Orbital wavefunctions many-electron molecular

The development of localized-orbital aspects of molecular orbital theory can be regarded as a successful attempt to deal with the two kinds of comparisons from a unified theoretical standpoint. It is based on a characteristic flexibility of the molecular orbital wavefunction as regards the choice of the molecular orbitals themselves the same many-electron Slater determinant can be expressed in terms of various sets of molecular orbitals. In the classical spectroscopic approach one particular set, the canonical set, is used. On the other hand, for the same wavefunction an alternative set can be found which is especially suited for comparing corresponding states of structurally related molecules. This is the set of localized molecular orbitals. Thus, it is possible to cast one many-electron molecular-orbital wavefunction into several forms, which are adapted for use in different comparisons fora comparison of the ground state of a molecule with its excited states the canonical representation is most effective for a comparison of a particular state of a molecule with corresponding states in related molecules, the localized representation is most effective. In this way the molecular orbital theory provides a unified approach to both types of problems. 

One approach to the treatment of electron correlation is referred to as density functional theory. Density functional models have at their heart the electron density, p(r), as opposed to the many-electron wavefimction, F(ri, r2,...). There are both distinct similarities and distinct differences between traditional wavefunction-based approaches (see following two sections) and electron-density-based methodologies. First, the essential building blocks of a many-electron wavefunction are single-electron (molecular) orbitals, which are directly analogous to the orbitals used in density functional methodologies. Second, both the electron density and the many-electron wavefunction are constructed from an SCF approach which requires nearly identical matrix elements. 

A is a parameter that can be varied to give the correct amount of ionic character. Another way to view the valence bond picture is that the incorporation of ionic character corrects the overemphasis that the valence bond treatment places on electron correlation. The molecular orbital wavefunction underestimates electron correlation and requires methods such as configuration interaction to correct for it. Although the presence of ionic structures in species such as H2 appears counterintuitive to many chemists, such species are widely used to explain certain other phenomena such as the ortho/para or meta directing properties of substituted benzene compounds under electrophilic attack. Moverover, it has been shown that the ionic structures correspond to the deformation of the atomic orbitals when they are involved in chemical bonds. 

The natural orbital concept, as originally formulated by Per-Olov Lowdin, refers to a mathematical algorithm by which bestpossible orbitals (optimal in a certain maximum-density sense) are determined from the system wavefunction itself, with no auxiliary as sumptions or input. Such orbitals inherently provide the most compact and efficient numerical description of the many-electron molecular wavefunction, but they harbor a type of residual multicenter indeterminacy (akin to that of Hartree-Fock molecular orbitals) that somewhat detracts from their chemical usefulness. 

The term e-, corresponds to the orbital energy of the electron ascribed by /<1). The molecular orbital wavefunctions, /J, are eigenfunctions of the Hartree-Fock Hamiltonian operator,, and can be chosen to be orthogonal causing many integrals in the expression to vanish. 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

There are m doubly occupied molecular orbitals, and the number of electrons is 2m because we have allocated an a and a spin electron to each. In the original Hartree model, the many-electron wavefunction was written as a straightforward product of one-electron orbitals i/p, i/ and so on... 

Hartree-FockWavefunction. The simplest quantum-mechanically correct representation of the many-electron wavefimction. Electrons are treated as independent particles and are assigned in pairs to functions termed Molecular Orbitals. Also known as Single-Determinant Wavefunction. 

Section treats the spatial, angular momentum, and spin symmetries of the many-electron wavefunctions that are formed as antisymmetrized products of atomic or molecular orbitals. Proper coupling of angular momenta (orbital and spin) is covered here, and atomic and molecular term symbols are treated. The need to include Configuration Interaction to achieve qualitatively correct descriptions of certain species electronic structures is treated here. The role of the resultant Configuration Correlation Diagrams in the Woodward-Hoffmann theory of chemical reactivity is also developed. 

Under the Hartree-Fock (i.e., HF) approximation, the function of in variables for the solutions of the electronic Hamiltonian is reduced to n functions, which are referenced as molecular orbitals (MOs), each dependent on only three variables. Each MO describes the probability distribution of a single electron moving in the average field of all other electrons. Because of the requirements of the Pauli principle or antisymmetry with respect to the interchange of any two electrons, and indistinguishability of electrons, the HF theory is to approximate the many-electron wavefunction by an antisymmetrized product of one-electron wavefunctions and to determine these wavefunctions by a variational condition applied to the expected value of the Hamiltonian in the resulting one-electron equations,... 

The pz orbitals have not been used so far. They are antisymmetric perpendicular to the (x,y)-plane and therefore orthogonal to the above combined sp and sp2 orbitals. pz orbitals from neighboring atoms combine independently to form molecular orbitals with their wavefunction delocalized over many atoms. The corresponding electrons are the 7T-electrons, which dictate to a large extend the linear and nonlinear optical properties. 

From molecular orbital theory, a many-electron wavefunction, may be defined by a determinant of molecular wavefunctions, i/ ,. The i[i, may in turn be expressed as a linear combination of one-electron functions, that is, (fj, = where cM, are the molecular orbital expansion coefficients... 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

One-electron picture of molecular electronic structure provides electronic wavefunction, electronic levels, and ionization potentials. The one-electron model gives a concept of chemical bonding and stimulates experimental tests and predictions. In this picture, orbital energies are equal to ionization potentials and electron affinities. The most systematic approach to calculate these quantities is based on the Hartree-Fock molecular orbital theory that includes many of necessary criteria but very often fails in qualitative and quantitative descriptions of experimental observations. 

Another possibility to obtain direct information from such a collision system is the observation of Molecular orbital (MO) X-rays resulting from electronic de-excita-tions between the molecular levels during the collision under emission of noncharacteristic photons. The result of our many-particle calculation is given in Fig. 10 where the spectrum of the collision system 20 MeV CP on Ar is compared with the experiment. In this calculation the radiation field was coupled to the system by first order perturbation theory but the wavefunctions were taken from the solution ot the time-dependent relativistic DV-Xa calculations . 

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