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

Abstract

The relativistic effects on absolute magnetic shielding tensors [σ(Z: Z = Te, Se and S)] are explicitly evaluated for various tellurium, selenium and sulfur species using the DFT(BLYP)-GIAO method. Calculations of σ(Te), σ(Se) and σ(S) are performed under the spin–orbit ZORA relativistic (Rlt-so) and nonrelativistic (Non) conditions with the Slater-type basis sets of the quadruple zeta all electron with four polarization functions (QZ4Pae). Structures optimized at the MP2 level under nonrelativistic conditions are employed for the evaluations. While the range of the relativistic effects on the total shielding tensors for Te (Δσ^t(Te)_Rlt-so = σ^t(Te)_Rlt-so − σ^d+p(Te)_Non) is predicted to be −55 to 658 ppm, that for Δσ^t(S) is predicted to be 5 to 32 ppm, except for Me₂SBr₂ (TBP), where Δσ^t(S)_Rlt-so = −29 ppm. The range for Δσ^t(Se) is 2 to 153 ppm. The magnitudes of the relativistic effects on σ^t(Te), σ^t(Se) and σ^t(S) are about 25 : 5 : 1. The applicability of σ^t(Te)_Rlt-so to analyze δ(Te)_obsd is also examined, mainly with the OPBE//OPBE method under the spin–orbit ZORA relativistic conditions with QZ4Pae, in addition to the above method.

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Introduction

NMR spectroscopy is established as a powerful tool to investigate molecular-level structures and dynamics in modern chemical sciences.¹ However, we often worry about the relativistic effects on NMR parameters, in order to better understand the observed values with the physical meanings. It is inevitable to consider the relativistic effects² when absolute magnetic shielding tensors of nuclei N(σ(N)) are evaluated, especially for the N of heavier atoms.^3,4 The relativistic effects on σ(N)^5,6 can be evaluated successfully under the zeroth-order regular approximation (ZORA),⁷ as ZORA predicts more accurately for lower energy valence electrons than deep core states.^7a,8

It is expected that the total values of σ(N) (σ^t(N)) can be expressed as the sum of diamagnetic (σ^d(N)), paramagnetic (σ^p(N)), and spin–orbit (σ^so(N)) terms, as expressed in eqn (1),⁹ when evaluated considering the relativistic effects. In this paper, the calculation conditions for σ(N) are given just after σ(N) as the subscript. Therefore, σ^t(N)_Rlt-so means the total values of σ(N) calculated under the spin–orbit ZORA relativistic conditions (Rlt-so). The expectation shown in eqn (1) will be rationalized when σ^t(N) is evaluated under the Rlt-so conditions.

The framework of the ZORA Hamiltonian is expressed by eqn (2),⁸ where V(r) is the effective Kohn–Sham potential given by the sum of the nuclear, Hartree and exchange-correlation potentials in the external magnetic field B within DFT (density functional theory), σ are the Pauli matrices, and c is the speed of light. π and K(r) are given by eqn (3) and (4), respectively. The first two terms in eqn (2) form the basis of the scalar relativistic approximation [σ^d+p(N)], and the third term represents the spin–orbit coupling term [σ^so(N)]. As a result, eqn (1) is rationalized based on eqn (2) by neglecting the fourth term for the coupling between the spin and magnetic field.

σ^t(N) = σ^d(N) + σ^p(N) + σ^so(N) = σ^d+p(N) + σ^so(N) (1)

H_ZORA = V(r) + π(K(r)/2)π + (K²(r)/4c²)σ[∇V(r) × p] − (K(r)/c)σp (2)

π = p + (1/c)A(r); B = ∇ × A(r) (3)

K(r) = {1 − (V(r)/2c²)}⁻¹ (4)

¹²⁵Te and ⁷⁷Se NMR parameters^10–12 are the typical cases for the relativistic effects to be considered. We recently reported the relativistic effects on σ(Se) evaluated explicitly and separately by the scalar ZORA and spin–orbit ZORA relativistic terms for various selenium species, employing the structures optimized at the B3LYP^13a,b level under nonrelativistic conditions (Non) (B3LYP_Non).¹⁴ The observed NMR chemical shifts of Se (δ(Se)_obsd) are well explained by σ^t(Se), considering both the scalar ZORA and spin–orbit ZORA relativistic terms. The scalar ZORA relativistic terms calculated under the scalar ZORA relativistic conditions (σ^d+p(Se)_Rlt-sc) are very close to those calculated under the spin–orbit ZORA relativistic conditions (σ^d+p(Se)_Rlt-so), although the spin–orbit ZORA relativistic terms (σ^so(Se)_Rlt-so) can only be obtained under the Rlt-so conditions. Therefore, the effect evaluated under the Rlt-so conditions will be employed for the discussion.

The magnitudes of the relativistic effects are evaluated as the differences from the corresponding values calculated under nonrelativistic conditions [Δσ(Se)_Rlt = σ(Se)_Rlt − σ(Se)_Non]. The relativistic effects on σ(Se) of the scalar ZORA and spin–orbit ZORA relativistic terms, as calculated under the Rlt-so conditions, are abbreviated as Δσ^d+p(Se)_Rlt-so (= σ^d+p(Se)_Rlt-so − σ^d+p(Se)_Non) and σ^so(Se)_Rlt-so, respectively. The values were reported to be −127 to −26 ppm (downfield) and 95 to 221 ppm (upfield), respectively, and the effect on σ^t(Se) (Δσ^t(Se)_Rlt-so = Δσ^d+p(Se)_Rlt-so + σ^so(Se)_Rlt-so) was determined to be 2 to 153 ppm for various selenium species (40 species) under the Rlt-so conditions.¹⁴ It is worthwhile commenting that σ^so(Se)_Rlt-so should be contained when δ(Se)_obsd is analyzed considering the relativistic effects, since σ^d+p(Se)_Rlt-so would not improve the applicability of σ(Se) so much as to analyze δ(Se)_obsd. Indeed, σ^d+p(Se)_Non often reproduce δ(Se)_obsd better than the case of σ^d+p(Se)_Rlt-so. The physical meanings of δ(Se)_obsd should also be explained by considering both the scalar ZORA and spin–orbit ZORA relativistic effects.

It is worth asking, what are the relativistic effects on σ(Te) and σ(S)? The effects on σ(S) would be negligibly smaller, whereas that on σ(Te)¹⁵ must be much larger, relative to the case of σ(Se).¹⁴ It must also be of great interest to discuss the trends of the effect on σ(Te), σ(Se), and σ(S) for the group of 16 elements. δ(Te)_obsd are reported to be proportional to δ(Se)_obsd when the values of structurally equivalent compounds are compared. This may show the existence of some linear correlations between σ(Te) and σ(Se), not only for those evaluated under nonrelativistic conditions, but also for those evaluated under relativistic conditions.

Herein, we present the results of the calculations of the relativistic effects on σ(Z) and the components, σ^d(Z), σ^p(Z), σ^d+p(Z), σ^so(Z), and σ^t(Z) (where Z = Te and S), for various tellurium and sulfur species (40 species for each Z), together with the Se derivatives (see Table 1 for Z = Te). The values are evaluated explicitly and separately by the scalar ZORA and spin–orbit ZORA relativistic terms. The calculation method for σ(Te) and σ(S) is set up equal to that for σ(Se),¹⁴ mainly for convenience of comparison. The trends of the effect on σ(Te), σ(Se), and σ(S) are also discussed. The applicability of σ^t(Te)_Rlt-so to analyze δ(Te)_obsd is examined carefully, employing the σ^t(Te)_Rlt-so of 46 organic tellurium species, other than those in Table 1, (see Table 2, although a few species are in common), calculated similarly at the OPBE level under the Rlt-so conditions on the optimized structures using the same method (OPBE_Rlt-so//OPBE_Rlt-so), in addition to those with BLYP_Rlt-so//OPBE_Rlt-so and BLYP_Rlt-so//MP2_Non.

Table 1 The σ^d(Te), σ^p(Te), σ^d+p(Te), σ^so(Te), and σ^t(Te) values calculated at the BLYP level with the QZ4Pae basis sets under the nonrelativistic (Non) and spin-orbit ZORA relativistic (Rlt-so) conditions for various tellurium species^a,b,c,d

