Orbital Decomposition of the Carbon Chemical Shielding Tensor in Gold(I) N‐Heterocyclic Carbene Complexes

Abstract

The good performance of N‐heterocyclic carbenes (NHCs), in terms of versatility and selectivity, has called the attention of experimentalists and theoreticians attempting to understand their electronic properties. Analyses of the Au(I)–C bond in [(NHC)AuL]^+/0 (L stands for a neutral or negatively charged ligand), through the Dewar–Chatt–Duncanson model and the charge displacement function, have revealed that NHC is not purely a σ‐donor but may have a significant π‐acceptor character. It turns out, however, that only the σ‐donation bonding component strongly correlates with one specific component of the chemical shielding tensor. Here, in extension to earlier works, a current density analysis, based on the continuous transformation of the current density diamagnetic zero approach, along a series of [(NHC)AuL]^+/0 complexes is presented. The shielding tensor is decomposed into orbital contributions using symmetry considerations together with a spectral analysis in terms of occupied to virtual orbital transitions. Analysis of the orbital transitions shows that the induced current density is largely influenced by rotational transitions. The orbital decomposition of the shielding tensor leads to a deeper understanding of the ligand effect on the magnetic response properties and the electronic structure of (NHC)‐Au fragments. Such an orbital decomposition scheme may be extended to other magnetic properties and/or substrate‐metal complexes.

(NHC) carbon chemical shielding tensor decomposition into orbital contributions. The current density is largely influenced by rotational transitions involving occupied and virtual orbitals localized on the (NHC) carbon atom with a ₁ and b ₁ symmetries, respectively.

Introduction

N‐heterocyclic carbenes (NHCs) have emerged as substrates for transition metal catalysts due to their electron‐donating features1, 2 NHCs have, unlike traditional carbenes, a neutral divalent carbon stabilized by electron donation from one or more adjacent nitrogen atoms into the empty carbon 2p orbital. When NHCs are coordinated to a metal center, NHCs behave as strong σ‐donor ligands, even stronger than alkyl phosphines, which in many cases lead to a greater stability of the pre‐catalysts or catalysts.3 The NHC‐metal bond stability has made NHCs quite attractive, not only for experimentalists in regard to the ligand design, but also for theoreticians who attempt to understand their electronic properties.4, 5, 6, 7 The nature of the NHC‐Au bond in [(NHC)AuL]^+/0 complexes strongly depends on the electronic structure of L and NHC. NHC is not a pure σ‐donor but may have π‐acceptor character, being entirely negligible {as found in [(NHC)AuCO]⁺} or accounting up to half of the σ‐donor character {as found in [(NHC)AuCl]}.8 A more comprehensive overview of the chemistry of N‐heterocyclic carbenes is given in ref.,9 and recently [(NHC)AgX] compounds were synthesized.10 The bonding properties of the carbene–Au bond (also Ag/Cu were considered) in [(NHC)AuCl] and metal‐biscarbenes have been investigated using an energy decomposition analysis. It was found that the carbene–Au bond has a large electrostatic component, and the orbital part consists of approximately 20 % π backbonding.11 A further extensive computational study12 showed that also higher coordination numbers are within reach, and that the π^*‐back‐donation can serve as a design criterion.

A charge displacement study of different [(NHC)AuCl] complexes showed that the NHC bonding properties are quite robust against variation of the structure. This study was extended with [(NHC)PPh] adducts and it was shown that the ³¹P NMR chemical shift only qualitatively correlates with the π acceptor properties of the NHCs.13 A recent computational study of various [(NHC)AuCl] complexes focused on Au NMR found a correlation between the π‐accepting ability of these complexes and the ¹⁹⁷Au chemical shift.14 Furthermore, they also found that the paramagnetic shielding tensor of the carbene‐C atom is the main shielding term which shows variation by changes in the (NHC) moiety. More NMR studies were performed for [(NHC)‐Se] and [(NHC)‐P] complexes, and a correlation was established between the Se and P chemical shifts and the ability of the (NHC) ligand to accept π‐electron density.15 No correlation between the σ‐donating contribution and the shielding was found.

