Molecular bond length

The usefulness of quantum-chemical methods varies considerably depending on what sort of force field parameter is to be calculated (for a detailed discussion, see [46]). There are relatively few molecular properties which quantum chemistry can provide in such a way that they can be used directly and profitably in the construction of a force field. Quantum chemistry does very well for molecular bond lengths and bond angles. Even semiempirical methods can do a good job for standard organic molecules. However, in many cases, these are known with sufficient accuracy a C-C single bond is 1.53 A except under exotic circumstances. Similarly, vibrational force constants can often be transferred from similar molecules and need not be recalculated. 

There is a fundamental difference between the properties of the molecular bond length in the STIRAP and APLIP processes that is overlooked by the average bond distance. In APLIP the wave function is always localized around the average internuclear distance, since it is constrained by the electronic forces exerted by Uup(x,t) x(t) is also stable since it corresponds to the minimum of a potential, and it is quasi-static, since the rate of change of Uup(x,t) can be arbitrarily controlled. Therefore, it truly corresponds to the classical notion of a bond length. 

Figure 1 The ubiquitous elbow potential energy surface showing for the dissociation of a diatomic molecule on a surface. This is a function of die molecular bond length and the molecule-surface distance. The reactants are intact molecules, while die products are the atoms chemisorbed separately on the surface. The two extreme cases are shown, an early barrier for which the initial vibration of the molecule is ineffective in overcoming the barrier, and a late barrier for which vibration assists in the dissociation process.

Figure 10 A typical trajectory showing rotational excitation accompanying vibrational de-excitation (i.e. a vibration to rotational energy transfer) [71]. The top panel shows the evolution in the Z (molecule-surface distance) and r (molecular bond length) coordinates. In the lower panel, the motion is projected onto the r — 0 (molecular bond orientation) plane. Coupling of vibrations and rotations occurs because the molecule attempts to dissociate at an unfavourable bond angle.

A brief mention of nomenclature is necessary here. First, note that the barrier at A is described as being in the entrance channel. This is merely a convenient name for an activation barrier in which the molecular bond length is nearly unchanged from its gas-phase value, (r — r ) By the same token, an exit channel barrier is one in which the molecular bond length is significantly stretched from its gas-phase value, (r — r " ) r . These are subjective criteria, useful mainly as limiting cases. Realistic PES will almost certainly fall in between each limit. 

The major role for e-h pair processes is likely in vibrational excitation and deexcitation for weakly interacting molecule-surface systems. In this regard, it is important to distinguish e-h pair processes from the adiabatic coupling of molecular and electronic degrees of freedom. For example, if the molecular bond length varies as the molecule approaches a surface as in... 

In the Nj/WfllO) system, remains less than unity even at kinetic energies well above those of the minimum barrier. By contrast, for the Hj/NiflOO) system considered previously, Sq approaches unity at high kinetic energies. To understand this behavior more fully, we have determined the location of the molecule in the surface unit cell when dissociation occurred. It was possible to define such a position because the molecular bond length oscillated around its gas-phase value until increasing quickly and monotonically during the irreversible dissociation. The rapid increase in the... 

Obstacles to modelling the evolution of quantum state populations under multiple collisions primarily arise from the complexity of standard collision theory. An accurate PES is needed for all potential collision partners in a gas mixture and some species will be in highly excited states. State-to-state collision calculations are highly computer intensive for even the simplest of processes and, without a major increase in computational speed, are not suited to multiple, successive calculations. By contrast, the AM method is fast, accurate and calculations for atoms and/or diatomic molecules require only readily available data such as molecular bond length, atomic mass, spectroscopic constants and collision energy. 

Figure A3.9.8. An elbow potential energy surface representing the dissociation of a diatomic in two dimensions-the molecular bond length and the distance from the molecule to the surface.

As the molecular bond lengths became known, correlations were sought between the geometry of the surface and catalytic activity. There developed the multiplet theory of Balandin which was applied successfully to dehydrogenation catalysts. It also provided an adequate explanation of the work of Maxted and others on catalytic poisons and of the behavior of the different plane faces of crystals. There is no inherent conflict between the interpretations based on geometry and those based on the electronic potential of the surface. The two effects are probably complementary. More knowledge is, however, required about the influence of the electronic potential on the decomposition of complex molecules, before a decision can be made on their relative significance. 

Figure 2.1. Molecular bond lengths and -angles for (a) thiophene and b) 2,2 -bilhiophene. The data are from microwave spectra for thiophene [1], and ifom electron diffraction in gas phase for bithiophene [5]. The bithiophene molecule is here shown in a planar antisyn conformation, but exists in two conformcr states in the gas phase the jyw-like form with a torsion angle of 36(5)°, and the aniisyn-hke form with torsion angle 148(3)".

Fig. 3. Calculated total energies (upper row), molecular bond lengths (middle row) and Mulliken atomic charges (lower row) for the I-type (left column) and T-type (right column) approaches of the H2 molecule to the Nl single atom, using the RHF procedure.

Fig. 7. Landscape-wise, calculated molecular bond lengths for threefold-type (left column), bridge-type (middle column) and top-type (right column) approaches of...

Fig. 11. Calculated total energies (upper row) and molecular bond lengths (lower row) for the EEl (outer) approach (left column), T/EE2 approaches (middle column) and N-approach (right column) of the O2 molecule to the Pt cluster, using a DFT procedure.

R. Shepard, G. S. Kedziora, H. Lischka, I. Shavitt, T. Miiller, P. G. Szalay, M. Kallay, and M. Seth, The Accuracy of Molecular Bond Lengths Computed by Multireference Electronic Stmcture Methods, Chem. Phys. 349, 37-57 (2008). 

Actually, these variations in bond lengths are of crystal-chemical nature and can be understood within the known concepts. As an example, let us consider a linear triatomic system Ii- l2- -Is. Table 4.2 lists the inter-molecular distances D vs. the intra-molecular bond length d obtained by averaging the experimental data [54, 55, 68-73]. Since the D vs d curve (Fig. 4.3) shows a hyperbolic behavior [58, 74], it can be described by the equation of O Keeffe and Brese [75]... 

Pyykko (1979b) used the Dirac-Hartree-Fock one-centre expansion method for the monohydrides to calculate relativistic values for the lanthanide and actinide contraction, i.e. 0.209 A for LaH to LuH and 0.330A for AcH to LrH. The corresponding nonrelativistic value derived from Hartree-Fock one-center expansions for LaH and LuH is 0.191 A, i.e., for this case 9.4% of the lanthanide contraction is due to relativistic effects. The experimental value of 0.179 A would suggest a correlation contribution of-14.4% to the lanthanide contraction if one assumes that the relativistic theoretical values are close to the Dirac-Hartree-Fock limit, which is certainly not true for the absolute values of the bond lengths themselves. Moreover, it is well known that for heavy elements relativistic and correlation contributions are not exactly additive. Corresponding nonrelativistic calculations for AcH and LrH have not been performed and experimental data are not available to determine relativistic and electron correlation effects for the actinide contraction. Table 8 summarizes values for the lanthanide and actinide contraction derived from theoretical or experimental molecular bond lengths. It is evident from Ihese results... 

Indeed, a principal application of rotational (and vibrational-rotational) spectroscopy is determination of molecular bond lengths (and bond angles in polyatomics. Chapter 5). 

Among the parameters that have to be known for the calculation of thermodynamic properties are the rotational constants. If these are unknown then the three principal moments of inertia of the molecule have to be calculated from assumed molecular bond lengths and angles. For this purpose there are two documented computer programs (Brinkmann and Burcat, 1979 Ehlers and Cowgill, 1964). 

---