In Kohn's Nobel Lecture, he claimed that:

"In general the many-electron wave function $$\Psi(\mathbf{r}_1,\ldots,\mathbf{r}_N)$$ for a system of $$N$$ electrons is not a legitimate scientific concept, when $$N\geq N_0$$ where $$N_0\simeq10^3$$."

He explains this in two ways: the first is that $$\Psi$$ cannot be calculated with sufficient accuracy, and the second that is cannot be stored with sufficient accuracy. As far as I understand, what he had in mind here was comparing traditional wave function methods in quantum chemistry (e.g. configuration interaction) with density functional theory, which is based on the density rather than the wave function.

My question concerns Monte Carlo techniques. Quantum Monte Carlo and stochastic extensions to traditional quantum chemistry techniques, for example Full Configuration Interaction Quantum Monte Carlo, could be called wave function methods in that the central quantity is the wave function, and are being almost routinely used for very high accuracy solid state calculations with large $$N$$. Additionally, their nice scaling properties suggest that they will be able to exploit future parallel computer resources effectively.

So the question is: how should we view quantum Monte Carlo techniques in view of Kohn's statement? Could we say that these techniques allow us to bypass the issue Kohn identified with wave functions by only sampling the wave function rather than calculating/storing it?

• There are deeper reasons than calculation issues as to why wave functions for large numbers of particles are useless, c.f. Landau Vol. 5 Statistical Physics, Sec. 5: due to the extremely high number of particles, the separation between energy levels becomes so small that the notion of a stationary state (a state s.t. the energy has a definite value) loses meaning, because a measurement fundamentally perturbs the energy so that the uncertainty in the energy measurement is just too big measure E accurately. The resolution is density matrices etc... Jun 26 '20 at 23:04
• @bolbteppa could you move this to an answer, I think you have something good here. Jun 26 '20 at 23:09
• @bolbteppa I totally agree with Cody. I think this comment should really be an answer. Welcome to our brand new site !!!! By the way there was some discussion about whether or not people should be discouraged away from putting "answers" in comments: mattermodeling.meta.stackexchange.com/q/74/5. Eventually all these comments (yours, Cody's, mine) will probably get deleted if the chain becomes huge. Jun 27 '20 at 2:31
• @bolbteppa I would certainly be interested in seeing your comment converted to an answer.
– Tyberius
Jun 27 '20 at 17:09
• @bolbteppa I encourage you to elaborate and post as an answer. A deeper theoreitical insight like the one you present will be very helpful to a lot of folks! Jun 28 '20 at 2:45

Kohn is easily one of my favorite humans of all time, and he was a role model to whom I looked up in great admiration for most of my academic life; in fact before this site was created, I proposed that we name it after him.

However I completely disagree with the sentence that you have quoted. Keep in mind that even though the Nobel Lecture was in 1999, Kohn was born in 1923, so I was not alive for much of his life, and I don't know what possible connotations might have surrounded the word "legitimate" back in those days; but certainly the way we use the word "legitimate" nowadays, and every dictionary definition of legitimate I've seen, would indicate that he might have been speaking in hyperbole.

Let me address now, some of the specific matters in your question:

## We have accurate wavefunctions for systems with far more than $${\small N=10^3}$$

In this answer I recently mentioned that CCSD(T) with local orbital methods have calculated wavefunctions for systems with up to 1023 atoms; in this case it was a lipid transfer protein (PDB: 1N89) for which I'd estimate the number of electrons is about 10,000. Kohn may have written $$N\simeq 10^3$$ instead of $$N = 10^3$$, but the order of magnitude turned out to be wrong 20 years after that quote. Surely the order of magnitude will also increase again.

## Accuracy

Energy differences with CCSD(T) or even LNO-CCSD(T) in the 4-zeta basis set that they used in the above example, are likely to be accurate to within 1.5 kcal/mol for a molecule like this, whereas DFT is unlikely to give you anything with an error less than 4 kcal/mol unless you use hybrids (which by definition use wavefunctions). A good energy difference doesn't necessarily mean an accurate wavefunction, but coupled cluster wavefunctions are not bad at all (otherwise you wouldn't be able to calculate accurate properties like polarizabilities).

