Energy level shifts in two-step spin–orbit coupling ab initio calculations Goar Sánchez-Sanz a , Zoila Barandiarán a , Luis Seijo b,⇑ a Departamento de Quı ´mica, Universidad Autónoma de Madrid, 28049 Madrid, Spain b Instituto Universitario de Ciencia de Materiales Nicolás Cabrera, Universidad Autónoma de Madrid, 28049 Madrid, Spain article info Article history: Received 23 April 2010 In final form 17 August 2010 Available online 21 August 2010 abstract We point out a problem with two-step spin–orbit ab initio calculations in which the energy levels of spin– orbit free Hamiltonians are shifted as a means to including dynamic correlations at low cost in small spin–orbit configuration interaction calculations. The usual shifts driven by the energy order of the states can lead to anomalous results when avoided crossings exist with significant change of wave function character, which take place at different nuclear positions in the configurational spaces of the first and the second steps. In these cases, the shifts of the spin–orbit free energy levels must be assigned according to the characters of the wave functions. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Two-step spin–orbit coupling ab initio methods have been pro- posed – and are presently under use – as a means of achieving a reasonable balance between accuracy and economy in electronic structure calculations in which both dynamic correlation and spin–orbit coupling are important effects [1–3]. These methods ap- peared as an answer to the problem associated with the fact that spin symmetry breaking leads to much larger configuration inter- action (CI) matrices to handle with spin–orbit Hamiltonians than with spin-free Hamiltonians. They are based on two ideas: first, that dynamic electron correlation and spin–orbit coupling are lar- gely uncoupled in a vast majority of atomic, molecular, and solid state systems, and, second, that these two effects can be considered in two different steps of the electronic structure calculation be- cause they pose very different demands in terms of electronic con- figuration space. This has been recognized and implemented in early two-step perturbation theory/CI methods [4], in which a first step consisting of a conventional correlated calculation with a spin-free Hamiltonian was followed by a second step, where the spin–orbit Hamiltonian was used and the correlation effects from the first calculation were conveniently transferred under some for- mal conditions of the wave functions, like contracted CI. Later, on the basis of the above ideas, Llusar et al. [1] formulated a simple, general, effective Hamiltonian to be used in the second step, in spin–orbit (relatively small) CI calculations with all types of many-electron basis sets, either contracted or uncontracted CI, determinantal, double group adapted configuration space, etc. This effective Hamiltonian is simply made of the original Hamiltonian plus an operator that shifts the eigenstates of the spin-free part of the Hamiltonian, within any configuration space, to their origi- nal energies. This was called spin-free-state shifting operator. Vallet et al. [2] and Malmqvist et al. [3] adopted the use of such a shifting operator in a two-step uncontracted determinantal effec- tive Hamiltonian spin–orbit CI method (implemented in EPCISO) [2] and a two-step restricted-active-space state-interaction spin– orbit method, RASSI-SO [3] (implemented in MOLCAS) [5]. In this Letter, we point out that the usual calculations of the shifting constants in the spin-free-state shifting operator, that is, the energy shifts of the spin–orbit free levels, which are driven by their energy order within each irreducible representation, can lead to anomalous results when avoided crossings exist with sig- nificant change of character of the wave functions at each side. In these cases, the shifts of the spin–orbit free energy levels must be assigned according to the character of the wave functions rather than according to their energy order. 2. Method and results Basically, in two-step spin–orbit methods an effective Hamilto- nian is used which is made of three operators: the spin-free Ham- iltonian b H SF , the spin–orbit coupling Hamiltonian b H SO , and the energy shift operator b H shift , b H eff ¼ b H SF þ b H SO þ b H shift : ð1Þ b H SF þ b H SO is the regular Hamiltonian of the electronic system, b H, and b H shift is defined as b H shift ¼ X iSM S Cc d iSC jU P iSM S Cc ihU P iSM S Cc j; ð2Þ with d iSC ¼½E G iSC  E G GS ½E P iSC  E P GS : ð3Þ 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.08.051 ⇑ Corresponding author. E-mail address: luis.seijo@uam.es (L. Seijo). Chemical Physics Letters 498 (2010) 226–228 Contents lists available at ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett