Molecular Mechanics Illustrations of its application Philip J. Cox School of Pharmacy, Robert Gordon's Institute of Technology, Aberdeen, AB9 IFR Scotland Molecular mechanics calculations, also known as force field calculations, are now of considerable importance in organic chemistry. These calculations have been used to investigate molecular conformations, thermodynamic properties, and vibrational spectra. The method treats a molecule as a collection of particles held together by simple harmonic forces. These forces may be described in terms vf ootential enerav -.functions which in turn sum to give the overall molecular potential energy or steric energy, E, of the molecule. In its simplest form the Westheimer equation is E=Es+E*+E,+Enb
where E, is the energy of bond deformation (stretching or compression), Ea is the energy of bending, E , is the torsional energy, and Enb is the energy of nonbonded interactions. Each of these potential energy functions represents a molecular deformation from an arbitrary reference geometry. If the strain free bond length of a Csp3 - Csp3 bond is taken as 15.20 nm, any deviation from this value will lead to an increase in potential energy. Such a potential energy term describing bond deformation would be
where KI = force constant, 1 = bond length, lo = strain free bond length and the summation is over all the bonds in the molecule. The potential energy term related to valency angles, 0, takes the form
where Fe and Fi are force constants and A0 = 0 - 00, 00 being the appropriate strain-free value. For torsion angles, w , the potential energy term is E,
+
% F, (1 scoa no)
= m
where F, = force constant (in this case the barrier to free rotation), n = periodicity of F, and s = + I (minimum energy when staggered), s = -1 (minimum energy in eclipsed conformation). Finally the potential energy term related to interatomic non-honded distances, r, may take the form En* =
x F, (-21ns + exp [12(1-u)])
where a = rl(r; + r;) and r;, r; are van der Wads interaction constants.
Other potential energy functions such as those which take into account out-of-plane bending and Coulombic and solvent interactions may also be used where appropriate. Several parameters such as force constants and strain-free aeometric values are required to prime the force field; these are obtained by considrring various ot~st.rvntionssuch as results of diffraction and thermodynamic experiments for a large
number of suitable molecules. The initial values of the parameters obtained are often only rough estimates so they are adjusted by either a trial-and-error method ( I ) or by leastsquares calculations (2). The aualitv of the derived force field isthen judged by its ability to reproduce data with an accuracy rivalline that achieved bv the ex~erimentalmethods. force fields have been derived independently, those of Lifson and Warshel (2) and Bovd ( 3 ) are referred to as "consistent" force fields as they haie been parameterized to reproduce vibrational frequencies as well as aeometric and thkrmodynamic data. ~ l l i n g e r(4) and ~ c h c y e r(5) have produced force fields that neglect vibrational frequencies but give excellent geometrical results. A superlative force field for the calculation of geometric and thermodynamic properties of alkanes and non-coniueated alkenes has been developed by White and Bovill(6);tLe values of some of the parameters used are shown in Table 1. For successful use an initial set of three-dimensional atom co-ordinates of the molecule under investigation is required. This trial set is progressively modified during the calculations to minimize the steric enerev. Of the various conformational isomers that may be possrble for a given molecule the assumption is that the a m f m n e r of l(~wwitstericenwgy wprrsents the most fawrnhle conformation d t h e i s ~ h t e driderule. It is in~portantto appreciate that rnulecular r,~echanirsis basically an empirical methud and the final molecul,tr tn~x1t.l obtained relates-to a hypothetical motionless state at absolute zero. However, it is possible to convert steric energy to heat of formation (at say 25°C) by addition of group enthalpy increments. Two fully saturated molecules, n-butane and cyclohexane, have been chosen to illustrate the use of molecular mechanics. Table 1. Molecular Force Field for Alkanes
an;
or kJ m ~ i - ' K - ~or kJ mai-': distances Constants have units of kJ m~l-hm-~ are in nm and angles in degrees. Bond deformation
1' 2 &
10
Csp3-H Cs~+-Csd
13.86
11.00 15.20
13.26 % Fo
F'o
H-Cw-H
0.0301
0.0402
108.2
Cs9-Cw-H Csp34sd-Csd
00368
0.0402 0.0402
109.0
Angle bending
0.0502
700
-
% F, 0.4602 0.4602 0.2632
Torsional strain H4sp3-Cw-H H4sp3-Csfl-Csp3 Csp3-Csp3-Csp3-Csp3
ZOa
109.1 109.2 110.4
s
n
1.0 1.0 1.0
3.0 3.0 3.0
van der Waais
interactions H-H
F.
r.
rr
0.0669
31.0
0.0
The superscripts on Aa refer to the number of carbon atoms anached to the central atom.
Volume 59
Number 4
A ~ r i l1982
275
Table 2.
W?
H&
We WI
'CH,
H
WI W5
Figure 1. Newman projections of *butane.
0 .
