Fine Structures of 1H-Coupled 13C MAS

Fine Structures of 1H-Coupled 13C MAS NMR Spectra for Uniaxially Rotating ... for [2-13C]decane molecules enclathrated in deuterated urea channels. 1...
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10854

J. Phys. Chem. 1996, 100, 10854-10860

Fine Structures of 1H-Coupled 13C MAS NMR Spectra for Uniaxially Rotating Molecules in Deuterated Surroundings: Conformations of n-Alkane Molecules Enclathrated in Urea Channels Atsushi Kubo, Fumio Imashiro, and Takehiko Terao* Department of Chemistry, Graduate School of Science, Kyoto UniVersity, Kyoto 606-01, Japan ReceiVed: February 27, 1996X

It is theoretically shown that spectral fine structures due to the intramolecular 1H-1H dipolar couplings and the 13C-1H J-couplings can be observed in uniaxially rotating molecules magnetically isolated from the surroundings when the 13C spectrum is observed under magic-angle spinning without 1H-decoupling. Such fine structures were observed for [2-13C]decane molecules enclathrated in deuterated urea channels.

1. Introduction Fine spectral structures due to J-couplings in solution state NMR have extensively been used to analyze the conformations of solute molecules.1-3 In solid samples, however, it is usually impossible to obtain such fine structures because of line broadening by dipolar interactions. Yet, if we use a combination of 1H-1H dipolar decoupling and magic-angle spinning (MAS), we can observe 13C-1H J split 13C spectra.4-8 On the other hand, several research groups found that highly resolved NMR spectra of dipolar-coupled spins can be obtained by MAS in systems such as inclusion compounds, liquid crystals, and a onedimensionally dipolar-coupled material of fluoroapatite.9-15 In these systems, all the intramolecular or intrachain dipolar interaction tensors share a common principal-axis frame, so that their dipolar broadening can be removed by MAS.16,17 The spinning sideband intensities of one- and two-dimensional MAS NMR spectra of such dipolar-coupled spin systems have theoretically been analyzed.16-19 However, the 1H-coupled 13C cross-polarization (CP) MAS NMR spectrum measured for the urea/tridecane inclusion compound17 is still broad probably due to the intermolecular 1H-1H dipolar interactions so that splitting by 13C-1H J-coupling is not apparent. If the broadening by intermolecular dipolar interactions can be reduced further, we would observe a new type of fine structure caused by the interference of the 13C-1H J-couplings and the intramolecular 1H-1H dipolar interactions. In the first half of this paper, we present a theory of such new fine structures of 1H-coupled 13C MAS NMR spectra for uniaxially rotating molecules magnetically isolated from the surroundings. Fine structures of 1H-coupled 13C MAS spectra were observed for uniaxially rotating n-alkane molecules enclathrated in deuterated urea channels. The conformations of alkanes in the urea channels have attracted physicochemical interest. Earlier 1H NMR studies20,21 revealed that the alkane molecules undergo uniaxial rotation in the urea channels. In our previous studies,22,23 the 13C-1H dipolar spectra of n-alkanes in urea inclusion compounds were separately observed for the 2-, 3-, and innerCH2 and CH3 groups using the switching-angle sample-spinning (SASS) NMR method, and the analysis with the aid of MM2 molecular mechanics calculations led to the conclusion that the n-alkane molecules are in dynamical disorder of various conformations in addition to the uniaxial rotation; they had long * Corresponding author. E-mail: [email protected]. FAX: 075-751-2085. X Abstract published in AdVance ACS Abstracts, June 1, 1996.

S0022-3654(96)00588-6 CCC: $12.00

been taken to be in the all-anti conformation since 1952.24 After our study, some investigations on this subject by IR and Raman experiments25-29 appeared; they concluded that there are no or very few gauche conformations in the n-alkane molecules enclathrated in the urea channels. On the other hand, deuteron NMR and molecular dynamics studies supported our conclusion.30,31 In the second half of this article, we will show that our previously proposed dynamical disorder model23 accounts for the temperature dependence of 1H-coupled 13C CPMAS NMR spectra for the magnetically isolated n-decane molecules enclathrated in urea channels. 2. Experiments 2-13C labeled n-decane ([2-13C]decane) was prepared as follows: octyl aldehyde was reacted with [1-13C]ethylmagnesium bromide prepared from [99.3%-1-13C]ethyl bromide. The obtained alcohol was brominated by PBr3 and was reduced to [2-13C]decane by use of LiAlH4. Urea was completely deuterated by 5-times repetition of dissolving in D2O and removing the solvents. The inclusion compound was prepared by the recrystallization of 14 mg of [2-13C]decane, 146 mg of [99.3%2H ]decane, and 600 mg of [2H ]urea from 2.5 mL of [2H ]22 4 1 methanol. To avoid the exchange of deuterons in urea for protons in the water vapor of the air, the sample was transferred to a glass tube (6 mm o.d. 10 mm length) under the N2-gas atmosphere and was sealed. The rotor involving this glass tube could be spun very stably: A sample spinning frequency was controlled by 2.000 kHz with an accuracy of (2 Hz at 298 K and (5 Hz at 193 K. The NMR experiments were performed on a Chemagnetics CMX300 spectrometer operating at 300.46 MHz and 75.56 MHz for 1H and 13C, respectively. 1H-coupled 13C MAS NMR spectra of the urea/n-decane inclusion compound were recorded using a normal cross-polarization technique but without 1H highpower decoupling. The spectra were obtained at 298 ( 3 and 193 ( 5 K. We transferred the sample, in a glovebag under nitrogen atmosphere, from one glass tube to another larger one after room temperature experiments before low-temperature experiments, intending to improve the signal-to-noise ratio by increasing the sample volume. The decane signals decreased after this procedure, probably due to the partial decomposition of the sample. However, the 1H-coupled 13C CPMAS spectrum was unchanged, so that this sample sufficed for low-temperature experiments. © 1996 American Chemical Society

