Calculation of hyperfine coupling constants of muonated radicals

Calculation of hyperfine coupling constants of muonated radicals

Volume 165,number 6 CALCULATION CHEMICAL PHYSICS LETTERS OF HYPERFINE Alexandre LOPES DE MAGALHAES COUPLING CONSTANTS 2 February 1990 OF MUONA...

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Volume 165,number 6

CALCULATION

CHEMICAL PHYSICS LETTERS

OF HYPERFINE

Alexandre LOPES DE MAGALHAES

COUPLING

CONSTANTS

2 February 1990

OF MUONATED

RADICALS

and Maria Jo50 RAMOS

Departamento de Quimica, Faculdade de Ci0ncia.x Vniversidade do Porte, 4000 Porte, Portugal

Received 5 September 1989; in final form 1I November 1989

Comparison of muon hyperfine coupling constants with the corresponding proton ones for the analogous H radical reveals considerable isotopic shifts. On isotopic substitution the bond length expectation values vary and muonic isotopic effects should be larger than with conventional isotopes such as deuterium. INDO calculations using a geometry based on MNDO-RHF calculations (with fixed C-Mu bond length expectation values in a Morse potential) have been carried out, giving improved values for hypefine coupling constants and for corresponding shifts upon isotopic substitution by Mu in several radicals.

1. Introduction Muonium-substituted organic radicals can be formed when a beam of pf mesons falls on liquid unsaturated hydrocarbons. The technique that enables the study of these radicals has been named muon spin rotation, uSR. Comparison of muon hypet-fine coupling constants with the corresponding proton ones, of the analogous H radicals, reveals considerable isotopic shifts. These isotopic shifts must be dynamic effects since, within the Born-Oppenheimer approximation, the wavefunctions used for the calculations are exactly the same for the muonic or the protonic species. Programs such as MNDO that perform semi-empirical calculations to obtain molecular properties have been parameterised to reproduce experimental data. In particular, when a geometry optimisation is carried out one obtains bond length values which represent a dynamic average over vibrational modes - rather than equilibrium internuclear distances, as is the case with ab initio programs, corresponding to minima on the potential energy surface. When isotopic substitution, such as hydrogendeuterium, is involved the bond length expectation values should vary; however, that variation is small enough to be neglected and there is no special parameterisation available for this in semi-empirical programs. Muonium is, however, a very light isotope which makes it a rather special one. Isotope effects, 528

should be larger than with conventional isotopes such as deuterium. Bond length expectation values for the diatomic species C-X (X= Mu, H, D) in a Morse potential [ 11, using accurate wavefunctions and the Morse parameters given by Johnston and Parr [2], have been calculated [ 3,4]. The following values for the bond length have been obtained: C-Mu = 0.1197 nm, C-H = 0. I 14 1 nm and C-D ~0.1134, for an equilibrium bond length of 0.1113 nm. This makes the C-Mu bond length 4.9% longer than the C-H one which, in turn, is only 0.6% longer than the C-D bond length. It is expected that this larger isotope effect in the bond length associated with muonium will produce a larger perturbation in the molecule, in particular in delocalised species which possess orbitals that extend through the whole system. A possible way of simulating this isotopic perturbation is by increasing the C-H bond length of H species, analogous to the muonic one, by 4.9% and keeping it fixed while the remaining geometry parameters are relaxed in a geometry optimisation calculation using the MNDO (RHF version ) semi-empirical program. INDO calculations, using the MNDO optimised geometries, give improved values for the corresponding hyperfine coupling constants. The cyclohexadienyl, 1,I -dimethylallyl and vinyl radicals have

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CHEMICAL

Volume 165. number 6

been studied and the corresponding cussed here.

