Volume 106, number 3
AN AUTOREGFCESSIVE MBRATIONALDEPHA!5lNG Richard
20 April 1984
CHEMICAL PHYSICS LETTERS
ANALYSIS
E. WILDE
Department
of &emistty,
Terns Tech Unirersir_v. Lubbock,
Texas 79409. USI
Received 23 January 1984
An autogressive time series analysis of vibrationalmemory functions has enabled the viintionaldephasing process to be separated into its homogeneous and inhornogeneous parts. The analysis has been applied to the vr bands of SiDa and PD3.
1. Introduction
2. Autoregressive
There have been several attempts [l4] recently to understand the effects of homogeneous and inhomogeneous processes in vibrational dephasing and to separate these effects experimentally. George et al. [ I] considered cases where the homogeneous and inhomogeneous vibrationalcorrelation functionsdecay on different time scales so that the total correlation function (obtained by Fourier transforming an isotropic spontaneous Raman band) can be written as c(t) = Ch(f)Cfi(t)_ It is then possibe to obtain the homogeneous (lorentzian) line shape from stimulated Raman scattering measurements and the inhomogeneous (gaussian) line shape by deconvoluting the isotropic spontaneous Raman band. Oxtoby [3] has questioned some of the results obtained by this method, pointing out that, if processes are present that relax on an intermediate time scale, spurious results can be obtained. Schweizer and Chandler [4] have developed a theory of linewidths that takes into account both the homogeneous effects arising from the shortrange, rapidly varying forces and the inhomogeneous broadening arising from the longrange, slowly varying forces. This paper reports a novel approach to the separation of homogeneous and inhomogeneous effects using a resolution of the vibrationalmemory function. This approach is based upon time series analysis [S] and, in particular, upon an autoregressive model.
Recently, Endo and Endo [6] resolved a velocity autocorrelation function into a sum of damped exponential and damped cosine functions using an autoregressive analysis. Because any autocorrelation function can be represented by a time series [5], it is possible to treat the vibrationalmemory function by an autoregressive process. For this purpose, consider a random process F(r)‘that is linearly dependent on p previous values of the process whose input is white
166
analysis
noise ut: P
Fr=ar
+ ~
(t=
F,_iQi
If the corelation
(F,F,)
1, 2, . ..) _
(1)
is formed
(2) Let B be a backwardshift
operator
such that
BK, = K,_ 1 . Using this operator,
(3) eq. (2) can be written
(1  #lB  Q2B2 _._$,Bp)K,
as
= @(B)Kt = 0 .
(4)
The function Q(B) can be treated as a polynomial in B with roots G’ (i = i, 2, _.., p). Hence, @(B) can be factored as
0 0092614/84/S 03.00 0 Elsevier Science Publishers (NorthHolland Physics Publishing Division)
B.V.
Volume
106,
number
O(B) =
fjl(1  CjB) .
CHEMICAL
3
PHYSICS
(5)
If the stochastic process F(r) is stationary, the elements K,_i of eq. (2) form a positive definite matrix and the roots 51 of d(B) must lie outside the unit circle, i.e. IG,FlI > 1. The general solution of eq. (2) is
(f9 The coefficients Ai are determined by the initial p conditions. If a root Gr’ is real and positive, the term A& behaves as a damped exponential function. If a root is real and negative, the term A jCik alternates in sign as it damps out with increasing k. As a practical matter, these rapidly oscillating terms can be ignored, since they contribute negligibly to the time series. If a pair Of roots Gi‘, Gl is complex the sum A.Ce + AiG,h behaves as a d:mped cosine finction. In kt:al practice, p equations of the form of eq. (2) are solved for the @r.Then the characteristic polynominal Q(B) = 0 is solved for the roots Gi‘. Finally, the Ai of eq. (6) are determined. This is known as a pth order autoregressive (AR) analysis. The order p is chosen to achieve parsimony. If desired, eq. (6) can be rearranged to the form A’, = z bi exp(q+) i
+ Bi) .
