Assessment of the Performance of DFT Functionals Using Off-Diagonal Hypervirial Relationships
Francesco Ferdinando Summa, Guglielmo Monaco, Riccardo Zanasi, Paolo Lazzeretti
AAssessment of the Performance of DFT FunctionalsUsing Off-Diagonal Hypervirial Relationships
Francesco F. Summa, Guglielmo Monaco, Riccardo Zanasi ∗ Dipartimento di Chimica e Biologia “A. Zambelli”, Universit`a degli studi di Salerno, viaGiovanni Paolo II 132, Fisciano 84084, SA, Italy
Lazzeretti Paolo
Istituto di Struttura della Materia, CNR, via del Fosso del Cavaliere 100, 00133 Roma, Italy
Abstract
Off-diagonal hypervirial relationships, combined with quantum mechanical sumrules of charge-current conservation, offer a way for testing electronic excited-state transition energies and moments, which does not need any external refer-ence. A number of fundamental relations were recast into absolute deviationsfrom zero, which have been used to assess the performance of some popular DFTfunctionals. Extended TD-DFT calculations have been carried out for a poolof molecules chosen to the purpose, adopting a large basis set to ensure highquality results. A general agreement with previous benchmarks is observed.
Keywords:
Transition Moments, TRK Sum Rules, MagnetizabilityTraslational Invariance, Charge-Current Conservation, ElectricDipole Polarizability, Optical Activity
1. Introduction
Owing to the impressive use of density functional theory (DFT) in last yearsfor quantum chemistry calculations on many-electron systems, the quality as-sessment of DFT functionals has become a rather popular and important ac-tivity [1, 2, 3, 4, 5, 6]. Unlike what has been done so far, here we present an ∗ Corresponding author: Riccardo Zanasi,
Email Address: [email protected]
Preprint submitted to Chemical Physics Letters February 16, 2021 a r X i v : . [ phy s i c s . c h e m - ph ] F e b ssessment of the performance of DFT functionals on the basis of excited-statestransition moments and energies, which could reveal useful in the choice of func-tionals for the calculation of second-order molecular properties arising from theinteraction with radiation.DFT is usually considered as a ground-state theory. However, as provenby Hohenberg and Kohn (for systems with a nondegenerate ground state) theground-state electron probability density ρ ( r ) determines the external potentialand the number of electrons [7, 8]. Once the external potential and the numberof electrons are specified, the electronic wave functions and allowed energiesof the molecule are determined as the solutions of the electronic Schr¨odingerequation. Hence, ground-state wave function and energy as well as all excited-state wave functions and energies are determined by ρ ( r ) [9]. However, theextraction of such information remained unsolved until the presentation of theformally exact formulation of DFT for time-dependent (TD) systems given byRunge and Gross [10]. Actually, electronic excited-states reveal themselves fromthe interaction with radiation, and the development of the linear response TD-DFT via approximated functionals has provided direct access to the excited-state information searched for [11, 12, 13, 14].The accuracy of a calculation of molecular properties depending on the elec-tronic response to a perturbation could be established via direct comparisonwith corresponding experimental data. However, this is not always an easy taskto accomplish because the experimental data can be affected by effects, suchas intermolecular interactions, which can be poorly described by the theoret-ical model, and the computations themselves can carry an uncertainty due tofiniteness of the basis set. As far as it concerns the assessment of DFT predic-tions, the comparison with higher level calculations is often employed [15, 16], apractice which should require the adoption of difficult reference calculations forobtaining a representative and robust grading of the methods. The fundamentalneed for instruments assessing the quality of a theoretical prediction a priori ,i.e., without external references, is evident. For testing excited-states informa-tion resulting from a TD-DFT calculation, these instruments can be supplied2y some internal quantum mechanical conditions, such as the fulfillment of theoff-diagonal hypervirial relationships, Thomas-Reiche-Kuhn sum rules for theoscillatory strengths [17, 18, 19] and other sum rules for the charge-currentconservation [20], which are equivalent to the translational invariance of themagnetizability [21].In the next section we will focus on the definition of a number of absolute de-viations that do not need any external reference. Then, results will be presentedfor a selection of functionals chosen among the most popular for the calculationof molecular response properties. Testing molecules of moderate size have beenchosen to allow accurate calculations of the full set of TD-DFT excited statesrequired for the linear response, adopting a fairly large set of basis functions.
