Journal of Molecular Structure (Theochem), 304 (1994) 45-5 1

0166-1280/94/$07.00 0 1994 - Elsevier Science Publishers B.V. All rights reserved 45

Computationa imidazoles

1 study of

Michael Meyer

i midazole and methyl

GBF, Gesellschaft fiir Biotechnologische Forschung, Department of Molecular Structure Research, Mascheroder Weg 1, D-38124 Braunschweig, Germany

(Received 23 July 1993; accepted 10 August 1993)

Abstract

A semiempirical and ab initio study of the structures of imidazole, methyl imidazoles and the corresponding protonated and deprotonated ions is presented. The energies of the different isomers and tautomers are discussed and the protonation and deprotonation energies are given. Dipole moments and the barriers hindering internal rotation of the methyl group were determined. The heats of formation were derived from an isodesmic reaction - _ scheme and semiempirical calculations.

Introduction

Imidazole (CsH4N2 (IA)), is a five-membered

heterocyclic system with basic and acidic proper-

ties. The IA ring of histidine plays an important

role in proton transfer processes at the active site

of certain enzymes. Thus IA may be used as a

simple model to study enzymatic modes of action

with semiempirical and ab initio methods [l]. The

structure of this molecule is available from micro-

wave spectroscopy [2] and computational studies

[3]. Nitrogen nuclear coupling constants, the

dipole [2] and molecular quadrupole moments [4],

the heat of formation [5] and the proton affinity [5,

61 have been determined experimentally. Less is

known about the methyl imidazoles (C4H6N2

(MIA)). The microwave spectra are more difficult

to analyse because the methyl group gives rise to a

torsional fine structure in addition to the nitrogen

nuclear quadrupole hyperfine structure. Further-

more, experimental investigations of 4-MIA and

5-MIA are complicated by the tautomeric equi-

librium of these molecules. So far the proton

affinities have been determined and the structures

have been calculated using a minimal basis set [6].

SSDI 0166-1280(93)03457-I

A study of IA, the MIA and the corresponding

protonated and deprotonated ions is presented

here. The structures were calculated using semiem-

pirical and ab initio methods based on the

Hartree-Fock (HF) theory. The accuracy of the

calculated structure of IA was checked by compar-

ing it with the experimental results. The structures

of the ions and the substituted molecules were

compared with the parent species IA. These calcu-

lations give insight into the methyl torsional angles,

referred to the lowest energy, which are difficult to

estimate from simple chemical arguments. The

influence of different basis sets and of correlation

effects on the energies was investigated in order to

determine the stabilities of the different tautomers

and isomers and to calculate the protonation and

deprotonation energies. The heats of formations of

the MIA were derived from ab initio calculations

and experimental data using isodesmic reaction

schemes and from semiempirical calculations.

Computational details

The ab initio calculations were carried out using

the GAUSSIAN 90 [7] and GAMESS [8] programs. All

46

geometries were optimized at the HF level, employ-

ing the 6-31G* split valence basis [9] with polariz-

ation functions on the heavy atoms. This level of

theory is generally considered to be sufficient to

provide consistent molecular geometries [lo].

Single-point calculations at the HF/6-3 lG*

geometries were carried out using the triple split

6-3 11 G** basis set with additional polarization

functions on hydrogen atoms [l 11. For the calcu-

lation of the deprotonation energies, the 6-3 1 + G*

basis set supplemented with diffuse functions on

the heavy atoms [12] was applied. For an estimate

of correlation effects, Moller~Plesset perturbation

theory of second order (MP2) was used with the

HF/6-31G* structures. For the parent species,

MP3 and MP4(SDTQ) calculations of the energies

were used to estimate the influence due to single,

double, triple and quadruple excitations on the

energies.

All semiempirical calculations were carried out

using the MNDO [14]. AMI [15] and PM3 [16]

methods of MOPAC 5.0 [13].

Results and discussion

Molecular structures

The structure of IA obtained with the HF/6-

31G* method is given in Table I together with

the experimental one derived from microwave

spectroscopy [2]. The computed structure is close

to the results of Mb et al. [3], who obtained a better

reproduction of the bond lengths and larger devi-

ations of the angles with the 6-31G basis. The mean

absolute deviation between the experimental

and calculated heavy atom bond lengths of the

imidazole ring is 0.014A and the error in the

angles is 0.2’. The corresponding errors of the

semiempirical methods MNDO, AMI and PM3

are 0.024, 0.028 and 0.024 A and I. 1, 0.6 and

2.5’, respectively. All semiempirical methods over-

estimate the heavy atom bond lengths, whereas the

ab initio calculations underestimate them. N-H

and CH bonds obtained via ab initio calculations

are a little too short.