cos(o+
LETTERS
ond moment M, of the band comes primarily the wings, and the contribution of the kernel
Cinh(f)=
dC(t)/dr
3.0
memory 4.5
In order to use an autoregressive analysis for the purpose of resolving a vibrationalmemory function into its homogeneous and inhomogeneous terms according to ,
= 
Table 1 Autoregessive
(cm‘)
+&h(t)
(9)
jK(r)C(r i)
K(r)
is related
dr .
to
(10)
Eq. (9) was solved for C (I) using values of M, = 3.0, Th 4.5, 17.0 and I 10.0 cm. Then eq. (10) was solved for K,(l), and Kinh(t) in turn was resolved by a 6th order AR process. The results are listed in table 1. In all cases only three cosine terms are necessary in order to achieve parsimony_ The frequencies w,+ll,) (i = 1, 2,s) are functions ofAl and increase as the value of M2 increases. For two different second moments, AI, and Mi, the ratios Wi(,~J,)/Wi(~J:) are nearly constant for each term. It is therefore apparent that the memory function for pure inhomogeneous broadening can be resolved into three damped cosine terms. the frequencies of these terms increasing with increasing values 0f nf,. T\;o molecules having spontaneous Raman spectra
h12
= Khil)
exp(M7r2/2).
In general, the memory function C(t) by the relation [7]
and PD,.
Rex&)
from to AIT
can be ignored. For these bands, the speed of mod&tion tends toward the fastmodulation limit. In the case of pure inhomogeneous broadening, the correlation function is
(7) is the form in which we choose to display rhe memory function. This analysis will now be applied to the experimental vibrationalmemory functions of SiD,
analysis of experimental
1984
(7)
Eq.
3. Autoregressive functions
20 April
17.0
analysis
Term
for pure inhomoSeneous
b Lr
1
;Jnd/ps,
110.0
e oN
1
0.701
0.105
0.94
0.090
2 3
0.298 0.025
0.089 0.078
0.70
0.324 0.673
1.15
1
0.677
0.091
0.29
0.062
2 3
0.305 0.303
0.078 0.067
0.84 1.37
02’3 0.458 0.081
1
0.685
0.232
0.56
2
0.306
0.200
1.63
0.293
3
0.030
0.179
1.68
0.623
1 2
0551 0.386
0.114 1.064
1.06 3.17
0.107 0.370
3
0.136
0.958
5.27
0.826
(8)
we restrict the analysis to those motionallynarrowed, spontaneous Raman bands that have a lorentzian kernel and gaussian wings such that the value of the sec
broadening
167
Volume 106. number 3
CHEMICAL
Table 3
Table 2 Autogressive analysis for the VI band of SiD, at 148 K Term
Autoregressive analysis for the ~1 band of PD3 at 173 6
t)
&J
B
TI
w
L9
kl)
(radlps)
(rad)
(Ps‘)
(dps)
(rad)
0.315 0.263 0.140 0.181
0.340 0328 0.370 0.239
0.27 0.85 l43 2.06
0.111 0.191 0286 0.085
0.089 0.192 0.207 0.186
0.88
0.035
0127
0.139
2.60
0.466
0.210 0.126 0.002
b
1 2 3 4 5
20 April 1984
PHYSICS LETTERS
that satisfy our criteria are SiD4 and PD,. The M2 value of the v1 band of SiD4 at 148 K is reported [8] to be 4.5 cmw2. For purposes of the present study, the experimental intensities lisD(o) used in ref. [S] were Fourier transformed by wellknown methods to produce a viirationalcorrelation function. Eq. (10) was then solved [7] for Kexptl (f)_ An 11 th order AR process with a lag of 0.90 ps gave the results listed in table 2. One rapidlyoscillating, damped exponential term contriiuted negligibly to the total and is not in_ eluded in table 2. It is immediately apparent that the fist three terms of table 2 have nearly the same fre
0363 0331 0.311 0.329 0229 0.105
1.81 2.65 3.43 4.08 5.78
0.508
0.047 0.011
0.129 0.380 0.110
quencies as the terms in table 1 for M2 = 4.5 cm*. We therefore set the sum of these three terms equal to Ka(r)_ The sum of the last two, highfrequency terms comprise Kb(t). A plot of the normalized mem, ory functions is shown in fig. 1. Endo and Endo [6] interpret the frequencies literally_ It is tempting to associate the frequencies in table 2 with the frequencies of the long and shortrange forces, but we have no
l.Og :.
