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

In summary, the conversation discusses the convergence and divergence of MP rows in quantum chemistry and raises the question of the "mathematical cornerstone" or "philosophical cornerstone" of perturbation theory. The speaker also shares two examples of equations and their convergence rates, with some people debating whether they are attributed to perturbation theory. Overall, the conversation highlights the complexity and ongoing research in this field.
  • #1
Spathi
Gold Member
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TL;DR Summary
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?
 
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  • #2
"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?
 

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

What is the Moller-Plesset perturbation theory?

The Moller-Plesset perturbation theory is a mathematical method used in quantum mechanics to calculate the energy of a many-electron system. It is based on the assumption that the total energy of a system can be broken down into a sum of smaller, more easily solvable parts.

Why is it considered the “philosophical cornerstone” of quantum chemistry?

The Moller-Plesset perturbation theory is considered the “philosophical cornerstone” of quantum chemistry because it provides a rigorous mathematical framework for calculating the energy of a many-electron system. It is also the foundation for many other perturbation theories and computational methods used in quantum chemistry.

What is the difference between first and second order Moller-Plesset perturbation theory?

The first order Moller-Plesset perturbation theory (MP1) only considers interactions between the electrons and the nucleus, while the second order Moller-Plesset perturbation theory (MP2) takes into account electron-electron interactions as well. This makes MP2 a more accurate method for calculating the energy of a system, but it also requires more computational resources.

What are the limitations of the Moller-Plesset perturbation theory?

The Moller-Plesset perturbation theory is based on the assumption that the total energy of a system can be broken down into smaller parts, which may not always be accurate. It also becomes increasingly difficult to calculate higher order perturbations, making it less accurate for larger systems. Additionally, it does not take into account relativistic effects, which can be important for heavy elements.

How is the Moller-Plesset perturbation theory used in practical applications?

The Moller-Plesset perturbation theory is used in practical applications to calculate the energies and properties of molecules, such as bond lengths and vibrational frequencies. It is also used in conjunction with other computational methods to improve accuracy and efficiency. Additionally, it can be used to analyze chemical reactions and predict the stability of different molecular structures.

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