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Performance of the DLPNO-CCSD and Recent DFT Methods in the

Calculation of Isotropic and Dipolar Contributions to 14N

Hyperne Coupling Constants of Nitroxide Radicals

Oleg Gromov ( aalchm@gmail.com )

Moskovskij gosudarstvennyj universitet imeni M V Lomonosova Himiceskij fakul'tet https://orcid.org/0000-0002-4119-8602

Research Article

Keywords: EPR spectroscopy, nitroxide radical, DFT, hyperne coupling, DLPNO-CCSD

Posted Date: March 11th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-279592/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License

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Abstract

In the present study, the performance of a set of density functionals: BP86, PBE, OLYP, BEEF, PBEpow, TPSS, SCAN, PBEGXPBE, M06L,

MN15L, B3LYP, PBE0, mPW1PW, B97, BHandHLYP, mPW1PW, HSE06, B98, TPSS0, PBE1KCIS, SCAN0, M06, M06-2X, MN15, CAM-

B3LYP, ωB97x, B2PLYP, and the B3LYP/N07D and PBE/N07D schemes in the calculation of the

14

N anisotropic hyperne coupling

(HFC) constants of a set of 23 nitroxide radicals is evaluated. The results are compared with those obtained with the DLPNO-CCSD

method and experimental HFC values. Harmonic contribution to the

14

N HFC vibrational correction was calculated at the

revPBE0/def2-TZVPP level and included in the evaluation. With the vibrational correction, the DLPNO-CCSD method yielded HFC

values in good agreement with the experiment (MAD = 0.3 G for the dipole-dipole contribution and MAD = 0.8 G for the contact

coupling contribution). The best DFT results are obtained using the M06 functional with mean absolute deviation (MAD) = 0.2 G for

the dipole-dipole contribution and MAD = 0.7 G for the contact coupling contribution. In general, vibrational correction signicantly

improved most DFT functionals' performance but did not change its overall ranking.

Introduction

Density functional theory (DFT) is now the most used method in quantum chemical modeling as it affords the best balance of cost

and accuracy for the time being. Though DFT itself is an exact theory connecting the molecular system's many-electron density with

its energy via “exact” density functional, it relies on approximate functionals in practical applications. Hence many general-purpose

and particular property tailored functionals were developed for the last few decades [1]. New functionals are supposed to perform

better than the older ones. However, the common practice shows that one should not always prefer some newly developed techniques

over the previous ones thoughtlessly. Indeed as shown in the paper by Medvedev et al. [2], the common strategy of DFT functional

development tailoring molecular energetics and geometry might result in DFT functionals poorly describing electron density. While

energy-based evaluation of DFT functionals’ performance readily delivers natural metrics, electron density-based evaluation metrics

are more ambiguous; see for discussion Marjewski et al. [3] and the references therein. Though electron density also delivers some

natural and experimentally observable metrics such as (deviation in calculated) dipole moment (and higher moments) as used by Hait

and Head-Gordon [4]. Interestingly Hait and Head-Gordon found open-shell species to be the most dicult cases for DFT. Some open-

shell species, namely radicals (including some coordination compounds), deliver another electron density-based natural metric:

hyperne coupling (HFC) [5]. One can extract some information on spin density distribution in the radical from HFC directly measured

by Electron Paramagnetic Resonance (EPR) spectroscopy and use it in DFT functionals performance evaluation [5–7].

EPR spectroscopy is a widely used method in studying organic radicals and d-block metal complexes [8]. To interpret experimental

EPR spectra, one usually uses an effective spin Hamiltonian describing the interaction between an electron spin (

Ŝ

), magnetic nuclei

spins (

Î

), and an external magnetic eld (

B

):see formula 1 in the supplementary les.

The rst term is the Zeeman interaction between the electron spin and the external magnetic eld, with

g

being a 3x3 tensor. The

hyperne coupling tensors

A

i

describe the interaction of the unpaired electron and the magnetic moments of the nuclei

Î

i

. The hyperne

coupling tensor can be divided into the isotropic (

a

iso

) and the traceless spin-dipole contributions:see formula 2 in the supplementary

les.

where the isotropic Fermi contact termsee formula 3 in the supplementary les.

readily delivers the spin density at the nucleus N (ρ(R

N

) ), while

A

dip

tensor depends on the whole spin density distribution.

