High-level multireference methods in the quantum-chemistry program system COLUMBUS: Analytic MR-CISD and MR-AQCC gradients and MR-AQCC-LRT for excited states, GUGA spin–orbit CI and parallel CI density¤ Hans Lischka,*a Ron Shepard,*b Russell M. Pitzer,*c Isaiah Shavitt,*cd Michal Dallos,a ThomasMu ller,a Pe ter G. Szalay,*e Michael Seth,f Gary S. Kedziora,g Satoshi Yabushitah and Zhiyong Zhangi a Institute for T heoretical Chemistry and Structural Biology, University of V ienna, Wa hringerstrasse17, A-1090 V ienna, Austria. E-mail: Hans.L ischka=univie.ac.at b T heoretical Chemistry Group, Chemistry Division, Argonne National L aboratory, Argonne, IL 60439, USA. E-mail: shepard.=tcg.anl.gov c Department of Chemistry, T he Ohio State University, 100 W est 18th Avenue, Columbus, OH 43210, USA. E-mail: pitzer=chemistry.ohio-state.edu, shavitt=chemistry.ohio-state.edu d Department of Chemistry, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA e Department of T heoretical Chemistry, Eo tvo s L ora nd University, P.O. Box 32, H-1518 Budapest, Hungary. E-mail: szalay=para.chem.elte.hu f Department of Chemistry, University of Calgary, University Drive 2500, Calgary, Alberta, Canada T 2N 1N4 g Department of Chemistry, Northwestern University, Evanston, IL 60208, USA h Department of Chemistry, Faculty of Science and T echnology, Keio University, 3-14-1 Hyoshi, Kohoku-ku, Y okohama 223-8522, Japan i PaciÐc Northwest National L aboratory, P.O. Box 999, Richland, W A 99352, USA Received 5th October 2000, Accepted 31st October 2000 First published as an Advance Article on the web 18th December 2000 Development of several new computational approaches within the framework of multi-reference ab initio molecular electronic structure methodology and their implementation in the COLUMBUS program system are reported. These new features are: calculation of the analytical MR-CI gradient for excited states based on state-averaged MCSCF orbitals, the extension of the MR-ACPF/AQCC methods to excited states in the framework of linear-response theory, spinÈorbit CI for molecules containing heavy atoms and the development of a massively-parallel code for the computation of the one- and two-particle density matrix elements. Illustrative examples are given for each of these cases. 1. Introduction of a calculation as compared to the SR case, especially in the selection of appropriate reference spaces. The popularity of quantum chemistry ab initio methods in An MR treatment usually starts with a multiconÐguration computational chemistry has increased rapidly in recent years SCF (MCSCF) calculation, which describes the nondynamical due to the development of new methods and of corresponding electron correlation by including the nearly degenerate elec- efficient computer programs that take advantage of modern tron conÐgurations (and often many others) in the MCSCF computer technology. The most popular methods by far have wavefunction. Dynamical electron correlation can then be been of the single-reference (SR) type, including approaches included in several ways. The classical method for that such as the HartreeÈFock self-consistent Ðeld (SCF) method, purpose is conÐguration interaction1,2 (CI), using the Ritz MÔllerÈPlesset perturbation theory, coupled-cluster theory variation principle. The most common form of this approach and density-functional theory (DFT). However, the SR is MR-CI singles and doubles (MR-CISD), in which the approaches often fail when nondynamical electron correlation expansion space for the wavefunction is constructed from (near-degeneracy e†ects) is important, as usually happens in single and double substitutions of occupied orbitals by virtual the description of bond stretching and dissociation, elec- orbitals in the individual conÐguration state functions (CSFs) tronically excited states and many open-shell cases. Multi- of the reference wavefunction. The truncation of the expansion reference (MR) methods can deal e†ectively with such space to single and double substitutions, while physically problems, but their successful application requires signiÐ- motivated, is usually mandated by the very steep increase in cantly more careful planning and understanding of the details the number of CSFs, and the consequent computational e†ort, with increases in the substitution level. ¤ Presented at the Third European Conference on Computational The CI method is very Ñexible and robust, but the greatest Chemistry, Budapest, Hungary, September 4È8, 2000. disadvantage of truncated CI is its lack of extensivity, i.e., its 664 Phys. Chem. Chem. Phys., 2001, 3, 664È673 DOI: 10.1039/b008063m This journal is( The Owner Societies 2001 incorrect scaling with the size of the system.3,4 A variety of this implementation has been included in the current version correction methods have been developed for the approximate of COLUMBUS,35 and has been applied to a number of treatment of this problem, but the use of extensivity correc- systems containing very heavy atoms. tions is by no means as straightforward in MR-CI as in the The