16058
J. Phys. Chem. 1996, 100, 16058-16065
The Force Field of Bromoform: A Theoretical and Experimental Investigation Marı´a P. Ferna´ ndez-Liencres, Amparo Navarro, Juan J. Lo´ pez, and Manuel Ferna´ ndez Departamento de Quı´mica Fı´sica y Analı´tica, Facultad de Ciencias Experimentales, UniVersidad de Jae´ n, Paraje Las Lagunillas, E-23071 Jae´ n, Spain
Viktor Szalay Research Laboratory for Crystal Physics, Hungarian Academy of Sciences, P.O. Box 132, 1502 Budapest, Hungary
Teresa de los Arcos, Jose´ V. Garcı´a-Ramos, and Rafael M. Escribano* Instituto de Estructura de la Materia, CSIC, Serrano 121-123, E-28006 Madrid, Spain ReceiVed: March 12, 1996; In Final Form: June 21, 1996X
The potential function of bromoform has been determined in a joint effort, comprising ab initio and genetic algorithm calculations and a refinement to observed and anharmonicity corrected wavenumbers. To this end, the infrared and Raman spectra of HCBr3, DCBr3, and H13CBr3 have been recorded in the gas and liquid phases, those of the last species for the first time. In the analysis of the spectra, a number of combination and overtone bands have been assigned, besides the fundamental vibrations, leading to the determination of some anharmonicity constants. This is also the first experimental determination of such constants for the deuterated and 13C molecules. A Fermi interaction in the spectrum of DCBr3 affecting ν4 and ν3 + ν5 has been interpreted, and the corresponding cubic potential constant has been evaluated. The value of some of the force constants determined in the ab initio and genetic algorithm calculations has been of key importance to allow convergence in the refinement process. The experimental frequencies are well reproduced by the force fields.
1. Introduction It is interesting to combine recently available theoretical methods with more traditional procedures based on fittings to experimental data, in order to determine the force field of a molecule. The molecule of bromoform, HCBr3, is a good candidate for this purpose. This molecule presents special interest in several fields. Bromoform is widely applied in industrial chemistry as a reagent in organic synthesis, and also in therapeutic treatments, due to its sedative, hypnotic, and antitussive characteristics.1 From the academic point of view, it is an interesting species, being a simple five atom molecule of C3V symmetry with a C-H bond; thus, it has been used as a model for the C-H chromophore, of special importance in the development of the local mode theory,2 and it has also been the subject of a study of the curvilinear internal coordinate model for C-H stretching and bending vibrations.3 The vibrational spectrum of bromoform has been the subject of several studies. Meister et al.4 obtained the IR and Raman spectra of normal and deuterated bromoform in the liquid phase and calculated a modified valence force field for this molecule, including several constraints, based on their data. Shimanouchi also collected these data in his tables.5 Galasso et al.6 carried out a similar task, obtaining a new potential function for this molecule. The Raman spectrum has also been recorded at different conditions by Teixeira-Dias,7 showing the existence of important intermolecular interactions in the liquid at room temperature. The spectra and phase transitions of crystalline bromoform were studied by Burgos et al.,8 and the band intensities of the infrared spectra were also the subject of a work by Ratajczak et al.9 More recently, Mikulec and Cerny10 have registered the infrared * E-mail:
[email protected]. FAX: (341) 585-5184. X Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)00750-2 CCC: $12.00
spectrum of HCBr3 in an argon matrix at 21 K. The most comprehensive work on the force field of this molecule is that of Bu¨rger and Cichon,11 who recorded the spectra of the normal and deuterated isotopomers in both liquid (in the far-infrared region, up to ∼650 cm-1) and gas phases and, using also previously available data, refined the force field in the general valence approach, including several constraints which will be described later. To study Fermi resonances in the overtone spectra of the C-H chromophore in bromoform, a series of thorough and careful papers have been published,2,12,13 which report observation of the high-resolution spectrum, focused on the C-H stretching and bending vibrations and their overtones, up to the sixth polyad 61-64 (16 400 cm-1), the analysis of the experimental spectra, the calculation of effective Hamiltonian and potential parameters, and a discussion of the intramolecular vibrational relaxation of HCBr3. These works include observation of H13CBr3 bands in the spectrum of the natural abundance sample2 and a prediction of D and 13C isotope shifts12 in terms of effective Hamiltonian parameters of the local mode theory. These predictions are compared to our observations below. The spectrum in the near-infrared region has also been experimentally recorded,14 and an alternative interpretation of the Fermi resonances has been proposed.15 We undertook this study with the aim of enlarging the available knowledge on the vibrational spectra and force field of this molecule. This research presents both experimental and theoretical contributions: the experimental part consists of the gas-phase infrared spectra and liquid-phase infrared and Raman spectra of the H, D, and 13C isotopomers of bromoform; the theoretical contribution includes an ab initio calculation of the force field, the application of the so-called genetic algorithm to the determination of an initial force field, and a refinement © 1996 American Chemical Society
Force Field of Bromoform
Figure 1. Part of the infrared spectrum of liquid H13CBr3, recorded with a Perkin-Elmer 1760X Fourier transform spectrometer, at a resolution of 1 cm-1, in a liquid cell with KBr windows and 0.015 mm optical path. Saturation is allowed to permit a better observation of the weaker bands.