results are dis-

2. Results and discussion 2.1. The muonated cyclohexadienyl radicals The MNDO optimised geometries of the C,H, and C6F6H radicals are given in table 1. The C6H7 structure is predicted to be planar with the exception of the methylene protons which lie on a plane perpendicular to the one defined by the ring; for the fluorinated radical, a small out-of-plane deformation of 5.48” of the methylene carbon is observed. The isotopic substitution of H by Mu was simulated as discussed above. The resulting optimised geometries are also given in table 1. It is seen that there is a shortening of the non-substituted methylene bond length of 0.3% in both cases as well as a shortening of the C-C bonds to the methylene carbon of approximately 0.2% also in both radicals, as compared to the Table 1 MNDO-RHF

optimised

geometries

2 February

PHYSICS LETTERS

1990

non-substituted radical. This shortening of bond lengths in the Mu species must be associated with the increase in charge density located on that atom as compared to the corresponding H radical. This increase seems to encourage the C atom bound to the Mu atom to compensate for its momentary decrease of charge density by removing it from the three neighbouring atoms and thus shortening their bond lengths. Table 2 shows the INDO hyperfine coupling constants of the four cyclohexadienyl radicals calculated with the respective MNDO optimised geometries. Also shown in table 2 are the experimental values for the same hypertine coupling constants. It is convenient to consider the “reduced” muon coupling for the experimental muon hypertine coupling constants, i.e. the measured coupling divided by the muon/proton magnetic moment ratio (3.1833). Previous INDO calculations on cyclohexadienyl radicals, yielding values for hyperfine coupling constants, do not provide satisfactory agreement with experimental data [ 5-81. The value shown in table

for radicals C6H,, C6H6Mu, C,F,H and C6F6Mu Radical

(X=Y=H)

C6H6Mu (X=H; YEMU)

C6F6H (X=F;Y=H)

C6F6Mu (X=F;Y=Mu)

1.117 1.117 1.503 1.370 1.427 1.088 1.092 I .090

1.114 1.172 1.500 1.370 1.427 I .088 1.092 1.090

1.351 1.132 1.546 1.396 1.457 1.311 1.321 1.316

1.347 1.187 1.542 I .396 1.457 1.311 1.321 1.316

105.97 113.38 122.45 120.89 119.96 118.51 121.22 109.30

105.21 113.58 122.44 120.86 119.91 118.60 121.19 110.23

108.95 112.39 122.83 120.69 120.29 118.51 121.19 110.69

108.57 112.36 122.94 120.65 120.2 1 118.55 121.10 111.38

W,

bond lengths (A)

bond angles (de&!)

1 2 3 4 5 6 7 8

529

CHEMICAL PHYSICS LETTERS

Volume 165, number 6

2 February 1990

Table 2 INDO hypefine coupling constants of radicals CeH,, C,H,Mu, C,F,H and C,F,Mu, in mT. MDNO geometries Radical

W nuclei ~ ortho meta

para

methylene

Protons or fluors (C&I, and C6H6Mu) (CsF6H and C,F,Mu) 9 ortho meta para methylene muon

proton

fluorine

C6H7

INDO exp. ‘)

1.738 -

- 1.347

2.173

- 1.627

- 1.015 ( - )0.904

0.535 0.280

- 1.184 (-)1.325

C,H,Mu INDO exp. b,

1.729 1.41

- 1.343 -1.26

2.148 1.93

- 1.772 -1.22

- 1.014 -0.897

0.53 1 0.267

- 1.170 1.291

C,F,H INDO exp. 1)

1.868 -

- 1.621

2.319

-1.311

3.732 2.41

-2.020 0.58

4.943 3.76

C6FbMU INDO exp. al

1.861 _

- 1.617 _

2.303 _

- 1.463 _

3.689 2.395

-2.025 -0.580

4.911 3.728

‘) Ref. [6].

‘) Ref. [lo].

2 seem to be the best agreement between theory and experiment reported so far for these radicals. The MNDO-RHF method, therefore, seems to have been a good choice to calculate the optimised radical geometries [ 3 ] . Perhaps easier to compare are the isotopic shifts of the muonated radicals relative to the corresponding protonic radicals given in table 3. There is good agreement, in general, between the calculated data

6.917 4.785 8.707 5.765

6.502 4.499 5.320 1.93

6.656 2.246

-

13.378 12.62 12.859 12.731

and experimental. However, although the values for the methylene nuclei are very good for the C6H,Mu/ C6H7 case, those for C6F6Mu/C6F6H are further apart. The trends, in general, seem to conform with experiment; the effect is predicted to be negative and larger for the para than for the ortho hydrogen in both cases. For the small meta couplings, relative experimental errors are large, and comparison with calculated shifts is not meaningful. A large shift is pre-

Table 3 isotopic shift in % for absolute value of the hyperfine coupling constants of the muonated cyclohexadienyl radicals relative to the corresponding protonic radicals (values obtained from table I ) Muonated radical

onho

c6Hg.M~‘) INDO exp. &F,Mu ” INDO exD.

meta

para

methylene

ortho

meta

para

methylene muon

proton -6.0 -6.0

-0.5

-0.3

-1.2

+8.9

-0.1 -0.8

-0.7 -4.6

-1.2 -2.6

+25.9 + 20.5

-0.4

-0.2

-0.7

+11.6

-1.2 -0.6

+0.2 0.0

-0.6 -0.8

+25.1 + 16.4

_

‘) See also ref. [ 3,4].

