The calculation of time-correlation functions for molecular collisions

THE CALCULATION OF TIME-CORRELATION FUNCTIONS FOR MOLECULAR COLLISIONS

Eduardo VILALLONGA Department of Chemistry, Princeton University, Princeton, NJ 08544, USA and David A. MICHA Quantum Theory Project, Departments of Chemistry and of Physics, University of Florida, Gainesville, FL 32611, USA

NORTH-HOLLAND

PHYSICS REPORTS (Review Section of Physics Letters> 212. No. 6 (1992) 329—31>S. North-Holland

PH YS CS REPORTS

THE CALCULATION OF TIME-CORRELATION FUNCTIONS FOR MOLECULAR COLLISIONS* Eduardo VILALLONGA I)eparmntent of Chemistry. Princeton (Iniversity. Princeton \J 95544, (S.

and

David A. MICHA Quantum Theory Project, I)epartou’nts’ of ( ‘hemistrr and of Physics. University of Florida. ( aine.ri il/c. H. “26/1, 1 ‘5.4 Editor: (Ii. HF. Dicrckscn

Received October 1991

(ontents

I . Introduction 2. The basic formalism of collisional time-correlation june-

33! ~

tions

2.1. Time-correlation functions and collisional cross see90115

2.2. Cumulant expansions for short-time events 2.3. Molecular interaction potentials and transition operators 2.4. Intramolecular motions and the separation of time scales 3. The calculation of cross sections for impulsive collisions 3.1. The multiple scattering expansion 3.2. The impulsive collision limit 3.3. Collisional excitation of harmonic vibrations 3.4. Collisional excitation of anharmonic vibrations

333

337 3~9

3411 343

343 344 347 341)

3.5 - Slow rotational motion: short—time expansions 3.6. Applications to scattering by polyatomics 4. The calculation of cross sections in the short-wavefen~th limit 4. I . Separation of relative from internal motions 4.2. Separation of time scales for intramolecular motioit~ 4.3. Collisional TCFs in the short-wavelength limit 4.4. Classical relative motion 4.5. Quantal internal motions 4.6. Applications to molecular collisions 5. (‘onclusioit 5.1 Advantages of calculations based on collisional I (‘K 5.2. Extensions of the theory and prospects for future calculations References

354 356

3(~4

36(~ 366 369 35(1

364 3~4

355

1 h,oract A general tormalism relating cross sections to collisional time-correlation functions is applied to the calculation of energy transfer in nioleculai collisions. Basic aspects of the formalism arc reviewed, followed b~’a detailed description of methods suitable for the calculation of quantal state-to-state cross sections. These methods arc particularly useful in the description of scattering by large systems. or of systems with high total energies where expansions in target basis sets are unsuitable. We review applications of quantal time—correlation functions (lCFs) to vibrational—rotational energy transfer iii molecular collisions. DouhI~ differential cross sections (in scattering angles and transferred energy) are obtained from Fourier transforms of ‘l’CFs of the transition operator bs means of two complementary treatments: multiple-scattering expansions for impulsive energy transfer, and a semiclassical limit for short translational wavelengths. The TCFs are evaluated by operator algebra. yielding efficient computational procedures that incorporate large numbers of vibrational—rotational transitions. The treatments are thus well suited to scattering by complex polyatomie targets. Examples~~ire given for inelastic scattering of atoms from diatomie and linear triatomic molecules at hyperthermal collision energies. and the calculated results are compared with experimental measurements. These methods are also applicable to scattering by solid surfaces and by adsorhates. and can include temperature effects. The review lists 14>) references to related work on the formalism, methods of calculation and applications

*

Originally commissioned for Computer Physics Reports by F.A. (3iantureo.

()37U-t573:92/~l5.O1)© 1992 Elsevier Science Publishers By. All rights rescrvef

E. Vilallonga and D. A. Micha, The calculation of time-correlation functions for molecular collisions

331

1. Introduction The calculation of collisional cross sections for phenomena involving atoms and molecules is particularly difficult because many quantum states of the colliding partners are coupled by the interaction forces. Even in cases involving electronically adiabatic phenomena, where one can assume that the electronic states of the system remain the same while the nuclei move, one must yet deal with the coupling of translational, rotational and vibrational degrees of freedom of the nuclei. The interaction forces furthermore depend intricately on the molecular orientations and on the atomic displacements within molecules, and change extensively with the atomic composition of molecules. It is therefore usually impossible to invoke physical considerations to make a preliminary selection of the quantum states that are relevant to the collision. We describe here the computational aspects of an alternative approach, based on the time evolution of operators for scattering, and on their timecorrelation functions, which eliminates the need for basis set expansions. In the standard approach to the quantal theory of scattering, the states of the whole system are expanded in a basis set which must contain at least the quantum states of the collision partners that are energetically accessible during the collision, and should also include states of the whole system that are relevant to the intermediate stages of the collisions [1—3].The expansion coefficients are functions of the relative position of the collision partners, and their asymptotic values provide the scattering amplitudes and cross sections. When the expansion is replaced in the Schrodinger equation, it leads to a set of coupled partial differential equations for the expansion coefficients, that are known as the coupled-channel equations of scattering theory. A similar approach is followed for molecular collisions, using products of internal states of the collision partners, that describe electronic and rotational— vibrational motions, as the basis set [1,4—8].This is a large set which grows enormously when the collision partners are polyatomic molecules; convergence studies with respect to basis set sizes must furthermore be undertaken to obtain reliable results. As a consequence, it is usually impossible to do standard coupled-channel calculations for interactions of polyatomic systems. To estimate the basis set size in an atom—polyatomic collision, let f~ be the number of vibrational degrees of freedom of the polyatomic, tmax the largest vibrational quantum number for each normal mode, and ~max the largest rotational quantum number; then the number of coupled channels for a linear polyatomic is fvUmax(Jmax + 1)2, and substituting typical values of f~ = 4, Umax = 5, and ~max 20 one finds 8 820 coupled states. The numerical solution of so many coupled partial differential equations for the scattering functions is affected by problems of numerical accuracy and error accumulation, and can seldom be undertaken. The output of such calculation would in any case be long tables of numerical values out of which only a few numbers would be relevant to a particular study. Coupled-channel calculations also become impractical when the total energy of the system is large, for example in collisions involving molecules prepared in vibrationally excited states (where Umax would be quite large). Relaxation to lower vibrational levels can then lead to populations of rotational levels with a very large ~max~ Similar problems, but of a more fundamental nature, arise when one tries to describe scattering by extended systems such as solid surfaces and by adsorbates. In these cases the basis sets must include an infinite number of states, which furthermore are not available except for a few systems and then only for the low excitation energies. Scattering by extended systems makes it necessary to introduce alternative approaches such as the one to be covered. The alternative approach introduces time-dependent scattering operators which depend on the internal degrees of freedom of the colliding partners. Instead of solving for a set of coupled partial