J. Phys. Chem., Vol. 100, No. 40, 1996 16059
Figure 2. Raman spectrum of liquid-phase bromoform up to 1200 cm-1 recorded on a Jobin-Yvon U-1000 spectrometer, at a resolution of ∼1 cm-1. The 514.5 nm line of an Ar+ laser is used as exciting source, working with a power of 300 mW at the sample.
TABLE 1: Symmetry Coordinates Used in This Work (Taken from ref 11) and Geometry of Bromoform16
of the force constants of this molecule to reproduce the vibrational data. 2. Experimental Section Samples of normal and deuterated (99% D) bromoform were obtained from Aldrich. The 13C 99% isotopically enriched sample has been purchased from Cambridge Isotopes Laboratory. The gas-phase infrared spectra have been recorded on a Bruker IFS66 Fourier Transform spectrometer at a resolution of ∼0.1 cm-1. We used a 10 cm gas cell with KBr windows. To transfer the gas into the cell, a liquid sample was placed in a vacuum line together with the cell. After evacuating the line, bromoform was allowed to expand and the cell was filled at the bromoform vapor pressure. The liquid-phase infrared spectra have been recorded on a Perkin-Elmer 1760X Fourier transform spectrometer, at a resolution of 1 cm-1, in a liquid cell with KBr windows and several optical path lengths. The Raman spectra were recorded on a Jobin-Yvon U-1000 spectrometer, using the 514.5 nm line of an Ar+ laser as exciting source, working with a power of 300 mW at the sample. The spectra were taken directly of the liquid samples. Figures 1 and 2 show standard portions of the IR and Raman spectra of liquid bromoform recorded in this work, reproduced here to give an indication of their quality and signal-to-noise ratio, whereas Figures 4 and 5 show details of the infrared spectrum of the gas, presented in connection to specific points which are discussed below. For the measurement of band centers of some very weak overtone or combination bands, smoothing techniques were applied in some cases to the interferograms. The uncertainty in the corresponding measurements is therefore larger than that of the stronger peaks. 3. Ab Initio Calculation The equilibrium geometry of bromoform has been taken from the joint analysis of microwave and electron diffraction data of Tamagawa and Kimura16 and is shown in Table 1. We have used the GAUSSIAN 92 package of programs17 for the ab initio calculation, working at the lanl2dz level, to obtain the force field in Cartesian coordinates. We have chosen the lanl2dz basis, since it is one of the few medium-sized commercial bases
which includes parameters for the bromine atom as well as relativistic corrections and nonstandard effective core potentials for atoms from Na to Bi in the periodic table of elements.18-20 The force constants in Cartesian coordinates were converted to force constants in the symmetry coordinate representation, which is the system of coordinates that we have used in the rest of the calculations. We have chosen the symmetry coordinates of Bu¨rger and Cichon,8 reproduced in Table 1 to avoid misunderstandings, since they do not coincide with those of Aldous and Mills,21 to which we shall refer later when discussing the Hybrid Orbital Force Field model. The corresponding internal coordinates are the usual ones for XY3Z molecules, where R and β refer to the BrCBr and HCBr angle deformations, respectively.
16060 J. Phys. Chem., Vol. 100, No. 40, 1996
Ferna´ndez-Liencres et al.
TABLE 2: Ab Initio and Genetic Algorithm (GA) Calculationsa F11 F12 F22 F13 F23 F33 F44 F45 F55 F46 F56 F66
ab initio
search space
GA force field
6.478 -0.048 3.157 -0.0076 0.325 1.069 2.474 -0.420 1.074 0.457 -0.067 0.735
4 e F11 e 6 -0.05b 2 e F22 e 4 -0.05 e F13 e 0.5 -0.05 e F23 e0.5 1 e F33 e 3 2 e F44 e 4 -0.5 e F45 e 0.5 1 e F55 e 3 F46 ) -F45 -0.067b 0 e F66 e 1
5.5758 -0.05 2.8402 -0.1378 0.4392 1.3549 2.4987 -0.4047 1.0266 0.4047 -0.067 0.6211
a First column: ab initio force field; second column: definition of search space for genetic algorithm calculation; third column: the best force field estimated in this GA calculation; swse ) 1.0; fobj ) 0.815. All force constants in mdyn/Å. b Values kept fixed in GA calculations.