2. Method
Let’s consider a molecule with n electrons of mass m e, charge − e , positions r k , canonical and angular momenta ˆ p k and ˆ l k = r k × ˆ p k , respectively. Let’sconsider also the origin of the coordinate system in the center of positive charges.Eigenvalues of the Born-Oppenheimer Hamiltonian ˆ H are E and E j , which areenergies of the stationary states Ψ and Ψ j , respectively (subscript 0 denotingthe ground reference state and j any excited state). Capital letters are used todenote total electron operators:ˆ R α = (cid:88) k r kα , ˆ P α = (cid:88) k ˆ p kα , ˆ L α = (cid:88) k ˆ l kα , (1)where ˆ P α = i m e¯ h (cid:104) ˆ H , ˆ R α (cid:105) . (2)Transition moments of ˆ R α and ˆ P α from ground to excited stationary statesare related by the off-diagonal hypervirial relation (cid:68) (cid:12)(cid:12)(cid:12) ˆ R α (cid:12)(cid:12)(cid:12) j (cid:69) = i m e ω − j (cid:68) (cid:12)(cid:12)(cid:12) ˆ P α (cid:12)(cid:12)(cid:12) j (cid:69) , (3)3here ω j = h ( E j − E ), which is fulfilled if the state functions are exact eigenfunctions of a model Hamiltonian and satisfy the hypervirial theorem forthe position operator [22]. The TD Hartree-Fock (TD-HF) method fulfils eq.(3)in the limit of a complete basis set calculation [23, 24] and it is generally assumedthat TD-DFT does the same, see for example Ref. [16]. Indeed, there is a verywell know connection among current density conservation, gauge invariance ofmagnetic properties and hypervirial relationship as proven by Ghosh and Dharafor Kohn-Sham-like approach [25, 26]. Excitation energies, as well as transitionmatrix elements of the position, linear and angular momentum operators, aremain results of any TD calculation and it would appear inappropriate to havelarge deviations among the oscillator and rotational strengths determined in theso-called length, velocity or mixed gauges (the latter applies only to oscillatorstrengths, of course).The Thomas-Reiche-Kuhn (TRK) [17, 18, 19] sum rule for the position op-erator in tensor notation is [27, 28]( ˆ R α , ˆ R β ) = m e¯ h (cid:88) j (cid:48) ω j (cid:60) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ R α (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ R β (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) = nδ αβ . (4)Using eq.(3) one gets alternative mixed velocity-length and velocity-velocityformulations:( ˆ P α , ˆ R β ) = 1¯ h (cid:88) j (cid:48) (cid:61) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ P α (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ R β (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) = nδ αβ , (5)( ˆ P α , ˆ P β ) − = 1 m e¯ h (cid:88) j (cid:48) ω j (cid:60) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ P α (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ P β (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) = nδ αβ , (6)which can be briefly summarized as nδ αβ = ( ˆ P α , ˆ P β ) − = ( ˆ P α , ˆ R β ) = ( ˆ R α , ˆ R β ) . (7)Additional sum rules for the translational invariance of the magnetizabiliytensor, which are intimately connected to the charge-current conservation con-4ition of Sambe [20], are [21, 27, 28] (cid:68) (cid:12)(cid:12)(cid:12) ˆ R β (cid:12)(cid:12)(cid:12) (cid:69) (cid:15) γαβ = (cid:16) ˆ P γ , ˆ L α (cid:17) − = − (cid:16) ˆ P α , ˆ L γ (cid:17) − = (cid:16) ˆ R γ , ˆ L α (cid:17) = − (cid:16) ˆ L γ , ˆ R α (cid:17) , (8)where (cid:15) γαβ is the Levi-Civita skew-symmetric tensor and( ˆ R γ , ˆ L α ) = − h (cid:88) j (cid:48) (cid:61) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ R γ (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ L α (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) , (9)( ˆ P γ , ˆ L α ) − = 1 m e¯ h (cid:88) j (cid:48) ω j (cid:60) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ P γ (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ L α (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) . (10)Allowing for the off-diagonal hypervirial relation (3), the frequency depen-dent electric dipole polarizability can be written in three formalisms [28, 29], ina way similar to that used for TRK sum rules (4)-(6). They are α RRαβ ( ω ) = e ¯ h (cid:88) j (cid:48) ω j ω j − ω (cid:60) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ R α (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ R β (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) , (11) α P Rαβ ( ω ) = e m e¯ h (cid:88) j (cid:48) ω j − ω (cid:61) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ P α (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ R β (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) , (12) α P Pαβ ( ω ) = e m e¯ h (cid:88) j (cid:48) ω j ( ω j − ω ) (cid:60) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ P α (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ P β (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) , (13)whilst the frequency dependent optical activity tensor in two formalisms is [28,29] ˆ κ RLαβ ( ω ) = − e m e¯ h (cid:88) j (cid:48) ω j − ω (cid:61) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ R α (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ L β (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) , (14)ˆ κ P Lαβ ( ω ) = e m e¯ h (cid:88) j (cid:48) ω j ( ω j − ω ) (cid:60) (cid:16)(cid:68) (cid:12)(cid:12)(cid:12) ˆ P α (cid:12)(cid:12)(cid:12) j (cid:69) (cid:68) j (cid:12)(cid:12)(cid:12) ˆ L β (cid:12)(cid:12)(cid:12) (cid:69)(cid:17) . (15)A direct measure of the ability of any density functional to provide a gooddescription of excited states can be obtained adopting self-consistent absolutedeviations defined by means of the above equations without resorting to any5xternal references. Considering first eq. (3), we define the absolute deviationfor the off-diagonal hypervirial relation, which collects at the same time allvector components and computed electronic transitions, as (atomic units)AD HV = (cid:88) α (cid:88) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:68) (cid:12)(cid:12)(cid:12) ˆ R α (cid:12)(cid:12)(cid:12) j (cid:69) − (cid:104) |∇ α | j (cid:105) E j − E (cid:12)(cid:12)(cid:12)(cid:12) , (16)which should be as small as possible for a precise calculation. Moreover, fromeq. (7), we define the three following absolute deviations testing the TRK sumrules AD RR = (cid:88) α (cid:88) β | ( R α , R β ) − nδ αβ | , (17)AD PR = (cid:88) α (cid:88) β | ( P α , R β ) − nδ αβ | , (18)AD PP = (cid:88) α (cid:88) β | ( P α , P β ) − − nδ αβ | . (19)Once again they should be as small as possible for accurate calculations. Con-dition (8) provides two further absolute deviationsAD RL = (cid:88) γ (cid:88) α (cid:12)(cid:12)(cid:12) ( R γ , L α ) − − (cid:15) γαβ (cid:68) (cid:12)(cid:12)(cid:12) ˆ R β (cid:12)(cid:12)(cid:12) (cid:69)(cid:12)(cid:12)(cid:12) , (20)AD PL = (cid:88) γ (cid:88) α (cid:12)(cid:12)(cid:12) ( P γ , L α ) − − (cid:15) γαβ (cid:68) (cid:12)(cid:12)(cid:12) ˆ R β (cid:12)(cid:12)(cid:12) (cid:69)(cid:12)(cid:12)(cid:12) . (21)A slightly different way to assess the off-diagonal hyperviral relation (3) isto check the equivalence of the computed electric dipole polarizability in thevarious formalisms provided by eqs. (11)-(13). This can be done defining thefollowing absolute deviationAD α = (cid:88) β (cid:88) γ (cid:0)(cid:12)(cid:12) α RRβγ − α P Rβγ (cid:12)(cid:12) + (cid:12)(cid:12) α RRβγ − α P Pβγ (cid:12)(cid:12) + (cid:12)(cid:12) α P Rβγ − α P Pβγ (cid:12)(cid:12)(cid:1) . (22)In case of chiral molecules also the optical activity tensor calculated using eqs.614) and (15) can be used to the purpose by means ofAD ˆ κ = (cid:88) β (cid:88) γ (cid:12)(cid:12) ˆ κ RLβγ − ˆ κ P Lβγ (cid:12)(cid:12) . (23)Of course, also AD α and AD ˆ κ should be as small as possible.
