United-atom limits

The fact that the separated-atom and united-atom limits involve several crossings in the OCD can be used to explain barriers in the potential energy curves of such diatomic molecules which occur at short intemuclear distances. It should be noted that the Silicon... 

In the united atom limit—that is, when all bond lengths are equal to zero— the ideal gas free energy functional is of course just given by... 

V — —This is clearly unphysical except in the united atom limit. However, generally it does not affect the result of interest to the chemists. 

In the united-atom limit, R = 0, the positive root is k = 2, which corresponds to the exact ground-state energy ofthe He ion (in Hartrees) ... 

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... 

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... 

The other approach, proposed slightly later by Hund[9] and further developed by Mulliken[10] is usually called the molecular orbital (MO) method. Basically, it views a molecule, particularly a diatomic molecule, in terms of its united atom limit . That is, H2 is a He atom (not a real one with neutrons in the nucleus) in which the two positive charges are moved from coinciding to the correct distance for the molecule. HF could be viewed as a Ne atom with one proton moved from the nucleus out to the molecular distance, etc. As in the VB case, further adjustments and corrections may be applied to improve accuracy. Although the imited atom limit is not often mentioned in work today, its heritage exists in that MOs are universally... 

Regularities in analogous transitions in molecules having the same united atom limit... 

In Tables 5 and 6 we compare the spectral intensities, in the form of oscillator strengths, for 3s-3p and 3p-3d transitions (in the united atom notation) between states of the same and different molecular symmetries within each corresponding molecular point group, in the Rydberg radicals CH5, H3O, H2F, and NeH, all ofwhich have sodium as the united atom limit. 

There are two reasons why so much is unknown. First, at high densities three (and even four) body forces are important. This is particularly so when chemically reactive atoms are present. Then, even for two-body forces, the strongly repulsive regime is not well understood and, in addition, close in, as one approaches the united atom limit, there is considerable promotion of molecular orbitals. This is a universal mechanism for electronic excitation which means a breakdown of the Born-Oppenheimer approximation for close collisions. 

The D-ND for R 00 can be recognized from theoretical considerations. However, because of the strong dependence of the Cl mixing on R, it is not possible to predict quantitatively its importance at equilibrium or further in, where, normally, the united atom limit also plays an important role. For the sake of completeness, it is useful to quantify the above. I write the two-term superposition wavefunction as (I omit the dependence of the orbitals on the parameter). 

Table II. Rydberg states of H30 at C3y geometry excitation energies (second column) and the corresponding quantum defects (third column). Data for the united atom limit (sodium) are also included (the last column).

Here we can make an important point. At the united atom limit Ouls becomes 2pa (a dumbbell). Since then the principal quantum number increases to 2, in a sense this would become a Rydberg orbital. Mulliken [7] called this Rydbergiza-tion and we see that Rydberg or non-Rydberg depends on the distance, that is geometry. At actual molecular distances, Ouls can then be reasonably considered as intermediate Rydberg/non-Rydberg. 

We consider the QDO method to be a useful tool for estimating transition probablities between molecular Rydberg states. A drawback of our method is its empirical character in that it requires the quantum defects for the states involved in transitions as an input. However, the well established correspondence between the energy levels of a Rydberg molecule and those of its united atom limit (9), as well as the regular behavior of the quantum defects along a Rydberg series are very helpful in their estimations. 

Note that in designating the molecular orbitals representing these two electronic states as above, the Is and 2p symbols refer to the united-atom limit. 

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