Covalent terms

Consider now the behaviour of the HF wave function 0 (eq. (4.18)) as the distance between the two nuclei is increased toward infinity. Since the HF wave function is an equal mixture of ionic and covalent terms, the dissociation limit is 50% H+H " and 50% H H. In the gas phase all bonds dissociate homolytically, and the ionic contribution should be 0%. The HF dissociation energy is therefore much too high. This is a general problem of RHF type wave functions, the constraint of doubly occupied MOs is inconsistent with breaking bonds to produce radicals. In order for an RHF wave function to dissociate correctly, an even-electron molecule must break into two even-electron fragments, each being in the lowest electronic state. Furthermore, the orbital symmetries must match. There are only a few covalently bonded systems which obey these requirements (the simplest example is HHe+). The wrong dissociation limit for RHF wave functions has several consequences. 

The classical VB wave function, on the other hand, is build from the atomic fragments by coupling the unpaired electrons to form a bond. In the H2 case, the two electrons are coupled into a singlet pair, properly antisymmetrized. The simplest VB description, known as a Heitler-London (HL) function, includes only the two covalent terms in the HF wave function. 

Figure 8. 2 shows the coefficients of the three structures in the total three-term wave function. As expected, the covalent term predominates at all distances, but... 

Equation (22) governs, as Eq. (12) for metals, the atomic volume vs. Z curves for these compounds (see Fig. 5 of Chap. A). Although the f-f metallic part has a parabolic behaviour vs. Z, the covalent part introduces a skewness in the parabola. Also the position of the minimum is drawn from the centre of the series towards its beginning. The skewness and the distance of the minimum from the centre increase as the strength of the covalent term increases. 

It can be seen that the first two terms are the same as the valence bond wavefunction, and there are an additional two terms. The first two are commonly called the covalent terms because each has the electrons associated with both centers. The final two are known as ionic terms because each places two electrons at one center (i.e. H+ H and H H ). Valence bond theory generally neglects ionic terms of this type, whereas in MO theory the covalent and ionic terms are treated equally. 

The dissociation constant KD, which is often derived from spectral properties in carbanion chemistry, therefore includes a covalent term that corresponds to Kx in carbocationic chemistry. As one would not expect equal reactivity of benzhydryl chloride and benzhydryl cations, one also should not expect equal reactivity for benzhydryl lithium and benzhydryl anions. As one realizes that the terms contact ion-pair and dissociation have a different meaning in carbocation and carbanion chemistry, the apparent discrepancies quoted above, will disappear. 

In some detailed calculations of relaxation rates due to dipolar interaction for the ruthenium(iii) complex RuCNHafg , Waysbort and Navon (285, 286) have allowed for covalency by including the distribution of unpaired spin density on the metal and ligand orbitals. They found the effect of this covalency term on the Ru d orbitals to be small but the spin delocalization to the ligands increases significantly compared with that calculated using a point dipole approximation. The new results are in better agreement with available experimental data. 

Consider now the behaviour of the HF wave function 0 (eq. (4.18)) as the distance between the two nuclei is increased toward infinity. Since the HF wave function is an equal mixture of ionic and covalent terms, the dissociation limit is 50% H+H and 50% H H. In the gas phase all bonds dissociate homolytically, and the ionic contribution... 

W3.VC function is identical lu RIIFl I liis will normally be tfac C3.se negr the CQuilihnuni. distance—As the bond is stretched th LTfF functlnn allows each of the electrons contains the covalent terms, which is the correct dissociation behaviour, but also ... 

The first two terms in the numerator describe configurations where both electrons are on A and B, respectively. These are called ionic terms. The last two ones describe the sharing of electrons in a symmetrized way, these are the covalent terms. For X = 1 (homopolar bond), the coefficients of the ionic and... 

These early GPU-based MD implementations are characterized by significantly oversimplifying the mathematics in order to make implementation on a GPU easier, neglecting, for example, electrostatics, covalent terms, and heterogenous solutions. This has resulted in a large number of GPU implementations being published but none with any applicability to "real-world" production MD simulations. It is only within the last year (2009/2010) that useful GPU implementations of MD have started to appear. 

Once the bond orders have been calculated, they are used to compute bond energies, angles, and torsions. These terms are also used in non-reactive force fields, but their use in reactive simulations requires some modifications. All covalent terms are pre-multiplied by the bond orders involved this ensures that whenever a bond is broken, all the terms involving it vanish smoothlv. Also, the equilibrium angle in covalent-angle terms depends upon the bond orders... 

ReaxFF also includes other terms—covalent terms to describe the effect of lone pairs, conjugation, under-coordination, etc. The reader is referred to the original publications for a description of these terms [13, 27][13]. 

The EDA results of the group-10 homoleptic complexes TM(EMe)4 (E = B-Tl, TM = Ni, Pd, Pt) are given in Table 13.23. The data indicate that the statement concerning significant Fe ER Tr-backdonation holds also true for other transition metals. In TM(EMe)4 (TM = Ni, Pd, Pt), the contribution of TM —> EMe ir-backdonation is between 33-49% of the covalent term However, the bonding... 

Eq. (19) but with a negative sign for the covalent terms. At infinite separation we want only these terms to survive. We can achieve this if we simply put C2 = — Cj in Eq. (26). On the other hand we know that Vicis dominated by the HE configuration for geometries close to equilibrium. Thus, by introducing the expansion coefficients as new variational parameters we can describe the wave function for all distances. 

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