MaxEnt

A successful method to obtain dynamical information from computer simulations of quantum systems has recently been proposed by Gubernatis and coworkers [167-169]. It uses concepts from probability theory and Bayesian logic to solve the analytic continuation problem in order to obtain real-time dynamical information from imaginary-time computer simulation data. The method has become known under the name maximum entropy (MaxEnt), and has a wide range of applications in other fields apart from physics. Here we review some of the main ideas of this method and an application [175] to the model fluid described in the previous section. 

The Maximum Entropy (abbreviated MaxEnt) method has been used in the field of accurate charge density studies for some time now (see Section 2.2) it has the potential to overcome some of the limitations of traditional multipolar modelling, but great care must be taken not to apply it outside the range of validity of its own foundations. 

In the last section of the paper, we discuss a Bayesian approach to the treatment of experimental error variances, and its first limited implementation to obtain MaxEnt distributions from a fit to noisy data. 

A second approach which is not subject to the limitations imposed by the choice of a parametrised model of the density, is the MaxEnt method. The appeal of the method is evident when counting the increasing number of applications to charge density studies that have appeared in the crystallographic literature in the last ten years see among the most recent ones [17-20], and the works cited in relevant sections of reviews... 

All of the studies published so far have been aiming at the reconstruction of the total electron density in the crystal by redistribution of all electrons, under the constraints imposed by the MaxEnt requirement and the experimental data. After the acceptance of this paper, the authors became aware of valence-only MaxEnt reconstructions contained in the doctoral thesis of Garry Smith [58], The authors usually invoke the MaxEnt principle of Jaynes [23-26], although the underlying connection with the structural model, known under the name of random scatterer model, is seldom explicitly mentioned. 

When it is employed to specify an ensemble of random structures, in the sense mentioned above, the MaxEnt distribution of scatterers is the one which rules out the smallest number of structures, while at the same time reproducing the experimental observations for the structure factor amplitudes as expectation values over the ensemble. Thus, provided that the random scatterer model is adequate, deviations from the prior prejudice (see below) are enforced by the fit to the experimental data, while the MaxEnt principle ensures that no unwarranted detail is introduced. 

Since 1993, a number of studies have been devoted to assessing the limitations of the MaxEnt method when applied to charge density studies, especially in conjunction with uniform prior-prejudice distributions. We summarise here the main points that have arisen from these model studies. 

Uneven distributions of residuals. The MaxEnt calculations in presence of an overall chi-square constraint suffer from highly non-uniform distributions of residuals, first reported and discussed by Jauch and Palmer [29, 30] the error accumulates on a few strong reflexions at low-resolution. The phenomenon is only partially cured by devising an ad hoc weighting scheme [20,31, 32]. Carvalho et al. have discussed this topic, and suggested that the recourse to as many constraints as degrees of freedom would cure the problem [33]. 

Errors in the low-density regions of the crystal were also found in a MaxEnt study on noise-free amplitudes for crystalline silicon by de Vries et al. [37]. Data were fitted exactly, by imposing an esd of 5 x 10 1 to the synthetic structure factor amplitudes. The authors demonstrated that artificial detail was created at the midpoint between the silicon atoms when all the electrons were redistributed with a uniform prior prejudice extension of the resolution from the experimental limit of 0.479 to 0.294 A could decrease the amount of spurious detail, but did not reproduce the value of the forbidden reflexion F(222), that had been left out of the data set fitted. 

Finally, recent work of Iversen et al. has carefully examined the bias associated to the accumulation of the error on low-order reflexions, and attempted a correction of the MaxEnt density [39]. The study, based on a number of noisy data sets generated with Monte Carlo simulations, has produced less non-uniform distribution of residuals, and has given quantitative estimate of the bias introduced by the uniform prior prejudice. For more details on this work, we refer the reader to the chapter by Iversen that appears in this same book. 

Two-channel MaxEnt techniques have also been used in the study of magnetization and spin densities [34, 35] and to interpret unpolarised neutron diffraction data [36]. 

None of the studies mentioned in Section 2.2 has explicitly addressed the main issue of the redistribution of core electron densities under MaxEnt requirements in the absence of high-resolution observations. This is indeed the key to explaining the unsatisfactory features encountered so far in the applications of the method to charge density studies. 

By its very definition, the MaxEnt method is optimally suited to flexibly reconstruct distributions whose main features are well represented in the available data, that is either in the observations or in the prior structural knowledge. When this is the case, the missing structure can be reasonably approximated by a collection of randomly and independently distributed constituents (by missing structure here we mean all those structural details which are not completely defined by the prior knowledge). 

It is therefore clear that MaxEnt redistribution of all electrons, using a uniform prior prejudice and carried out in the absence of very high-resolution diffraction measurements, cannot be expected to reproduce a physically acceptable picture of atomic cores. The reconstruction of total electron densities from limited-resolution diffraction measurements amounts to a misuse of the MaxEnt method, especially when the prior prejudice is uniform. 

In this section, we briefly recall the MaxEnt equations and the functional form of the MaxEnt probability distribution the formulation is the one obtainable for randomly and independently distributed electrons, in the presence of a subset of electrons whose distribution is assumed to be known. The latter structure will be denoted as fragment . 

To deal with all the observations h e H in compact form, the unitary structure factor components can be arranged in a vector Urand, and the components of the constraint functions collected in a vector C(x). The MaxEnt distribution of electrons (x) then takes the form... 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

Thanks to the particular choice made for the NUP, taken equal to a superposition of spherical atoms, it is for the first time possible within the present approach to compute MaxEnt deformation maps in a straightforward manner. Once the Lagrange multipliers X have been obtained, the deformation density is simply... 

