By expanding the spatial orbitals into atomic orbitals and manipulating them properly, we have [Pg.441]

B3.1.5.2 THE LINEAR COMBINATIONS OF ATOMIC ORBITALS TO FORM MOLECULAR ORBITALS EXPANSION OF THE SPIN ORBITALS [Pg.2169]

In many crystals there is sufficient overlap of atomic orbitals of adjacent atoms so that each group of a given quantum state can be treated as a crystal orbital or band. Such crystals will be electrically conducting if they have a partly filled band but if the bands are all either full or empty, the conductivity will be small. Metal oxides constitute an example of this type of crystal if exactly stoichiometric, all bands are either full or empty, and there is little electrical conductivity. If, however, some excess metal is present in an oxide, it will furnish electrons to an empty band formed of the 3s or 3p orbitals of the oxygen ions, thus giving electrical conductivity. An example is ZnO, which ordinarily has excess zinc in it. [Pg.717]

SCF approximation. The indices //, v, A, and o denote four atomic orbital centers, so that the number of such orbitals that needs to be calculated increases proportionally scales with ) N, where N is the number of AOs, This was an intractable task in 1965, so Pople, Santry, and Segal introduced the approximation that only integrals in which = v and J. = o (i.e., li)) would be considered and that, further- [Pg.382]

H at m energy of 1.2 eV in the center-of-mass frame. By using an atomic orbital basis and a representation of the electronic state of the system in terms of a Thouless determinant and the protons as classical particles, the leading term of the electronic state of the reactants is [Pg.231]

With 4) containing a normalization factor and all permutations over the atomic orbital wave functions i (1 = 1,2,... 2n). Likewise, if all electron pairs were exchanged in a cyclic manner, the product wave function, 4>b, has the fonn [Pg.391]

THE CRUDE BORN-OPPENHEIMER ADIABATIC APPROXIMATION 545 Next, we expand them into the atomic orbitals [Pg.439]

Molecules. The electronic configurations of molecules can be built up by direct addition of atomic orbitals (LCAO method) or by considering molecular orbitals which occupy all of the space around the atoms of the molecule (molecular orbital method). [Pg.152]

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the Is atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is [Pg.437]

LCAO method A method of calculation of molecular orbitals based upon the concept that the molecular orbital can be expressed as a linear combination of the atomic orbitals. [Pg.236]

There are two mechanisms by which a phase change on the ground-state surface can take place. One, the orbital overlap mechanism, was extensively discussed by both MO [55] and VB [47] formulations, and involves the creation of a negative overlap between two adjacent atomic orbitals during the reaction (or an odd number of negative overlaps). This case was temied a phase dislocation by other workers [43,45,46]. A reaction in which this happens is [Pg.344]

This, the well-known Hellmann-Feynman theorem [128,129], can then be used for the calculation of the first derivatives. In nonnal situations, however, the use of an incomplete atom-centered (e.g., atomic orbital) basis set means that further terms, known as Pulay forces, must also be considered [130]. [Pg.268]

Molecular dipole moments are often used as descriptors in QPSR models. They are calculated reliably by most quantum mechanical techniques, not least because they are part of the parameterization data for semi-empirical MO techniques. Higher multipole moments are especially easily available from semi-empirical calculations using the natural atomic orbital-point charge (NAO-PC) technique [40], but can also be calculated rehably using ab-initio or DFT methods. They have been used for some QSPR models. [Pg.392]

The intensity of shading at any point represents the magnitude of 1, i.e. the probability of finding the electron at that point. This may also be called a spherical charge-cloud . In helium, with two electrons, the picture is the same, but the two electrons must have opposite spins. These two electrons in helium are in a definite energy level and occupy an orbital in this case an atomic orbital. [Pg.54]

Population anaiysis methods of assigning charges rely on the LCAO approximation and express the numbers of electrons assigned to an atom as the sum of the populations of the AOs centered at its nucleus. The simplest of these methods is the Coulson analysis usually used in semi-empirical MO theory. This analysis assumes that the orbitals are orthogonal, which leads to the very simple expression for the electronic population of atom i that is given by Eq. (53), where N

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. [Pg.376]

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