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J. Phys. Chem., Vol. 100, No. 26, 1996 10855

3. Theory Two methods are employed to calculate the time-evolution operator. Firstly, we calculate it by multiplying the instantaneous time-evolution operators in order of time (multistep method). Secondly, we calculate it by treating the 13C-1H J-coupling and the 1H isotropic chemical shifts as perturbation for the larger 13C-1H and 1H-1H dipolar interactions. The former and the latter methods will be explained in sections 3.1 and 3.2, respectively. A large matrix size and a long CPU time are needed to calculate MAS NMR spectra for such multispin coupled systems. Especially, for powder samples, the calculation has to be repeated many times by changing the Euler angles with small steps. It was reported that the average over the Euler angle γ can be replaced by the Fourier transformation with respect to time in calculating the spinning sideband intensities of the MAS NMR spectrum of an isolated spin (I ) 1/2).32-34 In the Appendix, we will show by using Floquet theory35-39 that a similar technique can be used in calculating the average over the angle γ of a MAS NMR spectrum from a multispin coupled system. This method greatly reduces the CPU time. 3.1. Calculation by a Multistep Method. For calculating 1H-coupled 13C MAS NMR spectra of uniaxially rotating molecules, we assume the following internal interaction Hamiltonian for an observed 13C (I) spin and surrounding 1H (S) spins:

H(γ;ωrt) ) HCS + HJ + HDSI + HDSS

(1)

where HCS is the isotropic chemical shift interactions of 1H spins, and HJ the J-coupling between the directly bonded 1H and 13C spins. HDSI and HDSS are the intramolecular dipolar interactions between the directly bonded 1H and 13C spins, and between the 1H spins, respectively. The individual Hamiltonians are

HCS ) ∑ωkSkz

(2a)

HJ ) ∑JkIzSkz

(2b)

HDSI ) f(t)∑2bkIzSkz ) f(t)HDSI*

(2c)

k

Figure 1. Euler transformations (a) from the laboratory-fixed coordinate system (xL, yL, zL) to the rotor-fixed coordinate system (xR, yR, zR) and (b) from the rotor-fixed coordinate system to the crystal-fixed coordinate system (xC, yC, zC). zL and zR are chosen to be parallel to the static field and the sample rotation axis, respectively. ΘM is an angle between the static field and the sample rotation axis. ωr is a spinning frequency. Note that ωrt in a and γ in b are the rotation angles around the same axis zR.

1 f(t) ) (1 - 3 cos2 ΘM(t)) 2 1 ) - {sin2 β cos 2(ωrt + γ) - x2 sin 2β cos(ωrt + γ)} 2 (3) where we assumed that all the dipolar interaction tensors have the same orientation of their principal axes owing to motional averaging. ΘM is the angle between the magnetic field and the molecular rotation axis, and (-β and -γ) are the polar angles representing the principal axis orientation of the dipolar interaction tensor with respect to the rotor-fixed frame as shown in Figure 1. Calculating the trace for the I spin in eq A6 as shown in the Appendix, we obtain the following equation:

S+(γ;ωrt) ) TrS{U-(γ;ωrt) U+(γ;ωrt)-1}

where U( are the diagonal matrix elements of the time evolution operator U for mI ) (1/2. In this section, we calculate U( by multiplying the instantaneous time-evolution operators in order of time. As shown in the Appendix, we only have to calculate U( for γ ) 0.

