Spherical harmonics symmetry properties

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... 

While we have chosen to proceed here by reducing representations for the full group D3h, it would have been simpler to take advantage of the fact that D3h is the direct product of C3u and C where the plane in the latter is perpendicular to the principal axis of the former. The behaviour of any atomic basis functions with respect to the C3 subgroup is trivial to determine, and there are only two classes of non-trivial operations in C3v. In more general cases, it is often worthwhile to look for such simplifications. It is seldom useful, for instance, to employ the full character table for a group that contains the inversion, or a unique horizontal plane, since the symmetry with respect to these operations can be determined by inspection. With these observations and the transformation properties of spherical harmonics given in the Supplementary Notes, it should be possible to determine the symmetries spanned by sets of atomic basis functions for any molecular system. Finally, with access to the appropriate literature the labour can be eliminated entirely for some cases, since... 

In either case it is helpful to have a table of transformation properties of spherical harmonics like that given in the Supplementary Notes. We shall illustrate the procedure by finding symmetry-adapted basis functions arising from an s function on each F atom in BF3, using full matrix projection operators. We already know that the three atomic basis functions transform as a 0 s, and since the behaviour with respect to the horizontal plane is already known, we can, without loss of generality, work with the subgroup C3u only. We denote the s functions on Fi, F2, and F3 as si, S2, and s3, respectively, and apply our C3u projection operators... 

Eqs. (6.4) and (6.5) lead to the cylindrical symmetry of the final photofragment angular distribution W(0f, pf) in the form of a dumbbell and a toroid, which are symmetrical with respect to the E-vector (the 2-axis). The distribution W(0f,ipf) is proportional to a differential photodissociation cross-section in the laboratory frame, f(0f,(fif) = daph/dO. For a proper description of its symmetry properties it is usually [376, 402] expanded in a set of spherical harmonics Ykq The cylindric symmetry in this case means that only spherical functions Too and Y20 appear with non-zero coefficients, and then... 

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... 

Taking into account the properties of spherical harmonics [70], Clebsch-Gordon coefficients [71], and spherical Bessel and Hankel functions [70], it is possible to show that the mode functions in (18) obey the following condition of symmetry ... 

The irreducible tensor method was originally developed by G. Racah in order to make possible a systematic interpretation of the spectra of atoms. In the present paper this method has been extended to irreducible sets of real functions that have the same transformation properties as the usual real spherical harmonics. Such an extension is particularly useful in the discussion of the spectra of molecules which belong to the finite point groups or to the continuous groups with axial symmetry. There are several reasons for this. 

This potential is invariant under the symmetry properties of the metal complex. As a result, the operator part reduces to the totally symmetric components of the spherical harmonics. Moreover, interactions with d electrons imply that I must be limited to four, and to six for / electrons. In the case of an octahedral field, the subduction relations for spherical harmonics (see Sect. C.l) indicate that a totally symmetric A g component can be subduced only from = 4 and = 6. Filling in the angular positions of the ligands in an octahedron then yields... 

Another feature that emerges from these plots is the loss of nodal structure. Because the spin-up and spin-down components of each spinor have nodes in different places, the directional properties of the angular functions are smeared out compared with the properties of the nonrelativistic angular functions. Only for the highest m value does the spinor retain the nodal structure of the nonrelativistic angular function, and that is because it is a simple product of a spin function and a spherical harmonic. The admixture of me and me + I character approaches equality as I increases and as me approaches zero, resulting in a loss of spatial directionality. The implications of this loss of directionality for molecular structure could be significant, particularly where the structure is not determined simply from the molecular symmetry or from electrostatics. 

The mixing of AOs into MOs is restricted only by the nodal properties of the orbitals and by symmetry the 3s orbital of a chlorine atom may not contain contributions from any of the p gaussians, because the s—p overlap between AO s centered on the same atom is zero. This can be easily checked by mentally overlapping the two spherical harmonics in Fig. 3.2, where the (-1—1-) overlap is equal and of opposite sign to the (H—) overlap. In the same way, the pz AOs of the ethylene carbon atoms do not overlap with any of the s-type orbitals in the rest of the molecule, and mix as a separate subset of AO s into the n-MOs [5]. These restrictions ultimately stem from the angular momentum of electrons. 

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. 

In cases where there are no electronically driven distortions, the orbital description provides no better account of the chemistry than the bond valence model. Rather it tends to make an essentially simple situation more complex. For example, consider the phosphate and nitrate anions, and NOJ. In orbital models the P atom is described as sp hybridized and the N atom as sp hybridized, but these descriptions are just representations of the spherical and cylindrical harmonics appropriate to the observed geometries. They provide no explanation for why P is four but not three coordinate, or why N is three but not four coordinate. The bond valence account given in Chapter 6 is simpler, more physical, and more predictive. The orbital description is merely a rather complicated way of saying that the ions obey the principle of maximum symmetry but implying that the constraints are related in some unspecified way to the properties of one-electron orbitals rather than to the ionic sizes. 

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