Density Functional Theory with Fractional Orbital Occupations

Density functional theory with fractional orbital occupations

Cite as: J. Chem. Phys. 136, 154104 (2012); https://doi.org/10.1063/1.3703894 Submitted: 23 January 2012 . Accepted: 29 March 2012 . Published Online: 17 April 2012

Jeng-Da Chai

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J. Chem. Phys. 136, 154104 (2012); https://doi.org/10.1063/1.3703894 136, 154104

Density functional theory with fractional orbital occupations Jeng-Da Chaia) Department of Physics, Center for Theoretical Sciences, and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan (Received 23 January 2012; accepted 29 March 2012; published online 17 April 2012)

In contrast to the original Kohn-Sham (KS) formalism, we propose a density functional theory (DFT) with fractional orbital occupations for the study of ground states of many-electron systems, wherein strong static correlation is shown to be described. Even at the simplest level represented by the local density approximation (LDA), our resulting DFT-LDA is shown to improve upon KS-LDA for multi-reference systems, such as dissociation of H2 and N2, and twisted ethylene, while performing similar to KS-LDA for single-reference systems, such as reaction energies and equilibrium geometries. Because of its computational efﬁciency (similar to KS-LDA), this DFT-LDA is applied to the study of the singlet-triplet energy gaps (ST gaps) of acenes, which are “challenging problems” for conventional electronic structure methods due to the presence of strong static correlation effects. Our calculated ST gaps are in good agreement with the existing experimental and high-level ab initio data. The ST gaps are shown to decrease monotonically with the increase of chain length, and become vanishingly small (within 0.1 kcal/mol) in the limit of an inﬁnitely large polyacene. In addition, based on our calculated active orbital occupation numbers, the ground states for large acenes are shown to be polyradical singlets. © 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.3703894]

I. INTRODUCTION sonable computational cost seems to be the first step toward finding an efficient and accurate electronic structure method As the problem of solving the N-electron Schrödinger for large systems. equation quickly becomes intractable as the size of a system The SIEs of KS-DFAs lead to drastic failures for prob- increases, the development of efficient and accurate electronic lems such as barrier heights of chemical reactions, band gaps structure methods for large systems, continues being the sub- of solids, and dissociation of symmetric radical cations.11 In ject of intense current interest. Over the past two decades, TDDFT, SIE causes failures for problems such as Rydberg Kohn-Sham density functional theory (KS-DFT) (Refs. 1 and excitations in molecules and long-range charge transfer exci- 2) has become one of the most popular theoretical approaches tations between two well-separated molecules.12 The SIEs of for calculations of electronic properties of large ground-state KS-DFAs can be greatly reduced by hybrid DFT methods,13 systems (up to a few thousand electrons), due to its favor- incorporating some of the exact Hartree-Fock (HF) exchange able balance between cost and performance.3–7 Recently, its into the KS-DFAs. Over the past 20 years, several hybrid time-dependent extension, time-dependent density functional functionals, such as global hybrid functionals14, 15 and long- theory (TDDFT), has also been actively developed for treat- range corrected (LC) hybrid functionals,16–20 have been de- ing electron dynamics and excited states of large systems with veloped to improve the accuracy of E [ρ], extending the ap- considerable success.8, 9 xc plicability of KS-DFT to a wide range of systems. Although the exact exchange-correlation (XC) functional The proper treatment of noncovalent interactions requires E [ρ] in KS-DFT has not been known, functionals based xc the accurate description of dynamical correlation effects at on the standard density functional approximations (DFAs), medium and long ranges, which cannot be properly captured such as the local density approximation (LDA) and gener- by KS-DFAs.21 In particular, for dispersion-dominated (van alized gradient approximations (GGAs), can accurately de- der Waals (vdW)) interactions, KS-LDA tends to overesti- scribe short-range XC effects (due to the accurate treatment mate the binding energies, while KS-GGAs tend to give in- of on-top hole density), and are computationally favorable for sufficient binding or even unbound results. The NCIEs of large systems.3–7 Although KS-DFAs have been successful KS-DFAs can be efficiently reduced by the DFT-D (KS-DFT in many applications, due to the lack of accurate treatment with empirical dispersion corrections) schemes,22–25 which of nonlocality of XC hole, KS-DFAs can exhibit the follow- have shown generally satisfactory performance on a large ing three types of qualitative failures: (i) self-interaction er- set of noncovalent systems.26, 27 The dispersion corrections ror (SIE), (ii) noncovalent interaction error (NCIE), and (iii) can also be computed less empirically by the exchange-hole static correlation error (SCE). In situations where these fail- dipole moment method28 or by the local response disper- ures occur, KS-DFAs can produce erroneous results.10 There- sion method.29 Alternatively, a fully nonlocal density func- fore, resolving the qualitative failures of KS-DFAs at a rea- tional for vdW interactions (vdW-DF) (Ref. 30) can also be adopted to reduce the NCIEs of KS-DFAs. Currently, perhaps a)Electronic mail: [email protected]. the most successful approach to taking into account nonlocal

