##
1Analysis of an algebraic model for the chromophore vibrations of CF_{3}CHFI
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Analysis of an algebraic model for the chromophore vibrations of CF

_{3}CHFI
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We extract the dynamics implicit in an algebraic fitted model Hamiltonian for the hydrogen chromophore’s vibrational motion in the molecule CF

_{3}CHFI.
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The original model has four degrees of freedom, a conserved polyad allows the reduction to three degrees of freedom.

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For most quantum states we can identify the underlying motion that when quantized gives the said state.

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Most of the classifications, identifications and assignments are done by visual inspection of the already available wave function semiclassically transformed from the number representation to a representation on the reduced dimension toroidal configuration space corresponding to the classical action and angle variables.

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The concentration of the wave function density to lower dimensional subsets centered on idealized simple lower dimensional organizing structures and the behavior of the phase along such organizing centers already reveals the atomic motion.

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Extremely little computational work is needed.

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## Introduction

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In recent years we have developed methods to investigate algebraic models (spectroscopic Hamiltonians) for the vibrations of molecules.

^{1–5}
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The Hamiltonians reproduce and encode by their construction the experimental data.

^{6–12}
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In the mentioned examples the system is reducible to two degrees of freedom.

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As the next logical step we have turned to systems where after all possible reductions three degrees of freedom remain.

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A first example of this kind has been treated in

^{ref. 13}, in the present paper we treat as further example the molecule CF_{3}CHFI for which in^{ref. 14}an algebraic model has been presented for the motion of the H atom (chromophore).
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This model has four degrees of freedom and it has one conserved polyad which can be used to reduce it to three degrees of freedom.

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From known wave functions represented in the toroidal configuration space of action/angle variables

^{15}we show that we can visually sort most of the states into ladders of states with similar topology.
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Each ladder has a relatively simple spectrum.

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Complexity arises from their interleaving.

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We also can recognize underlying classical lower dimensional organizing structures by the fact that for these inherently complex functions the eigenstate density (magnitude squared) is concentrated around them and the phase has simple behavior near them.

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The position of these structures in angle space reveals the nature of the resonance interaction causing the topology.

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It also allows the reconstruction of the motion of the atoms which underlie this particular quantum state.

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By counting nodes in plots of the density and of the phase advances in corresponding plots of the phase quantum excitation numbers can be obtained.

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In many cases, along with the polyad quantum number itself, we thereby obtain a complete set of quantum numbers for a state even though the corresponding classical motion is nonintegrable.

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These classification numbers can be interpreted as quasi-conserved quantities for this particular state or for the ladder of states based on the same dynamic organizational element.

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Since the Hamiltonian used here has the same functional structure as the one used in

^{ref. 13}and since also the methods of investigation are the same, we mainly present results for the present molecule and refer to^{ref. 13}for the detailed explanations of the method.
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## The model

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For the description of the chromophore dynamics of the molecule CF

_{3}CHFI we use the model set up in .^{ref. 14}
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It is based on four degrees of freedom as required by the nature of the observed overtone spectroscopy.

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The index s stands for the stretch of the H atom, the index a labels its bend in the HCCF

_{(4)}plane, the index b labels its bend perpendicular to this plane, and the index f stands for the transpositioned C–F stretch.
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For the exact atomic motion belonging to these four degrees of freedom see Fig. 2 in .

^{ref. 14}
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The algebraic Hamiltonian fitted to experiment and to calculations on a fitted potential surface has a natural decompositioninto a diagonal part(here the indices

*j*and*m*run over the four degrees of freedom a, b, f, s) and an interaction partwhere*a*_{j}and*a*_{j}^{†}are the usual harmonic destruction and creation operators of degree of freedom*j*.
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In eqn. (3) the indices

*j*and m run over the three degrees of freedom a, b, f.
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For the coefficients we use the ones given in column 6 in Table 1 in

^{ref. 14}where we see that roughly*ω*_{s}≈ 2*ω*_{a}≈ 2*ω*_{b}≈ 2*ω*_{f}giving rise to the Fermi resonances anticipated by the terms of eqn. (3) and the Darling–Dennison resonances anticipated by the terms of Eqs.
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(4–6).

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This Hamiltonian has as a conserved quantity, the polyad

*P*, defined aswith the corresponding operator given as*P̂*by replacing*n*_{j}→*a*†*j**a*_{j}.
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One can now study each polyad separately.

