Deuterium Isotope Effects

Z. Phys. Chem. 217 (2003) 1549–1563  by Oldenbourg Wissenschaftsverlag, München

Interpretation of Hydrogen/Deuterium Isotope Effects on NMR Chemical Shifts of [FHF] − Ion Based on Calculations of Nuclear Magnetic Shielding Tensor Surface By Nicolai S. Golubev ∗, Sona M. Melikova, Dimitri N. Shchepkin, Ilja G. Shenderovich, Peter M. Tolstoy, and Gleb S. Denisov Institute of Physics, St. Petersburg State University, 198504 St. Petersburg, Russian Federation

Dedicated to Prof. Dr. Hans-Heinrich Limbach on the occasion of his 60 th birthday (Received April 1, 2003; accepted September 4, 2003)

Hydrogen Bonding / Hydrogen Difluoride / H/D Isotope Effect / NMR / Shielding Surfaces Using ab initio calculations of 1 H, 19 F magnetic shielding tensors of hydrogen difluoride ion as the functions of three coordinates of symmetry, an attempt is made to estimate the contributions of different vibrational isotope effects to H/D NMR isotope shifts referred in the literature. It is shown that the contributions of the amplitudes of proton stretching and bending vibrations dominate whereas the contribution of the totally symmetric vibration can be neglected. Different signs of the H/D isotope effects on hydron and fluorine chemical shifts are caused by very strong angle dependence of the fluorine magnetic shielding. The agreement of the calculated and measured values is nearly quantitative. An unusually strong paramagnetic deshielding of hydrogen for the equilibrium geometry of the [FHF] − ion is noted.

1. Introduction H/D isotope effects on nuclear magnetic shielding are widely used in the studies of strong hydrogen bonding as connected with the important question of the shape of proton potential function in a hydrogen bridge [1–5]. In particular, * Corresponding author. E-mail: [email protected]

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N. S. Golubev et al.

the sign of the primary isotope effect, i.e., the difference between the deuteron and proton NMR chemical shifts, is usually considered as an indication to single- or double-well character of the potential function of longitudinal proton movement [6, 7]. The general theory of NMR isotope effects was given by Jameson and Osten [8] in terms of the averaging of magnetic shielding function over vibrational wavefunctions of a molecule which are different for its isotopomers. Taking into account only one normal vibration, namely, the stretching X–H vibration, is often sufficient to explain qualitatively the NMR isotope effects on hydron and a heavy nucleus, X. Namely, the substitution of the proton for a heavier isotope leads to an effective (vibrationally averaged) X–D bond contraction as compared to X–H and, consequently, some magnetic shielding of both nuclei. However, this treatment is too simple for quantitative evaluations. In practice, the expansion of the nuclear shielding function over the powers of internal coordinates is used [9–11]. For hydrogen bonded complexes the implication of at least a three-atomic model, X–H . . . Y, is required [12, 13]. The most convenient three-atomic complex with a strong hydrogen bond is hydrogen (deuterium) difluoride ion, [FHF] − ([FDF] − ), studied in detail theoretically as well as experimentally. In particular, the two-bond scalar spinspin coupling constant transmitted through the hydrogen bond, 2hJFF , was calculated as a function of vibrational coordinates [14, 15]. The free ion is linear and centrosymmetric (D∞h ) and exhibits the shortest and strongest hydrogen bond (45.8 kcal/mol, [16]). A number of papers are devoted to quantum-mechanical calculations of the three-dimensional potential surface of [FHF] − and the solution of the anharmonic vibrational problem [17–24]. A very strong anharmonic coupling of the three normal modes has been noticed. In particular, this manifests itself in high intensities of combination bands, ν 2 + nν1 and ν3 + nν1 , n = 1 − 4, where ν 1 (Σg ) is the totally symmetric vibration of heavy atoms, ν 2 (Πu ) and ν3 (Σu ) are the bending (doubly degenerate) and longitudinal (antisymmetric) proton vibrations, respectively. In contrast to the case of asymmetric hydrogen-bonded complexes, the anharmonic coupling between two longitudinal modes of [FHF] − is negative, namely, the excitation of the ν 3 vibration leads to some increase in the separation of the heavy atoms, R. Therefore, some contraction of the F–F distance in the ground vibrational state, R0 , of [FDF] − as compared to [FHF] − should be expected due to a lower deuteron zero-point amplitude. This anomalous Ubbelohde effect was found theoretically by Almlöf [25] and experimentally by Kawaguchi and Hirota [26–28]. The high resolution rotation-vibrational spectra of both ions have been measured in the gas discharge which allowed to determine molecular spectroscopic and geometric constants. In particular, the ground state rotational constant, B0 , is equal to 0.334183 cm −1 for [FHF] − and 0.335787 cm −1 for [FDF] − and, therefore, the R0 value in [FDF] − is shorter by 0.0054 Å [26]. Another unusual feature of [FHF] − is an essentially quartic character of the one-dimensional proton po-

