Spin-flip pair-density functional theory: A practical approach to treat static and dynamical correlations in large molecules
SSpin-flip pair-density functional theory: A practical approach to treat static anddynamical correlations in large molecules
Oinam Romesh Meitei and Nicholas J. Mayhall ∗ Department of Chemistry, Virginia Tech, Blacksburg, Virginia 24061, USA
We present a practical approach to treat static and dynamical correlation accurately in largemulti-configurational systems. The static correlation is accounted for using the spin-flip approachwhich is well known for capturing static correlation accurately at low-computational expense. Unlikeprevious approaches to add dynamical correlation to spin-flip models which use perturbation theoryor coupled-cluster theory, we explore the ability to use the on-top pair-density functional theoryapproaches recently developed by Gagliardi and co-workers (JCTC, 10, 3669, 2014). Externalrelaxations are carried out in the spin-flip calculations though a restricted active space frameworkfor which a truncation scheme for the orbitals used in the external excitation is presented. Theperformance of the approach is demonstrated by computing energy gaps between ground and excitedstates for diradicals, triradicals and linear polyacene chains ranging from naphthalene to dodecacene.Accurate results are obtained using the new approach for these challenging open-shell molecularsystems.
I. INTRODUCTION
An accurate computational description of large molec-ular systems with multi-configurational characters orstrongly correlated systems still remains a challenge dueto the lack of a general approach that delivers high ac-curacy while remaining computationally affordable.The most widely used approach for treating multi-configurational systems is the complete active space self-consistent field (CASSCF) method in which the full con-figurational interaction (CI) is solved within a chosenactive space with the orbitals and the CI coefficientsoptimized . The active space is typically selected witha prior knowledge of the important orbitals contributingto the target chemical problem or the electronic state (al-though recent advances are making it possible to algorith-mically select active space orbitals ). When supplied awell-defined active space, the method has been shownto reliably recover static correlation. However in doingso, the bulk of the dynamical correlation is absent, pre-cluding any hope for quantitative predictions. A generalstrategy adopted to recover the dynamical correlation isto use the complete active space second-order perturba-tion theory (CASPT2) approach. As dynamical corre-lation is well-described by perturbation theory, CASPT2provides significant accuracy improvements compared toCASSCF, albeit with increased computational cost andmemory requirements. The CASSCF method itself islimited to 20 electrons in 20 orbital space with the cur-rent state-of-the-art computational resources. With amotivation to enable computation using larger activespaces, several approaches have been developed suchas various flavors of selected configuration interaction(SCI) approaches , the density matrix renormaliza-tion group (DMRG) , generalized active-space self-consistent field (GASSCF) method , the 2-RDM drivenCASSCF (v2RDM-CASSCF) method or the full CIquantum Monte Carlo (FCIQMC) method . Withthese reduced cost methods, larger active spaces can be used which begin to account for dynamical correlation aswell, but generally a large amount of dynamical correla-tion is still missing.An alternative approach for treating multi-configurational systems is the spin-flip (SF) method .The SF method is a relatively simple approach whichuses a well-defined single-determinant reference onwhich spin-flipping excitations are carried out to accesselectronic states starting from a high-spin configura-tion. The SF methods have been further extended toremain spin-pure, including external relaxations thougha restricted active space framework (RAS- n SF) .Recently, we have also proposed a redox SF approachthat simultaneously accounts for both spin and spatialdegeneracies which the SF approach alone cannothandle . The SF approaches have favorable com-putational scaling for practical applications to largemulti-configurational systems. In addition to allowingtraditionally single-reference methods to be applied tomulti-reference problems, (e.g., coupled-cluster theory),spin-flip also reduces the dependence on user input,since the active-space is defined automatically once thereference spin state has been chosen. As a result, priorknowledge of the orbitals that are important for thetarget electronic state is not necessarily required. Here,the active space is simply chosen as the singly occupiedorbitals. The external excitation space in RAS- n SFare simply the doubly occupied and the virtual orbitalsalthough knowledge of the important contributingorbitals can be utilized as well. Another advantage ofthe SF approach is the treatment of the electronic statesincluding the ground state on equal footing. However,because the RAS- n SF approach excludes all doubleexcitations outside of the active space, it can onlydeliver qualitative accuracy due to the lack of dynamicalcorrelation.