NMR spectroscopy, in terms of chemical shifts,16, 17, 18, 19 has been used for understanding the structural and electronic properties of NHCs. The NHC carbon chemical shielding tensor of 1,3,4,5‐tetramethylimidazol‐2‐ylidene (L) is largely anisotropic, as found experimentally by Arduengo et al.20 That is, the shielding tensor is dominated by a single component, the σ ^yy component (Figure 1) (for a review on the theoretical NMR spectroscopy of NHC, ref.21). The anisotropic character of the shielding tensor is preserved in [(NHC)AuL]^+/0 complexes, as found theoretically by Marchione et al.22 For a wide range of ligands, the σ ^xx and σ ^zz components remain nearly constant while the σ ^yy component significantly changes. This component lies perpendicular to the (NHC)–Au bond, and largely dominates the isotropic shielding constant, σ. The bonding and shielding tensor components have been mapped through the Dewar–Chatt–Duncanson (DCD) model23, 24 and the CD‐NOCV method.25 The CD‐NOCV method combines the charge displacement (CD) function26 with the natural orbital chemical valence (NOCV) approach.27 The NHC→Au σ‐donation component correlates fairly well with the σ ^yy component, and consequently with σ. However, the reasons why the σ ^yy component dominates over the other components of σ are not completely clear yet and only a qualitative picture has been given, mainly based on the Ramsey formula.28 Questions like which orbitals govern σ and what is the influence of L remain unanswered. Clearly, if the anisotropic character is understood, the correlation between the NHC→Au σ‐donation and the σ ^yy component may be rationalized. Thus, an analysis of the shielding tensor given in terms of orbital contributions is highly desirable.

Figure 1.

Principal component of the shielding tensor for NHC. The axes orientation identify the orthonormal reference system centered on the NHC carbon, in which the 3 × 3 chemical shielding tensor is diagonal. Image adapted from ref.20

Here, a detailed analysis of the shielding tensor in terms of the current density is presented. Among the different current density approaches, the continuous transformation of the current density ‐ diamagnetic zero (CTOCD‐DZ)29, 30, 31, 32, 33, 34 as implemented in the SYSMO code35 is used. The advantage of using CTOCD‐DZ is that it allows a natural decomposition of the shielding tensor into orbital contributions and a spectral analysis.36 Within CTOCD‐DZ the first‐order current density [j_n ⁽¹⁾(r)] is a sum of orbital contributions.

$$\begin{array}{l}{j}^{(1)}(r)={\displaystyle \sum _{n=1}^{N/2}{j}_n}(r)\mathrm{where}{j}_n(r)\mathrm{is:}\\ {\mathrm{j}}_n(r)=\frac{ie\hslash }{m}{\displaystyle \int ({\psi }_n\nabla {\psi }_n^{(1)*}+{\psi }_n^{(1)*}\nabla {\psi }_n)}dr\end{array}$$ (1)

and where the first‐order correction to the occupied orbital (ψ_n), denoted ψ_n ⁽¹⁾, is:31

$${\psi }_n^{(1)}=\frac{e}{2m}\left[d\times {\displaystyle {\sum }_{p>N/2}{\psi }_p}\frac{〈{\psi }_p\left|\widehat{p}\right|{\psi }_n〉}{{ϵ}_p-{ϵ}_n}\right].B+\frac{e}{2m}\left[d\times {\displaystyle \sum _{p>N/2}{\psi }_p}\frac{〈{\psi }_p\left|\widehat{l}\right|{\psi }_n〉}{{ϵ}_p-{ϵ}_n}\right].B$$ (2)

where N represents the number of electrons, ψ_n and ψ_p represent the occupied and virtual orbitals, respectively, B represents the external magnetic field and p and l are linear and angular momentum operators, respectively.

By substituting ψ_n ⁽¹⁾ in Equation 1, the CTOCD‐DZ j_n(r) is then expressed in two terms, the conventionally called orbital diamagnetic contribution j_n ^(d)(r) and the orbital paramagnetic contribution [j_n ^(p)(r)]. The first‐order correction ψ_n ⁽¹⁾ and therefore also j_n(r) are determined by the accessibility of states through translational transitions [by means of the linear momentum operator (p̂)] and rotational transitions [by means of the angular momentum operator l̂(0)].31 A transition ψ_n→ψ_p contributes to the induced current and thus the shielding constant if the direct product of Γ(ψ_n) × Γ(Î) × Γ(ψ_p) spans the totally symmetric representation, Γ₀, of the point group to which the molecule belongs (where Î represents the operators p̂, l̂).31

Results and Discussion

A series of complexes with C _2v equilibrium geometry having the general formula [(NHC)AuL] or [(NHC)AuL]⁺ (where NHC stands for imidazol‐2‐ylidene and L represents the auxiliary ligand) was studied. This series comprises (NHC), [(NHC)H]⁺, [(NHC)Au]⁺, [(NHC)AuH], [(NHC)AuCNH]⁺, [(NHC)AuBr], [(NHC)AuCl], [(NHC)AuCO]⁺, [(NHC)AuXe]⁺ and [(NHC)AuPyrrolyl] (Figure 2).

