Dependence of NMR isotropic shift averages and

MOLECULAR PHYSICS, 2000, VOL. 98, NO. 6, 329 ± 342

Dependence of NMR isotropic shift averages and nuclear shielding tensors on the internal rotation of the functional group X about the C± X bond in seven simple vinylic derivatives H2C5 CH± X M. BAADEN1 , P. GRANGER 1 and A. STRICH 2 * 1

2

UMR 7510, Laboratoire de ReÂ sonance MagneÂ tique NucleÂ aireÐ RM3 UMR 7551, Laboratoire de Chimie Quantique, UniversiteÂ Louis Pasteur, 4 rue B. Pascal, 67000 Strasbourg , France

(Received 6 July 1999 ; accepted 12 October 1999 ) The `Gauge Including Atomic Orbitals’ (GIAO) approach is used to investigate the question of intramolecular rotation. Ab initio GIAO calculations of NMR chemical shielding tensors carried out with GAUSSIAN 94 within the SCF± Hartree± Fock approximation are described. In order to compare the calculated chemical shifts with experimental ones, it is important to use consistent nuclear shieldings for NMR reference compounds like TMS. The in¯ uence of rotating functional groups X ÐÐ CH3 , CHO, NO2 , NH2 , CONH2 , COOH or C6 H5 on the shielding tensors in seven vinylic derivatives H2 CÐÐ CH± X is studied; the molecules are propene, acrolein, nitroethylene, ethyleneamine, acrylamide, acrylic acid and styrene. We observe a marked dependence of nuclear shielding and chemical shift on the torsional movement. Di erent Boltzmann averages over the conformational states are considered and compared for gas phase, liquid and solid state NMR. Their applicability to model cases for rigid or freely rotating molecules and for ® xed molecules (e.g. polymers or proteins) with rapidly rotating groups is discussed and simple calculation models are presented. On the basis of this work it can be concluded that intramolecular rotation clearly a ects the observed averages. E ects of up to 2 ppm have been observed for isotropic chemical shifts, and up to 17 ppm di erence have been observed for individual tensor components, for example, of the carboxylic 13 C atom in acrylic acid. The variation of the shielding tensor on a nucleus in a ® xed molecular backbone resulting from an attached rotating group furthermore leads to a new relaxation mechanism by chemical shift anisotropy.

1. Introduction The nuclear magnetic resonance (NMR ) shielding tensor [1] is in¯ uenced by several factors: molecular structure, temperature, electric gradients and ® elds and the environment. Its observation thus leads to precious information about phenomena on the molecular level. Recent enhancements in computing power and storage capacities, as well as in computational methods, have led to a rapid evolution in the ® eld of ab initio NMR calculations [2, 3]. The accuracy of computational results allows one to ® ne-tune calculation methods and to take into account even subtle e ects. Our goal was to investigate the torsional dependence of a molecule’s shielding tensor and the resulting average e ect on general shielding properties. We decided to simplify the problem by studying the intramolecular rotation about one speci® c bond in a series of seven vinylic derivatives.

* Author for correspondence. e-mail: strich@ chimie.ustrasbg.fr

A systematic comparison of the e ect of intramolecular rotation has been accomplished for propene, acrolein, nitroethylene, ethyleneamine, acrylamide, acrylic acid and styrene. The torsional dependence of the chemical shielding has formerly been studied on several classes of chemical compounds and examples can be found in the literature [1, 4± 7]. For example it is well known that the chemical shift non-equivalencies observed for the C¬ shielding in proteins are dominated by torsion angle contributions [7, 8]. Former studies generally report on the relation between chemical shifts and speci® c conformations of a molecule, but do not consider the in¯ uence of an ongoing intramolecular rotation. Our approach was to investigate how such an internal motion could a ect the observed chemical shifts and nuclear shielding tensors, depending on the global motion of the molecule and the time-scale of the rotation. In a recent mixed experimental and theoretical study by Malkin and co-workers, the relation between structure and NMR shielding tensors and coupling constants

Molecular Physics ISSN 0026 ± 8976 print/ISSN 1362± 3028 online # 2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals/tf/00268976.html

330

M. Baaden et al.

in a monosaccharide has been analysed and discussed in great detail [9]. Boltzmann averages are not reported, but were computed from low-energy conformers and were comparable with chemical shifts obtained from the single minimum within 1%. Stahl and co-workers report on Boltzmann averaging for 13 C NMR shifts [10, 11] and carbon ± carbon coupling constants [12] in some hydrocarbons and alcohols. In their investigation , all dihedral angles were varied and the lowest lying conformers were selected for averaging. Thus the e ect of rotation about one single bond cannot be assessed from their results. This issue will be addressed by the present work. The gauge problem [13, 14] rendered the computation of shielding tensors di cult for a long time, as commonly used basis sets for quantum mechanical calculationsÐ in contrast to complete onesÐ do not guarantee the invariance of the results with respect to the chosen gauge [15, 16]. The ® rst calculation methods, including the treatment of the gauge problem, appeared about 20 years ago, see for example HoÈller and Lischka [17, 18]. In this work we made use of the `Gauge Including Atomic Orbitals’ (GIAO) method [19, 20], which has recently become a widely used technique leading to gauge-independen t results [21, 22]. 2. Theory The nuclei in a molecule are subject to the magnetic in¯ uence of the core and valence electrons as well as to e ects such as ring currents or shielding anisotropy of neighbouring atoms. The resulting NMR chemical shift of the nucleus, characterized by the chemical shielding tensor r, is de® ned as the derivative of the molecular energy E with respect to the nuclear magnetic moment l and the magnetic ® eld B: rˆ

