Cluster-in-Molecule Local Correlation Approach for Periodic Systems

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Quantum Electronic Structure

Cluster-in-Molecule Local Correlation Approach for Periodic Systems Yuqi Wang, Zhigang Ni, Wei Li, and Shuhua Li J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b01200 • Publication Date (Web): 28 Mar 2019 Downloaded from http://pubs.acs.org on March 31, 2019

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Journal of Chemical Theory and Computation

Cluster-in-Molecule Local Correlation Approach for Periodic Systems Yuqi Wang, Zhigang Ni, Wei Li*, Shuhua Li*

School of Chemistry and Chemical Engineering, Key Laboratory of Mesoscopic Chemistry of MOE, Institute of Theoretical and Computational Chemistry, Nanjing University, Nanjing 210023, P. R. China

In this article, the cluster-in-molecule (CIM) local correlation approach for periodic systems with periodic boundary condition has been developed, which allows electron correlation calculations of various crystals computationally tractable. In this approach, the correlation energy per unit cell of a periodic system can be evaluated as the summation of the correlation contributions from electron correlation calculations on a series of finite-sized clusters. Each cluster is defined to contain a subset of localized Wannier functions (WFs) (for the occupied space) and projected atomic orbitals (for the virtual space), which can be derived from a periodic Hartree-Fock calculation. Electron correlation calculations on clusters at second-order Møller-Plesset perturbation theory (MP2) or coupled cluster singles and doubles (CCSD) can be performed with well-established molecular quantum chemistry packages. We perform illustrative calculations at MP2 and CCSD levels on several types of crystals (Neon lattice, carbon monoxide and ammonia crystals, two ionic liquid crystals and diamond). The results show that CIM is a powerful framework for accurate electron 1

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correlation calculations of crystals.

1. Introduction Electron-correlation problem is one of the major concerns in electronic structure calculations of condensed phase systems. At present density functional theory (DFT) plays a dominant role in this field. Although DFT has achieved great success for general systems, it is not accurate enough for many important applications. Some of its weaknesses include: highly problematic description of van der Waals interactions,1, underestimation of transition state energies2 and band-gap.3 Wavefunction-based electron correlation methods based on the Hartree-Fock (HF) wavefunction have gained impressive success for molecules. Extensions of these methods to periodic systems provide an alternative approach to DFT for electronic structure calculations of solid-state materials. Canonical second order Møller-Plesset pertubation theory (MP2) and various coupled cluster (CC) methods for periodic systems have been established, in which crystalline orbitals (COs) are in the form of Bloch functions, expanded by Gaussian-type atomic orbitals (GTOs)4-12 or plane waves (PWs).13-21 The Laplace form of canonical MP2 was also reported.22-24 Hybrid basis techniques were developed including the ‘Gaussian and Plane Wave’ (GPW)25, 26

technique for MP2, the ‘pseudized’ Gaussian (pGTO)27 technique for CC. Full

configuration interaction quantum Monte Carlo(FCIQMC)28 has been reported, being the most accurate approach. Beyond the correlation energy, correlated band structures 2


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have also been calculated by CC theory.29-31 The main bottleneck of these methods is their steep scaling with the system size and the number of sampled Bloch vectors, which limits their applications to systems with small scale of unit cells. Based on the fast decaying behavior of electron correlation with the inter-orbital distance, local-correlation approaches have been established for large molecules. The original idea comes from Pulay’s work32-35 and many successful approaches along this direction have appeared.36-65 In these direct local-correlation methods, electron correlation models are reformulated in terms of localized occupied orbitals and non-orthogonal local virtual orbitals so that large number of components can be treated approximately or ignored in solving the corresponding equations for the total system.53 Different from these methods, various orbital-based local correlation fragmentation methods66-90 have also been developed, in which the correlation energy of a large system is decomposed as the sum of contributions from a series of small clusters (containing a subset of localized orbitals), each of which can be solved separately with conventional electron correlation methods. The most significant advantage of these fragmentation methods over direct local correlation methods is that much larger systems can be treated on ordinary workstations. Direct local-correlation methods for systems with periodic boundary condition (PBC) have also been reported, and was reviewed recently.91 The local-MP2 (LMP2)92-95 method was implemented in the Cryscor program.96, 97 The key idea is to use localized Wannier functions (WFs)98 for the occupied space, in a manner like localized molecular orbitals (LMOs) in molecules. Further development of this approach includes the incorporation of the 3



density-fitting technique,99-101 F12 explicit correlation,102 orbital specific virtuals (OSVs) based LMP2,103 and a direct-local-ring CCD.104,

105

Among fragment-based

local correlation methods, MP2 within the Divide-Expand-Consolidate (DEC)87-90 framework was implemented for periodic systems.106 In addition, some non-periodic cluster-based approaches have been developed for treating the correlation energy of solids, which include incremental methods,68-70 “unit-cell-in-cluster” approach,105 embedding techniques,107, 108 and atom-based fragmentation approaches.109-112 The cluster-in-molecule (CIM) approach proposed in 2002 by our group76 has been demonstrated to be a practical and powerful orbital-based fragmentation method for correlation energy calculations of large molecules.75, 77-85 In the CIM approach, the correlation energy of a large system is evaluated as the sum of contributions from all occupied LMOs, each of which can be obtained by solving electron correlation equations (at different theory levels) of a relatively small cluster. Here a cluster means a subset of occupied and virtual LMOs distributing over a certain region. Several electron correlation methods within the CIM framework have been developed, including MP2, CC doubles (CCD), CC singles and doubles (CCSD), CCSD with noniterative triples corrections [CCSD(T)], and so on. The CIM approach at a certain theory level X (MP2, CCSD, …) is usually denoted as CIM-MP2, CIM-CCSD, etc. As electron correlation calculations on different clusters are independent, a very high degree of parallelization can be achieved for the CIM approach. In recent years, significant progress of CIM includes the introduction of quasi-canonical molecular orbitals (QCMOs) in 2009,78 the multi-level scheme in 2010,79 the combination with 4


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highly efficient density-fitting and Cholesky decomposition techniques,113 the combination with natural orbitals (NOs)114-117 and an efficient strategy of constructing various clusters.118 In this work, we extend CIM to the electron correlation calculation of condensed systems with periodic boundary condition (PBC) at the MP2 and CCSD level. The corresponding method will be denoted as PBC-CIM-X (X=MP2, CCSD) in this article. In constructing various clusters of periodic systems, localized Wannier functions (WFs) expressed with GTOs119, 120 are used to represent the occupied space. Projected atomic orbitals (PAOs) are used to represent the virtual space. Due to their locality, WFs and PAOs can be truncated to a finite subset of atoms. Taking advantage of translation symmetry of periodic systems, only clusters corresponding to a unit cell are required to be built. Then, the correlation energy of a periodic system per unit cell can be obtained by summing the contributions from electron correlation calculations on a series of clusters. Although clusters constructed for periodic systems are quite similar to those constructed for molecules, the PBC-CIM method is aimed to calculate the correlation energy of a periodic system. Our preliminary calculations on several crystals demonstrate that PBC-CIM is a powerful approach for electron correlation calculations of periodic systems. The remaining part of this article is organized as follows. In Methodology, we will briefly review the basic principles of CIM and discuss how to implement the PBC-CIM method. In Results and Discussion, we show the results of PBC-CIM for some typical crystals to demonstrate its accuracy and capability. In Conclusions, we 5



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offer a brief summary and some remarks.