In the previous paper [6], we evaluated the well-established and widely used functionals. During the last decade, many new functionals

arrived. For example, the strongly constrained and appropriately normed semilocal density functional (SCAN) yields at the cost of a

non-hybrid mGGA functional [2] electron densities better than the B3LYP functional. While the B3LYP functional is well tested [5, 7, 9–

14], the SCAN functional performance data are insucient. It has been shown [5, 15] that SCAN, while being competitive to the

commonly used functionals in some cases, is not, in general, the best choice for HFC calculation. In some cases, SCAN and SCAN0

functionals even predicted qualitatively incorrect electronic structures of transition metal complexes [15]. For that reason, it is

interesting to expand this research on the SCAN performance on compounds of p-elements. The B98, PBE1KCIS, HSE06, and SCAN

functionals are listed in the paper by Medvedev et al. among the best functionals for electron density and are included in the present

research.

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The DFT calculations results can be compared either with the experimental values or with the values calculated using some accurate

ab initio

method. The former requires taking into account solvation and vibrational averaging effects and calculating accurate

geometry[16]. The latter way, which requires, for example, coupled-cluster calculation of the reference values, is therefore preferred,

especially if the test set consists of small species, easily treated using accurate

ab initio

methods. Both choices of the reference are

consistent. For example, the coupled cluster methods as shown by Barone and Puzzarini [16] for small X

2

NO Systems (X = F, Cl, Br, I),

bring HFC constants in excellent agreement with the experiment, given vibrational, media, and basis set effects are taken into account

in CCSD(T) calculation. A routine CCSD calculation of substantially large radical systems was recently made possible in the domain-

based local pair-natural orbital coupled-cluster with singles and doubles (DLPNO-CCSD) framework [17]. Hence it is possible to use

both references even for large radical systems. Besides, it is interesting to evaluate the DLPNO-CCSD method performance for "real-

life" nitroxide radicals.

Calculation Details

ORCA 4.2.0 program package [18] was used to perform all calculations. The nitroxide radicals' geometries were optimized at the

UKS/B2GP-PLYP/def2-TZVPP [19, 20] level with the COSMO solvation model [21]. The B2GP-PLYP functional was chosen for

geometry optimization according to a recent benchmark study [22].

14

N HFCC were calculated using some newer density functionals available in Libxc [23]. DFT functionals used are: strongly

constrained and appropriately normed semilocal density functional (SCAN) [24] (we didn't nd signicant differences between the

results of SCAN and revised SCAN functional [25] hence the results of the calculations using the latter are not presented), hybrid

SCAN0 [26] (also similar to the results of the revised version), bayesian error estimation functional (BEEF) [27], power series extension

of the PBE exchange functional [28] (PBEpow exchange functional combined with the PBE [29] correlation functional), PBE-GX

functional [30] (combined with the PBE correlation functional), MN15 and MN15-L functionals [31, 32], PBE1KCIS hybrid [33] (with

KCIS correlation functional [34]), screened Coulomb potential hybrid functional (HSE06) [35] and B98 hybrid [36]. We compared these

functionals with the functionals used in our previous paper [6]: GGA functionals BP86 [37, 38], PBE [29], OLYP [39]; the meta-GGA

functionals TPSS [40], M06L [41]; the hybrid GGA functionals B3LYP [42], PBE0 [43], mPW1PW [44], B97 [45], BHandHLYP [46]; the

hybrid meta-GGA functionals TPSS0 [40], M06 [47], M06-2X [47]; the range-separated functionals CAM-B3LYP [48], ωB97x [49], the

double-hybrid functional B2PLYP (with relaxed density) [50]. The B3LYP/N07D and PBE0/N07D functional/basis combinations [51, 52]

were also used. All DFT calculations were performed using unrestricted Kohn-Sham formalism.

Basis sets for coupled-cluster and DFT calculations were constructed similarly to the one used by Datta and Gauss [53]. DFT

calculations were performed using uncontracted cc-pCVQZ basis set [54] augmented with four tight even-tempered s-functions

(denoted cc-pCVQZ-Juc by analogy with an aug-cc-pVTZ-Juc basis set of Sauer et al. [55]). For further discussion on the basis sets for

HFC calculations, see Jakobsen and Jensen [56]. The RIJCOSX [57] approximation with auto-generated auxiliary basis sets [58] and

integration grid Grid7 (the tightest in ORCA) and Gridx9 were used throughout except for the calculation of the vibrational correction. In

the case of dubious results, calculations were performed without RI or RIJCOSX approximations (for example, calculations using the

MN15, SCAN, and SCAN0 functionals). The COSMO solvation model [21] with toluene as a solvent was used as non-polar solvents

implicit solvation models were shown to yield reasonable solvent shifts [16, 59].

