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Multiconfiguration Pair-Density Functional Theory is Free from Delocalization Error Junwei Lucas Bao, Ying Wang, Xiao He, Laura Gagliardi, and Donald G. Truhlar J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b02705 • Publication Date (Web): 01 Nov 2017 Downloaded from http://pubs.acs.org on November 3, 2017
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The Journal of Physical Chemistry Letters Oct. 30, 2017 – prepared for JPC Lett.
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Multiconfiguration Pair-Density Functional Theory Is Free From Delocalization Error Junwei Lucas Bao,† Ying Wang,‡ Xiao He,‡ Laura Gagliardi,*,† and Donald G. Truhlar*,† †Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455-0431, United States ‡School of Chemistry and Molecular Engineering, State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China Supporting Information Placeholder ABSTRACT: Delocalization error has been singled out by Yang and coworkers as the dominant error in Kohn-Sham density functional theory (KS-DFT) with conventional approximate functionals. In this letter, by computing the vertical first ionization energy for well separated He clusters, we show that multiconfiguration pair-density functional theory (MC-PDFT) is free from delocalization error. To put MC-PDFT in perspective, we also compare it with some Kohn-Sham density functionals, including both traditional and modern functionals. Whereas large delocalization errors are almost universal in KS-DFT (the only exception being the very recent corrected functionals of Yang and coworkers), delocalization error is removed by MC-PDFT, which bodes well for its future as a step forward from KS-DFT.
ToC graphic
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Kohn-Sham density functional theory (KS-DFT) has become the most widely used tool in theoretical and computational chemistry and physics. Despite the continuing successful efforts in developing better exchange-correlation (XC) density functionals, KS-DFT still has known difficulties in several areas, including inherently multi-configurational systems, transition metal chemistry, 3 , 4 the energy gap between spin states, 5 and electronic excitation 6 and charge transfer.7 It is often stated that all the deficiencies of KS-DFT ultimately stem from self-interaction errors, and much has been learned by taking this perspective, but Cohen et al.,8 have shown that the kind of error usually ascribed to selfinteraction error “is best characterized as the delocalization error, as it captures the physical nature of the problem.” Delocalization error in KS-DFT can be attributed to an inaccurate estimation of the energy for a noninteger number of electrons,9,10,11 which leads to too low an energy for delocalized electron distributions and sometimes a qualitatively wrong density.8,12,13
Yang and coworkers 14 have elucidated this argument by discussing how a calculation may be self-interaction free but still suffer from errors that are often associated with selfinteraction error, and they stated that “Delocalization error is one of the dominant errors of mainstream density functional approximations (DFAs). It is responsible for many failures of DFT calculations.” Delocalization error predicts too low an energy for electronically delocalized distributions, and consequently this error can cause computed reaction barriers to be too low, spin states of transition metals to be in the incorrect order, band gaps of semiconductors to be too small, and electric conductance of electronic devices to be overestimated.8,15 Therefore it is useful to have a test of DFT that isolates the delocalization error and makes it manifest with the least distracting complications. Yang and coworkers have shown that one powerful such test is based on the theoretical concept of fractional charge.16,17 In particular, they showed that one can measure delocalization energy by calculating the ionization energy of a well separated cluster of He atoms, which are so far separated from one another as to be effectively noninteracting. The experimental ionization energy would be the
