Article pubs.acs.org/JCTC Choice of Optimal Shift Parameter for the Intruder State Removal Techniques in Multireference Perturbation Theory Shu-Wei Chang and Henryk A. Witek* Department of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, Hsinchu, Taiwan *S Supporting Information ABSTRACT: An extensive critical evaluation of intruder state removal techniques (aka shift techniques) applicable to multireference perturbation theory (MRPT) shows that the magnitude of the shift parameter σ does not influence the spectroscopic parameters of diatomics to a significant degree, provided that the shift is chosen to be sufficiently large. In such case, typical variation of spectroscopic parameters over a wide range of shift parameters is smaller than 0.005 Å for equilibrium distances, 30 cm−1 for harmonic vibrational frequencies, and 0.1 eV for dissociation energies. It is found that large values of σ not only remove intruder states but they also bring the MRPT energies and properties closer to experimental values. The presented analysis allows us to determine optimal values of the shift parameters to be used in conjunction with various versions of MRPT; these values are recommended to replace the ad hoc values of σ suggested in MRPT manuals in calculations for diatomics. Transferability of the optimal shift parameters to larger molecular systems and to other basis sets than aug-cc-pVTZ is anticipated but remains to be formally established. 1. INTRODUCTION Andersson and Roos by introducing a downward energy shift applied to the zeroth-order energy of the reference state,33 Multireference perturbation theory (MRPT) is a collection of fi robust quantum chemical techniques that can be used on a which can be mathematically justi ed as a repartitioning of the routine basis to calculations on ground and excited states of original Hamiltonian into new zeroth-order Hamiltonian and small and medium size molecules, determination of potential perturbation operators. It is intuitively obvious that such a energy surfaces and reaction paths, explication of reaction downward energy shift (i.e., transferring a fraction of the mechanisms, analysis of open-shell systems, and many other zeroth-order energy of the reference state to the perturbation applications which require more than one Slater determinant to operator) would remove the existing quasidegeneracies, but it is − describe properly the character of the wave function.1 3 At entirely possible that it would introduce instead other present, MRPT can be considered as the only feasible and degeneracies or quasidegeneracies. Such behavior was indeed commercially available way of treating systems whose physically observed. For many systems, a small downward shift (up to 0.1 realistic description calls for using more than 12 active orbitals au) was found to be too small to guarantee physically in the description of the zeroth-order wave functions, such as meaningful characteristics of MRPT energies and wave transition metal clusters, multicenter organometallic systems, functions, which could be obtained only with substantially larger repartitionings. This situation is well illustrated by our high-lying excited molecular states, etc. MRPT is not only a 34 well-established computational tool; it also constitutes a field of recent work on the ground state potential energy surface of the manganese dimer, for which we have determined the vigorous methodological development aiming, besides others, (0) at enhancing its accuracy,4 finding optimal partitioning zeroth-order energies Eα of the ground state along the − schemes5 9 leading to better accuracy and better perturbation internuclear coordinate and plotted it against the zeroth-order − series convergence,10 14 designing alternative algorithmic energy spectrum of the singly and doubly excited intermediate − schemes,15 23 and eliminating its various unwelcomed draw- states. It turned out that for two out of eight possible classes of 24−31 excitations, i.e., for the one- and two-electron active → virtual backs. (0) The most serious drawback of MRPT is the existence of excitations, Eα is completely embedded in the lower portion of intruder states,32 which are defined as some of the intermediate the dense spectrum of the zeroth-order Hamiltonian of the states k in the perturbation expansion with zeroth-order intermediate states k. A downward energy shift allows for (0) bringing the reference state α below the dense spectrum, but in energies Ek degenerated (or quasidegenerated) with the (0) zeroth-order energy Eα of the reference state α. It is easy to some situations the corresponding shift magnitude has to be see that such quasidegeneracies result in exceedingly small quite substantial. perturbation energy denominators possibly leading