SiD,
r’.. 0.8  “,, l
06

06
. .. . . . 
.
.
.
0.2
.
.
.
0.4
&J j___( , ; , 2
3
t(ps)
4
5
. .
.
.* . .% . o 
. .: . . 0. 0. 88.
.., 
0 . . 
.
.
.
.
0 .
.
, 6
7
F&.. 1. Memory functions of the v1 bandof Siia at 148 K; Kh(t) (01, &h(r) (+).and KexptlU) (0). 168
. 
. 0.. a . *&a% . *f DO. .* . .. . 0 '.;_*•
zoo
1

.
0.2
.
.
04
. 0
=. I
t
1.01
I
I
1
I
i
1
2
3
4
5
t (PS)
1
6
Fig. 2. Memory functions of the YI band of PD3 at 173 K; Kb(r) (0). &b(r) (+), and Ke;e?tptl(r) (0).
CHEhlICAL PHYSICS LETTERS
Volume 106, number 3 justification
for this.
The vl band of PD at 173 K has a reported [9] AIlz value of 16.7 cm 9 . As above AlexPtt (I) was resoIved with a 13th order AR process employing a lag of 0.5 ps. The results are listed in table 3. It is seen that AIcxPtl (I) can be resolved into one damped exponenetial (zerofrequency) and sh damped cosine terms.
The highestfrequency
term
makes
is somewhat
troublesome,
since
it does
4. Conclusion This paper reports the first attempt to resolve a vibrationalmemory function by means of an AR proObviously,
any isotropic
Raman
Acknowledgement This research was supported by a grant from the Robert A. We!& Foundation. Computer time was provided by Texas Tech University.
not
arise in the pure inhomogeneous case (table I), and we hesitate to incorporate it with the highfrequency terms. We have decided to include it with the lowfrequency, inhomogeneous terms because of its zerofrequency nature. This resolution is shown in fig. 2.
cedure_
homogeneous processes at the memoryfunction stage, where the effects are additive, rather than at the correlationfunction stage, where the effects are multiplicative, or in the spectralfrequency domain, where the effects are convolved.
a negligible
contribution to Kespt 1 (t) and can be ignored. AS above, it is apparent that terms 24 reflect the inhomogeneous processes and terms 5 and 6 reflect the homogeneous processes. The damped exponential term
20 April 1984
band can be
analyzed by this method, but the bands must be motionally narrowed if a successful resolution into homogeneous parts is to be made. Unlike the method [ 1,‘_I using spontaneous Raman and pulsedlaser techniques. our AR analysis separates the homogeneous and in
References [l]
S.M. George, 73 (1980)
H. Auaeter
and C.B. Harris, J. Chem. Phys.
5673.
[2 J A. Laubereau, in: Proceedings of the 7th International Conference on Raman Spectroscopy, ed. W.F. Murphy (NorthHolland, Amsterdam, 1980) p. 450. [3] D.W. 04oby. J. Chem. Phys. 74 (1981) 5371. 141 KS. Schwcizer and D. Chandler, J. Chem. Phys. 76 (1982) 2296. [S] C.E.P. Box and Cdl. Jenkins, Time series analysis (HoldenDay. San Francisco, 1976). [6] H. Endo and Y. Endo, Prozr. Theoret. Phys. 66 (1981) 794. (71 BJ. Berne and G.D. Harp, Adnn. Chem. Phys. 17 (1970)
63.
181 R.E. Wilde, J. Chem. Phys. 75 (1981) 58S3. 191 R.E. Wilde and S.S. Cohen, J. Chem. Phys. 70 (1979) 4557.
169