U((0;ωrnδt) ) exp{-iH((0;ωr(n - 1) δt) δt}

k

k

{

HDSS ) f(t)∑dj,k k

}

1 2SjzSkz - (Sj+Sk- + Sj-Sk+) ) f(t)HDSS* 2 (2d)

with

bk )

〈 〈

dj,k )

〉 〉

γIγSp2 (3 cos2 θk - 1) 2rk3 2 2

γS p 2rjk

3

(3 cos2 θjk - 1)

(2e)

... exp{-iH((0;0) δt}

where rk and rjk are the internuclear distances between the carbon and the kth proton and between the jth and the kth protons, respectively. θk and θjk represent the angles between the molecular axis and the corresponding internuclear vectors. The symbol 〈 〉 denotes an average over all thermally accessible nuclear positions. In eqs 2e and 2f, we expressed the dipolar Hamiltonian as a product of a time-dependent factor f(t) and a time-independent operator HDXY*. The angular factor f(t) is given by

(5)

where H( are the diagonal matrix elements of H for mI ) (1/ 2. Then U((0;2π) are diagonalized as follows:

U((0;2π) ) T( exp{-iΛ(Tr}T(-1

(6)

where Tr is the sample-spinning period. T( and Λ( are obtained from this equation. Equation 5 can be rewritten in the form of eq A3. The diagonal matrix elements P((0;x) of P for levels mI ) (1/2 can be calculated from U((0;x), T( and Λ( by the following equation:

P((0;x) ) U((0;x)T( exp{iΛ((x/ωr)}T(-1 (2f)

(4)

(7)

G(x) in eq A8 is written as

G(x) ) T+-1P+(0;x)-1 P-(0;x)T-

(8)

Using the Fourier coefficients Fm of G(x), the spectrum is given by the following equation:

〈S+(γ;ωrt)〉γ )

|〈k|Fm|l〉|2 exp{i(Λ+k,k - Λ-l,l + mωr)t} ∑ k,l,m (9)

If the

13C

chemical shift anisotropy should be involved in the

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Kubo et al.

calculation, eq 8 is multiplied by exp{-i∆ωCSAφ(x)}, where ∆ωCSA is the difference between the least-shielded principal value and the isotropic value of the 13C chemical shift tensor, and φ(x) is the integral of f(t):

φ(x) ) -

1 1 2 sin β sin 2x - x2 sin 2β sin x 2ωr 2

(

)

(10)

3.2. Perturbation Calculation. The time-independent terms of the 1H isotropic chemical shifts and the 13C-1H J-couplings are rather small compared with the time-dependent dipolar interactions. Thus we can treat these two interactions as perturbation to the dipolar interactions. The propagator representing the time evolution due to the dipolar interactions is given by

TABLE 1: Isotropic Interaction Energies in the 13C1H11H2 Spin System no.

spin eigenstate |mC,mH1,mH2〉

energya (freq unit)

1 2 3 4 5 6 7 8

|R,R,R〉 |R,R,β〉 |R,β,R〉 |R,β,β〉 |β,R,R〉 |β,R,β〉 |β,β,R〉 |β,β,β〉

J/8π + (ω1 + ω2)/4π J/8π + (ω1 - ω2)/4π -J/8π - (ω1 - ω2)/4π -J/8π - (ω1 + ω2)/4π -J/8π + (ω1 + ω2)/4π -J/8π + (ω1 - ω2)/4π J/8π - (ω1 - ω2)/4π J/8π - (ω1 + ω2)/4π

a J is the J-coupling between C and H , and ω (ω ) is the isotropic 1 1 2 chemical shift of H1 (H2).

(11)

this is the simplest example in which the effect of the 1H-1H dipolar coupling appears on the spectrum. Using the result of the perturbation calculation shown in section 3.2., the FID signal sampled at integral multiples of the rotation period is

The perturbation Hamiltonian in the interaction representation is given by

S+(t) ) 2 cos(Jt/2) + 2(1 - δ) cos(rJt/2) + 2δ cos{r(ω1 - ω2)t} (19)

U0(γ;ωrt) ) exp[-i(HDSI* + HDSS*){φ(ωrt + γ) - φ(γ)}]

Hint(t) ) U0(γ;ωrt)-1(HCS + HJ)U0(γ;ωrt)

(12)

The main dipolar Hamiltonian is diagonalized as follows: -1

HDSI* + HDSS* ) TDΛDTD

where r and δ are given by the following equations:

(bCH1 - bCH2)2 + |A0|2dH1H22 x r) D

(13)

Equation 11 can be rewritten by using eq 13.

U0(γ;ωrt) ) TD exp[-iΛD{φ(ωrt + γ) - φ(γ)}]TD-1 (14)

D ) x(bCH1 - bCH2)2 + dH1H22

(20b)

A0 ) (1/2π)∫0 exp{iDφ(γ)} dγ

(20c)

2π

We expand exp{iΛDφ(x)} into the Fourier series

exp{iΛDφ(x)} ) ∑FmD exp(imx)

(15)

m

where FmD are operators or matrices. The zero-order average Hamiltonian of eq 12 can be written as the product of the γ-dependent and γ-independent terms:

H h int(0) ) 1/Tr∫0 Hint(t) dt ) B(γ)HAB(γ)-1

(16a)

B(γ) ) TD exp{-iΛDφ(γ)}

(16b)

HA ) ∑FmDTD-1(HCS + HJ)TD(FmD)+

(16c)

