Long-range corrected density functional through the

Published on 06 March 2018. Downloaded by National Institute of Science Education & Research on 06/05/2018 05:42:00.

PCCP View Article Online

PAPER

Cite this: Phys. Chem. Chem. Phys., 2018, 20, 8991

View Journal | View Issue

Long-range corrected density functional through the density matrix expansion based semilocal exchange hole† Bikash Patra,

* Subrata Jana and Prasanjit Samal

The exchange hole, which is one of the principal constituents of the density functional formalism, can be used to design accurate range-separated hybrid functionals in association with appropriate correlation. In this regard, the exchange hole derived from the density matrix expansion has gained attention due to its fulfillment of some of the desired exact constraints. Thus, the new long-range corrected density functional proposed here combines the meta generalized gradient approximation level exchange functional designed from the density matrix expansion based exchange hole coupled with the ab initio Received 31st January 2018, Accepted 6th March 2018

Hartree–Fock exchange through the range separation of the Coulomb interaction operator using the

DOI: 10.1039/c8cp00717a

assessment and benchmarking of the above newly constructed range-separated functional with various

standard error function technique. Then, in association with the Lee–Yang–Parr correlation functional, the well-known test sets shows its reasonable performance for a broad range of molecular properties, such as

rsc.li/pccp

thermochemistry, non-covalent interaction and barrier heights of the chemical reactions.

I. Introduction The density functional formalism1 has made a remarkable impact on the application of quantum mechanics to address complex many-body effects due to its reliable accuracy and computational affordability. Although the formulation is exact in principle, in practical calculation we need to approximate a small fraction of the total energy functional, known as the exchange–correlation (XC) functional. The success of density functional theory (DFT) depends on the accuracy of the designed XC functional. Several approximations of the XC functional have been reported in the past several decades since the advent of the Hohenberg–Kohn–Sham DFT with continuously increasing accuracy.2–20 These proposed density functionals are distinguished through Jacob’s ladder21 of density functional approximations according to the accuracy and ingredients involved. On the lowest rung of Jacob’s ladder, there is the local density approximation (LDA),22,23 which uses only electron density as its main ingredient. One rung higher than the LDA are the generalized gradient approximations (GGA),8,24,25 which are designed using a reduced density gradient as an additional ingredient. Next are the meta-GGA14,15,18,20 approximations,

School of Physical Sciences, National Institute of Science Education and Research, Homi Bhava National Institute, Bhubaneswar 752050, India. E-mail: [email protected] † Electronic supplementary information (ESI) available: Result of the individual species for each case. See DOI: 10.1039/c8cp00717a

This journal is © the Owner Societies 2018

which are proposed using the Kohn–Sham kinetic energy density (KS-KE) along with the reduced density gradient. All these semilocal functionals are very successful in describing atomization energies,26–30 equilibrium lattice constants,26,31,32 equilibrium bond lengths,26,33 surface properties,31 cohesive energies31,32 etc. Although these semilocal functionals have enjoyed noticeable success, they fail badly to explain the Rydberg excitation,34 charge transfer excitations,35 reaction barrier heights36 and oscillator strengths.37–39 The missing non-locality40 and absence of the many-electron self-interaction errors (MESI)41 are the two major drawbacks of the semilocal formalism. Another serious difficulty is that the XC potential of the corresponding semilocal functional shows incorrect asymptotic behavior42 and decays faster than 1/r. The non-locality40 of the exchange functional, which is missing in semilocal functionals, is essential to describe the long-range charge transfer, barrier heights and dissociation limit of the molecules. Although the most popular hybrid B3LYP43 and similar ones7,27 resolve these problems to some extent, they are still far from accuracy in several cases. Dramatic failures of the hybrid functionals are observed especially in describing the dissociation limits, barrier heights and phenomena relating to the fractional occupation. Apart from that, in hybrid functionals, the XC potential decays like c/r, where c is the fraction of mixed Hartree–Fock exchange. One of the interesting findings is the mixing of Hartree–Fock (HF) exchange with that of the semilocal functional by using the range separation of the Coulomb interaction operator (either as long or short-range). The range

Phys. Chem. Chem. Phys., 2018, 20, 8991--8998 | 8991

View Article Online

Published on 06 March 2018. Downloaded by National Institute of Science Education & Research on 06/05/2018 05:42:00.

