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Proton dynamics in the hydrogen bond. The inelastic neutron scattering spectrum of potassium hydrogen carbonate at 5 K
Chemical Physics 124 (1988) North-Holland, Amsterdam
425-437
PROTON DYNAMICS iN THE HYDROGEN BOND. THE INELASTIC NEUTRON SCATTERING SPECTRUM OF POTASSIUM HYDROGEN CARBONATE AT 5 K F. FILLAUX Laboratoire de Spectrochimie Infrarouge et Raman, Centre National de la Recherche Scientifique, 2 rue Henri-Dunant. 94320 Thiais, France
J. TOMKINSON
and J. PENFOLD
Rutherford Appleton Laboratory, Chilton. OX10 9BH, UK Received
29 January
1988
We report the inelastic neutron scattering (INS) spectrum from KHC03 powder (5 K). The spectrum has been analysed by the frequency and intensity normal coordinate refinement program CLIMAX. The final force field indicates that the proton bending modes in the OH...0 bond are independent of other deformation modes, but that the stretching modes interact strongly. This is interpreted in terms of a significant ionic character for the hydrogen bond. Based upon simple calculations for its position and intensity an INS mode at z 220 cm- I is tentatively assigned to a tunneling transition for the proton.
1. Introduction
bonded compounds and has not yet shown its full potential. Nowadays, new INS spectrometers have a large frequency range (o-4000 cm- ’ ) with good resolution and signal-to-noise ratios (under favorable conditions INS can provide spectra of a quality equal to that obtained from infrared and Raman experiments) . In this paper, the INS spectrum of polycrystalline potassium hydrogen carbonate ( KHC03, 5 K) is analysed. This compound was selected for several reasons. The crystalline structure [ 4- 10 1, infrared and Raman spectra [ 1 l-l 81 have all been thoroughly studied. The scattering length for all the atoms except the proton is negligible. Furthermore, in the crystalline state, this compound forms centrosymmetric dimers; and it is believed that the proton transfer along the hydrogen bond occurs within a double well potential [ 18 1.
Infrared and Raman spectroscopy have been used extensively to investigate hydrogen bonded systems [ l-31 and a large amount of data on different types of hydrogen bonds has been accumulated. It is widely accepted that the physical origin of the observed phenomena (frequency shifts; intensities, bandshapes, etc. ) is due to the dynamics of the proton. These are usually described in terms of anharmonicity and strong non-linear coupling of the vibrational modes. Moreover, beside the purely mechanical effects, electrical anharmonicity and electrical coupling contribute to the infrared and Raman bandshapes, and intensities. The electrical factors may change from one system to an other and it is not straightforward to analyse a given spectrum in terms of its real proton dynamics. However, the inelastic neutron scattering (INS) intensities of hydrogenous compounds are totally determined by the mean-square displacements of the proton. INS spectroscopy is therefore much more specific to proton dynamics than either infrared or Raman. INS spectroscopy has been used on a few hydrogen 0301-0104/88/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
2. Experimental A sample of polycrystalline KHC03 (obtained by recrystallization from a slightly warmed concentrated solution) of about 5 cm by 2 cm by 0.2 cm was B.V.
F. Fillaux et al. /Proton dynamics in the hydrogen bond
426
contained in an aluminium can at 5 f 0.2 K. The INS spectrum was obtained on the TFXA spectrometer, ISIS at Rutherford Appleton laboratory, Chilton UK. TFXA is an inverted geometry time-of-flight spectrometer. The spectrum is collected in reflection (19,= 135’ ) and the final neutron energy is fixed (E+ 4 meV). The spectrometer has excellent energy resolution (AEJE, x 2%) and traces a line in ( Q, o) space such that Q*(A-*) xE(meV)/2. More details of the instrument have been published elsewhere [ 191. The data were actually collected over a period of several days in 1986, equivalent to present day capabilities of 1 day (or 2000 uA h). The data were transformed into an energy transfer scale, with the intensities proportional to S( Q, 0). This spectrum was corrected for background scattering by subtracting the spectrum of the empty aluminium can in the cryostat.
3. Crystal structure and symmetry The KI-IC03 crystal structure has been determined by X-ray [ 4-8 ] and neutron diffraction [ 9, lo] (fig. 1). The crystal is monoclinic, space group P2, /a (C$, ), with four KI-IC03 entities per unit cell. The centrosymmetric dimers (HCOS ) :- , occupying Ci sites, have relatively strong O-H...0 hydrogen bonds
K HCO,
rg.
1. Crystalline
structure
of the KHC03
crystal after ref.
[ 91.
characterized by an O...O distance of 2.6 1 A. According to Lucazeau and Novak [ 171, the intramolecular forces within HCO, are much stronger than intermolecular forces and the internal and external vibrations are well separated. The hydrogen bond interaction is much stronger than the other intermolecular forces in the crystal. Therefore the hydrogen bond vibrations, which are also external modes, can be distinguished from other lattice modes. Assuming a quasi C,, symmetry for the dimers, internal vibrations for the four KHC03 entities in the unit cell are represented by 9A,+9B, (Raman active) and 9A, + 9B, (infrared active) symmetry species. Under the Ci symmetry, these vibrations should be grouped into A, and A, species [ 16 1. The 12 hydrogen bond vibrations give 3A,+ 3B,+ 3A,+3B,. Finally, the 2 1 lattice vibrations are represented approximately by 15 translational (T’ ) (3A,+3B,+ 5A, + 4B,) and 6 rotational (R’ ) vibrations (3A,+ 3B,). Comparison of infrared, Raman and INS spectra is complicated by the differing wavelength in the optical experiments and the importance of momentum transfer in INS. The representations of vibrations in terms of the molecular symmetry species do not apply to INS spectra,.which reflect the density of vibrational states. Fortunately internal modes are not usually strongly dispersed, and can be considered as independent of the momentum transfer. INS frequencies should then be similar to those observed in infrared and Raman. The INS spectrum is also likely to be less sensitive to interdimer coupling than infrared and Raman. Therefore the single centrosymmetric dimer approximation symmetry C2,, will be used. The 18 internal vibrations are then represented by 7A, + 7B, species for in-plane modes, 2B, + 2A, for out-of-plane modes. The 6 hydrogen bond vibrations give 2A,+ 1B, for in-plane and 1B, + 2A, for out-ofplane motions. They can be approximately described as symmetric and antisymmetric antitranslations (T,, T,, T,) or antirotations (R,, R,, R,) of the two CO:- ions (fig. 2). Qualitatively, taking into account the mass of the carbonate ion (M=60 amu) and its moments of inertia (I, = 39.1 amu A*, I,= 40.8 arnuA2,ZW=79.9amuA2),theT, (A,,v(O...O))and R, (B,, v(O...O) ) modes should be at similar frequencies. This is because they both involve large O...O internuclear distance variation and similar masses or
F. Fillaux et al. /Proton dynamics in the hydrogen bond
0
w
T;
Tu
BuLM
‘(OF
Ag
vo.. .o
Ag
80.-o
Bu L.M
TV
Rw
Bu vo ..o
R’w
0 Ag
L.M
Au L.M
Tw
q.j-2
Ru
4000
Bg yo. ..o
‘-(.p Au yo...o
Au ro
o
100
200 Energy
300 400 transfer (me’./)
500
Fig. 3. Inelastic neutron scattering spectrum of the KHCOx powder crystal at 5 K.
out -of-plane
Rv
Energy tronsfer(cm-‘1 2000 3000
1000
‘(Oh
T;
T’w
427
R’v
Bg L M
R’u
Bg I_ M
Fig. 2. Schematic representation of the low-frequency hydrogen bond and lattice modes ofthe ( HCO,):- centrosymmetric planar dimer. LM: lattice mode.
moments of inertia. The R,(A,, ~(0...0)) and R, (A,,, y(O...O)) are expected at higher frequencies than the T, (B,, y(O...O)) and R, (A,, 6(0...0)) modes. Finally, besides the K+ translational motions, the lattice modes correspond to translation ( 1A,,T:, , + 2B,, T:, TV ) and rotation ( 1A,,RW, + 2B,, KU, R:. ) of the ( HC03):dimers.
4. INS spectrum and band assignment The INS spectrum of KHC03 at 5 K is presented in fig. 3 and the proposed assignments (table 1) are consistent with the infrared, Raman [ 15-l 71 and previous INS data [ 201. The INS spectrum is dominated by the intense y(OH) mode (933,983 cm-‘). Also present are the weaker 6 (OH ) ( x 1400 cm- ’ ), v(O...O) (219, 225 cm-‘) and 6(C03) (621, 639 cm-’ ) bands. All these vibrations involve large displacements of the proton. Above 1600 cm- ’ the intensity is due mainly to overtones and combinations.
The v(OH) bands, in the 1800-3500 cm-’ region, are hidden or embedded in this “background” and they cannot be identified unambigously. The rather small full width at half-height (fwhh) of the components for the v(O...O) (16 cm-‘), 6(C03) (20 cm - ’ ) and y ( OH ) ( 25 cm- ’ ) confirm the negligible frequency dispersion of these modes. Band splitting is ascribed to intradimer coupling, except for the 62 1 and 639 cm-’ features which are due to the &(CO,) and 6,( C03) modes, respectively (see below). The hydrogen bond vibrations located at 2 19 and 255 cm-’ (v(O...O)), 196 cm-’ (~(0...0)), 170 cm-’ (y(O...O)), 153 cm-’ (6(0...0)), and finally 145 cm-’ (y(O...O)), show quite different INS intensities. The bending modes are much weaker than the v(O...O). This is especially true for the z(O...O) (196cm-‘) andy(O...O) (170cm-‘) whicharevery weak in our spectrum, whereas their Raman counterparts are quite strong [ 171. Taken together with the large intensity of the y( OH) mode, this suggests that the out-of-plane vibrations of the proton and of the hydrogen bond are more or less independent. Below 120 cm-’ where most of the lattice modes are expected, the bands are broad and the spectrum is typical of the density of states for the compound. Strong INS intensity is anticipated for the translational and rotational vibrations of the dimers whereas translational motions of K+ should be weak, as far as these motions can be separated. Therefore, the INS bands at 114, 108, 90, 74 cm-’ which have strong Raman counterparts at 116, 98 and 87 cm-’ [ 171, should correspond mainly to rotational vibrations of the dimers.
