A method for the iterative algebraic generation of the numerically accurate two-component Hamiltonian for the use in relativistic quantum chemistry is presented. The separation of the electronic and positronic states of the Dirac Hamiltonian is accomplished by the algebraic solution for the Foldy–Wouthuysen transformation. This leads to the two-component formalism whose accuracy is primarily limited by the choice of basis functions. Its performance is tested in calculations of the most sensitive 1s1/2 energy for increasing values of the nuclear charge. These calculations show that the electronic part of the Dirac eigenspectrum can be obtained from the two-component theory to arbitrarily high accuracy. Moreover, if needed, the positronic states can be separately determined in a similar way. Thus the present method can be also used for the evaluation of quantum electrodynamic corrections in the finite basis set approximation.
A series of nonsingular two-component relativistic Hamiltonians is derived from the Dirac Hamiltonian by first performing the free-particle Foldy᎐Wouthuysen transformation and then a block-diagonalizing transformation. The latter is defined in terms of operators which can be determined iteratively through arbitrary order in ␣, leading to transformed Hamiltonians with the two-component block accurate through ␣ 2 k , k s 1, 2, 3, . . . . These Hamiltonians give relativistic energies which differ from Dirac's energies only in terms higher than ␣ 2 k . Their relation to other Ž nonsingular methods of relativistic quantum chemistry the Douglas᎐Kroll method, the . regular Hamiltonian schemes is discussed. By removing the spin-dependent operators, the derived Hamiltonians can be written in spin-free one-component form. The computational effort involved is essentially the same as in the case of the Douglas᎐Kroll scheme and amounts to relatively easy modification of the core Hamiltonian.
Different generalized Douglas-Kroll transformed Hamiltonians (DKn, n=1, 2,...,5) proposed recently by Hess et al. are investigated with respect to their performance in calculations of the spin-orbit splittings. The results are compared with those obtained in the exact infinite-order two-component (IOTC) formalism which is fully equivalent to the four-component Dirac approach. This is a comprehensive investigation of the ability of approximate DKn methods to correctly predict the spin-orbit splittings. On comparing the DKn results with the IOTC (Dirac) data one finds that the calculated spin-orbit splittings are systematically improved with the increasing order of the DK approximation. However, even the highest-order approximate two-component DK5 scheme shows certain deficiencies with respect to the treatment of the spin-orbit coupling terms in very heavy systems. The meaning of the removal of the spin-dependent terms in the so-called spin-free (scalar) relativistic methods for many-electron systems is discussed and a computational investigation of the performance of the spin-free DKn and IOTC methods for many-electron Hamiltonians is carried out. It is argued that the spin-free IOTC rather than the Dirac-Coulomb results give the appropriate reference for other spin-free schemes which are based on approximate two-component Hamiltonians. This is illustrated by calculations of spin-free DKn and IOTC total energies, r(-1) expectation values, ionization potentials, and electron affinities of heavy atomic systems.
Very accurate quantum mechanical calculations of the pure vibrational spectrum of the molecular ion are reported and compared with newly obtained pure vibrational transitions extracted from the available experimental data. The calculations are performed without assuming the Born-Oppenheimer approximation regarding separability of the nuclear and electronic motions and include the first order relativistic mass-velocity and Darwin corrections. For the two lowest transitions, whose experimental energies are established with the highest precision, the calculated and the experimental results show very good agreement.
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