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Intermolecular nuclear spin—spin coupling and scalar relaxation. A quantum-mechanical and statistical-mechanical study for the aqueous fluoride ion
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
2 July 1982
INTERMOLECULAR NUCLEAR SPIN-SPIN COUPLING AND SCALAR RELAXATION. A QUANTUM-MECHANICAL AND STATISTICAL-MECHANICAL STUDY
FOR THE AQUEOUS FLUORIDE ION Aatto LAAKSONEN, J ozef KOWALEWSK1 Department of Physical Chemistry, Arrhenius Laboratory, University of Stockholm, S-106 91 Stockholm, Sweden
and Bo JONSSON Division of Physical Chemistry 2, The Lund Institute of Technology, Chemical Center, P.O.B. 740, S-220 07 Lund, Sweden
Received 17 April 1982; in final forln 30 April 1982
Coupling between 19F and magnetic nuclei (a H, 170) in water molecules in the first hydration sphere of F- ate calculated using Monte Carlo simulations. The simulations are based on intelmolecular potentials from the literature and on variations of coupling constants with geometry obtained by coupled Hattree-Fock calculations. Average coupling constants are ~ 20-40 Hz. Intermolecular scalar relaxation in aqueous solutions is discussed.
1. Introduction Knowledge about indirect nuclear spin-spin coupling constants between nuclei residing in different molecular species (intermolecular coupling) is very scarce. To our knowledge, the only paper dealing with this problem theoretically is that of Wasylishen and Barfield [ 1 ], who investigated, by means of finite-perturbation INDO calculations, the coupling between 19F in the HF molecule and proximate protons in water and methane molecules. For certain geometrical configurations, the authors found the couplings to be of sizeable magnitudes. The coupling of this type cannot be observed experimentally as splittings of resonance lines because the lifetimes of particular geometrical arrangements are extremely short in solution. A special case (which however can hardly be denoted as intermolecular) arises in the kinetically stable metal complexes, where a large body of experimental data exists for the coupling constants between the nucleus of the central atom and the nuclei in the ligands [2]. An intermedi412
ate situation is present in labile complexes, where no splittings can be seen but where the coupling interaction, modulated by chemical exchange, may become a relaxation mechanism for the nuclear spins (scalar relaxation of the first kind [3]). Under certain circumstances this relaxation mechanism has been claimed to be efficient. For example, Holz and coworkers [4] reported a study of lt3Cd relaxation in the Cd 2+ ion in D 2170 solutions and found the scalar relaxation due to the coupling between l l 3 c d and 170 to be the dominant relaxation mechanism. They were also able to derive the numerical value of the coupling constant from the spin-lattice relaxation time measurements. A possible minor contribution of this type was also reported for the Pb 2+ ion in an aqueous medium [5,6]. In this paper, we report non-empirical calculations of the intermolecular spin-spin coupling between the 19F nucleus in the fluoride ion and the magnetic nuclei (1H, 170) in the water molecule for a variety of geometric arrangements of the ion around the water molecule. Further, we use the Monte Carlo (MC) sim0 009-2614/82/0000-0000/$ 02.75 © 1982 North-Holland
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CHEMICALPHYSICS LETTERS
ulation technique to calculate the average values of coupling constants between 19F and the proton or 170 in the coordination sphere of the fluoride ion in liquid water. Finally, we propose a few simple motional models, derive the relation between the average coupling constants and the relaxation rates and discuss the possibility that intermolecular scalar relaxation is operative in aqueous ion solutions.
2 July 1982
H/~-J0 Fig. 1. Definition of the geometrical parameters R, 0 and 4~.
2. Methods of calculation
2.1. Calculation o f nuclear spin-spin coupling constants Recent work on hydrogen fluoride and related system [7,8] has been based on the combined use of large gaussian basis set coupled Hartree-Fock (CHF) calculations of the Fermi contact (FC), orbital (OB) and spin-dipolar (SD) contributions and finite perturbation-configuration interaction (FP CI) calculations of the correlation correction to the FC term. For the HF molecule, the results are: J CFC H F = 478 ttz OB SD [7],JcH F = 197 Hz [ 7 ] , J c H F =--10 HZ [7], FC CI = --127 Hz [8], giving a total calculated value JFP of 538 Hz, to be compared with the experimental 530 Hz [91. Considering the size of the F - . H 2 0 system and the fact that the present prqiect required computing more than one hundred coupling constants, we have decided to restrict the calculations to the FC term and to the CHF level of approximation. The results, though probably not sufficiently good for quantitative purposes, should be reliable enough for a qualitative discussion [10]. The calculations have been performed using the self-consistent perturbation method of Blizzard and Santry [ 11,12] and the program system MOLECULE [13]. For fluorine and oxygen, the basis set consisted of 11 s and 7p gaussian-type functions, contracted to 5s and 3p. The exponents and contraction coefficients were taken from Salez and Veillard [14]. For hydrogen atoms, the basis set consisted of 5s and lp contracted to 3s and lp [15]. All calculations were performed with the water molecule constrained to its equilibrium geometry. The position of the fluoride ion with respect to the water molecule may be described with three parameters, R, 0, and 9, defined in fig. 1. The calculations of the coupling con-
stants have been performed for a total of 78 geometries corresponding to selected combinations of the following values of R, 0, and 9: R = 2.0, 2.25, 2.50, 2.75, 3.0, 3.50 au, 0 = 0, 30, 60,120, 150, 180 °, q~= 0, 26, 52, 67, 90 ° . The calculated coupling constants were subsequently fitted to a fourteen-term polynomial in reciprocal distances between F - and the three nuclei in the water molecule: j=~
k m 1 ~1 klm Cklm(rHlrH2R ) '
(1)
where rill and rH2 in eq. (1) refer to the distance between the fluorine nucleus and the proximate and the distant proton, respectively.
