Assessing the performance of the recent non-empirical semilocal density functionals on describing the lattice constants, bulk moduli and cohesive energies of alkali, alkaline-earth, and transition metals
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l Assessing the performance of the recent non-empirical semilocal density functionalson describing the lattice constants, bulk moduli and cohesive energies of alkali,alkaline-earth, and transition metals
Subrata Jana, ∗ Kedar Sharma, and Prasanjit Samal School of Physical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar 752050, India School of Physics, Indian Institute of Science Education and Research,Maruthamala, Vithura, Thiruvananthapuram 695551, India (Dated: July 10, 2018)The bulk properties (lattice constants, bulk moduli, and cohesive energies) of alkali, alkaline-earth,and transition metals are studied within the framework of the recently developed meta-GGA (meta-Generalized Gradient Approximation) semilocal exchange-correlation functionals. To establish theapplicability, broadness and accuracy of meta-GGA functionals we also put the results of PBE(Perdew-Burke-Ernzerhof) and PBEsol (PBE reparametrized for solids) functionals. The interestingfeature of the present paper is that it measures the accuracy of the recently developed TM (Tao-Mo) and TMTPSS (TM exchange with Tao-Perdew-Staroverov-Scuseria (TPSS) correlation) andSCAN (Strongly Constrained and Appropriately Normed) functionals on describing aforementionedproperties. The present systematic investigation shows that the TM is accurate in describing thelattice constants while for cohesive energies and bulk moduli the accuracy is biased towards thePBE and TPSS functionals.
I. INTRODUCTION
Since its advent, the Kohn-Sham (KS) density func-tional theory (DFT) is a de facto standard theoreti-cal framework for studying the electronic structures ofsolids and materials. The accuracy of the KS densityfunctional depends on the accuracy of the approximateexchange-correlation functionals. Due to the reason-able computational cost with comparatively well bal-anced accuracy the semilocal nature of the exchange-correlation functionals i.e., the local density approx-imations (LDA) , generalized-gradient approximation(GGA) and meta-generalized gradient approxima-tion (meta-GGA) are widely used for the bulkproperties of solids . The mainstream of semilo-cal density functionals are developed based on the con-straint satisfaction or modelling the ex-change hole or association both the properties . Usu-ally, benchmarking the density functional approxima-tions against the experimental features are common prac-tice to measure the accuracy and applicability of the ap-proximation, in particular when a new functional is in-troduced. Indeed, the systematic evaluation of the prop-erties of a density functional approximation guide usersto properly choose a functional for describing the mate-rial properties. Also, the behavior of the functionals forthe wide range of systems makes it easier to improve thedrawback of the functional.The present paper seeks to assess the performance ofthe recently proposed meta-GGA functionals at the accu-racy of the bulk properties of the transition metals. Morespecifically, we consider the lattice constants (or equilib-rium shortest distances), bulk moduli, and cohesive en-ergies of transition metals. Regarding the performanceof different level of density functional approximations forthe bulk properties of transition metals, it has been stud- ied earlier within the framework of GGA, meta-GGA andhybrid functionals theory. In ref. Janthon et. al. stud-ied the transition metals within the framework of GGAbased functionals. In ref. Janthon et. al. further ex-plore the behavior of transition metals by including meta-GGA level functionals. Besides, Hass et. al. , Csonka et.al. , Tran et. al. , Schimka et. al. , Hao et. al. , andZhang et. al. partially cover the lattice constants, co-hesive energies and bulk moduli of alkali, alkaline-earth,and transition metals. In this paper, we put togetherthe lattice constants, cohesive energies and bulk mod-uli of all the alkali, alkaline-earth, and transition met-als within the framework of recently developed meta-GGA functionals. Our comparison meta-GGA function-als contain Tao-Perdew-Staroverov-Scuseria (TPSS) ,revised TPSS (revTPSS) , Minnesota 2006 local func-tional (M06L) , meta-GGA made simple (MS0, MS1,and MS2) functionals, Strongly Constrained andAppropriately Normed (SCAN) , and Tao-Mo meta-GGA functional (TMTPSS and TM) functionals.Arguably, the recent advances in the meta-GGA func-tionals show that the accuracy of the functionals canbe further improved by imposing more exact constraintson the functional construction. The motivation of thepresent paper flows from the appealing features and ac-curacy of the recently developed meta-GGA functionals.The recent development of the meta-GGA functionalsshows that the SCAN functionals developed by Sun et.al. and TM functional developed by Tao et. al. quiteaccurate in describing several bulk properties of solids.Though the SCAN functional has been studied for bulkproperties of few metals but remains untested exten-sively for the alkali, alkaline-earth, and transition met-als. Also, the recently developed TM functional remainsuntested for those properties. In the present paper, wepresent the benchmark calculations of the bulk propertiesof the 3 d , 4 d and 5 d transition metals, alkali, and alka-line earth metals. In particular, due to different bondingnature of the alkali, alkaline-earth and transition metals,they are considered as the difficult cases within semilo-cal exchange-correlation functionals. Though they areconsidered mainly as metals and metallic bonding domi-nates, but the weak van-der-Waals interactions in closedsemi-core states also play important role . Thesemake the semilocal functionals difficult to describe ac-curately all the bonding nature and often in the bench-marking calculations the 3 d , 4 d and 5 d transition metals,alkali and alkaline earth metals are excluded.It was shown in ref. that the PBE functional doesnot perform in a satisfactory way in describing the lat-tice constants of all these metals. It was shown thatthe PBE lattice constants for 3 d metals are slightly toosmall, whereas, the lattice constants reported for 4 d and5 d metals using PBE are too large. It was also shownin ref. that the situation improves through the in-clusion of kinetic-energy density term in the functionalconstruction. Due to the one electron free correlationof the revTPSS functional, it performs reasonably forthe transition metal lattice constants. All these previ-ous studies motivate us to assess the accuracy of