However, it is worth considering key assumptions. The first is imbedded in the model and the second in approximate solutions.
Computational quantum chemistry does raise some questions that physicists rarely seem to think about. They are hard and scary.
1. Rigid orbitals.
Each lattice site is associated with some sort of atomic or molecular orbital. A beauty of the models is one does not have to know exactly what this orbital is.
Now consider different energy eigenstates of the model. These only differ in orbital occupations and the different coefficients in superposition of Slater determinants (or creation and annihilation operators). The localised orbitals do not change.
Similarly, when one changes lattice or vibrational co-ordinates in a Holstein or Su-Schrieffer-Heeger model, one assumes the orbitals associated with the lattice sites don't change.
2. Strong correlations can be captured with just a few Slater determinants.
Physicists like simple variational wave functions such as Gutzwiller, RVB, ...
Yet if one considers the treatment of simple molecules by high level quantum chemistry methods one finds two things need to be taken into account to obtain an accurate description of strong electron correlations.
A. Orbital relaxation can be significant.
If one considers the orbitals (whether localised or delocalised) that appear in many-body wave function for different eigenstates (e.g. singlet, triplet, low lying excited states) one sees these orbitals can be quite different. This is why one has methods such as the Breathing Orbital Valence Bond method, reviewed here. It discusses specific examples of how the quality of wave function can be improved by using different orbitals in covalent and ionic parts of a wave function and as bond lengths change.
If one wants to think about parametrising an effective Hamiltonian for the cuprates, such as the t-J model this paper is relevant.
Heisenberg exchange enhancement by orbital relaxation in cuprate compounds
A.B. van Oosten, R. Broer, W.C. Nieuwpoort
Chemists are very good at performing calculations that involve even hundreds of thousands of Slater determinants (configurations).
Consider the case of benzene. If one wants to get accurate energies for the ground and low-lying excited states. One finds that if one works with delocalised molecular orbitals one needs to include literally hundreds of thousands of orbitals.
However, if one works instead with localised orbitals and valence bond theory one finds one can use a handful of Slater determinants. For example, the ground state can be described in terms of the five structures below. Eighty per cent of the weight is in the first two. Their superposition forms the legendary resonating valence bond (RVB) structure. However, I want to stress that 20 per cent of the weight is in the rest, and this is not a trivial amount.
I doubt these issues matter if one just uses effective model Hamiltonians to obtain physical insights, describe qualitative behaviour and trends, and semi-quantitative comparisons with experiment. However, at some point one is not going to be able to get detailed quantitative description of real materials.