## Storage

The wavefunction in the above case is stored via cluster amplitudes, for which we can store billions (in fact trillions in the largest cases) of them. Since cluster amplitudes appear in the argument of an exponential, we actually get CI coefficients for 100% of the non-zero determinants. So storage of a big wavefunction is no problem when you use a compact representation. Sure there will be a point at which classical computers can no longer accurately store quantum wavefunctions, but there will also be a point at which electron densities can't be stored either, so in that sense why not call the density "illegitimate" too? Furthermore, not being able to "store" the wavefunction is only a problem if using a classical storage device, whereas if you use qubits instead of bits the statement no longer has legs.

## Quantum Monte Carlo (QMC)

FCIQMC, VMC, DMC, AFQMC, and similar methods, are all wavefunction methods. They are wavefunctions methods, whereas you wrote that they "could be called wave function methods, in that...". They do in fact manage to represent wavefunctions in huge Hilbert spaces, often by taking advantage of sparsity of the wavefunction, but using a compact representation as in the case of coupled cluster, means that you don't even need a "sparse" representation or a "stochastic sampling", you can represent the whole wavefunction by storing only the argument of the exponential. I am not trying to take shine away from QMC; I say all this as a contributor to a major FCIQMC code, and having used FCIQMC in papers with the inventor of FCIQMC and also separately on my own and even on this paper I put on arXiv only two days ago. FCIQMC has its place as one of the best methods for ultra-high accuracy in large multi-reference systems, but it is not needed in the proof that Kohn's statement is wrong: coupled cluster can be very accurate on even bigger systems if they don't have too much multi-reference character.

## Number of electrons is actually a red herring

The problem with wavefunctions when using classical computers has less to do with the number of electrons and more to do with its structure:

• Bosonic wavefunctions don't suffer from the fermionic sign-problem, so you can represent them with a Hartree product, and might be able to store a larger wavefunction than you can store a density! Kohn's statement is about electronic wavefunctions, but this bullet serves as a reminder that Kohn's argument is not so much about fundamental physics and ontology, than it is about "computability" (you probably already agreed with this, so this bullet is more for other people).
• Fermionic wavefunctions involving only one determinant (which can still be accurate for a very single-reference system) are very simple: instead of the $$\binom{M}{N}$$ type scaling for the number of determinants in a full CI expansion, you only have one term.
• Fermionic wavefunctions involving many determinants but only static correlation, can be represented by matrix product states and calculated using the polynomially scaling DMRG.
• Fermionic wavefunctions involving many determinants but only dynamic correlation, can be represented by coupled cluster ansatze which are also polynomially scaling.
• Fermionic wavefunctions involving infinitely many determinants can also be represented by a compact representation.

The problem is more about how many digits you need for each of the CI coefficients. Then you quickly see that the computational complexity of $$2^N$$ vs $$N^3$$ is irrelevant and what really matters is something more subtle, which is: how complicated is the wavefunction, not how many electrons are there.

50 electrons in a CAS(50,50) is currently an absolutely brutal calculation but 10,000 electrons in a CCSD(10000,44000) was done easily in the paper listed above. So there are cases where $$N=50$$ electrons is harder than $$N=10^4$$, and in those highly multi-reference cases, good luck getting an accurate energy with a single-reference method like DFT!

• Interesting that my sections always grow longer and longer, when in fact I intend for them to get shorter and shorter. Jun 26 '20 at 17:31
• +10 Thank you so much for this comprehensive answer! Jun 26 '20 at 17:37
• I do not think Kohn was talking about accuracy in energy. I think it is much more about how the error propagates in the calculations, and how the orbital coefficients become extremely correlated. I often see very different looking wavefunctions as the results of calculations on the same systems, which has very different orbital coefficients when compared to each other but produce similar energies and properties. In that sense, one can say these wavefunctions are not a very insightful representation of the system.
– Greg
Jun 27 '20 at 16:03
• @Tyberius, Greg: I agree. The last section of my answer says that what matters is not the number of electrons but the specific situation. You give a specific example of M non-interacting He atoms, or some systems with identical geometries where tiny changes of SCF give huge changes in wfn... in those situations getting an accurate wfn represented in classical storage (bits) is extremely hard. 50 electrons would be extremely hard. However what about a single-reference system where a single determinant gives a very accurate description of the system? 10000 electrons is now easy. Jun 27 '20 at 20:42
• @Nike Dattani, that's an important distinction, my apologies! Nov 7 '20 at 1:19