I The “philosophical cornerstone” of the Moller-Plesset perturbation theory

[Spathi]

TL;DR

Simple analogies as illustrations to the Moller-Plesset perturbation theory.

In quantum chemistry, the MP rows (MP2, MP3, MP4, etc) can converge both quickly and slowly, and for some cases (e.g. CeI4 molecule) they even diverge instead of converging.
My question is quite philosophic: what is the “mathematical cornerstone”, or “philosophical cornerstone” of the perturbation theory, and whether it can be shown with some simple samples. If yes, maybe this information will help us predict whether in quantum chemistry the MP rows will diverge for some molecule not yet investigated.

I have asked this question on some web forums, and got some answers. Let’s consider the salvation of two equations:

1)
x+sin(x)=3000
If we write the following:

x=3000-sin(x)

We can set x0=0 and get the following iterations:

0
3000
2999,78081002572
2999,5739029766
2999,39713977695
2999,26623684759
2999,18383222963
2999,13904100976

This series converge after 40 iterations.

2)
6000=(x−1)(x−3000)+sin(x)

We transform this equation into the following:

x=(6000-sin(x))/(x-3000)+1

Choosing x0=0 we get the following convergence:

0
-1
-0,999613952344155
-0,999614140048658
-0,999614139957402
-0,999614139957447
-0,999614139957447
-0,999614139957447

So, this series converges within 6 iterations.

Some people said that the second example illustrates the cornerstone of the perturbation theory, while the first one does not. Some other people said that both these examples are not really attributed to the perturbation theory. Can you suggest your opinion?

 

[mfb]

"Converges within x iterations" isn't a thing. Apart from some corner cases you just get increasingly accurate approximations no matter how many iterations you take. You can study how fast that convergence is. Do you e.g. get a fixed number of additional digits per iteration? Does it grow quadratic?