Coordinates parametrized

Tx represent the mass, frequency and distance traversed by the heavy atoms that control the distance between the H-donor and acceptor (such that Ar = ro — r ). According to this model, the configuration at which tunneling occurs is comprised of three coordinate systems (i) the environmental or solvent coordinate parametrized by the reaction driving force (AG°) and environmental reorganization (A) ... 

To do this one must introduce the concept of vibrational adiabati-city during a collision process. This notion is not new. The foundation was laid by Marcus from both a classical and a quantal point of view. The basic idea is that one picks out a reaction coordinate parametrized by u and assumes that motion along this coordinate is much slower than motion along the other degrees of freedom denoted by v. 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

The separation of nuclear and electronic motion may be accomplished by expanding the total wave function in functions of the election coordinates, r, parametrically dependent on the nuclear coordinates... 

Note that the last terra in expression (2) of V does not depend on electronic coordinates, and therefore it may be neglected in fifo. The adiabatic Hamiltonian still depends parametrically on R, and so is the electronic wave funcion 4>). If we expand the nuclear coordinates or some of the nuclear coordinates with respect to a given configuration, that is, if we define... 

Since the form of the electronic wave functions depends also on the coordinate p (in the usual, parametric way), the matrix elements (21) are functions of it too. Thus it looks at first sight as if a lot of cumbersome computations of derivatives of the electronic wave functions have to be carried out. In this case, however, nature was merciful the matrix elements in (21) enter the Hamiltonian matrix weighted with the rotational constant A, which tends to infinity when the molecule reaches linear geometry. This means that only the form of the wave functions, that is, of the matrix elements in (21), in the p 0 limit are really needed. In the above mentioned one-elecbon approximation... 

As usually indicated by the semicolon, both the wave functions and eigenvalues [ (R)] depend parametrically on the internal nuclear coordinates. 

Space Curves Space curves are usually specified as the set of points whose coordinates are given parametrically by a system of equations x =f t), y = g(t), z = h t) in the parameter t. 

A number of empirical tunneling paths have been proposed in order to simplify the two-dimensional problem. Among those are MEP [Kato et al. 1977], sudden straight line [Makri and Miller 1989], and the so-called expectation-value path [Shida et al. 1989]. The results of these papers are hard to compare because slightly different PES were used. As to the expectation-value path, it was constructed as a parametric line q(Q) on which the vibration coordinate q takes its expectation value when Q is fixed. Clearly, for the PES at hand this path coincides with MEP, since is a harmonic oscillator. 

This treatment differs from the usual approach to molecule-radiation interaction through the inclusion of the contribution from the electric field from the beginning and by not treating it as a perturbation to the field free situation. The notation 7/ei(r R, e(f)) makes the parametric dependence of the electronic Hamiltonian on the nuclear coordinates and on the electric field explicit. 

The solution of the Schrodinger equation with Helec is the electronic wave function xTelec and the electronic energy Eelec. depends on the electron coordinates, while the nuclear coordinates enter only parametrically and do not explicitly appear in Pelec. The total energy Etot is then the sum of Eelec and the constant nuclear repulsion term, M M... 

Nyquist plot Bode plot Nyquist plot is a frequency parametric plot of the magnitude and the argument of the open-loop transfer function in polar coordinates. Bode plot is magnitude vs. frequency and phase angle vs. frequency plotted individually. 

In calibration, the number of degrees of freedom depends on the number of parameters estimated by the given model. In case of the two-parametric model (Eq. 6.6) v = n — 2, in case of linear calibration through the coordinate origin (a = 0) v = n — 1, and in case of a three-parametric nonlinear calibration model... 

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