@ 2E @l ¢ @B

…

†

:

lˆBˆ0

… 1†

The principal values ¼ii and principal directions of this second-order, asymmetrical tensor [23] can be determined by diagonalizing its symmetrical part. They are accessible by measurements on single crystals or on powder samples [24]. The resulting shielding constant ¼ is measured on an absolute scale and de® ned with respect to the bare nucleus. A compilation of chemical shielding tensor data has been established by Duncan [25]. General information about NMR parameters can be found in [13]. Further equations and de® nitions used in this work follow the ones given by Haeberlen [26]. The ¼ii obey the inequalities: j¼11 ¡ ¼isoj 5 j ¼33 ¡ ¼isoj 5 j¼22 ¡ ¼isoj:

… 2†

The shift anisotropy D ¼ is de® ned by:

D ¼ ˆ ¼11 ¡ ¼iso :

The asymmetry parameter ² is de® ned by: ¼22 ¡ ¼33 ²ˆ : D ¼

…3 † …4 †

3. Methods A variety of methods with appropriate handling of the gauge problem are available. Main methods include the GIAO approach [21, 22], Schindler and Kutzelnigg’s `Individual Gauge for Localized Orbitals method’ (IGLO) [27± 29], the `Localized ORbital local oriGin method’ (LORG) of Hansen and Bouman [30, 31], and an extended `Individual Gauges for Atoms in Molecules’ approach by Keith and Bader [32, 33]. 3.1. Ab initio calculations: the GIAO method In order to avoid gauge dependence in our calculations, we chose the GIAO method and used gauge invariant atomic orbitals. This technique was introduced by Ditch® eld [21, 22] and relies on the use of London orbitals [19, 20]. A detailed comparison with other methods is given in the literature by Wolinski et al. [22] and Facelli et al. [34]. The GIAO type methods are invariant with respect to the choice of the gauge for any basis set size. They are preferred for small to average size molecules, as every atomic orbital has its own local gauge origin. 3.2. Advantages and limitations of ® rst principles calculations The calculation techniques are well established and yield results in good agreement with experiment. A precision of §5 ppm and better can be obtained for 13 C chemical shifts in reasonably large molecules such as sugars [35]. E ects of nuclear motion have to be taken into account to further improve agreement between calculated and experimental values [35]. Another problem may arise in condensed matter systems such as solids, where the calculation has to be extended to a crystal-size system. Some e orts in this direction have so far been undertaken [36, 37]. 3.2.1. Sensitivity of the chemical shift to nuclear motion The shielding constants depend on intramolecular distances, bond angles and the dihedral angles which de® ne the molecule. Rovibrational motion leads to a variation of these parameters, introducing a temperature and mass dependence (isotope shift). The experimentally observed nuclear shielding constant is not the equilibrium value for the minimum energy conformation but rather an average taken over the rotational and vibra-

GIAO for simple vinylic derivatives H2 CÐÐ tional states of the molecule depending on the present isotopes [35]. 3.2.2. Accounting for molecular motion: the rovibrational average The rovibrational average is described in the literature [38± 40]. The NMR parameters are calculated for a number of geometries near the equilibrium con® guration. On the basis of these values the average on the potential hypersurface can be calculated in order to obtain a theoretical value comparable to gas-phase measurements obtained for the zero density limit. The observed shielding e ect depends on the occupied rovibrational state. Thus it depends on the temperature. The magnitude of the e ect can attain up to 18 ppm for the ¯ uorine nucleus [41± 43], º13 ppm for oxygen [38], º3 ppm for carbon [44] and º13 ppm for phosphorus [45]. 3.3. Calculation conditions Ab initio calculations have been carried out with the GAUSSIAN 94 program [46] on an IBM RS 6000, model 43 P, under AIX Version 4.1. The geometrical structure was optimized within the density functional theory (DFT) [47] using Becke’s B3LYP functional [48] and the 6-31G* basis set. From this geometry the nuclear shielding tensors were calculated at Hartree± Fock level of theory with the same basis set. This procedure has been suggested recently by Cheeseman et al. [49]. After consideration of correlated methods such as MP2 or DFT, which are known to be of crucial importance for the treatment of vinyl cations for example [50], we assumed that electron correlation e ects were not signi® cant for the compounds and e ects under investigation. The employed GIAO± SCF method yields good results for 13 C, but correlated methods, as described in [51, 52], should indeed be used to obtain highly accurate chemical shifts for other heavy nuclei such as 15 N and 17 O [49]. Table 1.

CH - X

331

The basis set convergence for GIAO results is much faster than with other methods like LORG [27] or IGLO [30], as shown by Wolinski et al. [22]. In all their results for ® rst-row elements the GIAO isotropic shielding decreased monotonically with increasing basis set quality. They further show that basis sets such as 631G ¤ provide very satisfactory results for relative chemical shifts for organic molecules. This is particularly true for the compounds investigated in this paper, as they only contain ® rst-row elements, and as none of the molecules has a particular electronic con® guration which necessitates treatment of correlation e ects. The convergence behaviour would be more important for secondrow atoms, as the results for these appear to be more sensitive to basis set truncation than ® rst-row atoms.

4. Results The chemical shielding tensors calculated with the GAUSSIAN 94 program are quantum mechanical entities. They are calculated for an isolated rigid molecule in vacuo at 0 K, thus neglecting the zero-point contribution, which can be of several percent as in the case of the water molecule [53]. Nevertheless these values can be compared to experimental ones. The computation of absolute shielding constants carried out for NMR reference molecules (table 1) yields values in good agreement with published theoretical results [49]. The calculated absolute values are within a range of approximately 6% compared to experimental results where they are known. 4.1. Reference molecules If the principal values ¼ii of the shielding tensor r for a nucleus in a molecule are known, the chemical shift ¯ii can be calculated by equation (5). 6 ¯ii ˆ …¼ref iso ¡ ¼ii † £ 10 ;

where ¼ref iso is the isotropic shielding coe cient of the reference. In order to carry out this calculation, the absolute shielding tensor of the reference molecule for

Isotropic values of calculated absolute shielding constants for the and 15 N nuclei (in ppm relative to the bare nucleus).