530

Protons

“C nuclei

_

_

fluorine

-3.9 +0.9

2 February 1990

PHYSICSLETTERS

CHEMICAL

Volume 165, number 6

Me*C=C-CH,Mu. Both have been studied experimentally [ 9 ] _

dieted for the methylene “C coupling in both &H,Mu and C6F,Mu experimental data are not available for the unsubstituted radicals C6H7 and C6F6H.

The isotopic substitution

of H by Mu was simu-

Exposing 1,l -dimethylallene to a beam of spin-po-

lated here as with the previous radicals. The resulting optimised geometries of the two H radicals and the muonated ones are given in table 4. As far as the I,1 -dimethylallyl radical is concerned the substitution of H by Mu shortens the C-C bond adjacent to the -CH,Mu group by 0.3% and the other two C-H bonds of the same group by approximately 0.1%. In

larised positive muons results in the formation of two radicals: the muonated 1, I-dimethylallyl radical, Me&=CMu-CH2, and the muonated vinyl radical,

the case of the muonated vinyl radical most bond lengths are affected by small changes. Table 5 shows the INDO hypertine coupling con-

2.2. The muonated l,l-dimethylallyl and vinyl radicals

Table 4 MNDO-RHF

optimised

geometries

for radicals MezCCHCHI,

bond lengths (A)

Me$ZCMuCH,,

a b C

d

*

;y;;@es e

Me*CCCHs and Me2CCCH2Mu MezCCCHJ (XsH)

Me2CCCH2Mu (XrMu)

1.110 1so9 1.310 1.438

1.110 1.509 1.310 1.432 1.110 1.165 1.109

4

1.111

3 4

109.47 109.47 109.47

5

111.35 121,10

Me2CCHCH2 bon; lengths

(X=H)

MelCCMuCH2 (X=Mu)

B h i

1.110 1.110 1.502 1.502 1.415 1.099 1.370 I .089 1.086

1.110 1.111 1.502 1.500 1.418 1.153 1.362 1.089 1.087

1 2 3 4 5 6 7 8

111.58 111.63 122.57 120.21 128.10 115.57 125.01 121.22

111.58 111.65 122.40 120.16 129.2 1 115.04 124.99 121.47

a b : ;

bond angles (deg)

111.35 121.10 109.9 1 108.54 107.20

531

2

7

Radical

8

9

3.270 3.346

3.350

3.371

0.145

0.144

-0.247

3.201

3.090

-0.246

0.159

0.156

-0.241

-0.243

3.190

3.150

-0.328

-0.333

-

0.639

-

7.231

0.696

-

9.053

-

Mu

-1.192

-1.224

2.184

1.895

8

-1.145

-1.181

2.200

2.556

9

-1.019

-1.005

2.126

2.166

-0.326

-0.248

‘H

7

-0.247

-0.251

6

-0.331

4

-0.249

3

10

5

-0.961

3.325

13

2.448

- 1.574

- 1.565

1.690 3.260

2.418

-

-1.715

12

geometries

-0.942

2.146

2.185

11

in mT. MNDO

2

and Me$ZCCH,Mu,

1

Me2CCCH3 ‘%I nuclei

Me2CCMuCH,

Protons

INDO hypertine coupling constants of radicals Me2CCHCH2,

Table 5

1.740

2.146

2.229

-

- 1.564

14

3

G3

2 4 E

F

e

CHEMICAL PHYSICS LETTERS

Volume 165, number 6

2 February 1990

Table 6 Comparison of coupling constants, in mT, for Me2CCCHJ, Me2CCHCHZ and related muonated radicals (temperature values in parentheses)

