O⊃C2 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.
Equation (131) produces Table 13, but even after combining splittings from each subclass, repetition exists. We show the two steps to convert Table 13 into the transformation matrix just described. First we assume that only nm levels may interact with themselves, e.g., that a 12 cluster may not interact with a 02 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. Table 13 shows this for the 02 cluster. Looking at the A1 level in the 02 cluster, one can see that the subclasses 1, rn, ρn make a vector {1, 4, 4, 2, 1} while the Rn, in subclasses make a vector {4, 2, 4, 1, 1}. These vectors are reordered versions of each other. Thus only one is needed. The A2 level in the 12 cluster shows the same similarity, but the Rn, in now contain a negative sign.
By using only the minimum number of splitting parameters and including only a single cluster gives a condensed version of Table 13 that acts as a transformation that inputs symmetry-based tunneling values and outputs energy levels. Such a table is shown in Table 14. A simple inverse of the matrix in Table 14 will produce the transformation giving tunneling parameters for a given set of cluster energy splittings, as shown in Table 15.
There are multiple ways to use Tables 14 and 15. Among the most useful is to use the columns of Table 14 as a predictor of possible splitting patterns. Using the inverse matrix to find spectroscopic tunneling parameters from cluster splittings may also become a useful and automated process.
An example demonstrates this process for a model (4, 6)-octahedral-Hecht spherical-top Hamiltonian Equation (134) with varying spectroscopic parameters. The terms T[4] and T[6] model rotational distortions written in an octahedral basis of fourth and sixth order respectively in J. The parameter θ is varied to explore the different relative contributions of T[4] and T[6] while keeping them normalized. Because T[4] and T[6] each have octahedral symmetry, Equation (134) represents all possible octahedral pure rotational Hamiltonians up to sixth order.
H=BJ2+cos(θ)T[4]+sin(θ)T[6]
As noted in Section 3 cluster structure location and the RES shape will change significantly as the Hamiltonian parameters change in Equation (134) as Figure 29 (a copy of Figure 6) shows by plotting rotational energy levels of Equation (134) for changing θ with corresponding RES at points along the θ axis. RES plots in the figure demonstrate how the phase-space changes as θ varies.
RES diagrams in Figure 29 along with the cluster degeneracy indicate where in the parameter-space C2 clusters exist. The lowest 02(C2)↑O cluster in Figure 29 for θ between 18° and 132° labels a kaleidoscope of 12 waves each with C2 local symmetry. Its superfine levels are magnified about 100 times in the central inside plot of Figure 30 which has been adjusted to show level splittings but not whole cluster shifting. (The θ-dependent cluster center-of-energy is subtracted.) The locally antisymmetric 12(C2)↑O clusters contain quite similar superfine structure but with A2 replacing A1 and T1 switched with T2.
At certain θ-points in Figure 30 levels of different symmetry cross and one of three distinctive splitting patterns emerge. These points occur periodically as indicated by vertical lines that are (starting form left side) solid, dotted, dashed, dotted, solid, dotted, dashed, solid, and so forth across the plot. The three distinctive εα-energy level patterns for species α=(A1, E, T1, T2, T2) are given by vectors εdash=(0, 0, 0, 1,−1), εdot=(2, −1, 1, 1, −1) and εsolid=(2, −1, 1, 0, −1), respectively. These repetitious patterns seem to persist even outside of the marked-off sections to the very ends of the C2 cluster region at θ≃18° and θ≃132° where they grow slightly but maintain their respective superfine ratio patterns and degeneracy. The matrix in Table 15 transforms each of the three εα-vectors in Figure 30 into a vector of O-defined sub-class tunneling amplitudes gr. These are evaluated for clusters at several values of parameter θ used in T[4,6] Hamiltonian Equation (134). Proportioned values of the tunneling amplitudes gr for the three distinctive cases are listed in the inset legend of Figure 30 based on Tables 14 and 15.
Dotted-line and solid-line curve patterns appear alternately flipped in sign. Dotted-line patterns have a crossing (T1,T2) pair while the solid-line patterns have a crossing (T2,E) pair.
Solid-line patterns appear to be centered on quasi-hyperbolic avoided-level-crossing episodes involving the pair of repeated T2 tensor species of O. The ordering (A1, T1, T2, E, T2) of solid-line superfine level patterns reflects Bohr-like orbital ordering (s, p, d, f, ..) of orbital momentum and occurs only when there is just one non-zero sub-class of tunneling parameter, namely that of sub-class (r34 or equivalent Rxy) that affects tunneling between nearest-neighbor C2 valleys.