0021-9606/2012/136(15)/154104/17/$30.00136, 154104-1 © 2012 American Institute of Physics 154104-2 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012) dynamical correlation effects is provided by the double hy- less attention has been paid to the latter (representation of brid (DH) methods,31–33 which mix some of the HF ex- ground-state density) than to the former (ground-state energy change and some of the nonlocal orbital correlation energy functional). Due to the search over a restricted domain of from the second-order Møller-Plesset perturbation theory34 densities, the exact ground-state density of interest may not into the KS-DFAs. DH functionals have shown an over- be obtained within the framework of KS-DFT, in which case all satisfactory accuracy for thermochemistry, kinetics, and even the exact KS-DFT will fail (i.e., the exact XC functional noncovalent interactions. In addition, the sharp increase in may not be differentiable at the exact ground-state density).7 HF exchange in typical DH functionals has also greatly re- Noticeably, some of these situations occur for systems with duced the SIEs relative to KS-DFAs and conventional hybrid vanishingly small HOMO-LUMO gaps (SC systems), indi- functionals. cating the close relationship between strong static correlation Systems with strong static (nondynamical) correlation ef- effects and representations of the ground-state density. There- fects, such as bond-breaking reactions, diradicals, conjugated fore, to accurately describe SC systems, it seems intuitive to polymers, magnetic materials, and transition metal com- focus on devising an appropriate representation for the ex- pounds, belong to the class of strongly correlated (SC) sys- act ground-state density and a DFT associated with such a tems (multi-reference systems). Such a system is usually char- representation. acterized by a small (or vanishing) energy gap between the A ground-state density ρ(r) is said to be interacting v- highest occupied molecular orbital (HOMO) and the lowest representable, if it can be obtained from a ground-state wave unoccupied molecular orbital (LUMO), the HOMO-LUMO function of an interacting N-electron Hamiltonian for some gap. Despite their computational efficiency, the accurate treat- external potential v(r). The exact ρ(r) can be obtained by ment of SC systems poses a great challenge to KS-DFAs and the full configuration interaction (FCI) method at the com- hybrid DFT methods.10, 35 The DH methods may also be inad- plete basis set limit (i.e., interacting v-representable),37 and equate for SC systems, as the second-order perturbation en- can be compactly expressed in terms of the natural or- ergy components diverge to minus infinity for systems with bitals (NOs) {χ i(r)} and natural orbital occupation numbers 38 vanishing HOMO-LUMO gaps. Within the framework of KS- (NOONs) {ni}, DFT, fully nonlocal XC functionals, such as those based on ∞ random phase approximation (RPA), are believed to be es- 2 ρFCI(r) = ni |χi (r)| , (1) 6, 7 sential for the accurate treatment of SC systems. However, i=1 compared with KS-DFAs, hybrid functionals, and DH functionals, RPA-type functionals are computationally very de- where {ni}, obeying the following two conditions: manding for large systems, and their applications to SC sys- ∞ tems are too scarce to make a firm judgment on their accuracy. ni = N, 0 ≤ ni ≤ 1, (2) Recently, the SIEs of RPA-type functionals have been shown i=1 to be severe, even for a simple one-electron system such as are related to the variationally determined coefficients of the + 36 H2 . FCI expansion. As shown in Eq. (1), the exact ρ(r) can be Aiming to reduce the SCEs of KS-DFAs without extra represented by orbitals and occupation numbers, highlighting computational cost, we propose a DFT with fractional orbital the importance of ensemble-representable densities. occupations, rather than a fully nonlocal XC functional in KS- By contrast, in KS-DFT,2 ρ(r)isassumed to be nonin- DFT. The rest of this paper is organized as follows. In Sec. II, teracting pure-state (NI-PS) vs -representable, as it belongs to we briefly describe the rationale for fractional orbital occu- a one-determinantal ground-state wave function of a nonin- pations. In Sec. III, we describe the formulation of this DFT teracting N-electron Hamiltonian (KS Hamiltonian) for some and explain how strong static correlation is described by this 39–41 local potential vs (r) (KS potential). Correspondingly, the DFT. At the simple LDA level, the performance of our result- KS orbital occupation numbers should be either 0 or 1. As the ing DFT-LDA is compared with KS-LDA for several single- Aufbau principle (i.e., filling the KS orbitals in order of in- and multi-reference systems in Sec. IV. Based on physical ar- creasing energy) should be obeyed, ρ(r) can be expressed as guments and numerical investigations, the optimal parameter the sum of the densities of the lowest N occupied KS orbitals for this DFT-LDA is defined in Sec. V. Our conclusions are 3 {φi(r)}, given in Sec. VI. N 2 ρKS-DFT(r) = |φi (r)| . (3) II. RATIONALE FOR FRACTIONAL ORBITAL i=1 OCCUPATIONS As ground-state densities of most nondegenerate atomic and For the exact DFT, the exact ground-state energy can molecular systems (e.g., closed-shell systems with sizable be obtained, only when the exact ground-state density is HOMO-LUMO gaps) are likely to be NI-PS vs -representable, available to insert into the exact ground-state energy func- the commonly used XC functionals in KS-DFT are reli- tional. Therefore, the development of a generally accurate ably accurate for these systems (assuming that the SIEs and DFT method should involve not only an accurate approxima- NCIEs of these functionals are not severe). However, if a tion for the ground-state energy functional but also an appro- one-determinantal ground-state wave function is insufficient priate representation (possibly in terms of orbitals and occu- to represent ρ(r), these XC functionals may not be reliably pation numbers) of the ground-state density. However, much accurate, as they are all developed based on general properties 154104-3 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012) of systems where the KS wave function is a one-determinantal representable ground-state density can be distinctly different wave function.7 from the NOs in order to incorporate the effect of the config- As shown by several researchers,40–44 there are some uration mixing on the ground-state density. They concluded reasonable ground-state densities that are not NI-PS vs - that the ground-state density of a system with strong static representable. Clearly, such densities cannot be obtained correlation effects is likely to be non-NI-PS vs -representable, within the framework of KS-DFT. To remedy this situa- in which case an ensemble representation (via fractional or- tion, KS-DFT has been extended to ensemble DFT (E-DFT), bital occupations) of the ground-state density is crucial. Ar- wherein ρ(r)isassumed to be noninteracting ensemble (NI- guments in support of this are also available in Refs. 43 E) vs -representable, as it belongs to an ensemble of pure de- and 48. terminantal states of the noninteracting KS system.45, 46 Cor- The idea of simulating strong static correlation effects by respondingly, ρ(r) can be expressed as fractional occupation numbers (FONs) in DFT has spurred the 48–52 ∞ development of the DFT-FON method, the spin-restricted 53, 54 ρ (r) = g |φ (r)|2, (4) ensemble-referenced KS method, and the fractional-spin E-DFT i i 10, 35 i=1 DFT method, with great success for some SC systems. The practical implementation of E-DFT and related meth- where the occupation numbers {gi}, obeying the following ods has, however, been impeded by a few factors as fol- two conditions: lows. First, a double-counting of correlation effects may oc- ∞ cur, when the conventional approximate XC functional is = ≤ ≤ gi N, 0 gi 1, (5) evaluated with an ensemble density in Eq. (4), rather than i=1 a NI-PS vs -representable density in Eq. (3). By taking into are given by account the double-counting effects, the XC functional in E- ⎧ DFT may need to be re-derived. Second, the computational ⎨⎪ 1, for i <F cost of E-DFT is more expensive than that of KS-DFT, which = = makes E-DFT less practical for the study of large ground-state gi ⎪ xi , for i F (6) ⎩ systems. Third, the computational cost of analytical nuclear 0, for > . i F gradients (if available) for E-DFT is more expensive than Here, i is the orbital energy of the i-th KS orbital φi(r), F that for KS-DFT, which makes it a formidable computational is the Fermi energy, and xi is a fractional number (between task to perform geometry optimization of large molecules for 0 and 1). As can be seen, fractional orbital occupations can E-DFT. occur only for the orbitals at the Fermi level. In view of the above difficulties, we focus on the In 1998, the close relationship between strong static cor- representation of the ground-state density from the exact the- relation effects and non-NI-PS vs -representable densities was ory in Eq. (1). Although the exact orbital occupation num- 42 1 + observed by Baerends and co-workers, who studied the g bers for interacting electrons are intractable for large sys- ground state of the C2 molecule (a system where the ground- tems due to the exponential complexity, the approximate ones state wave function is nondegenerate but has a strong multi- can, however, be properly simulated. Based on the statistical reference character), and investigated the possibility that the properties of strongly correlated eigenstates, Flambaum et al. ground-state density ρ(r) may be NI-E vs -representable (as argued that the distribution of occupation numbers (the mi- assumed in E-DFT), rather than NI-PS vs -representable (as crocanonical averaging of the occupation numbers) for a assumed in KS-DFT). In their study, ρ(r) was first obtained finite number of interacting Fermi particles (with two-body from the highly accurate ab initio CI wave function method, interaction) practically does not depend on a particular many- wherein ρ(r) can be represented by the NOs {χ i(r)} and body system and has a universal form that can be ap- NOONs {ni}inEq.(1). An iterative method for the con- proximately described by the Fermi-Dirac distribution with struction of the KS orbitals and the KS potential from the renormalized parameters (i.e., orbital energies, chemical po- CI density was then adopted,47 combined with the constraint tential, and temperature).55–57 Statistical effects of the inter- of integer occupations of the KS orbitals. The ρ(r)ofC2 action have been shown to be absorbed by introduction of the was found to be NI-PS vs -representable at a bond distance effective temperature. shorter than the equilibrium distance, while being non-NI- In view of the close relationship between strong static PS vs -representable at the longer bond distances, leading to correlation effects and representations of the ground-state non-Aufbau solutions with unoccupied KS orbitals having density as well as the close relationship between the distri- energies lower than those of the highest occupied KS or- bution of occupation numbers for interacting electrons (in the bitals (i.e., holes below the Fermi level). On the other hand, sense of statistical average) and the Fermi-Dirac distribution when the ρ(r)ofC2 was represented by the ensemble so- for noninteracting electrons, in this work, the ground-state lution during the above iterative procedure, no holes below density ρ(r) of a system of N interacting electrons (at zero the Fermi level were found, and the corresponding KS or- temperature) in the presence of an external potential vext (r), bitals were close to the NOs. Based on their results, Baerends is assumed to be noninteracting thermal ensemble (NI-TE) vs - and co-workers42 argued that the KS orbitals (generated from representable, as it is represented by the thermal equilibrium Aufbau solutions) for a NI-PS vs -representable ground-state density of an auxiliary system of N noninteracting electrons at density are comparable to the NOs, while the KS orbitals a fictitious temperature θ ≡ kBTel (where kB is the Boltzmann (generated from non-Aufbau solutions) for a non-NI-PS vs - constant, Tel is the temperature measured in absolute 154104-4 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012) temperature, and θ is the temperature measured in energy In KS-DFT,1, 2 F[ρ] is usefully partitioned as units) in the presence of a local potential v (r). Correspond- s = + + + − − ingly, ρ(r) can be expressed as F [ρ] Ts [ρ] EH [ρ] (T [ρ] Vee[ρ] Ts [ρ] EH [ρ])

∞ = Ts [ρ] + EH [ρ] + Exc[ρ]. (12) 2 ρ(r) = fi |ψi (r)| , (7) Here, T [ρ] is the noninteracting kinetic energy at zero i=1 s temperature, where the occupation number fi is the Fermi-Dirac function 2 e ρ(r)ρ(r ) f ={1 + exp[( − μ)/θ]}−1, (8) E [ρ] ≡ drdr (13) i i H 2 |r − r| which obeys the following two conditions: is the Hartree energy, and Exc[ρ] ≡ (T [ρ] + Vee[ρ] ∞ − Ts [ρ] − EH [ρ]) is the XC energy deﬁned in KS-DFT. In = ≤ ≤ fi N, 0 fi 1, (9) KS-DFT, Ts[ρ], the big unknown in terms of the density, i=1 is exactly treated by the use of KS orbitals. However, as

i is the orbital energy of the i-th orbital ψi(r), and μ is the previously discussed, the basic ansatz of KS-DFT (i.e., NI-PS chemical potential chosen to conserve the number of electrons vs -representability of the given ρ(r)) can be violated for SC N. systems, in which case even the exact KS-DFT will fail to In Sec. III, we demonstrate how a DFT associated with convey reliably accurate results.7 the NI-TE vs -representable ρ(r)inEq.(7), can be formu- To make progress, a different representation of the lated. In other words, for a given fictitious temperature θ,the ground-state density is adopted in TAO-DFT, wherein ρ(r)is remaining “renormalized parameters” ({i} and μ) and represented by the thermal equilibrium density of an auxiliary the {ψi(r)}, can be self-consistently determined to represent system of N noninteracting electrons at a fictitious tempera- the ρ(r)inEq.(7), which then determines the ground-state ture θ in the presence of some local potential vs (r). Aiming energy of the system. Strong static correlation is shown to be to achieve this representation, in contrast to the original KS described by a term related to the θ and {fi} in this DFT. partition, F[ρ] is partitioned into the following set of terms: To avoid any possible confusion with KS-DFT, E-DFT, F [ρ] = Aθ [ρ]+E [ρ] + (T [ρ] + V [ρ]−Aθ [ρ]−E [ρ]) and finite-temperature DFT (FT-DFT),58 we refer to this DFT s H ee s H = θ + + + as thermally-assisted-occupation DFT (TAO-DFT). We wish As [ρ] EH [ρ] (T [ρ] Vee[ρ] to develop TAO-DFT with the following characteristics. − T ρ − E ρ + T ρ − Aθ ρ r s [ ] H [ ]) ( s[ ] s [ ]) It is developed for ground-state systems at zero tem- = θ + + + As [ρ] EH [ρ] Exc[ρ] Eθ [ρ]. (14) r perature. It represents the ground-state density with orbitals and Here, Ts[ρ], EH[ρ], and Exc[ρ] are the same as those defined occupation numbers. θ r in KS-DFT, As [ρ] is the noninteracting kinetic free energy It may be used together with existing XC functionals ≡ − θ = θ=0 at temperature θ, and Eθ [ρ] Ts [ρ] As [ρ] As [ρ] in KS-DFT. − θ r As [ρ] is the difference between the noninteracting kinetic It reduces to KS-DFT in the absence of strong static free energy at zero temperature and that at temperature θ. r correlation effects. Substituting Eq. (14) into Eq. (10) and minimizing the It treats single- and multi-reference systems in a more E[ρ] with respect to ρ(r) (subject to the constraint that the r balanced way than KS-DFT. number of electrons be N), yields the following Euler equation It has similar computational cost as KS-DFT (e.g., en- for the ground-state density ρ(r), ergy and analytical nuclear gradients). δAθ [ρ] ρ(r) μ = s + v (r) + e2 dr δρ(r) ext |r − r| III. TAO-DFT δE [ρ] δE [ρ] + xc + θ , (15) A. Self-consistent equations δρ(r) δρ(r) Consider a system of N interacting electrons moving in where μ is the chemical potential of the system. θ an external potential vext (r) at zero temperature. Based on the To bypass the exact functional form of As [ρ] (the big HK theorems,1 the ground-state energy E[ρ], a functional of unknown in terms of the density) needed in Eq. (15), consider the ground-state density ρ(r), can be written as an auxiliary system of N noninteracting electrons moving in a local potential vs (r) at temperature θ. Based on Mermin’s E ρ = F ρ + ρ r v r dr, 58 θ [ ] [ ] ( ) ext ( ) (10) theorems, the grand-canonical potential s of this reference system, a functional of the thermal equilibrium density ρs(r), where the universal functional can be written as = + F [ρ] T [ρ] Vee[ρ] (11) θ = θ + − s [ρs ] As [ρs ] ρs (r)[vs(r) μs ]dr, (16) is the sum of the interacting kinetic energy T[ρ] and the electron-electron repulsion energy Vee[ρ]. where μs is the chemical potential of the reference system. 154104-5 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

