Hat potential

Figure 6. Mexican-hat potential energy surface of AuCk, after Ref 14.

A quantitative treatment of the Jahn-Teller effect is more challenging (46). A major issue is that many theoretical models explicitly or implicitly assume the Bom—Oppenheimer approximation which, for octahedral Cu(II) systems in the vibronic coupling regime, cannot be correct (46,51). Hitchman and co-workers solved the vibronic Hamiltonian in order to model the temperature dependence of the molecular structure and the attendant spectroscopic properties, notably EPR spectra (52). Others, including us, take a more simphstic approach (53,54) but, in either case, a similar Mexican hat potential energy description of the principal features of the Jahn-Teller effect in homoleptic Cu(II) complexes emerges (Fig. 13). 

Fig. 13. Top Schematic representation of the two components of the Jahn-Teller-active vibrational mode for the E e Jahn-Teller coupling problem for octahedral d9 Cu(II) complexes. Bottom Resulting first-order Mexican hat potential energy surface for showing the Jahn-Teller radius, p, and the first-order Jahn-Teller stabilization energy, Ejt.

Figure 10. (a) The Qe and Qe components of the g vibrational mode, (b) the Mexican hat potential energy surface for copper(II) in an octahedral ligand field, and (c) a cross-section through the warped... 

Spectral function (22) for transverse vibration Projection of the normalized angular momentum on the symmetry axis of the hat potential Length of the hydrogen bond (see Fig. 1)... 

Reduced mass of the water molecule Number of librations of a dipole during lifetime of the hat potential... 

The librational fraction is discussed in the context of the concepts of the water structure The hat potential models the defects of the water (ice) structure and rigid polar molecules reorient relatively freely in these defects. In the case of water the lifetime Tor of this fraction (on the order of 10 13 s) is several times greater than that of the H-bond. 

The equation of motion of a reorienting dipole, governed by the hat potential, is nonlinear, so the law of periodic libration is rich in high harmonics, especially at high temperature. Due to a nonlinear equation of motion, the form of the librational band is far from being Lorentzian. 

An important feature of the LIB state (it is confirmed by experimental data) presents a strong isotopic shift of the loss-peak frequency vor. Since in the chosen (hat) potential a polar molecule librates almost freely, vor is determined by the moment of inertia 7or(H20), which comprises about half of 7or(D20). Therefore, vor(H20) Z2vor(D20). 

The mixed model of water comprising the librational (LIB) and vibrational (VTB) fractions will be used. We employ four molecular mechanisms a, b, c, and d. The first one, a, refers to the LIB state, in which a permanent dipole reorients in the hat potential formed by tom or weak hydrogen bonds (HB). The last three specific mechanisms (b, c, d), governed by vibrating hydrogen bonds, refer to the VIB state. 

Namely, the LDL is in some respect similar to our LIB fraction regarding its connection with the HB network and dominance in contribution to the low-frequency spectrum (described by the first term in (39)), in particular, to the static permittivity 8S. However, in contrast to RAK, it is hardly reasonable to bring this fraction into correlation with the HB network itself due to the almost free libration of the dipoles in an intermolecular (hat) potential. On the other hand, it is reasonable to assign the VIB fraction to the HB network, which in our simplified calculation scheme is modeled by a dimer of oppositely charged water molecules connected by a hydrogen bond. Thus, in our opinion... 

Thin lines in Figs. 5c-h and 6c-h refer to specific contributions due to nonharmonic reorientation of a permanent dipole in the hat potential (1), harmonic longitudinal vibration of HB nonrigid dipole (2), harmonic reorientation of a permanent HB dipole (3), and nonharmonic transverse vibration of a nonrigid HB dipole (4). 

Figure 12 Fitted parameters of the molecular model (a, b) Effective transverse intensity (a) and ratio of transverse frequency factors (b) solid curves refer to optimization o dielectric and dashed lines of Raman spectra, (c) Lifetimes of the hat potential, of transverse vibration, and of elastic vibration (from top to bottom), (d) form factor of the hat potential.(e) contributions of transverse (dashed line) and of elastic (solid line) vibrations to the static permittivity, (f) total vibration contribution to the static permittivity.

Figure 21 Profile of the hat potential expressed in degrees of K. Calculation for ice at the temperature —7°C (a) and for water at 27°C (b).

It is interesting to estimate the mean angular turn 0T of a rigid dipole, occurring in the hat potential during the lifetime time Tor. This turn is characterized by the number m of librations... 

We remark that the potential widths 2/3 are almost equal in the cases of water and ice. A rigid dipole, rotating in the hat potential, lives in water longer than in ice, since during the lifetime Tor(water) such a dipole performs about two librations (mor 2), while during Tor(ice) it performs about a half of one libration. 

In the frequency range 400-1000 cm-1 the a(v) and s" v) ice spectra are determined mostly by reorientation of rigid dipoles in the hat potential whose... 

It appears that in the structure, corresponding to the hat potential, hydrogen bonds are strongly bent or broken. Correspondingly, the law of motion of a dipole, governed by such a potential, substantially differs from the law determined by a harmonic-like potential. 

In order to pass on to pure elastic reorientations [we have in mind that the latter occur in the parabolic (and not in the hat) potential], we let/and w tend to infinity. Using (138a), (130b), and (130d), we find analogously to (150) and (151) the corresponding spectral function L ... 

The following three circumstances are characteristic for the hat-potential model ... 

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