The Valence-Electron Approximation

TTie optimum values of and C2 are found by the variation method this leads to the usual secular equation. We have ionic-covalent resonance, involving the structures H—F and H F . Tlie true molecular structure is intermediate between the covalent and ijonic structures. A term CsIIshI h corresponding to the ionic structure H F could also Ije included in the wave function, but this should contribute only slightly for HE For molecules that are less ionic, both ionic structures might well be included. 

Cesium has the lowest ionization potential, 3.9 eV. Chlorine has the highest electron affinity, 3.6 eV. (The electron affinity of fluorine is 3.45 eV.) Thus, even for CsCl and CsF, the separated ground-state neutral atoms are more stable than the separated ground-state ions. There are, however, cases of excited states of diatomic molecules that dissociate to ions. 

The simplest approach is to regard the core electrons as point charges coinciding with the nucleus. For Cs2 this would give a Hamiltonian for the two valence electrons that is identical with the electronic Hamiltonian for H2. If we then go ahead and mini- 

To overcome the deficiencies of the Hartree-Fock wave function (for example, improper behavior R oo and incorrect values), one can introduce configuration interaction (Cl), thus going beyond the Hartree-Fock approximation. Recall (Section 11.3) that in a molecular Cl calculation one begins with a set of basis functions Xi, does an SCF calculation to find SCf occupied and virtual (unoccupied) MOs, uses these MOs to form configuration (state) functions writes the molecular wave function i/ as a linear combination 2/ of the configuration functions, and uses the variation method to find the ft, s. In calculations on diatomic molecules, the basis functions can be Slater-type AOs, some centered on one atom, the remainder on the second atom. 

The number of MOs produced equals the number of basis functions used. The type of MOs produced depends on the type of basis functions used. For example, if we include only s AOs in the basis set, we get only a MOs, and no ir, S. MOs. 

Suppose we want to treat Cs2, which has 110 electrons. In the MO method, we wonld start by writing down a 110 X 110 Slater determinant of molecnlar orbitals. We would then approximate the MOs by functions containing variational parameters and go on 

Beeause eleetrons are mueh lighter than nuclei, we can use the Bom-Oppenheimer approximation to deal with moleeules. This approximation takes the molecular wave function i/f as the produet of wave functions for electronic motion and for nuclear motion = iAei( i a)iAiv( a), where and are the electronic and nuclear coordinates, respec- 

Quantum chemists use atomic units, in which energies are measured in hartrees and lengths in bohrs [Eqs. (13.29) and (13.30)]. 

The eleetronie Schrddinger equation for Hj can be solved exactly to give wave functions that are eigenfunctions of the operator for the component of electronic orbital angular momentum along the intemuclear axis. The letters r, rr, 5, / . denote A = m values of 0,1,2, 3. respeetively, where mh is the eigenvalue. 

Approximate wave funetions for the two lowest electronic states of H2 are N lSa + lij) andA (li a - li, ), whereIx andlx are l AOscenteredonnucleiaand, respectively. 

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Both groups of methods use the valence-electron approximation, i.e. all core electrons are ignored. It is assumed that core electrons are sufficiently invariant to differing chemical environments so that changes in their orbitals as a function of environment are of no chemical consequence, energetic or otherwise. The valence atomic orbitals are represented by a so-called Slater-type orbital (STO). The mathematical form of a normalized STO (in atom-centered polar coordinates) is... 

As the valence-electron approximation is supposed to be introduced, the term Zm/Rm in (6.16) means the electron interaction with the core of atom M in the point-charge approximation. As AO Xmm( ) can be considered as an eigenfunction of opera tor (—4+ m) and introducing the point-charge approximation for Zm/Rn N / M), we obtain the matrix elements of operator (6.15) in the form [222]... 

Whereas the tight-binding approximation works well for certain types of solid, for other s. items it is often more useful to consider the valence electrons as free particles whose motion is modulated by the presence of the lattice. Our starting point here is the Schrodinger equation for a free particle in a one-dimensional, infinitely large box ... 

The MP2 and CCSD(T) values in Tables 11.2 and 11.3 are for correlation of the valence electrons only, i.e. the frozen core approximation. In order to asses the effect of core-electron correlation, the basis set needs to be augmented with tight polarization functions. The corresponding MP2 results are shown in Table 11.4, where the A values refer to the change relative to the valence only MP2 with the same basis set. Essentially identical changes are found at the CCSD(T) level. 

Saunders states. Assuming that the valence electrons at the top of the band have the average hybrid character 3d34s4p2, the interaction energy of one of these valence electrons and an atomic electron, assumed to be approximately a 3d electron, is found to be —2707 cm-, or —0.334 ev, with probable error about 10%. 

The basic idea of the pseudopotential theory is to replace the strong electron-ion potential by a much weaker potential - a pseudopotential that can describe the salient features of the valence electrons which determine most physical properties of molecules to a much greater extent than the core electrons do. Within the pseudopotential approximation, the core electrons are totally ignored and only the behaviour of the valence electrons outside the core region is considered as important and is described as accurately as possible [54]. Thus the core electrons and the strong ionic potential are replaced by a much weaker pseudopotential which acts on the associated valence pseudo wave functions rather than the real valence wave functions (p ). As... 

The same reasoning may hold for the virtual absence of transition metal dicarbides with (C2) anions. On the other hand, stable multinary compounds in A-M-(BNx), AE-M-(BNx), or Ln-M-(BNx) systems can exist with an electron-rich transition metal (M), similar to the known ternary dicarbides AM(C2) with A=alkali and M = Pd, Pt [25]. In these cases the valence electrons provided from A or AE metals fill the bonding (BNx)" or (C2) levels, and the transition metal retains an approximate d ° configuration. 

In electrocatalysis, the reactants are in contact with the electrode, and electronic interactions are strong. Therefore, the one-electron approximation is no longer justified at least two spin states on a valence orbital must be considered. Further, the form of the bond Hamiltonian (2.12) is not satisfactory, since it simply switches between two electronic states. This approach becomes impractical with two spin states in one orbital also, it has an ad hoc nature, which is not satisfactory. 

For iron compounds, (yzz)iat and iat are amplified by approximately (1 —yc ) 10 as compared to the point charge contributions 14z and rj obtained from (4.42a) and (4.42b). Nevertheless, the lattice contribution is usually least significant for most iron compounds because it is superseded by a strong EFG from the valence electrons details will be found in Chap. 5. 

Fig. 7.7 Schematic diagrams for common electron configurations of Ni " complexes in the one-electron approximation. The resulting valence electron contributions V z are obtained from Table 4.2...

A theoretical interpretation relating the valence electron concentration and the structure was put forward by H. Jones. If we start from copper and add more and more zinc, the valence electron concentration increases. The added electrons have to occupy higher energy levels, i.e. the energy of the Fermi limit is raised and comes closer to the limits of the first Brillouin zone. This is approached at about VEC = 1.36. Higher values of the VEC require the occupation of antibonding states now the body-centered cubic lattice becomes more favorable as it allows a higher VEC within the first Brillouin zone, up to approximately VEC = 1.48. 

The structures of the prototype borides, carbides, and nitrides yield high values for the valence electron densities of these compounds. This accounts for their high elastic stiffnesses, and hardnesses. As a first approximation, they may be considered to be metals with extra valence electrons (from the metalloids) that increase their average valence electron densities. The evidence for this is that their bulk modili fall on the same correlation line (B versus VED) as the simple metals. This correlation line is given in Gilman (2003). 

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