Species	σ^dNon	σ^pNon	σ^d+pNon	σ^dRlt-so	Δσ^dRlt-so	σ^pRlt-so	Δσ^pRlt-so	σ^d+pRlt-so	Δσ^d+pRlt-so	σ^soRlt-so	σ^tRlt-so	Δσ^tRlt-so
a Structures optimized at the MP2Non level of Gaussian 03 being employed.b σ(Te) (in ppm) are denoted by σ in Table.c ΔσRlt-so = σRlt-so − σNon.d Δσ^tRlt-so = σ^tRlt-so − σ^d+pNon = Δσ^d+pRlt-so + σ^soRlt-so.
H2Te (C2v)	5360.1	−1856.4	3503.6	5302.6	−57.5	−2006.3	−149.8	3296.3	−207.3	743.5	4039.9	536.2
HTe^− (C∞v)	5365.2	−993.7	4371.5	5308.4	−56.8	−1102.6	−108.9	4205.8	−165.7	823.9	5029.7	658.2
H3Te^+ (C3v)	5353.7	−2167.7	3186.1	5296.4	−57.3	−2292.2	−124.5	3004.3	−181.8	639.0	3643.3	457.2
H4Te (C2v)	5356.4	−1756.7	3599.7	5299.2	−57.2	−1872.3	−115.6	3426.9	−172.8	741.6	4168.5	568.8
H5Te^− (C4v)	5350.4	−1983.0	3367.4	5293.9	−56.5	−2036.2	−53.1	^−3257.7	−109.6	557.2	3814.9	447.6
H5Te^+ (C4v)	5348.2	−2359.8	2988.5	5289.8	−58.5	−2447.1	−87.3	2842.6	−145.8	645.0	3487.6	499.2
H6Te (Oh)	5342.6	−2169.8	3172.8	5285.1	−57.5	−2231.4	−61.5	3053.8	−119.0	659.0	3712.8	540.0
MeTe^− (Cs)	5366.0	−2252.0	3114.0	5309.3	−56.7	−2436.1	−184.1	2873.2	−240.8	888.5	3761.8	647.8
MeTeH (Cs)	5361.3	−2312.2	3049.0	5304.3	−57.0	−2479.5	−167.3	2824.8	−224.3	741.1	3565.9	516.9
Me2Te (C2v)	5362.5	−2719.1	2643.4	5305.9	−56.6	−2890.6	−171.5	2415.3	−228.1	725.2	3140.5	497.1
EtTeH (Cs)	5360.9	−2392.9	2968.0	5304.2	−56.7	−2568.4	−175.5	2735.8	−232.2	716.4	3452.2	484.2
Et2Te (C2v)	5362.9	−2911.9	2450.9	5306.4	−56.5	−3105.8	−193.8	2200.6	−250.4	686.3	2886.8	435.9
MeTeTeMe (C2)	5364.1	−2736.1	2628.0	5307.3	−56.7	−3110.7	−374.6	2196.7	−431.3	884.9	3081.6	453.6
Me3Te^+ (C3)	5357.0	−2954.6	2402.4	5300.2	−56.8	−3121.3	−166.7	2178.9	−223.5	634.0	2812.8	410.5
Me4Te (C2v)	5357.5	−2399.5	2958.0	5300.6	−56.9	−2549.8	−150.3	2750.8	−207.2	718.2	3469.0	511.0
Me5Te^− (Cs)	5351.5	−2465.2	2886.4	5294.6	−56.9	−2558.7	−93.6	2735.9	−150.5	560.8	3296.7	410.3
Me5Te^+ (Cs)	5352.3	−2978.8	2373.5	5293.5	−58.8	−3111.9	−133.1	2181.5	−192.0	626.7	2808.3	434.8
Me6Te (Ci)	5348.9	−2542.4	2806.5	5287.2	−61.7	−2638.7	−96.3	2648.5	−158.0	649.5	3298.0	491.5
H2TeF2 (C2v)	5349.6	−3146.3	2203.4	5292.5	−57.2	−3351.6	−205.3	1940.8	−262.5	507.1	2447.9	244.6
H2TeO (Cs)	5354.2	−3280.6	2073.6	5297.0	−57.2	−3497.8	−217.2	1799.2	−274.4	563.2	2362.4	288.8
H2TeO2 (C2v)	5354.2	−3433.9	1920.3	5296.3	−57.9	−3656.5	−222.6	1639.8	−280.5	549.0	2188.8	268.5
H4TeO (C2v)	5349.8	−2705.3	2644.5	5291.5	−58.3	−2834.0	−128.7	2457.5	−187.0	598.0	3055.5	411.0
H2TeF2O (C2v)	5349.0	−3262.8	2086.2	5292.1	−56.9	−3480.2	−217.4	1812.0	−274.2	550.9	2362.8	276.6
Me2TeF2 (C2v)	5350.7	−3498.9	1851.8	5294.3	−56.4	−3741.6	−242.7	1552.7	−299.1	506.8	2059.5	207.7
(CF3)2TeF2 (C2v)	5351.9	−3472.5	1879.4	5295.6	−56.2	−3732.1	−259.6	1563.6	−315.8	432.1	1995.7	116.3
Me2TeCl2 (C2v)	5354.4	−3119.7	2234.7	5298.0	−56.3	−3322.4	−202.7	1975.6	−259.1	572.0	2547.6	312.9
Me2TeBr2 (C2v)	5356.8	−3034.2	2322.7	5299.6	−57.2	−3251.7	−217.5	2048.0	−274.7	542.0	2590.0	267.4
Me2TeO (Cs)	5356.3	−3591.0	1765.3	5299.1	−57.2	−3828.0	−237.0	1471.1	−294.2	570.0	2041.1	275.8
Me2TeO2 (C2v)	5355.9	−3681.1	1674.8	5298.7	−57.2	−3936.7	−255.6	1362.0	−312.9	555.5	1917.4	242.6
Me2TeF2O (C2v)	5350.4	−3494.5	1855.9	5293.7	−56.7	−3739.8	−245.3	1553.9	−302.0	549.3	2103.1	247.2
F2TeO (Cs)	5351.5	−4143.6	1207.9	5294.8	−56.7	−4530.0	−386.4	764.7	−443.1	388.1	1152.9	−55.0
Cl2TeO (Cs)	5353.9	−4335.1	1018.8	5298.2	−55.7	−4735.4	−400.2	562.9	−455.9	438.8	1001.7	−17.1
F2TeO2 (C2v)	5351.2	−3492.0	1859.2	5293.5	−57.7	−3756.0	−264.1	1537.4	−321.8	508.7	2046.1	186.9
TeF4 (C2v)	5345.9	−3671.4	1674.4	5289.1	−56.8	−3983.2	−311.7	1305.9	−368.5	331.0	1636.9	−37.5
TeCl4 (C2v)	5349.1	−4157.4	1191.7	5294.9	−54.3	−4562.3	−404.8	732.6	−459.0	469.5	1202.1	10.5
TeF5^− (C4v)	5340.6	−3426.1	1914.5	5284.7	−55.9	−3709.5	−283.4	1575.2	−339.3	347.2	1922.4	7.9
TeF5^+ (C4v)	5345.6	−3240.8	2104.8	5288.2	−57.4	−3509.7	−268.9	1778.4	−326.4	640.9	2419.4	314.5
HTeF5 (C4v)	5342.2	−3128.6	2213.7	5282.4	−59.8	−3385.6	−257.1	1896.8	−316.9	629.9	2526.6	313.0
MeTeF5 (Cs)	5341.4	−3234.9	2106.4	5283.9	−57.5	−3494.5	−259.6	1789.4	−317.0	601.6	2391.0	284.6
TeF6 (Oh)	5340.1	−3047.1	2293.0	5283.6	−56.5	−3301.1	−254.0	1982.5	−310.5	649.6	2632.1	339.1

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Table 2 The σ^d+p(Te:M)_Non:r, σ^d+p(Te:M)_Rlt-so:r and/or σ^t(Te:M)_Rlt-so:r values calculated at the BLYP and OPBE levels under the nonrelativistic (Non) and spin–orbit ZORA relativistic (Rlt-so) conditions with the QZ4Pae basis sets for various tellurium species,^a together with the δ(Te)_obsd values^b

