Valence-state wave function

Intramolecular cohesion evidently is a holistic quantum-mechanical property, which cannot be reduced to pairwise interactions. Molecules are formed during the interaction of atoms and/or radicals in their valence states, to be understood in terms of valence-state wave functions of appropriate symmetry. 

If the valence-state wave functions are written in their simplest, hard-sphere form,... 

The only remaining problem is calculation of the electron-density function, which cannot be done classically. However, for molecules in condensed phases the influence of the environment introduces another simplification. It has been shown that valence-state wave functions of compressed atoms are simpler, than hydrogenic free-atom functions. Core levels are largely unaffected and a nodeless valence-state wave function, which allows chemical distortion of electron density, can be defined. We return to this topic at a later stage. 

Bond length The Heitler-London method allows the calculation of all first-order diatomic interactions using valence-state wave functions as defined in terms of characteristic ionization radii. 

The valence electron of a promoted atom readily interacts with other activated species in its vicinity to form chemical bonds. The mechanism is the same for all atoms, since the valence state always consists of a monopositive core, loosely associated with a valence electron, free to form new liaisons. Should the resulting bond be of the electron-pair covalent type, its properties, such as bond length and dissociation energy can be calculated directly by standard Heitler-London procedures, using valence-state wave functions (section 5.3.4). 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

In the PP theory, the valence electron wave function is composed of two parts. The main part is the pseudo-wave function describing a relatively smooth-varying behavior of the electron. The second part describes a spatially rapid oscillation of the valence electron near the atomic core. This atomic-electron-like behavior is due to the fact that, passing the vicinity of an atom, the valence electron recalls its native outermost atomic orbitals under a relatively stronger atomic potential near the core. Quantum mechanically the situation corresponds to the fact that the valence electronic state should be orthogonal to the inner-core electronic states. The second part describes this CO. The CO terms explicitly contain the information of atomic position and atomic core orbitals. 

With regard to the former, one would like to include as many important configurations as possible. Unfortunately, the definition of an important configuration is often debatable. One popular remedy is the full-valence complete active space SCF (CASSCF) approach in which configurations arising from all excitations from valence-occupied to valence-virtual orbitals are chosen. [29] Since this is equivalent to performing a full Cl within the valence space, the full-valence CASSCF method is limited to small systems. Nevertheless, the CASSCF approach using a well-chosen (often chemically motivated) subspace of the valence orbitals has been shown to yield a much improved depiction of the wave function at all points on a potential surface. Furthermore, the choice of an active space can be adjusted to describe excited state wave functions. 

Analysis of the valence-band spectrum of NiO helped to understand the electronic structure of transition-metal compounds. It is to be noted that th.e crystal-field theory cannot explain the features over the entire valence-band region of NiO. It therefore becomes necessary to explicitly take into account the ligand(02p)-metal (Ni3d) hybridization and the intra-atomic Coulomb interaction, 11, in order to satisfactorily explain the spectral features. This has been done by approximating bulk NiO by a cluster (NiOg) ". The ground-state wave function Tg of this cluster is given by,... 

The highest-order spherical harmonics used to expand the valence level wave functions were 1 = 3 in the extramolecular region, 1 = 2 in the Re and P spheres, and 1 = 0 in the H spheres. All ground-state SCF calculations converged to better than 0.0004 Ry for each variance level. Core levels were relaxed in the ReHg2-... 

The ground-state wave function of cytosine has been calculated by practically all the semiempirical as well as nonempirical methods. Here, we shall discuss the application of these methods to interpret the experimental quantities that can. be calculated from the molecular orbitals of cytosines and are related to the distribution of electron densities in the molecules. The simplest v-HMO method yielded a great mass of useful information concerning the structure and the properties of biological molecules including cytosines. The reader is referred to the book1 Quantum Biochemistry for the application of this method to interpret the physicochemical properties of biomolecules. Here we will restrict our attention to the results of the v-SCF MO and the all-valence or all-electron treatments of cytosines. 

The BOVB method is aimed at combining the qualities of interpretability and compactness of valence bond wave functions with a quantitative accuracy of the energetics. The fundamental feature of the method is the freedom of the orbitals to be different for each VB structure during the optimization process. In this manner, the orbitals respond to the instantaneous field of the individual VB structure rather than to an average field of all the structures. As such, the BOVB method accounts for the differential dynamic correlation that is associated with elementary processes like bond forming/breaking, while leaving the wave function compact. The use of strictly localized orbitals ensures a maximum correspondence between the wave function and the concept of Lewis structure, and makes the method suitable for calculation of diabatic states. 

The most interesting feature of Fig.3 is the sharp peak in the weight of the metallic structures around 1.7 A. Each metallic structure has two polarized H2 molecules as a result of the transfer from a molecular valence bond to a new bond to a neighbor molecule (Fig.2). Our results therefore support a hypothesis, previously made [20], that the ground state wave function will have a component of charge-transfer states at pressures around 150 GPa. Moreover, our results indicate that small variations of the intermolecular separation around 1.7 A (as a result of a structural modification, for instance) can induce sizeable changes in the polarization of the H2 molecules. This is fully consistent with the spontaneous polarization predicted by Edwards and Ashcroft [21], and it provides an explanation for this phenomenon in terms of the chemical bonding in the solid. 

An important qualitative description of the spectral behavior of class II compounds was presented by Robin and Day. This simple model has found apphcabihty to the discussion of the spectra of numerous mixed valence compounds in which some delocalization occurs. In this model, it is assumed that the ground-state wave function contains the function, a, which describes mixing of the wave function for site A with the wave function of site B. 

Goodgame, M. M., and W. A. Goddard III (1985). Modified generalized valence-bond method a simple correction for the electron correlation missing in generalized valence-bond wave functions prediction of double-well states for Crj and Moj. Phys. Rev. Lett. 54, 661-64. 

In the preceding sections we have outlined the requirements a cluster has to fulfill in order to dissociatively chemisorb H in summary, the cluster first has to contain at least one atom with a d occupation including at least one open d-orbital. Second, there has to be at least one open shell valence (s-character) orbital in the cluster wave-function. If there is only one open shell orbital, a dihydride or possibly a molecularly chemisorbed state will be formed. If there are at least two open shell orbitals, atomically chemisorbed hydrogen atoms of the type found on surfaces will be formed. The formation of the latter state is normally more exothermic. Finally, if these requirements are not fulfilled by the ground state wave-function of the cluster, excitation to a low lying state which satisfies the requirements and which has an excitation energy less than the exothermic ty 20 kcal/mol) will lead to... 

To make the connection with MO theory, consider the SOPP approximation of valence bond theory and write the ground state wave function as in Eq. (4). For the MO approximation and are the same and this leads to... 

Write simple valence bond wave functions for the diatomic molecules Li2 and C2. State the bond order predicted by the simple VB model and compare with the LCAO predictions in Table 6.3... 

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