Computational Chemistry 4.24 - Density Matrix

Short lecture on the density matrix in Hartree-Fock theory.

The spatial probability density function of a quantum mechanical particle is its wavefunction multiplied by its complex conjugate. For an electron, this wavefunction is a spatial orbital. For a molecule with N electrons, we add the electron density from all electrons to get the total electronic density function. When using a finite atomic orbital basis set to represent the spin orbitals, we expand the density in terms of the linear combinations which represent each orbital. This expression can be factored to be the double sum of all basis functions multiplied by all complex conjugates multiplied by a matrix element. This matrix is the density matrix, which for a given basis set completely specifies the charge density, and it is invariant to the choice of orbitals. We find density matrix elements by multiplying the Hermitian adjoint of the coefficient matrix by itself. This allows us to express the Fock operator in terms of the density matrix, making it independent of the choice of orbitals.

Notes Slide: https://i.imgur.com/sh0zZPe.png

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