GerritJan Linker

Total Electron Density Probability density: In the Born interpretation, the probability density is ψ(r1,r2,r3,...,rN)ψ(r1,r2,r3,...,rN). The probability density is the probability of finding electron 1 in a volume element dr around r1, electron2 in a volume element dr around r2, ... Assumptions: ψ is a physically permissible, normalised wavefunction. Integration over all space yields the number of electrons N=∫dr1dr2dr3...drN ψ(r1,r2,r3,...,rN)ψ(r1,r2,r3,...,rN) The chance of locating particle one in a volume element dr around r1, irrespective the location of the other variables is: dx1∫dr2dr3...drN ψ(r1,r2,r3,...,rN)ψ(r1,r2,r3,...,rN) The chance to locate any particle from a collection of N independent particles is: ρ(x1)dx1=Ndx1∫dr2dr3...drN ψ(r1,r2,r3,...,rN)ψ(r1,r2,r3,...,rN) Integration over all spin variables (collectively s) gives the charge density in a pont in space: P=∫ρ(x)ds The density matrices are a representation, in the basis set of the electron density at a point in space. The total density can be subdivided into the probability of finding a spinup (alpha) electron plus the probability of finding a spindown (beta) electron.