θ Minimization of the s [ρs ] with respect to ρs(r), gives and entropy contribution the following Euler equation for the thermal equilibrium den- ∞ sity ρ (r): θ s − Sθ [{f }] = θ [f ln(f ) + (1 − f )ln(1− f )] k s i i i i i θ B i=1 δAs [ρs ] μs = + vs (r). (17) (26) δρ (r) s of noninteracting electrons at temperature θ. Based on Comparing Eq. (17) with Eq. (15), shows that both minimiza- Eqs. (10) and (14), the total ground-state energy E[ρ] can be tions have the same solution ρs(r) = ρ(r), if we choose vs (r) evaluated by (up to a constant) as = θ { } + ρ(r ) δE [ρ] δE [ρ] E[ρ] As [ fi ,ψi ] ρ(r)vext (r)dr v (r) = v (r) + e2 dr + xc + θ . s ext |r − r| δρ(r) δρ(r) + EH [ρ] + Exc[ρ] + Eθ [ρ]. (27) (18) To sum up, in TAO-DFT, the partition of F[ρ]in Alternatively, as Aθ [ρ] can be expressed exactly in terms s Eq. (14) and the exact treatment of Aθ [ρ]inEq.(24),are of orbitals and occupation numbers (see below), Eq. (17) can s shown to yield a set of self-consistent equations for the NI- also be handled, in an exact manner, by solving the one- TE v -representable ρ(r)inEq.(20). Note that these equa- electron Schrödinger equations for the potential v (r), given s s tions resemble the finite-temperature KS equations,2, 58 so the by implementation of TAO-DFT can be easily achieved using ex- ¯2 isting FT-DFT codes with a slight modification (i.e., replacing − ∇2 + = vs (r) ψi (r) i ψi (r), (19) the XC free energy in FT-DFT to the sum of E [ρ] and E [ρ] 2me xc θ in TAO-DFT). Hence, the computational cost of TAO-DFT is and construct similar to that of KS-DFT or FT-DFT. Similar to FT-DFT,2, 58 ∞ due to the explicit inclusion of Fermi-Dirac occupation func- ρ(r) = ρ (r) = f |ψ (r)|2, (20) θ θ s i i tion in TAO-DFT, the entropy contribution (− S [{fi }]) kB s i=1 in Eq. (26) is essential to make the total ground-state en- 59 where the occupation number fi is the Fermi-Dirac function ergy functional E[ρ] variational (e.g., making the nuclear gradients of E[ρ] equal to the Hellmann-Feynman ={ + − }−1 fi 1 exp[(i μ)/θ] , (21) forces60). and the chemical potential μ is chosen to conserve the number of electrons N, B. Spin-polarized formalism ∞ −1 {1 + exp[(i − μ)/θ]} = N. (22) For a system with Nα up-spin electrons and Nβ down-spin i=1 electrons, the standard computational approach is the spin- The formulation of TAO-DFT leads to a set of self-consistent polarized (spin-unrestricted) formalism, wherein the funda- equations, Eqs. (18)–(22). mental variables are the up-spin density ρα(r) and down-spin To obtain a self-consistent ground-state density in TAO- density ρβ (r) of the ground-state density DFT: (i) Choose a trial ρ(r) to construct vs (r)byEq.(18); (ii) ρ(r) = ρα(r) + ρβ (r) = ρσ (r). (28) solve Eq. (19), which gives {i, ψi(r)}; (iii) find μ by solving σ =α,β Eq. (22); (iv) determine {fi}byEq.(21) and new ρ(r)by Eq. (20). This process is coupled with Eq. (18) to achieve self- In analogy to the two-Fermi-level picture of spin-polarized consistency. When converged, the entropy functional reads KS-DFT,48, 61 spin-polarized TAO-DFT can also be formu- ∞ lated with the two-chemical-potential picture, wherein two θ { } =− + − − Ss [ fi ] kB [fi ln(fi ) (1 fi )ln(1 fi )]. noninteracting auxiliary systems at the same fictitious tem- i=1 perature θ are adopted: one described by the spin function (23) α and the other by function β, with the corresponding ther- The exact noninteracting kinetic free energy Aθ at the ficti- s mal equilibrium density distributions ρs, α(r) and ρs, β (r)ex- tious temperature θ, can be expressed in terms of {f , ψ }, i i actly equal to ρα(r) and ρβ (r), respectively, in the original θ interacting system at zero temperature. Similar to the previ- θ { } = θ { } − θ { } As [ fi ,ψi ] Ts [ fi ,ψi ] Ss [ fi ], (24) ous derivations (but using the spin-polarized extensions of kB the HK theorems48, 61 and the Mermin theorems62, 63 for the which is the sum of the kinetic energy physical and auxiliary systems, respectively), one-electron 2 ∞ Schrödinger equations for electrons with σ -spin (σ = α θ ¯ ∗ 2 T [{fi ,ψi }] =− fi ψ (r)∇ ψi (r)dr or β), can be obtained as follows (i runs for the orbital s 2m i e i=1 index): ∞ ¯2 = fi i − ρ(r)vs (r)dr (25) 2 − ∇ + vs,σ (r) ψi,σ (r) = i,σ ψi,σ (r), (29) i=1 2me 154104-6 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012) with a local potential Spin-unpolarized (spin-restricted) TAO-DFT can be for- mulated by imposing the constraints of ψi, α(r) = ψi, β (r) and 2 ρ(r ) v (r) = v (r) + e dr fi, α = fi, β to spin-polarized (spin-unrestricted) TAO-DFT. s,σ ext |r − r|

