Harmonic-oscillator functions

Xvf (Ra - Ra,e) Xvi> will be non-zero and probably quite substantial (because, for harmonic oscillator functions these "fundamental" transition integrals are dominant- see earlier) ... 

Let us consider an example, that of the derivative operator in the orthonormal basis of Harmonic Oscillator functions. The fact that the solutions of the quantum Harmonic... 

To solve this equation, an appropriate basis set ( >.,( / ) is required for the nuclear functions. These could be a set of harmonic oscillator functions if the motion to be described takes place in a potential well. For general problems, a discrete variable representation (DVR) [100,101] is more suited. These functions have mathematical properties that allow both the kinetic and potential energy... 

An alternative to using a superposition of Gaussian functions is to extend the basis set by using Hermite polynomials, that is, harmonic oscillator functions [24]. This provides an orthonormal, in principle complete, basis set along the trajectory, and the idea has been taken up by Billing [151,152]. The basic problem with this approach is the slow convergence of the basis set. 

In this approximation the nuclear wavefunctions are a product of N harmonic oscillator functions, one for each normal mode ... 

In the harmonic approximation the functions Xi and Xf are products of harmonic oscillator functions. We therefore specify the initial state by a set of quantum numbers n — (ni, ri2,..., n/v), and those for the final state by m = (mi,m2,..., tun)- So the nuclear wavefunctions are henceforth denoted by Xi,n and Xf,m- Equation (19.21) tells us how to calculate the rate of transition from one particular initial quantum mode n to a final quantum state m. This is more than we want to know. All we are interested in is the total rate from any initial state to any final state. The ensemble of reactants is in thermal equilibrium therefore... 

Now it becomes apparent why it was useful to replace the delta function by its Fourier transform. The wavefUnctions Xin are products of harmonic oscillator functions, the Hamiltonians Hi and H/ are sums of harmonic oscillator terms. Therefore the terms in the brackets factorize in the form ... 

In another example of differences in complexity, the bondstretching energy in CHARMM is calculated with a harmonic oscillator function. MM3 solves the problem described by French, Tran and Perez in this book for MM2 s cubic stretching function by using a quartic function for bond stretching. Additional complexity in MM3 is described in Ref. 12. 

The spherical harmonics are normalized.) The harmonic-oscillator function S(q) is very small for q less than — Re, so that we can write the lower limit in (4.41) as — oo without serious error. Also S is real. Equation (4.41) becomes... 

The harmonic-oscillator functions are either even or odd, so that S2 is even. Thus the integrand of the first integral in (4.42) is an odd function,... 

It should be noted that (4.28) is only an approximation for the nuclear wave function. The perturbation terms (4.36) will mix into the nuclear wave function small contributions from harmonic-oscillator functions with quantum numbers other than v. These anharmonicity corrections to the vibrational wave function will add further to the probability of transitions with At) > 1. 

As well as shifting the expected positions of vibrational bands, Fermi resonance affects their intensities. For example, the wave function of the (02°0) level of C02 has a considerable contribution from the harmonic-oscillator function corresponding to the (10°0) level since the latter is a fundamental level, the transition from the ground level (00°0) to the (02°0) level is much more intense than would be expected for an overtone band. This transition is Raman active, but IR inactive the Raman spectrum of C02 shows two very strong bands at 1285 and 1388 cm-1, corresponding to transitions to the (02°0) and (10°0) levels from the ground level. 

The system used for amplification and detection of an ESR signal is such that the first derivative of the absorption line is recorded. The shape of an ESR line in solution is usually Lorentzian [Equation (3.86)]. The Lorentzian shape resembles a Gaussian (except that it falls off more slowly). Differentiation of the Gaussian shape (3.89) gives — 2cd(v — v0)exp[-d(v - v0)2], which has the form of the u=l harmonic-oscillator function [(1.133) and (1.137)] with x — y v0. Thus the first-derivative of an absorption resembles Fig. 1.1b with the origin at v0. (See also Problem 8.22.)... 

