Cartesian atomic orbitals

The 2p and 2py orbitals are neither better nor worse than our W/ = +1 orbitals, but they reflect a different choice of how we quantize our quantum states. Its that choice that seems so weird about the quantum mechanics here, and its origins are back in the uncertainty principle (Section 1.5). Although we cannot know exact values for all the parameters of our atom simultaneously, we can choose which ones we do want to know. When we use the m/ = 1 angular wavefunctions, we have states for which the z-component of the angular motion is well defined. When we use 2p and 2py, we have built Cartesian atomic orbitals that are pure real and that correspond to well-defined positions in space. 

Cartesian axes. You may wish to refer to the animation w of the atomic orbitals found on the Web site for this text. 

The variation of iff with the angles 9 and 0 is exactly the same as illustrated in Fig 3.14 in Section 3.4. The angular dependence of atomic orbitals is often represented in another way, using the relation between spherical polar and cartesian coordinates. For example, the cosd function appropriate to l = 1 and m = 0 can be expressed as... 

Here, M is taken to be a member of the 3d series. The 3d, 4s and 4p orbitals are deemed to constitute its valence orbitals. L is a ligand which has available for bonding to M only a lone pair, oriented such that o overlap is possible with M orbitals having non-zero electron density along the M-L axes. Fig. 8.1 shows the labelling of the Cartesian axes, with respect to which the M atomic orbitals are labelled. The L lone pairs are... 

Figure 2.2 Boundary surface of atomic orbitals. The boundaries represent angular distribution probabilities for electrons in each orbital. The sign of each wave function is shown. The d orbitals have been classified into two groups, t2g and eg, on the basis of spatial configuration with respect to the cartesian axes. (Reproduced and modified from W. S. Fyfe, Geochemistry of Solids, McGraw-Hill, New York, 1964, figure 2.5, p. 19).

The first step in the symmetry determination of the dynamic properties is the selection of the appropriate basis. Appropriate here means the correct representation of the changes in the properties examined. In the investigation of molecular vibrations (Chapter 5), either Cartesian displacement vectors or internal coordinate vectors are used. In the description of the molecular electronic structure (Chapter 6), the angular components of the atomic orbitals are frequently used... 

The theory (7, 8, 9,10,11,12) will be outlined for molecules having n atoms with a total of P valence shell electrons. We seek a set of molecular orbitals (LCAO-MO s), that are linear combinations of atomic orbitals centered on the atoms in the molecule. Since we shall not ignore overlap, the geometry of the molecule must be known, or one must guess it. The molecule is placed in an arbitrary Cartesian coordinate system, and the coordinates of each atom are determined. Orbitals of the s and p Slater-type (STO) make up the basis orbitals, and as indicated above we restrict ourselves to the valence-shell electrons for each of the atoms in the molecule. The STOs have the following form for the radial part of the function (13,18) ... 

Matrix dements between atomic orbitals can be defined in terms of spherical harmonics based on an axis along the inteinuclear separation, as indicated at (a) in the upper left. There, tine lower state is an hi = 0 p orbital and the upper state is an m = 0 d orbital, which has angular dependence as 3 f(2 /r ) — l/3]/2. For other d orbitals (hi 0), it is easy to visualize other cartesian forms, such as the symmetry z.v/c shown at (e) or the xy/r shown at (f), which lead to the same values for the matrix elements as do the spherical harmonics. There is only one independent sd matrix element for it, the different cases indicated at (c) in the upper right can be related by algebraic manipulation or the transformations described in Eq. (19-21). The two independent pd matrix elements are shown at (a) and (b) and the three independent dd matrix elements arc shown at (d), (e), and (f). Signs of the wave functions arc chosen such that all but and Vjj, are expected to be negative. 

You may need to draw a visualization of atomic orbitals, usually the s, p and d orbitals. This can be simplified by the use of Cartesian co-ordinates which allow a three-dimensional representation on paper. This is neither easy to replicate or often necessary. A simplified approach is, for example, to replace the spherical s orbital with a circle (Fig. 42.7). Similarly, the three p orbitals can be represented in two dimensions by the use of correct labelling of the axes (Fig. 42.7). Finally, the same approach can be replicated on the five d orbitals (Fig. 42.7). It is worthwhile remembering that the d , d,. and d, -orbitals do not reside on the axes (.x, y or z), but in the plane of their respective axes. In addition, the dy2, .2 orbital occupies the. v- and y-axes and the d-2 orbital occupies the z-axis. 

