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Symmetry-Adapted Perturbation Theory Based on Multiconfigurational

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Symmetry-Adapted Perturbation Theory Based on Multiconfigurational Symmetry-adapted perturbation theory based on multiconfigurational wave function description of monomers Michał Hapka,∗,y,z Michał Przybytek,z and Katarzyna Pernaly yInstitute of Physics, Lodz University of Technology, ul. Wolczanska 219, 90-924 Lodz, Poland zFaculty of Chemistry, University of Warsaw, ul. L. Pasteura 1, 02-093 Warsaw, Poland E-mail: [email protected] Abstract We present a formulation of the multiconfigurational (MC) wave function symmetry- adapted perturbation theory (SAPT). The method is applicable to noncovalent interactions between monomers which require a multiconfigurational description, in particular when the interacting system is strongly correlated or in an electronically excited state. SAPT(MC) is based on one- and two-particle reduced density matrices of the monomers and assumes the single-exchange approximation for the exchange energy contributions. Second-order terms are expressed through response properties from extended random phase approximation (ERPA) equations. SAPT(MC) is applied either with generalized valence bond perfect pairing (GVB) or with complete active space self consistent field (CASSCF) treatment of the monomers. We discuss two model multireference systems: the H2 ··· H2 dimer in out-of-equilibrium geome- tries and interaction between the argon atom and excited state of ethylene. In both cases SAPT(MC) closely reproduces benchmark results. Using the C2H4*··· Ar complex as an ex- ample, we examine second-order terms arising from negative transitions in the linear response function of an excited monomer. We demonstrate that the negative-transition terms must be accounted for to ensure qualitative prediction of induction and dispersion energies and de- velop a procedure allowing for their computation. Factors limiting the accuracy of SAPT(MC) are discussed in comparison with other second-order SAPT schemes on a data set of small single-reference dimers. 1 Introduction Both approaches have been extensively devel- oped and benchmarked for systems which can Quantum chemistry offers two complementary be adequately described with single determi- arXiv:2104.06891v1 [physics.chem-ph] 14 Apr 2021 approaches to study noncovalent interactions— nants.4,5 With the supermolecular method it the supermolecular approach and energy de- is now possible to reach even submicro- or mi- composition methods. The former is concep- crohartree accuracy for small systems.6–10 At tually simple and capable of providing most the same time performing realistic energy de- accurate potential energy surfaces, e.g., for composition for enzymes exceeding 3,000 atoms interpretation of experiments carried out in is also within reach.11 Energy decomposition the cold- and ultracold regimes.1–3 Decompo- schemes are not only interpretative tools, but sition methods allow insight into the nature can also provide potentials for quantitative pre- of the interaction by partitioning the inter- dictions. The symmetry adapted perturbation action energy into well-defined contributions. 1 theory (SAPT)12,13 has been applied to gener- numerical instabilities and algebraic complex- ate potential energy surfaces for scattering cross ity. Single-reference coupled-cluster (CC) ap- sections calculations, predictions of spectra and proaches introduced by Piecuch and co-workers, bulk matter properties, as well as the develop- e.g. the CC(P ; Q) formalism,31,32 may be a ment of force fields for biomolecules (see, e.g., viable alternative, as indicated by studies of Refs. 14–18). interactions involving stretched intramonomer In contrast to the rich toolbox dedicated to covalent bonds.33,34 Encouraging results have single-determinantal wave functions, describing recently been obtained for strongly correlated intermolecular interactions in complexes which interacting systems from multiconfigurational demand multiconfigurational (MC) wave func- random phase approximation theory combined tions remains a challenge. The multiconfigura- with generalized valence bond method.34–36 tional treatment is often mandatory for transi- The multiconfiguration density functional the- tion metal complexes, open-shell systems, elec- ory (MC DFT)37,38 methods corrected to in- tronically excited states or systems dominated clude long-range dynamic correlation via per- by static correlation effects. From the stand- turbation theory,39 the adiabatic connection point of weak intermolecular forces, proper rep- formalism40 or semi-empirical dispersion mod- resentation of static correlation warranted by els41 are also worth mentioning, but their accu- expansion in multiple electron configurations racy for noncovalent interations remains to be is not sufficient. The main difficulty