Calculation of 125Te NMR Chemical Shifts at the Full Four

Article pubs.acs.org/JPCA

Calculation of 125Te NMR Chemical Shifts at the Full Four-Component Relativistic Level with Taking into Account Solvent and Vibrational Corrections: A Gateway to Better Agreement with Experiment Irina L. Rusakova,* Yuriy Yu. Rusakov, and Leonid B. Krivdin A.E. Favorsky Irkutsk Institute of Chemistry, Siberian Branch of the Russian Academy of Sciences, Favorsky St. 1, 664033 Irkutsk, Russian Federation S Supporting Information *

ABSTRACT: Four-component relativistic calculations of 125Te NMR chemical shifts were performed in the series of 13 organotellurium compounds, potential precursors of the biologically active species, at the density functional theory level under the nonrelativistic and four-component fully relativistic conditions using locally dense basis set scheme derived from relativistic Dyall’s basis sets. The relativistic effects in tellurium chemical shifts were found to be of as much as 20−25% of the total calculated values. The vibrational and solvent corrections to 125Te NMR chemical shifts are about, accordingly, 6 and 8% of their total values. The PBE0 exchange-correlation functional turned out to give the best agreement of calculated tellurium shifts with their experimental values giving the mean absolute percentage error of 4% in the range of ∼1000 ppm, provided all corrections are taken into account.

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INTRODUCTION Nowadays the synthesis of organotellurium compounds is rapidly progressing in parallel with the developing of their practical applications in many fields including biochemistry,1,2 radio-pharmaceutics,3 and microelectronics.4 Studied in this paper are 13 most-known organotellurium compounds, potential precursors of their biologically active derivatives. Indeed, a good many of organotellurium compounds provide different types of biological activity resulting in a creation of the well-known pharmaceuticals and drugs. Tellurium was regarded a poison for many years until nontoxic organotellurium compounds with high biological activity were found. Thus, organotellurides and diorganoditellurides have antioxidant activity being used as caspase and cathepsin inhibitors. Organotellurane compounds were proved to be toxic against promastigotes and amastigotes. Cadmium telluride nanoparticles were found to be fluorescent enabling their use as quantum dots in imaging and diagnosis. Some unsymmetrical 2-naphtyl-diorganyltellurium dichlorides were most effective organotelluranes with Gram-negative antibacterial effect. The antioxidant effects of organotellurides and diorganoditellurides, the immunomodulatory effects of the nontoxic inorganic tellurane and organic telluranes resulted in the creation of protease inhibitors. The effects of inorganic tellurium complexes are primarily caused by their specific redoxmodulating activities enabling the inactivation of cysteine proteases, inhibition of specific tumor survival proteins, and obstruction of tumor production. Multifunctional tellurium nontoxic molecules protect dopaminergic neurons and improve motor function in animal models of Parkinson’s disease. A © XXXX American Chemical Society

more detailed description of the biological activity of the tellurium-based medical products together with the references on the original publications can be found in the comprehensive review by Sekhon.5 Unfortunately, molecular size of the tellurium pharmaceuticals and drugs excludes the possibility of the relativistic calculation of tellurium chemical shifts of such biologically active compounds, at least at the modern state of computational capabilities. However, present publication is a first and solid step to the relativistic computation of the real-life organotellurium compounds, potential precursors of biologically active drugs. The accurate prediction of the tellurium spectrum is of great importance in tellurium magnetic resonance spectroscopy. In view of the expected noticeable relativistic effects in 125Te NMR chemical shifts, relativistic calculations of these parameters in a wide series of organotellurium compounds are of prime importance. In this respect, much progress has been made recently owing to the development of the relativistic density functional theory (DFT).6,7 Pioneering accurate calculations of tellurium shielding constants and chemical shifts at the twocomponent relativistic ZORA-LDA level8,9 in combination with gauge-including atomic orbitals (GIAO) formalism10−12 have been reported by Ruiz-Morales et al.13 for the series of diverse organic and inorganic tellurium compounds with 125Te NMR chemical shifts covering the range of ∼3000 ppm. One of the Received: April 5, 2017 Revised: June 2, 2017

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DOI: 10.1021/acs.jpca.7b03198 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

For example, according to the recent paper by Antušek,24 characteristic value of the relativistic effect on 129Xe NMR shielding constants amounts to ∼20% to the total value. As for the 119Sn atom, one could expect the relativistic effect on its shielding constant to be ∼20−25%25 in the absence of other heavy atoms in close vicinity. A vivid progress in computer technologies and computational NMR gives the opportunity to raise the accuracy of modern relativistic quantum chemical calculations by means of employing the four-component fully relativistic approaches to shielding constants (and accordingly, to chemical shifts) of heavy atoms. Generally speaking, all two-component relativistic methods are derived from the four-component Dirac equation via the elimination of the small component (this procedure is known as ESC) scheme with further expansion of the kinematic parameter, which is the part of the operator providing the relationship between the upper and lower components of the four-component wave functions. Consequently, they are only the approximate relativistic methods, involving a rather complicated formalism that may suffer from singularities in both the variational and perturbational approach.26 The aspects of the two-component methods are nowadays thoroughly investigated; see, for example, papers 27−29 and references cited therein. As is well-known,30 ZORA Hamiltonian, which was employed by Hayashi et al.14 to calculate 125Te chemical shifts, provides a good relativistic approximation for valence orbitals and outer core orbitals with comparatively small energy, even in the near-nucleus region of a heavy atom. It is well-known that for the core orbitals of heavy atoms with very large energy, ZORA results in substantial errors.30 Douglas− Kroll−Hess (DKH) method, used by Hada et al.,19 represents the different way of transforming the Dirac Hamiltonian to the two-component form. However, this is done to the lowest order resulting in the comparable results with ZORA for many molecular properties, in particular, NMR shielding constants. However, DKH method is significantly more accurate for core orbitals. Though two-component methods allow substantial central processing unit (CPU) savings, four-component methods based on Dirac−Coulomb Hamiltonian provide more accuracy because of the absence of the approximations, to which one ultimately must resort to during the separation of upper and lower components of the four-component spin or within any two-component method. As a matter of fact, until nowadays practical calculations of chemical shifts of heavy nuclei via the four-component fully relativistic approaches are very time-consuming. They are used much more rarely than quasi-relativistic two-component ones, the former mostly only for the calibration of the latter. In the present communication, relativistic corrections to 125Te magnetic shielding constants and chemical shifts were calculated at the four-component fully relativistic GIAO− DFT level. For the correct description of tellurium chemical shifts, we also took into consideration some less essential factors, such as solvent and vibrational effects, maintaining the balance between the level of approximation and computational effort. So that, the main goal of the present communication consists in a qualitative investigation of the methodology of calculation of tellurium shielding constants and chemical shifts together with a practical guideline to the versatile computational schemes for 125Te NMR chemical shifts at the fourcomponent relativistic level taking into account as many accuracy factors as possible.

