1. Henshall Royal College of Advanced Technology Salford. Lancashire
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Calculation of Molecular Geometry by Vector Analysis Application to six-membered alicyclic rings
consideration of molecular properties, especially those involving energetics, often requires an accurate knowledge of the molecular geometry. E. J. Corey and R. A. Sneenl describe a procedure for the calculatiou of the molecular geometry of alicyclic rings, based upon simple vector algebra. The usefulness of the method can be greatly improved by the simple expedient of calculating unit vectors along the bond directions. The atomic coordinates relative to any arbitrary origin are then given by the product of the bond length and the unit vector from that chosen origin directed toward the atom under consideration. The cyclohexaue molecule in the "chair" form is chosen to illustrate the method. Consider the bonds C1-C2, Cl-C6 (see figure), and let the direction cosines
+ n2 = 1.
Solution of these equations leads to the values 1 = 4 2 3 ; n = &/3. The unit vectors i - 1/3j e 8 , and ~ are hence respectively, m 2/2/3k and - 1/3j 4 / 3 k , or more - 1/3, &/3) and (- d@, compactly, - 1/3, @/3). Again, by invoking molecular sym~ %,I. metry, e4,3= ~ B , Ie;d ,= Having thus established the unit vectors for the C C bonds, those for the C-H bonds readily follow.First, by tetrahedral symmetry, e t , ~= e 6 , ~= ( d 2 / 3 , 1/3, .\/2/3). Then if ez,lo= (11, ml, n ~ )the , following equations hold: m2
.\/mi
(a,
+
+
The solution of these equations, with the correct choice of signs on extracting the square roots involved, leads to el,lo = (0, -1/3, - 2 4573). The other C-H unit vectors follow easily by symmetry arguments. All the unit vectors relative to the chosen cartesian frame are tabulated below.
Table of Unit Vectors Figure 1.
Cyclohexone molecule, "choir" form.
of C1-Cs relative to a cartesian frame disposed as indicated be (l,m,n); i.e., the unit vector e s , ~= li rnj nk, where i, j, k, form a right handed set of orthogonal unit vectors along the cartesian x, y, z directions respectively. Then by reflection symmetry mj+ nk. On the assumpthe unit vector e z , = ~ -1i tion of tetrahedral angles, 7 , the scalar product of these unit vectors leads to er.,.ee.,= (-l,m,n).(l,m,n)= -lZ + m Z + nZ = cosr = - 'h
+
+
+
Inspection of the table shows that the unit vectors satisfy the requirements both of normalization and tetrahedral symmetry. By the choice of axes, esr = e ~ =, j,~ since CZ-C3, The coordinates of any atom relative to any arbiCsCs lie in the y-direction; hence also trary origin, subject only to the condition that the ea,,.el.8= (-l,m,n).(O,l,O) = m = cosr = -'/k cartesian frame a t that origin parallels the original Finally, from the definition of direction cosines, l2 reference frame, are now easily obtainable. Thus, relative to an origin in C1, the co-ordinates of C2,C6,H,, H8 are simply rc-o rc-c e~,a,rc-H el,,, and TC-H e1.8 r c ~ rc-E , the C-C and C-H bond respectively, with ' COREY,E. J., AND SNEEN,R. A., J. Amer. Chem. Soc., 77, lengths. 2505 (1855); see also SCOTT, MACKENZIE, THIS JOURNAL, The extension to other systems is readily apparent. 43, 27 (1066).
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Journal o f Chemical Education