2. Methodology Before discussion, the meanings of index notations used in this article are summarized in Table 1.

Table 1. Summary of index notations u, v, λ, σ…

Basis functions, i.e. AOs

i, j, k, l…

1. WFs

2. Occupied non-localized orbitals

i’, j’, k’, l’…

Occupied localized orbitals

a, b, c, d…

Virtual non-localized orbitals

a’, b’, c’, d’…

Virtual localized orbitals

g, h, m, n…

Cells

2.1. Basic Principles of CIM In wave-function correlation methods, correlation energy can be considered as the sum of contributions from every occupied orbital: occ

E   Ei

(1)

i

In MP2 and CCSD, the correlation energy contribution from orbital i can be written as: Ei   VijabTijab

(2)

jab

where Vijab is an MO two-electron integral

ij ab , and Tijab is its associated

6


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amplitude. Equation (1) and (2) hold true both in the representation of canonical and localized orbitals. In the CIM framework, the contribution from an occupied orbital is calculated by solving electron correlation (MP2 or CC) equation for a cluster containing a subset of localized orbitals, which is built for this occupied orbital:

Ei{ P} 



j , a ,b{ P}

VijabTijab

(3)

In solving MP2 or CC equations for a cluster, quasi-canonical molecular orbitals (QCMOs) instead of localized orbitals will be used. LMOs are transformed to QCMOs by diagonalizing the Fock matrix in occupied and virtual space separately:

FR = Rε

(4)

i   Ri 'ii

(5)

i'

where F is the Fock matrix under the representation of LMOs, R is the unitary matrix which transforms LMOs ( i ) to QCMOs ( i ), ε is a diagonal matrix whose elements are QCMOs’ orbital energies. With QCMOs, eq. (3) can be reformulated as:

Ei’ { P} 



j , a ,b{ P}

Vi 'abj Ti 'abj

(6)

with Vi 'abj   RiiT'Vijab

(7)

Ti 'abj   RiiT'Tijab

(8)

i

i

With QCMO technique, MP2 can be solved with a non-iterative way, and the convergence of iteratively solving the CC equations is accelerated. In addition, the existing MP2 or CCSD code can be easily modified for electron correlation calculations of clusters. Density fitting technique can also be directly 7



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applied to speed up MP2 or CCSD calculations, with a similar scheme adopted for canonical MP2 or CCSD methods. 2.2. CIM with Periodic Boundary Condition It is straightforward to extend the CIM framework to periodic systems. As in molecules, every localized WF in a unit cell has significant interactions with only its spatially neighboring orbitals (both inside and outside this cell). Thus, the correlation energy per unit cell is just the sum of the correlation energy contributions from all localized WFs in a reference cell, as shown below: E

occ

 E

(9)

i

iref

where the summation of i runs in the reference cell, different from that in molecules. The way of building various clusters for periodic systems is quite similar to that for molecules. Here, an updated strategy for molecules developed by us

118,

will be slightly modified for periodic systems (with the incorporation of translational symmetry of periodic systems). First, a periodic HF calculation is done, and then a set of localized WFs are built using the efficient strategy proposed previously119. The WFs used in this article are expressed in the linear combination of atomic orbitals (LCAO) form:

Wi (h)   Cuih (g ) u (g ) u

(10)

g

where Wi (h) means the ith WF in the cell h,  u (g ) is the uth atomic orbital( or basis function) in the cell g, and Cuih (g ) is the corresponding coefficient. Since 8


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WFs are mutually translational invariant, only WFs in the reference cell need to be considered, whose expression is simplified as:

Wi   Cui (g ) u (g ) u

(11)

g

For a WF in the cell h, its coefficients can be easily obtained from its corresponding WF in the reference cell with this relationship: Cuih (g )  Cui (g  h)

(12)

Formally, a WF is expressed with infinite number of AOs in the total crystal. However, due to the locality of WFs, merely a subset of AOs in a finite array of unit cells around the reference cell needs to be considered. Once localized WFs are available, the procedure of constructing clusters can be roughly divided in 3 steps. The first one is the construction of MO-domain, denoted as I, for every WF. Though WFs are used instead of MOs, for the sake of consistency, we still use the name MO-domain. Each WF can be assigned to an atom with the largest Mulliken population, called as the central atom. The Mulliken population on atom A in the reference cell for the ith WF is calculated as follows:

Pi A   Cui (0)Cvi (h)Suv (h) uA v

(13)

h

where Suv (h) is the overlap integral between the uth basis function in the reference cell and vth basis function in the cell h. For each central atom, an MO-domain is constructed, which contains WFs centered on it and their spatially neighboring WFs. In other words, an MO-domain also corresponds to a subset of 9



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atoms, and the orbitals on the central atom are called the central orbitals. A distance threshold ξ is used to select the neighboring atoms (or WFs). If a larger MO-domain fully contains some other MO-domains, only a larger MO-domain will be reserved, with more than one central orbitals. In the second step, for each MO-domain I, a AO-domain, denoted as Ω(I), is constructed. Accurate description of all WFs in an MO-domain needs not only AOs in atoms selected in the first step, but also some additional AOs centered on other neighboring atoms (buffer atoms) outside the MO domain, which together constitute an AO-domain. This is determined by Boughton-Pulay (BP) projection121, which is a tool to approximately represent a localized orbital with a subset of AOs in a least-square manner. As suggested by previous works, the principle of constructing a suitable AO-domain is that the following functional (called the residue of BP projection) should be less than 0.01 for each WF: R[Wi ]  min  (Wi  Wi ) 2 d  0.01 Wi   Ci (g )   (g ) 

(  g  ( I ))

(14)

g

where Wi is a localized WF and W

is its truncated approximation in an

AO-domain Ω(I). The latest updated scheme to construct an AO-domain is to first truncate the projected atomic orbitals (PAOs) to a subset of atoms by BP projection. With an identical definition as that in molecular cases, PAOs for periodic systems (in the reference cell) are expressed as:

    X  (g )   (g ) 

g

X  (g )     g,0 

1  P (m  g)S (m) 2 v m 10


(15)

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where Puv is an element of density matrix, which is available from a periodic HF calculation. Due to the translational symmetry, only the truncation of PAOs in the reference cell needs to be calculated. The following procedure is used to obtain a truncated PAO for each PAO (in an iterative way): R[   ]  min  (      ) 2 d  

    X  (g )   (g ) 

(  g  (   ))

(16)

g

where   denotes a truncated PAO of the original PAO   , and X  (g ) is the expansion coefficient within a subset of atoms, denoted as the PAO domain

(   ) . The size of (   ) is gradually increased to include more and more atoms around its center, until the residue of BP projection on these atoms is less than a threshold, denoted as η. For PAOs outside the reference cell, their truncated PAOs are directly determined with translational symmetry. Once all PAO domains for all PAOs centered in an MO-domain have been obtained, we combine all the atoms of these PAO domains to constitute the AO-domain. In nearly all cases so far we have encountered, when η is set as 0.05, the condition that BP-projection produces a residue less than 0.01 for all occupied orbitals mentioned above is successfully satisfied for both molecules and periodic systems. Taking a 2D boron nitride (BN) sheet as an example, we have shown illustrative pictures of an MO-domain and its corresponding AO-domain in Figure 1. .

11



(a) structure

(b) Wannier functions

(c) MO-domain

(d) AO-domain

Figure 1. (a) Structure and unit cell of the boron nitride sheet; (b) WFs of the boron nitride sheet (only B-N σ-type ones are shown for clarity), the central orbital is colored in purple, others in blue; (c) The atoms (or its associated WFs) in orange color constitute an MO-domain; (d) The atoms in blue color constitute an AO-domain.