332

L - Vilallonga and 1). A - Md-ha, The calculation of time-correlation Junctions fur mnolecular collisions

differential equations. one must obtain solutions to the equations of motion of operators describing properties of the system, such as the atomic positions. In this way the complexity of the problem grows only with the number of atoms, or degrees of freedom, of the system. which is much lower than the number of quantum states required in a basis set. The approach to he described is particularly well suited for collisional energy transfer involving polyatomics or solid surfaces, where collisional excitations may involve collective (or normal) modes, local distortions, or combinations of both. The central objects of the calculations are time-correlation functions (TCFs) of the transition operators of scattering theory [21.which we shall call c’ollisional TCFs. These are very similar to more familiar TCFs, such as the dipole—dipole TCF giving the intensities of molecular spectra [9, 10]. or the Van Hove TCF relating to neutron inelastic scattering [11]. Collisional TCFs are however more general because they provide an exact formalism to calculate rates, and contain as special cases the other familiar TCFs in the literature [12, 13]. Collisional TCFs have two additional advantages. They provide a very convenient way of combining the theories of scattering and of many-body systems, and they provide an alternative physical picture which allows comparison of time scales such as collision times. periods of internal motion, and intramolecular relaxation times. Calculations in our many-body treatment require mathematical techniques involving operator algebras and operator differential equations. For example a sequence of atomic encounters can he described with expansions of scattering operators into many-atom terms, within a multiple-scattering formalism [141that leads to a many-atom expansion of the collisional TCFs. Another general tool is provided by short-time expansions of the TCFs in terms of cumulants. suitable for describing the sudden excitation of slow internal degrees of freedom [15]. Asymptotic expansions of TCFs have been found useful when exponential operators contain a large parameter in the exponent [16]. as happens in TCFs for rotational excitations. These general aspects are briefly considered in the following sections in connection with the calculational approaches and applications. Details of the formalism of collisional TCFs can be found in our published articles and in a review in preparation [17]. Several other related aspects of TCFs can he mentioned, but will not be covered here to concentrate instead on calculational methods and applications of collisional TCFs. An earlier alternative approach in terms of superoperators [18] suggests ways of extending the formalism to include phenomena where the total energy is not conserved due to interactions with external fields or media. It has led to different TCFs which however have not been used in calculations. Information-theory concepts can be combined with TCFs [10] to develop useful expressions for collisional problems [19]. Collisional TCFs can also he expressed as overlaps of time-dependent transition amplitude functions that satisfy differential equations and behave like wavepackets. This approach to the calculation of TCFs was developed for Raman scattering [20] and has more recently been extended using collisional TCFs for general interactions of photons with molecules [211and for systems coupled to an environment [22—251. This approach has so far been only applied to the interaction of photons with molecular systems. Flux—flux TCFs [26—281 have been applied to reactive collision and molecular dynamics problems, hut their connection to collisional TCFs have not yet been studied. We shall concentrate in this contribution Ofl energy transfer in electronically adiabatic phenomena involving collisions of atoms with diatomics and with polyatomics. We shall not deal with collisions involving electronic excitations. The formalism can be written down for these cases but not much has yet been done to develop the computational methods required in applications. This is in great part due to the lack of information on interaction-potential energies of electronically excited states and on their couplings due to nuclear motions, for polyatomic systems. Similarly, the formalism can be extended to include rearrangement collisions. Little is known however about interaction potentials for reactions

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

333

involving polyatomics, and a general computational treatment has not been developed in the context of collisional TCFs. The formalism of collisional dissociation of molecules can nevertheless be readily treated as an extension of energy transfer [29], and many of the developments in this review could be applied to dissociation phenomena. The theoretical and computational methods to be described allow the incorporation of experimental averages into the calculation of transition rates and cross sections, and can be closely related to experimental measurements of final velocity distributions using time-of-flight techniques, and of final state distributions using spectroscopic techniques [30—34]. The formalism of collisional TCFs can be conveniently applied to interactions involving polyatomics and solid surfaces, using a many-atom description of the collision partners [30, 31, 35]. Interaction potentials are decomposed into atom-pair, atom-triplet, etc., terms, and the collision can be understood as a sequence of atom—atom encounters by analogy to the three-atom approaches to atom—diatom collisions [36,37]. It is then possible to describe scattering involving polyatomics, solid surfaces and molecules adsorbed at solid surfaces. Two physical limits of the general formalism have proved very useful in applications. They are the impulsive collision limit, which can be derived from the multiple scattering expansion, and the semiclassical limit which derives from a short-wavelength or eikonal expansion. These are considered in detail in separate sections which include several of our applications to scattering by diatomics and polyatomics. Related aspects of collisional TCFs have been developed by other authors. A study includes distortion potentials to describe slow collisions of atoms by polyatomics [38]. Wavepackets have been used to calculate cross sections in a related time-dependent approach applied to surfaces and are promising for molecules [39, 40]. Several other approaches have been explored for collisions involving polyatomics. In particular we want to mention reviews emphasizing a quasiclassical treatment of energy transfer into polyatomics [41], a semiclassical coupled-channels approach for polyatomics [42, 43], quantal treatments where slow (usually rotational) degrees of freedom are treated in a sudden-collision approximation [44], and approaches based on the solution of the time-dependent Schrödinger equation for scattering wavepackets [45—50]. No attempt will however be made to review the extensive literature on molecular collisions. This has been periodically done in publications of reviews and workshop lectures [51—55].

2. The basic formalism of collisional time-correlation functions 2.1. Time-correlation functions and collisional cross sections Let us consider two scattering partners A and B with internal states identified by quantum numbers aA and aB. Their initial relative momentum is given by P = Ilk so that the total initial state is k, a) with a = (aA, aB). The coupling of internal and relative degrees of freedom during the collision leads to transitions k, a)—+~k’,a’) and a net energy transfer after the collision. The quantum dynamics of these phenomena is described by the time-independent Schrödinger equation with scattering boundary conditions, or alternatively by the operator equations of scattering theory. The Hamiltonian for the system is H=H0+V,

I~Iü=K,~+H1~, HIfltHA+HB

,

(2.1.la,b,c)

.3.34

F. Vilallonga and 1)4. Mu-/ta. i/te calculation of mute—correlation fiiniiion,v fuir unolecular collision,

where Krei refers to the kinetic energy operator for relative motion. H~1to the sum of the Hamiltonian operators for the internal motions of A and B, and V is the coupling potential energy between the internal and relative degrees of freedom of the system. Asymptotic eigenstates and eigenenergies satisfy Hk,a)E~k,a).

(2.l.2a,h)

EE~+E,

where the total energy E is the sum of the relative kinetic energy plus the internal energy. The state-to-state or detailed rate R for the transition k. a) R(k. a~k’. a’) = (2~/~(k’.a’~T~k. a)~(E E’),

—~

k’, a’) is given by [2]

—

(2.l.3a)

1—H,) ‘. (2.l.3h.c) T= T+VGT. G(E)=(E where T is the transition operator of scattering theory. satisfying the Lippman—Schwinger equation. and G 1 is the propagator for free motion. The total initial and final energies E and E’ must be equal as is indicated by the Dirac delta function. This rate can he transformed into a time-correlation function using the relations

6(E

—

E’)

=

(2~)

I

f

dtexp[i(E

—

E’)t/~].

exp(iH~~~t/h)~a) = exp(iE~,t/h)~a),

(2.l.4a) (2.1 .4h)

which lead to

R(k,

~

a

=

k’. a’) =

~2

J

dt exp(ift/h) K a’~Tkk(t)~a) ~(a’~Tk.k(0)~a).

(k’~T~k). Tk.k(t)

=

exp(iH,il/~)7’~.~ exp(—iH1~~tIh).

(2.1 .Sa)

(2. l.Sh. c)

In this expression. F = Ek Ek. is the amount of kinetic energy transferred into the internal degrees of freedom, a positive quantity when the target is collisionally excited. The detailed rate is therefore given by the Fourier transform of the product of the (a’, a) matrix element of the operators Tk.k at times I and i~= 0. This operator depends only on the internal variables, and evolves in time in accordance with the Heisenberg picture of quantum mechanics. For collision partners prepared with a statistical distribution w11 in an experiment that measures only the k to k’ transition, for example in a time-of-flight experiment, the total or inclusive rate is given by R(k~k’)=~w,, ~ R(k, a~k’,a’).

(2.1.6)

This expression can be transformed using the completeness of the set of final internal states, to obtain [12]

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

R(k~k’) = h2 F(t; k’, k) =

I

dt exp(irt/h) F(t),

w~(a

Tkk(t)tTk.k(0)~a)

335

(2.1.7a)

(2.1 .7b)

,

which shows that the rate is given by the Fourier transform of the time-correlation function of transition operators. The function F(t) will be called the collisional TCF, and the double average with respect to the quantal states and the statistical distribution, will be indicated by double brackets, so that F(t; k’, k)

=

((Tk,k(t)tTk,k(0)))

(2.1.8)

.