2.2. Monte Carlo simulations In order to obtain the intermolecular coupling constant one has to perform some kind of statisticalmechanical averaging. This can be done with either the Monte Carlo simulation technique or via a solution of the equations of motion for the interacting particles ( F - + N H 2 0 ). We have chosen to work with the Monte Carlo (MC) simulation technique of Metropolis et at. [16]. The intermolecular potentials, F - - H 2 0 and H 2 0 - H 2 0 , necessary for the MC simulations were taken from the literature [17-19]. They are both based on accurate quantum-chemical calculations. The F - - H 2 0 pair potential could have been obtained from the present calculations, but since Kistenmacher et al. [19] used a slightly larger basis set, their potential was chosen. The MC simulations were done with the F - . ( H 2 0 ) N cluster in a spherical cavity with F - fixed at its center. The radius of the sphere was 7.5 A in all simulations, which in the largest calculation (N = 30) gives a formal 413
density of ~ 0 . 6 g cm - 3 . In a previous simulation [20] it was shown that the field gradient fluctuation at the Li + nucleus in a water cluster was largely independent of the sphere radius. It seems reasonable to assume that this is true for the spin-spin coupling too, which has a very short-range character. The short-range nature of the coupling also indicated that it may be sufficient with a limited number of water molecules in the cavity to obtain a good statistical average of the coupling constants. The statistical errors were estimated with the method described by Wood [21]. The displacement and rotation parameters were chosen so as to give an acceptance ratio of ~50%.
5 6
ui
a •f ~.. - ~ : ~ - L °
';Lf
The calculated values of the coupling constants for an illustrative sample of geometries are presented in table 1 and fig. 2 together with the corresponding total energies. It is interesting to note that the calculated J(HF) in this study are similar to those reported by Wasylishen and Barfield [1] for the HF.H 2 0 system. Further it can be seen that in the vicinity of the equilibrium configuration (which agrees quite well with earlier theoretical work [ 2 1 - 2 3 ] ) the value of the coupling constant between 19 F and the proximate proton is rather large and negative. For the shorter H - F distance (rFH 1 = 1.043 A) the coupling constant becomes positive, i.e. approaches the value in the HF molecule (where r H F is 0.917 A).
Table 1 Calculated fluorine-proton and fluorine-oxygen-17 coupling constants (Hz) for a selected sample of geometrical configurations of the F-'H20 system R (A)
0
~
JFHprox
JFttdist
JFO
2.0 2.5 3.0 2.5 2.5 2.0 2.5 3.0 2.0
0 0 0 90 180 0 0 0 60
0 0 0 0 0 52 52 52 52
-112.2 -29.4 -8.2 -4.2 +6.6 +40.4 -79.0 -26.4 -140.9
-112.2 -29.4 -8.2 -4.2 +6.6 -20.3 a) a) -1.0
-126.4 -18.2 -4.7 -52.5 -89.4 -257.0 -65.7
414
-264.6
I;
t
i
i
3. Results
/~
It
o
a) Not calculated.
2 July 1982
CHEMICAL PffYSICS LETTERS
Volume 89, number 5
t
t
i I
Fig. 2. The plots of the total energy (solid line) and the coupling constants J ( H F ) ( - - - ) and J ( 0 F ) ( - . - . - ) versus the f l u o r i n e - o x y g e n distance for 0 = 0 and q~ = 52 ° .