Interpretation of Hydrogen/Deuterium Isotope Effects on . . .

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Fig. 1. Schematic representation of different vibrational contributions to NMR isotope effects.

tential function, which results in a positive anharmonic constant of the ν 3 vibration [29]. For the D ∞h structure of the ion, hydrogen bond contraction means, at the same time, the decrease of two F–H mean distances by the value ∆r 0 = 1/2∆R0 . Since proton shielding usually decreases with X–H bond stretching, it would be natural to expect some upfield hydron chemical shift as a result of H/D replacement. The experimental data for solutions of hydrogen difluoride salts in polar solvents [30, 31] exhibit the downfield isotope shift by +0.32 ppm for H,D and the upfield one by −0.37 ppm for 19 F. Qualitatively, the positive sign of the primary NMR isotope effect, δ(D) − δ(H), in single-well, no-barrier hydrogen bonds was explained in Refs. [6, 7] in terms of a decrease in the amplitude of the ν 3 vibration as a result of H/D replacement. The origin of the opposite effect on 19 F is impossible to rationalize when considering only the stretching vibration. The aim of the present work is to analyze the contributions of three normal vibrations to the NMR isotope shifts in [FHF] − with the help of ab initio calculations of 1 H and 19 F nuclear shielding functions. We can limit ourselves to the ground vibrational state because the lowest frequency is as high as 600 cm −1 . In this approach, we neglect the anharmonic coupling of normal modes assuming that its influence on zero point eigenfunctions is small [23]. Indeed, the comparison of the estimated zero point amplitudes with the experimental ones determined by means of neutron diffraction on KHF 2 [32] has confirmed this assumption. According to [8], the isotopic effect on chemical shift can be expanded to powers of three internal coordinates of symmetry: r as = r 1 − r 2 , r s = r 1 + r 2 and ϕ, the HFF

1552 angle (see Fig. 1):


∂δ ∆ r s 0 + δ(D) − δ(H) = ∂r s e 2 1 ∂ 2δ ∂δ + ∆ r + ∆ r as 0 + s 0 2 ∂r s2 e ∂r as e 2 2 ∂ δ 1 ∂ 2δ ∆ r as 0 + ∆ ϕ2 0 + . . . . + 2 2 2 ∂r as e ∂ϕ e

(1)

