Harmonic wave function

Figure 3. Coherence, induced between two states (lowest and first-harmonic wave functions) and the nature of the hybrid superposition, which evolves with time.

The rates for the C-H stretching vibrations are underestimated, which probably is a result of using harmonic wave-functions or of coupling to the continuum. 

A very peculiar dynamics has been revealed in the Ca(OH)2 crystal by means of inelastic neutron scattering technique [26]. It has been found that anharmonic terms must be included, which mix the vibrational states of the OH and lattice modes. In particularly, the lattice modes have successfully been represented as the superposition of oxygen and proton synchronous oscillators, and it appears that the proton bending mode Eu is strongly coupled to the lattice modes. The contribution of the proton harmonic wave functions has been taken as the zero-order approximation. 

A single particle in a symmetrical two-well potential is presented by the symmetric (0+) and antisymmetric (0 ) substates. The difference between two corresponding energies Eo- — o+ = too represents the tunneling splitting. In the case of the high potential barrier, the corresponding wave functions are symmetrical and antisymmetrical combinations of the harmonic wave functions (106) centered at the minima of two wells, X0 ... 

The appropriate scattering functions, which characterize symmetric and antisymmetric combinations of harmonic wave functions, are [119]... 

For a single particle in a symmetrical double minimum potential the ground state splits into two sublevels, symmetric 0+) and antisymmetric 0—), respectively. The tunnel splitting is /wj — E0+ = hioo . If the potential barrier is sufficiently high, the wave functions can be written within the two-state approximation as symmetrical and antisymmetrical combinations of harmonic wave functions centered at the potential minima op... 

Problem 49-9. Using the surface-harmonic wave functions mentioned in the footnote at the end of Section 35c, derive Equation 49-25, applying either the ordinary second-order perturbation theory or the method of Section 27a. 

In the case of a system with one degree of freedom no other dynamical quantity (except functions of H only, such as H2) is represented by a diagonal matrix with more degrees of freedom there are other diagonal matrices. For example, the surface-harmonic wave functions for the hydrogen atom... 

Taking anharmonicity into account is a much more difficult task than normal mode analysis. Note (see Fig. 7.11) that in the anharmonic case the wave function becomes asymmetric with respect to x = 0 as compared to the harmonic wave function. 

For harmonic wave function, the properties of the Hermite polynomials are such that the selection rules for vibrational transitions are... 

The radial dependency of harmonic wave functions in spherical coordinates is described by the differential equation (C.8) ... 

The vibrational part of the molecular wave function may be expanded in the basis consisting of products of the eigenfunctions of two 2D harmonic oscillators with the Hamiltonians ffj = 7 -I- 1 /2/coiPa atid 7/p = 7p - - 1 /2fcppp,... 

These new wave functions are eigenfunctions of the z component of the angular momentum iij = —with eigenvalues = +2,0, —2 in units of h. Thus, Eqs. (D.l 1)-(D.13) represent states in which the vibrational angular momentum of the nuclei about the molecular axis has a definite value. When beating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in... 

The problem is heated in elementary physical chemishy books (e.g., Atkins, 1998) and leads to a set of eigenvalues (energies) and eigenfunctions (wave functions) as depicted in Fig. 6-1. It is solved by much the same methods as the hamionic oscillator in Chapter 4, and the solutions are sine, cosine, and exponential solutions just as those of the harmonic oscillator are. This gives the wave function in Fig. 6-1 its sinusoidal fonn. 

The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3,...,oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is... 

Figure 1.13 Plot of potential energy, V(r), against bond length, r, for the harmonic oscillator model for vibration is the equilibrium bond length. A few energy levels (for v = 0, 1, 2, 3 and 28) and the corresponding wave functions are shown A and B are the classical turning points on the wave function for w = 28...

Figure 1.13 shows the potential function, vibrational wave functions and energy levels for a harmonic oscillator. Just as for rotation it is convenient to use term values instead of energy levels. Vibrational term values G(v) invariably have dimensions of wavenumber, so we have, from Equation (1.69),... 

Just as the electrical behaviour of a real diatomic molecule is not accurately harmonic, neither is its mechanical behaviour. The potential function, vibrational energy levels and wave functions shown in Figure f.i3 were derived by assuming that vibrational motion obeys Hooke s law, as expressed by Equation (1.63), but this assumption is reasonable only... 

However, unlike electrical anharmonicity, mechanical anharmonicity modifies the vibrational term values and wave functions. The harmonic oscillator term values of Equation (6.3) are modified to a power series in (u + ) ... 

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

We use s, p, and d partial waves, 16 energy points on a semi circular contour, 135 special k-points in the l/12th section of the 2D Brillouin zone and 13 plane waves for the inter-layer scattering. The atomic wave functions were determined from the scalar relativistic Schrodinger equation, as described by D. D. Koelling and B. N. Harmon in J. Phys. C 10, 3107 (1977). 

On application of the ordinary methods of perturbation theory, it is seen that the first-order perturbed wave function for a normal hydrogen atom with perturbation function f r)T, tesseral harmonic, has the form ] ioo(r)-HKr)r(i>, tesseral harmonic as the perturbation function. The statements in the text can be verified by an extension of this argument. 

Harmonic IR spectra of C3H2 calculated at the RHF/6-311++G(d,p), MP2/6-31 1++G(d,p) and MP4/6-31 1++G(d,p) levels are reported in Table 3. The results are nicely converging as electronic correlation is progressively included in the wave function. Excellent agreement between theory and experiment is thus obtained at the MP4 level, which allows for a correct treatment of simultaneous correlation effects in coupled vibrations. The only discrepancies which could show up, would proceed from anharmonicity, as illustrated by the CH stretching vibrations which are found shifted to higher frequencies than anticipated. 

The harmonic wave may also be described by the sine function... 

The representation of tp(x, t) by the sine function is completely equivalent to the cosine-function representation the only difference is a shift by A/4 in the value of X when t = 0. Moreover, any linear combination of sine and cosine representations is also an equivalent description of the simple harmonic wave. The most general representation of the harmonic wave is the complex function... 

The function ip x, t) also represents a harmonic wave moving in the positive x-direction. 

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