The direct product enables one to find the symmetry of a wave function when the symmetries of its factors are known. For example, consider In the harmonic-oscillator approximation, the vibrational wave function is the product of 3N—6 harmonic-oscillator functions, one for each normal mode. To find the symmetry of we first examine the symmetries of its factors. Let the distinct vibrational frequencies of the molecule be vx>v2,..., vk,...,vn, and let vk be <4-fold degenerate let the harmonic-oscillator... 

For the predissociative C2N2 (C- TIU) state in the collinear approximation, the nuclear wavefunction is approximated by the product of three harmonic oscillator functions describing the normal modes vibrations. The frequencies and normal coordinates of the three linear stretching vibrations were obtained from ab initio MCHF calculations. The validity of the harmonic approximations is supported from experimental data (8) where absorption spectra of C2N2 is found to give a set of equidistant bands. 

Here, instead of the usual delta function d(Eg, — Eg — ftv), which supplies only discontinuous spectral lines, a Lorentz function with a line width 29(normalized to one) is introduced for all vibrational levels labeled by n and m of electronic states j and k which contribute to the transition rate of Eq. (9) [58]. M is the transition operator, in general, pertinent to an electric dipole transition. The nuclear wavefunctions are approximated by products of harmonic oscillator functions... 

In the spectral region below the electronic band-to-band transitions, the Cauchy approximation (transparency region, (3.24)) or the damped harmonic oscillator function (both transparency as well as absorption region) are often utilized as MDF approaches. 

The DF spectra of wurtzite-structure ZnO within the VIS-to-VUV spectral region contain CP structures, which can be assigned to band-gap-related electronic band-to-band transitions Eq with a = A, B,C and to above-band-gap band-to-band transitions E13 with (3 = 1,..., 7. The F -related structures can be described by lineshape functions of the 3DMo-type (3.9 and 3.10), the CP structures with (3 = 3,4 by lineshape functions of the 2DMo-type (3.11), and the CP structures with (3=1,2,5,6,7 can be described by Lorentzian-damped harmonic oscillator functions (3.13). The CP structures Eq are supplemented by discrete (3.14) and continuum (3.16) excitonic contributions. Tables 3.9 and 3.10 summarize typical parameters of the CPs Eq and E, respectively, of ZnO [15]. 

Here vr, rs> denotes an unperturbed harmonic oscillator function, and we assume neither vibration is degenerate. For example, typical values of rrs might be of the order 30 cm-1 if the separation between the unperturbed vibrational levels (vr = 2, v, = 0) and (vr = 0, vs = 1) were also about 30 cm-1 the interaction would result in a pushing apart of the energy levels of about 7 cm-1 each way, giving an observed separation of about 44 cm-1. 

These values were computed by fitting the calculated uH(z) curves in the neighborhood of zm to perturbed harmonic oscillator functions [incidentally, these... 

The set of quantum numbers of a level also serves to define the corresponding wave-function, which in the usual approximation is written as a product of one-dimensional harmonic oscillator functions,... 

Fig. 2.27. (a) The experimental potential curve for H. (full curve) the dotted curve represents a Morse fit to the data, and the dashed curve shows a harmonic-oscillator function with the force constant taken at r = (0.7414 A). The first... 

The Franck-Condon factor is given by the squared overlap integral of displaced harmonic oscillator functions (Hermite functions). It can be related [154, p. 113] to the so-called Huang-Rhys parameter (or factor) S according to... 

The function < )n is reasonably approximated as a harmonic-oscillator function whose force constant k and equilibrium position qo depend on Q. Such an approximation signifies that the potential energy in expression (1) can be written... 

Ikeda et al.35 considered solutions of Eqs. (3.34) and (3.35) by the variation method with two-dimensional harmonic oscillator functions in Cartesian and polar coordinates, respectively, as basis functions. Some of the eigenvalues are plotted in Fig. 3.4 as a function of the parameter B. The case of pure pseudorotation corresponds to large negative values of B on the right hand side of the figure. 

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