Singlet−Triplet Energy Gaps for Diradicals from Fractional-Spin

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J. Phys. Chem. A 2011, 115, 76–83

Singlet-Triplet Energy Gaps for Diradicals from Fractional-Spin Density-Functional Theory Daniel H. Ess,*,† Erin R. Johnson,*,‡ Xiangqian Hu,§ and Weitao Yang*,§ Department of Chemistry and Biochemistry, Brigham Young UniVersity, ProVo, Utah 84602, United States, School of Natural Sciences, UniVersity of CaliforniasMerced, 5200 North Lake Road, Merced, California 95343, United States, and Department of Chemistry, Duke UniVersity, Durham, North Carolina 27708, United States ReceiVed: September 28, 2010; ReVised Manuscript ReceiVed: NoVember 4, 2010

Open-shell singlet diradicals are difficult to model accurately within conventional Kohn-Sham (KS) densityfunctional theory (DFT). These methods are hampered by spin contamination because the KS determinant wave function is neither a pure spin state nor an eigenfunction of the S2 operator. Here we present a theoretical foray for using single-reference closed-shell ground states to describe diradicals by fractional-spin DFT (FSDFT). This approach allows direct, self-consistent calculation of electronic properties using the electron density corresponding to the proper spin eigenfunction. The resulting FS-DFT approach is benchmarked against diradical singlet-triplet gaps for atoms and small molecules. We have also applied FS-DFT to the singlet-triplet gaps of hydrocarbon polyacenes. 1. Introduction For open-shell, diradical molecules density-functional theory (DFT) has become a widely used quantum methodology due to its balance between speed and accuracy1-4 versus expensive and very accurate multireference ab initio methods.5-7 Treatment of diradical species8 is usually done using a broken-symmetry unrestricted formalism where opposite-spin electrons have separate Kohn-Sham (KS) orbitals that are allowed to polarize. This type of implementation is advantageous because it typically results in energies that are significantly lower than a singlereference, doubly-occupied restricted calculation. However, a significant drawback of this type of calculation is so-called spin contamination of the “DFT wavefunction”.9 Spin contamination is the result of the admixture of multiple spin states; the KS determinant wave function in DFT is not a pure spin state and is not an eigenfunction of the S2 operator. The amount of spin contamination depends highly on the exact system and the approximate 〈S2〉 value is typically 1.0 for diradicals, which indicates an equivalent mixture of singlet and triplet states. Although there is no doubt that the energy associated with an unrestricted calculation is a significant improvement over a restricted type calculation it is unclear how much spin contamination diminishes the accuracy of an open-shell calculation.10 Several procedures have been proposed to eliminate or correct spin contamination from electronic energies, most of which are post self-consistent field (SCF) processes.11 These schemes were typically proposed for Hartree-Fock or Moller-Plesset perturbation methodology but have found use in DFT methodology.12 The spirit of these spin-correction methods is to eradicate the effects of spin contamination in unrestricted calculations while retaining the benefit of the polarized spin. Schlegel and co-workers proposed and popularized the use of spin projection * To whom correspondence should be addressed. E-mail: [email protected] (D.H.E.); [email protected] (E.R.J.); [email protected] (W.Y.). † Brigham Young University. ‡ University of CaliforniasMerced. § Duke University.

and an approximated s+1 state annihilation operator for correlated ab initio perturbation theories.13 Yamaguchi and coworkers have also pioneered projection procedures that scale the contaminated energy by a fractional portion of the 〈S2〉 values of the contaminated and uncontaminated s+1 states.14,15 Although these spin projection schemes can give improved energies there are some known failures to this type of methodology, most notably the failure to reproduce bond dissociation potential curves.16 Alternative to unrestricted formalisms, Filatov and Shaik17 have recently proposed a spin-restricted, ensemble-referenced18 KS DFT method that does not contain spin contamination. Baerends and co-workers have also compared one-determinantal19 and ensemble KS solutions for CH2 and C2 using a fractional occupancy approach.20 Previous to this work, fractional occupancy approaches were also developed by Slater et al.,21 Dunlap and Mei,22 Averill and Painter,23 Wang and Schwarz,24 and others.25 These types of fractional occupancy approaches are forerunners to our methodology presented here. Most recently, Tully and co-workers have proposed the use of constrained DFT to control spin contamination in a SCF process.26 In this work we provide a theoretical framework for using single-reference closed-shell KS DFT ground states to describe diradicals by using fractional-spin (FS) occupancy.27,28 The FS concept was introduced by Cohen, Mori-Sa´nchez, and Yang as the proper way to describe the spin density of degenerate/ near-degenerate systems, such as in the dissociation of chemical bonds, and as a tool to analyze the deviation of approximate functionals from the exact conditions established for fractionalspin densities.27 This FS-DFT methodology is now used to compute singlet-triplet energies for Slipchenko and Krylov’s29 atom and molecule benchmark set of diradical30 species and is also applied to trimethylenemethane (TMM) and hydrocarbon polyacenes. Only pure density-functionals, PBE and LSDA, are considered to minimize the effect of nondynamical correlation error.27 Comparison is made to results obtained using the Ziegler, Rauk, and Baerends (ZRB) energy correction31 as well as the Yamaguchi spin projection technique.14,15 In this work we show that

