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Visualization of EPR Hyperfine Structure Coupling Pathways Vladimir G. Malkin, Olga L. Malkina, and Georgy Mikhailovich Zhidomirov J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b01833 • Publication Date (Web): 14 Apr 2017 Downloaded from http://pubs.acs.org on April 21, 2017

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The Journal of Physical Chemistry

Visualization of EPR Hyperfine Structure Coupling Pathways Vladimir G. Malkin,*,1 Olga L. Malkina,1,2 and Georgy M. Zhidomirov 3 Institute of Inorganic Chemistry, Slovak Academy of Sciences, Dúbravská cesta 9, SK-84536 Bratislava, Slovakia

1

Department of Inorganic Chemistry, Comenius University, SK-84215 Bratislava, Slovakia

2 3

G.K. Boreskov Institute of Catalysis, Siberian Branch of the Russian Academy of Sciences, Prospect Lavrentieva 5,

630090 Novosibirsk, Russia * Corresponding author; e-mail: [email protected]

1

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Abstract The close relation between the EPR hyperfine coupling constant and NMR indirect spin-spin coupling constant is well known. For example, the Karplus-type dependence of hyperfine constants on the dihedral angle, originally proposed for NMR spin-spin coupling, is widely used in pNMR studies. In the present work we propose a new tool for visualization of hyperfine coupling pathways based on our experience with visualization of NMR indirect spin-spin couplings. The plotted 3D-function is the difference between the total electron densities when the magnetic moment of the nucleus of interest changes its sign and as such is an observable from the physical point of view. It has been shown that it is proportional to the linear response of the spin-density to the nuclear spin (i.e. magnetic moment). In contrast to the widely used visualization of spin-density our new approach depicts only the part of the electron cloud of a molecule that is affected by the interaction of the unpaired electron(s) with the desired nuclear magnetic moment. Since visualization of NMR spin-spin couplings and hyperfine interaction is based on the same ideas and done using similar techniques it allows a direct comparison of the corresponding pathways for the two phenomena so as to analyze their resemblance and/or dissimilarity.

2


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I. INTRODUCTION EPR spectroscopy and classical NMR spectroscopy are considered as complementary tools dealing with open and close shell systems correspondingly. Yet they are governed by the same physical laws and thus deeply interrelated. One of the most notable examples is the long-known relation between hyperfine coupling constants and NMR indirect spin-spin coupling constant often leading to correlation between them.

1,2,3

The first seminal papers on the relationship between EPR hyperfine

coupling constant (HFCC) and NMR indirect spin-spin coupling constant (J) were published about 50 years ago. Currently this analogy is widely used in EPR and especially in pNMR communities. dependence

7,8

4,5,6

1,2,3

In particular the Karplus-type

of hyperfine constants is extensively used in pNMR studies for 3D structure determination in solutions. 6 Also,

the question about hyperfine coupling pathways - the interaction of an unpaired electron, sitting (mostly) upon a particular atom, with the magnetic moment of a selected nucleus (which is analogous to NMR coupling pathways) – has become topical. However, despite all efforts, the interpretation of the HFCC and its relation to the details of electronic structure and geometry are still challenging. 4,5,6 Interest in this topic has been renewed with recent advances in NMR of paramagnetic compounds (pNMR). In the last few years significant progress in quantum-chemical methods for calculations of pNMR shifts has been achieved

9

bringing new

opportunities for interpretation of the experimental results. One of the most interesting questions in the analysis of pNMR shifts concerns the pathways of the contact and pseudo-contact shifts. Thus, Clément et al. suggested a way of decomposing pathways of hyperfine shifts into contributions from different transition metal centers based on DFT periodical calculations. 10,11