most important aspects of this work is raising a question about the rate of cancellation of the relativistic effects for the tellurium chemical shifts. Thus, for example, for Te(CH3)4 the relativistic part of the absolute tellurium shielding constant accounts to ca. 170 ppm resulting in the tellurium chemical shift scale of ∼11 ppm. However, for [TeCl6]2− the situation turns out to be reversal: relativistic effects on tellurium shielding constant result in 57 ppm, while the relativistic correction for the tellurium chemical shift is as much as ∼125 ppm. So far, this question is still open, and more wide statistical sampling is needed. The mean absolute percentage error (MAPE) of tellurium chemical shifts in the series of 16 molecules, studied in the abovementioned paper,13 equals to ∼75% in relation to experimental results, while the mean absolute error (MAE) is ∼180 ppm. Another recent paper by Hayashi et al.14 demonstrates an extensive insight into relativistic effects in 125Te NMR chemical shifts. The authors considered tellurium shifts in as many as 45 compounds covering the range of ∼4000 ppm. Relativistic effects were evaluated using two-component spin−orbit ZORA Hamiltonian within the GIAO-BLYP15,16 and GIAO-OPBE17 approaches. The agreement of calculated total tellurium chemical shifts with their experimental values was found to be sufficiently good. In particular, for the structures optimized within the relativistic two-component ZORA-OPBE method, the MAPEs for the tellurium chemical shifts presented in this paper14 are 20% for ZORA-OPBE and 23% for ZORA-BLYP. Other two-component quasi-relativistic approaches to calculation of tellurium shielding constants are found to be much less utilized than those based on ZORA formalism. In particular, Hada et al.18,19 used quasi-relativistic two-component method based on the second-order Douglas−Kroll−Hess transformed relativistic Hamiltonian20,21 (the latter including spin-free relativistic, one- and two-electron spin−orbit and relativistic magnetic interaction terms) within the framework of the generalized unrestricted Hartree−Fock method22,23 to calculate 125Te magnetic shielding constants and chemical shifts in the series of TeH2, TeF6, Te(CH3)4, and Te(CH3)2Cl2. The relativistic effects were found to amount to as much as 1300−1500 ppm for the tellurium shielding constants and to be in the range from −20 to 160 ppm for the tellurium chemical shifts. The MAPE for chemical shifts in this series accounts to 32% as compared to experimental data. As one can judge from the above-mentioned papers,13,14,18,19 the two-component spin−orbit ZORA-DFT approach is able to provide only some not very high accuracy of calculated 125Te NMR chemical shifts in relation to experiment. Although the MAPEs in the works mentioned above are not very high, and the methods employed are quite different, yet one can roughly estimate the measure of the relativistic effects on the NMR shielding constant of tellurium atom that one could expect to be reproduced in this work at a more accurate level of theory. Thus, according to Hayashi et al.,14 relativistic correction to shielding constants of tellurium atom varies in the range of ∼10−20% (excluding some specific molecules) relative to the total relativistic values, while Hada et al.19 reports on as much as ∼30%. Some reasonable question, which could arise at this point, concerns the role of relativistic effects on similar NMR atoms, like xenon, 129Xe (Z = 54), or tin, 119Sn (Z = 50), which are close to tellurium in the periodic table. It is noteworthy that 129 Xe NMR spectroscopy computational tools nowadays are being rapidly developed, allowing one to qualitatively estimate typical relativistic effects on 129Xe NMR shielding constants. B


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METHODS Nonrelativistic geometry optimizations of 1−14 were performed with GAMESS31,32 code at the MP2 level of theory using dyall.av3z basis set33,34 on tellurium atom and dyall.av2z basis set33,34 on the rest of atoms within the locally dense basis sets (LDBS) scheme;35−37 see below for more details. According to our previous results,38 the so-called “relativistic geometrical factor” becomes non-negligible for the compounds containing heavy elements only beginning with 6th period. Thus, in the case of the considered tellurium compounds 1−14, the “relativistic geometrical factor” is expected to be unessential, so that time-demanding relativistic geometry optimizations were not performed herewith. The nonrelativistic DFT calculations of tellurium shielding constants were performed within the GIAO approach10−12 with Dalton program package.39 At the stage of geometry optimization and nonrelativistic calculation of tellurium shielding constants, solvent effects were taken into account within the integral equation formalism of polarizable continuum model (IEF-PCM).40,41 All relativistic calculations of shielding constants were performed using full four-component Dirac−Kohn−Sham (DKS) Hamiltonian in DIRAC program package.34 In DKS Hamiltonian, the Gaunt−Breit interaction term42−45 has not been included. The (SS|SS) integrals were replaced by interatomic SS correction calculated as a classical repulsion term of small component atomic charges.46 The unrestricted kinetic balance condition in combination with GIAO formalism was used to recover the magnetic kinetic balance condition.47 In most part of nonrelativistic and relativistic calculations of tellurium shielding constants, dyall.av4z basis set33,34 was used for tellurium, while dyall.av3z was employed for the atoms coupled to tellurium (α atoms) and dyall.av2z for the rest of atoms, which is called throughout the article as “LDBS-3” scheme. Zero-point vibrational corrections48,49 to tellurium shielding constants of 2, 3, 4, 7, 12, and 13 were calculated at the nonrelativistic DFT-B3lYP15,50 level of theory with Dalton program package39 using dyall.av2z basis set on all atoms. Throughout the paper, dimethyl telluride Te(CH3)2 (1) was used as a reference compound for 125Te NMR chemical shifts.