Here ϕ2 = ϕx2 + ϕ2y is the proton deviation from the F . . . F axis, ∆r i 0 and ∆r i2 0 are the isotopic effects on the internal coordinates and their squares averaged in the ground vibrational state. The linear term with respect to ∆r as is equal to zero for the symmetric configuration, D ∞h , but becomes important for the disturbed ion deprived of the inversion center. The quadratic term with respect to ∆r s is non-zero, but the reduced mass of the totally symmetric vibration is not changed as a result of H/D replacement and, in considering isotope effects, we can neglect it. The linear term with respect to the angle is zero for every linear molecule. Now the problem is reduced to the calculation of the expectations of different powers of the coordinates for the ground states of [FHF] − and [FDF] − , and the estimation of the chemical shift function derivatives with respect to the same coordinates. On calculating the isotropic magnetic shielding functions, we found that proton and fluorine chemical shifts depend but very weakly on the fully symmetric coordinate, r s , and, consequently, the observed NMR isotope effects cannot be determined by the Ubbelohde effect. The interpretation of this phenomenon becomes clear when considering different components of the shielding tensor. The r s -dependencies of the longitudinal (purely diamagnetic) and transverse (essentially paramagnetic) proton shieldings nearly compensate each other. In this paper, an attempt is made to explain qualitatively such behaviour by introducing the paramagnetic contribution of low-lying excited electronic state arising from the excitation of an electron from a non-binding atomic orbital (lone pair) to the anti-binding molecular orbital of the hydrogen bond. However, the applicability of the obtained regularities to formally symmetric complexes other than [FHF] − is under question because the paramagnetic deshielding must be individual for every molecule.

2. Calculations and discussion 2.1 Computation method The optimized geometric parameters of the [FHF] − ion and potential energy as functions of r s , r as and ϕ were calculated at the MP2/6-311++G** level using the Gaussian 98 package [33]. The theoretical dependencies of the 1 H and 19 F


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Fig. 2. The calculated dependence of the potential energy (MP2/6-311++G**) of [FHF]− on two longitudinal coordinates of symmetry, U = f(∆r s , r as ), for the linear configuration, ϕ = 0.

chemical shifts on the coordinates of symmetry were calculated at the same level using the GIAO procedure included in the Gaussian 98 package.

2.2 Potential surface The fully optimized (MP2/6-311++G**) [FHF] − ion is linear and symmetric with the F–H and F–F distances equal to r opt = 1.1375 Å and Ropt = 2.2750 Å, correspondingly. The latter distance is in a good agreement with the experimental value, Re = 2.2777 Å [26, 27]. Fig. 2 depicts the two-dimensional potential surface of the linear [FHF] − ion in two longitudinal coordinates of symmetry, ∆r s = r s − Re and r as . It is seen that the inversion center in the equilibrium plane (r as = 0) is unstable with respect to hydrogen bond stretch which leads to the rise of a barrier along the r s coordinate. The potential energy surface (PES) can be described by combination of Morse potentials in the r 1 and r 2 internal coordinates: 2 2 U = D 1 1 − e−br1 + D 2 1 − e−br2 + D 12 1 − e−br1 1 − e−br2 .

(2)

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The Morse potential parameters D 1 = D 2 =12 719 cm −1 , D 12 =16 355 cm −1 and b = 2.361(2) Å −1 were obtained by fitting the PES in the ranges ∆r s = −0.28 ÷ 0.72 Å and |r as | = 0 ÷ 1.63 Å. One can obtain the reaction coordinate of the proton motion from the condition of minimum of potential (2): ∂U = 2D 1 1 − e−b∆rs /2 e−bras /2 + ebras /2 + D 12 = 0 . ∂r as

(3)

So, if the proton motion is fast enough to follow the fluorine nuclei (Born– Oppenheimer approximation), the relation between the coordinates will be: D 12 −br as /2 br as /2 eb∆rs /2 . e +e = 1+ (4) 2D 1 Near the equilibrium point (r as , ∆r s ≈ 0): 2D 1 br as2 2 2D 12 + . 1− ∆r s = b 2D 1 + D 12 2D 1 + D 12

(5)