therecently developed meta-GGAs in predicting the afore-mentioned bulk properties. It is noteworthy to men-tion that the SCAN functional includes the intermedi-ate van-der-Waals (vdW) interaction, therefore it will bean interesting study to assess its performance for alkaliand alkaline-earth materials, where the bonding is influ-enced by the vdW interaction in the semi-core states.Regarding the TM functionals, it was shown that boththe TM and TPSS correlation perform differently withthe TM exchange. But the accuracy of the TMTPSSand TM has not been measured against the 3 d , 4 d and5 d transition metals, alkali, and alkaline earth metals.In this paper, we put all the modern meta-GGA densityfunctionals (TPSS, revTPSS, M06L, MS0, MS1, MS2,SCAN, TMTPSS, and TM) together with GGA (PBE and PBEsol ) based semilocal functionals to assess theperformance of alkali, alkaline-earth, and transition met-als.This paper is organized as follows: In the following,we will describe our computational set up along with thetest set used for our calculations. Following that, wewill study the lattice constants, bulk moduli and surfaceenergies of the transition metals. We will conclude bydiscussing and comparing our results. II. COMPUTATIONAL SETUP
All computational studies are performed using theplane wave code based on the projector-augmentedmethod Vienna
Ab Initio
Simulation Package(VASP) . The Bulk calculations are performedwith 16 × ×
16 Gamma-centered k − points. Re- garding the atomic calculations of cohesive energiesa simulation box of 20 × ×
20 ˚A has been usedwith 1 × × k − points. The spinpolarization calculations are performed for atoms. Anenergy convergence criterion of 10 − has been set forbulk calculations, whereas, the atomic simulations areperformed with energy convergence criterion of 10 − .It is noteworthy to mention that all calculations ofthe meta-GGA functionals are performed by startingfrom the PBE wavefunctions and change densities.The energy cutoff 500 eV to 700 eV is used for bulkcalculations, whereas 700 eV to 1000 eV energy cutoff isused for the atomic calculations.The results present in TABLEs are arranged by sepa-ration out the 3 d transition metals, 4 d transition metals,5 d transition metals, alkali metals (K, Rb, and Cs) andalkaline-earth metals (Ca, Sr and Ba). Under the ambi-ent condition, all the alkali metals, alkaline-earth metals,and transition metals show f cc , bcc , or hcp structures.Only exceptions are Mn, La, and Hg. These materialsshow complicated hexagonal (La), rhombohedral (Hg)and cubic unit cell with 58 atoms (Mn). Due to thevery different structures of La, Hg, and Mn, they arealso discussed separately in all TABLEs.To compare the overall accuracy of all the functionalswe present mean-error (ME), mean absolute error (MAE)of the 3 d , 4 d , 5 d along with alkali and alkaline metals.The total ME and total MAE is also given in the lastcolumn of each TABLE. III. RESULTS AND DISCUSSIONSA. Equilibrium Inter-atomic Shortest Distances
All the lattice constants of alkali, alkaline-earth, andtransition metals are determined at their ambient condi-tions and non-magnetic phases. The experimental valuespresented in TABLE I are subtracted for the zero-pointvibrational effects (ZPVE). TABLE I presents the ZPVEcorrected equilibrium inter-atomic distances along withthe benchmark results of all the functionals. The equi-librium inter-atomic distances depend on the equilibriumlattice constants according to the different lattice struc-tures. For the details of the relation between the inter-atomic distances and equilibrium lattice constants, thereaders are suggested to go through the ref. . In FIG. I,we also show the percentage deviation of our calculatedvalues. Though the calculations for the PBE, PBEsol,M06L, TPSS, and revTPSS are reported in past for sev-eral solids, but in our present study, we recalculate allthe solids along with recently developed meta-GGAs. Inthis section, we will discuss the functional performancesaccording to the data present in TABLE I. TABLE I: Equilibrium shortest distances δ (in Picometre (pm)) of differ-ent solid structures using PBE, PBEsol, TPSS, revTPSS, M06L, MS0,MS1, MS2, SCAN, TMTPSS and TM functionals. The experimental ref-erence values are collected from references , where the correctiondue to the zero-point vibrational effects (ZPVE) are taken into account.For elements Mn, La and Hg the ZPVE corrected values are not availableand we reported only the experimental values taken from reference .Metals PBE PBEsol TPSS revTPSS M06L MS0 MS1 MS2 SCAN TMTPSS TM Expt.Sc 330.1 326.3 328.4 327.7 328.0 331.2 331.5 328.9 329.6 328.7 327.8 324.4Ti 292.3 288.8 290.4 289.2 291.0 291.9 292.1 290.4 290.8 291.2 290.1 288.9V 258.0 254.4 256.1 255.2 257.3 255.5 255.8 255.2 255.6 256.8 256.1 260.6Cr 245.6 242.4 244.0 243.0 244.3 242.7 243.0 242.7 243.2 244.3 243.5 248.5Fe 238.6 234.9 236.6 235.5 236.3 235.0 235.3 235.1 235.1 236.8 236.0 245.0Co 245.1 240.9 242.9 241.6 243.0 240.9 241.3 241.1 241.2 243.0 242.0 248.8Ni 248.2 243.6 245.2 243.2 241.8 243.8 244.2 243.7 243.7 244.9 243.6 248.4Cu 257.0 251.5 253.0 250.9 248.3 250.5 251.3 250.2 249.4 252.1 251.1 254.4Zn 263.5 261.1 263.2 261.4 263.4 260.5 260.9 259.7 258.8 261.0 260.5 264.5ME -0.6 -4.4 -2.6 -4.0 -3.3 -3.5 -3.1 -4.1 -4.0 -2.7 -3.6MAE 3.2 4.8 3.9 4.8 4.6 5.7 5.4 5.4 5.6 4.2 4.7Y 363.3 358.5 363.8 361.6 366.6 366.7 366.6 363.9 365.7 365.5 362.6 354.8Zr 323.6 318.0 321.4 319.2 324.0 321.4 321.5 320.5 321.1 323.1 320.2 317.4Nb 287.8 284.1 286.7 285.2 288.2 285.5 285.7 285.6 286.4 287.2 286.1 285.4Mo 272.9 269.9 271.9 270.3 271.9 270.2 270.4 270.4 271.1 271.7 270.6 272.1Tc 274.4 271.2 273.1 271.3 272.2 271.0 271.2 271.3 272.0 272.8 271.5 270.5Ru 271.6 268.1 270.5 268.1 269.0 266.9 267.2 267.7 267.4 270.0 268.3 264.2Rh 270.4 266.2 268.9 263.6 264.2 263.0 263.3 263.5 263.8 264.3 263.9 253.2Pd 278.6 273.2 276.1 272.9 277.2 272.8 273.2 273.0 273.8 275.9 273.6 274.5Ag 293.4 285.7 289.1 285.0 291.4 285.7 286.2 285.3 286.1 288.4 285.6 287.7Cd 302.0 306.2 301.3 298.4 311.8 313.9 298.2 297.8 296.3 299.6 298.0 295.9ME 6.2 2.5 4.7 2.0 6.1 4.1 2.8 2.3 2.8 4.3 2.5MAE 6.2 3.9 4.8 3.3 6.1 5.3 3.7 3.5 3.5 4.4 3.4Hf 319.5 315.2 317.1 315.3 320.4 316.1 316.5 315.4 315.1 316.6 315.9 312.6Ta 286.6 283.3 284.8 283.1 287.2 282.9 283.2 282.9 282.9 284.4 283.7 285.6W 274.7 272.0 273.3 271.7 273.5 271.4 271.7 271.6 271.8 272.8 272.1 273.8Re 277.2 274.4 273.4 272.5 273.0 271.4 271.8 272.2 271.1 273.2 272.8 256.2Os 275.4 272.4 274.1 271.8 272.5 270.9 271.2 271.6 270.6 273.2 271.9 267.1Ir 273.8 270.5 272.6 270.1 271.4 268.5 268.9 269.6 267.6 271.7 270.2 271.0Pt 280.5 276.1 278.9 275.6 278.8 274.3 274.6 275.1 274.8 278.0 275.8 276.6Au 293.9 287.7 291.1 286.9 291.6 285.8 286.3 286.4 286.8 