Quantum corrections to the Thomas-Fermi approximation - the Kirzhnits method

The method of Kirzhnits for the calculation of quantum corrections to the Thomas–Fermi approximation is reviewed. An equation is derived whose iteration gives quantum corrections to the density matrix directly in terms of gradients of the potential. This is then used to calculate the correct form of the quantum correction to power 4 in the gradient operator. The validity of the quantum correction or gradient expansion method is examined by comparing the results of linear response theory using truncated gradient expansions with the exact Lindhardt result.