An appealing approach to recover dynamical correla-tions for practical applications to large molecular sys-tems is the combination of multi-configurational wave- a r X i v : . [ c ond - m a t . s t r- e l ] F e b function methods with a density functional theory (DFT)based method. Here, the idea is to use a qualita-tively correct multi-configurational wavefunction to cap-ture the static correlation and include a DFT-based de-scription of the dynamical correlation. The idea wasfirst introduced by Lie and Clementi where they demon-strated that the DFT results can be substantially im-proved by adding the correlation energy obtained froma multi-configurational wavefunction . However, theapproach does not separate the static and dynamicalcorrelation, thereby double counting the correlation en-ergy. Several approaches have emerged in this linewhich attempts to overcome the deficiencies and im-prove over the previous works . Notable exampleinclude the approach by Colle and Solvetti , CAS-DFT by Savin and co-workers , modified CAS-DFTapproaches by Grafenstein and Cremer , Yamaguchiand co-workers and CAS-DFT using on-top pair den-sity by Gusarov and co-workers . A nice review ofthe available approaches are provided in Ref 63. Themain challenge in combining multi-configurational wave-function with a DFT-based description of the dynamicalcorrelation is due to the double counting of electron corre-lation. The multi-configurational wavefunction generallyincludes some part of the dynamical correlation withinthe active space. The second complication is the “sym-metry dilemma” in KS-DFT . The spin densities of themulti-configurational wavefunctions are not compatiblewith standard density functionals for low spin states.Recently, a new approach has been proposed byGagliardi and co-workers to include DFT-based corre-lation energy to a multi-configurational wavefunctionwhich addresses both the double counting of correlationenergy as well as the “symmetry dilemma”. Themethod referred to as multi-configurational-pair den-sity functional theory (MC-PDFT) only uses the multi-configurational wavefunction to compute the classicalCoulomb and the kinetic energy while the rest of theexchange-correlation energy and correction to the ki-netic energy is obtained from DFT using the on-toppair density functionals. The success of the method hasbeen demonstrated in conjunction with CASSCF ,GASSCF , DMRG and v2RDM-CASSCF .In this work, we propose an alternate approach tocombine MC-PDFT with the RAS- n SF approaches andour recently developed SF-IP/EA approach. The goalis to recover dynamical correlation energy in the SF ap-proaches using MC-PDFT for practical applications tolarge multi-configurational systems at a considerably lowcomputational cost. A brief description of the RAS- n SFapproaches and the MC-PDFT method is provided inSection II A and II B. A practical scheme is also presentedto further reduce the computational cost in the MC-PDFT calculation. Computational details are providedin Section III. The performance of the approach is demon-strated by computing energy gaps between spin states ofchallenging biradicals, triradicals and polyacenes rangingfrom naphthalene to dodecacene. The results are pre- sented in Section IV A, IV B and IV C. Finally, a sum-mary is provided in Section V.
II. METHODSA. RAS- n -SF and combined IP/EA approach We give a short overview of the RAS- n -SF and thecombined IP/EA approaches. For further details of themethods, we refer the reader to Refs. 29, 72, and 44.The spin-flip (SF) approach, proposed by Krylov, pro-vides an efficient way to model a large number of multi-configurational problems using only a single reference de-terminant. The key idea in SF approach is that while thevarious S z multiplets of high spin states have identicalelectronic energies, states with maximum S z have a sin-gle determinant representation, while lower S z multipletsare highly multi-configurational. SF approach leveragesthis degeneracy, by optimizing the orbitals for the sin-gle configurational high spin (S z = S) state, then usesspin-flipping excitations to access the target S z manifoldof states. Because both high-spin and low-spin statesappear in the target S z space, the high-spin states andlow-spin states are treated on an equal footing.As an example, the spin-degeneracy in the