# How exactly are valence orbitals combined in split valence basis sets such as 3-21G?

For hydrogen, the 3-21G basis set (in CFOUR format) is

H:3-21G
3-21G Split-valence basis set

1
0
2
3

0.5447178000D+01 0.8245472400D+00 0.1831915800D+00

0.1562849787D+00 0.00000000
0.9046908767D+00 0.00000000
0.00000000 1.0000000

My interpretation is that this defines two orbitals: $$\phi_1 = 0.1562849787g(0.5447178) + 0.9046908767g(0.8245472400)$$

and

$$\phi_2 = 1.0g(0.18319158)$$

But how are these two orbitals combined to form an orbital for the lone electron?

## 1 Answer

They are combined into the final orbitals $$\psi_i$$ through a linear combination, $$\sum_j c_{ij} \phi_j$$. Determining the coefficients $$c_{ij}$$ of that combination is the entire goal of the calculation.

• Ah, so essentially $\phi_1$ and $\phi_2$ get their own coefficients in the molecular orbitals, rather than sharing the same one if we only had $\phi=\phi_1 + \phi_2$ as our atomic orbital? May 23 at 1:20
• Exactly! - Also, you're correct in calling the result a molecular orbital even though the calculation is on a single atom. It's just a special case, but in general, the combinations of your atom-centered basis functions indeed gives you the delocalized MOs. May 23 at 20:13