Several Atomic Cores

Now the only choice for f fy and f B which will exactly reduce this equation to the form of equation (22) in deriving Pett are those orbitals which would be found by an all-electron calculation on the molecule, i.e. 

Clearly any attempt to base FeK on such molecularly defined cores defeats the aims of pseudopotential theory. However, the approximate invariance of atomic cores to molecule formation implies that, of the total of Na electrons which could be associated with the centre A in an atomic calculation, nx are core electrons and n K will contribute to the molecular valence set. Thus we can define a one-centred Fock operator  

The Fock operator for the valence orbitals may therefore be approximated by a simple generalization of equation (22), 

This Fock operator has been derived starting from the assumption of a Hartree-Fock valence function PV [equation (14)]. However, it can be seen that the coupling of the valence electrons has little influence on the core electrons, so that the many-electron valence hamiltonian may be similarly approximated as 

The first and last terms in equation (34) consist of the valence-electron components of the all-electron hamiltonian of equation (1), and the remaining terms constitute the pseudopotential represented symbolically in equation (2). Further, if we assume that the interaction of the two cores A and B can be approximated by a point charge potential (see Kahn et al.2i for errors in this assumption), 

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A full quantum mechanical calculation of the simplest dimer ion is a formidible task (—e.g., [F—H... H—F] with 21 electrons). We are forced to examine the matter using various approximations. Using a molecular orbital approach we can view the problem as one of five electrons associated with the several atomic cores. For [F—H... H—F], the amplest MO could be built from 2 P orbitals on F and Is orbitals on H, as in Table I. By symmetry, the orbital coefficients for the two F atomic orbitals will be identical in magnitude as will those of the two H atomic orbitals. The orbital coefficients, cit, can thus be chosen to be positive definite if the sign convention in Table I is used. The MO s should be... 

Molecules with Several Atomic Cores.—From the above discussion it is seen that, in principle, the effective hamiltonian for atomic valence electrons is dependent on the valence state of the atom, this dependence arising from the valence contribution to the all-electron Fock operator F. In practice this dependence is very weak unless the atom is multiply ionized, and can usually be safely neglected, so that a single effective hamiltonian can suffice for many valence states. However, for a molecular system in which there is more than one core region additional approximations must be introduced to maintain a simple form of the effective hamiltonian. For two atomic cores defined in terms of orbital sets and and a valence set < F) the equation equivalent to (21) is... 

To cope with several atomic cores in the same molecule. 

Molecular ion An ion with several atomic cores surrounded by an outer layer of electrons Molecular ions differ from molecules because they have either too many or too few electrons to cancel the nuclear charges... 

The use of RECP s is often the method of choice for computations on heavy atoms. There are several reasons for this The core potential replaces a large number of electrons, thus making the calculation run faster. It is the least computation-intensive way to include relativistic effects in ah initio calculations. Furthermore, there are few semiempirical or molecular mechanics methods that are reliable for heavy atoms. Core potentials were discussed further in Chapter 10. 

A complete description of droplet generator and of several atomization methods appears in a previous paper [8]. Simple air-stripping or piezoelectric drop generators were employed. The core liquid typically consisted of a polyanion solution, while the receiving bath contained a polycation(s) solution and, in many instances, a divalent cation. 

In theoretical terms, the total electron density in a molecule is easily expressed in terms of the occupied molecular orbitals. Additional information is gained from the m.o. approach especially regarding the electronic energy for ground and excited states and the detailed features (e.g. phase) of individual m.o.s. Molecular orbitals are mathematical functions that can be constructed as linear combinations of orbitals of the contributing atoms, in a process where the atoms lose their individuality, except for the respective nuclei and, perhaps, the core electrons. The valence electrons are described by functions which, in general, extend to several atoms or even to the whole molecule. 

Using pseudopotentials has several major beneficial consequences (i) Only the valence electrons must be treated explicitly, thus the number of equations to be solved [Eqs. (13)] can be reduced drastically (ii) the pseudoorbitals are very smooth near the atomic core, and thus Tout can be reduced drastically and (iii) important relativistic effects of the core electrons of heavy elements such as the 5d elements can be included in nonrelativistic calculations. The major downsides are that the potential v(r) in Eq. (3) must be replaced with a more complicated and computationally expensive nonlocal pseudopotential and, more importantly, that the transferability of the pseudopotential, i.e., its accuracy in different bonding environments, may not be perfect. Developing highly transferable pseudopotentials that can be used at as low an cut as possible is a major current topic of research. 

In molecules, of course, we may well be concerned with several spatially separated atomic cores which can be simulated on the separate atoms by the above methods. The way to proceed is obvious but is the obvious method valid ... 

To include several electrons outside an atomic core. 

The rippertene diterpenes are of current interest due to the condensed tetracyclic 16-carbon atom core and the presence of several quaternary carbons. The following synthesis starts with isopulegol and passes through a silylated diazomethane-induced regioselective ring expansion (Scheme 50) [87]. 

Several attempts have been made to alleviate the problems of the atom cores dominating the MQSM by emphasizing the role of the chemically more interesting outer electron density. Because no physical ground exists to... 

An analysis of all crystal structures available in the Cambridge Structural Database for hydrates of /8-lactams reveals that the water molecules in the crystal structures are not in sufficient proximity (within the sum of van der Waals distance) to the reactive center on the /8-lactam core to bring about reaction. For cases where water molecules are in close proximity to the reactive centre, further rationalization can be made in terms of the relative orientation of reactants and the steric hindrance of the initial approach of the water, or the low degree of conformational flexibility in the crystal. This type of calculation on cefadroxil monohydrate (reaction scheme shown in Figure 3.6) which contains a water molecule within van der Waals distance of the reactive /8-lactam carbonyl group, has shown that several atoms would lie outside of the molecular surface formed by the rest of the crystal, indicating steric clashes. 

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