Rotation of Atomic Orbitals

As calculated, the overlap integrals represent the overlap of atomic orbitals that are aligned along the z-axis. In general, this will not be the case, and the diatomic overlap integral matrix must be rotated in order to represent the actual orientation used. The rotation operation can be constructed in a completely general way, or it can be constructed using trigonometric functions, as shown here.

The rotation matrices are well known, but the method by which they are constructed is by no means simple. Consider the general case of one atom at the origin, and a second atom at some point p(x,y,z). Let the angle from the second atom to the first atom to the z z axis be b, and the angle of the projection of the second atom onto the xy plane to the y axis be a. Then, the p-orbital transform is as shown in the following table:

Angular Dependence of the p-orbitals
	p(x)	p(y)	p(z)
π+	cos a cos b	-sin a	cos a sin b
π-	sin a cos b	cos a	sin a sin b
σ	-sin b	0	cos b

The rotation matrices for the higher harmonics are fairly difficult to construct. In this work, several mathematical tools will be used. These are:

Mathematical Tools for use with Spherical Harmonics

: 1. All odd integrals, i.e., integrals of the type <xⁿ>, where n is an odd integer, are zero.
: 2. <x²> = <y²> = <z²> ≠ 0
: 3. <x²y²> = <x²y²> = <y²z²> = <x²z²> ≠ 0
: 4. <x⁴> = <y⁴> = <z⁴> ≠ 0
: 5. All integrals of the type <aⁿb^mc^l> where a, b, and c are different members of the set (x,y,z) are equal.
: 6. The normalization condition for the spherical harmonics is: ∫₀²^π ∫₀^π(Y_l^m(θφ))²sin(θ)dθdφ = 1
: 7. cos²(θ) + sin²(θ) = 1
: 8. cos(2θ) = 2cos²(θ) - 1
: 9. sin(2θ) = 2sin(θ)cos(θ)
: 10. All atomic orbitals are assumed to be normalized: , i.e., <ψ²> = 1
: 11. x = r.sin(θ).cos(φ), y = r.sin(θ).sin(φ), z = r.cos(φ)

When working with the angular components, it is normally easier to use the Cartesian symbols x, y, and z instead of the trigonometric forms. This avoids the potentially confusing expressions that involve the angles a, b, θ, and φ.

Consider the rotation of d(x²-y²). The rotation components involved are: R(x) = (cosa.cosb)x + (sina.cosb)y + (-sinb)z, and R(y) = (-sina)x + (cosa)y .

The rotation matrix element <d(x²-y²)|R|d(x²-y²)> can be derived using the relationship:

R\|d(x²-y²)>	=	(cos²a.cos²b)x² + (2cosa.cosb.sina.cosb)xy +
		(sin²a.cos²b)y² + (-2sina.cosb.sinb)yz + (sin²b)z²+
		(-sin²a)x² + (2sina.cosa)xy + (-cos²a)y²

In this expression, the integral over all odd terms vanish, therefore:

<d(x²-y²)\|R\|d(x²-y²)>	=	<x²(cos²a.cos²b - sin²a)x²> +
		<x²(sin²a.cos²b - cos²a)y²> +
		<x²(sin²b)z²> +
		<y²(-cos²a.cos²b + sin²a)x²> +
		<y²(-sin²a.cos²b + cos²a)y²> +
		<y²(-sin²b)z²>

The integral <x⁴> has the same value as <y⁴>, and <x²y²> has the same value as <x²z²>, etc., therefore this expression can be simplified. Collecting together terms of the type <x⁴> gives

cos²a.cos²b - sin²a -sin²a.cos²b + cos²a

(cos²b + 1).(2cos²a - 1)

Collecting terms of the type <y²x²> gives

sin²a.cos²b - cos²a + sin²b - cos²a.cos²b + sin²a - sin²b

-(cos²b + 1).(2cos²a - 1)

The rotation matrix element is thus:

<d(x² - y²)|R|d(x² - y²)> = <x⁴ -2x²y² +y⁴>½(cos²b + 1).(2cos²a - 1)

As a result of the fact that the angular components of the atomic orbitals are normalized, i.e., <x⁴-2x²y² +y⁴>=1:

<d(x² - y²)|R|d(x² - y²)> = (2cos²a - 1)cos²b + ½(2cos²a - 1)sin²b

which is the form used in the calculation of the d-rotation matrix. The full set of d-orbital rotation matrix elements is:

	Angular Dependence of the d-orbitals
	δ+ ≡ d(x² - y²)	π+ ≡ d(xz)	σ ≡ d(2z²-x²-y²)	π- ≡ d(yz)	δ- ≡ d(xy)
d(x²-y²)	(2cos²a - 1)cos²b + ½(2cos²a - 1)sin²b	-cosa.sinb.cosb	√¾.sin²b	-sina.sinb.cosb	2.sina.cosa.cos²b + sina.cosa.sin²b

d(xz)	(2cos²a - 1)sinb.cosb	cosa(2cos²b - 1)	-√3.sinb.cosb	sina (2cos²b - 1)	2.sina.cosa.sinb.cosb
d(2z²-x²-y²)	√¾.(2cos²a - 1)sin²b	√3.cosa.sinb.cosb	cos²b - ½sin²b	√3.sina.sinb.cosb	√3.sina.cosa.sin²b
d(yz)	-2sina.cosa.sinb	-sina.cosb	0	cosa.cosb	(2cos²a - 1)sinb
d(xy)	-2sina.cosa.cosb	sina.sinb	0	-cosa.sinb	(2cos²a - 1)cosb