the virtue of the local nature of NMR shielding constants. Indeed, according to Ramsey,51 in the simplest nonrelativistic framework the paramagnetic term represents the linear response function, which numerator contains the multiplication of two matrix elements. The integral kernel of the first matrix element contains the angular momentum operator, which represents the orbital Zeeman interaction, while the integrand of the second matrix element contains the paramagnetic spin− orbit operator, which has a short-range nature because of its dependence on 1/r3, where r is the distance between the electron i and NMR atom under consideration. The diamagnetic term had been shown by Ramsey51 to be the functional of 1/r, which also argues in favor of its short-range nature. In relativistic representation the locality remains. Generally speaking, because of the locality of NMR shieldings, the roughening of the description of the outlying regions should not result in the substantial loss of accuracy.52 In the present study, we investigated the performance of a number of different LDBS and FBS schemes in comparison with the dyall.av4z FBS scheme. All basis set schemes in this study are constructed from the uncontructed augmented Dyall’s basis sets. Such choice originates in the fact that Dyall’s basis sets are optimized within the framework of a full fourcomponent Dirac formalism, which mostly suits the aims of the present study of 125Te NMR shielding constants at the full fourcomponent relativistic level. Moreover, augmented basis sets are shown to provide more converged results than those without augmentation for the NMR shielding constants of selenium,53 a lighter analogue of tellurium. To investigate the performance of different LDBS schemes, we calculated 125Te NMR shielding constants in two conformationally labile tellurium-containing compounds, namely, diethyl telluride (2) and divinyl telluride (3), using the nonrelativistic GIAO-PBE054,55 approach. All properties of the labile compounds 2 and 3 presented in this paper including full molecular energies and shielding constants were conformationally averaged. As was mentioned by Adamo and Barone,55 hybrid functional PBE0 obtained from the parameter-free GGA functional PBE56 by 25% admixture of Hartree−Fock (HF) exchange represents probably the most suitable functional for the calculation of NMR chemical shifts of the elements of the most part of periodic table. Moreover, according to the findings of Adamo and Barone,55 PBE0 functional is competitive with the low-order post-HF techniques (e.g., MP2) in view of the fact that it gives significantly improved results in the presence of substantial correlation effects. The LDBS schemes tested in the present study are given in Table 1 in a descending order of their size and quality. All four schemes imply dyall.av4z basis set on tellurium, while the rest of the atoms are specified with either dyall.av3z or dyall.av2z sets.

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RESULTS AND DISCUSSION Convergence of Basis Set Schemes. As a matter of fact, very time-consuming four-component relativistic GIAO−DFT calculations of 125Te NMR shielding constants with full basis set (FBS) schemes based on relativistically oriented Dyall’s basis sets33 with large valence splitting can be successfully performed only for very small tellurium compounds. The socalled FBS schemes are those where for each atom in molecule the same type of basis set is used. In practice, such FBS approaches in relativistic GIAO−DFT calculations are not affordable for the real-life organotellurium compounds. Going on to the larger molecular systems or to ones containing heavier elements requiring an adequate description within the framework of relativistic theory leads to the growth of computational cost, which could rapidly exceed the computational limit. A reasonable alternative between the accuracy of calculation and computational effort originates in the idea of the so-called locally dense basis sets,35−37 which implies the use of large high-quality basis sets on a particular atom or a group of atoms (in this case, tellurium) and essentially smaller lowerquality basis sets elsewhere in the molecule. This is justified by

Table 1. LDBS Schemes Used in the Nonrelativistic GIAOPBE0 Calculation of 125Te NMR Shielding Constants of Diethyl and Divinyl Tellurides

C

LDBS scheme

Te

Cα

Cβ

H

LDBS-1 LDBS-2 LDBS-3 LDBS-4

dyall.av4z dyall.av4z dyall.av4z dyall.av4z





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The Journal of Physical Chemistry A In the most noneconomic LDBS-1 scheme, all non-tellurium atoms are specified with the triple-ζ basis sets, while in the most economic LDBS-4 scheme they are specified with the double-ζ basis sets. For each FBS and LDBS scheme, the number of basis set functions, total molecular energy, calculated 125Te shielding constant, and absolute error are given in Table 2. Table 2. Comparative Performance of Different Full Basis Set (FBS) and Locally Dense Basis Set (LDBS) Schemes in the Calculation of 125Te NMR Absolute Shielding Constants of Diethyl Telluride (2) and Divinyl Telluride (3) Considered at the Nonrelativistic GIAO-PBE0 Levela 125

compound 2

3

FBS or LDBSb

No. of AO

total energy (E), au

Te NMR shielding constant (σ), ppm

FBS-1 LDBS-1 FBS-2 LDBS-2 LDBS-3 LDBS-4 FBS-3 FBS-1 LDBS-1 FBS-2 LDBS-2 LDBS-3 LDBS-4 FBS-3

1226 826 763 666 605 550 428 1014 710 647 614 556 498 376

−6772.337 093 33 −6772.335 427 00 −6772.334 385 42 −6772.328 995 17 −6772.323 223 52 −6772.317 572 38 −6772.305 070 39 −6769.866 130 02 −6769.864 600 24 −6769.863 338 68 −6769.860 984 31 −6769.854 831 25 −6769.847 927 63 −6769.833 923 46

2561 2560 2566 2560 2560 2557 2581 2398 2398 2402 2399 2398 2395 2422

absolute error (Δσ), ppm 0 −1 +5 −1 −1 −4 +20 0 0 +4 +1 0 −3 +24

a

Figure 1. 125Te NMR shielding constants (in ppm) of diethyl telluride (2) and divinyl telluride (3) calculated at the nonrelativistic GIAOPBE0 level using different computational schemes.