Figure 2.

Molecular structures of (NHC), [(NHC)H]⁺ and [(NHC)AuL]^+/0 complexes.

These systems represent a suitable subset of those studied in ref.,22 and as it will be demonstrated, their symmetry properties play a key role in the rationalization of their DCD bonding and the shielding tensor components.

This section begins by briefly summarizing the bond analysis for the systems under study, which has been presented elsewhere.22 Such analysis is based on the donation and back‐donation components of the DCD model and on the electron density rearrangement (Δρ) upon formation of [(NHC)AuL]^+/0, from (NHC) and [AuL]^+/0. The DCD components are disentangled through the NOCV method, taking as a reference the occupied orbitals of (NHC) and [AuL]^+/0 suitably orthogonalized to each other and renormalized (for details, see ref.25). Δρ is decomposed into NOCV pairs (Δρ_k) which may be easily ascribed to the DCD components, on the basis of their local symmetry. For the systems under study, each NOCV pair belongs to one of the four irreducible representations of the C _2v point group (a ₁, a ₂, b ₁ and b ₂).

The top panel of Figure 3 shows the components of Δρ for [(NHC)AuCl]. Δρ ₁, the more significant component of Δρ, has regions of charge accumulation and depletion on the [AuCl] fragment and the NHC carbon lone‐pair, respectively. Δρ ₁, characterized by a cylindrical symmetry, is ascribed to the σ‐donation component (σ don) (Δρ_k with a ₁ symmetry). Δρ ₂ and Δρ ₄ are ascribed to the in‐plane and out‐of‐plane π‐back‐donation components (Δρ_k with b ₁ and b ₂ symmetries, respectively). The in‐plane and out‐of‐plane π‐back‐donation components are denoted in Figure 3 by π _∥ and π _⊥, respectively. Δρ ₃ is ascribed to the σ‐back‐donation component (σ back), which has been observed in a similar complex, [(S)AuCl] [where S represents 2‐(1‐hexynyl)dimethylaniline].37

Figure 3.

NOCV‐CD analysis for [(NHC)AuCl]. Top panel: contribution to deformation density of the four most significant NOCV pairs of [(NHC)AuCl], with [AuCl] and (NHC) fragments. Isodensity surfaces (±0.0015 e/au³) are superimposed to the molecular structure of the complex. Red surfaces (negative values) identify charge depletion regions; purple surfaces (positive values) identify charge accumulation regions. The small green sphere represents the chlorine atom. Bottom panel: CD curves. Red dots indicate the atomic nuclei on the z axis, where C_NHC represents the (NHC) carbon atom. A dotted vertical line marks the boundary between the [AuCl] and (NHC) fragments in which relative values of CT (CTσ _don, CTπ _⊥, CTπ _∥, CTσ _back) are obtained (for further details, see Computational Details of ref.22).

Quantitative information of Δρ has been obtained through the CD function. The results are given in Table 1. The data show that L mainly affects the CTσ _don (CT values range from 0.48 to 0.25 e) and CTπ _⊥ (CT values range from 0.00 to –0.12 e) while CTπ _∥ and CTσ _back give a systematic small contribution to the CT (whose values range from –0.02 to –0.04 e).

Table 1.

CT decomposition into CTσ _don, CTπ _⊥, CTπ _∥, CTσ _back, in e, for [(NHC)AuL]^+/0.22

System	CTσ don	CTπ ⊥	CTπ ∥	CTσ back	CT
[(NHC)Au]+	0.480	–0.043	–0.026	–0.026	0.385
[(NHC)AuH]	0.246	–0.075	–0.032	–0.019	0.120
[(NHC)AuCNH]+	0.326	–0.020	–0.020	–0.031	0.255
[(NHC)AuBr]	0.315	–0.104	–0.039	–0.033	0.140
[(NHC)AuCl]	0.313	–0.107	–0.040	–0.040	0.126
[(NHC)AuCO]+	0.341	–0.002	–0.017	–0.032	0.290
[(NHC)AuXe]+	0.409	–0.043	–0.025	–0.034	0.307
[(NHC)AuPyrrolyl]	0.302	–0.118	–0.035	–	0.149

Next, the anisotropic character of the shielding tensor is verified. The results are presented in Table 2. For a given system, the isotropic shielding constant, σ, is obtained by averaging the three components of the shielding tensor. For completeness, the experimental chemical shift, when available, is also reported.

