An arbitrary order Douglas-Kroll method with polynomial cost

A new Douglas-Kroll transformation scheme up to arbitrary order is presented to study the convergence behavior of the Douglas-Kroll series and the influence of different choices of parametrization for the unitary transformation. The standard approach for evaluating the Douglas-Kroll Hamiltonian suffers from computational difficulties due to the huge number of matrix multiplications, which increase exponentially with respect to the order of truncation. This makes it prohibitively expensive to obtain results for very high order Douglas-Kroll Hamiltonians. The highest order previously presented is 14th order, but it is not enough to obtain accurate results for systems containing heavy elements, where the Douglas-Kroll series converges very slowly. In contrast, our approach dramatically reduces the number of matrix multiplications, which only increase with a polynomial scaling. With the new method, orders greater than 100 and machine accuracy are possible. This fast method is achieved by employing a special transformation to all Douglas-Kroll operators and our algorithm is very simple. We demonstrate the performance of our implementation with calculations on one-electron systems and many-electron atoms. All results show very good convergence behavior of the Douglas-Kroll series. Very small differences are found between the different parametrizations, and therefore the exponential form, which is the simplest and fastest, is recommended.