Particle-on-a-Sphere

The problem of a confined electron in the valence state is identical to that of a particle confined to a line segment, which is controversial because it does not predict the classical situation (p = hk) in a limit of large quantum number. For a barrier that is high but finite, the electron begins to move like a free particle before it reaches the classical limit and then shows the correct behaviour. 

Apart from this detail it follows that a spherically confined particle has 

Xe defines the first zero at radius a = A 7r for spherical Bessel functions, je(x), x — kr  

In terms of the causal model the kinetic energy in every stationary state with me = 0 is zero hence the total calculated energy is pure quantum potential energy. To confirm this, recall that Vq must be constant for the confined particle, i.e. 

Equation 1 is simply Schrodinger s amplitude equation for the confined particle, provided that 

This model problem will be used to describe the rotational motion of molecules and the electron motion in atoms. Consider a particle of mass m free to rotate on the surface of a sphere with a constant radius r. The potential on the sphere is zero hence, the Hamiltonian will have only the kinetic energy operator for each coordinate. 

The second derivative over each coordinate is called the del-squared operator . The motion of the particle is not separable in Cartesian 

The del-squared operator must now be converted to spherical coordinates. 

The K contains the operations with the angular variables 0 and )) and is called the legendrian. 

The Hamiltonian for a Particle-on-a-Sphere can now be written in terms of spherical coordinates. 

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I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

Buckyballs A Simple Quantum Mechanical Particle on a Sphere Model (Chem. Phys. Lett. 1993, 205, 200-206. "A particle-on-a-sphere model for C60 ")... 

The particle on a ring is unduly restrictive to correspond immediately to a real situation. We now consider an obvious extension—the particle on a sphere—... 

The allowed wavefunctions for the particle on a sphere are called spherical harmonics, generally written... 

The physical significance of the particle-on-a-sphere wavefunctions is important in connection with the hydrogen atom. Comparing the energy formula (eqn 3.57) with the classical result (eqn 3.44) shows that the angular momentum is... 

The most direct application of particle-on-a-sphere result is to the rotational motion of diatomic molecules in a gas. As with vibrations (see Section 3.2), the real situation looks a little more complicated, but can be solved in a similar way. A molecule actually rotates about its centre of mass the coordinates 8 and can be used to define its direction in space. If we replace the mass in Schrodinger s equation by the reduced mass given by eqn 3.22, and let r be the bond length, then the moment of inertia is... 

The first term is called the radial wavefunction, and describes the in-out motion of the electron. It is written with the subscripts to show that the mathematical form of the function depends on the values of n and l we shall look at some examples below. The second term in eqn 4.18 is exactly the same spherical harmonic that arises in the particle-on-a-sphere problem. It... 

It is interesting that the normalized memory function for , which has non-Gaussian structure, lies very dose to the normalized correlation function /< I > for the angular momentum vector, while the memory function for the angular momentum is very neariy Gaussian. The former correspondence is not surprising, since the motion of a linear molecule is equival it to that of a particle on a sphere, so that the Kubo-Shimizu consideration of a plane rotation may apply with only the numerical change which makes the relevant correlation function in the exponent of Kubo s formula. 

Putting (6.29) into (6.24). the allowed energy levels for a particle on a sphere are found to be... 

The theoretical models which have been used to describe the bonding in cluster compounds of the main group and transition metal elements are reviewed. The historical development of these models is outlined and special emphasis is placed on those studies which have led to the elucidation of structure-electron count correlations. Theoretical treatments of cluster bonding are based on localised, delocalised (molecular orbital) or free electron methods derived from the solution of the Schrodinger equation for a particle on a sphere. A detailed analysis of the Tensor Surface Harmonic method, as an example of a free electron model, is presented. Group theoretical consequences of the model are also presented. 

It is therefore, apparent that, despite the fact that the vertices of a 3-connected polyhedron may all lie on the surface of a single sphere, the arrangement of these vertices represents a poor coverage of the sphere s surface. Thus, these polyhedra do not emulate the sphere sufficiently well for their skeletal MO s to be good approximations to the particle on a sphere solutions. In particular, the low number of close atom-atom contacts (i.e. cluster edges) is insufficient to enable a distinction to be made, in all cases, of the bonding characteristics of the even (L") and odd parity (L31) tangential orbitals. 

Smirnov L. P., Deryaguin B. V., On inertialess electrostatic Deposition of Aerosol Particles on a Sphere at Flow of viscous Fluid around the Sphere, Colloid J., 1967, No. 3 (in Russian). 

A, where the Laplacian A represented in the spherical coordinates is given in Appendix H available at booksite.elsevier.com/978-0-444-59436-5 on p. e91. Since / is a constant, the part of the Laplacian that depends on the differentiation with respect to / is absent In this way, we obtain the equation (equivalent to the Schrddinger equation) for the motion of a particle on a sphere ... 

A rigid rotator is a system of two pointlike masses, mi and m2, with a constant distance R between them. The Schrodinger equation may be easily separated into two equations, one for the center of mass motion and the other for the relative motion of the two masses (see Appendix 1 available at booksite.elsevier.com/978-0-444-59436-5 on p. e93). We are interested only in the second equation, which describes the motion of a particle of mass /r equal to the reduced mass of the two particles, and the position in space given by the spherical coordinates R, 9, (/>, where 0kinetic energy operator is equal to - A, where the Laplacian A represented in the spherical coordinates is given in Appendix H available at booksite.elsevier.com/978-0-444-59436-5 on p. e91. Since is a constant, the part of the Laplacian that depends on the differentiation with respect to R is absent. In this way, we obtain the equation (equivalent to the Schrodinger equation) for the motion of a particle on a sphere ... 

Magnitude of the z-component of the angular momentum of a particle on a sphere ... 

We saw in Section 9.4c that in certain cases a wavefunction can be separated into factors that depend on different coordinates and that the Schrodinger equation separates into simpler versions for each variable. Application of this separation of variables procedure to the hydrogen atom leads to a Schrodinger equation that separates into one equation for the electron moving around the nucleus (the analog of the particle on a sphere) and an equation for the radial dependence. The wavefunction is written as... 

A richer zoo of magic numbers can be obtained by filling the sphere. Then the qualitative types (a)-(d) in Figure 15.1 can be imagined. Case (a) is that of the thin attractive shell or particle on a sphere. Case (b) is a wine bottle potential, with a repulsive excess at the centre. Case (c) is the classic problem of a particle in a sphere. Case (d) is a parabolic one, possibly flattened at the bottom. 

Problem (a) Confirm that Ym is an eigenfunction of the Particle-on-a-Sphere Hamiltonian, and (b) that it is normalized. 

Since r is constant for the Particle-on-a-Sphere, dx wiU be in terms of0 and only. 

Confirm that Ylo and Yi i as given in Table 3-1 are a) eigenfunctions of the Particle-on-a-Sphere model problem b) normalized and c) orthogonal. 

Because the radial and angular parts are separable and the molecule rotates freely in space, the angular part of Equation 6-6 is identical to the Particle-on-a-Sphere model problem developed in Section 3.2. Hence, the angular functions Yim are the spherical harmonics (Equation 3-19). The solution of the A operator applied to Y is known and given in Equation 3-20. 

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