Often interpretation of the EPR hyperfine coupling is based on plotting the electron spin-density obtained from quantum-

chemical calculations - at present mostly from Density Functional theory (DFT) (see for example refs. 12,13). Such plot shows the distribution of the spin-density throughout the whole molecule showing the signs and values of the hyperfine constants for all magnetic nuclei in the whole structure. However, these plots do not allow one to comment on a particular pathway. This leaves significant space for guesswork and calls for development of new tolls for identification of the hyperfine constant pathways. With this in mind we propose a new way of visualization of the hyperfine pathways using an analogy between EPR hyperfine coupling and indirect NMR spin-spin coupling constants. The discussed resemblance between HFCC and spin-spin couplings is rooted in the identical physical mechanisms responsible for propagation of spin-polarization via the electronic structure of a molecule. While formation of an open shell system certainly leads to a different electronic structure in comparison with its closed-shell counterpart (precursor) to say nothing about possible changes of the geometry of the system, in the majority of cases it does not change dramatically either the geometry of the molecule or the electronic structure and hybridization patterns further than 1 or 2 bonds away from the center possessing the unpaired electron(s).

2,14

Thus the usual considerations based on the Dirac vector model

15

for long-

range spin-spin couplings can be applied to predict and analyze the signs and values of hyperfine constants. Also the tools developed for interpretation of spin-spin couplings such as visualization of NMR spin-spin coupling pathways can be used for hyperfine pathways as well. In the past an approach for visualization of NMR indirect spin-spin coupling pathways based on quantum-chemical calculations was developed by the two of the present authors and applied to different systems. 16,17,18,19 This approach allows one to unambiguously visualize the parts of the electronic structure that are affected by the interaction of two nuclear

3



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magnetic moments. This method is based on the calculation and visualization of the difference between the electron densities obtained from two separate calculations: one with the two nuclear magnetic moments being parallel and another with the nuclear moments antiparallel. In those calculations the interaction of the nuclear magnetic moments with electrons was implemented using double finite perturbation theory (Fermi-contact term only) as described in reference

16

(see also ref. 20

for the first implementation of the double-perturbation theory). In the present work we decided to extend this idea to visualization of the interaction between the unpaired electron(s) and a selected nuclear magnetic moment (i.e. for visualization of a selected hyperfine interaction pathway). It is based on the usually overlooked fact (which will be shown in the next section) that the spin density is slightly different when the nuclear spin is up and when it is down. In other words we will visualize the linear response of the spin-density to a small perturbation caused by the Fermi-contact interaction of the electron spin with the nuclear magnetic moment. This response obviously depends on the position of the nucleus and the location of the unpaired electron(s) in the system. One can expect that the electron spin-density in the remote (from the nucleus of interest and the center possessing the unpaired electron) parts of the molecule will be negligibly affected by the selected hyperfine interaction (i.e. will remain unchanged regardless of the orientation of the nuclear magnetic moment). Only the spin-density in the vicinity of the interaction pathway should be different and reveal the hyperfine coupling pathway. The paper is organized as follows: in section II we will describe the basic idea of the approach for the visualization of the hyperfine coupling pathway. In section III the details of the calculations will be given. In section IV we will present and discuss the results obtained for a series of organic radicals, a transition metal complex and a hydrogen bonded system. For selected cases visualization of the related NMR spin-spin couplings will be presented as well. In the final section we will summarize the outcome of this work and give some outlook for future developments.

II. METHOD Throughout this paper the Hartree system of atomic units is used. In what follows below we will restrict ourselves to the nonrelativistic framework and consider only the isotropic hyperfine coupling constant (HFCC) although the extension to the anisotropic part of the hyperfine tensor is straightforward. The isotropic hyperfine constant on nucleus M can be expressed as the derivative of the total electron energy with respect to the nuclear spin in the presence of the Fermi-contact interaction of the electron spin with the magnetic moment of the nucleus

∂E ∂I M

= I M =0

∂Ψ(I M ) ˆ 0 H Ψ(0) ∂I M

+ Ψ(0) I M =0

∂ Hˆ (I M ) = Ψ(0) Ψ(0) ∂I M

∂ Hˆ (I M ) Ψ(0) ∂I M

I M =0

(1)

I M =0

This is a consequence of the Hellmann-Feynman theorem. wavefunction, IM is nuclear spin,

I M =0

∂Ψ(I M ) + Ψ(0) Hˆ 0 ∂I M

21

Here E is the total electron energy, Ψ is the electron

Hˆ ( I M ) and Hˆ 0 are the perturbed and unperturbed Hamiltonians, Ψ( I M ) and Ψ(0)

