d2y/dx12 + d2y/dy12 + d2y/dz12 + d2y/dx22 + d2y/dy22 + d2y/dz22 + (8p2m/h2)(E - V)y = 0
Here x1, y1 and z1 are the coordinates of the first electron and x2, y2, and z2 the coordinates of the second. The potential energy term expands to:
V = -e2(1/rA1 + 1/rA2 + 1/rB1 + 1/rB2 + 1/r12 + 1/rAB)
where the nuclei are labelled A and B.
The problem, since this equation cannot be solved analytically, is to find a function y which will give the lowest energy for the molecule. A first approximation might be to to use:
y1 = yA(1).yB(2)(Here yA(1) and yB(2) would be the 1s orbitals of hydrogen, but, in principle, this mathematical treatment could be adapted for an electron pair bond between any two atoms by using the appropriate atomic or hybrid orbitals.)
This would be appropriate for two widely separated atoms but gives an energy only about 6% of the experimental value when the internuclear distance rAB is optimized at 0.90 Å, which is also too long contrasted with the experimental value of 0.74 Å. In fact, the function with the electrons exchanged:
y2 = yA(2).yB(1)is equally reasonable so mathematics requires us to use the linear combinations:
y+ = C1(y1 + y2) andThe second of these y_ does not yield a minimum, but the first y+ gives a bond energy which is about 72% of the observed at rAB = 0.87 Å, so the approximation is getting better.y_ = C1(y1 - y2)
The final major improvement is to allow for ionic canonical structures HA+ HB- and HA- HB+. This generates two more terms to be included in the combinations:
y3 = yA(1).yA(2)
y4 = yB(1).yB(2)then
y+' = C2(y1 + y2) + C3(y3 + y4)This gives a bond energy 80% of the experimental value at 0.77 Å. Further, (but more abstract) tinkering with the wave function leads to even better results.
| Valence Bond Theory Results for H2 | |||
|---|---|---|---|
| Wave Function | Bond Energy | Bond Length | |
| 435 kJ mol-1 | 0.74 Å | ||
| 25 | 0.90 | ||
| 301 | 0.87 | ||
| No minimum | No minimum | ||
| 335 | 0.77 | ||