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Mexican hat

An example that is closely related to organic photochemishy is the x e case [70]. A doubly degenerate E term is the ground or excited state of any polyatomic system that has at least one axis of symmetry of not less than third order. It may be shown [70] that if the quadratic tenn in Eq, (17) is neglected, the potential surface becomes a moat around the degeneracy, sometimes called Mexican hat, The polar coordinates p and <(>, shown in Figure 20, can be used to write an expression for the energy ... [Pg.356]

It was shown by several workers that in this case the first-order Jahn-Teller distortion is due to an ej vibration, and that the second-order distortion vanishes. Therefore, in terms of simple Jahn-Teller theoi, the moat around the symmetric point should be a Mexican hat type, without secondary minima. This expectation was borne out by high-level quantum chemical calculations, which showed that the energy difference between the two expected C2v structures ( A2 and Bi) were indeed very small [73]. [Pg.359]

There exist different pairs of wavelets and scaling functions. One such pair is shown in Fig. 4. This is the Mexican hat pair (Daubechies, 1992), which draws its name by the fact that the scaling function looks like the... [Pg.184]

Gaussian and the wavelet like the Gaussian second derivative ( Mexican hat function). [Pg.185]

Fig. 44.23. Some common neighbourhood functions, used in Kohonen networks, (a) a block function, (b) a triangular function, (c) a Gaussian-bell function and (d) a Mexican-hat shaped function. In each of the diagrams is the winning unit situated at the centre of the abscissa. The horizontal axis represents the distance, r, to the winning unit. The vertical axis represents the value of the neighbourhood function. (Reprinted with permission from [70]). Fig. 44.23. Some common neighbourhood functions, used in Kohonen networks, (a) a block function, (b) a triangular function, (c) a Gaussian-bell function and (d) a Mexican-hat shaped function. In each of the diagrams is the winning unit situated at the centre of the abscissa. The horizontal axis represents the distance, r, to the winning unit. The vertical axis represents the value of the neighbourhood function. (Reprinted with permission from [70]).
In the Mexican Hat function (Figure 3.20), the weights of the winning node and its close neighbors are adjusted to increase their resemblance to the sample pattern (an excitatory effect), but the weights of nodes that are... [Pg.75]

Figure 6. Mexican-hat potential energy surface of AuCk, after Ref 14. 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). [Pg.16]

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. 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.
A recent study of 23 Cu(II)-bispidines complexes with 19 different ligands (58) is further proof that LFMM provides an excellent description of the whole of the Mexican hat PE surface. Depending on the ligand, all three elongation axes can be located. In each case, theory and experiment agree (Fig. 15). [Pg.19]

Figure 67 (a) The adiabatic potential energy surfaces (Mexican hat) (b) the normal coordinates and (c) th projection of the potential energy surface (a) warped by the inclusion of higher order terms viewed down the prinrip axes of (a), with R]T = radius of the minimum potential432... [Pg.692]

Figure 76 A section through the potential energy surfaces of the Mexican hat for (a) equivalent ligands and (b)... Figure 76 A section through the potential energy surfaces of the Mexican hat for (a) equivalent ligands and (b)...
Linear coupling (termed the Mexican hat ) a continuous slab of the energy minima exists at the Jahn-Teller radius... [Pg.185]

One well-known example of the physical system where the described approach can be applied is the Aj-E optical transition in trigonal centers with the E-e-type Jahn-Teller interaction. In this case the AP in the electronic E-state has a shape of the Mexican hat [11 -14]. If the E-e-type Jahn-Teller interaction is strong, then the Mexican hat is large and the AP has a long minimum line along the trough. In this... [Pg.136]

In this communication we will give a description of the vibronic E-e interaction in an optical center in a crystal near one of the minima of the trough of the deformed (due to the quadratic vibronic coupling) Mexican-hat-type AP. We will also present a derivation of the nonperturbative formula describing the temperature dependence of the ZPL in the case of an arbitrary change of the elastic springs on the electronic transition. Then we will study a case when the excited state is close to the dynamical instability. Finally, we will apply the obtained general results to the ZPLs in N-V centers in diamond. [Pg.138]

As a rule, the oc cos(3 )- term is small. With account of this term, the difference of the potentials of the E- and A t -electronic states near a minimum/maximum of the deformed Mexican hat has a form... [Pg.139]

If c remarkably differs from zero then there are three equivalent minima and three equivalent saddle points on the trough of the deformed Mexican hat [see equation (3)], which are described by different signs in equation (3) for V. Therefore one can observe two ZPL, which corresponds to transitions into the vicinity of a minimum and into vicinity of a saddle point. The frequency difference of these lines is v2 c j(yv c2)- The lower frequency line corresponds to the optical transition into a minimum, which is dynamically stable. The higher frequency line corresponds to the optical transition into a saddle point, which is dynamically unstable. The temperature dependence of these lines at low temperatures are described by equations (15) and (16) with w= — (wj 9Id) (the sign + in the brackets corresponds to the line with higher frequency). The width of the line with lower frequency at T = 0 is equal to zero the width of the higher frequency line at T = 0 differs from zero it is described by equation (17) with wcr — w = 9 c. ... [Pg.146]


See other pages where Mexican hat is mentioned: [Pg.358]    [Pg.358]    [Pg.361]    [Pg.115]    [Pg.47]    [Pg.185]    [Pg.192]    [Pg.212]    [Pg.75]    [Pg.76]    [Pg.76]    [Pg.464]    [Pg.464]    [Pg.467]    [Pg.49]    [Pg.83]    [Pg.116]    [Pg.691]    [Pg.698]    [Pg.700]    [Pg.324]    [Pg.107]    [Pg.308]    [Pg.138]    [Pg.139]    [Pg.202]    [Pg.278]    [Pg.394]   
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