Kenneth Hedberg, who helped interpret the electron diffraction photographs; and from Dr. This is due to the radiation fields of the induced moments in the other molecules.
Y~ is the vector position of the particle T of the molecule with respect to the molecular center of mass. The notations ~ and Im indicate that the real and imaginary parts (respectively) of the quantity are to be taken.
I Rrsl •
Equations (27 – 32) provide a system of 2N equations for determining the electric and magnetic moments induced in each molecule for the system, as specified by the internal quantum states and positions of each molecule. The field vectors in (40) and (41) depend on the positions of all the molecules, and we still need to average these expressions over the positions of the molecules.
The operations curl, grad, etc. (orV) without any identifier will refer to vector differentiation with respect to r;.
We have neglected changes of the distribution functions due to the presence of the light wave. The two major approaches involved in the discussion are the neglect of the fluctuation of such orientation.
Depending on the accuracy with which it is possible to consider each of the electrons of the molecule as associated with a particular group, equations (112) and (114) are equivalent to equation (109). The terms gK, one for each of the groups, correspond to the interaction of the electric and magnetic moments induced in the same group.
POLARIZABILITY THEORIES OF OPTICAL ACTIVI'.rY; APPLICATION TO THE DETERMINATION OF THE ABSOLUTE CONFIGlJRATIONS OF. He considered the atoms to be isotropic. The induced moment in each atom is in the direction of the total electric field acting on that atom. Gray's treatment is interesting only because it was the first of the polarizability theories.
Nothing was said about the location of the oscillators or the nature of the coupling. The moment induced in each group is treated as a sum of ter:ns corresponding to the order of interaction with the moments of the other groups. The first-order term corresponds to the moment induced in D due to the part of the moment of each of the other molecules induced by the external field alone.
Because attempts to assign absolute configurations to several organic compounds on the basis of this first-order term in polarizability theory must be.
The zero-order frequencies will then be given by 116) We will use the subscript zero to represent the set of. It is then easily seen that if a complex function is a solution of the Schrodinger equation with a given It is easily seen that due to the properties of our zeroth-order wavefunctions as given by (118), the zeroth-order term of (114) calculated with these wavefunctions vanishes.
Calculation of g<0>correct to first-order terms, using the fact that the group wavefunctions (and thus the matrix elements of the group dipole moment operators) are real, gives It is instructive to note that the polarization tensor of the molecule, in the zero-order approximation, is given by 121). This is the first-order expression for g~ in terms of the polarity tensors of the substituent groups and the geometry of the molecule.
Equation (124) is the Expression Bx for the first-order term of the polarization theory, applicable to any molecule to which the plausible assumptions already introduced can be applied.
The degree of depolarization ~ of Rayleigh light scattering perpendicular to the direction of the incident beam, the latter being unpolarized, is related(3 2 ) to the principal polarizabilities by the formula. In the case of chloroform, however, it can be expected that the greater polarizability of the chlorine will result in the polarization being greatest perpendicular to the symmetry axis. Again, the measurements of the Kerr constant or the depolarization of scattered light, or both, for suitable molecules, lead to information regarding the polarization tensors of the molecules.
Thus, in the case of cylindrically symmetric molecules, the ellipsoid of polarizability can be determined, preferably with the qualitative application of Silberstein's theory. The starting point from this to the polarizability ellipsoid of the groups that can be considered to make up the molecule requires further assumptions. Furthermore, this will also be true if, in the case of free rotation around the valence bonds, the coplanarity of the three lines remains.
In addition to the characteristics discussed, the calculation of g(o) as given by equation (125) requires knowledge of the structure.
ECO I
HCOH I
It is also possible to continue in the alternative version of the theory described previously. Because of the twofold symmetry axis of 2,3-epoxybutane, it is easily seen that the latter two are complete. The calculation just described leads to the assignment of the configuration given in Figure 1 to the left turner.
Figure 3 also shows the projection formula of the same isomer according to the Fischer convention. The determination of V ( ~ ) is discussed in more detail there, in particular the range of variation of V ( ~ ) within which satisfactory interpretations of the dipole moment and electron diffraction data can be obtained. Here the discussion will be limited to stating the general characteristics of the problem and the results obtained.