The third step is to perform MP or CC calculations on various clusters. For electronic correlation calculations on a given cluster, we need to provide a set of orthonormal occupied and virtual orbitals. This is done by truncating the localized WFs and PAOs into the corresponding AO-domain with a BP projection described 12


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above:

R[Wi (g )]  min  (Wi (g )  Wi (g )) 2 d Wi (g )   Ci (m)   (m) 

(ig  I ,  m  ( I ))

(17)

m

R[   (g )]  min  (   (g )    (g )) 2 d

  (g )   X  (m)   (m) 

( g  I ,  m  ( I ))

(18)

m

Then, a canonical orthogonalization is performed on the truncated PAOs to generate a set of orthonormal virtual orbitals (the redundancy of truncated PAOs is eliminated), as shown below: Wa   C a (m)   (m) 

(  m  ( I ))

(19)

m

where Wa is a virtual orbital in the AO-domain. With a suitable value of η, occupied orbitals from Eq. (17) and virtual orbitals from Eq. (19) are demonstrated to be nearly orthogonal. Now we discuss the construction of the Fock matrix in the MO basis, which is essential for obtaining QCMOs and their orbital energies for each cluster. Since the AO-basis Fock operator in a cluster is assumed to be exactly identical to the Fock operator in the total system, the occupied and virtual blocks of the Fock matrix in the MO basis are calculated as:

Wi (g ) Fˆ W j (h)   Cgi (m)F (n  m)Chj (n)  mn

( ig, jh  I ,  m, n  ( I ) ) Wa Fˆ Wb   Cga (m)F (n  m)Chb (n)  mn

(  m, n  ( I ) )

(20)

(21)

where the AO-basis Fock matrix, F (n  m) , is directly available from periodic 13



HF calculations. Next, by diagonalizing the MO-basis Fock matrix for the occupied and virtual subspaces, separately, one can obtain a set of orthonormal occupied and virtual QCMOs for subsequent canonical MP or CC calculations. To summarize the discussions above, the accuracy of a CIM calculation of a system with periodic boundary condition at a certain correlation level X (denoted as PBC-CIM-X) is dependent on two parameters ξ and η . The default values of these two parameters for general systems will be discussed. In general, the values of these two parameters should be provided for each CIM calculation. 2.3. Computational Details Periodic HF calculations and transformation of WFs into localized WFs are performed with CRYSTAL14.122 Details of Cartesian coordinates and related computational parameters for all systems under study are provided in the Supporting Information. Formally infinite lattice summations in this work are truncated under the criteria of convergence, with a threshold of 10-10. MP2 and CCSD calculations on all clusters constructed in this work are performed with the open-source quantum chemistry code PYSCF-1.4.7.123 with a few local modifications. Unless specifically mentioned, core orbitals are frozen, both in localization of WFs and in correlation energy calculations. For some periodic systems, we have also performed MP2 and CCSD calculations with the generalized energy-based fragmentation method for systems with periodic boundary conditions (denoted as PBC-GEBF).111, 112 Two parameters for PBC-GEBF calculations are set to be 4.0 Å (the cutoff radius) and 6 (the maximum number of 14


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monomers in a subsystem). The PBC-GEBF method has been established to provide quite accurate descriptions for a wide range of molecular crystals and ionic liquid crystals. PBC-GEBF calculations are performed with LSQC package developed by our group.124 If not specially mentioned, the cutoff radius in the construction of MO-domains (ξ) and the threshold in the BP projection of PAOs (η) are set to be 5.5Å and 0.05 respectively, which are the suggested parameters from our previous CIM calculations on molecules.82 The choice of these two parameters is a good compromise between accuracy and computational cost.

3. Results and Discussion 3.1. Neon Lattice We perform PBC-CIM-MP2 and PBC-CIM-CCSD on a model system, Neon lattice. The purpose of this calculation is to compare our results with the previously reported DEC results, and to demonstrate the size-consistency of our approach. In Table 2, we compare the results of PBC-CIM-MP2 on the 1-dimensional (1d), 2-dimensional (2d) and 3-dimensional (3d) Neon lattice with the results of extended DEC-MP2

(X-DEC-MP2)

reported

recently.106

Both

PBC-CIM-MP2

and

X-DEC-MP2 calculations are performed with the 6-31G basis set at compressed geometries (to increase the correlation energy). The results of PBC-CIM-CCSD at the same geometry and basis set are also presented.

15



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Table 2. Comparisons of Correlation Energies per Unit Cell from CIM and DEC Calculations on Neon Lattices. The Unit Cell of all these Lattices Contains One Neon Atom. Electron Correlation Calculations are all Performed Without Freezing the Core Orbitals. Lattice

X-DEC-MP2a

PBC-CIM-MP2

PBC-CIM-CCSD

1Db

-0.114363

-0.114363

-0.115187

2Dc

-0.114470

-0.114469

-0.115262

3Dd

--

-0.114558

-0.115333

aThe

values are from ref.106, and the value for 3D lattice was not available; lattice parameter is 4.7 Bohr; cThe lattice is quadratic, with the lattice parameter being 4.7 Bohr; dThe lattice is rectangle, with the lattice parameters being 4.7, 4.8 and 4.9 Bohr. bThe

From Table 2 we can clearly see that the results of X-DEC-MP2 and PBC-CIM-MP2 for 1D and 2D Neon lattices are almost identical. For this type of van der Waals bonded crystals, PBC-CIM-CCSD correlation energies don’t show a significant improvement over PBC-CIM-MP2 values, with the difference of less than 0.5 kcal/mol. In other words, electron correlation descriptions at the MP2 level are good enough for Neon lattices. With a larger def2-TZVP basis set, we also perform PBC-CIM-MP2 and PBC-CIM-CCSD calculations on a 1D Neon lattice (atoms being separated with 20 Å), and compare the results with those of a single Neon atom. The results are shown in Table 1S of the Supporting Information. As expected, at this geometry PBC-CIM-MP2 (or PBC-CIM-CCSD) correlation energy per Neon atom is identical to the corresponding MP2 (or CCSD) value of a single Neon atom, demonstrating the 16


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size-consistency of the PBC-CIM approach. As mentioned by the authors, the currently implemented X-DEC-MP2 is computationally very demanding.106 The key difference between DEC and CIM is that the correlation energy of a periodic system is obtained from calculations on atomic-fragments and pair-fragments in the DEC framework, and from clusters in the CIM framework. In the present PBC-CIM-MP2 method, the size of clusters is much smaller than that of pair-fragments in the X-DEC-MP2 method, and the virtual orbital space is also treated differently. The much less computational cost of the PBC-CIM-MP2 method should come from the smaller size of clusters. The following calculations will show that our PBC-CIM approach is relatively cost-effective, and is computationally feasible for some real crystals. 3.2. Carbon Monoxide and Ammonia Crystals We perform PBC-CIM-MP2 and PBC-CIM-CCSD calculations for two molecular crystals (CO and NH3), whose geometries are shown in Figure 2. The def2-SVP basis set is chosen. For NH3, PBC-CIM-CCSD result is obtained with ξ=4.0 and η= 0.05, and a correction term, which is the difference between PBC-CIM-MP2 energy (with ξ=5.5 and η=0.05) and the corresponding value with ξ=4.0 and η=0.05. For molecules, the correlation energy obtained with this hybrid scheme was demonstrated to be a good approximation to that from a CIM-CCSD calculation (with ξ=5.5 and η=0.05).118 At the same geometry, we also performed PBC-GEBF-X (X=MP2, CCSD) with the same basis set for comparison. Correlation energies per unit cell obtained with these two methods and the information of the largest cluster in 17



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PBC-CIM-X calculations are shown in Table 3 and Table 4, respectively. One can see from Table 3 that for both molecular crystals, the CIM-MP2 (or CCSD) correlation energies per unit cell are very close to the corresponding GEBF-MP2 (or CCSD) values. However, the CIM method is capable of treating various crystals, and the GEBF method is usually applicable to molecular crystals or ionic crystals.

(a)

(b)

Figure 2. (a) The crystal structure of CO with 4 molecules in the unit cell; (b) The crystal structure of NH3 with 4 molecules in the unit cell.

Table 3. PBC-CIM-MP2 and PBC-CIM-CCSD Correlation Energies (in a.u.) per Unit Cell of CO and NH3 Crystals with the def2-SVP Basis Set. The Corresponding PBC-GEBF Results are also Provided for Comparison. Crystals

CIM-MP2

GEBF-MP2

CIM-CCSD

GEBF-CCSD

CO

-1.117295

-1.116867

-1.151406

-1.151103

NH3

-0.781606

-0.782734

-0.838531

-0.839461

18


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Table 4. Information of the Largest Clusters in PBC-CIM Calculations of CO and NH3 Crystals. The Number of Atoms (Natom), Basis Functions (Nbas), Occupied (Nocc) and Virtual Orbitals (Nvir) are provided. Natom

Nbas

Nocc

Nvir

CO

65

910

68

382

NH3

172

1193

76

563

For NH3 crystal, we also calculate the lattice energy at PBC-CIM-X (X=MP2, CCSD) level with the same def2-SVP basis set. In addition, a similar cc-pVDZ basis set is also used for CIM-MP2 calculation. The basis set superposition error (BSSE) correction is included in calculating the lattice energy. The results are collected in Table 5, together with the experimental value. Obviously, the lattice energies estimated with both methods show a large deviation from the experimental value, mainly because only a medium-sized double-zeta basis set (def2-SVP or cc-pVDZ) is used in PBC-CIM calculations.

