CAPITULO II MARCO TEÓRICO Y CONCEPTUAL
2. BASES TEÓRICAS
The Schrödinger equation is not a completely correct description of the interaction of electrons and nuclei as it ignores effects due to special relativity. The Dirac equation is a relativistically correct equation on the other hand and the Schrödinger equation can be understood as an approximation to it by taking the speed of light to be
infinite.2 Due to the complexity of the Dirac equation it is rarely affordable but for the
smallest molecules, but popular approximations to it have been derived that can capture most of the effects of relativity.
Relativistic effects are known to become important to account for, for the lower transition metal rows, but are often ignored for the 1st transition metal row (copper
being an exception). Relativistic effects have previously been shown to be negligible for 57Fe EFG tensors167 according to Zeroth-‐Order Regular Approximation (ZORA)
168,169 calculations, an approximation the relativistic Dirac equation. 170,170170 Recent
developments in Douglas-‐Kroll-‐Hess relativistic approximations171 that correct for
the picture-‐change effect on electric field gradients170 (an artifact arising due to the
mismatch between a nonrelativistic operator being used with a quasirelativistic wavefunction/density), have allowed us to explore this issue by performing Douglas-‐ Kroll-‐Hess (DKH) relativistic calculations using the Orca code.147
In a recent study, DKH electric field gradients were calculated for the hydrogen halides.170 By using spin-‐free Dirac-‐DFT results as reference values for the molecule
test set, the convergence and performance of the DKH approximation could be
systematically explored. The dependence of the results on the n-‐th order of the DKHn Hamiltonian as well as the order of the picture-‐change correction were explored and it was found that DKH2 (second order DKH) with picture-‐change corrections was a generally accurate approximation to capture scalar relativistic effects on EFGs. The picture-‐change correction was found to be crucial for accurate EFG calculations.
We carried out such DKHn calculations (and non-‐relativistic calculations for
comparison) using the revTPSSh functional and the decontracted def2-‐QZVPP basis set using the Orca code. DKHn calculations up to n=3 were carried out. Calculations were carried out with and without picture-‐change corrections. The DKH2 calculations were found to be ~1.5 times more expensive than nonrelativistic calculations while DKH3 calculations (including picture-‐change corrections) were ~1.7 times more expensive.
Table 7 Decontracted basis EFG calculations with the revTPSSh functional and different relativistic approximations: Vzz values in au,a compared to gas-‐phase MW data.
Comp. NRb DKH1 DKH1-‐pcc DKH2 DKH2-‐pcc DKH3-‐pcc Exp. 18 -‐0.355 -‐0.358 -‐0.362 -‐0.358 -‐0.362 -‐0.362 -‐0.344 19 -‐0.353 -‐0.372 -‐0.378 -‐0.372 -‐0.378 -‐0.378 -‐0.217 20 0.586 0.589 0.588 0.589 0.588 0.588 0.570 21 -‐0.871 -‐0.880 -‐0.880 -‐0.880 -‐0.880 -‐0.880 -‐0.877 22 -‐1.226 -‐1.201 -‐1.203 -‐1.203 -‐1.204 -‐1.204 -‐1.182 23 -‐0.532 -‐0.520 -‐0.534 -‐0.520 -‐0.535 -‐0.535 -‐0.356 24 -‐0.303 -‐0.317 -‐0.320 -‐0.317 -‐0.319 -‐0.319 -‐0.390 MAEd 0.068 0.064 0.067 0.064 0.067 0.067 MEd -‐0.037 -‐0.038 -‐0.042 -‐0.038 -‐0.042 -‐0.042 MaxEd 0.176 0.164 0.178 0.164 0.179 0.179 Sloped 1.008 0.999 0.998 1.000 0.999 0.999 aGaussian-‐style, decontracted def2-‐QZVPP basis set and BP86/AE1 geometries.
b NR: Non-‐relativistic.
c Picture-‐change correction used.
dMean absolute error (MAE), mean error (ME), maximum error (MaxE) and slope from a linear regression of
DFT vs. experimental data.
