Correlation Matrix Renormalization Theory: Improving Accuracy with

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Correlation matrix renormalization theory: improving accuracy with two-electron density-matrix sum rules Chen Liu, Jun Liu, Yongxin Yao, Ping Wu, Cai-Zhuang Wang, and Kai-Ming Ho J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00570 • Publication Date (Web): 26 Aug 2016 Downloaded from http://pubs.acs.org on August 27, 2016

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Journal of Chemical Theory and Computation

Correlation matrix renormalization theory: improving accuracy with two-electron density-matrix sum rules C. Liu,† J. Liu,† Y. X. Yao,∗,† P. Wu,‡ C. Z. Wang,† and K. M. Ho†,‡ †Ames Laboratory–US DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA ‡Hefei National Laboratory for Physical Sciences at Microscale, International Center for Quantum Design of Functional Materials (ICQD) and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: [email protected] Abstract We recently proposed the correlation matrix renormalization (CMR) theory to treat the electronic correlation effects [Phys. Rev. B 89, 045131 (2014) and Sci. Rep. 5, 13478 (2015)] in ground state total energy calculations of molecular systems using Gutzwiller variational wavefunction (GWF). By adopting a number of approximations, the computational effort of the CMR can be reduced to a level similar to Hartree-Fock calculations. This paper reports our recent progress in minimizing the error originating from some of these approximations. We introduce a novel sum-rule correction to obtain a more accurate description of the intersite electron correlation effects in total energy calculations. Benchmark calculations are performed on a set of molecules to show the reasonable accuracy of the method.

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1

Introduction

One of the outstanding problems of ab initio calculations is to develop a method with efficient scaling and quantitative accuracy for descriptions of correlated electron systems. The KohnSham density functional theory, KS-DFT, 1,2 is nowadays the working horse to calculate and predict physical properties of many materials, but fails to describe the strong electron correlation effects. Wavefunction-based quantum chemistry methods can be very accurate, and the efficiency has been greatly improved recently, e.g., by using density-matrix renormalization group. 3–5 However, they are still limited to finite systems. The quantum Monte Carlo (QMC) methods have also advanced significantly in recent years, and show-case studies have been available for realistic correlated materials. 6–8 Nevertheless, like advanced quantum chemical calculations, the computational load of QMC remains very heavy and restricts the size of systems which can be studied. Meanwhile, hybrid approaches which merge DFT with manybody techniques, like DFT+onsite Coulomb interaction (DFT+U), 9,10 DFT+dynamical mean-field theory, 11–13 and DFT+Gutzwiller, 14–19 are in practice highly successful in describing real correlated materials. However, the inclusion of adjustable screened Coulomb parameters limits the predictive power. Furthermore, the choice of double-counting term, which subtracts the DFT-level local onsite correlation contributions, remains open and still under active investigations. 20–24 In previous work, 25,26 we developed a computationally efficient method for ground state total energy calculations of correlated electron systems, namely, the correlation matrix renormalization method, to address the above challenges. The method adopts the Gutzwiller type variational wavefunctions and directly evaluates the expectation value of the ab initio manyelectron Hamiltonian—without the necessity of introducing screened Coulomb parameters and free of double-counting issues. The computational effort of the CMR method scales as N 4 or better with respect to the system size N , like Hartree-Fock. The accuracy of the previously proposed CMR method relies on a functional of the one-electron renormalization factors, which was obtained by numerically fitting to a set of 2 ACS Paragon Plus Environment

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exact results for each element in the chosen reference systems. However, the exact reference points, e.g., the binding energy curves, the onsite reduced density matrix as a function of bond length of dimers, are usually not readily available and involve heavy high-level quantum chemistry calculations. The numerical fitting can also be tedious. In this paper, we propose an alternative way to reach similar accuracy for the CMR method without the need to fit to any pre-obtained exact reference points. Our scheme is based on an analytically solvable system: the minimal basis hydrogen molecule H2 . Through comparison of the exact analytical solution for this simple system with the approximate results of the CMR method, we propose a modification of the one-electron renormalization factor and introduce an additional sum-rule term to reduce the error coming out of the approximations in the CMR method. Benchmark calculations are carried out on hydrogen clusters and multiple molecules composed of the light elements in the first two rows of the periodic table. To compare with largebasis full configuration interaction (FCI) calculations and experimental data, we make use of the quasi-atomic minimal basis set orbitals (QUAMBO) we previously developed. 27 They have been shown to be good approximations to the multi-configurational self-consistent fielddetermined correlating orbitals, recovering a large percentage of the correlation energy. 27