Species (no.)	σ^d+p(Te:M)Non:r	σ^d+p(Te:M)Rlt-so:r	σ^t(Te:M)Rlt-so:r	σ^t(Te:M)Rlt-so:r	σ^t(Te:M)Rlt-so:r	δ(Te:M)obsd	Solvent/comment
Basis set: evaluation of σ(Te:M)	OPBENon	OPBERlt-so	OPBERlt-so	BLYPRlt-so	BLYPRlt-so
Basis set: structural optimization	OPBERlt-so	OPBERlt-so	OPBERlt-so	OPBERlt-so	MP2Non
a σ^d+p(Te:M)r = −(σ^d+p(Te:M) − σ^d+p(Te:Me2Te)) and σ^t(Te:M)r = −(σ^t(Te:M) − σ^t(Te:Me2Te)) in ppm.b See also ref. 26 and 27.c OTf^− is a counteranion.d This work.e BF4^−/PF6^− is counteranion.f PF6^− is counteranion.
TeMe2 (C2v:1)	0.0	0.0	0.0	0.0	0.0	0.0	Neat/ref. 28–30
TeP(iPr)3 (C1:2)	−1145.0	−1119.3	−1201.7	−1168.4	−1360.7	−1000.3	Ref. 31
Te(SiMe3)2 (C2v:3)	−1012.0	−1041.8	−1053.0	−1078.0	−1106.5	−842	Ref. 32
TeH2 (C2v:4)	−778.0	−813.1	−820.9	−927.0	−899.4	−621	Ref. 33 and 35
TePMe3 (C1:5)	−900.7	−841.8	−916.2	−883.6	−881.4	−513.4	Ref. 31
Cyclo-(CMeNMe)2C=Te (C2v:6)	−271.1	−197.6	−257.1	−296.5	−301.6	−168	Ref. 35
TeMe4 (C2v:7)	−132.0	−166.0	−116.0	−188.2	−145.2	−67.0	C6D6/ref. 36
MeTeEt (av:8)	121.0	140.8	149.1	164.9	156.2	165	CDCl3/ref. 37
MeTeEt (Cs:8a) (as 0.0)	89.3	100.0	117.1	129.0	120.0
MeTeEt (C1-g:8b) (−2.9)	152.7	181.6	181.0	200.8	192.3
Cyclo-C6H4(CH2)2Te (C2:9)	211.6	242.1	241.7	237.1	123.6	269	(CD3)2NCHO/ref. 29 and 38
EtTeEt (av:10)	250.2	290.1	306.7	337.6	309.1	356	CDCl3/ref. 37
EtTeEt (C2v:10a) (as0.0)	207.4	232.2	265.4	291.4	253.6
EtTeEt (C1-g:10b) (−2.7)	250.5	291.5	308.6	341.8	323.8
EtTeEt (C2-gg:10c) (−7.1)	292.7	346.6	346.2	379.7	349.7
MeTeTeMe (C2:11)	−36.2	154.6	5.1	40.9	58.9	49	CDCl3/ref. 34
spiro-Te2C5H8O (C1:12)	37.1	177.4	88.3	85.1	51.9	57.1	Ref. 39
EtTeTeEt (av:13)	28.9	238.3	101.2	180.2	149.2	166	CD2Cl2/ref. 40
EtTeTeEt (C2:13a) (as 0.0)	84.9	293.8	160.2	180.2	118.2
EtTeTeEt (C1-g:13b) (−4.4)	11.8	229.4	96.9		148.5
EtTeTeEt (C2-gg:13c) (−8.6)	−10.0	191.8	46.4		181.0
Cyclo-Te(CH2CH2CH2)2Te (C2v:14)	73.6	115.6	110.2	149.9	354.1	164	Ref. 41
Cyclo-Te(C(tBu)CH)2CH2 (Cs:15)	140.8	192.7	225.5	173.2	162.1	257	Ref. 42
PhTeMe (Cs:16)	314.7	369.6	383.4	404.7	404.5	329	CDCl3/ref. 37
Cyclo-C6H4TeCH2CO (Cs:17)	160.9	184.8	219.6	172.8	221.4	383	CDCl3/ref. 43
PhTeTePh (av:18)	719.3	619.4	360.0	407.4	447.4	420	CDCl3/ref. 44
PhTeTePh (C2:18a) (as 0.0)	1326.3	904.4	514.1	595.3	629.9
PhTeTePh (C2:18b) (−2.1)	112.3	334.4	205.9	219.4	264.9
Cyclo-Te(C6H4)2O (Cs:19)	247.4	281.2	339.3	305.5	355.7	424	CDCl3/ref. 43
Me3Te^+ (C3:20)	308.5	302.7	392.7	307.4	327.7	408^c	Ref. 45
Cyclo-Te(C(tBu)CH)2CO (C2v:21)	239.5	295.0	335.6	273.1	261.3	445	Ref. 42
Cyclo-Te(C6H4)2CO (Cs:22)	222.0	248.1	325.3	281.3	316.6	468	CDCl3/ref. 43
Cyclo-Te(C6H4)2CH2 (Cs:23)	345.0	398.5	455.5	417.2	447.7	512	CDCl3/ref. 43
TeF6 (Oh:24)	458.1	532.6	611.7	445.9	508.4	545	Neat/ref. 28 and 29
PhTeCH = CH2 (C1:25)	475.5	547.9	593.7	588.7	590.1	615	Ref. 46
PhTePh (C2:26)	573.3	667.9	703.7	713.2	662.1	688	CDCl3/ref. 34 and 44
CF3TeTeCF3 (C2:27)	497.0	769.5	663.5	786.5	664.8	686	CDCl3/ref. 47
Cyclo-C6H4TeCH=CH (C1:28)	499.9	574.9	628.9	561.3	590.2	727	CDCl3/ref. 48
Me2TeCl2 (C2v:29)	495.0	526.4	682.5	639.5	592.9	734	Ref. 49
Me2TeBr2 (C2v:30)	402.1	449.0	633.3	610.7	550.5	649	Ref. 49
PhTeCl2Me (C1:31)	561.3	609.8	774.9	735.5	648.8	809.7^d	CDCl3/this work
PhTeBr2Me (C1:32)	480.6	548.1	729.5	709.5	603.0	744.6^d	CDCl3/this work
Cyclo-Te(CH)4 (C2v:33)	523.0	613.4	671.3	591.6	623.0	782	(CD3)2CO/ref. 50
Cyclo-Te(C6H4)2Te (C2v:34)	771.7	876.0	978.1	955.5	919.7	888	Ref. 51
Te(tBu)2 (C2:35)	912.7	1060.0	1091.7	1157.4	831.8	992	Toluene-d8/ref. 52
Bicyclo-Te(CH2CH2CH2O)2 (C2:36)	905.1	1010.8	1194.4	1118.7	1027.6	1096	Ref. 53
Cyclo-[Te(CH2CH2CH2)Te]^2+ (C2v:37)	998.1	1215.9	1271.6	1263.6	1517.6	1304^e	Ref. 41
Cyclo-Te^+(C(tBu)CH)2CH (C2v:38)	1088.6	1237.1	1351.2	1271.6	1220.2	1304^f	Ref. 42
CF3TeCF3 (C2v:39)	1102.5	1258.7	1342.7	1501.4	1279.8	1368	CD3CN/ref. 54
CF3TeF2CF3 (C2v:40)	948.1	1046.0	1346.7	1268.9	1144.8	1187	CD3CN/ref. 42, 53 and 55
CF3TeCF2Cl (Cs:41)	1251.5	1426.2	1528.6	1680.6	1466.4	1566	CD3CN/ref. 54
Cyclo-F2CTe2CF2 (C2:42)	2122.6	2316.0	2369.2	2530.6	2379.3	2321.7	Ref. 56
Cyclo-C6H4(CMe2)2C=Te (C2:43)	2797.3	2930.1	2893.6	2964.5	2905.9	2858	CDCl3/ref. 55
TeCl6^2− (Oh:44)	1202.4	1441.9	1717.2	1778.7	1813.7	1531	Ref. 29 and 57
TeBr6^2− (Oh:45)	1225.8	1581.9	1888.0	2008.1	1930.8	1348	Ref. 29
Cyclo-Te4^2+ (C2v:46)	2092.8	2414.4	2542.8	2681.6	2911.4	2665	Ref. 58

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The relativistic effects on σ(Z: Z = Te and S) will be discussed employing Δσ(Z)_Rlt-so (= σ(Z)_Rlt-so − σ(Z)_Non), similar to the case of Z = Se. The Δσ(Z)_Rlt-so values are analyzed by decomposing the contributions into Δσ^d(Z)_Rlt-so (=σ^d(Z)_Rlt-so − σ^d(Z)_Non), Δσ^p(Z)_Rlt-so (=σ^p(Z)_Rlt-so − σ^p(Z)_Non), Δσ^d+p(Z)_Rlt-so (=σ^d+p(Z)_Rlt-so − σ^d+p(Z)_Non = Δσ^d(Z)_Rlt-so + Δσ^p(Z)_Rlt-so), σ^so(Z)_Rlt-so and Δσ^t(Z)_Rlt-so (=σ^t(Z)_Rlt-so − σ^d+p(Z)_Non = Δσ^d+p(Z)_Rlt-so + σ^so(Z)_Rlt-so), according to eqn (1). The first three terms correspond to the diamagnetic, paramagnetic, and (diamagnetic + paramagnetic) terms of the relativistic effects on σ(Z), where σ(Z)_Rlt-so shows the values being calculated under the Rlt-so conditions. σ^so(Z)_Rlt-so is the spin–orbit ZORA relativistic term, which can only be calculated under the Rlt-so conditions. The symbols and definitions for σ(Z: Z = Te, Se and S) and Δσ(Z: Z = Te, Se and S) used in this paper and the previous one,¹⁴ evaluated under the Non, Rlt-sc, and Rlt-so conditions, are listed in Table S1 of the ESI.† Indeed, σ^d+p(Z)_Rlt-sc and Δσ^d+p(Z)_Rlt-sc correspond to σ^t(Z)_Rlt-sc and Δσ^t(Z)_Rlt-sc, respectively, if evaluated at the Rlt-sc level, but they will not be denoted as the latter in this series of investigations. σ^t(Z)_Rlt-so and Δσ^t(Z)_Rlt-so will be used only for the total values calculated at the Rlt-so level.

Calculation method

The σ(Z) (σ^d(Z), σ^p(Z), σ^so(Z), and σ^t(Z): Z = Te, Se, and S) values were calculated under relativistic and nonrelativistic conditions for various species containing tellurium, selenium, and sulfur nuclei, respectively. The calculation method for σ(Te) and σ(S) was set up equal to that for σ(Se), as reported recently.¹⁴ The ADF 2013 program^16–18 was employed for the calculations. The GIAO (Gauge-Independent Atomic Orbital) method¹⁹ was applied to evaluate σ(Z: Z = Te, Se and S) at the DFT level of the Becke density functional with the Lee–Yang–Parr correlation functional (BLYP)^13b,c (the DFT(BLYP)-GIAO method). Calculations of σ(Z) were performed at the BLYP level under the Non and Rlt-so conditions (BLYP_Non and BLYP_Rlt-so, respectively). The Slater-type basis sets of the quadruple zeta all-electron with four polarization functions (QZ4Pae: 3 × 1s, 3 × 2s, 3 × 2p, 5 × 3s, 4 × 3p, 3 × 3d and 2 × 4f for S, 4 × 1s, 3 × 2s, 4 × 2p, 3 × 3s, 3 × 3p, 4 × 3d, 4 × 4s, 4 × 4p, 2 × 4d and 3 × 4f for Se and 5 × 1s, 3 × 2s, 5 × 2p, 3 × 3s, 3 × 3p, 3 × 3d, 3 × 4s, 3 × 4p, 4 × 4d, 3 × 4f, 4 × 5s, 4 × 5p and 1 × 5d for Te) were applied for the calculations.^20,21 The relativistic effect on σ^t(Z) was analyzed separately by σ^d(Z), σ^p(Z), σ^d+p(Z), and σ^so(Z). The σ(S) values are given without scaling (cf.: EMPI approach²²).

