Results Grouping

Atom Grouping

With the default settings, MDANSE will group results by atom type, so that partial results are summed into datasets, with one dataset per atom type present in the system. The total result is calculated from these partial results using the Weighting Scheme. Let’s consider a system consisting of a mixture of water and ethanol. With atom grouping, the partial coherent intermediate scattering functions are obtained for each unique pair of atoms in the system: CC, CH, CO, HH, HO, and OO. For example, the C and H partial coherent intermediate scattering functions is

(27)\[F_{\text{coh},\text{CH}}(\mathbf{q},t) = \frac{1}{N \sqrt{c_{\text{C}}c_{\text{H}}}} \sum\limits_{j \in \text{C}} \sum\limits_{k \in \text{H}} \mathrm{Re}\left[F_{jk}(\mathbf{q},t)\right],\]

(28)\[F_{jk}{(\mathbf{q},t) = \left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{k}\left( t \right)} \right\rbrack} \right\rangle}\]

so the summations run over all carbon and hydrogen atoms in the system. The total is a weighted sum of all partial components

(29)\[ F_{\text{coh}}(\mathbf{q},t) = W_{\text{CC}} F_{\text{coh}, \text{CC}}(\mathbf{q},t) + W_{\text{CH}} F_{\text{coh}, \text{CH}}(\mathbf{q},t) + \cdots.\]

Similarly the partial incoherent intermediate scattering functions are obtained for each unique atom in the system e.g. C, H, and O. For exmaple, the C partial incoherent intermediate scattering functions is

(30)\[F_{\text{inc},\text{C}}(\mathbf{q},t ) = \frac{1}{Nc_{\text{C}}} \sum\limits_{j \in \text{C}} \mathrm{Re}\left[F_{jj}(\mathbf{q},t)\right]\]

so the summation run over all carbon atoms and the total is a weight sum of all partial components

(31)\[ F_{\text{inc}}(\mathbf{q},t) = W_{\text{C}} F_{\text{inc}, \text{C}}(\mathbf{q},t) + W_{\text{H}} F_{\text{inc}, \text{H}}(\mathbf{q},t) + W_{\text{O}} F_{\text{inc}, \text{O}}(\mathbf{q},t),\]

see Weighting Scheme for further details.

Molecule Grouping

MDANSE can also group results by molecule type, so that they are summed for all atoms of each type on each type of molecule, again we will consider a system of water and ethanol. With molecule grouping, the partial coherent intermediate scattering functions are obtained for each unique pair of molecules and their atoms types: [EtOH][EtOH]_CC, [EtOH][EtOH]_CH, [EtOH][H2O]_HO, and etc. Where [EtOH][H2O]_HO are all pairs of H and O atoms where H are the hydrogen atoms in ethanol and O are the oxygen atoms in water. The [EtOH][H2O]_HO partial coherent intermediate scattering function is

(32)\[F_{\text{coh},\text{HO}}^{[\text{EtOH}][\text{H2O}]}{(\mathbf{q},t) = \frac{1}{N \sqrt{c_{\text{H}}^{\text{EtOH}}c_{\text{O}}^{\text{H2O}}}}} \sum\limits_{j \in (\text{EtOH}\, \cap\, \text{H})} \sum\limits_{k \in (\text{H2O}\, \cap\, \text{O})} \mathrm{Re}\left[F_{jk}(\mathbf{q},t)\right]\]

where \(c_{\text{H}}^{\text{EtOH}} = N_{\text{H}}^{\text{EtOH}} / N\) and \(c_{\text{O}}^{\text{H2O}} = N_{\text{O}}^{\text{H2O}} / N\). Here, \(N_{\text{H}}^{\text{EtOH}}\) and \(N_{\text{O}}^{\text{H2O}}\) are the total number of atoms of hydrogen in ethanol and oxygen in water respectively, and \(N\) is the total number of atom in the system. The molecular coherent intermediate scattering functions between ethanol and water is proportional to the weighted sum of the partial terms

(33)\[\begin{split}\sqrt{c_{\text{EtOH}} c_{\text{H2O}}}F_{\text{coh}}^{[\text{EtOH}][\text{H2O}]}(\mathbf{q},t) = &W_{\text{CH}}^{[\text{EtOH}][\text{H2O}]} F_{\text{coh},\text{CH}}^{[\text{EtOH}][\text{H2O}]}(\mathbf{q},t) \\\\ &+ W_{\text{CO}}^{[\text{EtOH}][\text{H2O}]} F_{\text{coh},\text{CO}}^{[\text{EtOH}][\text{H2O}]}(\mathbf{q},t) + \cdots\end{split}\]

where \(c_{\text{EtOH}} = N_{\text{EtOH}} / N\) and \(c_{\text{H2O}} = N_{\text{H2O}} / N\). Here, \(N_{\text{EtOH}}\) and \(N_{\text{H2O}}\) are the total number of atoms in ethanol and water respectively. For coherent scattering lengths

