Vibrational Spectra of Polyatomic Molecules

Abstract

The idea of normal modes has been introduced to solve the problems related to vibrational spectra of polyatomic molecules both from classical and from quantum mechanical points of view. Selection rules for Raman and infrared spectra have been determined and extensive discussions on the vibrational spectra of symmetric and asymmetric linear triatomic molecules are presented. The idea of Wilson G matrix has been introduced to solve the vibrational problems. The ideas of internal coordinates, symmetry coordinates and s-vectors have also been introduced. Formulas of s-vectors have been derived for stretching, bending, wagging and torsional vibrations. Vibrational problem is solved for nonlinear triatomic molecule by Wilson GF matrix method. Besides these, Fourier Transform spectroscopy and its fundamental basis have been discussed.

Appendices

Appendix

s-Vectors for Torsional Vibrations

Analytically, the dihedral (torsion) angle τ (see Fig. 5.8) can be defined as

$$ \cos \tau \, = \,\frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{23} \times \vec{e}_{34} )}}{{\sin \phi_{2} \sin \phi_{3} }} $$

(5.137)

where $\frac{{(\vec{e}_{12} \times \vec{e}_{23} )}}{{\sin \phi_{2} }}$ and $\frac{{(\vec{e}_{23} \times \vec{e}_{34} )}}{{\sin \phi_{3} }}$ are the unit vectors perpendicular to the planes 1, 2, 3 and 2, 3, 4, respectively. By differentiation of Eq. (5.137), small variation of τ is obtained.

$$\begin{aligned} - \sin \tau \Delta \tau & = \frac{{(\Delta \vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin \phi _{3} }}\, + \frac{{(\vec{e}_{{12}} \times \Delta \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin \phi _{3} }} \\ & \quad + \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\Delta \vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin \phi _{3} }} \\ & \quad + \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \Delta \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin \phi _{3} }} \\ & \quad - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin \phi _{3} }}\left[ {\frac{{\cos \phi _{2} \,\Delta \phi _{2} }}{{\sin \phi _{2} }}\, + \,\frac{{\cos \phi _{3} \,\Delta \phi _{3} }}{{\sin \phi _{3} }}\,} \right] \\ & \quad = T_{1} \, + \,T_{2} \, + \,T_{3} \, + \,T_{4} \, + \,T_{5} \, + \,T{}_{6} \\ \end{aligned}$$

(5.138)

where

$$ \Delta \tau \, = \,\sum\limits_{\alpha = 1}^{4} {\vec{s}_{t\alpha } \cdot \vec{\rho }_{\alpha } } \,\,\, $$

(5.138a)

In deriving the expressions for the $\vec{s}\,\,\,$ vectors, the terms (T_is, i = 1–6) are needed to be determined.

$$\begin{aligned} T_{1} & = \frac{{(\Delta \vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin \phi _{3} }} \\ & = \frac{1}{{\sin \phi _{2} \sin \phi _{3} }}\left\{ {\frac{{r_{{12}} \Delta \vec{r}_{{12}} - \,(\Delta r_{{12}} )\vec{r}_{{12}} }}{{r_{{12}}^{2} }} \times \vec{e}_{{23}} } \right\} \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} ),\,\,\,\,{\text{from}}\,\,\,(5.67) \\ & = \frac{1}{{r_{{12}} \sin \phi _{2} \sin \phi _{3} }}\,[\{ (\vec{\rho }_{{2\,\,}} - \vec{\rho }_{1} ) \\ & - \{ \vec{e}_{{12}} \cdot (\vec{\rho }_{{2\,}} - \vec{\rho }_{1} )\} \,\vec{e}_{{12}} \} \times \vec{e}_{{23}} ] \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} ),\,\,\,\,\,{\text{from}}\,\,\,(5.69)\,\,\,{\text{and}}\,\,\,(5.71)\, \\ & = \frac{1}{{r_{{12}} \sin \phi _{2} \sin \phi _{3} }}\left[ \begin{gathered} \{ - \vec{\rho }_{1} \times \vec{e}_{{23}} \, + \,(\vec{e}_{{12}} \cdot \vec{\rho }_{1} )(\vec{e}_{{12}} \times \vec{e}_{{23}} )\} \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} ) \hfill \\ + \{ \vec{\rho }_{2} \times \vec{e}_{{23}} \, - \,(\vec{e}_{{12}} \cdot \vec{\rho }_{2} )(\vec{e}_{{12}} \times \vec{e}_{{23}} )\} \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} ) \hfill \\ \end{gathered} \right] \\ & = \frac{1}{{r_{{12}} \sin \phi _{2} \sin \phi _{3} }}\left[ \begin{gathered} \vec{\rho }_{1} \cdot \,\{ \, - \,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} ) \hfill \\ + \vec{e}_{{12}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} \hfill \\ + \,\vec{\rho }_{2} \cdot \,\{ \,\,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} ) \hfill \\ - \vec{e}_{{12}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} \hfill \\ \end{gathered} \right] \\ \end{aligned}$$