14

N hyperne couplings of nitroxide radicals were also calculated using the domain-based local pair-natural orbital coupled-cluster

with singles and doubles (DLPNO-CCSD) method [17, 60, 61] with DLPNO-HFC1 set of the thresholds and quasi-restricted orbitals

(QRO) from unrestricted B3LYP/COSMO calculation as a reference. COSMO could not be used in DLPNO-CCSD calculation directly.

However, the introduction of solvation at the reference level yields results similar to those obtained using an additive scheme of

Barone [16]. T

1

diagnostic parameter values were always lower than 0.02. DLPNO-CCSD calculations were performed using

uncontracted cc-pCVTZ basis set [54] augmented with four tight s-functions from aug-cc-pVTZ-J [55] basis set.

Vibrational corrections to isotropic hyperne coupling constants were calculated using the perturbative approach [62–66]. Only

harmonic corrections, given by:see formula 4 in the supplementary les.

14

N hyperne coupling constants of 23 nitroxide radicals (Fig. 1) used as the reference were taken from Lebedev et al. [68]. The choice

of particular radicals was discussed in our previous paper [6].

14

N isotropic hyperne coupling constants (

a

iso

) were calculated from

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the hyperne coupling tensors at low temperatures. The labels used for nitroxide radicals by Lebedev et al. were adopted for use in the

present paper.

Results And Discussion

3.1 DLPNO-CCSD calculation and ZPVC

The experimental data by Lebedev et al. [68] were acquired at low temperatures, and vibrational averaging effects are reduced to some

extent. For example, Barone et al. [65] estimated ZPVC to

14

N

a

iso

of the 2,2,5,5-tetramethylpyrrolidin-1-yl)oxydanyl radical (Proxyl,

pyrrolidine series) to be 0.3 G at 0K (harmonic contribution is 0.2G and anharmonic contribution is 0.14G). We omitted vibrational

corrections in our previous [6] due to this relatively low value. On the other hand, Barone et al. [59] estimated vibrational correction to

14

N

a

iso

of the Proxyl radical at 298K to be up to 2.6G. Recently Auer et al. calculated ZPVC of 2,2,3,4,5,5-

hexamethylperhydroimidazol-1-oxyl radical (HMI, imidazolidine series) at the revPBE0 level (0.9G) and the DLPNO-CCSD level (1.3G).

In the present study,

14

N HFCCs were calculated at the DLPNO-CCSD level (table 1). The relative error of the DLPNO-CCSD calculation

to experiment varies from 7% up to 22%. Moreover, the correlation of calculated and experimental

14

N

a

iso

(Fig. 4, black) is not linear.

These observations, along with literature data, indicated the importance of vibrational correction. Indeed, the inclusion of perturbative

vibrational correction (revPBE0/def2-TZVPP) reduces relative errors of the DLPNO-CCSD calculation to 3-8%, and the correlation of the

calculated and the experimental

14

N

a

iso

(Fig. 4, red) readily becomes almost linear. The remaining errors of 0.3-1.3G (MAD = 0.8G)

originate from several small contributions. First, DFT calculation probably underestimates vibrational correction (see Auer et al. [67]).

On the other hand, the inclusion of only the harmonic contribution probably leads to the overestimation of vibrational correction,

especially in the

3-imidazoline-3-oxide-oxyl

series. Secondly, deviation of DLPNO-CCSD results from canonical CCSD results [5] and the

lack of triples correction [16] further draw calculated values away from the experiment. Finally, error from incompletion of the basis set

and approximate way of the solvent's inclusion in the calculation may also contribute to deviation from the experiment. Some of these

errors are covered in the papers by Barone et al. [16, 69–71]. The divergence between DLPNO-CCSD calculation and experimental spin-

dipole contribution to hyperne coupling tensor (

A

dd

z

), which is already relatively small (MAD = 0.5G), further benets from the

inclusion of the vibrational correction (MAD = 0.3G). In general DLPNO-CCSD method with the perturbative harmonic vibrational

correction (a

iso

MAD = 0.8G,

A

dd

z

MAD = 0.3G) outperforms most DFT methods (vide infra) and should be preferred even in routine

usage.

