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Electron Diffraction Theory and Methods
Electron Diffraction Theory and Methods Istva´n Hargittai and Magdolna Hargittai, Budapest University of Technology and Economics, Budapest, Hungary & 2010 Elsevier Ltd. All rights reserved.
Gas-Phase Electron Diffraction Theory and Methods The eight-decade-old gas-phase electron diffraction technique of molecular structure determination was originally initiated in an industrial laboratory by the chemist Herman F. Mark and the physicist Raimund Wierl, who published a brief report in 1930. They determined the geometry of a series of simple molecules based on a visual evaluation of intensity distributions of the scattered electrons. Then, the technique was taken over by the California Institute of Technology and other academic laboratories, and Linus Pauling and Lawrence Brockway introduced a visually more appealing approach by producing the so-called radial distribution – a misnomer for the probability distribution function of the intramolecular interatomic distances. The technique distinguished itself by yielding direct information on molecular structure and has remained one of the principal experimental sources of such data. Gas-phase electron diffraction is a source of internal coordinates of the molecular geometry, such as bond lengths, bond angles, and angles of torsion and of information related to molecular dynamics, such as mean amplitudes of vibration, conformational abundances (and thus energy patterns between conformers), and even barriers to internal rotation. One of the roots of gas-phase electron diffraction is in the theory worked out for the diffraction of X-rays by randomly oriented rigid systems of electrons, where it was found that the interference effects do not cancel in the scattered intensity distribution. This is well demonstrated by the diffuse character of the interference patters produced in a gas-phase electron diffraction experiment in Figure 1. Another of the roots of the technique is in Louis de Broglie’s discovery about the wave nature of moving electrons. Ignoring the relativistic corrections (the higher the electron energy the poorer such an approximation is), l ¼ h/mv, where l is the electron wavelength, h is the Planck constant, m is the electron mass, and v is the electron velocity.
electron wavelength and the potential U used for accelerating the electrons is l ¼ h/(2meU)1/2, where e is the elementary charge. The pressure in the vapor beam is in the order of several tens of mmHg. Molecules having atoms with higher atomic numbers are stronger scatterers than molecules consisting of light atoms. The sample needs to be pure, but impurities may hinder the analysis in varying degrees depending on their ability to scatter electrons and on the distribution of their internuclear distances. Impurities with very different atoms and interatomic distances than the principal molecule disturb the emerging results less than those whose scattering is similar to the main target of the experiment. The higher the molecular symmetry the more reliable is the structure determination. Here, the complementary nature of gas-phase electron diffraction and microwave spectroscopy can be sensed, because the latter is disadvantageous for molecules with no electric dipole moment. At the beginning of the applications of gas-phase electron diffraction, the photographs of the diffraction patterns were evaluated visually, utilizing the exaggerating ability of the naked eye in distinguishing maximum and minimum interferences on the steeply falling background of the atomic electron scattering. Eventually, the introduction of a technical device, the so-called rotating sector, made a more quantitative determination of the molecular
Experiment In a typical gas-phase electron diffraction experiment for molecular structure determination, a beam of fast electrons is scattered by a beam of vapor molecules. The electron beam is usually of energy in the range of 40–100 keV; and it needs to be monochromatic, that is, of very little scatter in energy. The relationship between the
Figure 1 Gas-phase electron diffraction pattern of adamantane, C10H16.