Dashed-line pattern level curve slopes appear to alternate (+) and (−) signs and exhibit maximum separation of repeated T2-species surrounding a degenerate (A1, T1, E)-sextet crossing midway in between. Such triple-point crossings are quite remarkable. They appear repeatedly in Figure 30 and persist even at low-J as seen for J=4 in Figure 8c. Higher 12(C2)↑O clusters show similar triple points made of (A2, T2, E)-sextets.
Such crossings are quite ironic if we recall that it was (A1T1E), (A2T2E), and (T1T2) clusters noted by Lea, Leask, and Wolf [20] and later Dorney and Watson [21] that led to a theory involving induced representations K4(C4)↑O including 04(C4)↑O=A1⊕T1⊕E, 24(C4)↑O=A2⊕T2⊕E, and ±14(C4)↑O=T1⊕T2. (Recall C4 columns of Equation (117) and reciprocity Equation (106)). This theory uses an inter-C4-axial tunneling model [22,23] with a single ad.hoc. tunneling parameter that predicts a 2:1-splitting ratio for (ATE) clusters. C4-axial tunneling cluster splitting dies exponentially as body momentum-K approaches J (Recall Figure 26) and thus C4(ATE) levels never actually cross.
However, the C2(ATE) levels in Figure 30 clearly do so and with quite the opposite 1:2-spltting ratio. It is ironic that the more elegant ortho-complete multi-path tunneling models, while useful in exposing these crossings, seem at a loss to explain them, particularly given that they were first noted by Lea, Leask, and Wolf so very long ago!
It would be easy to write off such (ATE) triple-crossings and particularly the (T1T2) or (ET) double-crossings as “accidental” degeneracy. Indeed, all but the latter occur for special values of a complete set of sub-class parameters. However, Figure 30 clearly shows that each type of crossing belong to a periodic structure that is unlikely to be just an accident.
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.
9. Examples of Rovibronic Energy Eigenvalue Surfaces (REES) and J-Clusters
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 Equation (8) through Equation (10b) of Sections 1 and 2. The first semiclassical analysis of fundamental coupling in high-J octahedral molecules was done for ν2E[38] and ν3T1[39] bands in 1978 and for overtone ν2 + ν3 “hot-bands” in 1979 [40].
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 J-vector. However, the more general approach differs in that it builds an N-by-N matrix of such formulas that takes account of quantum rovibronic coupling between N vibronic (or vibrational) states, that is, a 2-by-2 matrix for the ν2E system, a 3-by-3 matrix for the ν3T1 system, and a 5-by-5 matrix for the ν2 + ν3 system.
The resulting N eigenvalues provide points on N nested Rovibronic Energy Eigenvalue Surfaces (REES) for each direction of the semiclassical J-vector. Visualization of P, Q, and R state mixing in ν3T1 bands by 3-sheet REES was done using the high-resolution 3D-graphics at Los Alamos in 1987 and reported in 1988 [25]. Interesting features of the ν3T1 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 ν3/2ν4 dyad of CF4 that involves a 9-sheet REES.
9.1. Rotor-With-Gyro Model of Internal Rotation
A first application by Ortigosa and Hougen [17] of REES to visualize molecules with internal rotation is related to a simple rotor-with-gyro model [25,41] based on the three lowest rank tensors possible, namely the scalar (rank-0), the vector (rank-1), and the tensor (rank-2). The prolate symmetric top RES in Figure 1 is an example of a scalar-tensor combination. A vector RES lacks J-inversion symmetry, that is, time reversal symmetry, so it is forbidden for normal molecules that have no intrinsic dynamic chirality such as embedded spin S. We consider how to include an S in a way that preserves overall T symmetry.
Total momentum J=R+S is the sum of rotor momentum R and gyro spin S. J is conserved in lab frame but R and S are not. If gryo is body-frame-fixed by frictionless bearing then rotor gyro-coupling does no work and is an ignorably constant HRS. S and |J| are conserved in body frame but J and R are not.
HR+S(bod-fixed)=ARx2+BRy2+CRz2+HS+HRS
Replacing bare-rotor momentum R=J-S gives the following with a new constant spin energy H′RS.
The simplest classical theory of the rotor-R-gyro-S momentum dynamics involves superimposed RES plots, one for +S and one for -S in Figure 31; A composite RES with T symmetry. If J and +S align (anti-align) then |R|=|J-S|, rotor energy Equation (135), and rotor-gyro relative velocity are minimized (maximized) (Thus, gyro-compass alignment with Earth rotation is seen to be relativistic quantum effect!).
