Relativistic DFT Calculations of Cadmium and Selenium Solid-State NMR Spectra of CdSe Nanocrystal Surfaces

Abstract

Solid-state NMR spectra have been used to probe the structure of CdSe nanocrystals and propose detailed models of their surface structures. Density functional theory (DFT)-optimized cluster models that represent probable molecular structures of carboxylate-coordinated surface sites have been proposed. However, to the best of our knowledge, ¹¹³Cd and ⁷⁷Se chemical shifts have not been calculated for these surface models. We performed relativistic DFT calculations of cadmium and selenium magnetic shielding tensors on model compounds with previously measured solid-state NMR spectra with (i) the four-component Dirac-Kohn–Sham (DKS) Hamiltonian and (ii) the scalar and (iii) spin–orbit levels within the ZORA Hamiltonian. Molecular clusters with Cd and Se sites in varying bonding environments were used to model CdSe (100) and CdSe(111) surfaces capped with carboxylic acid ligands. Our calculations identify the observed ¹¹³Cd isotropic chemical shifts δ(iso) of −465, −318, and −146 ppm arising from CdSeO₃, CdSe₂O₂, and CdSe₃O surface groups, respectively, with very good agreement with experimental measurements. The ¹¹³Cd chemical shifts linearly decrease with the number of O-neighbors. The calculated spans (δ₁₁ – δ₃₃) encompass the experimental values for CdSe₃O and CdSe₂O₂ clusters but are slightly larger than the measured value for CdSeO₃ clusters. Relativistic DFT calculations predicted a one-bond ¹¹³Cd–⁷⁷Se scalar coupling of 258 Hz, which is in good agreement with the experimental values of 250 Hz. With a dense coverage of carboxylic acid ligands, the CdSe (100) surface shows a distribution of Cd–Se bond lengths and J-couplings. Relativistic DFT simulations thus aid in interpretation of NMR spectra of CdSe nanocrystals and related nanomaterials.

Introduction

Semiconductor nanocrystal (NC) quantum dots (QDs) have novel tunable electronic and optical properties that depend on their size, shape, and surface passivation.¹⁻⁴ Additionally, QD and other NC solar cells exhibit multiple exciton generation (MEG)⁵⁻⁸ and are hence appealing for photovoltaic applications. NCs are often capped by ligands such as carboxylates, oleates, or phosphonates to prevent NC agglomeration. Furthermore, the extraction of photoexcited carriers from QDs is also critically controlled by their passivating ligands,⁹ a feature that is key to the performance of QD solar cells.¹⁰ For NCs that are a few nanometers in diameter, a large fraction of the atoms reside on surface sites. Surface ligands on NCs control their nucleation, growth, colloidal stability, and chemical reactivities.¹¹⁻¹⁷ Additionally, the surface sites strongly influence the electronic and optical properties of the NC. The importance of surface structure on the optical properties of NCs is well known, and undercoordinated surface atoms with localized electronic states significantly decrease both exciton lifetimes and photoluminescence quantum yields (PLQYs)^13,18,19 For example, carboxylate-capped CdSe NCs suffer a drop in PLQY when the ligand coverage drops below 3 ligands nm^–2.^1,15 A direct correlation between specific surface effects and nonradiative recombination processes has been proposed.¹⁴ It has also been hypothesized that edge and vertex sites on NC surfaces have different ligand binding energies and may be more active in ligand exchange and dissociation reactions.¹⁶ However, direct experimental probes of the molecular structure of NC surfaces are lacking. Hence, it is challenging to directly correlate the molecular level structure of NC surfaces to the photophysical properties.

Solid-state nuclear magnetic resonance (SSNMR) is a powerful tool to determine the structure of NC surfaces and sites on the surface where ligands can bind.²⁰⁻²² Additionally, dynamic nuclear polarization (DNP) is a promising method to enhance sensitivity and permit detection of the small fraction of atoms on the surface of NCs.²³⁻²⁷ Recently, DNP-enhanced ¹¹³Cd SSNMR spectroscopy was utilized to study the surfaces of zinc blende CdSe NCs with platelet or spheroidal morphologies, which were capped with carboxylic acid ligands.²⁸ The ⁷⁷Se and ¹¹³Cd cross-polarization magic angle spinning (CPMAS) NMR spectra displayed distinct signals from the surface atoms of the NCs and bulk atoms residing below the surface. 2D ¹¹³Cd CP-magic angle turning (MAT) spectra of CdSe nanoplatelets revealed the ¹¹³Cd isotropic chemical shifts (δ_iso) of −465, −318, and −146 ppm with spans (Ω) of 402, 472, and 364 ppm, respectively. The peak with an isotropic shift of −318 ppm was the most intense. By correlating NMR spectra with the measured infrared (IR) spectra,²⁹ it was suggested that the δ_iso features of −465, −318, and −146 ppm may be assigned to CdSe₃O, CdSe₂O₂, and CdSeO₃ coordination environments for Cd on the surface of the NC. The dominant configuration was suggested to arise from CdSe₂O₂ from a chelating termination on the CdSe(100) NC surface.²⁸ Although these NMR spectral assignments are very reasonable and consistent with measurements, quantum chemical calculations of ¹¹³Cd or ⁷⁷Se magnetic shieldings are lacking. Previous calculations of ¹³C NMR and infrared spectra of the carboxylate ligands were performed and compared with experimental ¹³C NMR spectra.²⁹

