Simulation of Indirect ¹³C–¹³C J-Coupling Tensors in Diamond Clusters Hosting the NV Center ^†

Abstract

The full tensors ⁿJ_KL (K,L = X,Y,Z), describing n-bond J-coupling of nuclear spins ¹³C in H-terminated diamond-like clusters C₁₀H₁₆ (adamantane) and C₃₅H₃₆, as well as in the cluster C₃₃[NV⁻]H₃₆ hosting the negatively charged NV⁻center, were simulated. We found that, in addition to the usually considered isotropic scalar ⁿJ-coupling constant, the anisotropic contributions to the ⁿJ-coupling tensor are essential. We also showed that the presence of the NV center affects the J-coupling characteristics, especially in the case of ¹³C–¹³C pairs located near the vacancy of the NV center.

1. Introduction

In the past decade, there was rapid progress in the development of quantum magnetic sensing technologies based on nitrogen-vacancy (NV) color centers in diamond (e.g., see [1,2] for recent reviews). A magnetometer based on a single NV center can have nanometer-scale spatial resolution and exceptional sensitivity (up to ~Hz) allowing the detection of target single ¹³C nuclear spins or coupled ¹³C–¹³C pairs located within the diamond, which can be used as long-lived quantum memory [3]. Moreover, an NV-based magnetometer allows to distinguish (by chemical shifts) inequivalent nuclear spins of molecules located at diamond surface [4]. This enables a new exciting application area of single-spin nuclear magnetic resonance (NMR) to investigate important issues ranging from determination of molecular structures of inorganic/biological compounds up to medical imaging for therapeutic matters. In these respects, predicting high-resolution NMR characteristics for studied spin systems is essential. Among them, the characteristics of indirect nuclear spin–spin coupling (J-coupling) that arise due to second-order hyperfine interactions with electrons from chemical bonds connecting nuclei are important. Generally, a second-rank tensor ⁿJ_KL(K,L = X,Y,Z) is required to fully describe J-coupling between two nuclei [5]. However, until recently, most high-resolution NMR experiments were focused on measuring only isotropic scalar constant ⁿJ_iso = SpⁿJ_KL/3 because the anisotropic parts of the J-tensor were averaged out to zero by fast molecular motion in solution-state NMR or fast magic-angle spinning (MAS) in solid-state experiments [6,7,8]. Meanwhile, in the case of crystalline solids, the constituent atoms are located in a certain order determined by the crystal structure so that many important NMR interactions are orientation dependent and the information about anisotropic NMR interaction tensors becomes essential [5,8]. In particular, both the symmetric ¹J_iso = (¹J_ZZ + ¹J_XX + ¹J_YY)/3 (=70 Hz) and the asymmetric Δ¹J = ¹J_ZZ − (¹J_XX + ¹J_YY)/2 (=90 Hz) parts of the J-coupling tensor ¹J_KL for the nearest-neighbor (N–N) ²⁹Si nuclear spins have been recently determined in single-crystal silicon [9]. This was achieved by measuring the NMR lineshapes which are sensitive to the value of Δ¹J, at four different crystallographic orientations relative to the applied magnetic field. In diamond with ¹³C nuclear spins, which is of the main interest here, an analogous experiment was undertaken many years ago [10] but no J-coupling effects were studied there due to low sensitivity. In addition, as far as we know, there were no theoretical works on the quantum-chemical calculation of the J-coupling characteristics of ¹³C nuclear spins in diamond. To fill this gap, we are presenting here the results of the simulation of full tensors J_KL (K,L = X,Y,Z) and describing the J-couplings of nuclear spins ¹³C in small H-terminated diamond clusters, as well as in a cluster hosting the NV-color center.