As one can see in Table 2, the less economical LDBS-1 scheme gives very accurate results with absolute errors Δσ of 1 and 0 ppm for compounds 2 and 3 as compared to the FBS-1 scheme implying dyall.av4z basis set on all atoms. At the same time, LDBS-1 is much more economic using for compounds 2 and 3, accordingly, 400 and 304 basis set functions less as compared to FBS-1 scheme. However, even more economical LDBS-2 and LDBS-3 schemes perform at the same level of accuracy with Δσ not exceeding 1 ppm in absolute value and giving further advantage in the number of basis set functions; see Figure 1. The value of Δσ of the FBS-2 scheme, which one could expect to be close to that of the LDBS-1 one, had actually reached as much as 5 and 4 ppm for compounds 2 and 3, respectively. In absolute value, such deviations are very similar to those of the LDBS-4 scheme for both compounds. From the formal point of view, the LDBS-4 scheme performs slightly worse than LDBS-3 or LDBS-2 resulting in the gain of 55 and 58 basis functions for compounds 2 and 3, respectively, as compared to LDBS-3. This is not too much for the DFT calculations, either relativistic or nonrelativistic. In our opinion, for the sake of reliability, it is better to choose slightly more accurate but less economic LDBS-3 rather than LDBS-4 scheme. As can be seen in Figure 1, FBS-3 scheme provides the worst results with absolute errors of as much as 20 ppm for compound 2 and even 24 ppm for compound 3, which could be thought of as the unconverged ones. On the basis of these

results, in all further calculations we will use the most reliable LDBS-3 scheme. Choosing the Exchange-Correlation Functional. To accomplish benchmark calculations of 125Te NMR chemical shifts as well as to make the reliable conclusions about the roles of relativistic, solvent, and vibrational effects, we first examined the performance of different exchange-correlation functionals, both at nonrelativistic and four-component fully relativistic levels of density functional theory. The representative series of organotellurium open-chain and heterocyclic compounds 1−14 considered in this paper is shown below (Chart 1). The 125Te NMR chemical shifts of 2−14 referenced to 1 calculated using KT2,57 KT3,58 PBE,56 PBE0,54,55 B3LYP,15,50 and BLYP15,16 DFT functionals within the LDBS-3 scheme at both nonrelativistic and fully four-component relativistic levels are compiled in Table 3. Absolute isotropic shielding constants were converted into 125Te NMR chemical shift scale referenced to dimethyl telluride (δ, parts per million) using approximate formula δ = σMe2Te − σ derived from the precise equation δ = (σMe2Te − σ)/(1 − 1 × 10−6 σMe2Te), as recommended by IUPAC.59,60 Herewith, σ and σMe2Te are the 125Te NMR isotropic absolute shielding constants of, accordingly, the compound under consideration and dimethyl telluride, the latter calculated within the FBS-1 scheme, either at nonrelativistic or fully four-component relativistic level of theory. Solvent corrections were evaluated as the difference of 125Te NMR chemical shifts calculated using the IEF-PCM model40,41 and those calculated in gas phase at the nonrelativistic level. As can be seen in Table 3, nonrelativistic DFT calculations overestimate tellurium chemical shifts by ∼18−28% of their

All properties in the table are conformationally averaged. bFor the specification of LDBS schemes, see Table 1. Notations FBS-1, FBS-2, and FBS-3 imply, accordingly, dyall.av4z, dyall.av3z, and dyall.av2z on all atoms.

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As can be seen in Table 4, paramagnetic relativistic corrections Δσpara (1252−1625 ppm) dominate over diamagnetic relativistic ones Δσdia (579−601 ppm in absolute value) resulting in the total shielding relativistic corrections of ∼661− 1030 ppm in the whole series of 1−14 (which is ∼25−35% of the total relativistic values of tellurium absolute shielding constants). It is noteworthy that diamagnetic relativistic corrections are practically unchangeable along this series (standard deviation is only ca. 5 ppm), while paramagnetic relativistic parts span over the range of 373 ppm with standard deviation of 97 ppm. Indeed, on the one hand, it is well-known that diamagnetic shielding is a ground-state core property,67,68 being found to be of small influence of valence electronic structure, never mind whether the relativistic or nonrelativistic representation is used. On the other hand, paramagnetic term and spin−orbit corrections are mostly determined by the core tails of the valence orbitals.68 For this reason, paramagnetic and spin−orbit terms are heavily dependent on the chemical environment. This is the cause of noticeable changes of paramagnetic shielding along the series. On the basis of these results, we conclude that it is paramagnetic term that determines the difference of total relativistic corrections in the organotellurium compounds under consideration. In the 125Te NMR chemical shifts scale (Table 5), relativistic corrections are much more moderate varying in the range of 17−369 ppm, which gives in average 26% of the total relativistic values of tellurium chemical shifts. Note that total relativistic corrections to 125Te NMR chemical shifts are deshielding (Δδrel), which is because Δσrel for dimethyl telluride used as a standard is the largest in the series. As can be seen in Table 5, relativistic corrections to tellurium chemical shifts (Δδrel) vary in a wide range from 17 to 369 ppm, with standard deviation of 160 ppm (the mean value is 126 ppm). Consequently, there is no evident cancellation of relativistic effects, which may be the result of subtraction of tellurium shielding constants of the compounds under consideration and that of a standard. It is interesting to note that in the course of the present study, an accompanying accessory data dealing with 13C NMR shielding constants of the carbon atoms in α-position relative to 125 Te atom were obtained. In the presence of heavy atom, the NMR parameters of the light atoms could be shifted because of the relativistic phenomenon, known as HALA effect.69−75 The latter was thoroughly investigated during the last decades and most recently, see, for example, recent papers by Rusakov et al.,76,77 Casella et al.78 Thus, we found it interesting to report on the typical values of the effect of 125Te atom on the shielding constants of the α-carbons, which amounts to as much as ∼20 ppm; see Table S1 in Supporting Information file. This value goes along with the data, reported on in our previous work,38 that was devoted to the short-range HALA effect on 13C NMR chemical shifts, initiated, in particular, by heavy chalcogens. Rovibrational Corrections, Solvent Effects, and Comparison with Experiment. For the most adequate comparison of the present theoretical results with experimental data, rovibrational averaging of the calculated 125Te NMR shielding constants must be performed. Unfortunately, calculations of vibrational corrections are very demanding, requiring the calculation of parameters defined as the third (cubic forceconstants and shielding gradients) and fourth (shielding Hessians) derivatives of the electronic energy. The computational time drastically increases with the number of atoms in