Paper

PCCP

separation technique12,44–47 also enables us to change the range of the semilocal and HF part. Using this method, one can mix the semilocal form of DFT in long-range or short-range depending on the requirements. It has been shown that for molecules the long-range corrected HF is more effective44–47 and for solid-state calculations the short-range HF exchange12 is superior from the computational point of view. In fact, these ideas were used to design the long-range corrected LC-oPBE45 and LC-BLYP44 which reproduce intriguing results and establish superiority over the B3LYP functional in many cases. In some functionals like CAM-B3LYP44 and LC-oPBEh,46 both the HF exchange and DFT counterpart are incorporated over the whole range by using the generalized parametrization of the Coulomb interaction operator. So far, the range-separated functionals are designed using the semilocal exchange hole constructed from the spherically averaged exchange hole2,5,20,48 or using the reverse engineering technique.49–51 The CAM-B3LYP functional is designed using the LDA exchange hole by passing the inhomogeneity through the modification of the Thomas-Fermi wave-vector.52 Later, the LC-oPBE, LC-oPBEh and HSE06 functionals were designed using the PBE exchange hole. The present work aims at designing the first ever long-range corrected meta-GGA level exchange energy functional using the recently developed density matrix expansion (DME) based exchange hole. The most successful meta GGA exchange hole proposed by Tao-Mo (TM)20 using the DME satisfies correct uniform density limit, correct small u expansion and in addition the convergence for large u limit. With all these exact criteria, it is always interesting to design a range-separated functional using this DME semilocal exchange hole. The present attempt is in that direction. We have used the semilocal exchange hole in the short-range limit and the long-range is fixed with the HF exchange. The exchange energy functional proposed here is tested with LYP4 correlation. The present paper is organized as follows. In the next section, we briefly describe the theoretical background of the range-separated functional and then our methods for designing the long-range corrected meta-GGA level range separated functional. In the next section our functional is benchmarked with several molecular test sets.

II. Theoretical background In the range-separated density functional formalism, the   1 Coulomb interaction Vee ri ; rj ¼ between the pair of electrons rij at ri and rj can be separated into a long-range (LR) and a shortrange (SR) part53,54 as,     1  g rij g rij 1 ¼ þ : (1) rij rij rij |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflffl{zfflffl} SR

LR

From various possible choices of the function g,55–57 the most suitable from both the physical and computational point of view is, g = erf(mrij), where m is a parameter. For this choice, the

8992 | Phys. Chem. Chem. Phys., 2018, 20, 8991--8998

Fig. 1 Shown are the full, long and short range parts of the coulomb interaction for range separation parameter m = 0.33 versus electron– electron distance |ri  rj|.

first term of eqn (1) approaches to 0 as rij - N and the second pffiffiffi term goes to 2m= p when rij - 0. The parameter m can be treated as a cutoff between SR and LR parts as shown in Fig. 1. Thus in the range-separated KS scheme, the ground state energy of an electronic system is expressed as58,59  E nD  h io LR   SR (2) E ¼ min cT^ þ V^ ne þ V^ ee c þ EHxc rc ; c

ˆ LR where V ee is the operator for the long-range part of the electron–electron interaction, ESR Hxc is the short-range Hartreeexchange–correlation energy functional, and c is the multideterminant wavefunction. The minimizing wavefunction cLR min minimizes the long-range interacting effective Hamiltonian and gives the exact density of the interacting many-body systems. As the long-range interaction vanishes for m = 0, we get back the standard KS system. And on the other hand for m = N, the wavefunction-based formulation of electronic structure is recovered. Up to now, the above theory is exact. Now if we replace the multi-determinant wavefunction in eqn (2) by a single Slater determinant wavefunction f, then the ground state energy of the range-separated hybrid scheme60 becomes     0 ERSH ¼ f0 T^ þ V^ ne f0 þ EH ½r0  LR SR ½f0  þ Exc ½r0 ; þ Ex;HF

(3)

where f0 is the Slater determinant that minimizes the energy functional and obtains the associated density r0. After applying range separation only on the exchange part, the above expression can be written as,     0 LR ERSH ¼ f0 T^ þ V^ ne f0 þ EH ½r0  þ Ex;HF ½f0 

(4)

þ ExSR ½r0  þ Ec ½r0 : The functional so-obtained is called the long-range corrected functional (LC). In the above expression the long-range part of the exchange interaction is,   occ ðð  erf mrij 1XX LR Ex;HF fis  ðri Þfjs  rj ½f0  ¼  2 s i;j rij (5)    fjs ðri Þfis rj dri drj ;

This journal is © the Owner Societies 2018

View Article Online

Published on 06 March 2018. Downloaded by National Institute of Science Education & Research on 06/05/2018 05:42:00.

PCCP

Paper

where fis is the ith s-spin molecular orbital. Whereas, the short-range exchange functional is defined by,      ðð 1 rðri Þ 1  erf mrij rx ri ; rj dri drj ; (6) ExSR ½r ¼ rij 2 where rx is the exchange hole which is conventionally defined as rx(ri,rj) = |r1(ri,rj)|2/2r(ri), with r1(ri,rj) as the first-order reduced density matrix. Therefore, knowing the exchange hole, one can design an exchange energy functional.