Table 7 shows the results of these calculations. The results reveal that there is sometimes an overestimation of relativistic effects when not including the picture change correction. Overall, the picture-‐change corrected relativistic effect is rather small and does not significantly affect the mean absolute error of EFGs for our test set (although some molecules seem to be more sensitive than others). This suggests that relativistic calculations for 3d transition metal EFGs are generally not needed as the DFT functional error will overshadow any error resulting from the neglect of
relativistic effects.
3.3.4. Summary
1. The electric field gradient in the gas-‐phase does not seem overly sensitive to geometric effects (BP86/TPSS geometry optimisations with a moderate basis set (AE1) seem to be sufficient).
2. There is a strong basis set dependence and normal contracted basis sets are unreliable for EFG computations. In order to minimise basis set errors in EFG calculations, a large decontracted quadruple-‐zeta basis set on the metal atom is recommended, while much smaller (double-‐zeta) normal contracted basis sets can be used for the ligand atoms with good accuracy.
3. Relativistic effects on EFGs can be easily checked for nowadays as the
additional computational cost is not excessive. Relativistic effects seem to be small for 1st transition metal row EFGs and can probably be ignored for the
most part.
4. The DFT functional is the greatest source of uncertainty in these calculations and most likely responsible for the remaining deviations with experiment. The (rev)TPSS and (rev)TPSSh hybrid functionals seem to be the most reliable functionals for EFG computations of 3d transion metal complexes.
5. Finally it must be noted that this test set of EFGs is rather small and any error in the experimental coupling constant or quadrupole moment can significantly skew the results. The experimental quadrupole coupling constants have small error bars, however, some of the quadrupole moments have sizeable
uncertainties, e.g. over 30 % for 53Cr.
Recently a new quadrupole moment of -‐4.8 fm2 instead of -‐5.2 fm2 has been
suggested for 51V.172 We note, however, that using this value for Q would
increase the deviation for the EFG at the V nucleus for all tested functionals as all of them underestimate the absolute value for this EFG while the new Q would increase the absolute value. Hence, this would affect the comparison between methods very little.
6. Our DFT functional recommendations should be further explored in the future. It is likely that test sets from solid-‐state experiments are necessary as
microwave NQCC determinations for transtion metal complexes are few. This will inevitably require reliable methods to take solid-‐state effects into account.
Finally, we give a table with estimated DFT functional errors for selected first
transition metal row nuclides (in MHz) based on the computed mean absolute errors in EFG Vzz values in Table 6 and Eq. 49. Later we will compare computed and
experimental 51V and 59Co NQCC data in the solid-‐state where this table will become
useful.
Table 8 Estimated DFT errors for NQCC (in MHz ) of the first transition metal row nuclides for selected functionals.a
Comp. B3LYP PBE TPSS TPSSh revTPSS revTPSSh
45Sc 6.0 5.0 4.2 4.2 3.8 3.5 47Ti 8.2 6.8 5.8 5.7 5.2 4.8 49Ti 6.7 5.6 4.7 4.7 4.2 3.9 51V 1.4 1.2 1.0 1.0 0.9 0.8 53Cr 4.1 3.4 2.9 2.8 2.6 2.4 55Mn 8.9 7.5 6.3 6.2 5.7 5.2 57Fe 4.3 3.6 3.0 3.0 2.8 2.5 59Co 11.4 9.5 8.0 7.9 7.2 6.6 61Ni 4.4 3.7 3.1 3.1 2.8 2.6 63Cu 6.0 5.0 4.2 4.2 3.8 3.5 65Cu 5.5 4.6 3.9 3.9 3.5 3.2 67Zn 4.1 3.4 2.9 2.8 3.5 2.4
a Mean absolute NQCC errors for each nuclide, calculated according to Eq. 49, using the mean absolute error of Vzz
from Table 6 as Vzz value and Q for each nuclide is taken from the latest set of quadrupole moments from
Pyykkö.144