(34)\[W_{\text{HO}}^{[\text{EtOH}][\text{H2O}]} = 2\frac{\sqrt{c_{\text{H}}^{\text{EtOH}}c_{\text{O}}^{\text{H2O}}} \ \mathrm{Re} \left[ b_{\mathrm{coh},\text{H}}^{\dagger} b_{\mathrm{coh},\text{O}}\right] }{c_{\text{C}}^{\text{EtOH}}c_{\text{C}}^{\text{EtOH}} b_{\mathrm{coh},\text{C}}^{\dagger} b_{\mathrm{coh},\text{C}} + \cdots}.\]

The total coherent intermediate scattering functions is a weighted sum of all molecular terms

(35)\[F_{\text{coh}}(\mathbf{q},t) = \sqrt{c_{\text{EtOH}} c_{\text{EtOH}}}F_{\text{coh}}^{[\text{EtOH}][\text{EtOH}]}(\mathbf{q},t) + \sqrt{c_{\text{EtOH}} c_{\text{H2O}}}F_{\text{coh}}^{[\text{EtOH}][\text{H2O}]}(\mathbf{q},t) + \cdots\]

where the weights are simply the square roots of the product of the atom concentrations.

Similarly the partial incoherent intermediate scattering functions are obtained for each molecule and its atom types: [EtOH]_C, [EtOH]_H, [EtOH]_O, [H2O]_H, and [H2O]_O. For example, the partial incoherent intermediate scattering functions for ethanol’s carbon atoms is

(36)\[F_{\text{inc},\text{C}}^{\text{EtOH}}(\mathbf{q},t ) = \frac{1}{Nc_{\text{C}}^{\text{EtOH}}} \sum\limits_{j \in (\text{EtOH}\, \cap \, \text{C})} \mathrm{Re}\left[F_{jj}(\mathbf{q},t)\right]\]

and the molecular incoherent intermediate scattering functions for ethanol is

(37)\[c_{\text{EtOH}}F_{\text{inc},\text{EtOH}}(\mathbf{q},t ) = W^{\text{EtOH}}_{\text{C}} F_{\text{inc},\text{C}}^{\text{EtOH}}(\mathbf{q},t ) + W^{\text{EtOH}}_{\text{H}}F_{\text{inc},\text{H}}^{\text{EtOH}}(\mathbf{q},t ) + \cdots.\]

with incoherent scattering length weights of

(38)\[W_{\text{C}}^{\text{EtOH}} = \frac{c_{\text{C}}^{\text{EtOH}}\vert b_{\text{inc},\text{C}} \vert^{2}}{c_{\text{C}}^{\text{EtOH}}\vert b_{\text{inc},\text{C}} \vert^{2} + \cdots}.\]

The total incoherent intermediate scattering functions is a weighted sum of all molecular terms

(39)\[F_{\text{inc}}(\mathbf{q},t ) = c_{\text{EtOH}}F_{\text{inc},\text{EtOH}}(\mathbf{q},t ) + c_{\text{H2O}}F_{\text{inc},\text{H2O}}(\mathbf{q},t )\]

where the weight are the atom concentrations of the atoms in ethanol and water. Similarly to the atom grouping, the total results with molecule grouping are a weighted sum of atomic or molecular terms. In MDANSE, either scaled or unscaled results can be plotted and may be more useful for the specific results that has been calculated.

Other Groupings Schemes

In this section includes some analysis calculations with a group setting which does follow the above mechanism.

Root Mean Square Deviation

Grouping in the root mean squared deviation (RMSD) analysis calculation works similarly to the above mechanism. In the RMSD calculation, grouping is done before the square root function is applied to the mean square displacement. This means that grouping with the RMSD analysis will gives the same result as the RMSD analysis when only those atom in the group selected. For our water and ethanol system the molecule grouping the RMSD of water is

(40)\[\text{RMSD}_{\text{H2O}}(t) = \sqrt{ \frac{1}{Nc_{\text{H2O}}} \sum\limits_{j \in \text{H2O}} \vert \mathbf{r}_{j}(t) - \mathbf{r}_{j}(t_{\mathrm{ref}}) \vert^{2} }\]

and RMSD of the hydrogen atoms in water is

(41)\[\text{RMSD}^{\text{H2O}}_{\text{H}}(t) = \sqrt{ \frac{1}{Nc^{\text{H2O}}_{\text{H}}} \sum\limits_{j \in (\text{H}\, \cap \, \text{H2O})} \vert \mathbf{r}_{j}(t) - \mathbf{r}_{j}(t_{\mathrm{ref}}) \vert^{2} }.\]

Root Mean Square Fluctuation

The root mean square fluctuation (RMSF) analysis does not average results but instead calculates them for individual atoms and groups them together. With the atom option, the RMSF results will be split up into datasets, with one dataset per each atom type. With the molecule option, the RMSF results will be split up into groups of datasets, with one group per molecule type. A group will contain results for each atom type in the molecule, as well as the results for all atoms in the molecule.