2. Methods and Materials

In principle, theoretical foundations of J-coupling are well established [11,12,13,14] and there has been considerable progress in calculating the J-coupling characteristics for many simple molecules (e.g., see [12,15,16,17]) including ¹³C–¹³C pairs [12,16,17,18,19]. However, most software packages were previously limited by calculation of the scalar J-coupling constants. Only recently has it become possible to calculate full J-coupling tensors. Here we have used for the purpose the latest version 5.0.2 of the ORCA package. To model a diamond crystal, we used H-terminated carbon clusters. First, in order to test the opportunities of the package, we calculated the J-tensors for all possible pairs ¹³C–¹³C in the diamond-like adamantane molecule C₁₀H₁₆ (see Figure 1a), for which the isotropic J-coupling constant ¹J_iso for N-N nuclear spins ¹³C was experimentally measured to be 31.4 ± 0.5 Hz [20]. Having obtained the value of ~29.9 Hz for them (see Figure 2a, below), which was quite close to the above experimental one, we performed similar calculations for the H-terminated carbon cluster C₃₅H₃₆ (Figure 1b), as well as for the similar cluster C₃₃[NV⁻]H₃₆ hosting the NV color center (Figure 1c). It should be noted that the choice of such small clusters was due to the fact that, as is known [14,15,16], the calculations of the J-coupling characteristics are very computationally demanding for even modest-sized molecules. We optimized the cluster geometry using the ORCA 5.0.2 software package with the B3LYP/def2/J/RIJCOSX level of theory and then simulated the n-bond J-coupling tensors ⁿJ_KL(Ci,Cj) for all possible ¹³Ci–¹³Cj pairs in the clusters using the B3LYP/TZVPP/AUTOAUX/decontract level of theory. The functional B3LYP in combination with TZVPP basis is recommended for NMR calculations by ORCA [21,22]. The last two keywords provide a general-purpose auxiliary basis for simultaneously fitting Coulomb, exchange and correlation calculationsas well as calculations of integrals for Fermi-contact interaction which require very tight s-functions. Both of them are needed for correct calculations of all contributions to the J-coupling tensor. The package returns matrices describing the diamagnetic, paramagnetic, Fermi-contact, spin-dipolar and spin-dipolar/Fermi contact cross-term contributions to the total ⁿJ_KL tensor in the coordinate systems indicated in Figure 1. Using them and taking into account the known coordinates of carbon atoms belonging to some definite ¹³Ci–¹³Cj pair in the cluster, one can find respective J-coupling matrices in the other coordinate system. In particular, for neighboring nuclear spins ¹³C, separated by a single bond in diamond (~1.54 Å), the total ¹J_KL matrix becomes diagonal with J_XX ≈ J_YY in the coordinate system in which the Z-axis is directed along this bond [9]. In this case it is conventional to describe an axial J-coupling tensor in terms of two parameters: the scalar constant ¹J_iso and the asymmetric part Δ¹J. Since the magnitude of the J-coupling decreases rapidly with bond order, we will mainly consider here only N–N nuclear spins.

3. Results and Discussion

In the case of adamantane, we first calculated the isotropic J-coupling constants ⁿJ_iso for all possible pairs Ci–Cj with the numbers i and j shown in Figure 1a. The simulation was performed in the arbitrarily chosen coordinate system, in which the origin was on the C1 atom, the X axis was directed from the C1 atom to the C2 atom and the Y and Z axes were directed as it is shown in Figure 1a. The results of calculations are illustrated graphically in Figure 2a, which shows, in the form of a bar graph, the calculated values of the isotropic constants ⁿJ_iso(Ci,Cj) for pairs of ¹³C nuclear spins withnumbers Ci and Cj, indicated in Figure 1a. In the molecule there are 12 pairs (C1–C2, C1–C4, C1–C6, C2–C3, C3–C9, C3–C10, C4–C5, C5–C8, C5–C10, C6–C7, C7–C8, C7–C9) wherein carbon atoms are nearest-neighbors separated by single C–C bond. For these one-bond pairs, the values of the ¹J_iso constants were in the range of 29.8–29.92 Hz, i.e., were close to the experimentally measured [20] value of 31.4 ± 0.5 Hz. For all these pairs, the calculated total matrices ¹J_KL(Ci,Cj) were close to diagonal, since the isotropic Fermi-contact interaction made the main contribution to them. Moreover, taking into account the symmetry of the N–N Ci–Cj pairs about their midpoint in the transformed coordinate system, in which the Z axis is directed along some Ci–Cj bond, it is possible to transform the J-coupling matrices to their simplest diagonal form [9]. As an example, we considered here the C1–C2 pair, in which both nuclear spins are located on the X axis (see Figure 1a) so that the transformation of the calculated matrices to the new coordinate system, where the Z axis is directed along the C1–C2 bond, is carried out simply by rotation counterclockwise by 90° around the Y axis. For the C1–C2 pair, the partial matrices in the thus transformed coordinate system are presented below in

Table 1. The total J-coupling matrix ¹J_KL(C1,C2) and partial contributions to it (in Hz) for the ¹³C1–¹³C2 pair in the adamantane molecule calculated in the transformed coordinate system, having the Z axis along the C1–C2 bond.