III. Proposed range separated functional We construct a range-separated hybrid functional based on the recently developed DME based exchange hole.20 The aforementioned DME-based semilocal exchange hole has the following form rx ðr; uÞ ¼ 

9r j12 ðkuÞ 105j1 ðkuÞj3 ðkuÞ  G 2 k2 u2 k4 u2

(7) 3675j32 ðkuÞ H;  8k6 u4



  1 jrrj2 t  tunif   t  tunif þ where G ¼ 3 l2  l þ 2 72r 2 2 jrrj jrrj and H ¼ ð2l  1Þ2 . This semilocal 7ð2l  1Þ2 18r r exchange hole uniquely recovers (i) the correct uniform density limit and (ii) the desired small u expansion proposed by Becke,5 and (iii) its large u limit also converges. However, to take care of inhomogeneity present in the system, the modified momentum vector k is set to be k = fkF, where kF is the Fermi momentum of the homogeneous electron gas and f is fixed by the sum rule of the exchange hole by extrapolating its small gradient and large gradient limits and is obtained to be f = [1 + 10(70y/27) + by2]1/10, where y = (2l  1)2p, p = |rr|2/(2kFr)2 and b is a parameter. As dictated by Tao-Mo,20 the values of the parameters l and b are taken to be 0.6866 and 79.873 respectively. Finally, the SR semilocal exchange is given by ð 1 ExSR ½r ¼ rðrÞðM þ N þ QÞd3 r; (8) 2

with a = m/2k. Most of the previously proposed LC functionals use the global system independent screening parameter. However, there are studies which use a position-dependent screening parameter,61 but those are nontrivial to implement. The value of screening parameter m is taken as 0.33 which is the same as given in the CAMB3LYP functional. We then calculate the mean absolute error of the atomization energies of the AE626 database using both LYP and TPSS correlation separately. We find that the errors for LYP and TPSS correlation are 4.94 kcal mol1 and 11.16 kcal mol1 respectively. That is why we employ the one-electron self-interaction free LYP62 correlation functional for the rest of the paper.

IV. Numerical demonstration The newly constructed range-separated functional given in eqn (8) is implemented by locally modifying the NWChem-6.663 program. We have used the same value of m = 0.33 in all parts of the code. A medium grid is used for the evaluation of the exchange–correlation contribution to the density functional. All the results of the benchmark calculations reported in this work are obtained self-consistently using the NWChem program. Spin unrestricted calculation has been done for open shell systems. The differences between theoretical and experimental results or deviations are reported as mean errors (MEs) and mean absolute errors (MAEs). For all the cases under study, we compared the results of the new range-separated functional (LC-TMLYP) with four other popularly known rangeseparated functionals, i.e. CAM-B3LYP,44 HSE06,12 LC-oPBE45 and LC-oPBEh.46 The dataset we have used for benchmarking the LC-TMLYP functional is summarized in Table 1. Atomic energy

where M¼

245p H 48k4

 8   1 þ a 8a þ 256a3  576a5 þ 3840a7  122 880a9 7

  1  exp  2 þ 24a3 35 þ 224a2  1440a4 þ 5120a6 4a   pffiffiffi 1 þ 2 p 2 þ 60a2 erf ; 2a (11)

Q¼ 

9pr 2k2





  8 pffiffiffi 1 1  1 a perf þ 2a  4a3 exp  2  3a þ 4a3 3 2a 4a (9)

N ¼ 



  35p 1 2 2 4 exp  G 1 þ 24a 20a  64a 3k4 4a2  (10) pffiffiffi 1 2 4  3  36a þ 64a þ 10a perf 2a

This journal is © the Owner Societies 2018

For the functionals under study, we calculate the total energies of atoms from H to Cl (AE17).26 The 6-311++G(3df,3pd) basis set is used for all atoms except Helium, for which the aug-cc-pVQZ basis set is used. Calculated total energies are compared with the accurate non-relativistic values.65 The CAM-B3LYP has the smallest MAE among the tested functionals. The LC-TMLYP functional gives a result which is close to the CAM-B3LYP result. LC-oPBEh has the largest MAE in this case. Atomization energy We calculate the atomization energies (AEs) of 148 molecules for the G2/97 test set using the MP2(full)/6-31G* optimized

Phys. Chem. Chem. Phys., 2018, 20, 8991--8998 | 8993

View Article Online

Weakly interacting systems remain a challenging class of materials to describe accurately within the DFT approaches in practice, as the non-local nature of the weak interactions such as van der Waals interactions is absent in semilocal functionals. However, introducing kinetic energy in the semilocal density functional can substantially reduce the error of the LSDA and GGA functionals for predicting the binding energy of weakly interacting systems.26 We test our meta-GGA range-separated functional for the NCCE31/05 database. We have taken the equilibrium geometries for single point calculation from the