δExc[ρα,ρβ ] δEθ [ρα,ρβ ] C. Analytical nuclear gradients + + . (30) δρ (r) δρ (r) σ σ The analytical computation of nuclear gradients is cru- Here, Exc[ρα, ρβ ] is the same as the XC energy cial for the efficient optimization of molecular geometries. In 48, 61 2, 58 defined in spin-polarized KS-DFT, and Eθ [ρα,ρβ ] light of the similarity of TAO-DFT and FT-DFT, analytical ≡ − θ = θ=0 − θ Ts [ρα,ρβ ] As [ρα,ρβ ] As [ρα,ρβ ] As [ρα,ρβ ]is nuclear gradients for TAO-DFT can be easily obtained from the spin-polarized version of Eθ [ρ]. The σ-spin density can those for FT-DFT with a slight modification (mentioned pre- be constructed by viously). Therefore, the computational cost of the analytical ∞ nuclear gradients for TAO-DFT is similar to that for KS-DFT 2 ρσ (r) = fi,σ |ψi,σ (r)| , (31) or FT-DFT. For nonorthogonal atomic orbital representations i=1 (e.g., Gaussian-type orbitals), the generalized Pulay force for a noninteracting thermal ensemble64 should be included to where the occupation number f is the Fermi-Dirac function i, σ add the effect of the basis-set dependent response to the ana- −1 fi,σ ={1 + exp[(i,σ − μσ )/θ]} , (32) lytical nuclear gradients for TAO-DFT. and a chemical potential μσ is chosen to conserve the number D. Local density approximation of σ -spin electrons Nσ , ∞ In TAO-DFT, as the exact Exc[ρ] and Eθ [ρ], in terms −1 of the ground-state density ρ(r), remain unknown, DFAs for {1 + exp[(i,σ − μσ )/θ]} = Nσ . (33) both of them (denoted as TAO-DFAs) are needed for practi- i=1 cal applications. The performance of TAO-DFAs depends on The formulation of spin-polarized TAO-DFT yields two sets the accuracy of DFAs and the choice of the fictitious tem- (one for each spin function) of self-consistent equations, perature θ. In this work, we adopt the LDA (the simplest Eqs. (29)–(33),forρα(r) and ρβ (r), respectively, which are DFA) for both the Exc[ρ] and Eθ [ρ] in TAO-DFT (denoted coupled with ρ(r)byEq.(28). as TAO-LDA). As TAO-LDA is exact for a uniform electron To obtain self-consistent spin densities (and the ground- gas, it provides a good starting point for more accurate and so- state density) in spin-polarized TAO-DFT: (i) Choose trial phisticated TAO-DFAs. Besides, TAO-LDA is readily avail- spin densities ρα(r) and ρβ (r) to compute the ground-state LDA able, as Exc [ρ] can be directly obtained from that of KS- density ρ(r)byEq.(28); (ii) for the σ-spin (σ = α or β) elec- 65, 66 LDA LDA, and Eθ [ρ] can be obtained with the knowledge trons, construct vs,σ (r)byEq.(30); (iii) solve Eq. (29), which LDA,θ of As [ρ]asfollows: gives {i, σ , ψi, σ (r)}; (iv) find μσ by solving Eq. (33); (v) de- ELDA[ρ] ≡ ALDA,θ=0[ρ] − ALDA,θ [ρ]. (36) termine {fi, σ }byEq.(32) and new ρσ (r)byEq.(31).This θ s s process is coupled with Eq. (28) to achieve self-consistency. LDA,θ Here, Perrot’s parametrization of As [ρ] (in its spin- When converged, Aθ , the sum of the kinetic energy and en- LDA s,σ unpolarized form) (Ref. 67) is adopted to obtain Eθ [ρ](in tropy contribution of noninteracting σ-spin electrons at the its spin-unpolarized form) by Eq. (36). For completeness of fictitious temperature θ, is given by LDA,θ this work, As [ρ] (after correcting some typos in Ref. 67) θ { } is explicitly shown here (in atomic units), As,σ [ fi,σ ,ψi,σ ] 2 ∞ LDA,θ = LDA,θ ¯ As [ρ] as (r)dr, (37) =− f ψ∗ (r)∇2ψ (r)dr 2m i,σ i,σ i,σ √ e i=1 LDA,θ ≡ ≡ 2 −3/2 where as (r) θρ(r)f (y) and y (π / 2)θ ρ(r). ∞ The function f(y) was parametrized separately for the two re- + + − − 3π θ [fi,σ ln(fi,σ ) (1 fi,σ )ln(1 fi,σ )] gions y ≤ y and y ≥ y (y ≡ √ ),67 0 0 0 4 2 i=1 = − + ∞ f (y) lny 0.8791880215 0.1989718742y −2 2 = {fi,σ i,σ + θ [fi,σ ln(fi,σ ) + (1 − fi,σ ) + 0.1068697043 × 10 y i=1 − 0.8812685726 × 10−2y3 + 0.1272183027 × − }− − − ln(1 fi,σ )] ρσ (r)vs,σ (r)dr, (34) × 10 1y4 − 0.9772758583 × 10 2y5 + × −2 6 − and the total ground-state energy E[ρα, ρβ ] in spin-polarized 0.3820630477 10 y 0.5971217041 TAO-DFT is evaluated by −3 7 × 10 y , for y ≤ y0, (38) = θ { } + E[ρα,ρβ ] As,σ [ fi,σ ,ψi,σ ] ρ(r)vext (r)dr σ =α,β f (y) = 0.7862224183u − 0.1882979454 1 −1 −3 + EH [ρ] + Exc[ρα,ρβ ] + Eθ [ρα,ρβ ]. (35) × 10 u + 0.5321952681u 154104-7 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

+ 0.2304457955 × 101u−5 − 0.1614280772 and (14))isgivenby 2 −7 2 −9 θ ×10 u + 0.5228431386 × 10 u ≈ + + − θ { } Vee[ρ] EH [ρ] Exc[ρ] Eθ [ρ] Ss [ fi ]. (43) kB − 0.9592645619 × 102u−11 On the right-hand side of Eq. (43), the first term is the + × 2 −13 0.9462230172 10 u Hartree energy, and the sum of the remaining terms (Exc[ρ] θ θ 2 −15 + Eθ [ρ] − S [{fi }]) should properly describe the XC en- − 0.3893753937 × 10 u , kB s ergy defined in the exact wave function theory. 2/3 for y ≥ y0 with u ≡ y . (39) Here, we explain how strong static correlation is de- = LDA,θ=0 scribed by TAO-LDA, with arguments similar to the above. The θ 0 case, As [ρ], is the same as the Thomas- Fermi kinetic energy density functional,3, 4 Suppose that the exact ρ(r)inEq.(1) can be reasonably represented by Eq. (20) (with the orbitals {ψi(r)} and their oc- = LDA,θ 0 = 5/3 cupation numbers {fi} from TAO-LDA). When applying the As [ρ] CF ρ (r)dr, (40) above arguments for TAO-LDA (i.e., choosing a θ so that {ni} 3 2 2/3 ≈ ≈ where CF = (3π ) . {fi}, which gives {χ i(r)} {ψi(r)}), T[ρ] can still be prop- 10 θ { } For spin-polarized (spin-unrestricted) TAO-LDA, the erly simulated by Ts [ fi ,ψi ] (see Eq. (42)), while Vee[ρ]is LDA only approximated by corresponding spin-polarized forms, Exc [ρα,ρβ ] (available from that of spin-polarized KS-LDA; Refs. 65 and θ ≈ + LDA + LDA − θ { } 66) and ELDA[ρ ,ρ ], should be adopted. From the spin- Vee[ρ] EH [ρ] Exc [ρ] Eθ [ρ] Ss [ fi ]. θ α β kB θ 68 scaling relation of As [ρα,ρβ ] (same as that of Ts[ρα, ρβ ] ), (44) LDA Eθ [ρα,ρβ ] can be conveniently expressed by its spin- On the right-hand side of Eq. (44), the first two terms (EH[ρ] LDA LDA unpolarized form Eθ [ρ], and Exc [ρ]) are the same as those defined in KS-LDA, the LDA LDA ≡ LDA,θ=0 − LDA,θ third term Eθ [ρ] locally accounts for the difference be- Eθ [ρα,ρβ ] As [ρα,ρβ ] As [ρα,ρβ ] θ tween the exact Ts and As (at the LDA level), and the last θ θ θ θ 1 LDA,θ=0 LDA,θ=0 term is the entropy contribution (− S [{fi }] ≈− S [{ni }] = (A [2ρ ] + A [2ρ ]) kB s kB s 2 s α s β = ∞ + − − θ i=1[ni ln(ni ) (1 ni )ln(1 ni )]) (see Eq. (26)). 1 Due to their local treatment, ELDA[ρ] and ELDA[ρ] − (ALDA,θ [2ρ ] + ALDA,θ [2ρ ]) xc θ 2 s α s β are not expected to properly describe nonlocal XC effects (e.g., long-range dynamical correlation and strong static cor- 1 LDA,θ=0 LDA,θ = {(A [2ρα] − A [2ρα]) relation). However, as the entropy contribution is a fully 2 s s nonlocal density functional ({fi} are implicit density function- + LDA,θ=0 − LDA,θ } (As [2ρβ ] As [2ρβ ]) als), it may describe nonlocal correlation effects. There is cer- 1 tainly a close relationship between the entropy (defined by the = {ELDA[2ρ ] + ELDA[2ρ ]}. (41) 2 θ α θ β NOONs {ni}) and correlation energy of a system. A famous example is given by the Collins conjecture69 that the correlation energy of a system is proportional to the Jaynes (in- E. Strong static correlation from TAO-LDA { } =− ∞ 70 formation) entropy SJaynes[ ni ] i=1 ni ln(ni ). Inter- The ground-state density ρ(r) of a strongly correlated estingly, the entropy contribution in Eq. (44) is proportional θ { } ≈ θ { } system containing a sufficiently large number of electrons, to the Gibbs (thermodynamic) entropy (Ss [ fi ] Ss [ ni ] =− ∞ + − − can be represented by Eq. (1) (with the exact NOs {χ i(r)} kB i=1[ni ln(ni ) (1 ni )ln(1 ni )]), with the con- θ and NOONs {ni}). Assume that the ρ(r) can also be repre- stant of proportionality being explicitly given by (− ). Note kB sented by Eq. (20) (with the orbitals {ψi(r)} and their occu- that the similarity of information entropy and thermodynamic pation numbers {fi} from the exact TAO-DFT). Note that for entropy in a many-body quantum system (with strong in- such a NI-TE vs -representable ρ(r), its “internal variables” teractions) has been shown in Ref. 71, based on statistical {fi} and {ψi(r)} in Eq. (20), can still be tuned by changing arguments. θ θ the fictitious temperature θ.Ifaθ is chosen so that {ni} ≈ {fi} As the entropy contribution (− S [{fi }]) in Eq. (44) es- kB s (in the sense of statistical average, as mentioned previously), sentially provides no contributions for a single-reference sys- we have {χ i(r)} ≈ {ψi(r)} (as both Eqs. (1) and (20) repre- tem ({fi} ≈ {ni} are close to either 0 or 1), and significantly sent the same ρ(r)). In fact, the NI-TE vs -representability of lowers the total energy of a multi-reference system ({fi} ρ(r) is likely to be fulfilled for this θ, due to the similarity of ≈ {ni} are fractional for active orbitals, and are close to ei- Eqs. (1) and (20). ther 0 or 1 for others), we expect that this term (absent in Consequently, the exact kinetic energy of interacting KS-LDA) plays a crucial role in simulating strong static cor- θ { } electrons T[ρ] can be properly simulated by Ts [ fi ,ψi ] relation (rather than dynamical correlation). θ θ θ θ (as appeared in A [{fi ,ψi }] = T [{fi ,ψi }] − S [{fi }]), s s kB s namely, IV. NUMERICAL INVESTIGATIONS = { } ≈ θ { } θ T [ρ] T [ ni ,χi ] Ts [ fi ,ψi ], (42) OF AN OPTIMAL VALUE due to their similar expressions,5 while the electron-electron The fictitious (reference) temperature θ for TAO-DFT, repulsion energy Vee[ρ] = F [ρ] − T [ρ](seeEqs.(11) controlling the orbital occupation numbers {fi}, is closely 154104-8 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

TABLE I. Statistical errors (in kcal/mol) of the reaction energies of 30 chemical reactions, Ref. 20, calculated by TAO-LDA (with various θ (in mHartree)). The θ = 0 case corresponds to KS-LDA.