Accordingly, to provide an insight into the Cd and Se bonding environments at NC surfaces, we present relativistic ab initio simulations of the electronic properties and NMR spectra. The ab initio calculations aid interpretations of the observed NMR spectra and the local bonding environments. Furthermore, we compare different levels of relativistic calculations on the calculated ¹¹³Cd and ⁷⁷Se magnetic shieldings and chemical shifts.

Results and Discussion

Computational Methods and Magnetic Shielding Calibration Curves

Prediction of electronic properties and NMR parameters for lighter elements are routinely performed with density functional theory (DFT) using Gauge Including Projector Augmented Waves (GIPAW)-based solutions of the ZORA Hamiltonian³⁰ with pseudopotentials and a plane wave basis for electronic wave functions. The well-established pseudopotential plane wave approach implemented in the CASTEP package³¹ and other plane wave DFT codes has been very successful for calculating the NMR parameters of lighter elements. However, the plane wave pseudopotential approach is inadequate for heavier elements as relativistic effects are needed to accurately calculate the shieldings of heavy atoms (including Cd and Se). Thus, we have compared results from two relativistic approaches provided by the four-component (4-c) Dirac-Kohn–Sham (DKS) equation in ReSpect^32,33 and the ZORA Hamiltonian with spin–orbit/scalar effects in the Amsterdam Density Functional (ADF)³⁴⁻³⁷ packages. The ZORA Hamiltonian with spin–orbit effects has been very successful in simulating the NMR parameters of heavy atoms including Tl–Pt complexes^38,39 and Ce^IV complexes.⁴⁰

The solution of the four-component DKS equation with relativistic spin–orbit couplings is provided by the ReSpect electronic structure package with atom-centered Gaussian orbital basis sets and gauge-included atomic orbitals (GIAO) for magnetic shielding calculations. Utilizing the Dirac equation is considered more accurate than incorporating relativistic corrections to the Schrodinger equation.³² Furthermore, utilizing a fully relativistic formalism provides more accurate NMR parameters not only for heavy atoms but also for the neighboring light atoms that are affected through relativistic spin–orbit effects.^32,41 A special feature in the 4-c DKS theory implemented in ReSpect is the restricted magnetically balanced (RMB) basis sets for the small component of the 4-c wave function.^42,43 The RMB basis avoids strong basis set dependence and yields good agreement with experimental data on shielding of halide molecules.⁴² The use of the RMB basis was extended to incorporate the GIAO approach,^43,44 and implemented in the ReSpect package.

The solution of the ZORA Hamiltonian with scalar relativistic and spin–orbit effects in the ADF package is an alternative approach.³⁴⁻³⁶ Here, we use both computational approaches to develop calibration relations to correlate the computed relativistic magnetic shieldings of ¹¹³Cd and ⁷⁷Se and compare them with chemical shift tensor parameters from experimental solid-state NMR data.

¹¹³Cd Calibration Curves

Holmes and Schurko (HS)⁴⁵ performed comprehensive calculations of the ¹¹³Cd magnetic shielding tensors of 30 sites in Cd-molecular structures with all electron methods, DFT functionals, and relativistic treatments and compared them with measured chemical shift tensors to obtain calibration relations. We utilize this comprehensive database of measured chemical shifts to calculate magnetic shieldings using different theoretical methods in this paper. We directly used the geometry-optimized coordinates of Cd-molecules and calculated the magnetic shielding tensors in (i) the four-component DKS formalism (in ReSpect), (ii) the scalar relativistic level with the ZORA Hamiltonian (in ADF), and (iii) with spin–orbit coupling within ZORA Hamiltonian (in ADF), offering a direct comparison between these methods. In order of decreasing isotropic shielding, the molecular structures ranged from Cd coordinated with O8, O6, O5, N2O4, N2O2I2, Cl6, O2N4, S4, Se4, and C2 (Supporting Information: Figures S1 and S2). The measured chemical shifts as a function of the calculated isotropic magnetic shieldings (σ(iso)) are plotted in Figure 1a. Additional information on the molecular structures, charges, shielding tensors, and shielding anisotropy is given in Table S1. The 91-atom Cd[N(Pri₂PSe)₂]₂ with Cd tetrahedrally coordinated by 4 Se was the largest molecule calculated (with 1947 large component basis functions in the ReSpect DKS formalism).

Figure 1.