290.3 287.2 287.0ME 6.5 2.7 4.4 2.1 4.8 1.4 1.8 1.9 1.4 3.8 2.5MAE 6.5 4.0 4.8 3.8 4.9 4.2 4.1 4.0 3.9 4.3 3.8K 457.5 452.5 463.5 459.0 427.8 463.5 464.0 459.2 458.9 446.1 446.0 451.4Rb 490.1 483.7 497.1 492.6 451.6 499.6 502.4 493.7 491.7 483.7 483.7 483.0Cs 534.8 520.3 542 536.5 482.2 546.9 549.3 538.6 539.9 523.5 523.5 523.0ME 8.3 -0.3 15.1 10.2 -31.9 17.5 19.4 11.4 11.0 -1.4 -1.4MAE 8.3 1.5 15.1 10.2 31.9 17.5 19.4 11.4 11.0 2.2 2.2Ca 394.3 385.8 391.1 390.0 378.7 394.3 395.5 392.7 393.5 388.6 388.1 392.9Sr 426.4 418.7 426.6 425.1 414.9 431.2 433.2 427.3 430.3 424.2 423.3 427.1Ba 423.7 423.6 433.5 431.5 430.8 440.2 441.6 435.9 435.9 433.5 431.2 433.2ME -2.9 -8.4 -0.7 -2.2 -9.6 4.2 5.7 0.9 2.2 -2.3 -3.5MAE 3.9 8.4 0.9 2.2 9.6 4.2 5.7 1.0 2.2 2.5 3.5Mn 230.7 227.1 228.4 227.7 228.6 227.4 227.7 227.4 227.6 228.6 230.2 224.0La 376.9 365.4 373.3 369.2 385.9 373.9 374.7 372.1 379.5 376.6 373.4 373.9 Hg 323.8 300.6 308.5 299.5 308.9 297.6 298.6 297.6 300.2 305.2 299.5 301.0ME 10.8 -1.9 3.8 -0.8 8.2 0.0 0.7 -0.6 2.8 3.8 1.4MAE 10.8 4.0 4.2 3.3 8.2 2.3 2.3 2.9 3.3 3.8 2.7TME 4.4 -0.7 3.1 0.6 -0.9 2.4 2.5 1.0 1.3 1.4 0.0TMAE 5.9 4.3 5.0 4.3 8.1 5.8 5.6 4.5 4.5 3.9 3.6 d transition metals : The inter-atomic distances of3 d elements are presented at the top of the TABLE I. The3 d elements contain with Sc, Ti, V, Cr, Fe, Co, Ni, Cu,and Zn. Let us start our discussion with the popularlyused GGA functional PBE. The PBE functional performfairly good throughout the series. However, It overesti-mates the inter-atomic distances of Sc, Ti, V, and Cu,and underestimations the inter-atomic distances for Cr,Fe, and Co. But it gives a fairly good inter-atomic dis-tance for Ni. Regarding the performance of the PBEsol,it underestimates the inter-atomic distances of all theelements except Sc and Ti. For Sc, the PBEsol over-estimates the inter-atomic distance, whereas, very goodinter-atomic distance is obtained for Ti. The underesti-mation percentage of all the elements are fairly large forPBEsol compared to PBE. Especially, for Fe, a fairlylarge underestimation is observed. Now, we considerthe meta-GGA functionals. Regarding the performancefor the TPSS and revTPSS functionals, the revTPSSfunctional lower the inter-atomic distances compared toTPSS for all the 3 d elements by almost 1 pm to 3 pm.TPSS overestimates the inter-atomic distances for Sc, Ti,but it follows the different trend as the d bands becomefilled. For V, Cr, Fe, Co, Ni, Cu and Zn TPSS underes-timates the inter-atomic distances. The underestimationtendency becomes less intense as the d band almost filled.Now, concerning the performance of M06L, it follows thesame trend as TPSS does except for Ni and Cu. For Niand Cu, M06L underestimates the inter-atomic distancesmore than TPSS and revTPSS do. Now, for the meta-GGA made simple functionals (MS0, MS1, and MS2), allperform equivalently and overestimate the inter-atomicdistances for Sc and Ti, but underestimate the inter-atomic distances form V to Zn. The underestimationand overestimation percentage of all the “MS” function-als are fairly large compared to the traditional TPSS andrevTPSS functionals. Now, we consider the performanceof the two recently proposed functionals SCAN and TM(TMTPSS and TM). The performance of SCAN quitedisappointing as it follows the same trend of “MS” func-tionals. The MAE of SCAN functional indicates that itperforms better than MS0 but less accurate than MS1and MS2. Now, concerning TMTPSS and TM function-als both are less accurate than TPSS in predicting theinter-atomic distances of 3 d metals. In fact, both showthe same tradition as it is observed using other meta-GGAs. Overall consideration shows that the PBE func-tionals perform fairly well for all the 3 d elements andreports the best MAE compared to the more advancedmeta-GGA functionals. Overall tendency of meta-GGAs shows that TPSS is the best among all the meta-GGAfunctionals. We do not observe improvement in the inter-atomic distances using the more advance functionals likeSCAN and TM based functionals.4 d transition metals : Unlike the 3 d elements, inthis case, the PBE functional overestimates the the inter-atomic distances of all the elements except Mo. A sizableoverestimation in the inter-atomic distances is observedfor Y, Zr, Tc, Ru, Rh, Ag and Cd. For Nb, the over-estimation is observed within the limit of ≈ d transition metals, the TM func-tional performs better than TMTPSS functionals. For 4 d transition metal elements, all semilocal functionals showoverestimation tendency in predicting the inter-atomicdistances except for few cases.5 d transition metals : We observe the same trendas it is observed for the 4 d elements. A fairly sizableoverestimation is observed for PBE functional. PBEsolreduces the MAE for the PBE functional. In case ofmeta-GGA functionals, the revTPSS, SCAN, and TMfunctionals perform equivalently. Using the “MS” meta-GGAs we obtain almost equivalent MAE. Interestingly,the “MS” meta-GGA functionals overall overestimate theinteratomic distances for few cases, whereas, underesti-mation in the inter-atomic distances are observed for oth-ers. A similar tendency is observed for the SACN func-tional. Both the TM and TMTPSS functionals quitereasonably predict the inter-atomic distances for the Ta,W, Os, Ir, Pt, and Au. For other elements overestima-tion is observed. Overall we obtain least MAE using therevTPSS and TM functionals. Alkali metals :
The alkali metals contain elementsK, Rb and Cs. We separately discuss these metals be-cause different kinds of interactions affect the bonding ofthese metals. Though they are considered as prototypi-cal metals but a contribution from the semi-core p and s states originate van-der-Waals bonding which affectthe lattice constants or equilibrium shortest distances. Asizable error in equilibrium shortest distances is observed (a) (b) (c) (d) (e) (f) FIG. 1. Histograms of relative error in the interatomic shortest distances. The percentage deviation is plotted along Y-axis.The numbering of the figures are as the order of the solids presented in TABLE I. from the PBE functionals. The PBE functionals overes-timate the equilibrium shortest distances for all the al-kali metals. The PBEsol improves the performance andyields the least MAE for the alkali metals. Regardingthe performance of the meta-GGAs, all functionals over-estimate the equilibrium shortest distances except theM06L, TMTPSS and TM functionals. The M06L mas-sively underestimates the equilibrium shortest distancesof alkali metals, whereas, TMTPSS and TM agree verywell with the experimental values. Both TMTPSS andTM functionals produce much better results than SCANmeta-GGA functional, though the SCAN contains inter-mediate van-der-Waals interactions.