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The expansion of the eigenfunctions

*Ψ*_{j}into number statesis gotten from the diagonalization of the*H*-matrix in number state basis.
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To construct the corresponding classical Hamiltonian in action/angle variables

*I*/*ϕ*we use the substitution rules ,after bringing all operators into symmetrical order, then the classical action*I*_{j}corresponds to*n*_{j}+ 1/2.
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## Semiclassical wave functions and their analysis

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Semiclassically we represent a number state (basis state in our case) as a periodic plane wave on the configuration torus of the coordinates

*ϕ*_{j}:Then from eqn. (8) we get That is, the expansion of the eigenstate into number states from eqn. (8) is converted into a Fourier decomposition on the torus.
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Dimension reduction can now be carried out by noting that, using the polyad of eqn. (7) to eliminate the fast degree of freedom s, the

*j*th eigenstate can be written By the last equation the reduced wave function*χ*_{j}^{P}is defined.
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This shows that the eigenfunctions in a given polyad

*P*have a common phase factor dependence on*ϕ*_{s}and really only depend on the reduced wave function*χ*_{j}^{P}which is a function of the three anglesor*j*= a,b,f.
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These three angles make up a three dimensional toroidal configuration space

*T*^{3}upon which*χ*_{k}^{P}is situated.
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The angle transformation of eqn. (14) plus the trivial transformationcan be supplemented to give a canonical transformation by introducing the new actions again for

*j*= a,b,f and*θ*is a cyclic angle, therefore the conjugate action*K*can be treated as parameter and the system is reduced to one with three degrees of freedom.
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The atomic motion belonging to a trajectory in reduced phase space is gotten as follows.

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First we integrate the Hamilton equations for

**the cyclic angle; second we undo the canonical transformation for all variables.
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And third we assume a harmonic model for the transition from action/angle variable to position variables in each degree of freedom.

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We call this reconstruction the “lift” (for details see

^{ref. 13}).
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The wave functions are manifestly complex function and in the following we use their representation by absolute value and phase as Their inspection is key to our analysis.

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Their magnitudes (or densities) and phases are both plotted in the cube whose sides are associated with the

*ψ*_{j}on the 0 to 2π range.
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Periodic boundary conditions are used so as to associate properly with the 3D configuration torus.

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Symmetry properties of the system will show up in the following.

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The Hamiltonian is invariant under a simultaneous shift of all angles

*ψ*_{j}by π.
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Therefore the reduced configuration torus

*T*^{3}covers the original configuration space twice and all structures show up in double, even though they really exist only once.
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In addition the system is invariant under a shift of any angle

*ψ*_{j}by 2π and it is invariant under a reflection where all angles go over into their negatives.
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The basis functions have constant density and therefore any eigenstate dominated by a single basis function has a density without sharp localization.

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Resonances as for example 2

*ω*_{a}≈*ω*_{s}will be seen to cause localization about a line*ψ*_{a}= constant.
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This follows as

*ψ*_{a}=*ϕ*_{a}−*ϕ*_{s}/2 =*constant*when differentiated with respect to time gives the frequency relation.
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Hence it is seen that resonances are associated with localization and the fact that d

*ψ*_{j}/d*t*= 0.
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It can be claimed that by using angle coordinates that slow to zero velocity at resonance we here assure that the wave function will “collapse” onto and about a lower dimensional subspace called the organization center.

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This in reverse gives a way to recognize the influence of resonances, namely localization of the wave function on the configuration torus.

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Each linearly independent locking of angles therefore reduces by 1 the dimension of the subset of configuration space around which the wave function is concentrated.

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The nodal structures will be visible and countable in directions perpendicular to the organization element and clearly will be associated with a localized direction.

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The count of such transversal nodes supplies for each direction of localization a transverse quantum number

*t*, that replaces an original mode quantum number*n*which has been destroyed by the resonant interaction.
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The wave function in all the localized directions can be considered as qualitatively similar (a continuous deformation) of an oscillator state of the corresponding dimension.

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The transverse quantum numbers

*t*_{k}are the corresponding oscillator excitation numbers.
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The phase functions

*Φ*according to eqn. (18) are smooth and close to a plane wave in subsets parallel to the organization center (usually they are subsets of high density) and have jumps by π and singularities in other parts of the configuration space.
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Therefore phase advances along fundamental cycles inside such distinguished subsets are well defined and are necessarily integer multiples of 2π.

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They provide longitudinal quantum numbers

*l*_{k}which also replace the interacting mode quantum numbers.
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## The motion behind individual quantum states

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The configuration space is a three-dimensional torus which we can represent as a cube with identified opposite boundary points.

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Three dimensional (perspective) plots of the semiclassical state functions look too often like large globes of ink and were not informative.