13

C,

17

O

Calculated isotropic value Nucleus 13 17 15 a b

C O N

Molecule TMS ® gure 1 …b† TMS ® gure 1 …a† H2 O CH3 NO2

…5 †

This work 195.4 200.3 a 323.9 7215.9

Obtained for the minimum energy conformation. No literature data known.

Literature

Experiment

‰49Š

188.1 ‰44Š

323.1 ‰49Š

344.0 ‰54Š

195.1 b

b

332

M. Baaden et al.

d(Si-C)=1.897 Å d(C-H)=1.097 Å

d(Si-C)=1.911 Å d(C-H)=1.096 Å

SI

C

C H

o

Ð (H-C-H)=107.52

(a) Figure 1.

H SI

Ð (H-C-H)=107.22

o

(b)

The two examined structures of TMS. One has the C± H bonds in antiparallel orientation to the central Si± C bond (a), the other has the C± H bonds parallel (b).

a given nucleus must be known. Generally speaking these molecules are water for the 17 O nucleus, tetramethylsilane (TMS) for 13 C, 29 Si, 1 H and nitromethane for 15 N. Our calculated isotropic values of the shielding tensors for the 17 O, 13 C and 15 N nucleus are given in table 1. Experimental data and calculation results taken from the literature are added [44, 49, 54]. The oxygen shielding in water nicely coincides with recent high-level rovibrationally corrected ab initio results and with the proposition of a new absolute shielding reference for the 17 O nucleus [53] given by Vaara et al. who suggest a value of 324 § 1:5 ppm as absolute oxygen shielding for 1 H 2 17 O in the gas phase (300 K). Particular attention should be paid to the case of TMS. For this reference compound in proton and carbon-13 NMR, two structures of Td symmetry exist, as illustrated in ® gure 1. The one in ® gure 1 (a) is the global minimum, and the structure of ® gure 1 (b) is a fourth-order saddle point. The energy of the global minimum (® gure 1 (a)) , taken from the B3LYP calculation, is 27.3 kJ mol¡1 lower than the one for structure (b). The corresponding absolute shielding value for 13 C is 200.3 ppm. We chose this value as the reference chemical shift as it corresponds to a true minimum structure. Cheeseman et al. [49] report a 13 C chemical shielding of 195.1 ppm for TMS, whereas our calculation for the non-minimum structure of ® gure 1 (b) reproduced a value of 195.4 ppm. We veri® ed that the small disagreement is due to a di erence in the representation of the d orbitals. The literature data was obtained by using 6 d functions whereas our calculation was carried out with 5 d type functions. This comparison revealed that many

authors only considered the non-minimum structure (® gure 1 (b)) as a reference compound. We thus performed test calculations on some simple molecules (C2 H4 , C2 H6 , CH3 F and CH3 COCH3 ) in order to decide which isomer should be used as reference. The calculated absolute shielding values were in excellent agreement with computational data given in the literature, but comparison with experiment showed that the choice of the energetically most stable TMS structure as reference leads to an important improvement of the chemical shift values for our test molecules. For example a systematic di erence of about 5 ppm was introduced with respect to the results of Cheeseman et al. [49], leading to better agreement with experiment. The calculated 13 C chemical shifts from this work/Cheeseman et al. [49]/experiment [44] are respectively C¤2 H4 : 121.7/116.6/123.6 ppm, C¤2 H6 : 9.5/4.6/7.2 ppm, C¤ H 3 F: 64.7/59.9/71.3 ppm, C¤ H 3 COC¤ H 3 : 28.8/23.2/30.1 ppm and CH3 C¤ OCH3 : 200.1/194.9/201.2 ppm. These results and related work carried out in our laboratory [55] lead us to suggest that calculated nuclear shielding, obtained with simple methods and basis sets, should not be compared to absolute experimental values directly. One should rather take into account relative values calculated with respect to the right reference chemical shift and compare those to experimental ones. 4.2. Calculation of NMR parameters for rotating molecules The in¯ uence of intramolecular rotation of a group X of atoms around a bond has been studied for a series of vinylic derivatives H2 CÐÐ CH± X (X = CH3 , CHO, NO2 ,

GIAO for simple vinylic derivatives H2 CÐÐ

CH - X

333

Table 2. Selected calculated B3LYP/6-31G* versus experimental (in parenthesis) geometrical parameters for the seven vinylic Ê , angles in degrees). In all structures (except X ˆCH3 ) , all the atoms lie in the vinylic derivatives H2 CÐÐ CH± X (distances in A plane. H2 CÐÐ CÐÐ

Xˆ C

C± X CCX X± O

CH3 1.334

(1.336 ) 1.503

(1.501) 125.1

(124.3 )

CHO 1.338

(1.36 )

1.475

(1.45) 121.1

(122.1 ) 1.215

(1.22)

X± N

NO2 1.327

(1.337) 1.466

(1.470) 120.9

(123.8 )

CH± X NH2 1.342

(1.335 ) 1.378

(1.401 ) 127.1

(124.5 )

1.230

COOH

1.334

1.335

(1.326 )

(1.34)a

1.497

1.483

(1.503) 120.5

(122.0 )

(1.47)a 120.5

(120.0 )b

1.225

(1.218)

C6 H5 1.339

(1.330 ) 1.473

(1.450 ) 127.7

1.214/1.358

(1.222)

(1.24/1.33 )a

1.371

(1.344)

X± C Exp.

