= Solution
{c}
The <two-mode approximation for a Bose gas> retains the real localized <orbitals>:
$$
\boxed{\hat\psi^\dagger(\mathbf r)\simeq\varphi_L(\mathbf r)\hat a_L^\dagger+\varphi_R(\mathbf r)\hat a_R^\dagger.}
$$
It is valid when the lowest doublet is well separated from higher <energy levels>, and the <temperature> and interaction-induced changes of the <orbitals> are small on that separation. More explicitly, if $\Delta$ is the <energy gap> to discarded <orbitals>, sufficient low-<energy> conditions are $k_BT,t,NU_0I\ll\Delta$, where $I=\int d^3r\,|\varphi_L|^4$. A deep <double-well potential> suppresses interwell interaction overlaps. Large $N$ alone does not justify the truncation: increasing $N$ at fixed coupling can violate the interaction condition.
Let $\epsilon=(E_0+E_1)/2$ and use the even and odd <eigenfunctions> to evaluate the projected one-particle <Hamiltonian operator>:
$$
\langle L|h_0|L\rangle=\langle R|h_0|R\rangle=\epsilon,\qquad
\langle L|h_0|R\rangle=\frac{E_0-E_1}{2}=-t.
$$
Thus
$$
\hat H_0=\epsilon(\hat n_L+\hat n_R)-t(\hat a_L^\dagger\hat a_R+\hat a_R^\dagger\hat a_L),\qquad
\boxed{t=\frac{E_1-E_0}{2}>0.}
$$
The same <quantum tunnelling> matrix element is $t=-\int d^3r\,\varphi_Lh_0\varphi_R$. Reflection symmetry makes $\int|\varphi_L|^4=\int|\varphi_R|^4=I$. Neglecting mixed <contact interaction> integrals leaves
$$
\hat H_{\rm int}=\frac{U_0I}{2}\sum_{j=L,R}(\hat a_j^\dagger)^2\hat a_j^2
=\frac{U_0I}{2}\sum_{j=L,R}\hat n_j(\hat n_j-1).
$$
The last step is the <normal-ordered boson pair count>, following from the <canonical commutation relation>. Consequently the <two-site Bose-Hubbard model> has precisely the requested coefficients
$$
\boxed{w=\frac{U_0}{2}\int d^3r\,|\varphi_L(\mathbf r)|^4,\qquad
\mu=w-\frac{E_0+E_1}{2},}
$$
and
$$
\boxed{\hat H=-t(\hat a_L^\dagger\hat a_R+\hat a_R^\dagger\hat a_L)
+w(\hat n_L^2+\hat n_R^2)-\mu(\hat n_L+\hat n_R).}
$$
Here $\mu$ is the coefficient obtained by rewriting the fixed-<particle number> <Hamiltonian operator>, rather than an independently prescribed thermodynamic <chemical potential>. Adding a reservoir term $-\mu_{\rm res}\hat N$ would replace it by $\mu+\mu_{\rm res}$.
The negligible-overlap instruction applies to the mixed <contact interaction> integrals while the small <quantum tunnelling> matrix element is retained. Exactly disjoint smooth <orbitals> for a local $h_0$ would also give $t=0$; setting every overlap-related matrix element identically to zero would discard the physical splitting one is trying to describe.
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