A quantum theory of multiple RES involves mixing extreme cases |J ± S|. An elementary quantum gyro-spin is a two-state spin-1/2 with a 2-by-2 Hamiltonian matrix found by inserting quantum spin S=σ/2 matrices into Equation (136) to give Equation (137). Gyro-rotor dynamics involves REES obtained from eigensolutions of the following 2-by-2 matrix for each body-based J-vector Euler orientation (β, γ ).
Eigensolutions of matrix form Equation (137) transform classical RES Figure 31 into quantum REES Figure 32 that has conical intersections or avoided crossing points replacing lines of classical surface intersections in the former Figure 31. Also, individual sheets of REES have J-inversion symmetry (or T symmetry) that individual RES lack. Where the RES of Figure 31 are well separated their shape is not so different from that of REES in Figure 32. Differences show up near the intersection lines where the two RES approach resonance. In this resonance region the REES is deformed extremely from rank-1 or rank-2 tensor shape of the separate RES, and there arises greater mixing of the extreme |±S| base-states.
9.2. REES of CF<sub>4</sub> in ν<sub>3</sub>/2ν<sub>4</sub> Dyad
The first practical REES application includes 9-sheet displays of the ν3/2ν4 dyad of CF4 recently shown by Boudon etal.[2]. This large scale numerical analysis may be summarized by a revealing plot of dyad eigenlevels as a function of J = 0 to 70 in Figure 33. This includes colored lines representing the REES values for J located on C4 axes (shaded red), C3 axes (shaded blue), or C2 axes (shaded green).
Each of the symmetry axes may take turns as central loci for clusters of their type of local symmetry C2, C3, or C4, or else, they may sit on a REES separatrix or saddle point between two or more different types of clusters. A third option involves C1 clusters that have no rotation axis point but are likely to belong to vertical xyz-plane reflection symmetry Cv = [1, ρz] or diagonal-plane reflection symmetry Cd = [1, i3]. These label clusters of 24 levels associated with 24 equivalent REES hills or valleys.
A final option involves true-C1 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 J.
A common ordering noted before on the left hand side of Figure 29 (pure T[4]) and in Figure 26 (16μ region of SF6) is (C3-valley→C2-saddle→C4-hill). It is present in the lowest REES band of Figure 33. An inverted version of the common ordering appears clearly in the 2nd band whose REES is cubic in Figure 34.
A cutaway view at J = 57 of the first five REES sheets shows glimpses of the first two REES deep inside of Figure 34. The second sheet has cubic topography similar to the inverted T[4] RES on the right hand side of Figure 29 (pure (−)T[4]). However, the first and lowest REES for J = 57 is practically spherical with all 2J + 1=115 levels and clusters crushed in Figure 33 into near degeneracy!
After the first two REES sheets the cluster topography become more complicated with multiple conical intersections and avoided crossing points.
On the 5th sheet of the (J = 57)ν3/2ν4 REES are found examples of C1-local symmetry valleys as shown in Figure 34. (The upper four sheets are made invisible.) Each C1 loop occupies an area that is comparable to the minimum uncertainty (J = K)-cone shown on vertical C4 axis of the figure and a nascent 24-level cluster of type 12(C2)↑O should be present in the level spectrum.
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.
10. Summary and Conclusions
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 vs. extrinsic symmetry duality (The “Mock-Mach-Principles” Equation (95) and Equation (94) of wave relativity.) leads to generalizing character relations between group classes and irreducible representation into sub-character relations between sub-classes and induced representations Equation (131) and (132). These provide ortho-complete parameter relations (Tables 11–15) for complex tunneling path lattices that determine molecular fine, superfine, and hyperfine spectra. The methods may be extensible to fluxional atomic and molecular systems.
AppendixA. Classical <italic>D</italic><sub>3</sub> Modes: Local <italic>C</italic><sub>2</sub> and <italic>C</italic><sub>3</sub> Symmetry Examples
Local symmetry theory applies to classical vibrational modes as well as to quantum tunneling. Examples of classical D3 modes given below help clarify global-vs-local symmetry and geometry of group projection. For example, D3 modes defined by local C2(i3) in Figure 20 are to be compared with quantum waves in Figure 19. Each mode ket |jkα〉 has the same coefficients Djkα* (g) for projections in Equation (78) as the waves do, but the mode shapes clearly display a vector geometry.