COLUMBUS program project28,29,35 was started in SR case. We shall mention just a few of the many extensivity 1980 with the aim of developing an efficient and Ñexible tool correction approaches. The simplest and most commonly used for MR-CISD calculations, with emphasis on the multi- are the Davidson correction5h7 and its MR extension,8,9 reference character of the computational approaches. An early which provide a correction to the computed energy, but not to version included MR-LCCM, and the MR-ACPF and the wavefunction or to computed properties. These correc- MR-AQCC methods were added as soon as they became tions are obtained as relatively cheap byproducts of SR-CISD available. The implementation was constructed as a set of and MR-CISD calculations, respectively, but are of limited individual programs for the computation of AO integrals and reliability.10 In order to address the previously mentioned for the various stages of the SCF, MCSCF and CI procedures. single reference limitations, multireference extensions of These programs communicated with each other through Ðles. coupled cluster theory (see, e.g., ref. 11È15) and perturbation The modular and open structure facilitated the implementa- methods (see, e.g., ref. 16 and 17) have also been developed. tion of new methods and features and interaction with other However, there is still no widely applicable and generally program systems. It is the purpose of this work to present new available MR-coupled-cluster procedure. Various approx- methodological developments, their implementation in imations have been introduced, mostly as modiÐcations to the COLUMBUS, and illustrative applications. An important MR-CISD procedure, to deal with this problem. Of these aspect of the current version concerns excited state calcu- methods we shall mention MR-LCCM12 (linearized coupled lations, including implementation of analytic gradients for cluster), MR-ACPF18 (averaged coupled-pair functional) and MR-CISD and MR-AQCC calculations based on state- MC-CEPA19 (coupled electron-pair approximation). In many averaged MCSCF orbitals and the treatment of extensivity applications the MR-ACPF approach has been quite suc- corrections by means of MR-AQCC and MR-AQCC-LRT cessful, but it tends to overestimate the e†ect of the omitted (linear-response theory).36 These features signiÐcantly increase higher excitations, sometimes producing anomalous results.20 the capabilities for accurate geometry optimizations of excited The MR-AQCC (averaged quadratic coupled-cluster) states. The other major new development is the spinÈorbit CI method,20,21 which is closely related to MR-ACPF, was for- implementation, which opens the way for completely new mulated in an attempt to overcome these problems, and applications. In addition, the e†ort to develop a version of usually produces more reliable results. The usefulness and COLUMBUS for massively parallel computers has continued, e†ectiveness of MR-ACPF and MR-AQCC are due to a good adding an efficient parallel version of the one- and two- balance of simplicity and satisfactory treatment of the basic particle density-matrix program to the previously available size-extensivity e†ect. The availability of analytic gradients for parallel modules. This feature is very important for efficient these models (see below) is another important point in their analytic gradient calculations. favor. For a review of these and related methods see, e.g., ref. 22. The availability of analytic gradient methods is crucial for 2. Synopsis of COLUMBUS the geometry optimization of molecules. Analytic gradient methods are often unavailable for MR calculations because of The MCSCF and CI wavefunctions are speciÐed using the the much greater complexity of the formalisms23h27 compared graphical unitary group approach (GUGA).37h40 This allows to the SR case. The situation becomes particularly compli- the treatment of any pure spin state with no practical cated if the invariance properties of the wavefunction with restriction with respect to the number of open shells in the respect to orbital transformations di†er in the MCSCF and individual CSF expansion terms. Mixing of di†erent spin MR-CI wavefunctions. Such situations will arise, for example, states through the spinÈorbit interaction is included in the if orbitals are frozen at the CI stage or if the MR-CI reference spinÈorbit capability. The reference space and CI expansion wavefunction includes only a selected subset of the CSFs in space are deÐned by means of a distinct row table (DRT), the original MCSCF wavefunction. Additional requirements which is a tabular representation of a distinct row graph.39,40 arise in calculations of electronically excited states. For a bal- The graph comprises two parts: the internal part describes the anced description of a series of states, state-averaged MCSCF occupations and spin couplings of the internal orbitals (the calculations are very useful in order to provide a common set orbitals occupied in at least one of the reference of orbitals for all states of interest.