RHF dissociation

UHF and RHF dissociation curves for H2. (Figure adapted from Szabo A,NS Ostlund 1982. Modem antum Chemistry. Introduction to Advanced Electronic Structure Theory. New York, McGraw-Hill.)... 

Illustrating how Cl Accounts for Electron Correlation, and the RHF Dissociation Problem... 

As Bartlett/18/ has pointed out, for a closed shell single determinant Hartree-Fock (HF) function as the starting point for MBPT or CC theory, separability into neutral fragments is not automatically guaranteed, since sometimes the UHF rather than the RHF dissociates properly. When the criteria (a) and (d) above are both satisfied, the model may be called... 

Illustrating how Cl Accounts for Electi-oii Coirelalionr and the RHF Dissociation Problem------------------------------... 

As another illustration of the performance and, in particular, the conditional convergence of the M0ller-Plesset series, we have in Figure 5.20 plotted the restricted MP2, MP3, MP4 and MP50 dissociation curves for the hydrogen molecule in the cc-pVQZ basis. For comparison, we have also plotted the FCI and RHF dissociation curves in the same basis. 

Choose LHH(spin Unrestricted Hartree-Fock) or RHF (spin Restricted Ilartree-Fock) calculations according to your molecular system. HyperChem supports UHF for both open-sh el I and closed-shell calcii lation s an d RHF for cUised-shell calculation s on ly, Th e closed-shell LHFcalculation may be useful for studyin g dissociation of m olectilar system s. ROHF(spin Restricted Open-shell Hartree-Fock) is not supported in the current version of HyperChem (for ah initio calculations). 

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. 

It should be noted that dative bonds, like metal complexes and charge transfer species, in general have RHF wave functions which dissociate correctly, and the equilibrium bond lengths in these cases are normally too long. 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

Figure 11.4 RHF, UHF and PUHF dissociation curves for H2O near the instability point...

Figures 11.9 and 11.10 compare the performance of the CCSD and CCSD(T) methods, based on either an RFIF or UHF reference wave function. Compared to the RMP plot (Figure 11.7), it is seen that the infinite nature of coupled cluster causes it to perform somewhat better as the reference wave function becomes increasingly poor. While the RMP4 energy curve follows the exact out to an elongation of 1.0A, the CCSD(T) has the same accuracy out to - 1.5 A. Eventually, however, the wrong dissociation limit of the RHF wave also makes the coupled cluster methods break down, and the energy starts to decrease.

Table 1. shows the total energies obtained using the RHF method for 1. LCAO minimal basis set STO-IG for the sake of comparison with FSGO, 2. FSGO in its symmetric and broken symmetry solutions and, 3. LCAO minimal basis set STO-3G in order to allow a safer comparison with the quality of the subminimal basis used in the FSGO technique. The dissociation curves are given in Figure 1. 

As a final comment, it is interesting to note that this FS(K) study of the hydrogen molecule offers a new and simple illustration of the behavior of sophisticated Hartree-Fock schemes like UHF, PHF and EHF. Furthermore, it provides a very efficient numerical example of instabilities in the standard Hartree-Fock method. It is important to see that the UHF, PHF and EHF schemes all correct the wrong RHF behavior and lead to the correct dissociation limit. However, the UHF and PHF schemes only correct the wave function for large enough interatomic distances and the effect of projection in the PHF scheme even results in a spurious minimum. The EHF scheme is thus the only one which shows a lowering of the energy with respect to RHF for all interatomic distances. 

Unfortunately, the failure of RHF is so severe that none of the correlated methods based upon it are able to overcome it (except, of course, full Cl). The MP2 curve diverges to negative infinity at the dissociation limit because of a near-degeneracy between the highest occupied and lowest unoccupied molecular... 

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