θ 0 1 3 5 7 10 15 20 30 40

MSE − 0.41 − 0.72 − 0.94 − 1.13 − 1.32 − 1.59 − 1.96 − 2.25 − 2.73 − 3.04 MAE 8.51 8.27 7.75 7.36 7.09 6.95 7.53 8.92 12.28 15.21 rms 11.10 10.89 10.31 9.76 9.38 9.16 9.75 11.25 15.20 19.03 Max(−) − 18.31 − 17.43 − 15.65 − 15.73 − 15.92 − 16.55 − 18.63 − 22.61 − 30.65 − 39.72 Max(+) 35.68 35.59 33.88 32.18 30.50 28.01 23.95 20.08 17.09 25.45 related to the strength of static correlation. At the LDA has similar performance to KS-LDA.73 This is unsurpris- level, an immediate question is how the θ for the resulting ing, as these systems do not have much static correlation, TAO-LDA should be chosen. As previously argued, the en- the exact NOONs should be close to either 0 or 1, which θ θ ∞ tropy contribution, − S [{fi }] = θ = [fi ln(fi ) + (1 − can be well simulated by the orbital occupation numbers kB s i 1 fi )ln(1− fi )], can be responsible for strong static correla- of TAO-LDA (with a sufficiently small θ). Consequently, θ { } θ=0 { } tion effects, especially when the {fi} (tunable by the θ) prop- Ts [ fi ,ψi ] (see Eq. (42)) is close to Ts [ ψi ] (KS kinetic LDA erly simulate the exact NOONs {ni}. To numerically inves- energy), and Eθ [ρ] (see Eq. (36)) and the entropy contribu- θ θ ∞ tigate this conjecture, the performance of TAO-LDA (with θ tion (− S [{fi }] = θ = [fi ln(fi ) + (1 − fi )ln(1− fi )]) kB s i 1 = 0, 1, 3, 5, 7, 10, 15, 20, 30, and 40 mHartree) is exam- have insignificant contributions to the total energy, relative to LDA ined for both single-reference systems (reaction energies and Exc [ρ] (see Eq. (44)). equilibrium geometries) and multi-reference systems (dissociation of H2 and N2, twisted ethylene, and singlet-triplet (ST) energy gaps of linear acenes). The limiting case where θ = 0 2. Equilibrium geometries for TAO-LDA is especially interesting, as this reduces to KS- Geometry optimizations for TAO-LDA are performed on LDA. Therefore, it is important to know how well KS-LDA the equilibrium experimental test set (EXTS),74 consisting of performs here to assess the significance of the extra parameter 166 symmetry unique experimental bond lengths for small θ for TAO-LDA. to medium size molecules. As the ground states of these All calculations are performed with a development ver- molecules near their equilibrium geometries can be well de- sion of Q-Chem 3.2.72 The error for each entry is defined scribed by single-reference wave functions, TAO-LDA (with as (error = theoretical value − reference value). The nota- a θ smaller than 10 mHartree) is also found to perform simi- tion used for characterizing statistical errors is as follows: larlytoKS-LDA,73 as shown in Table II. mean signed errors (MSEs), mean absolute errors (MAEs), root-mean-square (rms) errors, maximum negative errors (Max(−)), and maximum positive errors (Max(+)). Results B. Multi-reference systems are computed using the 6-311++G(3df,3pd) basis set, unless noted otherwise. 1. Dissociation of H2 and N2

H2 dissociation, a single-bond breaking system, is partic- ularly challenging for KS-DFT. Figure 1 shows the potential A. Single-reference systems energy curves (in total energy) for the ground state of H2, calculated by both the spin-restricted and spin-unrestricted 1. Reaction energies formalisms of the HF theory and KS-DFT (with LDA The accurate prediction of reaction energies is usually (Refs. 65 and 66) and B3LYP (Refs. 13 and 14) function- one of the major criteria in the assessment of the perfor- als), where the exact potential energy curve is calculated mance of electronic structure methods. The reaction energies by the coupled-cluster theory with iterative singles and dou- of 30 chemical reactions (a test set described in Ref. 20)are bles (CCSD).75 Due to the symmetry constraint, the spin- used to examine the performance of TAO-LDA. As shown restricted and spin-unrestricted potential energy curves, cal- in Table I, TAO-LDA (with a θ smaller than 10 mHartree) culated by the exact theory, should be the same. Therefore,

TABLE II. Statistical errors (in Å) of EXTS,74 calculated by TAO-LDA (with various θ (in mHartree)). The θ = 0 case corresponds to KS-LDA.

θ 0 1 3 5 7 10 15 20 30 40

MSE 0.004 0.004 0.004 0.005 0.005 0.005 0.006 0.008 0.015 0.030 MAE 0.013 0.013 0.013 0.013 0.013 0.013 0.013 0.014 0.021 0.036 rms 0.017 0.017 0.017 0.017 0.017 0.018 0.019 0.021 0.037 0.080 Max(−) − 0.091 − 0.091 − 0.091 − 0.091 − 0.091 − 0.092 − 0.095 − 0.101 − 0.110 − 0.110 Max(+) 0.081 0.081 0.080 0.080 0.080 0.080 0.078 0.083 0.222 0.581 154104-9 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

-0.7 -0.9 -0.75

-0.8 -0.95 -0.85 θ -0.9 =0 θ=1 -1 θ -0.95 =3 θ=5 θ=7 -1 θ=10 -1.05 θ=15 -1.05 Exact θ=20

HF (restricted) Total Energy (Hartree) θ=30 Total Energy (Hartree) -1.1 HF (unrestricted) θ=40 B3LYP (restricted) θ=50 -1.1 -1.15 B3LYP (unrestricted) LDA (restricted) (θ in mHartree) -1.2 LDA (unrestricted)

-1.25 -1.15 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 R (angstrom) R (angstrom)

FIG. 1. Potential energy curves (in total energy) for the ground state of H , FIG. 3. Potential energy curves (in total energy) for the ground state of H2, 2 = calculated by both the spin-restricted and spin-unrestricted formalisms of the calculated by spin-restricted TAO-LDA (with various θ). The θ 0 case HF theory and KS-DFT (with LDA and B3LYP functionals). The exact po- corresponds to spin-restricted KS-LDA. tential energy curve is calculated by the CCSD theory.

parisons, where the zeros of energy are set at the respective the difference between the dissociation limits of the spin- spin-unrestricted dissociation limits. As can be seen, LDA restricted and spin-unrestricted potential energy curves, can still has the smallest SCE, when compared with other approx- be used as a quantitative measure of SCEs of approximate imate methods. Popular hybrid functionals (B3LYP, M06- methods.10, 35 Spin-restricted KS-DFT yields the proper spin 2X, and ωB97X-D) perform very well near the equilibrium symmetry and spin densities but has much too high total en- geometry (dominated by single-reference character), but fail ergy (leading to a noticeable SCE (Refs. 10 and 35)), due to drastically at the larger R (dominated by multi-reference char- the lack of strong static correlation. On the other hand, spin- acter). B2PLYP (a popular DH functional) leads to an unphys- unrestricted KS-DFT artiﬁcially breaks the correct space- and ical divergence at the dissociation limit, due to the vanish- spin-symmetries to simulate strong static correlation, yielding ing HOMO-LUMO gap appeared in the energy denominator a reasonable energy but wrong spin densities.10 Similar results of its second-order perturbation energy components. Clearly, are also found for the HF theory. Among the three approxi- hybrid and DH functionals, the most popular schemes for re- mate methods, HF has the largest SCE due to the complete ducing the SIEs and NCIEs of KS-DFAs, respectively, can neglect of static (and also dynamical) correlation. The SCE of perform poorly for multi-reference systems due to their inac- LDA is smaller than that of B3LYP or HF. In Fig. 2, the poten- curate treatment of strong static correlation effects.10, 35 tial energy curves (in relative energy) for the ground state of To evaluate the performance of the present method, the H2, calculated by the spin-restricted HF theory and KS-DFT potential energy curves for the ground state of H2, calcu- (with LDA,65, 66 PBE,76 B3LYP,13, 14 M06-2X,15 ωB97X-D,24 lated by spin-restricted TAO-LDA, are shown in Figs. 3 (in and B2PLYP (Ref. 31) functionals), are presented for com- total energy) and 4 (in relative energy). Near the equilibrium geometry, where the multi-reference character is insigniﬁ- cant, the performance of TAO-LDA (with a θ smaller than 160 140 120 100 40 80 60 20 40 0 20 0 -20 θ=0 -20 Exact θ=1 θ=3 HF -40 -40 B3LYP θ=5 Relative Energy (kcal/mol) LDA θ=7 -60 θ PBE -60 =10 -80 M06-2X θ=15 ωB97X-D θ=20 -100 -80 θ=30 B2PLYP Relative Energy (kcal/mol) -120 θ=40 θ=50 1 2 3 4 5 6 7 8 9 10 -100 (θ in mHartree) R (angstrom) -120