(a) Experimental chemical shifts, δ(iso) for ¹¹³Cd in cadmium molecules relative to dimethyl cadmium (Me₂Cd) as the reference, as a function of the calculated isotropic magnetic shieldings, σ(iso), of ¹¹³Cd in cadmium molecules. Calculations with 4-c DKS theory (red squares) are compared with scalar ZORA (blue diamonds) and spin–orbit ZORA (green squares). The straight lines represent the best fits. The coordination of the Cd atom is shown on the right for the various compounds. (b) Experimental span (Ω = δ₁₁ – δ₃₃) for ¹¹³Cd in cadmium molecules as a function of the calculated span with 4-c DKS theory and spin–orbit ZORA. The straight line indicates perfect agreement.

We have augmented the HS database, with geometry-optimized dimethyl cadmium and diethyl cadmium, for which the measured chemical shifts in the gas phase and neat forms⁴⁶ are tabulated (Table S1). This provides shielding for weakly shielded Cd in a C2 coordination environment (Table S1). Details of the computational methods are given in the Methods section. We use dimethyl cadmium taken as the standard ¹¹³Cd chemical shift.²³ The experimentally measured ¹¹³Cd chemical shifts for CdSe NCs discussed below were also referenced relative to dimethyl cadmium.

The calculated isotropic shieldings ranged from σ(iso) = (4850, 4500) ppm (ReSpect, ADF SO) for the highly shielded O8-coordinated Cd in cadmium nitrate tetrahydrate (Cd(NO₃)₂·4H₂O) with a measured chemical shift δ(iso) = −103 ppm; to the least shielded Me₂Cd with σ(iso) = (3908, 3494) ppm; measured δ(iso) = 705 ppm. The ReSpect DKS shieldings are the largest, with the spin–orbit ADF ZORA shieldings ∼360 ppm lower, followed by the ZORA-scalar relativistic shieldings, which are ∼520 ppm lower than the spin–orbit result. This trend is similar to previously calculated ¹¹⁹Sn shieldings in Sn(Me)₄,⁴⁷ which found scalar ZORA and spin–orbit ZORA shieldings to be 466 and 916 ppm lower, respectively, than the 4c DKS value. For all the three computational methods, the calculated isotropic chemical shifts [δ(iso)] linearly vary with the calculated isotropic shielding [σ(iso)] with the best fit calibration lines given by δ(iso) = C – ασ(iso) (with C and α in Table 1). ReSpect DKS, spin–orbit ZORA ADF, and scalar ZORA ADF calculations yield slopes of 0.91, 0.836, and 0.829, respectively. Consequently, the spin–orbit and scalar ZORA lines are nearly parallel to each other.

Table 1. Calibration Parameters (C, α) in the Relation δ(iso) = C – ασ(iso) for Predicted Isotropic Chemical Shifts δ(iso) from Calculated Isotropic Shielding σ(iso) for the Different Methods^a.

113Cd (reference Me2Cd)	C (ppm)	α	mean residual difference (ppm)
ReSpect 4-comp DKS	3526.4	0.908	37
scalar ZORA (ADF)	2470.3	0.836	49
spin–orbit ZORA (ADF)	2876.1	0.829	21
77Se (reference Me2Se)
RESPECT 4-comp. DKS	1787.5	0.896	46
spin–orbit ZORA (ADF)	1753.7	0.925	27
scalar ZORA (ADF)	1621.4	0.951	23
DFT-GIPAW (CASTEP)	1453.6	0.891	23
Griffin et al. [ref (48)]	1626.0	1.02
DFT-GIPAW (CASTEP)

^aCalculations for ¹¹³Cd and ⁷⁷Se are listed in the upper and lower portions of the table. The DFT ZORA and results from ref (48) are included for ⁷⁷Se.

The measured chemical shift tensors δ_ii, have been tabulated.⁴⁵ Accordingly, we plot the experimental spans (δ₁₁–δ₃₃) as a function of the calculated span (Figure 1b). The same plot, but for the Haeberlen anisotropy, is shown in Figure S3.