Tr

m

The propagator representing the time evolution by the zeroorder average Hamiltonian can be written as

U(γ;ωrt) ) U0(γ;ωrt) exp{-iH h int(0)t} ) B(ωrt + γ)TA exp(-iΛAt)TA-1B(γ)-1 (17) where ΛD is a diagonal matrix obtained by the transformation HA ) TAΛDTA-1. Equations A7, A9, and A10 can be used again to obtain spectra for a powdered sample. We replace Λ by ΛA and P(0;x) by B(x) in eq A8 and obtain the following equation:

G(x) ) TA-1B(x)-1I+B(x)TA

(20a)

(18)

The spectrum is given by eq A10. 4. Results and Discussion 4.1. Model Calculation for a Uniaxially Rotating CH1H2 System. In this section, we will consider the 1H-coupled 13C MAS NMR spectrum of a uniaxially rotating CH1H2 system;

δ)

dH1H22(bCH1 - bCH2)2(1 - p)2 D4r2

p ) Re{A0 exp[-iDφ(γ)]}

(20d) (20e)

As seen from eq 19, the spectrum generally consists of the six lines shifting by (J/4π, (rJ/4π, and (r(ω1 - ω2)/2π from the center, where ω1 (ω2) is the resonance frequency of H1 (H2). The latter four lines appear, when dH1H2 is not zero. This spectrum can be simply explained as follows: If we consider only the isotropic interactions of the proton isotropic chemical shifts and the 13C-1H J-coupling, the energies for the three-spin system are given as listed in Table 1. The allowed transitions are (1,5), (2,6), (3,7), and (4,8), yielding only (J/ 4π lines. When the dipolar interactions between the two protons are included, levels 2 and 3, and also levels 6 and 7, are mixed together. As a result, the additional transitions (2,7) and (3,6) are allowed, giving the extra lines at (r(ω1 - ω2)/2π. The mixing also causes shifts of the energy levels, so that some (J/ 4π lines shift to (rJ/4π. Such an effect of the mixing of proton spin states has been reported for a rigid CH2 group in a stationary sample.40 Figure 2 shows the calculated 1H-coupled 13C MAS NMR spectra of a CH1H2 system rotating around an axis perpendicular to the plane, where only H1 is assumed to have a 13C-1H J-coupling of 0.12 kHz. The spectra were calculated by the multistep method described in section 3.1. We also calculated the spectra using the perturbation method shown in section 3.2, which almost the same as those calculated by use of the multistep method. In the present case, the size of the perturbation Hamiltonian given by eq 12 is much smaller than the spinning frequency, so that the zero-order average Hamiltonian can provide the accurate spectra. However, the CPU time is a little longer and the program is slightly more complicated in the perturbation method.

n-Alkane Molecules in Urea Channels

J. Phys. Chem., Vol. 100, No. 26, 1996 10857

Figure 3. Experimental 1H-coupled 13C CPMAS NMR spectra of the urea/n-decane inclusion compound: (a) at low temperature (193 ( 5 K) and (b) at room temperature (298 ( 3 K). 16 000 (21700) scans of the signals were accumulated with a contact time of 3 ms and with a recycle delay of 10 s (4 s) for the measurements at 298 K (193 K).

Figure 2. Calculated 1H-coupled 13C MAS NMR spectra of a CH1H2 system rotating around an axis perpendicular to the plane. The spectra were calculated using the multistep method described in section 3.1 on the following assumptions: Only H1 has a C-H J-coupling of 120 Hz. The C-H dipolar couplings are -6 kHz for dCH1 and -2 kHz for dCH2. The isotropic chemical shift of H2 is zero. The dipolar interaction dH1H2 between the two protons and the isotropic chemical shift ω1/2π of H1 is (dH1H2/2π, ω1/2π) ) (2 kHz, 0 kHz) for a, (4 kHz, 0 kHz) for b, and (4 kHz, 0.2 kHz) for c. The spinning speed is 2 kHz, and the line shapes are Lorentzian with T2 of 50 ms. Each sideband is expanded and plotted in the region from -0.3 to 0.3 kHz from the center of the sideband.