Diamagnetic contribution:			Paramagnetic contribution:			Fermi-Contact contribution:
[−0.80	0	0.00	[0.21	0	0.00	[28.91	0	0
0	−0.85	0.08	0	−0.08	−0.03	0	28.91	0
0.00	−0.06	2.53];	0.00	0.04	−1.81];	0	0	28.91];
Spin-Dipolar contribution:			SD/FC cross-term contribution:			Total coupling tensor:
[0.54	0	0.00	[5.01	0	−0.00	[33.87	0	−0.00
0	0.59	−0.07	0	4.98	−0.06	0	33.55	−0.08
0.00	0.08	2.34];	−0.00	−0.06	−9.99];	−0.00	0.01	21.97]

.

One can see from these data the relative contributions of various interactions. They also show that the total matrix ¹J(C1,C2) is, as expected [9], near-diagonal with ¹J_XX(C1,C2) ≈ ¹J_YY(C1,C2) so that for this pair the asymmetric part of the J-coupling tensor is Δ¹J= ¹J_ZZ − (¹J_XX + ¹J_YY)/2 = −11.74 Hz. Similar data can be obtained for other pairs of N–N nuclear spins in the adamantane molecule. Figure 2a also shows that the isotropic constants ²J_iso and ³J_iso for more distant nuclear spins are only few hertz or less.

The results of similar calculations of isotropic constants ⁿJ_iso, performed for all possible pairs ¹³C–¹³C in the clusters C₃₅H₃₆ and C₃₃[NV⁻]H₃₆, are illustrated by bar graphs shown in Figure 2b,c, respectively. As one can see from Figure 1b, in the case of the cluster C₃₅H₃₆ we choose the coordinate systems in which the origin was taken at the C₂ carbon atom and the Z axis was directed from the C2 to the C1 atom. In this cluster, there are 595 different ¹³C–¹³C pairs with 52 of them being N–N carbons. Among these N–N pairs, 13 have their bonds near-parallel to the chosen Z axis. These bonds are shown in red in Figure 1b. For the remaining 39 N–N pairs, shown in blue in Figure 1b, the angles between their bonds and the Z axis were approximately equal to the tetrahedral angle 109.47° (or 180°–109.47°). Respectively, in the case of the cluster C₃₃[NV⁻]H₃₆, the origin of the coordinate system was taken on the N atom of the NV center and the Z axis coincided with the NV center axis. In this cluster, there are 45 N–N ¹³C–¹³C pairs, 12 of them having bonds directed near-parallel to the Z axis. Again, these 12 pairs are shown in red in Figure 1c and the other ones are shown in blue. As one can see from Figure 2b, for the cluster C₃₅H₃₆, simulated one-bond constants ¹J_iso were in the range of 29.8–30.0 Hz (for specific values see

Table 2. Diagonal elements ¹J_KK(Ci,Cj) of total J-coupling tensors (in Hz) and corresponding values of the isotropic constants ¹J_iso calculated for N–N ¹³Ci–¹³Cj pairs with their bonds near-parallel to the Z-axis of the coordinate systems shown in Figure 1b,c. The left panel shows the data for the cluster C₃₅H₃₆; the right panel for the cluster C₃₃[NV⁻]H₃₆. Atoms C4, C5 and C6 shown in bold in the right panel are the nearest neighbors of the vacancy of the NV center in the cluster C₃₃[NV⁻]H₃₆.

Cluster C35H36					Cluster C33[NV−]H36
Pair Ci,Cj	1Jxx	1Jyy	1Jzz	1Jiso	Pair Ci,Cj	1Jxx	1Jyy	1Jzz	1Jiso
C2,C1	33.45	33.45	22.41	29.77	C4,C7	39.93	41.41	30.08	37.14
C6,C9	33.66	33.44	22.52	29.87	C5,C10	42.06	39.12	30.01	37.06
C7,C12	33.49	33.60	22.51	29.87	C6,C13	39.93	41.41	30.08	37.14
C8,C15	33.49	33.60	22.51	29.87	C8,C16	35.88	35.50	23.99	31.79
C18,C10	32.71	32.79	21.56	29.02	C11,C17	35.68	35.71	24.02	31.80
C20,C11	32.71	32.79	21.57	29.02	C9,C18	35.53	35.89	24.00	31.81
C19,C13	32.86	32.67	21.58	29.04	C14,C19	35.53	35.89	24.00	31.81
C22,C14	32.71	32.82	21.58	29.04	C12,C20	35.68	35.71	24.02	31.80
C21,C16	32.85	32.66	21.57	29.03	C15,C21	35.88	35.50	23.99	31.79
C23,C17	32.71	32.82	21.58	29.03	C22,C28	35.41	35.18	23.79	31.46
C30,C24	32.44	32.32	20.90	28.55	C24,C29	35.07	35.53	23.80	31.47
C31,C26	32.44	32.31	20.90	28.55	C26,C30	35.40	35.18	23.79	31.46
C32,C28	32.25	32.51	20.91	28.56