The parameters Cki m ofeq. ( I ) are listed in table 2. The fit was numerically good, with the standard deviation in J(FH) of 3.3 Hz and the standard deviation in J ( F O ) of 3.5 Hz. As has been mentioned in section 2, the Monte Carlo simulation was used for two purposes. First, we calculated the radial distribution functions around the fluoride ion, cf. fig. 3, and found that the number of water molecules in the first hydration sphere of the ion is close to 5. The cluster structure is in good agreement with the experimental findings of Hertz and Rfidle [24]. Second, we calculated the average values of J(FHprox), J(FHdist ) and J ( F O ) (and the squares of the coupling constant) for the nuclei in the first hydration sphere. In this step, the first hydration sphere in each configuration in the MC averaging procedure is defined as extending to the R value of 3.2 A, which roughly correspond to the first minimum in the GFO radial distribution function. The results of the averaging obtained for the different numbers of water molecules included in the simulation are shown in table 3. It can be seen that four water nrolecules do not fill the first coordination sphere and that the differences in the simulations involving 10 and 30 water molecules are insignificant. Also, the effect of decreashag the system temperature to 273 K is minor and any temperature dependence o f the relaxation rates lies almost solely in the dynamic parameters. Similar behav-
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
2 July 1982
Table 2 Least-squares fitted coefficients in the polynomial o f eq. (l) for the fluorine-proton and fluorine-oxygen-17 coupling constants. All distances are in Angstr6m units J(H-F)
J(O-F) Ckl m
k
I
m
Ckl m
k
l
m
16007.7 3191.89 -8860.35 -3694.15 -2544.15 -4359A9 -3041.43 2853.39 -770.330 1742.97 -227.540 -1033.25 -227.997 -7595.07
1 1 2 1 2 0 0 3 0 0 4 0 0 0
1 2 l 0 0 2
1 1 0 2 1 1
-3831.06 2992.44 405.514 2758.64 -1480.66 8945.36 -4903.11 30.3579 -4412.74 -839.044 1105.26 -303.130 -7519.15 523.201
1 1 2 1 2 0 0 3 0 4 6 0 0 0
1 2 1 0 0 2 1 0 3 0 0 4 6 0
1 0 0 2 1 1 2 0 0 0 0 0 0 4
1
2
0 3 0 0 4 0 6
0 0 3 0 0 4 0
iour has been f o u n d for q u a d r u p o l e relaxation o f Li + in w a t e r [25].
GFX! 2O f~ 16
I
12
i
8
l I I
/,I
I
4. Discussion OFH . . . . . .
....... The s p i n - l a t t i c e and s p i n - s p i n relaxation rates for a nucleus I ( 1 / T l l and 1/7'21, respectively) undergoing scalar relaxation o f the first kind due to m o d u l a t i o n o f the indirect nuclear s p i n - s p i n coupling with another nucleus S have been discussed by Abragam [3]. F o r a spin 1, in this case the fluoride ion, coupling to several o t h e r spins in the surrounding water molecules, the t i m e - d e p e n d e n t part o f the h a m i l t o n i a n reads
'
i
\
~\
\1
I~ ,\
i
,
\
1.5
" - / -IJ
/
\
~" ~ p v ~ .. . . . . . . . . . . .
,
2.5
3.5
~.--,'~
/.5
RFX, .~
Fig. 3. Radial distribution functions for the aqueous fluoride ion.
Table 3 Average values of J(FX) and j2 (FX) for the nuclei in the first coordination sphere of the fluoride ion. Simulation performed using 200000 configurations for equil~ration followed by 200000 configurations for analysis Numberofwater molecules in the MC simulation 4 10 30 10
Temperature (K)
300 300 300 273
Number of H 20 molecules within 3.2 A from F4 5.2 5.3 5.3
a) Statistical error estimate +-10%.
Average coupling constant a) (Hz)
Average square of coupling constant b) (Hz 2)
JFH 1
JFH 2
JFO
J~H 1
-39 -27 -30 -33
-6 -3 -3 -5
-20 -21 -21 -19
2.7 1.2 1.2 1.3
x × × x
103 103 103 103
J~H 2
J~o
49 11 11 22
0.48 0.41 0.43 0.30
x x x x
103 103 103 103
b) Statistical error estimate +-20%. 415
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
where ,]IS A a r e the spin-spin coupling constants and the sum is in principle over all nuclei in the system. Following the notation of Abragam [3], the time dependence of the magnetization becomes
2 July 1982
fective in relaxation and that the proximate protons, the oxygens and the distant protons can, on the time scale of the exchange process; be treated as equivalent within each group. The spin-lattice relaxation rate can then be written as
1/rlZ --
J}Sjl(
O, to, s 2 ,
(7)
o~
d(Iz)/dt = -tr{f~
° dr
[/~;(t - r), [/~l*(t),
Iz]](o*(t)-Oo)}
= -tr{/}(a* - o 0 ) ) ,
(3)
where a is the density matrix. Inserting eq. (2) into (3) gives
B= EA EB [i'SA, [i'SB,i~]]IodrJzSA(t--r)JISB(t). (4)
The water dynamics, i.e. the exchange of water molecules in the hydration sphere of F - , modulates J. If this modulation is rapi i and if the spins SA and SB are uncorrelated, then the cross terms in eq. (4) vanish, giving
B= ~[i.SA, [I'SA,Iz]] f~ JISA(t- r)JISA(t) dr .
