Nuclear curvature

Fig. 36.23. Illustration of nuclear curvature and relaxation vibrational effects.

Fig. 1-16. Moseley plot for Ka2 lines. The curvature at high Z is due to a change in the effective nuclear charge (Z — 1). The insert shows the atomic number Z to be more fundamental than the atomic weight M. X-rays made possible the first experimental determinations of Z. Crosses = atomic weight dots = atomic number.

The rate data display curvature for Pco, and the authors supposed that the rate limiting step was preceded by CO addition to the metal complex. Therefore, they proposed that the rate limiting step was reaction of either MCO+ or M2CO +, M = Rh(CO)2(4-pic)x + with H20. Interestingly, the turnover frequency increased as the concentration decreased. The authors ascribed this behavior to a higher activity for Rh complexes with lower nuclearity ( e.g., mononuclear). They proposed a mechanistic scheme (Scheme 43) whereby mononuclear and dinuclear complexes exhibit parallel catalytic cycles, joined by an equilibrium for monomer-dimer formation. 

The strong curvature of the Eyring plots at temperatures below 35°C (see ref. 19a, Table V and Figure 6) points to hydrogen-bonded interactions between catalyst molecules, as already observed in 1969 by Uskokovic (52) in the H nuclear magnetic resonance (NMR) spectrum of dihydroquinine. 

A simple diagram depicting the differences between these two complementary theories is shown in Fig. 1, which represents reactions at zero driving force. Thus, the activation energy corresponds to the intrinsic barrier. Marcus theory assumes a harmonic potential for reactants and products and, in its simplest form, assumes that the reactant and product surfaces have the same curvature (Fig. la). In his derivation of the dissociative ET theory, Saveant assumed that the reactants should be described by a Morse potential and that the products should simply be the dissociative part of this potential (Fig. Ib). Some concerns about the latter condition have been raised. " On the other hand, comparison of experimental data pertaining to alkyl halides and peroxides (Section 3) with equations (7) and (8) seems to indicate that the simple model proposed by Saveant for the nuclear factor of the ET rate constant expression satisfactorily describes concerted dissociative reductions in the condensed phase. A similar treatment was used by Wentworth and coworkers to describe dissociative electron attachment to aromatic and alkyl halides in the gas phase. ... 

Fig. 5. The pseudo-Jahn-Teller effect in ammonia (NH3). (a) CCSD(T) ground state potential energy curve breakdown of energy into expectation value of electronic Hamiltonian (He), and nuclear-nuclear repulsion VNN. (b) CASSCF frequency analysis of pseudo-Jahn-Teller effect showing the effect of including CSFs of B2 symmetry is to couple the ground and 1(ncr ) states to give a negative curvature to the adiabatic ground state potential energy surface for the inversion mode.

A stable nuclear configuration on a potential energy surface is associated with a point for which there is zero slope in any direction and for which there is no direction in which the curvature is negative or zero. Such points are uniquely defined in any system of internal coordinates but we shall see that some other characteristic features of a surface are dependent on the choice of coordinate. 

In theory, a properly developed force field should be able to reproduce structures, strain energies, and vibrations with similar accuracies since the three properties are interrelated. However, structures are dependent on the nuclear coordinates (position of the energy minima), relative strain energies depend on the steepness of the overall potential (first derivative), and nuclear vibrations are related to the curvature of the potential energy surface (second derivative). Thus, force fields used successfully for structural predictions might not be satisfactory for conformational analyses or prediction of vibrational spectra, and vice versa. The only way to overcome this problem is to include the appropriate type of data in the parameterization process 501. 

In the above formula, Q is the nuclear coordinate, p, and I/r are the ground state and excited electronic terms. Here Kv is provided through the traditional Rayleigh-Schrodinger perturbation formula and K0 have an electrostatic meaning. This expression will be called traditional approach, which has, in principle, quantum correctness, but requires some amendments when different particular approaches of electronic structure calculation are employed (see the Bersuker s work in this volume). In the traditional formalism the vibronic constants P0 dH/dQ Pr) can be tackled with the electric field integrals at nuclei, while the K0 is ultimately related with electric field gradients. Computationally, these are easy to evaluate but the literally use of equations (1) and (2) definitions does not recover the total curvature computed by the ab initio method at hand. 

