Symmetry-Adapted Perturbation Theory Based on Multiconfigurational

Symmetry-adapted perturbation theory based on multiconﬁgurational wave function description of monomers

Michał Hapka,∗,†,‡ Michał Przybytek,‡ and Katarzyna Pernal†

†Institute of Physics, Lodz University of Technology, ul. Wolczanska 219, 90-924 Lodz, Poland ‡Faculty 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 example, 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 develop 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 developed 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 |ΨAi wave function and monomer B in for high-spin restricted open-shell Hartree-Fock I state J described with the |ΨBi 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, Hˆ = Hˆ + Hˆ , a complete SAPT formalism applicable to inter- 0 A B the zeroth-order wave function takes a product actions involving multireference systems. First form |Ψ(0)i = |ΨAΨBi. In this work we assume steps in this directions have been already taken. I J that |Ψ(0)i is nondegenerate. Recently, we have devised multiconfigurational The intermolecular interaction operator, Vˆ , 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). ˆ 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- i∈A j∈B i∈A β∈B j∈B α∈A X X ZαZβ action operator and can be applied with any + rαβ wave function model which gives access to one- α∈A β∈B and two-electron reduced density matrices of X X 1 X B X A AB the monomers. Following the developments of = + v (ri) + v (rj) + V rij Ref. 52 and 53, the linear response properties i∈A j∈B i∈A j∈B required for second-order terms are accessed by (1) solving extended random phase approximation where i and j run over N and N electrons in (ERPA)54,55 equations. Both first- and second- A B monomers A and B, respectively, vA and vB are order exchange terms are derived assuming the one electron potentials, and V AB is the nuclear- single-exchange approximation,56 also known as nuclear repulsion term. the S2 approximation. We discuss the perfor- In the symmetrized Rayleigh-Schrödinger mance of SAPT(MC) combined either with gen- (SRS) formulation57 of SAPT the interaction eralized valence bond perfect pairing (GVB) energy is expanded with respect to Vˆ while or complete active space self consistent field enforcing the antisymmetry of |ΨAi and |ΨBi (CASSCF) description of the monomers. I J wave function products. The general expres- This work is organized in five sections. In Sec- sion for energy contribution in the n-th order tion 2 we present formulas for first- and second- in Vˆ takes the form order energy contributions in the ERPA-based variant of multiconfigurational SAPT. Special A B ˆ ˆ (n−1) (n) hΨ Ψ |V AΨ i attention is given to calculations of induction E = I J RS SRS hΨAΨB|AˆΨAΨBi and dispersion energies for complexes involving I J I J (2) n−1 (k) (n−k) electronically excited molecules. Section 3 con- P E hΨAΨB|Vˆ AˆΨ i − k=1 SRS I J RS tains details of our implementation and com- A B ˆ A B hΨI ΨJ |AΨI ΨJ i putations. Results for the model multireference and single-reference dimers are presented where Aˆ is the antisymmetrizer exchanging

3 (n) electrons between the monomers; |ΨRS i denotes are presented assuming singlet spin states of the n-th order component of the wave function ex- monomers which implies that αα and ββ blocks pansion, which is identical in both SRS and of 1-RDM are equal. in the conventional Rayleigh-Schrödinger (RS) perturbation theory, i.e., the expansion is based 2.1 First-order energy contribu- only on simple products of zero-order functions. The difference between the SRS and RS energy tions is defined as the exchange energy. The RS en- The polarization component of the first-order ergy contributions are often referred to as po- (1) SAPT energy is the electrostatic energy, Eelst = larization components. For convenience, we use A B ˆ A B hΨI ΨJ |V |ΨI ΨJ i. This energy contribution the SAPT acronym when referring to SRS. expressed in terms of 1-RDMs takes the form The SAPT(MC) formalism presented in this (1) X X work includes interaction energy components E = 2 n vB + 2 n vA ˆ elst p pp q qq through the second-order in V p∈A q∈B X pq AB (5) SAPT (1) (1) (2) (2) + 4 npnqvpq + V Eint = Eelst + Eexch + Eind + Eexch−ind (3) p∈A (2) (2) q∈B + Edisp + Eexch−disp A(B) A(B) (1) (1) where vpq = hϕp|v |ϕqi are matix ele- where Eelst and Eexch are first-order electro- rs ments of one-electron potentials and vpq denote static and exchange energy contributions, re- rs (2) (2) regular two-electron Coulomb integrals vpq = spectively, Eind and Eexch−ind are the second- −1 order induction and exchange-induction ener- hϕp(r1)ϕq(r2)|r12 |ϕr(r1)ϕs(r2)i. (2) (2) Evaluation of the exact expression for the gies, respectively, and Edisp and Eexch−disp denote the dispersion energy and its exchange first-order exchange energy counterpart, respectively. hΨAΨB|Vˆ AˆΨAΨBi All formulas are in the natural-orbitals (NOs) E(1) = I J I J − E(1) (6) exch A B ˆ A B elst representation. We use the following index con- hΨI ΨJ |AΨI ΨJ i vention: greek µ and ν indices denote electronic requires access to many-particle density ma- states of monomers, p q r s denote natural σ σ σ σ trices of the monomers. For the Hartree- spinorbitals, while pqrs pertain to natural or- Fock wave function many-particle density ma- bitals denoted by ϕ(r). Throughout the work trices are readily available as antisymmetrized the NOs are assumed to be real-valued. In the products of the one-particle density matrix.58 representation of natural spinorbitals the one- At the SAPT(DFT) level of theory one uses electron reduced density matrix (1-RDM) is di- approximate 1-RDMs of the monomers based agonal on the Kohn-Sham determinants and employs the same exchange expression as in the wave- 1 D † † E γpq = Ψ|aˆ aˆp +a ˆ aˆp |Ψ = npδpq (4) 59,60 2 qα α qβ β function SAPT. It is possible to significantly simplify the † where aˆpσ and {aˆpσ } are creation and annihi- structure of Eq. (6) by allowing only for single- lation operators, respectively, and np are natu- exchange of electrons between the monomers in ral occupation numbers from the h0, 1i range, the antisymmetrizer56,61 summing up to half a number of electrons, P n = N/2. (1) 2 A B ˆ (1) ˆ A B p p Eexch(S ) = hΨI ΨJ | V − Eelst P|ΨI ΨJ i All presented SAPT(MC) energy contribu- (7) tions are given in a spin-summed form. The where the single-exchange operator Pˆ collects expressions for the polarization energy compo- ˆ all permutations, Pij, interchanging the coordi- nents are valid for arbitrary spin states of the monomers. The exchange energy contributions