A
(5)
For ions of higher valency, the exchange is slower and splittings due to the spin-spin coupling can in principle occur. If all nuclei SA have a common rapid exchange rate described by a single exponential, then the relaxation rates become TI~1
=~
81r2 3 l+(wi
re
2
2_A(_A+l)_l~5) 1 , ~ , , ~ ,./2
_~OsA ) r e
T211= ~A 4@2[re + I + (coI_coSA)2r2]SA(SA +1) X J2SA= ~A J~ISAf2(c°'re'SA)'
(6b)
where coI and cosA are the Larmor frequencies and r e the correlation for the exchange process. For the sake of simplicity we assume that only the water molecules in the first hydration sphere are el416
where the summation is over groups with equivalent nuclei and N~ is the number of equivalent nuclei in each group. In order to proceed further, we must specify the motional processes in the F - - w a t e r cluster. This matter has been discussed extensively by Hertz and co-workers [26]. For simplicity we limit ourselves here to two dynamic processes: the exchange of water molecules between the first hydration sphere and the bulk solution (time constant re) and the flipping motion of the water molecules in the first hydration sphere causing the proximate and distant protons to exchange places (time constant ri). The relaxation rates due to 19 F_I H coupling can easily be derived for the following limiting cases. Only expressions for Ti-/1 are given. (i) r e ~ r i, flipping motion such slower than the exchange T 1 / = N(JF2Hprox + J2Hdist)fl (60, re, S H)
~_,NJ2Hproxfl "
(8)
This limiting case corresponds to the short-lived or rigid long-lived hydration sphere of Hertz et al. [26]. The last approximate equality is based on the fact -2 >: ,-;-2- that JFHprox >)>JFHdist" (ii) r i :~ re, flipping motion much faster than exchange. The coupling to the distant and proximate protons is averaged to Jave: 1
2
(9)
This case corresponds to the long-lived non-rigid coordination sphere of Hertz et al. [26]. For the relaxation due to the fluorine-oxygen coupling mechanism (i) may be applied. Hertz and co-workers [26] have found that the hydration sphere is most likely non-rigid and probably short-lived, i.e. a situation intermediate between
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
the two limiting cases above. Furthermore, they report the exchange process to be very rapid (r e 10 -11 s at 25°C). Such small values for re, in combination with our average coupling constants, would result in 19F scalar relaxation rates negligibly small (of the order of 10 - 6 s--1) compared to the dominant dipolar relaxation processes (of the order of Is-l). In a more general perspective, however, we believe that the calculations presented here support the conclusions reached by Holz and co-workers [4], who propose J(CdO) = 248 Hz based on the scalar relaxation rate of 0.244 s- 1 andT e of 3 X 10 - 9 s. If an average FO coupling of the order of 20 Hz arises in a situation where the fluoride ion is coordinated to the water molecules via hydrogen bonds, it is reasonable to expect substantially larger couplings for the cases of metal ions coordinated directly to the oxygen atoms.
References [1] R.E. Wasylishenand M. Barfield, J. Am. Chem. Soc. 97 (1975) 4545. [2] R.K. Harris and B.E. Mann, eds., NMR and the periodic table (Academic Press, New York, 1978). [3] A. Abragam, The principles of nuclear magnetism (Oxford Univ. Press, London, 1961). [4] M. Holz, R.B. Jordan and M.D. Zeidler, J. Magn. Reson. 22 (1976) 47. [5] R.M. Hawk and R.R. Sharp, J. Magn. Reson. 10 (1973) 385. [6] R.M. Hawk and R.R. Sharp, J. Chem. Phys. 60 (1974) 1522.
2 July 1982
[7] M.F. Guest, V.R. Saunders and R.E. Overill, Mol. Phys. 35 (1978) 427. [8] J. Kowalewski, A. Laaksonen, B. Roos and P. Siegbahn, J. Chem. Phys. 71 (1979) 2896. [9] J.S. Muenter and W. Klemperer, J. Chem. Phys. 52 (1970) 6033. [10] A. Laaksonen and J. Kowalewski, J. Am. Chem. Soc. 103 (1981) 5277. [11] A.C. Blizzard and D.P. Santry, J. Chem. Soc. Commun. (1970) 87. [12] A.C. Blizzard and D.P. Santry, J. Chem. Phys. 55 (1971) 950;58 (1973) 4714. [13] J. AhnlOf, USIP Report 74-29 (December 1974). [14] C. Salez and A. Veillard, Theoret. China. Acta 11 (1968) 441. [15] F.B. van Duijneveldt, IBM Technical Report RJ 945 (1971). [16] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087. [17] O. Matsuoka, E. Clementi and M. Yoshimine, J. Chem. Phys. 64 (1976) 1351. [18] J. Fromm, E. Clementi and R.O. Watts, J. Chem. Phys. 62 (1975) 1388. [19] H. Kistenmacher, H. Popkie and E. Clementi, J. Chem. Phys. 59 (1973) 5842. [20] S. Engstr6m and B. Jtinsson, Mol. Phys. 43 (1981) 1235. [21] W.W.Wood, in: Physics of simple liquids, eds. H.W.V. Temperley, J.S. Rowlinson and G.S. Rushbrooke (North-Holland, Amsterdam, 1968). [22] G.H.F. Diercksen and W.P. Kraemer, Chem. Phys. Letters 5 (1970) 570. [23] G.H.F. Diercksen, W.P. Kraemer and B.O. Roos, Theoret. Chim. Acta 36 (1975) 249. [24] H.G. Hertz and C. R~idle, Ber. Bunsenges. Phys~:. Chem. 77 (1973) 521. [25] S. Engstr6m, B. J6nsson and ]3. J~Snsson,to be published. [26] H.G. Hertz, G. Keller and H. Versmold, Bet. Bunsenges. Physik. Chem. 73 (1969) 549.