Table 2.

NHC carbon shielding tensor components and isotropic shielding constant, σ, of (NHC), [(NHC)H]⁺ and [(NHC)AuL]^+/0. Computed values are referenced to (NHC) (chosen as arbitrary zero) as Δσⁱⁱ = σⁱⁱ _NHC – σⁱⁱ _{[NHCAuL]+/–} and expressed in ppm. δ _exp is the experimental isotropic chemical shift of [(NHC)^IPrAuL]^+/0 relative to TMS; measured in solution where a, b, c and d represent d₆‐benzene, DMSO, CDCl₃ and CD₂Cl₂, respectively

System	Δ σ xx	Δ σ yy	Δ σ zz	Δ σ	δ exp
(NHC)	0.000	0.000	0.000	0.000	220.6a[38]
[(NHC)H]+	1.184	285.330	–16.041	–90.158	132.2b[38]
[(NHC)Au]+	16.900	256.951	–47.148	–75.567
[(NHC)AuH]	19.098	116.545	–20.950	–38.231	204.9a[39]
[(NHC)AuCNH]+	16.770	184.368	–25.193	–58.648	178.3c[40]
[(NHC)AuBr]	25.074	160.993	–29.120	–52.315	179.0c[41]
[(NHC)AuCl]	25.610	156.694	–30.612	–50.564	175.5c[42]
[(NHC)AuCO]+	16.945	193.790	–25.628	–61.712	174.6d[43]
[(NHC)AuXe]+	17.743	210.412	–35.931	–64.074
[(NHC)AuPyrrolyl]	24.182	140.876	–31.171	–44.629

Calculated and experimental chemical shifts – with a linear regression coefficient, R², of 0.91‐ follow the same trend, despite that spin‐orbit and solvent effects were not considered in the computations {for a larger series of [(NHC)AuL]^+/0 complexes, see Table 2 and Figure SI of ref.22}. The σ ^yy component varies in a wide range while the σ ^xx and σ ^zz components remain almost constant. The systems are oriented in such a way that the z axis is the symmetry axis, the y axis lies in the plane of the (NHC) ring and the x axis is perpendicular to the yz plane (Figure 1). The shielding tensor components are comparable within the series because in all the cases the corresponding component axes coincide. The component orientations are dictated by local C _2v symmetry at the (NHC) carbon.

The σ ^yy component originates from the action of the external magnetic field, B, in the y direction (B_y), which induces a current on the xz plane. The xz plane contains the (NHC) carbon lone pair and its formally empty p_x orbital, participating in the donation and back‐donation to and from [AuL]^+/0, respectively. Thus, σ ^yy is an ideal probe of the electronic properties of [(NHC)AuL]^+/0.

There is a strong correlation between the σ ^yy and σ‐donation components.22 There is a less strong correlation between the σ ^yy component and the total CT (CT is a measure of the net acidity of the [AuL]^+/0 fragment). The correlation is weakened by a significant π _⊥ back‐donation, nevertheless, this bonding component does not have a direct effect on the shielding tensor. The reason why only the σ‐donation component modifies the σ ^yy component is not obvious and motivated the following analysis.

Symmetry properties may be used to decompose the isotropic shielding into orbital contributions.44 Transitions due to the coupling between ψ_n and ψ_p via the operators p̂ and l̂, contribute to σ.31 These transitions satisfy the selection rules Γ(ψ_n) × Γ(Î) × Γ(ψ_p) = Γ₀ with Î = p̂,l̂.

Here, σ is decomposed into orbital contributions belonging to the a ₁, a ₂, b ₁ and b ₂ irreducible representations. For each irreducible representation, orbital contributions are grouped, leading to sets of ψ_n(a ₁), ψ_n(a ₂), ψ_n(b ₁) and ψ_n(b ₂). The occupied‐orbital‐symmetry decomposition is presented in Table 3. For a given system, the isotropic shielding constant, σ, is obtained by adding all the symmetry‐contributions.

Table 3.