The structures of tellurium, selenium, and sulfur species were optimized using the Gaussian 03 program package,²³ of which the NMR parameters are given in Tables 1, S2–S6,† Fig. 1–3 and S1–S3.† The (7433111/743111/7411/2 + 1s1p1d1f) type basis set²⁴ was employed for Te with the 6-311+G(3df,3pd) types²⁵ for the other nuclei in the optimizations (the all-electron basis set system). The optimizations were performed at the Møller–Plesset second-order energy correlation (MP2) level^20,21 and the DFT (B3LYP)^13a,b level under the Non conditions (MP2_Non and B3LYP_Non, respectively). It was found that the calculations of σ(Z: Z = Te, Se and S) were unsuccessful for a few structures, maybe due to some problem in the integration processes. In such cases, the processes can be successful by employing the structures of lower symmetry. The relativistic effects on σ(Z: Z = Te, Se, and S) were mainly calculated by employing the optimized structures at the MP2_Non level.

Fig. 1 Relativistic effects on σ(Te) for various tellurium compounds calculated at the BLYP level with QZ4Pae under Non and Rlt-so conditions: , , and stand for the total term (Δσ^t(Te)_Rlt-so = Δσ^d+p(Te)_Rlt-so + σ^so(Te)_Rlt-so), the scalar term (Δσ^d+p(Te)_Rlt-so), and the spin–orbit term (σ^so(Te)_Rlt-so), respectively. Values are from Table 1, evaluated on the structures optimized at the MP2_Non level.

Fig. 2 Relativistic effects on σ(Se) for various selenium compounds calculated at the BLYP level with QZ4Pae under Non and Rlt-so conditions: , , and stand for the total term (Δσ^t(Se)_Rlt-so = Δσ^d+p(Se)_Rlt-so + σ^so(Se)_Rlt-so), the scalar term (Δσ^d+p(Se)_Rlt-so), and the spin–orbit term (σ^so(Se)_Rlt-so), respectively. Values are from Table S2,† evaluated on the structures optimized at the MP2_Non level.

Fig. 3 Relativistic effects on σ(S) for various sulfur compounds calculated at the BLYP level with QZ4Pae under Non and Rlt-so conditions: , , and stand for the total term (Δσ^t(S)_Rlt-so = Δσ^d+p(S)_Rlt-so + σ^so(S)_Rlt-so), the scalar term (Δσ^d+p(S)_Rlt-so), and the spin–orbit term (σ^so(S)_Rlt-so), respectively. Values are from Table S3,† evaluated on the structures optimized at the MP2_Non level.

To examine the applicability of σ^t(Te)_Rlt-so to analyze δ(Te)_obsd, σ^t(Te)_Rlt-so, together with σ^d+p(Te)_Rlt-so and σ^so(Te)_Rlt-so, was evaluated similarly at the OPBE level under the spin–orbit ZORA relativistic conditions using the optimized structures and with the same method (OPBE_Rlt-so//OPBE_Rlt-so) for 46 tellurium species. Table 2 summarizes the results. The values were also calculated at the BLYP level under the spin–orbit ZORA relativistic conditions using the optimized structures at the OPBE_Rlt-so and MP2_Non levels for the 46 tellurium species (BLYP_Rlt-so//OPBE_Rlt-so and BLYP_Rlt-so//MP2_Non, respectively). The σ^d+p(Te)_Non values were also calculated with the OPBE_Non//OPBE_Rlt-so and BLYP_Non//MP2_Non methods for convenience of comparison. Partial optimizations at the OPBE_Non level were applied to obtain the structures with observed bond length(s) around Te in some tellurium compounds, to examine the applicability in more detail.

Results and discussion

Relativistic effects on σ(Te), σ(Se), and σ(S) evaluated with the QZ4Pae basis sets

Table 1 presents the σ^d(Te)_Rlt-so, σ^p(Te)_Rlt-so, σ^d+p(Te)_Rlt-so, σ^so(Te)_Rlt-so, and σ^t(Te)_Rlt-so calculated under the Rlt-so conditions with QZ4Pae, employing the structures optimized at the MP2_Non level. Table 1 also shows the σ^d(Te)_Non, σ^p(Te)_Non, and σ^d+p(Te)_Non obtained similarly under the Non conditions. The data for σ(Se) and σ(S) calculated under the Rlt-so conditions with QZ4Pae, employing the structures similarly optimized at the MP2_Non level, are summarized in Tables S2 and S3, respectively, of the ESI.† Those for σ(Te), σ(Se), and σ(S) evaluated similarly, employing the structures optimized at the B3LYP_Non level are given in Tables S4–S6, respectively, of the ESI,† for convenience of comparison.

Before discussion of the effect on each term, the relativistic effects on σ(Z) are overviewed. As shown in Table 1, the range for Δσ^d+p(Te)_Rlt-so is evaluated to be −459 to −110 ppm (downfield shifts), whereas that for σ^so(Te)_Rlt-so is 331 to 889 ppm (upfield shifts) for the Te species. Consequently, the range for Δσ^t(Te)_Rlt-so is predicted to be −55 to 658 ppm. Fig. 1 plots the data for Δσ^d+p(Te)_Rlt-so, σ^so(Te)_Rlt-so, and Δσ^t(Te)_Rlt-so, as given in Table 1. On the other hand, the ranges for Δσ^d+p(Se)_Rlt-so, σ^so(Se)_Rlt-so, and Δσ^t(Se)_Rlt-so are evaluated to be −127 to −26 ppm, 95 to 221 ppm, and 2 to 153 ppm, respectively, as shown in Table S2.† Fig. 2 summarizes the Δσ^d+p(Se)_Rlt-so, σ^so(Se)_Rlt-so, and Δσ^t(Se)_Rlt-so values.

In the case of σ(S), the ranges for Δσ^d+p(S)_Rlt-so, σ^so(S)_Rlt-so, and Δσ^t(S)_Rlt-so are evaluated to be −12 to 1 ppm, 11 to 38 ppm, and 5 to 32 ppm, except for Me₂SBr₂ (C_2v). The values for Me₂SBr₂ (C_2v) are −8.6, −19.9, and −28.6 ppm, respectively. While the Δσ^d+p(S)_Rlt-so value drops in the range, σ^so(S)_Rlt-so and Δσ^t(S)_Rlt-so seem smaller than the ranges by around 45 ppm, which would be the reflection of the second relativistic effect from heavier Br atoms. Fig. 3 summarizes the data for Δσ^d+p(S)_Rlt-so, σ^so(S)_Rlt-so, and Δσ^t(S)_Rlt-so, for which the values are given in Table S3.† The relativistic effects on σ(S) are small, and therefore, can be neglected for the usual purposes of analysis, although we must be careful when heavier atoms are attached to the atom. The relative ranges of the relativistic effects on σ^t(S)_Rlt-so, σ^t(Se)_Rlt-so, and σ^t(Te)_Rlt-so are around 1 : 5 : 25.

The relativistic effects on each term of σ(Te) are discussed next. The effects on σ^d(Te) cause small downfield shifts under the Rlt-so conditions. The Δσ^d(Te)_Rlt-so values are almost constant, at −58.0 ± 3.7 ppm for the species in Table 1. The magnitudes are about 2.6 and 9.8 times larger than those of Δσ^d(Se)_Rlt-so (−22.2 ± 2.0 ppm) and Δσ^d(S)_Rlt-so (−5.9 ± 4.4 ppm), respectively. The relativistic effects on σ^d(Te), σ^d(Se) and Δσ^d(S) could be almost entirely neglected when the relative values from a standard compound, such as MeZMe (σ^d(Z: Z = Te, Se and S)_r), are employed for analysis.

The plot of σ^p(Te)_Rlt-so versus σ^p(Te)_Non gives a very good correlation, although the data for MeTeTeMe seem to deviate slightly to the downside of the correlation line. The plot is shown in Fig. S4.† The plot is analyzed as a linear correlation with y = ax + b, where a and b are the correlation constants and the y-intercept, respectively, with the square of the correlation coefficient, R². The correlation is given in entries 1 and 1′ of Table 3 for all data and for data except those of MeTeTeMe, respectively. The a values are very close to 1.10. The a values larger than 1.00 must be a reflection of larger downfield contributions from the relativistic effects on σ^p(Te) for higher coordinated tellurium species as a whole.