4 nates of electrons i and j the interacting monomers. The induction and dispersion energies involve transition energies ˆ X X ˆ P = − Pij (8) and one-electron reduced transition densities i∈A j∈B (1-TRDMs). The SRS components, exchange- induction and exchange-dispersion energies, re- Neglecting multiple exchange of electrons is quire both 1-TRDMs and two-electron reduced known as the S2 approximation and allows one transition densities (2-TRDMs). to express the first-order exchange energy using In this work we approximate the transition only 1-RMDs and two-electron reduced density properties of the interacting monomers by solv- matrices (2-RDMs) of the monomers.62 Follow- ing the Extended Random Phase Approxima- ing the density-matrix-based formulation of Ref. tion52,54,63 equations (independently for each 62 we obtain monomer) (1) 2 (1) X q2 Eexch(S ) = 2 Eelst − VAB npnq Sp AB Xν −N 0 Xν p∈A = ων , q∈B BA Yν 0 N Yν X (11) −2 n n vA Sq + vB Sq + vqp p q pq p qp p pq with a diagonal metric matrix p∈A q∈B X A B t rt ∀p>q Npq,rs = (np − nq)δprδqs (12) −4 nt Npqrt (vpr Sq + vpq) r>s pqr∈A t∈B The ERPA equations may be formed as a sym- X B A t rt metric real eigenproblem using electronic Hes- −4 nt Npqrt (vpr Sq + vpq) pqr∈B sian matrices, A + B and A − B. For ground t∈A state calculations the Hessian matrices are pos- X A B rs −8 Npqru Ntusq vpt (9) itive definite (see, e.g., Refs. 64 and 65 for ex- pqr∈A plicit ERPA equations in the GVB and CAS sut∈B frameworks, respectively). In the case of ex- q cited state wave functions the Hessian matrices where Sp = hϕp|ϕqi denotes the overlap in- tegral, and we have introduced intermedi- may have negative eigenvalues corresponding to ates containing contractions of the 2-RDM, de-excitation modes in the ERPA propagator66 † † (see a more detailed discussion in section 2.2.3). Γp q r s = hΨ|aˆ aˆ aˆq aˆp |Ψi, with the σ σ0 σ00 σ000 rσ00 sσ000 σ0 σ overlap integrals Apart from transition energies ων which correspond to the poles of the ERPA eigenproblem, A X ¯A w B X ¯B b two quantities that are required in second-order Ntuvw = ΓtuvaSa ,Ntuvw = ΓtuvbSw a∈A b∈B SAPT are 1- and 2-TRDM of the monomers. (10) The 1-TRDM is defined as ¯ where Γpqrs is the spin-summed 2-RDM, ¯ γν = Ψ|aˆ† aˆ |Ψ (13) Γpqrs = Γpαqαrαsα + Γpβ qαrβ sα . Since we assume pσqσ qσ pσ ν monomers in singlet states, the ββββ+αβαβ block is equal to its αααα+βαβα counterpart. Note that for singlet states αα and ββ blocks are equal: γν = γν = γν . The general pαqα pβ qβ pq definition of 2-TRDM reads 2.2 Second-order energy contri- ν † † butions Γ = hΨ|aˆ aˆ aˆq aˆp |Ψνi (14) pσqσ0 rσ00 sσ000 rσ00 sσ000 σ0 σ 2.2.1 Transition properties from Ex- The 1-TRDM are expressed through the ERPA tended Random Phase approximation Second-order SAPT energy components may be expressed through transition properties of

5 eigenvectors as65,67 ing partner, the latter deﬁned as

ν Z ρB(r0) ∀p>q γqp = (np − nq)[Yν]pq (15) ˆ B 0 ΩB(r) = v (r) + dr (20) ν |r − r0| ∀q>p γqp = (np − nq)[Xν]qp (16) B and the formula for the half of spin-summed where ρ is the one-electron density of the 2-TRDM reads53 monomer B (analogous expression holds for ˆ ΩA). The total induction energy formula is now Γ¯ν = Γν + Γν pqrs pαqαrαsα pβ qαrβ sα conveniently expressed as X ¯ X ¯ = [Xν]ptΓtqrs + [Xν]qtΓptrs 2 P γA,µ ΩB t

r t>s 2 X X P γB,ν ΩA + [Y ] Γ¯ + [Y ] Γ¯ X rs∈B rs rs ν tp tqrs ν tq ptrs − t>p t>q ωB ν6=J ν X ¯ X ¯ − [Yν]rtΓpqts − [Yν]stΓpqrt ˆ t

The polarization components of SAPT in the 2 P A B [Z ]pqΩ second order are the induction and dispersion (2) X p>q∈A µ pq E = −4 energies. The induction energy is given as ind ωA µ∈A µ 2 2 (22) hΨAΨB|Vˆ |ΨAΨBi P [ZB] ΩA (2) X I J µ J X r>s∈B ν rs rs E = − − 4 ind ωA ωB µ6=I µ ν∈B ν 2 hΨAΨB|Vˆ |ΨAΨBi (18) X I J I ν − ωB where ν6=J ν = E(2)(A ← B) + E(2)(B ← A) ∀ [ZA] = N YA − XA ind ind p>q∈A µ pq pq µ pq µ pq A B B B B where ωµ (ων ) are transition energies from the ∀r>s∈B [Zν ]rs = Nrs Yν rs − Xν rs state I (J for the monomer B) to µ (ν) (23)

A A A ωµ = Eµ − EI (19) The pertinent expression for the dispersion energy is68,69 (2) The Eind(A ← B) term arises from the per- 2 manent multipole moments on B changing the P A,µ B,ν qs (2) pq∈A γpq γrs vpr wave function of monomer A. The Eind(B ← (2) X rs∈B E = − (24) A) term describes a corresponding change in disp A B ωµ + ων monomer B due to the perturbing ﬁeld of A. µ6=I,ν6=J Eq. (18) may be recast using contractions between 1-TRDMs of one monomer and the electrostatic potential of its unperturbed interact-

6 which in the ERPA form reads52 In Eqs. (29) and (30) signs of the transition energies in the denominators are written explic- 2 P A B qs itly. p>q∈A[Zµ ]pq[Zν ]rsvpr (2) X r>s∈B E = −16 Approximate methods which are based on disp ωA + ωB µ∈A,ν∈B µ ν single-excitation operators, and for which the (25) linear response is directly related to an orbital Hessian matrix, are likely to miss de-excitations 2.2.3 Excited state case: explicit contri- in the linear response function computed for the 70 butions to dispersion and induc- excited state of interest. Consequently, the tion energies from de-excitations second term in Eq. (29) would not be accounted for. Since this term involves transitions to the Consider a dimer AI B0 in an excited state, low-lying states, it is anticipated to give a non- which dissociates into a monomer A in the state negligible contribution to the dispersion energy. I > 0 denoted in this section as AI (for simplic- A viable way to account for the de-excitations ity it is assumed that states of A are not degen- from the AI state in dispersion energy calcula- erate) and a monomer B in the ground state, tions is by considering linear response proper- denoted as B0. While all transition energies, cf. ties of states J lower than I. After exploiting Eq. (19), corresponding to B0 are positive the relations connecting response properties of the states I and J B0 ∀ν>0 ων > 0 (26) AI A A ωJ = EJ − EI for the monomer A they take either negative or (31) = − EA − EA = −ωAJ positive values for transitions to states lower or I J I higher than I, respectively AI ,J † γpq = ΨI |aˆqaˆp|ΨJ ∗ ∗ (32) AI † AJ ,I ∀µI ωµ > 0 (28) one immediately writes the µ = J component of Eq. (29) as Let us rewrite the dispersion energy expression, Eq. (24), in a form in which we explicitly iso- 2 P γAJ ,I γB0,νvqs late terms involving negative transitions (de- pq∈AJ pq rs pr X rs∈B0 I→J − ≡ ε (AJ B0) excitations) AJ B0 disp ν>0 − ωI + ων (2) (2) (33) Edisp(AI B0) = Edisp (AI B0) + The dispersion energy for the AI B0 dimer can 2 now be written as P AI ,µ B0,ν qs pq∈A γpq γrs vpr X rs∈B − I−1 AI B0 (2) (2) X I→J − ωµ + ων E (AI B0) = E (AI B0) + ε (AJ B0) µ0 disp disp+ disp (29) J=0 (34) (2) It should be emphasized that Eq. (34) is fully where by E (AI B0) we denote the disper- disp+ equivalent to Eq. (24) if exact response proper- sion energy arising from the positive part of the ties are employed. The crucial diﬀerence be- monomer A linear response function spectrum tween the expressions in Eqs. (29) and (34) 2 is that in the former contributions to the dis- P AI ,µ B0,ν qs pq∈A γpq γrs vpr persion energy from negative excitations follow (2) X rs∈B E (AI B0) = − from the linear response of the state AI , while in disp+ ωAI + ωB0 µ>I,ν>0 µ ν the latter they are obtained from the response (30) of states AJ which are lower in energy than AI . The ERPA model applied to excited-state

7 reference wave function either completely ogous manner misses negative excitations or reproduces them I−1 with poor accuracy. As a result, ERPA- (2) (2) X I→J E (AI B0) = E (AI B0) + ε (AJ B0) approximated dispersion energy, Eq. (25), com- ind ind+ ind J=0 puted for the excited-state dimer AI B0 will (38) lack important contributions from dexcitations. The E(2) (A B ) term is obtained from Eq. (22) ind+ I 0 The way around this problem is to employ the where the sum with respect to µ runs through alternative formula for the dispersion energy positive transitions (ωA > 0). The εI→J (A B ) presented in Eq. (34) in the ERPA approxima- µ ind J 0 (2) terms are given as tion. This requires computing the E (AI B0) disp+ 2 term according to Eq. (25) and expressing P AJ B0 p>q∈A ZI Ωpq I→J I→J J pq the approximated εdisp (AJ B0) terms through ε (AJ B0) = 4 ind AJ ERPA transition properties |ωI | (39) I→J εdisp (AJ B0) = and follow from solving ERPA equations for the 2 monomer A in states lower than I (from the P AJ B0 qs p>q∈AJ ZI pq Zν rs vpr ground state, J = 0, up to J = I − 1). Ev- X r>s∈B0 − 16 idently, contributions to the induction energy AJ B0 ν>0 − ωI + ων from negative excitations always take a positive (35) sign.