The ab initio force field is reproduced in Table 2, and a discussion of the results is presented below. 4. Application of the Genetic Algorithm Technique The determination of the force field of a molecule by fitting to experimental data amounts to solving an optimization problem of many variables. Similar to most force field refinement programs, our computer code22 is based on a local optimization method, the method of nonlinear least squares. In this method the choice of the initial parameters is crucial to obtain convergence. Even when convergence is achieved, one must be aware that there may exist other sets of force fields, corresponding to several minima of the function to be optimized, which reproduce the experimental data satisfactorily. This problem may be minimized if a judicious choice of the initial values of the force constants is made. However, finding an appropriate initial force field can be very difficult for large molecules or for molecules containing heavy atoms, since the quantum chemically (e.g., by ab initio calculations) predicted force field of such molecules may have a poor accuracy. To avoid these difficulties, we have carried out in this work, besides the ab initio calculation already mentioned, genetic algorithm (GA) calculations, which are expected to provide an initial estimate of the force field that may lead to the best possible fit. The genetic algorithm is a search and optimization technique that borrows its ideas of operation from the evolution of living systems.23 It is one of the methods capable, at least in principle, of locating the global minimum (or maximum) on a multidimensional surface. Recently, there have been a number of successful applications of genetic algorithms to diverse problems in chemical physics, e.g., to the global geometry optimization of clusters,24 to parametrize NDDO wave functions,25 to optimize control of quantum systems,26 and to the calculation of bound states and to local density approximations.27 The present work is the first one, as far as we know, to employ the genetic algorithm to determine the force field of a molecule. The GA program that we have used28 is, in essence, a C code realization of the simple genetic algorithm by Goldberg.29 The fitness (objective) function has been defined as
fobj ) exp(-swse/5)
(1)
where the sum of weighted squares of errors, swse, defined as
swse ) ∑[(νobsd - νcalcd)iwi]2
(2)
Figure 3. Evolution curves in the genetic algorithm runs. Curve a (solid line): The evolution of the fitness value of the fittest member of the population. Curve b (dashed line): The evolution of the population’s average fitness.
measures the quality of a force field in terms of the deviation between the corresponding calculated spectrum and the experimental one. We have restricted the search space as indicated in the second column of Table 2. The value of each force constant has been decoded as a 32-bit bitstring. Thus the genetic operators, selection, crossover, and mutation act on bitstrings of length 12 × 32 ) 384, each one of them representing a complete force field of bromoform. Curve a in Figure 3 depicts the evolution of the fitness values of the fittest member of a population of 20 members throughout 300 evolution steps, and curve b shows how the average fitness of the same population evolves. In the GA run that produced these curves, the probability of crossover and mutation was set to 0.9 and to 0.01, respectively. The space explored by the genetic algorithm includes force fields leading to an swse value of order of magnitude ∼104 and may include force fields leading to an swse value close to zero. Despite this large span of values, the genetic algorithm has found force fields giving very small swse. The force field of the smallest swse value, found by the GA run corresponding to the evolution curves shown in Figure 3, is given in the third column of Table 2. Since this technique is very useful in finding a global minimum (maximum), but is inefficient in pinpointing the value of the minimum (maximum) with great accuracy, the GA force field has been used later as an initial force field for our nonlinear least squares refinements. The corresponding refined force constants are shown in Table 9, and the results are discussed in section 5.2. 5. Results and Discussion 5.1. Spectra and Assignments. Tables 3-5 list the more important observed features in the IR and Raman spectra of the H, D, and 13C isotopomers of bromoform, together with their proposed assignments. Besides the fundamentals (3A1 + 3E, all IR and Raman active), we have measured a number of overtone and combination bands. Their interpretation enables the determination of some of the anharmonicity constants or combinations of them; the corresponding results are collected in Table 7. A number of other very weak bands were also observed, but when they could not be confirmed by comparison among the isotopic species, or when their assignment was dubious, they were left out of the tables. It is interesting to make a few remarks on the observations of the spectra of these molecules. The ν1 band, corresponding
