Potential eneigy

Addition of a [ Icnn (Fig. 4-13) lo the. V, V2. and V Icmi (Fig. 4-1 T) has Ihc effect t.)f raising the eciilral maximum and liie twt.) symmetrical minima witii(.)tit ciianging the energy of tlie stable ami forni, so producing a potential eneigy function that shows tlie cjualitative form of Fig, 4-1 2, In this context, tlie V j term is called a low-order torsional term. 

Fig. S.l. Potential eneigy diagram for nucleophilic substitution by the ionization (S l) mechanism.

At both minima and saddle points, the first derivative of the energy, known as the gradient, is zero. Since the gradient is the negative of the forces, the forces are also zero at such a point. A point on the potential eneigy surface where the forces are zero is called a stationary point All successful optimizations locate a stationary point, although not always the one that was intended. 

The reaction path from the initial state to the final state of an elementary step is represented by the potential eneigy curves of the initial and final states of a reacting particle as shown in Fig. 7-6, where the reaction coordinate x denotes the position of a reaction particle moving across a compact double layer on the electrode interface. 

Fig. 7-g. Potential eneigy curves for an elementary step of reaction G = particle energy, x = reaction coordinate P = electrochemical potential of particles p- = electrochemical potential of an activated particle - dft-F = step aiSnity,... 

Fisher, J.J. Koyanagi, G.K. McMahon, T.B. The C2H7 Potential Eneigy Surface a Fourier Transform Ion Cyclotron Resonance Investigation of the Reaction of Methyl Cation with Methane. Int. J. Mass Spectrom. 2000, 795/796,491-505. 

R. Schinke In the case of HNO and HO2, we calculated the number of states and simply extrapolated this number into the continuum. We believe that this is the best what can be done, provided a global potential eneigy surface and full dimensionality dynamics calculations for this potential are available. Because of the much smaller number of states, for HCO this procedure is less well defined. In our final analysis (Ref. 33 of our paper in this volume) we tested the extrapolation from the bound to the continuum region and an estimation of the density of states based on a Dunham expansion of the term energies and found that both recipes give essentially the same result. 

Figure 9. Schematic representation of upper portion of potential eneigy surface for merging of substitution mechanisms. A Sjsj 1 mechanism. No nucleophilic solvation in transition state ion pair intermediate (possibly nudeophilically solvated) B Sn2 (intermediate). Transition state is nudeophilically solvated by solvent (SOH) intermediate is a nudeophilically solvated ion pair (see Fig. 8) C Classical Sn2. No energy minimum. In curves A and B, the second transition state may be of higher energy than the first in cases where internal return is important.

Assume that the surroundings are at 298 K, the kinetic and potential eneigy changes are negligible, and this is a steady process. 

Certain writers, instead of keeping a special name for the quantity prefer to consider the product EU, which they call the potential energy of the S3rstem in the state X] the potential eneigy is then a quantity of the same kind as and or a quantity measured in units of work we may say that the potential energy is the equivalent Oj the internal energy. The quantity ... 

FIGURE 30. Potential eneigy curves for a neutral molecule M, and its radical cation M+ in the ground and first excited state ( eq ai the equihbrium distances with respect to an arbitrary coordinate q along which the three geometries differ). Note the shift in the M+ /(M+ ) energy difference A on going from ijeq of M (A = A/y from the PE spectrum of M) to ijeq of M+ (A corresponds to Amax from the EA spectrum of M+ )... 

Alexander, M. H. (1993) Adiabatic and Approximate Diabatic Potential Eneigy Surfaces for the B... H2 van der Waals Molecule, J. Chem. Phys. 99, 6014-2026. 

Kinetic and Potential Eneigies of Harmonic Oscillator Computed from Eqs. (4.49) and (4.50)... 

Figure 5.7. Franck-Condon factors for radiationless transitions between different potential energy curves of a diatomic molecule a) for a large and b) for a small energy gap, such as those observed, for instance, between S and S, or between Sj and S respectively, and c) for the case that the potential eneigy curves (e.g., S, and T,) cross.

The observed vibration frequencies of a molecule depend on two features of the molecular structure the masses and equilibrium geometry of the molecule and the potential eneigy surface, or force field, governing displacements from equilibrium. These are described as kinetic and potential effects, respectively for a polyatomic molecule the form and the frequency of each of the 3N—6 normal vibrations depend on the two effects in a complicated way. The object of a force field calculation is to separate these effects. More specifically, if the kinetic parameters are known and the vibration frequencies are observed spectroscopically, the object is to deduce the potential eneigy surface. A major difficulty in this calculation is that the observed frequencies are often insufficient to determine uniquely the form of the potential energy surface, and it is necessary to use data on the frequency shifts observed in isotopically substituted molecules or data on vibration/rotation interaction constants observed in high resolution spectra in order to obtain a unique solution. 

Figure 2,1. The PES for a diatomic molecule. The potential eneigy increases if the bond length q is stretched or compressed away from its equilibrium value q. The potential energy at qt (zero distortion of the bond length) has been chosen here as the zero of energy.

Figure 2.2. Actual molecules do not sit still at the bottom of the potential eneigy curve, but instead occupy vibrational levels. Also, only near qe, the equilibrium bond length, does the quadratic curve approximate the true potential energy curve.

Figure 5.40. The distribution of electron density (charge density) p for an atom the nucleus is at the origin of the coordinate system, (a) Variation of p with distance from the nucleus. Moving away from the nucleus p decreases from its maximum value and fades asymptotically toward zero, (b) Variation of —p with distance from the nucleus —p becomes less negative and approaches zero. The —p picture is useful for molecules (Fig. 5.41) because it makes clearer analogies with a potential eneigy surface, (c) A 4D picture (p vs. x, y, z) of the variation of p in an atom the density of the dots (number of dots per unit volume) indicates qualitatively electron density p in various regions.

Example Spatial Oscillator.—A massive particle is restrained by any set of forces in a position of stable equilibrium (t.g. a light atom in a molecule otherwise consisting of heavy, and therefore relatively immovable atoms). The potential eneigy is then, for small displacement, a positive definite quadratic function of the displacement components. The axes of the co-ordinate system (x, y, z) can always be chosen to lie along the principal axes of the ellipsoid corresponding to this quadratic form. The Hamiltonian function is then... 

In Ref. [4] we have studied an intense chirped pulse excitation of a molecule coupled with a dissipative environment taking into account electronic coherence effects. We considered a two state electronic system with relaxation treated as diffiision on electronic potential eneigy surfaces with respect to the generalized coordinate a. We solved numerically equations for the density matrix of a molecular system under the action of chirped pulses of carrier frequency a> with temporal variation of phase... 

Figure 13. The schematic representation of the potential eneigy profile of the ff-meta-thesis reactions.

FIG. 14.14. (a) The liquid-drop model potential eneigy curve, (b) Same, but modified by shell corrections. 

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