The use of vicinal HH coupling constants in rotational isomerism studies, I

JOURNAL

OF MAGNETIC

RESONANCE

43, 28-39

(1981)

The Use of Vicinal H-H Coupling Constants in Rotational Isomerism Studies, I

Department

of Chemistry,

VICTORM.

S.GIL

University

of Aveiro,

Aveiro

3800,

Portugal

AND

ANT~NIO J. C. VARANDAS Department

of Chemistry,

University

of Coimbra,

Coimbra,

Portugal

Received August 28, 1980 By using simple model potential energy functions for internal rotation of ethane derivatives, associated with the Karplus relationship for vicinal H-H coupling as a function of the dihedral angle, an investigation was made to find out how serious an approximation it is to reduce the whole conformational problem to a case of rapid equilibrium between the three staggered conformers, especially as far as the calculated energy differences for these are concerned. A classical view of internal rotation was adopted and only systems for which two staggered conformers are isoenergetic were studied: CH,X-CH,X, CH,X-CH,Y, CH,X-CHX,, CH*YCHX,, CHX,-CHX,, CHX,-CHY,, CHXY-CHXY (meso form). It is found that the errors are temperature dependent and can be quite large, the calculated AE values being always smaller than the real ones. Simple rationalizations are offered for these and other findings.

A customary simplifying procedure when using vicinal H-H (and other) coupling constants to obtain conformational information on molecules possessing rapid internal rotation, especially substituted ethanes, is to reduce the conformation problem to a case of rapid equilibrium between the most stable rotational isomers, namely, the three staggered conformers in the case of ethane derivatives:

Effective relative populations, r, g,, g,, for these conformers are calculated so that the observed rotationally averaged coupling constants .I are reproduced as 28 0022-2364/81/040028-12$02.00/O Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.

VICINAL

H-H

COUPLING

weighted averages of J values attributed

CONSTANTS

29

to each conformer:

Jobs = t ‘J, + g, ‘.I,, + gz ‘J,,.

[II

From the populations, information on the ordering of the energies E of the three staggered rotamers is obtained (entropy changes are often neglected) as well as on their differences. A particular case occurs when only two different E minima exist, namely, as a result of symmetry; e.g., for E,, = E,,, = E, (g, = gz = g), one has E, - Et = RT In tlg. PI The same procedure has previously been used with other physical quantities that are conformation dependent, namely, electric dipole moments (I ). Usually recognized as the main source of uncertainty affecting the AE values calculated from [l] is the decision on which J,,Q values to use; these depend, in a manner not known with sufficient accuracy, on the geometric arrangement of the coupled nuclei (namely, the dihedral angle) and on the nature and orientation of substituents (2). In this work we investigate how serious is the approximation of assuming that only the conformations having minimal energy are populated, especially as far as the calculated AE values are concerned. We shall compare “true” AE values between the minima of model potential energy functions with the corresponding apparent values, A E ’ , obtained by the discrete average approach represented by Eqs. [ 11 and [2]. The discrete average approach is expected to be the more satisfactory the more pronounced the energy wells are (up to the point beyond which an average J value is no longer observed: it would be correct in the case of a potential energy 6 function, in which case, however, internal rotation would be absent). Obviously, an observed coupling constant will not depend only on the values of the energy minima in the potential energy function; in particular, it depends also on the energy barriers which separate these minima. By ignoring the dependence of./ on the behavior of the complete potential curve, the AE’ values obtained using [l] and [2] will in general deviate from the true energy differences, and the temperature dependence expected for Jobson the basis of a discrete approach will in general also deviate from the observed dependence. This means, in turn, that the apparent AE’ values become temperature dependent. There are reports of such dependence in the literature (3,4) but the above contribution to the explanation has, to our knowledge, been ignored. Also, when entropy changes become important and free energies are used instead of AE, such dependence of AE’ on temperature will affect the enthalpy and entropy differences calculated from the observed temperature dependence of free energies. The importance of corrections for the torsional vibrational effects in the determination of accurate AE values for tram and gauche conformations in substituted ethanes from H-H coupling constants was early recognized by Gutowsky and co-workers (5) in a theoretical study of the temperature dependence of vicinal H-H coupling using simple potential functions. Although a few calculations of