10.1021/jp109280y  2011 American Chemical Society Published on Web 12/09/2010

Singlet-Triplet Energy Gaps for Diradicals

J. Phys. Chem. A, Vol. 115, No. 1, 2011 77

SCHEME 1: Singlet and Triplet Wavefunction Spin Combinations

when used directly as the many-electron wave function, has a corresponding electronic energy of

ES ) 1/2〈ψ10 |H|ψ10〉 + 1/2〈ψ30 |H|ψ30〉

(7)

which is a linear combination of the energies of the correct singlet and triplet eigenstates. Therefore, the correct energy of the open-shell singlet eigenstate is

〈ψ10|H|ψ10〉 ) 2ES - ET the FS method with pure density-functionals gives results that are more accurate than spin projection and the ZRB approach. Moreover, FS-DFT allows direct, self-consistent computation of electronic properties using the density corresponding to the correct spin eigenfunction. 2. Theory and Methodology Consider a system with two degenerate (or near degenerate, see Scheme 1), singly occupied atomic or molecular orbitals, which we will denote a and b. This system is described by four wave functions, consisting of one singlet state (eq 1) and three degenerate triplet states (eqs 2-4). There are also two wave functions, corresponding to closed-shell singlet states that are higher in energy

ψ10 ) 1/√2(|a(v)b(V)| - |a(V)b(v)|)

(1)

ψ30 ) 1/√2(|a(v)b(V)| + |a(V)b(v)|)

(2)

ψ31 ) |a(v)b(v)|

(3)

3 ψ-1 ) |a(V)b(V)|

(4)

3 are single Slater determinants, The triplet states ψ13 and ψ-1 which are straightforward to model in DFT. However, the openshell singlet state, ψ01, is a linear combination of two determinants and presents a challenge for modeling within the conventional KS DFT framework. As a result, calculations on open-shell singlet states of atoms or molecules generally use the lowest-energy, single-determinant solution

ψs ) |a(v)b(V)|

(5)

However, this is a broken symmetry state that is not an eigenfunction of the total spin operator. It does not give the same electron density or even the same energy as the correct open-shell singlet wave function ψ01. Indeed

ψS ) 1/√2ψ10 + 1/√2ψ30

(6)

which is a linear combination of the singlet and triplet states, with equal weighting, for our example of two degenerate, singly occupied orbitals. This broken-symmetry KS wave function,

(8)

where ET is the energy of any of the three degenerate triplet states. This implies that the singlet-triplet gap, ∆EST, is

∆EST ) 2(ES - ET)

(9)

and is twice the value computed from conventional electronic structure calculations using the lowest-energy single-determinant reference states. This discrepancy was pointed out by Ziegler, Rauk, and Baerends,31 and application of the resulting factorof-two correction was shown to give much improved electronic transition energies relative to experimental data. However, such a posteriori energy correction precludes self-consistent calculation of the correct electronic structure for the open-shell singlet eigenstate. An alternative is to turn to fractional-spin occupations in density-functional theory. Fractional-spin was introduced in DFT recently to characterize the conditions of the exact densityfunctional for describing strongly correlated systems, such as the dissociation of chemical bonds with zero spin density.27 In practical calculations with fractional orbital occupancy, we follow the extension27 of Janak’s32 work and extend the σ-spin first-order reduced density matrix to fractional occupancy δ by the definition

Fσ(r′, r) ) Σiniφi(r′)φ*(r) i where ni ) 1 for i < f, ni ) δ for i ) f, and ni ) 0 for i > f and f is the index of the fractionally occupied orbital. Here we use FS-DFT to describe open-shell singlet states with zero spin density. The density of the open-shell singlet eigenstate is

F(ψ10) ) 1/2F[a(v)] + 1/2F[b(V)] + 1/2F[a(V)] + 1/2F[b(v)] ) 1/2F[a(vV)] + 1/2F[b(vV)] (10) assuming that the singly occupied orbitals a and b are orthonormal. An ensemble can be constructed to give the same density as this ψ01 eigenstate, provided that fractional-spin occupations are used. This ensemble is an admixture of two Slater determinants |a(v)b(V)| and |a(V)b(v)| with the density given by

