Spherical harmonics coordinates

The radial functions, R depend only upon the distance, r, of the electron from the nucleus while the angular functions, (6,(p) called spherical harmonics, depend only upon the polar coordinates, 6 and Examples of these purely angular functions are shown in Fig. 3-11. 

Cauchy function 276 Cauchy s ratio test 35-36 central forces 107,132-135 spherical harmonics 134-135 spherical polar coordinates 132-133 chain rule 37, 57, 160 character 153,195,197 orthogonality 197, 204 tables 198-200... 

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical harmonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

To find the irreducible representations of 0(3) it is necessary to find a set of basis functions which transform into their linear combinations on operating with the elements of 0(3). The set of 21 + 1 spherical harmonics Y[m(d, ), where l = 0,1, 2... and —l[Pg.91]

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. 

Related to the elliptic integral of the third kind are the Lame functions, which arise in the generalisation of spherical harmonics to confocal ellipsoidal coordinates. Applications of these in molecular electrostatics can be found... 

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. 

The factor r1 enters because the Cartesian spherical harmonics clmp are defined in terms of the direction cosines in a Cartesian coordinate system. The expressions for clmp are listed in appendix D. As an example, the c2mp functions have the form 3z2 — 1, xz, yz, (x2 — y2)/2 and xy, where x, y and z are the direction cosines of the radial vector from the origin to a point in space. 

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

Fig. A.I. Real spherical harmonics. The first one, Y , is a constant. The coordinate system attached to the unit sphere is shown. The two zonal harmonics, IT and II, section the unit sphere into vertical zones. The unshaded area indicates a positive value for the harmonics, and the shaded area indicates a negative value. The four sectoral harmonics are sectioned horizontally. The two tesserai harmonics have both vertical and horizontal nodal lines on the unit sphere. The corresponding "chemists notations," such as (3z — r ), are also marked.

These same rotation matrices arise when the transformation properties of spherical harmonics are examined for transformations that rotate coordinate systems. For example,... 

The one-electron s, p, and d orbitals frequently used to explain observed stereochemistries are a convenient but arbitrary means of decomposing the electron density into spherical harmonics. They represent nothing more than a suitable basis set for a quantum mechanical calculation. When assigned solely on the basis of the observed geometry, they convey no very profound information about the bonding processes at work. It is much simpler and more informative to say that an atom is tetrahedrally coordinated than to say that it is sp hybridized, just as it is easier to say that it forms three equatorial or two axial bonds than to say it is sp or sp hybridized, respectively. Only in the case of the electronically distorted ions discussed in Chapter 8 does an orbital description provide a meaningful rationale for the observed stereochemistry. 

Although we have shown above that / can be a half-integer, we shall see below in the discussion of spherical harmonics that, when the function is a function of the spatial coordinates xyz, the half-integer values are not allowed. Spin functions are not spatial functions, however, and half-integer values of / do occur for them. 

In our discussion of spherical harmonics we will use an expression of the three-dimensional Laplacian in spherical coordinates. For this we need spherical coordinates not just on but on all of three-space. The third coordinate is r, the distance of a point from the origin. We have, for arbitrary (x, y, zY e... 

A set of five real d-orbitals defines a function space. In spherical polar coordinates r, 0, and , they consist of a common radial function times a combination of spherical harmonics 7 (0, ), l 2 and m =0, 1, 2. The combinations of the five spherical harmonics are chosen such that the orbitals are real. 

The angular functions 7<(0, ), called spherical harmonics, are common to all atoms. They are listed in Table 11-2.1 (in this table the normalizing constants have been omitted) together with the well-known symbols for the orbitals to which they correspond, i.e. s, pr, pr, p0, etc. The subscripts in these symbols are directly related to the angular functions if, for example, the angular function is ain 0 sin 2, then changing to the Cartesian coordinates x, y, and z where x r sin Q cos y = r sin 6 sin and z — r cos 0 gives us ... 

The parity of atomic states is important in spectroscopy. A radial function is an even function [see (1.113)] the spherical harmonic Y(m is found to be an even or odd function of the Cartesian coordinates according to whether / is an even or odd number. For a many-electron atom, it follows that states arising from a configuration for which the sum of the / values of all the electrons is an even number are even functions when 2,/, is odd, the state has odd parity. 

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... 

In the Born-Oppenheimer approximation, the molecular wave function is the product of electronic and nuclear wave functions see (4.90). We now examine the behavior of if with respect to inversion. We must, however, exercise some care. In finding the nuclear wave functions fa we have used a set of axes fixed in space (except for translation with the molecule). However, in dealing with if el (Sections 1.19 and 1.20) we defined the electronic coordinates with respect to a set of axes fixed in the molecule, with the z axis being the internuclear axis. To find the effect on if of inversion of all nuclear and electronic coordinates, we must use the set of space-fixed axes for both fa and if el. We shall call the space-fixed axes X, Y, and Z, and the molecule-fixed axes x, y, and z. The nuclear wave function of a diatomic molecule has the (approximate) form (4.28) for 2 electronic states, where q=R-Re, and where the angles are defined with respect to space-fixed axes. When we replace each nuclear coordinate in fa by its negative, the internuclear distance R is unaffected, so that the vibrational wave function has even parity. The parity of the spherical harmonic Yj1 is even or odd according to whether J is even or odd (Section 1.17). Thus the parity eigenvalue of fa is (- Yf. 

The Onk and Q k are operators which are respectively linear combinations of spherical harmonics and expansions in terms of Cartesian coordinates, 1 = 2 for d-orbitals, 3 for /-orbitals. The parameters B k and A k are, of course, specialized forms of the general form given in equation (2), but including the evaluation of the relevant radial integrals. 

Spherical harmonics are derived from solutions of Laplace s equation ih spherical coordinates using the method of separation of variables—i.e., a solution of the form... 

In Table 2.2 we have listed the first few spherical harmonics, for the s, p, and d states. It is worth noting that some authors introduce a factor of [—l]m in defining the associated Legendre polynomials, producing a corresponding difference in the spherical harmonics.2 There are i-m nodes in the 6 coordinate, and none in the <(> coordinate. 

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