Occupied‐orbital‐symmetry decomposition into σ _k components of the (NHC) carbon isotropic shielding constant, σ, of (NHC), [(NHC)H]⁺ and [(NHC)AuL]^+/0 complexes; ψ_n(a ₁), ψ_n(a ₂), ψ_n(b ₁), ψ_n(b ₂). All values are expressed in ppm

System	ψn(a 1)	ψn(a 2)	ψn(b 1)	ψn(b 2)	σ
(NHC)	29.367	2.878	–24.840	–63.862	–56.458
[(NHC)H]+	121.986	3.399	–22.882	–68.803	33.700
[(NHC)Au]+	115.801	4.521	–23.166	–78.026	19.109
[(NHC)AuH]	74.776	4.160	–30.173	–66.991	–18.227
[(NHC)AuCNH]+	95.007	4.336	–28.114	–69.039	2.190
[(NHC)AuBr]	89.767	4.318	–30.362	–67.866	–4.143
[(NHC)AuCl]	88.055	4.324	–30.587	–67.687	–5.894
[(NHC)AuCO]+	98.592	4.364	–28.438	–69.234	5.254
[(NHC)AuXe]+	103.262	4.425	–26.626	–73.444	7.616
[(NHC)AuPyrrolyl]	82.341	4.201	–31.310	–67.061	–11.829

Table 3 shows that only transitions involving ψ_n(a ₁) significantly change, while transitions involving ψ_n(a ₁), ψ_n(b ₁) and ψ_n(b ₂) marginally change. The former correlate with σ, with a R² of 0.99 (Figure 4). ψ_n(b ₁) and ψ_n(b ₂) are related to the π _⊥ and π _∥ back‐donation components, respectively. The fact that ψ_n(b ₁) and ψ_n(b ₂) do not significantly contribute to the variation of σ is consistent with the fact that the π _∥ and π _⊥ back‐donation components have only negligible influence on the chemical shift.

Figure 4.

Correlation between the ψ_n(a ₁) component and the isotropic shielding constant, σ, of the (NHC) carbon atom for (NHC), [(NHC)H]⁺ and [(NHC)AuL]^+/0 complexes.

A further orbital partitioning into translational and rotational contributions is reported in Table 4 (Equation 2); for completeness translational and rotational contributions involving ψ_n(a ₂), ψ_n(b ₁) and ψ_n(b ₂) are also reported. For a given system, the isotropic shielding constant, σ, is obtained by adding all the translational and rotational contributions.

Table 4.

Occupied‐orbital‐symmetry decomposition into σ_k^i,jB_xyz components of the (NHC) carbon isotropic shielding constant, σ, of (NHC), [(NHC)H]⁺ and [(NHC)AuL]^+/0 complexes; i,j = trans,rot and k = ψ_n(a ₁), ψ_n(a ₂), ψ_n(b ₁), ψ_n(b ₂) (for simplicity B_xyz is omitted). R² is the linear regression coefficient between each component and σ. All values are expressed in ppm

System	σ ψₙ(a₁) trans Bj	σ ψₙ(a₁) rot Bj	σ ψₙ(a₂) rot Bj	σ ψₙ(a₂) rot Bj	σ ψₙ(b₁) trans Bj	σ ψₙ(b₁) rot Bj	σ ψₙ(b₂) trans Bj	σ ψₙ(b₂) rot Bj	σ
(NHC)	301.371	–272.009	8.923	–6.046	23.515	–48.354	–7.451	–56.409	–56.458
[(NHC)H]+	317.754	–195.768	10.456	–7.058	23.592	–46.474	–2.669	–66.135	33.700
[(NHC)Au]+	330.956	–215.154	36.082	–31.561	52.681	–75.868	70.385	–148.411	19.109
[(NHC)AuH]	316.692	–241.000	34.935	–30.774	47.947	–78.120	111.694	–178.686	–18.227
[(NHC)AuCNH]+	295.939	–201.049	36.237	–31.901	49.510	–77.624	175.887	–244.926	2.190
[(NHC)AuBr]	322.906	–233.136	37.280	–32.961	60.461	–90.823	142.460	–210.327	–4.143
[(NHC)AuCl]	314.567	–226.512	35.459	–31.135	55.774	–86.362	142.288	–209.975	–5.894
[(NHC)AuCO]+	309.706	–211.115	36.069	–31.705	51.188	–79.627	150.729	–219.963	5.254
[(NHC)AuXe]+	318.783	–215.388	39.396	–34.970	48.172	–74.799	146.397	–219.841	7.616
[(NHC)AuPyrrolyl]	289.687	–207.435	39.435	–35.234	48.279	–79.589	212.977	–280.039	–11.829
R2	0.21	0.76	0.05	0.04	0.04	0.01	0.01	0.01

Table 4 shows only a very weak correlation of the σ _ψₙ(a₁) ^rot B_j component with the isotropic shielding, with a R² of 0.76 (see also Figure 5).