M. Me,wr:J. Mol. Smut. (Theochcmj 304 (1994) 45-51

Table 1

Experimental and ab initio structures of imidazole and its

protonated and deprotonated ions

C~H~INI” GH,Nl C~HSNZ + C,H,NY

Bond length (k)

NlLC2 1.364 1.349 1.313 1.327

C2-N3 1.314 1.289

N3-C-4 1.382 1.372 1.382 1.357

C4GC5 1.364 1.350 1.340 1.372

C5-Nl 1.377 1.372

N1 HI 0.998 0.993 I .ooo

Bond anglr ideg)

N 1 -C2-N3 112.0 112.2 108.0 117.0

C2-N3-C4 104.9 105.3 109.5 102.3

N3-C4-C5 110.7 110.5 106.5 109.2

C4-C5 Nl 105.5 105.2

CS-Nl C2 106.9 106.8

’ Experimental value [2].

The Cl,. symmetric structures of the protonated

cation and the deprotonated anion derived from IA

are also compiled in Table 1. The protonation of

the N3 nitrogen atom of IA causes an increase in

the endocyclic angles X2N3C4 and /C5N 1 C2 and

a decrease in iNlC2N3. A removal of the hydro-

gen at Nl of IA leads to the opposite structural

changes in the anion. The bonds Nl-C2 and C2-

N3 of both ions are shorter than they are in the

neutral molecule. The heavy atom geometries of

the MIA are given in Table 2. The effects of methyl

substitution on the ring geometry are very small.

Substitution at carbon atoms leads to a slight elon-

gation of the imidazole ring along the bond to the

methyl carbon atom and a corresponding narrow-

ing of the ring. The endocyclic angle between the

carbon atoms, where the substitution takes place,

and the adjacent atoms is decreased. Both bond

lengths to the neighbouring atoms of the ring

are increased. Similar substitution effects have

been derived from the experimental structures of

benzene and toluene [ 171.

The complete structures of all MIA are shown in

Fig. 1 to indicate the torsional angles of the methyl

groups referring to the lowest energies. On the basis

of a previous calculation without polarization

M. Meyer/J. Mol. Struct. (Theochem) 304 (1994) 45-51

Table 2 Energies Structures of l-, 2-, 4- and 5-methylimidazoP

41

l-MIA 2-MIA 4-MIA 5-MIA

Bond length (A) Nl-C2 1.349 C2pN3 1.291 N3-C4 1.369 C4&C5 1.352 C5pNl 1.372

NlLC, 1.444 c2P& c4-c, c5w&

Bond angle (deg) NlPJ2pN3 112.9 C2pN3-C4 105.0 N3%C4-C5 110.3 C4&C5-Nl 105.8 C5-Nl X2 106.0

C,-NlWZ5 126.9 C,+ZZpNl c,-c4-c5 c,-c5wZ4

1.353 1.356 1.352 1.292 1.289 1.286 1.374 1.376 1.374 1.347 1.352 1.352 1.375 1.375 1.376

1.495 1.496

1.494

111.3 112.2 112.1 105.8 105.7 105.2 110.5 109.7 111.0 105.1 105.7 104.5 107.3 106.6 107.2

122.9 129.0

132.5

a HF/6-31G*.

functions [6], it was concluded that methyl

groups in the Q position to the basic nitrogen atom

N3 have a minimum-energy conformation that

contains one hydrogen atom in the ring plane

near to the nitrogen. However, the calculations

with the enlarged basis show (in agreement with

MNDO, AM1 and PM3) that this is only valid

for 2-MIA, and not for 4-MIA. As shown in

Fig. 1, the in-plane hydrogen atom of the methyl

group of 4-MIA is directed towards C5 and not to

N3.

The total energies of all ionic and neutral

molecules are listed in Table 3. The ions derived

from 4-MIA and 5-MIA are identical. The proton-

ation and deprotonation energies, defined as the

energy difference between the protonated and the

deprotonated species, are listed in Table 4. At

the HF level, the calculations with the 6-31G* and

6-31 lG* basis sets lead to similar results for

energy differences. The protonation energies of

IA are lower than those previously calculated

without polarization functions [3,6]. MP2 correc-

tions reduce the protonation and deprotonation

energies further to 234.0 kcalmol-‘. From the

data of IA it may be seen that both energy differ-

ences oscillate with the order of the perturbation

treatment. The HF method provides an upper limit

and the MP2 method provides a lower limit for MP

calculations of higher order. Hence, only the

former methods are applied to MIA. When calcu-

lating deprotonation energies, diffuse functions

on the heavy atoms must be taken into account.

Additional diffuse functions on hydrogen do not

influence the energies significantly.