Force Field of Bromoform
J. Phys. Chem., Vol. 100, No. 40, 1996 16061
TABLE 3: Observed Vibrational Spectrum of H12CBr3 (in cm-1) observed wavenumber IR gas ν
IR liquid I
669.7 694.3 745.4
vs w vw
1148.2 1827.2 2269.5 3048.2
s m w m
4180.8 5964.1
w w
Raman liquid
ν
ν
I
assignment
155.0 223.0 373.5 443.5 537.5 654.5 692.5
s vs vw vw s m vw
1074.0 1140.0
vw vw
3018.0
vw
ν6 ν3 ν3 + ν6 2ν3 ν2 ν5 ν2 + ν6 ν2 + ν3 ν3 + ν5 2ν2 ν4 ν4 + ν5 (E) 2ν4 (A1) ν1 ν1 + ν2 ν1 + ν4 2ν1
I
540.0 655.0 694.0 744.5 874.0
m vs s w vw
1142.0 1804.0 2256.0 3020.0 3558.0
vs vw w vs vw
TABLE 4: Observed Vibrational Spectrum of D12CBr3 (in cm-1) observed wavenumber IR gas ν
IR liquid I
Raman liquid
ν
ν
I
assignment
155.0 225.0 521.0 633.0 675.5 850.0
s vs m w vw vw
ν6 ν3 ν2 ν5 ν2 + ν6 ν4 ν3 + ν5 2ν2 ν3 + ν4 + ν5 2ν3 + 2ν5 ν1 ν1 + ν4 2ν1
I
646.9 674.9 852.4 868.0
vs m m m
522.0 635.0 674.0 844.0 859.0 1046.0
m vs m vw vs vw
1715 1750 2271.9 3118 4477.0
vw vw m w w
2251.0
s
2250.0
vw
TABLE 5: Observed Vibrational Spectrum of H13CBr3 (in cm-1) observed wavenumber IR gas ν
IR liquid I
648.3 674.0 724.0
vs m w
1146.5 1786.6 2265.0 3039.0 4168.4 5958.5
s vw vw m vw vw
ν
Raman liquid I
523.0 634.0 676.0 721.0
m vs m sh
1139.0 1765.0 2252.0 3011.0
s vw w s
ν
I
155.5 223.5 522.5 632.0
s vs s m
1049.0 1141.5
vw w
3009.5
m
assignment ν6 ν3 ν2 ν5 ν2 + ν6 ν2 + ν3 2ν2 ν4 ν4 + ν5 2ν4 ν1 ν1 + ν4 2ν1
to the C-H stretch, is seen in the spectra of the liquid at lower wavenumber than in the spectra of the gas: 28 and 20 cm-1 lower in the IR spectra of H and D species, respectively. This shift is due to strong intermolecular interactions in the liquid phase, possibly of hydrogen bonding type, which are found to vary with temperature.7 Other bands show much smaller displacements from the liquid to the gas; in particular, the ν4 band, associated to the HCBr bending, presents a displacement of 8 and 6 cm-1 for the H and D species, respectively. A strong Fermi interaction appears in the spectrum of DCBr3,
Figure 4. Fermi resonances in the mid-infrared spectrum of gas-phase DCBr3, recorded with a Bruker IFS66 Fourier transform spectrometer at a resolution of ∼0.1 cm-1, in a 10 cm gas cell with KBr windows. (top) In the region around 860 cm-1, the ν4 band (right) shares a good part of its intensity with the ν3 + ν5 band. (bottom) In the region around 1730 cm-1, two features are observed, which are tentatively assigned as ν3 + ν4 + ν5 (right) and 2ν3 + 2ν5. In this case, some remaining water lines have been eliminated by computer subtraction techniques, and the spectrum has been smoothed using a Savitzky-Golay algorithm.
illustrated in Figure 4. The V4 ) 1 and V3 ) V5 ) 1 levels in DCBr3 are close and interact through a vibrational resonance of Fermi type between two states of E symmetry species. Thus, the ν3 + ν5 band borrows intensity from the ν4 fundamental, and is observed as a fairly strong feature in this region of the spectrum (upper curve of Figure 4). From the separation of the two peaks and their relative intensity, it is possible to estimate the unperturbed band origin of each band and the cubic force constant responsible for the perturbation,30 using a simple 2 × 2 effective Hamiltonian matrix as follows. We use as basis functions products of the usual harmonic oscillator functions, represented by the corresponding vibrational quantum numbers |V3,V4(l4),V5(l5)〉, where lt is the vibrational angular momentum quantum number. The diagonal elements are then the unperturbed band origins and the matrix element of the interaction is31,32
〈0,1((1),0|H3,0|1,0,1((1)〉 ) 2-3/2K′3,4,5
(3)
where the operator for this specific perturbation is
H3,0 ) 1/6∑K′k,l,mqkqlqm ) 1/2K′3,4,5q3(q4+q5- + q4-q5+) (4) and K′3,4,5 is the cubic force constant (in cm-1):
16062 J. Phys. Chem., Vol. 100, No. 40, 1996
Ferna´ndez-Liencres et al.
