A Nuclear Magnetic Resonance and Relaxation Study of Dimethoxyborane

Abstract

Proton and boron-11 c.w. nuclear magnetic resonances have been studied in solid H¹¹B(OCD₃)₂ and H¹⁰B(OCD₃)₂. For ¹¹B, only the $-\frac{1}{2}\to \frac{1}{2}$ transition, broadened by second order quadrupolar effects and by proton dipolar interaction can be seen; from the spectrum at several rf frequencies, the quadrupolar coupling constant |e²qQ/h| was found to be 3.0 ± 0.2 MHz (± always refers to rms errors). In H¹⁰B(OCD₃)₂, the proton line shapes at 53 and 10 MHz are considerably different; this may be interpreted as due to changes in the directions of ¹⁰B nuclear quantization. Nuclear magnetic relaxation studies have been made in the liquid phase. From the ¹⁰B and ¹¹B relaxation times, the activation energy for molecular reorientation was found to be 8.7 ± 0.4 kJ/mol (2.1 ± 0.1 kcal/mol). Consistent values for |e²qQ/h| were obtained from relaxation measurements in liquid phase and from c.w. spectra in solid phase. The temperature dependence of proton relaxation times deviates significantly from the activation energy model at higher temperatures, where spin-rotation interactions may be important. Proton transverse relaxation times (T₂) have also been measured and are consistent with the Allerhand-Thiele theory.

1. Introduction

In the past few years, several experimental wide-line NMR studies [1–3]¹ of two-spin systems $\left(I=\frac{1}{2},S>\frac{1}{2}\right)$ have been reported. The early experimental work [1, 2] was initiated in an attempt to determine metal-hydrogen bond distances in transition-metal carbonyl hydrides. If one assumes, as everyone did at that time, that the Van Vleck moment theory [4] is valid for such systems, then in some cases metal-hydrogen bond distances can be obtained from the low-temperature proton wide-line NMR data. In studies of HMn(CO)₅ and HCo(CO)₄, values of 1.3 and 1.2 A, respectively, were obtained for the metal-hydrogen bond distances. The subsequent observation in this laboratory that the proton line shapes for HCo(CO)₄ and H¹⁰B(OC²H₃)₂ are frequency dependent indicates that factors other than the direct dipole-dipole interaction can play an important role in determining the proton line shape and second moment of such systems. Since that time it has been shown theoretically [3, 5, 6] that this is indeed the case. In this paper, we have examined the various interactions that may contribute to proton NMR line shapes and moments by studying the systems H¹¹B(OC²H₃)₂ and H¹⁰B(OC²H₃)₂ rather carefully. These molecules are not ideal, since the principal z-axis of the electric field gradient tensor does not coincide with the H-B internuclear vector. Consequently, detailed line shape calculations are not possible. Nevertheless, they do afford us the opportunity to determine some of the NMR and NQR parameters by at least two independent techniques. In this way, one can assess experimentally the accuracy of some of the newer pulse techniques to determine such parameters as spin-spin coupling constants J_ij and electric quadrupole coupling constants e²qQ/h (in liquid samples), and one can check the validity of the approximations made in the analysis of two-spin systems and some of the predictions made by the theory. If these new pulse techniques can be shown to be valid and reliable, they can then be used with confidence to determine, for example, J_H-Mn in HMn(CO)₅; in such two spin $\left(I=\frac{1}{2},S>\frac{1}{2}\right)$ systems, conventional c.w. techniques (high resolution and wide-line NMR) give no information about the coupling constants. In this paper, we would like to call attention to the various interactions which can contribute to the proton NMR line shapes and moments, to show how the magnitudes of these interactions can be determined experimentally using pulsed and wide-line NMR techniques and to demonstrate the validity of these techniques.

2. Theory

The second moment M₂ of the proton NMR absorption spectrum in two-spin systems (e.g., HMn(CO)₅, HCo(CO)₄, H¹¹B(OC²H₃)₂, etc.) with spins $I=\frac{1}{2}$ and $S>\frac{1}{2}$ arises from several sources [1, 2, 7]:

(a). The direct dipole-dipole interactions (Van Vleck’s result [4]).

For a two spin system, this is given by:

$$M_2^{d-d}=\left(4/15\right){\hslash }^2{\gamma }_s^{2}S\left(S+1\right)r^{-6}$$ (1)

(b). The chemical-shift anisotropy interaction.

This will be small for protons and may be neglected.