CONH2

1.406

[56]

[57]

[58]

NH2 , CONH2 , COOH, C6 H5 ). The calculated geometrical parameters for the minimum energy structure are in good agreement with experimental geometries [56± 63], and typical bond distances and angles are given in table 2. We calculated shielding tensors for di erent dihedral angles ³, measured between the vinylic plane and the plane de® ned by the attached functions. The same procedure was applied for all compounds. At each dihedral angle ³, the nuclear shielding was calculated for two geometrical structures. The ® rst one was obtained by relaxing all geometrical parameters (except ³) during the optimization procedure (see section 5.1.2). In the second calculation, presented in section 5.1.3, all bond distances, angles and dihedrals except ³ were taken from the minimum energy structure. Both protocols have formerly been used for the treatment of dihedral angle e ects on shielding [7]. De Dios and Old® eld [7] have compared results for both types of calculation. They concluded that the e ect of geometry optimization on the torsional dependence of the isotropic shielding was negligible for their systems. Another important point concerns the averaging of anisotropic properties such as the shielding tensor. It requires the de® nition of an appropriate reference frame with respect to the moving molecule, and will be discussed in section 5.1.3. Figure 2 shows the minimum energy structures of the molecules examined. Isotropic chemical shift values for the nuclei of the vinylic part of all molecules of the series are given in table 3. An excerpt of the calculated principal values is given in table 4.

[59]

[60]

a

[61], b [62]

(1.390 )

[63]

Concerning acrolein, H2 CÐÐ CH± CHO, we have studied the in¯ uence of intramolecular rotation around the RC± CHO bond. Several shielding tensors were calculated at a 158 increment of the dihedral angle ³, see ® gure 3. The symmetry of acrolein implies the computation of the nuclear shielding for conformers with 08 µ ³ µ 180 8 only. Tables 5, 6 and 7 contain the calculated values as well as the literature data. Figure 4 gives the conformational energy as a function of the dihedral OC± CH angle ³ for the case of geometry optimized rotamers. The planar conformation of ® gure 2 (b) is the most stable. For propene we considered the rotation around the CC± CH bond. Dihedral angles vary from 08 to 1208 with an increment of 158. The values for 1208 µ ³ < 360 8 can be deduced by symmetry. Intramolecular rotation in nitroethylene , ethyleneamine and styrene occurs around the CC± NO, CC± NH and CC± Ph bond respectively. Only dihedrals from 08 to 908 have to be considered. In acrylamide and acrylic acid, the dihedral angles CC± CN and CC± CO range from 08 to 180 8. In the case of acrylamide the OÐÐ C± NH2 function is kept planar. This can be justi® ed by calculating the rotational barrier in formamide (see table 8 for all rotational barriers ). The 104.6 kJ mol¡1 barrier for OÐÐ CH± NH2 is signi® cantly superior to the rotational barrier in acrylamide which is 23.5 kJ mol¡1 . These values correspond to the energy di erence between the extrema of the conformational energy over the whole dihedral angle range. It has to be added that a rotation with simultaneous pyramidalization of the nitrogen through transition structures [64] is more probable. This would lead to a

334

M. Baaden et al. 7 89 9

5

6 3

3

2

6 2

8

1

1

3

4

6 1

4

5

4

5

5 3

6 1

2

8

7

4

7

8

7

(a)

(b)

(c)

(d)

1-3=C; 4-9=H

3=O

4,5=O; 3=N

3=N

5

4

9

9

10 3

4

6 1

9

5

3

6

2

8

1 7

2

8

(e)

(f)

5=O; 4=N

4,5=O

10 4

2

8 3 5 11

2

1 7 6

7

15 12 13 14 16

(g)

Figure 2. The seven studied vinylic derivatives: minimum energy structures (planar) of (a) propene, (b) acrolein, (c) nitroethylene, (d) ethyleneamine, (e) acrylamide, ( f ) acrylic acid and (g) styrene. The atom labels serve as reference for the text. In all these structures the value of the dihedral angle Y (de® ned in ® gure 3) is equal to 0. Table 3.

Comparison between rigid conformation in the most stable geometry …¼min iso † and intramolecular rotation averaged values …h¼iso i†. Isotropic shift valuesa for seven vinylic derivatives. 13 C Isotropic shift value in ppme

Molecule

Atomb

c ¼min iso

h¼iso id

1 H Isotropic shift value in ppme

jh¼iso i ¡ ¼min iso j

Propene

1 2

84.62 68.95

84.30 69.08

Nitroethylene

1 2

57.79 72.95

57.48 73.50

0.31 0.56

Ethyleneamine

1 2

64.37 122.63

64.32 121.27

0.05 1.35

Styrene

12 13

65.83 87.09

65.68 86.67

0.14 0.43

Acrolein

1 4

66.44 63.64

66.88 64.49

0.44 0.85

Acrylamide

1 2

76.69 65.12

75.60 66.67

1.09 1.55

Acrylic acid

1 2

78.33 61.45

77.40 63.18

0.94 1.73

a

0.31 0.12

Absolute values. The numbering corresponds to the atom numbers given in ® gure 2. c Value obtained for the minimum energy conformation. d The average was calculated on fully geometry optimized rotamers. e Relative to the bare nucleus. b

Atomb

c ¼min iso

h¼ iso i d

4 5 6 6 7 8 6 7 8 15 14 16 5 7 8 6 7 8 6 7 8

27.48 26.60 27.36 25.21 26.40 25.22 28.97 29.16 26.46 26.66 27.21 25.91 26.08 26.06 26.29 25.47 26.46 26.87 25.36 26.21 26.45