428,
F. Fillaux et al. /Proton dynamics in the hydrogen bond
Table I Infrared, Raman and INS band frequencies (in cm-’ Raman data are reported from refs. [ 15,171 Infrared 90 K
unit)
and assignments
Raman 30 K
for the KHC03
crystal at low temperature.
Infrared
and
INS SK
42
A,
87
B,
98 110 116 120 138 142
A, A, B, A8 A, B,
147 163
A, B,
196 222
A, As
250 639
B, A,+B,
6,(COx) y(CO3)
74 90
136
108 114 A.
y(O...O)
152 170
Ag B,
&(O...O) y(O...O)
219
Au AE
T(O...O) v(O...O)
255 627 662
B, A,+B. A,+B.
v(O...O) 6,(CO3) &(CO,)
940 990
B, A,
Y(OH) y(OH)
145 6(0...0)
170
y(O...O)
v(O...O) 255 661 831
835
A,+B,
937
A,+B,
1036
A,+B,
v,(CO3)
1291
A,
v,(CO,)
987 1010
1375 1405
v,(CO,)
1625 1650 1695
v:(COx) 2y(C03)
1450 1688 1712
A, A,
vb(CO3)
2y(OH)
1050
v,(CO,)
1385 1415 1440
v,(CO,)+8(OH)
1620 1680
G(CO3) S(OH)+S(O...O)
1830
2y(OH)
2040
VOH)+y(OH)
1880 1940 2080
5. Calculation of the INS spectrum
The observed INS spectrum is reported here in terms of the scattering law (S( Q, w) ). The calculation of S( Q, o) for KHC03 on TFXA was performed using the program CLIMAX. This program has been described in detail elsewhere [ 2 11. It will not be described further here except to underline that it produces S( Q, o) intensities taking full account of the
Debye-Waller factors for fundamentals, overtones and combinations. This version of the program is slightly limited by two features; firstly its dependance upon harmonic frequencies and its non treatment of phonon wings. Anharmonicity results in harmonic overtones being misplaced (this is not a problem for the present analysis). Whilst the effect of phonon wings have been minimised by working at 5 K.
F. Fillaux et al. /Proton dynamics in the hydrogen bond
429
(namelyv(OH),&(OH),y(OH))andthehydrogen bond coordinates. However it was not possible to adjust the calculation to fit the observed intensities. This result shows that the picture of protons moving in a local field dominated by covalent bonds to CO!entities is incorrect. The proton motion is best regarded as decoupled from the hydrogen bond vibrations.
6. Force field calculation 6.1. The isolated dimer model
Most of the modes involving K+ motions are at very low frequency and poorly understood. Such that previous force field calculations [ 16 ] were carried out on an isolated planar centrosymmetric dimer of symmetry CZh. The structure obtained by projection of the atomic coordinates in the crystal [ 9 ] on the mid plane of the dimer is presented in fig. 4. The symmetry coordinates are shown in table 2. There are four redundant coordinates: two 6(COX), one 6(0...0) (B,) and one t( O...O) ( Bg). The valence force field (table 3, I) was refined on the basis of infrared, Raman and INS frequencies. Whereas the frequencies are well represented, the calculated relative INS intensities (table 4, I) are in total disagreement with the experimental data. Most of the calculated intensity arises from the hydrogen bond vibrations, whereas the y (OH ) and 6 (OH) are very weak (fig. 5, I). The discrepancy is unacceptably large. Many attempts were therefore made to introduce different coupling constants between the internal coordinates
6.2. The crystal-field model One possible way to account for the decoupling of the proton motion is to consider that the hydrogen atom is moving in a field due mainly to its crystalline environment. Owing to the long-range character of atom-atom and electrostatic interactions, it would not be possible to select any dominant contribution of a particular atom, or set of atoms. Indeed a summation over many unit cells in the crystal would be necessary to calculate the true crystal field experienced by the proton. Such a calculation is beyond the scope of the present paper and a simple phenomenological model will be used. The crystal field is supposed to average over all the atomic vibrations and
A
V
“M4 I .I I .
M2
----o _+
_---
_+--
I
-4~ 01
,--+r.
I
---
f-4009
C
G
--
0,
c7
9
IiS---I
I’ I
I ‘. I lk 1, M3
Fig. 4. Projection of the atomic coordinates [ 91 on the mid plane of the ( HC03)$- dimer and atom numbering for molecular coordinate definition. The heavy masses M,, Mz (stretching), M,, M, (bending) represent the fixed crystal field.
430
F. Fillaux et al. /Proton dynamics in the hydrogen bond
Table 2 Coordinate definitions torsion; 0: out-of-plane fig. 2 Internal No. I
2 3 4 5 6 7 7’ 8 8’ 9 10 11 12 13 14 15 16 17 17’ 18 18’ 19 20 21 21’ 22 22’ 23 24 25 26 21 28
for the (HCO,):bending. Primed
dimer. The atom numbering numbers describe the proton
Symmetry
coordinates definition c2 C7 c2 C7 c2 c7 04 Ml 09 M2 04 09 04 09 03 08 01 06 H5 H5 HI0 HlO 04 09 H5 H5 HlO HlO 03 08 01 09 03 08
correspond coordinates
01 06 03 08 04 09 H5 H5 HlO HlO 06 01 c2 c7 c2 C7 c2 C7 04 M3 09 M4 06 01 04 M3 09 M4 c2 C7 C2 c7 01 06
No.
s s
03 08 01 06 04 09 C2 C7 C7 c2 C2 Ml C7 M2 04 09 C7 c2 04 09
01 M2 06 Ml 01 06 06 04 06 01
S S S S S S S S S S B B B B B B B S B S B B T T T T 0 0 T T T T
to be fixed with respect to the crystal axes. For the purpose of calculation, it can be represented by heavy masses to which the protons are linked through forces governing the stretching and bending vibrations. These masses (M,, M2, M3, M4 in fig. 4), arbitrarily set to 1000 amu, were located on the inertial axes of the dimer, symmetrically with respect to the center of mass, and either along, or perpendicular, to the OH bond directions. To avoid unreasonably large force constants, due to the very long H5...M1 and HI0...M2
to that in fig. 2. S: stretching; B: in-plane bending; T: with respect to the heavy masses M,, Mz, MS, M, in
Group coordinates
coordinates Definition 1
2 3 4 5 6 7 7 8 8 9 10 11 12 13 14 15 16 17 17 18 18 19 20 21 21 22 22 23 24 25 26 27 28
2 A,
I -2B.
v(C-0)
3 3
4 A, -4B,
v(C=O)
5 5
6 A, -6B,
v(C-OH)
7 8 A, 7’ 8’A, 7 -8B, 7’ -8’B, 9 lOA, 9 -lOB, 11 l2A, 11 -12B, 13 14A, 13 -14B. 15 16A, 15 -16B, 17 18A, 17’ 18’A, 17 -18B. 17’-18’B. 19 20A, 19 -20 B, 21 22B, 21’ 22’B, 21 -22 A, 21’-22’A, 23 24 B, 23 -24 A, 25 26 B, 25 -26 A, 27 28B, 27 -28 A.
v(O-H)
v(O...O) 8(O=C-OH) S(O=C-0) 8(0-C-O)
8(0-H)
6(0-O)
do-HI
so3 1 r(O...O) y(O...O)
distances, the OH in-plane bending is now represented by the HS...M3 and HI0...M4 stretching coordinates, instead of the H5MIM2 and H,,M,M, bending coordinates. A new valence force field was refined assuming negligible force constants linking the proton to the CO:- ions (table 3, II). Calculated INS intensities now involve only the y ( OH ) ,6 (OH ) and v( OH) modes (table 4, II). Comparison of calculated and experimental spectra (fig. 5, II) shows that while the y( OH) and 6( OH) intensities are reason-
431
F. Fillaux et al. /Proton dynamics in the hydrogen bond Table 3 Refined force constants in mdyne/A for the KI-ICO, dimer. masses (M) representing the fixed crystal field
No.
6 7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
I
dimer.
II, III and IV: proton
oscillating
with respect to heavy
Force constants I
1
I: isolated
v(O-H) v(OH...M) v(C=O) v(C-0) v(C-OH) 6(C-0-H) &(OH...M) 6( HO-C-O) 6( o-C=O) 8(0-C-O) v(O...O) 6(0...0)
v(OH)/v(OH) v(CO)/v(CO) v(C-O)/v(C-0) v(c=o)/v(c=o) v(C-OH)/v(C-OH) s(C-0-H)/s(C-O-H) 6(OH...M)/&(OH...M) G(HO-C=O)/S(HO-GO) s(o-c=o)/6(0-c=O) s(HO-C-O)/G(HO-C-O) v(0...0)/v(0...0) v(OH)/v(O...O) 8(OH)/s(O...O) G(OH...M)/v(CO)
II 3.8 1700 6.37155 6.57870 5.90691 0.84691 2.62049 1.37956 2.27012 0.5 1893 0.37535 0.14682 2.03063 2.0035 1 -0.61209 -0.14708 0.04467 - 0.85656 0.33046 0.31442 0.05783 0.0 0.0
y(O-H) Y(0Hv.M)
0.24652
Y(C0) T(O...O) Y(O...O) y(OH...M)/Y(OH...M)
1.12962 0.19747 0.27274
Y(co)lY(co) Y(oH)lY(CO)
y(OH)lv(O...O) y(OH)/~(0...0)
IV
III
0.22805 0.0 0.0
ably represented, the v(O...O) bands do not appear and the height of the v (OH ) band is obviously overestimated. The real force field is thus likely to lie between these two simple models. 6.3. Thefinal model the observed INS intensities are well represented assuming that the v (OH ) dynamics is dominated by the O-H force constant while the bending modes are
4.07043 10.14405 5.17230 5.12270 1.17412 2.21393 1.61871 1.59373 0.51200 0.35580 0.15471 1.84746 0.01167 0.32383 0.10820
3.81700 9.88602 5.61895 5.34867 1.18000 2.04487 1.46726 1.65352 0.52988 0.36080 0.14780 1.70298 0.10862 0.322 13 0.01425
3.83890 9.91723 5.46318 5.37951 1.17713 2.16647 1.51702 1.6367 1 0.51588 0.38774 0.14624 1.73128 0.32509 0.32134 -0.33317
0.04570 0.28690 0.2749 1 -0.12270 0.05357 0.0 0.0
0.04635 0.06849 0.27240 -0.06650 0.06727 0.0 0.0 0.0
0.04487 - 0.00044 0.41059 -0.17081 0.06576 -0.10000 0.0 0.05000
5.46900 0.67953 0.22295 0.21301 0.28407 -0.06446 0.0 0.0 0.0
5.46924 0.68111 0.21684 0.07890 0.28407 -0.06349 0.0 0.0 0.0
5.46924 0.68111 0.21684 0.07890 0.28407 -0.06349 0.0 0.0 0.0
essentially governed by the crystal field (fig. 5, III, tables 3 and 4, III). Besides fundamental transitions below 1500 cm-’ the rest of the spectrum is dominated by overtones and combination bands. It is remarkable that the intensities of the two spikes at 630 and 640 cm-’ due to the CO:- bending modes, are well represented without introducing any additional coupling constant. This arises from the fact that the bending motions of the carbonate ion are mixed with the hydrogen bond vibrations. This result is very
F. Fillaux et al. /Proton dynamics in the hydrogen bond
432
Table 4 Experimental and calculated frequencies (cm-‘) and INS relative intensities with respect to heavy masses (M) representing the fixed crystal field Observed frequency
Assignment
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 4 5 6 7
v(OH)
B”
v(OH)
A, A,
v(C=O) v(C=O)
6COI-I) s(OH) v(C-0) v(C-0) v(C-OH) v(C-OH) 6(C=O) 6(C=O) S(OC0) S(OC0) v(O...O) v(O...O) 6(0...0) Y(OH) Y(OH) Y(C0) Y(COz) t(o...o) y(O...O) y(O...O)
B”
A, B” B”
A, A, B” J%
A, B”
4 4 BU B” A”
B, B, AU AU A”
5
2680 2580 1688 1638 1440 1385 1415 1291 1036 1010 665 660 630 625 244 219 153 987 937 835 831 196 170 145
Calculated
frequency
for the KHCOz dimer.