(5.139a)

Similarly, the next three terms of T₂, T₃ and T₄ can be determined as

$$T_{2} \, = \,\frac{1}{{r_{{23}} \sin \phi _{2} \sin \phi _{3} }}\,\,\,\,\left[ \begin{gathered} \vec{\rho }_{2} \cdot \,\{ \,\,\vec{e}_{{12}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} ) \hfill \\ + \vec{e}_{{23}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} \hfill \\ + \,\vec{\rho }_{3} \cdot \,\{ \, - \,\vec{e}_{{12}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} ) \hfill \\ - \vec{e}_{{23}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} \hfill \\ \end{gathered} \right]$$

(5.139b)

$$T_{3} \, = \,\frac{1}{{r_{{23}} \sin \phi _{2} \sin \phi _{3} }}\,\,\,\,\left[ \begin{gathered} \vec{\rho }_{2} \cdot \,\{ \,(\,\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times \vec{e}_{{34}} \hfill \\ + \vec{e}_{{23}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} \hfill \\ + \vec{\rho }_{3} \cdot \,\{ \, - (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times \vec{e}_{{34}} \hfill \\ - \vec{e}_{{23}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} \hfill \\ \end{gathered} \right]$$

(5.139c)

$$T_{4} \, = \,\frac{1}{{r_{{34}} \sin \phi _{2} \sin \phi _{3} }}\,\,\,\,\left[ \begin{gathered} \vec{\rho }_{3} \cdot \,\{ - \,(\,\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times \vec{e}_{{23}} \, + \,\, \hfill \\ + \vec{e}_{{34}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} \hfill \\ + \,\vec{\rho }_{4} \cdot \,\{ \,(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times \vec{e}_{{23}} \hfill \\ - \vec{e}_{{34}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} \hfill \\ \end{gathered} \right]$$

(5.139d)

T₅ and T₆ may be determined in the following manner.