Its Fourier transform can be defined by

f

=

F(t)

=

dt exp(iwt) F(t),

J

(2~)~

dw exp(—iwt) F(w)

(2.1.9b)

and satisfies several properties. In particular, from the definition of F(t) one finds F(t; k’, k)

=

~(Tkk(0)tTkk(—t)))

,

F(—t) = F(t)*

,

(2.1.10)

which leads to F(a)*

=

F(w)

(2.1.11)

and ensures real values for the rate insofar R(k—t. k’) = h2F(e/h; k’, k).

(2.1.12)

Expressions can be derived for the rates of elastic and inelastic collisions. Starting from the expanded form of F(t; k’, k) we can obtain the elastic TCFs restricting the summation over a’to states with the energy E~= E~.One then finds F(et)(ku, k)

=

~ (a

( Tkk)tP(Ea ) ~

a)

,

(2.1.13)

where P is a projection operator over the space of internal states with energy E~.This TCF is then independent of time, and leads to a rate for elastic scattering given by R(et)(k__~.k’) = (2~Ih)6(e)F~t)(k1, k)

,

(2.1.14)

339

F. Vtlallonga and D. A. Micha. 7he calculation of time-correlation Junction.s for molecular collusmon.~

which as expected is different from zero only when the energy transfer is null. It therefore involves only final wavevectors of the same magnitude as the initial ones. It is clear from this expression that the elastic rate involves the statistical average of the absolute value square of transition matrix elements. instead of the square of the average occasionally found in the literature. Inelastic TCFs and rates are given by the difference of their values for the total and the elastic expressions. Finally, an alternative expression for the rate used in some of our work follows from the definition of the TCF C(t) = F(—t). which gives

R(k~k’) = Il ~

j

dtexp(—iEt/h) C(t).

(2.l.lSa)

C(t) = K(Tk,k(0)Tk.k(t)~.

(2.l.15h)

Experimental results are generally expressed in terms of cross sections. Indicating with J(k) and p’(k’) the incoming flux and the final density of translation states respectively, and with .11 the orientation angles of k’ with respect to k, the state-to-state or partial cross section increment for scattering within the solid angle increment ~11at the total energy E is given by d3k’ p’(k’)R(k,a~k’,a’).

(2.1.16)

.sj2

Constructing translational states so that p(k) = p’(k’) and 11, the partial differential cross section is

1 and changing the integration variables to ~

=

du(a—*a’;k) f2ir\~ k’ dQ =~_~_)MJdE-~-R(k.a_~sk.a). ,

,

(~.I.l7)

where M is the reduced mass for the relative motion of A and B. The total or inclusive double differential cross section with respect to angles and the energy transfer follows immediately by adding over final states and averaging over initial ones, which gives du(k) Id~dfl

=

(2ir/l~)3M(k’/k)R(k—~ k’).

(2.1.18)

showing that the total rate is proportional to the physically measured cross section. Elastic differential cross sections follow from the TCF for elastic phenomena. Integrating the double differential cross section over transferred energies, we obtain (dO~/dI2)et= (21T/h)4M2Fd-’~(k’,k)

.

(2.1.19)

where now the initial and final wavevectors have equal magnitude. The calculation of these cross sections must be approached in ways appropriate to the nature of the interaction potentials and the values of kinematic parameters such as initial and final velocities and scattering angles.

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

337

2.2. Cumulant expansions for short-time events Characteristic functions may be studied in terms of their moments and cumulants [12]. Expanding F(t) in a power series centered at t = 0 one finds F(t)/F(0) =

~

(—it/h)~p~~In!

(2.2.1)

,

where p~is the nth order moment of F(w) IF(0). This is seen by expanding eq. (2.1.9b) around t = 0 and equating coefficients of equal powers of t in that equation and the one above. The result is

=

J

h~(2~1 dw w~F(w)IF(0),

(2.2.2)

or

=

J

dw w~(w),

~(w)

=

J

[h~F(w)]( dw F(w))

0.

(2.2.3a, b)

An alternative expansion in terms of cumulants,

(2.2.4)

F(t)/F(0)=exp(~ (_itIh)flK~In!),

where K~ is the nth order cumulant, is frequently more convenient in studies of F(t) for large I. The relation between moments and cumulants follows by re-expanding this equation in powers of t and comparing to the moment expansion to obtain forn1, The two lowest cumulants are then

,<~= ~

~=1•

(2.2.5a,b)

and K

When only these are different from zero, the function F(w) becomes a Gaussian distribution. This is in fact a distribution frequently encountered in impulsive collision processes where the duration of collisions is short compared with the periods of internal motions. Then one can make a short-time expansion of ln[F(t) IF(0)] and keep only two terms in the cumulant expansion. Moments can also be given a more explicit form in our case. Returning to eq. (2.1.9b) and using the cyclic property of the trace, t exp(—iHtIFl) T exp(iHtIIl))) (2.2.6) F(t) = ((T where we are again omitting the indices k and k’. Introducing the Liouville operator L defined by 2

=

/~2—

~.

,

LA

=

HA

—

AH,

(2.2.7)

335

E. Vtlallonga and 1). A - Micha, The calculation of time—correlation .1 unctions fdr molecular collusio,ts

it follows that

F(t) =

~(T~exp(—iLt/h)T~.

(2.2.8)

and by repeated differentiation with respect to I and comparison with the moment expansion one finds ~‘

(k’, k)

=

K((T~))IL~iT~) ~ K(T~)T~~ i.

(2.2.9)

where we have restored all indices. This expression shows that moments (as well as cumulants) are static properties. The moments from eq. (2.2.4) are not exactly equal to moments of cross sections for varying energy transfer F = hw because cross sections relate to F(~/Il)specifically for k’ = k,~.Hence an integral over F/h would involve also the dependence of k on However, for quasielastic scattering, where F is small compared to Ek and Ek~.we can replace k,’. by k’ in F(F/h). Then letting k = k, we find that ~,,(k’,.k,) is the nth moment of the normalized cross section. or ~.

~

k)

=

(J

d~dE~2)

~J

dF F”

dEdui’

(2.2.10)

In this case cumulants and moments are interrelated in such a way that K1 = = (F). k2 = p.-~ = 2), etc.. where (F) is the average transferred energy and ((SF)) is the square of (~) (~) = Here ((~F)and in what follows, a quantity between brackets means that it has been averaged its dispersion. over the F(w) distribution. When the collision time is short compared to the periods of internal motion, one can make a short-time approximation of ln[F(t)/F(0)1 and terminate the expansion in eq. (2.2.4) after two terms. Then the expansion is reduced to —

—

F(t) /F(0)

2K~/2!J (2.2.11) 1 + (—it/h) Substituting this expression into eq. (2.1.7a). and using eq. (2.1.18) one obtains the differential cross section for scattering into solid angle df2 with energy transfer dF.

dEdul

=

=

exp[(—it/Il)K

(~)4M2F(O)

.

~

(2)~

((~F)~) H 2exp(_~K~?~,

(2.2.12)

where k’ and F(0) depend on r and 12. An additional simplification occurs when the distribution is sharply peaked at a certain value of F, for example Fm~ so that ~F is small. Then the pre-exponential factor can be taken equal to its value at F,~, and an integration of eq. (2.2.12) can be carried out for —x < ~
d~ni(2 ~

=

~ ~)

-i.:

((~))

( (F (s))~ exp~- 2((~F)2) )~

(.2.l3)

This result tells us that, quite generally, a collision whose duration is short compared with the periods of internal motion of the target will lead to a Gaussian distribution of the transferred energy.

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

339

2.3. Molecular interaction potentials and transition operators Potential energies for the nuclear motions in a polyatomic system can be obtained from the Born—Oppenheimer separation of electronic and nuclear motions, for each adiabatic electronic state. Their values E can be separated into asymptotic contributions giving internal potential energies VA and VB, and a remainder term V describing the interaction potential. The total potential energy E is a function ofall the atomic positions in the collision partners, RA = {Ra; a = 1,. NA) and RB = {Rb~b = 1,. NB}. It can be conveniently expressed in terms of the relative position vector R = RA RB between the centers of mass of A and B, and their internal atomic position vectors ra = Ra RA, rh = Rb RB, collected in rM = (rA TB) = {rn; n = 1, NA + NB}. We write .

