Atomic limit

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... 

Solovyev IV, Dederichs PH, Anisimov VI (1994) Corrected atomic limit in the local-density approximation and the electronic structure of d impurities in Rb. Phys Rev B 50(23)46861... 

The two most popular basis sets consist of either Slater-type orbitals8 (STO s) or Gaussian functions. When using STO s one or more are placed on each nucleus - the more the better. The so-called minimal basis set consists of only those STO s which correspond to the occupied a.o. s in the seperated atom limit. Instead of using Slater s rules to determine orbital exponents they may be varied in order to minimize the energy. Once this optimization has been done for a small molecule the values so established can be used in bigger problems. The basis can be improved by adding additional STO s for various nuclei, e.g. with different orbital exponents. If every minimal basis a.o. is represented by two such STO s a "double Q" set is obtained. The only restriction on the number and type of STO that can be added, seems to be computer time. 

PCMODEL Version 8 became available in 2002. New features in version 8 include support for different and improved force fields along with the MMX, MM3, MMFF94, Amber, and Oplsaa force fields currently supported. The atom limit has been increased to 2500 atoms, and support for reading and writing PDB and SDF files has been added. Transition-metal complexes can be built with explicit sigma bonding, lone-pair coordination, and pi-system coordination. Parameters are available for all transition metals. 

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... 

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

Finally, chain transfer is undesirable except when it is used intentionally to limit molecular weight by adding good chain transfer agents such as carbon tetrachloride. Here transfer of a chlorine atom limits the size of one chain and at the same time initiates formation of a new chain by the trichloromethyl radical. Instead of (3), (3), (3), etc., we get (3), (3), (7), (8), (3), (3), (7), (8), etc., with a lower average chain length. 

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. 

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... 

The f-f overlapping in light actinides may cause broadening of the 5 f wave functions into 5 f bands. On the other hand, from Am on, this overlapping having decreased, this effect occurs much less. It follows that physical properties which depend from 5f orbitals may be better understood, in one case, in the band Umit, in the other case, in the atomic limit. 

In many actinide solids, as we shall see, the experimentally determined magnetic properties are explained well by assuming the permanent magnetic moment due to Hund s rules. The f-electrons are considered atomic, and their interaction with the environment is through crystal field forces or weak exchange forces with conduction electrons. Here, the magnetic properties are explained in the atomic limit. 

In the atomic limit, the intra-atomic Coulomb correlation is given by ... 

Obviously, the clusters NaAry (c), NaAry (c), NaAr4 (c), NaAry (d), and NaAre (c), which correspond to the simple van der Waals long distance addition of a excited sodium atom to an argon cluster, are characterized by 3p states and 3s states only weakly shifted with respect to the isolated atom limit. Their emission lines are therefore very close to the atomic line. 

Fig. 5.6 Schematic illustration of the behaviour of the energy bands as the strength of the Kronig-Penney barrier varies between zero and infinity. The so-called free-atom limit in fact corresponds to a free or isolated square well.

R(min) — Rar(max) — 0.8. If it is taken into account that only CH2 and CH groups can form alicyclic rings and that these rings contain not more than four and not less than two C atoms, limits for Cun eye 1 (alicyclic C atoms per unit) can be calculated. Results are exinite R[Pg.342]

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.

SU+). Intergroup energies are reasonably well predicted by the calculation for the first two groups (for which spectroscopic data are available). Te values are obtained by introducing correlation schemes, but the procedure is not very satisfactory for the third group since the necessary wave-functions become ionic in character at the dissociated atom limit. However, the calculations on the third group do predict ... 

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