Similarities and differences of the Lagrange formalism and the intermediate state representation in the treatment of...

When dealing with approximate wave functions, molecular properties can be computed either as expectation values or as derivatives of the energy with respect to a corresponding perturbation. In this work, the intermediate state representation (ISR) formalism for the computation of expectation values is compared to the Lagrange formalism following a derivative ansatz, which are two alternative approaches of which neither one can be considered superior in general. Within the ISR formalism, terms are included up to a given order of perturbation theory only, while in the Lagrange formalism, all terms are accounted for arising through the differentiation. Similarities and differences of the Lagrange and ISR formalism are illustrated using explicit working equations for selected methods and analyzing numerical results for a range of coupled-cluster as well as algebraic-diagrammatic construction (ADC) methods for excited states. The analysis explains why the ADC(3/2) method is able to yield a large amount of the orbital-relaxation effects for p-h states in contrast to ADC(2) although the same second-order ISR is used to represent the corresponding operator.