27.43 26.60 27.35 25.31 26.43 25.23 28.92 29.11 26.44 26.69 27.20 25.88 26.10 26.12 26.26 25.65 26.52 26.79 25.54 26.26 26.45

jh¼iso i ¡ ¼min iso j 0.05 0.00 0.01 0.10 0.03 0.01 0.05 0.05 0.02 0.04 0.01 0.03 0.02 0.05 0.03 0.19 0.06 0.08 0.18 0.06 0.00

1 …13 C†

1 …13 C†

12 …13 C†

1 …13 C†

1 …13 C†

1 …13 C†

Nitroethylene

Ethyleneamine

Styrene

Acrolein

Acrylamide

Acrylic acid

757.63 84.11 180.79 742.49 100.63 114.26 755.05 86.74 161.39 753.81 87.48 163.17 748.21 84.45 164.60 742.29 107.16 163.30 739.77 107.64 165.40

h¼ii ic 757.35 83.60 180.79 741.69 102.03 113.01 755.00 86.50 161.59 752.94 88.28 162.05 748.19 82.94 164.56 741.19 108.58 162.68 739.20 108.75 165.45

d ¼min ii

0.28 0.51 0.00 0.79 1.40 1.25 0.05 0.25 0.21 0.86 0.81 1.12 0.02 1.52 0.03 1.09 1.42 0.62 0.57 1.11 0.05

jh¼ii i ¡ ¼min ii j

2 …13 C†

2 …13 C†

4 …13 C†

13 …13 C†

2 …13 C†

2 …13 C†

1 …13 C†

Atomb 733.60 113.28 173.58 747.77 78.66 189.58 32.06 152.04 179.99 725.70 102.61 182.69 765.20 72.15 186.36 764.14 73.17 188.22 769.88 69.76 186.83

h¼ii ic

C¤ H2 tensor e

732.44 113.57 172.82 748.67 77.57 189.94 34.36 153.26 180.26 725.46 103.43 183.67 766.61 71.89 185.64 765.25 71.82 188.73 771.22 68.20 187.36

d ¼min ii

1.16 0.28 0.76 0.90 1.09 0.36 2.30 1.22 0.27 0.24 0.82 0.99 1.41 0.26 0.72 1.10 1.35 0.52 1.34 1.57 0.53


Eigenvalues in ppm

RXHCÐÐ

3 …13 C†

3 …13 C†

2 …13 C†

6 …13 C†

3 …15 N†

3 …15 N†

3 …13 C†

Atomb

b

Absolute values. The numbering corresponds to the atom numbers given in ® gure 2. c The tensor average was calculated from structures with the same minimum energy geometry except for the dihedral angle. d Value obtained for the minimum energy conformation. e Relative to the bare nucleus.

a

2 …13 C†

Propene

Atomb

e

Eigenvalues in ppm

RHC¤ ÐÐ CH2 tensor

198.21 173.93 172.97 93.39 7300.70 7412.96 264.08 211.81 189.23 182.82 44.37 734.34 108.71 5.68 778.63 754.3 43.77 117.64 748.88 52.89 101.34

h¼ii ic

CH2 tensor

197.78 172.97 171.96 105.38 7310.02 7412.19 263.54 188.68 188.68 192.68 35.41 733.84 111.30 10.36 786.23 755.56 42.45 121.67 766.24 67.03 104.97

d ¼min ii

0.42 0.96 1.01 11.99 9.32 0.77 0.54 0.55 0.55 9.86 8.96 0.50 2.59 4.68 7.60 1.26 1.32 4.03 17.36 14.14 3.63


Eigenvalues in ppme

RX ¤ HCÐÐ

a Comparison between rigid conformation in the most stable geometry …¼min ii † and intramolecular rotation averaged values …h¼ii i†. Tensor parameters for the series of vinylic derivatives.

Molecule

Table 4.

GIAO for simple vinylic derivatives H2 CÐÐ CH - X 335

336

M. Baaden et al.

O

q

H

Figure 3. Illustration of the rotational movement of acrolein and de® nition of the torsional angle ³. Table 5.

H

Comparison of calculateda

13

CH2 C*HCHO (1)

CH2 CHC*HO (2) C*H 2 CHCHO (4)

H

C magnetic properties of acrolein with the literature data (in ppm).

¯iso

D ¼

Comments

133.9 126:3 ¡ 135:6 137:7 § 0:2 136.4 188.5 184:8 ¡ 192:2 192:1 § 0:5 136.6 118:3 ¡ 136:6 136:7 § 0:2 136.0

7114.6 ¡120:9= ¡ 114:6

Figure 2 (b)c

Nucleus b

H

d

Exp. ‰65Š Exp. ‰66Š Figure 2 (b)c

99.5 99.5/114.9

d

Exp. ‰67Š Figure 2 (b)c

7130.3 ¡130:3=125:8

d

Exp. Exp.

‰65Š ‰66Š

¯iso is the calculated isotropic shift, D ¼ is the shift anisotropy, see the de® nition in section 2. a With respect to TMS as reference molecule (200.3 ppm were used for the absolute chemical shielding of 13 C) . b The numbers in parentheses correspond to the atom number given in ® gure 2 …b†. c Value observed for the minimum energy conformation. d Minimum/maximum of the observed values for all calculated rotamers. Table 6. Average isotropic shift values (Boltzmann’s law) for acrolein (in ppm relative to the bare nucleus). a

D ¼iso ˆ

D ¼iso .

