Q74 · CSIR-NET Chemistry, June 2016

Paper: CSIR-NET June 2016 · Subject: Physical Chemistry · Chapter: Quantum Chemistry · Topic: Variation Method · Marks: 2 · Difficulty: Medium

Choosing some Hamiltonian H and an orthonormal basis, a linear variation is carried out to get approximately energies $\overline{\mathrm{E}}_{\mathrm{j}}$ With 2 basis functions, one obtains $\mathrm{E}_{1}(2) \leq \mathrm{E}_{2}(2)$. Taking 3 basis functions, similarly three ordered energies $\mathrm{E}_{1}(3) \leq \mathrm{E}_{2}(3) \leq$ are found. The relation which holds from the following is?
(a)$\overline{\mathrm{E}}_{1}(2) \leq \overline{\mathrm{E}}_{1}(3) \quad \overline{\mathrm{E}}_{2}(2) \leq \overline{\mathrm{E}}_{2}(3)$
(b)$\overline{\mathrm{E}}_{1}(3) \leq \overline{\mathrm{E}}_{1}(2) \quad \overline{\mathrm{E}}_{2}(2) \leq \overline{\mathrm{E}}_{2}(3)$
(c)$\overline{\mathrm{E}}_{1}(2) \leq \overline{\mathrm{E}}_{1}(3) \quad \overline{\mathrm{E}}_{2}(3) \leq \overline{\mathrm{E}}_{2}(2)$
(d)$\overline{\mathrm{E}}_{1}(3) \leq \overline{\mathrm{E}}_{1}(2) \overline{\mathrm{E}}_{2}(3) \leq \overline{\mathrm{E}}_{2}(2)$
Answer
Answer: D ✓ checked by 4AB · confidence high

The source book printed no answer; this one was worked out and checked — see the explanation.

Explanation
Enlarging the variational space can only lower (or keep) each ordered root (Hylleraas–Undheim–MacDonald): E₁(3) ≤ E₁(2) and E₂(3) ≤ E₂(2).

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