Spin-orbit coupling constants in a multiconfiguration linear response approach Olav Vahtras and Hans Agren Department of Quantum Chemistry, UppsalaUniversity, Box 518, S-751 20 Uppsala, Sweden Poul JBrgensen and Hans Jdrgen Aa. Jensen Department of Chemistry, Aarhus University,DK-8ooO Aarhus C, Denmark Trygve Helgaker Department of Chemistry Universityof Oslo,Box 1033, Blindern, N-0315 Oslo3, Norway Jeppe Olsen Theoretical Chemistv, Chemistry Centre, University of Lund, Box 124, S-221 00 Lund, Sweden (Received2 August 1991; accepted 23 October 199 1) Spin-orbit coupling constants between singlet and triplet states are evaluated as residues of multiconfiguration linear response functions. In this approach,the spin-orbit coupling constantsare automatically determinedbetween orthogonal and noninteracting states.Sample calculations are presented for the X 38; 4 ‘2: transition in 0, and the ‘A i -3B, transition in CH, . The convergence of the coupling constantsis examined as a function of basisset and level of correlation. An exotic behavior is observed in the correlation of the ‘A 1 state for CH, when increasingthe active space, demonstratingan intricate coupling between the dynamic and static correlation. In general, the results indicate that reliable spin-orbit coupling constants between valence states may be obtained with a 4s3p2d lfbasis set for first row atoms and a modest active orbital space. 1. INTRODUCTION The magnetic interactions in a molecule can in many quantum chemical applications be viewed as perturbations of the nonrelativistic Born-OppenheimerHamiltonian. The effects of the magnetic interactions then appearas splittings of and transitions between the nonrelativistic Born-Oppen- heimerelectronic levels.The leading relativistic corrections to the electrostatic interactions in a many-electron system, iirst derived by Breit,ie3 yield in the Pauli approximation4 an electronicspin-orbit interaction operator of the form (in atomic units) I& =$ I& yi I 9,%-~2sj~ , [ 1 (1) iA IA g II where i,j refer to electronsandA to nuclei. r0 is the position of particle i relative to particlej and 1, = rii X pi is the orbital angular momentum of particle i with respectto the position of particle j. The particle asymmetry in the two-electron part, the differing sizeof the spin-own-orbit and spin-other- orbit interaction, can be seen asa consequence of the Thom- as precession.’a is the fine-structureconstant. One classof problems where the spin-orbit interaction becomes interestinginvolvesspin-forbidden transitions, e.g., between singlet and triplet states. The evaluationof a transi- tion amplitude PwscJ3w~s)A (2) where I’Y) is the singlet stateand 13Y (M,)) theM, compo- nent of the triplet state,is not a trivial matter if the two states are constructed from different sets of molecular orbitals, e.g., if the statesare determined from separate multiconfi- guration self-consistent field (MCSCF) calculations on the two states.A common remedy is to usethe same molecular orbitals in configuration statefunction (CSF) expansions of both states,either by using the orbitals optimized for one of the state&* or by applying some averaging procedure.’ An elegant evaluationof the transition amplitudeswhen the two statesare constructedfrom mutually nonorthogonal setsof molecular orbitals has beenproposed by Malmqvist. lo A different approach is to evaluate transition ampli- tudes from multiconfiguration linear response (MCLR) functions.” It has previously been shown how transition amplitudes and second-order propertiesfor large configura- tion spacescan be evaluated for perturbations of singlet character’* and for perturbationsof triplet character-i3 for a singlet referencestate. We here apply this method to the spin-orbit interaction. We first introduce the necessary formalism, for MCLR in general and for the spin-orbit interaction in particular. The evaluationof the spin-orbit integralsis described in Sec. III. Samplecalculations are presented for 0, and CH, in Sec. IV, and the last section contains some concluding re- marks. II. SPIN-ORBIT RESPONSE FUNCTIONS A. Notation and background A property Mof a molecular system variesin time when a perturbation V(t) is applied to the system.To first order, this change is given as (M)z(OIMIO) + due-‘“‘((M;Y)),, (3) where Vis the Fourier transform of V(t) : V(w) =-L 2~ (4) 2118 J. Chem. Phys. 96 (3),1 February 1992 0021-9606/92/032118-09$06.00 0 1992 American institute of Physics Downloaded 06 Jun 2003 to 129.242.5.30. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp