Chapter 12, Problem 12B.5P

Interpretation Introduction

Interpretation:

The version of Karplus equation that fits the given data better is to be predicted.

Concept introduction:

The coupling constant of a compound is represented as $J$, where N represents the number of nuclei involved in the coupling.  The coupling constant of $J{}_{\text{HH}}$ depends on the dihedral angle denoted by $\varphi$.  The equation relating both is known as Karplus equation.

Answer to Problem 12B.5P

The version of Karplus equation which fits the given data is shown below.

  $J{}_{\text{HH}}=\text{A}+\text{B}{\cos }^2{\varphi }_{\text{HH}}+\text{C}\cos 2{\varphi }_{\text{HH}}$

Where,

• J is the coupling constant.
• $\varphi$ is the dihedral angle.
• A and B are Karplus constants.

Explanation of Solution

The Karplus equation given is shown below.

  $J{}_{\text{HH}}=\text{A}{\cos }^2\varphi +\text{B}$        (1)

Where,

• J is the coupling constant.
• $\varphi$ is the dihedral angle.
• A and B are Karplus constants

The equation (12B.14) is shown below.

  $J{}_{\text{HH}}=\text{A}+\text{B}{\cos }^2{\varphi }_{\text{HH}}+\text{C}\cos 2{\varphi }_{\text{HH}}$        (2)

Where,

• J is the coupling constant.
• $\varphi$ is the dihedral angle.
• A, B, C are Karplus constants

The given system is ${\text{XYCHCHR}}_{\text{3}}{\text{R}}_{\text{4}}$.  In first case when ${\text{R}}_3={\text{R}}_4=\text{H}$, the structure of staggered conformation is shown below.

Figure 1

In the above conformation, $\varphi =60°$ between ${\text{H}}_{\text{2}}$ and ${\text{H}}_{\text{3}}$.

Substitute this value in equation (1).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}{\cos }^2\left(60°\right)+\left(-1\text{Hz}\right)\\ =0.75\text{Hz}\end{array}$

Substitute this value in above equation (2).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}+\left(-1\text{Hz}\right)\cos \left(60°\right)+\left(5\text{Hz}\right)\cos \left(120°\right)\\ =4\text{Hz}\end{array}$

In the above conformation, $\varphi =60°$ between ${\text{H}}_{\text{2}}$ and ${\text{H}}_4$.

Substitute this value in above equation (1).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}{\cos }^2\left(60°\right)+\left(-1\text{Hz}\right)\\ =0.75\text{Hz}\end{array}$

Substitute this value in above equation (2).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}+\left(-1\text{Hz}\right)\cos \left(60°\right)+\left(5\text{Hz}\right)\cos \left(120°\right)\\ =4\text{Hz}\end{array}$

In the above conformation, $\varphi =180°$ between ${\text{H}}_{\text{2}}$ and ${\text{H}}_1$.

Substitute this value in above equation (1).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}{\cos }^2\left(180°\right)+\left(-1\text{Hz}\right)\\ =6\text{Hz}\end{array}$

Substitute this value in above equation (2).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}+\left(-1\text{Hz}\right)\cos \left(180°\right)+\left(5\text{Hz}\right)\cos \left(360°\right)\\ =13\text{Hz}\end{array}$

The average of $J{}_{\text{HH}}$ calculated from equation (1) is $2.5\text{Hz}$.  The average of $J{}_{\text{HH}}$ calculated from equation (2) is $7\text{Hz}$.  The experimental value is $7.3\text{Hz}$.  Therefore equation (2) is more precise.

In second case when ${\text{R}}_3={\text{CH}}_3$ and ${\text{R}}_4=\text{H}$, the structure of staggered conformation is shown below.

Figure 2

In the above conformation, $\varphi =60°$ between ${\text{H}}_{\text{2}}$ and ${\text{H}}_{\text{3}}$.

Substitute this value in above equation (1).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}{\cos }^2\left(60°\right)+\left(-1\text{Hz}\right)\\ =0.75\text{Hz}\end{array}$

Substitute this value in above equation (2).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}+\left(-1\text{Hz}\right)\cos \left(60°\right)+\left(5\text{Hz}\right)\cos \left(120°\right)\\ =4\text{Hz}\end{array}$

In the above conformation, $\varphi =180°$ between ${\text{H}}_{\text{2}}$ and ${\text{H}}_1$.

Substitute this value in above equation (1).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}{\cos }^2\left(180°\right)+\left(-1\text{Hz}\right)\\ =6\text{Hz}\end{array}$

Substitute this value in above equation (2).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}+\left(-1\text{Hz}\right)\cos \left(180°\right)+\left(5\text{Hz}\right)\cos \left(360°\right)\\ =13\text{Hz}\end{array}$

The average of $J{}_{\text{HH}}$ calculated from equation (1) is $3.37\text{Hz}$.  The average of $J{}_{\text{HH}}$ calculated from equation (2) is $8.5\text{Hz}$.  The experimental value is $8\text{Hz}$.  Therefore equation (2) is more precise.

In third case when ${\text{R}}_3={\text{CH}}_3$ and ${\text{R}}_4={\text{CH}}_3$, the structure of staggered conformation is shown below.

Figure 3

In the above conformation, $\varphi =180°$ between ${\text{H}}_{\text{2}}$ and ${\text{H}}_1$.

Substitute this value in equation (1).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}{\cos }^2\left(180°\right)+\left(-1\text{Hz}\right)\\ =6\text{Hz}\end{array}$

Substitute this value in equation (2).

  $\begin{array}{c}J{}_{\text{HH}}=7\text{Hz}+\left(-1\text{Hz}\right)\cos \left(180°\right)+\left(5\text{Hz}\right)\cos \left(360°\right)\\ =13\text{Hz}\end{array}$

The average of $J{}_{\text{HH}}$ calculated from equation (1) is $6\text{Hz}$.  The average of $J{}_{\text{HH}}$ calculated from equation (2) is $13\text{Hz}$.  The experimental value is $11.2\text{Hz}$.  Therefore equation (2) is more precise.