Spherical harmonics definition

The vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

It is not possible to ascribe a definite value of the orbital angular momentum to a photon state since the vector spherical harmonic YjM may be a function of different values of . This provides the evidence that, strictly speaking, it... 

FIG. 3.5 Definition of the normalization coefficients for the spherical harmonic functions. Relations such as yimp — Am d,mp are implied by the direction of the arrows. 

The linear relationships between the traceless moments 0 and the spherical harmonic moments lmp are obtained by use of the definitions of the functions clmp. For example, for the quadrupolar moment element xx, we obtain the equality (3x2 — l)/2 = a (x2 — y2)/2 + b(3z2 — 1). Solution for a and b for this and corresponding equations for the other moments leads to... 

An element of an electrostatic moment tensor can only be nonzero if the distribution has a component of the same symmetry as the corresponding operator. In other words, the integrand in Eq. (7.1) must have a component that is invariant under the symmetry operations of the distribution, namely, it is totally symmetric with respect to the operations of the point group of the distribution. As an example, for the x component of the dipole moment to be nonzero, p(r)x must have a totally symmetric component, which will be the case if p(r) has a component with the symmetry of x. The symmetry restrictions of the spherical electrostatic moments are those of the spherical harmonics given in appendix section D.4. Restrictions for the other definitions follow directly from those listed in this appendix. 

For each nonnegative integer f, the space of spherical harmonics of degree f (see Dehnition 2.6) is the vector space for a representation of 50(3). These representations appear explicitly in our analysis of the hydrogen atom in Chapter 7. Recall the complex scalar product space L (S ) from Definition 3.3. 

Recall the vector space y of spherical harmonics from Definition 2.6 y is the set of restrictions to of homogeneous harmonic polynomials. Recall also the definition of spanning (Defiiution 3.7). The set y of spherical harmoiucs spans... 

Now we are ready to define the spherical harmonic functions. In Section 1,6 we gave examples for t = 0, 1, 2 here is the general definition. 

Symmetry Properties. Under inversion, for R being replaced by -/ , we have Qfm — (—1 YQem- A dipole is odd under inversion and a quadrupole is even. From the properties of spherical harmonics and the definition of the spherical harmonics, it is easy to see that Q m — (—1 )mQ(-m-If Q = a, P, y designates the Euler angles of the rotation carrying the laboratory frame X, Y, Z, into coincidence with the molecular frame, x, y, z, the body-fixed multipole components Q(m are related to the laboratory-fixed Q(m, according to... 

Spherical harmonics, 29, 38 Spherical polar coordinates, 21 Spherical top definition of, 199 degeneracy for, 210 degenerate vibrational modes in, 276 energies of, 209-210 microwave spectrum of, 217,225 polarizability, 202 wave functions of, 209, 211... 

The rotational coordinates are Q 2 and Q 5. The rotational motion can be visualized by mapping the trough onto the surface of a 2D sphere the rotation is governed by the usual polar coordinate definitions, 6 and . This is also shown in equation (7) which has the usual form for a rotator with spherical harmonic solutions Ylm. The solutions will be written in the form I i//lo, hn ). For the high spin states case, it was found that l must be odd in order to obey the Pauli s exclusion principle and preserve the antisymmetric nature of the total wavefunctions at any point on the trough under symmetric operations [26]. In the current case, similar arguments show that l must be even. This is because the electronic basis is even under inversion and the whole vibronic wavefunction must also be even under inversion. A general mathematical proof can be found in Ref. [23],... 

The functions Qim(9) and consequently the spherical harmonics Yim(6, associated Legendre polynomials, whose definition and properties are presented in Appendix E. To show this relationship, we make the substitution of equation (5.42) for cos 6 in equation (5.51) and obtain... 

All of the information that was used in the argument to derive the >2/1 arrangement of nuclei in ethylene is contained in the molecular wave function and could have been identified directly had it been possible to solve the molecular wave equation. It may therefore be correct to argue [161, 163] that the ab initio methods of quantum chemistry can never produce molecular conformation, but not that the concept of molecular shape lies outside the realm of quantum theory. The crucial structure-generating information carried by orbital angular momentum must however, be taken into account. Any quantitative scheme that incorporates, not only the molecular Hamiltonian, but also the complex phase of the wave function, must produce a framework for the definition of three-dimensional molecular shape. The basis sets of ab initio theory, invariably constructed as products of radial wave functions and real spherical harmonics [194], take account of orbital shape, but not of angular momentum. 

Definition (B.4) states that T is subjected to the same transformation law as the spherical harmonics of rank... 