Figure 3.2 Comparison of the local Cartesian and transformed coordinate systems at a general point on the unit sphere the standard spherical harmonic functions as the angular parts of the appropriate atomic orbitals are defined with reference to the local Cartesian set ex(j), ey(j) and ez(j) for each atomic position (j) with radius vector R on the unit sphere. Then the transformation of equation 3.1 is applied to construct the new local coordinate system cr(j), and JT (j).

Although the five d orbitals of an isolated atom are isoenergetic, they can be divided into two groups according to the orientation of the corresponding axes with respect to the x,y and z cartesian axes orbitals dxy, dy and dxz make a t group, whereas d 2 and d 2 2 form an e group. 

The two electron one center integrals are given as =Gss, =GSp, =Gpp, =Gp2 and =Hsp, while Uw is taken to be Ussor Upp. ForGp2 the p and p indices represent different Cartesian p orbitals on the same center. The nuclear energy is expressed as the summation over the individual nuclear-nuclear interactions for atoms i andj. 

Beginning with the second shell, each shell also contains a p subshell, defined by f = 1. Each of these subshells consists of a set of three p atomic orbitals, corresponding to the three allowed values of m (-1, 0, and +1) when f = 1. The sets are referred to as 2p, 3p, 4p, 5p,... orbitals to indicate the main shells in which they are found. Each set of atomic orbitals resembles three mutually perpendicular equal-arm dumbbells (see Eigure 5-22). The nucleus defines the origin of a set of Cartesian coordinates with the usual x, y, and z axes (see Eigure 5-2 3 a). The subscript x, y, or z indicates the axis along which each of the three two-lobed orbitals is directed. A set of three p atomic orbitals may be represented as in Eigure 5-2 3b. 

Surfaces are incomplete solids, in that the surface atoms have no nearest neighbors in one of the six Cartesian directions. This means that there are dangling atomic orbitals, both filled and empty, which are not being used. This applies, of course, to a clean surface. Which atomic orbitals are at the surface, and their orientation, depend on an arbitrary choice for a coordinate system and on which crystal plane forms the surface. In any case, the atomic orbitals form linear combinations called surface orbitals. The interaction is usually weaker than in the bulk of the solid. 

For a basis set of (re) atomic orbitals which are singly noded in the plane perpendicular to the radial vector (see Fig. 16b), the required Harmonics are the Vector Surface Harmonics146). The two p11 (or d") atomic orbitals at each cluster vertex (i) behave as a pair of orthogonal unit vectors which are tangential to the surface of the sphere, jrf and jif are defined as Jt-symmetry orbitals on the ith cluster atom, pointing in the direction of increasing 0 and cj> respectively, as shown in Fig. 18. At the poles of the cluster sphere these vectors may be related to Cartesian vectors as follows ... 

The nonorthogonal basis functions Xx( )are referred to as atomic orbitals (AOs) and are often taken to be Cartesian Gaussian-type orbitals (GTOs) of the (unnormalized) form ... 

For r = 1, V = y so that y is the value of the solid harmonic in the surface of the unit sphere at points defined by the coordinates 6 and surface harmonics of degree /. The associated Legendre polynomials Pf(cosd) have l — m roots. Each of them defines a nodal cone that intersects a constant sphere in a circle. These nodes, as shown in Figure 20-5, are in the surface of the sphere and not at r = 0 as assumed in the definition of atomic orbitals. Surface harmonics are obviously undefined for r = 0. The linear combinations i/ i i/i i define one real and one imaginary function directed along the X and Y Cartesian axes respectively, but these functions (denoted and ipy) are no longer eigenfunctions of L, but of or Ly instead. 

By convention, eachp atomic orbital is directed along one of the three Cartesian axes (Figure 1.10), and, in considering the formation of a diatomic X2, it is convenient to fix the positions of the X nuclei on one of the axes. In diagram 1.13, the nuclei are placed on the axis, but this choice of axis is arbitrary. Defining these positions also defines the relative orientations of the two sets of p orbitals (Figure 1.20). 

In Figure 4.6b we illustrate how the tetrahedral structure of CH4 relates to a cubic framework. This relationship is important because it allows us to describe a tetrahedron in terms of a Cartesian axis set. Within valence bond theory, the bonding in CH4 can conveniently be described in terms of an sp valence state for C, i.e. four degenerate orbitals, each containing one electron. Each hybrid orbital overlaps with the li atomic orbital of one H atom to generate one of four equivalent, localized 2c-2e C—H (T-interactions. 

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