. ,

.

. ,

—

—

—

.

E(R,rM)=VA(r~~)+VB(rB)+V(R,rM), V(R,rM)—~0forR—~x.

. .

,

(2.3.la,b)

The potential V can be written as a sum V(R,rM)=VSR(R,rM)+VLR(R,rM)

(2.3.2)

of a short-term VSR, which is repulsive at short distances R due to the repulsive energy of closed electronic shells, and a long-range term VLR which can be readily constructed from molecular multipoles and polarizabilities, and is attractive for neutral ground electronic states. The long-range interaction must be cut off at short distances R to avoid unphysical behavior at the origin, and remains small at all distances. It can therefore be treated as a perturbation in many cases. The short-range interaction is however very large and cannot be the subject of a perturbative treatment. When the atomic displacements from equilibrium are not large during the collision, it is sufficient to introduce equilibrium positions and displacements, r,~= d~+ u,, with the displacements u,~defined as usual so that the instantaneous center-of-mass position and Euler angles TM = (TA TB), TA = (a, /3 ~)A remain unchanged by the displacements. Short- and long-range potentials can then be written as functions of R, TM and UM, with the long-range term weakly dependent on the displacements. A suitable perturbation expansion can be based on the two-potential formulation of the Lipmann.— Schwinger equation [2] which gives in our case [30] T=

TLR +

TLR

=

~

=

W~t(VSR+

VLR + VLRGLRVLR,

W~P.?=

(E~ H

t

—

0

—

(2.3.3a)

VSRGVSR)WLR,

,

G

=

I+ ~

G~~VLR,

—

H)t

.

(2.3.3b, c) (2.3.3d, e)

VLR)

In this expression, the first term in T describes transitions due only to the long-range interaction, while the second term describes (from right to left) distortion of the inward motion by the long-range interaction, as given by the wave operator WLR, intermediate transitions due to short-range interactions (indirectly affected by long-range interactions present in the total propagator G), and distortion of the outward motion by long-range interaction. Alternatively, V can be written as a sum of atom-pair, atom-triplets, etc., terms,

341)

F. Vtlallonga and D. A. Micha. The calculation of tunic-correlation functions for unolecular collusion,

V= ~ v~t,(R,~, R,,) + ~ v,,,,t,(R,,, Re,. R1~)+~,

(2.3.4)

a.I,,b

(LI)

where Vat, is the interaction between atoms a and b, v,1t,1,. is the intrinsic three-atom interaction of a in A with b and b’ in B, and so on. These n-atom terms depend not only on the atomic positions but also on the orientation of the valence electronic states of each atom within the molecules. They furthermore contain both long- and short-range terms, such that their sums reconstruct the decomposition of V into long- and short-range terms. The decomposition of V into a sum over clusters of atoms has been found useful in the parametrization of a wide variety of potential-energy surfaces [56—591. This decomposition of the potential suggests a similar one for the transition operator. It can he done using the formalism of multicenter scattering [141,with a more general definition of the potential terms. Writing the potential as a sum over 2n-atom clusters C. (2.3.5) the transition operator can be written in the form of a final-channel decomposition 11 ~ T~

T”

=

T~+ T(.G, ~

~,

T~.= V~+ V(.G

T

0T( .

(2.3.6a, h, c)

1 describes here a transition where the last interaction occurs only within the cluster C’. The T” to similar operators except itself. It can be obtained by first calculating the simpler and term is coupled auxiliary operators T~.which only contain the interaction potential Ve and the free-motion propagator G 11 including the internal potential energies of A and B. These two examples show that the way one constructs the operators Tk.k appearing in the collisional TCFs depends on the potentials being used. Consequently their calculation must be based on different expansions and limits depending on the nature of the collision partners and on kinematical parameters. ,

2.4. Intramolecular motions and the separation of time scales Another important consideration in the calculation of transition operators and TCFs relates to the velocities of the internal degrees of freedom (df’s) of the collisions partners. It is frequently possible to classify these df’s into slow and fast ones, indicated with X and x respectively, depending on whether their velocities are small or large. In the case of polyatomics, bond-distance and bond-angle vibrational motions are fast, while rotations, librations, and torsional vibrations are slow. These degrees of freedom can be separately treated in calculations of TCFs. We modify here a previous treatment for semiclassical scattering [60,611, to deal with the fully quantal scattering case. Indicating with s, f and r the slow internal, fast internal and relative degrees of freedom, we decompose the internal Hamiltonian and the interaction potential by letting H1~~H,+H~+H51, H5=K5+V5,

H~=K~+l”),

V=Vt+l/~+l/~tr.

(2.4.la) (2.4.lh,c) (2.4.ld)

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

341

Our assumption is that the velocities of slow df’s are much smaller than velocities of fast df’s and of relative motion. For comparable masses, this means that the kinetic energy operators satisfy ((K5)) ‘~ ((Kr)) and ((Ks)) ~ Ek and that these inequalities also hold during the collision. We first consider the case where only slow df’s are present, and are excited by coupling to the relative motion. Let 7”~be the transition operator when K5 = 0. The slow variables X are then fixed, and only elastic scattering by the potential V~+ ~‘sr can take place. The full transition operator T can then be written as T= W~T5,

(2.4.2)

which defines a wave operator W~accounting for changes in the slow variables; this becomes the identity operator when the kinetic energy of slow motions is neglected. An explicit form for ~ can be derived from the Lippman—Schwinger equation for the operator T. For a given transition k—t. k’, and using the completeness of the set {~q)} of states for relative motion, we can write Tkk

=

f

3q W~?q(Ts)qk,

(2.4.3a)

d

W~= (k’~W~q)

(Ts)qk

,

(q~T 5~k) .

(2.4.3b, c)

The two operators in the factorized expression depend on the slow variables X. The one on the right is furthermore diagonal in the coordinate representation, so that (Ts)qk

=

f

dX X)Tqk(X)(XL

(2.4.4)

where ‘rqk is a transition amplitude function (off-energy shell, since Eq Ek) for scattering of the projectile by the target in configuration X. We next consider two special cases that lead to very simple TCFs. When the velocities of the slow variables are very small, we can replace K5 = 0 in the wave operator, which is then equal to the identity. The TCF becomes Fkk(t)

=

f f dX’

dX Tk(X’)(( P~.(t)P~(0))) Tk.k(X),

(2.4.5)

where P,,~.= X)(X~is the projection operator on the state X). In this case, the scattering amplitudes Tkk can furthermore be replaced by their on-energy-shell values, since Ek = Ek. The second case occurs when, in addition to neglecting the velocities of the slow variables, the remaining TCF can be obtained from short-time expansions as described in the section on cumulants, or from asymptotic expansions as shown for rotational motions in ref. [16]. This can lead to diagonal expressions in the X variables of the form ((P~.(t)P~(0))) = 6(X’

—

X)D(X; t),

where D is a position self-TCF, and the total TCF takes then the simple form

(2.4.6)

F - Vilallonga and 1). .4 .51k/ta. lhe calculation of time-correlation fiotction.s fIr molecular collt,sions

342

Fk,k(t)

=

f

dX rkk(X)~2D(X;t).

(2.4.7)

The Fourier transform of this expression leads to a very simple expression for the doubly differential cross sections, involving the transform D(X, F/h). This self-TCF acquires a specific form when

P~.(t)P~(0) = ~(X X’)P~exp[iy(X. —

t)]

(2.4.8)

.

where y is a function, as is the case for rotational motions. Then D can he obtained from the statistical distribution of slow variable values, wjX)

=

~ w,,~(aJX)V,

(2.4.9)

containing the original statistical distribution w, of the slow motion states. One finds D(X,

t)

=

wjX) exp(iy(X.

t)]

,

(2.4.10)

and the inelastic cross section depends on the transferred energy F only through the Fourier transform of this exponential. Continuing with the more general case where both slow and fast df’s are present. we consider the case where the fast variables are localized, as happens for vibrational df’s. We approximate the internal Hamiltonian by H,,~=H~+H, H=H+K(H~~.