Spectrum a was calculated assuming that ω1 ) ω2, so that the five lines should be observed at (J/4π, (rJ/4π, and 0 from the center of the center band or each sideband. However, at most three lines are seen in the center band or each sideband; the reason is that r is almost equal to 1 because a small dH1H2/ 2π value (2 kHz) was assumed. In spectrum b we assumed dH1H2/2π to be 4 kHz, leaving the condition of ω1 ) ω2 unchanged. The larger dH1H2 causes two effects: First, it makes δ larger so that the intensity of the line at ω ) 0 increases. Second, it reduces the value r so that the lines of (rJ/4π can be distinguished from those of (J/4π especially in the firstorder sidebands. The dH1H2 value of 4 kHz was again used for calculating spectrum c, but ω1 was set to ω2 + 0.2 kHz. As a result, the lines of (r(ω1 - ω2)/2π appear in the center band. The factor r also depends on the orientation of the dipolar tensor with respect to the static magnetic field, so that the powder average is expected to cause line broadening. Such broadening cannot be perfectly removed by a combination of uniaxial molecular rotation and magic-angle spinning. However, the orientation dependence is only included in φ in the integrals of eqs 20c and 20e. As shown in Figure 2, the broadening or the shift is usually small, r being nearly equal to 1. 4.2. 1H-Coupled 13C MAS NMR Spectra of the Urea/nDecane Inclusion Compound. Figure 3 shows the 1H-coupled 13C CPMAS NMR spectra of the urea/n-decane inclusion compound prepared as described in the experimental section. Spectra b and c were observed at 193 and 298 K, respectively. The strongest signal at the center of each spectrum is assigned to the 2-methylene. The spinning sidebands of the 2-methylene signal are clearly seen at integral multiples of a spinning

frequency of 2 kHz with fine structures. The signals of the other methylenes are observed at frequencies 590 Hz higher than those of the 2-methylene, while the methyl signal appears at the 860 Hz lower side.22 The signal of the deuterated urea appears at the 10 480 Hz higher side. No signals from enclathrated n-decane-d22 were observed because of the inefficiency of the intermolecular cross-polarization. They could be observed at frequencies about 100 Hz lower than the decaneh22 signals by a single-pulse excitation. For comparison, we calculated the spectra of the 2-methylene carbon, taking into account the seven protons of the methyl, 2-methylene, and 4-methylene groups. The protons of the 3-methylene were not taken into account, since the internuclear vectors from these protons to the 2-methylene carbon lie close to the magic-angle from the molecular rotation axis, and the dipolar couplings between these nuclei are relatively small (see Table 2). The dipolar couplings were calculated by using the atomic coordinates and the energies of the 25 decane conformations obtained previously by MM2 molecular mechanics calculations,23 on the assumption that the molecular rotation axis is parallel to the urea channel axis. They are listed in Table 2 for the following four cases: In case a, only the all-trans conformation is assumed to exist. In cases b, c, and d, we assume that the n-decane molecules are in dynamical disorder of the 25 conformations whose populations are given by the Boltzmann distribution at 193 K in b, 298 K in c, and the infinite temperature in d. The fractional populations of the three major conformers at both 193 and 298 K, which are obtained from the steric energies, are given in Table 3. The contents of molecules with at least one gauche bond are evaluated to be 44.9% at 298 K and 19.0% at 200 K. Most of the dipolar coupling constants increase or decrease to a large extent with the increase of gauche contents as seen in Table 2. For example, the 2-methylene carbon-proton dipolar coupling constant (13C2-1H4) increases by 13.6% when the temperature is decreased from 298 to 193 K. This increase is much larger than expected from the difference of the molecular vibrations41,42 between these temperatures: indeed the vibrationally averaged 13C -1H dipolar coupling constant increases only by 4.6% for 2 4 a temperature change from 300 to 77 K according to the calculation for crystal octane.42 The C-H dipolar coupling constants obtained by the previous SASS NMR study23 are slightly smaller than those obtained by MM2 calculations. The differences were attributed to the reduction of the dipolar interactions by the C-H vibrations.23 Bond distances determined by various methods are generally different from each other, because molecular vibrations affect measured values in different ways.42-44 In the MM2 calculation,

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Kubo et al.

TABLE 2: Dipolar Coupling Constants for a n-Decane Molecule Enclathrated in the Urea Channelsa (kHz) all-anti conformation (a)

Boltzmann-distributed conformationb (b) 193 K

(c) 298 K

(d) T ) ∞

0.720 -8.832 0.928 0.011

0.628 -7.063 0.551 -0.014

-3.915 0.583 -8.626 1.133 5.175 2.292 -10.031 2.639 -0.469 -0.302 -0.700 -0.649 -9.619

-3.408 0.673 -7.846 1.440 1.669 2.235 -8.986 1.670 0.297 0.008 0.513 -0.024 -8.602

13C -1H 2 k

kc 1 4 6 βd

0.984 -10.742 1.058 0.035

0.866 -10.032 1.026 0.040 1H -1H j k

(j,k)c (2,1) (4,1) (5,4) (6,1) (6,4) (6,5) (7,6) (β,1) (β,4) (β,5) (β,6) (β,7) (β,β)

-5.354 0.968 -10.163 0.768 7.157 1.981 -10.251 3.489 -0.980 -0.873 -0.990 -0.889 -10.206

-4.711 0.773 -9.564 0.910 6.541 2.118 -10.200 3.181 -0.772 -0.622 -0.944 -0.865 -10.047

a b and d are defined by eqs 2e and 2f, respectively, which were k jk calculated by using the atomic coordinates of the n-decane conformations obtained by MM2 molecular mechanics calculations23 and by assuming uniaxial rotation of the n-decane molecules around the urea channel axis. b Dynamic disorder of the Boltzmann-distributed conformations with the populations listed in Table 3 is assumed. c Protons are numbered 1, 2, and 3 for the methyl, 4 and 5 for the 2-methylene, and 6 and 7 for the 4-methylene. d β indicates the 3-methylene protons.