Fensmble ) cRF(|a(v)b(V)|) + cβF(|a(V)b(v)|)

(11)

where the mixing coefficients satisfy cR > 0, cβ > 0, and cR + cβ ) 1. In this work, we have set cR ) cβ ) 1/2, because of the symmetry of the states and to impose a closed-shell solution. With the fixed, symmetric fractional-spin occupation, the fractional-spin density is

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FFS ) 1/2F[a(vV)] + 1/2F[b(Vv)]

Ess et al.

(12)

which is identical to the density of the open-shell singlet eigenstate and we have the result

FFS ) F10

(13)

Therefore, if one knew the exact density-functional, calculations of any observable property using FFS would give results identical to those obtained from wave function theory using the exact open-shell singlet eigenfunction ψ01. An alternative conceptual view of this approach is to recognize that the same singlet density, F10, can be obtained from a single-determinant configuration if complex orbitals are used. For example, in an atom, where the degenerate pair of orbitals is px and py, then the singlet density can also be written in terms of the complex orbitals p1 and p-1 as

F10 ) 1/2F[px(vV)] + 1/2F[py(vV)] ) F[p1(vV)] - F[p-1(vV)] (14) This avoids the complication of applying DFT to multireference systems, where the exchange (or Fermi) hole does not obey the usual constraints.33 However, obtaining an appropriate singlereference configuration requires resorting to complex orbitals, which necessitates a current-density correction that will slightly lower the energies of the singlet states.34 Although the use of the fractional-spin DFT is advantageous, it is known to overestimate energies of systems due to neglect of nondynamical correlation in commonly used approximate functionals.27 To illustrate this we briefly review the fractionalspin analysis and static correlation issue in stretched H2 that was previously reported by Cohen, Mori-Sa´nchez, and Yang.27 If H2 is stretched to its dissociation limit, it becomes an openshell singlet with two electrons of opposite spin in two degenerate orbitals. Conventional DFT, without the use of fractional-spin, cannot properly dissociate spin-restricted H2 because the functionals implicitly model the exchange and correlation holes as localized around every reference point in space. The depth of the exchange hole is correctly -Fσ, the negative of the local spin-density at the reference point. However, in stretched H2, there is strong “multi-center” or “nondynamical” correlation with the opposite-spin electron. The resulting depth of the total exchange-correlation hole should be -F, the negative of the total density, which is twice the value of the spin density.35 Thus, local functionals do not predict sufficient correlation stabilization in stretched H2, resulting in a dissociation limit that is too high in energy. The situation is worse for Hartree-Fock (HF) theory,27 which completely neglects opposite-spin correlation effects. Therefore, HF gives a dissociation limit for spin-restricted H2 that is even higher than with local density-functionals. Hybrid functionals that include a fraction of HF exchange give a dissociation limit between local DFT methods and HF. Now consider only one of the H atoms in stretched H2. The density can be varied smoothly from the spin-polarized case (correct dissociation limit) to the spin-restricted case (incorrect dissociation limit) by allowing fractional-spins (Figure 1).27 If the fractional-spin is equal to 0 or 1, there is a single electron of up or down spin on this atom. A fractional-spin of 0.5 corresponds to the spin-restricted case (cR ) cβ ) 1/2). The energy difference between these two configurations is termed

Figure 1. Calculated, spin-restricted H2 potential energy curves (top) and hydrogen atom energies as a function of fractional-spin (bottom).

“static correlation” or “fractional-spin” error and results in a significant overestimation of the stretched H2 energy. The results in Figure 1 demonstrate that Hartree-Fock theory and hybrid functionals (such as PBE0) suffer particularly severely from fractional-spin error.27 Conversely, pure densityfunctionals, such as the local density approximation (LDA) and the PBE generalized gradient approximation, are shown to have the least fractional-spin error. For open-shell singlets at equilibrium geometries, the fractional-spin error should not be as severe as in the stretched H2 example. Thus, if a pure densityfunctional is employed, it may be possible to model open-shell singlets accurately with fractional occupancies via FFS, as in eq 12. This would allow direct, self-consistent modeling of electronic properties that are inaccessible using conventional single-determinant methods. The NH, NF, OH+, O2, CH2, and NH2+ singlet and triplet molecular geometries were optimized and confirmed to be minima by calculation of the Hessian using Gaussian 0336a and Gaussian 0936b with the (U)PBEPBE functional37 with the 6-31G(d,p) basis set. The Gaussian 03 local spin density approximation (LSDA) uses Slater exchange and the VWN5 correlation functional (SVWN).38 The Yamaguchi procedure14,15 was followed to produce spin-projected singlet energies (ESPS). In this procedure the difference in energy between the singlet and triplet energies is scaled by χ, which is a weighting ratio of singlet to triplet spin contamination values