Chart 1. List of Compounds

experimental values with the mean absolute errors varying in the range of 109−164 ppm, depending on the exchangecorrelation functional chosen. Taking into account relativistic effects results in an essentially better agreement with experiment, namely, MAPEs and MAEs are decreased to 6−19% or 36−69 ppm in chemical shift scale. Figure 2 demonstrates drastic decrease of MAPEs for each exchange-correlation functional on going from simple nonrelativistic model toward to the four-component fully relativistic DFT level. It could be seen in Figure 2 that most “accurate” nonrelativistic results are obtained when using KT2, PBE0, and B3LYP functionals with MAPEs being, respectively, 18, 19, and 19%. At the same time, at fully four-component relativistic level they also show the best behavior giving the MAPEs of, accordingly, 13, 6, and 10%. Among these three functionals, relativistic PBE0 and B3LYP calculations give the best agreement with experiment. As can be seen in Table 3, the smallest MAE is 36 ppm for B3LYP functional (MAPE = 10%), while the smallest MAPE is 6% for PBE0 functional (MAE = 40 ppm). In all further calculations we used PBE0 functional providing the best agreement with experiment. Relativistic Effects. Relativistic effects are expected to play an essential role in 125Te NMR shielding constants of the studied compounds. Herewith, we evaluated relativistic corrections (Δσrel) to 125Te NMR shielding constants (σ) in the series of organotelluriumum compounds 1−14 (including dimethyl telluride used as a standard) as the difference between their full Dirac’s four-component relativistic values and those calculated at the nonrelativistic level using the same method (DFT-PBE0) and the same basis sets (LDBS-3 scheme). The total relativistic corrections (Δσrel) together with their diamagnetic (Δσdia) and paramagnetic (Δσpara) parts are given in Table 4. E


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The Journal of Physical Chemistry A Table 3.

Te NMR Chemical Shiftsa of 2−14

125

calculated compound

solvent

KT2

KT3

2c 3c 4 5 6 7 8 9 10 11 12 13 14 MAE, ppm MAPE, %

C6D6 C6D6 C6D6 C6D6 CDCl3 PhCH3 C6D6 C6D6 CH2Cl2 C6D6 (CD3)2CO (CD3)2CO neat

295 459 331 188 771 692 808 482 81 592 555 661 510 125 18%

265 415 284 157 722 642 757 442 88 579 501 610 464 164 26%

2c 3c 4 5 6 7 8 9 10 11 12 13 14 MAE, ppm MAPE, %

C6D6 C6D6 C6D6 C6D6 CDCl3 PhCH3 C6D6 C6D6 CH2Cl2 C6D6 (CD3)2CO (CD3)2CO neat

332 564 436 243 929 856 1151 516 102 656 700 822 634 51 13%

292 516 390 207 871 812 1097 471 106 626 637 760 579 69 15%

125

Te NMR chemical shifts (δtotal), ppm PBE

PBE0

nonrelativistic level 305 291 454 470 232 278 131 158 820 776 716 705 845 817 502 461 83 36 673 633 599 668 695 774 538 597 124 109 22% 19% four-component relativistic level 342 329 568 583 342 385 191 213 984 942 914 935 1219 1186 546 491 125 64 742 650 762 829 872 955 681 736 46 40 14% 6%

B3LYP

BLYP

expb

293 457 243 130 784 682 818 500 68 634 615 731 551 122 19%

300 435 201 104 812 687 840 531 104 660 543 651 489 146 28%

374 534 338 215 993 858 1114 512 63 599 782 905 727

332 568 344 184 943 890 1201 533 96 672 771 909 685 36 10%

337 546 304 162 971 862 1226 575 138 733 697 823 621 63 19%

374 534 338 215 993 858 1114 512 63 599 782 905 727

a

In parts per million. Calculated at the nonrelativistic GIAO−DFT level with taking into account solvent corrections at the nonrelativistic level. Experimental data are taken from different sources: [61] for 2, 3, 4, and 5; [62] for 6; [63] for 7; [64] for 9 and 10; [65] for 11; [66] for 8, 12, 13, and 14. cConformationally averaged.

b

benchmark calculations in the series of compounds under consideration we chose only six not very demanding molecules 2, 3, 4, 7, 12, and 13 together with a standard 1. The results of the benchmark calculations of 125Te NMR chemical shifts of these representative organotellurium openchain and heterocyclic compounds against available experimental data are presented in Table 6. Total values of calculated 125 Te NMR chemical shifts (δtotal) were obtained as a sum of the basic nonrelativistic values (δnonrel) and relativistic (Δδrel), solvent (Δδsolv), and vibrational (Δδvib) corrections, while absolute percentage errors of the total calculated 125Te NMR chemical shifts are calculated as ε = |(δ − δexp)/δexp| × 100. All calculations of the zero-point rovibrationally averaged tellurium shielding constants49 were performed at the nonrelativistic DFT-B3LYP level using dyall.av2z full basis set scheme (denoted above as FBS-3) with the effective geometries as an expansion point. Using the effective geometry as the expansion point for the vibrational wave function instead of equilibrium geometry includes the anharmonicity of the potential.49 The effective geometries48 were evaluated within the same methods and basis sets as the property. At the evaluating of the vibrational corrections to tellurium chemical shifts, we used vibrational correction to tellurium shielding

