Chapter 4-1. RS Determination Method and Cross Product of Vectors
Recommended Post : 【Organic Chemistry】 Chapter 4. Stereochemistry
2. Author’s Proposed Principle
3. Proof of Author’s Proposed Principle
1. RS Determination Method
⑴ RS nomenclature is absolute position nomenclature.
① Absolute configuration
② Relative configuration : Experimental relationship indicating whether they are the same in position even if the absolute configuration is not known.
⑵ Basic principles
① Priority rules are the same as for EZ isomers.
② With the 4th atom placed behind, if the direction in which atoms 1, 2, 3 rotate is clockwise, it is R; counterclockwise, it is S.
③ R isomer : Derived from ‘rectus,’ meaning clockwise
④ S isomer : Derived from ‘sinister,’ meaning counterclockwise
2. Author’s Proposed Principle
⑴ Formula 1: If “1st vector × 4th vector = 2nd vector,” then S
Figure. 1. RS determination method using Formula 1
If the cross product direction is 3rd vector, it is R.
⑵ Formula 2: If “1st vector × 2nd vector = 3rd vector,” then S
⑶ Formula 3: If “3rd vector × 4th vector = 1st vector,” then S : Often, methyl is 3rd, and hydrogen is 4th.
⑷ Explanation of Formula 1: If the cross product direction of the position vectors of atoms 1 and 4, with the central carbon as the origin, is 2nd vector, the absolute configuration is S.
⑸ Memorize it as S because it’s special.
⑹ Note that the cross product does not perfectly match the tetrahedral orientation, as it outputs a vector orthogonal to the operand vectors.
3. Proof of Author’s Proposed Principle
Figure. 2. Situation of proving the author’s proposed principle
⑴ Let’s prove Formula 2.
⑵ It represents a chiral center of sp3 tetrahedral orientation with the center (blue circle) : The surrounding 4 atoms are A, B, C, O (black circles)
⑶ Assumption : A is given as 1st priority and B as 2nd priority according to CIP priorities
⑷ 1st vector = Position vector of A - Position vector of the center = (0.5, -0.5, -0.5)
⑸ 2nd vector = Position vector of B - Position vector of the center = (-0.5, 0.5, -0.5)
⑹ 3rd vector = Position vector of O - Position vector of the center = (0.5, 0.5, 0.5)
⑺ According to RS determination method, if A and B are 1st and 2nd priority, and O is 3rd priority while C is 4th priority, it is S configuration; if C is 3rd priority and O is 4th priority, it is R configuration.
⑻ 1st vector × 2nd vector = (0.5, 0.5, 0) (Note: ‘×’ denotes cross product)
⑼ Since the direction of 1st vector × 2nd vector and 3rd vector is similar, if O is 3rd priority, it is S configuration → Formula 2 holds.
⑽ The actual angle between 1st vector × 2nd vector and 3rd vector is only 35.26439°
v1 = (0.5, 0.5, 0), v2 = (0.5, 0.5, 0.5)
⇔ v1 ∙ v2 = 0.5 = |v1| |v2| cos θ = 0.25 √6
⇔ θ = arccos (0.5 / 0.25√6) = 0.6154797 rad = 35.26439°
4. Complementary Explanation
Input : 2023.11.07 10:43