3. Calculation details
Nineteen DFT functionals have been tested on the basis of the absolutedeviations defined in the previous section. They are: APFD [30], B3LYP [31,32, 33], B3P86 [31, 34], B3PW91 [31, 35, 36, 37], B97-2 [38], B97D3 [39], B97D[40], B98 [41], BLYP [42, 32, 33], CAM-B3LYP [43], SVWN [44, 45] (indicatedas LSDA for our purposes), MN12-SX [46], M06 [47], M06-2X [47], PBEP86[48, 49, 34], BHandHLYP [50], and LC-BLYP, LC-B97D, LC-BP86 [51].These functionals have been chosen considering the analysis by Medvedev etal [1], selecting some commonly used among those “yielding the best densities”(APFD, B3PW91, B98, BHandHLYP, B3P86, B972 and B3LYP), other amongthose “yielding the worst densities” [52] (SVWN, M06-2X, M06 and MN12-SX) and some other providing densities of intermediate quality (CAM-B3LYP,PBEP86 and BLYP). Two more functionals (B97D and B97D3) came from ourprevious experiences, and the three long-range corrected (LC) functionals havebeen included because long-range correction is a relevant correction in DFT asit helps providing good excitation energies [53].Calculations have been carried out for a set of molecules selected accordingto the following criteria: • the molecular size should not limit the calculation accuracy, which mustbe as large as possible in order to obtain reliable estimates of deviations(16)-(23); • all having the same number of electrons for a better comparison of theTRK sum rules in the different formalisms;7 all having the possibility to be fixed in a chiral conformation with a nonvanishing dipole moment.Accordingly, we have considered N = 6 isoelectronic molecules: CH -CH , CH -NH , CH -OH, NH -NH , NH -OH, HO-OH, with n = 18 electrons. Takinginto account the very large basis set employed for the calculations, see later,the chosen molecules fulfill the above criteria and appear well suited for thepurposes of the present investigation.For each functional F and for each molecule M, absolute deviations will beindicated as AD FX (M), where X stands for one of the testing conditions among:HV, RR, PR, PP, RL, PL, α , and ˆ κ . Mean absolute deviations have beencomputed for each DFT functional and each testing condition averaging theresults obtained for the N molecules and normalizing to the HF resultsMAD FX = N (cid:80) M AD FX (M)MAD HFX (24)MAD
HFX = 1 N (cid:88) M AD HFX (M) . Molecular geometries have been determined as follows. For each functional,internal coordinates have been optimized employing the largest pcSseg-4 of theJensen’s set [54], freezing the internal torsional angle of each molecule at the val-ues reported in Table 1. A total of 120 (19 functionals + HF times 6 molecules)geometries have been determined, which are reported in the Supporting Ma-terial, see Tables S1-S20. Torsional angles have been determined separately,taking the value that maximizes the largest component of the optical activitytensor of each molecule, along a rotation around the single bond connectingthe two heaviest atoms. This latter determination has been done adopting themuch smaller pcSseg-2 basis set [54] and using the B97-2 functional [38] for allmolecules.TD-DFT calculations of transition matrix elements (cid:68) (cid:12)(cid:12)(cid:12) ˆ R α (cid:12)(cid:12)(cid:12) j (cid:69) , (cid:68) (cid:12)(cid:12)(cid:12) ˆ P α (cid:12)(cid:12)(cid:12) j (cid:69) , (cid:68) (cid:12)(cid:12)(cid:12) ˆ L α (cid:12)(cid:12)(cid:12) j (cid:69) and transition energies ( E j − E ), for j = 1 , , . . . , NTRS, have beenperformed adopting the same large pcSseg-4 basis set of the geometry optimiza-8 able 1: Torsional angles frozen during geometry optimization and number of calculatedtransitions NTRS for the molecules here considered. CH -CH CH -NH CH -OH NH -NH NH -OH HO-OH θ ◦ ◦ ◦ ◦ ◦ ◦† NTRS 6624 5958 5265 5283 4599 3906 † Average value, since the molecular geometry has been fully optimized every time. tion, by means of the Gaussian 16 program package [55], using the TD=(full,sos)keyword. The pcSseg-4 basis set provides high quality molecular propertiesnearly coincident with the basis set limit, as documented previously [56]. De-spite the small size of the molecules, a fairly large amount of computer resourcesbecome necessary to accomplish the full set of calculations, much more thanwhat is normally available on a PC.Both the electric dipole polarizability and the optical activity tensors havebeen evaluated for the static case ω = 0.
4. Results and discussion
The full set of results for all molecules is collected in the Supporting Material.The discussion here is focused on a number of mean absolute deviations, whichrepresent the essential result of the analysis.Let us first comment on that none of the DFT functionals here consideredprovide any AD’s, except 6 over a total of 912 examined, less than those ob-tained at the HF level.[57] Anyway, this result cannot be regarded as an allegedsuperiority of the HF method (although this happens for some properties[6])because it depends on the fulfillment of the off-diagonal hypervirial relation (3)for the TD-HF state functions in the complete basis set limit [24]. We exploitthis in eq.(25) to put all the results on the same scale in order to make dataaggregations.Mean absolute deviations for the off-diagonal hypervirial relation, i.e., MAD
FHV ,calculated averaging the data in Table S21, are compared in Fig. 1. As can beobserved, the LC-functionals, B3P86 and BHandHLYP provide the smallest de-viations, whilst the Minnesota functionals M06, M06-2X, and MN12-SX give9 igure 1: Absolute deviations calculated according to eq. 16 and averaged over the molecularset. (cid:16) MAD
FRR + MAD
FPR + MAD
FPP (cid:17) shown in Fig. 2.
Figure 2: Normalized sum of the mean absolute deviations relative to the TRK sum rule inthe various formalisms. M06, M06-2X, and MN12-SX reach a MAD as high as 14, 36, and 34with respect to the HF, respectively.
A similar behavior can be observed also in the case of the sum rule forthe translational invariance of magnetizability. Once again, albeit to a lesser11xtent, the PL formalism shows large deviations for M06, M06-2X, and MN12-SX functionals, see Table S26, which are not shown by the RL formalism, seeTable S25, confirming the sensitivity of these functionals to the transition ener-gies that appear in eq.(10). The aggregated data given as the normalized sum (cid:16) MAD
FRL + MAD
FPL (cid:17) shown in Fig. 3, confirms the functional performanceof the hypervirial and TRK analysis.
Figure 3: Normalized sum of the mean absolute deviations relative to the translation invari-ance of magnetizability in the two formalisms. M06, M06-2X, and MN12-SX reach a MAD ashigh as 5, 3, and 8 with respect to the HF, respectively.