FIG. 2. Potential energy curves (in relative energy) for the ground state of 1 2 3 4 5 6 7 8 9 10 H2, calculated by the spin-restricted HF theory and KS-DFT (with various R (angstrom) XC functionals). The exact potential energy curve is calculated by the CCSD theory. The zeros of energy are set at the respective spin-unrestricted dissoci- FIG. 4. Same as Fig. 3 but in relative energy. The zeros of energy are set at ation limits. the respective spin-unrestricted dissociation limits. 154104-10 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

80 2

1.9 60 Reference 1.8 θ=0 40 θ=1 1.7 θ=3 θ=5 20 θ 1.6 =7 θ=10 0 θ=15 1.5 θ=20 θ=30 -20 unrestricted 1.4 θ=40 restricted restricted (without entropy contribution) θ=50 -40

Energy (kcal/mol) entropy contribution Occupation Number 1.3 (θ in mHartree) 1.2 -60

1.1 -80

1 -100

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 R (angstrom) R (angstrom)

FIG. 5. Occupation numbers of the 1σ g orbital for the ground state of H2 as FIG. 6. Potential energy curves (in relative energy) for the ground state of a function of the internuclear distance R, calculated by spin-restricted TAO- H , calculated by the spin-restricted (with and without the entropy contri- = 2 LDA (with various θ). The θ 0 case corresponds to spin-restricted KS- butions) and spin-unrestricted TAO-LDA (θ = 40 mHartree), where the ze- 37 LDA. The reference data are the FCI NOONs. ros of energy are set at the spin-unrestricted dissociation limit. The entropy contributions (in total energy) as a function of the internuclear distance R, calculated by spin-restricted TAO-LDA (θ = 40 mHartree), are also shown. 10 mHartree) is very similar to that of KS-LDA. At the dissociation limit, where the multi-reference character is pronounced, the SCE of TAO-LDA is shown to be reducible with the strong static correlation effects to make the spin-restricted the increase of θ value, at essentially no extra computational potential energy curve the same as the spin-unrestricted one cost! Interestingly, TAO-LDA (with a θ between 30 and 50 (as it should be). By contrast, the entropy contributions as a mHartree) performs very well, leading to a vanishingly small function of the internuclear distance R, calculated by spin- SCE. restricted TAO-LDA (θ = 7 mHartree), are still insufﬁcient To see how this is related to the ensemble representa- to simulate the strong static correlation effects (see Fig. 7), tion (via fractional orbital occupations) of the ground-state as the corresponding 1σ g orbital occupation numbers do not density, the occupation numbers of the 1σ g orbital (HOMO) match well with the FCI NOONs (see Fig. 5). for the ground state of H2 as a function of the internu- Similar results are also found for N2 dissociation, a triple- clear distance R, calculated by spin-restricted TAO-LDA, are bond breaking system. The potential energy curves for the presented in Fig. 5, where the reference data are the FCI ground state of N2, calculated by spin-restricted TAO-LDA, NOONs.37 At the equilibrium geometry (R = 0.741 Å), the are shown in Figs. 8 (in total energy) and 9 (in relative en- FCI NOON is 1.9643, indicating the absence of strong static ergy). As can be seen, spin-restricted TAO-LDA (with a θ correlation effects (with doubly occupied 1σ g orbital). How- between 30 and 50 mHartree) can dissociate N2 properly ever, the FCI NOON is 1.5162 at R = 2.117 Å, and 1.0000 (yielding a vanishingly small SCE) to the respective spin- at R = 7.938 Å, indicating the presence of strong static cor- unrestricted dissociation limits, which is closely related to relation effects. The 1σ g orbital occupation numbers of spin- the fact that the occupation numbers of the 3σ g (in Fig. 10) restricted TAO-LDA (with a nonvanishing θ) are very close and 1π ux (in Fig. 11) orbitals for the ground state of N2 to 2.0 (doubly occupied) near the equilibrium geometry, and as functions of the internuclear distance R, calculated by gradually reduced to 1.0 (singly occupied) at the dissociation limit. The larger the θ value is, the faster the corresponding

1σ g orbital occupation number approaches 1.0 at the larger R. 40 The 1σ g orbital occupation numbers of spin-restricted TAO- LDA (with a θ between 30 and 50 mHartree) are shown to 20 match well with the FCI NOONs, which is closely related to 0 the vanishingly small SCE of TAO-LDA (with the same θ). To examine the entropy contributions (in total energy) -20 unrestricted as a function of the internuclear distance R, calculated by -40 restricted restricted (without entropy contribution) spin-restricted TAO-LDA (θ = 40 mHartree), Fig. 6 shows entropy contribution the potential energy curves (in relative energy) for the ground Energy (kcal/mol) -60 state of H2, calculated by the spin-restricted (with and without -80 the entropy contributions) and spin-unrestricted TAO-LDA (θ = 40 mHartree), where the zeros of energy are set at the -100 spin-unrestricted dissociation limit. As can be seen, the en- -120 tropy contributions are insigniﬁcant near the equilibrium ge- 1 2 3 4 5 6 7 8 9 10 R (angstrom) ometry and pronounced at the larger R (approaching a negative constant at the dissociation limit), properly simulating FIG. 7. Same as Fig. 6 but for θ = 7 mHartree. 154104-11 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

-108 2

-108.1 1.9 Reference 1.8 θ=0 -108.2 θ=1 1.7 θ=3 θ=5 θ -108.3 1.6 =7 θ=10 θ θ=15 =0 1.5 -108.4 θ=1 θ=20 θ=3 θ=30 θ=5 1.4 θ=40 -108.5 θ=7 θ=50

θ=10 Occupation Number 1.3 Total Energy (Hartree) θ=15 (θ in mHartree) -108.6 θ=20 1.2 θ=30 θ=40 1.1 -108.7 θ=50 (θ in mHartree) 1 -108.8 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 R (angstrom) R (angstrom)

FIG. 11. Same as Fig. 10 but for the 1π ux orbital. FIG. 8. Potential energy curves (in total energy) for the ground state of N2, calculated by spin-restricted TAO-LDA (with various θ). The θ = 0 case corresponds to spin-restricted KS-LDA. spin-restricted TAO-LDA (with the same θ), match reasonably well with the corresponding MRCI NOONs (the reference data).77 150 To examine the entropy contributions (in total energy) as 100 a function of the internuclear distance R, calculated by spin- restricted TAO-LDA (θ = 40 mHartree), Fig. 12 shows the 50 potential energy curves (in relative energy) for the ground 0 state of N2, calculated by the spin-restricted (with and without θ=0 the entropy contributions) and spin-unrestricted TAO-LDA (θ -50 θ=1 θ=3 = 40 mHartree), where the zeros of energy are set at the spin- θ -100 =5 θ=7 unrestricted dissociation limit. As shown, the entropy contri- θ=10 -150 θ=15 butions are essentially responsible for simulating the strong θ Relative Energy (kcal/mol) =20 θ=30 static correlation effects to make the spin-restricted poten- -200 θ=40 θ tial energy curve the same as the spin-unrestricted one (as it =50 = -250 (θ in mHartree) should be). For spin-restricted TAO-LDA (θ 40 mHartree), the entropy contribution (−208.90 kcal/mol) at the dissoci- 1 2 3 4 5 6 7 8 9 10 ation limit of N , is considerably larger (about three times R (angstrom) 2 larger) than that (−69.64 kcal/mol) at the dissociation limit of FIG. 9. Same as Fig. 8 but in relative energy. The zeros of energy are set at H2, as the number of unpaired electrons (or singly occupied the respective spin-unrestricted dissociation limits.

200

2 150

1.9 100 Reference 1.8 θ=0 θ=1 50 1.7 θ=3 θ=5 θ 0 1.6 =7 θ=10 unrestricted θ=15 -50 1.5 θ=20 restricted θ=30 Energy (kcal/mol) restricted (without entropy contribution) 1.4 θ=40 -100 entropy contribution θ=50

Occupation Number 1.3 -150 (θ in mHartree) 1.2 -200 1.1 1 2 3 4 5 6 7 8 9 10 1 R (angstrom) 1 2 3 4 5 6 7 8 9 10 R (angstrom) FIG. 12. Potential energy curves (in relative energy) for the ground state of N2, calculated by the spin-restricted (with and without the entropy con- FIG. 10. Occupation numbers of the 3σ g orbital for the ground state of N2 as tributions) and spin-unrestricted TAO-LDA (θ = 40 mHartree), where the a function of the internuclear distance R, calculated by spin-restricted TAO- zeros of energy are set at the spin-unrestricted dissociation limit. The entropy LDA (with various θ). The θ = 0 case corresponds to spin-restricted KS- contributions (in total energy) as a function of the internuclear distance R, LDA. The reference data are the MRCI NOONs.77 calculated by spin-restricted TAO-LDA (θ = 40 mHartree), are also shown. 154104-12 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

