Molecular orbital hierarchy

An important characteristic of ab initio computational methodology is the ability to approach the exact description - that is, the focal point [11] - of the molecular electronic structure in a systematic manner. In the standard approach, approximate wavefunctions are constructed as linear combinations of antisymmetrized products (determinants) of one-electron functions, the molecular orbitals (MOs). The quality of the description then depends on the basis of atomic orbitals (AOs) in terms of which the MOs are expanded (the one-electron space), and on how linear combinations of determinants of these MOs are formed (the n-electron space). Within the one- and n-electron spaces, hierarchies exist of increasing flexibility and accuracy. To understand the requirements for accurate calculations of thermochemical data, we shall in this section consider the one- and n-electron hierarchies in some detail [12]. 

Quantum chemical methods may be divided into two classes wave function-based techniques and functionals of the density and its derivatives. In the former, a simple Hamiltonian describes the interactions while a hierarchy of wave functions of increasing complexity is used to improve the calculation. With this approach it is in principle possible to come arbitrarily close to the correct solution, but at the expense of interpretability of the wave function the molecular orbital concept loses meaning for correlated wave functions. In DFT on the other hand, the complexity is built into the energy expression, rather than in the wave function which can still be written similar to a simple single-determinant Hartree-Fock wave function. We can thus still interpret our results in terms of a simple molecular orbital picture when using a cluster model of the metal substrate, i.e., the surface represented by a suitable number of metal atoms. 

As in the abc hierarchy, each class composing the chemical behavior hierarchy contains only those methods and attributes that pertain to it. Concepts derived at a particular class are used in classes of higher specialization so that more sophisticated concepts can be deduced and discriminating properties elucidated at the proper level. For example, the class POMO contains methods for evaluating the POMO, whether it is the HOMO, the n-HOMO, or any other occupied molecular orbital. These attributes may then be used at more specialized level to expand on the features of an electronic state that gives rise to a particular conceptual behavior. Nucleophilicity illustrates this point, because it may arise from either a negatively charged ion or a neutral atom (i.e., nucleophilicity can... 

An important distinction between the abc and cb hierarchy is that the cb hierarchy of classes is not treated as task specialization by operators calling for the assignment of chemical behavior. Since cb is a set of potential behaviors, b, it is necessary to associate behaviors to a species that lies on the same class level [e.g., pertinent unoccupied molecular orbital (PUMO) and POMO] as well as on different, more specialized, levels of the same class (e.g., Lewis-base and POMO). This is necessary because the species dynamically assumes the requisite behavior—(weak) nucleophile or Lewis base—when it is required by the procedure... 

Different model chemistries, i.e., combination of levels of theory and basis sets are used in quantum chemical methods [Fig. 18.2], Levels of theory denote a hierarchy of procedures corresponding to different mathematical transformations and approximations. A basis set is a mathematical representation of the molecular orbitals within a molecule. 

The PPP model is the most complete model in this hierarchy (of Huckel, Hubbard and PPP), and it begins with the HF equations for the one electron ir-molecular orbitals of a conjugated molecule. Only IT orbitals are considered so all the a electrons are lumped together with the nuclear potential to form a "core" Hamiltonian, h ore carbon nucleus has one p atomic orbital... 

Election nuclear dynamics theory is a direct nonadiababc dynamics approach to molecular processes and uses an electi onic basis of atomic orbitals attached to dynamical centers, whose positions and momenta are dynamical variables. Although computationally intensive, this approach is general and has a systematic hierarchy of approximations when applied in an ab initio fashion. It can also be applied with semiempirical treatment of electronic degrees of freedom [4]. It is important to recognize that the reactants in this approach are not forced to follow a certain reaction path but for a given set of initial conditions the entire system evolves in time in a completely dynamical manner dictated by the inteiparbcle interactions. 

In the more general case S 0 and the molecular angular momenta can be coupled in various ways. It is of primary importance to ascertain to what extent the interaction of the spin momentum S with the orbital momentum L is comparable to the rotation of the molecule, as well as to the interaction of each of the momenta L and S with the internuclear axis. An attempt to establish a hierarchy of interactions yields a number of possible, certainly idealized, coupling cases between angular momenta, first considered by Hund and known as Hund s coupling cases. Here we will discuss the three basic (out of five) cases of coupling of momenta in a linear molecule. 

Historically, the first derivations of approximate relativistic operators of value in molecular science have become known as the Pauli approximation. Still, the best-known operators to capture relativistic corrections originate from those developments which provided well-known operators such as the spin-orbit or the mass-velocity or the Darwin operators. Not all of these operators are variationally stable, and therefore they can only be employed within the framework of perturbation theory. Nowadays, these difficulties have been overcome by, for instance, the Douglas-Kroll-Hess hierarchy of approximate Hamiltonians and the regular approximations to be introduced in a later section, so that operators such as the mass-velocity and Darwin terms are no... 

One establishes the hierarchy of the above CFD methods (see Table 3.5) through assessing the best correlation model between the two forms of energies, namely of the n-parabolic energy combining the electronegativity and chemical hardness with the pi-electrons (in molecular frontier orbitals) the present discussion follows (Putz, 2011c)... 

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