In particular, global x-vector modes |xxE1〉 and |xyE1〉 (left E1 column in figure) “point” along global x-direction while y-vector modes |yxE1〉 and |yyE1〉 (right E1 column) “point” along global y-direction. Each global pair [|xℓE1〉, |yℓE1〉](ℓ = x, y) is projected to be an i3-symmetric-antisymmetric pair like lab unit vectors [|x〉,|y〉] (Recall Equation (78)).
∣x〉=P03∣1〉2=(∣1〉+∣i3〉)/2,∣y〉=P13∣1〉2=(∣1〉-∣i3〉)/2
This exposes easy derivations of E-irrep DjkE1 (g)=〈j|g|k〉 in Equation (89). Irreps in Equation (87) such as DjkE1 (r) for 120°-rotation r simply contain direction cosines 〈j|r|k〉=êj●êr·k of rotated vectors [r|x〉,r|y〉] relative to original [|x〉,|y〉] (Note transpose of equation array to matrix array).
This also fixes local transformations. Local x-vector modes |xxE1〉 and |yxE1〉 (lower E1 row in figure) “point” along local x-axes that are local radial lines while local y-vector modes |xyE1〉 and |yyE1〉 (upper E1 row) “point” along local y-axes that are local angular lines. If global symmetry meets local anti-symmetry as in |xyE1〉 (or vice-versa in |yxE1〉), a zero appears on the i3-axis in Figure 20. Singlet modes |xxA1〉 and |yyA2〉 avoid such conflicts by being all one or the other.
For group-defined cases like Figure 20, symmetry arguments alone determine normal modes that usually require diagonalizing a K-matrix (below) just as tunneling states (Figure 19) usually require diagonalizing an H-matrix.
A.1. Comparing K-Matrix and H-Matrix Formulation
Classical modes are eigenvectors of force-field matrix K or operator K that is a linear function of spring constants (k0, etc. in Figure 35) for a harmonic approximate potential V (x) that is a quadratic K-form of coordinates xa based on sixD3-labeled axes x̂a or |a〉 shown in Figure 20. Each K component Kab=〈a|K|b〉 is a sum over spring k-constants that connect axis-xa to axis-xb multiplied by factor (k̂âx̂a)(k̂b̂x̂b) for projecting spring k’s end vectors k̂a and k̂b onto x̂a and x̂b at respective connections. (A straight-line spring has equal k̂a=k̂b. Curvilinear springs must only have k̂-ends with equal sense (→→) or (←←) of spring direction. Either direction gives the same Kab).
The classical equation of coupled harmonic motion is a Newtonian F= M·a relation of a n-dimensional force vector F, acceleration vector a, and mass operator M. The latter is a unit-matrix-multiple M·1 for the D3-symmetric case treated here. The driving force F is a (-)derivative of potential Equation (140) that becomes a K-matrix expression.
-M∂t2xa=∂V∂xa=∑bKabxb
It is instructive to compare this classical equation of motion to that of Schrodinger’s equation for quantum motion.
iℏ∂tψa=∑bHabψb
Squaring quantum time generator iħ∂t=H yields equations having classical form Equation (142) with K = H2 and M=ħ2.
-ℏ2∂t2ψa=∑bKabψbwhere:K=H2
The (H/ħ)-eigenvalues are quantum angular frequencies εm/ħ =ωm. The (K/M)-eigenvalues are classical squared angular frequencies km/M=ωm2. The former is Planck’s oscillator frequency relation ε= ħω. The latter is Hooke’s relation k/M=ω2. Apart from normalization, eigenvectors of quantum H are identical to those of classical K and either eigenvalue set corresponds to the respective energy spectrum.
A.2. Comparing K-Matrix and H-Matrix Eigensolutions for Local <italic>D</italic><sub>3</sub>⊃<italic>C</italic><sub>2</sub>(<italic>i</italic><sub>3</sub>)
The preceding relates eigensolutions Equations (92) and (93) of quantum Hamiltonian H-matrix in (71) with those of a classical K-matrix. In particular, eigenvectors of H found using D-matrices in Equation (89) or (139) also serve as mode-eigenkets in Figure 20 that diagonalize a D3⊃C2(i3)-locally-symmetric K-matrix. With this symmetry, K cannot couple radial (local-x) and angular (local-y) modes and is left with just four independent real group-based parameters ga=r0, r1, i12, and i3 allowed for D3⊃C2(i3)-symmetric H in Equation (93). These relate to four spring kh-constants in Figure 35a.
