Structure refinement by total energy minimization
Ľ. Smrčok
Institute of Inorganic Chemistry, Slovak Academy of Sciences, Dúbravská cesta 9, SK-845 36 Bratislava, Slovak Republic
uachsmrk@savba.sk
A restricted amount of structural information
extractable from X-ray or CW neutron powder diffraction patterns seriously
limits the accuracy of the structural data resulting from Rietveld
refinement, which, as a rule, follows a structure solution step. If the
information content of a powder pattern is insufficient and we are, in spite of
this fact still interested in accurate structure, it is time to turn out our
attention to other methods of refinement. Basically, there are two possible
routes. The first one in fact just modifies standard Rietveld
method by introducing various improvements (and also "improvements"),
which in many cases just mercifully mask the main drawback of the method, a low
number of accurate structure factors confronted to a large number of refined
atomic parameters.
Rietveld refinement relies on the idea that the parameters of
a (probability) function approximating diffraction profiles can be refined by a
non-linear weighted least-squares procedure simultaneously with the atomic
parameters. Due to the surviving and perhaps unsolvable problems with accurate
description of powder diffraction profiles and in particular with a small
number of structure factors compared to the number of refined atomic
parameters, the accuracy of the resulting structures is, more often than not, well-below
to that typical for the current single crystal standards. Although several
modifications of this procedure have been already proposed - restrained or
rigid body refinements, (e.g. [1-2] just among the
others) the crucial problem, i.e. low number of inaccurate structure factors
remains. As a result, accuracy of the structural parameters obtained in any
powder refinement is lower than (or at maximum identical to) accuracy of a
single crystal refinement based on the same numbers of independently collected
structure factors and refined parameters. Unfortunately, in a common practice
the latter case is the ideal point hardly reachable, as the accuracy of
structure factors from a powder pattern is always noticeably lower, especially
due to overlap of the individual diffractions caused by collapse of 3D
diffraction pattern to 1D, not to speak of various sample effects.
The second approach, on the other hand, totally
abandons the idea of repairing of irreparable and benefits from symbiosis of
powder diffraction with theoretical calculations, in this case with structure
optimization by energy minimization in the solid state. If we restrict ourselves only to the
solid-state DFT methods we realize, that the size of problems tractable by a
laboratory computer nowadays reaches thousands of atoms per unit cell, of
course in dependence on the level of approximations
used by the computational method. This number largely exceeds all widely
accepted limits for powder refinements, which frequently fail in providing
accurate results even for the structures containing much, much smaller numbers
of atoms. Moreover, since theoretical calculations are frequently done in the P1
space group, simultaneous optimization of the geometries of possibly
symmetrically equivalent units within a unit cell provides a good measure of
internal consistency of structure optimization and/or solution. The main “crystallographic” advantage of
this approach is that all the atoms in a structure are treated on equal
footing, i.e. not weighted by their scattering abilities. For organic
structures (in fact, not only for them) this approach provides direct access to
accurate geometry of hydrogen bonds, whose knowledge is in good deal of cases
essential for understanding of physicochemical properties of a compound.
Secondly, several "well-known" shapes or fragments (molecules or polyhedrons
in inorganic structures) are further optimized under the constraints imposed by
a crystal field. Last but not least, by exploiting the resulting wavefunction several interesting quantities like electronic
or vibrational densities of states, distribution of
the electron density in a crystal, stress tensors, etc., are easily accessible.
The approach to handling of the lattice parameters is twofold. Either they are
refined in advance from powder diffraction data and then kept fixed in the
course of optimization of the atomic coordinates, or their values are optimized
along with the atomic coordinates. Validity of both these approaches has been a
subject of irrational and useless discussions with many pros and cons. In
short, the main argument of the pros group is that without optimization of the
cell parameters "the optimized structure is not in the minimum"
(...of total energy), while the cons group proclaims that accuracy of the
lattice parameters obtained from a crystallographic experiment is as a rule
much higher than
that of those obtained by energy minimization and there is hence no need to
make it worse. (The author of this contribution pragmatically agrees with both
these groups knowing that one should always use the optimal method … .)
Such a combined approach to powder structure solution
and refinement was to our best knowledge pioneered by Dinnebier
with co-workers [3] in their study of a
rigid ferrocene-based macrocycle,
C48H44B4Fe2N8O4,
crystallizing in a relatively large monoclinic (C2/c) cell (V=4152.8Å3).
The initial structure models were generated by Monte Carlo method using the
synchrotron data and the atomic coordinates were subsequently refined by energy
minimization in the solid state. Refined structures were put to Rietveld refinement and the profile parameters were
improved, while keeping the atomic coordinates fixed. The study was soon
followed by the structure solution combined with the refinement by crystal
energy minimization of the red polymorph of tetrahexylsexitiophene
[4]. The powder pattern was collected with a laboratory diffractometer
and the structure solved by Monte Carlo technique. Nearly identical solution in
the space groups C2/m, C2 and P-1 were found, of
which the first was at last chosen.