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same as for a single He atom, but calculations with delocalization errors yield smaller and smaller ionization energies as the number of He atoms is increased because the hole created by the ionization process is incorrectly predicted to be delocalized over all the He atoms, with a fractional positive charge on each of them. The failure of KS-DFT for treating the He2+ cation has been identified before,16,18,19,20,21,22,23,24 and it is a special case of the radical cation problem. The well separated He cluster ionization test proposed by Yang and coworkers puts it in a broader context. Yang and coworkers recently developed a localized orbital scaling correction14 for eliminating delocalization error in a version of KS-DFT, and they showed using this test that this correction significantly reduces the delocalization error. Another possible way to improve on KS-DFT is to use multi-determinant representations in DFT,25,26,27,28,29,30,31,32,33,34 as in the recently proposed multiconfiguration pair-density functional theory35,36 (MC-PDFT). This theory works with self-interaction-free densities from wave function theory, but the energy is calculated using a density functional; MC-PDFT has been successfully applied in a variety of problems, including bond dissociation in transition metal compounds, 37 , 38 , 39 spin splitting, 40 main-group chemistry,38 electronic excitation, 41 and charge transfer. 42 Here, we will show that, while conventional KS-DFT has huge delocalization errors as measured by the He cluster ionization test, MC-PDFT with either translated35 or fully translated39 density functionals is immune to delocalization error. Overview of MC-PDFT theory. The Born-Oppenheimer energy in MC-PDFT is computed as the summation of nuclear repulsion energy VNN, electronic kinetic energy TM, electron-nuclear attraction energy VNe, classical electronelectron repulsion Coulomb energy Vee, and on-top energy EOT. The TM, VNe and Vee terms are evaluated using a variational multiconfiguration self-consistent-field (MCSCF) wave function, and EOT is evaluated by using an energy functional that takes the total density from the MCSCF oneparticle density matrix and the on-top pair density from the MCSCF two-particle density matrix. MC-PDFT eliminates the spin symmetry ambiguity43 of KS-DFT by using on-top pair density and not directly using spin-up or spin-down density; and it avoids double counting of the dynamic correlation energy by not using the MCSCF energy. The on-top energy functional is obtained, at the current stage of development, by translating a conventional KS XC density functional by replacing the α and β densities and their gradients with functions of the total density ρ and on-top pair density Π. The original35 translation involves ρ, grad ρ, and Π; the later39 full translation also involves grad Π. The original translated Perdew-Burke-Ernzerhof functional 44 is denoted tPBE; the full translation gives the fully translated PBE functional, denoted ftPBE. Similar translations of revPBE45 and BLYP46,47 give trevPBE, ftrevPBE, tBLYP, and ftBLYP. Background: MCSCF theory. The most popular kind of MCSCF calculation is complete-active space SCF (CASSCF) theory.48 The active spaces used for the clusters were chosen two ways: (a) the systematic correlatedparticipating-orbitals (CPO) scheme,37,40,49 in which we include the 1s orbital and its correlating orbital 1s' for each He
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atom such that for HeN the active space includes 2N electrons in 2N orbitals, and for HeN+ the active space includes (2N – 1) electrons in 2N orbitals (more generally the CPO scheme correlates each of the most important participants in the chemical problem under study with a corresponding correlating orbital, and the resulting active spaces should be better balanced than the alternative full-valence scheme); (b) the full valence (FV) active space, for which the HeN active space has 2N electrons in N orbitals (this neutral case is equivalent to restricted Hartree-Fock calculation), and for HeN+ the active space includes (2N – 1) electrons in N orbitals, which involves N configuration state functions (CSFs). As pointed out by McLean et al, 50 in order for small MCSCF wave functions to produce a correct potential curve for HeN+, one must allow the variationally optimized wave function to have a lower spatial symmetry than the symmetry of the nuclear framework. Thus in all cases we study here, we will find the variationally best solution of the CASSCF equations without constraining spatial symmetry, but all CASSCF wave functions are correct singlet or doublet spin eigenfunctions. Quantifying delocalization error using well separated He clusters. The model14 is as follows: Consider a planar HeN cluster (in the present work, N = 1, 2, 3, 4, 6, 8, and 16), in which all the He atoms are chemically equivalent and are evenly distributed on a circle. The Cartesian coordinates (x,y) of He atom i (i = 1, 2, …, N) in the cluster are R cos[(i − 1)α ], R sin[(i − 1)α ] , where R is the radius of the