(c). Indirect dipole-dipole interaction.

For a two-spin system $I=\frac{1}{2}$ and $S>\frac{1}{2},$ the nuclear spin I is quantized along the magnetic field H₀. On the other hand, the direction of quantization of the spin S lies between H₀ and the principal z axis of the electric field gradient tensor. For convenience, we define the parameter

$$\alpha =\left|4S\left(2S-1\right){\gamma }_s\hslash {\text{H}}_0/\left(e^{2}q\text{Q}\right)\right|.$$

In the limit α ⪡ 1, the spin S is quantized along the z axis of the field gradient tensor and for α ⪢ 1, S is quantized along H₀. As has been pointed out [3,6], both the line shape shape and second moment for the nucleus I change considerably in the region α ≈ 4. The direct (Van Vleck) interaction limit corresponds to α = ∞.

(d). The spin spin coupling interaction.

The contribution to M₂ is given [8] by

$${\text{M}}_2^{\text{s}.\text{c}.}=\frac{\text{S}\left(\text{S}+1\right)}{3}{J}^2.$$ (2)

For H¹¹B(OCD₃)₂ where the ¹¹B-H spin-spin coupling constant J_H-B = 164Hz, the contribution to the second moment is very small. J_H-Mn in HMn (CO)₅ and J_H-Co in HCo(CO)₄ are unknown, but could be rather large. In order to be significant, J would have to be greater than 1 kHz. Similarly, the contribution to M₂ from the anisotropic part of spin-spin coupling is also very small [6].

In 2-spin systems where the fine-structure has been partially or completely wiped out because of rapid quadrupole relaxation of one of the nuclei, the spin-spin coupling constant J_IS can, in principle still be obtained by pulsed NMR measurements. Following Abragam’s notation [8], the relaxation rates, R₁(R₁ = 1/T₁) and R₂ of the nucleus I(I = 1/2, S > 1/2) for dipolar relaxation are given by:

$$R_2^{\text{I}}=R_1^{\text{I}}={\gamma }_1^2{\gamma }_{\text{S}}^2{\hslash }^2\text{S}\left(\text{S}+1\right)\left\{\frac{1}{12}{J}^{\left(0\right)}\left({\omega }_{\text{I}}-{\omega }_{\text{S}}\right)+\frac{3}{2}{J}^{\left(1\right)}\left({\omega }_{\text{I}}\right)+\frac{3}{4}{J}^{\left(2\right)}\left({\omega }_{\text{I}}+{\omega }_{\text{S}}\right)\right\}$$ (3)

For ¹H = I, ¹¹B = S and ωτ_c ⪡ 1, we get:

$$R_2^{\text{I}}=R_1^{\text{I}}=\frac{5{\gamma }_{\text{H}}^2{\gamma }_{\text{B}}^2{\hslash }^2}{r_{\text{H}-\text{B}}^6}{\tau }_c\equiv \text{DD}$$ (4)

where γ_H and γ_B are the magnetogyric ratios of ¹H and ¹¹B, r_H-B = 1.25 × 10⁻⁸ cm, is the H-B bond distance [7], τ_c is the correlation time for the isotropic reorientation of the molecule.

The proton relaxation times arising from the indirect scalar coupling interactions are given by [8]

$$R_1^{\text{I}}=\frac{2A^2}{3}S\left(S+1\right)\frac{{\tau }_s}{1+{\left({\omega }_{\text{I}}-{\omega }_{\text{S}}\right)}^2{\tau }_s^2}$$ (5)

$$R_2^{\text{I}}=\frac{A^2}{3}S\left(S+1\right)\left\{\frac{{\tau }_s}{1+{\left({\omega }_{\text{I}}-{\omega }_{\text{S}}\right)}^2{\tau }_s^2}+{\tau }_s\right\}$$ (6)

where A is the spin-spin coupling constants in radians/s (2πJ = A), and τ_s in this case is just the relaxation time of the nucleus $\text{S}\left(R_1^{\text{S}}=R_2^{\text{S}}={\tau }_s^{-1}\right)$ which is dominated by quadrupole relaxation effects [8]:

$$R_1^{\text{S}}=R_2^{\text{S}}=\frac{3}{40}\frac{2S+3}{S^2\left(2S-1\right)}\left(1+\frac{{\eta }^2}{3}\right){\left(\frac{eq\text{Q}}{\hslash }\right)}^2{\tau }_c$$ (7)