ESPS ) ES + χ[ES - ET]

(15)

Singlet-Triplet Energy Gaps for Diradicals

J. Phys. Chem. A, Vol. 115, No. 1, 2011 79 TABLE 2: LSDA Open-Shell (OS, (U)LSDA), Open-Shell Spin Projected (SP), and Closed-Shell Fraction Spin (FS) Singlet-Triplet Gaps (∆EST)a (All Values in kcal/mol) C O Si NH NF OH+ O2 CH2 NH2+ MUE

TABLE 1: PBEPBE Open-Shell (OS), Factor of 2 Correction (ZRB), Open-Shell Spin-Projected (SP), and Fractional-Spin (FS) Singlet-Triplet Gaps (∆EST) (All Values in kcal/mol) exptl

OSa

ZRB

SPa

FSa

FSd

29.1 45.4 17.3 35.9 34.3 50.5 22.6 32.9 44.6

9.2 16.7 6.7 13.4 (13.5)b 11.1 (11.1)b 18.5 (18.5)b 9.0 (9.0)b 12.9 (12.8)b 16.2 (16.5)b 22.1 (22.1)b

18.4 33.4 13.4 26.8 22.2 37.0 18.0 25.8 32.4 9.5

18.4 33.4 13.5 26.9 (26.9)b 22.3 (22.3)b 37.0 (37.0)b 18.0 (18.0)b 25.8 (25.6)b 32.1 (33.0)b 9.5 (9.4)b

41.1 [41.2]c 33.9 [34.2]c 54.8 [55.0]c 26.2 [26.6]c 37.4 [37.4]c 47.8 [47.9]c 3.5 [3.6]c

34.0 50.0 21.5 41.0 34.1 54.8 26.5 36.2 47.5 3.7

a 6-311++G(2d,2p). pVQZ. d NUMOL.

b

Adiabatic singlet-triplet gaps.

χ ) (〈S2〉s/〈S2〉T)/[1 - (〈S2〉s/〈S2〉T)]

OSb

SPb

FSc

FSb

29.1 45.4 17.3 35.9 34.3 50.5 22.6 32.9 44.6

14.8 21.0 8.0 17.4 14.8 23.4 11.4 15.9 18.6 18.6

29.7 42.0 16.0 34.7 29.5 46.8 22.8 31.7 35.7 2.8

31.2 43.8 17.2 35.8 30.2 48.2 23.7 30.4 40.9 1.9

35.9 30.3 48.4 23.6 31.6 41.3 2.0

a Vertical singlet-triplet gaps. b 6-311++G(2d,2p). using the Perdew-Wang LSDA parameterization.

Figure 2. (U)PBEPBE/6-31G(d,p)-optimized structures.

C O Si NH NF OH+ O2 CH2 NH2+ MUE

exptl

c

aug-cc-

(16)

FS-DFT calculations were carried out in an in-house program (QM4D)39 as well as in the basis-set free NUMOL code.40 Molecular pictures in some of the figures were generated with CYLview.41 3.1. Benchmark Test Set. To validate the fractional-spin occupancy DFT (FS-DFT) approach we have benchmarked the PBE and LSDA singlet-triplet gaps (∆EST) for Slipchenko and Krylov’s test set that contains C, O, and Si atoms as well as NH, NF, OH+, O2, CH2, and NH2+ molecules (Figure 2).29 Table 1 gives the vertical and adiabatic PBE ∆EST values for these atoms and molecules. Column one gives the experimental values that Slipchenko and Krylov previously used to benchmark their ab initio spin-flip methodology.29 Column two gives the unrestricted broken symmetry open-shell energies. For the openshell ∆EST values the difference between the predicted gaps and the experimental gaps results in a mean unsigned error (MUE) of 22.1 kcal/mol. The vertical and adiabatic ∆EST values are very similar and give exactly the same MUE. This enormous error shows that spin polarization does not come close to compensating for the large amount of static (and dynamic) correlation energy associated with these diradicals species. In addition, each of these atoms and molecules show 〈S2〉 values of 1.0. This indicates that higher energy singlet states of each of these species are significantly contaminated by the lower energy triplet state leading to erroneously too low singlet-energy gaps. Indeed, the results of Ziegler, Rauk, and Baerends (ZRB) show that the open-shell ∆EST gaps should be too low by a factor of 2.31 Applying this correction factor still results in a systematic underestimation of the gaps but reduces the MUE