valencebonding orbitals (cid:12)(cid:12) σ π (cid:11) with the anti-bonding σ ∗ and π ∗ orbitals upon bond dissociation of N molecule canbe resolved using the SF approach. Using a high-spinheptet state ( (cid:12)(cid:12) σ π π ∗ σ (cid:11) ) with m s = 3 as the referencestate, the ground singlet state can be accessed using a3SF operator. | Ψ (cid:105) = (cid:88) i TABLE I: Truncation schemes used for the RAS1 and RAS3active space in the RAS(S) excitation in the natural orbitalbasis. The truncations are performed separately for the RAS1and RAS3 space.Scheme Abbreviation DescriptionI RAS-SS A state specific approach, truncatethe active space for each state sepa-rately. Each state has different num-ber of orbitals in the active spaceII RAS-SA A state average (SA) approach, trun-cate based on an average density ofthe participating states. Each statehas the same number of orbitals inthe active space.III RAS-eff An effective SA approach, the activespace is defined by the maximum ofthe number of orbitals in each statefor a given threshold (separately forRAS1 and RAS3). state. External relaxation effects are taken into accountusing RAS(S) by allowing the full set of singles excita-tions defined by (h, p, hp) . The approach withoutthe external effects is denoted by CAS- n SF. Overall, theRAS- n -SF and the combined IP/EA approaches providesqualitatively accurate descriptions of static correlation asdemonstrated in earlier works with the only disadvantagebeing the neglect of dynamical correlations. B. Multi-configurational pair-density functionaltheory We shortly review the MC-PDFT approach whichcombines multi-configurational methods with DFT-basedmethods without incurring double counting of correla-tion energy and the symmetry dilemma in the context ofKS-DFT . By using 1- and 2-RDMs from a multi-configurational calculation, the electronic energy in theMC-PDFT framework is given by E = (cid:88) pq h pq D pq + 12 (cid:88) pqrs g pqrs D pq D rs + E ot [ ρ, Π] (3)where h pq and g pqrs are the one- and two-electron in-tegrals respectively, D pq is the 1-RDM and E ot is theon-top energy with ρ and Π being that total densityand the on-top pair density respectively. The last twoterms are the classical Coulomb term and an on-top den-sity functional term which replaces the two-electron con-tribution in the usual electronic energy expression formany-electron systems. Here, the one-electron contribu-tion containing the kinetic and electron-nuclear potentialenergy as well as the classical electrostatic contribution are directly taken from the multi-configurational wave-function. The remainder exchange and the correlationcontributions are folded into the on-top pair density func-tional.The total density ρ and the on-top pair density, Π aredefined in terms of 1-RDM and 2-RDM obtained fromthe multi-configurational calculations respectively as ρ ( r ) = (cid:88) pq φ p ( r ) φ q ( r ) D pq (4)Π( r ) = (cid:88) pqrs φ p ( r ) φ q ( r ) φ r ( r ) φ s ( r ) D pqrs (5)The on-top pair density functional used within theMC-PDFT formalism is simply obtained with “trans-lated” existing exchange-correlation functionals em-ployed in standard KS-DFT. The derivation of the“translated” and the corresponding “fully translated”functionals are provided in Ref. 73.In the present RAS- n SF(-IP/EA)-PDFT strategies,the appropriate RDMs entering in Equation 3 are ob-tained from the RAS- n SF(-IP/EA) wavefunction pre-sented in the previous section. Because the computationof the 2-RDM in the combined spaces of RAS1, RAS2,and RAS3 would prevent application to large systems, wepropose the following truncation scheme which improvesefficiency while producing only small affects on the finalenergies:1. Perform RAS- n SF(-IP/EA) using the full space forthe external RAS(S) excitations.2. Diagonalize 1-RDMs of the RAS1 and RAS3 spacesseparately to obtain natural orbitals for the RASsubspaces. The RAS- n SF methods are invariantwith respect to orbital rotations within a RAS sub-space.3. Define a threshold for the orbital occupation num-ber to separately truncate the RAS1 and RAS3 ac-tive space for the RAS(S) excitation in the naturalorbital basis. The truncation schemes are tabulatedin Table I4. Using the truncated RAS1/RAS3 spaces repeatthe RAS- n SF(-IP/EA) calculation to obtain therequired one- and two-RDMs for the RAS- n