Structural Chemistry, Vol. 15, No. 5, October 2004 ( C  2004) Expectation Values in Spin-Averaged Douglas–Kroll and Infinite-Order Relativistic Methods 1 Dariusz Ke ¸dziera, 2 Maria Barysz, 2 and Andrzej J. Sadlej 2,3 Received January 3, 2004; revised January 28, 2004; accepted January 29, 2004 The change of picture for the r −1 operator which occurs on passing from the four component relativistic schemes to two component theories is investigated for the spin-averaged Douglas–Kroll approximation and the recently proposed infinite-order approach. For nuclei already with moderately large values of the nuclear charge the change of picture contribution is found to be relatively important. Its neglect significantly affects the calculated values of the total relativistic contribution to the expectation value of r −1 . A numerical method for the calculation of the total relativistic contribution to the expectation values of r −1 , which avoids the explicit use of the appropriately transformed r −1 operator, is devised and tested. Also the differences between the Douglas–Kroll approximation and the infinite-order scheme are investigated. KEY WORDS: Relativistic effects; expectation values; two-component methods; spin-averaged two-component methods; change of picture; finite perturbation schemes. INTRODUCTION The computational effort involved in relativistic cal- culations for many-electron systems [1–3] can be sig- nificantly reduced by passing from the four-component (Dirac) [1] to the two-component formalism [4]. In gen- eral this reduction assumes that the 4 × 4 Dirac hamil- tonian H D can be either exactly or approximately trans- formed into a block-diagonal form: H D → U † H D U =  h 11 h 12 h 21 h 22  , (1) where U is a 4 × 4 unitary transformation matrix (oper- ator) and h ij denotes 2 × 2 blocks of the transformed hamiltonian. The exact block-diagonalization of H D means that the off-diagonal blocks h 12 and h 21 become 1 This paper is dedicated to Professor Henryk Chojnacki on the occasion of his 70th birthday to honor his accomplishments in the development and applications of quantum chemistry, and his contributions to the promotion of theoretical chemistry in Poland. 2 Department of Quantum Chemistry, Institute of Chemistry, Nicolaus Copernicus University, PL 87 100 Torun, Poland. 3 To whom all correspondence should be addressed. e-mail: teoajs@ chem.uni.torun.pl 2 × 2 zero matrices. Then the upper diagonal block, h 11 = h + , (2) is the exact two-component hamiltonian for the elec- tronic (positive) [1,4] spectrum of H D . In most cases, however, the transformation (1) is carried out only ap- proximately; the nonzero off-diagonal blocks are simply neglected and the given h 11 is used to build approximate two-component theory. Further simplification of the two- component hamiltonians can be achieved by removing all spin-dependent terms. This is usually referred to as the spin-averaging of h 11 and brings about the correspond- ing spin-independent (scalar) one-component hamiltoni- ans (h) [5,6]. In recent years the relativistic methods based on the two- and one-component (spin-averaged, scalar) hamiltonians received a great deal of attention and were found to be highly efficient and accurate tools of the rela- tivistic quantum chemistry [4,6–9]. The traditional way of the derivation of different two-component hamiltonians for relativistic quantum chemistry follows from the transformation proposed by Foldy and Wouthuysen (FW) [1,10]. which is determined step-by-step as a power series expansion in the fine structure constant α; in atomic units α = 1/c, where c is the velocity of light (c = 137.0359895 au of 369 1040-0400/04/1000-0369/0 C  2004 Plenum Publishing Corporation