MP2 vs. HF Method Explanation for Beginners

[physicisttobe]

TL;DR

Computational physics

Hi everyone!

I am currently researching the difference between HF and MP2, but I have not found a detailed explanation that has helped me. I am not a computational researcher, which is why I have struggled to understand the difference between them. I am quite new in this field, so I need a simple explanation. What are the differences between MP2 and HF? Which optimization method is better and why? The explanations on the internet were too complicated for me. I really couldn´t understand the difference between these two methods. So, can you explain me that? I would be very pleased to receive your answers.

 

[berkeman]

High Frequency and Military Police 2? Well, where do I start...?

 

[physicisttobe]

Oh, sorry. HF means Hartree Fock and MP2 means Møller–Plesset perturbation theory.

 

[TeethWhitener]

(I’ll try to keep this I level; feel free to ask questions.)

They’re both approximations to the molecular Hamiltonian (or nuclear Hamiltonian, but I’m a chemistry guy, so we’re sticking with molecules).

The molecular Hamiltonian has three parts: a nuclear part, which can usually be parametrized under the Born-Oppenheimer approximation, a nuclear-electron interaction, and an electron-electron interaction.

The e-e interaction is not solvable analytically except for really special (usually cooked up) cases. In the HF approximation, which is probably the simplest approximation, we assume that each electron sees an average potential from all the other electrons (this is also called the mean field approximation). However, this is not very accurate, because electron motions can become correlated with each other (for example, it’s very unlikely that you’ll find two electrons within a femtometer of each other for very long). But this is a really hard problem. In fact, the vast majority of theoretical and computational chemistry has been focused on coming up with better and better ways to solve this problem.

Enter Moller-Plesset perturbation theory. It treats the true Hamiltonian as the HF “mean field” Hamiltonian plus a small perturbing potential. This potential can be expanded in a power series. It can be shown that the first order correction to the energy disappears, so the lowest order nonzero term is the second order term (the 2 in MP2). You can also calculate the MP2 correction to the HF energy explicitly (it ends up being a messy expression with a bunch of two-electron integrals). For more accurate treatments, you can go to higher order: MP3, MP4, etc. I don’t think I’ve seen higher than that because the expressions are even messier and the computational complexity shoots through the roof (I think MP3 scales at least as $O(n^6)$ with $n$ the number of spin orbitals and MP4 is even worse and it’s diminishing returns at that point).

 

[physicisttobe]

First of all, thank you very much for your explanation! I'm glad you're a chemist, because the questions I'm actually asking have something to do with computational chemistry. (By the way, I am studying chemistry). Anyway, I want to expand my knowledge in this field. So I am really glad that I can get chemical explanations here in this forum.

Back to the actual topic: I still haven't understood this phrase "we assume that each electron sees an average potential from all the other electrons (this is also called the mean field approximation). However, this is not very accurate, because electron motions can become correlated with each other."
What exactly do you mean by avergage potential? How can I imagine this?
And aren´t the electrons fixed in the mean field A.? I thought that we consider only one electron and that the rest of the electrons are not mobile. Therefore, they cannot correlate with each other?

 

[TeethWhitener]

What exactly do you mean by avergage potential? How can I imagine this?
And aren´t the electrons fixed in the mean field A.? I thought that we consider only one electron and that the rest of the electrons are not mobile. Therefore, they cannot correlate with each other?

In terms of the e-e interaction, each electron’s dynamics will be determined by the potential from all the other electrons in the system. This is a horribly complicated problem, so the mean field (HF) approximation says: let’s consider one electron at a time. We calculate the potential for that electron, but instead of looking at all the other electrons’ exact positions and momenta, we assume they have an average charge density and the single electron is only affected by the potential of that charge density. So the other electrons aren’t really fixed in this approximation. It’s more like their positions are averaged or smeared out.

You might wonder: we have to know the average density of the electrons to determine their dynamics, but we have to know their dynamics to obtain their average density. And you’d be right. We have to initialize the computational procedure with a (semi-) educated guess about what the electron density will be. For example, we could start by assuming the electrons are in hydrogen-like orbitals if we know absolutely nothing about the system (NB—in reality, we often use somewhat different starting guesses that are much more convenient computationally). Then we perform the HF procedure to get the dynamics, which we calculate an updated electron density from. Then we use this as the starting guess for a new round of HF calculations. Rinse and repeat until some convergence criterion is reached. This is known as a self-consistent procedure, and sometimes you’ll see the HF method (and others) referred to as SCF (self-consistent field) methods.

But as you can see, in this formalism, you’re not actually considering the interactions of individual electrons with respect to one another (so-called electron correlations), only the interaction of an electron with the average field of the rest of the electrons. So better approximations (sometimes called post-Hartree Fock) are needed to treat those correlations and get more accurate dynamics. MP2 is one type of post-Hartree Fock method. There are many many others.

 

[physicisttobe]

Thank you so much for your explanation! But I have further questions: What is the use of this HF approximation? In the practical course we drew molecules with GaussView and then optimized them with HF and MP2 in order to get the energy minimum. But what is the relation between the electron density and the minimum energy. I`m now trying to find a connection between your explanation and the optimization methods we applied in the course.

 

[TeethWhitener]

What is the use of this HF approximation?

It’s an approximate solution to the Schrödinger equation for atoms and molecules. It gives us energies an wavefunctions which we can then use to predict other properties.

In the practical course we drew molecules with GaussView and then optimized them with HF and MP2 in order to get the energy minimum.

If you were doing a geometry optimization, you calculated the energy of a certain geometry using HF as well as the energy gradient, then moved down the gradient to find a lower energy geometry. Every energy calculation step required the HF procedure. Even when you do MP2, the HF wavefunction is the zeroth order approximation onto which the MP2 perturbation is added.

But what is the relation between the electron density and the minimum energy.

This is a bit deeper question than you probably mean to ask. For the purposes of HF, the spatial distribution of electron density is associated with an electric field, which affects the motions of other electrons in the system. This enters into the equations as an averaged potential in the Hamiltonian.

But it should be clear that the energy of the system depends on electron density. Just imagine a system of a million electrons (and no protons) squeezed very close to each other (within a cubic angstrom or so). This system would have extremely high energy because of the very high negative charge density. That’s a simplified example, but more generally, there is a one-to-one mapping between the ground state energy of a system and its electron density. In fact, this fact is the basis for another computational approach known as density functional theory (DFT).

 

[physicisttobe]

Thank you so much for your explanation. That was a great help! If other questions arise, I`ll be in touch.