An autoregressive vibrational-dephasing analysis

An autoregressive vibrational-dephasing analysis

Volume 106, number 3 AN AUTOREGFCESSIVE MBRATIONALDEPHA!5lNG Richard 20 April 1984 CHEMICAL PHYSICS LETTERS ANALYSIS E. WILDE Department of &em...

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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. US-I

Received 23 January 1984

An autogressive time series analysis of vibrational-memory 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 [l--4] 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 vibrational-correlation 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 short-range, rapidly varying forces and the inhomogeneous broadening arising from the long-range, slowly varying forces. This paper reports a novel approach to the separation of homogeneous and inhomogeneous effects using a resolution of the vibrational-memory 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 vibrational-memory 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 backward-shift

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

0 009-2614/84/S 03.00 0 Elsevier Science Publishers (North-Holland 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

(f-9 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-‘, G--l 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 vibrational-memory function into its homogeneous and inhomogeneous terms according to ,

= -

Table 1 Autoregessive

(cm-‘)

+&h(t)

(9)

j-K(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 fast-modulation 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 vibrational-memory functions of SiD,

analysis of experimental

1984

(7)

Eq.

3. Autoregressive functions

20 April

17.0

analysis

Term

for pure inhomoSeneous

b L-r

1

;Jnd/ps,

110.0

e o-N

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 motionally-narrowed, 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

k-l)

(radlps)

(rad)

(Ps-‘)

(dps)

(rad)

0.315 0.263 0.140 0.181

0.340 0328 0.370 0.239

0.27 0.85 l-43 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 well-known methods to produce a viirational-correlation 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 rapidly-oscillating, 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, high-frequency 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 short-range 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 (zero-frequency) and sh damped cosine terms.

The highest-frequency

term

makes

is somewhat

troublesome,

since

it does

4. Conclusion This paper reports the first attempt to resolve a vibrational-memory 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 high-frequency 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 memory-function stage, where the effects are additive, rather than at the correlation-function stage, where the effects are multiplicative, or in the spectral-frequency domain, where the effects are convolved.

a negligible

contribution to Kespt 1 (t) and can be ignored. AS above, it is apparent that terms 2-4 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 pulsed-laser 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 (North-Holland, 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 (Holden-Day. 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