ˆ (I ) = 8πα are the perturbed and unperturbed wavefunctions. Here H M 3

2

∑ δ ( r ) sˆ iM

i

4 ACS Paragon Plus Environment

i

where the sum goes over all

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electrons, a » 1/137 is the fine-structure constant, ,

( ) is the

riM is the position of electron i relative to nucleus M, δ riM

Dirac delta function and sˆi is the spin operator for electron i. 22 The isotropic hyperfine coupling constant can be expressed as follows 23

Aiso ( M ) =

4π ββ gg S 3 e N e M Z

−1

ρ α − β ( RM )

(2)

where Aiso(M) is the isotropic hyperfine constant on nucleus M, be and bN are the Bohr magneton and the nuclear magneton respectively, ge and gM are the free-electron and nuclear g-values, S Z is the expectation value of the Z-component of the total effective electronic spin and

ρ α −β (RM ) is the spin density at the position RM of nucleus M.

Please note that the linear response of the wavefunction to the spin of nucleus M is not equal to zero:

∂Ψ(I M ) ∂I M

≠0

(3)

I M =0

As a consequence the linear response of the total electron density differs from zero too

∂ ρ (I M ) ∂I M I

M

∂Ψ * (I M ) = ∂I M I =0

⋅ Ψ(0) + Ψ * (0) ⋅ M =0

∂Ψ(I M ) ∂I M I

≠0

(4)

M =0

It was recognized already in the 1960s that the interaction of the nuclear magnetic moment with the electron spin via the Fermi-contact mechanism affects the spin-density in the proximity of the nucleus. 1 To get more insight let us consider the electron densities under the conditions of an EPR experiment – i.e. when in one case the nuclear spin is up and in another case it is down. Parameterizing the Fermi-contact operator by the perturbation parameter l we can expand the total electron densities in the two cases (the nuclear spin up is denoted by

λ ↑ and the nuclear spin down is denoted by λ ↓ ) in the Taylor

series with respect to λ . Neglecting the third and higher order terms we obtain the expression for the difference

Δρ between

the electron densities in those two cases

ρ (λ ↑ ) = ρ α (λ ↑ ) + ρ β (λ ↑ )

(5)

ρ (λ ↓ ) = ρ α (λ ↓ ) + ρ β (λ ↓ )

(6)

(

) (

Δρ ( λ ) = ρ ( λ ↑ ) − ρ ( λ ↓ ) = ρ α (0) + ρ β (0) + λ ρ α ' (0) − λ ρ β ' (0) − ρ α (0) + ρ β (0) − λ ρ α ' (0) + λ ρ β ' (0)

(

)

)

+O( λ 3 ) ≈ 2λ ρ α ' (0) − ρ β ' (0) = 2λ ρ (α −β ) ' (0) (7)

Δρ ( λ ) ≈ ρ (α −β ) ' (0) 2λ

(8)



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Here

∂ ρ α (I M ) ρ (0) = ∂I M α

'

where

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ρ α (I M ) is the electron density of a electrons in the presence of the nuclear spin

I M =0

I M (and the corresponding expressions for density of b electrons). Therefore the difference between the electron densities in the two events (the nuclear spin up/down) divided by twice the value of the perturbation parameter (representing the magnitude of the nuclear spin) is equal to the linear response of the spin-density (assuming that higher order terms with respect to the perturbation are small). Please note that for close shell systems as follows from Eq. 7 the linear responses of α and β electron densities will cancel each other and the total response of the density (and the spin-density) will be zero. Following the Hohenberg-Kohn theorem

