Quadratic potential functions

The matrix elements of the inactive electrons and the interaction between the active and inactive electrons can be approximated by expressing the corresponding potential surfaces as a quadratic expansion around the equilibrium values of the various internal coordinates, and by nonbonded potential functions for the interaction between atoms not bonded to each other or to a common atom ... 

The electron-diffraction method is not well suited for suggesting what kind of series or what kind of mathematical function should be chosen for a given molecular problem22, 23 In the case of C3O212c 24, 25 for example, the electron-diffraction method is able to exclude the possibility of a pure quadratic term, but is not able to distinguish between the two potential functions which have been suggested ... 

The ratio v /vq differs slightly from this harmonic ratio due to deviation of the true potential function from a quadratic form, as depicted in Fig. 1. A closer approximation to the solid curve can be had by adding cubic and higher anharmonic terms to U r), viz.,... 

In this paper, we will treat the nonlinear couplings, which are linear in the system coordinate but quadratic in the external heat bath coordinates, and show that the GHT is not equivalent to the multi-dimensional TST for the whole solution system if the potential function contains the nonlinear couplings. In Sec. II, we introduce the microscopic Hamiltonian (IIA), evaluate the GH rate expression (IIB), and, in Sec. IIC, we compare it with the multidimensional TST rate for the whole solution system. Finally, the main points are summarized in Sec. III. 

The most popular classification methods are Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), Regularized Discriminant Analysis (RDA), K th Nearest Neighbours (KNN), classification tree methods (such as CART), Soft-Independent Modeling of Class Analogy (SIMCA), potential function classifiers (PFC), Nearest Mean Classifier (NMC) and Weighted Nearest Mean Classifier (WNMC). Moreover, several classification methods can be found among the artificial neural networks. 

Symmetric single minimum quartic-quadratic potential function... 

During the late 1950 s and early 1960 s when the initial work on the ring puckering in trimethylene oxide was done, data were much harder to obtain. High-resolution far infrared spectroscopy was in its infancy and Raman spectra of puckering vibrations had not yet been obtained. Today there is a wealth of data available on trimethylene oxide that strikingly demonstrates the success of the simple onedimensional quartic-quadratic Hamiltonian [Eqs. (3.22), (3.27)]. At the same time, since the data are so extensive, the limitations of the simple one-dimensional potential function can be examined. 

In contrast to the case of cyclobutanone, the addition of two more adjustable parameters does not seem warranted in the case of silacyclobutane in that only a small improvement in the fit results. The barrier determined is 442 cm-1, within 2 cm"1 of the barrier determined from the simpler quartic-quadratic potential function. As pointed out by Pringle, the tendency is to weight the microwave data heavily because of the precision of the rotational data compared to that of the measurement of the vibrational intervals in the far-infrared or Raman spectrum. However, in doing so, one fails to recognize the limitations of the Hamiltonian. If the potential func-... 

At any rate, this failure of the simple two-parameter quartic-quadratic Hamiltonian with a constant reduced mass to reproduce simultaneously and precisely the inversion splittings and far-infrared or Raman data should not be considered a serious drawback. Attempts to use this as an indication of a real difference in the shape of the potential function fail to take into account other effects which have been neglected, among them the dependence of the reduced mass on the coordinate. 

Again, this latter effect is of some importance. Referring to Table 4.10 and Eq. (4.7), we see that the vibrational dependence of the quartic term in the effective potential function is quite small, indeed within the quoted uncertainty. For cyclobutane, the reduced quartic potential constant is 26.15 0.07 cm-1 for the ground state and 26.12 0.07 cm-1 for the first excited state of the i>14 mode. On the other hand, the effect on the quadratic term is more noticeable, as expected from Eq. (4.7). For the ground state of p14, it is - 8.87 0.03 cm-1 compared to - 8.76 0.04 cm-1 for the excited state. From these data, we may conclude that the sign of the coefficient of the interaction term Q24Z2 is positive. 

Carreira et al.86) studied the Raman spectrum of 1,3-cyclohexadiene vapor and observed a series of sharp Q branches probably due to Av = 2 transitions of the ringtwisting vibration (Fig. 4.17). The double-minimum potential function derived by fitting the data with the two-parameter quartic-quadratic Hamiltonian is shown in Fig. 4.18. The barrier height is 1099 50 cm-1, about twice as high as the highest... 

The relationship between the potential function K(R) and the observable spectroscopic parameters is summarized in Figure 2. The harmonic vibration frequencies are obtained as the eigenvalues of a secular determinant involving the quadratic force constants and the atomic masses and molecular geometry (the F and G matrices of Wilson s well-known formalism) by a calculational procedure discussed in detail by Wilson, Decius, and Cross.1 The eigenvectors determine the normal coordinates Q in terms of which the kinetic and quadratic potential energy terms are both diagonal (R = LQ). The various anharmonidty constants and vibration/rotation interaction constants are obtained in terms of the... 

It is also observed that oscillation-rotation bands with An = 2, 3, etc. occur this is to be correlated with the deviation of the potential function V(r) from a simple quadratic function. 

Paralleling the harmonic oscillator expansion of the potential function of a mechanical system, we next approximate the equilibrium entropy function 5(Tp A) by its quadratic order Taylor series expansion about A (the point at which 5 has its maximum). That is, we assume... 

The above results were obtained using a product-type trial wave function and then invoking the quantum mechanical boundary conditions in order to obtain a way of defining the parameter P0. If the intermolecular potential is quadratic in the coordinate R, then the TDSE can be exactly fulfilled by the trial function (1) (13). Since the potential is not quadratic, the trial function can only be a good approximation in a short time interval, and hence correction terms have to be included. This can be done formally by using the trial function (16)... 

The fact that the generalized coordinate / is a linear combination of all bath modes and that the potential is quadratic in the bath variables allows one to express the potential of mean force w[f ] in terms of a single quadrature over the system coordinate q. The detailed derivation is presented in Ref. 42, the main technical trick being the usual use of the Fourier representation of the Dirac 8 functions. The resulting expression is... 

In Refs. 44. 52. 55) the same potential function used to calculate <01 is also used to calculate the other tetrahedral frequencies o)t, (03 and (04. It is not clear whether V was redetermined for each type of normal mode V(Q ) should be calculated in the fixed nucleus approximation, shifting the atoms along their actual normal-mode trajectories for the various normal modes then a quadratic potential curve should be fitted. This could be explanation for the bad correlation between to calc and to exp which involve motion of the central x-atom and/or bending modes of the H-atoms (see later). 

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