SF(-IP/EA)-PDFT calculation. III. COMPUTATIONAL DETAILS In order to determine the accuracy of the RAS- n SF(-IP/EA)-PDFT approach presented in this work, we com-pare to existing accurate methods for the computation ofthe doublet-quartet gap of triradical and singlet-triplet n−1m−xylylenen−polyacene p−benzyneo−benzyne m−benzyneTMB TMB − FIG. 1: Schematic representations of the poly-radicals andpolyacenes considered in this work. gaps in aromatic biradicals and linear polyacenes and toexperimental results wherever available. The structuresof 1,3,5-trimethylenebenzene (TMB) and its negative ion(TMB − ) were optimized with UB3LYP/cc-pVDZ. Thestructure of meta-xylylene was taken from Ref. 74, the ortho -, para - and meta -benzyne radicals from Ref. 75, 76and the polyacenes from naphthalene to dodecacene fromRef. 69. A schematic representation of the poly-radicalsand polyacenes considered in this work is provides in Fig-ure 1.The doublet state of neutral 1,3,5-trimethylenebenzenetriradical was obtained by performing a single SF opera-tion on the quartet reference state with m s = . The sin-glet and triplet state of its negative ion was obtained byperforming 1SF-EA operation on the neutral high-spinquartet state obtained by oxidizing the target anionicstate. For the benzyne radicals as well as the polyacenes,the singlet states are obtained by performing only a sin-gle SF operation on the triplet reference state as done inRefs. 75 and 35.Augmented Dunning’s correlation consistent aug-cc-pVDZ basis set was employed for all the calculations .The RAS- n SF-PDFT and the RAS- n SF-IP/EA-PDFTcalculations were performed with tPBE, ftPBE, tBLYPand ftBLYP on-top density functional. Cholesky decom-position was used in all the two-electron integral calcula-tions with a decomposition threshold of 10 − a.u. TheRAS- n SF(-IP/EA) and the RAS- n SF(IP/EA)-PDFT calculations were performed using OpenMolcas andthe truncation scheme described in Section II B was com-puted using an in-house python plug-in code to Open-Molcas. TMB−(b) Reference + ...+ Reference + + ...+ Target statesTarget statesTarget states1SF−EA1SF(a) TMB(c) MX FIG. 2: Illustration of the operation of 1SF and 1SF-EA onthe high spin reference state for (a) TMB, (b) TMB − and(c) MX. The orbitals shown corresponds to the non-bondingorbitals and the spin-flips are indicated by red in color. IV. RESULTS AND DISCUSSIONA. All π -polyradicals The 1,3,5-trimethylenebenezene (TMB) presents as aprototypical high-spin all- π triradical . The triradicalsystem has an open-shell quartet ground state with thethree unpaired electrons each occupying the nondisjoint π non-bonding orbitals (NBO). Here, the three NBOsare nearly degenerate and most of the low-lying stateshave heavily multiconfigurational wavefunctions. Accu-rate theoretical studies are available for the energy gapbetween the ground state and the energetically lowestdoublet state . The doublet state is accesible withjust a single spin-flip operation on the high spin quartetstate ( m s = ), see the sketch in Figure 2(a).The negative ion of TMB (TMB − ) on the other handcannot be described solely by the spin-flip approach.Four electrons occupy the nearly degenerate three NBOsand it is ambigous to which of the three orbitals should bedoubly occupied. The ground state is a triplet state andthus is basically a ππ -diradical . The ground stateas well as the low-lying singlet states can be accessedusing 1SF-EA operation on the quartet reference stateobtained by oxidizing the anion. This is illustrated inFigure 2(b).Another prototypical all π -diradical is the meta -xylylene (MX) with two electrons distributed in twonearly degenerate NBOs. The singlet-triplet gap of MXhas been intensively studied both theoretically as well asexperimentally . The diradical has an open-shelltriplet ground state. The low-lying singlet state can beaccessed using a single SF operation on the triplet refer-ence state ( m s = 1), see the illustration in Figure 2(c). TMB RAS(S)-SF( eff )RAS(S)-SF(SS)RAS(S)-SF(SA) RAS(S)-SF-PDFT( eff )RAS(S)-SF-PDFT(SS)RAS(S)-SF-PDFT(SA) TMB − MX1e − − − − − − − Threshold E n e r g y g a p ( k c a l / m o l ) FIG. 3: Convergence of energy gap w.r.t. truncation of natu-ral orbitals, doublet-quartet gap of TMB (top), singlet-tripletgap of TMB − (center) and MX (bottom). Natural orbitalswith