$$\begin{aligned} T_{5} & = - \,\frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin \phi _{3} }}\,\,\,\frac{{\cos \phi _{2} \,\Delta \phi _{2} }}{{\sin \phi _{2} }}\, \\ & = - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin ^{2} \phi _{2} \sin \phi _{3} }}\cos \phi _{2} \,\,\left[ \begin{gathered} \,\frac{{ - \cos \phi _{2} \,\vec{e}_{{12}} \, - \,\vec{e}_{{23}} }}{{r_{{12}} \sin \phi _{2} }} \cdot \vec{\rho }_{1} \hfill \\ + \frac{{\cos \phi _{2} \,\vec{e}_{{23}} \, + \,\vec{e}_{{12}} }}{{r_{{23}} \sin \phi _{2} }} \cdot \vec{\rho }_{3} \, \hfill \\ + \frac{\begin{gathered} (r_{{12}} - r_{{23}} \cos \phi _{2} )( - \vec{e}_{{12}} ) \hfill \\ + \,(r_{{23}} - r_{{12}} \cos \phi _{2} )\vec{e}_{{23}} \hfill \\ \end{gathered} }{{r_{{12}} r_{{23}} \sin \phi _{2} }} \cdot \vec{\rho }_{2} \hfill \\ \end{gathered} \right] \\ & \left( {{\text{from}}\,\,\,{\text{Eq}}{\text{.}}\,\,5.63} \right) \\ & = - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin ^{2} \phi _{2} \sin \phi _{3} }}\cos \phi _{2} \left[ \begin{gathered} \frac{{(\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\,\vec{e}_{{12}} \, - \,\vec{e}_{{23}} }}{{r_{{12}} \sin \phi _{2} }} \cdot \vec{\rho }_{1} \hfill \\ + \frac{{ - (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\,\vec{e}_{{23}} \, + \,\vec{e}_{{12}} }}{{r_{{23}} \sin \phi _{2} }} \hfill \\ \end{gathered} \right. \cdot \vec{\rho }_{3} \\ & + \left. {\,\left( \begin{gathered} \frac{1}{{r_{{12}} \sin \phi _{2} }}\{ \vec{e}_{{23}} - (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\,\vec{e}_{{12}} \} \hfill \\ + \frac{1}{{r_{{23}} \sin \phi _{2} }}\{ - \vec{e}_{{12}} + (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\,\vec{e}_{{23}} \} \, \hfill \\ \end{gathered} \right)\, \cdot \vec{\rho }_{2} } \right] \\ & = - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin ^{2} \phi _{2} \sin \phi _{3} }}\cos \phi _{2} \,\,\,\left[ \begin{gathered} \frac{{\vec{e}_{{12}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )\,\,}}{{r_{{12}} \sin \phi _{2} }} \cdot \vec{\rho }_{1} \hfill \\ - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times \,\vec{e}_{{23}} \,}}{{r_{{23}} \sin \phi _{2} }} \hfill \\ \end{gathered} \right. \cdot \vec{\rho }_{3} \\ & + \,\left. {\,\left( { - \frac{{\vec{e}_{{12}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )}}{{r_{{12}} \sin \phi _{2} }}\,\, + \,\,\frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times \,\vec{e}_{{23}} }}{{r_{{23}} \sin \phi _{2} }}\,} \right) \cdot \vec{\rho }_{2} } \right] \\ \end{aligned}$$

(5.140a)

Similarly, it can be shown that

$$ \begin{aligned} T_{6\,} & = - \frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{23} \times \vec{e}_{34} )}}{{\sin \phi_{2} \sin^{2} \phi_{3} }}\cos \phi_{3} \left[ \begin{gathered} \frac{{\vec{e}_{23} \times (\vec{e}_{23} \times \vec{e}_{34} )}}{{r_{23} \sin \phi_{3} }} \cdot \vec{\rho }_{2} \hfill \\ - \frac{{(\vec{e}_{23} \times \vec{e}_{34} ) \times \,\vec{e}_{34} \,}}{{r_{34} \sin \phi_{3} }} \hfill \\ \end{gathered} \right. \cdot \vec{\rho }_{4} \\ & + \,\left. {\,\left( { - \frac{{\vec{e}_{23} \times (\vec{e}_{23} \times \vec{e}_{34} )\,\,}}{{r_{23} \sin \phi_{3} }}\,\, + \,\,\frac{{(\vec{e}_{23} \times \vec{e}_{34} ) \times \,\vec{e}_{34} \,}}{{r_{34} \sin \phi_{3} }}} \right) \cdot \vec{\rho }_{3} } \right] \\ \end{aligned} $$

(5.140b)

The $\vec{s}_{t1}$ vector, which is the scalar product multiplier of $\vec{\rho }_{1}$ (5.138a), is found to have contributions only from the terms T₁ and T₅. The contributions of these terms are.

