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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
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.
where F is the Fock matrix (which depends on the coefficients C due to electron-electron interactions), C is a matrix of coefficients, S is the overlap matrix of the basis functions, and is the (diagonal, by convention) matrix of orbital energies. In the case of an orthonormalised basis set the overlap matrix, S, reduces to the identity matrix.
Mulliken charges arise from the Mulliken population analysis [1] [2] and provide a means of estimating partial atomic charges from calculations carried out by the methods of computational chemistry, particularly those based on the linear combination of atomic orbitals molecular orbital method, and are routinely used as variables in linear regression (QSAR [3]) procedures. [4]
Unrestricted Hartree–Fock (UHF) theory is the most common molecular orbital method for open shell molecules where the number of electrons of each spin are not equal. While restricted Hartree–Fock theory uses a single molecular orbital twice, one multiplied by the α spin function and the other multiplied by the β spin function in the Slater determinant, unrestricted Hartree–Fock theory ...
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.
In chemistry, primarily organic and computational chemistry, a stereoelectronic effect [1] is an effect on molecular geometry, reactivity, or physical properties due to spatial relationships in the molecules' electronic structure, in particular the interaction between atomic and/or molecular orbitals. [2]
The s orbital is normalized and so the inner product s | s = 1. Also, the s orbital is orthogonal to the p i and p j orbitals, which leads to two terms in the above equaling zero. Finally, the last term is the inner product of two normalized functions that are at an angle of ω ij to each other, which gives cos ω ij by definition.