Fig. 9
Transformation of p-type atomic orbitals that is the combination of basis functions in different coordinate systems. The same p-type atomic orbital in orange color formed by the basic functions in red, gray, and blue colors under the two base coordinate systems \({{{\bf{B}}}}(\overrightarrow{{{{\bf{x}}}}},\overrightarrow{{{{\bf{y}}}}},\overrightarrow{{{{\bf{z}}}}})\) and \({{{\bf{E}}}}(\overrightarrow{{{{\boldsymbol{\sigma }}}}},\overrightarrow{{{{\boldsymbol{\gamma }}}}},\overrightarrow{{{{\bf{n}}}}})\) transformed by using linearity transformation matrix T[\(\overrightarrow{{{{\boldsymbol{\sigma }}}}}\),\(\overrightarrow{{{{\boldsymbol{\gamma }}}}}\),\(\overrightarrow{{{{\bf{n}}}}}\)]. The small black balls represent two bonded atoms A and B, \(\overrightarrow{{{{\boldsymbol{\sigma }}}}}\) vector represents the unit vector in the bond axis direction, whereas \(\overrightarrow{{{{\boldsymbol{\gamma }}}}}\) and \(\overrightarrow{{{{\bf{n}}}}}\) vectors are also the unit vectors in corresponding directions, and the three vectors are orthogonal to each other. The Cpx, Cpy, and Cpz denote the coefficients of basis functions in the global coordinate system B, whereas C′px, C′py, and C′pz represent the corresponding coefficients of basis functions in the local coordinate system E.