The Morse model predicts a quadratic rise of the hydrogen bond length with its asymmetrisation as measured by r as . By fitting the PES at the r as = 0 cross section we obtain: 2 (6) U(∆r s )ras =0 = (2D 1 + D 12 ) 1 − e−b∆rs /2 . Note that, however, the Morse fit of the potential surface, made in the vicinity of the equilibrium point, cannot describe correctly its asymptotical behaviour. In Fig. 3 the low-energy parts (in the vicinity of the potential minimum) of the surface cross-sections by the (∆r s = 0, ϕ = 0) and (∆r s = 0, r as = 0) planes are depicted together with calculated zero point energy levels. The polynomial fit of the curves shows that the account of anharmonicity is especially important for the ν 3 mode even for low vibrations. Thus, the energy expansion to the normal dimensionless coordinate, q 3 , is VH = 617q 32 + 351q 34 + 10.3q 36 for [FHF] − and VD = 447q 32 + 179q 34 + 3.8q 36 for [FDF] − , where the first coefficient has the meaning of one half harmonic frequency. The lowamplitude bending vibration is much more harmonic: VH = 658q 22 − 18.4q 24 ; VD = 472q 22 − 7.3q 24 .

2.3 Zero point state Let us calculate the coordinate expectations in expression (1). In general, one must solve the three-dimensional Schroedinger equation in order to obtain r s , r s2 , r as , r as2 and ϕ2 . In Born–Oppenheimer approximation by using the reaction coordinate of Eq. (4) one can reduce the three-dimensional problem to a one-dimensional one.


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Fig. 3. The dependencies of the potential energy on two vibrational proton coordinates, V = f(r as ), ∆r s = 0, ϕ = 0 (a) and V = f(ϕ), ∆r s = 0, r as = 0 (b).

Since the reduced mass of the fully symmetric vibration is insensitive to H/D isotopic replacement, in the one-dimensional approximation the mean value, ∆r s 0 = r s − Re 0 , turns out to be mass-independent. To estimate the zero point isotope effect on r s , the calculated dependencies of the optimized r s value on r as and ϕ were used. On expanding r s to hydron coordinates, we obtain: ∂∆r s R D 0 − R H 0 = ∆ r as 0 + ∂r as e 2 2 1 ∂ 2 ∆r s ∂ ∆r s + ∆ r as 0 + ∆ ϕ2 0 . (7) 2 2 ∂r as2 ∂ϕ e e In such a treatment, the Ubbelohde effect for the D ∞h geometry (deuterium bond contraction) can be explained by a decrease in zero point hydron vibration amplitude on deuteration. Note that, for asymmetric equilibrium geometry, C∞v , the linear term, describing the anharmonic deuteron displacement from the center, differs from zero, which can result in the conversion of the Ubbelohde effect to the normal one (deuterium bond stretching, typical for asymmetric H-bonds). Figure 4 shows the optimized r s value as the function of the anti-symmetric and doubly degenerate proton coordinates and illustrates the method of the account of the Ubbelohde effect, RD 0 − RH 0 , using Eq. (7). The dependence r s = f(ϕ) contains a flat area in the vicinity of the equilibrium point. This means that the anharmonic coupling with the ν 2 mode is substantial only for

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Fig. 4. The calculated mutual dependencies (MP2/6-311++G**) of the optimized geometric parameters of [FHF]− r s = f(r as ), ϕ = 0 (a) and r s = f(ϕ), r as = 0 (b).

highly excited states whereas, in the ground state with a low proton amplitude, it can be neglected. To calculate the ground state expectations of vibrational coordinates, r as , r as2 and ϕ2 , the one-dimensional anharmonic problems in symmetric potentials have been solved using the approach described in Ref. [34, 35]. In accordance with this procedure, the proton and deuteron potentials in dimensionless normal coordinates, qi =

√

γi si ,

γi = 4π 2 ch −1 µ i ωi ,

r as si = √ 2

(8)

where s i is an internal coordinate of symmetry, µ i is the reduced mass and ω i is the harmonic frequency, were calculated and fitted with even polynomials. (Mind that, in this representation, proton and deuteron potential functions are different). The Schroedinger equation was solved numerically in its dimensionless form [35], 1 Ψ (q) + (E − V )ψ(q) = 0 , 2 ω0 2 V(q) = q + β q 4 + . . . 2

(9)

by finding eigenfunctions as linear superpositions of a finite number of harmonic ones, N−1 cin ψn (q), N = 10 ÷ 20 . (10) Φi = n=0