Table 3 Correlations in σ(Z) or Δσ(Z) (Z = Te, Se, and S) for various species containing Z, evaluated at the BLYP level with the QZ4Pae basis sets, under Non and Rlt-so conditions^a

Entry	Correlation	a	b	R^2	n
a The constants (a, b, R^2) are the correlation constant, the y-intercept, and the square of correlation coefficient, respectively, in y = ax + b.b Neglecting the data of MeTeTeMe.c Neglecting the data of Me2ZBr2, MeZF5, HZF5, ZF5^+, and ZF6.
1	σ^p(Te)Rlt-so vs. σ^p(Te)Non	1.103	92.5	0.995	40
1′	σ^p(Te)Rlt-so vs. σ^p(Te)Non	1.105	103.2	0.997	39^b
2	σ^d+p(Te)Rlt-so vs. σ^d+p(Te)Non	1.102	−510.9	0.996	40
3	σ^t(Te)Rlt-so vs. σ^d+p(Te)Non	1.236	−232.6	0.992	40
4	σ^t(Te)Rlt-so vs. σ^d+p(Te)Rlt-so	1.119	348.3	0.990	40
5	σ^d+p(Te)Rlt-so vs. σ^d+p(Se)Rlt-so	1.331	568.0	0.963	40
6	σ^t(Te)Rlt-so vs. σ^t(Se)Rlt-so	1.460	787.6	0.969	40
7	Δσ^d+p(Te)Rlt-so vs. Δσ^d+p(Se)Rlt-so	3.783	−16.3	0.963	40
8	σ^so(Te)Rlt-so vs. σ^so(Se)Rlt-so	4.780	−140.4	0.969	35^c
9	Δσ^t(Te)Rlt-so vs. Δσ^t(Se)Rlt-so	4.576	−74.2	0.979	35^c

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How do Δσ^p(Te)_Rlt-so behave? Δσ^p(Te)_Rlt-so are plotted versus σ^p(Te)_Non. Fig. 4 shows the results. The plot is analyzed as three (linear) correlations, although tentative. The first group contains mono- and di-coordinated Te species, R₃Te⁺ and R₄Te (R = H and Me) form the second group, and the Te species in Table 1 other than the first and second groups belong to the third group. The correlations are given in Fig. 4. The relativistic effects on σ^p(Te) of MeTeTeMe (σ^p(Te:MeTeTeMe)_Rlt-so) shift it more downfield by ca. 190 ppm than that expected from σ^p(Te:MeTeTeMe)_Non of −2736 ppm. The results are similar to those predicted for Δσ^p(Se)_Rlt-so, although the plot of Δσ^p(Se)_Rlt-so versus σ^p(Se)_Non is analyzed assuming a cubic function.¹⁴ The results anticipate a the linear relation in the relativistic effects between σ(Te) and σ(Se).

Fig. 4 Plot of Δσ^p(Te)_Rlt-so versus σ^p(Te)_Non: , , and stand for the mono- and di-coordinated species, R₃Te⁺ and R₄Te species, where R = H and Me, and the other species in Table 1, respectively, with ○ for MeTeTeMe. The plot is analyzed as three linear correlations, which are given in the figure.

The σ^so(Te)_Rlt-so values cause upfield shifts of 331 to 889 ppm for the various tellurium species. Fig. 5 shows the plot of σ^so(Te)_Rlt-so versus Δσ^p(Te)_Rlt-so. To understand the structural dependence of the plot, the data are analyzed by dividing into four groups, G1–G4, although tentative. The data of HTe⁻ and MeTe⁻ belong to G1 (group 1), together with MeTeTeMe. The σ^so(Te)_Rlt-so values cause large upfield shifts of 824 to 888 ppm with the Δσ^p(Te)_Rlt-so range of −109 to −375 ppm. G2 consists of hydrogen telluride derivatives of the H_nTe* form, where * = null, + and −, which correspond to neutral, cationic, and anionic species, respectively, with n = 1–6. The alkyl derivatives of H_nTe* also belong to G2. The range of σ^so(Te)_Rlt-so in G2 is 557 to 744 ppm, with −194 < Δσ^p(Te)_Rlt-so < −53 ppm. G3 is formed with the data for H₂TeX₂, H₂TeO, H₂TeO₂, and the alkyl derivatives. Data for TeF₅⁺, RTeF₅⁻, RTeF₅, and TeF₆ are contained in G3 for convenience of explanation. The σ^so(Te)_Rlt-so values in G3 cover the range of 432 to 650 ppm, with −269 < Δσ^p(Te)_Rlt-so < −203 ppm, together with Δσ^p(Te)_Rlt-so = −129 ppm for H₄TeO. The σ^so(Te)_Rlt-so values for TeCl₄, Cl₂TeO, F₂TeO, TeF₄, and TeF₅⁻ are in G4. The plot shows a convex downward. The ranges of σ^so(Te)_Rlt-so and Δσ^p(Te)_Rlt-so are from 470 to 331 ppm and from −405 to −284 ppm, respectively. The results are very similar to those for σ^so(Se)_Rlt-so, which anticipates the linear relation again in the relativistic effects between σ(Te) and σ(Se).¹⁴

Fig. 5 Plots of σ^so(Te)_Rlt-so versus Δσ^p(Te)_Rlt-so. σ^so(Te)_Rlt-so are tentatively divided into four groups: G1 in red, G2 in black, G3 in blue, and G4 in pink.

Fig. 6 shows the plots of σ^d+p(Te)_Rlt-so and σ^t(Te)_Rlt-so versus σ^d+p(Te)_Non. The correlations are contained in entries 2 and 3 in Table 3, for which the a values are 1.10 and 1.24, respectively. The a value of 1.10 in entry 2 must be the reflection from the relativistic effects on σ^d+p(Te)_Rlt-so of more downfield shifts for higher coordinated species by about 10%, as a whole, especially for the effect on σ^p(Te)_Rlt-so (see entry 1 in Table 3). Similarly, a = 1.24 in entry 3 can be explained by more upfield shifts of σ^so(Te)_Rlt-so for the lower coordinated species by about 14%, as a whole, in addition to more downfield shifts in σ^d+p(Te)_Rlt-so (or σ^p(Te)_Rlt-so) for the higher coordinated species, as a whole. The correlation of σ^t(Te)_Rlt-so versus σ^d+p(Te)_Rlt-so is given in entry 4 of Table 3, for which the a value is 1.12, although the plot is not shown. The a value of 1.24 in entry 3 seems to be explained by the product between a = 1.10 (entry 2) and a = 1.12 (entry 4) (1.10 × 1.12 = 1.23).

Fig. 6 Plots of σ^d+p(Te)_Rlt-so () and σ^t(Te)_Rlt-so () versus σ^d+p(Te)_Non.

After elucidation of the relativistic effects on σ(Te), the next extension is to clarify the trends in the relativistic effects on σ(S), σ(Se), and σ(Te), as a series of the group 16 elements.

Trends in the relativistic effects on σ(Te), σ(Se), and σ(S)

The relativistic effects on each of σ(Te), σ(Se), and σ(S) are summarized in Table S7 of the ESI.† Table S7† also contains the correlations given in Table 3. The trends in the relativistic effects on σ(Te), σ(Se), and σ(S) will be discussed, taking the effect on σ(Se) as the standard.

Correlations of σ^d+p(Te)_Rlt-so versus σ^d+p(Se)_Rlt-so and σ^t(Te)_Rlt-so versus σ^t(Se)_Rlt-so are given in entries 5 and 6 of Table 3, respectively, for which the a values are 1.33 and 1.46, respectively. The increase in a of 1.33 to 1.46 must be the result of the larger relativistic effects on σ^so(Te)_Rlt-so relative to the case of σ^so(Se)_Rlt-so. The correlation of Δσ^d+p(Te)_Rlt-so versus Δσ^d+p(Se)_Rlt-so is given in entry 7 of Table 3 (a = 3.78). But, how is the correlation of σ^so(Te)_Rlt-so versus σ^so(Se)_Rlt-so? Fig. 7 shows the plot of σ^so(Te)_Rlt-so versus σ^so(Se)_Rlt-so. The plot gives a good correlation, except for the data of Me₂ZBr₂, MeZF₅, HZF₅, ZF₅⁺, and ZF₆ (Z = Te versus Se). The correlation of the plot is given in entry 8 of Table 3 (a = 4.78), although the deviating data are omitted in the correlation. The correlation for Δσ^t(Te)_Rlt-so versus Δσ^t(Se)_Rlt-so is shown in entry 9 of Table 3. The a value of 4.58 is close to 5, showing that the relativistic effects on σ^t(Te) are about five times larger than that on σ^t(Se), again. The results support the conclusion that the magnitude of the relativistic effects becomes smaller in the order of σ(Te) ≫ σ(Se) ≫ σ(S). The ratios are approximately 25 : 5 : 1, where the relativistic effects on σ(S) are summarized in Table S7,† although they are not discussed in detail here.

Fig. 7 Plot of σ^so(Te)_Rlt-so versus σ^so(Se)_Rlt-so. Data of Me₂ZBr₂, MeZF₅, HZF₅, ZF₅⁺, and ZF₆ (Z = Te versus Se) deviate from the correlation.

After clarification of the relativistic effects on σ(S), σ(Se), and σ(Te) as a series of group 16 elements, the next extension is to examine the applicability of σ^t(Te)_r to δ(Te)_obsd.

Applicability of σ^t(Te:M)_r to analyze δ(Te:M)_obsd

To confirm the reliability of the calculation method, it is necessary to demonstrate the applicability of σ^t(Te:M)_r to analyze δ(Te:M)_obsd. σ^t(Te:M)_r are defined as [−(σ^t(Te:M) − σ^t(Te:Me₂Te))], where σ^t(Te:M) stands for σ^t calculated for Te in a molecule M. The sign of σ^t(Te:M)_r is set up equal to that of δ(Te:M)_obsd in the definition employed in this paper. The δ(Te:M)_obsd values are reported for M of several compounds, albeit limited to those in Table 1. Therefore, the σ(Te) of 46 tellurium species (1–46) are calculated (see Table 2). Chart 1 gives illustrations of some of the structures. The conformational effect affects much on σ^t(Te), therefore, they must be carefully examined in the prediction of σ^t(Te)_r. Fig. 8 illustrates some of the conformers for M of MeTeEt (8: 8a and 8b), EtTeEt (10: 10a, 10b, and 10c), EtTeTeEt (13: 13a, 13b, and 13c), and PhTeTePh (18: 18a and 18b). The conformers of 10a, 10b, and 10c correspond to (trans, trans) of the C₂ symmetry, (trans, gauche) of C₁, and (gauche, gauche) of C₂ around the two CTeCC sequences, respectively, in 10. While the Te–Te bond is nearly on both the phenyl planes in 18a, it is almost perpendicular to the planes in 18b.