II, III and IV: proton
oscillating
(int.)
I
II
III
IV
2680.1(0.8533) 2580.1(1.033) 1693.9(1.775) 1638.8(2.150) 1441.5( 1.450) 1385.0(0.8441) 1408.5(2.814) 1290.1(4.058) 1036.5(7.410) 1010.1(7.548) 661.8(14.53) 660.0( 13.80) 627.7( 15.12) 624.9( 15.89) 244.7(40.82) 219.0(67.99) 147.9( 100.4) 988.7( 1.920) 935.5(4.443) 836.0( 11.64) 830.0( 11.83) 202.1(76.13) 171.0(100.4) 138.9(75.73)
2680.0( 90.48) 2580.0(95.40) 1688.1(0.0000) 1638.0(0.0000) 1440.0( 145.1) 1385.0( 150.7) 1415.6(0.0000) 1292.5(0.0000) 1036.9(0.0000) 1009.1(0.0000) 664.7(0.0000) 660.2 (0.0000) 625.4(0.0000) 624.5 (0.0000) 244.1(0.0000) 221.3(0.0000) 146.8 (0.0000) 987.0( 175.3) 937.0( 182.6) 835.7(0.0000) 831.0(0.0000) 210.3(0.0000) 156.4(0.0000) 139.0(0.0000)
2681.3(6.488) 2579.5(7.419) 1687.9(0.8031) 1638.0(0.9267) 1439.3(90.32) 1383.8(95.30) 1410.1(0.0153) 1328.5(0.0236) 1037.2(7.738) 1009.0(6.276) 664.8( 16.37) 660.3(23.48) 631.2(23.63) 623.5(25.11) 248.0(54.57) 220.9(51.58) 147.0(49.20) 987.0( 123.9) 937.0( 131.3) 835.0(0.0000) 831.0(0.0000) 196.0(0.0000) 120.6(0.0000) 95.2(0.0000)
2683.9(4.964) 2583.9 (5.766) 1687.9(0.5176) 1638.0(0.8507) 1440.3(64.09) 1384.9(69.21) 1415.0(45.54) 1336.9(26.47) 1035.9(6.133) 1010.0(7.230) 665.0( 15.77) 659.8(27.14) 630.3( 12.98) 624.9(23.59) 245.3(53.31) 220.3(47.87) 148.7(48.58) 987.0( 116.4) 937.0( 123.9) 835.0(0.0000) 831.0(0.0000) 196.0(0.0000) 120.6(0.0000) 95.2(0.0000)
helpful in assigning the observed vibrations. It is possible to adjust more accurately the calculated intensities with additional coupling constants in the force field (tables 3 and 4, IV, and fig. 5, IV). The 6( OH) region is slightly improved by a weak coupling of the v(C0) and 6(OH) coordinates; the v(O...O) intensities are quite sensitive to the v(OH), v(O...O) interaction; and the 6(C03) to the6(C03), 6(0...0) coupling. In principle, owing to the sensitivity of the fit to these coupling constants, it would be possible to obtain very accurate values from the spectrum. Unfortunately our data are too limited, both by noise and background uncertainties, to pursue these fine details. Therefore we suggest that force field III (table 3 ) can be considered as sufficient. 7. Discussion The absence of detectable INS intensities for the y(C03) and r(O...O) modes near 830 and 196 cm-’
respectively, and the very weak intensity in the y(O...O) region around 150 cm-’ show that the y(OH) mode is practically independent of the carbonate and hydrogen bond out-of-plane motions. This result holds only for a full separation of in-plane and out-of-plane vibrations, which may not be totally justified. (The true dimer structure in the crystal [ 9,10 ] deviates slightly from planarity.) Indeed, a weak interaction of the y(OH) at 987 cm-’ and v,(CO,) at 1050 cm-’ could account for the INS intensity of the latter which was not satisfactorily represented within the planar approximation. However, the influence of this coupling on the description of the y( OH) mode is negligible and the isolated character of this mode must be considered as one of the most firmly based conclusions of this work. The 6(OH) mode also appears quite isolated, although weak coupling with the CO3 stretching and 6(0...0) modes are observed. However, the v(OH) and v(O...O) modes are totally mixed.
433
F. Fillaux et al. /Proton dynamics in the hydrogen bond Energy lop0
Irans;;feom-‘)
3op0
i/ 0
100
200
Energy fransfer
300
(meV)
400
Fig. 5. Comparison of calculated (-•-) and experimental (-) INS spectra of KHCOX at 4.2 K. I, II, III, IV correspond to the force fields in table 3 with the same notations. Overtones and combination bands were calculated only in case IV.
sities of INS bands, which depend upon the eigenvectars, introduce new constraints upon models. Dynamical models which produce very poor fits to INS data can thus be quickly eliminated as being irrelevant. In this sense then, the force field proposed here is the most realistic presently available for KHC03. However, this force field is not unique and another choice of molecular coordinates might well give as good an agreement with the experimental data. For example, the masses Ml, Mz, M3, M4 could have been chosen differently. That said, however, after investigating several different alternatives, we believe that any analysis consistent with the INS intensities should lead to the same conclusion. Namely that the isolated dimer model proposed by Nakamoto and coworkers [ 161 is useful only for the description of the v (OH ) mode. The out-of-plane y ( OH ) mode is best regarded as independent of the internal and external vibrations of the carbonate dimer. Similarly for the 6 (OH) mode which is, however, slightly mixed with the CO3 stretching coordinates. These conclusions, based on the INS intensities, are largely independent of the choice of the vibrational coordinates. Within the valence bond approach [ 22-241, the real hydrogen bond is represented by a mixing of the three resonance forms: w, =OD-H...O, I,+ =O,
It is impossible to compare in detail the present valence force field with the Urey-Bradley force field determined by Nakamoto and co-workers [ 161. The two methods are much too different and their description of the normal modes is not sufficient to evaluate the corresponding INS intensities. However, the rather large force constants they have introduced for the hydrogen bond modes suggests that their model should lead to a poor match with the INS intensities. A more general point about force field calculations concerns the underdetermined nature of the mathematical problem. The number of force constants (N(N+ 1) /2 for N degrees of freedom) greatly exceeds the number (N) of observable frequencies. Therefore, no unique force field can be extracted from frequency data alone. Any of the force fields of table 3 would calculate acceptable eigenvalues. The inten-
,
v2=O~...H+...0A,
. ..H-O.+ ,
where 0, and OA represent the proton donor and acceptor entities. In the case of KHC03, the isolated dimer model of Nakamoto and co-workers [ 161 corresponds to a proton covalently linked to the oxygen. The w, resonance form, represented by the O-H force constants, is dominant, whereas the contribution of w2, represented by the H...O force constants, is much weaker in this model. Because of the directionality of the force constant associated with a covalent description of the hydrogen bond the proton will “ride” with the oxygen displacements. These riding modes have calculated INS intensities which are inconsistent with observations. Further, OH deformations should modify the CO:- geometry through electron orbital interactions. When this occurs it is usually signaled by significant vibrational coupling between the corresponding modes. The INS spectrum does not support this case.
