Rotational mechanism for vibrational relaxation in rigid media. Interaction potentials

Volume 49, number 1

CHEMICAL PHYSICS LETTERS

ROTATIONAL MECHANISM FOR VIBRhIONAL

1 July 1377

RELAXATION IN RICID MEDIA.

INTERACTION POTENTIALS Kar1 F. FREED,

Danny L. YEAGER

James Franck Institute and Department Chicago, Illinois 60637, USA

of Chemistty.

Universìty

of Chicago,

and Horïa METIU Department

of Chemìstry.

University

of Californìa.

Santa Barbara. California

93106.

USA

Received 13 April 1977

Using the assumption of pairwise additive forces we derive the interaction potential acting upon a substitutional impurity in a crystalline lattice. The angle dependent parts of the fcrce on thc impurity internal vibrations are represented in the form of a Fourier series. Numerical calculations of the Fourïer coefficients of the force are presented for a model system employing empirical argon-argon interactions. The higher Fourier coeftïcients are shown to decrease rapidly in conforrnity with the simple model potential used to describe the vibrational re!axation in a preceding paper.

1. Introduction

guest-host interactions. Many choose the interactions to be linear in the molecular oscillation coordinate, a

Vibrational relaxation processes of impurity molecules in condensed media represent one of the simplest, yet fundament& energy dissipation mechanisms in these media. Hence, there has recently been a considerable amount of experimental 11-41 and theoretical [5-101 effort devoted to their study. Previous theoretical studies have employed models in which the molecule3 vibrational energy is converted directly into lattice vibrational energy [5-9]_ The general qualitative predictions of such a theory are readily deduced. Because molecular vibrational entirgies are much larger than the lattice Debye frequency (x100 K) for solids like argon, it is necessary to create a large number of phonons in thïs process. The theory,

valid approximation in the smal1 oscillation, single quantum decay limit often encountered experimentally. The dependence on the heat bath is then introduced by a linear coupling or by an effective linearization or averaging of some partially unspecified “general” coupling. Despite the simplicity of these models, they have gained acceptance because they have been employed for a long time in the description of solid state phenomena. It is then hoped that these simple models would truly represent the salient physical features. In the case of phonon-models of vibrational relaxation Lin has used more realistic interactions to verify this hope. He employs exponential repulsive interactions between the atoms constituting the molecule and those of the lattice [7]. Very recent experimental results of Brus and Bondybey [2] have demonstrated the invalidity of the energy gap law of the lattice phonon models, and they have suggested that the molecular vibrational energy is fïrst converted into molecular rotational energy in the decay process. Legay and co-workers [s] show that the vibrational relaxation rates, k,,ib, are fairly tempe-

therefore,

yields an “energy gap law” with a rate de-

creasing exponentially with increasing molecular vibrational frequency, w. Because increasïng temperature produces excitarion of lattice phonons, these phonons (bosons) can stimulate the production of more lattice phonons, thereby substantially enhancing the decay of molecular vibrations with ìncreasing temperatures. These theories employ rather simple models for the

19

me solla), tnereby provmg the importante of the rotational mechanism. In another paper we provide a theory of kvibfrom such a rotational mechanism [ 101 which is capable of explaining Legay’s correlation, the observed temperature independente, as wel1 as including any lattice phonon contribution. The theory employs a model interaction of the form F(9) = Vo exp(or CON(P) m ~VoIo(oL)+2V

c r~(oL)cosivk~, O k=l

of different molecules in solid Ar and Kr matrices

c

,

2. Determïnation of the Fk

(2)

and the calculations are designed to investigate the dependence Of Fk on k. Qualitatively, this dependence can be deduced from simpk physical arguments as fol10~s: consider, Ïor simplicity, an oscillator-rotor at the origin in a square planar lattice (N = 4). The neighboring lattice atoms, at positions (0, +l), (kl, 0), (21, Ij, (21, - 1) give a four-fold symmetrie force, contributing to k = 0 and 1 in (2). Of the next shell of neighbors the atoms at (O,lr2), (112, O), (?2,2), (*2, -2) provide additional contributions to the k = 0 and 1 in terms in (2), but those at (22, l), (k2, -l), (Itl, 2), (21, -2) yield an eight-fold symmetrie force (k = 2). The mth shell terms, therefore, contribute to k = forces die off with in0, l,..., m. As the intermolecular creasing distance, it becomes clear that the Fk must decrease as k gets larger. Local latiice distortions in the neighborhood of the impurïty molecule yield additional low k contributior?s which are of import7ince in understanding the vib;ational portions of spectra, but it is the higher k terms which dominate the !ibrational relaxation because of the krge number of rotational quanta that must be created in the process. Thus, we 20

[3].