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Here we will resort to cuts of the cube (e.g. cut

*ψ*_{a}=*ψ*_{b}) and plot density and phase in such cuts.
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The appropriate choice of the cuts is found partially by trial and error and partially by the following considerations: In CF

_{3}CHFI a resonance with a rather close frequency is*ω*_{f}≈*ω*_{b}and its associated*k*_{b,f}in the Hamiltonian is largest.
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This suggests that modes b and f should be coupled in planes

*ψ*_{f}=*ψ*_{b}+ constant which assures*ω*_{f}^{eff}=*ω*_{b}^{eff}.
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The simultaneous Fermi coupling conditions

*ω*_{s}= 2*ω*_{a}and*ω*_{s}= 2*ω*_{f}give by similar reasoning the organizing line*ψ*_{a}=*c*_{1},*ψ*_{f}=*c*_{2}.
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For some states several organizing structures could be used and the same dynamics revealed.

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Our experience is to choose those corresponding resonances which seem more important in

*H*and which give longer ladders of states.
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### Scheme for a large number of states throughout the polyad

74

Starting at the bottom of the polyad we find 64 states that definitely lie in the Darling–Dennison

*ω*_{f}=*ω*_{b}resonance class in that they have densities localized about planes all denotable as*ψ*_{f}=*ψ*_{b}+ constant.
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They also have simple phase plots in these planes.

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At the lower end a few of them could also be organized by the

*ω*_{s}= 2*ω*_{f},*ω*_{s}= 2*ω*_{b}Fermi resonances, i.e. about a line in the “a” direction.
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But they occur when the stretch excitation is zero and this classification is not very physical.

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Some of the lowest states could also be assigned for all modes by quantum numbers

*n*_{j}and correspond to continuous deformations of basis functions.
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Although these

*n*_{j}are useful in the lift for obtaining actions these assignment misses the phase and frequency locking and the localization of the density.
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In addition we looked very carefully for primary tori in the classical phase space at the corresponding energy and did not find any.

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Therefore Einstein–Brillouin–Keller (EBK) quantization cannot be applied.

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Hence classical dynamics shows that the assignment by

*n*_{j}for*j*= a,b,f is unphysical.
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For the organization center

*ψ*_{f}=*ψ*_{b}+ constant. the modes “a” and “s” are uncoupled and the Darling–Dennison resonance couples “f” and “b” such that mode “b”, a true mode of the hydrogen atom motion, is driven by the generic source mode f which represents the effect of the motion of the rest of the molecule on the hydrogen atom motion.
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The organization plane has the topological structure of a two dimensional torus.

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Therefore it has two fundamental loops which are used for phase counts to get two longitudinal quantum numbers.

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One of these loops runs into the “a” direction and provides the excitation number

*l*_{a}=*n*_{a}.
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The other fundamental loop of the organization plane corresponds to the combination of a loop in the “b” direction with a loop in the “f” direction.

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Accordingly a phase count along this loop gives a number

*l*_{b+f}representing the excitation of the coupled b/f motion.
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Then the polyad number implies the value of

*n*_{s}as*n*_{s}=*P*−*n*_{a}/2 −*l*_{b+f}/2.
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Finally we need a transverse quantum number representing the degree of excitation perpendicular to the plane

*ψ*_{f}=*ψ*_{b}+ constant.
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What at first was confusing was that perpendicular structures appeared to be organized about one of two parallel planes at low energy and at higher energy about third and fourth planes lying in between.

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At low energy this observation and the fact that pairs of states with identical (

*n*_{s},*n*_{a},*l*_{f+b},*t*) assignments appeared, led to the discovery of a dynamic nearly symmetric double well or double “valley”.
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Two organizing planes at

*ψ*_{f}=*ψ*_{b}and*ψ*_{f}=*ψ*_{b}+ π, each run along one of the valleys.
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The

*ψ*_{f}=*ψ*_{b}valley is slightly deeper making the*ψ*_{b}→*ψ*_{b}+ π invariance only approximate.
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This asymmetric “double well” becomes also evident from an “accessibility diagram”.

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This is a plot of regions of configuration space, i.e. the values of

*ψ*_{a},*ψ*_{b}and*ψ*_{f}that satisfy*E*=*H*(*J*_{a},*J*_{b},*J*_{f},*ψ*_{a},*ψ*_{b},*ψ*_{f};*P*), at each*E*, for given*P*and any compatible actions.
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Around 11100 cm

^{−1}(well below the lowest quantum state of polyad 5 and near to the classical lower end of polyad 5) it was noted that only two slabs of configuration space were accessible; these are our wells.
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At the energy of the lowest quantum state all configuration space is accessible but the wells act as two attractive regions that cause quantum localization above them.