417
CHEMICAL PHYSICS LETTERS
2 July 1982
INTERMOLECULAR NUCLEAR SPIN-SPIN COUPLING AND SCALAR RELAXATION. A QUANTUM-MECHANICAL AND STATISTICAL-MECHANICAL STUDY
FOR THE AQUEOUS FLUORIDE ION Aatto LAAKSONEN, J ozef KOWALEWSK1 Department of Physical Chemistry, Arrhenius Laboratory, University of Stockholm, S-106 91 Stockholm, Sweden
and Bo JONSSON Division of Physical Chemistry 2, The Lund Institute of Technology, Chemical Center, P.O.B. 740, S-220 07 Lund, Sweden
Received 17 April 1982; in final forln 30 April 1982
Coupling between 19F and magnetic nuclei (a H, 170) in water molecules in the first hydration sphere of F- ate calculated using Monte Carlo simulations. The simulations are based on intelmolecular potentials from the literature and on variations of coupling constants with geometry obtained by coupled Hattree-Fock calculations. Average coupling constants are ~ 20-40 Hz. Intermolecular scalar relaxation in aqueous solutions is discussed.
1. Introduction Knowledge about indirect nuclear spin-spin coupling constants between nuclei residing in different molecular species (intermolecular coupling) is very scarce. To our knowledge, the only paper dealing with this problem theoretically is that of Wasylishen and Barfield [ 1 ], who investigated, by means of finite-perturbation INDO calculations, the coupling between 19F in the HF molecule and proximate protons in water and methane molecules. For certain geometrical configurations, the authors found the couplings to be of sizeable magnitudes. The coupling of this type cannot be observed experimentally as splittings of resonance lines because the lifetimes of particular geometrical arrangements are extremely short in solution. A special case (which however can hardly be denoted as intermolecular) arises in the kinetically stable metal complexes, where a large body of experimental data exists for the coupling constants between the nucleus of the central atom and the nuclei in the ligands [2]. An intermedi412
ate situation is present in labile complexes, where no splittings can be seen but where the coupling interaction, modulated by chemical exchange, may become a relaxation mechanism for the nuclear spins (scalar relaxation of the first kind [3]). Under certain circumstances this relaxation mechanism has been claimed to be efficient. For example, Holz and coworkers [4] reported a study of lt3Cd relaxation in the Cd 2+ ion in D 2170 solutions and found the scalar relaxation due to the coupling between l l 3 c d and 170 to be the dominant relaxation mechanism. They were also able to derive the numerical value of the coupling constant from the spin-lattice relaxation time measurements. A possible minor contribution of this type was also reported for the Pb 2+ ion in an aqueous medium [5,6]. In this paper, we report non-empirical calculations of the intermolecular spin-spin coupling between the 19F nucleus in the fluoride ion and the magnetic nuclei (1H, 170) in the water molecule for a variety of geometric arrangements of the ion around the water molecule. Further, we use the Monte Carlo (MC) sim0 009-2614/82/0000-0000/$ 02.75 © 1982 North-Holland
Volume 89, number 5
CHEMICALPHYSICS LETTERS
ulation technique to calculate the average values of coupling constants between 19F and the proton or 170 in the coordination sphere of the fluoride ion in liquid water. Finally, we propose a few simple motional models, derive the relation between the average coupling constants and the relaxation rates and discuss the possibility that intermolecular scalar relaxation is operative in aqueous ion solutions.
2 July 1982
H/~-J0 Fig. 1. Definition of the geometrical parameters R, 0 and 4~.
2. Methods of calculation
2.1. Calculation o f nuclear spin-spin coupling constants Recent work on hydrogen fluoride and related system [7,8] has been based on the combined use of large gaussian basis set coupled Hartree-Fock (CHF) calculations of the Fermi contact (FC), orbital (OB) and spin-dipolar (SD) contributions and finite perturbation-configuration interaction (FP CI) calculations of the correlation correction to the FC term. For the HF molecule, the results are: J CFC H F = 478 ttz OB SD [7],JcH F = 197 Hz [ 7 ] , J c H F =--10 HZ [7], FC CI = --127 Hz [8], giving a total calculated value JFP of 538 Hz, to be compared with the experimental 530 Hz [91. Considering the size of the F - . H 2 0 system and the fact that the present prqiect required computing more than one hundred coupling constants, we have decided to restrict the calculations to the FC term and to the CHF level of approximation. The results, though probably not sufficiently good for quantitative purposes, should be reliable enough for a qualitative discussion [10]. The calculations have been performed using the self-consistent perturbation method of Blizzard and Santry [ 11,12] and the program system MOLECULE [13]. For fluorine and oxygen, the basis set consisted of 11 s and 7p gaussian-type functions, contracted to 5s and 3p. The exponents and contraction coefficients were taken from Salez and Veillard [14]. For hydrogen atoms, the basis set consisted of 5s and lp contracted to 3s and lp [15]. All calculations were performed with the water molecule constrained to its equilibrium geometry. The position of the fluoride ion with respect to the water molecule may be described with three parameters, R, 0, and 9, defined in fig. 1. The calculations of the coupling con-
stants have been performed for a total of 78 geometries corresponding to selected combinations of the following values of R, 0, and 9: R = 2.0, 2.25, 2.50, 2.75, 3.0, 3.50 au, 0 = 0, 30, 60,120, 150, 180 °, q~= 0, 26, 52, 67, 90 ° . The calculated coupling constants were subsequently fitted to a fourteen-term polynomial in reciprocal distances between F - and the three nuclei in the water molecule: j=~
k m 1 ~1 klm Cklm(rHlrH2R ) '
(1)
where rill and rH2 in eq. (1) refer to the distance between the fluorine nucleus and the proximate and the distant proton, respectively.
