The Roothaan equations are a representation of the Hartree-Fock equation in a non orthonormalized basis set which can be of Gaussian type. In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative procedure to calculate the single determinant solution to the time-independent Schrödinger equation of a quantum chemical many-electron system in a Coulombic potential of fixed nuclei. ... In modern computational chemistry, quantum chemical calculations are typically performed within a finite set of basis functions. ...
The method was developed independently by Clemens Roothaan and George G. Hall in the early 1950s, and are thus sometimes called the Roothaan-Hall equations. The Roothaan equations can be written in the form of generalized eigenvalue problem In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
Where F is the so-called Fock matrix, C is a matrix of coefficients, S is the overlap matrix of the basis functions, and ε 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. In quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors. ...
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