(
)
circle, which is related to the distance d between two neighboring He atoms by R = d[2(1–cosα)]-1/2 with α = 2π N . The physical quantity to be computed is the vertical first ionization energy (which is denoted as IP). In this work, we consider d = 10 Å; at this distance, the He atoms are effectively not interacting with each other, and IP for all HeN clusters should be equal to the first IP of a ground-state He atom (which is 24.587 eV). We have checked this by computing the IP for the He2 cluster by using coupled cluster theory with large basis sets and up to full quadruple excitation (CCSDTQ) (notice that He2 only has 4 electrons, and thus CCSDTQ is equivalent to full CI); the IP of He2 computed this way is 24.579 eV, which is only 0.008 eV less than first ionization energy of a ground state He atom; this confirms that 10 Å is large enough that the experimental result is independent of N. We will see that if one uses KS-DFT to compute the IP of the HeN cluster, the error increases as the cluster size increases, until the error (which is huge for almost all functionals tested) roughly approaches convergence at He16. The ionization creates a positive hole in the system, and DFT tends to delocalize the hole over all N He centers. The signed error is computed as the difference between the computed IP and the experimental IP, which is taken as 24.587 eV for all sizes of HeN clusters. The delocalization error (DE) is defined as the variation of the signed error in the IP for He minus the signed error for well-separated He16. Computational methodology. We calculated the IPs by KS-DFT with several functionals, HF theory, CASSCF, complete active-space 2nd order perturbation theory, CASPT2,51,52 and MC-PDFT.
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Single-configuration calculations for background and comparison. In order to provide background for the MCPDFT results, we place them in the context of what can be predicted by previous density functional theory methods by also carrying out KS-DFT calculations with PBE, revPBE, BLYP, B3LYP,47,53,54 and several of more modern functionals, including M06-L,55 M06,56 M06-2X56 and M06-HF.57 A complete list of 35 functionals tested is given in the Supporting Information (SI). KS-DFT computations mentioned in the article proper are done by restricted Kohn-Sham for neutral clusters and unrestricted (i.e., spin-polarized) KohnSham (UKS) for cations; restricted open-shell calculations are given in the SI. Table 1. Delocalization error (eV) MC-PDFT 0.00a CASPT2 0.00 CASSCF 0.00 Hartree-Fock −1.38 M06-HF 0.67b M11 3.12b average 5.47c d B3LYP 6.31 PBEd 7.89 BLYPd 8.09 a same for all six on-top functionals and both kinds of active space b best and second best of the 35 KS functionals in the SI c average of 35 KS functionals in the SI d popular KS functional
Table 1 and tables in the SI list the delocalization errors for selected XC functionals in KS-DFT calculation, and they show that the delocalization error can be huge. As the size of the cluster gets larger, the IP computed by KS-DFT becomes smaller (this is true for all 35 functionals that we tested; see the SI). The PBE functional has a DE of 7.9 eV, and BLYP has DE = 8.1 eV, illustrating the enormous delocalization error that is built into KS-DFT. Even though UKS does break the symmetry for the cations (different α and β orbital energies), the IPs produced by RKS/UKS and RKS/ROKS calculations are very similar. For Hartree-Fock theory, on the other hand, the computed IP increases as N increase, and thus the delocalization error is -1.4 eV. Therefore, we might expect that mixing HF exchange into XC functionals will reduce the delocalization error; and, indeed, this is what we observed in the series (percentage of HF exchange is in parentheses) of M06-L (0), M06 (27), M06-2X (54), M06-HF (100), whose DEs are respectively 7.4, 5.9, 3.8 and 0.67 eV. M06-HF has the smallest DE among all the 35 functionals we tested in this work, and it is even smaller in magnitude than that for HF (0.67 eV vs. 1.38 eV), although by definition HF also has 100% HF exchange. The nonzero DE for these two methods with 100% HF exchange (and therefore no one-electron selfinteraction energy) confirms that delocalization error is indeed different from one-electron self-interaction (and, as discussed by Yang and coworkers,14 it is also different from and more fundamental than many-electron self-interaction). For other functionals such as LC-ωPBE 58 and ωB97X, 59