Chart 1 Some examined structures of tellurium compounds.

Fig. 8 Conformers of 8, 10, 13, and 18.

While the σ^d+p(Te:M)_Rlt-so:r and σ^t(Te:M)_Rlt-so:r values are evaluated by the OPBE_Rlt-so//OPBE_Rlt-so method, σ^d+p(Te:M)_Non:r are evaluated by the OPBE_Non//OPBE_Rlt-so method. The σ^t(Te:M)_Rlt-so:r values are similarly evaluated with BLYP_Rlt-so//OPBE_Rlt-so and BLYP_Rlt-so//MP2_Non. Table 2 presents the results for the compounds with the compound numbers. Table 2 also displays σ(Te:M)_r for some conformers in M of 8, 10, 13, and 18. Simple averaged values of σ^t(Te:M)_Rlt-so:r over the conformers are employed for discussion, in spite of the different energies for the conformers, since the energy differences are not so large and the population of each conformer may change depending on the conditions of measurements, such as depending on the solvent.

The systematic behavior is predicted for σ^t(Te:10)_Rlt-so:r. The value shifts more downfield by about 40 ppm for each process in the change from 10a to 10b then to 10c, if calculated with OPBE_Rlt-so//OPBE_Rlt-so. A similar trend is observed for the values evaluated with BLYP_Rlt-so//OPBE_Rlt-so and BLYP_Rlt-so//MP2_Non, although the shift values are not the same. The σ^t(Te:8)_Rlt-so:r value also goes more downfield by 60–70 ppm for the process from 8a to 8b, calculated with the three method.

The σ^t(Te:13)_Rlt-so:r value shifts more downfield from 13a (118.2 ppm) to 13b (148.5 ppm) then to 13c (181.0 ppm), similar to the case of EtTeEt (10), if calculated with BLYP_Rlt-so//MP2_Non. However, the value does more upfield, if calculated with OPBE_Rlt-so//OPBE_Rlt-so, which is just the opposite trend to those with BLYP_Rlt-so//MP2_Non. Much attention should be paid to the torsional angles of ϕ(CTeTeC) in 13, which decease in the order of 13a (83.6°), to 13b (72.6°), and to 13c (69.3°), if optimized with OPBE_Rlt-so. The optimized structures of 13a, 13b, and 13c with OPBE_Rlt-so must be responsible for the opposite trend. The inverse trend in σ^t(Te:13)_Rlt-so:r seems improved (by roughly half), if using the structures optimized with OPBE_Rlt-so, with ϕ(CTeTeC) fixed at 90.0°. The structural parameters and σ^t(Te:M)_Rlt-so:r calculated under some conditions are summarized in Table 4.

Table 4 Structural parameters of EtTeTeEt (13), optimized under MP2_Non and OPBE_Rlt-so, together with σ^t(Te)_Rlt-so:r evaluated at OPBE_Rlt-so

Compd	r(Te, Te) (Å)	r(Te, C) (Å)	ϕ(CTeTeC) (°)	σ^t(Te)Rlt-so:r (ppm)
a ϕ(CTeTeC) fixed at 90.0°.
Optimized with MP2Non
13a (C2)	2.6685	2.1450	−87.48	72.9
13b (C1)	2.6694	2.1443	−87.82	98.3
13c (C2)	2.6702	2.1444	−88.26	123.6
Optimized with OPBERlt-so
13a (C2)	2.6910	2.1839	−83.63	160.2 (204.1^a)
13b (C1)	2.6913	2.1813	−72.62	96.9 (189.8^a)
13c (C2)	2.6907	2.1818	−69.30	46.4 (145.3^a)

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Torsional angles ϕ(C_iTeTeC_i′) of 89.1° and 84.8° are predicted for 18a and 18b, respectively, when optimized with OPBE_Rlt-so. However, ϕ(C_iTeTeC_i′) = 76.8° is predicted for 18a, while a much smaller value of 43.1° is for 18b with MP2_Non, maybe due to the overestimation of the π–π interaction between the phenyl groups in 18. The partially optimized structure of 18b with ϕ(C_iTeTeC_i′) fixed at 90.0° is employed for the calculation of σ^t(Te:18)_Rlt-so:r, instead of the fully optimized 18. The σ^t(Te:18)_Rlt-so:r values of 514 and 206 ppm are predicted for 18a and 18b, respectively, with OPBE_Rlt-so//OPBE_Rlt-so. Table 2 lists the σ^t(Te:18)_Rlt-so:r, thus obtained. The simple average value of 360 ppm explains better the δ(Te:18)_obsd of 420 ppm than that of each conformer for 18. Namely, δ(Te)_obsd can be better explained by assuming the equilibrium between conformers, such as 18a and 18b, as expected. The populations would be almost equal or slightly in excess for 18b, judging from the energy difference between the two isomers: 18b is thus predicted to be more stable than 18a by 2 kJ mol⁻¹. The simple averaged values of σ^t(Te)_Rlt-so:r between 18a and 18b seem to explain the δ(Te:18)_obsd well, if calculated with BLYP_Rlt-so//OPBE_Rlt-so and BLYP_Rlt-so//MP2_Non.

Such equilibrium in solution would often be observed in tellurium species between some structures, other than 18, which must affect the δ(Te)_obsd. Four coordinated tellurium species is the typical example of this case. The weighed averaged values for the structures in the equilibrium are expected to explain the δ(Te)_obsd. The effect of the equilibrium on σ^t(Te)_Rlt-so:r is discussed in the ESI,† exemplified by TeF₄ and TeMe₄ (7) as evaluated with OPBE_Rlt-so//MP2_Non and OPBE_Rlt-so//OPBE_Rlt-so.⁵⁹

It is instructive to define the differences between the calculated and observed values, Δσδ(Te:M)_Rlt-so:r [=σ^t(Te:M)_Rlt-so:r − δ(Te:M)_obsd]. The applicability of the calculation methods employed here will be examined through discussion of Δσδ(Te:M)_Rlt-so:r. Fig. 9 shows the plot of Δσδ(Te:M)_Rlt-so:r, derived from data in Table 2. The plot is explained separately by three categories: category 1 (C(1)) contains TeP(iPr)₃ (2), Te(SiMe₃)₂ (3), TeH₂ (4) and TePMe₃ (5), of which δ(Te:M)_obsd and σ^t(Te:M)_Rlt-so:r appear very high field, while C(3) consists of the characteristic di-anionic and di-cationic species of TeCl₆²⁻ (44), TeBr₆²⁻ (45), and cyclo-Te₄²⁺ (46). C(2) forms with all compounds in Table 2, other than with C(1) and C(3) (1 and 6–43), which are rather more usual Te species.

Fig. 9 Plot of Δσδ(Te:M)_Rlt-so:r, evaluated with OPBE_Rlt-so//OPBE_Rlt-so, BLYP_Rlt-so//OPBE_Rlt-so, and BLYP_Rlt-so//MP2_Non.

As shown in Fig. 9, the magnitudes of Δσδ(Te:M)_Rlt-so:r change depending on the calculation method. The magnitudes seem larger in the order of OPBE_Rlt-so//OPBE_Rlt-so < BLYP_Rlt-so//OPBE_Rlt-so ≤ BLYP_Rlt-so//MP2_Non. This expectation is supported by the average values and the standard deviations for Δσδ(Te)_Rlt-so:r (x̄/ppm: σ/ppm). The calculated values for C(2) with the three methods are (−19.8:67.8), (−24.4:100.6), and (−53.2:90.4), respectively. σ^t(Te:M)_Rlt-so:r evaluated with OPBE_Rlt-so//OPBE_Rlt-so is discussed next.

Large negative Δσδ(Te:M)_Rlt-so:r values are predicted for C(1). While the value is extremely negative for 5 (−403 ppm), they are around −200 ppm for 2–4 in C(1). On the other hand, an extremely large positive value is predicted for 45 (540 ppm) and a large positive value for 44 (186 ppm), although a relatively large negative value for 46 (−122 ppm) in C(3). It seems difficult to predict the reliable σ^t(Te:M)_Rlt-so:r values for 5 and 45. The trend seems similar even when the other two methods are applied, although better methods should be searched for or the methods should be improved.

