Molecular orbital theory orthonormal

The Slater determinant is the central entity in molecular orbital theory. The exact -electron wave function of a stationary molecule in the Born-Oppenheimer approximation is a 4-dimensional object that depends on the three spatial coordinates and a spin coordinate of the N electrons in the system. This object is of course too complicated for any practical application and is, in first approximation, replaced by a product of N orthonormal 4-dimensional functions that each depend on the coordinates of only one of the electrons in the system. 

According to the ab initio molecular orbital theory methodology, atomic orbitals (set of functions, also called basis sets) combine in a way to form molecnlar orbitals that snrronnd the molecule. The molecular orbital theory considers the molecnlar wave function as an antisymmetiized product of orthonormal spatial molecular orbitals. Then they are constructed as a Slater determinant [56], Essentially, the calculations initially use a basis set, atomic wave functions [57, 58], to constract the molecular orbitals. The first and basic ab initio molecular orbital theory approach to solve the Schrodinger equation is the Hartree-Fock (HF) method [59, 60], Almost all the ab initio methodologies have the same basic numerical approach but they differ in mathematical approximations. As it is clear that finding the exact solution for the Schrodinger equation, for a molecular system, is not possible, various approaches and approximations are used to find the reliable to close-to-accurate solutions [61-68]. 

In the early years of quantum theory, Hiickel developed a remarkably simple form of MO theory that retains great influence on the concepts of organic chemistry to this day. The Hiickel molecular orbital (HMO) picture for a planar conjugated pi network is based on the assumption of a minimal basis of orthonormal p-type AOs pr and an effective pi-Hamiltonian h(ctT) with matrix elements... 

An X-ray atomic orbital (XAO) [77] method has also been adopted to refine electronic states directly. The method is applicable mainly to analyse the electron-density distribution in ionic solids of transition or rare earth metals, given that it is based on an atomic orbital assumption, neglecting molecular orbitals. The expansion coefficients of each atomic orbital are calculated with a perturbation theory and the coefficients of each orbital are refined to fit the observed structure factors keeping the orthonormal relationships among them. This model is somewhat similar to the valence orbital model (VOM), earlier introduced by Figgis et al. [78] to study transition metal complexes, within the Ligand field theory approach. The VOM could be applied in such complexes, within the assumption that the metal and the... 

In 1958, Nesbet extended Brueckner s theory for infinite nuclear mat-ter to nonuniform systems of atoms and molecules. By consideration of the CISD problem in which the electronic Hamiltonian is diagonalized within the basis of the reference and all singly and doubly excited determinants, Nesbet explained that Brueckner theory allows one to construct a set of orthonormal molecular orbitals for which the correlated wavefunction coefficients for all singly excited determinants vanish. Unfortunately, the construction of the set of orbitals that fulfill this Brueckner condition can be determined only a posteriori from the single excitation coefficients computed in a given orbital basis. As a result, the practical implementation of Brueckner-orbital-based methods has... 

Valence Bond theory using localized orthogonal orbitals For small couplings, i.e. nearly degenerate 5 = 0 and 5=1 states. Valence Bond (VB) theory provides a more intuitive starting point than the previous molecular orbital reasoning. For the Ms = 0 wave functions we make use of the local orthonormal orbitals and tjrb as defined in Eq. 3.10a and use them to construct two neutral determinants and... 

Contrary to the unperturbed Hartree-Fock theory, where the molecular orbitals are expanded in atomic one-electron basis functions Eq. (9.4), one normally expands the perturbed occupied spin orbitals in the set of orthonormalized unperturbed molecular spin orbitals V g ... 

In this subsection, we will briefly discuss how one may construct a basis

carrier space which is adapted not only to the treatment of the ground state of the Hamiltonian H but also to the study of the lowest excited states. In molecular and solid-state theory, it is often natural and convenient to start out from a set of n linearly independent wave functions = < > which are built up from atomic functions (spin orbitals, geminals, etc.) involved and which are hence usually of a nonorthogonal nature due to the overlap of the atomic elements. From this set O, one may then construct an orthonormal set tp = d>A by means of successive, symmetric, or canonical orthonormalization.27 For instance, using the symmetric procedure, one obtains... 

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