which have 100% Hartree-Fock exchange at long range, their delocalization error is still very high (respectively 3.2 and 3.5 eV). We have also done UKS calculations that started with an initial guess in which the hole was localized on one of the He atoms, followed by stability test of the solution (and if the solution is not stable, further optimization of the solution is carried out until a stable solution is reached); for none of the functionals (for details, please refer to SI) does the lowestenergy solution have zero or small DE. It is interesting to check whether or not the HeN+ system is an inherently multiconfigurational system so that the errors we observe in these single-configuration methods is indeed delocalization error rather than static correlation error. We checked the weight (square of the configuration interaction coefficient of the dominant configuration) in the configuration interaction expansion for HeN+ (from N = 1 to 6) and found that it remains approximately unchanged and is larger than 0.97. This confirms that we can attribute the errors to over-stabilized delocalized electronic distributions rather than to static correlation error. MC-PDFT eliminates the delocalization error in this fundamental test problem. The signed errors of the computed IP of tPBE, trevPBE, tBLYP, ftPBE, ftrevPBE, and ftBLYP do not vary as cluster size increases. In particular, with the CPO active spaces, the constant signed errors in the IP for the six on-top functionals are respectively 0.0, 0.2, 0.4, 0.0, 0.1, and 0.4 eV, and with the FV active spaces they are respectively 0.1, 0.0, 0.2, -0.1, 0.1 and 0.3 eV. Since the DE is the signed error for He minus the signed error for well-separated He16, and since the signed errors are constant, we see that all six on-top functionals have zero delocalization error with both active spaces. Thus MC-PDFT removes delocalization error. We plot the computed IPs by MC-PDFT and the corresponding KS-DFT methods for HeN clusters in Figure 1. For comparison, the signed errors for CASSCF, CASPT2, and CASPT2-0 are also constant, and they are respectively 0.7, -0.1, -0.1 for CPO active spaces and -1.1, -0.2, -0.2 for FV. The variational solutions for both CASSCF (used in MCPDFT) and KS-DFT have spatial symmetry lower than the nuclear framework. This allows both CASSCF and MCPDFT to avoid delocalization error, even with an eigenfunction of S2. However, KS-DFT, whether or not it breaks spin symmetry (i.e., for both UKS and restricted open-shell calculations) or spatial symmetry has large delocalization error. In fact, Tables S5 and S6 show that, with the exception of the M06-HF functional, the delocalization error is in the range 3–8 eV. Final perspective. From our previous studies, the major advantages of combining density functional theory with MCSCF wave functions in MC-PDFT theory are: (1) MCPDFT improves the accuracy of density functional theory for treating inherently multiconfigurational systems; (2) MCPDFT reduces cost and memory requirements as compared to CASPT2 (especially for systems with a large number of active electrons, for which cases CASPT2 is impractical) with similar (and sometimes better) accuracy. The present
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work adds to this that (3) MC-PDFT eliminates delocalization error, which plagues Kohn-Sham theory in many ways.
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FIGURE 1. Computed ionization energies (in eV) for well-separated HeN clusters by PBE and revPBE in UKSDFT, and by tPBE, trevPBE, ftPBE, and ftrevPBE in MC-PDFT (with CPO active space selection).
ASSOCIATED CONTENT Supporting Information. Computational details, symmetry considerations, localized soltutions in KS theory, absolute energies and computed ionization energies, delocalization errors, CASSCF orbitals. The Supporting Information is available free of charge on the ACS Publications website at DOI/
AUTHOR INFORMATION Corresponding Authors *E-mail:
[email protected] *E-mail:
[email protected] ACKNOWLEDGMENT This work was supported in part by the NSF Grant CHE1464536. J. L. Bao acknowledges a Doctor Dissertation Fellowship (DDF) provided by University of Minnesota.
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