⁷⁷Se Shielding Calibration Curves

Wasylishen and co-workers⁴⁹ have previously performed ZORA DFT calculations of ⁷⁷Se chemical shifts on a variety of organic and inorganic selenium compounds using ADF. Ashbrook and co-workers⁴⁸ recently used planewave GIPAW DFT calculations to predict ⁷⁷Se chemical shifts for a similar set of compounds. Maaninen et al.⁵⁰ utilized plane wave DFT-GIAO calculations to predict ⁷⁷Se chemical shifts for Se–N compounds. We performed calculations on a set of small Se-containing molecules [(CH₃)₂Se, H₂Se, C₄H₄Se, H₂SeO₃, SeOCl₂, and (Ph₂)Se₂] for which the chemical shifts have been measured.⁵¹ The molecular structures were relaxed with CASTEP plane wave DFT (utilizing GGA exchange–correlation with the PBE functional, ultrasoft pseudopotentials with Koelling–Harmon relativistic treatment, and 450 eV plane wave cutoff). The relativistic shieldings of these geometry-optimized structures were calculated (Figure S4). The experimental chemical shifts are relative to those of the reference compound (CH₃)₂Se. The computed shieldings range from 283 ppm for the weakly shielded SeOCl₂ to 2289 ppm for strongly shielded H₂Se (Figure 2 and Table S2). The calculated magnetic shieldings vary linearly with the experimentally measured chemical shifts (δ) with the calibration line δ = 1787.5–0.896 σ in four-component DKS (Figure 2). The spin–orbit ZORA level calibration equation is δ = 1753.7–0.925, very close to the 4-c DKS result. The ZORA-scalar and Griffin et al. results⁴⁸ are close (Figure 2) to each other. The ZORA-scalar result compares well with previous calibration curves (δ = 1863.4 – 1.01σ′) for Se–N compounds.⁵⁰ Relativistic methods show agreement between calculated and measured ⁷⁷Se chemical shifts in organic and inorganic selenium compounds.⁴⁹

Figure 2.

Experimental chemical shifts δ(iso) as a function of the calculated ⁷⁷Se isotropic magnetic shieldings in small molecules with Se. Our calculations with four-component Dirac-Kohn–Sham (red), ADF-scalar ZORA (green), and spin–orbit ZORA (black) are compared with our DFT-GIPAW calculations. The lines represent the best linear fits to the data. The experimental chemical shifts are relative to the reference compound (CH₃)₂Se.

We have also calculated shieldings with first-principles DFT calculations utilizing GIPAW for the geometry-optimized molecular structures using CASTEP. The nonrelativistic DFT-GIPAW shieldings (σ′) are uniformly ∼360 ppm lower than our calculated values, but the calibration line δ = 1453.5 – 0.891σ′ has nearly the same slope. The uniformly shifted calibration line is typical of calculated shielding with and without spin–orbit contributions. The calibration relations are summarized in Table 1.

We calculated the chemical shielding anisotropy (CSA), where CSA = σ₃₃ – 0.5(σ₁₁ + σ₂₂) with σ₃₃ > σ₂₂ > σ₁₁. The calculated CSA with relativistic 4-c DKS agrees very well with the DFT-GIPAW calculations (Figure S5) with differences of 20–90 ppm (Table S3), which suggests that the simpler DFT-GIPAW calculations may reasonably estimate chemical anisotropies.

It is of interest to compare the chemical shieldings on the protons that are near-neighbors of Se with the 4-c DKS and DFT-GIPAW. The calculated shieldings with these two approaches agree well (within ∼1 ppm) for most of the molecules considered, where the protons are second neighbors of Se. The notable exception is H₂Se, where the protons are strongly coupled nearest-neighbors of Se, showing a difference of 3 ppm between the two methods (Figure S6). This is consistent with the results from Pb/Sn hydrides, where very large NMR chemical shifts of up to +80 ppm were found for H atoms bonded to Pb or Sn.⁵² The extreme ¹H chemical shifts in the hydrides arise from the spin–orbit coupling at the heavy atom (HA) mixing triplet states into the ground-state wave function of the molecule, thereby introducing additional shielding at the light atom (LA), referred to as HALA effects.³²

Modeling ¹¹³Cd Solid-State NMR Spectra of NC Surfaces

Recently, DNP-enhanced ¹¹³Cd SSNMR spectroscopy was utilized to study surfaces of zinc blende CdSe NCs with platelet or spheroidal morphologies, which were capped with carboxylic acid ligands.²⁸ NMR spectra displayed distinct signals from surface atoms of the NCs and bulk atoms residing below the surface. 2D ¹¹³Cd CP- magic angle turning (MAT) spectra of CdSe nanoplatelets revealed the ¹¹³Cd isotropic chemical shifts δ(iso) of −465, −318, and −146 ppm with spans (Ω) of 402, 472, and 364 ppm, respectively (Figure 3). The peak with δ(iso)_iso = −318 ppm was the most intense.

Figure 3.

DNP-enhanced surface and core ¹¹³Cd and ⁷⁷Se solid-state NMR spectra of CdSe NCs obtained with an MAS frequency of 10 kHz. (a,b) 2D ¹¹³Cd CP-MAT spectra of CdSe nanoplatelets (a) and nanospheroids (b) where the dashed lines indicate distinct ¹¹³Cd NMR signals. Individual rows were extracted from the CP-MAT 2D spectrum showing ¹¹³Cd sideband manifolds associated with isotropic core and surface NMR signals. (c) ¹¹³Cd CP-CPMG and CP pulse cooling spectra of CdSe nanospheroids; ⁷⁷Se CP-CPMG spin diffusion spectra of CdSe nanospheroids, with spin diffusion times of 0 and 600 s. (d) Representative models of carboxylate-terminated CdSe{100} and {111} facets. Note: Models were obtained without DFT structural optimization. Reprinted (adapted) with permission from Chen et al. J. Am. Chem. Soc.2021, 143, 8747–8760. Copyright (2021) American Chemical Society.