8994 | Phys. Chem. Chem. Phys., 2018, 20, 8991--8998

15.69 17.54 15.44 23.07 9.53 15.69 1.15 4.43 21.21 4.40 1.96 1.67 1.57 1.42 2.03 1.96 0.65 1.05 1.12 1.80 5.37 13.65 20.07 5.92 5.35 5.37 9.22 16.05 0.14 5.35 9.08 6.42 5.52 9.38 6.63 9.08 6.42 5.52 9.38 6.63 0.45 0.56 0.78 0.71 0.45 0.27 0.45 0.70 0.46 0.24 0.051 0.470 0.086 0.077 0.084 0.128 0.012 0.105 0.470 0.089 0.086 0.123 0.077 0.084 0.047 0.077 0.052 0.027 0.068 0.022

MAE ME MAE ME MAE ME MAE

HC (kcal mol1) ABDE (kcal mol1)

ME MAE MAE ME MAE ME MAE ME

0.169 0.167 0.239 0.139 0.196 0.083 0.147 0.239 0.108 0.158 5.099 5.735 5.362 5.195 4.329 LC-TMLYP 0.341 0.382 4.494 LC-oPBEh 1.735 1.745 2.276 LC-oPBE 1.564 1.584 1.746 HSE06 1.322 1.334 4.610 CAM-B3LYP 0.140 0.332 1.199

Binding energy of weakly interacting systems

ME

Proton affinity (PA) is the energy released when a proton is added to a species at its ground state. We calculate the PA of the PA870,71 dataset with the MP2/6-31G(2df,p) level optimized geometry, where the 6-311++G(3df,3pd) basis set is used for all calculations. From Table 2, it is obvious that LC-TMLYP gives the smallest MAE of about 0.051 eV. All the investigated functionals overestimate PA except for CAM-B3LYP.

MAE

Proton affinity (PA)

MAE ME

Ionization potential (IP) and electron affinity (EA) is the energy difference between the ion and corresponding neutral atom or molecule, all at 0 K. We calculate IP for the IP1367 database and EA for the EA13/368 database. The QCISD/MG369 level optimized geometry and 6-311++G(3df,3pd) basis set have been taken for both cases. For IP, HSE06 performs best with the smallest MAE of 0.139 eV. For both IP and EA, the performance of LC-TMLYP is comparable in accuracy with the other rangeseparated functionals.

Functionals ME

Ionization potential (IP) and electron affinity (EA)

NCCE (kcal mol1)

geometry64 and 6-311++G(3df,3pd) basis set. The atomization energy of the molecule is defined as the energy difference between the total energy of a molecule and the sum of the energies of its constituent free atoms, all at 0 K. The calculated atomization energy is compared with that of CCSD(T),66 which is the gold-standard in quantum chemistry. Table 2 implies that CAM-B3LYP gives the smallest MAE in AE, followed by LC-TMLYP. The LC-oPBEh is the least accurate among the functionals considered for AE.

PA (eV)

26 64 26 26 26 26 26 26 26 26 26 26 26

EA (eV)

17 atomic energies (H-Cl) 148 atomization energies 13 ionization potentials 13 electron affinities 8 proton affinities 31 non-covalent interactions 38 hydrogen transfer barrier height 38 non-hydrogen transfer barrier height 13 thermochemistry of p system 12 alkyl bond dissociation energies 11 hydrocarbon chemistry 6 isomerization energies 7 difficult cases

IP (eV)

AE17 G2/97 IP13 EA13 PA8 NCCE31 HTBH38 NHTBH38 pTC13 ABDE12 HC7 ISOL6 DC7

G2 (kcal mol1)

Ref.

AE17 (eV)

Description

Deviations from reference values using different functionals are shown in terms of ME and MAE. All the calculations are performed self-consistently

Database

IsoL (kcal mol1)

The databases used for benchmarking calculations

Table 2

Published on 06 March 2018. Downloaded by National Institute of Science Education & Research on 06/05/2018 05:42:00.

Table 1

PCCP DC (kcal mol1)

Paper

This journal is © the Owner Societies 2018

View Article Online

Published on 06 March 2018. Downloaded by National Institute of Science Education & Research on 06/05/2018 05:42:00.