461
462
Electron Diffraction Theory and Methods
component of the intensity distribution possible. This greatly enhanced the scope and precision of structure determinations. Lately, in addition to the time-honored photographic techniques, digital techniques have also been employed. The scheme of a traditional experiment is presented in Figure 2. In some special experiments, mass spectrometric control has been added to the electron diffraction setup, which assists the determination and even optimization of vapor composition for establishing the best conditions for the scattering experiment. The scheme of such a combined experiment is presented in Figure 3. Theory A few expressions are presented to help the understanding of the origin of physical information derived from the experiment without demonstrating the way of arriving at them. First, the electron scattering intensity
Electron beam Nozzle
distribution as a result of electron scattering by randomly oriented rigid molecules is given as a function of a scattering angle variable, s, defined as s ¼ (4p/l) sin(y/2), where l is the electron wavelength and y is the scattering angle. Only elastic scattering is considered. The electron scattering intensity distribution for rigid molecules is given by I ðsÞ ¼ KI 0 R2
N X N X
f i ðsÞf j ðsÞsinðsr ij Þ=sr ij
i¼1 j ¼1
Here, I0 is the intensity of the incident electron beam, R is the distance between the scattering center and the observation point, K is a constant that can be expressed in terms of fundamental constants, N is the number of atoms in the molecule fi (s) and fj (s) are the scattering amplitudes of the ith and j th atom, respectively, and rij is the internuclear distance. This expression contains the scattering of electrons by the individual atoms in the molecule (i ¼ j) as well as the scattering of electrons by atomic pairs (iaj), which is relevant to molecular structure. Accordingly, I(s) may be considered as consisting of two components; one is Ia(s), the atomic component, and the other, Im(s), is the molecular component. Thus,
Vapor beam
I ðsÞ ¼ I a ðsÞ þ I m ðsÞ
and Photographic plate
I a ðsÞ ¼ KI 0 R2
N X
jf i ðsÞj2
i¼1
and Rotating sector
I m ðsÞ ¼ KI 0 R2 Figure 2 Principle of the gas-phase electron diffraction experiment.
N X N X
f i ðsÞ f j ðsÞ sinðsr ij Þ=sr ij
i¼1 j ¼1 ði a j Þ
Electron beam
Quadrupole analyzer Water cooling
GT-400 oil trap
Cold trap
Evaporator system Diffraction chamber
ODF-400
Photo-plate
diffusion pump To fore-pump
Figure 3
To high vacuum pump
Scheme of a combined gas-phase electron diffraction/quadrupole mass spectrometric experiment.
Electron Diffraction Theory and Methods
Atomic scattering amplitudes, fi (s), may be introduced at different levels of approximation (the asterisk denotes complex conjugate). A rough approximation is to assume fi (s) ¼ Zi, where Zi is the atomic number of the ith atom. In this case, the scattering by the electron cloud, which is especially important at low scattering angles, is ignored. A more adequate expression is of the form f i ðsÞ ¼ Zi F i ðsÞ
where Fi (s) is the contribution of the electron density distribution to the scattering amplitude. This expression is valid under the assumption of a weak scattering potential, so the electron wave encountering each volume element within the molecule is the unperturbed, incident wave. This is called the kinematic approximation, known as the first Born approximation when applied to electron diffraction. However, scattering potentials are mostly not weak, and an electron wave undergoes a significant phase shift as it propagates through the field of the scatterer. The phase shift increases with increasing atomic number and increasing scattering angle. The atomic scattering amplitudes used in modern structural work take the phase shift into account and are of the form f i ðsÞ ¼ j f i ðsÞjexp½Zi ðsÞ
where j f i ðsÞj is the modulus of the amplitude and Zi (s) is the phase angle in radians. If these complex scattering amplitudes are used, the expression for the molecular intensity is written as I m ðsÞ ¼ KI 0 R2
N X N X
gij ðsÞ sinðsr ij Þ=sr ij
i¼1 j ¼1 ði a j Þ
where gij ðsÞ ¼ j f i ðsÞj f j ðsÞcos½Zi ðsÞ Zj ðsÞ
This formalism (quasi-kinematic approximation) corresponds to the introduction of multiple scattering effects within the same atom (intraatomic multiple scattering). If the molecule is composed of light atoms, or atoms not differing too much in atomic number, then the phases Zi(s) are similar and the diffracted intensity is not much different from that expected from the kinematic approximation. If one or more atoms are substantially heavier than the others, the deviation from the kinematic theory cannot be ignored. Using the kinematic theory in structure analysis then introduces distortions in the geometry of the molecule. A classic example is rhenium hexafluoride, ReF6, where the peak corresponding to the Re–F distance is split in the radial distribution. Using the quasi-kinematic approximation explains the splitting of the Re–F peak, and leads to the expected, regular
463
octahedral geometry. The treatment of multiple scattering by different atoms in the same molecule (interatomic multiple scattering) requires a more sophisticated approach. However, it is stressed that molecules are not rigid bodies, and the vibrational motions may cause variations in internuclear distances of several picometers. The molecular component of the electron scattering intensity for the vibrating molecule is I m ðsÞ ¼ KI 0 R2
N X N X
g ij ðsÞ expðl 2ij s 2=2Þ sinðsr ij Þ=sr ij
i¼1 j ¼1 ði a j Þ
where lij is the mean vibrational amplitude along the line connecting the two nuclei, and rij is now an effective internuclear distance. This internuclear distance is usually denoted as ra, and is related to the vibrationally averaged internuclear distance at the temperature of the electron diffraction experiment, rg, by the expression r a E r g l 2 =r a