(1)

where F(q) is the force exerted on the oscillator, y is the orientation of the rotor, Nis a symmetry number of the lattice, and Vo and c: are parameters. (Ik(~j are hyperbolic Bessel functions.) The model (1) originates from the consideration of realistic molecule-lattice interactions as described ir. this paper. The truc force on the oscillator (for lattice atoms at their equilibrium positions) must be representable in the Fourier series OD

F(Y) = &q) Fk cos Nky

Our determination of (2) uses the familiar approximation of pairwise additive interactions between the atoms in the guest molecule and all the lattice atoms. Any smal1 corrections from three-body effects should not detract from the qualitative dependence of the Fk which are shown to follow (I). Smal1 values of Q!imply Fk which die off rapidly as k increases, and this is consistent with experimental data of kvibof a number

While it is possible to treat the case of a polyatomic molecule, it is simplest to tìrst limit consideration to a dïatomic molecule as the more general case follows readily. The center of mass of the diatomic molecule is assumed to reside at a lattice position which is taken as the origin of coordinates. The position of the atoms in the molecule is specified by r1 and r2, and the equilibrium bond vector r. lies in a !attice plane at an orientational angle cp. The force on the oscillator (evalua-

ted at its equilibrium positions)

position)

is (CR,-) are lattice atom

F(q)=-

C a[vlWz~-rlij/a~ I+ 0 + 5 (IR.!- rzl )l&-j >

(3)

where the case of a polyatomic molecule just involves a summation over contributions Vo,(lRi - r,l) from al1 atoms, Q, in the molecule, and a/ar is replaced by a derivative with respect to the appropriate normal coordinate, &?pVibrational-to-vibration relaxation would involve a2/aQI aQjtype terms, etc. T~us, (3) suffices to illustrate the be5avior of the general situation. The periodicity of the lattice [ 1 l] enables (3) to be rewritten as F(yj = -

f

2 0 [ &CP) exp(i P - q Yar

V,(dexp(iP- 311lrzo

,

(4)

with I.I the reciprocal lattice vectors p= 2sr(Z/al, mfa2, n/a3), where al .__, are the lattice constants and 1, m, n range over 0, +l, *2, .__, and ti(a) is the

atoms). Thus, we have p1 = ;(ro

+ r) = -r,,

where r.

has components (0, r. cos cp, r. sin (p). Performing the indicated differentiation in (4) and then setting r + 0 leads to F(v)

= -2~

X in b0 Ge

F(P) [(m/az) cos y + (n/a$

F

sin 501

[(miQ& cos
dependence

(5)

of V(JR 1) only on IR] implies that

V(j.t) lïke-wïse depends only on 1c~I = 2ni(Z/al)2 + (mla2)2 + (rz/a3)2] Ij2 = plmn E 2rrplmn. In order to determine the F’ in (2), it is necessary to evaluate tbe integrals Fk(p) = (27r)-1

7 dy F(q) cosNk cp , (6) 0 along with the summation over vz,, . A simple transformation of (5) enables (6) to be evaluated analytically as follows: We may write (m/a2)cos cp+ (n/fz,) X sin y as E(m/a2)2 + (d~)~

E [(m/a2)2

1 U2

(mla2) cos 40+ (n/a$ sin tg3 2 112 Kmla2)2 + (n/a3) 1

evaluation of Fk from (9), the evaiuation of the Z-summation by the use of th? Euler-Maclaurin summation formula [ 131. Because V(~I,,~) is an even function of 1, through the defïmtion of film,. 2nd because it vanishes as 1 -+ -, the correction terms in the summation formula vanish, so we merely require 7

dl i(i+,r,> A

Fk = -N

Co

where S nm = arc sin{(mla2) [(m/a2)2 + (t~/+)~] -IJ2 1.