PCCP

Paper

Minnesota database.69 The 6-311++G(3df,3pd) basis set is used for all the molecules except for the inert gas related molecules for which we use aug-cc-pVQZ. A summary of the performances of the tested functionals is listed in Table 2. All the investigated range-separated functionals perform very well. From Table 2, one can say that LC-TMLYP is the best choice in this case. Alkyl bond dissociation energies, hydrocarbon chemistry, isomerization energies of large molecules and difficult cases The alkyl bond dissociation energy database (ABDE12)69 contains two subsets ABDE4/05 and ABDEL8.71 The ABDE4 subset includes four bond dissociation energies of R–X organic molecules, where R = methyl and isopropyl, and X = CH3 and OCH3. The ABDEL8 subset contains eight molecules, with R = ethyl and tert-butyl and X = H, CH3, OCH3 and OH. B3LYP/6-31G(d) level optimized geometries have been taken from the Minnesota database.69 For hydrocarbon chemistry, isomerization energy and difficult cases, we employ the HC7,26 ISOL626 and DC726 databases, respectively. For all these cases the 6-311++G(3df,3pd) basis set is used. For alkyl bond dissociation energy LC-oPBE performs the best with MAE of 5.52 kcal mol1. In the case of hydrocarbon chemistry, LC-TMLYP, CAM-B3LYP and HSE06 give comparable results, while the performance of LC-oPBE and LC-oPBEh is not so good. For isomerization energies of large molecules, HSE06 achieves the smallest MAE, while other range-separated functionals also perform well. For difficult cases, CAM-B3LYP has the smallest MAE of 9.53 kcal mol1, followed by LC-oPBE and LC-TMLYP. The HSE06 functional does not perform well for difficult cases. Thermochemistry of p systems Molecules with p bonds are more dominated by multiconfigurational state functions than s-bonded molecules due to their small HOMO–LUMO gap, where HOMO and LUMO are defined as the highest occupied and lowest unoccupied molecular orbital respectively. We test the performance of the range-

Table 3 The MAE for pTC13 and its secondary databases for the different functionals are shown in each row. All values are given in kcal mol1

Functional

pTC13

pIE3/06

PA-CP5/06

PA-SB5/06

LC-TMLYP LC-oPBEh LC-oPBE HSE06 CAM-B3LYP

4.19 4.79 4.24 6.54 3.41

0.99 2.39 0.98 4.94 2.37

5.19 5.45 4.92 6.96 3.61

5.12 5.58 5.51 7.09 3.82

Table 4

separated functionals to explain the properties of p systems. The database (pTC13) of p systems contains three secondary databases – (i) pIE3/06 – isomeric energy differences between allene and propyne and higher homologs; (ii) PA-CP5/06 – proton affinities of five conjugated polyenes & (iii) PA-CP5/06 – proton affinities of five conjugated Schiff bases.72 We have taken MP2/ 6-31+G(d,p) level optimized geometries from the Minnesota database69 and the 6-311++G(3df,3pd) basis set is used for all the geometries. The CAM-B3LYP functional gives the smallest MAE with 3.41 kcal mol1, followed by the LC-TMLYP functional with MAE of 4.19 kcal mol1 (Table 3). Barrier heights of chemical reactions The semilocal density functionals very often fail to describe the reaction barrier height. Most of the time it gives the transition state to a lower energy state than reactants and products, giving rise to negative barrier height. In principle, one can associate this problem of semilocal functionals with self-interaction error (SIE), because the transition states have stretched bonds and as a result, SIE73 may be large. We calculate forward and reverse barrier heights of 19 hydrogen transfer reactions from the HTBH38/04 dataset and the same for 19 non-hydrogen transfer reactions from the HTBH38/0474,75 dataset. Furthermore, the set NHTBH38/04 is subdivided into a set of six heavy-atom transfer reactions, eight nucleophilic substitution reactions, and five association and unimolecular reactions. For all the calculations, the 6-311++G(3df,3pd) basis set is used. The equilibrium geometries and reference values are again taken from the Minnesota database.69 We observe in Table 4 that LC-oPBE gives the lowest MAE for hydrogen transfer reactions, and LC-TMLYP gives the minimum MAE for nonhydrogen transfer reactions. Overall LC-TMLYP and LC-oPBE give comparable results for barrier heights. Dissociation energy Dissociation energy is the amount of energy needed to break every chemical bond in a molecule by separating all its constituent atomic species. Local or semilocal density functionals fail to describe the dissociative nature of symmetrical radical cations76 like H2+, He2+ etc. The failure of the conventional density functional to explain the dissociative nature of oneelectron molecule H2+ is an indication of the self-interaction error. This is due to the fact that traditional density functionals suffer from delocalization error.77 In Fig. 2, we compare the dissociation curve of H2+ obtained from range-separated functionals with the Hartree–Fock result, which is exact in this case.