$$\begin{aligned} & \frac{{\{ \, - \,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\, + \,\,\vec{e}_{{12}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} }}{{r_{{12}} \sin \phi _{2} \sin \phi _{3} }} \\ & - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\cos \phi _{2} \,\{ \vec{e}_{{12}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )\} \\ & = \frac{1}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\,[\,\{ \, - \,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} ) \\ & + \,\,\vec{e}_{{12}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & + (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\{ (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )^{2} \vec{e}_{{12}} - (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\vec{e}_{{23}} \} \,], \\ & [{\text{since}}\,\,\,\,\cos \phi _{2} = \, - (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\,\,{\text{and}}\,\,\,\sin ^{2} \phi _{2} \, = \,\,\,(\vec{e}_{{12}} \, \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} )\,] \\ & = \,\frac{1}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\,[\,\{ \, - \,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\, + \,\,\vec{e}_{{12}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & + (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\{ (1 - (\vec{e}_{{12}} \times \vec{e}_{{23}} )^{2} )\vec{e}_{{12}} - (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\vec{e}_{{23}} \} \,] \\ & [{\text{since}}\,\,\,\cos ^{2} \phi _{2} \,\, = \,\,\,\left\{ { - (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\,} \right\}^{2} \, = \,1 - \sin ^{2} \phi _{2} = 1 - (\vec{e}_{{12}} \times \vec{e}_{{23}} )^{2} ] \\ & = \frac{1}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\,[\,\{ \, - \,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,\} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & + (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\{ (\vec{e}_{{12}} - (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\vec{e}_{{23}} \} \,]\, \\ & = \frac{1}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\,[\,\{ \, - \,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,\} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & + (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\{ \vec{e}_{{23}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} \} \,] \\ & = \frac{{\vec{e}_{{23}} \times }}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}[ - \,(\vec{e}_{{23}} \times \vec{e}_{{34}} )\,(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} )\, \\ & + (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )(\vec{e}_{{12}} \times \vec{e}_{{23}} )\} ]\, \\ & = \frac{{\vec{e}_{{23}} \times }}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}[(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times \{ (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} ] \\ & = \frac{{\vec{e}_{{23}} \cdot \{ (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} }}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\,(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & [{\text{since}}\,\,\,\vec{e}_{{23}} \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) = 0] \\ & = \frac{{\sin \tau }}{{r_{{12}} \sin ^{2} \phi _{2} }}(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & [{\text{since}}\,\,\,\,\,\sin \tau \,\, = \,\,\frac{{\vec{e}_{{23}} \cdot \{ (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin \phi _{3} }}\} ] \\ \end{aligned}$$

(5.141)

which, from Eq. (5.138a), gives

$$ \vec{s}_{t1} = - \frac{{(\vec{e}_{12} \times \vec{e}_{23} )}}{{r_{12} \sin^{2} \phi_{2} }} $$

(5.142)

Similarly, the $\vec{s}_{t2}$ vector is the scalar product multiplier of $\vec{\rho }_{2}$ (5.138) and this term will get contributions from the terms T₁, T₂, T₃, T₅ and T₆. First consider the contributions from the terms T₃ and T₆