Figure 5.

Correlation between the σ _ψₙ(a₁) ^rot B_jB_xyz component and the isotropic shielding constant, σ, of the (NHC) carbon atom for (NHC), [(NHC)H]⁺ and [(NHC)AuL]^+/0 complexes.

The rotational contributions may be further decomposed, restricting the lay axis of B to the y direction. The results are given in Table 5, where σ _ψₙ(a₁) ^rot,yy B_y represents the σ ^yy component of rotational transitions, 〈ψ_n(a ₁)|l̂_y(b ₁)|ψ_p(b ₁)〉, under the action of the external magnetic field B along the y axis. For completeness σ _ψₙ(a₁) ^rot,xx B_x and σ _ψₙ(a₁) ^rot,zz B_z contributions are also reported. For a given system, the orbital‐orbital rotational contribution, reported in Table 4, is obtained by averaging all the contributions [σ _ψₙ(a₁) ^rot,xx B_x and σ _ψₙ(a₁) ^rot,xx B_x represent the 〈ψ_n(a ₁)|l̂_x(b ₂)|ψ_p(b ₂)〉 and 〈ψ_n(a ₁)|l̂_z(a ₂)|ψ_p(a ₂)〉 transitions, respectively]. Note that each σ _ψₙ(a₁) ^rot,yy B_y contribution reported in Table 5 includes grouped ψ_n(a ₁)→ψ_p(b ₁) transitions. For instance, (NHC) and [(NHC)H]⁺ have 9 ψ_n(a ₁), [(NHC)Au]⁺ has 13 ψ_n(a ₁), [(NHC)AuCl] and [(NHC)AuCO]⁺ have 18 ψ_n(a ₁) and so on.

Table 5.

Decomposition of the σ _ψₙ(a₁) ^rot B_jB_xyz component into σ _ψₙ(a₁) ^rot,i B_j contributions of the (NHC) carbon for (NHC), [(NHC)H]⁺ and [(NHC)AuL]^+/0 complexes; i = xx, yy, zz; j = x, y, z. All values are expressed in ppm

System	σ ψₙ(a₁) rot,xx Bx	σ ψₙ(a₁) rot,yy By	σ ψₙ(a₁) rot,zz Bz	σ ψₙ(a₁) rot BjBxyz
(NHC)	–34.815	–493.434	–287.778	–272.009
[(NHC)H]+	–37.163	–248.160	–300.292	–195.768
[(NHC)Au]+	–36.277	–334.339	–274.535	–215.154
[(NHC)AuH]	–35.356	–473.243	–217.348	–241.000
[(NHC)AuCNH]+	–35.839	–444.944	–122.448	–201.049
[(NHC)AuBr]	–35.428	–455.745	–208.234	–233.136
[(NHC)AuCl]	–35.366	–451.421	–192.749	–226.512
[(NHC)AuCO]+	–36.070	–428.864	–168.109	–211.115
[(NHC)AuXe]+	–72.197	–383.149	–190.684	–215.388
[(NHC)AuPyrrolyl]	–37.368	–502.401	–94.498	–207.435

As observed in Table 5, the σ _ψₙ(a₁) ^rot,yy B_y contribution varies considerably. When excluding (NHC), σ _ψₙ(a₁) ^rot,yy B_y and σ fairly correlate, with a R² of 0.92 (Figure 6). The linear regression coefficients for σ _ψₙ(a₁) ^rot,zz B_z and σ _ψₙ(a₁) ^rot,zz B_z with σ are 0.02 and 0.42, respectively. This clearly shows that rotational transitions of the type ψ_n(a ₁)→ψ_p(b ₁) dominate the variation of σ.

Figure 6.

Correlation between the σ _ψₙ(a₁) ^rot,yy B_y component and the isotropic shielding constant, σ, of the (NHC) carbon atom for (NHC), [(NHC)H]⁺ and [(NHC)AuL]^+/0 complexes.