Using (7) in (6) leads to the representation c p(crr,,) Fk = - 1 ,n n

nornn y 0

X coskpsin(_iS,,)dq

sink, ,

sin(ip + S,,)] 03s)

where A,, = nropomn. Changing the integration variand using the properties of Bessel abletoe=
X [J,,,;(a,,d with JP ordinary

COS(~&,d

(A,,)

1

G%,)l ,

Bessel functions.

(9)

- JNk-1 64,,

VM(R)~= V. {exp [ -2ar(R-Ro)] which has a tìnite Fourier

(10)

)l

- exp

[-c@-R$ i .

transform

= 47rcuUo exp(o&-,)

X iexp(oRo)(4cY2 - Jjvk-

cos(NkS,,)

mc,’ +@‘r&&r~Omn s

as the expression to be evaluated. The primed summation implies that 0 f S,,, < 27r/N as the N such regions provïde identical contributions. Lattice sums with poteniials containing van der Waals, Rd6, interactions are unpleasant because Fourier transforms do not exist and numerical calculations are sensitive to the precise manner in which they are cut off at smal1 R [ 111. Thus we utilize the popular Morse potential

&(k) V(!%&Po,,m

>

with Vn the two-dimensional Fourier transform of V( IR 1) wïth R lying in the plane of the rotor. Hence, (9) b ecomes

x [%k+l

sin (p) ,

= &(2irp0,,&

-m

qQ. +(~i~~)-l

X (sin Snm cos
Fk = - t ,cn ,

uy airecr summanon ma nueglduuu 111~eg~uus wutx the summand develops a simple asymptotic form [ 111. Unfortunately, this is not possible with (9) because the angular dependence enters thrcugh the cos(NkSmn) factor whïch Yanishes identically when integrated over a period of 3nlNk. This oscillating term and the asymptotic oscihation of .ïNki l(Anm) for iarge A,, contribute to the decrease of the magnitude of Fk with larger k. The dependence of _yollm,,) on 12 enables US to introduce an approximation, that simplifies the

+ k2) -3/2 -(ar2 + k2)-3/2].

(12)

Using (12) and the asymptotic form of the Bessel functions [12] shows (10) to vary as 21

Volume 49, numbcr 1 Pò?g2

CHEMICAL

PHYSICS

cos(7r!-(Jp(),~,--ll/4)

for pOmn large. The radial density of points increasing as pomn. This provides an overall Pomln’ cos(rrgPom,* - x/4) slow, oscillatory convergence of the sum in (10). For this rezson we have chosen to perform the numerical computations for cases that are designed to supply favorable convergence as our basic aim in the numetical calculation is to study the dependence of Fk on k in the experimentally relevant range 1 < k < 4.

4. Numeri&

calculations

One simplification is introduced by using a simple cubic lattice with ci = nl = a7 = a3. Another occurs with the use of potentials and an impurity that ensurc repulsive interactions as this feature increases the total contributions from smal1 values of po,l,,z, so the tails provide less important relative contributions. The interaction (11) is taken to be that appropriate to an Ar-Ar interaction. The parameters, po = 1.994 X 1O- l4 erg, R o = 3.715 K, tio = 6.692, come from the Morse part of an empirical Morse-spline-van der Waals potential [ 141 - the precise farm of the attractive tail is unimportant for our calculations. Using a=Ro =ro makes th e calculations correspond to an Ar, substitutional impurity in a hypothetical simple cubic Ar solid. This leads to a somewhat repulsive potential compared to that corresponding to a smaller molecule as the substitutional impurity, but the .Ic-dependence of Fk should adequately represent the smal1 molecule less repulsive potenti& Indeed, calculations with a = r. = 1 bohr, representïng an extreme repulsion limit, provide more rapid convergente and different values of V. and 01in (1) from those obtained using the Ar-& separation as the lattice constant. Despite the simp;ifïcations introduced to speed convergence, the k = 3 sum employs over 250000 summation points with two significant figure accuracy, while the k = 1 and 2 cases have converged to an order of magnituäe more accuracy. T’he k = 1,2,3 summations include pojnts up to (m2 + n2)li2 equal to 280, 410,600, respe$ively, producing the values F, = 1.32 X IOS3 erg/cm, F2 = 3.17 X 1Oa4 erg/cm, and F3 = 4.11 X lom5 erg/cm. We can check the validity 22