Alkaline-earth metals :
Unlike alkali metals, a rea-sonably good performance is observed using the PBEfunctionals for the alkaline-earth metals. However, thePBEsol underestimates the equilibrium shortest dis-tances. Within meta-GGA functionals, TPSS and MS2perform quite well. A sizable underestimation in theequilibrium shortest distances is observed using theM06L functional. MS0 and MS1 overestimate the equi-librium shortest distances, whereas, the overestimation percentage comparatively inadequate for the SCAN func-tional. In this case, both the TMTPSS and TM func-tionals underestimate the equilibrium shortest distances.But, the TMTPSS quite close to the experimental equi-librium shortest distances. In this case, the SCAN func-tional agrees well with the experimental values thanTMTPSS and TM functionals.
Other transition metals :
Due to the complicatedstructure of the Mn, La, and Hg, we separate out these el-ements from others. Regarding the performance of PBE,it massively overestimates the equilibrium shortest dis-tances for all these metals. Reasonably good performanceis observed using PBEsol. For meta-GGAs, all performequivalently to predict the equilibrium shortest distancesexcept M06L. M06L overestimates the shortest distancesin a sizable order. The TPSS, MS0, MS1 perform equiv-alently. We obtain MAE 4 .
250 pm from the revTPSSfunctionals. The MS2 and SCAN also produce the sameamount of error in this case. Here, we obtain least MAEwith the MS0 and MS1 functionals. TM functional is thesecond best after MS0 and MS1 with MAE 2 . Overall performance :
Correspond to the overallranking we obtain the best MAE with the TM functional.Next, the performance of the TMTPSS is found to bebest. The performance of TM and TMTPSS is quitewell compared to the SCAN functional. The revTPSS,MS2 and SCAN functionals perform equivalently. TheMAE of TPSS is less compare to the MS0 and MS1. The M06L gives the largest MAE of order 8 . B. Bulk Moduli
The bulk modulus ( B ) is defined as the variation ofthe volume ( V ) due to the external pressure ( P ). In DFTthe bulk modulus is measured at the equilibrium latticeconstant a or volume ( V ) as, TABLE II: Bulk moduli ( B ) (in GPa) calculated using different solidstructures using PBE, PBEsol, TPSS, revTPSS, M06L, MS0, MS1, MS2,SCAN, TMTPSS and TM functionals. The experimental values cor-rected for the finite thermal corrections. All the corrected values aretaken from references . For elements Mn, La and Hg the finite tem-perature corrected values are not available and we reported only theexperimental values taken from reference . The the total mean error(TME) are reported without considering the Mn, La and Hg results andwith the values of Mn, La and Hg results.Metals PBE PBEsol TPSS revTPSS M06L MS0 MS1 MS2 SCAN TMTPSS TM Expt.Sc 52.6 55.4 54.6 55.6 62.6 54.2 53.6 56.0 59.8 61.8 57.4 55.6Ti 116.8 125.6 123.2 125.8 128.6 124.0 122.4 127.6 125.2 125.8 127.0 108.3V 187.8 204.0 201.8 205.6 198.0 200.8 199.6 207.0 203.8 201.8 181.8 158.9Cr 263.6 288.2 283.2 291.8 276.8 284.8 281.2 292.0 280.2 283.6 286.8 174.5Fe 166.6 208.6 310.4 320.6 300.0 316.2 217.8 294.2 316.6 312.6 316.0 169.8Co 212.6 285.8 280.2 291.4 263.6 294.2 288.0 268.8 297.6 244.2 289.6 193.0Ni 197.7 230.1 226.2 241.8 221.9 242.2 235.7 244.8 238.3 233.1 236.6 185.5Cu 137.1 163.3 156.5 170.5 151.9 158.9 146.9 155.9 152.4 161.2 164.2 140.3Zn 74.0 91.8 86.0 97.8 74.2 99.4 82.8 102.0 105.2 96.8 105.6 69.7ME 17.0 44.1 51.8 60.6 46.9 57.7 41.4 54.7 58.2 51.7 56.6MAE 19.1 44.2 52.1 60.6 46.9 58.0 41.8 54.7 58.2 51.7 56.6Y 39.6 42.0 40.2 40.2 44.4 37.4 37.2 36.6 36.4 38.4 41.4 41.7Zr 92.8 98.8 96.4 97.0 95.4 92.8 92.2 95.6 95.8 96.2 97.0 95.9Nb 172.0 186.8 183.4 187.6 169.0 181.6 180.4 183.4 180.8 181.2 183.2 172.0Mo 266.4 289.0 278.6 286.2 260.2 287.0 285.2 287.4 284.0 278.6 283.6 264.7Tc 301.4 330.6 317.0 329.0 292.2 334.6 331.2 331.2 324.2 319.0 325.2 303.1Ru 316.4 353.8 334.4 350.4 302.0 365.4 361.4 356.8 342.2 337.0 345.8 317.7Rh 254.8 295.8 276.7 292.4 235.0 298.4 294.7 292.9 290.3 275.9 284.0 288.7Pd 165.3 201.8 187.9 201.1 148.3 199.5 195.3 200.1 193.5 189.5 195.3 195.4Ag 86.1 112.3 102.3 113.0 89.1 103.7 100.9 110.7 105.4 109.3 113.0 103.8Cd 40.8 59.4 54.4 62.2 58.6 59.6 56.0 63.4 57.4 64.8 50.2 53.8ME -10.1 13.4 3.5 12.2 -14.3 12.3 9.8 12.1 7.3 5.3 8.2MAE 10.5 13.4 8.0 12.5 15.8 13.8 12.0 13.2 8.8 9.7 9.9Hf 110.0 116.6 114.2 116.4 114.2 113.8 112.8 117.0 116.6 117.4 118.2 109.7Ta 199.0 199.0 207.8 213.0 196.8 210.4 208.8 213.4 212.0 210.8 212.6 193.7W 309.6 309.6 324.6 336.0 310.6 338.8 332.2 336.0 331.4 330.2 334.4 312.3Re 370.6 399.6 390.8 406.8 392.8 417.8 410.2 408.4 412.6 398.8 404.8 368.8Os 405.6 443.6 426.8 450.8 409.0 466.6 461.0 455.6 459.6 440.6 449.4 424.6Ir 350.5 391.0 369.9 394.2 343.8 416.6 409.8 402.0 415.9 382.7 392.8 365.2Pt 245.0 285.8 264.6 284.1 224.2 305.9 298.8 294.4 244.1 271.2 280.4 284.2Au 131.1 164.1 157.6 162.7 127.7 168.2 172.5 167.8 158.2 153.1 159.5 174.8ME -14.0 9.5 2.9 16.3 -14.3 25.6 21.6 20.2 14.6 8.9 14.9MAE 15.8 12.9 12.1 19.4 22.2 27.3 22.2 21.9 28.8 17.6 19.6K 3.5 3.5 3.3 3.3 3.2 3.1 3.0 3.2 3.2 3.8 3.8 3.7Rb 2.8 2.9 3.7 3.5 4.7 3.6 2.8 3.3 3.1 3.1 3.1 2.9 Cs 2.0 2.0 1.8 1.9 4.1 2.0 2.0 2.0 2.0 2.1 2.1 2.1ME -0.1 -0.1 0.0 0.0 1.1 0.0 -0.3 -0.1 -0.1 0.1 0.1MAE 0.1 0.1 0.5 0.4 1.4 0.5 0.3 0.3 0.3 0.1 0.1Ca 16.8 17.2 16.8 17.1 20.9 18.2 17.5 17.7 17.6 18.4 18.5 18.4Sr 11.5 12.3 11.4 11.7 17.1 11.8 11.4 11.9 11.1 12.5 12.6 12.4Ba 8.7 9.3 8.4 8.7 11.8 7.7 7.6 8.3 8.1 9.2 9.2 9.3ME -1.0 -0.4 -1.2 -0.9 3.2 -0.8 -1.2 -0.7 -1.1 0.0 0.1MAE 1.0 0.4 1.2 0.9 3.2 0.8 1.2 0.7 1.1 0.1 0.1TME -1.9 18.3 15.8 24.1 5.4 25.6 19.3 23.4 21.5 17.9 21.5TMAE 12.3 19.3 19.7 25.1 23.4 26.7 20.6 24.3 25.6 21.3 23.2Mn 183.2 305.4 299.4 309.1 298.0 311.0 306.6 312.6 315.3 302.6 306.5 90.4La 24.2 26.4 25.8 25.4 28.0 22.8 22.6 23.6 24.2 25.6 24.2 26.6Hg 9.6 