#### Analyzing calculated NMR parameters: Why is this particular NMR parameter solarge / small /weird / ... ?

Most of our NMR parameter analyses are carried out with the ADF program and localized molecular orbitals, as described in detail in the following references. Selected applications: [273], [255], [235], [227], [208], [177], [143], [112], [101]. Method developments: [88], [76]. See also this link to the ADF program documentation on localized molecular orbital analysis functionality for NMR parameters with or without relativistic eﬀects included in a calculation.

Perhaps you know that some substituents of benzene activate the aromatic ring in certain positions for electrophilic substitution, while other substituents deactivate the ring. For example, the –NH${}_{2}$ substituent activates the ortho and para positions, while –NO${}_{2}$ deactivates the same positions and therefore directs substitutions to the weakly aﬀected meta position. These observations are explained in textbooks by the substituent causing an increase or decrease of the electron density in the $\pi$-aromatic system, and ﬁrst-principles calculations conﬁrmed these density changes [Fonseca Guerra et al., Phys. Chem. Chem. Phys. 18 (2016), 11624].

These increases or decreases of the $\pi$-density go along with changes of the carbon NMR nuclear magnetic shielding $\sigma$, which are then observed experimentally as a decrease or increase of the chemical shift $\delta$, respectively, because

 $\delta \left(\text{probe}\right)=\sigma \left(\text{reference}\right)-\sigma \left(\text{probe}\right)$

Here, $\sigma \left(\text{reference}\right)$ is the magnetic shielding of a nucleus of the same isotope in a reference compound, whereas ‘probe’ indicates our compound of interest. (In principle, the right hand side of the equation should be divided by $1-\sigma \left(\text{reference}\right)$, but shielding constants are usually on the order of a few 10 to a few thousand ppm and therefore the equation as written is a good enough approximation.) Therefore, observed trends in chemical shifts indicate equal and opposite trends in the NMR shielding.

Electrons (usually) shield the external ﬁeld of the NMR spectrometer from the nucleus. Indeed, across the periodic table the shielding constants of atoms increase from a few dozen ppm for H to several 10,000 ppm for heavy elements. Therefore, one often ﬁnds an argument in the literature that more electron density means more shielding. For example, in the benzene case, when the $\pi$-density in a certain position increases due to an activating substituent, the shielding in the NMR experiment also increases (which is observed by a decreasing chemical shift relative to unsubstituted benzene). For deactivating substituents the shielding trends are usually opposite. However, we showed in 2017 that the nuclear magnetic shielding eﬀects in substituted benzenes are associated with the $\sigma$-bonding framework of orbitals, not the $\pi$ orbitals [272]. So what is going on, and how do we know that the $\sigma$ orbitals cause the substituent eﬀects in benzene derivatives?

The nuclear magnetic shielding is a response property and not in a simple way related to the ground state electron density. The shielding depends on the ground state electronic structure and how this electronic structure responds to the presence of a magnetic ﬁeld. For a given isotope there may be large variations of the magnetic shielding among diﬀerent compounds, such that the general trend of increasing NMR shielding for increasingly electron-rich atoms in the periodic table should not be confused with simple electron density - shielding relationships. In a ﬁrst-principles molecular orbital (MO) calculation, such as Kohn-Sham (KS) theory (the most popular ﬂavor and generalization of density functional theory), an element of the paramagnetic contribution to the shielding tensor can be written like this:

 (1)

In the absence of relativistic eﬀects (i.e. in the absence of heavy elements in a compound), this contribution to the shielding usually determines the chemical shift variations observed for a given element in diﬀerent compounds. In the equation, ${\phi }_{i}$, ${𝜀}_{i}$ and ${\phi }_{a}$, ${𝜀}_{a}$ represent occupied and unoccupied KS orbitals and their energies, respectively. The superscript OZ refers to the Orbital Zeeman perturbation due to the external magnetic ﬁeld. The superscript PSO refers to the Paramagnetic nuclear Spin - electron Orbital perturbation due to the nuclear spin. The indices $u$ and $v$ indicate components of the external magnetic ﬁeld vector and the nuclear spin magnetic moment vector. Further, $\stackrel{^}{F}$ and $ĥ$ are the KS Fock operator and its one-electron part. The OZ and PSO perturbations of $\stackrel{^}{F}$ and $ĥ$ may be swapped. In order to get the isotropic shielding, for freely rotating molecules in solution, one needs to average ${\sigma }_{u,v=u}^{para}$ over the three axes $u\in \left\{x,y,z\right\}$ of the laboratory coordinate frame or a molecule-ﬁxed frame called the principal axis system (PAS). Further details about the tensor properties of the shielding are provided at this link.

The OZ and PSO matrix elements in the numerator of Equation (1) both involve orbital angular momentum operators, because magnetic moments associated with the electron motion are proportional to the angular momentum (for closed-shell molecules with light elements, the electron spin magnetic moments do not contribute to the shielding). In the PSO operators, the angular momentum is weighted by an inverse-cube of the electron-nucleus distance. Therefore, the PSO operators sample the electronic structure mainly only around the NMR probe nucleus of interest. The action of a component of the angular momentum operator on an atomic orbital (AO) can be represented by a rotation of the AO (e.g. 90 deg. for a 2p AO) or it gives zero (e.g. ${\stackrel{^}{L}}_{z}{p}_{z}$). In order for a matrix element $⟨\text{unocc}|{ĥ}_{v}^{\text{PSO}}|\text{occ}⟩$ to be large, one therefore has to consider the action of an angular momentum component $v$ on the part of the occupied orbital that is centered around the NMR nucleus, and whether the resulting ‘rotated’ orbital has a large overlap with an unoccupied orbital around the same nucleus. See my separate page on ‘orbital-rotation’ models for analyzing NMR shielding for examples of the action of a given OZ or PSO operator component on atomic orbitals.