lies in rigorously investigated. the recovery of the remaining dynamic corre- A separate challenge, parallel to advanc- lation both within and between the interacting ing supermolecular methods, is the develop- molecules. The latter effect, giving rise to the ment of energy decomposition schemes ap- attractive dispersion interaction, poses a par- plicable to multireference systems.42 On that ticular challenge due to its highly nonlocal and front, the SAPT formalism offers several im- long-ranged nature. Although many multirefer- portant advantages which make it one of ence methods restoring dynamic correlation ef- the most widely used and actively developed fects have been developed, neither has yet man- decomposition approaches.18 The interaction aged to combine the accuracy and efficiency energy components in SAPT have a clear required for noncovalent interactions. For in- physical interpretation and the most accu- stance, the popular multireference configura- rate variants of the method predict interaction tion interaction (MRCI) approach19 and mul- energies closely matching the coupled-cluster tireference perturbation theories20,21 are lim- singles-and-doubles with perturbative triples ited by the lack of triple excitations and trun- [CCSD(T)43,44] results. Compared to the su- cation of the perturbation series at the second permolecular approach, SAPT avoids the ba- order, respectively, and their application to in- sis set superposition error, since the interac- teractions is not trivial. First, MRCI is not size- tion energy is computed directly based only on consistent and requires approximate corrections monomer properties. Moreover, the interaction added a posteriori.22,23 In perturbation theories energy represented as a sum of energy contri- the fulfilment of the strict separability condi- butions is per se size-consistent. tion depends on the choice of the zeroth-order In more than forty years spanning the de- Hamiltonian.24,25 Second, perturbation theo- velopment of SAPT, applications going beyond ries, including complete active space (CAS) per- the single-reference treatment of the monomers turbation theory (CASPT2)20 and multirefer- are scarce. Exact full configuration interac- ence variants of the Møller-Plesset perturbation tion (FCI) wave functions are feasible only for theory,26 suffer from intruder states27 which model, few-electron dimers and have been em- have to be removed using one of the available ployed in studies of SAPT convergence.14,45–48 shift techniques.28 Intruder states are also a Reinhardt49 used valence bond (VB) wave significant problem in the development of mul- functions to represent the electrostatic inter- tireference coupled-cluster theories,29,30 next to action between monomers of a multireference 2 character and proposed an approximate, VB- in Section 4. In Section 5 we summarize our based approach for dispersion energy calcu- findings. lations. The spin-flip SAPT (SF-SAPT)50,51 formalism introduced by Patkowski and co- workers opened the possibility to treat mul- 2 Theory tireference, low-spin states based on single ref- Consider a weakly interacting dimer AB which erence description of the subsystems. First- dissociates into monomer A in state I described order spin-flip exchange energy expressions with the jΨAi wave function and monomer B in for high-spin restricted open-shell Hartree-Fock I state J described with the jΨBi wave function (ROHF) wave function have already been de- J (I and J refer either to ground or excited states rived and implemented,50,51 while extension to of the monomers). When the unperturbed the second-order is under way.18 Hamiltonian is chosen as the sum of Hamiltoni- The purpose of the present paper is to present ans of the isolated monomers, H^ = H^ + H^ , a complete SAPT formalism applicable to inter- 0 A B the zeroth-order wave function takes a product actions involving multireference systems. First form jΨ(0)i = jΨAΨBi. In this work we assume steps in this directions have been already taken. I J that jΨ(0)i is nondegenerate. Recently, we have devised multiconfigurational The intermolecular interaction operator, V^ , approaches for second-order dispersion52 and represents the perturbation and gathers all exchange-dispersion53 energy calculations. In Coulombic interactions between electrons and this work we complement these efforts and pro- nuclei of the interacting partners pose a variant of SAPT based on MC wave functions, which we refer to as SAPT(MC). ^ X X 1 X X Zβ X X Zα The method includes energy contributions up V = − − rij rβi rαj to the second-order in the intermolecular inter- i2A j2B i2A β2B j2B α2A X X ZαZβ action operator and can be applied with any + rαβ wave function model which gives access to one- α2A β2B and two-electron reduced density matrices of X X 1 X B X A AB the monomers. Following the developments of = + v (ri) +
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