to a good approximation. If the vibration is anharmonic, the term sin(srij) in the above equation should be replaced by sin[s(rij kij s2)], where kij is an asymmetry parameter related to the a constant of the Morse equation. In favorable cases, kij values can be determined from the electron diffraction data. Structure Analysis The experiment yields the total electron scattering intensity distribution, and it is a crucial step to extract the molecular component from it. In addition to the atomic component, there is also an inelastic scattering part of the intensity that is to be eliminated together with the consequences of extraneous scattering, depending on the particular construction of the experiment. All these can be considered together as making up the experimental background (inelastic þ atomic þ extraneous), Ib(s), and Im(s) ¼ I(s) Ib(s). The molecular contribution to the electron diffraction intensities – in short, molecular intensities – is the experimental data on which the determination of structural parameters is based, more often than not, using a least-squares refinement technique. Fourier transformation of the molecular intensities yields the so-called radial distribution, f (r) (a misnomer, because in reality it is related to the probability density distribution of the intramolecular internuclear distances). The molecular model, total intensities, molecular intensities, and radial distribution for tetramethylsilane are displayed in Figures 4–7 using a simple case study from our research group. Figures 6 and 7 show not only the experimental distributions but also the distributions calculated for the
464
Electron Diffraction Theory and Methods H
Si(CH3)4
C
H
H
H H
C Si
H
H
C H
Si H C H
Si H
C H
H
H
Si C
C
H 0
C
H
100
200
300
400
500
r, pm H Figure 7 Radial distributions for tetramethylsilane. The vertical scale is arbitrary. The continuous line is calculated for a model; the dots are calculated from the measured intensity data.
Molecular model of tetramethylsilane.
Figure 4
Si(CH3)4
0
50
100
150 200 s, nm−1
250
300
350
Figure 5 Total experimental intensities and background lines for tetramethylsilane. The vertical scale is arbitrary. The two curves refer to two different distances between the scattering point and the registration plane.
Si(CH3)4
0
50
100
150
200
250
300
350
s, nm−1 Figure 6 Molecular intensities for tetramethylsilane. The vertical scale is arbitrary. The continuous line is calculated and the dots refer to measurements.
best model of tetramethylsilane, which is characterized by the following bond lengths and bond angles, Si–C 187.7(4) pm, Si–H 111.0(3) pm, and Si–C–H 111.0(2) pm as well as Td symmetry and a staggered methyl conformation. The bond lengths and the bond angles here are so-called average parameters, see more on this in another entry of this encyclopedia. The radial distribution is convenient to visually inspect the validity of a model and to read off some principal internuclear distances, but the quantitative determination of all the parameters is done on the basis of the molecular intensities. The refinement of parameters usually starts from an initial set of parameters. The expression of the molecular intensities is a nonlinear relationship; a good choice of the initial parameters will ensure that the calculation reaches the global rather than a local minimum. Tetramethylsilane is a fortunate subject for electron diffraction, because of its high symmetry. Only a few parameters determine its geometry and almost all of their contributions are well resolved. Less symmetrical molecules usually have radial distributions on which the individual contributions are not so easily discernible. Even in such cases, the least-squares refinement of the parameters based on the molecular intensities may yield satisfactory results. For increasing molecular complexity, decreasing precision of the parameters may be the price to pay. In any case, the gas-phase data yield – in the usual terminology of X-ray crystallography – a one-dimensional Patterson function, which indicates the limitations of the technique toward more complex structures. It is desirable to learn as much as possible about the composition of the vapor sample used in the experiment from independent sources. The vapor composition may yet be another unknown to be determined in the analysis. In case of a conformational equilibrium of, say, two conformers of the same molecule the analysis may be more difficult but the results may be more rewarding at the same time. The analysis of ethane-1,2-dithiol at the temperature of 343 K revealed the presence of 62%
Electron Diffraction Theory and Methods
SH H
H
H
H SH
SH
H
SH
H
H g
H
62% a + 38% g
(S … S)a (S … S)g E
0
100
200
300 r, pm
400
465
for bond lengths, a few tenths of a degree for bond angles, and a few degrees for angles of torsion. The precision and generally, the reliability of structure determination may be enhanced considerably by concerted and combined application of electron diffraction experiments with spectroscopic measurements and quantum chemical calculations. In such cases, however, the question of accuracy has to be carefully considered as the physical meaning of structural information from different techniques varies. The combined application of techniques and the physical meaning of parameters are among the aspects discussed in another entry of this encyclopedia. See also: Fourier Transformation and Sampling Theory, Electron Diffraction Applications, Microwave and Radiowave Spectroscopy, Applications, Rotational Spectroscopy, Theory, X-Ray Crystallography of Macromolecules, Theory and Methods.