2.2. Monte Carlo simulations In order to obtain the intermolecular coupling constant one has to perform some kind of statisticalmechanical averaging. This can be done with either the Monte Carlo simulation technique or via a solution of the equations of motion for the interacting particles ( F - + N H 2 0 ). We have chosen to work with the Monte Carlo (MC) simulation technique of Metropolis et at. [16]. The intermolecular potentials, F - - H 2 0 and H 2 0 - H 2 0 , necessary for the MC simulations were taken from the literature [17-19]. They are both based on accurate quantum-chemical calculations. The F - - H 2 0 pair potential could have been obtained from the present calculations, but since Kistenmacher et al. [19] used a slightly larger basis set, their potential was chosen. The MC simulations were done with the F - . ( H 2 0 ) N cluster in a spherical cavity with F - fixed at its center. The radius of the sphere was 7.5 A in all simulations, which in the largest calculation (N = 30) gives a formal 413
density of ~ 0 . 6 g cm - 3 . In a previous simulation [20] it was shown that the field gradient fluctuation at the Li + nucleus in a water cluster was largely independent of the sphere radius. It seems reasonable to assume that this is true for the spin-spin coupling too, which has a very short-range character. The short-range nature of the coupling also indicated that it may be sufficient with a limited number of water molecules in the cavity to obtain a good statistical average of the coupling constants. The statistical errors were estimated with the method described by Wood [21]. The displacement and rotation parameters were chosen so as to give an acceptance ratio of ~50%.
5 6
ui
a •f ~.. - ~ : ~ - L °
';Lf
The calculated values of the coupling constants for an illustrative sample of geometries are presented in table 1 and fig. 2 together with the corresponding total energies. It is interesting to note that the calculated J(HF) in this study are similar to those reported by Wasylishen and Barfield [1] for the HF.H 2 0 system. Further it can be seen that in the vicinity of the equilibrium configuration (which agrees quite well with earlier theoretical work [ 2 1 - 2 3 ] ) the value of the coupling constant between 19 F and the proximate proton is rather large and negative. For the shorter H - F distance (rFH 1 = 1.043 A) the coupling constant becomes positive, i.e. approaches the value in the HF molecule (where r H F is 0.917 A).
Table 1 Calculated fluorine-proton and fluorine-oxygen-17 coupling constants (Hz) for a selected sample of geometrical configurations of the F-'H20 system R (A)
0
~
JFHprox
JFttdist
JFO
2.0 2.5 3.0 2.5 2.5 2.0 2.5 3.0 2.0
0 0 0 90 180 0 0 0 60
0 0 0 0 0 52 52 52 52
-112.2 -29.4 -8.2 -4.2 +6.6 +40.4 -79.0 -26.4 -140.9
-112.2 -29.4 -8.2 -4.2 +6.6 -20.3 a) a) -1.0
-126.4 -18.2 -4.7 -52.5 -89.4 -257.0 -65.7
414
-264.6
I;
t
i
i
3. Results
/~
It
o
a) Not calculated.
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CHEMICAL PffYSICS LETTERS
Volume 89, number 5
t
t
i I
Fig. 2. The plots of the total energy (solid line) and the coupling constants J ( H F ) ( - - - ) and J ( 0 F ) ( - . - . - ) versus the f l u o r i n e - o x y g e n distance for 0 = 0 and q~ = 52 ° .