AJ AJ The ZI and ωI are the I-th eigenvector and 2.2.4 Second-order exchange energy eigenvalue, respectively, of the ERPA equations contributions solved for the monomer A in the J-th state We begin with the general expressions for AJ AJ A ZI = ZI (ΨJ ) (36) the second-order induction and exchange- 2 71,72 AJ AJ A dispersion energies in the S approximation ωI = ωI (ΨJ ) (37) (2) 2 A B ˆ ˆ (1) To reiterate, the negative-energy transition Eexch−ind(S ) = hΨI ΨJ |V PΨindi I → J, which is either absent or erroneous (1) A B ˆ (1) − Eelst hΨI ΨJ |PΨindi in ERPA, is easily accessed through a positive- − E(2) hΨAΨB|P|ˆ ΨAΨBi energy transition J → I computation carried ind I J I J (40) (2) 2 A B ˆ ˆ (1) out for the states J < I. A similar ap- Eexch−disp(S ) = hΨI ΨJ |V PΨdispi proach has recently been applied to improve (1) (1) − E hΨAΨB|PˆΨ i the description of the correlation energy for elst I J disp (2) A B ˆ A B excited states within the adiabatic connection − EdisphΨI ΨJ |PΨI ΨJ i ERPA method.70 Notice that for the lowest ex- (1) (1) cited states, which are usually of interest, the where |Ψindi and |Ψdispi are the ﬁrst order in- I→J εdisp (AJ B0) terms have a negative sign, but duction and dispersion wave functions, respec- could in principle be positive for highly excited state I. The second-order induction energy for a dimer in the excited state, obtained with the ERPA approximation, Eq. (22), has to be corrected for the missing de-excitations in an anal-

8 tively in terms of one-electron orbital basis set were given in Ref. 73 for the exchange-dispersion en- A B A B ˆ A B (1) X |Ψµ ΨJ i hΨµ ΨJ |V ΨI ΨJ i ergy and in Ref. 74 for the exchange-induction Ψ = − ind A A contributions. During the development of Eµ − EI µ6=I the SAPT(CCSD) approach, Korona presented X |ΨAΨBi hΨAΨB|Vˆ ΨAΨBi − I ν I ν I J the density-matrix formulation of both second- EB − EB order exchange components.75,76 ν6=J ν J For ground-state single-determinant wave A B A B ˆ A B (1) X |Ψµ Ψν i hΨµ Ψν |V |ΨI ΨJ i Ψ = − function or Kohn-Sham determinant is it pos- disp EA + EB − EA − EB µ6=I,ν6=J µ ν I J sible to calculate second-order exchange terms (41) through all orders in the intermolecular overlap, as proven by Schäﬀer and Jansen.77,78 Recently, First calculations of second-order exchange Waldrop and Patkowski have derived expres- contributions in the single exchange approxima- sions for the third-order exchange-induction.79 tion for many-electron systems were perfomed The exchange-induction energy written in by Chałasiński and Jeziorski.72 The authors de- terms of density matrices and transition ener- rived general expressions in the form of a many- gies reads (the S2 notation is dropped for conve- orbital cluster expansion based on the induc- nience) tion and dispersion pair functions. Expressions

D E υ˜(r, r0) + VAB γA,µ(x, x0) γA,µ(x, x) |r − r0|−1 γB(x0, x0) (2) X NANB int E (A ← B) = − exch−ind ωA µ6=I,ν=0 µ A,µ 0 B 0 A,µ 0 −1 B 0 0 (1) X γ (x, x )γ (x , x) γ (x, x) |r − r | γ (x , x ) − E elst ωA µ6=I,ν=0 µ (2) A 0 B 0 + Eind γ (x, x )γ (x , x) where υ˜(r, r0) is the generalized interaction po- The 1- and 2-TRDMs in the position represen- tential tation are deﬁned as 0 γA,µ(x , x0 ) = 0 0 −1 υB(r) υA(r ) 1 1 ∀ r∈A υ˜(r, r ) = |r − r | + + Z r0∈B NB NA N ΨA(x ,...)∗ΨA(x0 ,...)dx dx ... dx (42) A I 1 µ 1 2 3 NA A,µ (45) and γint stands for the interaction density ma- 62,75 trix and A,µ 0 A,µ 0 0 γint (x, x ) = (43) Γ (x1, x2; x1, x2) = Z − γA,µ(x0, x)γB(x, x0) A ∗ A 0 0 NA(NA − 1) ΨI (x1, x2, ...) Ψµ (x1, x2, ...)dx3...dxNA Z − ΓA,µ(x, x0; x, x00)γB(x00, x0)dx00 (46)

Z The pertinent expressions for the E(2) (B ← − γA,µ(x00, x)ΓB(x0, x; x0, x00)dx00 exch−ind A) component follow by interchanging A and ZZ B indices. − ΓA,µ(x, x000; x, x00)ΓB(x0, x00; x0, x000)dx00dx000 In Ref. 53 we have derived the density- (44) matrix formula for the exchange-dispersion energy based on transition properties in the

9 B X B,ν rs X ¯B,ν b0 sr ERPA framework. The corresponding expres- Vν = npγrs v˜pp + npΓrsbb0 Sp v˜pb sion for the exchange-induction energy compo- p∈A p∈A rs∈B rsbb0∈B nent takes the form X B,ν ¯A r as + γrs Γpqaa0 Sa0 v˜pq (2) 0 Eexch−ind = pqaa ∈A rs∈B A A B B X Vµ uµ X Vν uν X ¯A ¯B,ν b0 s ab 4 + 4 + Γ 0 Γ 0 S S 0 v˜ ωA ωB pqaa rsbb q a pr µ∈A µ ν∈B ν pqaa0∈A rsbb0∈B A A B B ! (1) X uµ tµ X u t (51) − 4 E − V AB + ν ν elst ωA ωB µ∈A µ ν∈B ν where the eﬀective two-electron potential, (2) X q 2 Eq. (42) in the matrix representation is + 2Eind npnq(Sp) p∈A q∈B v˜rs = ϕ (r)ϕ (r0)|v˜(r, r0)|ϕ (r)ϕ (r0) (47) pq p q r s = vrs + N −1 hϕ |vB|ϕ i δ δ + Ss(1 − δ ) The intermediates in Eq. (47) read pq B p r qs XqXs q XqXs −1 A r + NA hϕq|v |ϕsi δprδXpXr + Sp(1 − δXpXr ) , A X A A B (52) uµ = [Yµ ]pq − [Xµ ]pq (np − nq)Ωpq p>q∈A (a ϕ orbital may belong either to monomer B X B B A p u = [Y ]rs − [X ]rs (nr − ns)Ω ν ν ν rs Xp = A or Xp = B). r>s∈B When 1- and 2-TRDMs in Eqs. (50)-(51) (48) are expanded according to Eqs. (15)-(16) and