Deviations from the experimental values of the barrier heights of the chemical reactions are shown for different functionals. All values are in kcal mol1

HTBH38

Non-hydrogen-transfer reactions of the NHTBH38 set

Hydrogen transfer (38) Heavy-atom transfer (12) Nucleophilic substitution (16) Unimolecular and association (10) Full NHTBH38 Functional ME LC-TMLYP LC-oPBEh LC-oPBE HSE06 CAM-B3LYP

0.88 3.28 0.57 3.45 3.00

MAE

ME

MAE

ME

MAE

ME

MAE

ME

MAE

2.17 3.31 1.26 3.48 3.29

2.49 4.45 0.19 2.68 5.47

2.63 4.45 2.11 12.37 5.47

0.05 0.08 2.95 1.34 0.88

1.17 0.95 2.95 1.53 1.19

0.64 0.10 1.41 0.55 0.54

2.04 2.26 2.34 1.83 1.76

0.59 1.39 1.55 0.14 2.24

1.86 2.40 2.52 5.04 2.69

This journal is © the Owner Societies 2018

Phys. Chem. Chem. Phys., 2018, 20, 8991--8998 | 8995

View Article Online

Published on 06 March 2018. Downloaded by National Institute of Science Education & Research on 06/05/2018 05:42:00.

Paper

Fig. 2 Dissociation curves of (a) H2+ and (b) NaCl obtained using the 6-311++G(3df,3pd) basis set. The zero level is set to be E(H) and E(Na) + E(Cl) respectively for both cases.

Although all the functionals give the same equilibrium bond length, the error increases as we increase the bond length away from the equilibrium bond length. LC-oPBE and LC-TMLYP give almost the same results while all the other functionals deviate too much from the HF result and the performance of HSE06 is not satisfactory for this case. We also consider the dissociation nature of a NaCl molecule, which is an ionic pair when the inter-atomic separation R is not very far from the equilibrium distance. It dissociates into neutral Na and Cl atoms at infinite inter-molecular separation because IPNa 4 EACl. The critical length78 Rc at which sudden charge transfer occurs is given by Rc = 1/(IPNa–EACl). From the experimental value79 of IPNa and EACl, we obtained Rc E 9.4 Å. The HF underestimated the critical length by giving Rc = 5.72 Å due to overestimation of the IPNa–EACl difference. LC-TMLYP and LC-oPBEh almost overlap and give Rc = 8.51 Å. From LC-oPBE and CAM-B3LYP we obtained Rc equal to 8.93 Å and 9.7 Å, respectively. The HSE06 functional completely fails in this case.

V. Conclusions An attempt has been made to develop a new long-range corrected meta-GGA exchange functional in density-functional theory. The formulation is based on Savin’s long-range correction scheme using the TM exchange hole in the short-range and the Hartree–Fock exchange integral in the long-range by

8996 | Phys. Chem. Chem. Phys., 2018, 20, 8991--8998

PCCP

separating the electron–electron interaction operator into short- and long-range by introducing a standard error function. We compare our results with four other popularly known range-separated functionals like CAM-B3LYP, HSE06, LC-oPBE, and LC-oPBEh. Among these functionals, CAM-B3LYP and LC-oPBEh mix both SR and LR Hartree–Fock exchange. In HSE06, a finite amount of HF exchange is used at SR but not in the LR limit, in order to scale down the computational cost of the non-local exchange integral for the extended system. The new LC-TMLYP functional is of LC-oPBE type where only one empirically fitted parameter m is involved. On the other hand, both CAM-B3LYP and LC-oPBEh have two extra parameters besides m, which determine how much HF exchange is mixed at SR and LR. The proposed functional retains all the exact constraints obeyed by TM and in addition to correct asymptotic exchange potential. This new range-separated meta-GGA functional performs well for the investigated molecular test sets and also for dissociation energy due to its improved long-range behavior. The LC-TMLYP functional also gives comparable results with other heavily parametrized range-separated functionals like oB97X80 (14 parameters) and M1181 (40 parameters). Finally, we conclude that in solid-state physics where the density tail is not that much important due to screened82 Coulomb interactions, the TM hole can be used for the long-range keeping HF treatment for the short-range.12 This approach is also computationally advantageous for extended system applications.

Conflicts of interest There are no conflicts to declare.

References 1 W. Kohn and L. J. Sham, Phys. Rev., 1965, 140, A1133. 2 J. P. Perdew and W. Yue, Phys. Rev. B: Condens. Matter Mater. Phys., 1986, 33, 8800. 3 A. D. Becke, Phys. Rev. A: At., Mol., Opt. Phys., 1988, 38, 3098. 4 C. Lee, W. Yang and R. G. Parr, Phys. Rev. B: Condens. Matter Mater. Phys., 1988, 37, 785. 5 A. D. Becke and M. R. Roussel, Phys. Rev. A: At., Mol., Opt. Phys., 1989, 39, 3761. 6 A. D. Becke, J. Chem. Phys., 1996, 104, 1040. 7 P. J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch, J. Phys. Chem., 1994, 98, 11623. 8 J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865. 9 T. V. Voorhis and G. E. Scuseria, J. Chem. Phys., 1998, 109, 400. 10 F. A. Hamprecht, A. J. Cohen, D. J. Tozer and N. C. Handy, J. Chem. Phys., 1998, 109, 6264. 11 M. Ernzerhof and G. E. Scuseria, J. Chem. Phys., 1999, 110, 5029. 12 J. Heyd, G. E. Scuseria and M. Ernzerhof, J. Chem. Phys., 2003, 118, 8207.

