Harmonic wavefunction

If the parent molecule is described by normal modes with coordinates q and momenta pk j the multi-dimensional wavefunction is simply a product of uncoupled one-dimensional harmonic wavefunctions ip iQk) (Wilson, Decius, and Cross 1955 ch.2 Weissbluth 1978 ch.27) and the corresponding Wigner distribution function reads... 

It is shown in the Appendix that the spherical harmonic wavefunctions, have the following important property when operated upon by the operator... 

The spherical harmonic wavefunctions can be applied to the rotational spectra of diatomic molecules, and also to electron and nuclear spin. In the latter case, the quantum numbers must be allowed to have half-integral values. 

Now the nuclear motion is harmonic, but the nuclei move only along the rigid coordinates. The soft motion, e.g. translation, is frozen. What happens in the order k At first sight, one may be tempted to assume that the cubic anharmonicity of Vj Q) in Eq. (44) contributes to this order. From a perturbational point of view, this is not the case, however. The harmonic wavefunctions are even functions of the coordinates, and the expectation values of all odd anharmonic terms vanish. These terms will contribute to the energy in second order perturbation theory, i.e. earliest in (note... 

From a perturbative standpoint, anharmonicity in the stretching potential mixes harmonic wavefunctions. If the anharmonicity were to produce a mixing of the zero-order n = l and n = 2 states, the nonzero transition integral between the harmonic n = 0 and n = l states would be distributed among the perturbed first and second excited states. This would make both the n = 0 to n = 1 and n = 0 to n = 2 transitions allowed in the anhar-monically perturbed states. Anharmonicity in the potential and nonlinearity in the dipole moment function make it possible to observe vibrational transitions other than An = 1. However, such other transitions are usually weaker or less intense because the harmonic nature of the potential and the linearity of the dipole moment function have the major role. 

Table 3-1. The normalized associated Legendre polynomials, 0jj ( )and spherical harmonic wavefunctions, up to/=3...

When the legendrian acts on the spherical harmonic wavefunctions, the result is as given in Equation 3-20. 

The spherical harmonic wavefunctions are eigenlimctions only of the z angular momentum operator and the overall angular momentum squared operator. 

The wavefunction for a particular vibrational/rotational eigenstate is a product of the radial and spherical harmonic wavefunctions. The v, J, and... 

It is usual to normalize the angular and radial parts separately. The radial functions listed in Table 8-1 are normalized along the r coordinate, and the spherical harmonic wavefunctions in Table 3-1 are also normalized. 

Due to the orthonormality of the spherical harmonic wavefunctions, the integral in Equation 8-30 is significantly reduced. Taking the limits on ri and T2 for when they are larger than one another, the first-order energy correction is determined by the following expression. 

Since an electron has an intrinsic spin, there must be a corresponding operator for the overall intrinsic spin angular momentum squared,. It is expected that the intrinsic spin eigenfunctions, Xsm, are analogous to the spatial spherical harmonic wavefunctions, Yi (6, ). The operators S, and 5 will be the only operators for which the intrinsic spin functions are eigenfunctions just like YUd. < >) are only eigenfunctions of i) and operators. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

If we further assume that the vibrational wavefunctions associated with normal mode i are the usual harmonic oscillator ones, and r = u + 1, then the integrated intensity of the infrared absorption band becomes... 

Table 1 Coefficients for 7[ (a ) for third harmonic generation (THG), degenerate four wave mixing (DFWM), electric field induced second harmonic generation (ESHG), and Kerr effect in methane at the experimental geometry rcH = 2.052 a.u. A CCSD wavefunction and the t-aug-cc-pVDZ basis were used. (Results given in atomic units, the number in parentheses indicate powers of ten.)...

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

Finally, it is a weU-known result of quantum mechanics" that the wavefunctions of harmonic oscillators extend outside of the bounds dictated by classical energy barriers, as shown schematically in Figure 10.1. Thus, in situations with narrow barriers it can... 

The selection rules illustrated above are general, as they depend only on the symmetry properties of the functions involved. However, more limiting, selection rules depend on the form of the wavefunctions involved. A relatively simple example of the development of specific selection rules is provided by the harmonic oscillator. The solution of this problem in quantum mechanics,... 

In the partial wave theory free electrons are treated as waves. An electron with momentum k has a wavefunction y(k,r), which is expressed as a linear combination of partial waves, each of which is separable into an angular function Yi (0. ) (a spherical harmonic) and a radial function / L(k,r),... 

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 ... 

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