The operator to approximate is with the maximal and minimal eigenvectors , . This is often referred to as propagation in imaginary time as it results from the above propagation operator by setting
If the operator is applied to a mixed state and the result renormalised, the state with the lowest energy in the mixed state is produced and all other states are filtered out at the rate , where and are the lowest eigenvalues. If a higher state is sought, the ground state has to be first produced and in the second run of the method projected out before every renormalization step. There are operators an even number, which produce the eigenstate closest to the energy directly without the need of producing the lower states first but I shall not go into any details about these .
The factor for the relaxation operator is not a phase factor anymore, and hence
This time, the coefficients go to zero exponentially as becomes greater than so the method is very fast except for very short times where this exponentially decaying tail becomes dominant in the calculation time.
The same result is produced by