However, a more explicit proof is obtained by rotating the D-matrices and the spherical harmonics appearing in the definition (1 b) of A ( , c, Q) by using Eq. A4, and subsequent application of the following relation ... 

The expansion of the electrostatic potential into spherical harmonics is at the basis of the first quantum-continuum solvation methods (Rinaldi and Rivail, 1973 Tapia and Goschinski, 1975 Hylton McCreery et al., 1976). The starting points are the seminal Kirkwood s and Onsager s papers (Kirkwood 1934 Onsager 1936) the first one introducing the concept of cavity in the dielectric, and of the multipole expansion of the electrostatic potential in that spherical cavity, the second one the definition of the solvent reaction field and of its effect on a point dipole in a spherical cavity. The choice of this specific geometrical shape is not accidental, since multipole expansions work at their best for spherical cavities (and, with a little additional effort, for other regular shapes, such as ellipsoids or cylinders). 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

In summary, when an atom has a definite value of L, it has indefinite values of and Ly, but it has the latent ability to develop a definite, but completely unpredictable, value of either or Ly, provided, for example, that it interacts with a suitably oriented Stern-Gerlach apparatus. In such a process, it would, of course, develop an indefinite value for L. As a generalization of this result, as equations (28-30) show, on arbitrary rotation any given spherical harmonic Yp(6, (f)), becomes a linear combination of spherical harmonics with the same I, and coefficients that depend on the angle and direction of rotation. 

By definition the components of the second-rank Cartesian tensor ax transform under rotation just like the product of coordinates xy (e.q., see Jeffreys, 1961) The motivation for what ensues springs from the observation that the spherical harmonics Ym (0, ft) (where 6, ft) are the polar and azimuthal angles of the unit vector (r/1 r )) can be written in terms of the coordinates (x, y, z) of the vector r, for example,... 

The basic definitions of the spherical harmonics as well as the details of the atomic centr2il field problem can be found in E. U. Condon uid G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, 1953). 

There are a tew definitions of the spherical harmonics in the literature [see E.O.Steinbom and KRuedenberg, Advan.Quantum Chem., 7,1 (1973)]. The Condon-Shortley convention often is used, and is related to the definition given above in the following way = em. Yj — (—l)" [f/], where sm = j... 

Other advantages of working in terms of spherical harmonic functions are that for cases with fibre symmetry, the Legendre addition theorem can be used, and affords considerable algebraic simplifications (see for example Ref. 25), and that for lower symmetries, the treatment can readily be generalised. It should be mentioned that the exact definitions of P2 cos9), etc., and p 9), can differ in different treatments due to the adoption of different normalisation procedures (see, for example. Chapter 5, Section 5.2).)... 

Our definition of spherical harmonics Yq k) and Clebsch-Gordan coefficients are in accord with the standard references. The quantity A(K, K ) is the product of radial integrals, and various factors which arise from the vector coupling coefficients of n equivalent electrons. The latter component of A(K, K ) is essentially embodied in the Racah tensor. Let... 

Our notation for these functions refers implicitly to generalized spherical harmonic functions which have been adapted for the crystal symmetry according to the previously discussed definition of an orientation (Sect. 3.1). This means that the functions, and consequently the ODF, f g), are invariant for... 

The form of the interaction of 4f moments in the lanthanides is further modified by both the crystal-field anisotropy and magnetostriction. The former is predominantly a single-ion interaction in the lanthanides and arises from the Coulomb coupling of the local spin moment (via the spin-orbit interaction) to the hexagonally symmetric charge cloud of the neighboring ions. Stevens (1952) was instrumental in providing the definitive description of this interaction via a series of operator equivalents of the spherical harmonics which describe in effect the quad-... 

Einally, we also remind that the definition of the spherical harmonics in the standard phase convention implies complex conjugation, as... 

From this definition, we argue that the densities of the nonbonding orbitals should reside in the vertices and edges of a tetragonal hosohedron. If we try to map them to the possible d orbitals (or spherical harmonics to be exact), we can see that the d y, d z, dyz, and d 2 orbitals can all contribute to the density. It should then be noted that all these orbitals are not used in M-LGO bonding in the molecular orbital picture (to be exact, d y, dxz, dy are indeed not participated, where dz2 has... 

Note, that this definition includes the Condon-Shortley phase (-1) , which is sometimes left for the sake of convenience (e.g. Bohren and Huffman, p. 90 Spiegel 1995, pp. 243-244), but has then to be included at other places of the spherical harmonics. 

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