This map can have negative as well as positive features, and yet its calculation involves only that of the positive map qmi, thus avoiding the issue of extending the MaxEnt method to two-channel problems. 

It appears from formula (6) that the prior-prejudice distribution mix) is a fundamental quantity in the calculation of the MaxEnt distribution of electrons, in that the latter is obtained by modulation of m(x). In all those regions where the modulating factor required to fit the observations is unity, the final picture is therefore always going to coincide with the prior expectation itself. For this reason, it is of the greatest importance that some of the prior information available about the system under study be conveyed into the calculation by means of a sensible choice for the prior-prejudice distribution. 

Not only is the choice of a uniform prior-prejudice distribution not sensible it also exposes the calculation to two main sources of computational errors, both connected with the functional form of the MaxEnt distribution of scatterers, and with its numerical evaluation namely series termination ripples and aliasing errors in the numerical sampling of the exponential modulation of mix). The next two paragraphs will illustrate these issues in some detail. 

The phenomenon can be illustrated by considering a model density q(x), from which diffraction data can be computed at arbitrarily high resolution. The (normalised) exponential factor needed to reconstruct q(x) by MaxEnt modulation of a chosen prior-prejudice distribution mix) can be written as... 

Figure 1 shows the average strength of the Fourier coefficients of log( (x)/m(x)), with q(x) a multipolar synthetic density for L-alanine at 23 K, and two different prior-prejudice distributions mix). It is apparent that the exponential needed to modulate the uniform prior still has Fourier coefficients larger than 0.01 past the experimental resolution limit of 0.463 A. Any attempt at fitting the corresponding experimental structure factor amplitudes by modulation of the uniform prior-prejudice distribution will therefore create series termination ripples in the resulting MaxEnt distribution. 

The exact amount of error introduced cannot immediately be inferred from the strength of the amplitudes of the neglected Fourier coefficients, because errors will pile up in different points in the crystal depending on the structure factors phases as well to investigate the errors, a direct comparison can be made in real space between the MaxEnt map, and a map computed from exponentiation of a resolution-truncated perfect m -map, whose Fourier coefficients are known up to any order by analysing log(<7 (x) tm (x)). 

Figure 2. L-alanine. Dynamic deformation density in the COO plane, (a) Model dynamic deformation density A Modei. (b) MaxEnt dynamic deformation density (Agj, (x)) map obtained with a non-uniform prior of spherical-valence shells. Map size 6.0A x 6.0A Contour levels from -1.0 to 1.0 eA 3, step 0.075 e A-f...

L-A la MaxEnt valence density from noise-free data... 

To check this prediction, a number of MaxEnt charge density calculations have been performed with the computer program BUSTER [42] on a set of synthetic structure factors, obtained from a reference model density for a crystal of L-alanine at 23 K. The set of 1500 synthetic structure factors, complete up to a resolution of 0.555 A [45], was calculated from a multipolar expansion of the density, with the computer program VALRAY[ 46],... 

The MaxEnt valence density for L-alanine has been calculated targeting the model structure factor phases as well as the amplitudes (the space group of the structure is acentric, Phlih). The core density has been kept fixed to a superposition of atomic core densities for those runs which used a NUP distribution m(x), the latter was computed from a superposition of atomic valence-shell monopoles. Both core and valence monopole functions are those of Clementi [47], localised by Stewart [48] a discussion of the core/valence partitioning of the density, and details about this kind of calculation, may be found elsewhere [49], The dynamic range of the L-alanine model... 

We stress here that any low-temperature valence density for a small organic molecule will have a comparably high dynamic range, so that even valence-only MaxEnt studies will always be likely to need a NUP if truncation ripples are to be avoided. 

The value of the mis deviation from the reference density can be deceptively low, due to the fact that in the intermolecular regions the model density is virtually the same as the one made of spherical-valence shells, which was used as a NUP. The agreement between the MaxEnt map and the reference model is very close in those regions. 

A second major source of computational difficulties associated with uniform prior-prejudice distributions is connected with the extremely fine sampling grids that are needed to avoid aliasing effects in the numerical Fourier synthesis of the modulating factor in (8). To predict the dependence of aliasing effects upon the prior prejudice, we need to examine more closely the way the MaxEnt distribution of scatterers is actually synthesised from the values of the Lagrange multipliers X. 

With this choice of constraint functions and Lagrange multipliers, we can rewrite formula (6) and express the MaxEnt distribution of electrons as... 

Expanding each of the exponential factors in a series of modified Bessel functions, the MaxEnt distribution can be written ... 

The MaxEnt distribution of scatterers qME, obtained for X = X, is also the one that maximises the a priori probability in (25) ... 

Under general hypotheses, the optimisation of the Bayesian score under the constraints of MaxEnt will require numerical integration of (29), in that no analytical solution exists for the integral. A Taylor expansion of Ao(R) around the maximum of the P(R) function could be used to compute an analytical expression for A and its first and second order derivatives, provided the spread of the A distribution is significantly larger than the one of the P(R) function, as measured by a 2. Unfortunately, for accurate charge density studies this requirement is not always fulfilled for many reflexions the structure factor variance Z2 appearing in Ao is comparable to or even smaller than the experimental error variance o2, because the deformation effects and the associated uncertainty are at the level of the noise. 

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