Here, we create computational models of the local Cd and Se environments present on CdSe NC surfaces passivated by carboxylic acid ligands. Similar structural models were developed by Zhang et al.²⁹ for CdSe surfaces and the calculation of ¹³C chemical shifts of carboxylic acid terminated CdSe. We note that Copéret and co-workers previously used [Cd(SeH₂)_4–n(OH₂)_n]²⁺ cluster models and ADF ZORA SO calculations to predict the ¹¹³Cd chemical shifts as the number of selenium and oxygen atoms were varied in the Cd coordination sphere.⁵³ They also measured δ(iso) = −310 ppm with Ω = 250 ppm for a carboxylate-capped CdSe NC.⁵³ Recently, Copéret and co-workers have measured ¹¹³Cd chemical shifts on amine-capped CdSe NCs and performed DFT simulations of surface conformations with cluster models.⁵⁴ They also obtained δ(iso) = −304 ppm with Ω = 420 ppm for a carboxylate-capped CdSe NC.⁵⁴ We have utilized our calibration relations to calculate the relativistic magnetic shielding tensors of ¹¹³Cd in CdSe surface models (Table 1).

For the CdSe(100) surface, we generated periodic unit cells passivated with one or three carboxylic acid ligands (Figure 4a,b). To simplify the calculations, acetate ligands were used in place of oleate. We geometry-optimized these structures in CASTEP under the constraint of keeping surface (Cd and Se) in their ideal positions (Figure 4a, charge, q = −1, and Figure 4b, q = −3). We extracted the smallest cluster containing a single acetate group (Figure 4a), capping each outer Se atom with two hydrogen atoms. Geometry optimization of the Se-bound protons resulted in a 14-atom cluster with Cd in CdSe₂O₂ coordination with charge q = +1 (Figure 4c). Similarly, the cluster with three acetate groups was extracted from the surface model (Figure 4b), resulting in a 34-atom cluster with hydrogen terminations (SeH₂ or SeH) and CdSe₂O₂ coordination for each Cd and charge q = +1 (Figure 4d).

Figure 4.

Geometry-optimized configurations of Cd on CdSe(100) and (111) surfaces. (a) CdSe(100) with a single acetate ligand and CdSe₂O₂. (b) CdSe(100) with three acetate ligands. (c) 14-Atom CdSe₂O₂ cluster. (d) 34-Atom CdSe₂O₂ cluster. (e) CdSe₃O cluster with a bridging acetate. (f) CdSeO₃ cluster with the bridging acetate group and (g) 56-Atom CdSe₂O₂ cluster.

For CdSe(111), we similarly started with a geometry-optimized surface model and generated a cluster with a single acetate group bridging two surface Cd, resulting in a 24-atom cluster with each Cd in CdSe₃O coordination and SeH₂ terminations and q = +3 (Figure 4e). We then constructed a model with each Cd bonded to three O and one Se with q = +1 (Figure 4f). Here, one acetate group bridges neighboring Cd atoms, while another acetate group is bonded to each Cd, resulting in a CdSeO₃ coordination geometry (Figure 4f).

The largest cluster utilized was a 56-atom cluster with CdSe₂O₂ coordination in which each subsurface Se was bonded to 4 Cd with q = −3 (Figure 4g). S-capping atoms were employed to reduce the computational workload.

We computed the chemical shielding tensors of CdSe_nO_4-n sites in each cluster with the three aforementioned relativistic methods (Table 2). Further geometry optimization performed with scalar/SO ZORA (ADF) did not significantly change ¹¹³Cd shieldings. From the isotropic shielding σ(iso), we infer the predicted chemical shifts using the calibration relations established for the different methods (Table 1). the

Table 2. Calculated Isotropic Shieldings σ(iso) and Chemical Shifts δ(iso) for ¹¹³Cd for the Different Models with Different Cd Coordination Environments Compared with Measurement^a.

model (from Figure 4)	no. atoms	calc. 4c DKS σ(iso) ppm	calc. 4c DKS δ(iso) ppm	calc. scalar ZORA σ(iso) ppm	calc. scalar ZORA δ(iso) ppm	calc. s-o ZORA σ(iso) ppm	calc. s-o ZORA δ(iso) ppm	exp, δ(iso) ppm
CdSeO3 (f)	29	4477	–479	3583	–462	4108	–465	–465
″	29	4440	–445	3549	–433	4073	–437	–465
CdSe2O2 (c)	14	4358	–371	3435	–338	3979	–358	–319
CdSe2O2 (d)	34	4355	–368	3392	–302	3966	–347	–319
		4286	–306	3350	–267	3893	–287
		4354	–367	3398	–307	3964	–346
CdSe2O2 in CdSe(S) (g)	56	4469	–471	3362	–276	3934	–321	–319
CdSe3O (e)	24	4092	–130	3145	–95	3703	–129	–146
″	24	4137	–170	3190	–132	3752	–170	–146
Se (bulk) in CdSe2O2 CdSe(S) (g)	56	2613	–554	2274	–543			–620
		2601	–543	2280	–549

^aTable note: s-o denotes spin–orbit.