(2.4.lla.h)

where we have averaged the internal (sf) coupling over a distribution of the fast eigenstates of H 1. Then we can write for the initial states a) = ~)~a1), a direct product of the eigenstates of the fast and the modified slow Hamiltonians and find for the time-evolution operator U,~~(t) = U,(t)Uf(t). We factor the full transition operator T as before, but now WN contains both slow and fast df’s. The TCF is given again by the same statistical average. Its time dependence comes from both slow and fast time evolution. This however can be separated into two contributions; one can first calculate the TCF of the fast df’s assuming that the slow ones are fixed, which can be obtained by letting again K~= 1) in the operator WH then one can allow for the time evolution of the slow df’s. We write W~(t)P~(t)~ Px(t)12k~q(X~ t) .

(2.4.12)

where the time dependence of 12 comes only from the fast df’s. The TCF for the fast df’s must first be calculated for given initial values.(X’, X). The treatment of the slow df’s can then be done as above. We therefore find that the total TCF is given by a nested expression, where an inner TCF describes fast—fast correlations and is obtained by averaging over fast motion states for fixed slow df’s (X’. X)~ this result is then incorporated into the final expression containing the time evolution of slow df’s and integrals weighted by scattering amplitudes for given values of X and X’. In specific models of elastic scattering, such as scattering from rigid bodies, or classical scattering, the wavevectors q and q’ become a function q = q’ = q~(b)of the initial wavevector k and of the impact

E. Vilallonga and D. A. Micha, The calculation of time-correlation functions for molecular collisions

343

parameters b, and energy is conserved so that Eq = Ek. The six-dimensional integrals over these wavevectors can then be replaced by a double integral over impact parameters or equivalently by integrals over the two scattering angles. A further stationary-phase approximation to these integrals gives then the specific impact parameters of the trajectories contributing to the k—+ k’ transition. This has been shown and implemented in a semiclassical treatment of molecular collisions [16, 62]. The nested TCF expressions can be readily applied to the separation of fast and slow motions in calculations of collisional energy transfer. For example, in a case involving rotational and vibrational motions, the slow variables are the Euler angles T and the fast variables are the displacements U; one would first calculate the inner TCF for vibrational motions while the orientations of the molecules are kept fixed. The result would then be weighted by a time-dependent function of the orientation angles describing the effect of rotations on the vibrational TCF for each initial vibrational state.

3. The calculation of cross sections for impulsive collisions 3.1. The multiple scattering expansion We consider scattering of an atom A by a polyatomic system B made up of atoms b = 1,. NB, SO that the position R of A refers to the center of mass of B. The interaction potential can be expanded as before into a sum over clusters, with the leading terms corresponding to the interaction of A with each atom b, given by an atom—atom potential 1~b~We assume that the interaction V can be correctly described by a sum over atoms in B, . .

,

NB

V(R,r8)=~vb(R,rb).

(3.1.1)

The corresponding final-channel decomposition of the transition operator is then T= ~ T~

T(b)

=

Tb +

TbGO

b’b ~

Tb

=

Ub + UhGOTh

(3.1.2a, b, c)

.

The (k’, k) matrix element of this transition operator is also a sum over the atoms in B, from which it follows that the collisional TCF is a sum over pairs (b, b’) of atoms in B, F(t; k’, k) = ~ F(t; k’, k)~’~ F(t; k’, k)~’~ = ~ ,

,

(3.1.3a, b)

and the rate is given by R(k-.-t’ k’) = h2 ~ Re[F~’ ~(r/h; k’, k)],

(3.1.4)

where we have used that p(bb)(w) = [F(b’b)(w)l* This formally exact expression shows that the cross sections are related to the correlation of the transition amplitude describing a final interaction with atom b’ at time t = 0, and the amplitude describing a final interaction with atom b at time t. Iteration of the coupled equations for the components of T gives the multiple-scattering expansion

F. Vtlallonga and D. A. Micha, The calculation of time-correlation functions for molecular collisions

344

t,1=

T

Tt,+~Tt,G

0Tt,+~,

(3.1.5)

where the second term includes all double collisions of A, first with each b’ and then with b; the following terms in the expansion describe sequences of threen-atom collisions where A interacts finally with b. This corresponds to the intuitive description of scattering by a many-atom system as a sequence of atom—atom collisions within the target. One expects on physical grounds that some atoms in the interior of the target will not contribute to a particular k—p k’ transition because they will be hidden by exterior atoms. This can happen because the interior atoms are shadowed by exterior ones as seen from the source of projectiles. or because they are eclipsed by exterior atoms as seen from the detector of scattered projectiles [631. This can he analytically described by writing 11’~ = w1’~T T” = Tt,W 1, (3. 1.6) ,

where the second member of this equation defines a wave operator describing the shadow effect Ofl h. while the third member defines a wave operator describing the eclipse effect on b. Replacing the second member in the atom-pair TCF. we write 1’~= ~([ Tt,W~ h]k,k(t)t[ Tt,W15~~]kk(0))~, (3.1 .7a) F(t; k’, k) = w6~(k)~ K((Tt, )k.k(t)’( T,, )kk(0)~w~ 1(k). (3.1 .7h) where we have defined the multiple-scattering coefficients w° describing the fraction of collisions where A starts with momentum Ilk, and undergoes a sequence of scatterings within the target to finally encounter atom b. Expressions for these coefficients can be derived from the theory of optical potentials, which has been developed for nuclear [14], atomic [64]and molecular collisions [65]. In a simple approximation. we let T,,W1”~k.a) T 1’~~(k) and find for the multiple-scattering coefficients the coupled 1jk, a)w equations ~,

w1’~1(k) = I + ~ (KKk~Tt,GTt,~k)~ K(k~Tt,~k)~ ~ )w1’

(k).

(3.1.8)

These coefficients can he estimated for molecular targets from geometrical considerations and can he expected to be very small for atoms in the interior of compact targets such as CF 4 and SF6, and in the inner layers of solid surfaces; they should instead be close to unity for exterior atoms and atoms in linear molecules such as N., and CO,. Assuming that the w-coefficients are known, one is yet left with the calculation of the Tt, transition operators. These are not just two-atom operators because the propagator G1 includes all the internal motions. It is possible however to introduce a very useful approximation when the A—b collisions are impulsive, that is, of short duration and involving large forces localized in space. 3.2. The impulsive collision limit

In the course of an impulsive collision of A with atom b at position Rt, and with velocity v1,, a large force F1, acts on it for a short time ~t. The position of h changes by the negligible amount v1, ~t. hut its

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

345

momentum changes by a significant amount of order Fb ~t. Consequently the kinetic energy of B in initial state a changes from (Kb)~to (Kb)a in the final state a’, and its internal energy changes by E E~ (Kb),, (Kb),, (3.2.1) —

—

—

.

Energy and momentum transfer into B occur through the A—b interaction while the other atoms remain undisturbed during the short collision time, providing only the force field in which b moves. The (NB + 1)-atom Hamiltonian and the propagator for free motion G0 can then be approximated by the A—b propagator [30] 14~ KR —

G0(E) =

gfl~(E)= (E E,, (Kb),,

—

Kb

—

E~)_l

(3.2.2a)

,

—

—

_

(Kb),,

—

(3.2.2b)

,

where the second line gives the energy of B after subtracting the kinetic energy of the stricken atom b. Consequently, Th(E)

—

tb(E)

=

Ub +

v~g 0~(E)tb(E).