A X A A r r Eq. (17), respectively, one arrives at the matrix tµ = [Yµ ]pq − [Xµ ]pq (np − nq)nrSpSq A B representation of the Vµ and Vν terms p>q∈A r∈B A X A h X rr B X B B p p V = − [X ]pq (np − nq) nrv˜ tν = [Yν ]rs − [Xν ]rs (nr − ns)npSr Ss µ µ pq r>s∈B p>q∈A r∈B p∈A X A B ABi (49) + nrPpqrr + (np − nq)Qpq − Rpq r∈B and X A h X rr − [Yµ ]pq − (np − nq) nrv˜pq A X A,µ rr X ¯A,µ r ar p>q∈A r∈B Vµ = nrγpq v˜qp + nrΓpqaa0 Sa0 v˜pq pq∈A pqaa0∈A X A B ABi r∈B r∈B + nrPqprr − (np − nq)Qqp − Rqp X A,µ ¯B b0 sr r∈B + γpq Γrsbb0 Sp v˜qb (53) pq∈A rsbb0∈B X ¯A,µ ¯B b0 s ab + Γpqaa0 Γrsbb0 Sq Sa0 v˜pr B X B h X rs Vν = − [Xν ]rs (nr − ns) npv˜pp pqaa0∈A rsbb0∈B r>s∈B p∈A (50) X B A BAi + npPpprs + (nr − ns)Qrs − Rrs p∈A X B h X rs − [Yν ]rs − (nr − ns) npv˜pp r>s∈B p∈A X B A BAi + npPppsr − (nr − ns)Qsr − Rsr p∈A (54)

10 where the influence of the perturbing field, is used in calculations of the dispersion and exchange A X A as A as A as Ppqrs = Naa0prv˜qa0 − Nqa0arv˜pa0 − Naqa0rvã0p dispersion energies in wave-function SAPT13 aa0∈A including the popular SAPT0 model.81,82 In A r + OpsSq both SAPT(DFT) and SAPT(CCSD) the cou- B X B b0s B b0r B rb0 pled level of theory has been shown to give Ppqrs = Nbb0rpv˜qb − Nsb0bpv˜qb − Nbsb0pv˜qb bb0∈B highly accurate second-order energy contribu- 60,75,76,83–86 + OB Ss tions. rq p Evaluation of the exchange-induction energy A X A r Qrs = OasSa requires construction of the T and W inter- a∈A mediates, Eq. (57) and Eq. (58), respectively, B X B b 6 Qpq = ObqSp which has the nOCC scaling (nOCC are orbitals b∈B with non-zero occupancy). Since the 2-RDM AB X ab qb0 b0 b matrix elements factorize unless all four indices Rpq = Tqabb0 v˜pb0 − Tapbb0 vãb + Waqb0bSp Sa a∈A correspond to fractionally occupied orbitals, the bb0∈B formal scaling with the sixth power is only with b0 b − Wpab0bSa Sq respect to the number of such orbitals. In com- BA X ba sa0 r b parison, the exchange-dispersion energy is more R = T 0 v˜ 0 − T 0 v˜ + W 0 S 0 S rs aa sb ra aa br ba a abs a a expensive, as it requires steps with a n3 n3 aa0∈A OCC SEC b∈B 53 scaling (nSEC = Ms2 + Ms3 ). It should also be b s − Wa0arbSa0 Sa noted that the bottleneck step in evaluation of (55) the first-order exchange energy, Eq. (9), engages three four-index quantities (the NA, NB inter- A B 6 with N and N intermediates given in mediates and integrals) which amounts to Ms2 Eq. (10), and the remaining intermediates de- scaling. Note that for GVB the 2-RDMs fac- fined as torize also in the active block54 which results in identical scaling as in the SAPT(HF) method. A X ¯A a00u Otu = Γtaa0a00 vãa0 Recently, we have demonstrated that the un- 0 00 aa a ∈A (56) coupled approximation in the ERPA framework B X ¯B b0b Otu = Γtbb0b00 v˜ub00 combined either with CASSCF or GVB descrip- bb0b00∈B tion of the monomers leads to a poor quality of the second-order dispersion energy.52,53 A X ¯A ¯B b b0 Ttuvw = Γtaua0 Γbvb0wSa0 Sa more accurate dispersion energy is obtained if aa0∈A the monomer response properties are expanded bb0∈B (57) up to the first order in the coupling parameter, X X ¯A b0 b ¯B = Γtaua0 Sa Sa0 Γbvb0w which we refer to as the semicoupled approxi- bb0∈B aa0∈A mation.52 The fully coupled ERPA scheme gives

X A B a0b0 best results for both dispersion and exchange- W = Γ¯ 0 Γ¯ 0 v˜ tuvw taua bvb w ab dispersion energies. In this work all second- aa0∈A 0 order energy components were obtained with bb ∈B (58) the coupled approximation. X X ¯A a0b0 ¯B = Γtaua0 vãb Γbvb0w The multiconfigurational SAPT method, bb0∈B aa0∈A comprising first- and second-order energy com- Both induction and exchange-induction terms ponents, is based on the chosen wave function in SAPT are routinely calculated in the cou- theory applied to description of monomers. It pled approximation,80 so that the response is important to notice that computation of all of monomer orbitals due to the perturbation SAPT terms requires only knowledge of the field of its interacting partner is accounted corresponding one- and two-electron reduced for. The uncoupled approach, which neglects density matrices of monomers. In the rest of

11 this work we use the notation SAPT(MC) for wave function. the proposed method, where MC indicates the The results obtained with CASSCF and GVB underlying multiconfigurational wave function treatment of the monomers are denoted as model employed to obtain reduced density ma- SAPT(CAS), and SAPT(GVB), respectively. trices. The results will be presented for two Pertinent calculations were performed in the multiconfigurational wave funcitons: CASSCF locally developed code. The necessary inte- and GVB approximations. For comparison, grals, 1- and 2-RDMs for CASSCF wave func- we also include the SAPT results following tions were obtained from a developer version of from the single-determinantal description of the Molpro program.87 The GVB calculations monomers, denoted as SAPT(HF). were carried out in the locally modified Dalton program88 and interfaced with our code. The MP2 natural orbitals were used as the starting 3 Computational details guess in both CASSCF and GVB calculations. For the H ··· H dimer, discussed in Sec- The ERPA equations applied to GVB or CAS 2 2 tion 4.1, we carried out reference calculations wave functions require dividing the orbital exact up to second-order in SAPT, using an space of each monomer into three disjoint sub- in-house code developed for interactions be- sets referred to as s , s and s . The cardi- 1 2 3 tween two-electron monomers and based on di- nalities of the subsets are represented by the rect projection onto irreducible representations M , M and M notation. For wave func- s1 s2 s3 of the symmetric S group.89 The pertinent re- tions of the CAS type the s set contains all 4 1 sults are denoted as SAPT(FCI) in this work. inactive orbitals, whereas s and s correspond 2 3 The augmented correlation-consistent orbital to the active and virtual orbitals, respectively. basis sets of double- and triple-zeta quality When ERPA is applied with the GVB refer- (aug-cc-pVXZ, X = D,T)90,91 were employed ence, the s set is defined as all orbitals which 1 throughout the work. Monomer calculations occupation numbers fulfil the n > 0.992 con- p were carried out in the dimer-centered basis set. dition. The s set includes all active orbitals, 2 In Section 4.3 we present results of i.e., strongly occupied orbitals with occupation SAPT(GVB) and SAPT(CAS) calculations for numbers 0.992 ≥ n ≥ 0.5 and their weakly oc- p benchmark data set of noncovalently bound cupied partners from the same geminal. The complexes introduced by Korona92 which we remaining orbitals are grouped in the s set. 3 refer to as the TK21 data set. The accu- The p, q indices of the [X ] and [Y ] vectors ν pq ν pq racy of individual SAPT energy components span the following range and interaction energies is verified against the SAPT(CCSD) benchmark. All CCSD p ∈ s2 ∧ q ∈ s1 calculations were performed with frozen core p ∈ s3 ∧ q ∈ s1 (59) electrons. The SAPT(HF), SAPT(DFT) and p ∈ s2 ∧ q ∈ s2 SAPT2+(CCD) results are also reported. The p ∈ s3 ∧ q ∈ s2 exchange-correlation PBE093,94 functional em- (analogous range is assumed for the pq and rs ployed in SAPT(DFT) was asymptotically- 95 indices of Apq,rs ± Bpq,rs). corrected using the GRAC scheme applied In ERPA the presence of degeneracies and with the experimental values of the ionization near-degeneracies in the p ∈ s2 ∧ q ∈ s2 potentials. The SAPT(HF), SAPT(DFT) and space, cf. Eq. (59), may lead to numerical in- SAPT(CCSD) calculations were performed in stabilities. To circumvent this, we discarded Molpro.87 The SAPT2+(CCD) results were pairs of orbitals [in practical terms, it means obtained with the Psi496 program. In the lat- discarding corresponding rows and columns in ter variant of SAPT, the interaction energy is the ERPA matrices, see Eq. (11)] applying the −2 |np −nq|/np < 10 condition for the GVB wave −6 function and |np − nq|/np < 10 for the CAS