24

that states that the electron density defines the energy we can say that the energy

splitting between the states with the nuclear spin up and down is defined by the corresponding difference in their densities. As a consequence, HFCC according to Eq. 8 is defined by the linear response of spin-density. Thus for visualization of HFCC pathways we propose to look at the difference in the electronic densities (divided by twice the perturbation parameter) obtained from two calculations corresponding to the cases when the electron and nuclear spins are either parallel or antiparallel. Note that the plotted function is an observable from the physical point of view. It remains to be seen whether it can be really measured in experiment. To emphasize the analogy with NMR spin-spin coupling and visualization of its pathways 16,17 let us reverse the usual picture of the hyperfine interaction and start with the nuclear magnetic moment of a selected nucleus: it interacts with the electron spin-density, slightly modifies it in the proximity of the nucleus and then these local changes propagate through molecule (as in the Dirac vector model for long-range spin-spin couplings 15). It is clear that only areas with sizeable spin-density would be affected. Though somewhat unusual this consideration is helpful for explaining the analogy and dissimilarity between hyperfine and spin-spin couplings and has been used in the past. 1 Whereas NMR spin-spin coupling is a second order property, HFCC is a first order property. Therefore visualization of the former naturally requires double finite perturbation theory while for the latter single finite perturbation theory is needed. Thus for visualization of the HFCC pathways we add only one Fermi-contact operator associated with the nucleus of interest since the electron spin has been already intrinsically included in quantum-chemical calculations. We will call the 3-D function

Δρ (r) for visualization of HFCC as the hyperfine structure deformation density (HFDD):

Δρ (r) =

ρ (r, λ ↑ ) − ρ (r, λ ↓ ) 2λ

(9)

where λ is the perturbation parameter that can be interpreted as the magnitude of the nuclear magnetic moment. In practical calculations instead of simulating experimental conditions and using the real value of the nuclear magnetic moment, the value of this parameter is chosen based on pragmatic considerations: the FC interaction should be big enough to be above the numerical noise and at the same time small enough to keep the perturbation scheme in the linear response regime. Numerical tests demonstrate that usually both of these conditions can be easily satisfied simultaneously. Note that the linear response of spin-density can be alternatively obtained directly by solving linear response equations. Since the isotropic hyperfine coupling is entirely defined by the spin density at the position of the nucleus, the traditional way to analyze HFCC is to plot the spin-density either in 3D or for some cross section of the molecule. The latter is especially


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useful for planar structures because in this case the plane can go through all the nuclei of the molecule. This plot would instantly give information about the signs (by color) and the relative values (by intensity of the colors) of hyperfine couplings on all nuclei of the system. An example of such visualization will be presented in the next section. While undoubtedly very useful such a picture does not provide any information about particular hyperfine coupling pathways responsible for interaction of the unpaired electron(s) with the magnetic moment of the nucleus of interest. In other words: all information in one glance but no individual details. Thus the approach proposed above and the traditional visualization of spin-density are answering different questions and therefore are complimentary.

III. COMPUTATIONAL DETAILS All calculations have been carried out at the density functional level of theory. The structure of the NAMI complex was taken from ref. 25. The geometries of all close-shell systems (see Supporting Information) were optimized with the G03 package 26 employing a hybrid B3LYP exchange–correlation functional

27, 28

and B3LYP/6-31G(d,p) basis sets.

29, 30

For better

comparison of NMR spin-spin coupling and HFDD pathways the structures of open-shell systems were obtained by removing a proton (without relaxation of the structure) from their close-shell counterparts. The calculations of the coupling deformation density (for visualization of NMR spin-spin coupling pathways) and HDD (for visualization of HFCC pathways) have been done with a local modified version of the deMon-KS code. calculations we employed the Perdew86 exchange-correlation functional used a quasirelativistic Wood-Boring effective pseudopotential

38

33, 34, 35,36

and IGLO-II basis set.

37

31,32

In these

For Ru we have

with the valence basis set [311111/2111/411].

38

Visualization has been done with AVOGADRO 1.1.1. 39 IV. RESULTS AND DISCUSSION To illustrate the aforementioned similarity between NMR spin-spin and EPR hyperfine coupling pathways we have decided to start with simple structures. Intuitively one can expect substantial similarity when the unpaired electron is well localized (at least in comparison with the length of the path to the nucleus of interest). This precondition can be more easily satisfied if one looks at long-range couplings. For this we have chosen an organic chain (nono-1,3,5,7-tetraene) and the corresponding radical obtained by the removal of a proton from a terminal carbon. We use the same geometry of the common part of those systems for better comparison. The structure of nono-1,3,5,7-tetraene is presented on Fig. 1. In the radical a proton (H12) was removed from the carbon (C1) at the bottom of the picture.