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For some of the lower states, the energy in the “mode f–mode b” lock is low enough that these states are primarily localized in the

*ψ*_{f}=*ψ*_{b}or the*ψ*_{f}=*ψ*_{b}+ π well.
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As energy increases the states are roughly speaking “above the barrier” of the “double well” with density on both sides and they fall into near symmetric (+) and antisymmetric (−) pairs (with the same

*n*_{s},*n*_{a},*l*_{a+b},*t*) reflecting the approximate invariance*ψ*_{b}→*ψ*_{b}+ π.
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Each member of these pairs has a slight density preference for one of the wells.

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The ladders with an even

*l*_{a}start with a + state and ladders with an odd*l*_{a}start with a − state.
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The lowest state in any ladder of constant

*l*_{a}always has most of its density near the plane*ψ*_{f}=*ψ*_{b}.
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104

In Table 1 this (±) classification label appears.

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In Table 1 we give the transverse quantum number as seen from the planes

*ψ*_{b}=*ψ*_{f}and*ψ*_{f}=*ψ*_{b}π as*t*.
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106

For higher states it sometimes becomes difficult to count

*t*.
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The reason for this can be traced to the fact that a slice transverse to the organizing plane should ideally reveal wave functions that represent oscillators.

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The density would then be the highest in two planes running parallel and on opposite sides of the organizing plane.

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109

Now consider that for higher states the density is localized about both planes (albeit with a preference for one plane).

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As excitation increases then the outer lobes of the transverse oscillator will move toward the planes in between (recall all features appear by symmetry in doubles) the two original planes i.e. the planes

*ψ*_{f}=*ψ*_{b}± π/2 on which the highest density will now accumulate.
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Hence it now make sense to count transverse nodes between these now bigger intensities.

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112

The corresponding transverse quantum number is called

*t*′.
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113

As always the transverse quantum number indicates to which extent the coupled motion goes out of exact phase lock.

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114

It is a measure of the width of the phase distribution.

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115

A feature that further complicates and hides the true interpretation of the states is mixing due to the accidental degeneracies of two or three states.

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The mixed final eigenstates that result are spotted by noting their near degenerate energies and their lack of almost all of the above discussed features.

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Novelty: None |
ConceptID: Obs14

117

Trial and error demixing using various weights yields states with the above features.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs15

118

The notation s/s′ or s/s″/s″ indicates in Table 1 that this particular state is a demixing of s + s′ etc.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs15

119

The dynamical importance of the

*ψ*_{b}=*ψ*_{f}+*c*planes is further confirmed by running many long classical trajectories that show that the motion, i.e. the flow of the trajectories, is mainly parallel to the*ψ*_{f}=*ψ*_{b}plane and in addition is mainly in the direction of the space diagonal i.e. guided by the condition*ψ*_{f}=*ψ*_{b}=*ψ*_{a}.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs16

120

This is not totally unexpected as

*ω*_{a}≈*ω*_{b}≈*ω*_{f}.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs16

121

This effect is reflected in the observation that in the organizing planes

*ψ*_{f}=*ψ*_{b}and*ψ*_{f}=*ψ*_{b}+ π the wave functions tend to have fibers of density running along the diagonal.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs16

122

In fact a diagonal classification could have served many states as the organizing structure and assignment might have been made relative to it.

Type: Hypothesis |
Advantage: None |
Novelty: None |
ConceptID: Hyp1

123

We chose the planes as the organizing structures as breaks in this diagonal fibration often made assignment less than clear.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod6

124

To illustrate our ideas let us consider two states, state

*s*= 7 and*s*= 43 at the bottom and the middle of the ladder respectively.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod6

125

Both states are organized about

*ψ*_{f}=*ψ*_{b}giving a DD*ω*_{f}=*ω*_{b}expectation.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod6

126

State 7 is a low state with its density running up the valley in the middle of the well as seen in Fig. 1c and 1a.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs17

127

Recall again that due to the symmetry property mentioned in the section 3 all features come in double.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs17

128

Fig. 1a looks down on the organizing plane and Fig. 1c looks sideways.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs18

129

In Fig. 2 the higher excited case is exhibited.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs19

130

The highest density is in the plane

*ψ*_{f}=*ψ*_{b}+ π/8.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs19

131

Fig. 2a and 2b look down on this plane of high density and Fig. 2c and 2d show a view from the side that shows transverse excitation.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs20

