Quantifying the error of the core-valence separation approximation

After only a few months of publishing our adcc paper, which introduces the novel algebraic-diagrammatic construction (ADC) code adcc, the package has already found mention in a number of related articles. One example is a manuscript on updates in the Psi4 quantum chemistry package, which features adcc as one of the highlighted community modules. As the paper discusses, the close integration of adcc into the Psi4 ecosystem now allows to start ADC calculations in adcc directly form Psi4's python frontend and its input files. This effectively extends Psi4 by all ADC capabilities adcc offers. Another example is a paper on computing complex polarisabilities for excited states, which especially emphasises the importance adcc has played for simplifying the implementation of the method.

On my end, I recently conducted a study with Thomas Fransson on the error of the core-valence separation (CVS) approximation. This approximation is very important for the simulation of X-ray absorption spectra using accurate wave-function methods like coupled-cluster or ADC. In the literature the error of the CVS approximation is widely accepted to be negligible compared to the error with respect to experiment. This statement is, however, based on previous investigations of the CVS error, which were limited by the methodologies to only small, non-representative basis sets.

In our work we present an iterative post-processing scheme in the ADC context, which is able to undo the CVS approximation and remove the CVS error. Our procedure is still basic (essentially just Rayleigh-Quotient iteration), but it allowed us to study the CVS error for a much larger range of systems and basis sets. This includes augmented and/or core-polarised triple-zeta basis sets as well as other bases prominent in the community to carry out simulations of core-excitations and X-ray spectra (read 6-311++G** and variants). Based on a representative compounds from elements of the second and third period we managed to confirm that the CVS error is not only small compared to experiment but additionally the spread across elements and compounds is even smaller, such that the impact on energy differences is small, too. This is an important finding, since most aspects of chemistry (spectroscopy, reaction barrier heights, thermochemistry) are dominated by energy differences and while relative errors in energies might be small, relative errors in energy differences can be much larger if the error spread across compounds is not uniform.

In particular our work identified the main contributions to the CVS error to originate from two classes of couplings, which are neglected by the CVS approximation and which moreover contribute with opposite sign. We demonstrate that basis sets providing a balanced description of core and valence regions of the electron density are also capable of describing these neglected couplings in a balanced fashion, thus providing not only a good description of the core-excitation process, but also a small CVS error in terms of absolute value and spread. Based on these findings we were able to conclude that especially tight polarising functions are key for describing core-excitations. Along our study we suggest appropriate modifications for the popular 6-311++G** basis to reduce its CVS error. The full abstract of our paper reads

For the calculation of core-excited states probed through X-ray absorption spectroscopy, the core-valence separation (CVS) scheme has become a vital tool. This approach allows to target such states with high specificity, albeit introducing an error. We report the implementation of a post-processing step for CVS excitations obtained within the algebraic-diagrammatic construction scheme for the polarisation propagator (ADC), which removes this error. Based on this we provide a detailed analysis of the CVS scheme, identifying its accuracy to be dominated by an error balance between two neglected couplings, one between core and valence single excitations and one between single and double core excitations. The selection of the basis set is shown to be vital for a proper description of both couplings, with tight polarising functions being necessary for a good balance of errors. The CVS error is confirmed to be stable across multiple systems, with an element-specific spread of only about ±0.02 eV. A systematic lowering of the CVS error by 0.02-0.03 eV is noted when considering excitations to extremely diffuse states, emulating ionisation.

Michael F. Herbst and Thomas Fransson. Quantifying the error of the core-valence separation approximation. Journal of Chemical Physics, 153, 054114 (2020). Part of JCP Emerging Investigators, received Editor's Pick. DOI 10.1063/5.0013538 [code]