Figure 2. Mean absolute percentage errors of 125Te NMR chemical shifts (ppm) for 2−14 calculated using different exchange-correlation functionals at three different levels: nonrelativistic level in gas phase (blue bars); nonrelativistic level with taking into account solvent effects within the IEF-PCM model (red bars) and four-component relativistic level with taking into account solvent effects within the nonrelativistic IEF-PCM model (green bars).

molecule or with the growth of the atomic numbers of elements, which electronic structure demands larger basis sets for appropriate description. Bearing this in mind, for the F


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The Journal of Physical Chemistry A Table 4. Relativistic Effects in

125

Te NMR Shielding Constantsa

nonrelativistic GIAO-PBE0

a

four-component relativistic GIAO-PBE0

relativistic corrections

com-pound

σnonrel

σdia

σpara

σrel

σdia

σpara

Δσrel

Δσdia

Δσpara

1 2b 3b 4 5 6 7 8 9 10 11 12 13 14

2853 2560 2398 2611 2721 2091 2286 2060 2416 2852 2008 2256 2157 2256

5430 5462 5455 5462 5433 5544 5563 5626 5470 5468 5430 5445 5461 5466

−2577 −2902 −3057 −2851 −2712 −3453 −3277 −3566 −3054 −2616 −3422 −3189 −3304 −3210

3883 3552 3315 3534 3696 2955 3086 2721 3416 3854 3021 3125 3006 3147

4835 4875 4868 4877 4854 4943 4971 5035 4881 4880 4844 4854 4871 4877

−952 −1323 −1553 −1343 −1158 −1988 −1885 −2314 −1465 −1026 −1823 −1729 −1865 −1730

1030 992 917 923 975 864 800 661 1000 1002 1013 869 849 891

−595 −587 −587 −585 −579 −601 −592 −591 −589 −588 −586 −591 −590 −589

1625 1579 1504 1508 1554 1465 1392 1252 1589 1590 1599 1460 1439 1480

In parts per million, without taking into account solvent corrections. bConformationally averaged.

Table 5. Relativistic Effects in 125Te NMR Chemical Shifts (ppm) without Taking into Account Solvent Corrections compound

nonrelativistic GIAO-PBE0 (δnonrel)

relativistic corrections to (Δδrel)

four-component relativistic GIAOPBE0 (δrel)

expa

293 455 242 132 762 567 793 437 1 845 597 696 597

38 113 107 55 166 230 369 30 28 17 161 181 139

331 568 349 187 928 797 1162 467 29 862 758 877 736

374 534 338 215 993 858 1114 512 63 599 782 905 727

b

2 3b 4 5 6 7 8 9 10 11 12 13 14

on all atoms within the FBS-1 scheme) and those obtained within a nonrelativistic Schrödinger picture with the same basis sets (see Table 5). Relativistic correction (Δδrel) is also defined as the difference between the relativistic correction to tellurium shielding constant of dimethyl telluride (standard) and that one to tellurium shielding constant of compound under consideration. Since NMR experiments are mostly performed in solution, solvent effects should also be considered. As was pointed out by Luthra and Odom,66 tellurium NMR chemical shift in reference standard, Me2Te, is approximately twice more solvent-dependent as compared to selenium chemical shift in selenium standard, Me2Se. In particular, according to this paper,66 in relation to neat Me2Te, the 125Te NMR chemical shifts of Me2Te diluted in benzene, acetone, and chloroform are, accordingly, −3.1, −12.7, and −18.8 ppm. Moreover, as follows from this work,66 that tellurium chemical shifts of some dialkyl tellurides diluted in different solvents could vary substantially, namely, by dozens of parts per millions. For example, according to experimental data presented in the paper,66 the difference between tellurium chemical shifts of diethyl telluride diluted in benzene and chloroform is as much as −27 ppm. This is far from being negligible as compared to experimental tellurium shift (374 ppm)61 demonstrating the importance of solvent effects. To approve this assumption and estimate solvent corrections we took into account solvent effects in 125Te NMR chemical shifts within the IEF-PCM scheme40,41 at the nonrelativistic level of theory. In the latter, the solvent effect is simulated as an apparent charge distribution spread on the cavity surface not taking into account the solute−solvent

a Experimental data are taken from different sources: [61] for 2, 3, 4, and 5; [62] for 6; [63] for 7; [64] for 9 and 10; [65] for 11; [66] for 8, 12, 13, and 14. bConformationally averaged.

constant of dimethyl telluride (1) of 110 ppm. The vibrationally averaged tellurium chemical shifts of conformationally labile dimethyl and diethyl tellurides (2 and 3) were obtained by averaging over the conformers as in the rest of calculations. As it is mentioned above, relativistic corrections (Δδrel) were obtained as the differences of relativistic tellurium chemical shifts (δrel) calculated at a fully four-component GIAO-PBE0/LDBS-3 level (for a standard, dyall.av4z was used Table 6. GIAO-PBE0 Benchmark Calculations of

125

Te NMR Chemical Shiftsa

compound

solvent

δnonrel

Δδrel

Δδsolv

Δδvib

δtotal

ε, %

expb

c

C6D6 C6D6 C6D6 PhCH3 (CD3)2CO (CD3)2CO

293 455 242 567 597 696

38 113 107 230 161 181

−2 15 36 138 71 78

20 −13 −61 −45 −22 −39

349 570 324 890 807 916

7 7 4 4 3 1

374 534 338 858 782 905

2 3c 4 7 12 13

a In parts per million, taking into account relativistic, solvent, and vibrational corrections. bExperimental data are taken from different sources: [61] for 2, 3, and 4; [63] for 7; [66] for 12 and 13. cConformationally averaged.

