Role of Spin-Dependent Terms in the Relationship among Nuclear

Letter pubs.acs.org/JPCL

Role of Spin-Dependent Terms in the Relationship among Nuclear Spin-Rotation and NMR Magnetic Shielding Tensors I. Agustín Aucar,† Sergio S. Gomez,† Claudia G. Giribet,‡ and Gustavo A. Aucar*,† †

Instituto de Modelado e Innovación Tecnológica, CONICET, and Departamento de Física, FCENA-UNNE, Avda. Libertad 5460, W3404AAS Corrientes, Argentina ‡ Instituto de Fı ́sica de Buenos Aires, CONICET, and Departamento de Fı ́sica, FCEN-UBA, Ciudad Universitaria, C1428EGA Buenos Aires, Argentina ABSTRACT: The broadly accepted procedure to obtain the experimental absolute scale of NMR magnetic shieldings, σ, is well-known for nonheavy atom-containing molecules. It was uncovered more than 40 years ago by the works of Ramsey and Flygare. They found a quite accurate relationship among σ and the nuclear spin-rotation constants. Its relativistic extension was very recently proposed, although it has an intrinsic weakness because a new SO-S twocomponent term needs to be considered. We show how to overcome this problem. We found that (νSY − νatom,S ) generalizes the SO-S term, where νSY = Y 3 (4) ⟨⟨[((r − rY) × α)/(|r − rY| )]; S ⟩⟩, r − rY is the electron position with respect to the position of nucleus Y, and S(4) is the four-component total electron spin. When including this new term, one finds that the best of our relativistic Flygare-like models fits quite well with the results of the most accurate method available at the moment. We also show that the difference among the parallel component of σ(Xe) in XeF2 and σ(Xe) of the free atom is almost completely described by that new term.

U

better matching? Are there some more subtle physical reasons for those differences? How do we go one step further in the development of more accurate relativistic Flygare-type models? We give some answers to these inquires. We shall show that the SO term is the leading order contribution of the complete spin-dependent (SD) contribution that must be included in Flygare-type models to obtain more accurate results. Ramsey first, and afterward Flygare, proposed a relationship among σY and MY tensors of a nucleus in a molecule, which allows to indirectly obtain “experimental” values of absolute shieldings using experimental SR constants1,8−10 together with highly accurate calculations. This nonrelativistic (NR) relation links the paramagnetic contribution to σY and the electronic contribution to MY according to

ntil the past few years, one of the best ways to obtain by experiments the absolute value of the nuclear magnetic shielding of a nucleus Y, σY, was to apply the long-standing Flygare’s relationship.1 This situation changes dramatically when the relativistic Dirac’s theory of electrons is applied to describe magnetic atomic and molecular properties. That famous Flygare’s relation is not valid any longer. The most important deviations are found when heavy-atom-containing molecules are considered, but they are also non-negligible in molecular systems that contain only light atoms. Wasylishen and coworkers had proven it experimentally by measuring the parallel component of the shielding tensors of xenon difluoride.2 Many years were devoted to the search of new strategies that could help to overcome this problem. The first important step forward was the finding of the proper relativistic expression of the nuclear spin-rotation tensor (M), recently published by two groups.3,4 Then, the next step shall be to look for a new Flygare-type relationship that should be valid within the relativistic framework. Even though, at first sight, a similar relation among the relativistic spin-rotation (SR) and shielding tensors seems difficult to obtain, few models were recently proposed.5−7 The most accurate of them has a two-component spin−orbit (SO) term.7 The application of those models to linear molecules gave results that are close to the current most exact values (obtained from four-component methods) but not in all cases (see ref 7). So, the new SO term seems to be one of few other unknown terms that must be considered. Where shall we look to get © XXXX American Chemical Society

σYNR‐para = =

mp gY mp gY

MYNR‐elec ⊗ I (MYNR − MYnuc) ⊗ I (1)

Atomic units were used in the last expression and shall be used throughout this work, where mp is the proton mass, gY is the nuclear g-value of nucleus Y, and I is the tensor of molecular moment of inertia with respect to the center of mass (CM) at Received: October 11, 2016 Accepted: November 28, 2016 Published: November 28, 2016 5188