500
Figure 8 Ethane-1,2-dithiol has a mixture of the anti (a) and gauche (g) forms at 343 K, with respect to the S–C–C–S framework. The bottom curve was calculated from experimental data.
of the anti form (a) and 38% of the gauche form (g) as far as the S–C–C–S framework was concerned. The radial distributions calculated for a set of models and the experimental distribution in Figure 8 serve as illustration. In the best cases, the gas-phase electron diffraction technique yields precisions of a few tenths of a picometer
Further Reading Cowley JM (ed.) (1992) Electron Diffraction Techniques, vols. 1 and 2. Oxford, UK: Oxford University Press. Domenicano A and Hargittai I (eds.) (1991) Accurate Molecular Structures: Their Determination and Importance. Oxford, UK: Oxford University Press. Hargittai I (2005) Looking back and ahead: Gas-phase electron diffraction at 75. Structural Chemistry 16: 1--3. Hargittai I and Hargittai M (eds.) (1988) Stereochemical Applications of Gas-Phase Electron Diffraction, Part A, The Electron Diffraction Technique. New York: VCH Publishers. Kuchitsu K (1972) Experimental errors in gas-phase electron diffraction. In: Cyvin SJ (ed.) Molecular Structures and Vibrations, pp. 148--170. Amsterdam: Elsevier.
Gas-Phase Electron Diffraction Theory and Methods The eight-decade-old gas-phase electron diffraction technique of molecular structure determination was originally initiated in an industrial laboratory by the chemist Herman F. Mark and the physicist Raimund Wierl, who published a brief report in 1930. They determined the geometry of a series of simple molecules based on a visual evaluation of intensity distributions of the scattered electrons. Then, the technique was taken over by the California Institute of Technology and other academic laboratories, and Linus Pauling and Lawrence Brockway introduced a visually more appealing approach by producing the so-called radial distribution – a misnomer for the probability distribution function of the intramolecular interatomic distances. The technique distinguished itself by yielding direct information on molecular structure and has remained one of the principal experimental sources of such data. Gas-phase electron diffraction is a source of internal coordinates of the molecular geometry, such as bond lengths, bond angles, and angles of torsion and of information related to molecular dynamics, such as mean amplitudes of vibration, conformational abundances (and thus energy patterns between conformers), and even barriers to internal rotation. One of the roots of gas-phase electron diffraction is in the theory worked out for the diffraction of X-rays by randomly oriented rigid systems of electrons, where it was found that the interference effects do not cancel in the scattered intensity distribution. This is well demonstrated by the diffuse character of the interference patters produced in a gas-phase electron diffraction experiment in Figure 1. Another of the roots of the technique is in Louis de Broglie’s discovery about the wave nature of moving electrons. Ignoring the relativistic corrections (the higher the electron energy the poorer such an approximation is), l ¼ h/mv, where l is the electron wavelength, h is the Planck constant, m is the electron mass, and v is the electron velocity.