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Table 1. The calculated and experimental parameters of the zero point state of [FHF]− and [FDF]− . E 0 (r s ) (cm−1 )

E 0 (r as ) (cm−1 )

R0 − Re (Å)

ϕ2 (rad2 )

r as2 (Å 2 )

FDF

294b

460a

482a

0.0205b

0.0262a 0.023c

0.0183a 0.019c

FHF

292b

651a

690a

0.0259b

0.0320a 0.029c

0.0261a 0.026c

2b

−191a

−208a

−0.0058a −0.0054b

−0.0058a −0.006c

−0.0078a −0.007c

∆ a

E 0 (ϕ) (cm−1 )

calc., this work; b exp., Refs. [26–28]; c exp., [32].

The problem is thus reduced to the diagonalization of a finite matrix with nonvanishing matrix elements being simple algebraic functions of indexes. The obtained data on zero point energies and coordinate expectations are collected in Table 1 and compared with the available experimental data. The experimental mean value RD e and RH e refers to the free ion in the gas phase [26–28], and the mean square amplitudes are derived from neutron diffraction experiments on single crystal KHF 2 [32]. The good coincidence of the measured and calculated values confirms the reliability of our potential surface.

2.4 Nuclear magnetic shieldings Fig. 5 depicts two-dimensional surfaces, δ(1 H) = f(r as , r s ) and δ(19 F) = f(r as , r s ) for the linear F a –H . . . F −b configuration. The conversion from the calculated shielding constants to chemical shifts with respect to conventional standards, TMS and CFCl3 , was made using the shielding constants of the standards calculated at the same level of theory, σ H (TMS) = 32.5 ppm, σ F (CFCl3 ) = 425.7 ppm. It is to be noted that, on the NMR time scale, the vibrational motion is always fast and, for comparison with experiment, the calculated chemical shifts of the two fluorine nuclei should be averaged, δ(19 F) = (δ(19 F a ) + δ(19 F b ))/2. The calculated chemical shifts of the equilibrium configuration, δ(1 H) = 17.4 and δ(19 F) = −154 ppm proved to be rather close to the experimental values [31], δ(1 H) = 16.6, δ(19 F) = −154.96 ppm. It is seen that, at a given r s , proton movement towards the nearest heavy atom along the anti-symmetric coordinate, r as , leads to rather strong proton and fluorine shielding, with a chemical shift maximum obtained at the central proton position. However, when moving along the r s coordinate, the proton shielding decreases only very slowly. This behaviour is unexpected, because the increase in the R separation with keeping the central proton position leads

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Fig. 5. The calculated (GIAO, MP2/6-311++G**) dependencies of 1 H (a) and chemical shifts of [FHF]− on two longitudinal coordinates, r as and r s at ϕ = 0.

19

F (b)

Fig. 6. The calculated dependencies of the effective charge on the hydrogen atom on two longitudinal coordinates, q M = f(r as ), ∆r s = 0, ϕ = 0 (a) and q M = f(∆r s ),r as = 0, ϕ = 0 (b).

to a strong decrease of the negative charge of the hydrogen atom which must result in its diamagnetic deshielding. In Fig. 6 the Mulliken charge located on hydrogen, q H , is plotted against r as and ∆r s on the same scale. It is seen that the charge dependence on ∆r s is much stronger. To understand the origin of the weak chemical shift sensitivity to this charge, let us consider the longitudinal and transverse components of the proton shielding tensor separately. The longitudinal (σ zz ) component of the tensor, purely diamagnetic for linear molecules [36], depends on vibrational coordinates in accordance with the electron density change, namely, the shielding falls with hydrogen moving away from heavy atoms along ∆r s , Fig. 7b. Proton deviation from the axis with increasing the angle leads also to diamagnetic deshielding, Fig. 7a. The