(3.2.3)

The propagator for non-interacting A—B has therefore been replaced by the propagator for A and b moving freely, while the (NB + 1)-atom transition operator is approximated by the much simpler two-atom operator tb, which involves only the A—b potential interaction. The (k’, k) matrix elements of this two-atom operator can be readily calculated [30]. We change wavevector variables from (k, kb) for A and b to the total and relative-motion wavevectors given by Kb

=k+ kh,

k/M—kblmb,

Kb/’P~b

respectively, where ~zb= MmbI(M find

+ mb)

(3.2.4a,b)

is the reduced mass of the A—b pair. Using these variables we

(k’k~tb(E)~kkb) = ~(K~ Kb)(, t~(rh)~scb) (3.2.5a) 2K~I[2(M + mb)], (3.2.5b) Tb = E E~ h where t~°is the transition operator for relative motion in the A—b center-of-mass frame. We next introduce the assumption of quasielastic scattering, whereby internal velocities can be neglected compared to relative ones. The relative wavevector and energy are then given by —

—

=

mbkl(M

,

—

+

mh),

~

=

(3.2.6a, b)

2I~Lb(hkIM),

and the two-body t-operator for the k, a) k’, a’) transition can be written as the product of a factor r describing the scattering of the free A—b pair times a factor describing the recoil transition from a to a’ due to the relative momentum transfer IlK = h(k k’) [30], —~

—

(k’, a~tb(E)Ik, a) ‘rb(k’,

k) = ~

=

Tb(k’,

k)(a’~-q~(K)~a)

(3.2.7a)

,

flb(K)

=

exp(—iic I~~) ,

(3.2.7b, c)

F. Vilallonga and D. A - Micha. The calculation of time—correlation functions for molecular co/li.sio,t.s

34h

where we have introduced the position operator ~t, = —i3/~kt,.and the last line eq. (3.2.7c) expresses the spatial Fourier transform of the density operator for atom b. The rate of energy transfer can now be written as a sum over pairs in B, collecting the previous results. We first find, using the factorized form of the I-operator, that F(t; k’, k)~”~ = r(I)*(k~, k)((?)t,(K, t)~t,(ic, 0)))i-~(k’, k), rt,(k’, k)w11~(k),

T~(k’. k)

=

~t,(K, t)

exp(iH

=

(3.2.8a) (3.2.8h)

111/h) 17t,(K) exp(—iH1,t/h)

(3.2.8c)

,

where the new T-factor is a quasielastic transition amplitude that incorporates multiple-scattering effects, and we have shown that the density operator evolves in time in accordance with the internal Hamiltonian. The rate is given by the Fourier transforms of these pair correlation functions, 11’1(k’. k)5S,,t,(K. R(k~ k’)

=

St,,,.(K, w)

=

F/h)r11’1(k’,

k)],

(3.2.9a)

~ Re[r

J

(2w)

dtexp(iwt) K(~t,(K,t)’~t,.(K, 0))~,

(3.2.9b)

where S is a dynamical form factor given by the TCF of atomic-density Fourier amplitudes. These are well known from the theory of inelastic neutron scattering [11]. They are found to play also an important role in atomic collisions with polyatomic targets, even though here the magnitudes of momentum transfer are much larger and the interaction potentials have much larger ranges. This similarity can be traced back to the importance of target atomic conformations for the description of all types of inelastic collisions. Proceeding to the cross sections, these can be written as a sum of the scattering contribution where b = b’ plus the contribution where they differ. We find that for h = b’, the form factor S contains exponentials with the large exponents. ic~(r,, r,,.)>1, and is rapidly oscillating, reflecting coherent scattering. For the usual experimental conditions, the statistical weights implicit in the double bracket will average these form factors into very small numbers which can be neglected. —

St,t,~O.

hh’

(3.2.10)

.

The only remaining terms will therefore give the incoherent scattering expression dodEdQ =

~ dut, dP,, ~ (~)1M2

~

~

(3.2.Ila)

.

T~(k’, k)~.

dP 1,

=

St,t,(K, ~

,

(3.2.1 lb. c)

where the atomic cross section duh/dI2 describes the quasielastic scattering by atom b corrected by multiple scattering shadowing. This atomic cross section can be obtained for example by calculating the

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

347

classical trajectories for the sequence of encounters of the projectile with the atoms in the target, and identifying the instances where its last encounter occurs with b. The double differential cross section follows then by multiplying atomic differential cross sections times probabilities of energy transfer from dynamical form factors for each of the stricken atoms. These form factors can readily be calculated in a number of important cases. 3.3. Collisional excitation of harmonic vibrations 3.3.1. Time correlation of normal vibrational modes

For most molecules, except H2, the time scales for rotation are usually much longer than vibrational periods, so the respective correlations are evaluated separately by the procedure of section 2.4. The vibrational correlation is calculated most conveniently by working in the body-fixed (BF) reference frame defined by the principal axes of the isolated target. The set of Euler angles F = (a, /3, y) specifies the orientation of the BF frame with respect to a space-fixed (SF) coordinate system; conventionally, a and /3 denote the azimuthal and polar angles, respectively, of the BF z axis, while y denotes rotation about this axis. The conditions of zero total linear and angular momenta in the BF frame define fRT = and 6 relationships among the vibrational displacements Ub, respectively, for linear and nonlinear molecules. For an N-atom polyatomic, this leaves fv = 3N fRT free internal displacements s, defined by the changes in bond lengths and bond angles [66]. In the frequent case of small displacements, one relates the Ub and the s, by the linear transformation u = Bs, the u and $ indicate column matrices of N elements U~, and of f~ elements s~,respectively. The N x f~ transformation matrix B is composed of elements BbI(T) which are vector coefficients that depend on the orientation of the BF frame. Indicating with IJ~and HR the purely vibrational and purely rotational components of the internal Hamiltonian, one then applies the separation procedure of section 2.4 for the present case of fI~= HR and H1 = H~.Letting F~(t,ic) denote the impulsive TCF ((r~(K,i~)’q~(K, 0))), one obtains the nested expression —

F~(t,K)

=

((exp[iK db(t)] F~(t,K; F) exp[—iic d~,(0)]))

F~(t,~ F)

.

=

~exp(i

~

(K Bb~)sI(t)) exp(—i

~

,

(K

Bh~)s~(0))).

(3.3.la) (3.3.lb)

The calculation of ~ is outlined next for the case of harmonic vibrations, and the following section incorporates anharmonicity. In general the Hamiltonian for harmonic vibrations is given by the quadratic form H~=

~(aTMa

+ STKS),

M11

=

~ mbBbI B~1,

~

=

~,

(3.3.2a, b)

where the superscript T indicates the transpose of a matrix and K is the f~ x fv diagonal matrix of harmonic force constants. The internal displacements are first expresse~4a~linear combipations of normal coordinates i.e. diagonal $ = GO,matrix whereofCvibrational is the solution of the well-known eigenvalue problem 2. Here wQ., is the eigenfrequencies w KG = MCw 1 given by the roots of the characteristic equation K = 0, and C is composed of the eigenvectors with the conventional normalization CTMC = I, where I is the identity matrix. The vibrational eigenfrequencies and eigenvec—

F. Vilallonga and D.A. Micha. The calculation of time-correlation functions for molecular collusion.,

348

tors are calculated by the standard techniques of normal-mode analysis [66]. Next, for each normal mode one introduces creation (a~) and annihilation (a1) operators [67]. so that = (h/ 2(a 2w1)t 1 + a). The vibrational Hamiltonian 2[a reduces to the well-known form H~,= ~ hw1(a~a1+ f,), from which it follows that Q1(t) = (h/2w,)’ 1 exp(—iw1t) + h.c.], where h.c. denotes the Hermitean conjugate of the preceding term. Since [a1, a~.]= the vibrational correlation function becomes factorized as the product of the correlation functions contributed by each normal mode, ~,.

F~(t,~ F)

=

[1F~(t,

F),

K;

(3.3.3a)

2(K .DhJ)[a, exp(—iw F~/~ = ~(exp{i(h/2w1)’

1t) + h.c.1} 2(ic .Dt,

1)(a,

exp[—i(h/2w1)”

1 +

a 1)]))

(3.3.3b)

,

where 0(F) = B(F)C is the N x f~,matrix describing the transformation from normal modes Q, in the BF frame to atomic displacements U~,in the SF frame. The ~ are calculated separately for each mode via the relation [68] exp(X) exp(Y) = exp(X + Y + ~[X, Y]) ~‘here2))1 X and are linear combinations of for YBoltzmann distributions of a1 and a~.The Bloch theorem ((exp(X + Y))) = exp[~(((X + Y) initial vibrational states [67] then gives F~(t,~ F) = exp(t’~.”~{coth a

1 [cos(w,t) 1(K,F)

—

1]

—

i sin(w11)})

,

(3.3.4a)

h(,c Dt, 2/2w

=

.