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2 2.1

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Methods Total Energy with CMR

In the form of second quantization, the full ab initio nonrelativistic Hamiltonian for an interacting many-electron system can be expressed as,

H=

X

EiΓ |Γi i hΓi | +

iΓ

+

0 X

tiαjβ c†iασ cjβσ

iαjβ,σ

1 2

0 X

u(iαjβ; kγlδ)c†iασ c†jβσ0 clδσ0 ckγσ ,

(1)

iαjβ kγlδ,σσ 0

where i, j, k, l are the atomic site indices, the α, β, γ, δ refer to orbital indices, and σ, σ 0 indicate the spin indices. Here, t is the one-electron hopping integral expressed as,

tiαjβ = hφiα |Tˆ + Vîon |φjβ i,

(2)

where Tˆ and Vîon are the operators for kinetic energy and electron-ion interaction, and u is the two-electron Coulomb integral expressed as, Z u(iαjβ; kγlδ) =

Z dr

dr0 φ∗iα (r)φ∗jβ (r0 )Uˆ (|r − r0 |)φlδ (r0 )φkγ (r)

(3)

with the Coulomb interaction operator Uˆ . In Eq. 1, the first term is a spectral representation of a site-wise local Hamiltonian, in which {Γi } are eigenstates of the local onsite many-body Hamiltonian Hi,loc , which can be written as

Hi,loc =

X αβ

tiαiβ c†iασ ciβσ +

X

u(iαiβ; iγiδ)c†iασ c†iβσ0 ciδσ0 ciγσ ,

(4)

αβγδ,σσ 0

and EiΓ is the energy for the local configuration |Γi i. The second and third terms in Eq. P0 1 describe the non-local one-body and two-body contributions respectively. means that


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the pure local on-site terms are excluded from the summation. Through the FCI approach which uses the exact many-body wavefunction, the exact total energy is written as,

EFCI =

X

0 X

EiΓ piΓ +

iΓ

D E tiαjβ c†iασ cjβσ

iαjβ,σ

FCI

0 D E 1 X † † + u(iαjβ; kγlδ) ciασ cjβσ0 clδσ0 ckγσ , 2 iαjβ FCI

(5)

kγlδ,σσ 0

where the first term is onsite energies that are determined by the diagonal elements {piΓ } of the reduced many-body density matrix, and the second and third components are determined by the nonlocal one-electron density matrix (1PDM) and two-electron density matrix or correlation matrix (2PCM), respectively. Here, for any operator O, hOiFCI is a short-hand notation for hΨFCI |O|ΨFCI i. In the CMR approach, the GWF is adopted to take care of the important local onsite correlations. The GWF can be constructed from a non-interacting wavefunction |Ψ0 i, being expressed as, ! |ΨGWF i =

Y X i

giΓ |Γi i hΓi | |Ψ0 i ,

(6)

Γ

where giΓ is the Gutzwiller variational parameter to optimize the occupation probability piΓ of the local configuration |Γi i. The total energy with the CMR approximation is then evaluated as,

ECMR =

X

EiΓ piΓ +

iΓ

0 X

D E tiαjβ c†iασ cjβσ

iαjβ,σ

CMR

0 D E 1 X u(iαjβ; kγlδ) c†iασ c†jβσ0 clδσ0 ckγσ . + 2 iαjβ CMR

(7)

kγlδ,σσ 0

In comparison with the exact expression in Eq. 5, the first term has the same form, since the local onsite terms can be treated exactly within the CMR method. The second and third 5 ACS Paragon Plus Environment