Table 7. Comparison between rigid conformation and intramolecular rotation for acrolein. Tensor parameters (in ppm relative to the bare nucleus). a Tensor components/ppm

h¼iso i

b ¼min iso

jh¼iso i ¡ ¼min iso j

C (1)

66.88

66.44

0.44

0.7

C (2)

11.94

11.81

0.14

1.1

C (4)

64.49

63.64

0.85

1.3

O (3)

7304.45

7304.20

0.16

0.1

H (5)

26.10

26.08

0.02

0.1

H (6 )

23.01

23.03

0.03

0.1

H (7)

26.12

26.06

0.05

0.2

H7c

H (8)

26.26

26.29

0.03

0.1

H8c

c

Nucleus

a

h¼iso i

%

The average was calculated for a temperature of 300 K. Each rotamer was fully geometry optimized except for its dihedral angle ³. b Value obtained for the minimum energy conformation. c The numbers in parentheses correspond to the atom number given in ® gure 2.

b

Nucleus

¼iso

C1c

66.44 66.95 11.80 11.92 63.64 64.43 26.08 26.10 26.06 26.12 26.29 26.26

d

C2c d

C4c d

H5c d d d

¼11 748.19 748.21 111.30 108.54 766.61 765.20 29.61 29.57 22.99 23.09 22.63 22.68

¼22 82.94 84.45 10.35 5.45 71.89 72.15 25.20 25.34 26.27 26.31 26.74 26.61

¼33 164.56 164.60 786.26 778.23 185.64 186.36 23.43 23.40 28.92 28.95 29.50 29.49

D ¼

²

7114.63 7115. 16 99.50 96.62 7130. 25 7129.64 3.53 3.47 73.07 73.03 73.66 73.58

0.71 0.70 0.97 0.87 0.87 0.88 0.50 0.56 0.86 0.87 0.76 0.81

² is the asymmetry parameter as de® ned in section 2. a Absolute values. b The numbers correspond to the numbers given in ® gure 2. c Values obtained for the minimum energy conformation. d Value obtained for the rotating molecule.


CH - X

337

40 35

E /k J.m ol

-1

30

20 15 10

Figure 4. Conformational energy of acrolein obtained by geometry optimization (B3LYP/631G*) for all parameters except the ® xed dihedral angle ³.

Table 8.

25

5 0

0

20

40 60 80 100 120 14 0 OC C H d ih ed r a l a n gle q /d eg r ees

Rotational barriers for the series of vinylic derivatives. Barrier/kJ mol¡1

Molecule

B3LYPb

Hartree± Fockc

Structurea

Propene Acrolein Nitroethylene Ethyleneamine Acrylamide Acrylic Acid Styrene Formamide

8.6 38.9 28.0 45.4 23.6 30.8 18.4 104.6

8.6 34.1 30.1 36.7 23.2 31.2 11.5 95.7

(a) (b) (c) (d) (e) (f ) (g)

a

See ® gure 2. Geometry optimization. c Magnetic properties calculation. b

rotational barrier of about 62.8± 71.2 kJ mol ¡1 [64] depending on the employed basis set. The COOH group in acrylic acid is also considered to be planar, with the hydrogen atom positioned between the two oxygen atoms.

5. Discussion We have chosen the model case of acrolein in order to explain our results for the series of vinylic derivatives. Table 5 gives the calculated chemical shifts for 13 C. We indicate the results for the minimum energy conformation as well as the range of variation. Experimental values are given for comparison [65± 67]. The agreement is excellent. In the following sections we will discuss the in¯ uence of rotational motion on such shielding constants.

16 0

18 0

5.1. T orsional dependence of NMR parameters and consequences for experimental observables Our calculations show the in¯ uence of the molecular conformation on nuclear shielding. In the case of acrolein this e ect is enhanced due to the CÐÐ O double bond. The variation of selected isotropic shielding values ¼iso and chemical shift anisotropies D ¼ (de® ned in (3)) for carbon-13 and oxygen-17 is illustrated in ® gures 5 (a)± (d). Average isotropic shift values obtained by the procedure described in section 5.1.2 are indicated as dotted lines in the same ® gures. From these plots it is obvious that a marked dependence of NMR parameters on the dihedral angle ³ exists. The indicated average isotropic shift is close to the value for the minimum energy conformation. In the case of C¤ H2 CHCHO (® gure 5 (c)) a change in the sign of D ¼ occurs at a dihedral angle of 1808, as the labelling of the principal axes changes according to convention (2). Thus ¼11 becomes ¼33 for ³ ˆ 180 8 and vice versa but no dramatic change in the ¼ii values takes place. For each compound the rotational barrier was calculated as the energy di erence between the most stable and the most labile conformer. The values are given in table 8 for the case of geometry optimized structures. In the following sections, we describe how the nuclei in the moving molecule interact with the magnetic ® eld. We di erentiate several cases: rigid molecules, e.g. in the solid state, molecules in vacuo or in solution which are freely rotating, and ® xed molecules with rapidly rotating groups. 5.1.1. A simple case: the rigid molecule Assuming that the molecule is rigid, the existence of di erent conformations can be neglected. Experiment-

338

M. Baaden et al. 75

-1 1 4

72

-1 1 6

16

11 4 11 1

14

10 8 12

69

-1 1 8

10 5 10

66

10 2

-1 2 0 iso

8

30

70

110

D s

150

99 iso

30

70

(a)

11 0

1 50

D s

(b) 120

80

90 60

76

-2 8 0

76 0

-3 0 0

30 0

72

74 0 -3 2 0

-3 0 68

-6 0 -9 0

64

-1 2 0 iso

30

70

110

150

D s

(c)

