Statistical description of multimodal atomic probability densities

Kuhs, Werner F.

doi:10.1107/s0108767383000240

Cited by 74 publications

(64 citation statements)

References 15 publications

(23 reference statements)

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“…Indeed, as well as describing thermal motion it can account for static disorder in a crystal structure (Kuhs, 1983) and can have contributions from systematic errors (Trueblood et al, 1996). In the present work we are interested solely in its ability to model thermal motion, in which case the structure-factor equation takes the form hFðQÞi ¼ P N n¼1 f nP P n ðQÞ expðiQ Á r n0 Þ; ð1Þ…”

Section: Modelling Thermal Motion In Crystallography 21 Debye-wallementioning

confidence: 99%

Using molecular-dynamics simulations to understand and improve the treatment of anharmonic vibrations. I. Study of positional parameters

Reilly¹,

Morrison²,

Rankin³

2011

Acta Cryst Sect A

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Molecular‐dynamics‐derived numerical probability density functions (PDFs) have been used to illustrate the effect of different models for thermal motion on the parameters refined in a crystal structure determination. Specifically, anharmonic curved or asymmetric PDFs have been modelled using the traditional harmonic approximation and the anharmonic Gram–Charlier series treatment. The results show that in cases of extreme anharmonicity the mean and covariance matrix of the harmonic treatment can deviate significantly from physically meaningful values. The use of a Gram–Charlier anharmonic PDF gives means and covariance matrices closer to the true (numerically determined) anharmonic values. The physical significance of the maxima of the anharmonic distributions (the most probable or mode positions) is also discussed. As the data sets used for the modelling process are theoretical in origin, these most probable positions can be compared to equilibrium positions that represent the system at the bottom of its potential‐energy surface. The two types of position differ significantly in some cases but the most probable position is still worthy of report in crystal structure determinations.

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Section: Modelling Thermal Motion In Crystallography 21 Debye-wallementioning

confidence: 99%

Using molecular-dynamics simulations to understand and improve the treatment of anharmonic vibrations. I. Study of positional parameters

Reilly¹,

Morrison²,

Rankin³

2011

Acta Cryst Sect A

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Section: Introductionmentioning

confidence: 99%

“…This is actually observed for H atoms, which generally vibrate vigorously owing to their small weight and their terminal position. In principle, one can resort to more sophisticated models of the p.d.f, at the price of refining more parameters (Johnson & Levy, 1974), but in practice this is performed only in exceptional cases (for examples, see Kuhs, 1983).Most published neutron crystal structures are based on conventional refinement, which leads to systematic shifts of observed H-atom positions along the X-H bond in the direction of X, i.e. to reductions of the observed bond length dxH.…”

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confidence: 99%

Distribution of observed C–H bond lengths in neutron crystal structures and temperature dependence of the mean values

Steiner¹

1993

Acta Cryst Sect A

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An analysis is given of C-H bond lengths and their apparent shortening by thermal vibrations in neutron crystal structures. D atoms were excluded. Mean bond lengths and the spread of observed values were compiled for six different chemical connectivities in three temperature ranges (room temperature, 80_ < T< 125 and T < 30 K). At low temperatures (T < 30 K), the mean values agree very well with spectroscopic data. Owing to thermal vibrations, the average observed C-H bond length in methyl groups at room temperature (1.057 A) is shortened by 0.03 A compared with that at T<30K (1.088A). The shortening reduces only slowly upon cooling and is still significant at temperatures around 100 K. For the other connectivities, in which the C-H orientations are more rigidly confined, the average shortening at room temperature is 0.01 to 0.02 A, and at T-100 K the mean observed bond lengths have already (or almost) reached the spectroscopic values. The distributions of the observed bond lengths are asymmetric with the maxima at distances greater than the average. The positions of the maxima are less affected by temperature changes than the average values, implying that at room temperature many bond lengths are only slightly affected by thermal motions. IntroductionThe appropriate crystallographic technique to determine X-H bond lengths (dxH) is neutron diffraction.Because of the apparent bond shortening caused by charge-density distortions [on the average 0.1 A for C-H (Allen, 1986)], X-ray data are generally unsuitable in this context. In the following, we concentrate on C-H bond lengths, dcH, for which the largest body of neutron diffraction data is available. For a given chemical type of C-H, neutron-determined dcH show a significant temperature dependence and, at any given temperature, the values of dcH vary considerably. These variations are (apart from experimental uncertainty) primarily due to thermal vibrations of the H atom. In contrast to O-H and N-H, the influence of hydrogen-bonding effects on dcH is only marginal (Steiner & Saenger, 1992) and will not be considered here any further.In crystals, neutron diffraction observes the time and lattice average of vibrating atomic nuclei. These represent three-dimensional probability density functions (p.d.f.s), which in most cases are Gaussian-like with more or less pronounced deviations from the ideal shape (Johnson & Levy, 1974). Deviations from a Gaussian shape are caused, for example, by curvilinear motion of librating atoms and bondstretching vibrations in anharmonic potentials. In conventional anisotropic refinement, the Gaussianlike p.d.f.s are approximated by ideal Gaussian functions (six thermal parameters), leading to systematic errors if a p.d.f, deviates significantly from the Gaussian function. This is actually observed for H atoms, which generally vibrate vigorously owing to their small weight and their terminal position. In principle, one can resort to more sophisticated models of the p.d.f, at the price of refining more parameters (Johnson & Levy,...

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“…If all terms up to infinity are included the Edgeworth expansion is not different from the Gram-Charlier (GC) expansion, which has been preferred for reasons discussed later by several authors (Zucker and Schulz 1982;Kuhs 1983;Scheringer 1985b). The Gram-Charlier series (up to sixth order) is given as Hijklm( u) +(1/6!…”

Section: Formalismmentioning

confidence: 99%

The Anharmonic Temperature Factor in Crystallographic Structure Analysis

Kuhs¹

1988

Aust. J. Phys.

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The most widely used formalisms to describe anharmonicity and disorder in crystallographic structure refinements are presented and their properties are reviewed. Their limitations are discussed and their range of applications is indicated. A comparative study on the lattice anharmonicity in Zns using different models is presented. The importance of measuring Bragg intensity data sufficiently far in reciprocal space is stressed. Some indications concerning the interpretation of the results are given and the decorrelation of electronic and atomic deformations is briefly discussed.

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Statistical description of multimodal atomic probability densities

Cited by 74 publications

References 15 publications

Using molecular-dynamics simulations to understand and improve the treatment of anharmonic vibrations. I. Study of positional parameters

Using molecular-dynamics simulations to understand and improve the treatment of anharmonic vibrations. I. Study of positional parameters

Distribution of observed C–H bond lengths in neutron crystal structures and temperature dependence of the mean values

The Anharmonic Temperature Factor in Crystallographic Structure Analysis

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