Slater-type orbitals description

The implementation of such a model mostly depends on the choice of the atomic orbitals. Linear combinations of Slater Type Orbitals arc natural and moreover allow a good description of one-center matrix elements even at large intemuclear distances. However, a complete analytical calculation of the two-center integrals cannot be performed due to the ETF, and time consuming numerical integrations [6, 7] are required (demanding typically 90% of the total CPU time). 

Slater type orbital (STO) functions will be presented here. They are well known fimctions since their description made by Slater around sixty years ago [43a] and heavily studied and used as atomic basis sets [43b,43c]. In this section the connection of STO with CETO functions is analyzed. 

The calculation of Mitroy started by calculating the Hartree—Fock approximation to the ground state 3s where we denote the states by the orbitals of the two active electrons in the configuration with the largest coefficient, in addition to the symmetry notation. The calculation used the analytic method with the basis set of Clementi and Roetti (1974) augmented by further Slater-type orbitals in order to give flexibility for the description of unoccupied orbitals. The total energy calculated by this method was —199.614 61, which should be compared with the result of a numerical Hartree—Fock calculation, —199.614 64. 

Consider a quantum mechanical description of the hydrogen molecular ion in its simplest version. Let us use the molecular orbital theory with the atomic basis set composed of only two Slater Type Orbitals (STOs) Isa and Is , centered on the nuclei a and b. The mean value of the Hamiltonian computed with the bonding (+) and antibonding (—) orbital (see p. 439 and Appendix D available at booksite.elsevier.com/978-0-444-59436-5) reads as... 

A basis set is used to express the unknown MOs in terms of a set of known functions. The more basis functions used, the more accurate the description of the MOs. Any type of function can be used, but most efficient are basis functions with a physical behavior (e.g., approach zero for large distances between electrons and nuclei) and which make the integral evaluation easy. Slater-type orbitals (STOs) are related to the exact solutions for the hydrogen atom, but Gaussian-type orbitals (GTOs) are preferable for computational ease ... 

For systems involving more particles (more electrons and more nuclear protons and neutrons), the number of variables and other factors immediately exceeds any ability to be calculated precisely. A solution is found, however, in a method that uses an approximation of the orbital description, known as a Slater-type orbital approximation, rather than a precise mathematical description. A third-level Gaussian treatment of the Slater-type orbitals, or STO-3G... 

Slater-type orbital A description of atomic orbitals that approximates electron-electron repulsion by scaling nuclear charges. 

The electronic configuration of the ground state is... 1a2 4b2 6ai 2bi on the basis of the MO description of OF2 (see p. 10). The total energy Ej = -273.37440 a.u. from an ab initio calculation with configuration interaction corresponds to a difference of 13.43 eV between the OF2 and OF ground states [3]. Ej was also calculated by an ab initio SCF-MO method with a minimal basis of Slater-type orbitals [10]. Electronically excited states for which energies were calculated follow Ai(6a7 ), 2Ai(5a7 ), B2(4b ), B2(3b ), A2(1a ), Bi(1b7 ) the superscript -1 denotes an electron removed from the respective orbital of OF2 [3]. For further discussion, see p. 14. 

The OBS-GMCSC method offers a practical approach to the calculation of multiconfiguration electronic wavefunctions that employ non-orthogonal orbitals. Use of simultaneously-optimized Slater-type basis functions enables high accuracy with limited-size basis sets, and ensures strict compliance with the virial theorem. OBS-GMCSC wavefunctions can yield compact and accurate descriptions of the electronic structures of atoms and molecules, while neatly solving symmetry-breaking problems, as illustrated by a brief review of previous results for the boron anion and the dilithium molecule, and by newly obtained results for BH3. 

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). 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

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