electron wavelength and the potential U used for accelerating the electrons is l ¼ h/(2meU)1/2, where e is the elementary charge. The pressure in the vapor beam is in the order of several tens of mmHg. Molecules having atoms with higher atomic numbers are stronger scatterers than molecules consisting of light atoms. The sample needs to be pure, but impurities may hinder the analysis in varying degrees depending on their ability to scatter electrons and on the distribution of their internuclear distances. Impurities with very different atoms and interatomic distances than the principal molecule disturb the emerging results less than those whose scattering is similar to the main target of the experiment. The higher the molecular symmetry the more reliable is the structure determination. Here, the complementary nature of gas-phase electron diffraction and microwave spectroscopy can be sensed, because the latter is disadvantageous for molecules with no electric dipole moment. At the beginning of the applications of gas-phase electron diffraction, the photographs of the diffraction patterns were evaluated visually, utilizing the exaggerating ability of the naked eye in distinguishing maximum and minimum interferences on the steeply falling background of the atomic electron scattering. Eventually, the introduction of a technical device, the so-called rotating sector, made a more quantitative determination of the molecular
Experiment In a typical gas-phase electron diffraction experiment for molecular structure determination, a beam of fast electrons is scattered by a beam of vapor molecules. The electron beam is usually of energy in the range of 40–100 keV; and it needs to be monochromatic, that is, of very little scatter in energy. The relationship between the
Figure 1 Gas-phase electron diffraction pattern of adamantane, C10H16.
461
462
Electron Diffraction Theory and Methods
component of the intensity distribution possible. This greatly enhanced the scope and precision of structure determinations. Lately, in addition to the time-honored photographic techniques, digital techniques have also been employed. The scheme of a traditional experiment is presented in Figure 2. In some special experiments, mass spectrometric control has been added to the electron diffraction setup, which assists the determination and even optimization of vapor composition for establishing the best conditions for the scattering experiment. The scheme of such a combined experiment is presented in Figure 3. Theory A few expressions are presented to help the understanding of the origin of physical information derived from the experiment without demonstrating the way of arriving at them. First, the electron scattering intensity
Electron beam Nozzle
distribution as a result of electron scattering by randomly oriented rigid molecules is given as a function of a scattering angle variable, s, defined as s ¼ (4p/l) sin(y/2), where l is the electron wavelength and y is the scattering angle. Only elastic scattering is considered. The electron scattering intensity distribution for rigid molecules is given by I ðsÞ ¼ KI 0 R2
N X N X
f i ðsÞf j ðsÞsinðsr ij Þ=sr ij
i¼1 j ¼1
Here, I0 is the intensity of the incident electron beam, R is the distance between the scattering center and the observation point, K is a constant that can be expressed in terms of fundamental constants, N is the number of atoms in the molecule fi (s) and fj (s) are the scattering amplitudes of the ith and j th atom, respectively, and rij is the internuclear distance. This expression contains the scattering of electrons by the individual atoms in the molecule (i ¼ j) as well as the scattering of electrons by atomic pairs (iaj), which is relevant to molecular structure. Accordingly, I(s) may be considered as consisting of two components; one is Ia(s), the atomic component, and the other, Im(s), is the molecular component. Thus,
Vapor beam
I ðsÞ ¼ I a ðsÞ þ I m ðsÞ
and Photographic plate
I a ðsÞ ¼ KI 0 R2
N X
jf i ðsÞj2
i¼1
and Rotating sector
I m ðsÞ ¼ KI 0 R2 Figure 2 Principle of the gas-phase electron diffraction experiment.
N X N X
f i ðsÞ f j ðsÞ sinðsr ij Þ=sr ij
i¼1 j ¼1 ði a j Þ
Electron beam
Quadrupole analyzer Water cooling
GT-400 oil trap
Cold trap
Evaporator system Diffraction chamber
ODF-400
Photo-plate
diffusion pump To fore-pump
Figure 3
To high vacuum pump
Scheme of a combined gas-phase electron diffraction/quadrupole mass spectrometric experiment.