434
F. Fillaux et al. /Proton dynamics in the hydrogen bond
Rather INS intensities support an alternative view that the hydrogen bond is best thought of in terms of the ionic form y2. Of course this can be seen from the intensity of the v (OH ) infrared band [ 15,17 1. Then owing to the more isotropic nature of the electrostatic field the proton-oxygen interactions depend mainly upon the Ob...H distance, and very much less upon angular changes. The v(OH) and v(O...O) modes are still allowed to interact as strongly as in the covalent form w,, but the O...O bending motions become dynamically independent of the proton motions (i.e. the “riding” is now absent). We have shown above that purely covalent force fields for the proton are inappropriate. However it would be misleading to overstress the utility of an ionic (crystal) force field. We must not forget that y(OH) is, to first order, well described by O...O forces (this can be seen in the excellent correlation between y (OH ) frequencies and O...O separations [ 3 ] ) . Beyond what has already been shown our methods are not capable of resolving the two contributing fields which control1 the dynamics of the proton. Indeed it is doubtful that complete analysis in terms of normal coordinates could ever be achieved. This we believe stems from the very anharmonic character of the hydrogen bond itself. The results of our INS work can be compared with previous elastic scattering results. The dynamical model allows us to calculate the total mean-square displacement of the protons, along the Cartesian axes. The total proton displacement approximately parallel to the O...O bond Vi (fig. 4) is Ui co.0378 A2. This can be compared with other estimates; (i) UT =0.042 A* [25]; and (ii) Vi ~0.029 A’ [lo], both obtained at room temperature. Our calculation has no contribution from external modes, it therefore represents a minimum acceptable value of Vi.This throws some doubt on the above value of Vi (ii) from ref. [ 10 1. This is perhaps not surprising since the derived value for the OH bond length is shorter than expected [26]. Increasing this bond length should result in different thermal parameters for the proton. If they became consistent with our results then, we expect Uf cz0.04 A*. (We shall avoid discussing UT values because we have adopted a planar molecular geometry. )
8. The tunnel effect and proton transfer in KHC03 dimers Carboxylic acid dimers have attracted much attention recently. Studies involving detailed analysis of NMR and NQR data [ 27,281, and calculations [ 291, have been interpreted in terms of thermally activated proton transfer. One of us (FF) has recently produced a double well potential for proton dynamics in KHC03 [ 181 (seetig. 6). Thepotential V(r) (cm-‘) is given by: V(r)=368r+438910r6 +5516 exp( -30.7r2)
,
(1)
with r as the off-center proton position (A). The wells are asymetric about the center, with an enthalpy difference of ~2.7 kJ mol-’ (213 cm-‘). This compares well with an estimated value from NQR of 3.4 kJ mol- [28]. It is appropriate to compare our INS observations with a prediction that can be made from eq. ( 1). On fig. 6 it is seen that there are four energy levels. Thus, although the protons are highly localised (and indeed in the dynamical analysis above we have assumed complete localisation) there is a small tunneling delocalisation. We should therefore expect to observe INS active transitions. The INS intensity of the O+ 1 transition is, to a first approximation, related to the overlap of the initial and final wavefunctions. (This is presented in the appendix.) It can be appreciated that because the wells are deep this over-
E Cd
6000 t
a=3 a=2
r + i
0 --+-
00
0.3
H
Fig. 6. Potential function for proton transfer along the hydrogen bond in the (HCO,)$dimer after ref. [ 181. (Y is the quantum number associated with the v(OH) mode.
F. Fillaux et al. /Proton dynamics in the hydrogen bond
lap is small and the corresponding intensity is low. We therefore predict a band of weak to moderate intensity at z 213 cm-‘. The bands observed in this region are shown in fig. 7. They are both associated with the O...O stretching modes. However the lowfrequency band at = 2 19 cm- ’ is highly asymmetric and disproportionately strong. All our dynamical analyses suggest that Z219/Z245x 1.O + 0.05, regardless of any reasonable coupling constants. This ratio can be obtained if we remove a sharp component from the low-frequency side of the band. This produces three bands; the two O...O stretching modes, now at 224 and 245 cm-‘, and the sharp component at 216 cm-‘. The relative intensity of this sharp band (fwhha 10 cm-‘) is Z2,6/Z224xO.5. We consider this to be of moderate intensity, in the context of KHC03, and tentatively assign the sharp band at 2 16 cm- ’ to a tunneling transition as discussed above. (The excellent frequency correspondence: 2 13 cm- ’ (predicted) and 2 16 cm-’ (observed), we regard as fortuitous.) As shown in the appendix, a proton situated at f 0.2 A from the O...O bond center is consistent with the observed intensities. This calculation shows that the tunneling frequency can be observed even for purely localised wavefunctions in the lower states. However, the present approach deserves several reserves, because of the many simplifications in the model, and in the band decomposition. ContriEnergy transfer
(cm-’
200
)
435
butions to the INS intensity arising from the possible mixing of the two wavefunctions (A. 1) and (A.2) have been neglected. Therefore, the difference with the value derived from the diffraction data ( z + 0.3 A) [9,10] may be not significant. The presence of tunneling at low temperature must influence the interpretation of the NMR and NQR data, which is presently treated in purely classical terms [ 27,29 1.
9. Conclusion The INS spectrum of the KHCO, powder crystal at 5 K shows strong features ascribed to the y(OH), g(OH), v(O...O) and 6(C03) vibrations, in good agreement with infrared and Raman data. Except for the latter, these bands are split into two components due to intradimer coupling. A force field description of the (HCO,)$cyclic dimer consistent with INS intensities indicates unusual proton dynamics which could not have been predicted from infrared and Ramn spectra alone. The bending modes of the proton are independent of both hydrogen bond and CO:- vibrations, but the stretching modes interact strongly. This suggests a pronounced ionic character for the hydrogen bond. Simple calculations, based upon wavefunction overlap and a double well potential, predict a moderately intense band at ~2220 cm-‘. This has been tentatively assigned.
320 J
Acknowledgement We should like to thank Dr. C. Carlile for help with the neutron experiment. Also, we thank the SERC (UK) for access to the ISIS neutron facilities and the CEA (France) for financial support. We should also like to thank Dr. G.J. Kearley (ILL) for supplying the CLIMAX program.
Appendix
Energy tronsfer(meV)
Fig. 7. The low-frequency spectrum of KHC03 decomposition in the O...O stretching region.
40 at 4.2 K and band
In the asymetric double well potential governing the proton transfer (fig. 6 and eq. ( 1) ), the E, -E. splitting is due to the energy difference of the two wells. The wavefunctions Q0 and 0, are almost entirely lo-
436
F. Fillaux et al. /Proton dynamics in the hydrogen bond
calised [30]; they mately, by harmonic
can be represented, approxifunctions centered at k ro: exp[ -OSa8(r-ro)*]
,
(A.1)
CD,(r)= (a~/~)‘/~ exp[ -OSa~(r+r,)‘]
,
(A.2)
@o(r)=(a;/x)“2
or:
1 Q: Q: '+2a&,,,o Z 2h.0 [---
>I .
with ai = 2nmH cv,/zi
(A.3)
where mH ( 1 amu) is the reduced mass for the proton motion, v. ( x 2000 cm-’ ) is the v( OH) frequency in the well, c is the velocity of light, and A is Plan&s constant over 27r. the INS intensity for the O+ 1 (tunneling) transition at v, is: StQt,
v,)a
xS( v+E,
j Q,(r) -E,)
exp(iQ,r)%(r) )
dr2 (A.4)
or:
v,)aexp[-2a~(r~+Q:/4a~)l xS(v+Eo-E,). (A.5)
&(Q,
The tunneling intensity may be compared with that of an isotropic harmonic oscillator at the same frequency, say v(O...O):
SO...O(Q,, ~,)a (Q~/~&...o) exp(-Q~/2&.o) XG(VS&-E,), (A.6) with a6...o =2xmo...ocv,lfi
,
(A-7)
where mo...o (30 amu) is the reduced mass for the antitranslation of the two carbonate entities. According to the band decomposition (fig. 7) the relative intensity for the tunneling and v(O...O) modes at 2 16 and 245 cm-’ is: I 216 -=
Z245
S(Qt, vt)xo 5 so...o(Qt, vo) . .
(A-8)
Then, eqs. (A.6 ) and (A. 8 ) give: Z
216 -z21 245
2af
mO...O
Q, mH (A.91
In the present experiment r. z k 0.205 A
(A. 10)
Q: x 2~2: and then: (A.
11)
References [ 1 ] P. Schuster, G. Zundel and L. Sandorfy, eds., The hydrogen bond. Recent developments in theory and experiments, Vols. 1-3 (North-Holland, Amsterdam, 1976). [2] H. Ratajczak and W.J. Orville-Thomas, eds., Molecular interactions, Vol. 1 (Wiley, New York, 1980). [ 3 ] A. Novak, Struct. Bonding 18 ( 1974) 177. [4] P. Herpin, Compt. Rend. Acad. Sci. 234 (1952) 2205. [ 51I. Nitta, Y. Tommiie and C. Hoe Koo, Acta Cryst. 5 ( 1952) 292. [6 ] I. Nitta, Y. Tommiie and C. Hoe Koo, Acta Cryst. 7 ( 1954) 140. [ 71 P. Herpin and P. Meriel, J. Phys. 25 ( 1964) 484. [8] B. Pedersen, Acta Cryst. B 24 ( 1964) 478. [9] J.O. Thomas, R. Tel&en and I. Olovsson, Acta Cryst. B 30 (1974) 1155. [ lo] J.O. Thomas, R. Tellgren and I. Olovsson, Acta Cryst. B 30 ( 1974) 2540. [ 111 L. Couture-Mathieu, J. Phys. Radium 11 ( 1950) 544. [ 121 L. Couture-Mathieu, J. Phys. Radium 15 (1954) 531. [ 131 P. Tarte, Hydrogen bonding (Pergamon, Oxford, 1959) p. 115. [ 141 Y.I. Ryskin, Opt. Spectry. 12 (1962) 287. [ 151 A. Novak, P. Saumagne and L.D.C. Bok, J. Chim. Phys. (1963) 1385. [ 161 K. Nakamato, Y.A. Sarma and K. Ogoshi, J. Chem. Phys. 43 (1965) 1177. [ 171 G. Lucazeau and A. Novak, J. Raman Spectry. 1 ( 1973) 573. [ 181 F. Fillaux, Chem. Phys. 74 (1983) 405. [ 191 J. Penfold and J. Tomkinson, internal report RAL-86-019, Chilton, UK. [ 201 K.P. Brierley, J. Howard, C.J. Ludmann, K. Robson, T.C. Waddington and J. Tomkinson, Chem. Phys. Letters 59 (1978) 467. [21] G.J. Kearley, J. Chem. Sot. Faraday Trans. II 82 (1986) 41; G.J. Kearley and J. Tomkinson, Inst. Phys. Conf. Ser. 81. Neutron scattering data analysis 1986, Ed. M.W. Johnson (Institute of Physics, Bristol, 1986) p. 169. [ 22 ] L. Pauling, in: The nature of the chemical bond (Cornell Univ. Press, Ithaca, 1960).
F. Fillaux et al. /Proton dynamics in the hydrogen bond [ 23 ] C.A. Coulson and U. Danielsson, Ark. Fys. 8 ( 1954) 239.
[ 241 A. Warshel and R.M. Weiss, J. Am. Chem. Sot. 102 ( 1980) 6218. [25] M. Arsic-Eskirja, KFA Jiilich internal report Jul-872-FF (1972). [26] M. Ichikawa, Acta Cryst. B 34 (1978) 2074. [ 27 ] T. Agaki, F. Imashita, T. Tesao, N. Hirota and S. Hayashi, Chem. Phys. Letters 139 (1987) 331.
437
[28] A. Cough, M.M.I. Haq and J.A.S. Smith, Chem. Phys. Letters 117 (1985) 389. [29] R. Meyer and R.R. Ernst, J. Chem. Phys. 86 (1987) 784. [30] R.L. Somorjai and D.F. Homig, J. Chem. Phys. 36 ( 1962) 1980.