DOI: 10.1021/acs.jpclett.6b02361 J. Phys. Chem. Lett. 2016, 7, 5188−5192

Letter

The Journal of Physical Chemistry Letters the equilibrium geometry. Equation 1 explicitly shows that the MY tensor can be expressed as the sum of two terms: one electronic and one nuclear. The MNR‑elec tensor is the linear Y response that takes account of two external potentials originated in (i) the total electronic orbital angular momentum centered in the molecular CM and (ii) the nuclear magnetic moment of nucleus Y. Besides, eq 1 is valid only when the gauge origin (GO) of the magnetic potential is located at that CM. Another finding of Flygare and coworkers was a relationship between the diamagnetic contribution to σ of a nucleus in a molecule, σNR‑dia , when the GO is placed at the nucleus position Y Y, and the shielding of that nucleus in the free atom, σatom,NR Y (which has only diamagnetic contributions within the NR domain). They first found that σNR‑dia can be approximated as Y and another tensor that only depends on the the sum of σatom,NR Y nuclear positions at equilibrium (the first term in equation 6 of ref 9, which includes our Mnuc as part of it; see ref 3).9,10 Y Applying this relation, they found that the average (isotropic) values are more accurate than the individual tensor elements. In further works, they improved the last relation by using an atom dipole model including two other terms: a dipole one, in general quite small, and a quadrupole term.1,11 It is important to highlight here that although Flygare found this relation taking the center of rotation and the GO positions in the SR and shielding tensors (respectively) at the position of nucleus Y,9 it also happens that when the CM is considered as the center of rotation for the SR tensor and also as GO for the magnetic potential in the shielding tensor, Flygare’s diamagnetic approach can be expressed as (it can be shown following mp the arguments of ref 3) σYNR‐dia ≅ σYatom,NR + g MYnuc ⊗ I .

This model includes terms that are formally the same as those of the original Flygare’s relationship plus the new SO term. The second term of eq 3 is the free atomic shielding, which includes both its “paramagnetic-like” (e-e) and “diamagnetic-like” (e-p) relativistic values.13 Again, isotropic values shall be more accurate than the individual tensor elements due to Flygare’s approach to diamagnetic contributions. One can expect that the model M-III shall fail when higher order effects are not vanishingly small. So, a more accurate model shall be obtained when the SO-S term is replaced by its four-component counterpart. In the special case of linear molecules, the first term of the right-hand side of eq 3 will not have parallel components (where the main line of the molecular geometry is parallel). In ref 7, we studied the accuracy of the M-III model. With such an aim we analyzed the (e-e) part of the perpendicular 1 ‐S components of σM‑III of eq 3. We studied how close 2 σ⊥SO Y ,Y and σ⊥(e, Y‐e) −

gY

M⊥(e, ‐Ye) − σYatom(e‐e) are in a series of linear

molecules. The last difference was coined ΔσII⊥,Y because it is II(e‑e) the difference between σ(e‑e) (see ref 7). Such a ⊥,Y and σ⊥,Y difference may be due to relativistic higher order terms, not included in the SO-S one. How do we get such terms? Let us start from the definition of the relativistic electronic contribution to the SR tensor.3 It is known that MYelec = MYL + MYS =

Y

Because the Flygare’s quadrupole term appears only for individual tensor elements of σNR‑dia (it does not contribute to Y the isotropic diamagnetic shielding), ignoring this term may introduce significant errors in the approximation of the tensor elements of σNR‑dia . So the isotropic diamagnetic shielding can Y accurately be reproduced as the sum of σatom,NR and Mnuc‑iso (in Y Y ppm), whereas only fair estimations will be obtained for the individual tensor elements if the quadrupole term is ignored.11 From these considerations, one obtains the well-known Flygare’s relationship σY = σYNR‐para + σYNR‐dia mp NR ≈ MY ⊗ I + σYatom,NR gY

m pI

gY 2m p c g + Y 2m p c

=

(r − rY ) × α ; (r − rCM) × p |r − rY |3 (r − rY ) × α (4) ;S |r − rY |3

⊗ I −1

⊗ I −1

gY ⎡ 1 L S ⎤ −1 ⎢ (νY + νY )⎥⎦ ⊗ I m p ⎣ 2c (4)

where r − rY are the electronic position operators and α stands for the Dirac matrices. The linear responses νLY and νSY are associated with MLY and MSY, respectively. The SO-S mechanism arises from MS(e‑e) .7 Y To generalize the SO-S term to a four-component expression, the same methodology used previously in ref 14 will be applied here. This scheme is related to the Dyall formalism,15 where the Dirac−Coulomb−Breit Hamiltonian is splitted into two Hamiltonians, that is, one that is spin-free (SF) and another one that is SO-dependent. From this separation, we are able to define a four-component SO mechanism for σY