132

From the phase plots we see the quantum numbers

*n*_{a}= 2 for state 7 (Fig. 1b) and*n*_{a}= 1 for state 43 (Fig. 2b) and*l*_{b+f}= 8 for state 7 and*l*_{b+f}= 7 for state 43.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs20

133

Using the polyad number

*P*= 5 this implies*n*_{s}=*P*− (*n*_{a}+*l*_{b+f})/2 = 0 for state 7 and*n*_{s}= 1 for state 43.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs21

134

Note: For state 43 phase simplicity only happens in the organizing plane.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs21

135

Fig. 2d shows no useful phase information.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs22

136

The plot shows jumps along lines and ramp singularities where the phase value depends on the direction of approach.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs22

137

The transverse structure of the states becomes evident in density plots in some appropriate plane transverse to the plane of high density.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs23

138

In Fig. 1c we see that state 7 has

*t*= 0 and in Fig. 2c we see that state 43 has*t*= 1.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs23

139

The number

*t*′ does not make sense for state 7 and the number*t*′ = 2 for state 43 is rather unclear from Fig. 2c.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs23

140

However the phase plot in Fig. 2d shows two lines of phase jumps near the lines

*ψ*_{a}=*ψ*_{b}± π/2 and these lines of phase jumps indicate nodal lines of the density.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs24

141

Since this is a somewhat indirect indication of the value

*t*′ = 2, in Table 1 the number*t*′ = 2 for state 43 is set in brackets.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs24

142

At this point 64 states for

*P*= 5 can be assigned clearly by the same scheme.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs24

143

An additional 38 states seem to resemble this picture.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs25

144

The resemblance is only seen as trends from the systematic following of the 64 other ones.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs25

145

For these, increasing but still smaller effects of many other resonances are taking their toll and they significantly degrade the assignable states into unassignable more heavily mixed quantum analogues of classical chaos.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs25

146

The Table 1 indicates which states have less certain assignments.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs26

147

By comparing the phase function at the

*ψ*_{f}=*ψ*_{b}(plane 1) and*ψ*_{f}=*ψ*_{b}+ π (plane 2) wells for each pair of states with the same quantum numbers the (+) and (−) assignment can be made.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res16

148

Even at this stage where other ladders discussed below are omitted spectral complexity arises from the interlacing of energy levels of different subladders.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res17

149

The roughly 100 states on the ladder which only demands the DD-f/b lock for its construction gives a complex spectrum and nontransparent energy spacings.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs27

150

In addition the individual ladders appear irregular by the mixing of states.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs27

151

A sequence of states which are pure excitations of a single organization element would form a regular ladder of energy levels looking like the spectrum of one anharmonic oscillator.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs28

152

However by the mixing the eigenstates are mixtures of various pure excitations and thereby their energies are shifted and thereby the ladders appear more irregular.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs28

153

The reader will note that some ladder states in Table 1 are missing.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs29

154

These states could not be found but their disappearance is confirmed using the anharmonic models expected energy spacing.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs29

155

Formally these states lie among the highly mixed (chaotic?) states we discuss below.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs29

156

They are mixings of more than two dynamically identifiable states and do not show the near degeneracy that our demixing process identifies.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res18

157

If we go back to displacements

*q*_{j}corresponding to the original degrees of freedom, then we find the following for the motion of the H atom in states based on the above described organization element: In projections into the (*q*_{s},*q*_{a/b/f}) planes a roughly rectangular region is seen whose sides are proportional to and .
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res18

158

The hydrogen trajectories motion interior to the rectangle will be quasiperiodic motion.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res19

159

The exact trajectory depends on the initial choices of

*ϕ*_{s}and*ϕ*_{a/b/f}and is really not important.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res19

160

In the (

*q*_{a},*q*_{b}) plane we find ellipses whose relative ranges (excursion from zero displacement) in the (*q*_{a},*q*_{b}) variables are .
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res20

161

The ellipses eccentricity depends on the organizing structures phase shift,

*viz.**ψ*_{a}=*ψ*_{b}+*δ*→*ϕ*_{a}=*ϕ*_{b}+*δ*.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res20

162

*δ*= 0 and π give zero angular momentum and something approaching a straight line motion while

*δ*= ±π/2 gives maximal angular momentum and elliptical motion which becomes circular if

*n*

_{a}=

*n*

_{b}.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res20

163

The

*t*value is the out of phase motion and causes fluctuations of the eccentricity.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res20

### Scheme for the upper end of the polyad

164

At the upper end of the polyad we take to start states 160 and 161.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs30

