Why does iteratively solving the Hartree-Fock equations result in convergence?

Question

In the Hartree-Fock self-consistent field method of solving the time-independent electronic Schroedinger equation, we seek to minimize the ground state energy, $E_{0}$, of a system of electrons in an external field with respect to the choice of spin orbitals, $\{\chi_{i}\}$.

We do this by iteratively solving the 1-electron Hartree-Fock equations, $$\hat{f}_{i}\chi(\mathbf{x}_{i})=\varepsilon\chi(\mathbf{x}_{i})$$ where $\mathbf{x}_{i}$ is the spin/spatial coordinate of electron $i$, $\varepsilon$ is the orbital eigenvalue and $\hat{f}_{i}$ is the Fock operator (a 1-electron operator), with the form $$\hat{f}_{i} = -\frac{1}{2}\nabla^{2}_{i}-\sum_{A=1}^{M}\frac{Z_{A}}{r_{iA}}+V^{\mathrm{HF}}_{i}$$ (the summation runs over nuclei, here, with $Z_{A}$ being the nuclear charge on nucleus A and $r_{iA}$ being the distance between electron $i$ and nucleus $A$). $V^{\mathrm{HF}}_{i}$ is the average potential felt by electron $i$ due to all the other electrons in the system. Since $V_{i}^{\mathrm{HF}}$ is dependent on the spin orbitals, $\chi_{j}$, of the other electrons, we can say that the Fock operator is dependent on it's eigenfunctions. In "Modern Quantum Chemistry" by A. Szabo and N. Ostlund, pp. 54 (the first edition) they write that "the Hartree-Fock equation (2.52) is nonlinear and must be solved iteratively". I have studied the details of this iterative solution as part of my research, but for this question I think they are unimportant, except to state the basic structure of the method, which is:

1. Make an initial guess of the spin-orbitals, $\{\chi_{i}\}$ and calculate $V_{i}^{\mathrm{HF}}$.
2. Solve the eigenvalue equation above for these spin orbitals and obtain new spin-orbitals.
3. Repeat the process with your new spin orbitals until self-consistency is reached.

In this case, self-consistency is achieved when the spin-orbitals which are used to make $V_{i}^{\mathrm{HF}}$ are the same as those obtained on solving the eigenvalue equation.

My question is this: how can we know that this convergence will occur? Why do the eigenfunctions of the successive iterative solutions in some sense "improve" towards converged case? Is it not possible that the solution could diverge? I don't see how this is prevented.

As a further question, I would be interested to know why the the converged eigenfunctions (spin orbitals) give the best (i.e. lowest) ground state energy. It seems to me that the iterative solution of the equation somehow has convergence and energy minimization "built-in". Perhaps there is some constraint built into the equations which ensures this convergence?

Cross-posted from the Physics Stack Exchange: https://physics.stackexchange.com/q/20703/why-does-iteratively-solving-the-hartree-fock-equations-result-in-convergence

Answer 1

The Hartree-Fock equations are the result of performing constrained Newton-Raphson minimization of the energy with respect to the parameter space of Slater determinants (I don't have my copy of Szabo-Ostlund at hand, but I believe this is pointed out in the derivation). Hence, HF-SCF will converge if your starting guess is in a convex region around a minimum. Elsewhere, it may or may not converge. SCF convergence fails all the time.

Answer 2

Density functional theory (DFT) also uses a one-particle approach similar to Hartree-Fock, although the effective potential is a little more involved. To achieve a global minimum, the problem is approached as a non-linear fixed point problem which, as Deathbreath said, can be solved via a constrained Newton-Raphson minimization. A common approach in the DFT community is to use Broyden's Method which if organized correctly (J Phys A 17 (1984) L317) requires only two vectors: the current input and output. (See Singh and Nordstrom, p. 91-92, for a quick overview of this method, or Martin, Appendix L, for a more complete overview of related techniques.) A more recent technique used in Wien2k attempts to overcome convergence difficulties with the Broyden method by employing a multi-secant method.(PRB 78 (2008) 075114, arXiv:0801.3098)

Answer 3

One may use the optimal damping algorithm ODA in the SCF cycle to obtain a real minimization algorithm. Then it always converges. (Related papers of Eric Cancès are also worth reading.)