Bond length contractions and

CH4 to PbR, Bond length contraction and increase of force constants... 

Figure 6. Influence of relativistic corrections to the electron-electron interaction on the bond length contraction and bond destabilization of the group 4 tetrahydrides XH. The percentage of results obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian wrt. those obtained with the Dirac-Coulomb (DC) Hamiltonian has been derived from Dirac-Hartree-Fock calculations [28,29].

Since bond length contractions and orbital contractions often are comparable in magnitude and direction it was assumed without further proof that the first were caused by the latter, cf. e.g. Pyykkd and Desclaux (1979a). However, in 1980 the so-called Dutch Revolution (Pyykkd 1988a) took place and it was shown (Snijders and Pyykkd 1980, Ziegler et al. 1980, 1981, Pyykkd et al. 1981) that both effects are indeed often... 

Fig. 12.1 DFT-derived BOLS relation from bulk fee crystals to atomic chains (reprinted with permission from [12]). The bond length contracts and the cohesive energy per bond in the chains are about 2-3 times that of the bulk fee crystals, which is in accordance with the BOLS correlation albeit the absolute amounts of variation...

Instead, we believe the electronic structure changes are a collective effect of several distinct processes. For example, at surfaces the loss of the bulk symmetry will induce electronic states with different DOS compared to bulk. As the particle sizes are decreased, the contribution of these surface related states becomes more prominent. On the other hand, the decrease of the coordination number is expected to diminish the d-d and s-d hybridization and the crystal field splitting, therefore leading to narrowing of the valence d-band. At the same time, bond length contraction (i.e. a kind of reconstruction ), which was observed in small particles [89-92], should increase the overlap of the d-orbitals of the neighboring atoms, partially restoring the width of the d-band. 

One-center expansion was first applied to whole molecules by Desclaux Pyykko in relativistic and nonrelativistic Hartree-Fock calculations for the series CH4 to PbH4 [81] and then in the Dirac-Fock calculations of CuH, AgH and AuH [82] and other molecules [83]. A large bond length contraction due to the relativistic effects was estimated. However, the accuracy of such calculations is limited in practice because the orbitals of the hydrogen atom are reexpanded on a heavy nucleus in the entire coordinate space. It is notable that the RFCP and one-center expansion approaches were considered earlier as alternatives to each other [84, 85]. 

More recently Ziegler, Snijders, and co-workers have suggest that the bond-length contraction is independent of the orbital contraction but rather it is a direct result of the Dirac Hamiltonian (63,64)- They observed that the contraction could be computed from the nonrelativistic molecular wave... 

To date, the best ab initio all-electron molecular calculations involving heavy elements are those of Lee and McLean, who published LCAO-MO SCF relativistic calculations on AgH and AuH (75). They reported relativistic bond-length contractions of 0.08 and 0.25 A, respectively, and increases in... 

The most important difference between particles inside the bulk and in the interfacial layer comes from the surrounding environment of the particles the particles inside the bulk are in an isotropic environment, while those in the interface are in an anisotropic environment thus, in the interlayer, the forces between the particles are unbalanced. To reduce the resulting surface pressure, some additional processes occur that must be taken into account. On clean surfaces (for example, on a solid surface in vacuum), these processes are the bond-length contraction or relaxation and reconstruction of the surface particles (Somorjai 1994). It results in significantly reduced spacing between the first and second layers compared to the bulk. The perturbation caused by this movement propagates a few layers into the bulk. The other effect is that the equilibrium position of the particles changes that is the outermost layers can have different crystal structure than the bulk. This phenomenon is the reconstruction. 

It should be pointed out that Schwarz (20),using double perturbation theory,has demonstrated that it is possible to rationalize the relativistic bond length contraction in terms of the attractive Hellmann-Feynman force due to the relativistic change in electron density.In such an approach it would be necessary to analyze and get a physical picture of the relevant density changes... 

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