G


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As shown in Figure 4, the MAPEs (ε averaged over the considered series of six compounds) amounts to, accordingly,

interactions at short distances (the so-called non-electrostatic effects), so that all solvent effects calculated within this scheme are constrained not to take into account any specific solvation effects. As it is mentioned above, solvent corrections (Δδsolv) were obtained as the differences of 125Te NMR chemical shifts calculated using the IEF-PCM scheme and those calculated in gas phase at the nonrelativistic level of theory within the GIAOPBE0 approach. Herewith, to evaluate the tellurium chemical shifts with taking into account solvent effects, we used the following tellurium shielding constants of dimethyl telluride evaluated at the nonrelativistic GIAO-PBE0/dyall.av4z (FBS-1) level: 2891 ppm for C6D6, 2933 ppm for (CD3)2CO, and 2893 ppm for C6H5CH3. Another interesting finding following from the data presented in Table 6 is that the relativistic correction (Δδrel) in average over six compounds amounts to 22% in relation to the total tellurium chemical shifts (δtotal). At that, it follows from Table 6 that solvent (Δδsolv) and vibrational (Δδvib) effects might be essential, reaching in absolute value as much as accordingly ca. 138 and 61 ppm. In average, Δδsolv and Δδvib amount to 8 and 6%, respectively. Shown in Figure 3 are the absolute percentage errors of 125Te NMR chemical shifts in the series of six representative

Figure 4. MAPEs of 125Te NMR chemical shifts of 2, 3, 4, 7, 12, and 13, calculated at the GIAO-PBE0 level with taking into account solvent, relativistic, and vibrational corrections in comparison with experiment.

24% for δnr, 16% for δnr + Δδsolv, 9% for δnr + Δδrel + Δδsolv, and only 4% for δt = δnr + Δδrel + Δδsolv + Δδvib. Note that the recent paper by S. Hayashi et al.14 on tellurium chemical shifts demonstrates much more moderate statistical parameters over the series of 45 compounds, namely, the MAPEs of their three calculation schemes, SO-ZORA-OPBE// SO-ZORA-OPBE, SO-ZORA-BLYP//SO-ZORA-OPBE, and SO-ZORA-BLYP//MP2 with QZ4 Pae-FBS (for details, see original paper) are, accordingly, 20, 23, and 25%. Concluding Remarks. In this paper, we have analyzed the main factors affecting the accuracy and computational cost of the calculation of 125Te NMR chemical shifts at the GIAO− DFT level (namely, methods, LDBS schemes, relativistic corrections, and solvent and vibrational effects) and proposed a versatile computational scheme for the tellurium chemical shifts of the medium-sized organotellurium compounds. In view of the “relativistic orientation” of the present study, we have thoroughly investigated the performance of different locally dense basis set schemes, constructed from “relativistic” Dyall’s basis sets. For the practical calculations of 125Te NMR chemical shifts of the medium-sized organotellurium compounds among a number of LDBS schemes investigated, we suggest the most efficient one, LDBS3, implying dyall.av4z basis set on tellurium atom, dyall.av3z basis set on all adjacent atoms, and dyall.av2z on the rest of the atoms. For diethyl and divinyl telluride, the LDBS3 scheme appeared to be two times smaller in the meaning of the number of basis set functions as compared to dyall.av4z full basis set, providing the deviation from the latter of not larger than 1 ppm. We have also investigated the performance of six exchangecorrelation functionals, namely, KT2, KT3, PBE, PBE0, B3LYP, and BLYP, at the nonrelativistic and relativistic levels through the comparison of calculated tellurium chemical shifts in a representative series of 13 organotellurium compounds with experimental data. The best results were achieved within the GIAO-PBE0/LDBS-3 scheme at the full four-component relativistic level of theory, taking into account solvent corrections, which is characterized by MAPE of as small as 6%.

Figure 3. Absolute percentage errors of 125Te NMR chemical shifts for the series of 2 (a), 3 (b), 4 (c), 7 (d), 12 (e), and 13 (f), calculated at the GIAO-PBE0 level with taking into account solvent, relativistic, and vibrational corrections in comparison with experiment.

compounds 2, 3, 4, 7, 12, and 13 calculated at four different levels: (a) at the nonrelativistic GIAO-PBE0 level in gas phase (δnr); (b) at the nonrelativistic GIAO-PBE0 level with taking into account solvent corrections within the IEF-PCM formalism (δnr + Δδsolv); (c) at the full four-component relativistic GIAO-PBE0 level with solvent corrections (δrel + Δδsolv = δnr + Δδrel + Δδsolv); and (d) at the full four-component relativistic GIAO-PBE0 level with solvent and vibrational corrections (δtotal = δrel + Δδsolv + Δδvib). As one can see in Figure 3, absolute percentage errors (ε) for each compound gradually decrease with taking into account more and more corrections, those of relativistic, solvent, and rovibrational effects. H


