Difference between revisions of "Molecular Structure"

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(Types of molecular structure)
(Types of molecular structure)
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| ''r''<sub>a</sub>
 
| ''r''<sub>a</sub>
| Effective average structure parameter type<ref group="gt" name="tav" /> resulting from fitting the theoretical model equation: ''M<sub>theo.</sub>(s)''=''const.'' Σ<sub>''i''≠''j''</sub><sup>''N''</sup> ''g''<sub>''ij''</sub>(''s'') exp(-0.5 ''l''<sub>''ij''</sub><sup>2</sup>''s''<sup>2</sup>) sin[''s''(''r''<sub>ij</sub> - κ<sub>ij</sub>''s''<sup>2</sup>)]/''r''<sub>ij</sub> to experimental scattering data ''M<sub>exp.</sub>''(''s'') with respect to the parameters  ''r''<sub>''ij''</sub>, ''l''<sub>''ij''</sub> and κ<sub>ij</sub> (distances, amplitudes and anharmonicity or "asymmetry" parameters). The resulting structure parameters ''r''<sub>''ij''</sub> are interpreted as "average structure parameters" of type "''r''<sub>a</sub>".  ''g''<sub>''ij''</sub>(''s'') is the interatomic scattering cross sections function (for which ab-initio calculation based tabulated values are usually used). ''N'' is the number of scattering centers ("atoms"). In general it is not possible to assign a set of atoms to positions in a Euklidian space '''R'''<sup>3</sup> such that values of a complete set of ''r''<sub>a</sub> parameters correspond quantitatively exactly to the values of the distances between the atom positioned in '''R'''<sup>3</sup> space. This is known also as the "shrinkage" problem, since due to zero point vibrational motions already for a single CO<sub>2</sub> in the vibrational ground state: 2 ''r''<sub>a</sub>(C-O) > ''r''<sub>a</sub>(O-O'). In general these geometrical inconsistencies are worst for three-atom moieties describing large angles close to 180°. Since the structure parameters are obtained from diffraction patterns with Fourier-Transform relation to real space of molecular geometry an alternative (empirically motivated definition) is ''r''<sub>a</sub> := 1/<1/''r''><sup>-1</sup>, where <> denotes averaging over the whole molecular ensemble in its specific state under observation.  
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| Effective average structure parameter type<ref group="gt" name="tav" /> resulting from fitting the theoretical model equation: ''M<sub>theo.</sub>(s)''=''const.'' Σ<sub>''i''≠''j''</sub><sup>''N''</sup> ''g''<sub>''ij''</sub>(''s'') exp(-0.5 ''l''<sub>''ij''</sub><sup>2</sup>''s''<sup>2</sup>) sin[''s''(''r''<sub>ij</sub> - κ<sub>ij</sub>''s''<sup>2</sup>)]/''r''<sub>ij</sub> to experimental scattering data ''M<sub>exp.</sub>''(''s'') with respect to the parameters  ''r''<sub>''ij''</sub>, ''l''<sub>''ij''</sub> and κ<sub>ij</sub> (distances, amplitudes and anharmonicity or "asymmetry" parameters). The resulting structure parameters ''r''<sub>''ij''</sub> are interpreted as "average structure parameters" of type "''r''<sub>a</sub>".  ''g''<sub>''ij''</sub>(''s'') is the interatomic scattering cross sections function (for which ab-initio calculation based tabulated values are usually used). ''N'' is the number of scattering centers ("atoms"). In general it is not possible to assign a set of atoms to positions in a Euklidian space '''R'''<sup>3</sup> such that values of a complete set of ''r''<sub>a</sub> parameters correspond quantitatively exactly to the values of the distances between the atom positioned in '''R'''<sup>3</sup> space. This is known also as the "shrinkage" problem, since due to zero point vibrational motions already for a single CO<sub>2</sub> in the vibrational ground state: 2 ''r''<sub>a</sub>(C-O) > ''r''<sub>a</sub>(O-O'). In general these geometrical inconsistencies are worst for three-atom moieties describing large angles close to 180°. Since the structure parameters are obtained from diffraction patterns with Fourier-Transform relation to real space of molecular geometry an alternative (empirically motivated definition) is ''r''<sub>a</sub> := 1/<1/''r''>, where <> denotes averaging over the whole molecular ensemble in its specific state under observation.  
 
| GED
 
| GED
 
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Revision as of 16:02, 15 September 2017

Molecular structure (molecular geometry) is a one of the most fundamental properties of molecules, which characterizes spatial positions of the atoms that constitute a molecule. This term is usually referred to the geometrical structure of molecules in gas phase.

