Difference between revisions of "Molecular Structure"
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| ''r''<sub>α</sub> = ''r''<sub>h0</sub> | | ''r''<sub>α</sub> = ''r''<sub>h0</sub> | ||
− | | 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.<ref group="gt" name="tavnp">In contrast to ''r''<sub>g</sub>, where subject to averaging are interatomic distances, here atomic positions are averaged. Therefore this structure is geometrically consistent.</ref> | + | | 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.<ref group="gt" name="tavnp">In contrast to ''r''<sub>g</sub>, where subject to averaging are interatomic distances, here atomic positions are averaged. Therefore this structure is geometrically consistent. Also for ''r''<sub>α</sub> there is a procedural definition, allowing for a analytical approximation to the actual value starting from computed structures and harmonic force fields, this is the so called "rectilinear approximation" where interatomic motions are decomposed into a radial component along the bond and two rectilinear components perpendicular to the bond. In this way rectilinear ''r''<sub>α</sub> corrections Δ =''r''<sub>e</sub>-''r''<sub>α</sub> can be obtained.</ref> |
| GED, MW, HRMS | | GED, MW, HRMS | ||
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Revision as of 16:30, 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 |
---|---|---|
r_{e} | Equilibrium structure. Appears as a result of Born-Oppenheimer approximation in quantum mechanics. | QC |
r_{g} | Thermally averaged structure.^{[gt 1]} r_{g} distances are centers of gravity for corresponding distribution functions P(r), and thus can be physically defined as r_{g} := <r>, where <> denotes averaging over the whole molecular ensemble in its specific state under observation. | GED |
r_{a} | Effective average structure parameter type^{[gt 1]} resulting from fitting the theoretical model equation: M_{theo.}(s)=const. Σ_{i≠j}^{N} g_{ij}(s) exp(-0.5 l_{ij}^{2}s^{2}) sin[s(r_{ij} - κ_{ij}s^{2})]/r_{ij} to experimental scattering data M_{exp.}(s) with respect to the parameters r_{ij}, l_{ij} and κ_{ij} (distances, amplitudes and anharmonicity or "asymmetry" parameters). The resulting structure parameters r_{ij} are interpreted as "average structure parameters" of type "r_{a}". g_{ij}(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^{3} such that values of a complete set of r_{a} parameters correspond quantitatively exactly to the values of the distances between the atom positioned in R^{3} space. This is known also as the "shrinkage" problem, since due to zero point vibrational motions already for a single CO_{2} in the vibrational ground state: 2 r_{a}(C-O) > r_{a}(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_{a} := 1/<1/r>, where <> denotes averaging over the whole molecular ensemble in its specific state under observation. From this point of view the former procedural definition of r_{a} can be interpreted as an approximation to this quantity, holding to that extent that the model equation for sM(s) holds. | GED |
r_{α} = r_{h0} | 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 |
r^{0}_{α}, r_{z} | Distances between average nuclear positions at 0 K, i.e. for ground vibrational state.^{[gt 2]} | GED, MW, HRMS |
r_{h1} | Distances between average nuclear positions calculated taking into account nonlinear relationships between internal and Cartesian coordinates.^{[gt 2]} | GED, MW, HRMS |
r_{a3,1} | Semi-experimental equilibrium structure obtained using corrections calculated by Shrink program utilizing cubic force fields. | GED, MW, HRMS |
r^{se}_{e} | Semi-experimental equilibrium structure obtained using a set of theoretically calculated corrections to experimental and/or operational data. | GED, MW, HRMS |
r_{s} | Effective structure derived from isotopic differences in rotational constants. | MW, HRMS |
r_{0} | Effective structure derived from rotational constants of zero-point vibrational level. | MW, HRMS |
r_{m} | 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", r_{m} is used to refer to the maximum of the peaks in the radial distribution curves P(r)/r. | MW, HRMS, (GED) |
r^{ρ}_{m} | Effective structure similar to r_{m} obtained by a slightly modified method of Harmony et al. | MW, HRMS |
- ↑ ^{1.0} ^{1.1} Interatomic distances are averaged over molecular vibrations for a given temperature. Generally such distances are geometrically inconsistent.
- ↑ ^{2.0} ^{2.1} ^{2.2} In contrast to r_{g}, where subject to averaging are interatomic distances, here atomic positions are averaged. Therefore this structure is geometrically consistent. Also for r_{α} there is a procedural definition, allowing for a analytical approximation to the actual value starting from computed structures and harmonic force fields, this is the so called "rectilinear approximation" where interatomic motions are decomposed into a radial component along the bond and two rectilinear components perpendicular to the bond. In this way rectilinear r_{α} corrections Δ =r_{e}-r_{α} can be obtained.
In some of the old GED papers the parameter types r_{g}(0) and r_{g}(1) are used. They are equivalent to normal r_{g} and r_{a} 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