Electron Diffraction Theory and Methods
Atomic scattering amplitudes, fi (s), may be introduced at different levels of approximation (the asterisk denotes complex conjugate). A rough approximation is to assume fi (s) ¼ Zi, where Zi is the atomic number of the ith atom. In this case, the scattering by the electron cloud, which is especially important at low scattering angles, is ignored. A more adequate expression is of the form f i ðsÞ ¼ Zi F i ðsÞ
where Fi (s) is the contribution of the electron density distribution to the scattering amplitude. This expression is valid under the assumption of a weak scattering potential, so the electron wave encountering each volume element within the molecule is the unperturbed, incident wave. This is called the kinematic approximation, known as the first Born approximation when applied to electron diffraction. However, scattering potentials are mostly not weak, and an electron wave undergoes a significant phase shift as it propagates through the field of the scatterer. The phase shift increases with increasing atomic number and increasing scattering angle. The atomic scattering amplitudes used in modern structural work take the phase shift into account and are of the form f i ðsÞ ¼ j f i ðsÞjexp½Zi ðsÞ
where j f i ðsÞj is the modulus of the amplitude and Zi (s) is the phase angle in radians. If these complex scattering amplitudes are used, the expression for the molecular intensity is written as I m ðsÞ ¼ KI 0 R2
N X N X
gij ðsÞ sinðsr ij Þ=sr ij
i¼1 j ¼1 ði a j Þ
where gij ðsÞ ¼ j f i ðsÞj f j ðsÞcos½Zi ðsÞ Zj ðsÞ
This formalism (quasi-kinematic approximation) corresponds to the introduction of multiple scattering effects within the same atom (intraatomic multiple scattering). If the molecule is composed of light atoms, or atoms not differing too much in atomic number, then the phases Zi(s) are similar and the diffracted intensity is not much different from that expected from the kinematic approximation. If one or more atoms are substantially heavier than the others, the deviation from the kinematic theory cannot be ignored. Using the kinematic theory in structure analysis then introduces distortions in the geometry of the molecule. A classic example is rhenium hexafluoride, ReF6, where the peak corresponding to the Re–F distance is split in the radial distribution. Using the quasi-kinematic approximation explains the splitting of the Re–F peak, and leads to the expected, regular
463
octahedral geometry. The treatment of multiple scattering by different atoms in the same molecule (interatomic multiple scattering) requires a more sophisticated approach. However, it is stressed that molecules are not rigid bodies, and the vibrational motions may cause variations in internuclear distances of several picometers. The molecular component of the electron scattering intensity for the vibrating molecule is I m ðsÞ ¼ KI 0 R2
N X N X
g ij ðsÞ expðl 2ij s 2=2Þ sinðsr ij Þ=sr ij
i¼1 j ¼1 ði a j Þ
where lij is the mean vibrational amplitude along the line connecting the two nuclei, and rij is now an effective internuclear distance. This internuclear distance is usually denoted as ra, and is related to the vibrationally averaged internuclear distance at the temperature of the electron diffraction experiment, rg, by the expression r a E r g l 2 =r a
to a good approximation. If the vibration is anharmonic, the term sin(srij) in the above equation should be replaced by sin[s(rij kij s2)], where kij is an asymmetry parameter related to the a constant of the Morse equation. In favorable cases, kij values can be determined from the electron diffraction data. Structure Analysis The experiment yields the total electron scattering intensity distribution, and it is a crucial step to extract the molecular component from it. In addition to the atomic component, there is also an inelastic scattering part of the intensity that is to be eliminated together with the consequences of extraneous scattering, depending on the particular construction of the experiment. All these can be considered together as making up the experimental background (inelastic þ atomic þ extraneous), Ib(s), and Im(s) ¼ I(s) Ib(s). The molecular contribution to the electron diffraction intensities – in short, molecular intensities – is the experimental data on which the determination of structural parameters is based, more often than not, using a least-squares refinement technique. Fourier transformation of the molecular intensities yields the so-called radial distribution, f (r) (a misnomer, because in reality it is related to the probability density distribution of the intramolecular internuclear distances). The molecular model, total intensities, molecular intensities, and radial distribution for tetramethylsilane are displayed in Figures 4–7 using a simple case study from our research group. Figures 6 and 7 show not only the experimental distributions but also the distributions calculated for the
464
Electron Diffraction Theory and Methods H
Si(CH3)4
C
H
H
H H
C Si
H
H
C H
Si H C H
Si H
C H
H
H
Si C
C
H 0
C
H
100
200
300
400
500
r, pm H Figure 7 Radial distributions for tetramethylsilane. The vertical scale is arbitrary. The continuous line is calculated for a model; the dots are calculated from the measured intensity data.
Molecular model of tetramethylsilane.