(2)

which is restricted to rigid rotor molecules in their equilibrium geometry. The approximation is generally more accurate for the isotropic than for each individual tensor element. As mentioned above, few recent works have shown that such a relationship is broken within the relativistic regime.3−6,12 In our search for generalizing Flygare’s NR relationship, we have proposed three increasingly accurate relativistic theoretical models.7 Such models were related to a few previous ones, such as those of Malkin and coworkers5 and Xiao and coauthors.6 However, the aim of the present work is to go one important step further. The most accurate of the above-mentioned models is the MIII one, which was defined as mp 1 σYM‐III = MY ⊗ I + σYatom + σYSO‐S gY 2 (3)

σYSD = σY − σYSF

(5)

where σSF Y is the shielding tensor obtained when considering the SF Hamiltonian. A similar definition can be applied to the linear response νSY of the second term of eq 4, νYS‐SD = νYS − νYS‐SF

Replacing 1 σSO‑S with Y 2

5189

(6) 1 S‐SD(e‐e) ν 2c Y

in eq 3, we obtain DOI: 10.1021/acs.jpclett.6b02361 J. Phys. Chem. Lett. 2016, 7, 5188−5192

Letter

The Journal of Physical Chemistry Letters σYM‐IV =

mp gY

MY ⊗ I + σYatom +

1 S‐SD(e‐e) νY 2c

important the S-SD term is. It also shows that the difference 1 ‐S between ΔσII⊥,Y and 2 σ⊥SO , Y arises from higher order terms.

(7)

1

1

S‐SD(e‐e) ‐S The replacement of 2 σ⊥SO greatly improves , Y by 2c ν⊥, Y the behavior of the M-III model. The subtraction between 1 S‐SD(e‐e) ν and ΔσII⊥,Y has nearly the same value for each 2c ⊥ , Y nucleus Y in different molecules. Such a subtraction is close (in ppm) to 0.1 when Y = F, 0.2 for Cl, 1 for Br, 9 for I, and 172 for At. So, the remaining difference seem to be of atomic nature. 1 ‐SD(e‐e) − We then evaluated the term 2c νYatom,S and found the following values (in ppm): 0.0029 for X = F, 0.0299 for X = Cl, 0.9324 for X = Br, 8.2424 for X = I, 1.1477 for X = Xe, and 171.7071 for X = At. They have a remarkable similarity with those given above. There is an error that comes from the fact that we have calculated σatom(e‑e)(Y−) instead of σatom(e‑e)(Y) (see Table 1). In all cases, σatom(e‑e)(Y−) are larger by few ppm than the values of

where only (e-e) rotations are considered in νS‑SD because they Y (and not the (e-p) ones) give rise to the SO-S term as a leading order correction.16 We should also mention here that for linear molecules the first term has null parallel component. Now we shall compare the performance of M-III and M-IV models on a set of linear molecules, HX, (X = F, Cl, Br, I, At), XF, IX (X = Cl, Br, I), and XeF2, and see which one is the best. Four-component calculations of both, σY and MY tensor elements were performed at the RPA level of approach of the polarization propagator formalism with Dirac−Coulomb Hartree−Fock wave functions, employing a developmental version of the DIRAC14 program package.17 Gaussian-type nuclear models were employed in all calculations. To fulfill Flygare’s prescription for the relation of eq 1, the GO was placed at the molecular CM for the calculation of σY. Furthermore, to perform a four-component calculation of σatom Y , an electron was added to the halogen free-atoms. Calculations of open-shell structures are not available in the DIRAC code. Experimental internuclear distances for molecules in their gaseous phase were considered and taken from refs 18 (all compounds but IF) and 19 (only IF). For HAt, an optimized internuclear distance was used (1.720284558 Å). All calculations were performed using large uncontracted Gaussian basis sets and the common gauge-origin (CGO) approach. The NR Dunning’s aug-cc-pCV5Z basis set was used for hydrogen, fluorine, and chlorine atoms,20 whereas for Br, I, Xe, and At, the dyall.acv4z21 basis set was employed. In all cases the restricted kinetic balance prescription (RKB) was used, although calculations of few selected molecules were also performed using the unrestricted kinetic balance prescription (UKB).13 This is because to evaluate the performance of our models, it is enough to consistently use RKB with a huge basis set and also because the differences found applying UKB are such that the accuracy of our models does not vary. In Figure 1 we show the performance of SO-S and S-SD contributions versus ΔσII⊥,Y. It gives us an insight into how