Since then, two different attitudes to co-existence of classic Rietveld refinement and refinement by total energy
minimization have developed. The first and (unfortunately …) the more
frequently used, gives more weight to standard
Rietveld refinement and uses theoretical calculation
only to refine positions of hydrogen atoms (if any) and/or for “validation” of
a refined structure (see for example [8-13]). Regrettably, in many cases
"validation" means that the authors just try to convince themselves
that their Rietveld effort was not a pure waste of
time. The second approach, on the contrary, is based on the presumption that it
is unlikely that Rietveld refinement with all atoms
relaxed can provide sufficiently accurate structural data for the structures
built from more than few atoms. Rietveld refinement
is therefore used only for estimation of isotropic displacement parameters (required
by many journals as if they could have any reasonable meaning, neutron TOF data
being an exception) and/or of selected profile parameters, like preferred
orientation correction. Like the first approach, also the second has been
successfully applied to both organic, metal-organic and inorganic compounds [14, 15-23]. Note,
however, that the lists of the papers belonging to any of these groups are not claimed to be complete and the author apologizes for unavoidable omissions.
The requirements for successful application of energy minimization in structure refinement are however stricter than those typical for standard Rietveld refinement. First of all, the model must be complete, i.e. no dangling bonds resulting from intentional or accidental omission of atoms are allowed. Partial occupancies of atomic positions are not possible and must be modeled by a supercell simulating distribution of atoms or vacancies. A good beginner’s guide to the art of solid state calculations is provided in the review paper by Gillan [24] or Hafner [25].
References
1. A.Immirzi, Journal of Applied Crystallography, 42, (2009), 362.
2. P.V.Afonine, R.W.Grosse-Kunstleve, A.Urzhumtsev, P.D.Adams, Journal of Applied Crystallography, 42, (2009), 607.
3. R.E.Dinnebier, L.Ding, M.Kuangbiao, M.A.Neumann, N.Tanpipat, F.J.L.Leusen, P.W.Stephens, M.Wagner, Organometallics, 20, (2001), 5642.
4. M.A.Neumann, C.Tedesco, S.Destri, D.R.Ferro, W.Porzio, Journal of Applied Crystallography, 35, (2002), 296.
5. A.Bhattacharya, K.Kankanala, S.Pal, A.K.Mukherjee, Journal of Molecular Structure, 975, (2010), 40.
6. V.Brodski, R.Peschar, H.Schenk, A.Brinkmann, E.R.H.Van Eck, A.P.M.Kentgens, B.Coussens, A.Braam, Journal of Physical Chemistry, B108, (2004), 15069.
7. U.Das, B.Chattopadhyay, M.Mukherjee, A.K.Mukherjee, Chemical Physics Letters, 501, (2011), 351.
8. J.A.Kaduk, Acta Crystallographica, B58, (2002a), 815.
9. J.A.Kaduk, Transactions of American Crystallographic Association, (2002b), 63.
10. J.A.Kaduk, Powder Diffraction 19, (2004), 127.
11. J.A.Kaduk, M.A.Toft, J.T.Golab, Powder Diffraction, 25 (2010), 19.
12. P.S.Whitfield, Y.Le Page, I.J.Davidson, Powder Diffraction, 26, (2011), 321.
13. P.S.Whitfield, L.D.Mitchell, Y.Le Page, J.Margeson, A.C.Roberts, Powder Diffraction, 25, (2010), 322.
14. A.J.Florence, J.Bardin, B.Johnston, N.Shankland, T.A.N.Griffin, K.Shankland, Zeitschrift für Kristallographie Suppl., 30, (2009), 215.
15. Ľ.Smrčok, V.Jorík, E.Scholtzová, V.Milata, Acta Crystallographica, B63, (2007), 477.
16. Ľ.Smrčok, M.Brunelli, M.Boča, M.Kucharík, Journal of Applied Crystallography, 41, (2008), 634.
17. Ľ.Smrčok, B.Bitschnau. Y.Filinchuk, Crystal Research and Technology, 44, (2009a), 978.
18. Ľ.Smrčok, M.Kucharík, M.Tovar, I.Žižak, Crystal Research and Technology, 44, (2009b), 834.
19. Ľ.Smrčok, R.Černý, M.Boča, I.Macková. B.Kubíková, Acta Crystallographica, C66, (2010), I16.
20. A.LeBail, Ľ.Smrčok, Powder Diffraction, 23, (2011), 292.
21. M.Oszajca, Ľ.Smrčok, H.Pálková, W. Łasocha, Journal of Molecular Structure, 1021, (2012), 70.
22. M.Šimúneková, P.Schwendt, J.Chrappová, Ľ.Smrčok, R.Černý, W.Van
Beek, Central European Journal of
Chemistry, 11, (2013), 1352.
23. Ľ.Smrčok, P.Mach, A.LeBail, Acta Crystallographica, B69, (2013), doi:10.1107/S2052519213013365.
24. M.J.Gillan, Contemporary Physics, 38, (1997), 115.
25. J.Hafner, Acta Materialia, 48, (2000), 71.