Weighting Scheme

Partial properties

In MDANSE, most properties are split by atom-type and the total result is a sum of these partial properties. For example, the partial coherent and incoherent intermediate scattering functions scaled with weight factors are

(10)\[\mathcal{F}_{\text{coh},\alpha\beta}(\mathbf{q},t) = \frac{W_{\alpha\beta}}{N \sqrt{c_{\alpha}c_{\beta}}} \sum\limits_{j \in \alpha} \sum\limits_{k \in \beta} \mathrm{Re} \left[ F_{jk}(\mathbf{q},t) \right],\]

(11)\[\mathcal{F}_{\text{inc},\alpha}(\mathbf{q},t ) = \frac{W_{\alpha}}{Nc_{\alpha}} \sum\limits_{j \in \alpha} \mathrm{Re} \left[ F_{jj}(\mathbf{q},t) \right],\]

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

where \(\alpha\) and \(\beta\) are the atom-types. \(W_{\alpha\beta}\) and \(W_{\alpha}\) are the weights of the atom-type pairs \(\alpha\beta\) and the atom type \(\alpha\). \(c_{\alpha} = N_{\alpha} / N\) and \(c_{\beta} = N_{\beta} / N\) are the concentrations of atoms of atom-types \(\alpha\) and \(\beta\). \(N_{\alpha}\) and \(N_{\beta}\) are the the number of atoms-type \(\alpha\) and \(\beta\), and \(N\) is the total number of atoms. In MDANSE, the real part of \(F_{jk}(\mathbf{q},t)\) is taken in Eqs. (10) so that it is the average over \(+\mathbf{q}\) and \(-\mathbf{q}\). The total is now a sum of the partial terms

(13)\[ F_{\text{coh}}(\mathbf{q},t) = \sum_{\alpha}\sum_{\beta \geq \alpha} \mathcal{F}_{\text{coh},\alpha\beta}(\mathbf{q},t), \qquad F_{\text{inc}}(\mathbf{q},t) = \sum_{\alpha} \mathcal{F}_{\text{inc},\alpha}(\mathbf{q},t).\]

For summation involving two atom-types only the unique pairs are included up, since the averaging over \(+\mathbf{q}\) and \(-\mathbf{q}\) will mean that \(\mathcal{F}_{\text{coh},\alpha\beta}(\mathbf{q},t) \approx \mathcal{F}_{\text{coh},\beta\alpha}(\mathbf{q},t)\). The factor of two for the off-diagonal terms is included in the weight factor.

Fig. 1 The total and partial DOS of water. Partial DOS are weighted so that the sum of partial DOS equals to the total.

The partial properties can also be scaled without the weights

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

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

so the total will now be a weighted sum of these partial terms

(16)\[ F_{\text{coh}}(\mathbf{q},t) = \sum_{\alpha}\sum_{\beta \geq \alpha} W_{\alpha\beta} F_{\text{coh},\alpha\beta}(\mathbf{q},t), \qquad F_{\text{inc}}(\mathbf{q},t) = \sum_{\alpha} W_{\alpha} F_{\text{inc},\alpha}(\mathbf{q},t).\]

In the MDANSE_GUI you have the option to plot either weighted (e.g. \(\mathcal{F}_{\text{coh},\alpha\beta}(\mathbf{q},t)\) and \(\mathcal{F}_{\text{inc},\alpha}(\mathbf{q},t)\)) or unweighted (e.g. \(F_{\text{coh},\alpha\beta}(\mathbf{q},t)\) and \(F_{\text{inc},\alpha}(\mathbf{q},t)\)) partial properties.

Fig. 2 The total and partial intermolecular PDF of water. Partial PDF are unweighted and only their weighted sum equals to the total PDF.

The weighted and unweighted options are more useful for different cases, for example, it might be more useful to use the weighted terms for the density of states (DOS) calculations (Fig. 1) while the unweighted terms might be more useful of the pair distribution function (PDF) calculations (Fig. 2).

Rescaled Weights

Single Atom-Type Weights

MDANSE weights are rescaled so that weights for DISF calculation using the b_incoherent will be

(17)\[W_{\alpha} = \frac{c_{\alpha} \vert b_{\mathrm{inc},\alpha} \vert^2}{\sum_{\gamma} c_{\gamma} \vert b_{\mathrm{inc},\gamma} \vert^2}\]

where \(\vert b_{\mathrm{inc},\alpha} \vert^2\) is the squared incoherent scattering length of the atom type \(\alpha\). Note that the weights were squared prior to the rescaling, see Squared Weights for details. By using the rescaled weights the total incoherent intermediate scattering functions becomes

(18)\[F_{\text{inc}}(\mathbf{q},t ) = \frac{1}{\sum_{\gamma} c_{\gamma} \vert b_{\mathrm{inc},\gamma} \vert^2 } \frac{1}{N} \sum\limits_{j} \vert b_{\mathrm{inc},\alpha} \vert^2 \ \mathrm{Re} \left[ F_{jj}(\mathbf{q},t) \right]\]