72 0

-3 4 0 -3 6 0

70 0 iso

30

70

11 0

1 50

D s

(d)

Figure 5. The isotropic values ¼iso (~, left-hand scale) and anisotropies D ¼ (*, right-hand scale) for 13 C (a)± (c) in acrolein as a function of the dihedral angle characterizing the rotation. Isotropic shielding constant for (a) CH2 C¤ HCHO (1), (b) CH2 CHC¤ HO (2), (c) C¤ H2 CHCHO (4). (d) gives the isotropic value and anisotropy for oxygen in acrolein: CH2 CHCHO¤ . Dotted lines correspond to average isotropic shift values as de® ned in section 5.1.2.

ally this could be the case of a low temperature NMR experiment or of a solid powder sample with molecular motions being reduced to a minimum. Under these conditions it should be su cient to take into account only the most stable structure. For acrolein, this is the planar conformation of ® gure 2 (b). The calculated ¯iso and D ¼ values are given in table 5. 5.1.2. Isotropic averaging: rapid molecular rotation in vacuo or in solution This case has to be considered for an ensemble of freely rotating molecules. Experimentally it can be observed by variable temperature NMR spectroscopy in the gas phase. Furthermore such a situation occurs in the liquid state, where the Brownian motion leads to a freely tumbling motion of the considered solute. As the molecules move rapidly within the ® xed ® eld, the in¯ uence of an average tensor is observed. Only the 0 rank scalar isotropic shift value ¼iso can be measured. Instantaneously, several rotamers with di erent shift values coexist and give rise to a distribution of ¼iso . A statistical average can be computed under the assumption that

collisions between molecules can be neglected. As an example we will explain the case of acrolein. In order to determine the average over the complete dihedral angle range from [0,360 [, we use the calculated shifts ¼iso ;i for the 13 conformers i ˆ 1;. . . ;13 with ³ ˆ 0, 15, 30;. . . ; 180 8 respectively. Each chemical shift is weighted with a normalized factor pi ˆ …m=N † exp …¡D Ei =kT †, obtained by Boltzmann’ s law after calculating the energy di erences D Ei with respect to the minimum energy structure. The contributions for angles 08 and 1808 are counted once (m ˆ 1) and for the intermediate P ˆ 1. angles twice (m ˆ 2), ful® lling the condition 13 iˆ1 pi N is the resulting normalization constant, k is the Boltzmann constant and the temperature was set to T ˆ 300 K for our calculation. Figure 4 shows a plot of D E versus the dihedral angle ³. We can then calculate the average isotropic shielding h¼iso i by equation (6 ): h¼iso i ˆ

13 X iˆ1

pi ¢ ¼iso;i :

…6 †

For this calculation it is preferable to deal with data of geometry optimized structures in order to use the

GIAO for simple vinylic derivatives H2 CÐÐ most accurate energy values for the determination of the populations. This implies the assumption that the rotation speed is such that the molecule can accommodate its geometry at intermediate structures (centrifugal distortion [68] and inertia e ects are neglected). The above computational procedure has been applied to the case of acrolein. The calculated isotropic values are given in table 6 and compared to the minimum energy conformation. The di erence is about 1% for carbon and a magnitude lower for oxygen and hydrogen. The results for the other molecules of the series of vinylic derivatives are given in table 3. It can be seen that the di erence between the two calculations can attain 1.7 ppm for 13 C in the case of acrylic acid and 0.2 ppm for 1 H in acrylamide. This illustrates well that rotation may lead to a noticeable variation of the chemical shift in the gas phase or in the liquid state. With respect to the rigid molecule (section 5.1.1) it can be concluded that the chemical shift for ¯ exible species in solution is not equal to one third of the trace of the values obtained for the solid state. 5.1.3. Anisotropic averaging: fast intramolecular rotation in a static environment This case applies when the intramolecular rotation of a speci® c group in a molecule is much faster than the movement of the whole molecule with respect to the magnetic ® eld. A similar phenomenon might be observed for a polymer with freely rotating functional groups or side chains. The series of vinylic compounds studied in this work can be taken as a simple model for such a macromolecule, where the vinylic part represents the rigid polymer backbone. An interesting example for molecular rotation with modi® cation of the experimentally observed shielding tensor has been pointed out by Fleischer et al. [69]. The 31 P shielding tensors in the P4 O6 S molecule have been measured and a complementary theoretical study seems to con® rm the rotation of the molecule about the PS-bond axis. Hexamethylbenzene is also known to rotate about its C6 axis [70]. On the NMR time-scale one can only observe average magnetic properties, corresponding to the con® gurations sampled during NMR measurement. The acquisition time is such that several ten thousand rotations occur before a signal can be processed [3]. In our case we assume that the rotating group is attached to a nearly ® xed molecule. The calculated average is thus no longer isotropic but contains the contribution of each rotamer via its individual anisotropic shielding tensor. First, an average tensor (Boltzmann’s law) has to be calculated to account for the average in¯ uence of the moving part, by applying a similar procedure as described in section 5.1.2 to the whole shielding tensor instead of its isotropic