The substitution of hydrogen with methyl groups

cause a small increase in the deprotonation protona-

tion energies of approximately 1 kcalmoll’ The

protonation energies rise by 4 kcal mol-’ . Hence IA

and 4-MIA seem to be interchangeable in model

studies of the protonation of histidine.

The calculated energies of the MIA are also given

in Table 3. The total energy of 4-MIA is only

0.16 kcal mol-’ smaller than the energy of the tauto-

merit species 5-MIA. Single-point calculations

Fig. 1, The structures of l-, 2-, 4- and 5-methylimidazole.

Tab

le

3

Tot

al

ener

gies

’ (H

, kc

al

mol

-‘)

IA

I-M

IA

2.M

IA

4.M

IA

S-M

IA

CxH

,Nl

CIH

~N:

C,H

,N;

CdH

,Nz

C,H

,N:

CdH

,N,

CdH

,N:

ChH

,N;

CIH

,N~

CdH

,N:

C,H

,N,

Czt

HaN

z

HF/

6-3l

G*

-224

.814

43

-225

.196

35

-224

.224

70

-263

.846

36

-264

.235

52

-263

.857

92

-264

.248

53

-263

26

562

-263

.855

59

-264

.243

09

-263

26

500

-263

.855

34

HF/

6-31

+

G*

-224

.822

82

-224

24

834

-263

86

561

-263

.288

36

-263

.863

98

-263

.286

99

-263

.863

40

HF’

b-31

f +G

* -2

24.8

2293

-

224

2484

0

HF

,6-3

1 I’

S**

-224

.869

86

-225

25

309

-224

28

080

-263

.908

49

-264

.299

45

-263

.922

23

~- 26

4.31

449

-263

.330

75

-263

.919

85

~264

.308

93

-263

.329

47

-263

.919

25

MP2

!6-3

lG*

-225

.538

40

-225

.91

I38

-224

.959

98

-264

.707

29

-265

.087

19

-264

.717

61

-265

.098

12

-264

.137

32

-264

.716

09

-265

.094

87

-264

.136

99

-264

.715

68

MP3

:6-3

I G

* -2

25.5

3 I I

1

-225

.909

17

-224

.946

65

MP4

/6-3

lG*

-225

.570

28

--22

5.94

551

-224

.988

41

a A

t th

e H

F/6-

3lG

* ge

omet

ry

M. Meyer/J. Mol. Struct. (Theochem) 304 (1994) 45-51 49

Table 4

Protonation and deprotonation energies

IA

Protonation energy (kcalmol-‘)

HF/6-31G* 239.1

HF/6-311G** 240.5

MP2/6-31G* 234.0

MP3/6-31G* 237.2

MP4/6-3 1 G* 235.5

Deprotonation energy (kcalmol-‘)

HF/6-31G* 370.1

HF/6-31 + G* 360.5 HF/6-31 + + G* 360.5

HF/6-311G** 369.6

MP2/6-31G* 363.0 MP3/6-31G* 367.2

MP4/6-31G* 365.1

l-MIA 2-MIA 4-MIA 5-MIA

244.2 245.1 243.2 243.3

245.3 246.1 244.2 244.5

238.4 238.8 231.1 231.9

311.1 310.6 310.4

362.2 362.1 361.1

311.2 370.5 310.1

364.1 363.4 363.1

confirm this order of the energies and yield

slightly larger energy differences of 0.38 kcal mol-’

(HF/6-31 lG**) and 0.27 kcalmoll’ (MP2/6-31G*).

The potential V(a) of a methyl top rigidly

rotating relative to a frame may be expressed as a

Fourier series of the torsional angle cy:

l’(o) = V,/2( 1 - cos 3~) + V6/2( 1 - cos 6cr) + .

(1)

The barrier height of the three-fold potential is

given by Vs. The term I’,, which modifies the

shape of the potential, is generally much smaller

than I’,. It is possible to estimate V’s by calculating

the energies using the structures indicated in Fig. 1

and then calculating the energies with the methyl

groups rotated by 180”. Barrier heights of 0.57,

0.53, 1.08 and 1.45 kcalmol-’ have been deter-

mined this way for l-, 2-, 4- and 5-MIA, respec-

tively. Similar barrier heights have been observed

experimentally for 2-, 4- and 5-methyl oxazole. The

barrier heights of these molecules are 0.7200 (35),

1.2237 (35) and 1.3663 (37) kcal mol-‘, respectively

]181. Except for 2-MIA, the conformations of the

methyl groups referring to the energy minima are

generally the same for the ions and neutral mole-

cules. As the methyl top with a three-fold symmetry

is rotating with respect to the frame of two-fold

symmetry for the ions derived from 2-MIA, indis-

tinguishable conformations are obtained at

intervals of 2rr/6. The potential function has

six-fold symmetry; V3 in Eq. (1) cancels and the

barriers are expected to be very low.