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Fig. 7. The longitudinal (σ zz ) and transverse (σ xx , σ yy ) components of proton shielding tensor as the function of ϕ at ∆r s = 0, r as = 0 (a) and as the function of ∆r s at r as = 0, ϕ = 0 (b).

transverse components, σ xx = σ yy , which can be essentially paramagnetic, exhibit strong shielding as hydrogen moves away from both fluorine atoms with increasing the hydrogen bonds length or the angle. The R dependencies of the longitudinal and transverse components compensate one another resulting in practically constant isotropic shielding. The angle dependence of the transverse components is stronger than that of the longitudinal one which results in increasing isotropic shielding with proton deviation from the axis. The diamagnetic and paramagnetic contributions to the r as -dependence of proton shielding, Fig. 8a, look similar, and the resulting isotropic shielding increases strongly with the rise of asymmetry along r as . Fig. 8b depicts the dependence of σ zz and σ xx , σ yy components of the nonaverage shielding tensor for the F a fluorine atom in the F a –H . . . F −b ion on the antisymmetric coordinate, r as , describing the transformation to F −a . . . H–F b . The diamagnetic shielding, σ zz , is practically constant. The σ xx = f(r as ) curve is non-monotonous and passes through a minimum near the center (but not exactly in the center). In Fig. 9 the average isotropic shielding of two fluorine nuclei is plotted against the angle coordinate. In the vicinity of the potential minimum, the proton deviation from the axis leads to fluorine deshielding and, in accordance with this curve, the decrease of the bending vibration amplitude as a result of H/D replacement must give rise to a negative isotope effect on the average fluorine chemical shift. The above complicated behaviour of the shielding functions can be understood qualitatively when assuming that there exists a strong source of paramagnetic deshielding in the structure [F . . . H . . . F] − , which is weak for F–H and absent for F − due to its spherical symmetry. We suppose that its source is

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Fig. 8. The longitudinal (σ zz ) and transverse (σ xx , σ yy ) components of proton shielding tensor as the function of ϕ at ∆r s = 0, r as = 0 (a) and fluorine shielding tensor as the function of r as at ∆r s = 0, ϕ = 0 (b).

Fig. 9. The isotropic fluorine shielding as the function of the angle ϕ at ∆r s = 0, r as = 0.

a low-lying excited electronic state, Π g , arising from the excitation of a nonbinding electron, p x,y (F), to the anti-binding molecular orbital of the hydrogen ∗ bond, z . In the first order of perturbation theory, the mixing of this electronic state with the ground state by the transverse component of the external magnetic field is determined elements of orbital angular mo by the

∗ matrix ∗ mentum p x |L y | z and p y |L x | z , and the energy difference between the


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Table 2. The contributions of different vibrations to the isotope effects on NMR chemical shifts of FHF − at the ground state. rs

ϕ2

r as2

∂ n δH /∂qin

2.5 ppm/Å

−0.2 ppm/Rad2

−76.5 ppm/Å 2

∂ n δF /∂qin

4.1 ppm/Å

144 ppm/Rad2

−98.2 ppm/Å 2

−0.052 −0.064 0.012 3.773 4.608 −0.835

−0.700 −0.998 0.298 −0.898 −1.281 0.383

∆δD , ppm ∆δH , ppm ∆δD − ∆δH , ppm ∆δF(D) , ppm ∆δF(H) , ppm ∆δF(D) − ∆δF(H) , ppm

0.051 0.065 −0.014 0.084 0.106 −0.022

exp

−0.701 −0.997 0.296 2.959 3.433 −0.474

0.32 −0.37

ground and excited states [36]. Representing the anti-binding molecular orbital,

∗

∗ , as a linear superposition of atomic functions = c p (F 1 z a ) + c 2 p z (F b ) + z z c3 s(H) + . . . , taking that p x (F a )|L y | p z (F a ) = i and neglecting the terms of the type p x (F a )|L y | p z (F b ) and p x (F a )|L y |s(H), for the paramagnetic fluorine and proton shieldings we obtain: 2 1 c1 eh σ xx (F a ) ≈ − , 2mc r F3 ∆E 2 1 c2 eh , σ xx (F b ) ≈ − 2mc r F3 ∆E 2 1 (c1 + c2 ) eh , σ xx (H) ≈ − 2mc r H3 ∆E