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term, however, are evaluated with the approximated 1PDM and 2PCM, which are based on the GWF. Following the idea of Gutzwiller approximation, 28–30 the single-electron term can be written as, D

c†iασ cjβσ

E CMR

D E jβ c†iασ cjβσ . ≈ ziασ

(8)

0

jβ jβ ziασ can be evaluated as ziασ = ziασ zjβσ if (iα) 6= (jβ) and 1 otherwise. ziασ is known as

the one-electron renormalization factor. Note that the most general form of the z-factor has been worked out and is quite complicated. 31 However, if the local onsite eigenstates {Γ} coincide with Fock states {F}, which is exactly true for single-orbital systems like hydrogen molecules and approximately holds if Coulomb and exchange integrals are dominant in the onsite Hamiltonian shown in Eq. 4, a much simpler and manageable form can be obtained as, GA ziασ

X √piF piF 0 |hFi |c†iασ |Fi0 i| p = . n0iασ (1 − n0iασ ) FF0

(9)

Here, n0iασ = hc†iασ ciασ i0 ≡ hΨ0 | c†iασ ciασ |Ψ0 i, and the Fock states occupation probabilities {piF } are the redefined variational parameters of {giF }. The Fock states can be generated as, |Fi i =

Y † F (ciασ )niασ |0i,

(10)

ασ

with nF iασ = 0 or 1, indicating whether there is an electron occupying a spin-orbital state contained in Fock state |Fi i. For simplicity, we use the above form of z-factor in the work presented here, i.e., the local many-body eigenstates {Γ} are replaced by the Fock states {F}. 29,30 Furthermore, by assuming the applicability of Wick’s theorem, we can have a factorized two-body Coulomb term like Hartree-Fock method (HF) with additional renormalizations, D

c†iασ c†jβσ0 clδσ0 ckγσ

E CMR

D E E D kγ lδ =ziασ zjβσ0 c†iασ ckγσ c†jβσ0 clδσ0 0 0 D E D E kγ † † lδ 0 − δσσ ziασ zjβσ ciασ clδσ cjβσ ckγσ . 0


0

(11)

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There remain two key issues for the CMR approach to be applied to real systems. Firstly, the best form of the z-factors in finite dimensions, e.g., 0, 1, 2 or 3-dimension, are not necessarily the one from the standard GA 31 shown above, which is only exact in infinite dimension. Secondly, the Wick’s theorem does not necessarily hold for GWF. Therefore the factorization of the 2PCM in Eq. 11 introduces additional errors. As a manifestation, the P CMR approximation violates the sum rule, that is, βσ0 hˆ nασ n ˆ βσ0 iCMR 6= Ne hˆ nασ iCMR .

2.2

Insight from Minimal Basis H2

In our early publications, 25,26 we proposed a functional form of the z-factors, f (z), for the renormalization of the 1PDM and 2PCM. It works well for a series of hydrogen clusters Hn and nitrogen clusters Nn , as well as ammonia NH3 . However, the functional f (z) can only be obtained by numerically fitting to a set of exact results (e.g., total energy, local reduced density matrix as a function of bond length of H2 and N2 ) for each element in the chosen reference systems. In this work, we propose a way to solve the above issues without fitting. For this purpose, we study a simplest, analytically solvable model—minimal basis hydrogen dimer H2 , which holds a transition from weak electron correlations to strong correlations as the bond length increases. The Hamiltonian for H2 with minimal basis is written as,

H=

2 X X σ

i=1

−

X

c†iσ ciσ

1X + u0 c†iσ c†iσ0 ciσ0 ciσ 2 σσ0

!

t(c†1σ c2σ + H.C.)