occupation number < threshold or > . − threshold aretruncated in the RAS- n SF(-IP/EA)-PDFT computation fol-lowing the truncation schemes (in parenthesis), SS, SA, eff asoutlined in Table I. Figure 3 plots the convergence of the different trunca-tion schemes described in the last part of Section II B.The natural orbitals computed from the 1-RDM of theRAS3 active space with occupation number smaller thana given threshold are truncated in the RAS(S) excita-tion scheme. The doubly occupied orbitals in the nat-ural orbital basis constructed from the 1-RDM of theRAS1 space is truncated with occupation number largerthan the difference of two and the given threshold. Inthe three truncation schemes namely, SS, SA, and eff ,the distinction between the SS and the eff schemes van- ishes at the convergence. The RAS(S)- n SF-PDFT con-verges with a threshold of 0.0001 for all the truncationschemes with the exception of SS truncation scheme inTMB − . Here, the difference in the energy gap betweenspin states of TMB − obtained using the SS and eff trun-cation scheme is 0.13 kcal/mol at the tightest thresholdconsidered which is well within the target accuracy ofthe presented approach. Furthermore, the SA trunca-tion scheme is slightly slower to converge for TMB − andMX, with the largest difference at the minimum thresh-old considered being 0.24 kcal/mol. Hereafter, we willconsider the eff truncation with a threshold of 0.0001 forthe RAS3 orbitals and 1.9999 for the RAS1 orbitals.The energy gap between spin-states of TMB, TMB − and MX computed using CAS-SF, CASSCF, RAS(S)-SF and the corresponding MC-PDFT method using thetPBE functional is provided in Table II. Literature val-ues for the energy gaps from previous electronic struc-ture calculations as well the experimental energy gapfor MX are also reported in therein. For the TMB andTMB − radicals, experimental values were not availableand so comparisons are made to the available literaturevalues. First, we consider the vertical doublet-quartet(DQ) energy gap of TMB radical. The application ofMC-PDFT on CAS-1SF which does not account for anyexternal relaxation effect decreases the DQ energy gap byalmost 15.0 kcal/mol. CAS-1SF-tPBE strongly underes-timates the DQ energy gap as compared to the previ-ously reported gaps from CASPT2 and DDCI methods.Contrary to this, CASSCF-tPBE increase the DQ en-ergy gap computed with CASSCF (with CAS(2,2)) by7.3 kcal/mol. Interestingly, the CASSCF-tPBE energygap is very close to a previously reported CASSCF valuewith a larger active space (CAS(9,9)). On the other handRAS(S)-1SF-tPBE reduces the DQ energy gap from thebare RAS(S)-1SF by almost 10 kcal/mol. Comparing tothe available DDCI and CASPT2 literature, the inclusionof dynamical correlation in the MC-PDFT frameworkgreatly improves the bare RAS(S)-1SF doublet-quartetenergy gap.Next, we consider the vertical singlet-triplet (ST) gapfor the anion of TMB. Here, CAS-1SF as well as CAS-1SF-tPBE predicts a singlet ground state as opposed tothe experimental evidences for a high spin triplet groundstate for the anion . The ST gap of bare CASSCFwith CAS(2,2) active space is increased by about 2.0kcal/mol with the corresponding MC-PDFT method. Incontrast to this, RAS(S)-1SF-tPBE decreases the ST gapcomputed using the bare RAS(S)-1SF by 1.7 kcal/mol.Here, the ST gap agrees well with an existing CASPT2result available in literature based on CASSCF orbitalwith CAS(10,9).For the MX diradical, the experimental ST energy gapis available and so comparisons can be made to the adia-batic ST energy gap from the various methods tabulatedin Table II. The CAS-1SF-tPBE strongly underestimatesthe ST energy gap which collectively with the results ob-tained for the TMB and TMB − radicals suggest that theone- and two-RDMs from the bare CAS-1SF without anyexternal relaxation effect is not accurate to be used inthe MC-PDFT equation. The CAS-1SF-tPBE even pro-duces qualitatively wrong ordering of spin-states in theTMB − radical. On the other hand, the CASSCF-tPBEbased on CAS(2,2) active space underestimates the ex-perimental ST gap of MX by 2.2 kcal/mol whereas theRAS(S)-1SF-tPBE underestimates the energy gap by al-most 4 kcal/mol. In literature, the different wavefunctionbased methods tend to overestimate the ST gap of MX,see Table II. LCCQMC based on a stochastic approachfrom