1)

v~”

1,

a,

=

hw,/2kBT,

(3.3.4c, d)

.

where kB is the Boltzmann constant and T, is the initial temperature of mode j. The generating function of the modified Bessel functions of the first kind (I,,) then leads to F~(t,~ F)

p,

(61

(K,

F)

exp(in,w1t)

=

exp(—n1a1

=

(6)

—

ii~

(3.3.5a)

p~.

coth a,) /,,(e1(6) /sinh a1)

.

(3.3.5h)

where the indices n are equal to the change in quantum number (initial minus final) of mode], and I,, is evaluated from the series [69] 1~

1,,(z)

=

(~z)”

(~z)21/[l!(InI+ 1)!].

(3.3.6)

t=II

A direct numerical implementation of the above would lead frequently to overflows due to large exponents and/or factorials. The overflows are avoided by grouping factors so that numerators and denominators are of comparable orders of magnitude, which yields the numerically more reliable expressions p~(ic,F)

=

exp(— i’ cosh a) [(vp)~”1/~nI!]Srni(P/sinh a)

,

(3.3.7a)

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

S,,(z) =

ct,

c1

=

2c ~z 11I[l(n + 1)],

349

(3.3.7b, c)

1for n ~0, otherwise p = [exp(2a) l]~,and we have starting with c0 = 1,j where p for = [1—exp(—2a)]~ omitted the indices and (b) the sake of clarity. The above summation is accumulated term by term, until it has converged to the desired relative error. In our calculations we have found that the series converges to within i05 with ten or fewer terms for the typical values K ~ 150 A. —

3.3.2. Poisson distributions of transferred vibrational energies Additional simplifications are possible for low initial temperatures. Whenever T 1 4 Ii WI/kB, then a1 ~ 1 and therefore sinh a1 cosh a3 ~(exp a1). In these cases the argument of the Bessel functions is much smaller than unity, so one may retain only the leading term in the expansion (3.3.6); hence [70] 1/~n~!, (3.3.8) n)a i’] t where we have set coth a — 1 for a 1, and we have omitted the indices j and (b) for brevity. For a fixed value of ii, i.e. for fixed K and F, the probability of vibrational de-excitation (n > 0) decreases exponentially as n increases. This trend is greatly emphasized as the initial temperature is lowered, because a ~ lIT, and vibrational de-excitation vanishes in the limit of T—~0, as one expects. Instead, the probability of vibrational excitation (n <0) follows a normalized Poisson distribution of n~values, independently of temperature for kBT/ 4kw 1, with the average value of n~equal to v. Such distributions have in fact been measured by several scattering experiments [71—73].One can also take advantage of the approximate expression for the p,, to estimate a priori which values of n will give transition probabilities that are numerically significant for the given K and F. This represents a significant computational advantage over basis-set expansions, where one would have to redo the calculation with progressively larger basis-sets to ensure numerical convergence. p~ ~exp[—($n~+

—

~‘

3.4. Collisional excitation of anharmonic vibrations The calculation of TCFs for anharmonic forces is not as straightforward as in the harmonic case, since the evolution of atomic displacements is usually not known in closed form. In fact, quantum dynamical calculations are notoriously difficult for anharmonic polyatomics. As a consequence anharmonicity has often been treated as a perturbation to purely harmonic motion by expanding the observables in a power series of the anharmonic couplings. Such perturbation series may converge relatively slowly, depending on the strength of the anharmonic couplings and the energy of the system. Alternatively, for small momentum transfers, such as in cold neutron scattering from solids, the ~b(K, t) have been expanded in a low-order power series of K [11]; the resulting approximations are then appropriate only to few-quantum transitions, whereas multi-quantum transitions are frequently observed in experiments [72,73]. Accordingly, we developed an algebraic treatment of molecular anharmonicity whereby the TCFs incorporate anharmonic couplings as well as K to infinite order. The algebraic calculations are well suited to symbolic manipulation programs, such as MACSYMA [74] and MATHEMATICA [75]. A wide variety of anharmonic couplings are possible in polyatomics, so here we outline the procedure along general lines, and refer the reader to refs. [76,77] for the details of its application to specific molecules.

F. Vilallonga and LI. A. Mkha, The calculation

35(1

of time-correlation functions for molecular collisions

3.4.1. Cumulant expansion of dynamical jbrm factors In order to abbreviate the notation, let x, denote the projection of the vibrational displacement u,,(t) in the direction of i, so that K• u,,(t) = KX,. The calculations are complicated by the fact that exp(iKx~)exp(—iKx11) ~ exp[i,(x, x11)]. Nevertheless, the exponents can be collected by introducing —

the superoperator 0 which, when acting on a product of powers of x and x,. orders all the powers of x, to the left of all the powers of x,. One can then write

F~(t,ic; F)

=

K(0 exp[iK(x~ x)]))

(3.4.la)

=

~ (iK)~((O(x,—x0)”~/n!.

(3.4.lh)

0

where the average of the ordered exponential is defined as usual in terms of the moment averages of the ordered exponent ((0(x, x,1)”)). Instead of truncating the above series, which would correspond to retaining only few-quantum transitions [Ill, we expand the logarithm of F~,.in a power series in K. This is readily done by means of cumulant techniques [78], which give [76. 77] F~(t,~ F) where the (((x, —

—

=

exp(~(iK)”~((x, x,,Y~~/n!).

(3.4.2)

—

x~))))~. are cumulant averages defined in terms of the moment averages by

x11)’))~= ((0(x,

—

x11)’))

—.. ‘~ ~

(

n

(n m

~—

1

((0(x,

x1)””))(((x,

—

.(

(3.4.3) This recursive definition is exploited most profitably by symbolic programming. One also takes advantage of the fact that ~(x’),~is independent of t due to the invariance of the trace of a product of operators with respect to a cyclic permutation of their order. For examples, the first few cumulants are —

x1t)~~ = 0,

~((x1 x~)~(. = 2( ~(x)~ —

—

~(x,x11~),

(3.4.4a, b)

x11) )~ = 3(~(x,x0)~ ~(x,x11)~). (3.4.4c) 1,whose evolution is highly complicated for anharmonic forces, are thus systematically The F~ obtained from the simpler displacement—displacement correlation functions (DCFs for brevity), such as ~(x’x~’)~.Each term of the cumulant expansion is of order of (K((x))’2)”, which implies fast convergence whenever K ((x2)) Ii 2 < 1. It is significant that this condition involves the unperturbed displacements, instead of their instantaneous values during the collision, which could be considerably larger. Many scattering experiments are carried out at initial temperatures T tlwJ/kB, so that only the 2)) .2 <1 4 ~ and consequently lowest vibrational states are populated initially. In those cases, K ((x F~(t,~ F) exp{- (([KS uh(0)1)) + (([K. ut,(t)][K. uh(O)]))}. (3.4.5) 3

—

2

2

—

The calculations can be systematically refined by including progressively higher order cumulants if necessary. The resulting F~includes the dynamical effects of intramolecular anharmonic forces to

E. Vilallonga and D. A. Micha, The calculation of time-correlation functions for molecular collLsions

351

infinite order, since they are contained implicitly within the evolution of the Ub as well as explicitly in the average over the initial distribution. Therefore the present treatment is more efficient than the more conventional perturbative expansions of Fv in powers of the anharmonic couplings. To calculate the necessary DCFs one must first choose a coordinate representation for the anharmonic force field. In order to simplify the ultimate numerical work, it is often convenient to employ the set of coordinates that diagonalizes the harmonic component of the vibrational Hamiltonian. For this purpose, the intramolecular potential Vv is written as the sum of purely harmonic (Vh) and anharmonic (Vaflh) components, — —

where

$

17 “h

.±.

ii

.