σ 20 1 XX + u1 (c†iσ c†iσ0 ciσ0 cjσ + c†iσ c†iσ0 cjσ0 ciσ + c†iσ c†jσ0 ciσ0 ciσ + c†jσ c†iσ0 ciσ0 ciσ ) 2 i,j=1 σσ0

+

(12)

20 20 1 XX 1 XX u2 (c†iσ c†iσ0 cjσ0 cjσ + c†iσ c†jσ0 ciσ0 cjσ ) + u3 (c†iσ c†jσ0 cjσ0 ciσ ), 2 i,j=1 σσ0 2 i,j=1 σσ0

where −t is the one particle inter-site hopping integral, is the 1s orbital energy level of



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hydrogen, u0 is the onsite Coulomb integral u(11; 11), u1 is u(11; 12), u2 is the inter-site exchange integral u(11; 22), and u3 is the inter-site Coulomb integral u(12; 12). Note that in general it holds u0 > u3 u2 > u1 . The exact total energy can be expressed explicitly as, Eexact =2 + 2du0 + 2zGA (−t + u1 ) (13) + u2 + u3 − 2du3 , where d is the on-site double occupancy, i.e., the occupation probability of the Fock state p | ↑↓i, and zGA =4 d(1/2 − d). The CMR energy is also given analytically as, ECMR =2 + 2du0 + 2z 2 (−t + u1 ) z4 1 + (3z 4 − 1)u2 + u3 − u3 . 2 2

(14)

From the comparisons between Eq. 13 and Eq. 14, we can find that the on-site treatment is indeed exact. Furthermore, the CMR approach produces correct limiting behavior: at small separation where the double occupancy d approaches 1/4 and z, zGA approach 1, the CMR energy approaches the exact result of E = 2 + 12 u0 + 2(−t + u1 ) + u2 + 12 u3 , while at large separation limit, d, z, zGA and u2 , u3 approach 0, and CMR approaches the correct atomic limit of E = 2. However, errors still exist in the intermediate range, which is a very important region where interesting physics like bond breaking process happens. Fortunately, the comparison based on the analytical formulations provides a way to minimize the error due to the CMR approximation. Evidently, the z-factor in CMR should √ be qual to zGA by matching the one-particle hopping component between CMR and FCI. It also fixes the error associated with exchange integral u1 simultaneously. However, the component involving the inter-site Coulomb integral u3 remains different. Viewing that the CMR approach treats the local onsite components accurately, it would be highly desirable if the intersite Coulomb contributions can be reorganized and absorbed into the local onsite Hamiltonian, which is treated exactly in the CMR formalism. This can be done for the H2


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case by introducing the following sum-rule term to the original Hamiltonian in Eq. 12, 2

Hs.r.

1 XX = (−u3 )ˆ niσ 2 i=1 σ

2 X X j=1

! n ˆ jσ0 − Ne

.

(15)

σ0

Note that the expectation value of the sum-rule term is always zero if evaluated exactly with respect to any many-body wave function, thus the additional of this term would not change the total energy of the system. Rather for the CMR approach, which violates the sum-rule but has accurate local onsite evaluations, the sum-rule provides a natural way to improve the accuracy of the method by shifting the non-local inter-site terms to local onsite terms. In the special case of minimal basis H2 , the major inter-site u3 term is exactly canceled out and replaced by an onsite u3 term, which is evaluated accurately. As a result, the CMR total energy can be written as, 3 2 − 1)u2 . ECMR = Eexact + (zGA 2

(16)

Here, the residual term is generally very small, because the inter-site exchange integral u2 is much smaller than the Coulomb integral u0 and u3 , and it decays very fast with respect to bond length. Furthermore, zGA approaches 1 at small atomic separation, so that the 2 prefactor (zGA − 1) also vanishes.

For general multi-orbital molecular systems, the sum-rule terms cannot completely cancel out the inter-site Coulomb interactions. We introduce the following sum-rule correction,

Hs.r.

X 1X λiασ n ˆ iασ n ˆ jβσ0 − Ne . = 2 iα 0 jβσ

(17)

Here λiα is determined by the weighted average of the relevant intersite two-electron Coulomb integrals, expressed as, P λiα = −

−6 u(iαjβ; iαjβ)Rij P −6 j6=i,βσ 0 Rij

j6=i,βσ 0

(18)

where Rij is the distance from atom j to atom i. The distance-dependent weighting factor, 9 ACS Paragon Plus Environment


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−6 Rij , is chosen for practical convenience. In fact, the exact form of the weighting factor is

not crucial, as long as it decays sufficiently fast. A more sophisticated way to derive the weighting factor is presented in the supporting information.

