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Because of its size, for example a protein can not be modelled from the quantum chemical basis in a reasonable amount ot time without approximations. A hydrogen atom though can, manually even. Where is the boundary between those, where approximations become necessary for the results to be sensible while taking a managable amount of time for the calculations to converge?

I suppose this question is dependent on parameters, such as what computer power is available and what is considered tolerable amount of time for the calculations. Let's say one year computation on Piz Daint, but feel free to choose the parameters to your liking.

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    $\begingroup$ I was not the one that gave the close vote, and in fact I was the only one so far taht gave an upvote, but some approximations are always made no matter what, because we don't have a general quantum theory of gravity or "grand unified theory" yet. If you're solving the ordinary Schroedinger equation you're neglecting relativistic effects, if you're solving the Dirac equation you're neglecting QED, if you're using QED you're neglecting gravity. If you're including gravity (with general relativity, for example) you're englecting quantum. $\endgroup$ – Nike Dattani May 11 '20 at 15:48
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    $\begingroup$ I would suggest you to rephrase to question asking about different levels of approximations we use for materials modelling $\endgroup$ – Thomas♦ May 11 '20 at 16:12
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    $\begingroup$ Even with the hydrogen atom, we make an often unstated approximation, when we assume the proton to be point-like. It's a very good approximation, but the proton has internal structure, so it's not point-like. If you want a truly approximation-free solution, the best thing we have is Muonium. Like the electron, muons are point-like charges, but have around one-third of proton mass. $\endgroup$ – ksousa May 11 '20 at 16:34
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    $\begingroup$ @BernhardWebstudio, perhaps you can include in your question, a specific approximation that you had in mind, rather than trying to cover "all possible approximations"? The close votes are piling up (neither of them from me), but the question can certainly be a good one if slightly improved! $\endgroup$ – Nike Dattani May 11 '20 at 16:44
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    $\begingroup$ Thanks for all the comments. I see that it does indeed not make sense to ask for no approximations whatsoever. I feel like editing this question would render all these comments obsolete – without knowing the actual consequences, wouldn't it make more sense to actually close the question and I reopen a new one once I have a selection of approximations I am willing to make? $\endgroup$ – BernhardWebstudio May 11 '20 at 17:56
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Very small.

The problem is, that for there to be no approximations, you must be solving the full many-body Hamiltonian for all particles. That means to solve just O$_2$ you have six electrons and one nucleus. Even using the Born-Oppenheimer approximation, that leaves you with an eigenvalue problem for 16 3D vectors.

To take a simpler case (with a discrete Hilbert space), say you want to solve the 1D $S=1/2$ Heisenberg model, that's a simple spin chain with $L$ sites. Each site has two possible states, so the Hilbert space for the whole chain is $2^L$. Right now the maximum length of the chain that can be treated without approximations is about 40 sites.

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