c

NUMOL,

to 9.5 kcal/mol (Table 1). We have also applied the Yamiguchi spin projection procedure after the SCF process using eqs 12 and 13 from section 2. Table 1 gives the spin-projected adiabatic and vertical singlet-triplet gaps. The MUE relative to experiment is 9.5 kcal/mol for vertical gaps and 9.4 kcal/mol for abdiabtic gaps, which is a similar error to that obtained with the factor-of-two correction. Although the error in ZRBcorrected and spin-projected ∆EST values is now less than half the open-shell values it is still unsatisfactorily too high to suggest the successful use of DFT for the prediction of singlet-triplet energy gaps in diradicals. As an alternative to the unrestricted broken symmetry results, section 2 outlined the possibility to use fractional-spin occupancy in a single-reference fully SCF process. After converging the SCF of a closed-shell solution, the final orbital occupations of the highest-occupied molecular orbital (HOMO) and lowestunoccupied molecular orbital (LUMO) were changed from 1.0 and 0.0, respectively, to 0.5 and 0.5 giving exactly eq 11, which should approximate the true singlet energy density (F10). The SCF process was then fully converged with this fixed orbital occupation. Table 1 gives the PBE ∆EST values using this fractional-spin occupancy procedure for 6-311++G(2d,2p), augcc-pVQZ, and the basis-set limit. The MUE for FS values is only 3.5 kcal/mol for the 6-311++G(2d,2p) basis set, 3.6 kcal/ mol for the aug-cc-pVQZ basis set, and 3.7 kcal/mol at the basisset limit. These DFT singlet-triplet gaps are without spin contamination error, and the relatively small remaining error is the result of deficiencies in using the approximate PBEPBE functional instead of the exact, but still unknown, functional. Yang and co-workers have previously shown that the deficiency in the PBE and other popular functionals is a result of error in determining static electron correlation.27 From Table 1 it is clear that spin projection systematically underestimates all of the singlet-triplet gaps. In contrast, FSDFT overestimates the singlet-triplet gaps, which is the expected effect of static correlation error with approximate functionals.27 The carbon, oxygen, and silicon atoms are overestimated by 4.9, 4.6, and 4.2 kcal/mol, respectively. For all of the molecules except NF, the singlet-triplet gaps are less overestimated. The overestimation may also be due to the neglect of current-density in the singlet states, if the complex orbital picture is used. For the atoms and diatomics considered in this work, the singlet-triplet gap lowering is computed to range from 1.9 kcal/mol for Si to 5.6 kcal/mol for OH+ with the jBR functional.33,40 Current density is neglected in all tabulated results; however, if a current-density correction was added to FS-DFT, the MUE for the atoms and diatomics decreases to 1.5 kcal/mol.

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Figure 3. Errors, relative to experiment, for the calculated singlet-triplet gaps (∆EST) at the basis-set limit (NUMOL) plotted as a function of fractional-spin from unrestricted open-shell (0 or 1) to half orbital occupancies (0.5). (All values are in kcal/mol.)

Figure 4. PBE TMM geometry for the triplet state.

Using the FS-DFT procedure to compute the ∆EST values with the LSDA showed slight improvement over the PBE functional due to the decrease in static correlation error. For the molecules in Figure 2 the MUE from experiment is 2.0 kcal/ mol with the 6-311++G(2d,2p) basis set and 1.9 kcal/mol at the basis-set limit (Table 2). The energies of the fractionally occupied carbon, oxygen, and silicon atom could not be evaluated with the QM4D program due to SCF convergence problems using the LSDA functional. In general the LSDA and PBE functionals show the same trends, but LSDA is slightly more accurate for molecular singlet-triplet gaps.27 Figure 3 plots the computed singlet-triplet gap error as a function of fractional-spin for carbon, oxygen, and silicon atoms at the basis-set limit. The plotted PBE and LSDA energies vary from unrestricted open-shell (0 or 1) to half orbital occupancies (0.5). These plots show that conventional, unrestricted calculations, which correspond to the integer points on either side of the graphs, consistently overstabilize the singlet states. The singlet state energies increase with change in fractional-spin, reaching a maximum when both degenerate highest-occupied orbitals have equal spin-up and spin-down occupation (nR ) nβ ) 1/2). This situation recovers the correct open-shell singlet electron density (in the case of two degenerate, partially occupied orbitals) and gives good agreement with the experimental singlet-triplet gaps. Again, this FS approach should be exact with the exact functional. The general overestimations of the gaps is likely a consequence of static correlation error inherent in the density-functional approximations used. 3.2. TMM. TMM is a diradical molecule with a non-Kekule´type bonding arrangement.42 The electronic structure of TMM has been studied extensively by Davidson and Borden,43 Cramer,44 Schaefer,45 Pitzer,46 Slipchenko and Krylov,29 and others.47 The ground state is a flat D3h symmetry triplet with degenerate C-C bond lengths (Figure 4). There are two lowlying excited singlet states, 1A1 and 1B2, with distorted geometries, but these states will be degenerate in the D3h geometry (vertical excitation). Cramer and Smith have compared unrestricted DFT geometries to multiconfiguration SCF (MCSCF) geometries.44