The parameters Cki m ofeq. ( I ) are listed in table 2. The fit was numerically good, with the standard deviation in J(FH) of 3.3 Hz and the standard deviation in J ( F O ) of 3.5 Hz. As has been mentioned in section 2, the Monte Carlo simulation was used for two purposes. First, we calculated the radial distribution functions around the fluoride ion, cf. fig. 3, and found that the number of water molecules in the first hydration sphere of the ion is close to 5. The cluster structure is in good agreement with the experimental findings of Hertz and Rfidle [24]. Second, we calculated the average values of J(FHprox), J(FHdist ) and J ( F O ) (and the squares of the coupling constant) for the nuclei in the first hydration sphere. In this step, the first hydration sphere in each configuration in the MC averaging procedure is defined as extending to the R value of 3.2 A, which roughly correspond to the first minimum in the GFO radial distribution function. The results of the averaging obtained for the different numbers of water molecules included in the simulation are shown in table 3. It can be seen that four water nrolecules do not fill the first coordination sphere and that the differences in the simulations involving 10 and 30 water molecules are insignificant. Also, the effect of decreashag the system temperature to 273 K is minor and any temperature dependence o f the relaxation rates lies almost solely in the dynamic parameters. Similar behav-
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2 July 1982
Table 2 Least-squares fitted coefficients in the polynomial o f eq. (l) for the fluorine-proton and fluorine-oxygen-17 coupling constants. All distances are in Angstr6m units J(H-F)
J(O-F) Ckl m
k
I
m
Ckl m
k
l
m
16007.7 3191.89 -8860.35 -3694.15 -2544.15 -4359A9 -3041.43 2853.39 -770.330 1742.97 -227.540 -1033.25 -227.997 -7595.07
1 1 2 1 2 0 0 3 0 0 4 0 0 0
1 2 l 0 0 2
1 1 0 2 1 1
-3831.06 2992.44 405.514 2758.64 -1480.66 8945.36 -4903.11 30.3579 -4412.74 -839.044 1105.26 -303.130 -7519.15 523.201
1 1 2 1 2 0 0 3 0 4 6 0 0 0
1 2 1 0 0 2 1 0 3 0 0 4 6 0
1 0 0 2 1 1 2 0 0 0 0 0 0 4
1
2
0 3 0 0 4 0 6
0 0 3 0 0 4 0
iour has been f o u n d for q u a d r u p o l e relaxation o f Li + in w a t e r [25].
GFX! 2O f~ 16
I
12
i
8
l I I
/,I
I
4. Discussion OFH . . . . . .
....... The s p i n - l a t t i c e and s p i n - s p i n relaxation rates for a nucleus I ( 1 / T l l and 1/7'21, respectively) undergoing scalar relaxation o f the first kind due to m o d u l a t i o n o f the indirect nuclear s p i n - s p i n coupling with another nucleus S have been discussed by Abragam [3]. F o r a spin 1, in this case the fluoride ion, coupling to several o t h e r spins in the surrounding water molecules, the t i m e - d e p e n d e n t part o f the h a m i l t o n i a n reads
'
i
\
~\
\1
I~ ,\
i
,
\
1.5
" - / -IJ
/
\
~" ~ p v ~ .. . . . . . . . . . . .
,
2.5
3.5
~.--,'~
/.5
RFX, .~
Fig. 3. Radial distribution functions for the aqueous fluoride ion.
Table 3 Average values of J(FX) and j2 (FX) for the nuclei in the first coordination sphere of the fluoride ion. Simulation performed using 200000 configurations for equil~ration followed by 200000 configurations for analysis Numberofwater molecules in the MC simulation 4 10 30 10
Temperature (K)
300 300 300 273
Number of H 20 molecules within 3.2 A from F4 5.2 5.3 5.3
a) Statistical error estimate +-10%.
Average coupling constant a) (Hz)
Average square of coupling constant b) (Hz 2)
JFH 1
JFH 2
JFO
J~H 1
-39 -27 -30 -33
-6 -3 -3 -5
-20 -21 -21 -19
2.7 1.2 1.2 1.3
x × × x
103 103 103 103
J~H 2
J~o
49 11 11 22
0.48 0.41 0.43 0.30
x x x x
103 103 103 103
b) Statistical error estimate +-20%. 415
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where ,]IS A a r e the spin-spin coupling constants and the sum is in principle over all nuclei in the system. Following the notation of Abragam [3], the time dependence of the magnetization becomes
2 July 1982
fective in relaxation and that the proximate protons, the oxygens and the distant protons can, on the time scale of the exchange process; be treated as equivalent within each group. The spin-lattice relaxation rate can then be written as
1/rlZ --
J}Sjl(
O, to, s 2 ,
(7)
o~
d(Iz)/dt = -tr{f~
° dr
[/~;(t - r), [/~l*(t),
Iz]](o*(t)-Oo)}
= -tr{/}(a* - o 0 ) ) ,
(3)
where a is the density matrix. Inserting eq. (2) into (3) gives
B= EA EB [i'SA, [i'SB,i~]]IodrJzSA(t--r)JISB(t). (4)
The water dynamics, i.e. the exchange of water molecules in the hydration sphere of F - , modulates J. If this modulation is rapi i and if the spins SA and SB are uncorrelated, then the cross terms in eq. (4) vanish, giving
B= ~[i.SA, [I'SA,Iz]] f~ JISA(t- r)JISA(t) dr .