19



Page 20 of 37

Table 5. Lattice energies (in kJ/mol) of NH3 Crystal Obtained from CIM-MP2 and CIM-CCSD Calculations with a def2-SVP (or cc-pVDZ) Basis Set, and from GEBF-MP2 Calculations with cc-pVDZ (or aug-cc-pVDZ) Basis Set.

CIM-MP2/def2-SVP

CIM-CCSD/def2-SVP

CIM-MP2/cc-pVDZ

24.28

21.49

24.12

GEBF-MP2/cc-pVDZ

GEBF-MP2/aug-cc-pVD

Expa

Z 26.44

35.83

37.57

a. The experimental lattice energy from ref. 125. We emphasize that diffusion functions are essential for getting an accurate description of lattice energy for molecular crystals. Our PBC-GEBF-MP2 results in Table 5 show that the enlargement of the basis set from cc-pVDZ to aug-cc-pVDZ can greatly improve the calculated lattice energy, with the deviation of about 2 kJ/mol (relative to the experimental value). Thus, it is necessary to employ larger basis sets with diffuse functions in PBC-CIM calculations in the future work. At present, the bottleneck to be solved is the convergence difficulty of periodic HF calculations with such Gaussian-type basis sets. Alternatively, a dual-basis approach was suggested for avoiding this issue, in which HF calculations are performed with a medium-sized basis set, and a larger basis set (with diffusion functions) may be employed for electron correlation calculations.126,127 It should be mentioned that single-excitation 20


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amplitudes are non-zero in MP2 calculations within the dual-basis approach. 3.3. Ionic Liquid Crystal Here the PBC-CIM-MP2 method is applied to calculate the correlation energies per unit cell for two kinds of ionic liquid (IL) crystals, ethylammonium chloride ([C2H5NH3][Cl], P21/m phase)128 and methylammonium nitrate ([CH3NH3][NO3], Pnma phase),129 whose geometries are shown in Figure 3. The basis sets chosen for these two ILs are 6-31G(d,p) for [C2H5NH3][Cl], and def2-SVP for [CH3NH3][NO3], respectively. In addition, we want to discuss the convergence behavior of CIM correlation energies with the cutoff distance (ξ). The CIM-MP2 correlation energies of these two systems at different ξ values (with η=0.05) are listed in Table 6. In addition, CIM-MP2 correlation energies of two molecular crystals (CO and NH3) discussed above at different ξ values are also collected in Table 2S for comparison. By taking the correlation energy obtained with ξ=6.0 Å as the reference value, we have shown deviations of CIM-MP2 correlation energies per unit cell at different ξ values in Figure 4 for these four crystals.

(a)

(b)

Figure 3. (a) Unit cell of IL-1 ([C2H5NH3][Cl]), with 2 ionic pairs in it; (b) Unit cell 21



Page 22 of 37

of IL-2 ([CH3NH3][NO3]), with 4 ionic pairs in it. Table 6. CIM-MP2 Correlation Energies of IL-1([C2H5NH3][Cl]) and IL-2([CH3NH3][NO3]) Crystals at Different Cutoff Radius(ξ) Correlation Energies (a.u.) ξ/Å IL-1a

IL-1b

IL-2

4.0

-1.215589

-1.287060

-4.261855

4.5

-1.221220

-1.295574

-4.270682

5.0

-1.225225

-1.301632

-4.280689

5.5

-1.227869

---

-4.285690

6.0

-1.229168

---

-4.288413

a. With the 6-31G(d,p) basis set; b. With the 6-311G(d,p) basis set.

Figure 4. Relative deviations of CIM-MP2 correlation energies as a function of ξ values for two IL crystals, [C2H5NH3][Cl] and [CH3NH3][NO3], and two molecular 22


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crystals (CO and NH3). The CIM-MP2 correlation energies obtained with ξ=6 Å are taken as the reference values for these four crystals.

As shown in Table 6 and Figure 4, we can see that when ξ is no more than 4.5 Å, the accuracy of CIM-MP2 correlation energy in an ionic liquid crystal is significantly lower than that in a molecular crystal. To analyze the reason behind this fact, we have examined the spatially decaying behavior of the density matrix for the ionic liquid crystal, [C2H5NH3][Cl], and the CO crystal. The results are listed in Table 3S. One can see that the density matrix in the [C2H5NH3][Cl] crystal shows a slower decaying behavior, relative to that in the CO crystal. This result indicates that occupied orbitals in the [C2H5NH3][Cl] crystal tend to be somewhat more delocalized than those in the CO crystal.130 The decaying behavior of the density matrix is closely related to the calculated band gap of periodic systems.131 The calculated band gaps for these two crystals are shown in Table 4S. One can see that the [C2H5NH3][Cl] crystal has a smaller band gap than that in the CO crystal. Thus, the slower decaying behavior of the density matrix in the [C2H5NH3][Cl] crystal is consistent with its smaller band gap. The relatively slower convergence of the correlation energy with the ξ value for ionic liquid crystals may result from their slightly more delocalized orbitals. However, when ξ is set to be about 5.50 Å, CIM-MP2 seems to provide satisfactory descriptions for both types of crystals. Our recent benchmark CIM calculations on a wide range of molecules has demonstrated that the accuracy of the CIM method with default thresholds is less dependent on the basis set size and correlation levels.118 It is 23



Page 24 of 37

expected that the accuracy of the PBC-CIM method should also remain unchanged if larger basis set and higher correlation levels are used. 3.4. Diamond Here the CIM-MP2 approach is applied to calculate the correlation energy per unit cell of diamond (Fd 3 m), and the result is compared with that from a previous LMP2 study.92, 99 The same basis set, 6-21G*, which was used in the previous LMP2 calculation, will be used here. For this covalently bonded crystal, our periodic HF calculation revealed that the density matrix exhibits a relatively slow decaying behavior. As a result, PAOs’ locality is much less than that in other crystals discussed above, and the AO-domains of this crystal are significantly larger than those in other types of crystals. To make CIM-MP2 calculations feasible, we set two parameters to be ξ=4.0 Å and 4.5 Å, and η=0.1 for this system. The CIM-MP2 correlation energy and the CPU time, together with those reported in a previous LMP2 study, are shown in Table 7.

Table 7. Comparison Between the Correlation Energy and Computational Time of Diamond Calculated with CIM-MP2 and LMP2 Methods

Correlation Energy/a.u.

CPU time/s

CIM(4.0, 0.1)

-0.253159

1.64×105 b

CIM(4.5, 0.1)

-0.259643

5.96×105 b

LMP2(2.0)a

-0.228655c

1.39×106 c

LMP2(4.0)a

-0.252379d

---

DF-LMP2(5.0)a

-0.260153e

1800e

a. The value of the cutoff radius for strong pairs in LMP2 is placed in the parenthesis. b. Timing was obtained on a single Intel Xeon 2.60GHz processor. 24


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c. The result is taken from ref 100. Timing was obtained on a single Intel Xeon 3.06GHz processor. Threshold on the coefficients for the truncation of WF and PAO is 0.0001. d. The result is taken from ref 92. Threshold on the coefficients for the truncation of WF and PAO is 0.001. Multipolar approximation is used to treat pairs from 4 to 14Å. e. The result is taken from ref 101. Timing was obtained on a single AMD64 Opteron 2.60GHz processor. Multipole expansion and extrapolation technique are used to treat pairs beyond 5 Å. From Table 7, we can see that CIM-MP2 can provide similar correlation energies as LMP2 if similar thresholds are employed. When similar correlation energies are obtained, the computational time of CIM-MP2 is less than that of LMP2. This implies that CIM may be a more cost-effective framework than LMP2 in the treatment of diamond. However, as shown by previous LMP2 calculations,92 for diamond very tight parameters need to be employed to obtain an accurate correlation energy. The same situation holds true for the CIM-MP2 method. As shown in Table 7, the density fitting technique can dramatically speed up LMP2 calculations.101 In the future, the density fitting technique and other accelerating techniques must be employed for the PBC-CIM method to make it a practical tool for electron correlation calculations of covalent bonded crystals.