They have pointed out that triplet DFT and triplet MCSCF geometries are very similar, whereas singlet DFT geometries are significantly different from singlet MCSCF geometries. Here we have used the PBEPBE/6-31G(d,p) triplet 3B2 geometry to compute the vertical singlet-triplet gap to the 1A1 state using FS-DFT. The trends in the calculated vertical singlet-triplet gaps of TMM (Table 3) agree well the results for atoms and small molecules in section 3.1. The usual open-shell treatment gives gaps that are too low, while the fractional-spin approach gives improved agreement with experiment. As noted earlier, LSDA performs somewhat better than PBE. These findings give us confidence in application of fractional-spin DFT to singlet-triplet gaps of larger conjugated hydrocarbons. 3.3. Application to Polyacenes. The band gap of oligoacenes or polyacenes is of considerable interest due to their possible use as semiconducting materials in field-effect transistors.48 It is well-known that as polyacenes increase in size the singlettriplet energy gap becomes increasingly small. Because of the small band gap the ground state for large polyacenes, such as octacene and nonacene, may have a closed-shell singlet, openshell singlet diradical, or triplet ground state, all of which have been suggested over the years.49 The singlet-triplet gaps of octacene and nonacene provide an example of structures that are too large to treat with very accurate multireference correlated methods but can be easily computed with FS-DFT. The octacene and nonacene UPBEPBE/6-31G(d,p) optimized singlet geometries are shown in Figure 5. For octacene the unrestricted singlet energy is 0.9 kcal/mol lower than the closedshell energy (Table 4). The lower energy unrestricted DFT solution was previously described by Bendikov et al.49 The ∆EST between the UPBEPBE singlet and triplet energy is -2.9 kcal/ mol with the 6-31G(d,p) basis set and -3.9 kcal/mol with the 6-311++G(2d,2p) basis set. These values are similar to the values of -5.8 and -2.9 kcal/mol49 calculated by Bendikov et al. using B3LYP and BLYP. Also in accord with the results reported by Bendikov et al. and others50 for the B3LYP and BLYP functionals this unrestricted solution contains significant spin contamination. The PBE 〈S2〉 value for octacene is 0.74. After projection the ∆EST value is -4.5 kcal/mol. However, with our fractional-spin method the PBEPBE/6-31G(d,p) method predicts a triplet ground state with a ∆EST value of 5.3 kcal/ mol. Similar to the set of atoms and molecules computed in section 3.1, the FS-DFT calculations carried out for octacene

Singlet-Triplet Energy Gaps for Diradicals

J. Phys. Chem. A, Vol. 115, No. 1, 2011 81

Figure 5. Optimized unrestricted singlet PBE structures of octacene and nonacene.

TABLE 3: Open-Shell (OS) and Closed-Shell Fractional-Spin (FS) Singlet-Triplet Gaps (∆EST) for Vertical Excitation of TMM with Calculations Carried out using NUMOL (All Values in kcal/mol) PBE LSDA

ref 29

OS

FS

16.1 16.1

9.2 6.9

21.3 15.4

TABLE 4: PBEPBE Open-Shell (OS), Open-Shell Spin-Projected (SP), and Fraction Spin (FS) Singlet-Triplet Gaps (∆EST) for Octacene and Nonacene (All Values in kcal/mol) OS octacene nonacene a

SP

(-3.9) -2.9 (-3.1)a -1.7b a

b

FS

(-6.0) -4.5 (-6.3)a -3.3b a

b

5.3a 5.1a

6-311++G(2d,2p). b 6-31G(d,p).