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Figure 1. Structure of nono-1,3,5,7-tetraene.

In the first papers on the topic

1,2

the analogy between HFCC on proton and proton-proton NMR couplings was discussed.

However taking into account that in the considered example the unpaired electron is located closer to the terminal carbon than to the position of the removed proton we have decided to explore the analogy of HFCC with spin-spin couplings of the terminal carbon with other nuclei. By analogy with the notation 7J(C1-C8) for spin-spin couplings we will adopt the notation 7

A(C1-C8) for the hyperfine coupling constant on C8 (assuming that the unpaired electron is located on C1). The

corresponding coupling pathways for 7A(C1-C8) and 7J(C1-C8) are presented on Fig. 2a and 2b, respectively. Since the proton and electron magnetic moments have opposite signs, an interexchange of the colors in Figs. 2a and 2b (red

↔

blue) is

observed. With this in mind it is clear that the two pictures show nearly identical pathways except for small differences on bonds next to carbon C1 possessing the unpaired electron. The latter can be explained by slight delocalization of the unpaired electron. The rest of the coupling pathway is mostly defined by the electronic structure of the involved bonds and their ability to propagate spin polarization through the molecule (i.e. the linear response of the spin-density as explained in the previous section). Interestingly, despite small differences both visualized coupling pathways are rather symmetric with respect to C1 and C8. This is particularly amazing for the hyperfine coupling pathway showing the interaction of the unpaired and slightly delocalized electron with the magnetic moment of C8 (located at the position of the nucleus in the Fermi-contact interaction)! These pictures not only illustrate the experimentally found analogy of the spin-spin and HFCC couplings but also validate the proposed method for visualization of hyperfine coupling pathways.


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Figure 2. The coupling pathways for (a) 7A(C1-C8) (isosurface value 0.00025) and (b) 7J(C1-C8) (isosurface value 0.002).

(a)

(b)

To see the dependence of the pathways on the number of bonds, let us consider the 6A(C1-C7) and 6J(C1-C7). The corresponding coupling pathways are shown on Figs. 3a and 3b, respectively. Note the reverse of colors for both Figs. 3a and 3b in comparison with Figs. 2a and 2b. We will return to this point later when we will discuss pathways for 3A(C1-C7) and 3

J(C1-C7) presented on Fig. 4. While the overall similarity between the visualized pathways is remarkable some additional

dissimilarities show up. These are particularly notable around the double bonds C5-C6 and C7-C8 (the latter is outside of the formal pathway). This confirms our expectation that since multiple bonds and lone pairs are easily polarizable their presence along and in the proximity of pathways may lessen the similarity.


(a)

(b)