4. Conclusions In this work, the CIM approach for electron correlation calculations of periodic systems has been developed and implemented. Within the CIM framework, the correlation energy per unit cell of periodic systems is evaluated as the summation of the correlation energy contributions from electron correlation calculations on a series of clusters, which are constructed in terms of localized WFs and projected atomic 25



Page 26 of 37

orbitals. The way of building clusters is very similar to that for molecules. Electron correlation calculations at different theory levels (MP2, CCSD, …) on clusters can be performed with slightly modified molecular quantum chemistry packages. Illustrative applications on several kinds of crystals have demonstrated that the CIM approach can provide satisfactory electron correlation energies for periodic systems. In comparison with quite expensive canonical MP2 or CCSD calculations,4,

5

the

computational cost of the PBC-CIM approach at the same theory level is much more cost-effective, because it scales linearly with the number of atoms in a unit cell of periodic systems. Applications of the CIM approach to various periodic systems still suffer from some challenges. First, it is still difficult to get converged HF solutions for crystals when larger Gaussian basis sets with diffusion functions are used. The use of such basis sets is essential for accurate descriptions of many crystalline materials, especially molecular crystals. Without converged HF solutions, construction of various clusters is not available. Second, for a large number of real crystals with moderate basis sets, the size of clusters may be beyond the range of full canonical MP2 and CCSD calculations. More efficient local correlation methods for electron correlation calculations of clusters should be employed. With these advances, the CIM approach will become an attractive theoretical tool for accurate electron correlation calculations of crystals.

ASSOCIATED CONTENT Supporting Information 26


Page 27 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Related

coordinates

and

lattice

parameters

(CO,

NH3,

[C2H5NH3][Cl],

[CH3NH3][NO3] and Diamond), computational parameters in HF and localization, results of the test of size-consistency, and CIM-MP2 correlation energies per unit cell at different ξ values for CO and NH3 crystals, the spatially decaying of the density matrix of CO and [C2H5NH3][Cl], band gaps of Ne, CO, NH3, [C2H5NH3][Cl] and [CH3NH3][NO3] crystals, are placed in the Supporting Information. This information is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] *E-mail: [email protected] ORCID Wei Li: 0000-0001-7801-3643 Shuhua Li: 0000-0001-6756-057X Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos. 21833002, 21333004 and 21873046). Part of the calculations in this work were performed with computational resources on an IBM Blade cluster system from the 27



High Performance Computing Center (HPCC) of Nanjing University. We dedicate the work to professor Jean-Paul Malrieu on the occasion of his 80th birthday.

REFERENCES (1) Johnson, E. R.; Wolkow, R. A.; DiLabio, G. A. Application of 25 density functionals to dispersion-bound homomolecular dimers. Chem. Phys. Lett. 2004, 394, 334-338. (2) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. Development and assessment of a new hybrid density functional model for thermochemical kinetics. J. Phys. Chem. A 2004, 108, 2715-2719. (3) Jones, R. O.; Gunnarsson, O. The density functional formalism, its applications and prospects. Rev. Mod. Phys. 1989, 61, 689-746. (4) Sun, J. Q.; Bartlett, R. J. Second-order many-body perturbation-theory calculations in extended systems. J. Chem. Phys. 1996, 104, 8553-8565. (5) Hirata, S.; Podeszwa, R.; Tobita, M.; Bartlett, R. J. Coupled-cluster singles and doubles for extended systems. J. Chem. Phys. 2004, 120, 2581-2592. (6) Hirata, S.; Shimazaki, T. Fast second-order many-body perturbation method for extended systems. Phys. Rev. B 2009, 80, 7. (7) Keceli, M.; Hirata, S. Fast coupled-cluster singles and doubles for extended systems: Application to the anharmonic vibrational frequencies of polyethylene in the Gamma approximation. Phys. Rev. B 2010, 82, 6. (8) Katouda, M.; Nagase, S. Application of second-order Moller-Plesset perturbation theory with resolution-of-identity approximation to periodic systems. J. Chem. Phys. 2010, 133, 9. (9) Ohnishi, Y. Y.; Hirata, S. Hybrid coupled-cluster and perturbation method for extended systems of one-dimensional periodicity. J. Chem. Phys. 2011, 135, 10. (10) He, X.; Ryu, S.; Hirata, S. Finite-temperature second-order many-body perturbation and Hartree-Fock theories for one-dimensional solids: An application to Peierls and charge-density-wave transitions in conjugated polymers. J. Chem. Phys. 2014, 140, 7. (11) Shiozaki, T.; Hirata, S. Communications: Explicitly correlated second-order Moller-Plesset perturbation method for extended systems. J. Chem. Phys. 2010, 132, 4. (12) Ohnishi, Y. Y.; Hirata, S. Logarithm second-order many-body perturbation method for extended systems. J. Chem. Phys. 2010, 133, 8. (13) Shepherd, J. J.; Henderson, T. M.; Scuseria, G. E. Range-separated brueckner coupled cluster doubles theory. Phys. Rev. Lett. 2014, 112, 5. (14) Marsman, M.; Gruneis, A.; Paier, J.; Kresse, G. Second-order Moller-Plesset perturbation theory applied to extended systems. I. Within the projector-augmented-wave formalism using a plane wave basis set. J. Chem. Phys. 28


Page 28 of 37

Page 29 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2009, 130, 10. (15) Gruneis, A.; Marsman, M.; Kresse, G. Second-order Moller-Plesset perturbation theory applied to extended systems. II. Structural and energetic properties. J. Chem. Phys. 2010, 133, 11. (16) Gruneis, A.; Booth, G. H.; Marsman, M.; Spencer, J.; Alavi, A.; Kresse, G. Natural orbitals for wave function based correlated calculations using a plane wave basis set. J. Chem. Theory Comput. 2011, 7, 2780-2785. (17) Shepherd, J. J.; Gruneis, A.; Booth, G. H.; Kresse, G.; Alavi, A. Convergence of many-body wave-function expansions using a plane-wave basis: From homogeneous electron gas to solid state systems. Phys. Rev. B 2012, 86, 14. (18) Shepherd, J. J.; Gruneis, A. Many-body quantum chemistry for the electron gas: convergent perturbative theories. Phys. Rev. Lett. 2013, 110, 5. (19) Gruneis, A.; Shepherd, J. J.; Alavi, A.; Tew, D. P.; Booth, G. H. Explicitly correlated plane waves: Accelerating convergence in periodic wavefunction expansions. J. Chem. Phys. 2013, 139, 13. (20) Schafer, T.; Ramberger, B.; Kresse, G. Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis. J. Chem. Phys. 2017, 146, 11. (21) Neufeld, V. A.; Thom, A. J. W. A study of the dense uniform electron gas with high orders of coupled cluster. J. Chem. Phys. 2017, 147, 6. (22) Ayala, P. Y.; Kudin, K. N.; Scuseria, G. E. Atomic orbital Laplace-transformed second-order Moller-Plesset theory for periodic systems. J. Chem. Phys. 2001, 115, 9698-9707. (23) Pino, R.; Scuseria, G. E. Laplace-transformed diagonal Dyson correction to quasiparticle energies in periodic systems. J. Chem. Phys. 2004, 121, 2553-2557. (24) Schafer, T.; Ramberger, B.; Kresse, G. Laplace transformed MP2 for three dimensional periodic materials using stochastic orbitals in the plane wave basis and correlated sampling. J. Chem. Phys. 2018, 148, 9. (25) Del Ben, M.; Hutter, J.; VandeVondele, J. Second-Order Moller-Plesset perturbation theory in the condensed phase: an efficient and massively parallel gaussian and plane waves approach. J. Chem. Theory Comput. 2012, 8, 4177-4188. (26) Del Ben, M.; Hutter, J.; VandeVondele, J. Electron correlation in the condensed phase from a resolution of identity approach based on the gaussian and plane waves scheme. J. Chem. Theory Comput. 2013, 9, 2654-2671. (27) Booth, G. H.; Tsatsoulis, T.; Chan, G. K. L.; Gruneis, A. From plane waves to local Gaussians for the simulation of correlated periodic systems. J. Chem. Phys. 2016, 145, 10. (28) Booth, G. H.; Gruneis, A.; Kresse, G.; Alavi, A. Towards an exact description of electronic wavefunctions in real solids. Nature 2013, 493, 365-370. (29) Furukawa, Y.; Kosugi, T.; Nishi, H.; Matsushita, Y. Band structures in coupled-cluster singles-and-doubles Green's function (GFCCSD). J. Chem. Phys. 2018, 148, 8. (30) McClain, J.; Lischner, J.; Watson, T.; Matthews, D. A.; Ronca, E.; Louie, S. G.; Berkelbach, T. C.; Chan, G. K. L. Spectral functions of the uniform electron gas via coupled-cluster theory and comparison to the GW and related approximations. Phys. 29



