“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

The Davidson correction: a patch, not a cure

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Lesson 5 of 5 standard ~5 min

Once you have seen CISD lose size consistency, the natural question is whether it can be rescued cheaply. The Davidson correction estimates the energy lowering the missing quadruples would have provided, from quantities CISD already has:

Δ E_{Q} \approx (1 - c_{0}^{2}) (E_{CISD} - E_{HF})

where c_0 is the coefficient of the HF determinant in the normalized CISD wavefunction. The logic: a small c_0 signals a wavefunction with lots of double-excitation character, which is exactly when disconnected quadruples (a double on top of a double) matter — so scale the correlation energy up by the missing weight 1 − c_0².

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The Davidson correction ΔE_Q ≈ (1 − c₀²)(E_CISD − E_HF) estimates the missing quadruples. What does a SMALL c₀ (HF-coefficient) signal?

Go deeper ↓Full Configuration Interaction

It works well enough that 'CISD+Q' was a standard method for years, and it largely restores size consistency. But it is an estimate bolted onto a flawed ansatz, not a fix of the ansatz itself. The structural repair — making size extensivity automatic — is the exponential of coupled cluster, the next unit. Davidson is the bridge: it concedes exactly what CISD lacks.