), i.e., very close to those obtained for the adamantane molecule. Conversely, in case of the cluster C₃₃[NV⁻]H₃₆ containing the NV center, there were several pairs of N–N ¹³C atoms, located near the vacancy of the NV center, for which the values of the ¹J_iso constants were slightly higher (~37.1 Hz) than for the other pairs (~31.5–31.8 Hz, see Table 2).

The above data on the isotropic constants ¹J_iso for the clusters C₃₅H₃₆ and C₃₃[NV⁻]H₃₆ have been obtained from total J-coupling matrices ¹J_KL calculated for these clusters. Generally, as in the case of adamantane, the matrices have diagonal elements which are much larger than the non-diagonal ones. These diagonal elements are shown in Figure 3a–f for the clusters C₃₅H₃₆ and C₃₃[NV⁻]H₃₆, respectively. In these Figures, the red bars give the values of the corresponding diagonal elements for those adjacent carbon pairs for which the C–C bond is directed almost parallel to the Z axis of the coordinate system used; whereas the blue bars are for pairs for which the C–C bond makes a tetrahedral angle with the Z axis. More specifically, the values of the diagonal elements ¹J_KK (K = X,Y,Z) of the J-coupling matrices of N–N ¹³C–¹³C pairs shown in red in Figure 3 are given below in Table 2 along with the corresponding values of the isotropic constants shown in Figure 2b,c.

One can see from Figure 3 and, more specifically, from Table 2 that for the Ci–Cj pairs which are near-parallel to the Z axis the values ¹J_XX(Ci,Cj) ≈ ¹J_YY(Ci,Cj) are about one and a half times larger than ¹J_ZZ(Ci,Cj). Moreover, the presence of the negatively charged NV⁻ center in the cluster C₃₃[NV⁻]H₃₆, which introduces additional electron density, leads to some increase in the diagonal elements ¹J_KK of the J-coupling matrices for all Ci–Cj pairs compared with the cluster C₃₅H₃₆. As follows from Table 2, such an increase in the ¹J_KK values is especially pronounced (~9%) for the C4,C7; C5,C10 and C6,C13 pairs, in which the atoms C4, C5 and C6 are the nearest neighbors of the vacancy of the NV center on which the electron density of the center is mainly localized [23]. A similar increase in J-coupling takes place for other pairs C4/5/6-Cj, for which the corresponding bonds make an angle of ~109.47° with the axis Z of the selected coordinate system.

4. Conclusions

For the first time, the total tensors describing the indirect interaction of ¹³C nuclear spins in adamantane molecule, H-terminated diamond cluster C₃₅H₃₆ and in the cluster C₃₃[NV⁻]H₃₆ hosting NV centers have been calculated by quantum chemistry methods. It is shown that the presence of the NV center leads to a change in characteristics of the indirect interaction of ¹³C nuclear spins. The results obtained are important for quantum information and sensor applications, in particular, for the creation of long-lived quantum memory based on singlet-state ¹³C–¹³C dimers in diamond [3], creation of nanoscale NV-based quantum sensors that ensure the detection of adsorbed molecules/radicals on the surface of nanostructured diamond [4] and the determination of their chemical structure. Such sensors can be used to study biological processes at the level of individual cells, membranes, nerve fibers, targeted drug delivery and control of such delivery. The data obtained can also be useful for studies of NMR in the zero-to ultralow-field regime [24,25,26,27], where the internal spin interactions are dominated in their natural environment. An analysis of the dynamics of multi-spin systems ¹⁴NV–¹³C–¹³C with the account of both direct dipole–dipole interactions of ¹³C nuclear spins and of their J-coupling, as well as modeling of the NMR spectra of such objects, will be presented elsewhere.