Rev. B 2016, 93, 6. (31) McClain, J.; Sun, Q. M.; Chan, G. K. L.; Berkelbach, T. C. Gaussian-based coupled-cluster theory for the ground-state and band structure of solids. J. Chem. Theory Comput. 2017, 13, 1209-1218. (32) Pulay, P. Localizability of dynamic electron correlation. Chem. Phys. Lett. 1983, 100, 151-154. (33) Saebo, S.; Pulay, P. Local configuration-interaction - an efficient approach for larger molecules. Chem. Phys. Lett. 1985, 113, 13-18. (34) Saebo, S.; Pulay, P. 4th-order moller-plessett perturbation-theory in the local correlation treatment .1. METHOD. J. Chem. Phys. 1987, 86, 914-922. (35) Saebo, S.; Pulay, P. The local correlation treatment .2. implementation and tests. J. Chem. Phys. 1988, 88, 1884-1890. (36) Schutz, M.; Hetzer, G.; Werner, H. J. Low-order scaling local electron correlation methods. I. Linear scaling local MP2. J. Chem. Phys. 1999, 111, 5691-5705. (37) Hetzer, G.; Schutz, M.; Stoll, H.; Werner, H. J. Low-order scaling local correlation methods II: Splitting the Coulomb operator in linear scaling local second-order Moller-Plesset perturbation theory. J. Chem. Phys. 2000, 113, 9443-9455. (38) Schutz, M. Low-order scaling local electron correlation methods. III. Linear scaling local perturbative triples correction (T). J. Chem. Phys. 2000, 113, 9986-10001. (39) Schutz, M.; Werner, H. J. Low-order scaling local electron correlation methods. IV. Linear scaling local coupled-cluster (LCCSD). J. Chem. Phys. 2001, 114, 661-681. (40) Schutz, M. Low-order scaling local electron correlation methods. V. Connected triples beyond (T): Linear scaling local CCSDT-1b. J. Chem. Phys. 2002, 116, 8772-8785. (41) Kats, D.; Korona, T.; Schutz, M. Local CC2 electronic excitation energies for large molecules with density fitting. J. Chem. Phys. 2006, 125, 15. (42) Hampel, C.; Werner, H. J. Local treatment of electron correlation in coupled cluster theory. J. Chem. Phys. 1996, 104, 6286-6297. (43) Werner, H. J.; Schutz, M. An efficient local coupled cluster method for accurate thermochemistry of large systems. J. Chem. Phys. 2011, 135, 15. (44) Ayala, P. Y.; Scuseria, G. E. Linear scaling second-order Moller-Plesset theory in the atomic orbital basis for large molecular systems. J. Chem. Phys. 1999, 110, 3660-3671. (45) Scuseria, G. E.; Ayala, P. Y. Linear scaling coupled cluster and perturbation theories in the atomic orbital basis. J. Chem. Phys. 1999, 111, 8330-8343. (46) Maslen, P. E.; Head-Gordon, M. Non-iterative local second order Moller-Plesset theory. Chem. Phys. Lett. 1998, 283, 102-108. (47) Head-Gordon, M.; Maslen, P. E.; White, C. A. A tensor formulation of many-electron theory in a nonorthogonal single-particle basis. J. Chem. Phys. 1998, 108, 616-625. 30


Page 30 of 37

Page 31 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(48) Maslen, P. E.; Head-Gordon, M. Noniterative local second order Moller-Plesset theory: Convergence with local correlation space. J. Chem. Phys. 1998, 109, 7093-7099. (49) Neese, F.; Wennmohs, F.; Hansen, A. Efficient and accurate local approximations to coupled-electron pair approaches: An attempt to revive the pair natural orbital method. J. Chem. Phys. 2009, 130, 18. (50) Neese, F.; Hansen, A.; Liakos, D. G. Efficient and accurate approximations to the local coupled cluster singles doubles method using a truncated pair natural orbital basis. J. Chem. Phys. 2009, 131, 15. (51) Riplinger, C.; Neese, F. An efficient and near linear scaling pair natural orbital based local coupled cluster method. J. Chem. Phys. 2013, 138, 18. (52) Liakos, D. G.; Sparta, M.; Kesharwani, M. K.; Martin, J. M. L.; Neese, F. Exploring the accuracy limits of local pair natural orbital coupled-cluster theory. J. Chem. Theory Comput. 2015, 11, 1525-1539. (53) Liakos, D. G.; Neese, F. Is it possible to obtain coupled cluster quality energies at near density functional theory cost? domain-based local pair natural orbital coupled cluster vs modern density functional theory. J. Chem. Theory Comput. 2015, 11, 4054-4063. (54) Guo, Y.; Riplinger, C.; Becker, U.; Liakos, D. G.; Minenkov, Y.; Cavallo, L.; Neese, F. Communication: An improved linear scaling perturbative triples correction for the domain based local pair-natural orbital based singles and doubles coupled cluster method DLPNO-CCSD(T). J. Chem. Phys. 2018, 148, 5. (55) Pinski, P.; Riplinger, C.; Valeev, E. F.; Neese, F. Sparse maps-A systematic infrastructure for reduced-scaling electronic structure methods. I. An efficient and simple linear scaling local MP2 method that uses an intermediate basis of pair natural orbitals. J. Chem. Phys. 2015, 143, 17. (56) Riplinger, C.; Pinski, P.; Becker, U.; Valeev, E. F.; Neese, F. Sparse maps-A systematic infrastructure for reduced-scaling electronic structure methods. II. Linear scaling domain based pair natural orbital coupled cluster theory. J. Chem. Phys. 2016, 144, 10. (57) Guo, Y.; Becker, U.; Neese, F. Comparison and combination of "direct" and fragment based local correlation methods: Cluster in molecules and domain based local pair natural orbital perturbation and coupled cluster theories. J. Chem. Phys. 2018, 148, 11. (58) Yang, J.; Kurashige, Y.; Manby, F. R.; Chan, G. K. L. Tensor factorizations of local second-order Moller-Plesset theory. J. Chem. Phys. 2011, 134, 13. (59) Kurashige, Y.; Yang, J.; Chan, G. K. L.; Manby, F. R. Optimization of orbital-specific virtuals in local Moller-Plesset perturbation theory. J. Chem. Phys. 2012, 136, 7. (60) Schutz, M.; Yang, J.; Chan, G. K. L.; Manby, F. R.; Werner, H. J. The orbital-specific virtual local triples correction: OSV-L(T). J. Chem. Phys. 2013, 138, 10. (61) Werner, H. J.; Knizia, G.; Krause, C.; Schwilk, M.; Dornbach, M. Scalable electron correlation methods i.: pno-lmp2 with linear scaling in the molecular size and 31