3

Results and Discussions

We have obtained a functional form of the z-factor and introduced the sum-rule term as a way for error cancellations in the CMR method, based on the analytical solution of the minimalbasis hydrogen dimer model. Here we first show the CMR numerical results for the reference H2 dimer, as well as H6 ring and H8 cube, which exhibit different bonding environments. We also address the basis-set convergence issue in H2 by utilizing the QUAMBO approach we developed a few years back. 27 The QUAMBO-based CMR method is further demonstrated to be applicable in multi-active-orbital molecular systems, including N2 , F2 , HF, CO, NH3 and CH4 .

3.1

Hn Clusters

Figure 1 shows the CMR ground state total energy and local onsite double occupancy curves of H2 , H6 ring and H8 cube with STO-3G minimal basis-set orbitals. The results from FCI, HF and LDA calculations are also included for comparison. The CMR approach indeed reproduces the exact results in the reference H2 very well. Remarkably, it also predicts the energies and local double occupancies in close agreement with FCI in H6 and H8 clusters, which have quite different coordination numbers. Noticeably, the single Slater determinant based approaches, like HF and LDA, show significant errors in the potential energies, especially at larger atomic separations. The reason is that the local double occupancy, which should be quickly suppressed as the bond length increases in FCI, is instead fixed at 0.25 by the 1s-orbital occupation of 0.5 per spin-channel in HF and LDA. The constant double occupancy contributes an energy term in proportion to the onsite Coulomb integral u0 , causing


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the large errors in HF and LDA. E (Har./atom)

−0.4 −0.45 −0.5 H6 ring

H2

−0.55 Double occupancy

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0.25 0.2

H8 cube

FCI CMR HF LDA

0.15 0.1 0.05 0

0

1

2

3 0

1

2

3 0

Bond length (Å)

1

2

3

Figure 1: Energy and double occupancy curves of hydrogen clusters. The upper plot is the potential energy curves of Hn clusters, showing the CMR results overlap with the FCI calculations. In contrast, the curves from HF and LDA have significant deviations from the exact result. The lower plot shows the corresponding results of double occupancy.

3.2

Large Basis through QUAMBO Approach

Although it is instructive to work with the STO-type minimal basis-set orbitals, it becomes mandatory to use converged large basis-set orbitals in order to compare with experiments or make quantitative predictions. However, it is not convenient to construct GWF directly using the large basis-set orbitals. To address this issue, we take advantage of the QUAMBOs, which has been shown to be good approximations to the multi-configurational self-consistent field-determined correlating orbitals, recovering large percent of correlation energy. 27 We here demonstrate the QUAMBO-based CMR calculations in H2 using the 6-311G(p) basisset, which contains 3 s-orbitals and 1 p-orbitals for each hydrogen atom. The 1s-QUAMBO, which is a linear combination of the basis-set orbitals, is first constructed based on large-basis HF calculation and subsequently further optimized to minimize the CMR energy. Figure 2 shows that the QUAMBO-based CMR binding energy curve of H2 , which lies 9 mhartrees higher than the large-basis FCI results at equilibrium position, overlaps very well with the QUAMBO-FCI curve. 11 ACS Paragon Plus Environment


−0.48

E (Har./atom)

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−0.5 −0.52 −0.54 FCI (large basis) FCI (QUAMBO) CMR (QUAMBO)

−0.56 −0.58 0

0.5

1

1.5

2

2.5

3

3.5

R (Å)

Figure 2: Energy curve of H2 with large basis. The CMR and FCI calculations were performed on QUAMBO constructed from 6-311G(p) basis set. The corresponding large-basis FCI results are also included.