TABLE 3: Populations of the Three Major Conformers of the n-Decane Molecules Enclathrated in the Urea Channelsa population/% conformer

193 K

298 K

AAAAAAA GAAAAAA AGAAAAA

81.0 13.8 3.1

55.1 23.7 9.6

a Calculated using the steric energies of the decane conformations obtained by MM2 molecular mechanics calculations.23 A and G denote anti and gauche conformations around the C-C bond.

the force fields are mainly determined on the basis of the electron diffraction data.43 The C-H distance obtained by MM2 calculations may be different from that obtained by solid state NMR. We define a reduction factor fr of the C-H dipolar coupling as the third power of a ratio of the latter to the former distance. Henry and Szabo calculated the effectiVe C-H internuclear distance of the 2-methylene group in crystal octane at 300 K to be 1.141 Å, taking account of the average of the dipolar coupling over all the vibrational modes.42 The fr value is obtained to be (1.116/1.141)3 ) 0.936, using the C-H bond length (1.116 Å) assumed in MM2 calculations.43 We apply the fr not only to the dipolar coupling for the directly bonded 13C2-1H but also to that between the protons bonded to a carbon and the chemical shift anisotropy of the 2-methylene carbon to simulate the spectrum for room temperature. Although, to be exact, the 13C1H dipolar coupling, the 1H-1H dipolar coupling, and the 13C chemical shift anisotropy ought to be influenced by molecular vibrations in different ways,44 we use the same fr for these interactions for simplicity. The vibrational correction for a spin pair separated by a long distance is unimportant and neglected, the size of the correction being inversely proportional to the third power of the distance. According to the calculation by Henry and Szabo,42 the dipolar coupling for the directly bonded 13C -1H in crystal octane increases by 4.6% as the temperature 2

Figure 4. Expanded 5th ∼ -5th order sidebands of the experimental spectra (thick lines) and of the calculated spectra (thin lines) for n-decane molecules enclathrated in urea. (a) Calculated spectrum for the all-anti conformation. (b) Experimental and calculated spectra for 193 K. (c) Experimental and calculated spectra for room temperature. (d) Calculated spectrum for the infinite temperature. The calculations were made by using the multistep method described in section 3.1 and the dipolar coupling constants listed in Table 2 with the following other parameters: the isotropic shift of methyl protons from that of the methylene protons is -108 Hz, the 13C chemical shift anisotropy of the rotating methylene is -12.3 ppm, the scalar coupling constant for the directly bonded 13C2-1H is 125 Hz, the convoluted Lorentzian line widths are 21 Hz for a, c, and d and 37 Hz for b, and the reduction factors fr (see text) are 0.936 for a, c, and d and 0.956 for b. Magicangle missetting was corrected (see text): θoff used in the calculations are θm + 0.11° for a, c, and d, and θm - 0.05° for b. The vertical bars show 10% of center band intensities. All the calculated spectra have the same center band intensity.

decreases from 300 to 77 K. From this fact, we can assume fr to be 0.956 at 193 K. The observed outer-line splittings of the individual sidebands are 284 and 233 Hz at 298 and 193 K, respectively. They do not agree with the expected splittings 2J ) 250 Hz, where the coupling constant of 125 Hz is the value which we measured for the directly bonded 13C2-1H in liquid n-decane. It may be caused by slight magic-angle missetting: It has been shown that the peaks of JCH split spectra observed under MAS and 1H-1H dipolar decoupling separate more for θ off < θm and get closer to each other for θoff > θm than the separation expected from the JCH value because of the 13C-1H dipolar coupling, where θm is a magic angle, and θoff an off-magic angle.8 In the present case, the sign of the dipolar coupling is changed owing to the uniaxial rotation of the n-alkane molecule so that the spectra observed at room and low temperatures are considered to have been measured at off-magic angles θoff > θm and θoff < θm, respectively. Indeed, the outer-line splittings of the room and low-temperature spectra could be well reproduced by assuming θoff ) θm + 0.11° and θoff ) θm 0.05°, respectively. Figure 4 shows the expanded 5th ∼ -5th order sidebands of the spectra observed at 298 and 193 K and of those calculated for the above four cases, in which at most 5 lines can be seen. This number of lines can be explained by neglecting the rather weak dipolar couplings between the methyl and the 2-methylene protons: We treat the 13C21H2/12C41H2 spin system by representing the proton spin states of each CH2 segment in terms of the triplet states T1, T0, and T-1 and the singlet state S0, where the subscript denotes the total 1H magnetic quantum number mH. The 13C transitions are allowed between the spin states with the same mH and the same reflection-symmetry representation of the proton spin states as shown in Table 4; lines appear at frequencies of 0, (J/4π, and (J/2π from the center of each sideband, explaining the results shown in Figure 4. The lines

n-Alkane Molecules in Urea Channels

J. Phys. Chem., Vol. 100, No. 26, 1996 10859 so that 〈mCS0|Hcp*|m′CS0〉 ) 0, where mC and m′C denote the carbon magnetic quantum numbers. Therefore, the 1H singlet state does not cause cross-polarization:

