ABSTRACT
CONTENTS 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 13.2 Unstable Modes and Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
13.2.1 Statistical Mechanics on the PEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 13.2.1.1 The Partition Function and the Im− ω Density of
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 13.2.1.2 The Composite Landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 13.2.1.3 The Functional Form of the Density of States . . . . . . . 261 13.2.1.4 The Escape Rate and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
13.2.2 The Random Energy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 13.2.2.1 The Fraction of Unstable Frequencies, Dynamics,
and Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 13.2.2.2 The Configurational Entropy Sc . . . . . . . . . . . . . . . . . . . . . . 269
13.3 Diffusive and Nondiffusive Unstable Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 270 13.3.1 Potential Energy Profile Based Methods . . . . . . . . . . . . . . . . . . . . . . . 271 13.3.2 Landscape Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
13.3.2.1 Escape Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 13.3.2.2 Saddle Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 13.3.2.3 Partial Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
13.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
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An elementary definition is that normal modes are dynamically independent linear combinations of coordinate displacements, representing a direction in the relevant configuration space. Each mode obeys a wave equation in time, that is, it executes harmonic motion; the time dependence of each original coordinate is expressed as a sum of harmonic oscillations with different frequencies. A harmonic system is rigorously described by normal modes. Physically
challenging problems are anharmonic. However, in many instances harmonic approximations are possible, and extremely valuable. The truncation of an expansion of the total potential energy, U(r), at second order in the displacements, δr, of the coordinates from a minimum of U is the standard procedure. The approximations are useful so long as the system remains close to theminimum, which, if the potentialwell is truly confining, canmean forever. The example most relevant to the current discussion is a crystal at low temperature. In a remarkable article [1], Rahman et al. made the harmonic expansion
of U about snapshot configurations of an atomic glass, taken from molecular dynamics simulation; such configurations are not minima of the potential. Consequently, the curvature of U is negative along some normal mode directions, leading to imaginary frequencies and unstable modes. Substitution of Im− ω into textbook harmonic formulae leads to unphysical divergences. The resolution, of course, is that the system does not remain near the expansion point, but moves away along the unstable directions, invalidating the ill-behaved expressions. The question of how to use these unusual modes remains. Rahman et al. simply ignored the small number of Im − ω and calcu-
lated what is probably the most important dynamical quantity in atomic fluids, the velocity correlation function, as if the systemwere truly harmonic; excellent agreement with simulation was obtained. In a harmonic system the self-diffusion constant, D, the integral of the velocity correlation, vanishes. However, if D is small, as in a glass, the true velocity correlation may be well represented by one with D = 0. In addition, they speculated that the number of Im − ω might be correlated with D, as both vanish in the harmonic case. Developing a theory of transport coefficients, for example, D or the shear
viscosity η, in supercooled, glass-forming liquids [2] is an outstanding problem in physical chemistry. The dynamic range between room temperature and the glass transition is approximately 15 decades! Both theory and simulation become problematic below the mode-coupling [3] temperature, Tc, whereD extrapolates to zero from above, and relaxation has slowed by typically 3 to 4 decades from room temperature. Rereading Rahman et al. in 1988,
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it appeared to me [4-6] that the unstable modes were an interesting, novel basis for a theory of transport, and my students and I subsequently spent considerable effort developing the idea. In thiswork, the Im−ω are employed with no reference to harmonic dynamics. Later, I realized that LaViolette and Stillinger [7] had discussed the number of unstable modes as an indicator of melting in 1985. Diffusion is now routinely described in terms of the unstable instantan-
eous normal modes (INM), as the normal modes of snapshot, nonminimum configurations have [8] come to be named. More generally, liquids at low temperature are described [9, 10] in terms of the potential energy landscape (PEL), the topology of the total potential energy, U(r), regarded as a surface over the 3N coordinates of a systemofN atoms. The stationary points of order K are important “waypoints” on the landscape. Minima haveK = 0, ordinary transition states have K = 1, and K > 1 corresponds to higher-order saddles. Reaction pathways for diffusion lead from minima to transition states, and thence to connected neighbor minima. AKthorder stationarypointhasK unstablemodes, establishing the intimate
connection between the PEL and the unstable INM. The connection extends to diffusion. For example, in a low temperature “hopping” mechanism, the system spends long periods near a local minimum, with occasional activated passages over transition states to neighbor minima. Obviously the activation is signaled by the presence of an Im − ω, where there had been none, and it might be hoped that the number of Im − ω could express the frequency of barrier crossing, and thence diffusion. There is no reason why the same ideas could not be applied to other aspects of dynamics, but so far the focus has been on diffusion, and the discussion here will maintain that priority. Furthermore, we will consider classical diffusion only, although the importance of the transition states, and hence Im − ω, must hold for the quantum case as well. In general, dynamical problems are more difficult than equilibrium prob-
lems. A central aim of nonequilibrium statistical mechanics is to express dynamics in terms of equilibrium averages. Most static (thermodynamic) averages at low temperature are dominated by contributions of configurations near the minima. Dynamics, however, is determined by the reaction pathways, transition states, and barriers on the PEL. How [11] can such information be found in a static average reflecting properties of the minima? The most fundamentally important aspect of the INM theory of diffusion is that, despite the obvious dynamical association, unstable mode properties are static averages that get no contributions from the minima by construction. In the following, I will describe the current state of the connection between
imaginary frequency, unstable modes, and relaxation dynamics in liquids. The basic idea that D can be expressed with the unstable modes has stood up to themost rigorous analysis. However, despite the efforts ofmanygroups, there is no agreement on a single definitive theory of the underlying physics. Much remains to be done.