Each σ _ψₙ(a₁) ^rot,yy B_y contribution depends on the matrix elements [〈ψ_n(a ₁)|l̂_y(b ₁)|ψ_p(b ₁)〉] and on the reciprocal of the energy difference between the occupied and virtual orbitals (∈_p – ∈_n). Ideally, the dominant transitions should be recognizable. However, for the systems under study several transitions contribute to σ _ψₙ(a₁) ^rot,yy B_y. For instance, selecting the ψ_n(a ₁)→ψ_p(b ₁) transitions which ratio between 〈ψ_n(a ₁)|l̂_y(b ₁)|ψ_p(b ₁)〉 and its corresponding ∈_p – ∈_n is greater than 1 (Equation 2), would include at most four transitions, as seen in Table 6. However, for some systems, this would leave out other non‐negligible transitions. For neutral complexes, selected ψ_n(a ₁)→ψ_p(b ₁) σ _ψₙ(a₁) ^rot,yy B_y transitions lead to around 50 % of the σ _ψₙ(a₁) ^rot,yy B_y contribution. For charged complexes, [(NHC)AuCO]⁺, [(NHC)AuXe]⁺ and [(NHC)Au]⁺ selected ψ_n(a ₁)→ψ_p(b ₁) σ _ψₙ(a₁) ^rot,yy B_y transitions cover a larger portion of the σ _ψₙ(a₁) ^rot,yy B_y contribution, about 75 %. In contrast, for [(NHC)AuCNH]⁺, selected ψ_n(a ₁)→ψ_p(b ₁) σ _ψₙ(a₁) ^rot,yy B_y transitions cover 47 % of the σ _ψₙ(a₁) ^rot,yy B_y contribution.

Table 6.

Largest σ _ψₙ(a₁) ^rot,yy B_y contributions to the σ ^yy tensor component of (NHC), [(NHC)H]⁺ and [(NHC)AuL]^+/0 in ppm, and their ∈_b₁ – ∈_a₁, in eV. 'Quota' stands for the ratio between 〈ψ_n(a ₁)|l̂_y(b ₁)|ψ_p(b ₁)〉 and its corresponding ∈_p – ∈_n, Δσ _ψₙ(a₁) ^rot,yy B_y stands for the difference between all σ _ψₙ(a₁) ^rot,yy B_y transitions and selected ψ_n(a ₁)→ψ_p(b ₁) σ _ψₙ(a₁) ^rot,yy B_y transitions. ∈_b₁

System	ψn(a 1)→ψp(b 1)	Quota	∈a₁	∈b₁	∈b₁ – ∈a₁	ψn(a 1)→ψp(b 1) σ ψₙ(a₁) rot,yy By	Δ σ ψₙ(a₁) rot,yy By
(NHC)	18 → 21	6.092	–4.818	0.039	4.779	–642.744	149.309
[(NHC)H]+	14 → 19	3.579	–16.837	–6.777	10.060	–289.038	40.858
[(NHC)Au]+	20 → 29	3.071	–15.463	–6.231	9.232	–212.058	–34.290
	26 → 29	2.953	–11.313	–6.231	5.082	–88.033
[(NHC)AuH]	20→ 29	3.373	–9.901	–1.319	8.582	–233.162	–240.081
[(NHC)AuCNH]+	20 → 37	2.106	–16.526	–4.701	11.825	–50.207	–233.784
	27 → 37	2.098	–12.790	–4.701	8.089	–160.953
[(NHC)AuBr]	35 → 46	3.444	–10.558	–1.712	8.846	–268.718	–187.027
[(NHC)AuCl]	26 → 37	3.371	–10.623	–1.675	8.948	–257.891	–193.53
[(NHC)AuCO]+	21 → 37	2.827	–16.893	–5.353	11.540	–58.638	–115.277
	27 → 37	2.223	–13.553	–5.353	8.200	–174.919
	27 → 42	1.930	–13.553	–3.547	10.006	–83.336
[(NHC)AuXe]+	29 → 42	3.050	–15.124	–5.628	9.496	–139.337	–69.361
	31 → 46	3.840	–12.956	–3.732	9.224	–154.450
[(NHC)AuPyrrolyl]	26 → 46	1.953	–12.122	–1.674	10.448	–41.721	–300.470
	31 → 46	1.133	–10.194	–1.674	8.520	–75.243
	31 → 52	1.254	–10.194	0.089	10.105	–47.015
	31 → 56	2.807	–10.194	1.265	11.459	–37.950