30

GIL

AND

VARANDAS

coupling constants making a full use of potential energy functions in connection with the conformation of some molecules have been performed (6,7), no detailed and systematic investigation is known of the effect the appropriate inclusion of all available conformations has on the calculated differences between the energies of the more stable conformations. CALCULATIONS

AND

RESULTS

At this stage, we have adopted a classical view of an internally rotating molecule, where all conformations are populated according to a Boltzmann distribution ignoring entropy effects. Hence, for an ethane derivative, 277 277 J= J(6) exp(-E(B)/RT)d0 exp( -E(f3)/RT)d0, [31 i0 Ii 0 where 0 is the dihedral angle between the H,-C,-C, and C,-C,-H, nuclear planes. ForJ(B) we take the Karplus relationship (8) and neglect substituent effects and their orientational dependence: J = A + B cos 8 + C cos 20

(A = 4.2, B = -0.5,

C = 4.5 Hz).

[41

In this paper we deal only with those more symmetrical systems where only two different E minima (for the three staggered conformations) occur, namely, CH,X-CH,X, CH2X-CH2Y, CH*X-CHX2, CHzY-CHX2, CHX,-CHX,, CHX,CHYz, CHXY-CHXY (meso form). For the interval 8 = O-120” the form of the E function has been taken as E(60”)(1 - cos 38)/2 and similarly for the intervals 120-240” and 240-360”. This is the form of the leading term in the potential energy function for ethane (9). For reasons of simplicity, it has been assumed that all eclipsed conformations have the same energy (for convenience taken as zero). Thus, depending on the system and (or) on the pair of H nuclei being considered, the types of E(8) curves are as shown in Fig. 1. When there is just one vicinal H-H coupling constant (the case of the tetrasubstituted molecules listed above) the curve is either of type (a) or type (b), because the two conformers where the pair H-H has a gauche orientation are isoenergetic. For the trisubstituted systems (CH,X-CHX, and CH,Y-CHX,), for example, X

: 1

3 *

2

Y

X

3 *

Y

X I

2

X

3

Y

X *

2

1

the appropriate energy function is (c) or (e) depending on the coupling constant being considered, if the first two conformers shown above are more stable than the third one, and (d) or(f) otherwise. For the disubstituted molecules (CH,X-CH,X and CH*Y-CHZX), if curve (a) (or (b)) is appropriate for one of the vicinal coupling constants, (c) (or (d)), obtained by a translation of 120” from a curve of type (a) (or (b)), is correct for the other.

VICINAL

H-H

COUPLING

31

CONSTANTS

0

-0 -m--f)

60'

1ao*

300'

e

EIBI

0

-c --__

60°

1600

3000

60°

180~

3000

,j

e

ElfI

0I.

a

FIG. 1. Types of potential energy functions used.

The integration [3] was performed numerically, as a function of the energy minima and temperature. The simulated “observed J values” thus obtained were then used to calculate the effective populations t , g,, g, (two of these being equal)

32

GIL

AND

VARANDAS

of the three rotational isomers by taking J (trans) and J (gauche) as given by Eq. [4]: 9.2 and 1.7 Hz, respectively. From these populations, apparent AE’ values were obtained. Finally these were compared with the corresponding correct energy differences AE, for variable temperature. The general scheme is illustrated below:

4

I

i

compare-

- - - --,

cl

J’

We have also calculated the percentage deviations between J and the apparent J’ values which would be obtained if only the staggered conformers, with energies given by the potential energy functions used, were populated (populations t;, g;, gh). It is noted that J and J’ may be regarded as corresponding to 6 functions for E(B) allowing for rotational averaging, characterized, respectively, by the differences BE’ and AE.

50

200

LOO

600

FIG. 2. The deviations D(AE) and D(J) type (b) differing in E, from each other.

