Surface-harmonic wave functions

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... 

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... 

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. Fock s projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

These discrepancies result (a) from the harmonic approximation used in all calculations [to,- (theory) > v, (exp)], (b) the known deficiencies of minimal and DZ basis sets to describe three-membered rings [polarization functions are needed to describe small CCC bond angles a>,(DZ + P) > w,(DZ) > to,(minimal basis)] and (c) the need of electron correlated wave functions to correctly describe the curvature of the potential energy surface at a minimum energy point [ [Pg.102]

Tanczos35 has extended the theory (for V-T and V-V transfer) to polyatomic molecules, and a detailed comparison with experiment was recently given by Stretton33. Considering each surface atom, energy transfer depends on how the intermolecular potential varies with the oscillation of the atom. In deriving the result for the diatomic molecule from the harmonic-oscillator wave functions, we substituted... 

Fig. 13.2. Illustration of the origin of reflection structures for polyatomic molecules. The potential energy surface is of the form (6.35) with e = 0. The wave-function of the parent molecule is simply the product of two harmonic oscillator wavefunctions. The heavy arrows illustrate the dissociation path.

The first function. Pi (r, t), corresponds to a time-harmonic wave with its wave-front (defined as the constant phase surface) described by a sphere expanding with time, i.e. Pi(r, t) is nothing else but a divergent spherical wave. The function P2(r, t) characterizes a convergent, i.e. arriving from infinity, spherical wave. 

Figure 3. Born Oppenheimer surfaces generated by the model electronic Hamiltonian in Equation (5) as the hydrogen is displaced from the origin in the -direction. The inset at the right schematically shows the model which electron is harmonically bound to a point at the origin of coordinates while the electron and proton interact via a Coulomb potential. The wave function is expanded as a linear combination of three basis functions, hydrogen Is, 2s and 2pz eigenstates.

Another factor of which a nonclassical theory must take account is the quantisation of the internal modes of D and A, and the consequent relaxation of the Bom-Oppenheimer constraint that the electron must transfer within a fixed nuclear framework. In classical theory, the vibrational modes of D and A are treated as classical harmonic oscillators, but in reality their quantisation is usually significant (that is, one or more of the vibration frequencies v is sufficiently high that the classical limit hv IcT does not apply). Electron transfer then requires the overlap, not only of the electronic wavefunctions of R and P, but also of their vibrational wavefunctions. It is then possible that nuclear tunnelling may assist electron transfer. As shown in Fig. 4.12, the vibrational wave-functions of R and P extend beyond the classical parabolas and overlap to some extent. This permits nuclear tunnelling from the R to the P surface, particularly in the region just below the classical intersection point. Part of the reorganisation of D and A, required prior to ET in the classical picture, may then occur simultaneously withET, by the nuclei tunnelling short (typically < 0.1 A) distances from their R to their P positions. 

For this purpose we change to the usual notation for the wave functions, in which the magnitude of the angular momentum about a particular axis may be recognized directly, viz. the general surface harmonic Yi(9, is replaced by the special function then... 

The capillary wave method is based on the generation of harmonic waves on the surface of a bulk volume of liquid [28], The wavelength of the ripples formed, A, is a function of the surface tension, which can be evaluated from expressions given by Kelvin ... 

Figure 13 Potential energy surfaces for electron transfer reactions. Harmonic oscillator potential energy functions for reactants and product are shown, including the nuclear wave functions, which are shaded. The dark shaded region indicates the magnitude of overlap of the nuclear wave functions, which is the Franck-Condon factor, (a) is the normal region, (b) is the activationless region and (c) is the inverted region as defined in the text. (Ref. 72. Reproduced by permission of Nature Publishing Group, www.nature.com)...

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