The low frequency density of states

The low frequency density of states and vibrational population dynamics of polyatomic molecules in liquids
Preston Moore
Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104
A.
Tokmakoffa)
Department of Chemistry, Stanford University, Stanford, California 94305
T.
Keyes
Department of Chemistry, Boston University, Boston, Massachusetts 02215
M.
D. Fayer
Department of Chemistry, Stanford University, Stanford, California 94305
~Received 10 March 1995; accepted 24 May 1995!
Instantaneous normal mode calculations of the low frequency solvent modes of carbon tetrachloride ~CCl4! and chloroform ~CHCl3!, and experiments on the vibrational population dynamics of the T1u CO stretching mode ~;1980 cm21! of tungsten hexacarbonyl in CCl4 and CHCl3 are used to understand factors affecting the temperature dependence of the vibrational lifetime. Picosecond infrared pump–probe experiments measuring the vibrational lifetime of the T1u mode from the melting points to the boiling points of the two solvents show a dramatic solvent dependence. In CCl4, the vibrational lifetime decreases as the temperature is increased; however, in CHCl3, the vibrational lifetime actually becomes longer as the temperature is increased. The change in thermal occupation numbers of the modes in the solute/solvent systems cannot account for this difference. Changes in the density of states of the instantaneous normal modes and changes in the magnitude of the anharmonic coupling matrix elements are considered. The calculated differences in the temperature dependences of the densities of states appear too small to account for the observed difference in trends of the temperature dependent lifetimes. This suggests that the temperature dependence of the liquid density causes significant changes in the magnitude of the anharmonic coupling matrix elements responsible for vibrational relaxation. © 1995 American Institute of Physics.
I.
INTRODUCTION
The population dynamics of the vibrations of a polyatomic solute molecule in a polyatomic liquid solvent can involve the internal vibrational modes of the solute, the vibrational modes of the solvent, and the low frequency continuum of solvent modes.1,2 An initially excited high frequency vibrational mode of a solute molecule can relax by transferring vibrational energy to a combination of lower frequency internal vibrations and solvent vibrations. However, in general, a combination of lower frequency vibrations will not match the initial vibrational frequency. Therefore, one or more quanta of the continuum will also be excited ~or annihilated! to make up for the mismatch in the vibrational frequencies and conserve energy.
The low frequency solvent continuum can be described in terms of instantaneous normal modes ~INMs!.3–7 For vibrational relaxation in liquids, the INMs play the same role as do phonons in crystals. In a crystal, the phonons provide a continuum of modes that can be created or annihilated to conserve energy in relaxation processes between high frequency vibrations. Because of this similarity, for simplicity we will refer to the INMs as phonons of the liquid or just phonons. It is important to recognize the important differ
a!Current address: TechnischeUniversita¨tMu¨nchen, Physik Department E11, James-Franck-Str., 85748 Garching, Germany.
ence between INMs and true crystalline phonons. In a crystal, all phonon modes are bound, giving real phonon frequencies. In a liquid, the structure is continually evolving. Not all of the modes are bound, so that INMs have both real and imaginary frequencies. The imaginary frequency modes are related to the structural evolution of the liquid.8
Vibrational relaxation involves a cubic or higher order anharmonic process. The ‘‘order’’ of the process refers to the number of quanta involved in the relaxation. In the simplest cubic anharmonic process, the initial excited vibration is annihilated, a lower frequency internal mode or solvent mode is excited, and a phonon is excited to conserve energy. For a high frequency mode to relax by a cubic process, there must be another high frequency mode close enough in energy for the energy mismatch to fall within the phonon bandwidth. In a quartic or higher order process, the initial vibration is annihilated, two or more lower frequency vibrations are created, and one or more phonons are created to conserve energy. Unless there is a coincidence or Fermi resonance in which energy can be conserved by the creation and annihilation of discreet vibrational modes alone, at least one mode of the low frequency continuum of states will be involved in vibrational population dynamics. The rate constants for vibrational population relaxation dynamics can be described in terms of Fermi’s golden rule. Therefore, it is clear that the density of states of the low frequency continuum of INMs will be important in vibrational dynamics.
J.