K′3,4,5 ) (1/hc)(∂3V/(∂q3 ∂q4 ∂q5))
(5)
The eigenvalues of this Fermi matrix can be matched to the observed wavenumbers, and the eigenvector elements give the ratio in which the basis functions are mixed and can therefore be related to the ratio of the intensities of the two bands. We have to assume that the intensity seen for ν3 + ν5 arises only from the interaction. We have calculated in this way the unperturbed band origins, which turn out to be ν40 ) 854.6 cm-1 and ν3,50 ) 866.5 cm-1, and the force constant, K′3,4,5 ) 22.29 cm-1. The first of these values is therefore used later as data for the force field refinement, instead of the mean of the observed peaks as was used in previous works. In the spectral region of the second excitation of these bands, it is possible to observe some weak band structure, amid water lines and a high level of noise. This region is shown on the lower half of Figure 4. The same cubic potential element of the Hamiltonian that links the V4 ) 1 and V3 ) V5 ) 1 states, links the V4 ) 2, V3 ) V4 ) V5 ) 1 and V3 ) V5 ) 2 states, two by two, with the following matrix elements:
A1 symmetry: -3/2
〈0,2(0),0|H3,0|1,1((1),1(-1)〉 ) 2
K′3,4,5 (6a)
〈1,1((1),1(-1)|H3,0|2,0,2(0)〉 ) 1/2K′3,4,5
(6b)
E symmetry: 〈0,2((2),0|H3,0|1,1((1),1((1)〉 ) 1/2K′3,4,5 (7a) 〈1,1((1),1((1)|H3,0|2,0,2((2)〉 ) 2-1/2K′3,4,5
(7b)
This means that the 2ν4 transition would share its intensity with ν3 + ν4 + ν5, which in turn would share it with 2ν3 + 2ν5. As indicated in these equations, each of the three excited levels can be of symmetry A1 (total lt ) 0) or E (total lt ) (2), and therefore, it might be possible to observe transitions to all six levels. In practice, it is difficult to assign the observed features of Figure 4 to these theoretically predicted transitions. The fact that the peak at higher wavenumber is stronger than the one at lower wavenumber is also puzzling, since the reverse is found in the first diad at 850 cm-1. Table 6 presents a scheme of the predicted unperturbed energy levels and the relevant anharmonicity terms involved. If the peak observed at 1715 cm-1 were to be assigned to 2ν4, a simple calculation shows that, after removing the effect of the Fermi coupling, we would need x44 + g44 to be positive and larger than 5 cm-1. Since the x44 constant, related to the C-D bending, is expected to be negative (see Table 7 for H12CBr3 and H13CBr3 values), the more likely conclusion is that the 2ν4 band is too weak to be observed, and that the features at ∼1715 and ∼1750 cm-1 should be assigned to ν3 + ν4 + ν5 and 2ν3 + 2ν5, respectively. Whereas the observations for the 12C molecules basically coincide with previous results,4,5,11 this is the first work, within our knowledge, where the complete vibrational spectrum of H13CBr3 has been reported. The general appearance of the spectrum is the same as that of the 12C species, with the expected wavenumber shifts for the vibrations where the C atom is relevant. Davidsson et al.2 assigned some peaks observed in the spectrum of natural abundance bromoform to the 13C species and listed also (footnote to Table 1 of ref 2) their predictions to these vibrations. Our spectrum confirms their assignments, except for 2ν4, which we observe at 2265 cm-1 (2258.8 cm-1 in ref 2), and 2ν1, where we measure the stronger peak at 5958.5 cm-1 (tentatively assigned at 5936.5 cm-1 and predicted at 5953 cm-1 in ref 2). This last portion of the spectrum of vaporphase H13CBr3 is shown in Figure 5.
TABLE 6: Observed and Predicted Unperturbed Energy of the Levels Involved in the Fermi Resonance of DCBr3 Depicted in Figure 4 and Relevant Anharmonicity Constants for the Excited States with Respect to Those of the First Diad energy, cm-1 obsd calcd assignment 1750 1733 2ν3 + 2ν5
terms 2(ν3+50) + 2x33 + 2x35 + 2x55
1715 1721 ν3 + ν4 + ν5 (ν40) + (ν3+50) + 5/2x34 + 3x45 2(ν40) + 2x4
1709 2ν4 868 852
-2g55 (A1) +2g55 (E) +g45 (A1) -g45 (E) -2g44 (A1) +2g44 (E)
0
866 ν3+5 855 ν40
TABLE 7: Anharmonicity Constants of H12CBr3, D12CBr3, and H13CBr3 (in cm-1) Evaluated from the Observed Spectra and the Assignments of Tables 3-5 H12CBr3 x11 x12 x14 x22 x23 x26 x33 x35 x36 x44 - g44 x45 + g45
-66.2a -2 -15.6 -0.5 -18.5 0 -1.3 -4 -4.5 -13.5 9.3
D12CBr3
H13CBr3
-63.9b
-33.4
-35.3c
-59.8
-65.2c
-21.8b
-8.5 1
-9.8c
-17.1 2 -25.5 -2.5
-18.5c
-2.1c
-14.0 -8.2
-3.8c
-0.6 1 -13.7b
a Boldface values indicate that the constant has been evaluated from gas-phase spectra. b Table 3, column a of Davidsson et al.;2 x11 ) -64.577 from Halonen and Kauppi.15 c Predicted values from Table 4 of Hollenstein et al.16
Figure 5. 2ν1 band origin in the gas-phase infrared spectrum of H13CBr3. A Savitzky-Golay smoothing has been performed.