Notice that by using this weight scheme the total DISF has the property that

(19)\[F_{\text{inc}}(\mathbf{q},t=0) = 1.\]

Double Atom-Type Weights (DCSF and CCF)

For the DCSF calculation using b_coherent, the weights are

(20)\[W_{\alpha\beta} = \left[2 - \delta_{\alpha\beta}\right] \frac{\sqrt{c_{\alpha}c_{\beta}}\ \mathrm{Re} \left[b_{\mathrm{coh},\alpha}^{\dagger}b_{\mathrm{coh},\beta} \right]}{\sum_{\gamma\delta} c_{\gamma}c_{\delta} b_{\mathrm{coh},\gamma}^{\dagger}b_{\mathrm{coh},\delta}}\]

where \(b_{\mathrm{coh},\alpha}^{\dagger}\) and \(b_{\mathrm{coh},\beta}\) are the coherent scattering lengths of the atoms of types \(\alpha\) and \(\beta\). (\(b_{\mathrm{coh},\alpha}^{\dagger}\) is the complex conjugate of \(b_{\mathrm{coh},\alpha}\), as neutron scattering lengths are complex numbers.) The total coherent intermediate scattering functions becomes

(21)\[F_{\text{coh}}(\mathbf{q},t) = \frac{\sqrt{c_{\alpha}c_{\beta}}}{\sum_{\gamma\delta} c_{\gamma}c_{\delta} b_{\mathrm{coh},\gamma}^{\dagger}b_{\mathrm{coh},\delta}} \frac{1}{N} \sum\limits_{jk} \mathrm{Re} \left[ b_{\mathrm{coh},\alpha}^{\dagger}b_{\mathrm{coh},\beta} \right] \mathrm{Re} \left[ F_{jk}(\mathbf{q},t) \right]\]

where \(b_{\mathrm{coh},j}^{\dagger}\) and \(b_{\mathrm{coh},k}\) are the coherent scattering lengths of atoms \(j\) and \(k\). Notice that the total intermediate scattering function (sum of the incoherent and coherent parts) will not be equal (or equal to the sum by some scaling factor) to the to the sum of intermediate scattering function from the DISF and DCSF calculations using the scaled weight scheme since they are not scaled in the same way.

Double Atom-Type Weights (Other)

For calculation other than the DCSF and current correlation function (CCF) a slightly different weight scheme must be used as their partials are normalized slightly differently. In MDANSE the partial static structure factor (SSF) is

(22)\[S_{\alpha\beta}(q) = 1 + \frac{4 \pi \rho}{q} \int\limits_{0}^{\infty} \mathrm{d}r \, \left[ g_{\alpha\beta}(r) - 1\right] r\sin(qr)\]

where

(23)\[\begin{split}g_{\alpha\beta}(r) = \frac{1}{N c_{\alpha} c_{\beta}}\frac{1}{4 \pi r^2} \frac{1}{\rho} \sum_{j \in \alpha} \sum_{\substack{k \in \beta \\ k \neq j}} \left\langle \delta(r - \vert \mathbf{r}_{k} + \mathbf{r}_{j} \vert ) \right\rangle\end{split}\]

are the partial PDFs. Using b_coherent the weights are

(24)\[W_{\alpha\beta} = \left[2 - \delta_{\alpha\beta}\right] \frac{c_{\alpha}c_{\beta} \ \mathrm{Re} \left[b_{\mathrm{coh},\alpha}^{\dagger}b_{\mathrm{coh},\beta} \right]}{\sum_{\gamma\delta} c_{\gamma}c_{\delta} b_{\mathrm{coh},\gamma}^{\dagger}b_{\mathrm{coh},\delta}}\]

notice that the concentrations \(c_{\alpha}c_{\beta}\) are not square-rooted, this is because the the partial SSF has a prefactor of \(1 / N c_{\alpha}c_{\beta}\) while DCSF and CCF calculations have a prefactor of \(1 / N \sqrt{c_{\alpha}c_{\beta}}\).

Squared Weights

For the DISF, GDISF, EISF, and VHF (self-part) calculations all weights are squared prior to being rescaled. For the DOS, PACF, VACF, and PPS calculations the weights are squared prior to being rescaled for only the b_coherent or b_incoherent weights. In most cases the rescaled weights with single atom-type weights will be

(25)\[W_{\alpha} = \mathrm{Re} \left[ \frac{c_{\alpha} w_{\alpha}}{\sum_{\gamma} c_{\gamma} w_{\gamma}} \right]\]

while in some cases when the weight are squared

(26)\[W_{\alpha} = \frac{c_{\alpha} \vert w_{\alpha} \vert^2}{\sum_{\gamma} c_{\gamma} \vert w_{\gamma} \vert^2}\]

where \(w_{\alpha}\) is some weight parameter for the atom-type \(\alpha\).