Alavi et. al. provides the ST gap of MX very closeto experimental result . The difficulty in achieving anST gap using wavefunction-based approaches compara-ble to the experimental result have been demonstratedin earlier works . An extensive study using differentwavefunction methods and DFT methods for the ST gapof MX radical can be found in Ref. 93. The authorstherein pointed out the importance of including dynam-ical correlation with high accuracy . In our case, whilethe bare RAS(S)-1SF which also includes some dynami-cal correlation overestimates the experimental ST gapby 4.3 kcal/mol, RAS(S)-1SF-tPBE underestimates theST by 3.9 kcal/mol. B. Benzyne radicals The ortho -, meta - and para -benzyne isomers havebeen used as benchmarks for new theoretical approachesand at the same time challenging because of the stronglycorrelated biradical electrons . The benzyne iso-mers have a closed-shell singlet state with the diradicalcharacter correlating with the distance between the un-paired electrons. As a result, the energy gap betweenthe ground singlet and the lowest excited triplet state de-creases following the ortho -, meta - and para - sequence.The adiabatic singlet-triplet gaps of the benzyne iso-mers computed using bare CAS-1SF, CASSCF, RAS(S)-1SF and the corresponding MC-PDFT methods are pre-sented in Table III along with the available experimentalvalues and selected literature values for comparison. TheRAS(S)-1SF and RAS(S)-1SF-PDFT ST energy gapswere computed using truncated natural orbitals outlinedin Section II B. A threshold of 0.0001 and 1.9999 wasused for the virtual and doubly occupied space respec-tively following the results from Section IV A in the ex-ternal RAS(S) excitation scheme. The different trunca-tion schemes, namely, SS, SA and eff resulted in simi-lar ST gaps and so only results from the eff truncationscheme is presented. The largest difference in the STgaps between the different truncation schemes was only0.22 kcal/mol. Also presented in Table III are the resultsfrom employing different on-top functionals, viz., tPBE,tBLYP and the fully translated variants. Overall the re-sults obtained from using tPBE performs better than theother functionals and so we focus our discussion only tousing the tPBE functional. TABLE II: Energy gaps between spin states of TMB b (ver-tical doublet-quartet gap), TMB − a (vertical singlet-tripletgap) and MX b (adiabatic singlet-triplet gap) obtained us-ing RAS(S)-1SF and RAS(S)-1SF-tPBE. The energy gap forTMB − was obtained using 1SF-EA operation. All resultswith aug-cc-pVDZ basis set and natural orbitals truncatedwith a threshold of 0.0001. Units are in kcal/mol.TMB TMB − MXCAS-1SF 17.6 -1.9 4.3CAS-1SF-tPBE 2.8 -6.1 1.7CASSCF c c − ZPE –– –– 9.9Literature 15.7 d ,11.2 e ,13.6 f g , 1.9 h i ,11.7 j ,11.3 k ,11.8 l , 9.5 ma Optimized geometry with UB3LYP/cc-pVDZ b Geometry from Ref. 74, UB3LYP/6-311G(d,p) c CAS(3,3) for TMB, CAS(4,3) for TMB − and CAS(2,2) for MX d CASSCF(9,9)/6-31G(d,p) from Ref. 84 e DDCI/6-31G(d) using localized orbitals and complete virtual or-bitals (fully variational) from Ref. 85 f CASPT2/6-31G(d,p) with CASSCF(9,9) orbitals from Ref. 84 g CASSCF(10,9)/ANO-L and CASPT2/ANO-L using the sameCASSCF orbitals from Ref. 89 h state averaged MS-CASPT2/ANO-L from Ref. 89 i CASSCF/6-31G(d) with CAS(8,8) from Ref. 96 j CASPT2/6-31G(d) with CASSCF orbitals with CAS(8,8) fromRef. 96 k EOM-SF-CCSD/6-31G(d) from Ref. 92 l Multireference second-order Møller-Plesset/aug-cc-pVTZ fromRef. 93 m LCCQMC/6-311++g(d,p) from Ref. 95 The CAS-1SF and the CASSCF methods are based onminimal active spaces, i.e., CAS-1SF does not accountfor any external relaxations while the CASSCF methodis based on CAS(2,2) active space. This is reflected in theST gaps obtained from CAS-1SF-tPBE as compared tothe experimental results (ZPE included) for ortho - and meta -benzyne radicals. In spite of the small active space,CASSCF-tPBE on the other hand performs well for the ortho -benzyne radical, within 1.0 kcal/mol of the exper-imental value whereas the ST gap is overestimated by3.4 kcal/mol for the meta -benzyne radical. Contrary tothis, RAS(S)-1SF-tPBE underestimates the experimen-tal ST gap by 3.2 kcal/mol for ortho -benzyne while itexactly matches the result for the meta -benzyne. Here,the ST gap of ortho -benzyne obtained from RAS(S)-1SFis