Dimensionless normal coordinate

Here the dimensionless normal coordinates of the tetragonal (e) vibrations are qu and qv those of the trigonal fa) vibrations are q, qn, and q. At this simple point several different nomenclatures exist. Hereafter, the (normal) vibrational coordinates Q relate to the dimensionless coordinates q by the formula q = Q(Mco/h)1/2, where M is the effective mass of the vibrator and co its frequency [88]. The force constant is K = a M. 

Here the nuclear coordinates Q are taken to be dimensionless normal coordinates of the neutral ground state characterized by subscripts x and y in case of the doubly degenerate modes (where cartesian displacement components are meant). The vertical IPs are denoted by Ex and EB, with an obvious labelling. TN and Vo collect the nuclear kinetic energy and unperturbed harmonic potential, with frequency a), for the z-th mode. The quantities Af and Af are the usual linear JT coupling constants. 

The V-B coupling Hamiltonian to first order in the three HOD dimensionless normal coordinates is Hv b = —2, c], l , where F, is the inter-molecular force due to the solvent exerted on the harmonic normal coordinate, evaluated at the equilibrium position of the latter. This force obviously depends on the relative separations of all molecules, and on their relative orientations. In the most rigorous quantum description of rotations, this term would depend on the excited molecule rotational eigenstates and of the solvent molecules. Instead rotation was treated classically, a reasonable approximation for water at room temperature. With this form for the coupling, the formal conversion of the Golden Rule formula into a rate expression follows along the lines developed by Oxtoby (2,53), with a slight variation to maintain the explicit time dependence of the vibrational coordinates (57),... 

The force constants in dimensionless normal coordinates are usually defined in wavenumber units by the... 

At last, a few steps are necessary to make the expansion of Fq. (5) amenable to quantum CPT procedures The dimensionless normal coordinates for the stretch degrees of freedom are expressed in terms of the ladder operators... 

The vibrational displacements are described in terms of dimensionless normal coordinates Qx, Qy of a degenerate vibrational mode of E symmetry. The electrostatic Hamiltonian is expanded at the reference geometry in powers of Qx, up to second order... 

Defining Q as the set of dimensionless normal coordinates g of the (model) harmonic ground state potential To(Q) with frequencies coi, we obtain [6,14] ... 

The vibronic coupling in the radical and radical cation of aromatic hydrocarbons is studied by photoionizing the corresponding anion and neutral molecules, respectively. The vibronic Hamiltonian of the final states of the ionized species is constructed in terms of the dimensionless normal coordinates of the electronic ground state of the corresponding (reference) anion or neutral species. The mass-weighted normal coordinates ) are obtained by diagonalizing the force field and are converted into the dimensionless form by [68]... 

To analyze the vibronic structures of the X, A and B electronic states Ph we constructed a vibronic Hamiltonian in a diabatic electronic basis which treats the nuclear motion in the X state adiabatically, and includes the nonadiabatic coupling between the A and B electronic states. The Hamiltonian terms of the dimensionless normal coordinates of the electronic ground state (XMi) of phenide anion is given by [19]... 

The model diabatic vibronic Hamiltonian of the Do — Di — D2 electronic manifold can be expressed in terms of dimensionless normal coordinates of N as [20]... 

The second derivatives of the third term are the force constants , Ftr, which enter into the GF-treatment as discussed above. Consequently we can transform to a more convenient expression in the dimensionless normal coordinates,... 

Here = (2 rrc)-1 A, Ag, higher order terms are constants chosen as a sort of mean value of the p-dependent quantities Xfc, > and so on, in order to minimize the perturbations from the corresponding terms in Vb. Notice, that now we use the constant A in the transformation to dimensionless normal coordinates,... 

Table 3. Derivatives of Bgg elements with respect to dimensionless normal coordinates, qk = (2 ffcwfr/h)1/2 Qk...

Substituting Eq. (5.14) into Eq. (5.7) and introducing dimensionless normal coordinates, the potential function becomes,... 

N02,239,247,248,256 and C102256 should be regarded as tentative since they are based on various model potentials (as indicated in the footnotes) and on much less extensive sets of data. Table 3 lists the values of the cubic constants in the valence-force coordinate space (fyt) and in the dimensionless normal coordinate space (kg/s") as obtained for... 

To obtain the perturbative solutions for the (J = 0) Hamiltonian we take as our starting point the vibrational Hamiltonian of Eq. (2), where the coordinate-dependent terms G, V, and V are expanded in a Taylor series in the dimensionless normal coordinates. The coordinates Q, and corresponding momenta P, are reexpressed in terms of harmonic oscillator raising and lowering operators aj and ah where... 

In the preceding subsection, the interference effects between nuclear WPs of DCP were numerically treated. In this subsection, to confirm the interference effects, we present the results of an analytical treatment in a simplified one-dimensional model shown in Fig. 6.10. Here, q is the dimensionless normal coordinate of the effective breathing mode. The potentials in the ground and two electronic excited states (b and c, which correspond to L and H, respectively) were assumed to be displaced and undistorted ones. At least two vibrational eigenstates in each electronic state are needed for consideration of both the electronic and vibrational coherences in the simplified model. Here, b0(c0) and bl(cl) denote the lowest... 

The vibrational potential energy is usually expanded in terms of dimensionless normal coordinates qr as [63A11, 84Gor]... 

One possible way to include mode-mode couplings in normal coordinates is by perturbation theory. The perturbation-theory expressions for the energies of a polyatomic system are usually given in terms of dimensionless normal coordinates, m=l, 2, . . . , F-1. These are-... 

Equations (29)-(3l) are for the case of nondegenerate vibrations the modifications in these equations for degenerate vibrations may be found eIsewhere. 6-68 por the discussion below we emphasize that eqs. (29)-(31) are based on a knowledge of the cubic and some of the quartic dimensionless normal coordinate force constants. 

By repeated application of the chain rule for derivatives, this potential energy can be transformed through quartic terms to the representation in dimensionless normal coordinates of eq. (28) in the standard way.Since the internal coordinates are curvilinear while the normal coordinates are not, this transformation is necessarily non-1 inear. 

The transformation from the internal coordinate force constants K3(s), Ka 3(s), etc. to the dimensionless normal coordinate force constants ki(s), kij(s), etc. can be accomplished through a set of several steps. For this purpose it is convenient to employ a dual notation for the atomic cartesian coordinates. Let be an index such that... 

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