GATE 2024 PH – Question 28
Trial functions $\phi_1=e^{-Z^{\prime}(r_1+r_2)}$ and $\phi_2=e^{-Z^{\prime}(r_1+r_2)}(1+g|\mathbf r_1-\mathbf r_2|)$ estimate helium ground energy $E_0$. With variational parameters $Z^{\prime},g$ and corresponding estimates $E_1,E_2$, which is true?
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Correct answer: (D) $E_1\ge E_0,E_2\ge E_0,E_1\ge E_2$
Explanation
**Variational principle.** For any trial wave function, the estimated energy is an **upper bound** to the true ground-state energy:
$$E_{trial}\geq E_0 .$$
So $E_1\geq E_0$ and $E_2\geq E_0$.
**Comparing the two trial families.** The second function is $\phi_2=\phi_1\,(1+g|\mathbf r_1-\mathbf r_2|)$. Setting $g=0$ gives $\phi_2=\phi_1$, so the first family is a special case of the second. Minimising over the larger set of parameters ($Z^\prime$ and $g$) can only give an energy that is **lower than or equal to** the one obtained with $Z^\prime$ alone:
$$E_2\leq E_1 .$$
So $E_1\geq E_0$, $E_2\geq E_0$, $E_1\geq E_2$: option **D**. (The added $g$ term builds in electron correlation, which brings the estimate closer to the true value.)