12 represented as dataset are given in the Supporting Informa- tion. SAPT2+(CCD) (10) (12) Eint = Eelst + Eelst,resp + E(10) + E(11) + E(12) exch exch exch 4 Results (20) (22) (20) (22) + Eind,resp + Eind + Eexch−ind,resp + Eexch−ind (2) (20) 4.1 Multi-reference ground-state + Edisp,CCD + Eexch−disp (60) system: H2-H2 We begin the analysis of multiconfigurational where the (ij) superscript refers to the ith- and SAPT with a model H ··· H dimer. We jth-order expansion in the intermolecular inter- 2 2 monitor the change of the interaction energy action operator and intramolecular correlation upon bond dissociation in one of the hydro- operator, respectively; the energy terms marked gen molecules. A quantitative description of with the “resp” index account for the orbital re- this system is challenging as it has to capture laxation effects. Except for the E(2) term, disp,CCD the balance between long-range dynamic cor- the interaction energy components grouped in relation and increasing nondynamic correlation Eq. (60) are identical to the SAPT297 approach. effects.34,106 The “+(CCD)” notation indicates that the dis- We examine the T-shaped structure of the persion energy is obtained in the coupled pair H ··· H complex in which one the covalent approximation including noniterative contribu- 2 2 H-H bonds is stretched from 1.37 a to 7.2 a tions from single and triple excitations, here re- 0 0 (see Figure 1 for a detailed description). In ferred to as CCD+ST(CCD)98,99 approach. SAPT(CAS) calculations each monomer is de- The accuracy of SAPT interaction energies scribed with a CAS(2,5) wave function. Note discussed in Section 4.3 is verified against that for two-electron monomers SAPT(GVB) is counterpoise-corrected100 (CP) supermolecular equivalent to SAPT(CAS) based on CAS(2,2) CCSD(T) results. To this end, we approx- wave functions. imate higher-order induction contributions at SAPT schemes based either on Hartree-Fock the Hartree-Fock level of theory101,102 or Kohn-Sham description of the monomers fail HF to predict the behavior of individual interaction δHF = E int energy components as the H−H bond is elon- (10) (10) (20) (20) − Eelst + Eexch + Eind,resp + Eexch−ind,resp gated and the complex gains a multireference (61) character (Figure 1). The SAPT(CSSD) approach initially remains in excellent agreement HF where Eint is the supermolecular Hartree-Fock with the SAPT(FCI) benchmark. The largest interaction energy. The δHF component was relative percent errors in SAPT(CCSD) near added to the SAPT interaction energy provided the equilibrium geometry (RH−H = 1.41 a0) oc- that the ratio of the sum of the induction and cur for the exchange-induction and exchange- exchange-induction energies to the total inter- dispersion energies which are overestimated action energy was larger than 12.5%, in agree- by ca. 4% and 7%, respectively. These dis- ment with the criterion selected in Ref. 103. crepancies in the single-reference regime re- Note that in Section 4.3 error statistics for to- sult from exclusion of certain cumulant con- tal interaction energies are reported for the S2 tributions in the second-order exchange expres- subset of the TK21 dataset which excludes six sions.75,76 After the H−H bond length exceeds largest dimers (see Refs. 92 and 104). 3.0 a0, the XCCSD-3 approximation underlying As an additional test, we performed SAPT SAPT(CCSD)107,108 starts to break down which calculations for the A24 dataset105 of Řezáč and translates into qualitative errors in all interac- Hobza. Since we observed the same qualitative tion energy components.34 trends as in the TK21 case, results for the A24 Both SAPT(GVB) and SAPT(CAS) predict

13 8 8 7 7 DISP H-H R distance is 6 6 EXCH-DISP H − H 0 5 R 0 5 / a / a d=6.21 a.u. d=6.21 H-H H-H 4 4 R R 3 3 . The basis set is aug-cc-pVTZ. 2 0 2 a 1 1 8 6 20 18 16 14 12 10 -160 -180 -200 -220 -240 -260 . In one of the monomers the 0 8 8 a 7 7 IND 6 6 EXCH-IND 0 5 0 5 / a / a H-H H-H 4 4 R R 3 3 2 2 1 1 -4 -5 -6 -7 -8 -9 7 6 5 4 3 2 1 -10 -11 8 8 7 7 ELST EXCH 6 6 HF FCI DFT CAS GVB CCSD 0 0 5 5 while in the other monomer the H–H bond is kept at a fixed distance of 1.44 / a / a 0 a H-H H-H 4 4 R R 3 3 2 2 dimer in the T-shaped configuration. The intermolecular distance is fixed at 6.21 1 1 2 H 60 -30 -40 -50 -60 -70 -80 -90

260 220 180 140 100

-100

h h SRS

RS ··· E / E E / E

µ 2 (n) (n) Figure 1: SAPTH energy contribution at the Hartree-Fock (HF), GVB, CASSCF (CAS), CCSD, DFT and FCI levels of theory for the varied from 1.37 to 7.20

14 the correct shape of the interaction energy this section. curves (Figures 1-2). The GVB-based vari- To access the excited states wave functions ant systematically underestimates the magni- of both the dimer and the ethylene molecule, tude of all SAPT contributions and SAPT in- we carried out three-state state-averaged CAS teraction energy. The exchange-induction en- calculations (SA-CAS) in the CAS(2,3) active ergy deviates most from the benchmark with space, i.e., two active electrons distributed on relative percent errors in the 10 − 20% range. π, σ, and π∗ active orbitals. In these calcu- Errors for the remaining components stay below lations the targeted π → π∗ state is the third 12% near the equilibrium and the accuracy im- state in the SA ensemble. Note that in both proves together with the increasing share of the SAPT(CAS) and supermolecular CASSCF cal- nondynamic correlation in the system (see also culations the Ar atom is represented with a sin- Tables S1-S3 in the Supporting Information). gle determinant. SAPT(CAS) is more accurate—errors with re- For ground state calculations we used super- spect to the SAPT(FCI) benchmark do not ex- molecular CCSD(T) results as benchmark. To ceed 3% not only in individual components, but obtain reference values for excited states, we also in the total interaction energy. The error adopted the procedure of Ref. 110 which com- of the SAPT(FCI) interaction energy with re- bines the CCSD(T) description of the ground FCI spect to supermolecular FCI (denoted as Eint state with excitation energies calculated at the in Figure 2) increases from 1.1% in the equilib- EOM-CCSD111 level of theory rium geometry to 15% at RH−H =7.2 a0. Est. EOM-CCSD(T) ∗ As we discussed in Ref. 52, further extension Eint (C2H4 ··· Ar) = of the active space in SAPT(CAS) is of little CCSD(T) ∗ E (C2H4 ··· Ar) + ωEOM−CCSD(C2H ··· Ar) beneﬁt for this system. Instead, to reach higher int 4 − ω (C H∗) accuracy one needs to move beyond the ERPA EOM−CCSD 2 4 scheme and solve full linear response equations, where the asterisk indicates a molecule in ex- i.e., include response not only from the orbitals, cited state, ωEOM−CCSD denotes the pertinent but also the wave function expansion coeﬃ- CCSD(T) EOM-CCSD excitation energy and Eint is cients. a ground state interaction energy. In Figure 3 we compare SAPT(CAS) in- ∗ 4.2 Excited-state system: C2H4- teraction energy curves with supermolecular Ar CAS(2,3) results and a coupled-cluster benchmark. As it has been rigorously shown in In this section we present SAPT(CAS) calcu- Ref. 104, supermolecular CAS interaction en- lations for the C2H4··· Ar dimer in which the ergy misses dispersion contributions if active ethylene molecule is either in the ground or orbitals are assigned only to one monomer, electronically excited state. We focus on the which is the case here. The CAS+DISP curves singlet excitation of a valence character with in Figure 3 represent CAS interaction energy the largest contribution from the π → π∗ tran- supplemented with the dispersion component sition.109 The C H ··· Ar complex is kept in (2) 2 4 taken from SAPT(CAS) calculations, EDISP = the C2v symmetry with the Ar atom located (2) (2) ∗ Edisp + Eexch−disp. For the π → π state both on the axis perpendicular to the C2H4 plane SAPT(CAS) and CAS+DISP interaction en- and bisecting its C-C bond (see also Table S4 in ergies were computed by explicitly accounting the Supporting Information for geometry of the for the de-excitation-energy terms according to C2H4 molecule). The interaction energy curves Eq. (34) and Eq. (38). The SAPT(CAS)∗ and presented in Figure 3 and Table 1 are obtained CAS+DISP∗ curves were obtained by neglect- by varying the distance R between the Ar atom I→J I→J ing the εdisp and εind terms. and the center of the mass of ethylene. Counter- Inspection of Figure 3 and Table 1 shows that 100 poise correction has been applied to all su- the C H ··· Ar complex in the ground state is permolecular interaction energies presented in 2 4