to com- The downward energy shift technique described above is pletely unphysical, very large energy corrections already at low usually referred to as the level shift technique or as the real shift fi (RS) technique.33 A viable alternative was developed by orders of perturbation theory. This problem was rst 35 recognized in MRPT by Andersson and Roos in their work Forsberg and Malmqvist, who added an imaginary shift to on the chromium dimer,33 for which they obtained a discontinuous potential energy surface plagued with local Received: October 6, 2011 singularities. The problem of intruder states was tackled by Published: September 10, 2012 © 2012 American Chemical Society 4053 dx.doi.org/10.1021/ct2006924 | J. Chem. Theory Comput. 2012, 8, 4053−4061 Journal of Chemical Theory and Computation Article the zeroth-order energy of the reference state. This resulted in a bond lengths, vibrational frequencies, and dissociation energies (0) complex value of Eα , which thus became well separated from underwent noticeable changes when the shift parameter was the purely real zeroth-order spectrum of the intermediate states. made larger; these changes were particularly pronounced when Consequently, all the degeneracies and quasidegeneracies were the studied electronic state had a weakly bound character. The completely removed at the cost of additional imaginary most unexpected finding associated with using the shift (0+1+2) component in the resulting MRPT energy, Eα . This techniques in MRPT was probably the prediction of a false obstacle was solved by neglecting the imaginary component ground state for the scandium dimer.46,47,49 MRPT calculations and retaining only the real part of the MRPT energy as the final performed with the (12o,6e) active space and small values of 3Σ− answer. This approach is usually referred to as the imaginary the shift parameter predicted u as the ground state, while shift (IS) technique.35 Yet another approach, usually referred to larger values of the shift parameter indicated that the ground 36 5Σ− as the intruder state avoidance (ISA) technique, was state is u . Analogous calculations, performed with a larger developed by Witek and collaborators. It was based on the active space (18o,6e) and independently with the second- and (0) observation that a modification of Eα affects the contribution third-order n-electron valence perturbation theory (NEVPT), to the perturbation series from all possible intermediate states, were free of the intruder state problem and unequivocally fi 5Σ− while it would be desirable to introduce the modi cation only showed that u is located a few kilocalories per mole lower 3Σ− 50 19,20,51 for the intruding intermediate states. Following this observa- than u . Note in passing that NEVPT is the only tion, it was suggested to modify the zeroth-order energies of the version of the commercially available37 popular variants of (0) intermediate states instead of modifying Eα ; the corresponding MRPT that seems to be completely free of the intruder state shift magnitude was made proportional to the intruding power problem owing to an enhanced, partially bielectronic definition of a given intermediate state computed as the inverse of the of the zeroth-order Hamiltonian developed by Dyall,52 while ff (0) − (0) zeroth-order energies di erence, Eα Ek . It is important to the other methods (CASPT2, MRMP, and MCQDPT) use a highlight here that all the intruder state removal shift multireference generalization of a purely one-electron Møller− techniques discussed above are not simply ad hoc corrections Plesset operator to determine the zeroth-order Hamiltonian to the second-order perturbation energy but constitute well- and the corresponding perturbation operator. defined perturbation schemes differing from the original In the present manuscript, we use a statistical approach to versions of MRPT by slightly different partitioning of the analyze in detail the dependence of various intruder-state original Hamiltonian into the zeroth-order operator and the removal MRPT techniques on the magnitude of the shift perturbation. It was explicitly demonstrated for ISA that the parameter. Our work is motivated by the above-mentioned repartitioning does not alter the infinite-order PT solutions, apparent discord between the conceptual problems associated provided that both series, MRPT and ISA-MRPT, are with using small values of σ and a common approach of MRPT convergent.36 practitioners, who, following manual suggestions, choose σ as The intruder state removal shift techniques were imple- small as possible. We aim, in particular, at resolving two − mented in quantum chemistry packages37 39 and are available important issues: (i) How big can the change be in for popular variants of MRPT.