A pair natural orbital based implementation of ADC(2)-x: Perspectives and challenges for response methods for singly and doubly excited states in large molecules

At the example of the extended algebraic-diagrammatic construction through second order, ADC(2)-x, we study the performance of a pair natural orbital (PNO) expansion for the description of partially or dominantly doubly excited states within a response theory framework. We find that with the presence of dominantly doubly excited states in the ADC(2)-x spectra the PNO truncation errors are between a factor of two and five larger than for strict second-order methods, where such excitations are only described in a zeroth-order approximation, but that otherwise the rate of convergence with the PNO selection threshold is similar. Furthermore, we analyze the reason for the red shift of excitation energies in ADC(2)-x and trace it back to an unbalanced account of electron correlation effects in the ground and the excited state by the term added in the extended method to describe double excitations correct through first order. The balance can be restored by including the contributions from one additional commutator which accounts in lowest order for the coupling between double excitations in the ground and the excited state. A perturbative correction to ADC(2) which includes both correction terms for doubly excited configurations gives excitation energies close to results from CCSD with, however, much lower computational expenses. The truncated pair natural orbital expansion for the ground and excited state doubles amplitudes opens a route to reduce the steep O(N-6) cost-scaling of the extended ADC(2) and other higher-order response methods without significant degradation of their accuracy or a priori restriction of the double excitation manifold. Already our proof-of-principles implementation exhibits a cost-scaling between O(N-3) and O(N-4), which can be further reduced with a stringent use of integral screening and local density fitting to a quadratic cost-scaling. (C) 2014 Elsevier B.V. All rights reserved.