J. Phys. Chem. 1995,99, 17532-17538
17532
Infrared Spectral Features Due to Very Rapid Fluxional Motion: Changes in the Infrared Carbonyl Stretching Spectra of Tricarbonyl(q4-norbornadiene)iron with Temperature James J. Turner,* Charles M. Gordon, and Steven M. Howdle Department of Chemistry, University of Nottingham, Nottingham NG7 2RD, England Received: March 16, 1995; In Final Form: August 23, 1995@
It is shown that the coalescence ‘of the v(C0) infrared bands of tricarbonyl(q4-norbomadiene)iron cannot be explained on an “intrawell” mechanism and must involve “interwell” dynamics. This leads to the general conclusion that vibrational spectroscopic evidence for chemical exchange (i.e. “interwell”) can be obtained by demonstrating that there is extra intensity between spectral bands that cannot be explained by simple overlap of individual bands.
I. Introduction
(ii)
(iii)
In a recent paper’ describing the temperature dependence of the v(C0) R spectrum of tricarbonyl(q4-norbomadiene)iron, (Fe(CO)3(q4-NBD)} 1, and its I3COisotopomers, it was argued W
0
2
I
2
C
that the results were best interpreted on a fast dynamic intramolecular exchange model in which the CO groups exchange positions via some concerted motion, such as Berry pseudorotation or a turnstile mechanism. The barrier for this motion must be very low (1-2 kcal mol-’). This is analogous to-but much faster than-those dynamic processes associated with coalescing NMR spectra. We acknowledged that other mechanisms could contribute to the IR band collapse. More recently Strauss2 has argued that all our observations can be interpreted without invoking intramolecular CO exchange. Looked at from the point of view of the potential function describing the intramolecular CO concerted exchange, the distinction lies between dynamics confined within the potential well (intrawell) and that involving jumps between the potential wells (interwell), as proposed in ref 1. In the intrawell model the CO groups do not exchange positions, whereas in the interwell model they do. We wish to challenge this assertion; we argue that the intrawell model proposed by Strauss cannot explain both our earlier observations and a more detailed recent examination of the collapsing spectra. Firstly (section 11) we look at simple qualitative arguments; secondly (section 111) we examine in some detail the basic mathematics which apply to exchange in both Nh4R and vibrational spectroscopy. In both cases it is shown that the intrawell model cannot explain the spectroscopic phenomenon observed for 1. In particular, chemical exchange involving interwell motion results in spectroscopic features which show up as extra intensity between bands over and above that expected simply from the overlap of individual bands. This “extra intensity” is a striking feature of the temperature-dependent v ( C 0 ) IR spectrum of 1. _____
_______
@Abstractpublished in Advance ACS Absrracts, November 1, 1995.
0022-365419512099-17532$09.00/0
Figure 1. Schematic representation of a possible exchange process based on the tumstile mechanism which will exchange the CO groups of Fe(C0)3(q4-norbomadiene),1. The molecule is viewed along an axis lying in the symmetry plane; the dotted line represents the diene, which has a plane of symmetry, AB.
11. A Qualitative Treatment A. Background. To set the scene, Figure 1, reproduced from Figure 1 of ref 1, describes a possible motion that results in interconversion of 1 between different arrangements by a tumstile mechanism; note that this is not a simple rotation and that, to maintain a zero angular momentum, the NBD group will rotate in the opposite direction. The v(C0) IR spectrum as a function of temperature is reproduced in Figure 2. In ref 1 this spectral collapse and that of spectra associated with I3CO-substituted 1 were interpreted largely on the basis of “jumps” between the minima of the torsional (turnstile) potential function, i.e. an interwell motion. The net result is that the CO groups change from nonequivalence (C,symmetry with three v ( C 0 ) bands, 2a’ and a”) at low temperature to equivalence (C3u symmetry with two bands, a and e) at high temperature. The altemative proposed by Strauss2 (spelled out in more detail in section 111) is best illustrated first by considering a much simpler system. B. A Model System. Suppose that a vibrating group XY (e.g. CO) exchanges via a torsional motion between two positions, A and B, in which it has different IR spectra. The potential function describing this exchange might be as illustrated in Figure 3. T h i s will produce similar effects to those shown in Figures 11 afld 12 of ref 3. We can predict three different spectral effects. ( a ) Very Slow Exchange. By this we mean that any exchange between A and B (interwell) or among the torsional levels within 0 1995 American Chemical Society
