The dissociation limit

Let us finally consider the hydrogen molecule in the dissociation limit. There are two possible outcomes of the dissociation - in the limit of infinite separation and no interactions, the elections may end up in the same atom ionic dissociation) or they may end up in different atoms (covalent dissociation). In the ionic dissociation, the wave function is a closed-shell singlet state in the covalent case, the electrons may be coupled to yield either a singlet or a triplet. 

State Kinetic Attraction Electron repulsion Nuc. rep. Total 

A Tiereas the covalent state is lower in energy and represents two noninteracting hydrogen atoms in the ground state, the ionic state represents the hydrogen anion. The energies of these states are given by 

Let us also consider the energies of the bonding and antibonding configurations at infinite separation. From (5.2.59) and (5.2.60), we obtain 

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Figure 3. Relaxed triangular plot [68] of the U3 ground-state potential energy surface using hyperspherical coordinates. Contours, are given by the expression (eV) — —0.56 -t- 0.045(n — 1) with n = 1,2,..,, where the dashed line indicates the level —0.565 eV. The dissociation limit indicated by the dense contouring implies Li2 X Sg ) -t- Li.

If r r" there may be appreciable intensity involving the continuum of vibrational levels above the dissociation limit. This results in a u" = 0 progression like that in Figure 7.22(c) where the intensity maximum is at a high value of u or it may be in the continuum. An example of this is the B Uq+ — transition of iodine. In the B and X states is 3.025 A and 2.666 A, respectively, leading to the broad intensity maximum close to the continuum, as observed in Figure 7.19. 

Section 6.13.2 and illustrated in Figure 6.5. The possible inaccuracies of the method were made clear and it was stressed that these are reduced by obtaining term values near to the dissociation limit. Whether this can be done depends very much on the relative dispositions of the various potential curves in a particular molecule and whether electronic transitions between them are allowed. How many ground state vibrational term values can be obtained from an emission spectrum is determined by the Franck-Condon principle. If r c r" then progressions in emission are very short and few term values result but if r is very different from r", as in the A U — system of carbon monoxide discussed in Section 7.2.5.4, long progressions are observed in emission and a more accurate value of Dq can be obtained. 

If the values of in the combining states are very different the dissociation limit of a progression may be observed directly as an onset of diffuseness. However, the onset is not always particularly sharp this is the case in the B Uq+ absorption system of iodine... 

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 2-configurational Cl wave function (7.4) allows a qualitatively correct description of the H2 molecule at all distances, in the dissociation limit the weights of the two configurations become equal. 

At the dissociation limit the UHF wave function is essentially an equal mixture of a singlet and a triplet state, as discussed in Section 4.4. Removal of the triplet state by projection (PUHF) lowers the energy in the intermediate range, but has no effect when the bond is completely broken, since the singlet and triplet states are degenerate here. 

The improvement brought about by extending the perturbation series beyond second order is very small when a UHF wave function is used as the reference, i.e. the higher-order terms do very little to reduce the spin contamination. In the dissociation limit the spin contamination is inconsequential, and the MP2, MP3 and MP4 results are all in... 

In comparison with the more standard Configuration Interaction (Cl) method, the one-particle Green s function approach offers the essential advantages, in the outlook of numerical applications on extended systems, of a stronger and systematic compactness (30) of the configuration spaces in high order approximations and of energy separability (5,31) in the dissociation limit (size-consistency). The latter is a necessary prerequisite ( ) for a correct (i.e. size-... 

Neglecting for simplicity the long-range character of the Coulomb force, the above summations yield (31) a bounded result (x) when extended to infinity. Bielectron integrals can thus be regarded as scaling like Nq", either in the thermodynamic limit (Nq °°), or (31) in the dissociation limit (aQ °°). 

At the equilibrium inter-atomic distance R, two paired electrons of occupy the bonding orbital with a closed-shell low-spin singlet (S = 0). When the bond length is further increased, the chemical bond becomes weaker. The dissociation limit of corresponds to a diradical with two unpaired electrons localized at each atom (Fig. 1). In this case, the singlet (S spin-antiparaUel) and triplet (T spin-parallel) states are nearly degenerate. Different from such a pure diradical with... 

Fig. 1 A schematic illustration of the in-phase and out-of-phase combinations of the atomic orbitals into the bonding and antibonding molecular orbitals, respectively. The dissociation limit of a H molecule corresponds to a pure diradical with degenerate singlet and triplet states...

Resonance Enhanced Photodissociation FeO States Below the Dissociation Limit... 

Unfortunately, predissociation of the excited-state limits the resolution of our photodissociation spectrum of FeO. One way to overcome this limitation is by resonance enhanced photodissociation. Molecules are electronically excited to a state that lies below the dissociation limit, and photodissociate after absorption of a second photon. Brucat and co-workers have used this technique to obtain a rotationally resolved spectrum of CoO from which they derived rotational... 

With respect to correlation, the behaviour of the hydrogen molecule studied in a subminimal FSGO basis set is still more striking than the one observed in a minimal basis set. By symmetry arguments, the single FSGO which describes the electron pair of the hydrogen molecme is centred at the middle of the H-H bond. As the intemuclear distance increases and ultimately when the molecule dissociates, such a description would lead to a physical nonsense. Indeed, at the dissociation limit, this would correspond to two protons (2H ) and an isolated pair of electrons (2e ). 

In all the other sehemes (UHF, PHF, EHF), the dissociation limit is the correct one corresponding to two neutral hydrogen atoms (2H-) each FSGO-hydrogen atom energy is thus obtained by the simple variational procedure ... 

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