A
(5)
For ions of higher valency, the exchange is slower and splittings due to the spin-spin coupling can in principle occur. If all nuclei SA have a common rapid exchange rate described by a single exponential, then the relaxation rates become TI~1
=~
81r2 3 l+(wi
re
2
2_A(_A+l)_l~5) 1 , ~ , , ~ ,./2
_~OsA ) r e
T211= ~A 4@2[re + I + (coI_coSA)2r2]SA(SA +1) X J2SA= ~A J~ISAf2(c°'re'SA)'
(6b)
where coI and cosA are the Larmor frequencies and r e the correlation for the exchange process. For the sake of simplicity we assume that only the water molecules in the first hydration sphere are el416
where the summation is over groups with equivalent nuclei and N~ is the number of equivalent nuclei in each group. In order to proceed further, we must specify the motional processes in the F - - w a t e r cluster. This matter has been discussed extensively by Hertz and co-workers [26]. For simplicity we limit ourselves here to two dynamic processes: the exchange of water molecules between the first hydration sphere and the bulk solution (time constant re) and the flipping motion of the water molecules in the first hydration sphere causing the proximate and distant protons to exchange places (time constant ri). The relaxation rates due to 19 F_I H coupling can easily be derived for the following limiting cases. Only expressions for Ti-/1 are given. (i) r e ~ r i, flipping motion such slower than the exchange T 1 / = N(JF2Hprox + J2Hdist)fl (60, re, S H)
~_,NJ2Hproxfl "
(8)
This limiting case corresponds to the short-lived or rigid long-lived hydration sphere of Hertz et al. [26]. The last approximate equality is based on the fact -2 >: ,-;-2- that JFHprox >)>JFHdist" (ii) r i :~ re, flipping motion much faster than exchange. The coupling to the distant and proximate protons is averaged to Jave: 1
2
(9)
This case corresponds to the long-lived non-rigid coordination sphere of Hertz et al. [26]. For the relaxation due to the fluorine-oxygen coupling mechanism (i) may be applied. Hertz and co-workers [26] have found that the hydration sphere is most likely non-rigid and probably short-lived, i.e. a situation intermediate between
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CHEMICAL PHYSICS LETTERS
the two limiting cases above. Furthermore, they report the exchange process to be very rapid (r e 10 -11 s at 25°C). Such small values for re, in combination with our average coupling constants, would result in 19F scalar relaxation rates negligibly small (of the order of 10 - 6 s--1) compared to the dominant dipolar relaxation processes (of the order of Is-l). In a more general perspective, however, we believe that the calculations presented here support the conclusions reached by Holz and co-workers [4], who propose J(CdO) = 248 Hz based on the scalar relaxation rate of 0.244 s- 1 andT e of 3 X 10 - 9 s. If an average FO coupling of the order of 20 Hz arises in a situation where the fluoride ion is coordinated to the water molecules via hydrogen bonds, it is reasonable to expect substantially larger couplings for the cases of metal ions coordinated directly to the oxygen atoms.
References [1] R.E. Wasylishenand M. Barfield, J. Am. Chem. Soc. 97 (1975) 4545. [2] R.K. Harris and B.E. Mann, eds., NMR and the periodic table (Academic Press, New York, 1978). [3] A. Abragam, The principles of nuclear magnetism (Oxford Univ. Press, London, 1961). [4] M. Holz, R.B. Jordan and M.D. Zeidler, J. Magn. Reson. 22 (1976) 47. [5] R.M. Hawk and R.R. Sharp, J. Magn. Reson. 10 (1973) 385. [6] R.M. Hawk and R.R. Sharp, J. Chem. Phys. 60 (1974) 1522.
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[7] M.F. Guest, V.R. Saunders and R.E. Overill, Mol. Phys. 35 (1978) 427. [8] J. Kowalewski, A. Laaksonen, B. Roos and P. Siegbahn, J. Chem. Phys. 71 (1979) 2896. [9] J.S. Muenter and W. Klemperer, J. Chem. Phys. 52 (1970) 6033. [10] A. Laaksonen and J. Kowalewski, J. Am. Chem. Soc. 103 (1981) 5277. [11] A.C. Blizzard and D.P. Santry, J. Chem. Soc. Commun. (1970) 87. [12] A.C. Blizzard and D.P. Santry, J. Chem. Phys. 55 (1971) 950;58 (1973) 4714. [13] J. AhnlOf, USIP Report 74-29 (December 1974). [14] C. Salez and A. Veillard, Theoret. China. Acta 11 (1968) 441. [15] F.B. van Duijneveldt, IBM Technical Report RJ 945 (1971). [16] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087. [17] O. Matsuoka, E. Clementi and M. Yoshimine, J. Chem. Phys. 64 (1976) 1351. [18] J. Fromm, E. Clementi and R.O. Watts, J. Chem. Phys. 62 (1975) 1388. [19] H. Kistenmacher, H. Popkie and E. Clementi, J. Chem. Phys. 59 (1973) 5842. [20] S. Engstr6m and B. Jtinsson, Mol. Phys. 43 (1981) 1235. [21] W.W.Wood, in: Physics of simple liquids, eds. H.W.V. Temperley, J.S. Rowlinson and G.S. Rushbrooke (North-Holland, Amsterdam, 1968). [22] G.H.F. Diercksen and W.P. Kraemer, Chem. Phys. Letters 5 (1970) 570. [23] G.H.F. Diercksen, W.P. Kraemer and B.O. Roos, Theoret. Chim. Acta 36 (1975) 249. [24] H.G. Hertz and C. R~idle, Ber. Bunsenges. Phys~:. Chem. 77 (1973) 521. [25] S. Engstr6m, B. J6nsson and ]3. J~Snsson,to be published. [26] H.G. Hertz, G. Keller and H. Versmold, Bet. Bunsenges. Physik. Chem. 73 (1969) 549.
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