8 ATOMIC ORBITAL THEORY z-axis,and is the angle between the x-axis and the projection of the radial line on the xy-plane.The relationship between the two coordinate systems is shown in Fig.1.15.An orbital centered on a single atom (an atomic orbital)is frequently denoted as(phi)rather than w(psi)to distinguish it from an orbital centered on more than one atom (a molecular orbital)that is almost always designated w. The projection ofon the =OB.and OBA is a righ tangle.Hence cos =z/r,and th oC,but OC= r sin 0.Hence. rsin coso.Similarly,sin=ylAB:therefore,y=AB sin sin0sin Accordingly,a point(x.y.z)in Cartesian coordinates is transformed to the spherical coordinate system by the following relationships: 7=re0e日 y=r sine sin x=r sin 0 cos &9gpaeelaoent Origin (0) Figure 1.15.The relationship between Cartesian and polar coordinate systems. 1.16 WAVE FUNCTION In quantum mechanics,the wave function is synonymous with an orbital. z-axis, and ϕ is the angle between the x-axis and the projection of the radial line on the xy-plane. The relationship between the two coordinate systems is shown in Fig. 1.15. An orbital centered on a single atom (an atomic orbital) is frequently denoted as φ (phi) rather than ψ (psi) to distinguish it from an orbital centered on more than one atom (a molecular orbital) that is almost always designated ψ. The projection of r on the z-axis is z
OB, and OBA is a right angle. Hence, cos θ
z/r, and thus, z
r cos θ. Cosϕ
x/OC, but OC
AB
r sin θ. Hence, x
r sin θ cos ϕ. Similarly, sin ϕ
y/AB; therefore, y
AB sin ϕ
r sin θ sin ϕ. Accordingly, a point (x, y, z) in Cartesian coordinates is transformed to the spherical coordinate system by the following relationships: z
r cos θ y
r sin θ sinϕ x
r sin θ cosϕ 1.16 WAVE FUNCTION In quantum mechanics, the wave function is synonymous with an orbital. 8 ATOMIC ORBITAL THEORY Z x y z r θ φ φ θ Origin (0) volume element of space (dτ) B A Y X C Figure 1.15. The relationship between Cartesian and polar coordinate systems. c01.qxd 5/17/2005 5:12 PM Page 8