(and nonacene, see below) used an ensemble admixture of two Slater determinants (eq 11, section 2) with R and β determinant coefficients set to 1/2. This ensemble of densities provides an approximate total density. For large systems with more than two degenerate states it is possible to use an ensemble of more than two Slater determinants. In such an ensemble the determinant coefficients would be determined by symmetry as in the present work. For nonacene the HOMO and LUMO orbitals are near degenerate. The next occupied orbital, HOMO-1, is ∼0.7 eV below the HOMO energy level. This indicates that there are only two degenerate orbitals that should be considered. The predicted ground state triplet is because the closed-shell singlet energy is raised using FS-DFT. This situation is similar to the atoms and small molecules tested in Section 3.1. For nonacene the unrestricted singlet ground state is again lower in energy than the closed-shell singlet by 1.3 kcal/mol. However, for nonacene the spin contamination is even worse than in octacene with a 〈S2〉 value of 1.0, which indicates an almost equal mixture of singlet and triplet states contributing to the singlet energy. The PBEPBE/6-31G(d,p) ∆EST value before spin projection is -1.7 kcal/mol with a predicted ground state singlet. After spin projection, the gap is -3.3 kcal/mol. The larger 6-311++G(2d,2p) basis set predicts unprojected and projected gaps of -3.1 and -6.3 kcal/mol, respectively. Applying the FS-DFT methodology to nonacene predicts a ground state triplet with a ∆EST value of 5.1 kcal/mol.

The assignment of triplet ground states for octacene and nonacene based on FS-DFT is in opposition to the current view from broken symmetry unrestricted DFT.49,51 Because the fractional-spin occupation calculations were carried out using the PBE functional there remains some static correlation error but it is free from spin contamination error. Although unrestricted calculations show disjoint orbitals,52 that is, opposite spin orbitals that are spatially distinct, the energy of the singlet solution is significantly contaminated, and there is no way to know whether this unphysical contamination “tips the balance” in favor of assignment of a singlet diradical. Liu and co-workers have commented that the spin contamination in spin-unrestricted calculations results in singlet-triplet gaps of polyacenes larger than octacene showing strange and irrational patterns and spinpolarized DFT also displays a similar evolution of gaps that is quite unphysical.50 Ultimately, the singlet-triplet splitting, exchange energy, and dynamic and static electron correlation energies combine to dictate the ground state of these small band gap materials. Our FS-DFT calculations suggest a triplet ground state and the predicted singlet-triplet gaps from unrestricted calculations should be further scrutinized. 4. Conclusion We have presented a theoretical foray for describing diradicals using fractional-spin DFT. In contrast to conventional unrestricted KS DFT, this single-reference closed-shell KS DFT approach provides a self-consistent calculation of electronic structure and properties that is the correct spin eigenfunction of the S2 operator. Most techniques that improve upon broken spin KS DFT use post-SCF spin projection techniques. FS-DFT is advantageous because energies and properties can be computed in a SCF process. The FS-DFT approach was benchmarked using the PBE and LSDA pure density-functionals for the singlet-triplet gaps of Slipchenko and Krylov’s atom and molecule set.29 For the PBE functional, the FS-DFT method gave a MUE of 3.7 kcal/mol at the basis-set limit. This is a significant improvement over standard broken spin and spinprojected methods that gave MUEs of 22.1 and 9.5 kcal/mol, respectively. Because the LSDA is the least sensitive to static correlation error, the FS-DFT singlet-triplet gap for this functional gave a MUE of only 1.9 kcal/mol. The FS-DFT and spin-projected approaches were applied to the singlet-triplet gaps of TMM, octacene, and nonacene. FS-DFT is able to