IR Spectral Features Due to Very Rapid Fluxional Motion
J. Phys. Chem., Vol. 99, No. 49, 1995 17533
"1 I
4
t * 2? t w
I
A
Vtor
0 1
I
0"
180"
Reactlon Coordinate
0 2070
2020
1970
1920
Wavenumber / cm-' Figure 2. IR spectra in the v(C0)region of Fe(CO)&f-norbomadiene), 1, in (a) liquid KrKe at -150 "C, (b) supercritical Xe at 25.5 "C, and
(c) supercritical Xe at 102 "C (the increase in absorbance from (b) to (c) is due to increased solubility). The spectra were recorded (32K data points; Happ-Genzel apodization; ca. 0.5 cm-' resolution) on a Nicolet 730 FTIR instrument with a 620 data station. the potential well (intrawell) is very slow. This is equivalent3 to setting the Kubo parameter K >> 1. (K = AT,, where A is "modulation amplitude", the root mean square of the frequency fluctuation; T, is the "modulation time", the characteristic time that frequency fluctuations persist.) In principle the spectrum will consist of a series of individual lines yoA, vlA,vzA ... yoB, vlB, v2B .... If the individual transitions cannot be resolved in the condensed phase, the spectrum will be the envelope of these transitions, with the shape determined by the frequencies of the transitions and by the relative populations of the levels. This is illustrated in Figure 4a, where we have assumed a particular set of anharmonicities to produce a simple pattern of lines. The spectrum will consist of two asymmetric bands and in addition may have a component between the two bands due to transitions vrABarising from levels above the barrier (see Figure 3). On raising the temperature, the shapes will change as the relative intensities of the components change, because of the change in Boltzmann population of the levels. In particular, the relative intensities of the transitions with high values of ni (the population of the ith torsional level) will increase. Unless the component between the band maxima (vrAB)becomes extremely intense, the spectrum will still show two maxima at the original voAand voBpositions because the transitions originating in levels noA and noB will remain the most intense. Thus, although there may be the effect of band broadening, there will be no movement of the band maxima toward each other. This conclusion is the same as that shown in Figure 11 of ref 3. This behavior is nothing like that experienced with 1, where there is quite definitely a moving together of the band maxima. (b) Intrawell Fast Dynamics. In this case there is no exchange between A and B (i.e. no interwell), but very rapid motion among the torsional levels (Le. intrawell) causes an averaging effect (Redfield mechanism) yielding two bands whose positions are given by the population-weighted means,
Figure 3. Schematic representation of the potential function for restricted rotation of the XY group between sites A and B, assumed to be 180" apart. The torsional levels are denoted by ut = 0, 1, 2, etc., and nl represents the population of level i for site j . For convenience
the ground-state potential function is drawn symmetric, with the difference in frequency in the two sites arising from the potential function differences in the excited vibrational state of XY. vgA is the frequency of the XY stretch in site A with the XY group in level 0, etc. Ignoring splitting of levels due to degeneracy, there will also be transitions such as vrAB,which represent transitions from energy levels well above the barrier, i.e. from free rotational levels. These have the superscript AB, since in the simplest approximation they will have the same frequency, which will be the mean of yoA and yoB.