Because of the shifts found between the spectra of liquid and gas samples, we have evaluated each anharmonicity constant from the measurements of either type, but never mixing values of one spectra with those of the other. This is indicated with different symbols attached to the values in Table 7. The accuracy of the estimated constants depends on that of the measurements of the bands used in their calculation. Some overtone or combination bands are observed as very weak and broad features, the band origin being estimated using smoothing techniques to reduce the noise level. Although the precision of these measurements is therefore poor and difficult to estimate with accuracy (it may be as low as 5 cm-1), we have nonetheless included the corresponding anharmonicity constants in the table for completeness. Overtones and combination bands of degenerate vibrations give rise to several components of different
Force Field of Bromoform symmetry33 (e.g., A1 + E for a first overtone), and only in a few cases has it been possible to identify them separately. This prevents the independent determination of the corresponding xss and gss anharmonicity parameters, and therefore, the relevant combination of these parameters, rather than the individual parameters, are listed in Table 7. The more complete previous estimation of anharmonicity constants of bromoform that we are aware of comes from the Fermi resonance works mentioned above2,12,13 and deals exclusively with the C-H stretching and bending modes, ν1 and ν4, respectively. These authors determined experimentally the constants of H12CBr3 with good accuracy, since they observed highly excited overtones of these vibrations, and predicted those of the D and 13C species. Their values are also shown in Table 7. It can be seen that the agreement with the results of this work is good in general. Discrepancies may be due to the fact that the constants of the former works have been obtained through an extensive treatment of a Fermi resonance affecting the C-H stretching and bending modes over several polyads of excited levels; this resonance has not been considered here, since its effects on the first excitations of these levels are small. 5.2. Force Field Calculations. The most recent study on the force field of bromoform is that of Bu¨rger and Cichon,11 who used as experimental data the wavenumbers of the fundamental vibrations of the normal and deuterated isotopomers. This amounts to only five independent data (when the product rule is subtracted from the six experimental frequencies) for each symmetry species, whereas the number of quadratic force constants is six. Thus, Bu¨rger and Cichon had to use some constraints among the force constants, and they chose the following: F12 and F13 were transferred from CH4;34 F46 ) -F45 from the Hybrid Orbital Force Field model;35 and F23 and F56 were evaluated through parametric estimation of the potential energy distribution.36 In the present work we have the 13C spectrum as additional source of data, but as discussed by Duncan et al.,30 these data do not usually provide sufficient information to allow all force constants to be refined simultaneously. We decided to adopt the Hybrid Orbital Force Field (HOFF) model proposed by Aldous and Mills,20 according to which F45 ) - F46 = -x2F23, F12 and F45 should be negative, and F23 should be positive, in terms of the symmetry coordinates used in this work. These relations are based on physical considerations, from the idea that the stretch-bend force constants should conform to the change in s-p hybridization on the carbon atom due to orbital following on the bending coordinate. For bromoform, we found that a fit in which the three parameters F12, F23, and F45 were allowed to refine freely led to large correlations among the refined parameters and to very high estimated uncertainty for some of the interaction force constants, indicating that the amount of independent information was still not large enough to fix all the individual parameters of the model. From the purely theoretical point of view, we carried out an ab initio calculation along the lines indicated above. The result of this calculation was in excellent agreement with the predictions of the HOFF model, as can be seen in the values of Table 2. We therefore fixed the F12, F23, and F56 force constants to the results of the ab initio calculation and refined the rest of the constants keeping the F45 ) -F46 restriction of the HOFF model. The force field refinements of the present work have been carried out using the program AJEF,22 which has the particularity that the inverse generalized method37 is employed to solve the least-squares system, allowing a better treatment than other usual methods in case of poor determinacy of some of the refining parameters. We have performed several fits, for different
