How to calculate the amount charge transfer from a natural population analysis? I am trying to reproduce the quantity referred to as the degree of charge transfer reported in this publication by Zhu et al. [Ref. 1] for the F4-TCNQ molecule. I am interested in section 3.2 in the main text about using the bond length as an estimate for the charge transfer amount. 



In figure 3b, the authors show the correlation between the degree of charge transfer, CT$_{pop}$ from their natural population analysis compared to the approximation of the bond lengths of F4-TCNQ from equation 1, shown below.

$$\rho = \frac{\alpha_{CT}-\alpha_{0}}{\alpha_{-1}-\alpha_{0}}$$
with $\alpha_i = l_3 (l_2+l_4)$ where $l$ is the bond length, and subscript $i=0, -1, CT$ denote the netural molecule, the anion, and the compelx, respectively.

It's not clear to me, how the authors determined the values on the y-axis in Figure 3b, and there are no further clarification or references to a methodology for the values within the text or supplement information. Both axes in Figure 3b use the F4-TCNQ anion as an arbitrary reference. 

I can calculate $\rho$ from the above equation, but I want to compare $\rho$ to the CT$_{pop}$ at the level of theory I am employing, verifying the accuracy of my calculations.  

Through reading and googling, I believe the CT$_{pop}$ refers to a population analysis such as a Mulliken or natural population (AKA Lowdin population), referenced in Mendez et al. [Ref. 2] (bottom of page 9 in the Theory footnote). I do not know how to obtain the degree of charge transfer from the resulting population analysis.

I have one reference from Joo et al. [Ref. 3] who gives more details on a similar method, writing on page 2, left column:



The calculated atomic charges were then used to obtain the degree
of charge transfer between the donor and acceptor molecules,
which may be defined as the sum of all atomic charges on the
donor part of the complex.



The above text leaves me to believe the method of determining the degree of charge transfer is as follows:


Calculate a population analysis of the DA (donor and acceptor) complex to determine the atomic charge
Determine the atomic charge of each of the fragments D and A by summing the atomic charge for the D and A fragments, respectively 
Report the atomic charge of fragments as the degree of charge transfer.


The last aspect is Joo et al. [Ref. 3] reports the value for the electron donor while Zhu et al. [Ref. 1] indicates the value for the electron acceptor, which is F4-TCNQ.

Does this procedure sound reasonable? Any thoughts or comments would be appreciated.



TL;DR
How can I reproduce the values shown in Figure 3b on the y-axis, the degree of charge transfer from a natural population analysis?

References


Zhu L; Kim E.-G; Yi Y; Bredas J.-L. Charge Transfer in Molecular Complexes with 2,3,5,6-Tetrafluoro-7,7,8,8- Tetracyanoquinodimethane (f4-Tcnq): A Density Functional Theory Study. Chemistry of Materials 2011, 23 (23), 5149–5159 DOI: 10.1021/cm201798x.
Méndez Henry; Heimel, G.; Winkler, S.; Frisch, J.; Opitz, A.; Sauer, K.; Wegner, B.; Oehzelt, M.; Röthel Christian; Duhm, S.; et al. Charge-Transfer Crystallites As Molecular Electrical Dopants. Nature Communications 2015, 6 (1) DOI: 10.1038/ncomms9560.
Joo B; Kim EG. Model-Independent Determination of the Degree of Charge Transfer in Molecular and Metal Complexes. Chemical Communications (Cambridge, England) 2015, 51 (81), 15071–15074 DOI: 10.1039/c5cc05274b.
I will evade answering the main question because I am not feeling expert enough, but I want to point out that Natural population analysis (NPA) is very different (and better) from the Loewdin one. Loewdin is computed from orbitals of a special basis set (although it comes from the density, it is still free-atom-derived basis set which is used to obtain the density matrix). NPA (part of NBO) is from density matrix which is adjusted to the particular molecular environment, hence better representing the "real" atomic population. NPA is available in the commercial NBO program, old version of which is also included with Gaussian package (maybe also in other packages, I don't know). You can also try AIM charges which are absolutely basis-independent (if Your density is good enough). 

Personally I would think the analysis You proposed is fine with NPA or AIM charges, but I state again that I am not an expert in the topic. 

See https://www.q-chem.com/qchem-website/manual/qchem43_manual/sect0061.html and also http://www.huntresearchgroup.org.uk/teaching/teaching_comp_chem_year4/L7_bonding.pdf .