In the case of C(2), the magnitudes of Δσδ(Te:M)_Rlt-so:r are much smaller, relative to the cases of C(1) and C(3). The magnitudes are within less than 100 ppm for most of tellurium species, as shown in Fig. 9. The magnitudes of Δσδ(Te:M)_Rlt-so:r larger than 100 ppm are as follows: c-C₆H₄TeCH₂C=O (17) (Δσδ(Te:17)_Rlt-so:r = −163 ppm), Te(C(tBu)CH)₂C=O (21) (−109 ppm), Te(C₆H₄)₂C=O (22) (−143 ppm), c-Te(CH)₄ (33) (−111 ppm), and CF₃TeF₂CF₃ (40) (159 ppm). The effect of the conjugation between Te and C=O through the π-system on σ^t(Te:M) would be less if estimated with OPBE_Rlt-so//OPBE_Rlt-so for M of 17, 21, and 22, so would be the 6π-ring system in 33 since the Δσδ(Te:M)_Rlt-so:r values are negative. On the other hand, the effect of the highly positively charged Te on σ^t(Te:40) seems overestimated with the positive Δσδ(Te:40)_Rlt-so:r, although Δσδ(Te:M)_Rlt-so:r are acceptable for M of TeF₆ (24) (67 ppm) and very good for Me₂TeX₂ (29 (X = Cl) and 30 (X = Br)) and PhTeX₂Me (31 (X = Cl) and 32 (X = Br)) (−15 to −35 ppm).

The σ^t(Te:M)_Rlt-so:r values calculated with OPBE_Rlt-so//OPBE_Rlt-so are plotted versus δ(Te:M)_obsd, separately by C(1), C(2), and C(3). Fig. 10 shows the plots. The plots are analyzed for C(1) and C(2), assuming the linear correlations. Table 5 presents the correlations (entries 1 and 2, respectively). δ(Te:M)_obsd are usually compared directly to σ^t(Te:M)_Rlt-so:r, in the assignment processes of the ¹²⁵Te NMR signals. The process must correspond to the forced pass of the origin in the correlation. The treatment for C(2) is shown in entry 3 in Table 5, where the a value is equal to 1.00. The results strongly support the good applicability of σ^t(Te:M)_Rlt-so:r evaluated with OPBE_Rlt-so//OPBE_Rlt-so in the assignment of δ(Te:M)_obsd for the usual tellurium compounds such as C(2). Entries 4–7 in Table 5 shows the correlations for C(1) and C(2) in the plots of σ^t(Te:M)_Rlt-so:r, calculated with BLYP_Rlt-so//OPBE_Rlt-so and BLYP_Rlt-so//MP2_Non, versus δ(Te:M)_obsd.

Fig. 10 Plots of σ^t(Te:M)_Rlt-so:r, evaluated with OPBE_Rlt-so//OPBE_Rlt-so, versus δ(Te:M)_obsd: data for C(1), C(2), and C(3) are shown by , , and , respectively.

Table 5 Correlations in the plots of σ^t(Te:M)_Rlt-so:r, σ^d+p(Te:M)_Rlt-so:r, and σ^d+p(Te:M)_Non:r versus δ(Te:M)_obsd for various tellurium species, evaluated with OPBE_Rlt-so//OPBE_Rlt-so, BLYP_Rlt-so//OPBE_Rlt-so, BLYP_Rlt-so//MP2_Non, and the related methods with the QZ4Pae basis sets^a,b

Entry	a	b	R^2	n	Applied to	Method
a The constants (a, b, R^2) are the correlation constant, the y-intercept, and the square of correlation coefficient, respectively, in y = ax + b.b Data for TeBr6^2− are neglected in the correlations.c Forced pass of the origin in the correlation with y = ax.
σ^t(Te:M)Rlt-so:r versus δ(Te:M)obsd
1	1.180	−95.7	0.929	5	C(1)	OPBERlt-so//OPBERlt-so
2	1.043	−49.1	0.990	39	C(2)	OPBERlt-so//OPBERlt-so
3^c	1.003	^c	^c	39	C(2)	OPBERlt-so//OPBERlt-so
4	1.177	−110.7	0.928	5	C(1)	BLYPRlt-so//OPBERlt-so
5	1.086	−83.1	0.983	39	C(2)	BLYPRlt-so//OPBERlt-so
6	1.321	−63.3	0.972	5	C(1)	BLYPRlt-so//MP2Non
7	1.022	−68.5	0.980	39	C(2)	BLYPRlt-so//MP2Non
σ^d+p(Te:M)Rlt-so:r versus δ(Te:M)obsd
8	0.991	−55.7	0.972	39	C(2)	OPBERlt-so//OPBERlt-so
9	0.971	−78.4	0.942	39	C(2)	BLYPRlt-so//MP2Non
σ^d+p(Te:M)Non:r versus δ(Te:M)obsd
10	0.936	−101.4	0.968	39	C(2)	OPBENon//OPBERlt-so
11	0.937	−148.3	0.957	39	C(2)	BLYPNon//MP2Non

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The σ^d+p(Te:M)_Rlt-so:r values calculated with OPBE_Rlt-so//OPBE_Rlt-so and σ^d+p(Te:M)_Non:r with OPBE_Non//OPBE_Rlt-so are also given in Table 2, while the values with BLYP_Rlt-so//MP2_Non are in Table S9, in the ESI.† Correlations for σ^d+p(Te:M)_Rlt-so:r versus δ(Te:M)_obsd for C(2) are given in Table 5 (entries 8 and 9) and those for σ^d+p(Te:M)_Non:r versus δ(Te:M)_obsd for C(2) are in Table 5 (entries 10 and 11). The R² values become smaller in the order of σ^t(Te:M)_Rlt-so:r (0.990) > σ^d+p(Te:M)_Rlt-so:r (0.972) > σ^d+p(Te:M)_Non:r (0.968), if calculated with OPBE_Rlt-so//OPBE_Rlt-so and OPBE_Non//OPBE_Rlt-so.

The (a, b, R²) values of entries 2, 5, and 7 in Table 5 are (1.043, −49.1, 0.990), (1.086, −83.1, 0.983), and (1.022, −68.5, 0.980), respectively. The correlation obtained with OPBE_Rlt-so//OPBE_Rlt-so seems better than the other two, although the three methods could be recommended for the purpose. Both scalar-ZORA and spin–orbit ZORA relativistic effects should be considered, since the consideration of only scalar-ZORA relativistic effects seems not to improve the correlations so much, relative to the case of the consideration of both effects (Table 5).

Effect of optimized structures on σ^t(Te:M)_r

The correlation for σ^t(Te:M)_Rlt-so:r versus δ(Te:M)_obsd must be excellent if it is analyzed, as y = ax + b with (a, b) = (1.00, 0.0) and R² are very close to 1.00. Therefore, what are the reasons for the deviations from the excellent case in the plot? The δ(Te:M)_obsd values measured in solutions must contain solute–solute and solute–solvent interactions, whereas the theoretically predicted σ^t(Te:M)_Rlt-so:r values correspond to those for single species in vacuum. Factors, other than those mentioned above, should also be examined for the better assignments of the NMR signals.

There must be some differences between the optimized and observed structures. Bond distances around Te in the optimized structures are examined here, although the angular and/or torsional angular dependence of δ(Te:M) would affect the chemical shifts. Bond lengths around Te for some compounds optimized with OPBE_Rlt-so are given in Table 6, together with the observed values (r(Te–X)_calcd and r(Te–X)_calcd, respectively, where X = C, Si, P, Te, F, Cl, Br, and O). The r(Te–X)_calcd values optimized with BLYP_Non, BLYP_Rlt-sc, and BLYP_Rlt-so are given in Table S10 in the ESI.†

Table 6 Evaluation of σ^t(Te:M)_Rlt-so:r values at the observed Te–X distances

Compound	r(Te–X)calcd (Å)	σ^t(Te:M)Rlt-so:r (ppm)	r(Te–X)obsd (Å)	σ^t(Te:M)po:r^a (ppm)	δ(Te:M)obsd (ppm)	X
a See text.b For r(Te–Ceq)c σ^t(Te)Rlt-so:r = 360 ppm and σ^t(Te)po:r = 439.6 ppm in the average.
Te(SiMe3)2 (3)	2.531	−1053.0	2.514	−1098.4	−842	Si
TePMe3 (5)	2.330	−916.2	2.357	−1188.8	−513.4	P
TeMe4 (7)^b	2.148	−116.0	2.126	−155.8	−67.0	C
MeTeTeMe (11)	2.681	5.1	2.712	96.0	49	Te
PhTeTePh (18a)^c	2.672	514.1	2.709	553.9	420	Te
PhTeTePh (18b)^c	2.672	205.9	2.709	325.2	420	Te
c-Te(C6H4)2CH2 (23)	2.104	455.5	2.132	538.5	512	C
TeF6 (24)	1.831	611.7	1.815	576.3	545	F
CF3TeTeCF3 (27)	2.225	663.5	2.181	546.1	686	C
Me2TeCl2 (29)	2.489	682.5	2.515	702.3	734	Cl
Me2TeBr2 (30)	2.654	633.3	2.683	649.0	649	Br
CF3TeF2CF3 (40)	1.949	1346.7	1.974	1234.3	1187	C, F
TeCl6^2− (44)	2.568	1717.2	2.545	1489.3	1531	Cl
TeBr6^2− (45)	2.743	1888.0	2.700	1632.1	1348	Br
c-Te4^2+ (46)	2.687	2542.8	2.668	2405.9	2665	Te

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The Δr(Te–X) [=r(Te–X)_calcd − r(Te–X)_obsd] values obtained with various methods are plotted for some compounds. Fig. 11 shows the results, separately for three categories of C(1), C(2), and C(3), similar to the case of Fig. 9. The Δr(Te–X) values are negative for almost all of the compounds examined and appear at the bottom in plots if evaluated at MP2_Non. However, the values are positive for all the compounds examined and they appear at the top of the plot if calculated with BLYP_Rlt-so, and are also positive for almost all of the compounds examined and appear at the next top when calculated with B3LYP_Non. The results imply that r(Te–X)_calcd are evaluated shorter than r(Te–X)_obsd for most of the compounds with MP2_Non, whereas r(Te–X)_calcd are evaluated longer than r(Te–X)_obsd with BLYP_Rlt-so and B3LYP_Non. On the other hand, the magnitudes of Δr(Te–X) are less than 0.01 Å for most of the compounds if evaluated with OPBE_Rlt-so and they appear in the intermediate area between those at the MP2_Non and B3LYP_Non levels. The data with OPBE_Non seem very close to those with OPBE_Rlt-so. The Δr(Te–X) values with OPBE_Rlt-so will be mainly discussed by classifying them into three categories, (i) −0.01 Å > Δr(Te–X), (ii) −0.01 Å < Δr(Te–X) < 0.01 Å and (iii) 0.01 Å < Δr(Te–X). The case of (ii) is desirable.