The total proton (or nucleus I) relaxation times then are given by

$$R_1^{\text{I}}=\text{DD}+\frac{2A^2}{3}S\left(S+1\right)\frac{{\tau }_s}{1+{\left({\omega }_{\text{I}}-{\omega }_{\text{S}}\right)}^2{\tau }_s^2}$$ (8)

$$R_2^{\text{I}}=\text{DD}+\frac{A^2}{3}S\left(S+1\right)\left[\frac{{\tau }_s}{1+{\left({\omega }_{\text{I}}-{\omega }_{\text{S}}\right)}^2{\tau }_s^2}+{\tau }_s\right].$$ (9)

For our experiments, ω ~ 10⁸s⁻¹, τ_s ~ 10⁻⁴ s, hence

$$R_1^{\text{I}}=\text{DD}$$ (10)

$$R_2^{\text{I}}-R_1^{\text{I}}=\frac{1}{3}{A}^{2}S\left(S+1\right){\tau }_s=\frac{1}{3}{A}^{2}S\left(S+1\right){\left(R_1^S\right)}^{-1}.$$ (11)

For very fast quadrupole relaxation of nucleus S (e.g., Mn(CO)₅ where $T_1^{Mn}\approx 90\mu s$), a simple spin-echo experiment will give an accurate value of $R_2^{\text{I}}.$ For slower quadrupole relaxation (e.g., H¹¹B(OCD₃)₂ where T₁B ≈ 3 ms), Carr-Purcell experiments are required to overcome the residual spin-spin coupling interaction.

The spin-spin coupling constant, J, then is given by:

$$2\pi J={\left[\frac{3\left(R_2^{\text{I}}-R_1^{\text{I}}\right)}{S\left(S+1\right)R_1^{\text{S}}}\right]}^{1/2}.$$ (12)

To obtain e²qQ/ℏ is also relatively easy. If we know the I-S bond distance and $R_1^{\text{I}},$ then from eq (4) we obtained τ_c. From the value of τ_c and $R_1^{\text{S}}$ we can obtain e²qQ/ℏ from eq (7) (in most of the simple cases with which we are concerned the asymmetry parameters, η, is approximately zero).

3. Experimental Procedure

The synthesis [10] of the molecules H¹¹B(OC²H₃)₂ and H¹⁰B(OC²H₃)₂ and a preliminary NMR study [7] have already been reported. The proton and ¹¹B c.w. line shape studies were performed using a modified marginal oscillator [11], a gas-flow cryostat and a commercial magnet, magnet power supply and field sweep system. The field sweep was calibrated by the audio side-band technique. The pulsed NMR experiments were done using a spectrometer designed and built in this laboratory. The π/2 pulse-widths were about 1 μs in duration for protons and the overall system recovery time was about 16 μs. The T₁ values at each temperature were determined via the π − π/2 pulse sequence; a least square fit of the data to the equation

$$\text{ln}\left[\text{I}\left(\infty \right)-\text{I}\left(t\right)\right]=\left(-t/T_1\right)+\text{K}$$ (13)

was used to obtain T₁. At least ten data points were used to obtain each T₁ value. A rms deviation of about 3 percent was typical for the T₁ values. The temperature of the gas-flow cryostat system was controlled by a servo-device which employed a platinum resistor as a sensing element. The temperature could be held constant to about 0.01 °C. The sample temperature was measured with a copper-constantan thermocouple placed near the sample coil. Since the samples were synthesized in a high vacuum system the presence of even minute amounts of dissolved oxygen can be discounted. The final samples for our experiments were distilled in the high-vacuum system into 10 mm quartz sample tubes and sealed off. (Borosilicate glass containers are not suitable in this case because of the intense ¹¹B resonance from the glass.)

4. Results and Discussion

4.1. Line-Shape Studies

Proton NMR spectra of H¹¹B(OCD₃)₂ were recorded from 9.7 MHz (0.23T = 2.3kG) to 53 MHz. All of them are quite similar to the previously recorded spectra [7] at 30 MHz. This reflects the fact that even at 0.23T, α ≅ 14 and consequently no appreciable indirect dipolar interaction is to be expected.

