Energy-variance minimization for ground and excited states in the Lipkin model and its application to molecular magnets

Abstract This work introduces a variational method for accurately determining both ground and excited states in systems described by the Lipkin model. Traditional variational techniques often struggle to capture excited states because they tend to converge to the lowest energy solution, a phenomenon known as variational collapse. To address this, we employ an energy-variance minimization approach, which treats all states on equal footing and prevents the method from favoring the ground state. Several functional forms for wave functions are explored, including a self-consistent mean-field ansatz, a coupled-cluster-inspired formulation, and an extended coupled-cluster ansatz. The latter approach improves accuracy by incorporating correlations into the reference state with minimal computational cost, leading to wave functions that closely approximate exact solutions. The proposed methodology is applied to a molecular magnet, specifically a cation $$\textrm{Fe}_8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Fe</mml:mtext> <mml:mn>8</mml:mn> </mml:msub> </mml:math> cluster, incorporating magnetic anisotropy. The results as functions of the applied magnetic field demonstrate the effectiveness of this approach in providing an accurate and efficient description of energy spectra of these systems.