IO00

as a function

16oJK

of 7 and AE,

for

T

various

E(0)

functions

of

VICINAL

50

200

400

H-H

600

COUPLING

33

CONSTANTS

1000

15OOK

T

FIG. 3. The deviations D(AE) and D(J) as a function of T and AE, for various E(8) functions of type (b) differing in E, from each other.

Figures 2 and 3 show the deviations D(AE) = (AE - AE’)IAE and D(J) = IJ - J’ 1IJ as a function of T, obtained with energy functions of type (b) (see Fig. 1) and keeping E( 180”) constant while varying E(60”) = E(240”) or vice versa. The appropriate D(AE) and D(J) curves for energy functions of type (a) are given in Figs. 4 and 5. It is found, as could be anticipated, that energy functions of types (a), (c), and (e), related by a translation of 120”, lead to identical D(AE) and D(J) curves, the same being true for types (b), (d), and (f). This means that, in the case when there are two vicinal coupling constants (systems CH,X-CH,X, CH,Y-CH,X, CHX,-CH,X, and CHX2-CH2Y) we get the same D(AE) and D(J) results irrespective of the vicinal pair used. Whereas Figs. 2 to 5 essentially show the effect of AE on D(AE) and D(J) as a function of T, Figs. 6 and 7 illustrate the variation of D(AE) and D(J), for a given AE value, with the depths of the potential wells. Addition of a constant to E(6) does not lead to any changes, as can be directly seen from Eq. [3]. Multiplication of a given E(8) by a constant A, thus changing simultaneously AE and the depth of the energy wells, leads to D(AE) and D(J) curves (Fig. 8) that can be related to the original curves by recognizing that, using [3],

J[H@,Tl = J[Wf3,hTl,

[51

hence AE’[E(@,T]

= AE’[hE(B),AT]lA

[61

34

GIL AND VARANDAS

AE=

1600

1200

600

COO Cd

0.60

0.50

0.40

o-30

0.20

0.10

0 50

200

4w

600

1000

1500

K

T

FIG. 4. The deviations D(AE) and D(J) as a function of T and AE, for various E(0) functions of type (a) differing in E, from each other.

and D(AE)[E(B),T]

= D(AE)[hE(B),hT].

[71

= J’[hE(B),AT]

PI

Similarly, J’[E(B),T] and, using [5],

WJ)[E(@,Tl = WO[Wf9,A~l.

[91

The crossing observed in Figs. 3 and 5 but not in Figs. 2 and 4 (the D(J) curves in these are almost coincident for low temperatures) can be related to that shown in Fig. 8 as follows. The curves of Figs. 3 and 5 correspond to a AE change obtained by keeping constant the value of the upper potential energy minima (urn) and varying the low one(s); the AE changes accompanying the multiplication of E( (3)by a factor (Fig. 8) result from a bigger variation of the lower energy minima (urn) with respect to that of the upper one(s). On the contrary, the curves that do not cross correspond to fixing the lower energy minima (urn) and changing the upper one(s). DISCUSSION

AND CONCLUSIONS

A first observation that can be made is that the apparent AE’ values (difference between the apparent energies of the less and the more stable staggered con-

-a----------__

-0

D(J)

-------KJ

200

50

LOO

600

IOOOK

1500K

1

FIG. 5. The deviations D(AE) and D(J) as a function of T and AE, for various E(B) functions of type (a) differing in E, from each other.

0.60 I 070.

060-

0 (AE) o.so-

0.40-

0.30.

0.20.

0.10.

0 (J! 0

50

200

FIG. 6. The effect onD(AE)

400

600

1000

15OOK

r

andD(J) of the depth of the potential wells (energy functions of type (a)). 35

GIL AND VARANDAS

36

c0.1 O-

----------___ -___

,-o-----Kg/

e-------------Q---

-=_*

+:-:=-

+*i---o- e-* 50 FIG.

I 200

7. The effect onD(AE)

Loo

600

- - _ ----==z=;--=g - - - - - -_- - --

moo

O(,)

15CQK

r

and D(J) of the depth of the potential wells (energy functions of type(b)).