within 0.1 kcal/mol of the experimental value. We notethat the corresponding bare RAS(S)-1SF method ac- TABLE III: Adiabatic singlet-triplet energy gap for ortho -, meta - and para -benzyne using the RAS-1SF and RAS-1SF-PDFT method. All results with aug-cc-pVDZ basis set andnatural orbitals truncated with a threshold of 0.0001. Unitsare in kcal/mol. o -benzyne m -benzyne p -benzyneCAS-1SF 16.7 2.3 0.6CAS-1SF-tPBE 30.2 14.1 3.4CASSCF a a b − ZPE c d ,32.6 e , 36.8 f g d ,19.0 e , 19.6 f ,20.6 g d , 5.8 e ,4.9 f , 4.0 ga using CAS(2,2) active space b expt. value from Ref. 97 c ZPE correction with SF-DFT/6-311G* from Ref. 75 d CASSCF using CAS(8,8) active space from Ref. 98 e CASPT2 using CASSCF(8,8) orbitals from Ref. 98 f ic-MRCCSD(T) based on CAS(2,2) from Ref. 99 g SF-CCSD(T) from Ref. 100 counts for some dynamical correlations but we emphasiseagain that the MC-PDFT method avoids double countingof electron correlation. Comparing to literature, we findthat for the ortho - and meta -benzyne radicals, RAS(S)-1SF-tPBE is more accurate than CASPT2 (based onCASSCF orbitals with CAS(8,8) active space) whichis often a method of choice for including dynamical cor-relations in modelling strongly correlated systems.The para -benzyne radical presents as a more challeng-ing case and has been the focus in comparing the ST gapobtained from various highly accurate theoretical meth-ods to the available experimental result .The discrepency between experimental and theoreticalresults have been discussed in earlier works on thediradical . K¨ohn and co-workers have pointed outthe possibility of wrong assignments of the singlet andtriplet states in the experiment . The ST gap forthe diradical presented in Table III further supports thispossiblity. The ST gap from CAS-1SF-tPBE in which thebare CAS-1SF does not account for any external relax-ation agrees well within 0.1 kcal/mol of the experimen-tal result in contrast to RAS(S)-1SF-tPBE. This wouldmean that CAS-1SF provides a more accurate one- andtwo-RDMS than the RAS(S)-1SF in the MC-PDFT equa-tion in contrast to the results obtained for the other rad- icals discussed before. Furthermore, the ST gap formRAS(S)-1SF agrees well with other literature values ob-tained from CASPT2 (based on CASSCF orbitals withCAS(8,8) active space) and ic-MRCCSD(T) . The STgap also agrees well with the CASSCF-tPBE. Length of acenes S − T g a p ( k c a l / m o l ) RAS(S)-1SFRAS-1SF-PDFTCCSD(T)JSD-LRDMCv2RDM-CASSCF-PDFTexpt.GAS-PDFT FIG. 4: Adiabatic singlet-triplet gap with increasing lengthof acenes. C. Polyacenes In the previous sections, we have shown that the RAS- n SF(-IP/EA)-PDFT approach can describe challengingmedium-sized biradical and triradical molecular systemswith good accuracy. In this section, we demonstrate theperformance of our approach for larger systems: the poly-acenes, ranging from naphthalene to dodecacene. Thepolyacenes have a singlet ground state, with the openshell character increasing as the number of the benzenering increases. The singlet-triplet (ST) gap in this casecorrelates with the acene length which exponentialy de-creases.Table IV reports the adiabatic singlet-triplet gapsof the polyacenes obtained from RAS-1SF and RAS-1SF-PDFT. Here, the on-top tPBE functional was em-ployed. Using the triplet state (m s =1) as the refer-ence, the singlet state is accessed by performing onlya single SF operation on the reference state. The ex-ternal RAS(S) excitations were carried out from allthe valence π orbitals. The adiabatic ST gaps areplotted in Figure 4 along with selected literature val-ues, namely, CCSD(T) extrapolated according to fo-cal point analysis , Monte Carlo (JSD-LRDMC) ,v2RDM-CASSCF-PDFT and GAS-PDFT (WFP-3 TABLE IV: Adiabatic singlet-triplet energy gap ( E triplet − E singlet ) for polyacenes. aug-cc-pVDZ basis set are used uptoHexacene and cc-pVDZ basis set for the remainder. RAS(S)-1SF-PDFT results are obtained using tPBE functional. Units arein kcal/mol. Acene CAS-1SF CASSCF a RAS(S)-1SF expt. expt. bare -PDFT bare -PDFT bare -PDFT − ZPE b Naphthalene 52.7 57.4 64.7 76.8 56.2 62.3 60.9 , 61.1 , 43.1 a using CAS(2,2) b ZPE correction with B3LYP/6-31G(d,p) from Ref. 70 partitioning) . Several other literature values are avail-able, a detailed analysis of the available ST gaps in liter-ature are provided in Ref. 113 and 