Ground state of hydrogen molecule using H-L
I am reading The Theory of Magnetism I, by <PERSON>. In Chapter 2, a hydrogen molecule is studied in the following way:
We have a Hamiltonian of a hydrogen molecule: $$H = H^0_1 + H^0_2 + H^\lambda \tag{1}$$
where:
$$ H^0_1 = \left ( \frac{p_1^2}{2m}-\frac{e^2}{r_{1a}} \right ), \quad H^0_2 = \left ( \frac{p_2^2}{2m}-\frac{e^2}{r_{2b}} \right ), \quad H^\lambda =\left( \frac{e^2}{R_{ab}}+\frac{e^2}{r_{12}}-\frac{e^2}{r_{1b}}-\frac{e^2}{r_{2a}} \right )$$
where (a,b) represent the nuclei of each hydrogen atom and the numbers 1,2 refer to their respective electron.
The eigenfunctions of $H_a \equiv H_1^0$ and $H_b \equiv H_2^0$ are: $\phi_a(\vec{r_1})$ and $\phi_b(\vec{r_2})$ and we can define $\Psi_I = \phi_a(\vec{r_1}) \phi_b(\vec{r_2})$ and $\Psi_{II} = \phi_a(\vec{r_2}) \phi_b(\vec{r_1})$
Defining $U$, $V$ and $l$ as:
$$ l \equiv \int d^3 r \phi_a^(\vec{r})\phi_b(\vec{r}); \quad U \equiv \int d^3r_1d^3r_2 \Psi_I^ \Psi_{II} H^\lambda \tag{2a}$$ $$V \equiv \int d^3r_1d^3r_2 |\Psi_{II}|^2 H^\lambda = \int d^3r_1d^3r_2 |\Psi_{I}|^2 H^\lambda\tag{2b} $$
Then a variational function: $\Psi = c_I \Psi_I + c_{II} \Psi_{II}$ is chosen. And in order to determine the ground state (in terms of $c_I$ and $c_{II}$)we solve the following variational problem:
$$E_{var} = \frac{\int d^r_1 d^3r_2 \Psi^ H \Psi}{\int d^r_1 d^3r_2 \Psi^ \Psi}, \quad \frac{\partial E_{var}}{\partial c_{I,II}} = 0 \tag{3}$$
Then he changes the variational problem to the following eigenvalue problem:
$$ \left ( \begin{matrix} V & U \ U^\dagger& V \end{matrix} \right ) \left ( \begin{matrix} c_I \ c_{II} \end{matrix} \right ) = (E-2E_0) \left ( \begin{matrix} 1 & l^2 \ (l^2)^* & 1 \end{matrix} \right ) \left ( \begin{matrix} c_I \ c_{II} \end{matrix} \right )\tag{4} $$
Where he finds 2 solutions:
$$ c_I = \pm c_{II} \qquad E_{\pm} = 2 E_0 + \frac{V\pm U}{1\pm l^2}\tag{5}$$
My question is: How can $E_-$ and $E_+$ be both minimum values of $E_{var}$? If $(c_I,-c_I)$ is the solution for $E_-$, then can't we change it slightly to get an energy $E = E_- +\delta E < E_+$? 474