Recognition of space-time curvature as the decisive parameter that regulates nuclear stability as a function of the ratio, Z/N, with unity and the golden mean, r, as its upper and lower limits, leads to a consistent model for nucleogenesis, based on the addition of -par tides in an equilibrium chain reaction. This model is also consistent with the limitations imposed by the number spiral. 

Bonds with r < dl < d[ become possible because of nuclear screening (increased bond order), which causes concentration of the bonding pair directly between the nuclei. The exclusion limit is reached at d = t and appears as a geometrical property of space. The distribution of molecular electron density is dictated by the local geometry of space-time. Model functions, such as VSEPR or minimum orbital angular momentum [65], that correctly describe this distribution, do so without dictating the result. The template is provided by the curvature of space-time which appears to be related to the three fundamental constants tt, t and e. 

Extrapolation of the hem lines to Z/N = 1 defines another recognizable periodic classification of the elements, inverse to the observed arrangement at Z/N = t. The inversion is interpreted in the sense that the wave-mechanical ground-state electronic configuration of the atoms, with sublevels / < d < p < s, is the opposite of the familiar s < p < d < f. This type of inversion is known to be effected under conditions of extremely high pressure [52]. It is inferred that such pressures occur in regions of high space-time curvature, such as the interior of massive stellar objects, a plausible site for nuclear synthesis. 

When the molecule is raised to the electronic excited state, the normal mode Q preserves the same decomposition with respect to the nuclear displacements, except for Jahn-Teller effects, which we exclude in this work. In these conditions, still in the harmonic approximation, two parameters, the equilibrium point and the potential curvature, will change in the excited state (in the particular case of non-totally-symmetric vibrations, the equilibrium point does not change). In the excited state e, the nuclear potential becomes... 

When applied to isotope effects, the main weakness of the reflection method is the assumption that the transition dipole moment is constant for all isotopologues. This weakness remains in the improved model presented below. Only ab initio calculations are able to go beyond this approximation. However, the dependence of the transition dipole moment along the nuclear coordinates can be introduced (numerically or analytically) in the model below, even if a less compact analytic form is expected. This paper is organized as follows in Section 2 the "standard" reflection model is improved by taking into account the curvature of the upper state potential (in addition to its slope). In Section 3, the quantum character of the final state is taken into account by replacing the Dirac function by an Airy function. In Section 4 the model is applied to the CI2 molecule. In Section 5 the model is adapted and applied to the O3, SO2 and CO2 triatomic molecules. Conclusions and perspectives are presented in Section 6. 

To conclude, most of the asymmetry observed in the singlet-singlet transition in the Abs. XS of CI2 can be ascribed to the curvature of the 77iu upper potential and not to the quantum effect described in Section 3, nor to the dependence of the transition dipole moment. However, this is probably not always the case and the three effects (the curvature effect (Section 2), the quantum effect (Section 3) and the nuclear dependence of the transition dipole moment) may contribute to the asymmetry of any XS is analyzed. It is important to note that the asymmetry due to the "quantum effect" detailed in Section 3 does not require an additional parameter in the fit. Consequently, the first (and easy ) step is to compare two fits of a a(E)/E both using Formula (27 ) the first fit with t constrained to f = 2 hco/ Ve/P) (Formula (26)) and a second fit with f as a free parameter. This comparison allows us to estimate if the quantum effect is dominant or not and then, to know whether another contribution is important, e.g., the curvature of the upper potential. In addition, the contribution of the hot bands discussed by Burkholder [4] and Alder-Golden [24] can also contribute to observed asymmetry. 

In an alternative approach, the reference curvature b is scaled by the diameter d(K) of the 3D nuclear configuration K. If r(K) = 0.5 d(K) is the radius of the smallest sphere that encloses all the nuclei of the given nuclear configuration K, then the scaled relative curvature parameter bK is defined as... 

The resulting curvature domains Do(bK)> D (bK). and D2(bK) are not invariant with respect to the size of the G(a) objects (this size is dependent on the contour parameter a), nevertheless, the scaling is specific for the size of the nuclear arrangement K, hence these shape domains provide a valid shape comparison of MIDCO s or other molecular surfaces of molecules of different sizes. This approach is simpler than the fully size-invariant approach using the reference curvature be, where a new scaling factor r(G(a)) is required for each new MIDCO G(a). 

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