15 -25 HF -50 GVB CAS -75 DFT CCSD -100 FCI EFCI h -125 int E µ /

int -150 E

-175

-200 ESAPT -225 eint

-250 1 2 3 4 5 6 7 8

RH-H / a0

Figure 2: SAPT interaction energy for the H2··· H2 dimer in the T-shaped conﬁguration (see FCI Figure 1 for geometry description). Eint denotes the supermolecular FCI interaction energy. The basis set is aug-cc-pVTZ. bound by the dispersion forces. The CAS in- Table 1: Components of interaction energy for teraction curve is mainly repulsive and features the C2H4··· Ar dimer in the ground (g.s.) and only a shallow minimum located at ca. 5.0 Å SAPT SAPT∗ excited states. Eint and Eint are the total and 0.03 mEh deep. Addition of the disper- interaction energies with and without, respec- sion energy in CAS+DISP builds up a van tively, the inclusion of contributions from the der Waals minimum 0.56 mEh deep localized at negative-energy transitions εI→J deﬁned in 4.0 Å, which is in reasonable agreement with the disp/ind Eqs (35) and (39). All values in mEh. CCSD(T) reference (0.48 mEh at Req = 4.0 Å).

The performance of SAPT(CAS) is excellent. ∗ The total SAPT(CAS) interaction energy at g.s. π → π the optimal monomer separation is equal to Component R=4.00 Å R=3.30 Å −0.49 mE and the entire interaction curve al- (1) h E −0.257 −2.486 most coincides with the benchmark. The dis- elst (1) persion energy is clearly the dominating attrac- Eexch 0.723 6.054 (2) tive contribution amounting to −1.03 mEh in Eind −0.474 −3.987 the minimum (see Table 1). + P I→J Computation of the second-order SAPT com- J εind 0.000 0.000 (2) ponents in the proposed SAPT(CAS) approach Eexch−ind 0.457 4.434 involves solving the ERPA equations. When (2) E −1.026 −4.561 the monomer reduced density matrices enter- disp+ P I→J ing ERPA equations correspond to an unstable J εdisp 0.000 −1.087 CAS solution, either near-instabilities or insta- (2) Eexch−disp 0.090 0.844 bilities may occur in the linear response. In SAPT∗ general, the SA-CAS calculation in a small ac- Eint −0.487 0.298 SAPT tive space bears the risk that wave functions Eint −0.487 −0.789 describing higher excited states are not sta-

16 0.8 2.0 CAS CAS SAPT(CAS) 1.5 SAPT(CAS)* 0.6 CAS+DISP SAPT(CAS) CCSD(T) CAS+DISP* 1.0 0.4 CAS+DISP Est. EOM-CCSD(T) 0.5 h 0.2 0.0 / mE

int 0.0 E -0.5 -0.2 -1.0

-0.4 -1.5

-0.6 -2.0 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 R / Angstrom R / Angstrom

Figure 3: Interaction energy curves for the C2H4··· Ar dimer in the ground state (left) and in the π → π∗ state (right). The basis set is aug-cc-pVTZ.

ble in ERPA. This is what we have encoun- and underbinds by as much as 0.8 mEh com- tered for the SA-CAS π → π∗ state of the pared to the CC reference (Table 1). The large studied C2H4··· Ar dimer (see Figures S1-S3). discrepancy between second-order SAPT and To avoid instabilities in the ERPA equations, the hybrid CAS+DISP approach reﬂects that which manifest in discontinuous interaction en- higher-order induction terms, present in the su- ergy curves, we applied a three-point cubic permolecular CAS and absent in SAPT, become extrapolation of second-order energy contribu- important already for the low lying π → π∗ va- tions based on the Dyall partitioning of the lence state. monomer Hamiltonian112,113 and expansion of Contributions from the negative-energy tran- the ERPA response properties in the coupling sitions in the linear response are essential for a 52,114 ∗ parameter. The details on the cubic ex- quantitative description of the C2H4 ··· Ar in- trapolation model are provided in the Support- teraction. Neglecting the εI→J terms in SAPT ing Information. reduces the well depth by a factor of two, cf. The interaction energy curves for the π → π∗ SAPT(CAS)* results in Figure 3. Similarly, state are shown in Figure 3. At the CASSCF comparing CAS+DISP with CAS+DISP* re- level of theory the interaction has a purely veals that a good agreement of CAS+DISP repulsive character. The CAS+DISP model with the coupled-cluster reference is possible gives a binding curve which remains in excellent only after inclusion of the de-excitation part of agreement with the coupled-cluster reference. the spectrum. The observed energy lowering I→J The employed CC method [Eq. (62)] predicts a comes solely from the εdisp terms, as the induc- 1.61 mEh deep minimum at the intermonomer tion counterparts vanish due to symmetry (Ta- separation of 3.3 Å. The CAS+DISP minimum ble 1). In the van der Waals minimum, the two 2→0 2→1 occurs at a slightly shorter distance of 3.2 Å and dispersion terms [εdisp and εdisp , cf. Eq. (34)] is 1.60 mEh deep. Note that the nearly per- sum up to −1.9 mEh which is a sizeable effect agreement with CC has to rest partially fect considering that positive-energy transitions on error cancellation, since CAS+DISP neglects amount to −4.6 mEh. contributions from negative excitations in the second-order exchange-dispersion energy (only I→J 4.3 Single-reference systems εdisp terms are included in the model). The interaction energy curve from SAPT(CAS) cal- In this section we analyze the performance culations deviates from both CAS+DISP and of the multiconﬁgurational SAPT schemes for CC results at the intermediate and short range. many-electron dimers of the TK21 dataset of SAPT(CAS) localizes the minimum at 3.6 Å Korona92 against benchmark SAPT(CCSD) re-