36.0 22.4 32.4 19.6 24.4 21.6 24.0 21.2 33.6 38.8 28.2ME 23.9 74.2 67.5 73.9 66.8 71.0 68.5 71.7 71.8 72.2 74.8MAE 37.9 74.3 71.9 74.7 72.5 76.1 75.6 76.5 78.1 72.9 76.4TME 0.2 23.0 20.1 28.3 10.5 29.4 23.4 27.4 25.7 22.4 26.0TMAE 14.5 23.8 24.0 29.3 27.5 30.8 25.1 28.7 30.0 25.6 27.7 B = − V (cid:16) ∂P∂V (cid:17) a = a . (1)Several equations of state (EOS) are available tofit the energy versus volume curve to obtain the bulkmoduli. However, in the present case, we use the Birch-Murnaghan equation of state to fit and obtain the bulkmoduli of alkali, alkaline-earth, and transition metals. Itis well known that determining the bulk modulus posses agreat challenge, in particular for the transition metals .The experimental values along with all the functionalsvalues are presented in TABLE II. The general trend ofthe arrangement of TABLE II is the same as it is done inthe case of the equilibrium shortest distances. The 3 d , 4 d and 5 d band elements are separated out. The values ofalkali metals and alkali earth metals are also shown sep-arately. The Mn, La, and Hg values are also separatedout. In Fig. (2) we also plot the percentage deviation ofall the metals considered in our work. The Fig. (2) isalso arranged according to the data presented in TABLEII.3 d transition metals : The PBE functional workswell for the 3 d transition metals and produces the leastMAE. PBE functional overestimates the bulk moduli forfew cases, whereas, underestimates in results are ob-served for others. However, the performance of the re-vised version of PBE i.e., PBEsol deviates more from theexperimental values and yield MAE 44 .
178 GPa. Re-garding the meta-GGAs, all functionals deviate from theexperimental values more or less. However, MS1 reportsbeing least MAE among the meta-GGA functionals. Weobtain equivalent performance using the TPSS, revTPSS,M06L, MS0, MS2, SCAN, TMTPSS and TM functionals.4 d transition metals : For the 4 d transition metals the TPSS functional performs best with MAE 8 . d metals and provides the MAE 13 . d metals.5 d transition metals : For the 5 d transition metalsthe performance of revTPSS, TPSS, and PBE are foundto be better compared to the others. The M06L under-estimates the bulk moduli for all, except Hf, Ta, and Re.The overall ME obtain from the M06L is found to benegative. The MS0, MS1, MS2, and SCAN functionalsperform equivalently and provide almost equivalent MEand MAE. The performance of TM functional indicatesthat it yields slightly greater values for bulk moduli com-pared to the TMTPSS functional. Comparing the per-formance of SCAN, TMTPSS and TM functionals, boththe TMTPSS and TM show improve performance thanSCAN functional. Alkali metals :
The alkali matters are considered asa “soft-matter” due to the smaller extent of their bulk (a) (b) (c) (d) (e) (f)
FIG. 2. Histograms of relative error in bulk moduli (in %) are presented. The numbering of the figures are as the order of thesolids presented in TABLE II. moduli. The bulk moduli of alkali metals vary only from2 . . . . Alkaline-earth metals :
Like the alkali metals, weobtain equivalent performance of all the functionals foralkaline-earth metals. In this case, the largest MAE is ob-tained from M06L. The performance of revTPSS, MS0,MS1, MS2, and SCAN are almost equivalent. The TPSSfunctional yields the MAE 0 . Other transition metals :
The bulk moduli of Mn,La, and Hg are problematic within (most of) the den-sity functionals. The experimental bulk modulus of Mnis 90 . Overall performances :
To evaluate the overall per-formance of all the functionals we present the ME andMAE with and without considering Mn, La and Hg tran-sition metals. Overall, the PBE functional yields the bestMAE (14 . . C. Cohesive Energies
The Cohesive energies are equivalent to the atomiza-tion energies in the case of the bulk solids. It is expressedas the energy per atom as,
TABLE III: Fixed lattice constant cohesive energies (in eV/atom) of dif-ferent solid structures using PBE, PBEsol, TPSS, revTPSS, M06L, MS0,MS1, MS2, SCAN, TMTPSS and TM functionals. All the finite tem-perature corrected experimental values are collected from references .For elements Mn, La and Hg the temperature corrected values are notavailable and we report only the experimental values taken from refer-ence .Metals PBE PBEsol TPSS revTPSS M06L MS0 MS1 MS2 SCAN TMTPSS TM Expt.Sc 4.20 4.58 4.47 4.58 5.08 4.39 4.31 4.49 4.37 4.69 4.81 3.93Ti 5.40 5.87 5.56 5.77 6.22 5.29 5.23 5.48 5.30 6.01 6.04 4.88V 5.25 5.83 5.51 5.76 6.28 5.09 5.01 5.43 4.96 5.89 5.93 5.34Cr 4.05 4.71 4.24 4.47 4.50 3.49 3.44 3.96 3.26 4.62 4.66 4.15Fe 4.89 5.61 5.32 5.59 5.03 5.16 5.08 5.51 5.13 5.68 5.80 4.32Co 5.07 5.83 5.72 6.04 5.71 5.76 5.67 6.01 5.86 6.19 6.30 4.47Ni 4.68 5.33 5.06 5.43 4.54 5.22 5.11 3.23 5.25 5.62 5.71 4.48Cu 3.48 4.03 3.75 4.09 3.06 3.80 3.71 4.09 3.87 4.32 4.38 3.51Zn 1.10 1.57 1.34 1.61 1.54 1.55 1.46 1.74 1.52 1.71 1.89 1.38ME 0.18 0.77 0.50 0.76 0.61 0.37 0.28 0.39 0.34 0.92 1.01MAE 0.30 0.77 0.51 0.76 0.71 0.57 0.52 0.71 0.62 0.92 1.01Y 4.21 4.60 4.43 4.57 5.06 4.36 4.30 4.49 4.42 4.70 4.81 4.42Zr 6.27 6.78 6.35 6.55 6.80 5.83 5.83 6.04 6.12 6.82 6.84 6.32Nb 6.79 7.47 7.14 7.40 8.63 6.85 6.75 7.07 6.56 7.45 7.49 7.47Mo 6.35 7.18 6.63 6.95 6.85 6.34 6.23 6.61 5.81 6.96 7.03 6.84Tc 6.90 7.85 7.17 7.59 6.53 7.28 7.16 7.57 6.72 7.61 7.78 7.17Ru 6.88 7.87 7.21 7.66 6.25 7.69 7.54 7.79 7.56 7.70 7.85 6.80Rh 5.70 6.66 6.00 6.40 5.40 5.9 5.81 6.18 5.58 6.41 6.55 5.76Pd 3.74 4.47 4.00 4.39 4.17 4.24 4.13 4.46 4.38 4.61 4.71 3.93Ag 2.52 3.08 2.73 3.03 3.24 2.79 2.70 3.10 2.88 3.29 3.35 2.96Cd 0.73 1.16 0.95 1.2 1.33 1.08 1.01 1.34 1.03 1.39 1.49 1.18ME -0.28 0.43 -0.02 0.29 0.14 -0.05 -0.14 0.18 -0.18 0.41 0.51MAE 0.29 0.43 0.18 0.30 0.45 0.34 0.34 0.36 0.42 0.41 0.51Hf 6.48 7.14 6.77 7.08 7.40 6.82 6.73 7.03 6.35 7.05 7.22 6.44Ta 8.58 9.23 8.67 9.03 