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

*B. Many Slater determinants are often required.*

*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.

So here's a strong claim. I presume it's wrong, but I don't see why (I'd be delighted to be enlightened).

ReplyDeleteOrbital relaxation isn't "important", because it doesn't change symmetry of the orbital. So it just leads to a renormalisation of the effective parameters.

So outside of the example you give, calculating parameters from first principles, you don't have to worry about it. Semi-empirical and empirically derived parameters should include this (to some degree). Say a DFT calculation should include this if you compare different 'ground state' energies (e.g. neutral vs ionic for Hubbard U, as was done for buckyballs). I agree it's more of a problem for, say, cRPA calculations - but then I would have thought any method that uses unoccupied Kohn-Sham orbitals (are their energies) is going to have some issue along these lines somewhere.

Hi Ben

DeleteThanks for the comment.

I have also been wondering about whether is just gets absorbed into the parameters.

However, I am pretty sure that when it comes to the many body-wave functions the orbital relaxation will matter. Hence, this will change observables involving matrix elements, e.g. optical absorption and neutron scattering cross sections.

There is a way to obtain relaxation of single-particle orbitals using a fixed set of rigid orbitals: just take their linear combination with the coefficients as the adjustable parameters.

ReplyDeleteThis requires the set of rigid orbitals to be complete, which is not feasible in computer implementation of models, at least for spatial orbitals, i.e., truncation to a finite set is required.

Is this what you mean by lack of flexibility in allowing the relaxation of orbitals?

As long as a reasonably complete set of rigid single-particle states are available, any many-body state can be constructed by appropriate superposition of Slater determinants, even without allowing for basis-orbital relaxation.

So, I guess it is a limitation of some condensed-matter models such as the Hubbard model, that they truncate the basis set to a small, and thus incomplete, set of localized orbitals at each site.

I don't think the Heisenberg model, being a model of localized spins with a finite Hilbert space suffers from the same limitation.

The physical picture of single-particle orbitals is only an approximate one in interacting quantum systems.

ReplyDeleteMaybe a more mathematical way of saying this is that the occupation numbers of any given single-particle (basis) orbital will be < 1 for interacting systems, as more than one Slater determinant contributes to the state.

Maybe more useful physical quantities than single-particle orbitals would be single-particle Green functions and spectral function, and 1-body reduced density matrices (1-RDM)...

In quantum chemistry, there is a way to obtain so-called natural orbitals by diagonalizing the (1-RDM), but I think (if I am not wrong) these are again just for physical intuition and not mathematically exact properties of the interacting system.

Thanks for the comment.

DeleteSingle-particle orbitals are certainly an approximation.

I would say that unlike molecular orbitals, natural orbitals are a physical observable, that is not mathematically ambiguous, and is independent of the representation of the many-body wave function.

I hope your trip is going well, Ross.

ReplyDeleteIt's maybe useful to point out that the configuration interaction coefficients (amplitudes of the many-body state over determinants or configuration state functions) and the orbital coefficients are redundant to some degree. To see this, note that for full configuration interaction calculations (where all possible Slater determinants over the orbital basis are included), the state and its energy are totally invariant to orbital transformations. On the other hand for a single-particle picture eg. Hartree-Fock, the state & energy are only invariant to transformations within the occupied or unoccupied orbital sets. For intermediate ansatze, where some but not all determinants are included, the invariance with respect to orbital transformations will generally be more complicated. The point is that orbital relaxation may or may not matter depending on what your many-body basis actually is. One suspects that the importance of the orbital relaxation will diminish as the flexibility of the many-body basis increases. For complete active space calculations, which are a rare case where the redundancies are simple, one finds that orbital transformations within each of the occupied, active, and unoccupied orbital spaces don't matter, but mixing between them does.

A common way to account for orbital relaxation in some many-body ansatz X is to use an X*singles ansatz, where all single excitations from the configurations in X are included. Since single excitations are redundant with pairwise orbital rotations, this allows one to describe orbital relaxation without actually relaxing the orbitals.

Don't know if this helps the discussion, but it is true.

Seth, thanks for the expert comment. It is helpful. It underscores that the notion that orbitals are really only defined relative some particular representation (sum of slater determinants) of the many-body wave function. Changing the orbitals may be equivalent to changing the coefficients of the slater determinant expansion.

DeleteThus, orbital relaxation may be in the eye of the beholder.