Is it Possible to Reconcile the Amount of Charge Transfer Defined in Real Space by the Charge Displacement Function and in Hilbert Space by Absolutely Localized Molecular Orbitals?

How should one define the number of electrons transferred or delocalized when hydrogen bonds or other intermolecular interactions are formed based on quantum chemistry calculations? While not directly an observable, there are many suggested answers. Two apparently reasonable approaches, the charge-displacement function (CDF) in real space, and the absolutely localized molecular orbital charge-transfer analysis (ALMO-CTA) in Hilbert space yield results that differ by nearly a factor of 10 for the water dimer. Which one is right and why? This paper addresses this question in several ways. Numerical tests reveal that the water dimer discrepancy can be reduced roughly 3-fold by using CDF with a redefined initial state such that remaining electron flows are only due to CT. An origin of the remaining discrepancy is illuminated by CT in the model H₂⁺ problem that reveals potential issues with measuring CT via the CDF at short-range, whereas long-range behavior is fully correct. Another model problem, the polarization of the He atom by a uniform electric field, reveals even more severe problems when using CDF to measure the number of electrons rearranged by polarization: the CDF should not be used for this purpose. Numerical results for the xenon-water complex support previous CDF results that indicate a small amount of CT occurs, with the CDF value about 4 times larger than the ALMO-CTA value of 0.9 me^-. Smaller, yet still nonzero CT is found for the neon-water complex by both CDF and ALMO-CTA. For ligand-metal bonding, CDF at best reveals net CT, while ALMO-CTA separates forward and backward electron flow. When this is accounted for, and polarization is separated out as well, there is reasonable qualitative agreement between the two measures for net M-CO CT in the Cr(CO)₆, Fe(CO)₆²⁺ and Co(CO)₆³⁺ complexes. Overall, these results for model systems and numerical comparisons for intermolecular interactions suggest that while caution is needed when measuring electron flows via the CDF, the CDF results for net electron flow due solely to CT are in qualitative agreement with the ALMO-CTA.