9.00 8.99 8.85 9.20 8.90 9.53 9.55 8.11W 8.47 9.27 8.83 9.22 9.77 9.31 9.12 9.46 9.09 9.41 9.45 8.83Re 7.79 8.77 8.22 8.65 7.74 8.89 8.68 8.95 8.51 8.71 8.91 8.06Os 8.30 9.36 8.80 9.10 8.23 9.53 9.34 9.53 9.08 9.26 9.49 8.22Ir 7.19 8.27 7.56 8.10 6.97 8.51 8.31 8.43 8.37 8.07 8.27 6.96Pt 5.42 6.27 5.74 6.18 5.81 6.31 6.15 6.42 6.17 6.31 6.46 5.87Au 3.03 3.72 3.27 3.60 3.57 3.59 3.45 3.81 3.55 3.83 3.93 3.83ME -0.13 0.71 0.19 0.58 0.27 0.70 0.54 0.81 0.46 0.73 0.87MAE 0.34 0.74 0.37 0.64 0.43 0.76 0.63 0.82 0.56 0.73 0.87K 0.86 0.92 0.92 0.94 0.66 0.86 0.84 0.88 0.84 0.99 0.99 0.94Rb 0.78 0.84 0.81 0.83 1.30 0.78 0.76 0.80 0.76 0.93 0.92 0.86Cs 0.72 0.78 0.74 0.77 1.35 0.73 0.71 0.76 0.70 0.88 0.88 0.81ME -0.08 -0.02 -0.05 -0.02 0.23 -0.08 -0.10 -0.06 -0.10 0.06 0.06MAE 0.08 0.02 0.05 0.02 0.42 0.08 0.10 0.06 0.10 0.06 0.06Ca 1.91 2.11 2.02 2.08 2.50 2.00 1.96 2.03 2.08 2.17 2.29 1.86Sr 1.61 1.81 1.76 1.83 2.27 1.79 1.74 1.84 1.82 1.96 2.06 1.73Ba 1.88 2.12 2.03 2.11 2.51 2.00 1.95 2.1 2.04 2.24 2.34 1.91ME -0.03 0.18 0.10 0.17 0.59 0.10 0.05 0.16 0.15 0.29 0.40MAE 0.07 0.18 0.10 0.17 0.59 0.10 0.05 0.16 0.15 0.29 0.40Mn 3.80 4.55 3.95 4.06 2.81 3.04 3.01 3.54 2.90 4.13 4.28 2.92La 4.30 4.79 4.51 4.56 5.03 3.83 3.82 4.06 3.72 4.58 4.61 4.47 Hg 0.15 0.54 0.22 0.43 0.55 0.43 0.32 0.64 0.39 0.67 0.75 0.62ME 0.08 0.62 0.22 0.35 0.13 -0.24 -0.29 0.08 -0.33 0.46 0.54MAE 0.51 0.68 0.49 0.47 0.25 0.32 0.35 0.35 0.33 0.46 0.54TME -0.06 0.53 0.19 0.44 0.33 0.22 0.12 0.34 0.11 0.57 0.67TMAE 0.29 0.55 0.31 0.47 0.50 0.45 0.41 0.51 0.44 0.57 0.67 E coh = E atom − E bulk N , (2)where E atom is the atomic energy and E bulk is the bulkenergy of unit cell having N atoms. Predicting the cohe-sive energies of transition metals posses great challengebecause of their “strongly” correlated nature . TABLEIII presents the performance of all the functionals alongwith the experimental values. We arrange TABLE IIIin the same manner as it is done in the case of latticeconstants and bulk moduli. The behavior of all the func-tionals are also plotted in the Fig. 3. The percentagedeviation of all the functionals is shown there. All thecohesive energies are calculated at the equilibrium latticeconstants of the functional.3 d transition metals : The PBE functional predictsthe cohesive energies of 3 d transition metals quite wellcompared to the other GGA and meta-GGA based func-tionals. Though PBE overestimates the cohesive ener-gies for Sc, Ti, Fe, and Co, but overall both the ME andMAE are found to be reasonably well predicted withinPBE functional. The PBEsol overestimates the cohesiveenergies of 3 d transition metals more. Therefore, increasein ME and MAE is observed with the PBEsol functional.Within the meta-GGA functionals, the performance ofMS1 is the best. MS0 quite closely follows the MS1 func-tional. Comparing the SCAN, TMTPSS and TM func-tionals, the SCAN functional performs better comparedto TM based functionals. The revTPSS overestimatesthe TPSS values more and yield more enhanced ME andMAE. The M06L yield the same MAE as it is obtainedfrom MS2.A noticeable observation is that revTPSS behavesclosely as PBEsol. This can be understood from theexplanation given in ref. . In metals, as the d bandstarted filling, the meta-GGA total charge density be-comes the sum of several one-electron orbitals . There-fore, revTPSS becomes PBEsol like. But, the improvedfunctionals like SCAN behaves more closely to PBE forSc, Ti. But the SCAN functional reasonably underesti-mates the cohesive energies of V, and Cr, and overesti-mates for Fe to Zn. A different tendency is observed forTMTPSS and TM functionals. Both overestimates thecohesive energies noticeably.4 d transition metals : Unlike previous observation,the PBE functional underestimates the cohesive energiesof 4 d transition metals. The PBEsol overestimates thecohesive energies of all metals except Nb. In this case,the TPSS predicts the best ME and MAE. The TPSS functional predicts reasonably good cohesive energies forY, Zr, Tc, and Pd. For others, it underestimates oroverestimates the values. In this case, also we observethat the revTPSS follows closely that of the PBEsol re-sults. The M06L functional overall overestimates the co-hesive energies except for few cases. The MS0, MS1 andMS2 functionals perform almost equivalently. The SCANfunctional are also closely following the “MS” functionals.We observe that the TMTPSS and TM functionals showPBEsol like tendency. Here, the TM functional overesti-mates the cohesive energies more which are produced byTMTPSS.5 d transition metals : For 5 d transition metals also,the PBE performs quite reasonably. PBE overestimatesthe cohesive energies for Ta, and Ir but performs wellfor Hf, W, Re, Os, Pt, and Au. The PBEsol shows thesame tendency as it is obtained in the cases of 3 d and4 d elements. The TPSS functional enhances the PBEresults, therefore, the increase in the MAE is observedcompared to the PBE results. Here, the revTPSS alsofollows closely the PBEsol results. The M06L functionalsoverestimate the half d fill transition metals but performwell for largely filled d transition metals. In this case alsothe “MS” functionals perform equivalently, though theresults of MS2 are found to be more enhanced than MS0and MS1. In this case, the SCAN functional performsquite well compared to the TMTPSS and TM function-als. Both the TMTPSS and TM functional overestimatethe results of all the 5 d metals, but the overestimationtendency is lesser than the SCAN functional. Alkali metals :
The alkali metals are often includedin the benchmarking calculations for the different func-tionals for the semiconductor in predicting the cohesiveenergies. All the GGAs and meta-GGAs (except M06L)perform quite reasonably for the cohesive energies of al-kali metals. The PBEsol and revTPSS show the bestperformance with MAE 0 .
