1549380323-Statistical Mechanics Theory and Molecular Simulation

Derivation of the GLE 575

whereRαβis an orthogonal matrix that diagonalizes the symmetric matrixH ̃αβvia
H ̃diag=RTHR ̃ , whereRTis the transpose ofRandH ̃diagcontains the eigenval-
ues ofH ̃ on its diagonal. Lettingkαdenote these eigenvalues and introducing the
transformation into eqn. (15.1.10), we obtain

Ubath(q,x 1 ,...,xn) =

∑

α

gαqxα+

1

2

∑

α

kαx^2 α, (15.1.12)

wheregα=

∑

βCβRβα. The potential energy in eqn. (15.1.12) is known as ahar-
monic bathpotential; it also contains a bilinear coupling to the coordinateq. We will
henceforth refer to the coordinateqas the “system coordinate.” In order to construct
the full Hamiltonian in the harmonic bath approximation, we introducea set of mo-
mentap 1 ,...,pn, assumed to be conjugate to the coordinatesx 1 ,...,xn, and a set of
bath massesm 1 ,...,mn. The full Hamiltonian for the system coordinate coupled to a
harmonic bath can be written as

H=

p^2

2 μ

+V(q) +

∑n

α=1

[

p^2 α

2 mα

+

1

2

∑n

α=1

mαω^2 αx^2 α

]

+q

∑n

α=1

gαxα, (15.1.13)

where the spring constantskαhave been replaced by the bath frequenciesω 1 ,...,ωn
usingkα=mαωα^2. We must not forget that eqn. (15.1.13) represents a highly idealized
situation in which the possible curvilinear nature of the generalized coordinates is
neglected in favor of a very simple model of the bath (Deutsch and Silbey, 1971;
Caldeira and Leggett, 1983).
A real bath is often characterized by a continuous distribution of frequenciesI(ω)
called thespectral densityordensity of states(see Problem 14.9).I(ω) is obtained by
taking the Fourier transform of the velocity autocorrelation function.^1 The physical
picture embodied in the harmonic-bath Hamiltonian is one in which a realbath is
replaced by an ideal bath under the assumption that the motion of the real bath is
dominated by small displacements from an equilibrium point described by discrete
frequenciesω 1 ,...,ωn. This replacement is tantamount to expressingI(ω) as a sum of
harmonic-oscillator spectral density functions. It is important tonote that the har-
monic bath does not allow for diffusion of bath particles. In general, aset of frequencies,
ω 1 ,..,ωn, effective massesm 1 ,...,mn, and coupling constants to the systemg 1 ,...,gn
need to be determined in order to reproduce at least some of the properties of the
real bath. The extent to which this can be done, however, depends on the particular
nature of the original bath. For the purposes of the subsequentdiscussion, we will
assume that a reasonable choice can be made for these parameters and proceed to
work out the classical dynamics of the harmonic-bath Hamiltonian.

15.2 Derivation of the generalized Langevin equation

We begin by deriving the classical equations of motion generated by eqn. (15.1.13).
Applying Hamilton’s equations, these become

(^1) The density of states encodes the information about the vibrational modes of the bath; however,
it does not provide any information about absorption intensities.