Hartree Fock method for Molecules

[brydustin]

Could someone please explain the general idea of the Hartree Fock method to optimize the geometry (and energy) of a molecule. I understand that in the atomic case, one uses atomic orbitals and optimizes these, but in a molecule (especially polyatomic, say water or benzene) I am a bit confused; I know that at the very basic level each molecular orbital is treated as a linear combination of the atomic orbitals and then the Hartree-potential is calculated and compared to the previous energy, but then how are the orbitals "transformed" so that the calculation converges?

 

[SpectraCat]

Could someone please explain the general idea of the Hartree Fock method to optimize the geometry (and energy) of a molecule. I understand that in the atomic case, one uses atomic orbitals and optimizes these, but in a molecule (especially polyatomic, say water or benzene) I am a bit confused; I know that at the very basic level each molecular orbital is treated as a linear combination of the atomic orbitals and then the Hartree-potential is calculated and compared to the previous energy, but then how are the orbitals "transformed" so that the calculation converges?

The Hartree-Fock method is a self-consistent field (SCF) method, where the electron electron repulsion term in the Hamiltonian is approximated to obtain an effective one-electron Hamiltonian for each electron. The approximation is that each individual electron interacts with the average field resulting from the rest of the electrons in the atom or molecule. This gives a set of coupled equations that are solved simultaneously using the Raleigh-Ritz variational method. Since it is a variational method, you are guaranteed that the solution will approach the true ground state energy of the system from above, and never go below it.

Now, to answer your question, optimizing the geometry of a molecule using the HF method is a multiple step process. First you guess a starting geometry. Then you generate a guess for the initial orbitals somehow, and run an SCF calculation to solve the electronic problem for the initial geometry you have chosen. Chances are, this is not a minimum energy geometry, which means that there are forces on the atoms in the molecule. You can solve for these forces by computing the gradient of the electronic potential experienced by the atoms in the molecule. Once you have the forces, you then have a prediction about how the atoms want to move, so you allow the geometry to relax along the predicted coordinates a little bit. Then you start from the top with another SCF calculation, calculate the new forces, adjust the geometry, and so on. You stop when the molecular geometry is at a minimum of the potential energy surface ... you know this has happened when the energy is minimized (any further change of the geometry will raise the energy of the molecule) and/or when the forces are zero (the gradient of the potential at a minimum is zero).

Ideally, you should then run an additional calculation of the second derivatives of the potential (the force constants) ... if the geometry you have found is truly a minimum, then the curvature along all internal degrees of freedom (think bond lengths, angles and dihedral angles) should be positive, and therefore all of the force constants should be positive. If that is not the case, then your second derivatives calculation will result in one or more imaginary frequencies (since the harmonic frequencies are proportional to the square roots of the force constants).

Hope this helps.

 

[brydustin]

Thank you for this very clear and detailed response, you would be surprised how complicated many authors make this topic (actually you are probably aware of this yourself), going straight into sticky equations without any rationality. Thanks!

 

[brydustin]

Just curious though...
Is there any consistent methodology for how much the molecule relaxes in each direction (per atom). I.e. I know there are 3N degrees of freedom (actually 3N - 5 or 3N - 6 after including translations, rotations, and other invariant transformations), but how much does the molecule relax (not just in what direction), and is it based on the geometry itself?

For example, I know that if I give a rediculous bond length but make the angle straight for a 3-atomic linear molecule then it will make bigger steps and converge quickly compared to if I am trying to optimize an oligomer (for example). Now, the question got a bit picky, sorry about that... I guess my question is: How does the computer know, how much to descend in the direction of quickest descent (gradient)?

 

[Einstein Mcfly]

Just curious though...
Is there any consistent methodology for how much the molecule relaxes in each direction (per atom). I.e. I know there are 3N degrees of freedom (actually 3N - 5 or 3N - 6 after including translations, rotations, and other invariant transformations), but how much does the molecule relax (not just in what direction), and is it based on the geometry itself?

For example, I know that if I give a rediculous bond length but make the angle straight for a 3-atomic linear molecule then it will make bigger steps and converge quickly compared to if I am trying to optimize an oligomer (for example). Now, the question got a bit picky, sorry about that... I guess my question is: How does the computer know, how much to descend in the direction of quickest descent (gradient)?

There are many methods for doing this, some of which are "faster" than others (in a per iteration sense) and some are "better" than others (in a "more likely to converge sense) and which to use depends on the problem at hand and what you're already tried. If you want to look them up, just plug in a couple of computational software package names along with "gradient" and see what they describe in their manuals. However, if you want to avoid mathematical formalism you may just want to accept "very well thank you" as the answer to "how does the computer know how to adjust the gradient according to the calculated forces".