(11)

where r F and r H are the distances from F and H nuclei to a lone pair and ∆E is the energy difference between the ground ( 1 Σg ) and excited (Π g ) electronic states. When moving along the r as coordinate (from F–H to F − , ∆r s = 0), the c1 coefficient falls to zero, c2 rises and the energy splitting passes through a minimum at the central proton position. This explains qualitatively the nonmonotonous and asymmetric character of the σ(F a ) = f(r as ) curve, Fig. 8b, which becomes symmetric after averaging the F a and F b shieldings, and the deep minimum in the σ(H) = f(r as ) plot (Fig. 8a). The way along rising R leads to the (F − + H+ + F− ) state and results in increasing the average r H distance, as well as in decreasing both coefficients, c1 and c2 . Consequently, the paramagnetic proton deshielding falls rapidly with rising the separation of two fluorine atoms.

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The above consideration is based on the fact that for the free F − ion the paramagnetic deshielding is forbidden by symmetry. For other systems, e.g., the complexes of the CN − ion, the whole pattern must be different. In Table 2 the final data on the chemical shift derivatives and the resulting contributions of different vibrational effects are collected. The primary isotope

effect, = ∆δ D − ∆δ H , is determined mainly by the quadratic term, namely, the decrease in the anti-symmetric proton vibration amplitude as a result of H/D replacement. The amplitude of the bending vibration provides but a small contribution of the same sign. The linear (with respect to R) term describing hydrogen bond shortening on H/D replacement gives rise to a small effect of the opposite sign. It should be emphasized that the main geometric isotope effect, namely, the Ubbelohde one, is opposite to the one of the primary NMR isotope effect and diminishes its absolute

value. The total one-bond secondary effect, = ∆δ F(D) − ∆δ F(H) , is an algebraic sum of several terms of the same order of magnitude. The sign of the effect is determined by the competition of two quadratic terms, namely, the bending and stretching hydron vibrational amplitudes, with the first effect to predominate. This can explain the mentioned above discrepancy between the signs of the primary and secondary isotope effects for [FHF] − . Although the coincidence of the calculated and measured values is surprisingly good, the account of anharmonic coupling of two proton vibrations can, in principle, change the result. Therefore, the solution of the multidimensional vibrational problem is required.

Acknowledgement This work was financially supported by Russian Foundation of Basic Research, grant 02-03-32668. The authors are very grateful to Prof. Dr. H.-H. Limbach who has initiated the experimental study of F − (HF) n clusters [31, 37] and motivated us to perform these calculations.

References 1. P. E. Hansen, Prog. Nucl. Magn. Reson. Spectrosc. 20 (1988) 207. 2. N. N. Shapetko, Yu. S. Bogachev, I. Y. Radushnova, and D. N. Shigorin, Dokl. Akad. Nauk SSSR 231 (1976) 409. 3. M. D. Fenn and E. Spinner, J. Phys. Chem. 88 (1984) 3993. 4. C. L. Perrin and J. B. Nielson, J. Am. Chem. Soc. 119 (1997) 12734. 5. C. L. Perrin, J. B. Nielson, and Y. J. Kim, Ber. Bunsenges. Phys. Chem. 102 (1998) 403. 6. L. J. Altman, D. Laungani, G. Gunnarsson, H. Wennerström, and S. Forsen, J. Am. Chem. Soc. 100 (1978) 8264. 7. G. Gunnarsson, H. Wennerström, W. Egan, and S. Forsen, Chem. Phys. Lett. 38 (1976) 96. 8. C. J. Jameson and H.-L. Osten, Annu. Rep. NMR Spectrosc. 17 (1986) 1. 9. A. D. Buckingham and R. M. Olegario, Mol. Phys. 92 (1997) 773.