We plot the calculated isotropic chemical shifts δ(iso) for ¹¹³Cd for the CdSe₃O, CdSe₂O₂, and CdSeO₃ clusters and compare with the experimental data derived from CdSe nanoclusters (Table 2 and Figure 5). The predicted δ(iso) values encompass the measured δ(iso) for each O-coordination, exhibiting very good agreement between the theory and experiment for all methods. The chemical shifts decrease linearly as the O-coordination around Cd increases (Figure 6) since the Cd is shielded by the electronegative O neighbors. The calculations are at most ∼20–50 ppm lower than the experimental values. This agreement also validates our scheme of building clusters from the periodic structures of CdSe surfaces. It suggests that small (14–34 atom) clusters with SeH₂ and SeH terminations are a reasonable representation of the local structure of CdSe NCs.

Figure 5.

(a) Measured ¹¹³Cd isotropic chemical shifts δ(iso) for CdSe nanoclusters passivated with carboxyl acid ligands compared with calculated values from four-component DKS (ReSpect) with increasing O-coordination (N) on Cd. The larger 34-atom CdSe₂O₂ cluster is shown as green circles. The line represents the best linear fit to the experimental data δ(iso) = 10.8–159.5N. Multiple points for each configuration represent the inequivalent Cd sites. (b) Scalar ZORA (ADF) calculated isotropic chemical shifts compared with measurement using the same conventions. (c) Spin–orbit ZORA (ADF) calculations compared with experimental measurements.

Figure 6.

Calculated spans Ω = δ₁₁ – δ₃₃ for the CdSe clusters with an increasing number of oxygen neighbors. Calculations are with (a) 4-c DKS, (b) scalar ZORA, and (c) spin–orbit ZORA methods.

The calculated spans (Ω = δ₁₁ – δ₃₃) bracket the experimental values for CdSe₃O and CdSe₂O₂ clusters but are slightly larger (80–160 ppm) than the measured value for CdSeO₃ (Figure 6). Calculated values in scalar ZORA (Figure 6b) are higher than the measured spans. The accompanying calculated chemical shielding anisotropy (CSA), δ₃₃ – 0.5(δ₁₁ + δ₂₂), of the Cd molecules and CdSe_nO_4–n clusters are shown in Figures S3 and S7. The calculated shieldings of the CdSe_nO_4–n clusters with the different theoretical methods are shifted by nearly constant values as shown in Figure S8, with the highest shieldings for 4-c DKS, followed by spin–orbit ZORA and scalar ZORA methods. This is similar to the trends found for Cd-molecules in Figure 1a.

⁷⁷Se NMR Spectra

The 56-atom cluster (Figure 4g) has a surface Cd atom bonded to two Se atoms, which are in turn coordinated with four Cd neighbors. These two Se sites have the same local bonding environment that should be present in surface Se in CdSe(100) NCs and therefore provide a representation of surface Se. With our ⁷⁷Se calibration relations (Table 1), the calculated ⁷⁷Se relativistic shieldings for these two sites lead to chemical shifts of −543 and −554 ppm (Table 2) in comparison with measured surface ⁷⁷Se chemical shifts of −620 ppm.²⁴ Although the first neighbor shell of Se in the 56-atom cluster (Figure 4g) and the CdSe(100) surface are the same, there are significant differences in the second- and third-neighbor coordination shells between the cluster and the surface. These differences may lead to the difference between the calculated and experimental δ(iso) values.

¹³C NMR Spectra

The calculated ¹³C isotropic shieldings σ(iso) (Table S4) with 4-c DKS (ReSpect) show that the C bonded to the acetate group (–COO) is weakly shielded (−13 to −16 ppm), whereas the C in the methyl group is strongly shielded (159–166 ppm). We utilized the ¹³C calibration relation δ(iso) = −167.3 – 0.9526·σ(iso) derived earlier for magnetic shielding calculations of small organic molecules correlated with measured chemical shifts.⁵⁵ We then obtained calculated chemical shifts of 177–184 ppm [average δ(iso) = 181 ppm] for C in the CdSe₂O₂–COO ligand (Table S4) that are in very good agreement with the measured value of 184 ppm.²⁹ We also obtain δ(iso) = 7–16 ppm [average δ(iso) = 12 ppm] for the C in the methyl group also in very good agreement with the measured value of 10–40 ppm²⁹ and consistent with known ¹³C chemical shifts of organic compounds.⁵⁶

J-Coupling

Indirect scalar (J-) couplings between ¹¹³Cd and ⁷⁷Se nuclear spins were previously measured with ⁷⁷Se–¹¹³Cd J-HMQC and J-resolved experiments.²⁸ These measurements revealed that for core Cd and Se sites, the J-coupling was on the order of 75–100 Hz, while for surface selenium and cadmium spin pairs, J-couplings of 100 and ca. 250 Hz were observed.