TABLE 4: 13C Transition Frequencies in the 13C21H2/ 12C 1H Spin System 4 2

mHa symmetryb 2 1

A′ A′ A′′

0

A′

A′′

proton spinc eigenstates mH2mH4 T1T1 T1T0 T0T1 T1S0 S0T1 T1T-1 T-1T1 T0T0 S0S0 T0S0 S0T0

energiesd (freq unit) mC ) +1/2

mC ) -1/2

J/4π J/4π 0 J/4π 0 J/4π -J/4π 0 0 0 0

-J/4π -J/4π 0 -J/4π 0 -J/4π J/4π 0 0 0 0

transitione freq J/2π J/2π, J/4π, 0 J/2π, J/4π, 0 (J/2π, (J/4π, 0

at frequencies (J/4π from the center of each sideband arise from mixing of the 2- and 4-proton spin states by the intersegment dipolar interactions. As the temperature increases, the intensities of the (J/4π lines of calculated spectra decrease as shown in Figure 4, reflecting a decrease of the dipolar coupling between the 2- and 4-proton spins (dH6H4 in Table 2) because of an increase of the gauche content. The whole intensity of each experimental spectrum in Figure 4 was adjusted so that the intensities of the (J/2π lines become close to those of the calculated spectrum. Then, the intensities of the 0 and the (J/4π lines observed are weaker than those of the calculated spectrum. The peak intensities of the experimental center bands are lower than those of the calculated ones by factors of 0.625 and 0.75 at low- and room temperatures, respectively. This feature was also found in our previous studies8,23 and was interpreted by the low cross-polarization rate in the proton S0 state contributing to the center band: in that state, the C-H dipolar interaction vanishes. A similar observation has also been reported in the J-cross-polarization experiments without 1H decoupling.45 Here we discuss the observed low intensities of the center bands more exactly. The FID signal obtained by 1H-coupled CPMAS NMR is given by

S+(t) ) Tr{S+U(t) Ucp(tcp)∑IkxUcp(tcp)-1 U(t)-1} (21) k

where Ucp(tcp) is the time evolution operator during a contact period. When the Hartman-Hahn condition is matched, Ucp(tcp) is given by 0

(22)

cp

Hcp* ) {∑bk(IkzSz + IkySy) - 1/2∑djk(2IjxIkx - IjzIkz - IjyIky)} (23) k

(24a)

For the 1H triplet states, the intensities of the cross-polarization signals are calculated similarly to those in the J-cross-polarization experiments in solution NMR.45,46 We obtain the following equations for the intensities of the lines at J/2π, 0, and -J/2π:

〈-1/2T1|Ucp(I1x + I2x)Ucp-1|1/2T1〉 ) (3b2/8D2) sin2[D{φ(γ) - φ(γ - tcp)}] (3ib/4D) sin[2D{φ(γ) - φ(γ - tcp)}] (24b)

0

a m and m are the 1H and 13C total magnetic quantum numbers. H C Only the case of mH > 0 is shown, because the energy of (-mH, mC) is the same as that of (mH, -mC). b A′ denotes the reflection-symmetric representations of the proton spin states in the 2/4 methylene groups, and A′′ denotes the antisymmetric representation. c T1, T0, and T1 represent the triplet states of the protons in each CH2 segment, and S0 denotes the singlet state, where the subscript denotes the total 1H magnetic quantum number. The singlet state is antisymmetric for the reflection, while the triplet states are symmetric. The total 1H spin state of the 13C21H2/12C41H2 group is symmetric if it contains zero or two singlet states and antisymmetric if one singlet state is involved.d J is the J-coupling constant for the bonded 13C and 1H pair. The isotropic chemical shifts for all the protons are assumed to be the same. e The 13C transitions are allowed between the spin states with the same m H and symmetry representation.