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Finally to demonstrate the attenuation of the similarity for short-range interactions we visualize the coupling pathways for 3A (C1-C4) and 3J(C1-C4) in Fig. 4a and 4b. The visualized pathways show that in both cases the same part of the structure is affected (Figs. 4a and 4b). Yet the hyperfine coupling pathway has less in common with the spin-spin coupling pathway than in the previous examples presented on Figs. 2 and 3. To rationalize the differences in the pathways presented on figures 4a and 4b let us first compare them with the pathways shown on Figs 2 and 3. As we noted before the color (i.e. sign) of the plotted functions changes when going from Figure 2 to 3 and the same holds for Figs. 3 and 4. This is easily seen when looking at the area of a particular bond on different plots. Here it is important to note again that we do NOT visualize the spin-density but rather the difference of total electron densities for cases when the nuclear magnetic moment is up and down for HFCC and for cases when the magnetic moments of the two interacting nuclei are parallel and antiparallel for J. The color on the bond next to the magnetic nucleus of interest (i.e. the nucleus whose magnetic moment changes its sign in the two calculations) remains the same –this “active” nucleus defines the color pattern as it should be since the derivative of the spin density in Eq. 8 is taken with respect to its magnetic moment. However since on Figs. 2, 3, and 4 the plots show pathways for different “active nuclei”, the color on particular bonds (like bond C1-C2) becomes different. Second, to rationalize the plots on Figs. 2, 3, and 4 let us consider the calculated data for HFCCs and spin-spin couplings (with carbon C1) for different carbons presented on Figure 5. We would like to note here that even the long-range HFCCs shown on Fig. 5 should be within the reach of modern EPR (see for example, refs. 40 and 41). It is clear that for long-range HFCC and spin-spin couplings (via 4 and more bonds) there are very systematic trends (alternation of the sign and monotonous change of the magnitude) for both couplings and the correlation between them. The trends are less pronounced (especially for spin-spin coupling) for three-bond (C1-C4) couplings and are completely broken for one- and two-bond couplings. A possible explanation for HFCC is that for short-range HFCC the role of the delocalized spin-density of the unpaired electron on the carbon nuclei (“direct mechanism”) becomes more significant in comparison with the propagation of spin-polarization via the structure of individual bonds (“indirect mechanism”). On the other hand, for spin-spin coupling it would be reasonable to assume that in this case direct (through space) interaction of the bond electrons has become a rival to the through-bond Dirac vector model mechanism. While it is certainly interesting to look on the trends and corresponding mechanisms in more details, this analysis lies outside of the scope of the present paper. Yet even at this stage it is clear that the difference in the visualized HFCC and spin-spin coupling pathways shows the changes in the dominant mechanisms responsible for particular couplings and at what point the similarity of HFDD and CDD breaks down.


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(a)

(b)

Figure 5. The calculated values of HFCC (in MHz) and indirect spin-spin coupling constants (in parentheses, in Hz).

0.81 ( -0.26) -1.67 ( 0.56)

1.89 ( -0.57)

-2.21 ( 0.88) 2.84 ( -0.62) -5.18 ( 7.29) 45.44 ( 1.92) 2.92 ( 79.25)



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As the next example we have chosen a monoprotonated form of N,N’-bis-(phosphanylidene)-1,8-diamine for which P-P spinspin coupling pathways were previously studied both experimentally

42

and theoretically.

43

The question about spin-spin

coupling pathways was interesting because this system exhibits several potential pathways and it was not obvious a priory, which one is realized. Both experimental and theoretical studies

42, 43

demonstrated that the major pathway goes through the

hydrogen bond whereas the p-structure of the rings is only marginally involved. This spin-spin coupling pathway is visualized in Fig. 6a. Fig. 6b shows the HFCC pathway for the corresponding open shell system obtained by removing a proton connected to the phosphorus on the left. Again a certain similarity can be seen between J and HFCC pathways. In both pathways the hydrogen bond plays the central role. Yet a few differences are seen as well: the left and right ends of the paths are symmetric for J coupling and asymmetric for HFCC; a significant part of the interaction involves the p-system for the latter; to some extent spin-delocalization likely even goes though the hydrogen bond resulting in different pictures around the right phosphorous atom. For comparison the spin-density in the system is presented on Fig. 6c. As we have mentioned earlier the spin-density does not bring insight into the HFCC pathway but gives important complementary information. Figure 6. (a) J(P-P) coupling pathway (isosurface value 0.02); (b) HFCC pathway for the phosphorous on the right (isosurface value 0.0007); and (c) spin-density (isosurface value 0.0001).

(a)

(b)

(c)


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Transition metal (TM) complexes often possessing unpaired electrons represent a very important class of compounds for many fields of research. In EPR and pNMR studies two important questions are often addressed with respect to TM complexes: the HFCC pathway and the related question about the separation of coupling pathways originated from unpaired electrons sitting on different transition metal atoms in the system. We leave the study of the second question for future developments and consider below only an example of a small transition metal complex (NAMI: trans-(dimethyl sulfoxide)(imidazole) tetrachlororuthenate(III)) studied previously both experimentally and theoretically.

25, 44, 45

The structure of the

complex was taken from ref. 44 and it is graphically presented in Fig 7.