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accurately predict the vertical singlet-triplet energy in TMM, whereas conventional unrestricted DFT underestimates this gap. Lastly, in contrast to unrestricted KS DFT that predicts diradical singlet ground states, FS-DFT predicts triplet ground states for large polyacenes. Acknowledgment. D.H.E. thanks BYU for financial support and the Fulton Supercomputing Center for computational support. E.R.J. thanks UC Merced for financial support. W.Y. acknowledges support from the National Science Foundation (CHE-06-16849-03). W.Y. also acknowledges that this material is based upon work supported as part of the UNC EFRC: Solar Fuels and Next Generation Photovoltaics, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0001011. All of us thank Ernest Davidson for helpful comments. Note Added after ASAP Publication. This Article was published ASAP on December 9, 2010. Due to a production error, additional changes were needed to Tables 1, 2, and 4, to energy values in section 3.3, and to text in other locations. The corrected version was posted on December 15, 2010. Supporting Information Available: PBE geometries and full ref 34. This information is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) (a) Borden, W. T.; Davidson, E. R. Annu. ReV. Phys. Chem. 1979, 30, 125. (b) Borden, W. T.; Davidson, R. R. Acc. Chem. Res. 1996, 29, 67. (c) Bally, T.; Borden, W. T. ReV. Comput. Chem. 1999, 13, 1. (d) Lim, M. H.; Worthington, S. E.; Dulles, F. J.; Cramer, C. J. Chemical Applications of Density-Functional Theory; ACS Symposium Series; American Chemical Society: Washington, DC, 1996, Chapter 27, pp 402-422, Vol. 629. (2) (a) Gunnarsson, O.; Lundqvist, B. I. Phys. ReV. B 1976, 13, 4274. (b) Dunlap, B. I. Phys. ReV. A 1984, 29, 2902. (c) Delley, B.; Freeman, A. J.; Ellis, D. E. Phys. ReV. Lett. 1983, 50, 488. (d) Weiner, B.; Trickey, S. Int. J. Quantum Chem. 1998, 69, 451. (e) Goddard, J. D.; Orlova, G. J. Chem. Phys. 1999, 111, 7705. (f) Goddard, J. D.; Chen, X.; Orlova, G. J. Phys. Chem. A 1999, 103, 4078. (3) For representative examples of the use of unrestricted DFT to treat diradicals see:(a) Goldstein, E.; Beno, B.; Houk, K. N. J. Am. Chem. Soc. 1996, 118, 6036. (b) Cramer, C. J.; Squires, R. R. J. Phys. Chem. A 1997, 101, 9191. (c) Beno, B. R.; Wilsey, S.; Houk, K. N. J. Am. Chem. Soc. 1999, 121, 4816. (d) Hrovat, D. A.; Beno, B. R.; Lange, H.; Yoo, H.-Y.; Houk, K. N.; Borden, W. T. J. Am. Chem. Soc. 1999, 121, 10529. (e) Gra¨fenstein, J.; Hjerpe, A. M.; Kraka, E.; Cremer, D. J. Phys. Chem. A 2000, 104, 1748. (f) Gra¨fenstein, J.; Hjerpe, A. M.; Kraka, E.; Cremer, D. J. Phys. Chem. A 2000, 104, 1748. (g) Brown, E. C.; Borden, W. T. J. Phys. Chem. A 2002, 106, 2963. (h) Bachler, V.; Olbrich, G.; Neese, F.; Wieghardt, K. Inorg. Chem. 2002, 41, 4179. (i) Khuong, K. S.; Houk, K. N. J. Am. Chem. Soc. 2003, 125, 14867. (j) Caramella, P.; Quadrelli, P.; Toma, L.; Romano, S.; Khuong, K. S.; Northrop, B.; Houk, K. N. J. Org. Chem. 2005, 70, 2994. (k) Tantillo, D. J.; Hoffmann, R.; Houk, K. N.; Warner, P. M.; Brown, E. C.; Henze, D. K. J. Am. Chem. Soc. 2004, 126, 4256. (l) Bethke, S.; Hrovat, D. A.; Borden, W. T.; Gleiter, R. J. Org. Chem. 2004, 69, 3294. (m) Khuong, K. S.; Jones, W. H.; Pryor, W. A.; Houk, K. N. J. Am. Chem. Soc. 2005, 127, 1265. (n) Abe, M.; Ishihara, C.; Kawanami, S.; Masuyama, A. J. Am. Chem. Soc. 2005, 127, 10. (o) Head-Gordon, M.; Rhee, Y. M. J. Am. Chem. Soc. 2008, 130, 3878. (4) Perdew, J. P.; Ruzsinszky, A.; Constantin, L. A.; Sun, J.; Csonka, G. I. J. Chem. Theory Comput. 2009, 5, 902. (5) Krylov, A. I. Acc. Chem. Res. 2006, 39, 83. (6) Beran, G. J. O.; Gwaltney, S. R.; Head-Gordon, M. Phys. Chem. Chem. Phys. 2003, 5, 2488. (7) Gherman, B. F.; Cramer, C. J. Inorg. Chem. 2004, 43, 7281. (8) For a general discussion on the definition of a diradical see: (a) Salem, L.; Rowland, C. Angew. Chem., Int. Ed. Engl. 1972, 11, 92. (b) Jung, Y.; Head-Gordon, M. Chem. Phys. Chem. 2003, 4, 522. (9) For a discussion on the evaluation of 〈S2〉 in DFT see: Wang, J.; Becke, A. D.; Smith, V. H., Jr. J. Chem. Phys. 1995, 102, 3477. (10) It should be noted that spin contamination affects unrestricted DFT much less than unrestricted Hartree-Fock. In addition, pure densityfunctionals are even less affected than hybrid functionals because of the

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