Kinetic energy of the molecule as a function of the displacements of the atoms from equilibrium positions. For Cartesian displacement coordinates ($x,y,z$), $$T=\sum _{\alpha }½m_{\alpha }({\dot{x}}_{\alpha }^2+{\dot{y}}_{\alpha }^2+{\dot{z}}_{\alpha }^2)$$ where the sum is over all atoms and ${\dot{x}}_{\alpha }$, etc., are the displacement velocities. For internal coordinates $$T=½{\mathit{P}}^t\mathit{G}\mathit{P}=½{\dot{\mathit{R}}}^t{\mathit{G}}^{-1}\dot{\mathit{R}}=½\sum _{ij}{G}_{ij}^{-1}{\dot{R}}_i{\dot{R}}_j$$ where ${\dot{\mathit{R}}}^t$ and $\dot{\mathit{R}}$ are the row and column vector, respectively, of the $\partial R_i/\partial t$, ${\mathit{P}}^t$ and $\mathit{P}$ are the row and column vector, respectively, of the momenta conjugate to the $R_i$, ${\mathit{G}}^{-1}$ is the inverse of the $\mathit{G}$ matrix and $G_{ij}^{-1}$ is the $i{j}^{\mathrm{t}\mathrm{h}}$ element of ${\mathit{G}}^{-1}$. The elements of the $\mathit{G}$ matrix are defined by $$G_{ij}=\sum _{\alpha }\frac{1}{m_{\alpha }}{B}_{i\alpha }{B}_{j\alpha }$$ where $B_{i\alpha }$ and $B_{j\alpha }$ relate the $i^{\mathrm{t}\mathrm{h}}$ and $j^{\mathrm{t}\mathrm{h}}$ internal coordinate to the ${\alpha }^{\mathrm{t}\mathrm{h}}$ Cartesian coordinate through the equation $\mathit{R}=\mathit{B}\mathit{X}$ in which $\mathit{X}$ is the column vector of the Cartesian coordinates.