Ucp(tcp) ) exp{-i∫-t f(τ) dτ Hcp*}

〈-1/2S0|Ucp(I1x + I2x)Ucp-1|1/2S0〉 ) 0

j>k

For the singlet state, any component of the 1H total spin is zero,

〈-1/2T0|Ucp(I1x + I2x)Ucp-1|1/2T0〉 ) (b2/4D2) sin2[D{φ(γ) - φ(γ - tcp)}] (24c) 〈-1/2T-1|Ucp(I1x + I2x)Ucp-1|1/2T-1〉 ) (3b2/8D2) sin2[D{φ(γ) - φ(γ - tcp)}] + (3ib/4D) sin[2D{φ(γ) - φ(γ - tcp)}] (24d) where b ) b1 ) b2 and D ) [(3d12/8)2 + b2/2]1/2. If the complex terms in (24b) and (24d), which induce the phase distortion of the (J/2π lines, are ignored, then the intensity ratio of the J/2π, 0, and -J/2π lines is 1.5:1:1.5. In our sample, however, the 2-methylene proton spin states can be mixed with the singlet state of the 4-methylene proton spin states by the intersegment dipolar interactions as shown in Table 4, enabling the cross-polarization of the 2-methylene carbon. The cross-polarization of the center lines can also be caused directly by the dipolar coupling with protons other than the 2-methylene protons, since this process is independent of the spin state of the 2-methylene protons. However, the 2-methylene proton singlet state should still present a lower cross-polarization rate than the triplet states, resulting in weak intensities of the center lines. Also we can similarly explain weak intensities of the (J/4π lines. As shown in Table 4, some of these lines are attributed to the transitions between the singlet S0 state and the triplet T(1 states of the 2-methylene protons, while the (J/2π lines are due to those between the triplet T(1 states. Therefore the (J/4π lines show weaker intensities than the (J/2π lines. The spinning-sideband patterns of the (J/2π lines which were calculated for a model of the dynamical disorder of Boltzmanndistributed conformations are in good agreement with the experimental patterns at both temperatures, as shown in Figure 4. On the other hand, the spectrum calculated for the all-anti conformation agrees badly with the experimental spectrum especially at room temperature: the outer spinning sidebands are more intense in Figure 4a than in Figure 4c. Although in this study no attempt was made to refine either the value of fr or the population of each conformer by comparing the experimental and calculated spectra in detail, the present results are sufficient to support our previous conclusion that the n-alkane molecules enclathrated in the urea channels are in dynamical disorder of various conformations with the Boltzmann distribution. In the present study we have theoretically and experimentally shown that we can observe fine structures in 1H-coupled 13C MAS spectra, if the molecules undergo uniaxially rapid rotation in solids, and if they can be magnetically diluted. Such fine structures involve detailed information on the molecular structure.

10860 J. Phys. Chem., Vol. 100, No. 26, 1996

Kubo et al.

Acknowledgment. We thank Dr. D. Kuwahara for having recorded the first experimental spectrum of this compound. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education.

computation time to great extent. Then, P(0;ωrt) and Q(0) can be obtained from U(0;ωrt), and the spectrum is calculated by using eqs A8-A10. References and Notes

Appendix An Efficient Method of the Powder Average for a MAS NMR Spectrum in a Multispin System In calculating a MAS NMR spectrum, the powder average has to be taken over the three Euler angles, which relates the crystal-fixed frame (C) to the rotor fixed frame (R) as shown in Figure 1. As shown previously for an isolated spin (I ) 1/2) subjected to a chemical shift anisotropy, the average over one (γ) of the Euler angles can be calculated by the Fourier transformation with respect to time.32-34 This arises from the fact that the angle γ is the initial position of the crystal-fixed frame for the rotation around the spinning axis. Here we will show that a similar technique can be used in calculating the MAS NMR spectrum of a dipolar-coupled multispin system. The spin Hamiltonian under MAS should satisfy

H(γ;ωrt) ) H(0;ωrt + γ)

(A1)

The time evolution operator U(γ;ωrt) also satisfies a similar relation:

U(γ;ωrt) ) U(0;ωrt + γ) U(0;γ)-1

(A2)

H is a periodic function of time; therefore, following the Floquet theory,35-39 we can write U as the product of a periodic function P and an exponential operator:

U(γ;ωrt) ) P(γ;ωrt) exp{-iQ(γ)t}

(A3)

From eqs A2 and A3 we obtain

P(γ;ωrt) ) P(0;ωrt + γ) P(0;γ)-1

(A4)

Q(γ) ) P(0;γ) Q(0) P(0;γ)-1

(A5)

The free induction decay S+(γ;ωrt) is given by

S+(γ;ωrt) ) Tr{I+U(γ;ωrt)I-U(γ;ωrt)-1}

(A6)

Using eqs A3-A5 and diagonalizing Q(0) ) TΛT-1, eq A6 can be rewritten as follows:

S+(γ;ωrt) )

exp{i(Λk,k - Λl,l)t}〈k|G(ωrt + γ)|l〉〈l|G(γ)+|k〉 ∑ k,l

(A7)

with

G(x) ) T-1P(0;x)-1I+P(0;x)T

(A8)

We expand G(x) into the Fourier series:

G(x) ) ∑Fm exp(imx)

(A9)

m

Equation A7 can be reduced to a simple form by averaging it over γ:

〈S+(γ;ωrt)〉γ )

|〈k|Fm|l〉|2 exp{i(Λk,k - Λl,l + mωr)t} ∑ k,l,m (A10)

For calculating a powder spectrum, only the time evolution operator U(γ;ωrt) for γ ) 0 has to be obtained, reducing the

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