To understand the origin of the observed J-couplings, we performed calculations on the simple CdSe₂O₂ acetate model (Figure 4c). We calculated the J-couplings with a hybrid functional within the ZORA framework and incorporating spin–orbit interactions. This yielded J-couplings of 257 Hz (Table S5) between the surface-Cd and the two back-bonded Se (Figure 7a), in excellent agreement with the measured values of 250–255 Hz. For comparison, the 4c DKS formalism for this CdSe₂O₂ acetate model yielded J(iso) = 327 Hz, ∼28% larger than the experiment (Table S5). These results suggest that the hybrid functional is a good representation of the indirect J-coupling between the nuclear spins. This conclusion is supported by extensive studies of DFT approximations that found hybrid functional families provided the most accurate results.⁵⁷

Figure 7.

(a) CdSe₂O₂ model showing the J-couplings between surface ¹¹³Cd and bonded ⁷⁷Se ( arrows). (b) Geometry-optimized structure of a CdSe₂O₂ model obtained from a CdSe(100) surface densely covered with acetate ligands with bond-lengths and J-couplings. (c) Geometry of CdSeO₃ on the CdSe(111) surface, showing bond lengths and J-couplings.

We also investigated the effect of disorder in the Cd–Se bond lengths in a larger 105-atom CdSe(100) model in which all the eight Cd-surface atoms had carboxyl acid ligands corresponding to a dense coverage of 5.4 ligands/nm², as inferred in experimental studies.²⁸ This larger model yielded several distorted surface Cd configurations with asymmetric Cd–Se and Cd–O bond lengths (Table 3 and Figure S9). The surface undergoes structural distortions that allow the spacing between the ligands to increase, resulting in locally distorted bonding geometries at the surface Cd sites (Table 3). The shorter/longer Cd–Se bond lengths result in a distribution of Cd–Se J-couplings, with shorter Cd–Se bonds yielding slightly larger calculated Cd–Se J-couplings of 280 and 289 Hz (Figure 7b and Table S5). These calculations suggest that distorted Cd environments on CdSe(100) cannot account for the second Cd–Se J-coupling of ca. 100 Hz observed in HMQC experiments.

Table 3. Bond Lengths at the Cd Sites of the CdSe(100) Surface Are Densely Packed with Acetate Ligands.

Cd site	CdSe bond lengths (Å)	Cd O bond lengths (Å)
Cd 1	2.643, 2.657	2.244, 2.358
Cd 5	2.577, 2.947	2.32, 2.30
Cd 8	2.559, 2.633	2.387, 2.48
Cd 9	2.60, 2.839	2.213, 2.405
Cd 12	2.587, 2.758	2.254, 2.336
Cd 13	2.629, 2.660	2.30, 2.304

Since CdSe NCs/nanoplatelets also have (111) facets, we also examined the J-couplings of surface Cd–Se in the CdSeO₃ bonding configuration expected on CdSe(111) (Figure 4f). Note that we have been unable to measure these J-couplings experimentally since the NMR signals from (111) facets are less intense than those from (100) facets. The calculated J-couplings ¹J(⁷⁷Se_surf–¹¹³Cd_surf) between surface Cd and its Se neighbor are 111 and 156 Hz (Figure 7c and Table S5). Future NMR experiments should address whether CdSe(111) does indeed have weaker J-couplings than CdSe(100). Furthermore, the ¹J(⁷⁷Se_surf–¹¹³Cd_surf) couplings between second-neighbor pairs in the larger 34-atom CdSe₂O₂ model (Figure 4d) were negligible (<10 Hz); thus, further neighbor Cd–Se pairs likely do not account for the second J(iso) = 85 Hz peak. Also, previously measured ¹J(⁷⁷Se–¹¹³Cd) ∼ 120–135 Hz⁵⁸ for the Cd[N(Pri₂PSe)₂]₂ molecule with tetrahedral Cd–Se₄ coordination is more similar to the measured ¹J(⁷⁷Se_core–¹¹³Cd_core) = 105 Hz from core Cd–Se sites.

We used calculations to also predict the two-bond ¹J couplings between surface Cd and C in CdSe₂O₂ on CdSe(100) (in the acetate group; Figure 4a). These calculations gave an average ¹¹³Cd–¹³C J-couplings of 69 and 76 Hz, using the hybrid ZORA and 4-c DKS calculations, respectively (Table S5). Again, these couplings have not been experimentally measured, but the calculations suggest that they should be measurable and should be the target of future studies. The predicted results for the wurtzite CdSe(010) surfaces are summarized in the Supporting Information (Figure S10 and Table S6).