17 sults. Additionally, we present SAPT(PBE0) used.52,53 Similar as in the first-order, the po- (2) (2) and SAPT2+(CCD) results. Although TK21 larization terms (Eind and Edisp) from multi- includes systems governed by the dynamic configurational SAPT compare favorably with rather than static correlation effects, our aim SAPT(HF), but do not match the quality of is to determine the level accuracy which could SAPT(DFT) or SAPT2+(CCD) results. The be expected of the studied mulitconfigurational discrepancy is more pronounced for the disper- SAPT if applied to multireference systems. sion energy where mean absolute errors from Note that in all SAPT calculations the ex- CCD+ST(CCD) and SAPT(DFT) calculations change terms were obtained in the S2 ap- are equal to 2.2% and 2.9%, respectively, com- proximation. The first-order exchange and pared to 6.4% obtained with SAPT(CAS) and second-order exchange-induction contributions 9.9% at the SAPT(GVB) level of theory. in SAPT(CCSD) include the cumulant contri- The second-order exchange-induction and butions.75,115 exchange-dispersion energies are more challeng- Figure 4 shows relative percent errors of ing than the polarization terms. SAPT(DFT) the individual SAPT energy components with (2) performs best recovering the Eexch−ind and respect to the SAPT(CCSD) reference (see (2) Eexch−disp contributions with the ∆abs val- also Tables S8-S11 in the Supporting Informa- ues of 7.5% and 4.3%, respectively. Both tion). Let us begin with the first-order energy SAPT(GVB) and SAPT(CAS) tend to under- terms. Both SAPT(GVB) and SAPT(CAS) estimate second-order exchange (∆abs values recover the electrostatic and exchange en- fall in the 10-14% range). It is worthwhile to ergies with similar accuracy—the mean ab- note that in the SAPT2+(CCD) scheme the solute errors (∆abs) for these contributions exchange-dispersion energy is calculated using fall in the 6-8% range. In contrast to the the uncoupled formula which leads to the mean electrostatic energy, the first-order exchange absolute error as large as 24% with respect to is systematically underestimated with mean the coupled SAPT(CCSD) reference. errors of −6.4% and −5.4% obtained with In Table 2 we examine the accuracy of to- GVB and CAS wavefunctions, respectively. tal SAPT interaction energies with respect to SAPT(GVB) affords smaller spread of errors the SAPT(CCSD) reference evaluated for the compared to SAPT(CAS), in particular for the S subset of the TK21 dataset. Error statis- (1) 2 Eelst component. The multireference treatment tics is given in terms of the mean error ∆, of the monomers constitutes an improvement the standard deviation σ, the mean absolute over the Hartree-Fock (single-determinantal) error ∆abs, and the maximum absolute error description, (the ∆abs values from the Hartree- ∆max. Both multiconfigurational SAPT ap- Fock-based SAPT calculations amount to ca. proaches reach similar accuracy. With ∆abs = 13% for both components) but it remains infe- 6.0% and σ = 7.4%, SAPT(GVB) remains in rior to both SAPT(DFT) and SAPT2+(CCD). slightly better agreement with the benchmark The second-order SAPT energy contributions than SAPT(CAS) (the respective values for the obtained with the SAPT(CAS) variant are con- latter are ∆abs = 6.9% and σ = 8.9%). The sistently more accurate than their SAPT(GVB) error statistics for multiconfigurational SAPT counterparts. In the TK21 dataset the largest matches SAPT(DFT) results where ∆abs and σ difference occurs for the induction energy, amount to 5.6% and 7.5%, respectively. This where SAPT(GVB) deviates from the bench- indicates a systematic error cancellation be- mark by 14.6% and 20.1% in terms of ∆abs tween attractive and repulsive energy contribu- and standard deviation, respectively, whereas tions in the ERPA-based SAPT, since for the the respective errors for SAPT(CAS) amount individual energy components SAPT(DFT) is to 7.8% and 9.5%. This confirms that the CAS- clearly closest to SAPT(CCSD). based ERPA provides better approximation for It is interesting to compare SAPT interaction both transition density matrices and transition energies against the supermolecular CCSD(T) energies than when GVB density matrices are

18 40 40 20 E(1) E(2) E(2) elst 30 ind disp 30 20 15 20 10 10 10 0 -10 0 5

∆ [%] -20 -10 -30 0 -20 -40 -50 -5 -30 -60 -40 -70 -10

HF HF HF GVB CAS GVB CAS GVB CAS PBE0 PBE0 PBE0

SAPT2+ SAPT2+ CCD+ST 40 50 40 E(1) E(2) E(2) exch 40 exch-ind exch-disp 30 30 30 20 20 20 10 10 10 0 0 ∆ [%] 0 -10 -10 -10 -20 -20 -30 -20 -30 -40

-30 -50 -40

HF HF HF GVB CAS GVB CAS GVB CAS PBE0 PBE0 PBE0

SAPT2+ SAPT2+ SAPT2+ Figure 4: Box plots of relative percent errors (∆) in the polarization SAPT components (top: electrostatic, induction and dispersion energies) and in the exchange SAPT components (bottom: exchange, exchange-induction and exchange-dispersion, all in the S2 approximation) for dimers of the TK21 data set. HF, GVB and CAS denote wave function description of the monomers. PBE0 stands for the asymptotically-corrected PBE0 functional. Relative percent errors are calculated Ei−Ei,ref according to the ∆i = · 100% formula. Errors are given with respect to the SAPT(CCSD) |Ei,ref | (2) reference. Errors for the Eexch−disp energy are reported for the S2 subset of TK21. The box and outer fences encompass 50% and 95% of the distribution, respectively.

19 Table 2: Summary of error statistics (in percent) for the SAPT interaction energy for dimers SAPT of the TK21/S2 dataset. Errors of the SAPT interaction energy (Eint ) are given with respect to SAPT(CCSD) results. Errors of the SAPT interaction energies corrected for the δHF term SAPT+δHF 100 (Eint ) are given with respect to CP-corrected supermolecular CCSD(T) results calculated in the same basis set. The 2+(CCD) notation refers to the SAPT2+(CCD) scheme. All exchange energy components are computed in the S2 approximation. The basis set is aug-cc-pVTZ.

† SAPT Eint HF GVB CAS PBE0 2+(CCD) ∆ −9.87 −0.80 −0.44 2.72 −6.95 σ 14.94 7.41 8.87 7.54 5.71 ∆abs 12.66 5.95 6.89 5.60 7.93 ∆max 33.66 12.32 14.74 16.93 14.78

‡ SAPT+δHF Eint HF GVB CAS PBE0 2+(CCD) CCSD ∆ −10.00 −2.71 −1.59 0.86 −7.91 −1.78 σ 10.38 6.22 7.44 7.09 5.48 3.57 ∆abs 11.33 5.89 6.40 4.72 8.74 3.22 ∆max 27.64 12.08 15.69 17.24 16.46 7.96 † errors with respect to SAPT(CCSD) ‡ errors with respect to supermolecular CCSD(T) reference. To this end, we approximate higher- less, the observed effect is small and relatively order induction effects with the δHF term, large errors compared to fully correlated SAPT Eq. (61). As expected, SAPT(CCSD) is schemes persist. This is best exemplified by the front runner (∆abs = 3.2%) followed by first-order energies which probe the quality of (1) SAPT(DFT) with mean absolute error of 4.7% the monomers density (Eelst) and density ma- (lower section of Table 2). Both SAPT(GVB) (1) trices (Eexch). In the second order the accuracy and SAPT(CAS) are less accurate—the ∆abs of SAPT(MC) is affected both by the missing value for the former reaches 5.9%, while for intramonomer correlation and approximations the latter it amounts to 6.4%. Still, the multi- in the ERPA response equations (see also dis- configurational SAPT variants outperform not cussion in Refs. 52 and 53). For the TK21 only the Hartree-Fock-based scheme (∆abs = dataset we observed that both SAPT(GVB) 11.3%), but also the SAPT2 model with the and SAPT(CAS) tend to underestimate second- CCD+ST(CCD) dispersion (∆abs = 8.7%). order contributions which leads to a fortuitous The relatively large errors of SAPT2+(CCD) error cancellation in the total interaction en- can be traced to poor representation of the ergy. When both TK21 and A24 datasets second-order exchange components. Recall that are considered (Table S24), SAPT(MC) pre- the presented SAPT results neglect exchange dicts interaction energies with mean absolute effects beyond the S2 approximation which errors and standard deviation below 8% and is expected to worsen the agreement between 10% , respectively, which is significantly bet- SAPT and CCSD(T) interaction energies. ter than Hartree-Fock based SAPT (∆abs ≤ To summarize, the examined SAPT(GVB) 18% and σ ≤ 20%) and comparable to the and SAPT(CAS) methods benefit from a par- SAPT2+(CCD) model (∆abs ≤ 10% and σ ≤ tial recovery of the intramonomer correlation 10%). effects by the underlying multiconfigurational wave function, as evidenced by a systematic improvement of all energy components with respect to the SAPT(HF) results. Neverthe-

