LCAO technique

Simple organic radicals such as the vinyl radical mentioned above were routinely treated using semi-empirical or rudimentary ab initio HF-LCAO techniques. 

The detailed first principles study of the three stable polymorphs has been performed recently using the LCAO technique The main drawback of that work is that no cell optimization was performed for anatase or brookite. The energy-volume curves that were used to calculate the bulk modulus, B, and its pressure derivative, B, have been produced by varying the volume with the c/a ratio and fractional atomic coor nates being fixed at experimental values which makes results unreliable. 

Since the first theoretical works of the sixties (1 ) on LCAO techniques in polymer quantum chemistry, the field has known a rapid development and standard SCF calculations on regular polymers are now routinely performed. In those methods, the translational symmetry is fully exploited (and consequently assumed) in order to reduce to manageable dimensions the formidable task of computing electronic states of an extended system. 

In calculations based on the MO-LCAO technique [32-34], the one-electron Kohn-Sham equations Eq. (11) are solved by expanding the molecular orbital wavefunctions V i(r) in a set of symmetry adapted functions Xj(r), which are expanded as a linear combination of atomic orbitals i.e. 

Since the first pioneering theoretical works on LCAO techniques in polymer quantum chemistry were written by Ladik and Andre in the 1960s [4-9] this field has rapidly... 

The virtual orbitals generated by the solution of the LCAO approximation to the Hartree-Fock equations are indeed an artifact of the LCAO technique and do not have any physical interpretation except as a residue of those features of the basis functions which are not suitable for the description of the single-determinant model of the electronic structure of the molecule. 

It is, however, natural to ask if these orbitals have any meaning do they have any physical interpretation for example, are they approximations to any orbitals associated with the system under investigation or are they just an artifact of the use of the LCAO technique In the case of the orbitals of the single determinant — the occupied orbitals — the LCAO expansion is a more or less good approximation to the solutions of the differential Hartree-Fock equation depending on the quality and length of the AO expansion. As the quality and size of the basis is improved, the occupied MOs presumably become better and better approximations to these orbitals. What about the unoccupied molecular orbitals ... 

This is certainly a standard valence bond (VB) way to view the electronic structure in N2, with the possible exception that we have taken a linear combination of lone pair orbitals. The three bonds between the nitrogen atoms would be Identified with the double occupation of lcr+ and the two components of ItTu from Figure 6.3. As we have described, the lone pair orbitals are best identified with lcr+ and 2cr+. The perhaps surprising result from our LCAO technique in Figure 6.3 is that the in-phase combination lies at higher energy than the out-of-phase combination. Furthermore, the latter is at an energy well below 7t in the valence bond picture. Which model more accurately resembles reality Photoelectron spectroscopy provides a direct experimental link to the orbital sequence. 

Since the first theoretical works of the sixties (1) on LCAO techniques in polymer quantum chemistry, the field has known a rapid development. 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

Table 1. shows the total energies obtained using the RHF method for 1. LCAO minimal basis set STO-IG for the sake of comparison with FSGO, 2. FSGO in its symmetric and broken symmetry solutions and, 3. LCAO minimal basis set STO-3G in order to allow a safer comparison with the quality of the subminimal basis used in the FSGO technique. The dissociation curves are given in Figure 1. 

In order to describe the hydrogen molecule by quantum mechanical methods, it is necessary to make use of the principles given in Chapter 2. It was shown that a wave function provided the starting point for application of the methods that permitted the calculation of values for the dynamical variables. It is with a wave function that we must again begin our treatment of the H2 molecule by the molecular orbital method. But what wave function do we need The answer is that we need a wave function for the H2 molecule, and that wave function is constructed from the atomic wave functions. The technique used to construct molecular wave functions is known as the linear combination of atomic orbitals (abbreviated as LCAO-MO). The linear combination of atomic orbitals can be written mathematically as... 

While in principle all of the methods discussed here are Hartree-Fock, that name is commonly reserved for specific techniques that are based on quantum-chemical approaches and involve a finite cluster of atoms. Typically one uses a standard technique such as GAUSSIAN-82 (Binkley et al., 1982). In its simplest form GAUSSIAN-82 utilizes single Slater determinants. A basis set of LCAO-MOs is used, which for computational purposes is expanded in Gaussian orbitals about each atom. Exchange and Coulomb integrals are treated exactly. In practice the quality of the atomic basis sets may be varied, in some cases even including d-type orbitals. Core states are included explicitly in these calculations. 

Direct calculation of the ionization potential by LCAO-Xa, HAM/3 and Green s function techniques or via the Koopmans theorem by ab initio techniques. 

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