02 eV/atom. In this case,the “MS” functionals and SCAN perform equivalently.The TMTPSS and TM also predict the cohesive energieswhich are quite close to the experimental results.
Alkaline-earth metals :
For alkaline-earth metals,PBE and TPSS perform quite remarkably. The PBEand TPSS show opposite tendency. PBE shows a smalldegree of underestimation (except Ca), whereas, TPSSshows a small degree of overestimation. The same behav-ior is observed for the case of PBEsol and revTPSS. Inthis case, also the “MS” functionals and SCAN performequivalently. We observe a noticeable amount of overes-timation in the performance of TMTPSS and TM func-tionals. The performance of SCAN functional is quite1 (a) (b) (c) (d) (e) (f)
FIG. 3. Histograms of relative error in cohesive energies (in %) are presented. The numbering of the figures are as the orderof the solids presented in TABLE III. better compared to the TM based functionals.
Other transition metals :
The cohesive energy ofMn is largely overestimated within all the GGA andmeta-GGA functionals except MS0, MS1 and SCANfunctionals. Overall the M06L, “MS” and SCAN func-tionals predict the best MAE for all these metals. TheTM functional predicts very well cohesive energies for Laand Hg. As usual the TM functional enhance the error ofTMTPSS more. Unlike other transition metals, in thiscase, we observe that the revTPSS and PBEsol resultsdeviate from each other.
Overall performances :
Overall, the PBE performsquite reasonably for the cohesive energies. Within themeta-GGA functionals, the TPSS performs quite welland predicts the overall best MAE within the meta-GGAs. The PBEsol and revTPSS perform equivalently.We also obtain the same degree of overall ME and MAEusing the M06L, MS0, MS1, MS2 and SCAN functionals.The TM functional deviates more from accuracy in pre-dicting the cohesive energies of alkali, alkaline-earth, andtransition metals. The errors obtained from the TM func-tional are often more enhanced than that of the TMTPSSfunctional.
IV. CHALLENGES OF ADVANCE META-GGAFUNCTIONALS
In TABLE IV we present the overall statistical anal-ysis and ranking of each functional best on their perfor-mance. In Fig.(4), we present the mean absolute per-centage of the individual functionals excluding the Mn,La and Hg. Based on these analyses it is indicative thatsimultaneously predicting both the structural and ener-getic properties of the transition metals within GGAs andmeta-GGAs functionals possess great challenge. Meta-GGA functionals belong to the third rung of the Ja-cob’s ladder. Within the meta-GGAs, the SCAN andTM functionals considered as one of the most advancedfunctionals but the performance of those functionals donot show any improvement over the PBE functionals forthe bulk moduli and cohesive energies. For the 4 d and 5 d transition metals, alkali metals and alkaline-earth metalsthe SCAN and TM functionals improve the performancebut for the 3 d transition metals PBE performs better.The SCAN, M06L and TM functionals capture mid-rangevdW interactions, which accounts improve performancefor some systems but remain difficult for other systems.Especially, for alkali metals, the TMTPSS and TM func-tionals show improvement over PBE, M06L, and SCANfunctionals. In the present study, we do not include anylong-range corrected vdW interactions into the function-als form. The inclusion of the long-range corrected vdW2 FIG. 4. Histograms of mean absolute percentage error (MAPE) in the shortest distances, bulk moduli and cohesive energiesare presented. In the MAPE we excluded the Mn, La and Hg.TABLE IV. Statistical analysis and ranking of different functionals in the Shortest Distances, Cohesive Energies and BulkModuli. PBE PBEsol TPSS revTPSS M06L MS0 MS1 MS2 SCAN TMTPSS TMTME 3.8 -0.6 3.1 0.8 -1.7 2.6 2.7 1.2 1.2 1.1 -0.1Shortest TMAE 5.4 4.4 5.1 4.3 8.1 6.1 5.9 4.6 4.6 3.9 3.7Distances TMAPE 1.9 1.6 1.8 1.5 2.9 2.2 2.1 1.6 1.6 1.4 1.3rank 8 4 7 3 11 10 9 4 4 2 1TME -1.9 18.3 15.8 24.1 5.4 25.6 19.3 23.4 21.5 17.9 21.5Bulk TMAE 12.3 19.3 19.7 25.1 23.4 26.7 20.6 24.3 25.6 21.3 23.2Moduli TMAPE 7.6 11.8 12.1 15.4 14.4 16.4 12.6 14.9 15.7 13.1 14.2rank 1 2 3 9 7 11 4 8 10 5 6TME -0.08 0.53 0.18 0.45 0.35 0.26 0.16 0.37 0.15 0.58 0.68Cohesive TMAE 0.26 0.54 0.29 0.47 0.53 0.46 0.41 0.52 0.45 0.59 0.68Energies TMAPE 5.58 11.59 6.22 10.09 11.37 9.87 8.8 11.16 9.66 12.66 14.59rank 1 9 2 6 8 5 3 7 4 10 11Average rank 3.3 5 4 6 8.7 8.7 5.3 6.3 6 5.7 6 interactions may improve the functional performance byincluding the long-range electron correlation effect.