J. Phys. Chem., Vol. 100, No. 40, 1996 16063 choices of data and weights. Thus, for the data, we have tried either observed or anharmonicity corrected wavenumbers; for the 13C isotopomer, either 13C/12C isotopic shift data or band origin data; and for the data uncertainties, either experimental uncertainty or around 1% of the observed wavenumber, as is sometimes chosen. The weights wi for the weighted leastsquares fits were always taken as the reciprocal of the squared uncertainty of each datum. We have also tested for every fit the result of releasing one or two more parameters. We present below the results of two fits which, in our opinion, better represent the available set of data and choices of types of data and uncertainties. Fit I corresponds to a case in which the observed wavenumbers νs were used for both 12C compounds, and 13C/12C isotopic shift data for H13CBr3, taking the estimated experimental uncertainty (around 0.1 cm-1 for the gas-phase spectra and 0.5 for the liquid-phase spectra) for all data, keeping F12, F23, and F56 fixed to the ab initio results, and making F46 ) -F45. Fit II represents a refinement in which anharmonicity corrected wavenumbers ωs have been used as data for all three molecules, and the GA force constants were used as initial force field for the fitting process, with the same restrictions on the parameters as for fit I. The anharmonicity corrections have been performed using a general expression for fundamental vibrations, derived from Herzberg’s treatment:38
ωs ) νs - xss(1 + ds) - 1/2∑xskdk - gssls2
(8)
k*s
where the xsk and the gss are the usual anharmonicity constants, dk is the degeneracy of the kth normal mode, and ls represents the vibrational angular momentum quantum number (ls ) 0 for A1 modes, ls ) 1 or -1 for degenerate fundamental modes). All available anharmonicity constants from Table 7 were used in the evaluation of the ω’s. The uncertainty on the “harmonic” wavenumbers was assumed to arise mainly from the lack of knowledge on the anharmonicity constants which could not be evaluated from the experimental data; these unknown terms were estimated to be not larger than 1% of the corresponding wavenumber in any case. The results of these fits are summarized in Tables 8 and 9. Table 8 lists the observed and anharmonicity corrected wavenumbers for all three species and the wavenumber shifts for 13C bromoform. We also include the uncertainty of each data in the refinement procedure and the observed calculated deviations resulting from the corresponding fits. Table 9 shows the best estimates of the force constants of fits I and II, with their estimated uncertainty, together with the previous results of Bu¨rger and Cichon.11 The reproduction of the observed data is in general quite satisfactory, better than 4 cm-1 for the wavenumber and shift data in the fit to observed wavenumbers. Some harmonic wavenumbers are reproduced with a lower precision, reflecting the fact already anticipated that the anharmonicity corrections have been made using an incomplete set of data. It is interesting to observe that the Fermi corrected value of the ν4 band of DCBr3 employed in this work is quite well reproduced by the refined force field of fit I and leads also to a better reproduction of the ν4 band of HCBr3 than in previous works. The role played by the 13C data turns out to be mainly that of reducing the uncertainty in the refined force constants. We found that these data, when weighted similarly to the other data, did not bring significant changes in the value of the final parameters as compared to a fit in which the 13C data were not included. The ab initio force constants yielded a poor fit to the anharmonicity corrected wavenumbers, whereas the GA force field led to a nice fit, as shown on Tables 8 and 9.
16064 J. Phys. Chem., Vol. 100, No. 40, 1996
Ferna´ndez-Liencres et al.
TABLE 8: Wavenumber Data Used in the Refinement of the Force Constants of Bromoform (in cm-1)a Fit I H12CBr3 ν1 ν2 ν3 ν4 ν5 ν6
D12CBr3
H13CBr3
ν
σ
∆
ν
σ
∆
ν
δ
σ
∆
3048.2 540.0 223.0 1148.2 669.7 155.0
0.1 0.5 0.5 0.1 0.1 0.5
-0.3 -3.4 -1.1 -1.2 -1.9 0.4
2271.9 521.0 225.0 854.6 646.9 155.0
0.1 0.5 0.5 0.1 0.1 0.5
0.4 3.6 1.1 1.6 2.1 0.7
3039.0 522.5 223.0 1146.5 648.3 155.0
9.2 17.5 0.0 1.7 21.4 0.0
0.1 0.5 0.5 0.1 0.1 0.5
-3.4 -0.4 -0.1 -0.9 -1.5 -0.1
Fit II H12CBr3 ω1 ω2 ω3 ω4 ω5 ω6
D12CBr3
H13CBr3
ω
σ
∆
ω
σ
∆
ω
σ
∆
3197.2 551.3 234.0 1166.0 671.7 157.0
15 5 2 15 7 2
7.3 0.1 -0.5 -1.6 -0.1 0.5
2347.0 530.0 234.0 865.0 647.4 155.3
10 5 2 10 7 2
-1.4 -1.2 -0.3 0.1 0.6 -0.9
3175.0 533.8 235.0 1165.0 648.3 156.8
15 5 2 15 7 2
-4.9 1.1 0.7 0.0 -0.5 0.4
a ν: observed wavenumbers; ω: anharmonicity corrected wavenumbers; δ: 13C/12C isotopic shifts; σ: estimated uncertainty for the fitting process; ∆: observed-calculated differences using the force constants of fits I and II.
TABLE 9: Force Constants of Bromoform in Symmetry Coordinate Representation (in mdyn/Å)a F11 F12 F22 F13 F23 F33 F44 F45 F55 F46 F56 F66
fit I
fit II
ref 11
4.947(14) -0.05 2.599(240) 0.628(48) 0.325 1.475(73) 2.392(46) -0.374(8) 0.982(90) 0.374(8) -0.067 0.600(2)
5.566(26) -0.05 2.866(22) -0.052(99) 0.439 1.373(12) 2.416(28) -0.382(14) 1.041(8) 0.382(14) -0.067 0.589(4)
5.080 0.175 3.773 0.310 0.300 0.940 2.232 -0.260 1.044 0.260 0.100 0.579
a Estimated uncertainties from fits I and II are given in units of the last digit. Values quoted without uncertainties have been held fixed to their ab initio or genetic algorithm value.