Equation (1) can be used in various ways to analyze NMR shielding trends. There is an additional positive ‘diamagnetic’ shielding contribution that tends to vary little among diﬀerent compounds (hydrogen being a notable exception), such that the total shielding is usually positive. The paramagnetic shielding contribution is usually negative and tends to vary strongly among diﬀerent compounds. The paramagnetic mechanism is particularly large if the two orbitals are close in energy, all else being equal. The PSO matrix elements in the numerator are large if the MO in question has large atomic orbital (AO) coeﬃcients at the atom of interest, and if the ‘magnetic ﬁeld rotated AO’ has good overlap with an unoccupied MO. This requires the unoccupied MO also to have large AO coeﬃcients at the atom of interest, because the PSO operator samples the regions around the NMR nucleus much more strongly than those around other atoms.

In the benzene substitution analysis, we employed Equation (1) with the usual ‘canonical’ MOs (CMOs), which are delocalized over the benzene ring both for the $\pi$ and for the $\sigma$ set, and localized MOs (LMOs). The LMOs are strongly delocalized for $\pi$ but correspond nicely to 2-center C-C and C-H, C-N etc. bonds within the $\sigma$ orbital framework (see Publication [170] for comments on localized vs. delocalized sets of MOs and other aspects of electron orbitals of molecules). Since Equation (1) and its LMO analog partition the shielding into a sum over the occupied MOs, it is straightforward to assign the substituent eﬀects on the carbon NMR shielding, relative to benzene, to $\sigma$ vs. $\pi$ orbitals. The analysis was very clear in that the observed eﬀects are associated with the occupied $\sigma$ orbitals, and speciﬁcally with the in-plane carbon 2p AOs that contribute to the $\sigma$ MOs.

In a nutshell, our explanation for the aryl carbon shielding changes upon substitution is as follows:

• The substituents increase or decrease the $\pi$ electron density at speciﬁc carbons in the ring. Let’s call this the $\pi$ deformation density (DD) pattern.

• As already found by Fonseca-Guerra et al. with the help of a Voronoi Deformation Density (VDD) analysis, and conﬁrmed by us, there is a mirror-image substitution-DD pattern in the $\sigma$ framework. I.e. if the $\pi$ density increases at a certain carbon due to substitution, the $\sigma$ density decreases, and vice versa.

• The $\sigma$ substitution-DD can be tied to an decrease or increase of 2s and in-plane 2p AO coeﬃcients in the relevant MOs.

• In the numerator of Equation (1), an in-plane 2p carbon AO is ‘rotated’ by an in-plane magnetic ﬁeld operator such that it overlaps with one or more of the unoccupied ${\pi }^{\ast }$ orbitals. The analysis showed that this $\sigma -{\pi }^{\ast }$ magnetic coupling determines nearly all of the change of the shielding due to substitution.

• If the $\pi$ density increases at a certain carbon, the in-plane 2p AO coeﬃcient in one or more of the $\sigma$ orbitals centered on that carbon decreases, due to the mirror-image patterns in the $\sigma$ and $\pi$ DD. This renders the $\sigma -{\pi }^{\ast }$ magnetic coupling in the negative paramagnetic shielding mechanism less eﬀective, and the total carbon shielding become more positive.

• A $\pi$ density depletion, due to a deactivating substituent, gives the opposite result, viz. a more eﬀective paramagnetic mechanism and a more negative shielding instead.

• The ﬁnal result looks as if an increased / decreased $\pi$ density has directly caused an increased / decreased carbon shielding. The actual chain of arguments needs to involve the $\sigma$ MOs, however.

• For a carbon in ortho position of a strongly electronegative deactivating substituent such as –NO${}_{2}$, the situation is a bit more complicated, and one needs to consider the direct polarization of the $\sigma$ framework by the nearby substituent. We showed that once this eﬀect is taken into consideration, the $\sigma$-${\pi }^{\ast }$ magnetic coupling model predicts correctly that such an ortho-carbon will have an increased shielding, due to a less eﬀective paramagnetic mechanism.

We have often found that an analysis of the shielding in terms of CMOs produces large per-orbital contributions of opposite signs, which may complicate an analysis greatly. NMR parameters such as shielding and J-coupling are primarily determined by the local electronic structure and its response around the probe nuclei, and an analysis in terms of LMOs is often easier. In the benzene substitution case, only three occupied LMOs centered on the carbon of interest determined the observed trends, namely the two C-C $\sigma$ bonds to the neighboring carbons and the C-H bond. There are other cases, however, when the primary driver of a shielding change is a change in the numerator of Equation (1). In this case, an analysis in terms of CMOs is much easier. Sometimes, one needs a combination of CMO and LMO pictures to arrive at a chemically intuitive explanation.

© 2017 – 2022 J. Autschbach. The material shown on this web page is in parts or wholly based on the results of research funded by grants from the National Science Foundation (NSF, grants CHE 0447321 (2005-2011, 0952253 (2010-2014), 1265833 (2013-2017)) and educational projects supported by these grants. Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reﬂect the views of the National Science Foundation.