165

A cut in

*ψ*_{b}= 0 (or any other value of*ψ*_{b}) reveals as we see in Fig. 3 for state 160 what looks like two columns of density localized around*ψ*_{a}= π and*ψ*_{f}= π.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs30

166

In comparison state 161 (not shown) only has one column.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs30

167

A cut in the plane

*ψ*_{b}=*constant*is called to see that the columns really exist.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs30

168

Fig. 3 shows that they do for state 160 and it also holds for state 161.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs31

169

Clearly for both these states the line

*ψ*_{a}= π,*ψ*_{f}= π is the organizing structure.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs31

170

Since it loops in

*ψ*_{b}, mode b is decoupled and the phase picture for both states shows (Fig. 3 for state 160, state 161 not shown) that as*ψ*_{b}→*ψ*_{b}+ 2π along the organizing line no wave fronts are crossed and hence there is no phase advance.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs31

171

Hence

*l*_{b}=*n*_{b}= 0.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs31

172

For the upper end of the polyad also classical dynamics shows a flow in b direction.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs32

173

As always the figures show all structures in two copies which in reality are only one structure.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs32

174

Now the

*ψ*_{a}=*ψ*_{f}= π organizing structure implies that the phase locks are*ϕ*_{a}=*ϕ*_{s}/2 and*ϕ*_{f}=*ϕ*_{s}/2.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs32

175

Time differentiation of these relations show that the frequency locks are of 2:1 Fermi type with the stretch mode locking with the bend mode a and the mode f.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res21

176

Clearly for the organizing structure

*ψ*_{a}=*ψ*_{f}then*ϕ*_{a}=*ϕ*_{f}is a valid phase lock and*ω*_{a}=*ω*_{f}is a DD 1 ∶ 1 frequency lock.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res22

177

Since all three types of terms appear in the Hamiltonian we can be assured that all three frequencies are in lock and that the quantum numbers

*n*_{a},*n*_{f}and*n*_{s}no longer exist.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res22

178

They must be replaced by three quantum numbers of the lock one of which can be

*P*and the other two can be taken as the number of nodes seen in the*ψ*_{a}= 0 cut along the antidiagonal as*t*_{1}= 1 and along the diagonal as*t*_{2}= 0.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res22

179

For state 161 since one column exists

*t*_{1}=*t*_{2}= 0.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res22

180

If energy decreases from the top, then the classical flow turns into the space diagonal direction rather soon and accordingly only the four highest quantum states show (partly after demixing) a clean fiber structure in b direction.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs33

181

The fiber is always sharp in the a direction but broad in the f direction.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs33

182

This demonstrates again an emerging b–f mixing in the system.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs34

183

The next seven states still show some tendency towards fibers in b direction (they are no longer really clean), but at the same time also show the beginning of fibers in the diagonal direction.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs34

184

Their assignment as states organized along b fibers is less clear but still they continue the trends seen in the four highest states (partly after demixing).

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs34

185

The corresponding assignments for the 11 highest states are compiled in Table 2.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs35

186

An idea of the internal motion of the deuterium and the CF stretch can be obtained from the actions, phase relations and the

*t*_{1}and*t*_{2}values.
Type: Hypothesis |
Advantage: None |
Novelty: None |
ConceptID: Hyp2

187

In all (

*q*_{i},*q*_{j}) planes the actions again define the maximal displacement of each*q*_{j}from its equilibrium.
Type: Hypothesis |
Advantage: None |
Novelty: None |
ConceptID: Hyp2

188

Clearly this range is meant in the sense that an average is made over the values of the two variables not represented in the plane.

Type: Hypothesis |
Advantage: None |
Novelty: None |
ConceptID: Hyp2

189

The plane (

*q*_{b},*q*_{j}),*j*= a,f,s will show rectangularly bound quasiperiodic motion.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs36

190

In the (

*q*_{s},*q*_{a/f}) plane a U shaped region is swept out.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs36

191

Here

*q*_{a/f}reaches its extreme values as*q*_{s}reaches its maximum displacement and*q*_{s}sweeps its range twice for each sweep of*q*_{a/f}.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs36

192

Increasing

*t*_{1}and/or*t*_{2}causes the lifted trajectory to oscillate in a tube about this basic U lift.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res23

193

Of course

*q*_{f}and*q*_{a}reach their maxima (minima) in phase as the three modes s, a and f are phase locked.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res23

### Patterns in the dense region of the polyad

194

In the middle of the polyad there are approximately 35 states with very complicated wave functions which we could not classify in any scheme.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs37

195

It is possible that further and closer analysis could reveal some systematics.