near-inverse-linear scaling in the number of processors. J. Chem. Theory Comput. 2015, 11, 484-507. (62) Ma, Q. L.; Werner, H. J. Scalable electron correlation methods. 2. parallel pno-lmp2-f12 with near linear scaling in the molecular size. J. Chem. Theory Comput. 2015, 11, 5291-5304. (63) Schwilk, M.; Usvyat, D.; Werner, H. J. Communication: Improved pair approximations in local coupled-cluster methods. J. Chem. Phys. 2015, 142, 5. (64) Schwilk, M.; Ma, Q. L.; Koppl, C.; Werner, H. J. Scalable electron correlation methods. 3. efficient and accurate parallel local coupled cluster with pair natural orbitals (PNO-LCCSD). J. Chem. Theory Comput. 2017, 13, 3650-3675. (65) Ma, Q. L.; Schwilk, M.; Koppl, C.; Werner, H. J. Scalable electron correlation methods. 4. parallel explicitly correlated local coupled cluster with pair natural orbitals (PNO-LCCSD-F12). J. Chem. Theory Comput. 2017, 13, 4871-4896. (66) Forner, W.; Ladik, J.; Otto, P.; Cizek, J. Coupled-cluster studies .2. the role of localization in correlation calculations on extended systems. Chem. Phys. 1985, 97, 251-262. (67) Forner, W. Coupled cluster studies .4. analysis of the correlated wave-function in canonical and localized orbital basis for ethylene, carbon-monoxide, and carbon-dioxide. Chem. Phys. 1987, 114, 21-35. (68) Stoll, H. Correlation-energy of diamond. Phys. Rev. B 1992, 46, 6700-6704. (69) Stoll, H. The correlation-energy of crystalline silicon. Chem. Phys. Lett. 1992, 191, 548-552. (70) Paulus, B. The method of increments - a wavefunction-based ab initio correlation method for solids. Phys. Rep.-Rev. Sec. Phys. Lett. 2006, 428, 1-52. (71) Rosciszewski, K.; Doll, K.; Paulus, B.; Fulde, P.; Stoll, H. Ground-state properties of rutile: Electron-correlation effects. Phys. Rev. B 1998, 57, 14667-14672. (72) Stoll, H.; Paulus, B.; Fulde, P. On the accuracy of correlation-energy expansions in terms of local increments. J. Chem. Phys. 2005, 123, 9. (73) Friedrich, J.; Hanrath, M.; Dolg, M. Fully automated implementation of the incremental scheme: Application to CCSD energies for hydrocarbons and transition metal compounds. J. Chem. Phys. 2007, 126, 7. (74) Friedrich, J.; Dolg, M. Fully Automated incremental evaluation of mp2 and ccsd(t) energies: application to water clusters. J. Chem. Theory Comput. 2009, 5, 287-294. (75) Li, W.; Li, S. H. Divide-and-conquer local correlation approach to the correlation energy of large molecules. J. Chem. Phys. 2004, 121, 6649-6657. (76) Li, S. H.; Ma, J.; Jiang, Y. S. Linear scaling local correlation approach for solving the coupled cluster equations of large systems. J. Comput. Chem. 2002, 23, 237-244. (77) Li, S. H.; Shen, J.; Li, W.; Jiang, Y. S. An efficient implementation of the "cluster-in-molecule" approach for local electron correlation calculations. J. Chem. Phys. 2006, 125, 10. (78) Li, W.; Piecuch, P.; Gour, J. R.; Li, S. H. Local correlation calculations using standard and renormalized coupled-cluster approaches. J. Chem. Phys. 2009, 131, 30. 32


Page 32 of 37

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(79) Li, W.; Piecuch, P. Multilevel extension of the cluster-in-molecule local correlation methodology: merging coupled-cluster and moller-plesset perturbation theories. J. Phys. Chem. A 2010, 114, 6721-6727. (80) Li, W.; Piecuch, P. Improved design of orbital domains within the cluster-in-molecule local correlation framework: single-environment cluster-in-molecule ansatz and its application to local coupled-cluster approach with singles and doubles. J. Phys. Chem. A 2010, 114, 8644-8657. (81) Li, W.; Guo, Y.; Li, S. H. A refined cluster-in-molecule local correlation approach for predicting the relative energies of large systems. Phys. Chem. Chem. Phys. 2012, 14, 7854-7862. (82) Guo, Y.; Li, W.; Li, S. H. Improved cluster-in-molecule local correlation approach for electron correlation calculation of large systems. J. Phys. Chem. A 2014, 118, 8996-9004. (83) Li, W.; Li, S. H. Cluster-in-molecule local correlation method for large systems. Sci. China-Chem. 2014, 57, 78-86. (84) Guo, Y.; Li, W.; Yuan, D. D.; Li, S. H. The relative energies of polypeptide conformers predicted by linear scaling second-order Moller-Plesset perturbation theory. Sci. China-Chem. 2014, 57, 1393-1398. (85) Li, W.; Ni, Z. G.; Li, S. H. Cluster-in-molecule local correlation method for post-Hartree-Fock calculations of large systems. Mol. Phys. 2016, 114, 1447-1460. (86) Flocke, N.; Bartlett, R. J. A natural linear scaling coupled-cluster method. J. Chem. Phys. 2004, 121, 10935-10944. (87) Ziolkowski, M.; Jansik, B.; Kjaergaard, T.; Jorgensen, P. Linear scaling coupled cluster method with correlation energy based error control. J. Chem. Phys. 2010, 133, 5. (88) Kristensen, K.; Hoyvik, I. M.; Jansik, B.; Jorgensen, P.; Kjrgaard, T.; Reine, S.; Jakowski, J. MP2 energy and density for large molecular systems with internal error control using the Divide-Expand-Consolidate scheme. Phys. Chem. Chem. Phys. 2012, 14, 15706-15714. (89) Kjaergaard, T. The Laplace transformed divide-expand-consolidate resolution of the identity second-order Moller-Plesset perturbation (DEC-LT-RIMP2) theory method. J. Chem. Phys. 2017, 146, 13. (90) Kjaergaard, T.; Baudin, P.; Bykov, D.; Kristensen, K.; Jorgensen, P. The divide-expand-consolidate coupled cluster scheme. Wiley Interdiscip. Rev.-Comput. Mol. Sci. 2017, 7, 16. (91) Usvyat, D.; Maschio, L.; Schutz, M. Periodic and fragment models based on the local correlation approach. Wiley Interdiscip. Rev.-Comput. Mol. Sci. 2018, 8, 27. (92) Pisani, C.; Busso, M.; Capecchi, G.; Casassa, S.; Dovesi, R.; Maschio, L.; Zicovich-Wilson, C.; Schutz, M. Local-MP2 electron correlation method for nonconducting crystals. J. Chem. Phys. 2005, 122, 12. (93) Pisani, C.; Maschio, L.; Casassa, S.; Halo, M.; Schutz, M.; Usvyat, D. Periodic local MP2 method for the study of electronic correlation in crystals: Theory and preliminary applications. J. Comput. Chem. 2008, 29, 2113-2124. (94) Usvyat, D.; Civalleri, B.; Maschio, L.; Dovesi, R.; Pisani, C.; Schutz, M. 33