20 5 Conclusions method is the only one among the existing SAPT approaches that offers the analysis of We have proposed a SAPT(MC) formalism ap- noncovalent interactions in systems involv- plicable to dimers in which at least one of the ing electronically excited molecules in singlet monomers warrants a multireference descrip- states. In this work we examined the role tion. In the approach the interaction energy of negative transitions in the linear response is expanded through the second-order terms in function of an excited subsystem in the de- the intermolecular interaction operator. For- scription of the second order components of mulas for the exchange energy contributions SAPT. In Section 2.2.3 a general protocol for are given in the single-exchange approxima- direct evaluation of negative-transition terms tion (the S2 approximation) and are valid for has been proposed and its implementation in ground and nondegenerate excited states of the the ERPA-approximation framework has been monomers in spin singlet states. While singlet presented. As an example of an excited-state states require spin-free reduced density matri- complex, we have selected the C2H4··· Ar dimer ces, extension to higher spin states is straight- and described it with a small CAS wave func- forward and involves spin-resolved components tion. While for the ground state of the system of RDMs. Response properties which enter the SAPT(CAS) remains in excellent agreement density-matrix-based SAPT formulas are ob- with supermolecular CCSD(T) results, the ex- tained by solving the extended random phase cited π → π∗ state of ethylene poses a signif- approximation (ERPA) eigenproblems for each icant challenge. First, we have demonstrated subsystem. Combined with ERPA equations, that second-order energy contributions related the presented variant of SAPT requires access to negative excitation energies are sizeable and only to one- and two-electron reduced density must be accounted for in SAPT. Second, even matrices of the monomers. Note that, contrary for the low-lying valence state of ethylene the to the supermolecular method, in SAPT the lack of higher-order induction terms and restric- dimer wave function is never computed which is tion to the S2 approximation significantly limit advantageous for multiconfigurational systems. the accuracy of SAPT(CAS) results. To illus- In this work we applied SAPT(MC) either with trate this, we have presented interaction energy CASSCF or GVB wave functions. curves obtained in a hybrid approach which re- Based on the model H2··· H2 dimer in which covers induction terms up to infinite order in one of the monomers undergoes dissociation, Vˆ . Indeed, a combination of supermolecular we have verified that SAPT(MC) is capable of CASSCF and second-order dispersion energy describing interactions in systems dominated from SAPT(CAS) calculations, which we refer by nondynamic correlation. The interaction to as the CAS+DISP approach,104 outperforms energy curve from SAPT(GVB) calculations SAPT for the π → π∗ state, and remains in has the correct shape and the largest devia- excellent agreement with the coupled-cluster tion from the FCI benchmark does not exceed reference. 13%. In the H2··· H2 dimer several active or- The CAS+DISP hybrid can be viewed as bitals are sufficient to recover both intra- and SAPT(CAS) supplemented with a CASSCF intermonomer correlation effects. SAPT(CAS), analogue of the δHF term, i.e., the δCAS correc- with only 5 active orbitals per monomer, pre- tion. These two methods become equivalent if dicts total interaction energy as well as indi- the δCAS term is computed from formula simi- vidual energy contributions with errors below lar to Eq. (61) with CAS supermolecular energy 3% with respect to the FCI results. In con- and SAPT(CAS) energy components. While trast, SAPT schemes based on single-reference there is no advantage of SAPT(CAS)+δCAS description of the monomers, SAPT(HF) and procedure over CAS+DISP when both employ SAPT(DFT), fail dramatically when entering the same CAS wave functions, using CAS func- the strongly-correlated regime. tions of different levels could be beneficial. Such The proposed multiconfigurational SAPT an approach would employ CAS in the mini-

21 mal active space to evaluate the δCAS term and damage to the accuracy involves density fit- higher level CAS for description of monomers in ting or Cholesky decomposition techniques rou- 116–118 SAPT(CAS). Similar to δHF correction, tinely applied in single-reference SAPT ap- 81,85,97,128–131 addition of δCAS would be recommended not proaches. only for excited-state complexes but also for Acknowledgement The authors would like ground state polar systems. to thank Grzegorz Chałasiński for helpful dis- To better characterize the performance of cussions and commenting on the manuscript. SAPT(MC) for many-electron systems, we K. P. and M. H. were supported by the Na- compared different SAPT schemes against a tional Science Centre of Poland under Grant standard single-reference data set of nonco- No. 2016/23/B/ST4/02848. valently bound dimers. The individual energy components from both SAPT(GVB) and SAPT(CAS) calculations are more accurate References than their SAPT(HF) counterparts. This holds also for total interaction energies, where we (1) Balakrishnan, N. Perspective: Ultracold observe a partial error cancellation between molecules and the dawn of cold controlled polarization and exchange terms in the sec- chemistry. J. Chem. Phys. 2016, 145, ond order. The correlated SAPT schemes in- 150901. cluded in the comparison, i.e., SAPT(DFT) and SAPT2+(CCD), are systematically better than (2) Dulieu, O., Osterwalder, A., Eds. Cold our multiconfigurational SAPT which should be Chemistry; Theoretical and Computa- attributed to two factors. One is that ERPA- tional Chemistry Series; The Royal So- based SAPT misses the majority of dynamical ciety of Chemistry, 2018; pp P001–670. correlation within the monomers as a result of (3) Pawlak, M.; Żuchowski, P. S.; employing GVB or CAS wave functions with Jankowski, P. Kinetic Isotope Effect small active spaces. Second is the quality of re- in Low-Energy Collisions between Hy- sponse properties (exitation energies and tran- drogen Isotopologues and Metastable sition density matrices) from ERPA equations. Helium Atoms: Theoretical Calculations 119 Unlike the full linear response (LR-MCSCF, Including the Vibrational Excitation of 120 equivalent to MCRPA ), ERPA includes re- the Molecule. J. Chem. Theory Comput. sponse of the orbitals only. 2021, 17, 1008–1016. The proposed formulation of multireference SAPT can be applied with wave func- (4) Řezáč, J.; Hobza, P. Benchmark Calcula- tion methods capable of handling large active tions of Interaction Energies in Noncova- spaces, such as density-matrix renormalization lent Complexes and Their Applications. group (DMRG),121,122 generalized active space Chem. Rev. 2016, 116, 5038–5071. (GAS)123,124 or v2RDM-driven CAS.125 An ef- (5) Patkowski, K. In Chapter One - Bench- ficient alternative is offered by range-separated mark Databases of Intermolecular Inter- multiconfigurational DFT.126,127 action Energies: Design, Construction, Without additional approximations SAPT(MC) and Significance; Dixon, D. A., Ed.; scales with the sixth power of the molecu- Annu. Rep. Comput. Chem.; Elsevier, lar size. The computational bottlenecks are 2017; Vol. 13; pp 3–91. the solution of the full ERPA eigenproblem and evaluation of the exchange-dispersion en- (6) Patkowski, K.; Cencek, W.; ergy formula,53 both involving steps with a Jankowski, P.; Szalewicz, K.; Mehl, J. B.; 3 3 nOCCnSEC cost, where nOCC = Ms1 + Ms2 and Garberoglio, G.; Harvey, A. H. Potential nSEC = Ms2 + Ms3 . energy surface for interactions between A feasible path to reduce the scaling and in- two hydrogen molecules. J. Chem. Phys. crease the efficiency of the method with no 2008, 129, 094304.

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