ψ_n(a ₁) and ψ_n(b ₁) involved in the σ _ψₙ(a₁) ^rot,yy B_y transitions are significantly delocalized, thus, a correlation between the electronic structure of the systems and the NMR parameters is not straightforward (Figure 7). The analysis of 〈ψ_n(a ₁)|l̂_y(b ₁)|ψ_p(b ₁)〉 may provide insights in this regard. The effect of the angular momentum operator l̂_y appears as a simple 90 degrees rotation of the orbital along the principal y axis.44 There is a significant contribution to σ only if the rotated orbital l̂_y(b ₁)ψ_p(b ₁) overlaps with an occupied orbital (ψ_n(a ₁)|). Figure 7 shows for (NHC), [(NHC)Au]⁺ and [(NHC)AuCl] the contour plots of ψ_n(a ₁) and ψ_n(b ₁) that give the most significant contribution to σ _ψₙ(a₁) ^rot,yy B_y. It is eye‐catching that across the systems ψ_n(a ₁) and ψ_n(b ₁) are very similar, particularly around the (NHC) carbon atom. The p_x character of the ψ_n(b ₁), being largely dominated by the electronic structure of (NHC), is expected to be preserved across the series. The p_z character of ψ_n(a ₁) is influenced by the σ‐acidity of the metal fragment. Note that for the selected transitions, ∈_p – ∈_n does not vary significantly. This suggests that the induced current density is mainly governed by the transition matrix elements rather than the orbital energy difference.

Figure 7.

Contour plots of ψ_n(a ₁) and ψ_n(b ₁) involved in the dominant ψ_n(a ₁)→ψ_p(b ₁) transition of (NHC), [(NHC)Au]⁺ and [(NHC)AuCl] complexes. Isodensity surfaces (±0.003 e/au³) are superimposed to the molecular structure of the complex.

Conclusions

In the framework of the CTOCD‐DZ formulation, the shielding tensor of the (NHC) carbon atom in [(NHC)AuL]^+/0 complexes was systematically analyzed. Emphasis was given to the anisotropic character of the shielding tensor, considering that only one of its three components, σ ^yy, significantly varies along the series. Progressive decomposition of the chemical shielding tensor into orbital contributions and a spectral decomposition revealed that its anisotropic character is governed by ψ_n(a ₁)→ψ_p(b ₁) rotational transitions. Along the series, not a single ψ_n(a ₁)→ψ_p(b ₁) rotational transition but a few transitions dominate the isotropic shielding. These transitions, so‐called σ _ψₙ(a₁) ^rot,yy B_y transitions, occur between occupied and virtual orbitals with strong p_z(a ₁) and p_x(b ₁) characters (symmetries), respectively. The σ‐donation component, associated to ψ_n(a ₁), correlates with the isotropic shielding. The π _∥ and π _⊥ back‐donation components associated to ψ_n(b ₁) and ψ_n(b ₂), respectively, have a negligible influence on the chemical shift because rotational transitions from ψ_n(b ₁) and ψ_n(b ₂) do not significantly contribute to the variation of the isotropic shielding. The carbon chemical shielding of [(NHC)AuL]^+/0 is mainly governed by the transition matrix elements rather than by the orbital energy difference, since the latter remains fairly constant. The presented orbital decomposition analysis contributes to clarify the relationship between the isotropic shielding and the nature of the [(NHC)–AuL]^+/0 bond, thus, it may be used for quantifying other magnetic properties for [(NHC)‐Au] based complexes or different substrate‐metal fragments.

Computational Methods

Geometry optimizations and electron densities of all structures were calculated using density functional theory (DFT). Specifically, the BLYP exchange correlation functional45, 46 in combination with the TZ2P basis set were used. Scalar and spin‐orbit relativistic effects were included via the zeroth‐order regular approximation (ZORA) as implemented in ADF.47, 48, 49, 50 The NMR constants were computed using the CTOCD‐DZ formulation,31 as implemented in the SYSMO code.35 An intermediate step between the optimized geometry and the actual calculation of the magnetic properties was needed. Starting from optimized structures, the perturbed Kohn–Sham orbitals31, 51, 52 were calculated, using GAMESS‐UK,53 with the BLYP exchange correlation functional45, 46 and the uncontracted (u) cc‐pV5Z basis set (for the basis set calibration, moving from Slater type orbitals within ADF to Gaussian type orbitals within GAMESS‐UK, see Supporting Information). For the calculations with GAMESS‐UK the scalar relativistic effects were included via effective core potentials, by using the energy‐adjusted pseudo potential developed by Figgen et al.54

Supporting information

Supporting Information