Approaching the theoretical limit in periodic local MP2 calculations with atomic-orbital basis sets: The case of LiH. J. Chem. Phys. 2011, 134, 6. (95) Martinez-Casado, R.; Usvyat, D.; Maschio, L.; Mallia, G.; Casassa, S.; Ellis, J.; Schutz, M.; Harrison, N. M. Approaching an exact treatment of electronic correlations at solid surfaces: The binding energy of the lowest bound state of helium adsorbed on MgO(100). Phys. Rev. B 2014, 89, 7. (96) Usvyat, D.; Maschio, L.; Pisani, C.; Schutz, M. Second order local moller-plesset perturbation theory for periodic systems: the CRYSCOR code. Z. Phys. Chemie-Int. J. Res. Phys. Chem. Chem. Phys. 2010, 224, 441-454. (97) Pisani, C.; Schutz, M.; Casassa, S.; Usvyat, D.; Maschio, L.; Lorenz, M.; Erba, A. CRYSCOR: a program for the post-Hartree-Fock treatment of periodic systems. Phys. Chem. Chem. Phys. 2012, 14, 7615-7628. (98) Marzari, N.; Mostofi, A. A.; Yates, J. R.; Souza, I.; Vanderbilt, D. Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 2012, 84, 57. (99) Maschio, L.; Usvyat, D.; Manby, F. R.; Casassa, S.; Pisani, C.; Schutz, M. Fast local-MP2 method with density-fitting for crystals. I. Theory and algorithms. Phys. Rev. B 2007, 76, 9. (100) Usvyat, D.; Maschio, L.; Manby, F. R.; Casassa, S.; Schutz, M.; Pisani, C. Fast local-MP2 method with density-fitting for crystals. II. Test calculations and application to the carbon dioxide crystal. Phys. Rev. B 2007, 76, 11. (101) Maschio, L.; Usvyat, D. Fitting of local densities in periodic systems. Phys. Rev. B 2008, 78, 4. (102) Usvyat, D. Linear-scaling explicitly correlated treatment of solids: Periodic local MP2-F12 method. J. Chem. Phys. 2013, 139, 15. (103) Usvyat, D.; Maschio, L.; Schutz, M. Periodic local MP2 method employing orbital specific virtuals. J. Chem. Phys. 2015, 143, 12. (104) Masur, O.; Schutz, M.; Maschio, L.; Usvyat, D. Fragment-based direct-local-ring-coupled-cluster doubles treatment embedded in the periodic Hartree-Fock solution. J. Chem. Theory Comput. 2016, 12, 5145-5156. (105) Schutz, M.; Maschio, L.; Karttunen, A. J.; Usvyat, D. Exfoliation energy of black phosphorus revisited: a coupled cluster benchmark. J. Phys. Chem. Lett. 2017, 8, 1290-1294. (106) Rebolini, E.; Baardsen, G.; Hansen, A. S.; Leikanger, K. R.; Pedersen, T. B. Divide-expand-consolidate second-order Moller-Plesset theory with periodic boundary conditions. J. Chem. Theory Comput. 2018, 14, 2427-2438. (107) Libisch, F.; Huang, C.; Carter, E. A. Embedded correlated wavefunction schemes: theory and applications. Accounts Chem. Res. 2014, 47, 2768-2775. (108) Sun, Q. N.; Chan, G. K. L. Quantum embedding theories. Accounts Chem. Res. 2016, 49, 2705-2712. (109) Hirata, S. Fast electron-correlation methods for molecular crystals: An application to the alpha, beta(1), and beta(2) modifications of solid formic acid. J. Chem. Phys. 2008, 129, 11. (110) Hirata, S.; Gilliard, K.; He, X.; Li, J. J.; Sode, O. Ab Initio molecular crystal structures, spectra, and phase diagrams. Accounts Chem. Res. 2014, 47, 2721-2730. 34


Page 34 of 37

Page 35 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(111) Fang, T.; Li, W.; Gu, F. W.; Li, S. H. Accurate prediction of lattice energies and structures of molecular crystals with molecular quantum chemistry methods. J. Chem. Theory Comput. 2015, 11, 91-98. (112) Fang, T.; Li, Y. Z.; Li, S. H. Generalized energy-based fragmentation approach for modeling condensed phase systems. Wiley Interdiscip. Rev.-Comput. Mol. Sci. 2017, 7, 13. (113) Nagy, P. R.; Samu, G.; Kallay, M. An integral-direct linear-scaling second-order Moller-Plesset approach. J. Chem. Theory Comput. 2016, 12, 4897-4914. (114) Rolik, Z.; Kallay, M. A general-order local coupled-cluster method based on the cluster-in-molecule approach. J. Chem. Phys. 2011, 135, 18. (115) Rolik, Z.; Szegedy, L.; Ladjanszki, I.; Ladoczki, B.; Kallay, M. An efficient linear-scaling CCSD(T) method based on local natural orbitals. J. Chem. Phys. 2013, 139, 17. (116) Nagy, P. R.; Kallay, M. Optimization of the linear-scaling local natural orbital CCSD(T) method: Redundancy-free triples correction using Laplace transform. J. Chem. Phys. 2017, 146, 15. (117) Nagy, P. R.; Samu, G.; Kallay, M. Optimization of the Linear-Scaling Local Natural Orbital CCSD(T) Method: Improved Algorithm and Benchmark Applications. J. Chem. Theory Comput. 2018, 14, 4193-4215. (118) Ni, Z.; W, Li.; S. Li.; Fully Optimized implementation of the cluster-in-molecule local correlation approach for electron correlation calculations of large systems. J. Comput. Chem. 2019, 40, 1130-1140. (119) Zicovich-Wilson, C. M.; Dovesi, R.; Saunders, V. R. A general method to obtain well localized Wannier functions for composite energy bands in linear combination of atomic orbital periodic calculations. J. Chem. Phys. 2001, 115, 9708-9719. (120) Jonsson, E. O.; Lehtola, S.; Puska, M.; Jonsson, H. Theory and Applications of Generalized Pipek-Mezey Wannier Functions. J. Chem. Theory Comput. 2017, 13, 460-474. (121) Boughton, J. W.; Pulay, P. Comparison of the boys and pipek-mezey localizations in the local correlation approach and automatic virtual basis selection. J. Comput. Chem. 1993, 14, 736-740. (122) Dovesi, R.; Orlando, R.; Erba, A.; Zicovich-Wilson, C. M.; Civalleri, B.; Casassa, S.; Maschio, L.; Ferrabone, M.; De La Pierre, M.; D'Arco, P.; Noel, Y.; Causa, M.; Rerat, M.; Kirtman, B. CRYSTAL14: A Program for the Ab Initio Investigation of Crystalline Solids. Int. J. Quantum Chem. 2014, 114, 1287-1317. (123) Sun, Q. M.; Berkelbach, T. C.; Blunt, N. S.; Booth, G. H.; Guo, S.; Li, Z. D.; Liu, J. Z.; McClain, J. D.; Sayfutyarova, E. R.; Sharma, S.; Wouters, S.; Chan, G. K. L. PYSCF: the Python-based simulations of chemistry framework. Wiley Interdiscip. Rev.-Comput. Mol. Sci. 2018, 8, 15. (124) Li, W.; Chen, C. H.; Zhao, D. B.; Li, S. H. LSQC: Low scaling quantum chemistry program. Int. J. Quantum Chem. 2015, 115, 641-646. (125) Otero-de-la-Roza, A.; Johnson, E. R. A benchmark for non-covalent 35



interactions in solids. J. Chem. Phys. 2012, 137, 10. (126) Wolinski, K.; Pulay, P. Second-order Moller-Plesset calculations with dual basis sets. J. Chem. Phys. 2003, 118, 9497-9503. (127) Maschio, L.; Usvyat, D.; Schutz, M.; Civalleri, B. Periodic local Moller-Plesset second order perturbation theory method applied to molecular crystals: Study of solid NH3 and CO2 using extended basis sets. J. Chem. Phys. 2010, 132, 9. (128) Migliorati, V.; Ballirano, P.; Gontrani, L.; Triolo, A.; Caminiti, R. Thermal and structural properties of ethylammonium chloride and its mixture with water. J. Phys. Chem. B 2011, 115, 4887-4899. (129) Bodo, E.; Postorino, P.; Mangialardo, S.; Piacente, G.; Ramondo, F.; Bosi, F.; Ballirano, P.; Caminiti, R. Structure of the molten salt methyl ammonium nitrate explored by experiments and theory. J. Phys. Chem. B 2011, 115, 13149-13161. (130) Maslen, P. E.; Ochsenfeld, C.; White, C. A.; Lee, M. S.; Head-Gordon, M. Locality and sparsity of ab initio one-particle density matrices and localized orbitals. J. Phys. Chem. A 1998, 102, 2215-2222. (131) Blount, E. I. Formalisms Of Band Theory. In Solid State Physics-Advances in Research and Applications. Seitz, F., Turnbull, D. Eds.; Academic Press: New York, 1962; Vol. 13, pp305-373

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For Table of Contents use only

Cluster-in-Molecule Local Correlation Approach for Periodic Systems

Yuqi Wang, Zhigang Ni, Wei Li*, Shuhua Li*

School of Chemistry and Chemical Engineering, Key Laboratory of Mesoscopic Chemistry of MOE, Institute of Theoretical and Computational Chemistry, Nanjing University, Nanjing 210023, P. R. China

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Cluster-in-Molecule Local Correlation Approach for Periodic Systems

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