orbitals) for N2 dissociation is more (three times more) than 2 that for H dissociation. 2 1.9 To sum up, when the orbital occupation numbers {fi}of 1.8 TAO-LDA are close to the exact NOONs {ni}, the strong 1.7 Reference static correlation effects are shown to be properly simulated θ=0 1.6 θ=1 by the entropy contribution of TAO-LDA. As this feature is θ=3 θ 1.5 =5 independent of the number of unpaired electrons in a sys- θ=7 θ 1.4 =10 tem, TAO-LDA seems to be promising for the study of large θ=15 polyradical systems, such as linear acenes (as will be shown θ=20 Occupation Number 1.3 θ=30 θ=40 later). 1.2 θ 1.1 ( in mHartree) 2. Twisted ethylene 1 0 10 20 30 40 50 60 70 80 90 The π (1b2) and π*(2b2) orbitals in ethylene (C2H4) Torsion Angle (degree) should be degenerate when the HCCH torsion angle is 90◦. Spin-restricted single-reference methods cannot handle such FIG. 14. Occupation numbers of the π (1b2) orbital for the ground state of twisted ethylene as a function of the HCCH torsion angle, calculated by a degeneracy properly and show an unphysical cusp in the tor- spin-restricted TAO-LDA (with various θ). The θ = 0 case corresponds to ◦ sion potential near 90 . In the calculations, we use the exper- spin-restricted KS-LDA. The reference data are the half-projected NOONs of CASSCF method (HPNO-CAS).79 imental geometry of C2H4 (RCC = 1.339 Å, RCH = 1.086 Å, ◦ 78 HCH = 117.6 ). Figure 13 shows the torsion potential energy curves (in relative energy) for the ground state of twisted ethylene as a function of the HCCH torsion angle, calculated by spin-restricted TAO-LDA, where the zeros of energy are To examine the entropy contributions (in total energy) as set at the respective minimum energies. Spin-restricted TAO- a function of the HCCH torsion angle, calculated by spin- θ = LDA (with a θ larger than 5 mHartree) is shown to be able restricted TAO-LDA ( 15 mHartree), Fig. 15 shows the to remove the unphysical cusp, though spin-restricted TAO- torsion potential energy curves (in relative energy) for the LDA (with a θ larger than 20 mHartree) is shown to yield a ground state of twisted ethylene as a function of the HCCH torsion barrier which is far too low. torsion angle, calculated by the spin-restricted (with and with- Figure 14 shows the occupation numbers of the π (1b ) out the entropy contributions) and spin-unrestricted TAO- 2 θ = orbital for the ground state of twisted ethylene as a function of LDA ( 15 mHartree), where the zeros of energy are set the HCCH torsion angle, calculated by spin-restricted TAO- at the respective minimum energies. As shown, the entropy LDA, where the reference data are the half-projected NOONs contributions are responsible for simulating the strong static of complete active space self-consistent ﬁeld (CASSCF) correlation effects to make the spin-restricted torsion poten- 79 tial energy curve the same as the spin-unrestricted one (as it method. As can be seen, the π (1b2) orbital occupation numbers of spin-restricted TAO-LDA (with a θ between 10 and 20 should be). mHartree), match reasonably well with the accurate NOONs, which is related to the accurate torsion potential energy curve, calculated by spin-restricted TAO-LDA (with the same θ). 100 90 unrestricted restricted 80 restricted (without entropy contribution) 100 entropy contribution 70 θ 90 =0 60 θ=1 θ 80 =3 50 θ=5 θ=7 40 (θ in mHartree) 70 θ=10 θ=15 30 60 θ=20 θ=30 20

θ=40 Energy (kcal/mol) 50 10

40 (θ in mHartree) 0 -10 30

Relative Energy (kcal/mol) -20 20 -30 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 10 Torsion Angle (degree)

0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 FIG. 15. Torsion potential energy curves (in relative energy) for the ground Torsion Angle (degree) state of twisted ethylene as a function of the HCCH torsion angle, calculated by the spin-restricted (with and without the entropy contributions) and spin- FIG. 13. Torsion potential energy curves (in relative energy) for the ground unrestricted TAO-LDA (θ = 15 mHartree), where the zeros of energy are state of twisted ethylene as a function of the HCCH torsion angle, calculated set at the respective minimum energies. The entropy contributions (in total by spin-restricted TAO-LDA (with various θ). The zeros of energy are set at energy) as a function of the HCCH torsion angle, calculated by spin-restricted the respective minimum energies. The θ = 0 case corresponds to KS-LDA. TAO-LDA (θ = 15 mHartree), are also shown. 154104-13 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

70 Experiment 65 DMRG (cc-pVDZ) DMRG (STO-3G) 60 B3LYP (6-31G*) 55 BLYP (6-31G*) LDA (6-31G*) 50 45 FIG. 16. Pentacene, consisting of 5 linearly fuzed benzene rings, is desig- 40 nated as 5-acene. 35 30 25 3. Singlet-triplet energy gaps of linear acenes 20 15 Linear n-acenes (C4n + 2H2n + 4), consisting of n linearly fused benzene rings (see Fig. (16)), have attracted great inter- Singlet-Triplet Energy Gap (kcal/mol) 10 est from many experimental and theoretical researchers due 5 0 to their fascinating electronic properties and technological 2 4 6 8 10 12 14 16 potential.80–95 The experimental ST gaps of n-acenes are only n-acene available up to pentacene,80–83 due to the increasing reactiv- FIG. 17. Singlet-triplet energy gap as a function of the acene length, cal- ity of the larger acenes. Recently, the calculated ST gaps have culated by spin-unrestricted KS-DFT (with LDA, BLYP, and B3LYP func- been in serious debate.84–94 Typically, delocalized π-orbital tionals), using the 6-31G* basis set. The experimental data are taken from systems, such as n-acenes, require high-level ab initio meth- Refs. 80 to 83, and the DMRG data are taken from Ref. 88. ods, such as the DMRG algorithm,88 to capture the essential strong static correlation effects. Based on the recent work of Chan and co-workers,88 the DMRG ST gaps as a function ST gaps calculated by spin-unrestricted TAO-LDA (with a of the acene length have been shown to decrease monoton- θ larger than 5 mHartree), are shown to decrease monotonically with increasing chain length. Based on a good fit to ically with the increase of chain length. The ST gaps cal- the DMRG ST gaps of the smaller n-acenes (up to 12-acene), culated by spin-unrestricted TAO-LDA (with a θ between 5 an exponential fitting function of the form a + b e−cn was and 10 mHartree), are in good agreement with the existing adopted for extrapolation of the ST gaps to the infinite chain experimental and high-level ab initio data.88, 92, 94 Due to the limit,88 yielding a finite ST gap (3.33 kcal/mol for the cc- symmetry constraint, the spin-restricted and spin-unrestricted pVDZ basis set, and 8.69 kcal/mol for the STO-3G basis set) energies for the lowest singlet state of n-acene, calculated by for polyacene (triplet above singlet). However, the extrapo- the exact theory, should be the same. To examine this prop- lated results have been shown subject to details of the fit.92 erty, spin-restricted TAO-LDA calculations (using the 6-31G* More importantly, it is unclear whether the ST gaps of the basis set) are also performed for the lowest singlet energies larger n-acenes (e.g., n ≥ 20) still decrease exponentially (as on the respective geometries that were fully optimized at the those of the smaller n-acenes) with increasing chain length, same level. The spin-unrestricted and spin-restricted TAO- which may significantly affect the extrapolated ST gap. Cal- LDA (with a θ larger than 5 mHartree) calculations are found culations on the larger acenes are necessary to address this, to essentially yield the same energy value for the lowest sin- which are, however, prohibitively expensive for the DMRG glet state of n-acene (i.e., no unphysical symmetry-breaking algorithm88 and other high-level ab initio methods.92, 94 effects occur). On the other hand, KS-DFT is computationally efficient but unable to handle such strong static correlation effects properly. To show this, spin-unrestricted KS-DFT calcula- 70 65, 66 96, 97 tions (with LDA, BLYP, and B3LYP (Refs. 13 and 65 Experiment θ=0 14) functionals) are performed, using the 6-31G* basis set (up 60 θ=1 θ 55 =3 to 16-acene), for the lowest singlet and triplet energies on the θ=5 50 θ=7 respective geometries that were fully optimized at the same θ=10 45 θ=15 level. The ST gap of n-acene is calculated as (ET − ES), the θ 40 =20 energy difference between the lowest triplet (T) and singlet 35 (θ in mHartree) (S) states of n-acene. As shown in Fig. (17), in contrast to the 30 88 DMRG results, the ST gaps calculated by spin-unrestricted 25 KS-DFT, are shown to unexpectedly increase beyond 10- 20 73 acene, due to unphysical symmetry-breaking effects. 15

To assess the performance of the present method, spin- Singlet-Triplet Energy Gap (kcal/mol) 10 unrestricted TAO-LDA calculations are performed, using both 5 0 the 6-31G* (up to 46-acene) and 6-31G (up to 74-acene) ba- 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 sis sets, for the lowest singlet and triplet energies on the re- n-acene spective geometries that were fully optimized at the same level.73 In Fig. 18, the calculated ST gaps as a function FIG. 18. Singlet-triplet energy gap as a function of the acene length, calculated by spin-unrestricted TAO-LDA (with various θ), using the 6-31G* of the acene length (using the 6-31G* basis set) are plot- basis set. The θ = 0 case corresponds to spin-unrestricted KS-LDA. The ex- ted. In contrast to the spin-unrestricted KS-LDA results, the perimental data are taken from Refs. 80 to 83. 154104-14 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

9 2 8 6-31G* 6-31G Reference θ 7 =0 1.8 θ=5 θ=7 6 θ=10 θ=15 θ=20 5 1.6 (θ in mHartree) 4 1.4

3 Occupation Number

2 1.2 Singlet-Triplet Energy Gap (kcal/mol) 1

0 1 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 0 4 8 12 16 20 24 28 32 36 40 44 n-acene n-acene

