Chapter 3
CONSTRAINTS

With ERIS, \(S\) in Eq. (2.1) is maximized by imposing a linear combination of the generalized \(F\) and \(G\) constraints.

3.1 F Constraint

Let \(N_F\) be the total number of reflections, \(F_\mathrm {o}(\bm {h}_j )\) the observed structure factor, \(F(\bm {h}_j )\) the calculated structure factor, and \(\sigma (\bm {h}_j)\) the standard uncertainty of \(|F_\mathrm {o}(\bm {h}_j )|\). The \(F\) constraint used in MEM analysis from diffraction data is formulated as \begin {equation} C_F' = \sum _{j=1}^{N_F} \left | \Delta F_j \right | ^2 - N_F , \label {eq:C_F'} \end {equation} where \(\Delta F_j\) is the normalized residual for the structure factor: \begin {equation} \Delta F_j = \frac {F_\mathrm {o}(\boldsymbol {h}_j) - F(\boldsymbol {h}_j)}{\sigma (\boldsymbol {h}_j)} . \end {equation} In the actual implementation of the \(F\) constraint in ERIS, \(C_F'\) is normalized with respect to \(N_F\):

3.2 G Constraint

Information about overlapped reflections in powder diffraction data can be introduced into MEM analysis through the \(G\) constraint [17]:

with \begin {equation} G_j = \left [ \frac {\displaystyle \sum _{i=1}^{L_j} m_i\left | F(\boldsymbol {h}_i) \right |^2}{\displaystyle \sum _{i=1}^{L_j} m_i} \right ]^{1/2} . \label {eq:G_j} \end {equation} In Eqs. (??) and (3.3), \(N_G\) is the total number of groups comprising overlapped reflections, \(G_{\mathrm {o}j}\) is the sum of the integrated intensities of overlapped reflections in group \(j\), \(L_j\) is the number of overlapped reflections in group \(j\), and \(m_i\) is the multiplicity of reflection \(i\). Combination of \(C_F'\) and \(C_G'\) affords an integrated constraint, \(C_2\): \begin {equation} C_2 = \frac {C_F' + C_G'}{N_F + N_G}. \label {eq:C_FG} \end {equation}

3.3 Generalized Constraint

The classical \(F\) and \(G\) constraints are based on \(\chi ^2\) statistics, whose use assumes that experimental errors in \(\left | F_\mathrm {o}(\boldsymbol {h}_j) \right |\)’s are random with a Gaussian (normal) distribution. Although the use of \(\chi ^2\) constraint is justified by the Gaussian distribution of errors, it is not a sufficient condition to ensure the Gaussian distribution of residuals between observed and estimated values. In the case of MEM analysis from diffraction data, some low-\(Q\) reflections tend to have very large \(\left | \Delta F_j \right |\)’s, which leads to a noisy distribution of electron or nuclear densities estimated by MEM. Such a characteristic in MEM analysis from diffraction data was theoretically explained by Jauch [18]. To reduce such an undesirable tendency of MEM, Palatinus and van Smaalen [19] proposed to use higher order moments of \(\Delta F_j\) as a constraint.

A probability distribution of a random variable \(x\) is characterized by the values of its central moments \(M_n\). For the normalized Gaussian distribution, the central moments are defined as \begin {equation} M_n(\mathrm {Gauss}) = \int _{-\infty }^{\infty } x^n (2\pi )^{-1/2} \exp \left ( -x^2/2 \right ) \mathrm {d}x . \label {eq:mn_Gauss} \end {equation} The moments of odd \(n\) are all zero while those of even \(n\) are \begin {equation} M_{2n}(\mathrm {Gauss}) = \prod _{i=1}^{n} (2i-1) . \label {eq:m2n} \end {equation} In the case of \(N\) samples of the variable \(x\), the central moments \(M_n\) can be computed by \begin {equation} M_{n} = \frac {1}{N}\sum _{i=1}^{N}x_i^n. \label {eq:mn} \end {equation} The generalized \(F\) constraint of order \(n\) is formulated as

The classical \(F\) constraint corresponds to the generalized constraint of order 2, \(C_{F2}\). The classical \(G\) constraint is also generalized as

The generalized \(F\) and \(G\) constraints of order \(n\), are combined to give a \(C_n\) constraint:

3.4 Linear Combination of Generalized Constraints

ERIS adopts a linear combination of the generalized \(F\) and \(G\) constraints with relative weights, \(\lambda _n\) (\(n = 2\), 4, 6, .....): \begin {equation} C = \sum _n \lambda _n C_n \label {eq:Cw} \end {equation} with \begin {equation} C_n = \frac {1}{(N_F + N_G) M_n(\mathrm {Gauss})} \left [ \sum _{j=1}^{N_F} w_j \bigl ( \left | \Delta F_j \right | \bigr )^n + \sum _{j=1}^{G_F} w_j \bigl ( \left | \Delta G_j \right | \bigr )^n\right ] - C_{\mathrm {w}_n}, \label {eq:CFn} \end {equation} where \(w_j\) is the weighting factor (see 4), and \(C_{\mathrm {w}_n}\) is the criterion for convergence. When \(w_j\) is unity, the ideal constraint is \(C_{\mathrm {w}_n} = 1\) for any even \(n\). Such a rigorous constraint seems to be hardly satisfied in actual problems. Accordingly, ERIS always determines \(C_{\mathrm {w}_n}\) automatically so as to satisfy Eq. (??) or (3.4) on use of the \(G\) constraint.

Information about central moments of the normalized residuals, \(\Delta F_j\) and \(\Delta G_j\), is output to file *.out, for example,

     C2  =  9.9992285E-01   ln(C2 ) =  -0.000077
     C4  =  4.0802412E+00   ln(C4 ) =   1.406156
     C6  =  2.8819103E+01   ln(C6 ) =   3.361038
     C8  =  2.0165493E+02   ln(C8 ) =   5.306558
     C10 =  1.2533774E+03   ln(C10) =   7.133597
     C12 =  6.8066503E+03   ln(C12) =   8.825655
     C14 =  3.2373854E+04   ln(C14) =  10.385106
     C16 =  1.3593284E+05   ln(C16) =  11.819916

The first column, C2–C16, lists the central moments of order \(n\) for \(|\Delta F_j|^n\) and \(|\Delta G_j|^n\) normalized by those for Gaussian distribution. The second column, ln(C2)–ln(C16), gives their natural logarithms.

If the distributions of \(\Delta F_j\) and \(\Delta G_j\) are Gaussian, values in the first column are all 1 while those in the second column are all 0. Normalized central moments larger than 1 imply that information in observed data is underestimated and not fully reconstructed by MEM. On the contrary, normalized central moments smaller than 1 imply the overestimation of data, which result in overfit to observed data.

In MEM analyses from X-ray diffraction data with traditional second-order constraints, the normalized central moments of order \(n > 2\) are typically larger than 1, as shown in the above example. In such a case, increase \(\lambda _n\) of order 4 or higher to put restraints on them and bind them to the ideal value. The central moment of a higher order may be smaller than 1 depending on the type and quality of data, particularly when \(N_F + N_G\) is relatively small. In such a case, set \(\lambda _n\) such that \(\lambda _2 = 1\) and \(\lambda _n = 0\) for \(n > 2\), which corresponds to the traditional \(\chi ^2\) constraint.

On the use of the ZSPA algorithm, substitution of Eq. (3.8) for \(C\) in Eq. (2.6) will give rise to a problem that only the \(C_n\) constraint with the highest order practically takes effect at the beginning of MEM iterations. This is because the highest order \(C_n\) constraint can be orders of magnitude larger than the \(C_2\) constraint at early cycles of MEM iterations. Because the ZSPA algorithm do not optimize the information entropy, and it is sensitive to calculation route, the final results will also be strongly constrained by the highest order \(C_n\). To overcome such a problem specific to the ZSPA algorithm, \(\lambda _n\) in Eq. (3.8) in the \(i\)-th iteration, \(\lambda _n^{(i)}\), is adjusted as follows. If \(\Delta F_j\)’s and \(\Delta G_j\)’s are kept close to Gaussian distribution during MEM iterations, the order of \(C_n\) in the \((i)\)-th iteration is roughly equal to \((C_2)^{n/2}\). Then \(\lambda _n^{(i)}\) is replaced by \begin {equation} \lambda _n^{(i)} = \frac {\lambda _n}{\left ( C_2 \right )^{n/2-1}}. \label {eq:lambdan} \end {equation}

At any rate, be sure to check whether or not 3D distribution of electron/nuclear densities is physically and chemically reasonable with VESTA [20].