This journal is © the Owner Societies 2018

View Article Online

Published on 06 March 2018. Downloaded by National Institute of Science Education & Research on 06/05/2018 05:42:00.

PCCP

13 R. Armiento and A. E. Mattsson, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 085108. 14 Y. Zhao and D. G. Truhlar, J. Chem. Phys., 2006, 125, 194101. 15 J. Tao, J. P. Perdew, V. N. Staroverov and G. E. Scuseria, Phys. Rev. Lett., 2003, 91, 146401. 16 J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin and J. Sun, Phys. Rev. Lett., 2009, 103, 026403. 17 J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou and K. Burke, Phys. Rev. Lett., 2008, 100, 136406. 18 J. Sun, A. Ruzsinszky and J. P. Perdew, Phys. Rev. Lett., 2015, 115, 036402. 19 A. V. Arbuznikov and M. Kaupp, J. Chem. Phys., 2014, 141, 204101. 20 J. Tao and Y. Mo, Phys. Rev. Lett., 2016, 117, 073001. 21 J. P. Perdew and K. Schmidt, AIP Conf. Proc., 2001, 577, 1. 22 S. H. Vosko, L. Wilk and M. Nusair, Can. J. Phys., 1980, 58, 1200. 23 J. P. Perdew and Y. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 45, 13244. 24 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 6671. 25 Z. Wu and R. E. Cohen, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 235116. 26 R. Peverati and D. G. Truhlar, Philos. Trans. R. Soc., A, 2014, 372, 20120476. 27 V. N. Staroverov, G. E. Scuseria, J. Tao and J. P. Perdew, J. Chem. Phys., 2003, 119, 12129. 28 P. Hao, J. Sun, B. Xiao, A. Ruzsinszky, G. I. Csonka, J. Tao, S. Glindmeyer and J. P. Perdew, J. Chem. Theory Comput., 2013, 9, 355. 29 L. Goerigk and S. Grimme, J. Chem. Theory Comput., 2010, 6, 107. 30 L. Goerigk and S. Grimme, J. Chem. Theory Comput., 2011, 7, 291. 31 Y. Mo, R. Car, V. N. Staroverov, G. E. Scuseria and J. Tao, Phys. Rev. B: Condens. Matter Mater. Phys., 2017, 95, 035118. 32 V. N. Staroverov, G. E. Scuseria, J. Tao and J. P. Perdew, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 69, 075102. 33 D. Jacquemin and C. Adamo, J. Chem. Theory Comput., 2011, 7, 369. 34 D. J. Tozer and N. C. Handy, J. Chem. Phys., 1998, 109, 10180. 35 A. Dreuw, J. L. Weisman and M. Head-Gordon, J. Chem. Phys., 2003, 119, 2943. 36 J. Zheng, Y. Zhao and D. G. Truhlar, J. Chem. Theory Comput., 2009, 5, 808. 37 N. N. Matsuzawa, A. Ishitani, D. A. Dixon and T. Uda, J. Phys. Chem. A, 2001, 105, 4953. 38 M. E. Casida and D. R. Salahub, J. Chem. Phys., 2000, 113, 8918. 39 H. Appel, E. K. U. Gross and K. Burke, Phys. Rev. Lett., 2003, 90, 043005. 40 J. P. Perdew, V. N. Staroverov, J. Tao and G. E. Scuseria, Phys. Rev. A: At., Mol., Opt. Phys., 2008, 78, 052513. 41 J. P. Perdew and A. Zunger, Phys. Rev. B: Condens. Matter Mater. Phys., 1981, 23, 5048.

