Restricting the Dirac-Fock energy to the Electron Subspace
The Dirac-Fock energy functional is the relativistic counterpart to the Hartree-Fock energy functional, and acts on four-component square integrable wave functions which arise from Slater determinants. This ensures the wave functions satisfy the particle statistics we expect from fermions. In normal atomic units, the Dirac-Fock energy \(\mathcal{E}\) acts on an \(N\) electron wave function \(\psi\in H^{1/2}(\mathbb{R}^3;\mathbb{C}^4)\wedge ...\wedge H^{1/2}(\mathbb{R}^3;\mathbb{C}^4)\) via (Esteban and Séré 1999),
\[\mathcal{E}(\psi):=\sum_{j=1}^N\left((\psi_j,D\psi_j)-\alpha Z\left(\psi_j,\frac{1}{|x|}\psi_j\right)\right)\]
\[+\frac{\alpha}{2}\int\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{\rho(x)\rho(y)-\text{tr}(R(x,y)R(y,x))}{|x-y|}dydx\]
Here, \((\cdot,\cdot)\) is the \(L^2\) inner product, \(\rho(x)\) is the usual particle density and \(R(x,y)=\sum_{j=1}^N\psi_j(x)\otimes\psi_j(y)^*\) (the star denoting conjugate transpose). \(\alpha\) is the fine structure constant, \(Z\) is the nuclear charge, and \(D\) is the Dirac operator in normal atomic units. Critical points of \(\mathcal{E}\) solve the Dirac-Fock equations, which is an eigenvalue problem when an \(L^2\) constraint (typically norm one) is imposed.
Spectral Projections
The Dirac operator has purely absolutely continuous spectrum \(\left(-\infty, -1\right]\cup \left[1,\infty\right)\). This is the primary source of numerical difficulties in dealing with functionals such as \(\mathcal{E}\), which is not bounded from below. A possible means to correct this is as follows: \(D\) induces spectral projectors which take the form (Thaller 2011):
\[P^{\pm}=\frac{1}{2}\left(1\pm\frac{D}{|D|}\right)\]
The Hilbert space \(H^+:=P^+L^2(\mathbb{R}^3;\mathbb{C}^4)\) denotes the positive spectral subspace, which corresponds to the “electron subspace.” In a sense, the orthogonal complement to this space denotes the space of positrons, and it is this mathematics that first led physicists to conjecture the existence of positrons. On \(H^+\), we have \((\psi,D\psi)>0\) and in fact when \(\psi\in H^{1/2}(\mathbb{R}^3;\mathbb{C}^4)\), we have \(||\psi||_{H^{1/2}}^2=(\psi,D\psi)\). Let \(H:=P^+H^{1/2}(\mathbb{R}^3;\mathbb{C}^4)\). By Kato’s inequality, one easily sees that \(\mathcal{E}\) is bounded below when restricted to \(H\). Thus, since the electron subspace is presumably the physically interesting space to us, it is natural to minimize \(\mathcal{E}\) on \(H\).
Sketch of the Minimization Procedure
The procedure here follows the same ideas as in (Lieb and Simon 1977). We’ve already mentioned that \(\mathcal{E}\) is bounded below on \(H\), which follows from Kato’s inequality for \(\psi\in H\):
\[(\psi,V\psi)\leq K(\psi,D\psi)\]
(\(K=\frac{\pi}{4}+\frac{1}{\pi}\)) and the fact that we have the estimate (Esteban and Séré 1999),
\[\int\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{\rho(x)\rho(y)}{|x-y|}dydx\leq NK(\psi, D\psi)\]
In particular, one has the estimate,
\[\mathcal{E}(\psi)\geq \left(1-\alpha Z\right)\sum_{j=1}^N||\psi_j||_{H^{1/2}}^2\]
which is positive with suitable restrictions on \(Z\). Hence, given a minimizing sequence \(\psi_n\in H\) with \(||\psi_n||_2\leq 1\) for \(\mathcal{E}\), we have a weakly convergent subsequence \(\psi_n\rightharpoonup \psi\) in \(H\). The goal is to show that \(\psi\) is, in fact, the solution to the minimization problem and that \(\psi\) is of unit norm. Relaxing a minimization problem in this sense (that is, assuming \(||\psi_n||_2\leq 1\)) is common in the literature, as one can often prove readily after getting a solution that the minimizing function does have the desired norm.
Lower-semicontinuity of \(\mathcal{E}\) actually follows fairly readily, giving us a minimizer. Given \(\varphi_n\rightharpoonup 0\), we actually have (Coti Zelati and Nolasco 2019) \((\varphi_n,V\varphi_n)\rightarrow 0\), so combining this, Kato’s inequality, and weak lower-semicontinuity of the norm yields weak lower-semicontinuity of \(\mathcal{E}\).
Similar arguments to (Lieb and Simon 1977) yield the appropriate norm constraint on the solution, and a simple application of the Rayleigh-Ritz principle (see (Reed and Simon 1978) theorem 13.6) yield infinitely many eigenvalues below 1 of the Dirac-Fock mean field operator (when restricted to \(H\)).
Evidently, extending (Lieb and Simon 1977) to the Dirac-Fock case when restricted to \(H\) is not terribly difficult. More challenging is the existence of critical points of \(\mathcal{E}\) with no restriction of \(\mathcal{E}\) to \(H\), in which case one needs to deal with the unboundedness of \(\mathcal{E}\). This is solved given suitable restrictions on \(N\) and \(Z\) in (Esteban and Séré 1999). In any case, Kato’s inequality allows one to very nicely extend the classical results of Lieb and Simon to the relativistic theory.