Eigenfunction harmonic

Table 3.1 Eigenfunctions Harmonic Oscillator H = — i//Jx) of the Hamiltonian Operator for the h2/2m) d2/dx2) + knX2...

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. 

It is customary to express the eigenfunctions for the stationary states of the harmonic oscillator in terms of the Hermite polynomials. The infinite set of Hermite polynomials // ( ) is defined in Appendix D, which also derives many of the properties of those polynomials. In particular, equation (D.3) relates the Hermite polynomial of order n to the th-order derivative which appears in equation (4.39)... 

From equations (4.34) and the orthonormality of the harmonic oscillator eigenfunctions n), we find that the matrix elements of a and are... 

Thus, the operators H and have the same eigenfunctions, namely, the spherical harmonics Yj iO, q>) as given in equation (5.50). It is customary in discussions of the rigid rotor to replace the quantum number I by the index J m the eigenfunctions and eigenvalues. 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

To find the perturbation corrections to the eigenvalues and eigenfunctions, we require the matrix elements for the unperturbed harmonic... 

Since equation (10.43) with F = 0 is already solved, we may treat V as a perturbation and solve equation (10.43) using perturbation theory. The unperturbed eigenfunctions S H q) are the eigenkets n) for the harmonic oscillator. The first-order perturbation correction to the energy as given by equation (9.24) is... 

The problem of evaluating the effect of the perturbation created by the ligands thus reduces to the solution of the secular determinant with matrix elements of the type rp[ lICT (pk, where rpj) and cpk) identify the eigenfunctions of the free ion. Since cpt) and cpk) are spherically symmetric, and can be expressed in terms of spherical harmonics, the potential is expanded in terms of spherical harmonics to fully exploit the symmetry of the system in evaluating these matrix elements. In detail, two different formalisms have been developed in the past to deal with the calculation of matrix elements of Equation 1.13 [2, 3]. Since t/CF is the sum of one-electron operators, while cpi) and cpk) are many-electron functions, both the formalisms require decomposition of free ion terms in linear combinations of monoelectronic functions. 

The spherical harmonics in real form therefore exhibit a directional dependence and behave like simple functions of Cartesian coordinates. Orbitals using real spherical harmonics for their angular part are therefore particularly convenient to discuss properties such as the directed valencies of chemical bonds. The linear combinations still have the quantum numbers n and l, but they are no longer eigenfunctions for the z component of the angular momentum, so that this quantum number is lost. 

The use of spherical harmonics in real form is limited by the fact that, for m 0 they are not eigenfunctions of Lz. They may be used to specify the angular distribution of electron density, but at the expense of any knowledge about angular momentum, and vice versa. 

The eigenfunctions of the L operators will be denoted by the spherical harmonics Y(m. 

The eigenfunctions of the M operators are the physically more meaningful quantities. They are denoted byYjM and called vector spherical harmonics. In terms of the spin variable a(= x, y, z), and written as functions of the unit vector n = k/k,... 

The solutions of the inner and outer fields can now be written as expansions in these spherical harmonic functions or vector eigenfunctions, once the incident irradiation and the boundary conditions are specified. 

The Sturmian eigenfunctions in momentum space in spherical coordinates are, apart from a weight factor, a standard hyperspherical harmonic, as can be seen in the famous Fock treatment of the hydrogen atom in which the tridimensional space is projected onto the 3-sphere, i.e. a hypersphere embedded in a four dimensional space. The essentials of Fock analysis of relevance here are briefly sketched now. 

Quartic terms cannot be neglected relative to cubic. It is true that they represent a higher order of the potential energy expression. However, first order terms of type j tpi P y>t dz, where the ipt are eigenfunctions of the harmonic potential energy and P represents deviations from anharmonicity, vanish when P is a cubic (or any odd powered) term but not when P is a quartic (or any even powered) term. 

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