Only 1st-row parameters gb=〈1|K|gb〉=K1b of the force matrix Kab are needed for the spring model in Figure 35a. That model includes kr(angular) and ki(radial) constants for internal connections between masses. The k3(angular) and k0(radial) constants represent external connections between each mass and an outside lab frame.
Generic group parameters gb=H1b, labeled [r0, r1, r2, i1, i2, i3] for the H-matrix in Equation (71), are now applied to gb=K1b. The gb are to be related to spring-constants kj using coordinate-spring projection cosine factors (k̂1 ● x̂1)( k̂b ● x̂b) in Equations (140) and (141). The usual harmonic limit assumes small vibrational amplitudes (xb≪1) for which direction of spring end vectors k̂1 or k̂b do not vary to 1st-order, and so, for lab-fixed x̂a the Kab are constants.
One may visualize each −K1b as the acceleration of x1 due to setting a (tiny) unit xb in Equation (142). Diagonal -K11 must be negative or else x1 blows up. Higher order anharmonic terms are needed to describe effects of rotating k̂b or x̂b and such models are likely to suffer from classical stochastic (chaotic) motion.
Substitution of generic ga from Equation (145) into reduced D3⊃C2(i3)-symmetric H-matrix in Equation (92) or (93) gives K-matrix eigenvalues Kℓℓα due to each spring ki, kr, k3, or k0 in Figure 35a separately or together. Modes in Figure 20 remain eigenmodes for all values of four spring constants ki, kr, k3, and k0 since none can mix local x-and-y-symmetry.
Any set of four K-matrix eigenvalues kA1, kA2, kxE, and kyE is arithmetically possible by adjusting the four spring constants. However, their arrangement in Figure 20 (this was drawn to match tunneling states in Figure 19) is impossible without negative k-values that would give classical instability. As shown below, free ring molecules often have A1-stretching modes among the highest frequencies. In contrast, tunneling amplitudes are often negative so their A1 states lie low. As a rule, fewer quantum nodes imply lower energy.
A.3. K-Matrix Eigensolutions for Broken Local Symmetry
In some ways the direct-k1-connection spring model of Figure 35b is quite the opposite of the D3⊃C2(i3) model just treated since it involves maximal (50-50) mixing of x and y local symmetry. Below are recalculated generic gb=〈1|K|gb〉 in terms of direct spring-constants k1 using (141) with projection cosines listed in Figure 35b.
Again, a substitution of generic ga from Equation (147) into reduced H-matrix Equation (93) gives a reduced K-matrix like Equation (146), but now the E-symmetry submatrix is not diagonal.
Figure 36 shows (50-50 ±)-mixing due to k1. It distinguishes genuine vector modes (|x,(−)E〉 or |y,(−)E〉) and the scalar breathing mode (|x,xA1〉) from non-genuine (low or zero-frequency) vector modes of pure x or y-translation (|x,(+)E 〉 or |y,(+)E 〉) and rigid rotation (pseudo-scalar |y,yA2〉). The i3-local symmetry is wiped out by direct connection-k1.
In order to reestablish approximate D3⊃C2(i3)-local-symmetry there needs to be a C2(i3)-“locale” provided by lab-grounded potential springs such as those with constants k3 and k0 in Figure 35a. Adding these in the form of Equation (146) to Equation (148) causes a transition between the two extremes. If the difference (k3 + 2kr − k0) between eigenvalues Equation (146) begins to dominate the off-diagonal component (3k1/4) of Equation (148), then mixed E-modes of Figure 36 begin to recover D3⊃C2(i3) locality seen in Figure 20.
Meanwhile the constant k3 that determines eigenvalue ky,yA2 does not affect locality for either of the singlet A1 or A2 modes. Singlet eigenvectors are non-negotiable as long as master symmetry D3 holds.
A.4. K-Matrix Eigensolutions for <italic>D</italic><sub>3</sub>⊃<italic>C</italic><sub>3</sub> Symmetry
Another choice for D3 local symmetry is the C3 subgroup of Equation (79) corresponding to a strong chiral perturbation by internal rotation, spin, or B-field. The E-submatrix in Equation (146) with zero generic reflection parameters (i1=i2=i3=0) may take a purely chiral C3 form if the generic rotation parameters r1 and r2=r1* are purely imaginary corresponding to velocity dependent force ( r1=ir and r2=−ir. Here K is assumed Hermitian self-conjugate as was H).