Orbital optimization

Suppose we would like to describe the effect of an orbital rotation of the underlying spinor basis on the normalized wave function \(\ket{\Psi}\). We then have to calculate the expectation value of the Hamiltonian in the rotated state: \[\begin{equation} \require{physics} E(\boldsymbol{\kappa}) = \ev{ \exp(\hat{\kappa} ) \hat{\mathcal{H}} \exp( -\hat{\kappa} ) }{\Psi} \thinspace . \end{equation}\]

This real-valued function \[\begin{equation} \begin{split} E : \thinspace \mathbb{C}^{M^2} &\rightarrow \mathbb{R} : \thinspace \boldsymbol{\kappa} &\rightarrow E(\boldsymbol{\kappa}) \end{split} \end{equation}\] is now considered as the subject of a minimization procedure. Up to second order in the orbital rotation generator parameters \(\kappa_{PQ}\), the energy function can be rewritten as: \[ \begin{equation} E(\boldsymbol{\kappa}) = \ev{ \hat{\mathcal{H}} }{\Psi} + \ev*{ \comm*{ \hat{\kappa} }{ \hat{\mathcal{H}} } }{\Psi} + \frac{1}{2} \ev*{ \comm*{ \hat{\kappa} }{ \comm*{ \hat{\kappa} }{ \hat{\mathcal{H}}} } }{\Psi} \thinspace , \end{equation} \] by applying the BCH expansion. Unfortunately, due to the anti-Hermiticity of the orbital rotation generator, the energy function \[\begin{equation} E(\boldsymbol{\kappa}) = \ev{ \hat{\mathcal{H}} }{\Psi} + \sum_{PQ}^M \kappa_{PQ} \ev*{ \comm*{ \hat{E}_{PQ} }{ \hat{\mathcal{H}} } }{\Psi} + \frac{1}{2} \sum_{PQRS}^M \kappa_{PQ} \kappa_{RS} \ev*{ \comm*{ \hat{E}_{PQ} }{ \comm*{ \hat{E}_{RS} }{ \hat{\mathcal{H}}} } }{\Psi} \thinspace . \end{equation}\] must be subject to a constrained optimization since not all variables \(\kappa_{PQ}\) may be varied freely (because of the requirement \(\kappa_{PQ}^* = -\kappa_{QP}\)).

Since the orbital rotation parameters \(\kappa_{PQ}\) are, in general, complex, there are two different, but related approaches to optimizing this energy function. (The similarities and differences are explained in utmost detail in this article, starting in chapter 6.) We may:

• do a complex optimization with respect to the conjugate pair \(( \boldsymbol{\kappa}_{\geq}, \boldsymbol{\kappa}^*_{\geq})\);
• do a real optimization with respect to the real parameters \(({}^{\text{R}} \boldsymbol{\kappa}_{>}, {}^{\text{I}}\boldsymbol{\kappa}_{\geq})\).