Separated-atom limits

In the united-atom limit, 5 = 0, this gives N = 2 while in the separated-atom limit, s we have —> 1. Beginning with a series of s-values, we can now immediately... 

In the separated-atom limit, —> °o, both roots yield k/ = I, which again gives the... 

At the united atom limit, R 0, both have small values, then rise to a maximum value, and hnally vanish at the separated atom limit, R 00. However, note that for / > 3 A the correlation between the two electrons is almost zero but the entanglement is maximal until around 4 A the entanglement vanishes for... 

In the separated-atom limit, R = oo, equation (83) also yields exact energies ... 

The approach used first, historically, and the one this book is about, is called the valence bond (VB) method today. Heitler and London[8], in their treatment of the H2 molecule, used a trial wave function that was appropriate for two H atoms at long distances and proceeded to use it for all distances. The ideal here is called the separated atom limit . The results were qualitatively correct, but did not give a particularly accurate value for the dissociation energy of the H—H bond. After the initial work, others made adjustments and corrections that improved the accuracy. This is discussed folly in Chapter 2. A cmcial characteristic of the VB method is that the orbitals of different atoms must be considered as nonorthogonal. 

Fig. 1. Unperturbed energy levels of the organic molecules A and B in the separated atom limit and the corresponding perturbed energy levels of the bimolecular collision complex.

Correlation diagram A diagram which shows the relative energies of orbitals, configurations, valence bond structures, or states of reactants and products of a reaction, as a function of the molecular geometry, or another suitable parameter. An example involves the interpolation between the energies obtained for the united atoms and the values for the separated atoms limits. 

Let us consider how independent /i(i ) 2 effects contribute to the v E) for the hydrogen halides, HX (X = I, Br, and Cl). The curves shown on Fig. 7.6 correspond to relativistic adiabatic potential energy curves (respectively 0 dotted, 0+ dashed, 1 and 2 solid) for HI obtained after diagonalization of the electronic plus spin-orbit Hamiltonians (see Section 3.1.2.2). The strong R-dependence of the electronic transition moment reflects the independence of the relative contributions of the case(a) A-S-Q basis states to each relativistic adiabatic II state. The independent experimental photodissociation cross sections are plotted as solid curves in Fig. 7.7 for HI and HBr. Note that, in addition to the independent variations in the A — S characters of each fl-state caused by All = 0 spin-orbit interactions, all transitions from the X1E+ state to states that dissociate to the X(2P) + H(2S) limit are forbidden in the separated atom limit because they are at best (2Pi/2 <— 2P3/2) parity forbidden electric dipole transitions on the X atom. In the case of the continuum region of an attractive potential, the energy dependence of the dissociation cross section exhibits continuity in the Franck-Condon factor density (see Fig. 7.18 Allison and Dalgarno, 1971 Smith, 1971 Allison and Stwalley, 1973). 

To illustrate the significance of measurements of internal state branching ratios, we will turn once again to the example of the photodissociation of the hydrogen halides, HX. The fine structure ratio is the branching ratio of X /X populations 2Pi/2/2P3/2- In the non-relativistic adiabatic representation, this branching ratio would be predicted to be zero because the only case (a) basis state which has a non-zero transition moment from the X1E+ state is the 1IIi state which correlates adiabatically with the X(2P3/2) +H(2S) separated atom limit. However, in the more realistic relativistic adiabatic representation, Afl = 0 3E/, 1n3 3ni, and 3ni 3Ei" spin-orbit matrix elements... 

Since (for fixed -I- nj P " i( f > n m = 0) is an even function ofthis alternative interpretation encounters no LSD or GGA spin-symmetry dilemma, In the separated-atom limit for H2, it correctly makes Px i r,r) = 0 for in the vicinity of either atom, since (by the Pauli exclusion principle) P"= (Hf, n u = 0) vanishes when either or vanishes. 

Neglect of off-diagonal elements leads to that the number operators Na commute with the total hamiltonian and that the bond orders (olor) vanish. It also follows that the expectation values (Nc) assume integer values that equal the normal number of occupied valence spin orbitals in an isolated atom, f.e., Nc) —> Identification with the separated atoms limit and comparison with... 

FVom the above discussion, it follows that the construction of parameterized hamiltonians for molecular orbital calculations may lead to certain operator relationships being violated in a limited basis. It is also clear that some of these operator relations can be restored at the expense of introducing various approximations in the evaluation of integrals. Atomic parameters may be derived from consideration of the separated atoms limit, while interatomic parameters are commonly associated with overlap integrals and possibly other functions of the interatomic distance. For instance, it is often assumed that when r is a spin orbital on atom A and s is one on atom B, a suitable form for the hopping term is... 

At the united atom limit (J2 = 0), the potential surface Woo has a single well, but at distances near the equilibrium bond length at R 2) double minima become prominent [3]. At large R these evolve into a pair of isolated wells in the separated atom limit. The critical point at which the symmetry breaking transition from single to double wells occurs is determined from the conditions = 0 and = 0, both evaluated at z = 0. At that point Rg = (27/16) / = 1.299038 Pc = (27/32)1/2 = 0.918559 Wg = -32/27 = -1.185185. Figure 1 shows the variation with R of the coordinates ram, and bm that correspond to the minima of Woo-... 

In the united-atom hmit, s = 0, this gives = 1/2 while in the separated-atom limit, s 00 we have —> 1. Beginning with a series of s-values, we can now immediately generate the corresponding values of bg andN, as shown in Table 1. Interestingly, the interelectron repulsion matrix element, T n, which seems at first sight to depend independently on the two parameters kg and R, can be shown to depend only on their product, s = kgR, (Appendix 1). The approximate functional dependence of this matrix element on s is given [19] by... 

The problems associated with the 4f shell may be circumvented by including the 4f orbitals in the PP core and using the experimental atomic excitation energy of Eu to fix the potential curves in the separated atom limit. With this procedure a molecular excitation energy of 0.46 eV is obtained when spin-orbit corrections are included. 

Figure 29. Electron energy spectra from He-He collisions at collision energies of (a) 200 eV and (b) 500 eV. Electrons are due to autoionization of quasimolecular states, correlating to two singly excited He atoms in the separated atom limit. The separated atom states are indicated in part (b).

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