114The RAS(S)-1SF-PDFT adiabatic ST gaps have agood agreement with the corresponding experimental val-ues as compared to the RAS(S)-1SF values. Although thebare RAS(S)-1SF includes some dynamical correlation ,RAS(S)-1SF-PDFT improves over the bare RAS(S)-1SFmethod.Here, a delicate balance between the dynamic and thestatic correlation exist in the singlet and triplet state .From Table IV, it can be seen that the larger acenes agreevery well with the experimental value whereever avail-able while for anthracene the difference is ∼ .Table IV also presents ST gaps obtained from bare CAS-1SF, CASSCF(2,2) and the corresponding MC-PDFTapproaches. In both the cases, including the PDFTcorrection improves upon the bare method for the STgaps. However, the RAS(S)-1SF-PDFT performs betterin comparison to CAS-1SF-PDFT and CASSCF-PDFTfor the polyacene chains. This shows that RAS(S)-1SFprovides a more accurate 1- and 2-RDMs entering theMC-PDFT equation (Equation 3, 4 and 5) than CAS-1SFand CASSCF based on minal CAS(2,2) active space.For the acenes where experimental values are not avail-able, the RAS-1SF-PDFT values are lower than the avail-able literature values shown in Figure 4 although in somecases good agreements can be seen. The CCSD(T) aswell as the Monte Carlo values tend to overestimate theavailable experimental values while it can be seen that the largest reported acenes with these methods agreeswell with the RAS-1SF-PDFT values. The values re-ported for the v2RDM-CASSCF-PDFT differ between1.8 to 5.7 kcal/mol while the difference is between 0.3 to3.1 kcal/mol with the GAS-PDFT method.Another feature that can be observed in Figure 4 isthe smoothly decaying exponential curve for the RAS-1SF-PDFT ST gaps which can be fitted to the form aexp ( − bx ) + c . The fit then can be used to estimate anextrapolated ST gap for polyacene with infinite length.Using the fitting formula, E ST ( x ) = 138 . e ( − . x ) + 0 . 10 (6)The adiabatic ST gap estimated for an infinitely longlinear polyacene chain is 0.10 kcal/mol. The value is invery good agreement with the ST gap of 0.18 kcal/molfrom CCSD(T)/cc-pV ∞ Z reported in Ref. 111 and withpp-RPA/cc-pVDZ with a value between 0.0 and 2.3kcal/mol reported in Ref. 116. The later however is es-timated using vertical ST gaps. The ST gap is directlyrelated to the HOMO-LUMO gap and the obtained resultsuggest a closure for the HOMO-LUMO gap for inifinitlylong polyacene chains. Note that the result is in con-trast to Ref. 71 where the ST gap for the infinite chain isreported to be 4.87 kcal/mol with the v2RDM-CASSCF-PDFT/cc-pVTZ method and Ref. 115 with a value of5.06 and 5.37 kcal/mol obtained as their best estimateand from SF-CCSD/6-31+G(d,p) method respectively.With GAS-PDFT/6-31+G(d,p) (WFP-3 partitioning),the ST gap is reported to be 1.9 kcal/mol which is roughlyhalf way between our value and the ones obtained fromv2RDM-CASSCF-PDFT and SF-CCSD methods .Although the theoretical results presented above showqualitative consistency, the basis sets used to computethe ST gaps differ for the various approaches and so com-parison with the available literature values does not ac-count for basis set effects. V. SUMMARY In this work, we have presented a practical approachfor treating large multiconfigurational molecular systemwith a low computational cost. The new method, spin-flip pair-density functional theory (SF-PDFT) uses aspin-flip or a redox spin-flip operator to account for thestatic correlation while the dynamical correlation is de-scribed with DFT using the MC-PDFT approach. TheSF-PDFT method improves upon the result of bare spin-flip approach, thereby capturing the missing dynamicalcorrelations in the spin-flip approach. In cases where thespin-flip results are already close to the available exper-imental results, the SF-PDFT only changes the spin-flip results slightly ascertaining the relaibility of SF-PDFT.The reason is because in such cases, the static correlationdominates the electron correlation. The method yieldsgood accuracy for energy gaps between ground and thelowlying excited states for challenging open-shell molec-ular systems. The applicability range of the method wasdemonstrated by computing the singlet-triplet gap in lin-ear polyacene chains ranging from naphthalene to dode-cacene. 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