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The overlap matrix is a square matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system, such as an atomic orbital basis set used in molecular electronic structure calculations. In particular, if the vectors are orthogonal to one another, the
The off-diagonal Hamiltonian matrix elements are given by an approximation due to Wolfsberg and Helmholz that relates them to the diagonal elements and the overlap matrix element. [2] = + K is the Wolfsberg–Helmholz constant, and is usually given a value of 1.75. In the extended Hückel method, only valence electrons are considered; the core ...
The method predicts how many energy levels exist for a given molecule, which levels are degenerate and it expresses the molecular orbital energies in terms of two parameters, called α, the energy of an electron in a 2p orbital, and β, the interaction energy between two 2p orbitals (the extent to which an electron is stabilized by allowing it ...
The interatomic overlap matrix elements , should be rather small or neglectable. If they are large it is again an indication that the tight binding model is of limited value for some purposes. Large overlap is an indication for too short interatomic distance for example.
where is the so-called Fock matrix (which is a function of ), is a matrix of coefficients, is the overlap matrix of the basis functions, and is the (diagonal, by convention) matrix of orbital energies. Unlike restricted Hartree–Fock theory for closed shell molecules, the form of the Fock matrix is not unique.
The other CSFs can be characterised by the number of spin orbitals that are swapped with virtual orbitals from the Hartree–Fock determinant. If only one spin orbital differs, we describe this as a single excitation determinant. If two spin orbitals differ it is a double excitation determinant and so on.
The localized orbital corresponding to one O-H bond is the sum of these two delocalized orbitals, and the localized orbital for the other O-H bond is their difference; as per Valence bond theory. For multiple bonds and lone pairs, different localization procedures give different orbitals.
To understand how to get the number of functions, consider the cc-pVDZ basis set for H: There are two s (L = 0) orbitals and one p (L = 1) orbital that has 3 components along the z-axis (m L = −1,0,1) corresponding to p x, p y and p z. Thus, there are five spatial orbitals in total. Note that each orbital can hold two electrons of opposite spin.