$$\begin{aligned} & \frac{{\{ \,(\,\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times \vec{e}_{{34}} \, + \,\,\vec{e}_{{23}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} }}{{r_{{23}} \sin \phi _{2} \sin \phi _{3} }} \\ & - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin ^{2} \phi _{3} }}\cos \phi _{3} \frac{{\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{r_{{23}} \sin \phi _{3} }} \\ & = \frac{{\{ \,(\,\vec{e}_{{12}} \cdot \vec{e}_{{34}} )\vec{e}_{{23}} \, - (\,\vec{e}_{{23}} \cdot \vec{e}_{{34}} )\vec{e}_{{12}} + \,\,\vec{e}_{{23}} [\vec{e}_{{12}} \cdot \{ \vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} ]}}{{r_{{23}} \sin \phi _{2} \sin \phi _{3} }} \\ & - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin ^{2} \phi _{3} }}\cos \phi _{3} \frac{{\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,\,}}{{r_{{23}} \sin \phi _{3} }} \\ & = \frac{{\{ \,(\,\vec{e}_{{12}} \cdot \vec{e}_{{34}} )\vec{e}_{{23}} \, - (\,\vec{e}_{{23}} \cdot \vec{e}_{{34}} )\vec{e}_{{12}} + \,\,\vec{e}_{{23}} [\vec{e}_{{12}} \cdot \{ \vec{e}_{{23}} (\vec{e}_{{23}} \cdot \vec{e}_{{34}} ) - \vec{e}_{{34}} \} ]}}{{r_{{23}} \sin \phi _{2} \sin \phi _{3} }} \\ & - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin ^{2} \phi _{3} }}\cos \phi _{3} \frac{{\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,\,}}{{r_{{23}} \sin \phi _{3} }} \\ & = \frac{{ - (\,\vec{e}_{{23}} \cdot \vec{e}_{{34}} )\{ \vec{e}_{{23}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )\} }}{{r_{{23}} \sin \phi _{2} \sin ^{3} \phi _{3} }}\sin ^{2} \phi _{3} \\ & - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin \phi _{2} \sin ^{2} \phi _{3} }}\cos \phi _{3} \frac{{\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,\,}}{{r_{{23}} \sin \phi _{3} }} \\ & = \,\frac{{\cos \phi _{3} }}{{r_{{23}} \sin \phi _{2} \sin ^{3} \phi _{3} }}[\,(\vec{e}_{{23}} \times \vec{e}_{{34}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\{ \vec{e}_{{23}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )\} \\ & - (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\{ \vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} ], \\ & {\text{since}}\,\,\,\,\,\cos \phi _{3} = \, - (\,\vec{e}_{{23}} \cdot \vec{e}_{{34}} )\,\,\,{\text{and}}\,\,\,\sin ^{2} \phi _{3} = \,\,(\vec{e}_{{23}} \times \vec{e}_{{34}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,\, \\ & = \frac{{\cos \phi _{3} }}{{r_{{23}} \sin \phi _{2} \sin ^{3} \phi _{3} }}\vec{e}_{{23}} \times \,\,[(\vec{e}_{{23}} \times \vec{e}_{{34}} ) \cdot \,(\vec{e}_{{23}} \times \vec{e}_{{34}} )(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & - (\vec{e}_{{23}} \times \vec{e}_{{34}} )(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot \,(\vec{e}_{{23}} \times \vec{e}_{{34}} )] \\ & = \frac{{\cos \phi _{3} }}{{r_{{23}} \sin \phi _{2} \sin ^{3} \phi _{3} }}\vec{e}_{{23}} \times \,[(\vec{e}_{{23}} \times \vec{e}_{{34}} ) \times \{ (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} ] \\ & = \frac{{\cos \phi _{3} (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{r_{{23}} \sin \phi _{2} \sin ^{3} \phi _{3} }}\vec{e}_{{23}} \cdot \{ (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} , \\ & {\text{since}}\,\,\,\vec{e}_{{23}} \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\, = \,0 \\ & = \sin \tau \,\frac{{\cos \phi _{3} (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{r_{{23}} \sin ^{2} \phi _{3} }},\quad {\text{see}}\,\,\,{\text{Eq}}{\text{.}}\,\,\,(5.141) \\ \end{aligned}$$

(5.143a)

Next consider the contributions from T₂ and the second term of the scalar multiplier of $\vec{\rho }_{2}$ in τ₅ (the first term will be considered along with T₁).

