Molecular orbitals defined

The Exclusion Principle is fundamentally important in the theory of electronic structure it leads to the picture of electrons occupying distinct molecular orbitals. Molecular orbitals have well-defined energies and their shapes determine the bonding pattern of molecules. Without the Exclusion Principle, all electrons could occupy the same orbital. 

A convenient orbital method for describing eleetron motion in moleeules is the method of molecular orbitals. Molecular orbitals are defined and calculated in the same way as atomic orbitals and they display similar wave-like properties. The main difference between molecular and atomic orbitals is that molecular orbitals are not confined to a single atom. The crests and troughs in an atomic orbital are confined to a region close to the atomic nucleus (typieally within 1-2 A). The electrons in a molecule, on the other hand, do not stick to a single atom, and are free to move all around the molecule. Consequendy, the crests and troughs in a molecular orbital are usually spread over several atoms. 

In the MOVB method, we use one Slater determinant with block-localized molecular orbitals to define individual VB configuration, called diabatic state. For example, the reactant state of the Sn2 reaction between HS- and CH3CI is defined as the Lewis bond structure of the substrate CH3CI ... 

In order to construct localized orbitals for molecules, it is necessary to define a measure for the degree of localization of an arbitrary set of molecular orbitals. The localized orbitals are then defined as that set of orthogonal molecular orbitals obtained by a transformation of the type given in Eq. (5), for which the measure of localization has the maximum value. It is clear that the resulting localized orbitals will depend, at least to some degree, upon the choice of the localization measure. In the present work the localized molecular orbitals are defined as those self-consistent-field orbitals which maximize the localization sum 14)... 

This expression is just the one which obtains for the Hartree product wave-function. The difference between this Hartree wavefunction and the Fock wavefunction of Eq. (1) is the absence of the antisymmetrizer j4 in that equation. This means that in the Hartree wavefunction each electron can be identified with a specific molecular orbital, whereas in the Fock wavefunction all electrons make use of all orbitals. The Hartree wavefunction is of course not a proper quantum mechanical wavefunction, since it is not antisymmetric in the electrons. Moreover, for the Fock wavefunction, it is in general not possible to reduce the interorbital exchange energy to zero. But the localized molecular orbitals, as defined here, represent that set of molecular orbitals for which the energy expression comes closest to the Hartree form, i.e. they come closest to being identifiable with electrons which are not exchanged among different orbitals. 

The inactive and active orbitals are occupied in the wave function, while the external (also called secondary or virtual) orbitals span the rest of the orbital space, defined from the basis set used to build the molecular orbitals. The inactive orbitals are kept doubly occupied in all configurations that are used to build the CASSCF wave function. The number of electrons occupying these orbitals is thus twice the number of inactive orbitals. The remaining electrons (called active electrons) occupy the active orbitals. 

Two examples of polyatomic calculations, on H20 and NH3, are outlined and explained in detail. In both cases the analysis starts from an assumed molecular structure of known symmetry. The transformation properties of the atomic orbitals on each atomic centre, under the symmetry operations of the group, are examined next. The atomic orbitals are defined as Is, 2s, 2pxi 2py and 2pz. Nothing can be more explicit - these are the occupied atomic orbitals of a many-electron atom. This configuration violates the exclusion principle9. Although the quantum numbers may not be needed,... 

A general form of linear combination of spherical Gaussians has been used for describing each molecular orbital. The multi-Gaussian function Ith molecular orbital is defined as... 

This is a local vertex invariant derived both from the H-JiUed molecular graph and the —> Graph of Atomic Orbitals and defined to distinguish the different atom types in the framework of the OCWLI approach [Toropov and Toropova, 2002a Toropov, Nesterov ef al., 2003a]. From the H-filled molecular graph, it is calculated as... 

In this chapter we shall describe how we can obtain wave functions that are of the FCI type but in a limited orbital space defined as the active orbitals of the system. But before developing such a model we shall describe how the molecular orbital concept can be extended to any type of wave functions. 

Here, Fj and F2 denote sets of orbitals that define molecular fragments or orbital spaces (i.e., core, a, n, lone-pair). The decision on the type of orbital and the atom to which it belongs can (automatically) be made on the basis of a MuUiken population analysis [56] of the LMO in question. Asymptotically, when fragments Fj and F2 are spatially well separated, the interfragment correlation energy is equal to the interaction energy, which can be further interpreted as their dispersion interaction... 

On the other hand, the linear combination of atomic orbitals - molecular orbital (LCAO-MO) theory, is actually the same as Hartree-Fock theory. The basic idea of this theory is that a molecular orbital is made of a linear combination of atom-centered basis functions describing the atomic orbitals. The Hartree-Fock procedure simply determines the linear expansion coefficients of the linear combination. The variables in the Hartree-Fock equations are recursively defined, that is, they depend on themselves, so the equations are solved by an iterative procedure. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method. 

Accordingly, the molecular orbitals defined by the eigenvalue equation H = e

molecular orbitals also as adiabatic states. The atomic orbitals are obtained from HQ = where Hq stands for the Hamiltonians Ha = T+ and... 

The second valence-only model starts from a molecular orbital viewpoint and was derived in the mid 1970s by Hay, Thibeault and Hoffmann (HTH) [2], approximately at the same time as the Kahn-Briat model. The magnetic orbitals are defined as linear combinations of orthogonal atomic-like orbitals... 

In the two-orbital mixing problem, we showed that when molecular orbitals are defined as linear combinations of atomic orbitals and are put into the Schrbdinger equation, followed by differentiation to minimize E, a series of simultaneous equations in the c s and E results. When mixing two orbitals, only two energies result along with two molecular orbitals. It is not much of a stretch to realize that there will be as many molecular orbitals with distinct energies as the number of atomic orbitals (or basis functions) we use to create the molecular orbitals. Moreover, there will be the same number of secular equations as the number of starting atomic orbitals or basis functions ( ), and hence the secular determinant will be n by n. 

Using the Born-Oppenheimer approximation, we describe the electronic wave-functions of the diatomic molecule by first assuming the nuclei to be separated at some constant distance R on the potential surface. Molecular orbital (MO) theory is the most widely used method for writing those electronic wavefunc-tions. Each molecular orbital wavefunction defines the distribution of a single electron over the entire molecule, just as an atomic orbital is a one-electron spatial wavefunction for an atom. 

---