σ⊥(e, Y‐e) −

m pI

M⊥(e, ‐Ye) −

gY

1 S‐SD(e‐e) ν 2c ⊥ , Y

and σ (e, Y‐e) −

1 S‐SD(e‐e) ν . 2c , Y

The

worst results were found for At, where the parallel and perpendicular components of mp (e‐e) 1 S‐SD(e‐e) (e‐e) σAt − g MAt ⊗ I − 2c νAt have values that are close Y

to 10 390 ppm, where σatom(e‑e)(At−) ≅ 10 563 ppm. Then, we propose a model that includes all of these considerations mp

σYM‐V =

gY

MY ⊗ I + σYatom +

1 S (νY − νYatom,S) 2c

(8)

For linear molecules this equation is written as σ⊥M, Y‐V =

m pI gY

M⊥, Y + σYatom +

σ M, Y ‐V = σYatom +

1 S (ν⊥, Y − ν⊥atom,S ,Y ) 2c

1 S (ν , Y − ν atom,S ) ,Y 2c

(9)

Our M-V model has a few important advantages: (i) the restrictions of considering only the SD and (e-e) parts of νSY are not needed any more; (ii) from calculations we find that νS(e‑p) Y ≅ νatom,S(e‑p) and νS‑SF(e‑e) ≅ νatom,S‑SF(e‑e) . Therefore, νSY − νatom,S Y Y Y Y ≅ νS‑SD(e‑e) − νatom,S‑SD(e‑e) and (iii) the application of the linear Y Y response within the elimination of small components (LRESC) 1 model to 2c (νYS − νYatom,S) gives, as the leading order 1

contribution, 2 σYSO‐S.7 In Table 1 we compare results of the (e-e) contributions to the shieldings of nuclei in ionized free atoms (except for Xe) and the differences between four-component calculations of shieldings of the nucleus in molecules, SR constants, and the SSD terms. Also, the differences when replacing the S-SD terms − νatom,S(e‑e) are shown. We see that the values of by νS(e‑e) Y Y tensor elements of mp 1 ‐e) − σY(e‐e) − g MY(e‐e) ⊗ I − 2c (νYS(e‐e) − νYatom,S(e ) are closer to 1

σ Y a− t o σY(e‐e) −

1

S‐SD(e − e) ‐S Figure 1. Comparison of the matching of 2 σ⊥SO with , Y and 2c ν⊥, Y ΔσII⊥,Y, where Y represents each nucleus of all the compounds studied 1 in the present work. The agreement between ΔσII⊥,Y and 2c ν⊥S,‐YSD(e‐e) is

Y

m ( e ‑ e )

mp gY

than the elements of 1 MY(e‐e) ⊗ I − 2c νYS ‐ SD(e‐e). So, the M-V model is

more accurate than the M-IV one. What is the behavior of the M-V model in the NR limit? We scaled the speed of light several times in all calculations of the (e-e) contributions, and its results are summarized as

1

‐S by far better than that with 2 σ⊥SO , Y for all Y in all compounds, except for At in HAt.

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DOI: 10.1021/acs.jpclett.6b02361 J. Phys. Chem. Lett. 2016, 7, 5188−5192

Letter

The Journal of Physical Chemistry Letters

Table 1. Comparison of σatom(e‑e)(Y−) with the Difference among the Calculated (e-e) Parts of Shielding, the SR Constant and the S-SD Term, or the νS − νatom,S Onea

a

Mol

Y

b

c

d

e

σatom(e‑e) Y−

HBr BrF IBr HI IF ICl IBr I2 XeF2 HAt

Br Br Br I I I I I Xe At

525.6652 525.2953 524.2390 1906.8889 1906.1397 1905.0742 1903.9301 1903.1371 2024.6561 10390.5182

526.6818 526.1843 525.2110 1915.2733 1914.2449 1913.3050 1912.1788 1911.4018 2025.7857 10562.5504

525.4341 525.8365 525.5713 1907.1891 1907.9397 1907.0292 1906.9541 1906.6010 2024.7760 10390.2688

526.4220 526.8936 526.5569 1915.5107 1916.4363 1915.4419 1915.3500 1914.9608 2026.0695 10562.0430

526.5790 526.5790 526.5790 1915.7296 1915.7296 1915.7296 1915.7296 1915.7296 2026.0618f 10563.0186

All values are given in ppm.

e (e‐e)

σ ,Y −

1 (ν S, Y(e‐e) 2c

b (e‐e)

σ⊥, Y −

m pI gY

M⊥(e, ‐Ye) −

1 S‐SD(e‐e) ν . 2c ⊥ , Y

c (e‐e)

σ⊥, Y −

m pI gY

M⊥(e, ‐Ye) −

1 (ν S(e‐e) 2c ⊥ , Y

‐e) − − νYatom,S(e ).

d

σ (e, Y‐e) −

1 S‐SD(e‐e) ν . 2c , Y

S(e‐e) f atom(e‑e) − − νYatom, ). σXe was calculated as such (not ionized) because it has a closed-shell structure.

