Left - right correlation

In this example, tlie two non-orthogonal polarized orbital pairs involve mixing the k and k orbitals to produce two left-right polarized orbitals as depicted in figure B3.1.7. Here one says that the n electron pair undergoes left-right correlation when the (n configuration is introduced. 

The optimum values of die oq and a coefficients are determined by the variational procedure. The HF wave function constrains both electrons to move in the same bonding orbital. By allowing the doubly excited state to enter the wave function, the electrons can better avoid each other, as the antibonding MO now is also available. The antibonding MO has a nodal plane (where opposite sides of this plane. This left-right correlation is a molecular equivalent of the atomic radial correlation discussed in Section 5.2. 

Figure 4.11 (left). Tablet hardness found for the sequence of 120 tablets. On the suspicion that there is a dependency by stamp, the data are grouped by stamp (Fig. 4.12 left), (right). Correlation between tablet hardness and weight. For the residual.s that are obtained after correction for the tablet weight, see Fig. 4.12 right. 

As the Kohn-Sham wave function has a delocalized exchange hole, and therefore lacks the left-right correlation, is expected to display different bond midpoint features to In fact, Vs,km is ro for two-electron systems, so that Tc [p] and Vc reduce in this case to... 

The GVB and SC methods provide wave functions that are, of course, much more compact than the corresponding valence—CASSCF one (e.g., only 14 spin-coupling modes for methane with the SC method, and a single one with the GVB method). Owing to this difference in size, the GVB and SC methods cannot be expected to include the totality of the nondynamical correlation, even if these two methods treat well, by definition, the left—right correlation for each bond of the molecule. Physically, this is because the various local ionic... 

The importance and physical nature of dynamic correlation is even better appreciated in the case of 3e bonds, a type of bond in which the electron correlation is entirely dynamic, since there is no left-right correlation associated with odd-electron bonds. As noted earlier, the Hartree Fock and simple VB functions for 3e bonds (hence, GVB, SC, or VBSCF) are nearly equivalent and yield about similar bonding energies. Taking the F2 radical anion as an example, it turns out that, compared to the experimental bonding energy of... 

A severe inconvenience of describing each bond of a polyatomic molecule by one covalent and two ionic components is that the number of VB structures grows exponentially with the size of the molecule. Coulson and Fischer [17] proposed a very elegant way to incorporate left-right correlation into a single and formally covalent VB structure of the HL type. To this end they used deformed or rather slightly delocalized orbitals as exemplified in eq 6 for H2. 

The GVB and SC methods take care of the left-right correlation for each bond of a polyatomic molecule. However, these methods do not include the totality of the "non-dynamical" correlation since the various local ionic situations are not interconnected with these methods. For example, the two ionic situations 1 and 2 below are expected to have different weights, 2 being more important than 1,... 

The importance of left-right correlation for the description of the bond is best appreciated in the case of the F 2 molecule. Here the experimental bonding energy is 38 kcal/mol, while the Hartree-Fock bond energy is negative, -36... 

Handy NC, Cohen A (2001) Left-right correlation energy, Mol Phys, 99 403 112... 

As the wavefunction approaches the Hartree-Fock limit one would expect the Ti-Cl bond distance to be shorter than the experiment because of the lack of bond-pair correlation. The bond-pair correlation added by the GVB wavefunction lengthened the Ti-Cl bond 0.021 A, because the GVB wavefunction adds only limited left-right correlation and none of the dynamical correlation. For most A-B bonds, the calculated bond lengths at the SCF level are too short, and the correlation added by a GVB calculation accounts for a major portion of the non-dynamical correlation error in the SCF wavefunction. But for Ti-Cl bonds, both the SCF and GVB calculations predict too long a bond distance because they do not include necessary dynamical atomic correlation of the Cl atoms. 

If we allow for correlation effects within each pair using the GVB-PP method we obtain the SOPP orbitals also shown in Fig. 4. The close relationship between the SOPP orbitals and the LMOs is apparent. Each pair in an LMO is either left-right correlated in the case of a bond or in-out correlated in the case of lone pairs. The dxy LMO or the corresponding pair from the GVB-PP calculation are the only orbitals to exhibit considerable delocalization. In the MO case, this behavior is referred to as back-bonding from the metal to the CO, but in the valence bond situation, our experience is that all... 

Similarly, in linear molecules, the axial (or left-right ) correlation and the "equatorial correlation are related to the coefficients t and t obtained from the projection of the position vector 7 on the intemuclear axis and the plan perpendicular to it, respectively. For the hydrogen molcule at the equilibrium distance, one has... 

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