Some of the refined force constants of Table 9 still present large uncertainties. This is a consequence of the uniformity of the data used in the refinement and can only be positively overcome through the introduction of data of a different nature, like Coriolis constants or centrifugal distortion constants. However, such data, which could be obtained from the analysis of high-resolution spectra, do not seem to be available for these molecules for the time being. In fact, the rotational structure of the spectra of bromoform could probably be resolved only if monoisotopic Br samples were used, since when natural abundance Br isotopes are present, four molecular species exist, with closely overlapping spectra. Spectra of monoisotopic bromoform species have not yet been recorded as far as we know. We have seen in Table 2 the force constants estimated through ab initio and genetic algorithm calculations. The ab initio force field gives, besides the typically large values for the diagonal force constants, very good estimations for the off-diagonal parameters. It is even surprising how well these constants follow the predictions of the HOFF model, including the x2 approximate ratio between F46 and F23 (the actual calculated ratio is 1.406). Since these off-diagonal force constants are difficult to determine from wavenumber data, we qualify the ab initio values as very valuable in this case. The GA calculation is not a purely theoretical method to determine the
force field, since it is based in a refinement to experimental data. The GA force field of Table 2 was obtained by fitting the force constants to the anharmonicity corrected wavenumbers. It is therefore not surprising that this force field is quite close to the refined value listed as fit II in Table 9. Since we found some difficulties in estimating the F12 and F56 force constants in the GA calculation, we decided to freeze them to their ab initio values, maintaining also the F45 ) -F46 restriction of the HOFF model. This final force field was used as the initial value for the standard least-squares refinement, with a reasonable success. Finally, it is interesting to note that the description of the normal modes in terms of symmetry coordinates is quite dependent on the actual force field values, even in such a simple case as this one. For instance, whereas the force field of Bu¨rger and Cichon allows a univocal direct correlation of the normal modes to the symmetry coordinates, the description of the ν2 and ν3 modes, provided by the L matrices calculated in this work, presents a heavy mixing of the C-Br symmetric stretching and bending coordinates in the A1 species. 6. Conclusions We have tried to optimize the general harmonic force field of bromoform by means of a joint investigation consisting in a theoretical and an empirical task. In the experimental part, we have obtained and analyzed the IR and Raman spectra of HCBr3, DCBr3, and H13CBr3 at moderate resolution. The analysis includes the interpretation of a Fermi resonance in the spectrum of DCBr3 and the calculation of some anharmonicity constants. In the theoretical part, we have performed ab initio and genetic algorithm calculations, and we have refined the force constants to reproduce the observed data. The inclusion of some of the theoretical force constants has enabled us to obtain a force field for bromoform in the Hybrid Orbital approximation, which reproduces the vibrational data with good accuracy. The inclusion of 13C data is essential to reduce the uncertainty in the refined force field parameters. A study of the highresolution spectra of monoisotopic Br samples of these molecules could provide valuable data for determining the complete quadratic force field of bromoform. We have found that genetic algorithms can be successfully employed in determining the force field of a molecule. In
Force Field of Bromoform particular, with the help of genetic algorithms, one can solve the often difficult question of finding an initial force field suitable for refinement. Works on determining the force field of more complicated, larger molecules using genetic algorithms are in progress.39 Acknowledgment. R.E. acknowledges DGICYT Project No. PB93-0138. V.S.’s research has been partially supported by Grant OTKA T007294; he is also grateful to Jane Sinor and Ferenc Toth for their support. T.A., R.E., and V.S. gratefully acknowledge Joint Project No. 12 between the Hungarian Academy of Sciences and the Consejo Superior de Investigaciones Cientificas (Spain). We are grateful to M. A. Moreno for technical help and to J. Ortigoso for carefully reading the manuscript. References and Notes (1) The Merck Index, 11th ed.; Merck & Co., Inc.: Rahway, NJ, 1989. (2) Davidsson, J.; Gutow, J. H.; Zare, R. N.; Hollenstein, H. A.; Marquardt, R. R.; Quack, M. J. Phys. Chem. 1991, 95, 1201 and references therein. (3) Kauppi, E. J. Mol. Spectrosc. 1994, 167, 314. (4) Meister, A. G.; Rosser, S. E.; Cleveland, F. F. J. Chem. Phys. 1950, 18, 346. (5) Shimanouchi, T. Tables of Molecular Vibrational Frequencies, NSRDS-NBS 39; 1972; Vol. 1. (6) Galasso, V.; De Alti, G.; Costa, G. Spectrochim. Acta 1965, 21, 669. (7) Teixeira-Dias, J. J. C. Spectrochim. Acta 1979, 35A, 857. (8) Burgos, E.; Halac, E.; Bonadeo, H. J. Chem. Phys. 1981, 74, 1546. (9) Ratajczak, H.; Ford, T. A.; Orville-Thomas, W. J. J. Mol. Struct. 1972, 14, 281. (10) Mikulec, J.; Cerny, C. Spectrochim. Acta 1987, 43A, 849. (11) Bu¨rger, H.; Cichon, J. Spectrochim. Acta 1971, 27A, 2191. (12) Hollenstein, H.; Lewerenz, M.; Quack, M. Chem. Phys. Lett. 1990, 165, 175. (13) Ross, A. J.; Hollenstein, H. A.; Marquardt, R. R.; Quack, M. Chem. Phys. Lett. 1989, 156, 455.
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