Figure 7. Structure of NAMI, trans-(dimethyl sulfoxide)-(imidazole) tetrachlororuthenate(III), new antitumormetastasis inhibitor.

The HFCC pathway for one of the protons (H1) of the methyl group is presented in Fig. 8. In this case only the metal, the sulfur and the area of the methyl group (i.e. the part of the complex between the unpaired electron and the H1 nucleus) are involved in the pathway. The contribution of the p-system of the ring and the chlorine atoms to the hyperfine coupling is practically negligible. On the other hand, the HFCC pathway for H10 (one of the protons of the ring) exhibits a more complicated character and involves almost all atoms in the NAMI complex (see Fig. 9a). Fig. 9b (side view) provides an explanation for the difference from the previous example: the p-system of the ring interacts with the off-plan chlorine atoms effectively transferring spin-polarization to them. Halogens (each possessing three lone pairs) are often easily spinpolarizable and, as has been generally our experience, typically show up on the spin-spin coupling pathways even if they are located outside of the formal path between the two interacting nuclei. Even more interesting is the strong influence of the psystem on HFDD that leads to polarization even of the sulfur atom and methyl groups on the other side of the transition metal. Again from the study of spin-spin coupling pathways one can expect that in such cases the replacement of atoms “visible” in the pathway picture by another chemical element may change the spin-spin coupling more significantly than the



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replacement of a “silent” atom in the coupling pathway. Likely a similar result can be found with respect to HFCCs and HFCC coupling pathways. However we will postpone this interesting study for further work. Figure 8. HFCC pathway for H1 (isosurface value 0.000018).

HFS on H1

Figure 9. HFCC coupling pathway for H10 (isosurface value 0.0001): (a) top view and (b) side view.

(a)

(b)


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V. CONCLUSION AND OUTLOOK In this paper a new method for interpretation of EPR hyperfine coupling constants by visualization of hyperfine coupling pathways has been presented. The plotted 3D-function is the difference between the total electron densities when the magnetic moment of the nucleus of interest changes its sign and as such is an observable from the physical point of view. It was shown that it is proportional to the linear response of the spin-density to the nuclear spin. In contrast to the commonly used visualization of spin-density the new approach depicts only that part of the electron cloud of a molecule that is affected by the interaction of the unpaired electron(s) with a particular nuclear magnetic moment. In combination with visualization of NMR spin-spin coupling pathways this approach allows one to address the old but still topical subject of similarities and distinctions between NMR indirect nuclei spin-spin coupling and EPR hyperfine coupling. Providing unique information about which parts of a molecule are involved in the interaction of the spin-density with the magnetic moment of a particular nucleus, the new approach can shed light onto HFCC and even predict which atoms/fragments should be replaced in order to affect the value of HFCC. This can be beneficial for interpretation of the contact shifts in pNMR studies. While in the present paper we consider only the Fermi-contact operator, this method can be straightforwardly extended to the anisotropic part of HFCC and hence to the analysis of pseudo-contact pNMR shifts. We also plan to extend the presented approach to the relativistic two- and four-component frameworks. Another possible line of development is implementation of the HFDD analysis in terms of molecular orbitals, which may be useful for distinguishing electron delocalization and spin-polarization effects. ASSOCIATED CONTENT Supporting Information Atomic Cartesian coordinates used in the calculations. This material is available free of charge via the Internet at http://pubs.acs.org. ACKNOWLEDGMENTS

We acknowledge EU financial support within the Marie Curie Initial Training Networks action (FP7-PEOPLE-2012ITN), project n. 317127 "pNMR". This work has received funding from the Grant Agency of the Ministry of Education of the Slovak Republic and Slovak Academy of Sciences VEGA (grant No. 2/0116/17) and from the Slovak Research and Development Agency (grant No. APVV-0510-12 and APVV-15-0726). This work was also supported by Russian Academy of Sciences and Federal Agency of Scientific Organizations (project V.45.3.6).



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TOC Graphics

HFS on H10

HFS on H10 (side view)


Visualization of Electron Paramagnetic Resonance ... - ACS Publications

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