425-437
PROTON DYNAMICS iN THE HYDROGEN BOND. THE INELASTIC NEUTRON SCATTERING SPECTRUM OF POTASSIUM HYDROGEN CARBONATE AT 5 K F. FILLAUX Laboratoire de Spectrochimie Infrarouge et Raman, Centre National de la Recherche Scientifique, 2 rue Henri-Dunant. 94320 Thiais, France
J. TOMKINSON
and J. PENFOLD
Rutherford Appleton Laboratory, Chilton. OX10 9BH, UK Received
29 January
1988
We report the inelastic neutron scattering (INS) spectrum from KHC03 powder (5 K). The spectrum has been analysed by the frequency and intensity normal coordinate refinement program CLIMAX. The final force field indicates that the proton bending modes in the OH...0 bond are independent of other deformation modes, but that the stretching modes interact strongly. This is interpreted in terms of a significant ionic character for the hydrogen bond. Based upon simple calculations for its position and intensity an INS mode at z 220 cm- I is tentatively assigned to a tunneling transition for the proton.
1. Introduction
bonded compounds and has not yet shown its full potential. Nowadays, new INS spectrometers have a large frequency range (o-4000 cm- ’ ) with good resolution and signal-to-noise ratios (under favorable conditions INS can provide spectra of a quality equal to that obtained from infrared and Raman experiments) . In this paper, the INS spectrum of polycrystalline potassium hydrogen carbonate ( KHC03, 5 K) is analysed. This compound was selected for several reasons. The crystalline structure [ 4- 10 1, infrared and Raman spectra [ 1 l-l 81 have all been thoroughly studied. The scattering length for all the atoms except the proton is negligible. Furthermore, in the crystalline state, this compound forms centrosymmetric dimers; and it is believed that the proton transfer along the hydrogen bond occurs within a double well potential [ 18 1.
Infrared and Raman spectroscopy have been used extensively to investigate hydrogen bonded systems [ l-31 and a large amount of data on different types of hydrogen bonds has been accumulated. It is widely accepted that the physical origin of the observed phenomena (frequency shifts; intensities, bandshapes, etc. ) is due to the dynamics of the proton. These are usually described in terms of anharmonicity and strong non-linear coupling of the vibrational modes. Moreover, beside the purely mechanical effects, electrical anharmonicity and electrical coupling contribute to the infrared and Raman bandshapes, and intensities. The electrical factors may change from one system to an other and it is not straightforward to analyse a given spectrum in terms of its real proton dynamics. However, the inelastic neutron scattering (INS) intensities of hydrogenous compounds are totally determined by the mean-square displacements of the proton. INS spectroscopy is therefore much more specific to proton dynamics than either infrared or Raman. INS spectroscopy has been used on a few hydrogen 0301-0104/88/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
2. Experimental A sample of polycrystalline KHC03 (obtained by recrystallization from a slightly warmed concentrated solution) of about 5 cm by 2 cm by 0.2 cm was B.V.
F. Fillaux et al. /Proton dynamics in the hydrogen bond
426
contained in an aluminium can at 5 f 0.2 K. The INS spectrum was obtained on the TFXA spectrometer, ISIS at Rutherford Appleton laboratory, Chilton UK. TFXA is an inverted geometry time-of-flight spectrometer. The spectrum is collected in reflection (19,= 135’ ) and the final neutron energy is fixed (E+ 4 meV). The spectrometer has excellent energy resolution (AEJE, x 2%) and traces a line in ( Q, o) space such that Q*(A-*) xE(meV)/2. More details of the instrument have been published elsewhere [ 191. The data were actually collected over a period of several days in 1986, equivalent to present day capabilities of 1 day (or 2000 uA h). The data were transformed into an energy transfer scale, with the intensities proportional to S( Q, 0). This spectrum was corrected for background scattering by subtracting the spectrum of the empty aluminium can in the cryostat.
3. Crystal structure and symmetry The KI-IC03 crystal structure has been determined by X-ray [ 4-8 ] and neutron diffraction [ 9, lo] (fig. 1). The crystal is monoclinic, space group P2, /a (C$, ), with four KI-IC03 entities per unit cell. The centrosymmetric dimers (HCOS ) :- , occupying Ci sites, have relatively strong O-H...0 hydrogen bonds
K HCO,
rg.
1. Crystalline
structure
of the KHC03
crystal after ref.
[ 91.
characterized by an O...O distance of 2.6 1 A. According to Lucazeau and Novak [ 171, the intramolecular forces within HCO, are much stronger than intermolecular forces and the internal and external vibrations are well separated. The hydrogen bond interaction is much stronger than the other intermolecular forces in the crystal. Therefore the hydrogen bond vibrations, which are also external modes, can be distinguished from other lattice modes. Assuming a quasi C,, symmetry for the dimers, internal vibrations for the four KHC03 entities in the unit cell are represented by 9A,+9B, (Raman active) and 9A, + 9B, (infrared active) symmetry species. Under the Ci symmetry, these vibrations should be grouped into A, and A, species [ 16 1. The 12 hydrogen bond vibrations give 3A,+ 3B,+ 3A,+3B,. Finally, the 2 1 lattice vibrations are represented approximately by 15 translational (T’ ) (3A,+3B,+ 5A, + 4B,) and 6 rotational (R’ ) vibrations (3A,+ 3B,). Comparison of infrared, Raman and INS spectra is complicated by the differing wavelength in the optical experiments and the importance of momentum transfer in INS. The representations of vibrations in terms of the molecular symmetry species do not apply to INS spectra,.which reflect the density of vibrational states. Fortunately internal modes are not usually strongly dispersed, and can be considered as independent of the momentum transfer. INS frequencies should then be similar to those observed in infrared and Raman. The INS spectrum is also likely to be less sensitive to interdimer coupling than infrared and Raman. Therefore the single centrosymmetric dimer approximation symmetry C2,, will be used. The 18 internal vibrations are then represented by 7A, + 7B, species for in-plane modes, 2B, + 2A, for out-of-plane modes. The 6 hydrogen bond vibrations give 2A,+ 1B, for in-plane and 1B, + 2A, for out-ofplane motions. They can be approximately described as symmetric and antisymmetric antitranslations (T,, T,, T,) or antirotations (R,, R,, R,) of the two CO:- ions (fig. 2). Qualitatively, taking into account the mass of the carbonate ion (M=60 amu) and its moments of inertia (I, = 39.1 amu A*, I,= 40.8 arnuA2,ZW=79.9amuA2),theT, (A,,v(O...O))and R, (B,, v(O...O) ) modes should be at similar frequencies. This is because they both involve large O...O internuclear distance variation and similar masses or
F. Fillaux et al. /Proton dynamics in the hydrogen bond
0
w
T;
Tu
BuLM
‘(OF
Ag
vo.. .o
Ag
80.-o
Bu L.M
TV
Rw
Bu vo ..o
R’w
0 Ag
L.M
Au L.M
Tw
q.j-2
Ru
4000
Bg yo. ..o
‘-(.p Au yo...o
Au ro
o
100
200 Energy
300 400 transfer (me’./)
500
Fig. 3. Inelastic neutron scattering spectrum of the KHCOx powder crystal at 5 K.
out -of-plane
Rv
Energy tronsfer(cm-‘1 2000 3000
1000
‘(Oh
T;
T’w
427
R’v
Bg L M
R’u
Bg I_ M
Fig. 2. Schematic representation of the low-frequency hydrogen bond and lattice modes ofthe ( HCO,):- centrosymmetric planar dimer. LM: lattice mode.
moments of inertia. The R,(A,, ~(0...0)) and R, (A,,, y(O...O)) are expected at higher frequencies than the T, (B,, y(O...O)) and R, (A,, 6(0...0)) modes. Finally, besides the K+ translational motions, the lattice modes correspond to translation ( 1A,,T:, , + 2B,, T:, TV ) and rotation ( 1A,,RW, + 2B,, KU, R:. ) of the ( HC03):dimers.
4. INS spectrum and band assignment The INS spectrum of KHC03 at 5 K is presented in fig. 3 and the proposed assignments (table 1) are consistent with the infrared, Raman [ 15-l 71 and previous INS data [ 201. The INS spectrum is dominated by the intense y(OH) mode (933,983 cm-‘). Also present are the weaker 6 (OH ) ( x 1400 cm- ’ ), v(O...O) (219, 225 cm-‘) and 6(C03) (621, 639 cm-’ ) bands. All these vibrations involve large displacements of the proton. Above 1600 cm- ’ the intensity is due mainly to overtones and combinations.
The v(OH) bands, in the 1800-3500 cm-’ region, are hidden or embedded in this “background” and they cannot be identified unambigously. The rather small full width at half-height (fwhh) of the components for the v(O...O) (16 cm-‘), 6(C03) (20 cm - ’ ) and y ( OH ) ( 25 cm- ’ ) confirm the negligible frequency dispersion of these modes. Band splitting is ascribed to intradimer coupling, except for the 62 1 and 639 cm-’ features which are due to the &(CO,) and 6,( C03) modes, respectively (see below). The hydrogen bond vibrations located at 2 19 and 255 cm-’ (v(O...O)), 196 cm-’ (~(0...0)), 170 cm-’ (y(O...O)), 153 cm-’ (6(0...0)), and finally 145 cm-’ (y(O...O)), show quite different INS intensities. The bending modes are much weaker than the v(O...O). This is especially true for the z(O...O) (196cm-‘) andy(O...O) (170cm-‘) whicharevery weak in our spectrum, whereas their Raman counterparts are quite strong [ 171. Taken together with the large intensity of the y( OH) mode, this suggests that the out-of-plane vibrations of the proton and of the hydrogen bond are more or less independent. Below 120 cm-’ where most of the lattice modes are expected, the bands are broad and the spectrum is typical of the density of states for the compound. Strong INS intensity is anticipated for the translational and rotational vibrations of the dimers whereas translational motions of K+ should be weak, as far as these motions can be separated. Therefore, the INS bands at 114, 108, 90, 74 cm-’ which have strong Raman counterparts at 116, 98 and 87 cm-’ [ 171, should correspond mainly to rotational vibrations of the dimers.
428,
F. Fillaux et al. /Proton dynamics in the hydrogen bond
Table I Infrared, Raman and INS band frequencies (in cm-’ Raman data are reported from refs. [ 15,171 Infrared 90 K
unit)
and assignments
Raman 30 K
for the KHC03
crystal at low temperature.
