Eigenvalues potential function

To determine the vibrational motions of the system, the eigenvalues and eigenvectors of a mass-weighted matrix of the second derivatives of potential function has to be calculated. Using the standard normal mode procedure, the secular equation... 

If the rotational quantum number J is zero, the molecule possesses no angular momentum arising from the motion of the nuclei nuclear motion is purely vibrational. The vibrational energy levels depend on the shape of the potential function 1/(7 ), most often of the well-known diatomic form. Near the minimum the potential function approximates to a parabola. The eigenvalues and functions are thus approximately those appropriate for a harmonic oscillator,... 

Let N denote the number of the degrees of freedom of a system. We also use the term the index of the saddle to indicate the number of negative eigenvalues of the Hessian matrix of the potential function at the saddle. 

After solving Eq. (3.45) we can obtain the eigenvalues and eigenvectors of the inversion—rotation states of ammonia in the ground vibrational state i.e. we can calculate - in the rigid bender approximation - the rotational dependence of the inversion splittings in the states of ammonia. Note that the / and k dependent terms in the Schrodinger equation [Eq. (3.45)] represent a modification of the double-minimum potential function fo(p) for each rotational state/, k (see further Sections 5.1 and 5.2). 

As was already mentioned in Section 3.4, we can calculate the vibration—inversion-rotation energy levels of ammonia by solving the Schrodinger equation [Eq. (3.46)]. We are of course primarily interested in the determination of the potential function of ammonia from the experimental frequencies of transitions between these levels (Fig. 11), Le. we must solve the inverse eigenvalue problem [Eq. (3.46)]. 

We have done basically two kinds of determination of the potential function of ammonia. In the rigid bender approximation, we solved the inverse eigenvalue... 

In the non-rigid bender approximation, we solved the inverse eigenvalue problem described by Eq. (5.4), i.e. we determined the potential function parameters given in Table 3 for NX3 (X = H, D, T). We have used the experimental infrared frequencies of transitions from the ground state to the i>2,2 2 > 2. and 41 2 inversion states and the zero-order frequencies of vibrations (Table 4). The zero-order frequencies have been obtained from the observed fundamental frequencies of NH3 [Ref. >], ND3 [Ref. °>], NTg [Refs." and [Ref.- 3)] corrected for... 

The existence of complex eigenvalues of the value matrix IV implies that the coefficients in Eqn. (11.15) are complex and rules out the existence of a real-valued potential function. Transient oscillations in the concentrations may occur, but in the limit of long times the system nevertheless converges toward the dominant eigenvector. The corresponding largest eigenvalue is real and positive, and hence all oscillations in concentrations have to fade out inevitably. 

Figure 15. Frequency distribution of the number of negative eigenvalues in the Hessian matrix of the potential function that are counted along running trajectories. The solid, dotted, and dot-dashed lines are, respectively for the energies —ll.Oe, -11.5e, and -12.0e. (Reproduced from Ref. 11 with permission.)...

For purposes of comparison, it is possible to classify the various types of potential functions which may be represented by the functional form used in Eq. (3.32) with a few simple considerations. The restrictions we shall make are always to locate the origin in the minimum, or if more than one, in the deepest minimum second minima or inflection points are restricted to negative values of the coordinate Z and the positive values of Z always represent the most rapidly rising portion of the function. These restrictions do not eliminate any unique shape of potential function. Any other functions described by Eq. (3.32) are related to those already included by a simple translation of the origin or by rotation about the vertical axis. These operations, at most, change the eigenvalues by an additive constant. The different types of potential functions are summarized in Table 3.1. 

The WKBJ method provides energy eigenvalues with very high precision for diatomic potential functions. 

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... 

Computations for molecules for which the potential function is symmetric are facilitated by the fact that the Hamiltonian can be factored into two matrices-one containing even values of the vibrational quantum number n, the other odd values. Experience has shown that in this case basis sets of 30 X 30 to 35 X 35 for each matrix will be sufficient to obtain accurate energy levels up to at least n = 15. The accuracy of the eigenvalues, of course, should be checked by comparison of results obtained from basis sets of different sizes. Reid16 outlines other... 

So far we have obtained parametrisations of the logarithmic derivative and potential functions which are appropriate when the Schrodinger equation is regarded as a differential equation, and which allow us to find and whenever E is given. In the ASA, however, Schrodinger s equation is treated as an eigenvalue problem subject to boundary conditions in the form of specified logarithmic derivatives at the sphere. Therefore, we need to find a parametrisation of the function E (D) inverse to D (E), valid around E. ... 

The first term represents the kinetic rotational energy. This equation is amenable to Mathieu s equation and analytical formulae for the eigenstates are known. However, with computer facilities available nowadays, it is faster to calculate numerically both eigenvalues and eigenfunctions [59]. If necessary, these calculations can be performed for potential functions including several Fourier terms. 

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