This journal is © the Owner Societies 2018

Paper

42 C.-O. Almbladh and U. von Barth, Phys. Rev. B: Condens. Matter Mater. Phys., 1985, 31, 3231. 43 A. D. Becke, J. Chem. Phys., 1993, 98, 5648. 44 T. Yanai, D. P. Tew and N. C. Handy, Chem. Phys. Lett., 2004, 393, 51. 45 O. A. Vydrov and G. E. Scuseria, J. Chem. Phys., 2006, 125, 234109. 46 M. A. Rohrdanz, K. M. Martins and J. M. Herbert, J. Chem. Phys., 2009, 130, 054112. 47 R. Baer, E. Livshits and U. Salzner, Annu. Rev. Phys. Chem., 2010, 61, 85. 48 J. Tao, Density Functional Theory of Atoms, Molecules, and Solids VDM Verlag, Germany, 2010. 49 M. Ernzerhof and J. P. Perdew, J. Chem. Phys., 1998, 109, 3313. 50 L. A. Constantin, J. P. Perdew and J. Tao, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 205104. 51 L. A. Constantin, E. Fabiano and F. Della Sala, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 88, 125112. 52 H. Iikura, T. Tsuneda, T. Yanai and K. Hirao, J. Chem. Phys., 2001, 115, 3540. 53 T. Leininger, H. Stoll, H.-J. Werner and A. Savin, Chem. Phys. Lett., 1997, 275, 151. 54 R. Pollet, A. Savin, T. Leininger and H. Stoll, J. Chem. Phys., 2002, 116, 1250. 55 A. Savin and H.-J. Flad, Int. J. Quantum Chem., 1995, 56, 327. 56 Y. Akinaga and S. Ten-no, Chem. Phys. Lett., 2008, 462, 348. 57 J.-W. Song, M. A. Watson, A. Nakata and K. Hirao, J. Chem. Phys., 2008, 129, 184113. 58 J. Toulouse, F. M. C. Colonna and A. Savin, Phys. Rev. A: At., Mol., Opt. Phys., 2004, 70, 062505. 59 J. Toulouse, I. C. Gerber, G. Jansen, A. Savin and ´ngya ´n, Phys. Rev. Lett., 2009, 102, 096404. J. G. A ´ngya ´n, I. C. Gerber, A. Savin and J. Toulouse, Phys. 60 J. G. A Rev. A: At., Mol., Opt. Phys., 2005, 72, 012510. 61 A. V. Krukau, G. E. Scuseria, J. P. Perdew and A. Savin, J. Chem. Phys., 2008, 129, 124103. 62 C. Lee, W. Yang and R. G. Parr, Phys. Rev. B: Condens. Matter Mater. Phys., 1988, 37, 785. 63 M. Valiev, E. Bylaska, N. Govind, K. Kowalski, T. Straatsma, H. V. Dam, D. Wang, J. Nieplocha, E. Apra, T. Windus and W. de Jong, Comput. Phys. Commun., 2010, 181, 1477. 64 L. A. Curtiss, K. Raghavachari, P. C. Redfern and J. A. Pople, J. Chem. Phys., 1997, 106, 1063. 65 S. J. Chakravorty, S. R. Gwaltney, E. R. Davidson, F. A. Parpia and C. F. p Fischer, Phys. Rev. A: At., Mol., Opt. Phys., 1993, 47, 3649. 66 R. Haunschild and W. Klopper, J. Chem. Phys., 2012, 136, 164102. 67 B. J. Lynch, Y. Zhao and D. G. Truhlar, J. Phys. Chem. A, 2003, 107, 1384. 68 Y. Zhao, N. E. Schultz and D. G. Truhlar, J. Chem. Theory Comput., 2006, 2, 364. 69 B. J. Lynch, Y. Zhao and D. G. Truhlar, see http://t1.chem. umn.edu/db/ for the Minnesota databases for chemistry and solid state physics.

Phys. Chem. Chem. Phys., 2018, 20, 8991--8998 | 8997

View Article Online

Published on 06 March 2018. Downloaded by National Institute of Science Education & Research on 06/05/2018 05:42:00.

Paper

70 S. Parthiban and J. M. L. Martin, J. Chem. Phys., 2001, 114, 6014. 71 Y. Zhao and D. G. Truhlar, J. Phys. Chem. A, 2006, 110, 10478. 72 Y. Zhao and D. G. Truhlar, J. Chem. Phys., 2006, 125, 194101. 73 Y. Zhang and W. Yang, J. Chem. Phys., 1998, 109, 2604. 74 Y. Zhao, B. J. Lynch and D. G. Truhlar, J. Phys. Chem. A, 2004, 108, 2715. 75 Y. Zhao, N. Gonzlez-Garca and D. G. Truhlar, J. Phys. Chem. A, 2005, 109, 2012. 76 H. Chermette, I. Ciofini, F. Mariotti and C. Daul, J. Chem. Phys., 2001, 114, 1447.

8998 | Phys. Chem. Chem. Phys., 2018, 20, 8991--8998

PCCP

´nchez and W. Yang, Science, 2008, 77 A. J. Cohen, P. Mori-Sa 321, 792. 78 J. P. Perdew, R. G. Parr, M. Levy and J. L. Balduz, Phys. Rev. Lett., 1982, 49, 1691. 79 D. R. Lide, CRC Handbook of Chemistry and Physics, CRC, Boca Raton, 2004. 80 J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 2008, 128, 084106. 81 R. Peverati and D. G. Truhlar, J. Phys. Chem. Lett., 2011, 2, 2810. 82 D. Pines, Elementary Excitations in Solids, Perseus Books, MA, 1999.

This journal is © the Owner Societies 2018