According to the reference mentioned in the question (Ref.1):
The term ‘trans-influence’, being a long-established concept of broad relevance in the realm of inorganic chemistry, was defined first in 1966 by Pidcock et al. as the ability of ligand L in a complex to weaken the metal–ligand bond trans to itself. This ground-state phenomenon should be distinguished from the kinetic phenomenon called the ‘trans-effect’, which is the effect of coordinated ligand L upon the rate of substitution reactions of the ligand in trans-position to L.
Note: Pidcock et al. 1966: Ref.2
Considering the high sensitivity of the $\ce{^{13}C}$-$\mathrm{NMR}$ isotropic shielding tensor elements to small structural/electronic changes, the authors of Ref.1 have published a reliable trans-philicity ladder for octahedral $\ce{[Cr(CO)5L]^{−/0/+}}$ complexes using $\ce{^{13}C}$-$\mathrm{NMR}$ isotropic shielding tensor elements. In $\ce{[Cr(CO)5L]^{−/0/+}}$ complex, $\ce{L}$ represents a wide variety of ligands (50 ligands) commonly used in coordination and organometallic chemistry. Briefly, all $\ce{^{13}C}$-$\mathrm{NMR}$ isotropic shielding tensor elements and other parameters have been calculated using PBE0/Def2-TZVP(Cr)∪6-31G(d,p)(E)/PCM and PBE0/Def2-TZVP(Cr)∪6-311++G(d,p)(E)/PCM computational protocols set in dichloromethane solution where the latter protocol is more sophisticated than the former.
I think, major drawback in this publication is the lack of experimental date to support the calculations. For instance, the authors admit that to the best of their knowledge, experimental data for $\delta \ \ce{^{13}C}$-$\mathrm{NMR}$ chemical shifts of $\ce{[Cr(CO)5L]^{−/0/+}}$ complexes are available only for the $\ce{Cr(CO)6}$ complex and the “free” $\ce{CO}$ ligand, which are $212$ and $\pu{184.4 ppm}$, respectively.
When compared the calculations of $\delta \ \ce{^{13}C}$-$\mathrm{NMR}$ chemical shifts of the $\ce{Cr(CO)6}$ complex and the “free” $\ce{CO}$ ligand employing the two computational protocols, the PBE0/Def2-TZVP(Cr)∪6-31G(d,p)(E)/PCM predicted $\delta \ \ce{^{13}C}$-$\mathrm{NMR}$ chemical shifts of $210.2$ and $\pu{186.1 ppm}$, respectively for two compounds, while the protocol PBE0/Def2-TZVP(Cr)∪6-311++G(d,p)(E)/PCM) predicted $\delta \ \ce{^{13}C}$-$\mathrm{NMR}$ chemical shifts of $226.8$ and $\pu{197.6 ppm}$, respectively for the same two compounds:
$$\begin{array}{l|cc}
\hline
\text{Compound} & \ce{\delta \ ^{13}C} \text{ (calculated)}^1 & \ce{\delta \ ^{13}C} \text{ (calculated)}^2 & \ce{\delta \ ^{13}C} \text{ (experimental)} \\
\hline
\ce{Cr(CO)6} \text{ (complex)} & \pu{210.2 ppm} & \pu{226.8 ppm} & \pu{212.0 ppm} \\
\ce{CO} \text{ ('free' ligand)} & \pu{186.1 ppm} & \pu{197.65 ppm} & \pu{184.4 ppm} \\
\hline
\end{array}\\
^1 \text{From protocol PBE0/Def2-TZVP(Cr)∪6-31G(d,p)(E)/PCM;} \\ ^2\text{ From protocol PBE0/Def2-TZVP(Cr)∪6-311++G(d,p)(E)/PCM.}
$$
Evidently, the GIAO/PBE0/Def2-TZVP(Cr)∪6-31G(d,p)(E)/PCM computational protocol is a better performer in the calculation of the $\ce{^{13}C}$-$\mathrm{NMR}$ spectra of $\ce{[Cr(CO)5L]^{−/0/+}}$ complexes than that of PBE0/Def2-TZVP(Cr)∪6-311++G(d,p)(E)/PCM one. Nevertheless, the differences of the calculated $\Delta\sigma \ \ce{^{13}C}$-$\mathrm{NMR}$ descriptors of trans-philicity for the complexes using either protocol were minimal.
Yet the authors have mentioned that:
It can be seen that the $\mathrm{NMR}$ trans-philicity ladders constructed by the two computational protocols are similar with some minor local reversed orders in the trans-philicity series of similar ligands. The PBE0/Def2-TZVP(Cr)∪6-31G(d,p)(E)/PCM computational protocol predicts for the $\ce{NCR}$ ligands the order: $\ce{NCH \gt NCPh \gt NCMe}$, while the PBE0/Def2-TZVP(Cr)∪6-311++G(d,p)(E)/PCM computational protocol predicts the order: $\ce{NCMe \gt NCPh \gt NCH}$. Consideration of the $\sigma$-donor and $\pi$-acceptor abilities of the $\ce{NCR}$ ligands supports the order predicted by the PBE0/Def2-TZVP(Cr)∪6-31G(d,p)(E)/PCM computational protocol.
Thus, I could argue that the difference in two protocols make this difference than actual situation. Unless we have experimental data to support the finding, it is just speculation.
Reference:
- A. C. Tsipis, "Building trans-philicity (trans-effect/trans-influence) ladders for octahedral complexes by using an NMR probe", Dalton Trans. 2019, 48, 1814-1822 (DOI: 10.1039/C8DT04562C).
- A. Pidcock, R. E. Richards, and L. M. Venanzi, "$\ce{^{195}Pt–^{31}P}$ nuclear spin coupling constants and the nature of the trans-effect in platinum complexes," J. Chem. Soc. A 1966, 1707–1710 (DOI: https://doi.org/10.1039/J19660001707).