Conclusions

We have performed all-electron relativistic DFT calculations of cadmium and selenium magnetic shielding tensors with (i) the four-component Dirac-Kohn–Sham (DKS) Hamiltonian, (ii) the scalar level ZORA Hamiltonian, and (iii) spin–orbit level within the ZORA Hamiltonian. Calculations for Cd-containing molecular clusters with varying bonding environments and coordinations (C2, C4, Se4, S4, N2O4, N4O2, N2O2I2, O6Cl2, O6, and O8) were used to generate a calibration relation between magnetic shielding tensors and chemical shifts for ¹¹³Cd. We similarly simulated magnetic shielding tensors in Se-containing molecules with different bonding environments (C2, H2, O3, Se, and C), which generated calibration relations between magnetic shielding tensors and chemical shifts for ⁷⁷Se. The four-component DKS Hamiltonian offered the highest shielding, followed by spin–orbit ZORA and scalar ZORA methods. The linear calibration for each method had similar slopes and was linearly shifted relative to each other for ¹¹³Cd and ⁷⁷Se.

Geometry-optimized cluster models were made to represent CdSe(100) and CdSe(111) surfaces capped with carboxylic acid ligands. We utilized our relativistic DFT calibration to identify the SSNMR observed ¹¹³Cd isotropic chemical shifts δ(iso) of −465, −318, and −146 ppm to CdSeO₃, CdSe₂O₂, and CdSe₃O surface groups. Relativistic DFT shows a linear decrease of ¹¹³Cd chemical shifts δ(iso) with the number of O-neighbors, as Cd becomes more shielded, with very good agreement between calculation and measurement. The calculated spans bracket the experimental values for CdSe₃O and CdSe₂O₂ clusters but are slightly larger (80–160 ppm) than the measured value for CdSeO₃ clusters. From the calculated surface ⁷⁷Se shieldings in a larger 56-atom cluster in which the surface Se have a similar bonding environment as in a CdSe(100) NC, we calculated δ(iso) for ⁷⁷Se of −542 to −557 ppm, which compare with the measured value of −620 ppm for surface ⁷⁷Se in CdSe NCs. Calculations also reproduced the one-bond ¹¹³Cd–⁷⁷Se J-couplings. The hybrid functional within the ZORA is most accurate for the ¹J couplings, although the shieldings within the hybrid formalism and the nonhybrid spin–orbit levels of theory are very similar. Taken together, the ability of relativistic DFT calculations to construct cluster models of surface species and accurately predict NMR parameters for heavy nuclei can be a powerful tool to probe the surface structure in a variety of inorganic semiconductor NCs.

Methods

The ReSpect all-electron calculations were performed with (i) the standard dyall-vdz 21s14p10d2f basis set with 127 Gaussian-type orbitals (GTOs) for Cd; (ii) [14s11p6d] basis for Se (77 GTOs); (iii) dyall-vdz 10s6p1d basis set for C (33 GTO’s), and (iv) upc-1 4s1P orbitals for H (7 GTO’s). The Cd[N(Pri₂PSe)₂]₂ 91-atom molecule had 1947 Cartesian GTOs—nearly the largest feasible structure in ReSpect. The nonrelativistic simulations were used as an input to the full four-component Dirac equation in ReSpect. The relativistic Dirac Kohn–Sham (DKS) formalism utilized the PBE0 density functional.¹⁵ Self-consistent field calculations converged within 30 steps for energies and chemical shielding tensors.

The ADF simulations utilized the TZ2P-J all-electron basis for all elements, generalized gradient approximation (GGA) and PBE density functionals for exchange and correlation. The scalar ADF and spin–orbit coupling calculations were performed with a ZORA Hamiltonian. CdSe fragments were geometry-optimized. The hybrid functional used a PBE0 functional with 25% Hartree–Fock exchange and was more computationally intensive. All relativistic DFT simulations were performed on parallel computing clusters using 8–64 cores. The spin–spin J-coupling was the sum of the diamagnetic, paramagnetic, Fermi contact (FC) and spin diffusion (SD) contributions, with the FC + SD contributions being the dominant one.

Glossary

Abbreviations

DNP

dynamic nuclear polarization

SSNMR

solid-state nuclear magnetic resonance

DFT

density functional theory

DKS

Dirac-Kohn–Sham

4c

four component

ZORA

zero-order regular approximation

ReSpect

relativistic spectroscopy DFT program package

ADF

Amsterdam Density functional package

GGA

generalized gradient approximation

TZ2P

triple-ζ with two polarization functions

HMQC

heteromolecular quantum correlation

PBE

Perdew–Burke–Ernzerhof

GGA

generalized gradient approximation

CP

cross-polarization

CPMG

Carr–Purcell–Meiboom–Gill

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.3c07680.

• Details of molecular structures (PDF)

The authors declare no competing financial interest.