D.. M. McEachern, Jr. and P. A. Lehmann Center for Research and Advanced Studies National Polytechnic Institute A.P. 14-740, Mexico 14, D.F., Mexico
Cakulating Dipole Moments and Atom Coordinates Using Molecular Models A shadow proiection method
calculating the dipole moment of a molecule from group moment contributions can be accomplished if the atom coordinates are known, because the group moments always lie along bonds or a t known angles to certain bonds. This calculation may fail to give the correct dipole moment in certain special cases if electron delocalization, not accounted for in the group moments, occurs in the molecule. I n these cases it is necessary to invoke, in addition, a 'Lmesomeric moment" to account for this effect. Atom coordinates are necessary not only in dipole moment calculations, but also in conformational calculations of energy minimization, non-bonded interactions, and as the most compact method to present all interatomic distances in a molecule. Methods for calculating atom coordinates in terms of bond distances and angles are available.' However, the solutions involve simultaneous equations which necessitate a digital computer for molecules of any complexity. We have devised a simple analog method which gives atom coordinates and the dipole moment easily and accurately, even for complicated molecules and conformations. The equipment needed is a set of molecular models2 (framework, not space-filling), a p r ~ j e c t o ran ,~ open box of transparent, rigid plastic constructed with three mutually perpendicular sides of about 50 X 50 cm which form one corner, a support made of the same plastic, and transparent tape. Figure 1 is a drawing of the box with the triangular plastic support mounted in it. The theory of this method can be found in most analytic geometry t e x h 4 A perpendicular projection of a bond, represented by a vector, onto a plane, produces its component in that plane. Further project~on of this component perpendicularly onto each axis defining the plane is equivalent to taking the dot product of the vector with respect to a unit vector along each of the two axes. The net result is equivalent to finding the two direction cosines of this vector. Perpendicular projections of a point done in the same manner produce its MACKENZII,:, S., J. CHEM.EDUC.,43, 27 (1966); HENSHALL, 600 (1966); COREY,E. J., AND SNEGN, R.A,, J . Am. Chem. Soc., 77, 2505 (1955). Dreiding Stereomodels were used because of their accuracy and convenient size. They are available in theU.8.A. from Rinco Instrument Co., Inc., P.O. Box 167, Greenville, 111. 62246. A 3 M averheed projector w i s used, manufactured by the 3M Co., Ino., P.O. Box 3800, St. Paul, Minn. 65101. Others can be found in a. list given by BARNAED, W. R., J. CHEM. EDUC.,45, 341 (1968). See, e.g., Snnm~,PERCEY F., GALE,ARTHUR S., and NEGLLBY, JOHNH.. "Nev Analvtic Geornetrv." (Alternate ed.). Ginn and CO., York, 1938;pp. 235-44
two coordinates. By using three perpendicular planes the three components of any vector or the three coordinates of any point are each determined twice. Difficulties encountered in using this method are: to preserve the model conformation, to mount the model in the box, to orient the light beam, model, and planes to obtain the two-dimensional projections, and to transfer the projections onto the axes. Each of these is solved easily using this method. With Dreiding models it is possible to preserve a conformation by hindering the normally free rotations, if necessary, by pinching the female bond entrances lightly with pliers to produce sufficient friction. This does not harm the models since rotation and disassembly are still possible. The model is mounted in any orientation with clear tape on a triangular plastic support whose corners have been cut where they will make contact with the sides of the box and whose center has been removed (Fig. 1). This sup-
Figure 1.
Open box with plortic support.
Figure 2.
Photograph of diphenyl ether model and one proiection.
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port is taped to the sides of the box so that no atoms will be projected through the plastic of the support, to avoid refraction error. I t is convenient, but not necessary, that no bonds be perpendicular to. any plane. Figure 2 shows a photograph of a model of diphenyl ether in a twist conformation, mounted near the interior corner of the box, together with the projection obtained by shining light perpendicularly to the right face of the box through the model. To obtain projections the sides of the box are successively lined up perpendicularly to the light beam as shown in Figure 3, which is a diagram of the general
Figure 3.
Diagram of
the general set-up.
set-up. I t is easiest to use a line on the table supporting the box which has previously been drawn perpendicularly to the beam and orient the box with respect to this line. The locations of atoms or the ends of bonds are marked on paper taped to the back of each face, using back projection. The axes and origin are conveniently made to coincide with the edges and corner of the paper. Atoms obscured by bonds or other atoms are located by touching them with a needle from two or three directions and marking the indicated intersection. A group dipole not lying betv-een two atoms can be represented by a rod of any length, taped to the model in the correct direction, such as in Figure 2, where a mesomeric moment, from C1 to between C1 and C4 is represented by glass rods.
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The transfer of the two-dimensional components or atom points to the axes is made with a 90" triangle after removal of the paper. The coordinates or components can then be measured on each axis. A check is furnished since each axis component or coordinate appears once in two different planes. For dipole moments the actual bond or rod length is taken proportional to the group moment. This length is divided into its axes projections and multiplied by the group moment to obtain the group's x, y, and z components. The components of each axis are summed and squared and the square root is taken of the sum of the squares to calculate the net dipole moment. The errors from this method are about zt0.15 Debye for dipole moments and 1 0 . 5 mm for atom coordinates, corresponding to zt0.02 1( for the models used. The method, of course, is only as accurate as the models used, and the above coordinate error is a measure of precision. The method does not take into account, for example, the variations found in C-C single bond lengths, unless these are built into the models. Variations of this type will affect dipole moment calculations to a negligible degree, but should be accounted for in very accurate atom coordinate calculations. The coordinate error is estimated while the dipole moment error comes from a comparison of a number of exact calculations with those of this method, and is, therefore, a measure of accuracy. Errors occur not only because of inaccuracies in the angles and bond lengths of the models (reflected in the dipole moment errorquoted but not in the coordinate error) but also because of inaccuracies in the 90" dihedral angles and in the planarity of the faces of the box, in the positioning of the light beam relative to the projection plane, and in the shadow recording, transfer, and measuring steps. These latter errors can be minimized by building the box with care and by aligning, recording, transferring to the axes, and measuring carefully. I n our case (projector-to-screen distance 7 m) variation in shadow size with model-toscreen distance is only 0.1% per cm between zero and 30 cm. Beyond this the shadows begin to be diffuse. This variation can be corrected for but was neglected here. The errors in this method are sufficiently small for use in many applications where atom coordinates are needed and for almost all applications in which dipole moments determined in solution are used.
