Nonorthogonal atomic orbitals

The determinantal wave function in Eq. (21) is built [23] from complex dynamical spin orbitals Even when the basis orbitals ut in Eq. (22) are orthogonal these dynamical orbitals are nonorthogonal, and for a basis of nonorthogonal atomic orbitals based on Gaussians as those in Eq. (24) the metric of the basis becomes involved in all formulas and the END theory as implemented in the ENDyne code works directly in the atomic basis without invoking transformations to system orbitals. 

We first briefly consider the photoexcitation of ethylene as a prototype of a re —> re isomerization reaction. In the following discussion, we will focus on analogies and differences with ethylene. For a minimal basis set description of the re-system of ethylene, only the two nonorthogonal atomic orbitals (AO) pCa and pc, localized on the two carbon atoms, need to be taken into account. All possible MO configurations (three singlets and three triplets) can be constructed as Slater determinants of the two AOs ... 

The advantages of VB theory are formal as well as conceptualJ In a formal sense, we start with a set of VB Configuration Wavefunctions (CW s), constructed from a set of nonorthogonal Atomic Orbitals (AO s), which ultimately yield the proper electronic states with no constraints placed on the way in which the CW s are combined to form these various electronic states. In a conceptural sense, we can define elementary bonds, "lone" electrons, and vacant orbitals (or holes) so that their interaction can be conveniently studied. 

This can be easily proven for the general case of an array of equal atoms and s type nonorthogonal atomic orbitals with only nearest neighbor interactions. The expressions for the molecular orbitals are ... 

We may re-define the active orbitals utilizing the invariance of the active orbital space. In the CASVB with nonorthogonal LMOs, we employ Rueden-berg s procedure of projected localized MOs [6-8] and obtain quasi-atomic CASSCF MOs that have maximal overlaps with atomic orbitals (AOs) of the free atoms. Consider an AO, Xa, centered on a nucleus A. Diagonalizing the matrix,... 

The nonorthogonal LMOs were determined in the same manner as in the previous subsection. The atomic orbitals used for the determination are two Is orbitals of the hydrogen atoms and 2p(o) orbitals of the halogen atom. All the overlaps between the atomic orbital (AO) and the nonorthogonal LMO are greater than 0.9 (0.9004 at minimum). The molecular orbitals are therefore well localized. 

The original VB method has been difficult to use in practice because of the nonorthogonality of the atomic orbital basis, but there has been a revival of interest in it recently. Gallup and co-workers116-118 have described a new technique for carrying out such calculations, and the results of applications to the first-row hydrides. Calculations were carried out using a minimal basis set constructed from gaussian lobe orbitals. The orbitals were scaled to their best atom value and also optimally scaled in the molecule. Atomic populations were also computed.118... 

The integrals are calculated in terms of the atomic orbitals (AOs) and are subsequently transformed to the orthonormal basis. In some cases it may be more efficient to evaluate the expressions in the nonorthogonal AO basis. We return to this problem when we consider the calculation of the individual geometry derivatives. For the time being we assume that the Hamiltonian is expressed in the orthonormal molecular orbital (MO) basis. The second-quantized Hamiltonian [Eq. (8)] is a projection of the full Hamiltonian onto the space spanned by the molecular orbitals p, i.e., the space in which calculations are carried out. 

In the spin-coupled description of a molecule such as SF6, the sulfur atom contributes six equivalent, nonorthogonal sp -like hybrids which delocalize onto the fluorine atoms. Each of these two-centre orbitals overlaps with a distorted F(2p) function and the perfect-pairing spin function dominates. Of course, using only 3s, 3px, 3p and 3pz atomic orbitals, we can at most form four linearly independent hybrid orbitals localized on sulfur, with a maximum occupancy of 8 electrons, as in the octet rule. However, the six sulfur+fluorine hybrids which emerge in the spin-coupled description are not linearly dependent, precisely because each of them contains a significant amount of F(2p) character. It is thus clear that the polar nature of the bonding is crucial. 

At this point, the Hiickel approximations are often imposed to simplify Eq. 5.14. These were introduced by the German physicist Erich Armand Arthur Joseph Hiickel (1896-1980). Even though atomic orbitals on neighboring atoms are nonorthogonal (they have nonzero overlap), it is possible to make the approximation that ... 

This does not preclude the use of atomic orbitals in a variational calculation of the electron states, but it docs change the form of the resulting equations. Let us look briefly then at the effects of that nonorthogonality (Harrison and Ciraci, 1974 see also Tejeda and Shevchik, 1976). 

Notice again that atomic orbitals on the same atom are orthogonal. The corrections of Eq. (B-11) do introduce intra-atomic nonorthogonalities, but only to second order in S.) We may also obtain the energy of the corrected orbitals. 

The nonorthogonal basis functions Xx( )are referred to as atomic orbitals (AOs) and are often taken to be Cartesian Gaussian-type orbitals (GTOs) of the (unnormalized) form ... 

The problem of negative populations is attributable to working with a nonorthogonal basis. A symmetric transformation of all the atomic orbitals to an orthogonal basis restricts the values of the atomic populations to between zero and two. While many transformations are possible, the most commonly used is the symmetrical transformation of Lowdin, leading to the Lowdin... 

The expressions for interatomic currents in many-electron formulation, unfortunately, are not as simple as Eq. (5), because of the exchange and overlap effects. (To define an atom, one needs to deal with the nonorthogonality of atomic orbitals of its neighbours, which complicates the formalism [6].)... 

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