Atomic structure orbital approximation

A complicating factor is tlrat each spitr density matrix element is multiplied by the corresponding basis function overlap at tire nuclear positions. The orbitals having maximal amplitude at the nuclear positions are tire core s orbitals, which are usually described with less flexibility than valence orbitals itr typical electronic structure calculations. Moreover, actual atomic s orbitals are characterized by a cusp at tire nucleus, a feature accurately modeled by STOs, but only approximated by the more commonly used GTOs. As a result, tlrere are basis sets in the literature tlrat systematically improve tire description of the core orbitals in order to improve prediction of h.f.s., e.g. IGLO-III (Eriksson et al. 1994) and EPR-III (Barone 1995). 

The starting point of the creation of the theory of the many-electron atom was the idea of Niels Bohr [1] to consider each electron of an atom as orbiting in a stationary state in the field, created by the charge of the nucleus and the rest of the electrons of an atom. This idea is several years older than quantum mechanics itself. It allows one to construct an approximate wave function of the whole atom with the help of one-electron wave functions. They may be found by accounting for the approximate states of the passive electrons, in other words, the states of all electrons must be consistent. This is the essence of the self-consistent field approximation (Hartree-Fock method), widely used in the theory of many-body systems, particularly of many-electron atoms and ions. There are many methods of accounting more or less accurately for this consistency, usually named by correlation effects, and of obtaining more accurate theoretical data on atomic structure. 

In the frozen atomic structure approximation, where the same orbitals are used in the initial and final states, this overlap matrix element yields unity. Hence, one obtains for the remaining matrix element... 

This result shows that the original matrix element containing the orbitals of all electrons factorizes into a two-electron Coulomb matrix element for the active electrons and an overlap matrix element for the passive electrons. Within the frozen atomic structure approximation, the overlap factors yield unity because the same orbitals are used for the passive electrons in the initial and final states. Considering now the Coulomb matrix element, one uses the fact that the Coulomb operator does not act on the spin. Therefore, the ms value in the wavefunction of the Auger electron is fixed, and one treats the matrix element Mn as... 

Since the same orbitals are used for the ground state and the hole state, no relaxation effects due to the change in the shielded nuclear charges are taken into account, and this model is called the frozen atomic structure approximation.) Due to the interpretation of els as the binding energy of one ls-electron, the differential equation for the Pls(r) orbital, equ. (7.66b), can be interpreted as a one-particle Schrodinger equation for the Is orbital (see equs. (1.3) and (1.4)) ... 

The Is and 2s orbitals which are affected by neither the photoionization nor the Auger process are omitted for simplicity.) If these wavefunctions are constructed from single-electron orbitals of a common basis set (the frozen atomic structure approximation), the photon operator as a one-particle operator allows a change of only one orbital. Hence, the photon operator induces the change 2p to r in these matrix elements ... 

The Bond Orbital Approximation is nevertheless essential to an understanding of the electronic structure of tetrahedral solids. The transformation from atomic states to bond orbitals has inserted a gap between occupied and empty energy levels, and approximate bands can be obtained in the Bond Orbital Approximation. There is no qualitative change in these bands if we reintroduce the matrix elements that the Bond Orbital Approximation neglects. In this sense, the Bond Orbital Approximation is conceptually correct and hence it is legitimate to discuss all properties of tetrahedral solids in terms of it. Quantitatively the approximation is not always adequate. Sometimes this drawback can be alleviated by suitable choice of parameters sometimes it is necessary to compute corrections. The remainder of this section will deal with the choice of parameters and corrections for Bond Orbital Approximations, though these arc quite inessential to a basic understanding of the concept. 

There are two guiding principles that direct the generation of the solutions to the Schrbdinger equation. First, in order to avoid the inconvenient wavefiuictions mentioned above, the method takes advantage of a technique analogous to the so-called orbital approximation in atomic structure. In this approximation, a many-coordinate function is expressed as a linear combination of products of one-electron wavefunctions. 

The periodic structure of the elements and, in fact, the stability of matter as we know it are consequences of the Pauli exclusion principle. In the words of A. C. Phillips Introduction to Quantum Mechanics, Wiley, 2003), A world without the Pauli exclusion principle would be very different. One thing is for certain it would be a world with no chemists. According to the orbital approximation, which was introduced in the last Chapter, an W-electron atom contains N occupied spinoibitals, which can be designated a, In accordance with the exclusion principle,... 

The self-consistent field (SCF) orbital approximation method developed by Hartree is especially well suited for applications in chemistry. Hartree s method generates a set of approximate one-electron orbitals, associated energy levels, s , reminiscent of those for the H atom. The subscript a represents the appropriate set of quantum numbers (see later in this chapter for a definition). The electronic structure of an atom with atomic number Z is then built up by placing Z electrons into these orbitals in accordance with certain rules (see later in this chapter for descriptions of these rules). 

This chapter begins with a description of the quantum picture of the chemical bond for the simplest possible molecule, Hj, which contains only one electron. The Schrodinger equation for Hj can be solved exactly, and we use its solutions to illustrate the general features of molecular orbitals (MOs), the one-electron wave functions that describe the electronic structure of molecules. Recall that we used the atomic orbitals (AOs) of the hydrogen atom to suggest approximate AOs for complex atoms. Similarly, we let the MOs for Hj guide us to develop approximations for the MOs of more complex molecules. 

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