Heats of formation

Ab initio calculations of the energies were com-

bined with experimental heats of formation (AHr)

of imidazole (35 kcal mall’), ethane (-20 kcal

mall’), methane (- 18 kcal mall’), methylamine

(-5 kcal mall’) and ammonia (- 11.02 kcal mol-‘)

[5] in order to determine the heats of formation of

the MIA from the following isodesmic reaction

schemes:

C3H4NZ + CH3CH3 + C4H6N2 + CH4

for 2-, 4- and 5-MIA

C3H4N2 + NH2CH3 --) C4H6N2 + NH,

for l-MIA

Differences in the zero-point energies between reac-

tants and products are neglected in this approach.

The energies of IA and its derivatives are listed in

Table 3, and the energies of the other molecules

used for the isodesmic reaction are compiled in

Table 5. The heats of formation shown in Table 6

50

Table 5 Total energies (H, kcalmol-‘)

M. Meyer:J. Mol. Strut. (Theochem) 304 (1994) 45-51

HF/6-3 lG* -19.22816 -40.19517 -95.20983 -56.18436 HF/6-31 lG** -79.25170 -40.20901 -95.24251 -56.21039 MP2/6-3 1 G* -79.50376 -40.33695 -95.51380 -56.35690

increase from 2-MIA to 4- and 5-MIA, for which

the calculated AH, values are almost identical. The

AH, of l-MIA is the highest of the substituted

molecules. An increase in the basis quality from

6-31G* to 6-31 lG** hardly affects AH,-. The

MP2 results for all molecules have lower values

than the HF data. The semiempirical methods

MNDO and PM3 yield heats of formation up to

4 kcal mall’ below the experimental value for IA.

Similarly, these methods give lower heats of for-

mation than the ab initio calculations for the

MIA. AM1 overestimates the heat of formation

of IA significantly. The calculated data for the

substituted molecules also seem too high. This

overestimation of AH, by AM1 is apparent in

case of oxazole too [19].

Dipole moment

The results of the ab initio calculations of the

dipole moments are summarized in Table 7.

These data show only a small change with respect

to an increase in the basis at the HF level. The MP2

values obtained using the z-vector method [20] are

close to the dipole moments obtained with the HF

Table 6 Heats of formation (kcal mol-‘)

IA” l-MIA 2-MIA 4-MIA j-MIA

HF/6-31G* ~ 37.0 26.8 28.1 28.4 HF/6-31 lG** - 36.9 26.9 28.4 28.8 MP2/6-3 1 G* - 33.5 25.2 26.2 26.4 MNDO 33.3 32.5 21.6 21.7 21.6 AMI 50.8 55.1 42.8 43.0 42.3 PM3 31.3 29.8 21.3 21.8 21.2

a Experimental value 35 kcalmol-’ [5].

method. For IA the calculated dipole moments are

in good agreement with the experimental value

(3.67 (5)D) obtained from Stark-effect measure-

ments [2]. Semiempirical calculations lead to

results of similar quality (MNDO 3.48 D; AM1

3.60D; PM3 3.86D). The substitution of methyl

groups at Nl and C5 increases the dipole

moment, whereas substitution at C2 and C4

decreases the dipole moment. The sequence of

the magnitude of dipole moments of IA, 2-MIA,

4-MIA and 5-MIA is the same as for the corre-

sponding methyl oxazoles [17].

Conclusion

Accurate structures of IA and its methyl deriva-

tives were determined. The heats of formation of

the methyl substituted species were obtained by

using isodesmic reaction schemes. The semiempiri-

cal MNDO and PM3 methods give good results for

IA. AM 1 predicts much higher heats of formation

for the type of molecules studied than any other

method. The predicted methyl group torsional

barriers may be used to assist in the assignment

of the microwave spectra of the MIA. The

sequence of the magnitudes of the dipole moments

and the torsional barriers resembles those of the

methyl oxazoles.

Table 7 Dipole moments (D)

IAa I-MIA 2-MIA 4-MIA 5-MIA

HF/6-31G* 3.86 4.17 3.72 3.60 4.03 HF/6-31 lG** 3.85 4.19 3.73 3.55 4.06 MP2/6-31G* 3.84 4.13 3.68 3.52 4.01

a Experimental value 3.67 (5) D [2].

M. Meyer/J. Mol. Struct. (Theochem) 304 (1994) 45-51

Acknowledgements

The calculations were carried out at the com-

puter centres of the GBF and Kiel University. 1

thank H. Hartwig at the Chemical Physics Depart-

ment of Kiel University for supporting this study.

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