It then follows that the sign of the contribution of the skewed isomer to the optical activity is the same as that of the average rotation in the relevant temperature range.
SECTION IV
Kuhn's (46) use of classical coupled oscillator theory in this section, although it is quite similar to the polarizability theories discussed in the previous section. There are several possible procedures in obtaining approximations of the wave functions, and for the purpose of the present discussion it will be simplest to outline these. The wave functions of this chromophoric electron are considered solutions of the Hartree equation.
The height projection formula is the isomer on the left according to the Fischer projection convention. However, with the wave function Gorin, Walter, and Eyring chose to include in the grounq extensions and. The electric moment in this plane is found to be due to the perturbation E' 1 , so that the term calculated by Gorin, Walter and Eyring is of the order H1H' 1 .
The success of the first-order perturbation theory depends on the closeness of the zero-order wave functions.
Furthermore there are probably other stable orientations of the ethyl group that have statistical probabilities of the same order of magnitude as that to which their treatment applies. Calculations very similar to those just described for sec-butyl alcohol have been applied by Gorin, Kauzmann, and Walter(30) to the sugars «-methylarabinopyranoside, p-methyl-. Calculations for «-methylarabinopyranoside were based on the enantiomorph shown in Figure 9, the conformation being the chair form in which the carbon-oxygen bonds of the hydroxyl groups attached to carbons 1, 2, and 3 are approximately perpendicular to the mean plane. of the ring, instead of the other chair shape in ~1ich these bonds are approximately in the mean plane of the ring.
HCH I
No attempt has been made to indicate the orientation with respect to internal rotation of the hydroxyl and methoxy groups, for which reference can be made to the original article. The projection formula in Figure 9 is actually the one assigned to the levorotatory isomer of this sugar, again supporting the correctness of Fischer's original assumption. Such agreement does not, of course, prove that the calculations are valid, especially given the above objections.
Similar agreement with relative configurational studies of organic chemistry was found by Gorin, Kauzmann, and Walter among the four sugars they investigated. A magnitude of the same order as that inferred for this bond from rotational dispersion measurements was obtained. Waser (49) has provided arguments leading to an absolute configuration of tartaric acid based on the correlation of the crystal structure of tartaric acid as determined by Beevers and Stern (50) and the observed crystal habit, using qualitative arguments for the expected rate. of crystal face growth as affected by the number and ease of formation of.
The result obtained by Waser for the absolute configuration of dextrorotatory tartaric acid is shown in Figure 10, where the thick lines are.
COOH I
A check of the consistency of the g-theory, or first-order polarizability theory, awaits the completion of the experimental work described in the previous section. The structure of the related molecule 1,2-dibromopropane was determined by Schomaker and Stevenson (41) using electron diffraction methods. The photographs used in the measurements and in drawing the visual curves were made with the gas near room temperature.
Perhaps the most striking feature of the radial distribution curves is the high value obtained for the C-Cl distance, about 0.04-0.06 a larger 0. Because the conditions of the experiment do not allow the observation of I(s) for s greater than A certain maximum it was necessary to introduce the convergence factor mentioned earlier, a new function rD'(r) being defined as . Distortion of the C-H peak is not unusual in molecules as complicated as this.
Now to proceed to the discussion of our use of the correlation method, we first list the various terms followed.
The amplitude attenuation increases as the range of distances within the range of high probability values of ~ increases. The parameters of the single cosine barrier near the _gauche minimum will be denoted by V0g and ~og•. The dipole moment of the vapor over the temperature range 340 -500° K has been determined by Oriani and SmythC64.
Under these assumptions, the dipole moment of the molecule is given as a function of by the equation. However, it appears that most of the dipole moment is contributed by the g_auche conformation, so that. To reasonably reproduce the observed temperature dependence of the dipole moment allowing a possible high experi.
The Raman and infrared spectra of the vapors were apparently not investigated. Smmnarizj_ng the results described in this section, an investigation of the electron diffraction structure. The functions appearing in the second equation are ordinary Bessel functions of the first order with an imaginary argument.