FIG. 19. Singlet-triplet energy gap as a function of the acene length, cal- FIG. 20. HOMO occupation numbers for the lowest singlet states of n- culated by spin-unrestricted TAO-LDA (θ = 7 mHartree), using both the 6- acenes as a function of the acene length, calculated by spin-restricted TAO- = 31G* and 6-31G basis sets. LDA (with various θ)/6-31G*. The θ 0 case corresponds to spin-restricted KS-LDA. Reference data are the NOONs of the active-space variational 2- RDM method.95 Figure (19) shows the ST gaps of n-acenes (n ≥ 8) as a function of the acene length, calculated by spin-unrestricted TAO-LDA (θ = 7 mHartree), using both the 6-31G* and absence of Peierls distortions98 in this limit, and the closure 6-31G basis sets.73 The effects of basis sets on the calculated of the fundamental gap.85, 92 ST gaps are shown to be insignificant for the larger acenes. The orbital occupation numbers of TAO-LDA provide in- The ground state of n-acene is found to remain a singlet as formation useful in assessing the possible polyradical char- chain length is increased. At the level of spin-unrestricted acter of n-acenes. Figure (20) shows the HOMO occupation TAO-LDA (θ = 7 mHartree)/6-31G, the ST gap of the largest numbers for the lowest singlet states of n-acenes as a func- acene studied here (74-acene) is 0.66 kcal/mol. In view of the tion of the acene length, calculated by spin-restricted TAO- slow convergence of the ST gaps with the increase of chain LDA/6-31G*, where the reference data are the NOONs of the length, the ST gaps of the larger n-acenes (n ≥ 20) are found active-space variational two-electron reduced density matrix to fit extremely well to a power-law function of the form (2-RDM) method.95 Here, HOMO is the (N/2)-th orbital, and a + bn−c, rather than the popular exponential function.88, 92, 94 LUMO is the (N/2 + 1)-th orbital, where N is the number As shown in Table III, nonlinear least-squares fitting of 3 dif- of electrons in n-acene. As can be seen, the HOMO occupa- ferent data sets (20- to 74-acene, 30- to 74-acene, and 40- tion numbers of spin-restricted TAO-LDA (with a θ between to 74-acene) of the ST gaps calculated by spin-unrestricted 5 and 15 mHartree), match reasonably well with the NOONs, TAO-LDA (θ = 7 mHartree)/6-31G, by means of the above which may suggest that the ST gaps calculated by TAO-LDA power-law fitting function, gives estimates of 0.08, 0.04, and (with a θ between 5 and 15 mHartree) should be reliably ac- 0.03 (kcal/mol), respectively, for the ST gaps of n-acenes in curate (due to the appropriate treatment of strong static cor- the polymer limit (n →∞). As the extrapolated ST gaps are relation effects via the entropy contribution), providing that rather insensitive to the choices of the fitting data sets, the ST these agreements are extendible for the larger acenes. gaps selected in the fitting data sets should have approached Figure (21) shows the active orbital occupation num- the asymptotic (large-n) behavior, decreasing as slowly as bers for the lowest singlet states of n-acenes as a function about n−1 with increasing chain length. In view of the mi- of the acene length, calculated by spin-restricted TAO-LDA nor dependence of the extrapolated results with the choices of (θ = 7 mHartree)/6-31G*.73 Here, for simplicity, HOMO, the fitting data sets, we can only conclude that in the polymer HOMO−1, ..., and HOMO−6, are denoted as H, H − 1, limit, the lowest singlet and triplet states should be degenerate ..., and H − 6, respectively, while LUMO, LUMO+1,..., within 0.1 kcal/mol (triplet above singlet), which supports the and LUMO+6, are denoted as L, L + 1, ..., and L + 6,

TABLE III. Singlet-triplet energy gaps (ST gaps) of n-acenes in the polymer limit (n →∞), obtained by nonlinear least-squares fittings of 3 different data sets (20- to 74-acene, 30- to 74-acene, and 40- to 74-acene) of the ST gaps calculated by spin-unrestricted TAO-LDA (θ = 7 mHartree)/6-31G, using a power-law fitting function of the form a + bn−c. Here, the coefficient of determination R2 is a statistical measure of the goodness- of-fit (R2 = 1, for a perfect fit).

Data set ST gap (kcal/mol) a (kcal/mol) b (kcal/mol) cR2

20- to 74-acene 0.08 0.077247 ± 0.004245 72.249 ± 0.737 1.1199 ± 0.0038 1.0000 30- to 74-acene 0.04 0.041992 ± 0.002988 64.485 ± 0.593 1.0806 ± 0.0032 1.0000 40- to 74-acene 0.03 0.028669 ± 0.001251 61.405 ± 0.284 1.0645 ± 0.0015 1.0000 154104-15 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

2 H-6 reference systems, while performing similar to KS-LDA for H-5 H-4 single-reference systems. 1.8 H-3 H-2 1.6 H-1 H 1.4 L L+1 VI. CONCLUSIONS 1.2 L+2 L+3 We have proposed TAO-DFT, a DFT with fractional or- L+4 1 L+5 bital occupations produced by the Fermi-Dirac distribution (in L+6 0.8 order to simulate the distribution of orbital occupation num-

Occupation Number 0.6 bers for interacting electrons). TAO-DFT offers an explicit

0.4 description of strong static correlation via the entropy contribution, a function of the fictitious temperature θ and orbital 0.2 occupation numbers {fi} (implicit density functionals). Even 0 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 at the simplest LDA level, the resulting TAO-LDA has been n-acene shown to perform reasonably well for multi-reference systems (due to the appropriate treatment of static correlation), when FIG. 21. Active orbital occupation numbers (HOMO-6, ..., HOMO-1, the {f } (related to the θ) are close to the exact NOONs {n }. HOMO, LUMO, LUMO+1,...,andLUMO+6) for the lowest singlet states i i of n-acenes as a function of the acene length, calculated by spin-restricted As this feature is independent of the number of unpaired elec- TAO-LDA (θ = 7 mHartree)/6-31G*. trons in a system, TAO-LDA seems to be very useful for the study of large polyradical systems. In our study, an optimal value of θ = 7 mHartree, has been defined based on respectively. As can be seen, the active orbital occupation physical arguments and numerical investigations. TAO-LDA numbers exhibit oscillatory behavior in the approach to unity (θ = 7 mHartree), though not optimal for all the systems stud- (singly occupied) with increasing chain length. The number ied, has been shown to consistently improve upon KS-LDA of fractionally occupied orbitals is shown to increase with the for multi-reference systems, while performing similarly to increase of chain length, which supports previous finding that KS-LDA for single-reference systems. Due to its computa- large acenes should exhibit polyradical character.88, 89 tional efficiency and reasonable accuracy, TAO-LDA has been To sum up, it seems plausible to believe the results ob- applied to the study of the ST gaps of acenes, which are chal- tained by TAO-LDA (θ = 7 mHartree) here, as the calculated lenging problems for conventional electronic structure meth- ST gaps are in good agreement with the existing experimen- ods. At the level of TAO-LDA (θ = 7 mHartree)/6-31G, the tal and high-level ab initio data,88, 92, 94 the calculated HOMO ST gap of polyacene has been shown to be vanishingly small occupation numbers match reasonably well with the accurate (within 0.1 kcal/mol), and large acenes should exhibit singlet NOONs, and no unphysical symmetry-breaking effects occur polyradical character in their ground states. for the lowest singlet states of n-acenes. As TAO-LDA is conceptually simple, computationally efficient, and easy to implement, it seems to be a promising method for the study of ground states of large single- and V. DEFINITION OF AN OPTIMAL θ VALUE multi-reference systems. However, as with all approximate electronic structure methods, some limitations remain. The In our study, TAO-LDA (with some fictitious tempera- optimal θ = 7 mHartree for TAO-LDA is system-independent ture θ) has been found to perform reasonably well for multi- (not fully optimized for each system), a system-dependent θ reference systems, when the orbital occupation numbers {f } i (related to the distributions of NOONs) is expected to enhance are close to the exact NOONs {n }. In such a situation, the i the performance of TAO-LDA for a wide range of systems. strong static correlation effects can be properly simulated by For single-reference systems, the performance of TAO-LDA the entropy contribution of TAO-LDA. However, for multi- is similar to that of KS-LDA. A possible TAO-DFA is ex- reference systems, the optimal θ for TAO-LDA has been pected to perform better than TAO-LDA for single-reference found to be highly system-dependent, ranging from 5 to 50 systems. Although the SCEs of TAO-DFAs are expected to mHartree, due to the different strengths of static correlation be less than those of KS-DFAs, the SIEs and NCIEs of TAO- and the diversities of the {n }. On the other hand, for single- i DFAs may remain enormous in situations where these fail- reference systems (in the absence of strong static correlation ures occur. A fully nonlocal TAO-DFT (i.e., nonlocal E [ρ] effects), TAO-LDA (with a θ smaller than 10 mHartree) has xc and E [ρ]) may be needed to resolve all the three qualitative been shown to perform similar to KS-LDA. θ errors (SIE, NCIE, and SCE). We are currently investigating For TAO-LDA, although it is impossible to choose a θ along these lines, and results may be reported elsewhere. that is optimal for all the systems studied, it is still useful to define one to provide an explicit description of orbital occupations. Here, the optimal θ value is defined as the largest θ ACKNOWLEDGMENTS value for which the performance of the TAO-LDA (with this θ) and KS-LDA is similar for single-reference systems. Based This work was supported by National Science Council of on our numerical investigations, an optimal value of θ = 7 Taiwan (Grant No. NSC98-2112-M-002-023-MY3), National mHartree, is finally chosen. TAO-LDA (θ = 7 mHartree) has Taiwan University (Grant Nos. 99R70304 and 10R80914-1), been shown to consistently improve upon KS-LDA for multi- and NCTS of Taiwan. We are grateful to the Computer and 154104-16 Jeng-Da Chai J. Chem. Phys. 136, 154104 (2012)

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