3.3

Molecules with Multiple Active Orbitals

We further performed calculations on several homonuclear and heteronuclear molecules in order to demonstrate the application of CMR method to multiple active-orbital molecular systems, using QUAMBOs constructed from aug-cc-pVTZ basis functions. 32 We considered three categories of molecules: homonuclear dimers including N2 and F2 , heteronuclear dimers including HF and CO, and polyatomic molecules including NH3 and CH4 . Shown in Figure 3 are the binding curves of the six molecules from QUAMBO-FCI and QUAMBO-CMR calculations. The available experimental curves are also plotted for comparison. Remarkably, QUAMBO-CMR produces energy curves in close agreement with the QUAMBO-FCI results, and both curves follow the experimental data reasonably well. It should be noted that the dissociation curve of N2 is a difficult case, since it involves dissociation to highly open-shell atoms with three unpaired electrons. It also has been proven to be difficult to obtain accurate potential energy curves for F2 fully by ab initio calculations. 33 The equilibrium bond lengths and binding energies are summarized in Table 1. One can see that QUAMBO-CMR and QUAMBO-FCI yield very close results, and have standard deviations of 0.024, 0.027 ˚ A for equilibrium bond length and 9, 11 mhartrees per atom for binding energy, respectively.


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0.1

HF

0

−0.1

E (Har./atom)

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−0.2 0.1

0

−0.1

−0.2

1

1.5

2

2.5 3

1

1.5 2 R (Å)

2.5 3

1

1.5

2

2.5

3

Figure 3: Potential energy curves of N2 , F2 , HF, CO, NH3 and CH4 . The CMR and FCI calculations are based on QUAMBOs constructed from aug-cc-pVTZ basis set. Included are available experimental data. 34–36

Table 1: Summary of equilibrium bond length Re and binding energy Eb from ab initio calculations on various molecules. QUAMBO-FCI and QUAMBO-CMR results are compared with experimental data, 37,38 reported with the standard deviation σ. The Re ’s for methane and ammonia are equilibrium bond lengths of C-H and N-H, respectively.

H2 N2 F2 CO HF CH4 NH3 σ

Re (˚ A) QUAMBO-FCI QUAMBO-CMR 0.744 0.744 1.100 1.118 1.467 1.467 1.133 1.121 0.909 0.942 1.115 1.112 1.012 1.033 0.024 0.027

Exp. 0.741 1.098 1.412 1.128 0.917 1.087 1.012

Eb (Har./atom) QUAMBO-FCI QUAMBO-CMR 0.073 0.076 0.164 0.160 0.022 0.030 0.192 0.186 0.105 0.100 0.126 0.132 0.106 0.110 0.009 0.011


Exp. 0.082 0.180 0.030 0.204 0.108 0.126 0.110


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4

Conclusions

In order to relieve the previously proposed CMR method from an inconvenient functional fitting procedure, we made two innovations into the existing CMR formalism. We introduced a fixed form of the renormalization factor that is the crucial quantity reflecting electronic correlation effects, by carefully studying the minimum basis H2 dimer under FCI and CMR. We also identified the main error source of CMR with this fixed form of z-factor, and concluded that it comes out of the intersite interactions. To minimize this error, we included into the CMR energy functional a sum-rule term to shift the intersite two-electron contribution onto the local onsite Hamiltonian that is accurately treated within the CMR formalism. The new CMR formalism has been applied to systems including Hn clusters, homonuclear and heteronuclear molecules with different basis sets for benchmark. We can see the agreements between CMR and FCI across different systems are of the order of 10 mhartree per atom, and the results are also reasonably close to the experimental data.

Acknowledgement The authors thank M. Schmidt and K. Ruedenberg for useful discussions. This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineering Division, including the computer time support from the National Energy Research Scientific Computing Center (NERSC) in Berkeley, CA. The research was performed at Ames Laboratory, which is operated for the U.S. DOE by Iowa State University under contract # DE-AC02-07CH11358. This work was also partially supported by the China Postdoctoral Science Foundation funded project (2015M570539), USTC Qian-Ren B (1000-Talents Program B) fund.


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Supporting Information Available This material is available free of charge via the Internet at http://pubs.acs.org/.

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