What do you mean by exact perturbation? This is my first doubt about your question. I can add that Perturbation Theory tell you how to add more corrections to an approximate result, but says nothing about how you will get corrections adding more terms. The behavior of the perturbation series is far from being well defined. You can get steady correction terms getting smaller every time, but you can get fluctuations or even divergence. It depends on the system, the perturbation nature, the exact eigenvectors and eigenvalues,...

Your question is not clear, will you please clarify your question, i.e. accurate result means in what sense?

I do not understand the question but I would say that, to be satisfied with any calculation, it is first necessary to give the desired accuracy limit (as well as the cost in time and computational cost) then we can compare the results obtained after two successive approximations to see if it is necessary to go further according to the desired accuracy. But we must first check that the different orders of approximations are convergent otherwise we can not talk about perturbative calculation.

@ Jared : Thanks

You generally require an infinite number of perturbative terms to perfectly describe a problem. However, this is not feasible, and you must decide on a truncation point.

The question about how many orders of perturbation to retain in the calculations to approach a more and more accurate solution depends on two points:

- the degree of non-linearity in the governing equations

- the size of the perturbation(s) compared to the equilibrium structure

Therefore, you must set up a test, if analytically and/or numerically feasible, and trace the relative change in the (eigen-)solutions of your problem as a function of extra perturbation orders added.

What do you mean by exact perturbation? This is my first doubt about your question. I can add that Perturbation Theory tell you how to add more corrections to an approximate result, but says nothing about how you will get corrections adding more terms. The behavior of the perturbation series is far from being well defined. You can get steady correction terms getting smaller every time, but you can get fluctuations or even divergence. It depends on the system, the perturbation nature, the exact eigenvectors and eigenvalues,...

@ Ramon: Thanks for nice explanation. Exact perturbation means perturbation treatment with standard perturbation theory. As it is clear, in fist order approximation , the perturbation is assumed to be small.

During perturbation analysis of a system, up to some extent first and standard perturbation theory results was more or less close, but after certain point (in terms of perturbation magnitude) when magnitude of perturbation is high, it started diverging with high rate of divergence. So, i wish to understand why it is so, adding more correction will given result with high accuracy, behavior of the perturbation series, reasons for high fluctuation etc..?

I see your problem, which is common in several problems. We did try to obtain at least kind of empirical answer to convergence in a now an old paper series:

1) E.Besalú,R.Carbó. "Generalized Rayleigh-Schrödinger Perturbation Theory in Matrix Form" Journal of Mathematical Chemistry 15 (1994) 397-406

2) E. Besalú, R. Carbó-Dorca "Rayleigh-Schrödinger Perturbation Theory: Practical Implementation of Matrix and Vector Formalisms of an Heuristic Sufficiency Convergence Criterion" J. Math. Chem. 22 (1997) 85-95

3) E. Besalú, R. Carbó-Dorca "Rayleigh-Schrödinger Perturbation Theory in Matrix Form" J. Chem. Edu. 75 (4) (1998) 502-506

At the light of these papers you can easily arrive to the conclusion that, in fact, PT is just kind of alternative diagonalization procedure.

Hope that at least those papers can help you in some way.

@ Ramon : Thank you very much for suggesting papers. Will go through and will try to resolve.

In non-relativistic QM, we use perturbation theory mostly when we cannot solve exactly the eigenvalue problem. Perturbation theory is an approximate method, and it is supposed to be more accurate when we go beyond first or second order, but indeed it diverges as you apply it infinite times. With some academical exceptions, it becomes a monotonically decreasing series, but that in general does not guarantee its convergence.

In the finite case you probably will obtain a more accurate answer, but you have to consider the assumptions made from the beggining of the problem, such as: the order of magnitude of the difference in energies of the original (to be perturbed) system compared with the maximum value of the perturbation, and so on.

Antonio is talking about the widespread beliefs on PT. There is no general argument proving that PT higher order terms will converge, nor that second order will give you more information than first, and so on. Might be he has not read my previous comments...

In this sense of complete summation of the series, and in the context of many body perturbation theory that's out of the question if there is even a theorem proving it. I was referring to plain Rayleigh-Ritz perturbation theory, and my papers are on that line of description of Schroedinger equation, in fact of any secular equation. However, I'm not sure (because it is not my field of work) if Coupled Cluster theory, even if carried out to infinite order, furnishes some energy result which is variational. I have a doubt about this issue. Moreover: another doubt arises how to carry a summation up to infinite order, if the series contains so complicated terms as in Coupled Cluster theory. I guess that, say, when you compute the energy of a peptide of 300 atoms (in case you can study with CC such large molecular structures) you cannot be sure that the, say again, fifth term is lesser than the sixth or bigger by chance... I do not see any interesting behaviour which might be predicted in advance... So, there appears kind of paradox: in CC theory you can get a convergent result in case you can carry the summation to infinite, but no one knows how to arrive up to this level, moerover no one can know (I'm not sure about this) wheter after some terms, the summation do not fluctuates in such a way that behaves as divergent locally... Difficult questions indeed...

Sure all this. However, which kind of large molecules have been computed up to now?

In PT of intermolecular interactions we use the first, second, and sometimes third order, and it works very well in practice, although we know from a formal analysis that the series diverges eventually. One should just know where to stop:-). In intermolecular interactions the main problem is non-fulfilling of the "smallness condition" for the perturbational operator. Natural and the only physically sensible operator is V=H-(HA+HB), where for the interaction of A and B: H is the total Hamiltonian, and HA, HB are the Hamiltonians of subsystems. One can easily see that V has a lower permutational symmetry wrt to electron permutations than the whole H, so V cannot be considered as "small". In spite of this, PT up to second order works fine, if one forces a proper permutational symmetry on approximate wave functions (it is the main idea of symmetry-adapted perturbation theory -SAPT). In our case the second order SAPT is a standard, and adding third order does not change much.

I also want to add some observations to the Jared contributions: from the Moeller-Plesset theories only MP2 is used in practice nowadays (with rare exceptions), although MP3 and MP4 are available in most quantum chemistry codes. Instead of MP3 and MP4 people prefer to use CCSD and CCSD(T), which require about the same amount of computational resources, but are free from PT convergence problems. For some cases the oscillatory behaviour of higher MPn corrections to the correlation energy is quite pronounced, see works of Jeppe Olsen, e.g.

https://www.researchgate.net/publication/234990965_On_the_divergent_behavior_of_MllerPlesset_perturbation_theory_for_the_molecular_electric_dipole_moment?ev=prf_pub

Article On the divergent behavior of Møller–Plesset perturbation the...

Dear Tatiana,

Unfortunately, I don't understand the relation between permutational symmetry and V being large or small? How can one explain it? Unfortunately, I haven't been able to find any papers or books explaining this.

Thanks in advance.