0.60

0.70

0.6t

ofArc) 0.5c

04

0.31

0.2(

011

I 50 FIG.

200

LOO

600

1wO

8. The effect on D(AE) and D(J) of the multiplication

15OOK

T

of E(B) by 2.

VICINAL

H-H

COUPLING

37

CONSTANTS

formers) are always smaller than the corresponding true AE differences (for the same temperature). A qualitative explanation of this is as follows. Going from a continuous treatment to a discrete effective average approach corresponds to a of population from the intermediate conformations into the three “transfer” minimal energy rotamers; as a result, a bigger increase in the effective population is produced in the case of the upper minima, which thus appears to be stabilized with respect to the lower minima. For an energy function of form (b) (and similarly for the other types), with, e.g., E(60”)IRT = E(300”)IRT = -4.00 and E(180”)lRT = -6.00 (AEIRT = 2.00), the ratio of the effective populationsp*(60”) = ~“(300”) andp*(180’) defined by the total percentage population for the intervals 0- 120” (or 240-360”) and 120-240” is 2400

p*( 180’) p*(6()o)

1

exp[3.00(1 - cos 30)]d0

120°

=

= 5.82,

120’

exp[2.00(1

[lOI

- cos 30>]d0

I 00

which leads to AE*/RT

= lnp*(180”)/p*(60”)

= 1.76.

These values are to be compared with AEIRT = 2.00 andp(180”)/p(60”)

[Ill = exp(AEl

RT) = 7.39.

In parallel with these results, we find that J < J’ for a curve of type (b), due to an enhanced effective contribution of the gauche orientations with respect to truns, on going from a S function of energy difference AE to a continuous treatment (or 6 function with energy difference AE’). This inequality was also found by Gutowsky and co-workers (5). For a function of type (a), J > J’ because, in this case, it is the trans conformation (being the staggered conformation of higher energy) that exhibits a greater relative increase of population. The numerical results corresponding to [lo] and [I I] only depend on the actual values of E(B)/RT for the staggered conformations and not on the type of E(8) curves given in Fig. 1. As could be expected, for a given temperature and a given AE value, the deviations D(AE) and D(J) diminish as the potential wells become deeper. This is illustrated in Figs. 6 and 7. In terms of the above discussion, it is recognized that, for the same AEIRT value, the deeper the potential wells the closer the changes in the effective populations of the staggered conformers brought about by “transferring” into them the populations of the intermediate conformations. For example, for an energy function of type (b), with E(6O”)IRT = E(3OO“)IRT = - 12.00 and E(180”)IRT = - 14.00 (AEIRT = 2.00), the ratio equivalent to [lo] is p*(lW)/p*(60”)

= 6.82

[=I

leading to AE*IRT

= 1.92.

[I31

These values are much closer to AEIRT = 2.00 and p(180°)/p(600) = 7.39 than

38

GIL

AND

VARANDAS

those from [lo] and [ 111. In the limit of a 6 function for E(0) (allowing for rotational averaging), one would have AE = AE’, and D(AE) and D(J) would vanish. As shown in Figs. 2 and 4, by keeping constant the lower energy minima (urn) and lowering the upper one(s), thus decreasing AE, one obtains a decrease of D(AE) and D(J). Indeed, the effects of “transferring” population to the three minima become more similar as one lowers the upper minima (urn). In the limit of AE = 0, the effective populations of the three minima become equal; hence, AE’ = AE = 0, and D(AE) and D(J) vanish. We note that, in the present case, the effect of decreasing AE and that of lowering one (or two) of the potential wells operate in the same direction: decreasing D(AE) and D(J). By fixing the upper energy minima (urn) and raising the lower one(s), thus decreasing AE (Figs. 3, 5), one gets a decrease in D(AE) at low temperatures and a slight increase at high temperatures; an opposite result is observed for D(J). We have already commented upon the corresponding crossing in the D(AE) and D(J) curves. In this case, there are two opposing effects. On one hand, AE decreases, thus contributing to a decrease in D(AE) and D(J); on the other hand, one (or two) of the wells rises, thus leading to an increase in D(AE) and D(J). The effect that dominates depends on the temperature (as well as on which D(AE) or D(J) deviation is considered). A similar discussion applies to Fig. 8, showing the effect of multiplying E(8) by a given constant. It is clear from Figs. 2 to 8 that D(AE) is maximal when T goes both to 0 and ~0 K. To rationalize this result, let us consider a function E(8) of type (a). By using equations like [2], D(AE) can be written in the form D(AE)