Type: Hypothesis |
Advantage: None |
Novelty: None |
ConceptID: Hyp3

196

These states are dispersed among the states mentioned in subsection 4.1.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs38

197

But interestingly a few simple states which do not fall into any scheme discussed so far are also dispersed in this region.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs38

198

In this subsection we present a few of them.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs38

199

Fig. 4 shows the structure of state 82, parts a and b give density and phase respectively in the plane

*ψ*_{f}= 0 and parts c ans d show the plane*ψ*_{b}=*ψ*_{a}+ 0.29 * 2π.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs39

200

Part a demonstrates some concentration around

*ψ*_{a}= ±π/2 and at the same time around*ψ*_{a}=*ψ*_{b}±π/2 with a transverse excitation*t*= 1.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs39

201

Plots of the phases in various planes show that planes

*ψ*_{a}=*ψ*_{b}+*constant*are the only planes with a simple phase structure.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs39

202

Fig. 4d shows it in the plane of highest density.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs40

203

The phase function comes very close to a plane wave with

*l*_{a+b}= 8 and*l*_{f}= 2.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs40

204

Together with the above mentioned transverse quantum number

*t*= 1 this gives the complete assignment of the state.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res24

205

In Fig. 4 notice that also the interaction between the degrees of freedom f and s has an effect, it concentrates the density mainly around

*ψ*_{f}= 0 and*ψ*_{f}= π.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs41

206

There are a few other examples (states 93, 111, 114, 122 and 136) which show such a clean phase function in planes

*ψ*_{b}=*ψ*_{a}±π/2 but do not show a clean phase function in any other plane.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs41

207

There are approximately 5 other states where the phase function is not as clean as in Fig. 4d but comes close to a continuous deformation of a plane wave only in a plane of this type.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs41

208

The motion of all these states is a circular motion of the H atom, two orthogonal bends with relative phase shift locked at ±π/2.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs41

209

Numerical results for state 80 are shown in Fig. 5, part a and b show density and phase respectively in the plane

*ψ*_{b}=*ψ*_{a}±*ψ*_{f}+ π/2 which is the plane of high density and thereby the organizing structure.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs42

210

Part c and d show density and phase respectively in the transverse plane

*ψ*_{f}= 0.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs42

211

In the plane of high density the phase function comes close to a plane wave with

*l*_{a+b}= 6 and*l*_{f+b}= 5.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs42

212

This is the only plane where the phase function is a continuous deformation of a plane wave.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs42

213

This case is something new in so far as there is a plane with a simple phase function and at the same time not a single

*n*_{j}has a definite value.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs42

214

There is a locking involving all modes without implying a locking in any pair of modes.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs43

215

A few other states (75,83,87,118) fall into the same scheme, i.e. they have simple phase functions only in a plane of the form

*ψ*_{b}=*ψ*_{a}+*ψ*_{f}+*constant*.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs43

216

The implications for the motion of the atoms are as follows.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res25

217

In the nonreduced variables the locking condition is

*ϕ*_{b}+*ϕ*_{s}/2 −*ϕ*_{a}−*ϕ*_{f}= ±π/2.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res25

218

This can be interpreted as a locking of the beat frequency between a pair of degrees of freedom to the beat frequency of the remaining pair of degrees of freedom.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res25

219

We have two possibilities to group the four modes into appropriate pairs.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res25

## Conclusions

220

We have shown how our previously developed methods can also be applied to systems with three degrees of freedom after reduction.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con3

221

For 75 percent of the states the strategy works successfully.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con3

222

We investigate the wave functions constructed on a toroidal configuration space.

Type: Object |
Advantage: None |
Novelty: New |
ConceptID: Obj4

223

We investigate on which subsets the density is concentrated and evaluate the phase function on this subset of high density and in addition the nodal structure perpendicular to it.

Type: Object |
Advantage: None |
Novelty: New |
ConceptID: Obj5

224

This allows us to obtain a set of excitation numbers and a complete assignment of the state.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res26

225

The excitation numbers can be interpreted as quasi-conserved quantities for this particular state.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res26

226

The success of the method is based on three properties of our strategy which more traditional approaches starting from potential surfaces do not have.

Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met1

227

First, we start from an algebraic Hamiltonian which naturally decomposes into an unperturbed

*H*_{0}and a perturbation*W*.
Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met1

228

Perturbation theory assures us that

*W*contains only the dominant interactions, leaving out those that do not affect the qualitative organization of the states.
Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met1

229

In addition having the ideal

*H*_{0}results in the classical Hamiltonian being formulated from the start in action and angle variables belonging to*H*_{0}.
Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met1

230

For a potential surface it is very complicated to construct an appropriate

*H*_{0}and the corresponding action/angle variables.
Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met1

231

Our knowledge of the correct

*H*_{0}allows the continuous deformation connection of some eigenstates of*H*with the corresponding eigenstates of*H*_{0}which was useful for many states.
Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met1

232

Second in the algebraic Hamiltonian the polyad (which for the real molecule is only an approximate conserved quantity) is an exactly conserved quantity and allows the reduction of the number of degrees of freedom.

Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met1

233

We always deal with one degree of freedom less than the number of degrees of freedom on the potential surface.

Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met1

234

Third, our strategy depends to a large extent on having a compact configuration space with nontrivial homology where the phase of the wave function is important and where the phase advances divided by 2π along the fundamental cycles of the configuration space provide a large part of the excitation numbers.

Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met1

235

In part also feature one is responsible for the importance of the phases.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res27

236

This is an essential difference to working in usual position space.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res27

237

In total we have found a semiclassical representation of the states which is actually much simpler to view than the usual position space one.

Type: Conclusion |
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ConceptID: Con4

238

We emphasize that no calculation is required to produce these states since theare gotten from eqn. (8).

Type: Conclusion |
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ConceptID: Con4

239

As discussed in

^{ref. 13}, when localization occurs it is often possible to lift the organizing structure back to molecular space in a qualitative manner and hence avoiding even the need to compute any trajectories.
Type: Result |
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ConceptID: Res28

240

Qualitative pictures of the motion upon which the states are quantized are then available.

Type: Result |
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ConceptID: Res28

241

The classical dynamics of the system shows rather little regular motion and is strongly unstable.

Type: Result |
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ConceptID: Res28

242

Then the question arises in which way the quantum states reflect anything from the classical chaos.

Type: Conclusion |
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ConceptID: Con5

243

We propose the following answer: In the middle of the polyad a large number of states do not show any simple organization pattern.

Type: Conclusion |
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ConceptID: Con5

244

For some of those states it might be that we simply have not found the right representation in which to see the organizational center.

Type: Hypothesis |
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ConceptID: Hyp4

245

However we doubt that this is the case for many states.

Type: Hypothesis |
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ConceptID: Hyp4

246

For many of the complicated states the wave function is really without a simple structure, it seems to be a complicated interference between several organizational structures.

Type: Observation |
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Novelty: None |
ConceptID: Obs44

247

Then immediately comes the idea to mix these states with their neighboring states to extract simple structures in the way as we did it for example for states 22 and 24.

Type: Observation |
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ConceptID: Obs44

248

For most of the complicated states this method does not lead to any success.

Type: Result |
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ConceptID: Res29

249

This indicates that underlying simple patterns are distributed over many states so that demixing becomes rather hopeless and maybe also meaningless.

Type: Conclusion |
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ConceptID: Con6

250

In the spirit of reference

^{16}this indicates that the density of states is already so high that many states fall into the energy interval over which states are considered almost degenerate and mix strongly.
Type: Conclusion |
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ConceptID: Con6

251

This is a strong difference to the systems which could be reduced to two degrees of freedom.

Type: Conclusion |
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ConceptID: Con6

252

There the density of states was much lower such that mixing only occurred occasionally between two neighboring states.

Type: Conclusion |
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ConceptID: Con6

253

This stronger mixing gives some clue as to how the quantum wave functions become complicated for classically chaotic systems and for sufficiently high quantum excitation.

Type: Conclusion |
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ConceptID: Con6

254

Finally there may come up the question why we do not use some of the common statistical procedures (like random matrix theory) to treat our system when effects of classical chaos become evident in a quantum system.

Type: Goal |
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ConceptID: Goa2

255

The answer is: We are not satisfied with only generic (universal) properties of the system or with the investigation to which extent such universal properties are realized by our system.

Type: Conclusion |
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ConceptID: Con7

256

We aim to investigate each individual quantum state one after the other and to work out its individual properties in order to arrive in the best case at a complete set of quantum numbers and in addition at a detailed description of the atomic motion underlying the individual state.

Type: Goal |
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ConceptID: Goa3

257

Previous examples have shown that our procedure provides a complete classification for simpler systems which can be reduced to two degrees of freedom.

Type: Conclusion |
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ConceptID: Con8

258

As the example presented in this paper shows, this program is doable with good success for a large number of states even in a system with originally four degrees of freedom, reducible only to three degrees of freedom and with a complicated classical dynamics.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con8