Radical

Me*C=C-CH3 Me,C=C-CH,Mu

0’ .I II INDO c’

experimental a1

9.053

2.844 (210 K) 2.597 (288 K)

lND0 c’

experimental d’

INDO ‘)

experimental d,

I .25

1.33 e)

I .09

1.10

7.23 1

Me,C=CH-CH, Me#Z=CMu-CHZ

a;la,d,bb

ap

0.639 0.696

0.401 (210 K) 0.426 (288 K)

0.356 (153K) 0.404 (356 K)

*) Extrapolated to 298 K. “a~=0.31413a,. “Thiswork. d’Ref. [9]. e, Proton analogue ofthis radical unknown; estimate ofthe a/p isotope ratio by comparison with ESR data for the I-methylvinyl radical (ref. [91)

stants of these same radicals calculated with the respective MNDO-RHF optimised geometries. Also shown in table 5 are the experimental values. The muon and corresponding proton hyperfine coupling constants are compared in table 6. Although the temperatures are different, so an exact comparison is impossible, it is seen that the calculated muon hyperfine coupling is very far off the experimental one in the case of the vinyl radial; however, for the muonated 1,l -dimethylallyl radical, the calculated muon hyperfine coupling is far more in agreement with the experimenta value. One possible explanation is the fact that the latter radical is a delocalised system and therefore the perturbation exerted by the longer C-Mu bond propagates through the radical; that is well characterised by the model used in this work. On the contrary, with the vinyl radical, the unpaired electron is localised and therefore the model seems to overrate the perturbation that the muonium atom will bring to the whole system. However, the values obtained for the ratio of the muon and proton couplings, a’,,/a,, are in good agreement with the ones obtained from experimental values, as can be seen from table 6.

3. Concluding remarks The proposed model is a simple one in which INDO calculations were carried out using a geometry based on MNDO-RHF calculations with fixed

C-Mu bond length expectation values in a Morse potential. This model has produced improved values for hypertine couplings constants and for corresponding shifts upon isotopic substitution by Mu in several radicals. However there are some weakness in the model. Muonic isotopic effects are larger than isotope effects with conventional isotopes such as deuterium. Moreover, one should be aware of the fact that extending a bond does introduce electronic changes which cannot be properly accounted for by existing parameterisation of the semi-empirical procedure, particularly in the case of hyperfine coupling constants. It seems, therefore, advantageous to attempt to find a new parameterisation suitable for muonium rather than use the one for protium, bearing in mind that the calculation of the expectation value of a particular hyperline coupling constant at the bond length’s expectation value does not necessarily follow. However, although the static approach of the proposed model can only be a first step to the understanding of isotopic shifts in hypert’ine coupling constants of muonated radicals, it seems to provide quite good values for rigid radicals. For the more flexible ones, the values are not so good. A more advanced treatment should calculate the expectation value of the hypcrfine coupling constant for the vibrations in an anharmonic potential directly. This should then also describe the temperature dependence of the couplings.

533

Volume 165,number 6

CHEMICAL PHYSICS LETTERS

Acknowledgement We thank Dr. Emil Roduner for helpful discussions. Financial support by the Instituto National de Investigaqgo

Cientifica

is gratefully

acknowledged.

References [I ] P.M. Morse, Phys. Rev. 34 (1929) 57. [2] H.S. Johnston and Ch. Parr, I. Am. Chem. Sot. 85 (1963) 2544.

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2 February 1990

[ 3 ] M.J. Ramos and E. Roduner, Rev. Port. Quim., in press. [ 41 E. Roduner and I.D. Reid, Isr. J. Chem., in press. [ 5] J.A. Pople, D.L. Beveridge and P. Dobosh, J. Am. Chem. Sot. 90 (1968) 4201. (6jM.B. Yim and D.E. Wood, J. Am. Chem. Sot. 97 (1975) 1004. [ 7) P. Bischof, J. Am. Chem. Sot. 98 (1976) 6844. [S] D.M. Chipman, J. Chem. Phys. 78 (1983) 4785. [ 9 ] C.J. Rhodes, M.C.R. Symons, E. Roduner and C.A. Scott, Chem. Phys. Letters 139 ( 1987) 496. [IO] E. Roduner, G.A. Brinkman and P.W.F. Louwrier, Chem. Phys. 73 (1982) 117.