Real spherical harmonic functions

Atomic density functions are expressed in terms of the three polar coordinates r, 6, and multipole formalism, the density functions are products of r-dependent radial functions and 8- and -dependent angular functions. The angular functions are the real spherical harmonic functions ytm (8, ), but with a normalization suitable for density functions, further discussed below. The functions are well known as they describe the angular dependence of the hydrogenic s, p, d,f... orbitals. 

A more detailed discussion of the complex and real spherical harmonic functions, with explicit expressions and numerical values for the normalization factors, can be found in appendix D. 

Combining terms with m — —l and m = l, gives the expansion in terms of the real spherical harmonic functions, which we will use to evaluate the Fourier transform of the real density functions ... 

D.1 Real Spherical Harmonic Functions and Associated Normalization Constants (x, y, and z are Direction Cosines)... 

E.1 Expressions for the Integrals over Products of Three Real Spherical Harmonic Functions... 

The integral over the product of three real spherical harmonic functions (Su 1993)... 

Table E.3 lists the products of the real spherical harmonic functions in terms of the density-normalized spherical harmonic functions dlmp.

TABLE E.3 Products of Two Real Spherical Harmonic Functions ylmp, with Normalization Defined in Appendix D ... 

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. 

D.4 Transformation of Real Spherical Harmonic Density Functions on Rotation of the Coordinate System... 

D.4.3 Rotation of the Real Spherical Harmonic Density Functions... 

As the real spherical harmonic density functions dlmp(0, 0) are directly related to the Y (9, ) terms, their rotation follows from Eq. (D.7) (Arfken 1970). The results are, for dlm+(0, 0),... 

Here P and Plm are monopole and higher multipole populations / , are normalized Slater-type radial functions ylm are real spherical harmonic angular functions k and k" are the valence shell expansion /contraction parameters. Hartree-Fock electron densities are used for the spherically averaged core and valence shells. This atom centered multipole model may also be refined against the observed data using the XD program suite [18], where the additional variables are the population and expansion/contraction parameters. If only the monopole is considered, this reduces to a spherical atom model with charge transfer and expansion/contraction of the valence shell. This is commonly referred to as a kappa refinement [19]. 

The use of real spherical harmonics is particularly bothersome. It has been demonstrated convincingly that the notion of geometrical sets of oriented real atomic angular momentum wave functions is forbidden by the exclusion principle. The use of such functions to condition atomic densities therefore cannot produce physically meaningful results. The question of increased density between atoms must be considered as undecided, at best. 

Commonly (in position space), hybrid orbitals are written in terms of single-center linear combinations of basis functions that axe themselves products of radial parts and real spherical harmonics. Let us consider... 

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. 

To relate Dff to the anisotropic motion of a molecule in a liquid crystalline solvent, we employ the function P(d, < ), defined as the probability per unit solid angle of a molecular orientation specified by the angles 6 and <3>, the polar coordinates of the applied magnetic field direction relative to a molecule-fixed Cartesian coordinate system. We expand P(0, ) in real spherical harmonics ... 

The real version of the irreducible tensor method, related to the complex representations as mentioned, is highly useful in the ligand-field theory, as will be shown in the second part of this paper. An additional reason for this is the fact that expansion theorems concerning functions and operators achieve apt forms when the tensor method is applied to real spherical harmonics. 

The irreducible representations may be classified according to whether j is an integer or half of an odd integer. We shall consider here the former, which are the potentially real representations 22 b, p. 287). These representations in contrastandard form 4) can be transformed into real form by a constant unitary matrix, i.e. the same matrix for every element in the group. The elements of the constant matrix will be chosen such that the contrastandard, self-conjugate sets which form the bases for the potentially real representations in complex standard form are connected to the sets of the usual real spherical harmonics which form the bases for the real standard representations. From the constant matrix, the vector coupling coefficients pertaining to the real functions will be deduced. 

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. 

Now since the product of powers of x etc. is merely compact notation for a product of powers of trigonometric functions, we can expand the real spherical harmonics in terms of i, y, z to generate a sum of integrals which are entirely products of powers of these angular variables, and use the fact that... 

There is a lot compressed in this expression so it is weU that we spend some time unravelling it First of aU, the subscripts t and u label the angular momenta of the real spherical harmonics and take the Im values 00,10,11c,11s,20,21c,21s,22c,22s,--- (the labels c and s stand for cosine and sine respectively). Qf is the real form of the multipole moment operator of rank t centered on A and expressed in the local-axis system of A, while T/J is the so-called T-tensor that carries the distance and angular dependence. The T-function of ranks h and I2 has a distance dependence where R is the separation between the centers of A and B. 

An unnormalized primitive real spherical harmonic Gaussian function centered at A with the exponent is defined as... 

Probability density representations of the real parts of the angular functions formed from linear combinations of spherical harmonic functions. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

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