Table 1 Popular analytical queries in MS

Moment of inertia

$I=\sum _{i=1}^n{m}_i{\mathbf{r}}_i$

Moment of inertia on z axis

$\begin{array}{lll}{I}_z& =& \sum _{i=1}^n{m}_i{r}_{\mathit{\text{zi}}}\end{array}$

Sum of masses

$\begin{array}{lll}M& =& \sum _{i=1}^n{m}_i\end{array}$

Center of mass

$\begin{array}{lll}C& =& \frac{I}{M}\end{array}$

Radius of gyration

$\begin{array}{lll}\mathit{\text{RG}}& =& \sqrt{\frac{I_z}{M}}\end{array}$

Dipole moment

$\begin{array}{lll}D& =& \sum _{i=1}^n{q}_i{\mathbf{r}}_i\end{array}$

Dipole histogram

$\begin{array}{lll}{D}_z& =& \sum _{i=1}^n\frac{D}{z}\end{array}$

Electron density

$\begin{array}{l}\mathit{\text{ED}}=\frac{\sum _{i=1}^n(e_i-q_i)}{\mathit{\text{dz}}\ast x\ast y}\end{array}$

Heat capacity

$\frac{3000\sqrt{T}\ast \mathit{\text{boltz}}}{2\ast \sqrt{T}-n\ast \mathit{\text{df}}\ast \mathit{\text{VarT}}}$

Isothermal compressibility

$\frac{\mathit{\text{VarV}}}{V_{\mathit{\text{avg}}}\ast \mathit{\text{boltz}}\ast T\ast \mathit{\text{PresFac}}}$

Mean square displacement

$\begin{array}{lll}\mathit{\text{msd}}& =& 〈(r_{t+{\Delta }_t}-r_t)〉\end{array}$

Diffusion constant

$\begin{array}{lll}{D}_t& =& \frac{6\ast \mathit{\text{msd}}\left(t\right)}{t}\end{array}$

Velocity autocorrelation

$\begin{array}{lll}{V}_{\mathit{\text{acor}}}& =& 〈(V_{t+{\Delta }_t}\ast V_t)〉\end{array}$

Force autocorrelation

$\begin{array}{lll}{F}_{\mathit{\text{acor}}}& =& 〈(F_{t+{\Delta }_t}\ast F_t)〉\end{array}$

Density function

Histogram of atom counts

Spatial distance histogram (SDH)

Histogram of all atom-to-atom distances

RDF

$\begin{array}{lll}\mathit{\text{rdf}}\left(r\right)& =& \frac{\mathit{\text{SDH}}\left(r\right)}{4\ast \pi \ast r^2\ast {\sigma }_r\ast \rho }\end{array}$