Table 2. Four-Component Calculations of the Isotropic Parameters, M, σatom, νS, νatom,S, and σ, at RPA Level of Approacha,b Mol

Y

HF ClF BrF IF

F F F F

XeF2

F

HCl ClF ICl HBr

Cl Cl Cl Br

BrF IBr HI

Br Br I

IF

I

ICl IBr I2 XeF2

I I I Xe

HAt

At

2 m pI M 3 gY ⊥, Y

σatom Y−

−57.9462 210.9556 291.9238 375.8094 [375.8150] −31.9061 [−31.9027] −192.3764 −1598.3155 25.7220 −482.9111 [−482.9190] −3569.1870 −388.0969 −902.1302 [−902.1781] −5630.2973 [−5630.9135] −2827.5506 −2184.7155 −1236.6301 −3609.8499 [−3609.9578] −146.9260 [−147.4289]

1 S,iso ν 2c Y

480.3801 480.3801 480.3801 480.3801 [482.3759] 480.3801 [482.3759] 1132.2008 1132.2008 1132.2008 3355.2831 [3439.9540] 3355.2831 3355.2831 6406.4582 [6803.6769] 6406.4582 [6803.6769] 6406.4582 6406.4582 6406.4582 6619.4386c [7020.7966]c 18778.8819 [19193.0912]

1 atom,S ν − 2c Y

2.0197 1.7950 −6.9508 −56.3174 [−56.4510] −2.2784 [−2.4125] 13.8434 13.2515 −7.3600 130.3628 [121.1226] 94.3739 156.9831 511.0723 [495.7096] 278.0488 [262.6546] 294.2667 310.5022 494.2027 562.4590 [546.6925] 2797.5863 [2761.4493]

2.0608 2.0608 2.0608 2.0608 [1.9266] 2.0608 [1.9266] 13.9431 13.9431 13.9431 133.3157 [124.0756] 133.3157 133.3157 541.2102 [525.8491] 541.2102 [525.8491] 541.2102 541.2102 541.2102 570.4629c [554.6929]c 3662.3453 [3626.2449]

σM−V,iso Y

σiso Y

422.3927 691.0699 763.2923 797.8113 [799.8132] 444.1348 [446.1340] 939.7247 −466.8064 1136.6197 2869.4190 [2954.0819] −252.8456 2990.8537 5474.1901 [5871.3594] 512.9994 [909.5689] 3331.9641 3991.0347 5122.8205 3001.5848 [3402.8383] 17767.1969 [18180.8666]

415.8361 683.9108 756.7222 790.7507 [793.7221] 438.4208 [440.4896] 934.1685 −473.7810 1129.3206 2865.2450 [2948.7345] −258.1816 2983.1190 5469.7790 [5866.5947] 507.5656 [903.8078] 3326.0317 3984.5477 5112.9888 2999.2927 [3400.2130] 17762.6035 [18175.7059]

RKB prescription was used in all calculations, but between brackets are the UKB calculations. bAll values in ppm. cσatom(Xe) and νatom,S(Xe) were calculated as such (not Xe−) because it has a closed-shell electronic structure. a

L S (1) For all studied systems, the SD values of σY, σatom Y , MY, MY L S atom,S (and therefore also νY and νY), and νY go to zero as c scales to ∞. This behavior is fine because they arise only within the relativistic domain. also goes to (2) The SF contribution to MSY, νSY, and νatom,S Y zero as c → ∞. Some remarks concerning the isotropic contributions of both properties need to be made. As was pointed out in ref 7, the MII and M-III models match each other when isotropic values of shieldings of nuclei in linear molecules are considered. This is zero for linear molecules,22 but occurs because σSO‑S,iso Y

1 S‐SD(e‐e),iso ν 2c Y

and

1 (ν S(e‐e),iso 2c Y

In

Table

σYM‐V,iso

=

2

2 m pI M 3 gY ⊥, Y

we +

− σYatom

show +

the

1 (ν S,iso 2c Y

−

values

of

− νYatom,S ).

and σ(e‑e) In addition, a good agreement between σM‑V(e‑e) ⊥,Y ⊥,Y and M‑V(e‑e) (e‑e) and σ∥,Y is found. The differences are of also between σ∥,Y