Polyad normal mode

It is well known that the tetrahedral frame of the CH4 molecule is easily distorted. If the tetrahedral frame of CH4 were robust, the purely rotational infrared spectra of CH4 would not exist. However, even at temperatures as low as room temperature, the CH4 molecule features hundreds of very weak, dipole-allowed rotovibrational lines at frequencies from 42 to 208 cm-1, the so-called groundstate to groundstate (gs—>gs) transitions. Moreover, more than 1500 weak, dipole-allowed transitions exist within the polyad system v /v — 1/2/1, at frequencies from 14 to 500 cm-1 [42]. These allowed transitions arise from distortions of the tetrahedral frame by rotation and the internal dynamics of the CH4 molecule, due to the coupling of normal modes of the flexible CH4 frame. Collisional frame distortion should probably be associated with unresolved gs— gs and similar polyad bands. Some evidence of such collision-induced bands of CH4 in CH4-X complexes has been pointed out [39-41]. Besides these collision-induced bands that presumably are due to collisional frame distortion of CH4, fairly significant, unexplained collision-induced bands also exist that are shaped by rotovibrational transitions of the collisional partner X = H2, N2, or CH4, and by double transitions of the bimolecular CH4-X complex [39-41]. 

The polyad quantum number is defined as the sum of the number of nodes of the one-electron orbitals in the leading configuration of the Cl wave function [19]. The name polyad originates from molecular vibrational spectroscopy, where such a quantum number is used to characterize a group of vibrational states for which the individual states cannot be assigned by a set of normal-mode quantum numbers due to a mixing of different vibrational modes [19]. In the present case of quasi-one-dimensional quantum dots, the polyad quantum number can be defined as the sum of the one-dimensional harmonic-oscillator quantum numbers for all electrons. 

When more than one anharmonic interaction term couples near-degenerate, zero-order levels, a simple vector orthogonalization technique can be used to generate a complete set of the dynamically important (i.e., approximately conserved) polyad quantum numbers (Fried and Ezra, 1987 Kellman, 1990). For example, in acetylene, HC = CH, where the ratios of normal mode frequencies u2 W3 W4 W5 are approximately 5 3 5 1 1, modes 1, 2, and 3 are stretching modes (respectively symmetric CH stretch, and CC stretch, and antisymmetric CH stretch), modes 4 and 5 are bending modes (trans-bend and cis-bend), each polyad is labeled by 3 polyad quantum numbers,... 

The 9 important anharmonic resonances destroy all of the 7 normal mode quantum numbers (iq, rq, v3, u4, I4, v3, I5), but three approximately conserved polyad quantum numbers remain,... 

Figure 9.13 Phase space trajectories and polyad phase spheres for polyad N = 3 of H2O. H2O is an example of a molecule with two identical coupled anharmonic bond stretch oscillators. The local mode (Iz, ip) phase space trajectories in part (a) contain the same information as the normal mode (lz,ip) trajectories in part (b). The corresponding polyad phase spheres in parts (c) and (d) are identical except for a rotation by tt/2 about the y axis (from Xiao and Kellman, 1989).

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

The normal mode representation of the phase space trajectories contains the same information as the local mode representation. However, the resonance region on the normal mode phase space map contains the local mode trajectories (la, lb 2a, 2b) and the stable fixed points Ca and C t,. The trajectories contained within the resonance zone are not free to explore the entire 0 < tp < n range whereas the trajectories outside the resonance zone do explore the 0 < ip < n range and are therefore classified as normal mode trajectories. The fixed point B (Iz = I = +2) is unstable, because it lies on a separatrix, and is located at the north pole of the normal mode polyad phase sphere. The stable fixed point A (7Z = — I = —2) is located at the south pole. 

Figure 9.16 shows the evolution of the local mode polyad phase spheres for H2O as I increases from 1 (N = vs + va = 1) to 3 (N = vs + va = 5). As I increases the lowest energy levels sequentially pass through the unstable fixed point B, depart from the normal mode resonance region, and become local mode states. At I = 1 (part (a)) there are only 2 levels and both are on the normal mode side of the separatrix. At I = 3 (part (e)) there are 6 levels and the lowest 4 of these have departed the resonance region and are local mode states. 

Figure 9.15 Evolution of the polyad phase sphere from the local mode to the normal mode limit as the strength of the 1 1 coupling term (antithetical to the local mode limit) is increased from 0 (part a) to oo (part f). As the coupling term, <5 in Eq. (9.4.174) increases from 0, first in part (c) one trajectory (level 4, at highest E) falls through the unstable fixed point into the normal mode region (antisymmetric stretch) eventually, in part (e), the resonance zone fills the entire phase sphere finally, in part (f), the normal mode limit is reached (from Xiao and Kellman, 1989).

The v4l4v5l5) normal mode basis states form polyads (see Section 9.4.5), which are labeled by the approximately good polyad quantum numbers, N and... 

Figure 9.18 Surfaces of section for the HCCH [Nj, l] = [4,0] and [8,0] polyads. The surface section for the [4,0] polyad shows that all of phase space is divided between cis-bend and trans-bend normal modes. The phase space structure for the [8, 0] polyad contains large scale chaos as well as at least two new qualitative behaviors (from Jacobson, et al., 1999).

Although we will not discuss in detail this particular aspect of anharmonic resonances, it is important to note that Darling-Dennison couplings are automatically included by the action of the Majorana operator. A practical way to convince ourselves of this inclusion is to diagonalize (either numerically or in closed form) the Hamiltonian matrix explicitly for the first two polyads of levels and then to convert, in normal-mode notation, the vibrational states obtained. As discussed in Ref. 11, the Hamiltonian (4.38) can also be written (neglecting Cj2 and Cj2 interactions) as... 

At this point it is useful to define creation (at) and annihilation (a) operators, which are analogous to angular momentum raising and lowering operators. These at, a operators profoundly simplify the algebra needed to set up the polyad Heff matrices, to apply some of the dynamics diagnostics discussed in Sections 9.1.4 and 9.1.7, and to transform between basis sets (e.g., between normal and local modes). They also provide a link between the quantum mechanical Heff model, which is expressed in terms of at, a operators and adjustable molecular constants (evaluated by least squares fits of spectra), and a reduced-dimension classical mechanical HeS model. 

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