For the H¹⁰B(OCD₃)₂ molecule, the situation is quite different. Although the details are obscured due to the larger spin quantum number (S = 3 for ¹⁰B) and the smaller magnetogyric ratio γ(¹¹B)/γ(¹⁰B) ≅ 3, there is a considerable difference in the proton spectra recorded at 10 and at 53 MHz; these spectra are shown in figure 1. The predicted trends [3, 6] and the observed spectra are in good general agreement. A detailed comparison of the theory and the experimental results is not possible since the magnetic dipole-dipole vector does not coincide with the principal axis of electric field gradient and the calculations are therefore intractable.

Figure 1. Proton absorption derivative spectra in solid H¹⁰B(OCD₃)₂.

At 52.8 (top) and 9.8 (bottom) MHz.

The quadrupolar coupling constant e²qQ/h for boron-11 may be determined directly from the NMR spectra in the solid phase [12]. The spectra at 18.8, 12.7, and 9.7 MHz are shown in figure 2. Only the central $\left(-\frac{1}{2}\to \frac{1}{2}\right)$ component, broadened by second order quadrupole splitting, can be seen. The spectra are in general agreement with the calculated line shape for a symmetric field gradient, indicating η ⪡ 1. However, fine structures due to the magnetic dipole interactions between ¹¹B and proton nuclei are also observed. Including the magnetic dipolar contributions, the total splitting (in Hz) is (25/192) ${\left(e^{2}q\text{Q}/h\right)}^2{\nu }_{\text{L}}^{-1}+\left(\xi /h\right),$ where ν_L is the rf oscillator frequency, ξ = γ_Bγ_Hℏ²r⁻³, ξ/h = 20 KHz. The dipolar contribution, ξ/h, is about 10 percent of the total splitting. From figure 2, we get |e²qQ/h| = 3.0 ± 0.2 MHz for ¹¹B in the solid phase of H¹¹B(OCD₃)₂. (In this paper, ± refers to rms deviations). Slightly lower values (|e²qQ/h| = 2.5 MHz) have been reported [13] for ¹¹B coupling constants at triangularly coordinated sites in borates.

Figure 2. ¹¹B spectra, absorption derivative, HB(OCD₃)₂ in quartz tube.

Top to bottom, 18.8, 12.7 and 9,7 MHz. Dashed lines indicate unshifted ¹¹B resonance position. Magnetic field decreases from left to right.

4.2. Relaxation Studies

Spin-lattice relaxation time (T₁) data at various temperatures in the liquid phase are summarized in figure 3. The proton relaxation times T_1H at 4.2 and 19 MHz are not very different. Also, we observe that T_1H for H¹⁰B(OCD₃)₂ is a factor of 2.5 ± 0.3 longer than T_1H for H¹¹B(OCD₃)₂ over the experimental temperature range 140–240 K, and that the ¹⁰B relaxation time T_1B-10 in H¹⁰B(OCD₃)₂ is a factor of 1.4 ± 0.2 longer than the ¹¹B relaxation time T_1B-11 in H¹¹B(OCD₃)₂.

Figure 3. Spin-lattice relaxation times T₁ (in s) versus inverse temperature on semilogarithm scale.

H¹¹B(OCD₃)₂: proton at 19 MHz, ○; proton at 4.2 MHz, x; ¹¹B at 19 MHz, ◬. H¹⁰B(OCD₃)₂: proton at 19 MHz, ●; proton at 4.2 MHz, ; ¹⁰B at 4.2 MHz, ▲.

We shall consider the nuclear magnetic relaxation due to molecular rotation only, the rotational correlation time τ_c being very short (τ_c ⪡ 10⁻⁸ sec). Then the proton relaxation time T_1H due to H-B dipolar interaction and the boron relaxation time due to electric quadrupole interaction are given by [8]:

$${\left(T_{1_{\text{H}}}\right)}^{-1}=\frac{4}{3}{\gamma }_{\text{B}}^2{\hslash }^2{r}^{-6}S\left(S+1\right){\tau }_c$$ (14)

$${\left(T_{1_{\text{B}}}\right)}^{-1}=\frac{3\left(2S+3\right)}{40S^2\left(2S-1\right)}{\left(\frac{e^{2}q\text{Q}}{\hslash }\right)}^2\left(1+\frac{{\eta }^2}{3}\right){\tau }_c$$ (15)

where γ_B, S, e²qQ/h and η are the magnetogyric ratio, spin, quadrupolar coupling constant and asymmetry parameter of the boron nucleus, and r is the H-B distance. From (15), T₁ for ¹⁰B is expected to be a factor of 1.54 longer than T₁ for ¹¹B, this ratio being independent of q, η and τ_c; experimentally, we find the ratio to be 1.4 ± 0.2. From (14), we expect T_1H in H¹⁰B(OCD₃)₂ to be a factor of 2.78 longer than T_1H in H¹¹B(OCD₃)₂, in reasonable agreement with the ratio 2.5 ± 0.3 observed experimentally.