CH - X

339

value only. Such averages and corresponding minimum energy values are given in table 7 for acrolein. Second, a powder average is taken over this average tensor, in order to account for the di erent possible orientations of the whole polymer with respect to the magnetic reference frame. The average shielding tensor taken over the rotational states can in principle be calculated after performing a unitary coordinate transformation on each rotamer, in order to reorient it into an equivalent position with respect to the other ones. Therefore a reference frame has to be de® ned, such that the mutual orientations of the rotamers correspond to the `real’ physical movement. Assuming that the vinylic (`polymer’ ) part is immobile during NMR acquisition, this task becomes obvious for non-geometry optimized structures, because the vinylic part can be superposed exactly for each rotamer. Unfortunately this is not the case when a geometry optimization has been carried out for each dihedral angle. Bond angles and distances of the molecule and in particular of the quasi-immobile vinylic part vary, and a straightforward superposition of the structures is no longer possible. A general reference frame exists for molecules subject to rovibrational movement. This so-called Eckart frame [71] has formerly been used for the calculation of rovibrational averages for various NMR parameters and for other anisotropic properties [38, 53]. One important point is that a well-de® ned equilibrium geometry is needed to ful® l the so-called Eckart conditions. An adaptation of the Eckart frame to a partly immobile molecule has not yet been undertaken, and remains a challenging task. All tensor averages in this work were calculated from non-geometry optimized rotamers. Table 4 shows the corresponding eigenvalues and compares them to the principal values for the minimum energy structure. For this purpose, we have considered the three immobile RHC¤ ÐÐ CH2 , RHC ÐÐ C¤ H2 and RX ¤ HC ÐÐ CH2 (X ÐÐ 13 C, 15 N) nuclei of each vinylic compound. Di erences up to 17 ppm on the X atom and up to 2.3 ppm on the carbons could be observed. The variation of the eigenvalues during torsional motion hints at a new relaxation mechanism by chemical shift anisotropy at least for heavier nuclei. Relaxation processes arise from molecular motions which create magnetic or electric modulations at the nucleus. The magnetic anisotropic relaxation mechanism arises from the rotation of a disymmetrical shielding tensor. Another magnetic modulation may also arise from an internal rotation which leads to the modulation of the magnetic shielding tensor without the rotation of the tensor axes. This is a new relaxation process which may be observed experimentally if this modulation is large enough. Our work

340

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shows that for carbon, nitrogen and oxygen such a modulation occurs when internal rotation exists. In the cases studied here, the e ect is too small compared to the usual e ect of the other mechanisms. Larger e ects, which perhaps can be observed, may exist with transition metals, since they have large shielding tensors and large anisotropies. On the basis of the computed average tensor a powder average can be calculated for all possible orientations of the macromolecule with respect to the magnetic ® eld. The calculation was carried out with a program based on the algorithm of Alderman et al. [72], developed in our laboratory. We have calculated powder spectra for the RHC ˆ C¤ H2 carbon in ® ve molecules of the series: acrolein, ethyleneamine, acrylamide , styrene and acrylic acid. Figure 6 (a) shows the superposition of two simulated spectra visualized by the FELIX 95 program [73] for the case of acrylic acid. One spectrum is derived from the tensor of the rotating molecule and the other from the tensor of the minimum energy structure. The di erence between both spectra is very small, which is also true for the other spectra. They have therefore been omitted. A remarkable di erence of 17 ppm between the two spectra is observed for the carboxylic 13 C nucleus in acrylic acid, as illustrated in ® gure 6 (b). To summarize, the nuclear shielding in the rotating molecule di ers from the one in the rigid molecule. In the majority of cases a noticeable di erence of at least some ppm is observed for the principal values of the shielding tensor. Consequently the powder spectra show a di erence between the molecule subject to intramolecular rotation and the minimum energy structure.

6. Conclusion and outlook Ab initio chemical shielding calculation methods have reached a precision that is generally su cient to describe experiment. In our work the error is below 10% for chemical shift values, which are taken with respect to a reference compound. Chemical shielding tensors are strongly in¯ uenced by molecular geometry and furthermore by conformational changes (cf. TMS) . In order to further improve agreement with experiment, several e ects, which are not calculated by quantum chemical programs, have to be taken into account: rovibrational movement, the in¯ uence of the environment and torsional movement. This work shows that isotropic values in solution are in¯ uenced by geometry variation due to intramolecular motion. We have furthermore shown that torsional motion leads to a variation of chemical shielding in both ® elds of liquid and solid state NMR. The results on intramolecular rotation lead to many new questions concerning the time-scale of the motion, the geometry variations of the intermediate structures and the in¯ uence of e ects such as centrifugal distortion. Comparison with experiment is desirable but di cult to undertake because an important number of e ects are superposed. A model compound with strongly preferred intramolecular rotation, which is accessible for experimental measurement, has to be found. The variation of the chemical shielding tensor also implies the existence of a formerly unknown relaxation mechanism by chemical shift anisotropy on a ® xed nucleus in a solid bound to a rotating group. This e ect depends on the range of the chemical shift variation, which may be experimentally checked with nuclei with sensitive chemical shift giving rise to larger e ects than for 13 C.

Figure 6. Theoretical powder spectra calculated with the algorithm of Alderman et al. [72] using 2048 frequency intervals and N ˆ 32 segments. Two simulated spectra for acrylic acid are shown, one for the RHCÐÐ C¤ H2 nucleus (a) and one for HOOC¤ ± CHÐÐ CH2 (b). The spectra for the rigid molecule (&, minimum energy structure) and the molecule subject to rapid rotation (°, tensor average over non-geometry optimized rotamers) are superposed.

GIAO for simple vinylic derivatives H2 CÐÐ Registry No. C3 H6 , 115-07-1 ; C3 H4 O, 107-02-8; C2 H 3 NO2 , 363864-0 ; C2 H5N, 593-67-9; C3 H5 NO, 79-06-1; C3 H4 O2 , 7910-7 ; C8 H 8 , 100-42-5; CH3 NO, 75-12-7 ; C4 H 12 Si, 75-763; CH3 NO2 , 75-52-5 ; H2 O, 7732-18-5 . We thank both referees of this paper for their constructive comments and criticisms, which helped us to clarify some aspects and to include additional useful information. References

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