V. CONCLUSIONS
In this paper, we assess the performance of the recentmeta-GGA density functionals along with the popularlyused GGA based functionals for the lattice constants,bulk moduli and cohesive energies of alkali, alkaline-earth, and transition metals. The present paper is ar-ranged by addressing the performance of the 3 d , 4 d , 5 d , alkali, alkaline-earth, and other transition metals. Dueto the complicated structure of the Mn, La, and Hg, wediscuss these three materials separately. Special atten-tion has been paid to the performance on the recentlyproposed SCAN, TMTPSS and TM functionals. Basedon these analysis, benchmark calculations and the levelof deficiencies of all the functionals one conclude that:(i) For the equilibrium shortest distances of the 3 d transition metals PBE results reasonably good comparedto all other GGAs and meta-GGAs. For 4 d and 5 d transition metals, alkali, and alkaline-earth metals thePBE results are too large except few cases. The PBEsol3and revTPSS follow essentially the identical results. Thelargest error is obtained from the M06L functional. Theperformance of MS0 and MS1 are identical, whereas, MS2and SCAN closely follow each other in most of the cases.We obtain the best performance with the TM functional.The TM perform even better than SCAN in most of thecases except alkaline-earth metals. We found the perfor-mance of TMTPSS is quite close that of the TM func-tional for 3 d transition metals.(ii) Regarding the bulk moduli, the PBE functionaloutperforms all other semilocal GGAs and meta-GGAsfunctionals in every case. Overall PBE underestimatesthe bulk moduli of 4 d transition metals, 5 d transitionmetals, alkali metals and alkaline-earth metals. For 3 d metals the PBE seems to overestimate the results. Thebulk moduli of Mn is a difficult case for all the GGAsand meta-GGAs functionals. In this case, the revTPSSresults also follows very closely the PBEsol results. Theperformance of all other meta-GGAs vary noticeablyfrom the PBE results. Concerning the performance ofTMTPSS and TM functional performance, both performwell then the SCAN functional.(iii) In predicting the cohesive energies of all the met- als, PBE outperforms all other functionals. Only for thealkali metals, the PBE results deviate from the experi-mental results. The revTPSS closely follow the PBEsolin all the cases. Overall consideration shows that the TMpredicts the largest mean absolute error. Though in thecase of the alkali metals, the TMTPSS and TM function-als perform quite well compared to the other cases, butoverall both the functionals overestimate the cohesive en-ergies. Overall, comparison of SCAN, TMTPSS and TMshow that the SCAN functional performs better compareto the TMTPSS and TM functional in estimation the co-hesive energies. VI. ACKNOWLEDGEMENT
S. J. would like to acknowledge the financial supportfrom the Department of Atomic Energy, Governmentof India. K.S. would like to acknowledge the financialsupport from the Department of Science and Technol-ogy, Government of India, during his summer intern inNISER. ∗ [email protected] W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965). J. P. Perdew and A. Zunger. Phys. Rev. B, J. P. Perdew and Y. Wang, Phys. Rev. B , 8800 (1986). A. D. Becke, Phys. Rev. A , 3098 (1988). C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B , 785(1988). J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson,M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev.B , 6671 (1992). A. D. Becke, J. Chem. Phys. , 1040 (1996). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996). R. Armiento and A. E. Mattsson, Phys. Rev. B , 085108(2005). Z. Wu and R. E. Cohen, Phys. Rev. B , 235116 (2006). Y Zhao, and D. G. Truhlar, J. Chem. Phys. 128, 184109(2008). 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. , 136406 (2008). L. A. Constantin, J. P. Perdew, and J. M. Pitarke, Phys.Rev. B 79, 075126 (2009). E. Fabiano, L. A. Constantin, and F. Della Sala, Phys.Rev. B 82, 113104 (2010). E. Fabiano, L. A. Constantin, and F. Della Sala, J. Chem.Theory Comput., 7 (11), pp 35483559 (2011). L. A. Constantin, A. Terentjevs, F. Della Sala, P. Cortona,and E. Fabiano, Phys. Rev. B 93, 045126 (2016). A. D. Becke and M. R. Roussel, Phys. Rev. A , 3761(1989). T. V. Voorhis and G. E. Scuseria, J. Chem. Phys. , 400(1998). Y. Zhao and D. G. Truhlar, J. Chem. Phys. , 194101(2006). J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria,Phys. Rev. Lett. , 146401 (2003). J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Con-stantin, and J. Sun, Phys. Rev. Lett. , 026403 (2009). L. A. Constantin, E. Fabiano, and F. Della Sala, J. Chem.Theory Comput., 9 (5), pp 22562263 (2013). J. Sun, B. Xiao, and A. Ruzsinszky, J. Chem. Phys. 137,051101 (2012). J. Sun, R. Haunschild, B. Xiao, I. W. Bulik, G. E. Scuseria,and J. P. Perdew, J. Chem. Phys. 138, 044113 (2013). J. Sun, J. P. Perdew, and A. Ruzsinszky, Proceedings ofthe National Academy of Sciences of the United States ofAmerica, 112 (3) 685-689 (2015) J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. , 036402 (2015). J. Tao and Y. Mo, Phys. Rev. Lett. , 073001 (2016). Y. Mo, G. Tian, R. Car, V. N. Staroverov, G. E. Scuseria,and J. Tao, Phys. Rev. B 95, 035118 (2017). Y. Mo, G. Tian and J. Tao, Chem. Phys. Lett. , 38 − A. E. Mattsson, R. Armiento, J. Paier, G. Kresse, J. M.Wills, and T. R. Mattsson, J. Chem. Phys. , 084714(2008). J. Sun, M. Marsman, A. Ruzsinszky, G. Kresse, and J. P.Perdew Phys. Rev. B , 121410(R). J. Sun, M. Marsman, G. I. Csonka, A. Ruzsinszky, P. Hao,Y. S. Kim, G. Kresse, and J. P. Perdew, Phys. Rev. B ,035117 (2011). A. Patra, J. E. Bates, J. Sun, and J. P. Perdew, Proceed-ings of the National Academy of Sciences, 114, 44, E9188-E9196 (2017). P. Janthon, S. M. Kozlov, F. Vines, J. Limtrakul, and F.Illas, J. Chem. Theory Comput. 9, 1631-1640 (2013). P. Janthon, S. (Andy) Luo, S. M. Kozlov, F. Vines, J.Limtrakul, D. G. Truhlar,and F. Illas, J. Chem. TheoryComput. 10, 3832-3839 (2014). P. Haas, F. Tran, and P. Blaha, Phys. Rev. B , 085104(2009). F. Tran, J. Stelzl, and P. Blaha, J. Chem. Phys. ,204120 (2016). G. I. Csonka, J. P. Perdew, A. Ruzsinszky, P. H. T.Philipsen, S. Lebgue, J. Paier, O. A. Vydrov, and J. G.ngyn, Phys. Rev. B 79, 155107 (2009). L. Schimka, R. Gaudoin, J. Klimes, M. Marsman, and G.Kresse, Phys Rev B 87, 214102 (2013). P. Hao, Y. Fang, J. Sun, G. I. Csonka, P. H. T. Philipsen,and J. P. Perdew, Phys. Rev. B , 014111 (2012). G. Zhang, A. M. Reilly, A. Tkatchenko, and M. Scheffler,New J. Phys. 20 063020 (2018). S. Jana, A. Patra, and P. Samal, (to be appeared), J.Chem. Phys. (2018). P. E. Bl¨ochl, Phys. Rev. B, 50:17953, (1994). G. Kresse and D. Joubert, Phys. Rev. 59 , 1758 (1999). G. Kresse and J. Hafner, Phys. Rev. B 47 , 558 (1993);ibid. 49 , 14 251 (1994). G. Kresse and J. FurthmAller, Comput. Mat. Sci. 6 , 15(1996). G. Kresse and J. FurthmAller, Phys. Rev. B 54 , 11 169(1996). G. Kresse and J. Hafner, J. Phys.: Condens. Matt. , 8245(1994). J. Tao, J. P. Perdew, and A. Ruzsinszky, Phys. Rev. B 81,233102 (2010). F. D. Murnaghan, Proc. Natl. Acad. Sci. U.S.A. , 244(1944). A. B. Alchagirov, J. P. Perdew, J. C. Boettger, R. C. Al-bers, and C. Fiolhais, Phys. Rev. B 63, 224115 (2001). A. B. Alchagirov, J. P. Perdew, J. C. Boettger, R. C. Al-bers, and C. Fiolhais, Phys. Rev. B 67, 026103 (2003).53