Fig. 11 Plots of Δr(Te–X) in the structures optimized with MP2_Non, OPBE_Non, OPBE_Rlt-so, B3LYP_Non, and BLYP_Rlt-so.

The magnitude of Δr(Te–C) is negligibly small for MeTeMe (1) (−0.0003 Å) if calculated with OPBE_Rlt-so. Fifteen Δr(Te–X) values belong to case (ii), seven to (i), and eight to (iii), among the 30 cases. The typical cases calculated with OPBE_Rlt-so are as follows: while Δr(Te–P) in TeP(iPr)₃ (2) satisfy the requirement of (ii) (0.002 Å), that in TePMe₃ (5) belongs to (i) (−0.027 Å). The Δr(Te–Si) values in Te(SiMe₃)₂ (3) belong to (iii) (0.017 Å). Indeed, a couple of Δr(Te–C_eq) belong to (iii) (0.022 Å), and another couple of Δr(Te–C_ax) to (ii) (−0.007 Å) in Me₄Te (7), but they belong to (ii) if the averaged value is examined (0.008 Å). While Δr(Te–Te) in MeTeTeMe (11) (−0.031 Å) and PhTeTePh (−0.037 Å) (18) are in the range of (i), that in CF₃TeTeCF₃ (27) is in the range of (ii) (0.002 Å), with Δr(Te–C) in (iii) (0.044 Å). Compounds with Δr(Te–X) outsides of (ii) are summarized in Table 6.

The Δr(Te–X) values with OPBE_Non are often very close to those with OPBE_Rlt-so, as illustrated in Fig. 11. The magnitudes of the differences in Δr(Te–X) between those with OPBE_Non and OPBE_Rlt-so are large for 3 (X = Si: 0.0395 versus 0.0173 Å). The relativistic effect would be substantial on r(Te–Si) in 3. While those in Δr(Te–X) are large for 7 (X = C_ax: 0.0418 versus 0.0221 Å and X = C_eq: −0.0424 versus −0.0066 Å), that of the average value are very small (−0.0003 versus 0.0078 Å).

The Δr(Te–X) values in the range of (i) and (iii) are used to adjust to the observed values. The optimized distances with OPBE_Non are very close to those with OPBE_Rlt-so. Therefore, the OPBE_Non method is applied to the partial optimization for the adjustments, where the r(Te–X) in question are fixed as the observed values. Then σ^t(Te:M)_Rlt-so:r are evaluated with OPBE_Rlt-so on the partially optimized structures with OPBE_Non. Table 6 presents the σ^t(Te:M)_Rlt-so:r thus evaluated for some selected species (as shown by σ^t(Te)_po:r).

The σ^t(Te:M)_Rlt-so:r values shift more high and low field as r(Te–X)_calcd become shorter and longer, respectively, for the species in Table 5, except for M of TePMe₃ (5) and CF₃TeF₂CF₃ (40). The Te=P bond in 5 behaves differently from the others in σ^t(Te:M). The dependence of σ^t(Te:M)_Rlt-so:r on the bond distance in M of 40 is just the opposite of that in CF₃TeTeCF₃ (27). The magnitude of Δσδ(Te:M)_Rlt-so:r is improved for M of 40 (160 ppm to 47 ppm), but not for 27 (−23 to −140 ppm). Indeed, the values are much improved for M of TeCl₆²⁻ (44) (186 to −42 ppm) and TeBr₆²⁻ (45) (540 to 284 ppm), but the value for 45 after the improvement is not acceptable for practical purposes. That for c-Te₄²⁺ (46) becomes worse (−122 to −259 ppm). The magnitudes of Δσδ(Te:M)_Rlt-so:r are improved for 9 compounds, but not for other 6 compounds as a whole, if the structures with the observed distances are employed for the evaluations. It would be difficult to predict reliable σ^t(Te:M)_Rlt-so:r for some species, such as 5 and 45, by the methods employed above.

The σ^t(Te:M)_Rlt-so:r values will change depending on the calculation methods, therefore, a method should be selected that is most suited for the purpose or otherwise a method of wider applicability should be selected. The optimized structures can be used to predict reliable σ^t(Te:M)_Rlt-so:r, if a suitable method is employed for the optimization. Observed structures may then give better results. It is recommended to employ those after a partial optimization fixing of some important parameters, such as bond lengths, to the observed values. The partial optimization could avoid some extreme deviations of the angular parameters in the observed structures affected by the surroundings, such as by the crystal packing effect.

Conclusion

The relativistic effect on σ^t(Z) and the components, σ^d(Z), σ^p(Z), σ^d+p(Z), and σ^so(Z) (Z = Te, Se, and S), were evaluated explicitly and separately by the scalar and spin–orbit ZORA relativistic terms. The structures of various tellurium, selenium, and sulfur species optimized at the MP2 level were employed for the evaluation of σ(Z). The σ(Z) values were calculated with the DFT(BLYP)-GIAO method under spin–orbit ZORA relativistic and nonrelativistic conditions with QZ4Pae. While the range of the relativistic effects on the total shielding tensors for Te (Δσ^t(Te)_Rlt-so = σ^t(Te)_Rlt-so − σ^d+p(Te)_Non) was predicted to be −55 to 658 ppm, that for σ^t(S) was 5 to 32 ppm, except for Me₂SBr₂ (TBP), of which Δσ^t(S)_Rlt-so = −29 ppm. The range for σ^t(Se) was 2 to 153 ppm. The magnitudes of the relativistic effect on σ^t(Te), σ^t(Se) and σ^t(S) were about 25 : 5 : 1.

The applicability of σ^t(Te:M)_Rlt-so:r to δ(Te:M)_obsd was examined in detail. The plot of σ^t(Te:M)_Rlt-so:r versus δ(Te:M)_obsd was explained separately by three categories: category 1 (C(1)), containing those with δ(Te:M)_obsd and σ^t(Te:M)_Rlt-so:r appearing very high field, C(3) consisting of the characteristic di-anionic and di-cationic species, and C(2) forms with all the compounds in Table 2 other than C(1) and C(3), which are rather more usual Te species. The correlation obtained σ^t(Te:M)_Rlt-so:r with OPBE_Rlt-so//OPBE_Rlt-so seemed better than those with BLYP_Rlt-so//MP2_Non, and BLYP_Rlt-so//OPBE_Rlt-so, although the three methods could be recommended as fitting the purpose. The structural parameters, such as ϕ(CTeTeC) in ditellurides, should be examined carefully, together with the bond length around the Te atoms. The partial optimization fixing of some important parameters to the observed values is recommended to obtain more reliable σ^t(Te:M)_Rlt-so:r.

Acknowledgements

This work was partially supported by a Grant-in-Aid for Scientific Research (no. 20550042, 21550046 and 23350019) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The support of the Wakayama University Original Research Support Project Grant and the Wakayama University Graduate School Project Research Grant is also acknowledged.

Notes and references

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59. Fig. S7† draws the C_2v, C₄ and T_d structures of 7, optimized with MP2_Non. Table S11† collects the σ^t(Te)_Rlt-so:r values for TeF₄ (C_2v), TeF₄ (C_4v), TeF₄ (T_d), 7 (C_2v), 7 (C₄) and 7 (T_d), together with some structural data and the relative contributions. The T_d structures are estimated to be so unstable, therefore, they need not be considered in σ^t(Te)_Rlt-so:r. The σ^t(Te)_Rlt-so:r values of TeF₄ (C_2v), TeF₄ (C₄), 7 (C_2v) and 7 (C₄) are predicted to be 1651, 1532, −116, and −242 ppm, respectively. TeF₄ (C_2v) and 7 (C₄) are predicted to contribute predominantly, relative to the case of TeF₄ (C₄) and 7 (C_2v), respectively. However, σ^t(Te)_Rlt-so:r of 7 (C_2v) seems to explain δ(Te)_obsd of 7 (−67 ppm) better than that of 7 (C₄). 7 (C_2v) would be predominantly in the solution, maybe due to the solvent effect.

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Footnote

† Electronic supplementary information (ESI) available: The σ(Z) values for Z = Se and S calculated employing the structures optimized at the MP2 level are collected in Tables S2 and S3, respectively. The σ(Z) values for Z = Te, Se and S calculated employing the structures optimized at the DFT (B3LYP) level are collected in Tables S4–S6, respectively. Plots of σ^d+p(Te)_Non,r and σ^d+p(Te)_Rlt-so,r versus δ(Te)_obsd are depicted in Fig. S1 and S2, respectively. The full-optimized structures given by Cartesian coordinates for examined molecules and adducts. See DOI: 10.1039/c4ra07818g

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