Question

Following trial wave functions $\phi_1 = e^{-Z' (r_1 + r_2)}$ and $\phi_2 = e^{-Z' (r_1 + r_2) (1 + g |\vec{r}_1 - \bar{\vec{r}}_2|)}$ are used to obtain variational estimates $E_1$ and $E_2$ for the ground state energy $E_0$ of the helium atom. Here, $Z'$ and $g$ are variational parameters, $\bar{\vec{r}}_1$ and $\bar{\vec{r}}_2$ are the position vectors of the electrons. Let $E_0$ be the exact ground state energy of the helium atom.$E_1$ and $E_2$ are the variational estimates of the ground state energy of the helium atom corresponding to $\phi_1$ and $\phi_2$, respectively, Which one of the following options is true?

• A. $E_1\leq E_0, E_2\leq E_0, E_1\geq E_2$
• B. $E_1\geq E_0, E_2\leq E_0, E_1\geq E_2$
• C. $E_1\leq E_0, E_2\geq E_0, E_1\leq E_2$
• D. $E_1\geq E_0, E_2\geq E_0, E_1\geq E_2$

Answer 1

Answer: (D)

$E_1\geq E_0, E_2\geq E_0, E_1\geq E_2$

Answer 2

The correct option is (D): $E_1\geq E_0, E_2\geq E_0, E_1\geq E_2$