F. Borremans and M. Anteunis State University-Gent 9000-Gent, Eelglum
I1 Diamond Rapid Calculation of Dipole Moment for Lattice-Like Molecules
The "classical" method of calculating the dipole moment of a molecule is based on the idea suggested by Thomson in 1923 (1). The dipole moment of a molecule is the vector sum of the individual hond moments
!J
- Gi
(1)
In general, the evaluation of ji according to eqn. (1) requires the determination of the direction cosines of the individual bonds, which necessitates the accurate calculation of the coordinates of the atoms in an ap~ropriateframe of reference. This is usually accomplished ;iacomputer programs (2) for the conversion of "internal" coordinates (bond lengths, bond angles, and dihedral angles) to atomic coordinates using a matrix formalism for successive coordinate transformations (3). We will refer to the latter as the "exact" or "sophisticated" treatment. The classical method suffers from serious limitations resulting in calculated dipole moments to be meaningful only within -+ 0.2 D, because 1) the validity of eqn. (1)is based on the assumption that bonds possess no net charge (4) 2) different authors recommend appreciably different bond moments and the supposed constancy of bond dipoles in different molecules is probably too drastic a simplification 3) "small" bond moments, e.g. the one associated with C-H, are often assumed to cancel out; hence it is a wide-spread custom to neglect them in the calculations (5) 4) experimental internal coordinates are of limited accuracy or are often not known for the ease at hand, necessitating the use of presupposed "standard" band lengths, valence angles, and torsion angles
Fgure 1. The tetrahedral directions a. b, c. and d, and the orthogonal coordinate axes x, y. and r.
5-
If the extra error introduced by idealizing the molecular
eeometrv is less than 0.2 D hiehlv -yield - .simolified models can calculated dipole moments of comparahle reliability with those obtained bv the exact method. The model we present here assumes that each hond in the molecule, and so its associated hond moment, is parallel to one of the four tetrahedral directions (as in a diamond skeleton). A bond moment is arbitrarily taken positive if directed as shown in Figure 1, and vice versa. For each of these four directions one adds algebraically the dipole moments of all individual bonds pointing in that direction. This yields four signed numbers a, b, c, and d corresponding to four tetrahedrally arranged vectors, whose sum is the dipole moment of the molecule. The squark of the dipole moment is equal to the sum of sauares minus two thirds of the sum of products in groups o i two of these four numbers.
The student is able to calculate hv hand or with a slide rule in two minutes the dipole moment, without recourse to a digital computer. The formula is easv to remember and the . calculation is unelaborate since a, b, c, and d are only twodigit numbers. Alternatively one can use the nomogram presented in Figure 2. The nomogram is self-explanatory and its construction is based on eqns. (3)-(5).
Figure 2. Nomogram for the evaluation of the dipole momant, illustrated for cis-5-lluoromathyl-2-methyl-l,3dioxane (entry 2 in the table). a = 2 X ilc-o 11~-~+~l~~=2X0.9+1.5+0.3=3.6:b=c=ilc-0+11~.-~-2 X I ~ L= 0.9 0.3 = 0.6:d = -3 X ir-c irH-c = -0.9 0.3 = -0.6. The line connecting point a b - c d = 4.2 on the left scale with b = 3.0 on the right scale, intersects the non-numbered vertical point s line in a point hom which a new line is drawn towards the point c d = 1.2. The intersection with the p-scale gives the dipole moment of the molecule in Debye (3.95 D).
+
-
-
+
-
+
+
-
Volume 53, Number 1, January 1976 1 23
Comparison of Calculated Dipole Moments "Rapid''a method Entry
1
Compound
"So~hirticated"~ method
(D)
(D)
p
2.67
2.64
P
3.59
3.63
p~
1.15
1.14
2.62
2.64
F
2
3
58" Figure 3. Geometry of the 1.3dioxane ring as obtained from X-ray diffradion 161.
Equation (2) can easily be derived hy projection of the four tetrahedrally directed vectors a, b, c, and d on three orthogonal cartesian axes x , y and z (Fig. 1). The x-axis is the common bisectrice of a 6 6 and cod. T h e y - and z-axes are orthogonal to the x-axis in the ab- and cd-plane, respectively. One finds
a
p,
= (C
-
p'
= a2
+ b2 + c2 + d2 +
(5)
d ) sin?
+ cd) cosa + bXe + d ) cos'?a
2(ab 2(a
(6)
For a being the tetrahedral angle, it is easy to prove by elementary goniometrics that
4 a Note that the bond moments for C-H culationr.
are included in there cai-
1,3-dioxane system as derived by X-ray analysis (6),reproduced in Figure 3. The bond moments used were: p(C-0) = 0.9 D; p(H-C) = 0.3 D; p(C-F) = 1.51 D (7) and p(CF3) = 2.03 D; p(CCl3) = 1.27 D (8). I t follows that deviations of a few degrees from the tetrahedral valence angles, or as much as 6' from the ideally staggered 60° torsion angles, do not affect the dipole moment calculations appreciably. Even more substantial deviations would fall within the 0.2 D error-limit. The method presented here saves much time and effort in the calculation of the dipole moment of molecules that possess approximately the diamond skeleton geometry. This should he extremely valuable both to teacher and student, especially in the lecture-room situation where the blackboard is nearer than the computer. Literature Clled
reducing eqn. (6) to eqn. (2). The table compares the dipole moments of some 1,3dioxanes bearing polar substituents, as calculated by the "rapid2'- and by the "exact" method. The latter uses internal coordinates abstracted from the ring geometry of the
24 / Journal of Chemical Education
11) Thumron. J. J., Phil. Mag., 46.513 (1923). 12) Nordiander,J.E., J.CHEM. EDUC.. 50,713 (1973). 181 Eyring. H., Phyr. Re". 39, 746 (1932): see also: Scott, R. A,,and Scherapa. H.A,,J. C h e m Phya.. 44,3056 (1966). 14) Buckingham.A. D.."PhysicalChemirtry,"A~ademic Press, 1970,Vol.IV.p.383. (51 Wi1cox.C. F..J. Amer Chen). Soc.. 82,&14 119601. (6)de Kok,A. 3.. and Rome-. C.,Rec. T r m . Chim. Ppva.Bos. 89,313 11970). (7) Smyth,C.P . , J Chsm. Phys., 4,20911937). 18) Minkin, V. I.,Osipov. 0, A..and Zhdsnov. Y. A.."DQole Moments in OrganieChemiLry"P1enum Press, N.Y.. 1971,p.91.
