Table 2 Measures used to compute the ternary (A) and quaternary (B) relations (multi-metrics) among atoms of a molecule

(A) Ternary measures (T _XYZ )

 Perimeter (M19–M20)

T _XTZ  = d _xy  + d _yz  + d _zx

 Triangle area (M21–M22)

\(\begin{aligned} T_{XYZ} & = \sqrt {s\left( {s - d_{XY} } \right)\left( {s - d_{YZ} } \right)\left( {s - d_{ZX} } \right)} \\ s & = \frac{{d_{XY} + d_{YZ} + d_{ZX} }}{2} \\ \end{aligned}\)

 Sides summation (M25–M26)

T _XTZ  = d _xy  + d _yz

 Bond angle (angle between sides) (m27–m28)

\(\begin{aligned} & A_{X} ,A_{Y} ,A_{Z} \;coordinates\;of\;three\;atoms\;of\;a\;molecule \\ & U = A_{X} - A_{Y} ,\;\;V = A_{Z} - A_{Y} \\ & T_{XYZ} = \alpha = \arccos \left( {\frac{U*V}{\left| U \right|*\left| V \right|}} \right) \\ \end{aligned}\)

(B) Quaternary measures (T _XYZ )

 Perimeter (M19–M20)

Q _XTZW  = d _XY  + d _YZ  + d _ZW  + d _WX

 Volume (M23–M24)

\(\begin{aligned} A_{X} ,A_{Y} ,A_{Z} ,A_{W} \;coordinates\;of\;four\;atoms\;of\;a\;molecule \hfill \\ Q_{XYZW} = \frac{1}{6}\left( {\begin{array}{*{20}c} {A_{Y1} - A_{X1} } & {A_{Z1} - A_{X1} } & {A_{W1} - A_{X1} } \\ {A_{Y2} - A_{X2} } & {A_{Z2} - A_{X2} } & {A_{W2} - A_{X2} } \\ {A_{Y3} - A_{X3} } & {A_{Z3} - A_{X3} } & {A_{W3} - A_{X3} } \\ \end{array} } \right) \hfill \\ \end{aligned}\)

 Sides summation (M25–M26)

Q _XTZW  = d _XY  + d _YZ  + d _ZW

 Dihedral angle (M29–M30)

\(\begin{aligned} & A_{X} ,A_{Y} ,A_{Z} \;coordinates\;of\;three\;atoms\;of\;a\;molecule\;in\;the\;plane\;A \\ & B_{W} ,B_{Y} ,B_{Z} \;coordinates\;of\;three\;atoms\;of\;a\;molecule\;in\;the\;plane\;B \\ & U_{A} = \left( {A_{X} - A_{Y} } \right) \times \left( {A_{Z} - A_{y} } \right) \\ & U_{B} = \left( {B_{W} - A_{Y} } \right) \times \left( {B_{Z} - A_{y} } \right) \\ & Q_{XYZW} = \alpha = \arccos \left( {\frac{{U_{A} *U_{B} }}{{\left| {U_{A} } \right|*\left| {U_{B} } \right|}}} \right) \\ \end{aligned}\)