The absolute deviations that measure the equivalence of the static electricdipole polarizability and optical activity tensor in the various formalisms, re-ported in Tables S27 and S28, and relative Figs. S7 and S8, do not change thepicture.Therefore, to best appreciate the uniformity of the results, we have puttogether all the MAD’s for the eight criteria in the same histogram chart shownin Fig. 4. As it can be seen, the functional performance turns out to be exactly12 igure 4: Mean absolute deviations for all criteria. able 2: Functional ranking based on the grading index values calculated via eq. (25). Therung of Jacob’s ladder[58] is also indicated as LDA (Local Density Approximation), GGA(Generalised Gradient Approximation), mGGA (meta-GGA), hGGA (hybrid-GGA). Functional Rung Index MBSPL a LDFS b HF Ab Initio 1LC-B97D hGGA 1.228LC-BP86 hGGA 1.365B3P86 hGGA 1.517 10BHandHLYP hGGA 1.643 6 1LC-BLYP hGGA 1.648B3PW91 hGGA 1.651 3APFD hGGA 1.676 2B97-2 hGGA 1.687 17 29B98 hGGA 1.690 5CAM-B3LYP hGGA 1.793 48B3LYP hGGA 1.876 55 16LSDA LDA 2.133 109 42B97D GGA 2.164B97D3 GGA 2.170PBEP86 GGA 2.298 70BLYP GGA 2.478 92 31M06 hmGGA 9.921 121 51MN12-SX hmGGA 16.579 128M06-2X hmGGA 18.770 114 45 a Electron density-based normalized error rank over 128 functionals fromMedvedev et al. [1]. b Magnetizability-based mean absolute deviation rank over 51 functionals fromLehtola et al. [5].the same regardless of the MAD. Encouraged by this rather nice result, we havedecided to work out a unique index for ranking the functionals here considered,summing all together the mean absolute deviations asI = 18 (cid:88)
X MAD FX . (25)The index is shown in Table 2 and the relative histogram chart constitutes thegraphical abstract of the paper.According to this result, the LC-B97D, LC-BP86, B3P86 and BHandHLYPfunctionals turn out to be good choices among the functionals here considered14or the calculation of the linear response properties to external electromagneticperturbations. A fairly good agreement with previous analysis, reported byMedvedev et al. [1] and Lehtola et al. [5], can be appreciated, as we have fa-cilitated adding more columns to Table 2, where the rank attributed by thesetwo different analyses is reported. Since Lethola et al. have considered mag-netizability as the benchmarking property, which is strictly connected with ourRL and PL testing conditions, we remark the good performance reported bythem for BHandHLYP and the poor performance of the Minnesota functionalsfamily, which is consistent with our findings and those in Ref. [16]. A lookat the second column of Table 2 shows that the worst-performing functionalsare the only ones of the meta-GGA family, including a kinetic energy densitycorrection. This kinetic energy correction is not gauge-invariant and to copewith the presence of the electromagnetic field it should be corrected with thecurrent density [59, 6]. Likely, the absence of gauge invariance is responsible forthe poor performance of the Minnesota functionals reported here.
5. Conclusions
A new independent DFT functional assessment has been proposed, whichdoes not need any external reference. Among the functionals taken into account,LC-B97D, LC-BP86, B3P86 and BHandHLYP turn out to be good choices forthe calculation of properties which require access to electronic excited-stateinformation.In agreement with a global ranking index I, which summarizes all the meanabsolute deviations deduced from off-diagonal hypervirial relationships com-bined with quantum mechanical sum rules of charge and current conservation,DFT functionals can be separated into three groups according to: i) 0 < I < < I < .
5, to be used with somecaution; iii) I (cid:29)
3, not recommended for the calculation of molecular propertiesresulting from an interaction with external fields.Eventually, we would like to emphasize that devising approximate function-15ls which reduce off-diagonal hypervirial relationship deviations could indicateone of the possible ways to improve functionals for specific task as the calculationof second order electromagnetic molecular properties.
Declaration of Competing Interest
The authors declare no conflict of interest.
Acknowledgements
The authors are thankful to Prof. K. Hirao for correspondence. Financialsupport from the MIUR (FARB 2018 and FABR 2019) is gratefully acknowl-edged.
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