Torslon Angles (degrees)
Chair
Boat
Twist-boat
55.2 -55.2 55.2
53.7 -53.7 0.0
-55.2 55.2 -55.2
53.7 -53.7 0.0
60.5 -29.3 -29.3 60.5 -29.3 -29.3
1) For every complete 360' rotation there is one anti form and two gauche forms; the anti form has a steric energy 2.74 kJ mol-1 lower than the gauche form. 2) The Cl-C2-C3-C4 torsion angle of the gauche form is not 60(nor -60') hut calculated to be 69.4". Of the four potentialenergy terms only E(w) is lower at 60' than at 69.4'. 3) Band lengths and valency angles are not constant but vary slightly with the conformations. For example C2-C3 is 15.27 nm when w = 180' and 15.36 nm when w = 0'. Similarlythe valencyangles C1-C2-C3 and C2-C3-C4 are both 111.7" when w = 180' and 116.6' when w = 0". These results compare to strain-free values of 15.20 nm and 110.4"; deformation is greatest in the most unstable form and least in the moat stable form. 4) The ratio of antkgauche conformers may be calculated. The group enthalpy increment8 for butane = -130.1 kJ mol-'; therefore, heat of formation (at 2 5 T ) for the anti form = -127.2 kJ mol-I and heat of formation (at 2 5 T ) for the gauche form = -124.5 kJ mol-1
AS' (anti-gauche) = Rln% = -5.763 AG' = AHo- TASo = 2.7 1718.5 = 1715.8 kJ mol-I
+
Figure 2. Variation of steric energy with Cl-C2-C3--C4
K=-- [anti] -e- -AGO [gauche] RT
torsion angle.
i.e., there are two molecules in the anti conformation for every molecule gauche at 2 5 T . Cyclohexane
T h e n-butane molecule is often used to show the existence of conformational isomers; these are shown in Figure 1.T h e anti form is such t h a t t h e C1-C2-C3-C4 torsion angle (w) is 180' whereas the gauche form is normally drawn with w = 60'; w is ODfor the unstable e c l i ~ s e dform. If t h e anale is moved through 360° and t h e steric energy calculated i t each step Figure 2 is obtained. (In practice the torsion angle magnitude is never greater than 180°,i.e., while looking down the C2-C3 bond w ispositioe if a clockwise rotation of C1 is required to eclipse C4 and negative if the rotation is anticlockwise, hence the value of 270' in Figure 2 corresponds t o w = -go0.) Several interesting observations can be made from these calculations.
Figure 3. The boat conformation.
276
Journal of Chemical Education
A model of this molecule is often used t o show boat and chair conformers. Another conformer of cyclohexane is t h e twist-boat; all three conformers are shown in Figures 3,4, and 5. Other conformations are also possible. T h e calculations show t h a t the boat ( E = 40.2 k J mol-') does not represent a conformer of minimum energy. T h e chair ( E = 14.4 k J mol-I) and twist-boat ( E = 36.8 k J mol-') are conformers of minimum energy and all the IR fundamentals which can be calculated (3N - 6 = 48) are real (for the boat one IR fundamental is imaginary). T h e final torsion &?gles obtained are shown in Table 2; wa and we were restricted to 0.0' for calculations on the boat conformer.
Figure 4. The chair conformation.
Figure 5. The twist-boat Conformation.
The bond lengths and angles vary with the conformation. In the stable chair conformer all carbon-carbon bonds are 15.30 nm whereas in the boat form the C1-C2 and C4-C5 hond lengths are elongated to 15.36 nm to reduce the van der Waals interaction between the two hydrogens occupying the "bowsprit" and "flagpole" positions. Although in comparison to the chair conformer there is an increase in van der Waal interaction, it is the overall increase in torsional strain which significantly contributes to the unfavorable boat conformers. Group entbalpy increments for cyclohexane sum to -132.7 kJ mol-' the heat of formation for the chair conformer is -118.3 kJ mol-'. This corresponds to an ohserved heat of formation of -123.4 kJ mol-I. The chair bond lengths, valency angles and torsion angles, involving carbon
atoms only, are calculated to be the same as the experimentally observed values. Finally, as the iterative minimization techniques used in molecular mechanics calculations are very extensive, it is necessary to have access to a computer. The main program used here was PECALC written by Dr. D. N. J. White of Glasgow University and modified for use on a DECSYSTEM-20 computer. Literature Cited
13,1i19761.
11) ~ l l i n ~ eN. r .L., do. ~ h . ~ or#. s . them., 12) Lifum.S. and W ~ ~ P ~ P I . A C . ,hJ m . l Phyhya.49.5116(1968).
0 , Chang,S..McNaIIy,D.,Shary-Tehrany,S..Hickey,M.J..andBoyd,R.. J A m m Chem. SU~..~Z.SLO i19701. ~ 14) ~ l i i n ~N.r ,~ . . , ~ r i b h M. ~ s ,T.,~ i ~ ~M. e rand , 93. I637 119711. (5) E ~ ~E. M., L ~ ~ ,J . D.. and schlever. P
werb. D. H., il A ~ W chem. . snc, R., J.
119731. 16) White,D.N. J, and Bovill. M . J., J. C. SPwkin11,1610119771.
Volume 59
Number 4
April 1982
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