Chem. Phys. 103 (9), 1 September 1995 0021-9606/95/103(9)/3325/10/$6.00 © 1995 American Institute of Physics 3325
Moore etal.: Vibrational population dynamics in liquids
In this paper, we consider the relationship between the density of states of the INMs of two liquids on the vibrational relaxation of a high frequency mode as a function of temperature. The vibrational relaxation dynamics of the T1u CO stretching mode of tungsten hexacarbonyl, W~CO!6,is studied in dilute solution in the liquids carbon tetrachloride, CCl4, and chloroform, CHCl3. Picosecond infrared ~ir! pump–probe experiments were used to measure the temperature dependence of the vibrational lifetime of the T1u mode.1 The temperature is varied from the melting points to the boiling points of the two solvents.
The ir pump–probe experiments show that the lifetime of the T1u mode in CCl4 becomes shorter as the temperature is increased from the melting point to the boiling point. One might be tempted to fit an activation energy to such a temperature dependence. However, when the lifetime is measured in CHCl3 from the melting point to the boiling point, it is observed that the lifetime actually becomes longer as the temperature is increased. This inverted temperature dependence is counterintuitive, yet it has also been observed in other solvents.9,10 This temperature dependence clearly demonstrates that a simple description in terms of an activation energy will not account for the change in behavior with solvent. As discussed below, a fully quantum mechanical treatment of vibrational relaxation of polyatomic molecules in polyatomic liquids2 cannot account for the inverted temperature dependence. If the relaxation pathway involves a number of high frequency vibrations, whether internal or from the solvent, and a phonon, then the temperature dependence will be determined by the thermal occupation numbers of the modes involved. Since these occupation numbers increase with temperature, the lifetime should get shorter with increasing temperature. The quantum mechanical treatment2 and other quantum mechanical11–14 and classical treatments15–17 of vibrational relaxation do not account for two aspects of the problem that are brought to the fore by the inverted temperature dependence: the temperature dependence of the phonon density of states and the temperature dependence of the magnitude of the anharmonic coupling matrix elements ~liquid density dependence!.
To gain insights into the temperature dependences of the vibrational lifetimes in the two solvents, the temperaturedependent low frequency INM spectra, ^r~v!&,ofCCl4 and CHCl3 were calculated. The calculations employed a detailed potential that included intermolecular and intramolecular components. While the low frequency intermolecular modes are of interest here, the potential is able to do a reasonable job of reproducing the vibrational spectrum of the liquids as well. The use of the full potential proved important. Calculations with an accurate Lennard-Jones ~LJ! potential do not generate the high frequency ‘‘rotational’’ part of the INM spectrum even though the LJ potential yields the correct melting point of the crystal. The real and imaginary components of the INM spectrum were calculated at three temperatures, near the melting point, at room temperature, and near the boiling point for both solvents. As the temperature is increased, there is a small decrease in the density of states across most of the real part of the spectra with a corresponding increase in the imaginary part.
mc-ref
FIG.1.Pump–probedataofthevibrationalrelaxationoftheT1uCOstretchingmodeofW~CO!6in~a! CCl4and~b! CHCl3atT5295K.Singleexponentialfitsconvolvedwiththe40pspulsesareshown.Thevibrationallifetimesforthesesamplesare700and370ps,respectively.
The paper is laid out in the following manner. In Sec. II, the experimental methods are briefly described, and the experimental results are presented. In Sec. III, the method for calculating the INMs is described, and the results are presented. In Sec. IV, we discuss the results of the experiments and the calculations.
II.
EXPERIMENTAL PROCEDURES AND RESULTS
Picosecond pump–probe experiments were performed with mid-ir pulses generated with a LiIO3 optical parametric amplifier ~OPA!. The laser system is a modified version of a system that has been described in detail previously.18 The description of the ir pulse generation and pump–probe data collection on the systems discussed here has been described in detail elsewhere.1
Figure 1 displays pump–probe data taken at 295 K.1 The data are for ~a! W~CO!6/CCl4 and ~b! W~CO!6/CHCl3. The calculated lines through the data are single exponential fits that include convolution with the pulse shape. The decays can be followed for greater than 4 factors of e. As can be seen from the figure, the data are of very high quality, and the fits are excellent. The vibrational lifetimes for these room temperature samples are 700 and 370 ps, respectively.
When pump–probe experiments are conducted