Spherical symmetry harmonics

Physicists are familiar with many special functions that arise over and over again in solutions to various problems. The analysis of problems with spherical symmetry in P often appeal to the spherical harmonic functions, often called simply spherical harmonics. Spherical harmonics are the restrictions of homogeneous harmonic polynomials of three variables to the sphere S. In this section we will give a typical physics-style introduction to spherical harmonics. Here we state, but do not prove, their relationship to homogeneous harmonic polynomials a formal statement and proof are given Proposition A. 2 of Appendix A. 

Physics texts often introduce spherical harmonics by applying the technique of separation of variables to a differential equation with spherical symmetry. This technique, which we will apply to Laplace s equation, is a method physicists use to hnd solutions to many differential equations. The technique is often successful, so physicists tend to keep it in the top drawer of their toolbox. In fact, for many equations, separation of variables is guaranteed to find all nice solutions, as we prove in Proposition A.3. 

The unperturbed eigenfunctions have complete spherical symmetry and Vc can be expanded in terms of the crystal field parameters, A , and the usual spherical harmonics Y (0, 97) to give... 

In this book, we adopt a form in which the function is expressed as a linear combination of spherical harmonics. This form is particularly appropriate for systems with near-spherical symmetry (such as Rydberg states or molecules which conform to Van Vleck s pure precession hypothesis [68, 69]) and is also consistent with the spirit of spherical tensors, which have the same transformation properties under rotations as spherical harmonics. The functional form of the ket rj, A) is written... 

Alternatively, the deformation ED can be described by a series of multipoles, that is, spherical harmonic density functions, the parameters of which can be refined by the LS technique. The core ED is described as spherically symmetrical, using the so-called k formalism. The k parameter expresses the isotropic expansion ( < > 1) or contraction (/c < 1) of a valence shell as a whole. The higher-order multipoles describe the deviations of the ED from spherical symmetry. 

From the practical point of view, the results that we show in this section are obtained using a numerical procedure similar to some classical references in the field [4]. As the system has spherical symmetry, we expand (Pi(r) in the spherical harmonic basis set and solve self-consistently the set of KS equations (Equation (1)). 

Sometimes there is a special and easier manner of obtaining hi. We exemplify this by the totally symmetric cubic harmonic of the degree 4. This harmonic must consist of a linear combination of x -f- y -f- z (of cubic symmetry) and (of spherical symmetry, and therefore also of cubic symmetry), since only two linearly independent homogeneous polynomia simultaneously are of cubic symmetry and of the degree 4. This means that the spherical harmonic must have the form... 

The compositions of the solutions to the 9 and 4> equations, by virtue of their relevance to all problems with spherical symmetry, are so important as to merit a special name, the spherical harmonics. These solutions bear the same relation to spherical problems that the Bessel functions do to those with cylindrical symmetry. The spherical harmonics, Yim 9, (/>), may be written as... 

The contribution of interelectron repulsion to diagonal matrix elements of the type shown in equation (56) is more complex. However, it turns out that in the case of spherical symmetry one can get rid of the m-summations by making use of the sum rule for spherical harmonics. The result is as follows ... 

We thus have a simple model (the aufbau or building-up principle of Bohr [1] and Stoner [2]) which correctly predicts the periodic structure of Mendeleev s table of the elements. More precisely, one should state that Mendeleev s table is the experimental evidence which allows us to use an independent electron central field model and to associate each electron in a closed shell with a spherical harmonic of given n and i, because there is no physical reason why a particular l for an individual electron should be a valid quantum number angular momentum in classical mechanics is only conserved when there is spherical symmetry. 

The transformations of the standard vector form the fundamental irrep of spherical symmetry. All other irreps can be constructed by taking direct products of this vector. In particular, the spherical harmonic functions can be constructed by taking fully symmetrized powers of the vector. The symmetrized direct square of the / -functions yields a six-dimensional function space with components z, x, yz, xz, xy. This space is not orthonormal the components are not normalized, and the first... 

In the following we consider in particular the multipole susceptibilities of spherical particles. The spherical symmetry entails that all orbitals split into radial functions times spherical harmonics and that the reaction potential of the particle to an external perturbation shows the same symmetry as the latter. A free atom is a priori spherical The inert gas atoms may be assumed spherical even in their respective lattices. 

More promising is to describe the deformation electron density by a series of spherical harmonic density functions (multipoles), which can be included into least-squares refinement. The inner (core) electron shells of an atom are presumed and the k parameter, which describes the isotropic expansion (ic <1) or contraction (/c > 1) of the valence shell as a whole. Multipole parameters of higher orders describe deviations of the electron density from spherical symmetry. They can be related to the products of atomic... 

Abstract This contribution reviews a selection of findings on atomic density functions and discusses ways for reading chemical information from them. First an expression for the density function for atoms in the multi-configuration Hartree-Fock scheme is established. The spherical harmonic content of the density function and ways to restore the spherical symmetry in a general open-shell case are treated. The evaluation of the density function is illustrated in a few examples. In the second part of the paper, atomic density functions are analyzed using quantum similarity measures. The comparison of atomic density functions is shown to be useful to obtain physical and chemical information. Finally, concepts from information theory are introduced and adopted for the comparison of density functions. In particular, based on the Kullback-Leibler form, a functional is constructed that reveals the periodicity in Mendeleev s table. Finally a quantum similarity measure is constructed, based on the integrand of the Kullback-Leibler expression and the periodicity is regained in a different way. 

Since the open-shell term in P does not possess spherical symmetry, the effective Hamiltonian will contain a non-spherical potential and as a result, even with initial orbitals of true central-field form (i.e. with spherical-harmonic angle dependence), the first cycle of an SCF iteration will destroy the symmetry properties of the orbitals—the solutions that give an improved energy will not be of pure s and p type but will be mixtures. This is a second example of a symmetry-breaking situation, akin to the spin polarization encountered in the UHF method. The resultant many-electron wavefunction will also lose the symmetry characteristic of a true spectroscopic state there will be a spatial polarization of the Is 2s core and the predicted ground state will no longer be of pure P type, just as in the UHF calculation there will be a spin polarization and the exact spin multiplicity of the many-electron state will be lost. Of course, the many-electron Hamiltonian does possess spherical symmetry (i.e. invariance under rotations around the nucleus), and the reason for the symmetry breaking lies at the level of the one-electron (i.e. IPM-type) model—the effective field in the 1-electron Hamiltonian is a fiction rather than a reality. 

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