$$ \begin{aligned} & \frac{{\{ \,\,\vec{e}_{12} \times (\vec{e}_{23} \times \vec{e}_{34} )\, + \,\,\vec{e}_{23} (\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{23} \times \vec{e}_{34} )\} }}{{r_{23} \sin \phi_{2} \sin \phi_{3} }} \\ & - \frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{23} \times \vec{e}_{34} )}}{{\sin^{2} \phi_{2} \sin \phi_{3} }}\cos \phi_{2} \,\frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \times \,\vec{e}_{23} }}{{r_{23} \sin \phi_{2} }} \\ & = \,\,\frac{{\{ \,(\vec{e}_{12} \cdot \vec{e}_{34} )\vec{e}_{23} \, - (\vec{e}_{12} \cdot \vec{e}_{23} )\vec{e}_{34} \} \, + \,\,\vec{e}_{23} [\vec{e}_{12} \cdot \{ \vec{e}_{23} (\vec{e}_{23} \cdot \vec{e}_{34} ) - \vec{e}_{34} \} ]}}{{r_{23} \sin \phi_{2} \sin \phi_{3} }} \\ & - \frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{23} \times \vec{e}_{34} )}}{{\sin^{2} \phi_{2} \sin \phi_{3} }}\cos \phi_{2} \,\frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \times \,\vec{e}_{23} }}{{r_{23} \sin \phi_{2} }} \\ & = \frac{{ - (\vec{e}_{12} \cdot \vec{e}_{23} )\vec{e}_{34} \, + \,\,\vec{e}_{23} (\vec{e}_{12} \cdot \vec{e}_{23} )(\vec{e}_{23} \cdot \vec{e}_{34} )}}{{r_{23} \sin \phi_{2} \sin \phi_{3} }} \\ & - \frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{23} \times \vec{e}_{34} )}}{{\sin^{2} \phi_{2} \sin \phi_{3} }}\cos \phi_{2} \,\frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \times \,\vec{e}_{23} }}{{r_{23} \sin \phi_{2} }} \\ & = \, - \frac{{\cos \phi_{2} \{ \vec{e}_{23} \, \times (\vec{e}_{23} \times \vec{e}_{34} )\} }}{{r_{23} \sin^{3} \phi_{2} \sin \phi_{3} }}\sin^{2} \phi_{2} \\ & - \frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{23} \times \vec{e}_{34} )}}{{\sin^{2} \phi_{2} \sin \phi_{3} }}\cos \phi_{2} \,\frac{{(\vec{e}_{12} \times \vec{e}_{23} ) \times \,\vec{e}_{23} }}{{r_{23} \sin \phi_{2} }} \\ & {\text{since}}\,\,\,\cos \phi_{2} = - (\vec{e}_{12} \cdot \vec{e}_{23} ) \\ & = - \frac{{\cos \phi_{2} \,\vec{e}_{23} \, \times }}{{r_{23} \sin^{3} \phi_{2} \sin \phi_{3} }}\,\{ \,(\vec{e}_{23} \times \vec{e}_{34} )(\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{12} \times \vec{e}_{23} ) \\ & - (\vec{e}_{12} \times \vec{e}_{23} )(\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{23} \times \vec{e}_{34} )\} \\ & {\text{since}}\,\,\,\,\,\sin^{2} \phi_{2} = (\vec{e}_{12} \times \vec{e}_{23} ) \cdot (\vec{e}_{12} \times \vec{e}_{23} ) \\ & = \frac{{\cos \phi_{2} \,\vec{e}_{23} \, \times }}{{r_{23} \sin^{3} \phi_{2} \sin \phi_{3} }}[(\vec{e}_{12} \times \vec{e}_{23} ) \times \{ (\vec{e}_{12} \times \vec{e}_{23} ) \times (\vec{e}_{23} \times \vec{e}_{34} )\} ] \\ & = \frac{{\cos \phi_{2} (\vec{e}_{12} \times \vec{e}_{23} )\,}}{{r_{23} \sin^{3} \phi_{2} \sin \phi_{3} }}[\vec{e}_{23} \, \cdot \{ (\vec{e}_{12} \times \vec{e}_{23} ) \times (\vec{e}_{23} \times \vec{e}_{34} )\} ] \\ & = \frac{{\cos \phi_{2} (\vec{e}_{12} \times \vec{e}_{23} )\,}}{{r_{23} \sin^{2} \phi_{2} }}\sin \tau ,\quad {\text{see}}\,\,\,\,{\text{Eq}}{.}\,\,(5.141) \\ \end{aligned} $$

(5.143b)

The remaining contributions come from T₁ and the first term of the scalar multiplier of $\vec{\rho }_{2}$ in T₅.