Table 1

14

N isotropic hyperne coupling constants of nitroxide radicals VI- LXXVIII (

a

iso

)

and the diagonal element of spin-dipole

contribution to hyperne coupling tensor (

A

dd

z

) calculated using the DLPNO-CCSD method and respective absolute errors (AE)

compared with the experiment. The vibrational correction (

Δa

iso

vib

and

ΔA

dd

z

vib

) is estimated at revPBE0/def2-TZVPP level

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Radical

Δa

iso

vib

(0K/140K)

a

iso

DLPNO-

CCSD

a

iso

DLPNO-

CCSD

corrected

a

iso

exp*

AE

iso

uncorrected

/corrected

ΔA

dd

z

vib

(0K/140K)

A

dd

z

DLPNO-

CCSD

A

dd

z

DLPNO-

CCSD

corrected

A

dd

z

Exp

AE

dd

uncorrected

/corrected

Piperidine series

VI 0.3/0.4 14.6 14.9 15.7 1.2/0.8 -0.1/-0.1 18.0 17.9 18.3 0.2/0.4

VII 0.3/0.3 14.6 14.9 15.8 1.3/1.0 -0.2/-0.2 18.5 18.3 18.4 0.1/0.1

VIII 0.4/0.5 14.4 14.9 16.1 1.7/1.3 -0.1/-0.1 18.3 18.2 18.7 0.4/0.5

X 0.3/0.3 14.5 14.9 15.9 1.4/1.0 -0.2/-0.2 18.3 18.2 18.1 0.2/0.1

XIII 0.3/0.3 14.5 14.8 15.8 1.4/1.0 -0.2/-0.2 18.4 18.2 18.5 0.1/0.3

XV 0.3/0.3 14.3 14.6 15.8 1.5/1.2 -0.2/-0.2 18.5 18.3 17.7 0.8/0.6

XIX 0.3/0.3 14.3 14.6 15.5 1.2/0.9 -0.1/-0.2 18.0 17.9 17.9 0.1/0.0

XX 0.3/0.3 14.4 14.7 15.7 1.3/1.0 -0.1/-0.2 18.0 17.8 18.1 0.1/0.3

Pyrroline series

XXXVII 1.3/1.6 12.0 13.6 14.7 2.8/1.1 -0.4/-0.4 19.7 19.3 19.3 0.4/0.0

Imidazoline series

XLIV 1.4/1.9 11.5 13.4 14.0 2.5/0.6 -0.4/-0.5 19.3 18.8 18.4 1.0/0.5

3-imidazoline-3-oxide-oxyl series

LII 2.3/2.9 11.0 13.9 14.1 3.2/0.3 -0.6/-0.7 19.2 18.4 18.1 1.1/0.4

LIII 1.1/1.6 11.5 13.1 13.9 2.4/0.8 -0.3/-0.4 19.1 18.7 18.6 0.5/0.1

LIX 1.9/2.5 11.0 13.5 14.0 3.0/0.5 -0.5/-0.6 19.2 18.5 18.2 0.9/0.3

LX 1.9/2.5 11.0 13.4 14.0 3.0/0.5 -0.5/-0.6 19.2 18.6 18.3 0.8/0.2

LXI 1.9/2.5 11.0 13.4 14.0 3.0/0.5 -0.5/-0.6 19.0 18.4 18.2 0.8/0.2

LXII 0.8/1.1 12.2 13.2 14.1 1.9/0.8 -0.2/-0.3 18.7 18.4 18.0 0.7/0.4

Pyrrolidine series

LXVII 0.9/1.4 12.1 13.5 14.0 1.9/0.5 -0.3/-0.4 19.5 19.1 19.0 0.5/0.1

LXVIII 1.1/1.6 11.5 13.1 14.0 2.5/0.9 -0.3/-0.4 19.3 18.9 19.3 0.0/0.4

Imidazolidine series

LXX 0.5/0.6 13.4 14.0 15.1 1.6/1.1 -0.2/-0.2 18.6 18.5 18.1 0.5/0.3

LXXI 0.5/0.6 13.5 14.0 14.7 1.2/0.7 -0.2/-0.2 18.4 18.2 18.1 0.4/0.2

LXXII 0.5/0.5 13.8 14.3 15.5 1.8/1.2 -0.2/-0.2 18.4 18.2 18.1 0.3/0.1

LXXIV 0.4/0.5 14.2 14.7 15.5 1.3/0.8 -0.2/-0.2 18.3 18.1 17.4 0.9/0.7

LXXVIII 0.8/0.9 12.5 13.4 13.9 1.4/0.5 -0.2/-0.3 19.1 18.8 18.7 0.4/0.1

* values calculated as an average of the diagonal elements (

A

n

) of the hyperne coupling tensors

3.2 DFT calculation

Our previous paper mentioned a slightly controversial trend: description of

14

N isotropic HFC in nitroxide radicals by DFT functionals

generally follows pattern noted by Medvedev et al. [2]. In contrast, the description of the dipole-dipole contribution to

14

N HFC is