uv
-I
B* b
V
Figure 4. Schematic representation of the spectra to be expected for the potential function in Figure 3. A simple assumption is that, because of anharmonicity, the excited-state potential function is such that yoA, vlA, vzA, etc., and yoB, vlB, vzB, etc., differ in frequency by a small but constant increasing or decreasing amount. If the individual lines can be resolved, then the spectrum will be as shown in (a), where the intensity of each line is determined only by the Boltzmann distribution. If the individual lines cannot be resolved, then the spectrum may appear as shown by the dotted line. If rapid Redfield-type relaxation occurs, then the spectrum will have the appearance of (b). as shown in Figure 4b, and whose widths are given by the meansquared amplitude of the variation from the mean, multiplied by a correlation times3 This is equivalent3 to setting the Kubo parameter K (r,2- r:)
+ r:) +
+ 1/28nsin ne(t)rarb
In these expressions,
ra = (1/&)(2r1 - r2 - r3) rb = (1/&)(r2 - r3) where rI, r2, and r3 are the C-H stretching intemal coordinates; thus under C3” symmetry ra and rt, would be the two components of the e symmetry coordinate. Since HOis separable in the ra and rt, coordinates, three zero-order vibrational eigenfunctions can be expressed as products of one-dimensionless harmonic oscillator wave functions;
0 = 100)
a = 110)
T2a
aOaO
and
wb
=-
1
T2b
- RbObO
where 1/(Tza) and 1/(T2b) describe other relaxation processes which might contribute to the linewidths. “The spectrum is thus nYo Lorentzians centered at Wa and Wb withfull widths at half maximum of 2Wa and 2 w b . ” (Note that W A and W B are temperature dependent, see eq 4,and the R terms are negative.) This is exactly what is expected on simpler arguments, as shown in Figure 4b. However, none of the correlation functions contain terms that incorporate exchange so it is not surprising that the overall profile does not show the characteristic “exchange term” which is responsible for the extra intensity between the peaks. In ref 6 , detailed calculations follow, and it is shown that, by reasonable choice of the appropriate parameters, the splitting and bandwidths can mimic the experimental observations on the solid hydrocarbons. We now tum to Strauss’s arguments2 on the changing IR spectrum of 1. The coupling between the diene group and the CO groups reduces the symmetry to C,. Precisely the same argument is followed for the v(C0) stretches as for the hydrocarbon v(CH) stretches, to arrive at an equation exactly the same as (4),where this time G = pc po, g = 0, and F and f are the energy-factored CO force constants. Assuming that the reduced mass for intemal rotation is 272 amu A2, giving a rotational constant, B, of 0.062 cm-I, and a barrier height of 1.5 kcal mol-’ for a 6-fold barrier, gives a torsional frequency of -22.7 cm-’. With some further simplifications the splitting becomes
+
(We believe there is actually a misprint here, and it should read w, - o
b
= Amo cos n e = Amo{ 1 - (n2/2)(B/v,)(coth d 2 ) ) )
b = 101)
corresponding to the degree of vibrational excitation in the ra and rt, coordinates. Applying the Redfield relaxation mechanism to this vibrational problem then requires the calculation of the appropriate ‘ ‘ R a a * ~terms. ~’’ In the present context this means Raoao,RaObO, Rmao, and RWW. Raoa0 and &bo are then evaluated in terms of the spectral densities “J”, and it is stated,6 “Both RaOm and RwaO depend entirely on correlation functions of the type sin &(t) [cos nO(t+z) - cos ne] and sin n e ( t ) [cos 38(t+z) - cos 381 Since these are products of odd and even functions of 8, their averages will be zero (over the even potential of HT). Thus, RaOm = RWao = 0 and the two modes ra and rt, have no ‘crossrelaxation’ terms.”
It then states that “coth(d2) is a monotonically decreasing function of a and so the splitting should decrease with increasing temperature, as is observed.2 The paper goes on to consider the bandwidths; the details are not spelled out, but it is suggested that the method of ref 6 can be used. The implication is that if we follow that method, we shall arrive at the same conclusion, namely, that we shall observe two Lorentzians which broaden and move together. It is argued that the width of the individual components will be determined by the relaxation within the potential well and not by any interwell dynamics. However “if extreme rates (of interchange) do occur, an occurrence we think unlikely, they would contribute to the observed band collapse in addition to the one-well mechanism”.2 It is our contention that this analysis deliberately omits the exchange contribution to the calculations and therefore predicts what we argued in section I. That is, by making appropriate assumptions, it is possible to produce a pattern of two bands broadening and moving together and eventually coalescing. What is not possible is to mimic the crucial part of the spectral
17538 J. Phys. Chem., Vol. 99, No. 49, I995
Turner et al.
observations, namely, the exchange contribution between the peaks, without explicitly including such a contribution in the mathematics. Thus, Strauss’s model cannot satisfy the experimental observations, and there must be other contributions to the spectral pattern; the most obvious is the exchange process as we argued in ref 1.
C. Effect of Exchange in Vibrational Spectroscopy. Finally we should briefly consider the arguments for and against the use of Bloch equations in vibrational spectroscopy. At first glance the Bloch equations and their equivalent formulations ought to be applicable to the coalescence of vibrational bands, but of course on a much faster time scale. Two bands, in either NMR or vibrational spectroscopy, will be indistinguishable when z < 1/2n(dv); with a separation in NMR of 10 Hz, this occurs when z < 0.016 s, but with two IR bands separated by 10 cm-l, t
