0, angular function

The Gaussian functions are multiplied by an angular function in order to give the orbital the symmetry of a s, p, d, and so on. A constant angular term yields s symmetry. Angular terms of x, y, z give p symmetry. Angular terms of xy, xz, yz, x —y, Az —2x —2y yield d symmetry. This pattern can be continued for the other orbitals. 

A basis set is a set of functions used to describe the shape of the orbitals in an atom. Molecular orbitals and entire wave functions are created by taking linear combinations of basis functions and angular functions. Most semiempirical methods use a predehned basis set. When ah initio or density functional theory calculations are done, a basis set must be specihed. Although it is possible to create a basis set from scratch, most calculations are done using existing basis sets. The type of calculation performed and basis set chosen are the two biggest factors in determining the accuracy of results. This chapter discusses these standard basis sets and how to choose an appropriate one. 

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. 

Figure 3-11. The shapes of the angular functions are determined only by the theory of angular momentum in spherical symmetry.

The angular functions presented in Table 1 are derived from the wavefunc-tions for one-electron systems, e.g. the hydrogen atom. However, they can be... 

In this form the equation is rather cumbersome and not easily solved, so it is customary to express it in spherical polar coordinates r, 6, and, (p, where r is the distance from the nucleus and 6 and (p are angular coordinates, rather than in the Cartesian coordinates x, y, and z. The relationship of the polar coordinates to the Cartesian coordinates is shown in Figure 3.5. In this form V = e2/r, and the equation is easier to solve particularly because it can be expressed as the product R(r)Q(9)(dimensional functions R, the radial function, and 0 and , the angular functions. Corresponding to these three functions there are three quantum numbers, designated n, /, and m. 

A plot of the angular function of 0< against the value of 9 in a plane through the z axis has the form of a plot of cos 9 and consists of two circles with a node in a line along the x axis... 

Figure 3.7 (a) Plot of the angular function (0) = constant for an s orbital in plane through the z axis, (b) Plot of (00) = constant (cos 8) for a p orbital in a plane through the z axis, (c) Three-dimensional plots of the angular function (00) for the s and p orbitals. (Adapted with permission from P. A. Cox, Introduction to Quantum Theory and Atomic Structure, 1996, Oxford University Press, Oxford, Figure 4.4.)... 

To obtain pictures of the orbital ip = R0< >, we would need to combine a plot of R with that of 0, which requires a fourth dimension. There are two common ways to overcome this problem. One is to plot contour values of ip for a plane through the three-dimensional distribution as shown in Figures 3.8a,c another is to plot the surface of one particular contour in three dimensions, as shown in Figures 3.8b,d. The shapes of these surfaces are referred to as the shape of the orbital. However, plots of the angular function 0 (Figure 3.7) are often used to describe the shape of the orbital ip = RQ because they are simple to draw. This is satisfactory for s orbitals, which have a spherical shape, but it is only a rough approximation to the true shape of p orbitals, which do not consist of two spheres but rather two squashed spheres or doughnut shapes. 

In the partial wave theory free electrons are treated as waves. An electron with momentum k has a wavefunction y(k,r), which is expressed as a linear combination of partial waves, each of which is separable into an angular function Yi (0. ) (a spherical harmonic) and a radial function / L(k,r),... 

The procedure given above is an excellent example of the utilization of the Claisen rearrangement to generate an angularly functionalized steroid. The vinyl ether and aldehyde were originally prepared by Burgstahler and Nordin.2 This procedure combines variations employed by Ireland and co-workers and, in addition, introduces the use of silica gel for the purification of the vinyl ether, thereby improving the reproducibility of the procedure. 

Carosati, E., Sciabola, S. and Cruciani, G. (2004) Hydrogen bonding interactions of covalently bonded fluorine atoms from crystallographic data to a new angular function in the GRID force field. Journal of Medicinal Chemistry, 47, 5114-5125. 

The projection of T,p on each of the radial unit vectors can be evaluated in terms of the basic angular functions which make up the vector spherical harmonics.(27) Although these functions are associated Legendre polynomials for an arbitrarily oriented donor dipole, for the case of full azimuthal symmetry shown in Figure 8.19 the angular functions are ordinary Legendre functions, P (i.e., w = 0). Under these circumstances,... 

The first term on the right in the sum of potentials in equation 9.12 is the electrostatic term (represented here as a simple dipole moment). Factor / is the angular function of dipole orientation ... 

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 aspherical density formalism of Hirshfeld is a deformation model with angular functions which are a sum over spherical harmonics. It will be described in more detail in section 3.2.6. All three models have been applied extensively in charge density studies (for a comparison, see Lecomte 1991). 

Symbol Wave Functions, Mlmc Normalization for Density Functions, Llmd ... 

Equations (4.30) and (4.31) have been developed and dehned within a time-dependent framework. These equations are identical to Eqs. (35) and (32), respectively, of Ref. 80. They differ only in that a different, more appropriate, normalization has been used here for the continuum wavefunction and that the transition dipole moment function has not been expanded in terms of a spherical harmonic basis of angular functions. All the analysis given in Ref. 80 continues to be valid. In particular, the details of the angular distributions of the various differential cross sections and the relationships between the various possible integral and differential cross sections have been described in that paper. 

This section of the appendix is based on Appendix B of Ref 80. It outlines the transformation of the space-fixed form of the continuum wavefunction, Eq. (4.3), to a body-fixed form. It differs from the previous development in that the angular functions used in the final equations are all parity-adapted. 

In order to transform to the body-fixed representation, we will need to relate the angular functions Wj (R,r) to angular functions defined relative to the body-fixed axes [L., J,K,M,p)QjK ), where J,K,M,p) are the parity-adapted total angular momentum eigenfunctions of Eq. (4.5) and x(0) normalized associated Legendre polynomials of the body-fixed Jacobi angle]. 

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