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(2) Ley, S. V.; Meerholz, C. A.; Barton, D. H. R. Diaryl telluroxides as new mild oxidising reagents. Tetrahedron 1981, 37, 213−223. (3) Knapp, F. F. In Selenium and Tellurium as Carbon Substitutes; Spencer, R. D., Ed.; Grune&Stratton: New York, 1981; Vol. 11, pp 345−391. (4) Cacho, V. D. D.; Siarkowski, A. L.; Morimoto, N. I.; Borges, B.-H. V.; Kassab, L. R. P. In Microelectronics Technology and Devices-SBMicro 2010; Pavanello, M. A., Claeys, C., Martino, J. A., Eds.; The Electrochemical Society: Pennington, NJ, 2010; Vol. 31, No. 1, pp 225−229. (5) Sekhon, B. S. Metalloid compounds as drugs. Res. Pharm. Sci. 2013, 8, 145−158. (6) Saue, T.; Helgaker, T. Four-component relativistic Kohn−Sham theory. J. Comput. Chem. 2002, 23, 814−823. (7) MacDonald, A. H.; Vosko, S. H. A relativistic density functional formalism. J. Phys. C: Solid State Phys. 1979, 12, 2977−2990. (8) Wolff, S. K.; Ziegler, T.; van Lenthe, E.; Baerends, E. J. Density functional calculations of nuclear magnetic shieldings using the zerothorder regular approximation (ZORA) for relativistic effects: ZORA nuclear magnetic resonance. J. Chem. Phys. 1999, 110, 7689−7698. (9) Van Wüllen, C. Relativistic all-electron density functional calculations. J. Comput. Chem. 1999, 20, 51−62. (10) Ditchfield, R. Molecular orbital theory of magnetic shielding and magnetic susceptibility. J. Chem. Phys. 1972, 56, 5688−5693. (11) Ditchfield, R. Self-consistent perturbation theory of diamagnetism. Mol. Phys. 1974, 27, 789−807. (12) Wolinski, K.; Hinton, J. F.; Pulay, P. Efficient implementation of the gauge-independent atomic orbital method for NMR chemical shift calculations. J. Am. Chem. Soc. 1990, 112, 8251−8260. (13) Ruiz-Morales, Y.; Schreckenbach, G.; Ziegler, T. Calculation of 125 Te chemical shifts using gauge-including atomic orbitals and density functional theory. J. Phys. Chem. A 1997, 101, 4121−4127. (14) Hayashi, S.; Matsuiwa, K.; Nakanishi, W. Relativistic effects on the 125Te and 33S NMR chemical shifts of various tellurium and sulfur species, together with 77Se of selenium congeners, in the framework of a zeroth-order regular approximation: applicability to tellurium compounds. RSC Adv. 2014, 4, 44795−44810. (15) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 785−789. (16) Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 3098−3100. (17) Swart, M.; Ehlers, A. W.; Lammertsma, K. Performance of the OPBE exchange-correlation functional. Mol. Phys. 2004, 102, 2467− 2474. (18) Hada, M.; Fukuda, R.; Nakatsuji, H. Dirac−Fock calculations of the magnetic shielding constants of protons and heavy nuclei in XH2 (X = O, S, Se, and Te): a comparison with quasi-relativistic calculations. Chem. Phys. Lett. 2000, 321, 452−458. (19) Hada, M.; Wan, J.; Fukuda, R.; Nakatsuji, H. Quasirelativistic study of 125Te nuclear magnetic shielding constants and chemical shifts. J. Comput. Chem. 2001, 22, 1502−1508. (20) Reiher, M.; Wolf, A. Exact decoupling of the Dirac Hamiltonian. II. The generalized Douglas−Kroll−Hess transformation up to arbitrary order. J. Chem. Phys. 2004, 121, 10945−10956. (21) Wolf, A.; Reiher, M.; Hess, B. A. The generalized Douglas−Kroll transformation. J. Chem. Phys. 2002, 117, 9215−9226. (22) Wan, J.; Fukuda, R.; Hada, M.; Nakatsuji, H. Quasi-Relativistic study of 199Hg nuclear magnetic shielding constants of dimethylmercury, disilylmercury and digermylmercury. J. Phys. Chem. A 2001, 105, 128−133. (23) Fukuda, R.; Hada, M.; Nakatsuji, H. Quasirelativistic theory for the magnetic shielding constant. I. Formulation of Douglas−Kroll− Hess transformation for the magnetic field and its application to atomic systems. J. Chem. Phys. 2003, 118, 1015−1026. (24) Antušek, A.; Pecul, M.; Sadlej, J. Relativistic calculation of NMR properties of XeF2, XeF4 and XeF6. Chem. Phys. Lett. 2006, 427, 281− 288.

A special attention in this study was focused on relativistic effects evaluated within the Dirac’s full four-component relativistic scheme at the GIAO-PBE0 level. It was found that relativistic paramagnetic correction Δσpara (varying from 1252 to 1625 ppm) dominates over diamagnetic relativistic correction Δσdia (varying from −601 to −579 ppm) resulting in a total shielding relativistic correction Δσrel of ∼661−1030 ppm, which is 25−35% of the total relativistic values of tellurium absolute shielding constants without taking into account solvent and vibrational corrections. It is noteworthy that diamagnetic relativistic corrections are practically the same along the considered series of compounds (standard deviation is ∼5 ppm) as compared to paramagnetic relativistic parts (standard deviation is ∼97 ppm). In the 125Te NMR chemical shifts scale, relativistic corrections are ∼17−369 ppm, which is 26% of their total relativistic values. For the benchmark series of six tellurium compounds, 2, 3, 4, 7, 12, and 13, the average relativistic corrections to tellurium chemical shifts amount to 22% in relation to their total values calculated at the full four-component relativistic DFT-PBE0 level with taking into account solvent and rovibrational corrections amounting to 8 and 6% in average. Within this series, the MAPE is shown to gradually decrease from 24% for the nonrelativistic level to 4% for the full four-component relativistic domain with all corrections being taken into account. The most essential result of the present communication is that proposed for 125Te NMR chemical shifts fully relativistic four-component computational scheme GIAO-PBE0/LDBS-3 (shielding constants and chemical shifts) + nonrelativistic MP2 (geometry optimization) with taking into account solvent and vibrational corrections results in an MAPE of as small as 4% in the range of 1000 ppm, which looks very promising as compared to any of the two-component computational schemes.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b03198. Differences of the relativistic values of NMR shielding constants with nonrelativistic ones of 13C carbon atoms in α-position relative to 125Te tellurium atom (in ppm) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] or [email protected]. ORCID

Irina L. Rusakova: 0000-0002-1089-3864 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support of The Federal Agency for Scientific Organizations, Russia (Project No. 0342-2014-0006) is greatly appreciated.

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REFERENCES

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Calculation of 125Te NMR Chemical Shifts at the Full Four

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