Probability Distribution Functions for Molecular Rotation: The rotation of atoms

Question

Probability Distribution Functions for Molecular Rotation: The rotation of atoms about a carbon- carbon single bond can be represented using Newman projections as shown: where the rotation angle theta can be represented as a probability distribution function (PDF). Below are two example PDFs (called rotamers) for molecules with different sets of chemical groups: R_1 and R_2. a) Given the PDFs as shorn, calculate the average rotation angle theta for rotamer A and rotamer B. Comment on these calculated values in 2 sentences or less. Are they what you would expect given the periodicity of this system (theta values range from 0 degree -to 359.999 .. degree)? b) Calculate the variance, sigma ^2 of the rotation angle theta rotamer A and rotamer B. Please comment on the meaning of the calculated sigma ^2 values with respect to each rotamer PDF in 2 sentences or less. c) Speculate on the chemical character of R_1 and R_2 for both rotamers given their PDFs.

Explanation / Answer

(a) Average rotation angle for Rotamer A =pixi = 0 * 1/12 + 60 * 1/4 + 120 * 1/12 + 180 * 1/4 + 240 * 1/12 + 300 * 1/4 = 165o

Average rotation angle for Rotamer B= pixi = 0 * 1/24 + 60 * 1/3 + 120 * 1/6 + 180 * 1/3 + 240 * 1/24+ 300 * 1/12= 1350

For rotamer A , the average rotation angle is 165o and for rotamer B, the average roation angle is 135o . As we can see that average angle for rotamer A is more than rotamer B.

(B) Variance for Rotamer A = (0 - 165)2 * 1/12 + (60 - 165)2 * 1/4 + (120 - 165)2 * 1/12 + (180 - 165)2 * 1/4 + (240 -165)2 * 1/12 + (300 - 165)2 * 1/4 = 10275

Variance for Rotamer B = (0 - 135)2 * 1/24 + (60 - 135)2 * 1/3 + (120 - 135)2 * 1/6 + (180 - 135)2 * 1/3 + (240 -135)2 * 1/24 + (300 - 135)2 * 1/12 = 6075

Here we can see that variance of angle is more for rotamer A with compare to Rotamer B.

(c) As for rotamer A has more variance and mean angle near to 180 degree, rotamer A is staggered in nature and Rotamer B is Eclipsed in nature.

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