Infrared
and
INS SK
42
A,
87
B,
98 110 116 120 138 142
A, A, B, A8 A, B,
147 163
A, B,
196 222
A, As
250 639
B, A,+B,
6,(COx) y(CO3)
74 90
136
108 114 A.
y(O...O)
152 170
Ag B,
&(O...O) y(O...O)
219
Au AE
T(O...O) v(O...O)
255 627 662
B, A,+B. A,+B.
v(O...O) 6,(CO3) &(CO,)
940 990
B, A,
Y(OH) y(OH)
145 6(0...0)
170
y(O...O)
v(O...O) 255 661 831
835
A,+B,
937
A,+B,
1036
A,+B,
v,(CO3)
1291
A,
v,(CO,)
987 1010
1375 1405
v,(CO,)
1625 1650 1695
v:(COx) 2y(C03)
1450 1688 1712
A, A,
vb(CO3)
2y(OH)
1050
v,(CO,)
1385 1415 1440
v,(CO,)+8(OH)
1620 1680
G(CO3) S(OH)+S(O...O)
1830
2y(OH)
2040
VOH)+y(OH)
1880 1940 2080
5. Calculation of the INS spectrum
The observed INS spectrum is reported here in terms of the scattering law (S( Q, w) ). The calculation of S( Q, o) for KHC03 on TFXA was performed using the program CLIMAX. This program has been described in detail elsewhere [ 2 11. It will not be described further here except to underline that it produces S( Q, o) intensities taking full account of the
Debye-Waller factors for fundamentals, overtones and combinations. This version of the program is slightly limited by two features; firstly its dependance upon harmonic frequencies and its non treatment of phonon wings. Anharmonicity results in harmonic overtones being misplaced (this is not a problem for the present analysis). Whilst the effect of phonon wings have been minimised by working at 5 K.
F. Fillaux et al. /Proton dynamics in the hydrogen bond
429
(namelyv(OH),&(OH),y(OH))andthehydrogen bond coordinates. However it was not possible to adjust the calculation to fit the observed intensities. This result shows that the picture of protons moving in a local field dominated by covalent bonds to CO!entities is incorrect. The proton motion is best regarded as decoupled from the hydrogen bond vibrations.
6. Force field calculation 6.1. The isolated dimer model
Most of the modes involving K+ motions are at very low frequency and poorly understood. Such that previous force field calculations [ 16 ] were carried out on an isolated planar centrosymmetric dimer of symmetry CZh. The structure obtained by projection of the atomic coordinates in the crystal [ 9 ] on the mid plane of the dimer is presented in fig. 4. The symmetry coordinates are shown in table 2. There are four redundant coordinates: two 6(COX), one 6(0...0) (B,) and one t( O...O) ( Bg). The valence force field (table 3, I) was refined on the basis of infrared, Raman and INS frequencies. Whereas the frequencies are well represented, the calculated relative INS intensities (table 4, I) are in total disagreement with the experimental data. Most of the calculated intensity arises from the hydrogen bond vibrations, whereas the y (OH ) and 6 (OH) are very weak (fig. 5, I). The discrepancy is unacceptably large. Many attempts were therefore made to introduce different coupling constants between the internal coordinates
6.2. The crystal-field model One possible way to account for the decoupling of the proton motion is to consider that the hydrogen atom is moving in a field due mainly to its crystalline environment. Owing to the long-range character of atom-atom and electrostatic interactions, it would not be possible to select any dominant contribution of a particular atom, or set of atoms. Indeed a summation over many unit cells in the crystal would be necessary to calculate the true crystal field experienced by the proton. Such a calculation is beyond the scope of the present paper and a simple phenomenological model will be used. The crystal field is supposed to average over all the atomic vibrations and
A
V
“M4 I .I I .
M2
----o _+
_---
_+--
I
-4~ 01
,--+r.
I
---
f-4009
C
G
--
0,
c7
9
IiS---I
I’ I
I ‘. I lk 1, M3
Fig. 4. Projection of the atomic coordinates [ 91 on the mid plane of the ( HC03)$- dimer and atom numbering for molecular coordinate definition. The heavy masses M,, Mz (stretching), M,, M, (bending) represent the fixed crystal field.
430
F. Fillaux et al. /Proton dynamics in the hydrogen bond
Table 2 Coordinate definitions torsion; 0: out-of-plane fig. 2 Internal No. I
2 3 4 5 6 7 7’ 8 8’ 9 10 11 12 13 14 15 16 17 17’ 18 18’ 19 20 21 21’ 22 22’ 23 24 25 26 21 28
for the (HCO,):bending. Primed
dimer. The atom numbering numbers describe the proton
Symmetry
coordinates definition c2 C7 c2 C7 c2 c7 04 Ml 09 M2 04 09 04 09 03 08 01 06 H5 H5 HI0 HlO 04 09 H5 H5 HlO HlO 03 08 01 09 03 08
correspond coordinates
01 06 03 08 04 09 H5 H5 HlO HlO 06 01 c2 c7 c2 C7 c2 C7 04 M3 09 M4 06 01 04 M3 09 M4 c2 C7 C2 c7 01 06
No.
s s
03 08 01 06 04 09 C2 C7 C7 c2 C2 Ml C7 M2 04 09 C7 c2 04 09
01 M2 06 Ml 01 06 06 04 06 01
S S S S S S S S S S B B B B B B B S B S B B T T T T 0 0 T T T T
to be fixed with respect to the crystal axes. For the purpose of calculation, it can be represented by heavy masses to which the protons are linked through forces governing the stretching and bending vibrations. These masses (M,, M2, M3, M4 in fig. 4), arbitrarily set to 1000 amu, were located on the inertial axes of the dimer, symmetrically with respect to the center of mass, and either along, or perpendicular, to the OH bond directions. To avoid unreasonably large force constants, due to the very long H5...M1 and HI0...M2
to that in fig. 2. S: stretching; B: in-plane bending; T: with respect to the heavy masses M,, Mz, MS, M, in
Group coordinates
coordinates Definition 1
2 3 4 5 6 7 7 8 8 9 10 11 12 13 14 15 16 17 17 18 18 19 20 21 21 22 22 23 24 25 26 27 28
2 A,
I -2B.
v(C-0)
3 3
4 A, -4B,
v(C=O)
5 5
6 A, -6B,
v(C-OH)
7 8 A, 7’ 8’A, 7 -8B, 7’ -8’B, 9 lOA, 9 -lOB, 11 l2A, 11 -12B, 13 14A, 13 -14B. 15 16A, 15 -16B, 17 18A, 17’ 18’A, 17 -18B. 17’-18’B. 19 20A, 19 -20 B, 21 22B, 21’ 22’B, 21 -22 A, 21’-22’A, 23 24 B, 23 -24 A, 25 26 B, 25 -26 A, 27 28B, 27 -28 A.
v(O-H)
v(O...O) 8(O=C-OH) S(O=C-0) 8(0-C-O)
8(0-H)
6(0-O)
do-HI
so3 1 r(O...O) y(O...O)
distances, the OH in-plane bending is now represented by the HS...M3 and HI0...M4 stretching coordinates, instead of the H5MIM2 and H,,M,M, bending coordinates. A new valence force field was refined assuming negligible force constants linking the proton to the CO:- ions (table 3, II). Calculated INS intensities now involve only the y ( OH ) ,6 (OH ) and v( OH) modes (table 4, II). Comparison of calculated and experimental spectra (fig. 5, II) shows that while the y( OH) and 6( OH) intensities are reason-
431
F. Fillaux et al. /Proton dynamics in the hydrogen bond Table 3 Refined force constants in mdyne/A for the KI-ICO, dimer. masses (M) representing the fixed crystal field
No.
6 7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
I
dimer.
II, III and IV: proton
oscillating
with respect to heavy
Force constants I
1
I: isolated
v(O-H) v(OH...M) v(C=O) v(C-0) v(C-OH) 6(C-0-H) &(OH...M) 6( HO-C-O) 6( o-C=O) 8(0-C-O) v(O...O) 6(0...0)
v(OH)/v(OH) v(CO)/v(CO) v(C-O)/v(C-0) v(c=o)/v(c=o) v(C-OH)/v(C-OH) s(C-0-H)/s(C-O-H) 6(OH...M)/&(OH...M) G(HO-C=O)/S(HO-GO) s(o-c=o)/6(0-c=O) s(HO-C-O)/G(HO-C-O) v(0...0)/v(0...0) v(OH)/v(O...O) 8(OH)/s(O...O) G(OH...M)/v(CO)
II 3.8 1700 6.37155 6.57870 5.90691 0.84691 2.62049 1.37956 2.27012 0.5 1893 0.37535 0.14682 2.03063 2.0035 1 -0.61209 -0.14708 0.04467 - 0.85656 0.33046 0.31442 0.05783 0.0 0.0
y(O-H) Y(0Hv.M)
0.24652
Y(C0) T(O...O) Y(O...O) y(OH...M)/Y(OH...M)
1.12962 0.19747 0.27274
Y(co)lY(co) Y(oH)lY(CO)
y(OH)lv(O...O) y(OH)/~(0...0)
IV
III
0.22805 0.0 0.0
ably represented, the v(O...O) bands do not appear and the height of the v (OH ) band is obviously overestimated. The real force field is thus likely to lie between these two simple models. 6.3. Thefinal model the observed INS intensities are well represented assuming that the v (OH ) dynamics is dominated by the O-H force constant while the bending modes are
4.07043 10.14405 5.17230 5.12270 1.17412 2.21393 1.61871 1.59373 0.51200 0.35580 0.15471 1.84746 0.01167 0.32383 0.10820
3.81700 9.88602 5.61895 5.34867 1.18000 2.04487 1.46726 1.65352 0.52988 0.36080 0.14780 1.70298 0.10862 0.322 13 0.01425
3.83890 9.91723 5.46318 5.37951 1.17713 2.16647 1.51702 1.6367 1 0.51588 0.38774 0.14624 1.73128 0.32509 0.32134 -0.33317
0.04570 0.28690 0.2749 1 -0.12270 0.05357 0.0 0.0
0.04635 0.06849 0.27240 -0.06650 0.06727 0.0 0.0 0.0
0.04487 - 0.00044 0.41059 -0.17081 0.06576 -0.10000 0.0 0.05000
5.46900 0.67953 0.22295 0.21301 0.28407 -0.06446 0.0 0.0 0.0
5.46924 0.68111 0.21684 0.07890 0.28407 -0.06349 0.0 0.0 0.0
5.46924 0.68111 0.21684 0.07890 0.28407 -0.06349 0.0 0.0 0.0
essentially governed by the crystal field (fig. 5, III, tables 3 and 4, III). Besides fundamental transitions below 1500 cm-’ the rest of the spectrum is dominated by overtones and combination bands. It is remarkable that the intensities of the two spikes at 630 and 640 cm-’ due to the CO:- bending modes, are well represented without introducing any additional coupling constant. This arises from the fact that the bending motions of the carbonate ion are mixed with the hydrogen bond vibrations. This result is very
F. Fillaux et al. /Proton dynamics in the hydrogen bond
432
Table 4 Experimental and calculated frequencies (cm-‘) and INS relative intensities with respect to heavy masses (M) representing the fixed crystal field Observed frequency
Assignment
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 2 3 4 5 6 7
v(OH)
B”
v(OH)
A, A,
v(C=O) v(C=O)
6COI-I) s(OH) v(C-0) v(C-0) v(C-OH) v(C-OH) 6(C=O) 6(C=O) S(OC0) S(OC0) v(O...O) v(O...O) 6(0...0) Y(OH) Y(OH) Y(C0) Y(COz) t(o...o) y(O...O) y(O...O)
B”
A, B” B”
A, A, B” J%
A, B”
4 4 BU B” A”
B, B, AU AU A”
5
2680 2580 1688 1638 1440 1385 1415 1291 1036 1010 665 660 630 625 244 219 153 987 937 835 831 196 170 145
Calculated
frequency
for the KHCOz dimer.