Figure 4
Si(CH3)4
0
50
100
150 200 s, nm−1
250
300
350
Figure 5 Total experimental intensities and background lines for tetramethylsilane. The vertical scale is arbitrary. The two curves refer to two different distances between the scattering point and the registration plane.
Si(CH3)4
0
50
100
150
200
250
300
350
s, nm−1 Figure 6 Molecular intensities for tetramethylsilane. The vertical scale is arbitrary. The continuous line is calculated and the dots refer to measurements.
best model of tetramethylsilane, which is characterized by the following bond lengths and bond angles, Si–C 187.7(4) pm, Si–H 111.0(3) pm, and Si–C–H 111.0(2) pm as well as Td symmetry and a staggered methyl conformation. The bond lengths and the bond angles here are so-called average parameters, see more on this in another entry of this encyclopedia. The radial distribution is convenient to visually inspect the validity of a model and to read off some principal internuclear distances, but the quantitative determination of all the parameters is done on the basis of the molecular intensities. The refinement of parameters usually starts from an initial set of parameters. The expression of the molecular intensities is a nonlinear relationship; a good choice of the initial parameters will ensure that the calculation reaches the global rather than a local minimum. Tetramethylsilane is a fortunate subject for electron diffraction, because of its high symmetry. Only a few parameters determine its geometry and almost all of their contributions are well resolved. Less symmetrical molecules usually have radial distributions on which the individual contributions are not so easily discernible. Even in such cases, the least-squares refinement of the parameters based on the molecular intensities may yield satisfactory results. For increasing molecular complexity, decreasing precision of the parameters may be the price to pay. In any case, the gas-phase data yield – in the usual terminology of X-ray crystallography – a one-dimensional Patterson function, which indicates the limitations of the technique toward more complex structures. It is desirable to learn as much as possible about the composition of the vapor sample used in the experiment from independent sources. The vapor composition may yet be another unknown to be determined in the analysis. In case of a conformational equilibrium of, say, two conformers of the same molecule the analysis may be more difficult but the results may be more rewarding at the same time. The analysis of ethane-1,2-dithiol at the temperature of 343 K revealed the presence of 62%
Electron Diffraction Theory and Methods
SH H
H
H
H SH
SH
H
SH
H
H g
H
62% a + 38% g
(S … S)a (S … S)g E
0
100
200
300 r, pm
400
465
for bond lengths, a few tenths of a degree for bond angles, and a few degrees for angles of torsion. The precision and generally, the reliability of structure determination may be enhanced considerably by concerted and combined application of electron diffraction experiments with spectroscopic measurements and quantum chemical calculations. In such cases, however, the question of accuracy has to be carefully considered as the physical meaning of structural information from different techniques varies. The combined application of techniques and the physical meaning of parameters are among the aspects discussed in another entry of this encyclopedia. See also: Fourier Transformation and Sampling Theory, Electron Diffraction Applications, Microwave and Radiowave Spectroscopy, Applications, Rotational Spectroscopy, Theory, X-Ray Crystallography of Macromolecules, Theory and Methods.
500
Figure 8 Ethane-1,2-dithiol has a mixture of the anti (a) and gauche (g) forms at 343 K, with respect to the S–C–C–S framework. The bottom curve was calculated from experimental data.
of the anti form (a) and 38% of the gauche form (g) as far as the S–C–C–S framework was concerned. The radial distributions calculated for a set of models and the experimental distribution in Figure 8 serve as illustration. In the best cases, the gas-phase electron diffraction technique yields precisions of a few tenths of a picometer
Further Reading Cowley JM (ed.) (1992) Electron Diffraction Techniques, vols. 1 and 2. Oxford, UK: Oxford University Press. Domenicano A and Hargittai I (eds.) (1991) Accurate Molecular Structures: Their Determination and Importance. Oxford, UK: Oxford University Press. Hargittai I (2005) Looking back and ahead: Gas-phase electron diffraction at 75. Structural Chemistry 16: 1--3. Hargittai I and Hargittai M (eds.) (1988) Stereochemical Applications of Gas-Phase Electron Diffraction, Part A, The Electron Diffraction Technique. New York: VCH Publishers. Kuchitsu K (1972) Experimental errors in gas-phase electron diffraction. In: Cyvin SJ (ed.) Molecular Structures and Vibrations, pp. 148--170. Amsterdam: Elsevier.