Since η² ⪡ 1, the quadrupolar coupling constant e²qQ/h may be calculated from the ratio T_1H/T_1B which is independent of τ_c. In H¹¹B(OCD₃)₂, we find T_1H/T_1B = 360 ± 40 over our experimental temperature, range in figure 3, T_1B being the ¹¹B relaxation time. Using B-H bond distance r = 1.25 A from the proton c.w. resonance data [8], we get e²qQ/h = 2.7 ±0.2 MHz for ¹¹B in H¹¹B(OCD₃)₂ from (14) and (15). This calculated value is expected to be slightly lower than the true ¹¹B quadrupolar coupling constant, since we have considered only the intra molecular rotational ¹¹B-H dipolar relaxation for protons in (14). From the second order quadrupolar splitting in polycrystalline solids, we found e²qQ/h = 3.0 ± 0.2 MHz. A previous estimate of 2.6 MHz has been given by Boden, Gutowsky, Hansen, and Farrar [9] from pulse measurements in liquids.

Rotational correlation times for liquids HB(OCD₃)₂ from T₁ data using (14) and (15) are summarized in figure 4, using the B-H bond distance r = 1.25 A and e²qQ/h = 2.7 MHz for ¹¹B. The agreement between various methods is satisfactory, indicating that (14) and (15) are in general correct, although the problem is somewhat oversimplified. We expect the boron nuclear relaxation to be dominated by quadrupolar interaction. The correlation times from the boron relaxation can be fitted to usual form:

$${\tau }_c={\tau }_0{e}^{V/RT}$$ (16)

where V is the activation energy, with V = (8.7 ± 0.4) × 10³ J/mol (2.1 ± 0.1 kcal/mol). Previous studies [9] over a narrower temperature range gave a somewhat lower activation energy, (5.5 ± 0.4) × 10³ J/mol.

Figure 4. Rotational correlation time (in s) in liquid HB(OCD₃)₂ versus inverse temperature on semilogarithm scale: from proton data in H¹¹B(OCD₃)₂, x; from proton data in H¹⁰B(OCD₃)₂, ⊙; from ¹¹B data in H¹¹B(OCD₃)₂, ◬.

On the other hand, the proton relaxation times in H¹¹B(OCD₃)₂ and H¹⁰B(OCD₃)₂ cannot be fitted to the activation energy from (16); whereas the fit is reasonably good in the low temperature region, the T₁ versus 1/T curve is considerably flatter than expected in the 10³/T > 5 region. Similar behaviors have been reported, for example, by Powles and co-workers [14], and their results have been interpreted through spinrotation interactions. This may well be the cause of the nonlinear high-temperature dependence of T_1H versus temperature for H¹¹B(OCD₃)₂.

Transverse proton relaxation times T_2H have been measured by the Carr-Purcell pulse sequence in the liquid phase of H¹¹B(OCD₃)₂, and the results are summarized in figure 5. At 147 K, our results are in good agreement with the fast quadrupolar relaxation limit [15] when t_cp ≧ T_1S:

$${\left(T_{2\text{H}}\right)}^{-1}={\left(T_{1\text{H}}\right)}^{-1}+\frac{1}{3}S\left(S+1\right)A^2{T}_{\text{1S}}\left(2T_{1\text{S}}/t_{cp}\right)\text{tanh}\left(t_{cp}/2T_{1\text{S}}\right)]$$

where S refers to ¹¹B, T_1S = 6.2 × 10⁻⁴ sec. and T_1H = 0.2 sec. from figure 3, A = 2πJ. From our data, we estimate the ¹¹B-H spin-spin coupling constant J = 170 Hz, in agreement with the previous high resolution studies [9].

Figure 5. 1/T₂ versus 1/t_cp for proton in H¹¹B(OCD₃)₂.

198 K, ⊙; 175 K, +; 167 K, ●; F55 K, *; 147 K, ▽.

In conclusion, we believe that these results demonstrate that pulsed NMR techniques can, in favorable cases be used to obtain reliable, reasonably accurate values for spin-spin and quadrupole coupling constants in liquids.