II, III and IV: proton
oscillating
(int.)
I
II
III
IV
2680.1(0.8533) 2580.1(1.033) 1693.9(1.775) 1638.8(2.150) 1441.5( 1.450) 1385.0(0.8441) 1408.5(2.814) 1290.1(4.058) 1036.5(7.410) 1010.1(7.548) 661.8(14.53) 660.0( 13.80) 627.7( 15.12) 624.9( 15.89) 244.7(40.82) 219.0(67.99) 147.9( 100.4) 988.7( 1.920) 935.5(4.443) 836.0( 11.64) 830.0( 11.83) 202.1(76.13) 171.0(100.4) 138.9(75.73)
2680.0( 90.48) 2580.0(95.40) 1688.1(0.0000) 1638.0(0.0000) 1440.0( 145.1) 1385.0( 150.7) 1415.6(0.0000) 1292.5(0.0000) 1036.9(0.0000) 1009.1(0.0000) 664.7(0.0000) 660.2 (0.0000) 625.4(0.0000) 624.5 (0.0000) 244.1(0.0000) 221.3(0.0000) 146.8 (0.0000) 987.0( 175.3) 937.0( 182.6) 835.7(0.0000) 831.0(0.0000) 210.3(0.0000) 156.4(0.0000) 139.0(0.0000)
2681.3(6.488) 2579.5(7.419) 1687.9(0.8031) 1638.0(0.9267) 1439.3(90.32) 1383.8(95.30) 1410.1(0.0153) 1328.5(0.0236) 1037.2(7.738) 1009.0(6.276) 664.8( 16.37) 660.3(23.48) 631.2(23.63) 623.5(25.11) 248.0(54.57) 220.9(51.58) 147.0(49.20) 987.0( 123.9) 937.0( 131.3) 835.0(0.0000) 831.0(0.0000) 196.0(0.0000) 120.6(0.0000) 95.2(0.0000)
2683.9(4.964) 2583.9 (5.766) 1687.9(0.5176) 1638.0(0.8507) 1440.3(64.09) 1384.9(69.21) 1415.0(45.54) 1336.9(26.47) 1035.9(6.133) 1010.0(7.230) 665.0( 15.77) 659.8(27.14) 630.3( 12.98) 624.9(23.59) 245.3(53.31) 220.3(47.87) 148.7(48.58) 987.0( 116.4) 937.0( 123.9) 835.0(0.0000) 831.0(0.0000) 196.0(0.0000) 120.6(0.0000) 95.2(0.0000)
helpful in assigning the observed vibrations. It is possible to adjust more accurately the calculated intensities with additional coupling constants in the force field (tables 3 and 4, IV, and fig. 5, IV). The 6( OH) region is slightly improved by a weak coupling of the v(C0) and 6(OH) coordinates; the v(O...O) intensities are quite sensitive to the v(OH), v(O...O) interaction; and the 6(C03) to the6(C03), 6(0...0) coupling. In principle, owing to the sensitivity of the fit to these coupling constants, it would be possible to obtain very accurate values from the spectrum. Unfortunately our data are too limited, both by noise and background uncertainties, to pursue these fine details. Therefore we suggest that force field III (table 3 ) can be considered as sufficient. 7. Discussion The absence of detectable INS intensities for the y(C03) and r(O...O) modes near 830 and 196 cm-’
respectively, and the very weak intensity in the y(O...O) region around 150 cm-’ show that the y(OH) mode is practically independent of the carbonate and hydrogen bond out-of-plane motions. This result holds only for a full separation of in-plane and out-of-plane vibrations, which may not be totally justified. (The true dimer structure in the crystal [ 9,10 ] deviates slightly from planarity.) Indeed, a weak interaction of the y(OH) at 987 cm-’ and v,(CO,) at 1050 cm-’ could account for the INS intensity of the latter which was not satisfactorily represented within the planar approximation. However, the influence of this coupling on the description of the y( OH) mode is negligible and the isolated character of this mode must be considered as one of the most firmly based conclusions of this work. The 6(OH) mode also appears quite isolated, although weak coupling with the CO3 stretching and 6(0...0) modes are observed. However, the v(OH) and v(O...O) modes are totally mixed.
433
F. Fillaux et al. /Proton dynamics in the hydrogen bond Energy lop0
Irans;;feom-‘)
3op0
i/ 0
100
200
Energy fransfer
300
(meV)
400
Fig. 5. Comparison of calculated (-•-) and experimental (-) INS spectra of KHCOX at 4.2 K. I, II, III, IV correspond to the force fields in table 3 with the same notations. Overtones and combination bands were calculated only in case IV.
sities of INS bands, which depend upon the eigenvectars, introduce new constraints upon models. Dynamical models which produce very poor fits to INS data can thus be quickly eliminated as being irrelevant. In this sense then, the force field proposed here is the most realistic presently available for KHC03. However, this force field is not unique and another choice of molecular coordinates might well give as good an agreement with the experimental data. For example, the masses Ml, Mz, M3, M4 could have been chosen differently. That said, however, after investigating several different alternatives, we believe that any analysis consistent with the INS intensities should lead to the same conclusion. Namely that the isolated dimer model proposed by Nakamoto and coworkers [ 161 is useful only for the description of the v (OH ) mode. The out-of-plane y ( OH ) mode is best regarded as independent of the internal and external vibrations of the carbonate dimer. Similarly for the 6 (OH) mode which is, however, slightly mixed with the CO3 stretching coordinates. These conclusions, based on the INS intensities, are largely independent of the choice of the vibrational coordinates. Within the valence bond approach [ 22-241, the real hydrogen bond is represented by a mixing of the three resonance forms: w, =OD-H...O, I,+ =O,
It is impossible to compare in detail the present valence force field with the Urey-Bradley force field determined by Nakamoto and co-workers [ 161. The two methods are much too different and their description of the normal modes is not sufficient to evaluate the corresponding INS intensities. However, the rather large force constants they have introduced for the hydrogen bond modes suggests that their model should lead to a poor match with the INS intensities. A more general point about force field calculations concerns the underdetermined nature of the mathematical problem. The number of force constants (N(N+ 1) /2 for N degrees of freedom) greatly exceeds the number (N) of observable frequencies. Therefore, no unique force field can be extracted from frequency data alone. Any of the force fields of table 3 would calculate acceptable eigenvalues. The inten-
,
v2=O~...H+...0A,
. ..H-O.+ ,
where 0, and OA represent the proton donor and acceptor entities. In the case of KHC03, the isolated dimer model of Nakamoto and co-workers [ 161 corresponds to a proton covalently linked to the oxygen. The w, resonance form, represented by the O-H force constants, is dominant, whereas the contribution of w2, represented by the H...O force constants, is much weaker in this model. Because of the directionality of the force constant associated with a covalent description of the hydrogen bond the proton will “ride” with the oxygen displacements. These riding modes have calculated INS intensities which are inconsistent with observations. Further, OH deformations should modify the CO:- geometry through electron orbital interactions. When this occurs it is usually signaled by significant vibrational coupling between the corresponding modes. The INS spectrum does not support this case.
434
F. Fillaux et al. /Proton dynamics in the hydrogen bond
Rather INS intensities support an alternative view that the hydrogen bond is best thought of in terms of the ionic form y2. Of course this can be seen from the intensity of the v (OH ) infrared band [ 15,17 1. Then owing to the more isotropic nature of the electrostatic field the proton-oxygen interactions depend mainly upon the Ob...H distance, and very much less upon angular changes. The v(OH) and v(O...O) modes are still allowed to interact as strongly as in the covalent form w,, but the O...O bending motions become dynamically independent of the proton motions (i.e. the “riding” is now absent). We have shown above that purely covalent force fields for the proton are inappropriate. However it would be misleading to overstress the utility of an ionic (crystal) force field. We must not forget that y(OH) is, to first order, well described by O...O forces (this can be seen in the excellent correlation between y (OH ) frequencies and O...O separations [ 3 ] ) . Beyond what has already been shown our methods are not capable of resolving the two contributing fields which control1 the dynamics of the proton. Indeed it is doubtful that complete analysis in terms of normal coordinates could ever be achieved. This we believe stems from the very anharmonic character of the hydrogen bond itself. The results of our INS work can be compared with previous elastic scattering results. The dynamical model allows us to calculate the total mean-square displacement of the protons, along the Cartesian axes. The total proton displacement approximately parallel to the O...O bond Vi (fig. 4) is Ui co.0378 A2. This can be compared with other estimates; (i) UT =0.042 A* [25]; and (ii) Vi ~0.029 A’ [lo], both obtained at room temperature. Our calculation has no contribution from external modes, it therefore represents a minimum acceptable value of Vi.This throws some doubt on the above value of Vi (ii) from ref. [ 10 1. This is perhaps not surprising since the derived value for the OH bond length is shorter than expected [26]. Increasing this bond length should result in different thermal parameters for the proton. If they became consistent with our results then, we expect Uf cz0.04 A*. (We shall avoid discussing UT values because we have adopted a planar molecular geometry. )
8. The tunnel effect and proton transfer in KHC03 dimers Carboxylic acid dimers have attracted much attention recently. Studies involving detailed analysis of NMR and NQR data [ 27,281, and calculations [ 291, have been interpreted in terms of thermally activated proton transfer. One of us (FF) has recently produced a double well potential for proton dynamics in KHC03 [ 181 (seetig. 6). Thepotential V(r) (cm-‘) is given by: V(r)=368r+438910r6 +5516 exp( -30.7r2)
,
(1)
with r as the off-center proton position (A). The wells are asymetric about the center, with an enthalpy difference of ~2.7 kJ mol-’ (213 cm-‘). This compares well with an estimated value from NQR of 3.4 kJ mol- [28]. It is appropriate to compare our INS observations with a prediction that can be made from eq. ( 1). On fig. 6 it is seen that there are four energy levels. Thus, although the protons are highly localised (and indeed in the dynamical analysis above we have assumed complete localisation) there is a small tunneling delocalisation. We should therefore expect to observe INS active transitions. The INS intensity of the O+ 1 transition is, to a first approximation, related to the overlap of the initial and final wavefunctions. (This is presented in the appendix.) It can be appreciated that because the wells are deep this over-
E Cd
6000 t
a=3 a=2
r + i
0 --+-
00
0.3
H
Fig. 6. Potential function for proton transfer along the hydrogen bond in the (HCO,)$dimer after ref. [ 181. (Y is the quantum number associated with the v(OH) mode.
