Spherical harmonic solutions

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],... 

Figure 11.25 (a) Shear rate ymax at which the first normal stress difference reaches its positive maximum Nimax. versus reduced concentration U/U predicted by the Smoluchowski equation using an exact spherical harmonic solution and using the Hinch-Leal approximate closure (see Baek et al. 1993b). These predictions use... 

The left-hand side is the operator for the Lj component of the angular momentum acting on the spherical harmonic function (see Further Reading section of this appendix). We can apply this operator to the general form of the spherical harmonic solutions ... 

The radial functions in Table A9.2 are already real functions and are common factors for the orbitals from which we will take linear combinations, and so we will concentrate on the spherical harmonic solutions of the angular equation in this section. 

In summary, separation of variables has been used to solve the full r,0,( ) Schrodinger equation for one electron moving about a nucleus of charge Z. The 0 and (j) solutions are the spherical harmonics YL,m (0,(1>)- The bound-state radial solutions... 

The solution

spherical harmonic Yim), where the radial part //(r) depends of the quantum number / but not of m (2). 

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. 

Expressions for the electric and magnetic fields can likewise be obtained. These plane-wave solutions are then expanded in terms of spherical harmonics... 

The solutions of the angular dependent part are the spherical harmonics, Y, known to most chemists as the mathematical expressions describing shapes of (hydrogenic) atomic orbitals. It is noted that Y is defined only in terms of a central field and not for atoms in molecules. 

A problem that arises in connection with the construction of the basis is that of finding what are the allowed values of the quantum numbers of the subalgebra G contained in a given representation of G. For example, what are the allowed values of Mj for a given J in Eq. (2.12). In this particular case, the answer is well known from the solution of the differential (Schrodinger) equation satisfied by the spherical harmonics (see Section 1.4), that is,... 

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 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... 

Not surprisingly, formalisms with very diffuse density functions tend to yield large electrostatic moments. This appears, in particular, to be true for the Hirshfeld formalism, in which each cos 1 term in the expansion (3.48) includes diffuse spherical harmonic functions with / = n, n — 2, n — 4,... (0, 1) with the radial factor rn. For instance when the refinement includes cos4 terms, monopoles and quadrupoles with radial functions containing a factor r4 are present. For pyridin-ium dicyanomethylide (Fig. 7.3), the dipole moment obtained with the coefficients from the Hirshfeld-type refinement is 62.7-10" 30 Cm (18.8 D), whereas the dipole moments from the spherical harmonic refinement, from integration in direct space, and the solution value (in dioxane), all cluster around 31 10 30 Cm (9.4 D) (Baert et al. 1982). 

The tip wavefunctions can be expanded into the spherical-harmonic components, T/, (0, ( )), with the nucleus of the apex atom as the origin. Each component is characterized by quantum numbers I and m. In other words, we are looking for solutions of Eq. (3.2) in the form... 

Consider now the field scattered by an isotropic, optically active sphere of radius a, which is embedded in a nonactive medium with wave number k and illuminated by an x-polarized wave. Most of the groundwork for the solution to this problem has been laid in Chapter 4, where the expansions (4.37) and (4.38) of the incident electric and magnetic fields are given. Equation (8.11) requires that the expansion functions for Q be of the form M N therefore, the vector spherical harmonics expansions of the fields inside the sphere are... 

The simplest possible case is a non-homonuclear diatomic (non-homonuclear because a dipole moment is required for a rotational spectrum to be observed). In that case, solution of Eq. (9.38) is entirely analogous to solution of the corresponding hydrogen atom problem, and indicates the eigenfunctions S to be the usual spherical harmonics (j)), with eigenvalues given by... 

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. 

Functions 0(0) < >() such that 0 and solve this equation are called spherical harmonic functions of degree I. can find solutions hy separating vari-... 

The angular part = Pf.m(cos6 )c"" of the solution (1.12) is a spherical harmonic function. It turns out that there is a nonzero whenever f is a nonnegative integer and m is an integer with m < , In Appendix A we will prove this and other facts about spherical harmonic functions. The number f is called the degree of the spherical harmonic. From Equation 1.10 we see that each spherical harmonic of degree f satisfies the equation... 

Proof. First we will show that only a finite number of solutions are of the form a 0, for a e J and Fa spherical harmonic function. Then we... 

Fix an eigenvalue E. Suppose we have a solution to the eigenvalue equation for the Schrodinger operator in the given form. I.e, suppose we have a function a 1 and a spherical harmonic function such that... 

The goal of this appendix is to prove that the restrictions of harmonic polynomials of degree f to the sphere do in fact correspond to the spherical harmonics of degree f. Recall that in Section 1.6 we used solutions to the Legendre equation (Equation 1.11) to dehne the spherical harmonics. In this appendix we construct bona hde solutions to the Legendre equation then we show that each of the span of the spherical harmonics of degree is precisely the set of restrictions of harmonic polynomials of degree f to the sphere. 

The technical conditions on f are quite reasonable if a physical situation has a discontinuity, we might look for solutions with discontinuities in the function f and its derivatives. In this case, we might have to consider, e.g., piecewise-defined combinations of smooth solutions to the differential equation. These solutions might not be linear combinations of spherical harmonics. 

Proof. Let V denote the set of solutions in 2(]R3) obtained by multiplying a spherical harmonic by a spherically symmetric function ... 

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