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10. R. D. Wigglesworth, W. Raynes, S. Kirpekar, J. Oddershede, and S. Sauer, J. Chem. Phys. 112 (2000) 736. 11. R. D. Wigglesworth, W. Raynes, S. Kirpekar, J. Oddershede, and S. Sauer, J. Chem. Phys. 112 (2000) 3735. (Erratum: J. Chem. Phys. 114 (2001) 9192). 12. M. V. Vener, Chem. Phys. 166 (1992) 311. 13. N. D. Sokolov, M. V. Vener, and V. A. Savel’ev, J. Mol. Struct. 222 (1990) 365. 14. J. E. Del Bene, M. J. T. Jordan, S. A. Perera, and R. J. Bartlett, J. Phys. Chem. A 105 (2001) 8399. 15. M. Pecul, J. Sadlej, and J. Leszczinski, J. Chem. Phys. 115 (2001) 5498. 16. P. G. Wenthold and R. R. Squires, J. Phys. Chem. 99 (1995) 2002. 17. V. C. Epa and W. R. Thorson, J. Chem. Phys. 92 (1990) 466. 18. V. C. Epa and W. R. Thorson, J. Chem. Phys. 92 (1990) 473. 19. V. C. Epa and W. R. Thorson, J. Chem. Phys. 93 (1990) 3773. 20. V. Spirko, A. Cejchan, and G. H. F. Diercksen, Chem. Phys. 151 (1991) 45. 21. K. Yamashita, K. Morokuma, and C. Leforestier, J. Chem. Phys. 99 (1993) 8848. 22. M. J. Bramley, G. C. Corey, and R. Hamilton, J. Chem. Phys. 103 (1995) 9705. 23. A. K. Rappe and E. R. Bernstein, J. Phys. Chem. A 104 (2000) 6117. 24. M. A. McAllister, J. Mol. Str. (Theochem.) 428 (1998) 39. 25. J. Almlöf, Chem. Phys. Lett. 17 (1972) 49. 26. K. Kawaguchi and E. Hirota, J. Chem. Phys. 84 (1986) 2953. 27. K. Kawaguchi and E. Hirota, J. Chem. Phys. 87 (1987) 6838. 28. K. Kawaguchi and E. Hirota, J. Mol. Struct. 352/353 (1995) 389. 29. G. L. Cote and H. W. Thomson, Proc. Roy. Soc. A 210 (1951) 206. 30. F. Y. Fujiwara and J. S. Martin, J. Am. Chem. Soc. 96 (1974) 7625. 31. P. Schah-Mohammedi, I. G. Shenderovich, C. Detering, H.-H. Limbach, P. M. Tolstoy, S. N. Smirnov, G. S. Denisov, and N. S. Golubev, J. Am. Chem. Soc. 122 (2000) 12878. 32. I. A. Ibers, J. Chem. Phys. 40 (1964) 402. 33. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, J. V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. G. Ochterski, G. A. Petersson, P. Y. Ayala, G. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, L. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P. M. W. Gill, B. G. Johnson, W. Chen, M. W. Wong, J. L. Andres, M. Head-Gordon, E. S. Replogle, and J. A. Pople, Gaussian 98 (Revision A.7), Gaussian, Inc., Pittsburgh, PA (1988). 34. R. L. Somorjai and D. F. Hornig, J. Chem. Phys. 36 (1961) 1980. 35. E. Heilbronner, H. H. Guenthard, and R. Gerdil, Helv. Chim. Acta 39 (1956) 1171. 36. J. D. Memory, Quantum Theory of Magnetic Resonance Parameters, McGraw-Hill, New York (1968). 37. I. G. Shenderovich, S. N. Smirnov, G. S. Denisov, V. A. Gindin, N. S. Golubev, A. Dunger, R. Reibke, S. Kirpekar, O. L. Malkina, and H.-H. Limbach, Ber. Bunsenges. Phys. Chem. 102 (1998) 422.