F. Fillaux et al. /Proton dynamics in the hydrogen bond
lap is small and the corresponding intensity is low. We therefore predict a band of weak to moderate intensity at z 213 cm-‘. The bands observed in this region are shown in fig. 7. They are both associated with the O...O stretching modes. However the lowfrequency band at = 2 19 cm- ’ is highly asymmetric and disproportionately strong. All our dynamical analyses suggest that Z219/Z245x 1.O + 0.05, regardless of any reasonable coupling constants. This ratio can be obtained if we remove a sharp component from the low-frequency side of the band. This produces three bands; the two O...O stretching modes, now at 224 and 245 cm-‘, and the sharp component at 216 cm-‘. The relative intensity of this sharp band (fwhha 10 cm-‘) is Z2,6/Z224xO.5. We consider this to be of moderate intensity, in the context of KHC03, and tentatively assign the sharp band at 2 16 cm- ’ to a tunneling transition as discussed above. (The excellent frequency correspondence: 2 13 cm- ’ (predicted) and 2 16 cm-’ (observed), we regard as fortuitous.) As shown in the appendix, a proton situated at f 0.2 A from the O...O bond center is consistent with the observed intensities. This calculation shows that the tunneling frequency can be observed even for purely localised wavefunctions in the lower states. However, the present approach deserves several reserves, because of the many simplifications in the model, and in the band decomposition. ContriEnergy transfer
(cm-’
200
)
435
butions to the INS intensity arising from the possible mixing of the two wavefunctions (A. 1) and (A.2) have been neglected. Therefore, the difference with the value derived from the diffraction data ( z + 0.3 A) [9,10] may be not significant. The presence of tunneling at low temperature must influence the interpretation of the NMR and NQR data, which is presently treated in purely classical terms [ 27,29 1.
9. Conclusion The INS spectrum of the KHCO, powder crystal at 5 K shows strong features ascribed to the y(OH), g(OH), v(O...O) and 6(C03) vibrations, in good agreement with infrared and Raman data. Except for the latter, these bands are split into two components due to intradimer coupling. A force field description of the (HCO,)$cyclic dimer consistent with INS intensities indicates unusual proton dynamics which could not have been predicted from infrared and Ramn spectra alone. The bending modes of the proton are independent of both hydrogen bond and CO:- vibrations, but the stretching modes interact strongly. This suggests a pronounced ionic character for the hydrogen bond. Simple calculations, based upon wavefunction overlap and a double well potential, predict a moderately intense band at ~2220 cm-‘. This has been tentatively assigned.
320 J
Acknowledgement We should like to thank Dr. C. Carlile for help with the neutron experiment. Also, we thank the SERC (UK) for access to the ISIS neutron facilities and the CEA (France) for financial support. We should also like to thank Dr. G.J. Kearley (ILL) for supplying the CLIMAX program.
Appendix
Energy tronsfer(meV)
Fig. 7. The low-frequency spectrum of KHC03 decomposition in the O...O stretching region.
40 at 4.2 K and band
In the asymetric double well potential governing the proton transfer (fig. 6 and eq. ( 1) ), the E, -E. splitting is due to the energy difference of the two wells. The wavefunctions Q0 and 0, are almost entirely lo-
436
F. Fillaux et al. /Proton dynamics in the hydrogen bond
calised [30]; they mately, by harmonic
can be represented, approxifunctions centered at k ro: exp[ -OSa8(r-ro)*]
,
(A.1)
CD,(r)= (a~/~)‘/~ exp[ -OSa~(r+r,)‘]
,
(A.2)
@o(r)=(a;/x)“2
or:
1 Q: Q: '+2a&,,,o Z 2h.0 [---
>I .
with ai = 2nmH cv,/zi
(A.3)
where mH ( 1 amu) is the reduced mass for the proton motion, v. ( x 2000 cm-’ ) is the v( OH) frequency in the well, c is the velocity of light, and A is Plan&s constant over 27r. the INS intensity for the O+ 1 (tunneling) transition at v, is: StQt,
v,)a
xS( v+E,
j Q,(r) -E,)
exp(iQ,r)%(r) )
dr2 (A.4)
or:
v,)aexp[-2a~(r~+Q:/4a~)l xS(v+Eo-E,). (A.5)
&(Q,
The tunneling intensity may be compared with that of an isotropic harmonic oscillator at the same frequency, say v(O...O):
SO...O(Q,, ~,)a (Q~/~&...o) exp(-Q~/2&.o) XG(VS&-E,), (A.6) with a6...o =2xmo...ocv,lfi
,
(A-7)
where mo...o (30 amu) is the reduced mass for the antitranslation of the two carbonate entities. According to the band decomposition (fig. 7) the relative intensity for the tunneling and v(O...O) modes at 2 16 and 245 cm-’ is: I 216 -=
Z245
S(Qt, vt)xo 5 so...o(Qt, vo) . .
(A-8)
Then, eqs. (A.6 ) and (A. 8 ) give: Z
216 -z21 245
2af
mO...O
Q, mH (A.91
In the present experiment r. z k 0.205 A
(A. 10)
Q: x 2~2: and then: (A.
11)
References [ 1 ] P. Schuster, G. Zundel and L. Sandorfy, eds., The hydrogen bond. Recent developments in theory and experiments, Vols. 1-3 (North-Holland, Amsterdam, 1976). [2] H. Ratajczak and W.J. Orville-Thomas, eds., Molecular interactions, Vol. 1 (Wiley, New York, 1980). [ 3 ] A. Novak, Struct. Bonding 18 ( 1974) 177. [4] P. Herpin, Compt. Rend. Acad. Sci. 234 (1952) 2205. [ 51I. Nitta, Y. Tommiie and C. Hoe Koo, Acta Cryst. 5 ( 1952) 292. [6 ] I. Nitta, Y. Tommiie and C. Hoe Koo, Acta Cryst. 7 ( 1954) 140. [ 71 P. Herpin and P. Meriel, J. Phys. 25 ( 1964) 484. [8] B. Pedersen, Acta Cryst. B 24 ( 1964) 478. [9] J.O. Thomas, R. Tel&en and I. Olovsson, Acta Cryst. B 30 (1974) 1155. [ lo] J.O. Thomas, R. Tellgren and I. Olovsson, Acta Cryst. B 30 ( 1974) 2540. [ 111 L. Couture-Mathieu, J. Phys. Radium 11 ( 1950) 544. [ 121 L. Couture-Mathieu, J. Phys. Radium 15 (1954) 531. [ 131 P. Tarte, Hydrogen bonding (Pergamon, Oxford, 1959) p. 115. [ 141 Y.I. Ryskin, Opt. Spectry. 12 (1962) 287. [ 151 A. Novak, P. Saumagne and L.D.C. Bok, J. Chim. Phys. (1963) 1385. [ 161 K. Nakamato, Y.A. Sarma and K. Ogoshi, J. Chem. Phys. 43 (1965) 1177. [ 171 G. Lucazeau and A. Novak, J. Raman Spectry. 1 ( 1973) 573. [ 181 F. Fillaux, Chem. Phys. 74 (1983) 405. [ 191 J. Penfold and J. Tomkinson, internal report RAL-86-019, Chilton, UK. [ 201 K.P. Brierley, J. Howard, C.J. Ludmann, K. Robson, T.C. Waddington and J. Tomkinson, Chem. Phys. Letters 59 (1978) 467. [21] G.J. Kearley, J. Chem. Sot. Faraday Trans. II 82 (1986) 41; G.J. Kearley and J. Tomkinson, Inst. Phys. Conf. Ser. 81. Neutron scattering data analysis 1986, Ed. M.W. Johnson (Institute of Physics, Bristol, 1986) p. 169. [ 22 ] L. Pauling, in: The nature of the chemical bond (Cornell Univ. Press, Ithaca, 1960).
F. Fillaux et al. /Proton dynamics in the hydrogen bond [ 23 ] C.A. Coulson and U. Danielsson, Ark. Fys. 8 ( 1954) 239.
[ 241 A. Warshel and R.M. Weiss, J. Am. Chem. Sot. 102 ( 1980) 6218. [25] M. Arsic-Eskirja, KFA Jiilich internal report Jul-872-FF (1972). [26] M. Ichikawa, Acta Cryst. B 34 (1978) 2074. [ 27 ] T. Agaki, F. Imashita, T. Tesao, N. Hirota and S. Hayashi, Chem. Phys. Letters 139 (1987) 331.
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[28] A. Cough, M.M.I. Haq and J.A.S. Smith, Chem. Phys. Letters 117 (1985) 389. [29] R. Meyer and R.R. Ernst, J. Chem. Phys. 86 (1987) 784. [30] R.L. Somorjai and D.F. Homig, J. Chem. Phys. 36 ( 1962) 1980.