Types of molecular structure

Abbreviations are:

  • QC — Quantum-chemical calculations
  • GED — Gas electron diffraction
  • MW — Microwave spectroscopy
  • HRMS — High resolution molecular spectroscopy
Type Description Methods
re Equilibrium structure. Appears as a result of Born-Oppenheimer approximation in quantum mechanics. QC
rg Thermally averaged structure.[gt 1] rg distances are centers of gravity for corresponding distribution functions P(r), and thus can be physically defined as rg := <r>, where <> denotes averaging over the whole molecular ensemble in its specific state under observation. GED
ra Effective average structure parameter type[gt 1] resulting from fitting the theoretical model equation: Mtheo.(s)=const. ΣijN gij(s) exp(-0.5 lij2s2) sin[s(rij - κijs2)]/rij to experimental scattering data Mexp.(s) with respect to the parameters rij, lij and κij (distances, amplitudes and anharmonicity or "asymmetry" parameters). The resulting structure parameters rij are interpreted as "average structure parameters" of type "ra". gij(s) is the interatomic scattering cross sections function (for which ab-initio calculation based tabulated values are usually used). N is the number of scattering centers ("atoms"). In general it is not possible to assign a set of atoms to positions in a Euklidian space R3 such that values of a complete set of ra parameters correspond quantitatively exactly to the values of the distances between the atom positioned in R3 space. This is known also as the "shrinkage" problem, since due to zero point vibrational motions already for a single CO2 in the vibrational ground state: 2 ra(C-O) > ra(O-O'). In general these geometrical inconsistencies are worst for three-atom moieties describing large angles close to 180°. Since the structure parameters are obtained from diffraction patterns with Fourier-Transform relation to real space of molecular geometry an alternative (empirically motivated definition) is ra := 1/<1/r>, where <> denotes averaging over the whole molecular ensemble in its specific state under observation. GED
rα = rh0 Distances between average nuclear positions. The averaging can be thought of to be carried out in a coordinate framework where all translational and rotational motions are projected out.[gt 2] GED, MW, HRMS
r0α, rz Distances between average nuclear positions at 0 K, i.e. for ground vibrational state.[gt 2] GED, MW, HRMS
rh1 Distances between average nuclear positions calculated taking into account nonlinear relationships between internal and Cartesian coordinates.[gt 2] GED, MW, HRMS
ra3,1 Semi-experimental equilibrium structure obtained using corrections calculated by Shrink program utilizing cubic force fields. GED, MW, HRMS
rsee Semi-experimental equilibrium structure obtained using a set of theoretically calculated corrections to experimental and/or operational data. GED, MW, HRMS
rs Effective structure derived from isotopic differences in rotational constants. MW, HRMS
r0 Effective structure derived from rotational constants of zero-point vibrational level. MW, HRMS
rm Effective structure derived from the mass-dependence method of Watson. Sometimes in the older literature e.g. Hargittai, Hargittai, "The molecular geometries of coordination compounds in the vapour phase", rm is used to refer to the maximum of the peaks in the radial distribution curves P(r)/r. MW, HRMS, (GED)
rρm Effective structure similar to rm obtained by a slightly modified method of Harmony et al. MW, HRMS
  1. 1.0 1.1 Interatomic distances are averaged over molecular vibrations for a given temperature. Generally such distances are geometrically inconsistent.
  2. 2.0 2.1 2.2 In contrast to rg, where subject to averaging are interatomic distances, here atomic positions are averaged. Therefore this structure is geometrically consistent.

In some of the old GED papers the parameter types rg(0) and rg(1) are used. They are equivalent to normal rg and ra types, respectively.[Kuchitsu1959]Author: Kuchitsu, Kozo
Journal: Bulletin of the Chemical Society of Japan
Number: 7
Pages: 748-769
Title: Electron Diffraction Investigation on the Molecular Structure of n-Butane
Volume: 32
Year: 1959
Link to Google Scholar