Constraining wave function models

Suppose we have a wave function model \(\require{physics} \ket{\Psi(\vb{p})}\), parametrized by the variables \(\vb{p}\) and we want it to have some kind of feature \(m(\vb{p})\): \[\begin{equation} m(\vb{p}^\star) = M \thinspace , \end{equation}\] where \(M\) is the targeted value for that feature. In general, we then have to solve the augmented set of equations \[\begin{equation} \begin{cases} & \vb{F}(\vb{p}^\star) = \vb{0} \\ & m(\vb{p}^\star) = M \thinspace , \end{cases} \end{equation}\] which may have no, one, or multiple solutions.

If we use Rayleigh energy function, we can reformulate the problem as the minimization of the energy function \[\begin{equation} E(\vb{p}) = \frac{ \ev{\hat{\mathcal{H}}}{\Psi(\vb{p})} }{ \braket{\Psi(\vb{p})} } \thinspace , \end{equation}\] subject to the constraint that the feature is attained. Such an optimization problem may be described using the method of Lagrange’s multipliers, which introduces the Lagrangian \[\begin{equation} \mathscr{L}(\vb{p}, \mu) = E(\vb{p}) - \mu ( m(\vb{p}) - M ) \end{equation}\] that is optimized instead of the just the energy function. The stationary equation for the Lagrange multiplier \(\mu\) recovers the original constraint: \[\begin{equation} \eval{ \pdv{ \mathscr{L}(\vb{p}, \mu) }{\mu} }_{\mu^\star, \thinspace \vb{p}^\star} = - ( m(\vb{p}^\star) - M ) = 0 \thinspace , \end{equation}\] while the stationary equations for the parameters \(\vb{p}\) now become: \[\begin{equation} \eval{ \pdv{ \mathscr{L}(\vb{p}, \mu) }{p_i} }_{\mu^\star, \thinspace \vb{p}^\star} = \eval{ \pdv{ E(\vb{p}) }{p_i} }_{\vb{p}^\star} - \mu^\star \qty( \eval{ \pdv{ m(\vb{p}) }{p_i} }_{\vb{p}^\star} ) = 0 \thinspace . \end{equation}\] A general optimization strategy would be to scan possible values for \(\mu\), solve this equation and check if the constraint is fulfilled. If both equations are simultaneously solved, we have found a solution \((\vb{p}^\star, \mu^\star)\), whose corresponding Lagrangian value is the energy \(\mathcal{E}\): \[\begin{equation} \mathscr{L}(\vb{p}^\star, \mu^\star) = E(\vb{p}^\star) = \mathcal{E} \thinspace , \end{equation}\] exactly because the feature is attained, i.e. the corresponding constraint is fulfilled.

In the special case of configuration interaction theory, and if the feature can be described as the expectation value over the feature operator \(\hat{m}\): \[\begin{equation} m(\vb{p}) = \frac{ \ev{\hat{m}}{\Psi(\vb{p})} }{ \braket{\Psi(\vb{p})} } \thinspace , \end{equation}\] the stationary equation due to the parameter derivative can be written it as \[\begin{equation} (\vb{H} \vb{c}^\star - E(\vb{c}^\star) \vb{c}^\star) - \mu^\star (\vb{M} \vb{c}^\star - m(\vb{c}^\star) \vb{c}^\star) = \vb{0} \end{equation}\] by introducing the relevant matrix representations in the associated Fock space of the configuration interaction. Recognizing the constraint, we may thus rewrite it as the (modified) eigenvalue problem \[\begin{equation} \vb{H}_{\text{mod}}(\mu^\star) \vb{c}^\star = E_{\text{mod}}(\vb{c}^\star, \mu^\star) \vb{c}^\star \thinspace , \end{equation}\] of the modified Hamiltonian \(\vb{H}_{\text{mod}}(\mu^\star)\) \[\begin{equation} \vb{H}_{\text{mod}}(\mu^\star) = \vb{H} - \mu^\star \vb{M} \thinspace , \end{equation}\] whose eigenvalues \(E_{\text{mod}}(\vb{c}^\star, \mu^\star)\) are related to the energy \(\mathcal{E}\) as \[\begin{equation} \mathcal{E} = E_{\text{mod}}(\vb{c}^\star, \mu^\star) + \mu^\star M \thinspace . \end{equation}\]