$$\begin{aligned} & \frac{{\{ \,\,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\, - \,\,\vec{e}_{{12}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{r_{{12}} \sin \phi _{2} \sin \phi _{3} }} \\ & + \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{\sin ^{2} \phi _{2} \sin \phi _{3} }}\cos \phi _{2} \frac{{\vec{e}_{{12}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )}}{{r_{{12}} \sin \phi _{2} }}\,\,\, \\ & = \frac{{\,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} )\, \\ & + \frac{\begin{gathered} - \vec{e}_{{12}} (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \hfill \\ - (\vec{e}_{{12}} \cdot \vec{e}_{{23}} )(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )\{ \vec{e}_{{12}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )\} \hfill \\ \end{gathered} }{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}, \\ & {\text{since}}\,\,\,\,\,\cos \phi _{2} = - \,(\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\,\,\,\,\,and\,\,\,\,\sin ^{2} \phi _{2} = (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & = \frac{{\,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & + \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\,[\,\vec{e}_{{12}} \{ \vec{e}_{{23}} \cdot \{ \vec{e}_{{12}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )\} \} \\ & - \,(\vec{e}_{{12}} \cdot \vec{e}_{{23}} )\{ \vec{e}_{{12}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )\} ] \\ & = \frac{{\,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\,(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & + \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\vec{e}_{{23}} \times [\vec{e}_{{12}} \times \{ \vec{e}_{{12}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )\} ] \\ & = \frac{{\,\vec{e}_{{23}} \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\,}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\,(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & - \frac{{(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}\{ \vec{e}_{{23}} \times (\vec{e}_{{12}} \times \vec{e}_{{23}} )\} \\ & {\text{since}}\,\,\,\,\,\vec{e}_{{12}} \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) = 0 \\ & = \frac{{\,\vec{e}_{{23}} \times \,}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}[(\vec{e}_{{23}} \times \vec{e}_{{34}} )(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \\ & - (\vec{e}_{{12}} \times \vec{e}_{{23}} )(\vec{e}_{{12}} \times \vec{e}_{{23}} ) \cdot (\vec{e}_{{23}} \times \vec{e}_{{34}} )] \\ & = - \frac{{\,\vec{e}_{{23}} \times \,}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}[\{ (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times \{ (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} ] \\ & = - \frac{{\,\vec{e}_{{23}} \cdot \{ (\vec{e}_{{12}} \times \vec{e}_{{23}} ) \times (\vec{e}_{{23}} \times \vec{e}_{{34}} )\} \,}}{{r_{{12}} \sin ^{3} \phi _{2} \sin \phi _{3} }}(\vec{e}_{{12}} \times \vec{e}_{{23}} ),\,\,{\text{since}}\,\,\vec{e}_{{23}} \cdot (\vec{e}_{{12}} \times \vec{e}_{{23}} ) = 0 \\ & = - \frac{{\,(\vec{e}_{{12}} \times \vec{e}_{{23}} )\,}}{{r_{{12}} \sin ^{2} \phi _{2} }}\sin \tau ,\,\,{\text{see}}\,\,{\text{Eq}}{\text{.}}\,\,(5.141) \\ \end{aligned}$$

(5.143c)

Thus substituting the terms from Eqs. (5.87, a, b and c) into the Eq. (5.82a), we get

$$ \vec{s}_{t2} = \frac{{r_{23} - r_{12} {\text{cos}} \phi_{2} }}{{r_{12} r_{23} {\text{sin}} \phi_{2} }}\frac{{\vec{e}_{12} \times \vec{e}_{23} }}{{{\text{sin}} \phi_{2} }} - \frac{{{\text{cos}} \phi_{3} }}{{r_{23} {\text{sin}} \phi_{3} }}\frac{{\vec{e}_{23} \times \vec{e}_{34} }}{{{\text{sin}} \phi_{3} }} $$

(5.144)

In a similar fashion, $\vec{s}_{t3}$ and $\vec{s}_{t4}$ can be determined. But without entering into the detailed calculation, the expressions for the respective vectors can be obtained by exchanging the indexes (1,4) and (2,3) of $\vec{s}_{t2}$ and $\vec{s}_{t1}$, respectively.

The s-vectors thus found are,

$$ \vec{s}_{t1} = - \frac{{\vec{e}_{12} \times \vec{e}_{23} }}{{r_{12} {\text{sin}}^{2} \phi_{2} }} $$

(5.145a)

$$ \vec{s}_{t2} = \frac{{r_{23} - r_{12} {\text{cos}} \phi_{2} }}{{r_{12} r_{23} {\text{sin}} \phi_{2} }}\frac{{\vec{e}_{12} \times \vec{e}_{23} }}{{{\text{sin}} \phi_{2} }} - \frac{{{\text{cos}} \phi_{3} }}{{r_{23} {\text{sin}} \phi_{3} }}\frac{{\vec{e}_{23} \times \vec{e}_{32} }}{{{\text{sin}} \phi_{3} }} $$

(5.145b)

$$ \vec{s}_{t3} = \left[ {\left( {14} \right)\left( {23} \right)} \right]\vec{s}_{t2} $$

(5.145c)

$$ \vec{s}_{t4} = \left[ {\left( {14} \right)\left( {23} \right)} \right]\vec{s}_{t1} $$

(5.145d)

where the expressions in brackets $\left( {14} \right)$ and $\left( {23} \right)$ indicate that the latter vectors can be obtained by permutation of the atom subscripts by 1 & 4, and 2 & 3 in the expressions for the first two vectors.