= 1 - (In t/g)l(ln t’lg’).

[I41

When T -+ 0, only the gauche conformations become populated; accordingly, t/g + 0 and t’/g’ + 0. Since t is always greater than t’ (note that we always have AE’ < AE), the ratio of the logarithms in Eq. [14] goes to zero, and D(AE) -+ 1. When T goes to infinity, the populations t, g, t’, g’ tend to a common value (l/3) and [ 141 goes to a constant. A similar reasoning applies to all other curves shown in Fig. 1. In parallel with the D(AE) behavior, Figs. 2 to 8 show that D(J) goes to zero as T + 0 and also when T + m. As mentioned above for a curve of type (a), when T -+ 0, g and g’ go to 1 and (by using equations like [ 11) J - J’ -+ 0; when T + 30, t,t’, g,g’ += l/3 and again J - J’ -+ 0. As seen from Figs. 2 to 5 and 8 the T value for which D(AE) is a minimum (Tmin) and that for which D(J) is a maximum (T,,,) increase with AE. For typical potential energy barriers, the former is found to go from 500 ? 80 K for AE = 3000 cal to 100 ~fr 30 K for AE = 500 cal, in a roughly linear variation with AE. The maxima in D(J) always occur for greater temperatures. In addition, Figs. 2 to 8 suggest that the extreme in one curve, D(AE) or D(J), corresponds to the inflection point in the other. The variation of the Tmin (or T,,,) values with AE can be partially understood from Eqs. [7] and [9] and Fig. 8. In fact, if AE is changed by a factor A, as a consequence of multiplying E(0) by A, then Tmin (or T,,,) becomes multiplied by A. For the same AE difference, Tmin is found not to vary appreciably with the potential energy wells, whereas T,,, increases significantly as they become deeper (see Figs. 6, 7).

VICINAL

H-H

COUPLING

39

CONSTANTS

A more quantitative analysis of Figs. 2 to 5 shows that experimental determinations of AE’ by the usual method, at room temperature, are affected by errors that can be as large as 40-50% for common systems. For high values of AE (AE 5 2000 cal) it is necessary to go to higher temperatures, up to a certain value (T&, in order to decrease the error; for lower values of AE (AE c 1200) the error decreases with temperature, down to a certain T value (Tmin).For an activation energy for internal rotation of about 5000 cal, the error at room temperature is 15% if AE s 500 cal; for activation energies of the order of 2000 cal, it is necessary that AE s 100 cal for the error to be less than 20%. The errors D(J) are found to be appreciably smaller than D(AE). We note that all these qualitative and quantitative conclusions remain unaltered if the coefficients in the Karplus equation [4] are multiplied by a given factor (it is noted that Eq. [4] gives J values that are about half of the experimental values). This study is being extended to other, less simple, potential functions E(8) as well to other physical properties, namely, dipole moments. ACKNOWLEDGMENTS This work is also a contribution of the Centro de Investigacao Quimica (INIC) at the University of Coimbra. V.M.S.G. thanks Dr. Beatriz Matias for help with the computations in the preliminary stage of the work. The principal results were presented at the 3rd Meeting of the Portuguese Chemical Society, Coimbra, April 1980. REFERENCES “The Structure of Molecules and Internal Rotation,” York, 1954, and references therein. J. ABRAHAM AND G. GATTI, J. Chem. Sot. (B), %l (1969).

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AND

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