Computational Chemistry 4.27 - Self-Consistent Field

Short lecture on the self-consistent field procedure in Hartree-Fock theory.
The self-consistent field procedure produces solutions to the Hartree-Fock-Roothaan equations for the best possible approximate molecular orbitals for a molecule within the Born-Oppenhiemer approximation and Hartree-Fock approximation for a particular choice of atomic orbital basis set and nuclear coordinate configuration. The high-level algorithmic steps for this procedure are as follows:
1) Specify nuclear charges and positions, number of electrons, and choice of basis set.
2) Calculate all overlap, kinetic energy, nuclear attraction, and two-electron integrals.
3) Diagonalize S and obtain the orthogonalization matrix X.
4) Obtain an initial guess density matrix (P).
5) Calculate the two-electron potential matrix G.
6) Calculate the Fock matrix as the sum of the core Hamiltonian and G matrices.
7) Transform the Fock matrix with the X matrix.
8) Diagonalize the Fock matrix to obtain the coefficient matrix (C) and orbital energy matrix (E).
9) Un-transform the C matrix with the X matrix.
10) Obtain the next density matrix from the last C matrix.
11) Determine if the density matrix is converged. If not, go back to step 5.
12) Use the final C matrix to calculate all desired molecular properties.
Notes Slide: i.imgur.com/zv22Qm6.png
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[@kbarracuda]

Bless you for uploading a whole new set of computational videos

[@TMPChem]

Glad to be back and breaking new ground.

[@meghashreemodak1905]

Thank you for your effort..I have seen many videos on Hartree Fock theory on youtube and also read some chapters of Szabo Ostlund... But your videos make me more comfortable in understanding the basics of hartree fock and inspire me to study more in this computational chemistry field.

[@lordofutub]

I would love to see you implement quantum theory in python as an example. Hope you haven't given up on these vids!

[@MrMultiMediat0r]

Those fairly fancy methods to bypass N^4 scaling for 2e- integrals. Do you think there'll be a video in the future? Like on density fitting for example. That would be cool

[@TMPChem]

Density fitting is certainly the most commonly used method to reduce N^4 scaling of Coulomb integrals down to N^3, though there are others like it and others still that make even more aggressive trade offs between accuracy and scaling which helps most in post-HF methods. However, all of that involves a nuanced discussion of the fundamentals, which we still have a long way to go to cover. Not that I don't think it might get there eventually, but it is a very long road before that would become a high priority, and I don't expect it to happen in the current year to say the least.

[@seitdrs2467]

Hi, you said quadratic and 'quordret' what does that mean?

[@TMPChem]

quartic --> 4th order --> depend on the 4th power of something. There are 4 indices in the two-electron integrals (ab|cd), each of which runs from 1 to K, therefore there are O(K^4) two-electrons integrals to compute for a basis set of size K.

[@chrisaxis5043]

Are there any example program for atoms to learn it? If there's any? Could you please tell?

[@TMPChem]

Hi Chris. I don't have a sample program for this (yet). I have plans to do so like I did for MM / MD, but this one is much more complicated, will take a lot more time, and I have the constraint of having a full-time day job. In the mean time, My recommendation would be to download a free quantum chemistry program (such as Psi4, or others), and run Hartree-Fock with a small basis set on a few small molecules. That output can give you a general feel for what goes on in these calculations and their results at a high level.

[@chrisaxis5043]

@@TMPChem thank you for the reply, the book you suggested modern quantum chemistry by ostlund and szabo contains a small program for molecule HeH+ in appendix B with output, I tried to study it but it contains some terms which was not included in the theory sessions so I was bit confused... I think I'll have to learn and focus more about computational stuff from now on... And thank you so much for this great content, I couldn't have found it anywhere else...

[@TMPChem]

Sure. The value of the integrals is a very complicated affair, not only because the integrals themselves get very complicated, but because many basis sets are composed of contractions of Gaussians which really blow up the number of required integrals. If you really want to know how to solve the integrals, those can be found in Molecular Electronic Structure Theory by Helgaker, but that's probably unnecessary. I would say treat the integrals as givens, plug them into the matrices (S, T, V, etc.), and perform the matrix operations from there.

[@vikikumarprasad9336]

@@TMPChem Can we anticipate to see ever videos based on Molecular Electronic Structure Theory by Helgaker that would build upon this chapter? It would be a really great resource for everyone as it would cover more detailed aspect of compchem (and useful for PhD comprehensive exams). I believe no one has done that before and would help lots of graduate students.

[@apburner1]

First.