LETTERS

1 July 1977

of the foxn (1) by comparing the ratios of Fk to ratios of hyperbolic Bessel functions. Equivalently, usïng F1 (to give Vo) and the assumcd form Fk = 2V,-#k (CU)yields for cy = 0.9 the “predicted” values of 3 = 2.88 X los4 erg/cm and Fi = 4.15 X 10-5 erg/cm which provides a good fit to the computed Fk. (For ci = 1 bohr we also obtain a good fit to Fk (IC= 1,2,3,4) for Q = 0.284.) If the face centered cubic lattice were employed, the greater atomic density would lead to more effective cancellations from the cos(AR8,,) factor in (9) at shorter distances than in the simple cubic limit. Thus, we would expect a smaller overall value of V. for thislattice. This feature combined with the decreased V. arising from smaller impurities than the bulky Ar2 (r. = 3.715 A) would provide the rather small interactions F(p) necessary to explain the slow vibrational relaxation rates of the smal1 nearly freely rotating impuríty molecules which have been studied experimentally.

5. Conclusïons We have introduced a full pairwise additive interaction between the guest molecule and the lattice atoms in order to determine the Fourier components, Fk,of the force on the molecular vibration that is responsible for the vibration to rotation energy transfer of an almost freely rotating molecule as a guest impurity The general expressions exhibit IFklas a decreasing function of k, a feature that is also evident on the basis of sïrnple physical considerations. The calculated results are nicely fit by the simple two-parameter model potential(1) that is reminiscent of the popular exponential repuldve interparticle interaction. This simple potential has been shown to provide a general qualitative description of experimental vibrational relaxation rates of a number of molecules in Ar and Kr matrices [ 101. The full interactions, given by (9) or its polyatomic generalizations, should enable considera tions of “fine structure” effects, smal1 deviations of particular systems from the model form of (1). One interesting case involves the increase in kvib in going from an Ar matrix to a Kr one. In the latter case it has been argued that long range attractive forces contribute to the increased anisotropy of F(q) [2]. It would therefore be of interest to consider a systematic experimental and theoretical study to determine the lattice dependence,

$51 G. Nitzatxand T. htner, Mal. Phys. 25 (f9733 113; G ~ï~~nandR.S~~ey,f.Chttm.Phys.60CI97~) 40% A. Nitzan, 5. Mnkamet and f. Joxtncr, J. Chem. Phys. 60 (1974) 3929; 63 (1975) 200. [ci] D.J. DiestXer and R.S. Wilson, J. Chem. Phys. 62 (1974) 1572; D.3. Dbstier, J, Chem. Phjrs. BO (1974) 2692; Topics Appì, Phvs. X5 Fl976) 1699; chem. Phys. Letters 39 (1976) 39. 171 GR. Fleming. O.L.J. Gijzeman and SH. Wn, J. Chem, Snc. Fztratiay ïf 70 (1974) 30; s.zI, Lin,J. Chf?M. Pk@%. 6% (1974) 38x0. $81 S.F. Fischer and A. Laubxeau, Chem. Fhys. Letters 35 (1975) 6.

[if_Jj K.F.Freed and H. Metìu, Chem. phys. Letters 48 L.E. Brus and V.E. Bondybey, J. Chem. Phys. 63 (1975) 786; V.E. Bondybey and LX. Brus, f. Cbern. Phys. (53 (19753 794; L Goodman and LE. Brrrs, 3. Cbm. Phys. 65 (1976) 3146; V.E. Brnxdybey, 1. Chem. Phys. 65 0976) 5138. F. Legay, in: ChemicaI und biochemieal applicati~n af

(1977) x2. 11 X] J.M. Ziman, Eiectrons and phonons (Oxtàrd 1972) ; J, Have and J.A. Krumh;au&, Phys. Rev. 92 119533 569; W.A. Stee& Surfíice Sci. 36 0973) 317. fL2] 1.3. Gradshìe~n and LM- Ryzhik, Tabfes of seriesand products (Academie Press, New York, 2965). f 23 ] M. Abramotitz and LA. Stqpn, Hatrdbook of matUmatic& functions (Dover, New York, 19653 p. 16. f14] J.M. Parsont P.E. Siska and Y.T. Lee, 3. Chem. PW. 56 11972) 1511.

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