Before relationships can be determined, some simple tools must be constructed:
Given that fxyz is normalized, <√(105/16π)xyz/r3√(105/16π)xyz/r3> = 1, then <x2y2z2/r6> = 16π/105.
Given that fz(x2-y2) is normalized,
| <√(105/16π)z(x2-y2)/r3√(105/16π)z(x2-y2)/r3> | = | 1 | |
| then | (105/16π)<(z2(x4-2x2y2+y4)/r6)> | = | 1 |
| but | <z2x4/r6> | = | <z2y4/r6> |
| therefore | (105/16π)<2z2x4/r6> - 2*16π/106 | = | 1 |
| therefore | <z2x4/r6> | = | 24π/105 |
Given that fz3 is normalized,
| <√(7/16π)z(5z2-3r2)/r3√(7/16π)z(5z2-3r2)/r3> | = | 1 | |
| then | (7/16π)<(z2(2z2-3x2-3y2)2/r6)> | = | 1 |
| therefore | (7/16π)<(4z6-12z4x2-12z4y2+18x2y2z2+9x4z2+9y4z2)/r6> | = | 1 |
| (7/16π)(<4z6/r6>-6*24π/105+18*16π/105) | = | 1 | |
| or | <z6/r6> | = | 24π/105 |
To summarize: <x2y2z2/r6> = 16π/105, <z2x4/r6> = 24π/105, <z6/r6> = 24π/105
Integrals involving real spherical harmonics are as follows:
f and f harmonics:
The orbitals are normalized, e.g.:
| <fxz2|fxz2> | = | <(21/32π)x(5z2-r2)/r3(21/32π)x(5z2-r2)/r3> | |
| = | (21/32π)<(x2(4z2-x2-y2)2/r6)> | ||
| = | (21/32π)<(x2(16z4+x4+y4-8z2x2-8z2y2+2x2y2)/r6> | ||
| = | (21/32π)<(16x2z4 +x6 +x2y4 -8x4z2 -8x2z2y2 +2x4y2)/r6> | ||
| = | 21/32π)(11*24π/105+24π/105 -8*16π/105) | ||
| = | 1 |
Most integrals are zero by inspection: if the product involves an odd power, the integral is automatically zero. The non-zero integrals are:
| <fx3|fxz2> | = | <(7/16π)x(5x2-3r2)/r3√(21/32π)x(5z2-r2)/r3> | |
| = | (7/16π)√(3/2)<(x2(2x2-3y2-3z2)(4z2-x2-y2)/r6)> | ||
| = | (7/16π)√(3/2)<(x2(-2x4+3y4-12z4+x2y2+11x2z2-9y2z2)/r6> | ||
| = | (7/16π)√(3/2)<(-2x6+3x2y4-12x2z4+x4y2+11x4z2-9x2y2z2)/r6> | ||
| = | (7/16π)√(3/2)(-2* 24π/105+3*24π/105-9*16π/105) | ||
| = | -√(3/8) | ||
| = | <fy3|fyz2> |
| <fx3|fx(x2-3y2)> | = | <√(7/16π)<x(5x2-3r2)/r3√(35/32π)x(x2-3y2)/r3> | |
| = | (7/16π)√(5/2)<(x2(2x2-3y2-3z2)(x2-3y2)/r3> | ||
| = | (7/16π)√(5/2)<(x2(2x4+9y4-9x2y2-3x2z2+9y2z2)/r6> | ||
| = | (7/16π)√(5/2)<(2x6+9x2y4-9x4y2-3x4z2+9x2y2z2)/r6> | ||
| = | (7/16π)√(5/2)(2*24π/105-3*24π/105+9*16π/105) | ||
| = | √(5/8) | ||
| = | <fy3|fy(y2-3x2)> |
| <fx(z2-y2)|fx(x2-3y2)> | = | <√(105/16π)x(z2-y2)/r3√(35/32π)x(x2-3y2)/r3> | |
| = | (35/16π)√(3/2)<x2(z2x2-3z2y2-x2y2+3y4)/r6> | ||
| = | (35/16π)√(3/2)<(x4z2-3x2y2z2-x4y2+3y4x2)/r6> | ||
| = | (35/16π)√(3/2)( 24π/105-3* 16π/105- 24π/105+3* 24π/105) | ||
| = | √(3/8) | ||
| = | <fy(z2-x2)|fy(y2-3x2)> |
| <fx(z2-y2)|fxz2> | = | <√(105/16π)x(z2-y2)/r3√(21/32π)x(5z2-r2)/r3> | |
| = | (21/16π)√(5/2)<x2(z2-y2)(4z2-x2-y2)/r6)> | ||
| = | (21/16π)√(5/2)(x2(4z4+y4+x2y2-x2z2-5y2z2)/r6)> | ||
| = | (21/16π)√(5/2)(4x2z4+x2y4+x4y2-x4z2-5*x2y2z2)/r6)> | ||
| = | (21/16π)√(5/2)(5*24π/105-5*16π/105) | ||
| = | √(5/8) | ||
| = | <fy(z2-x2)|fyz2> |
The complete unitary matrix relating the cubic and axial sets of f orbitals is as follows:
| Axial | ||||||||
| fz3 | fxz2 | fyz2 | fz(x2-y2) | fxyz | fx(x2-3y2) | fy(y2-3x2) | ||
| fx3 | 0 | -√(3/8) | 0 | 0 | 0 | √(5/8) | 0 | |
| fy3 | 0 | 0 | -√(3/8) | 0 | 0 | 0 | √(5/8) | |
| fz3 | 1.0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| Cubic | fx(z2-y2) | 0 | √(5/8) | 0 | 0 | 0 | √(3/8) | 0 |
| fy(z2-x2) | 0 | 0 | √(5/8) | 0 | 0 | 0 | √(3/8) | |
| fz(x2-y2) | 0 | 0 | 0 | 1.0 | 0 | 0 | 0 | |
| fxyz | 0 | 0 | 0 | 0 | 1.0 | 0 | 0 |