Vanh

~‘h

—

i

—

2

T

(2)

(2) ,

— —

02VV 3s 35 s=o

A ~

,.~t.ua, c ~,

indicates the column matrix of internal displacements s

1. The Vaflh can also be expanded in a power series of the s1 involving cubic K~, quartic K~J,,. ,couplings as appropriate to the given molecule. Following the procedure outlined in section 3.3.1, the (([KS Ua(0)][K Ua(t)])) are then expressed in terms of the coordinates that are normal modes with respect to Vh, so that T)) + (( Q(t)QT(0)))]DT}bb ic), (3.4.7) F~(t,~ F) = exp(K~{D[—((QQ where the subscripts bb indicate the (b, b) element of the matrix between the braces. As before, the matrix D corresponds to the transformation from coordinates in the BF frame to Ub displacements in the SF frame. It is important to keep in mind that the Q.(t) here do not execute simple harmonic motion, because their evolution is also determined by Vaflh~The ((Q(t)QT(O))) are then calculated from their associated Green functions, as follows [77]. . .

.

Q1

Q~

3.4.2. Decoupling of the Green function hierarchy For any two operators A and B, the double-time retarded (+) and advanced Green functions are defined in terms of their correlation functions by [79] ((A(t); B(t’)))’~5’1= ~[±(t t’)] (([A(t), B(t’)])), where ~ is the Heavyside step function. The Heisenberg equations 3Q1 01 = i[Hv, Q(t)]/k lead to the following equation of motion for the matrix of Green functions corresponding to the DCFs: (—)

~

(132/3t2 where

~0h is

+

w~)((0(t); QT(t’)))1’~

—

QT(t’)))~~ = —h~(t t’)I

((fanh(t)

—

(3.4.8)

,

the diagonal matrix of eigenfrequencies corresponding to Vh, and

—

~

11~his a column matrix with elements f1 = ~0Vaflh/t9Qjwhich are the anharmonic forces that act on the One could also formulate equations of motion for the force-displacement Green functions ((faflh(t); QT(tl)))(±) which would involve Green functions of higher-order derivatives of Vanh. One would thus develop an infinite hierarchy of coupled differential equations for each matrix element, which is computationally intractable unless it is closed by means of a decoupling approximation. One class of decouplings is particularly well-suited to the typical experimental situations; kBTV 4kw1, where the vibrational displacements are accordingly small. Anharmonic vibrations can then be represented on average by effective forces linear in the Q(t), which are implicitly defined by Q..

T(tI))) ~ ((fanh(t);

Q

—

A(( 0(t);

QT(t’))) ~

,

(3.4.9)

352

F. Vulallonga and L). A. Mic-ha. The calculation of time-correlation functions fIr molecular collision.,

where k is a real symmetric matrix of time-independent parameters A11.. The A11. are chosen to reproduce the dynamical features of the exact DCF during the brief collision times that are characteristic of impulsive energy transfer, as will be shown shortly. Alternatively, when the initial population of high vibrational states is non-negligible, the treatment is systematically generalized by means of nonlinear effective forces as follows: 1

=

((f,(t); Q1(t’)))

A~I2~Q

~

1t1

((

1(t); Q1(t’)))~+

/

~t,,i

A~ Q1(t)Q,~(t);Q1(t’))) ((

~ A~,,~(Q

+

1(t)Q,,,(t)Q5(t); Q 1(t’

)

~

+...

-

(3.4.10)

1,,,,,

1then form a closed set of The equationsordinary of motion resulting equations for the ~(Q1(t)Q,,(t)” ; Q1(t’))~ that are readily solved via second-order differential with constant coefficients Fourier transform techniques. The ((Q 1(t)Q,,(t)~ ; Q1(t’)))~required for a given molecule will in general depend on the relative strength of the real anharmonic couplings, i.e. within Van6~as well as on the geometries of the bonds, the initial temperature, etc. Hereafter we outline the procedure using the linear model as an example, since we have found it to be quite adequate for the usual experimental situations [76, 77]. The equations of motion of the Green functions, presently 2/3t2 + A)(( 0(t); Q’(t’)))1 = —h8(t t’)I (3.4.11) . .

—

*

—

,

(10

are solved via their Fourier transforms G1 1(w), which here become G~(w) = lirn h{2rr[(w

±u

2!

+

—

A])

-!

-

7)

For nonlinear decoupling schemes, the G~(w) are evaluated most conveniently by symbolic programming, and they involve polynomials of accordingly higher order of (w ±u 2 in their denominators. Analytic expressions are necessary for G1~(w)in order to take advantage 1) of the relation [791

=

~i x

f

dw exp[—iw(t

lim [G~(w

+

—

t’)

-

hw/(2kBT\,)]

i~) G~1(w i~)]1cosech[hw/(2KBT~,.)] —

—

(3.4.12)

.

for the Boltzmann distribution of initial vibrational states characterized by the temperature Tx,.. To evaluate the above limit, one introduces the real orthogonal matrix C that diagonalizes w~ A. —

(3.4.13) with the normalization OTO = I. Here C and the diagonal matrix o implicitly depend on the decoupling matrix A which will be specified shortly. Using the relation lim~~ 2 + ~2) = ir5(x), one achieves ijl(x 1~ an analytic representation for the DCFs, ((Q(t’)Q’(t)))

=

CJ(t’. t;~)CT,

(3.4.l4a)

E. Vilallonga and D. A. Micha, The calculation of time-correlation functions for molecular collisions

353

J(t’, t;~) = (h/2~i){coth~cos[~(t— t’)] +isin[i(t— t’)]} ,

(3.4.14b)

s~i= h~I2kBTv.

(3.4.14c)

The decoupling matrix A is chosen to reproduce the dynamical features of the exact DCFs that are most relevant to the collisions of interest. In the impulsive regime, the transfer of energy occurs in a brief time interval t — t’. Hence it is most appropriate to choose A, or equivalently w and C, to yield the exact values of the first few time derivatives of the DCFs evaluated at t = t’. The exact derivatives are obtained from [a~ (( Q(t’)Q~(t)))Iat’2]1..~,= (i/k)” (( Q([Hv,)”QT]))

where ([H~,)” indicates the n-times repeated commutator of H~with the operators on its right. The first derivative is independent of the force field whenever t = t’. Therefore, in the present linear model, one compares the second time derivatives to find that w and C are explicitly defined by W= ((QQT))w~—

hi coth c~= 2CTWC,

((Qf~~h)),

(3.4.15a, b)

where W corresponds to the average work performed by the intramolecular forces on the displacements. Nonlinear models can be systematically implemented, if necessary to reproduce progressively higher-order derivatives at t = t’. Once W, to and C have been evaluated numerically, A is obtained from eq. (3.4.13), which gives A=

(Oh

+

(3.4.16)

The latter step is actually unnecessary because A does not appear explicitly in the DCFs, but it shows that the decoupling coefficients in general depend on the initial temperature. The above algebraic procedure greatly reduces the numerical work necessary for anharmonic forces, which now involves: (1) calculation of the average-work matrix W, followed by (2) diagonalization of W, and (3) calculation of the effective frequencies The first step is by no means a simple task for large polyatomics; however, the computational cost is here proportional to the number of atoms, instead of the number of energetically open vibrational states. Since the trace of an operator is independent of the choice of basis set, the required averages can be calculated with harmonic-oscillator basis functions. Therefore, the procedure does not require knowledge of the anharmonic vibrational states, whose accurate calculation would be prohibitively costly for polyatomics. Moreover, this work needs be done only once for the given T~,since W involves neither the collision energy nor the scattering angles. In general W would have to be evaluated by numerical integration. However, for the temperatures T~< hwJIkB typical of scattering experiments, one usually has that Vh~4 Vanhi. It is then more efficient to apply thermal perturbation theory to the density operator for the initial distribution of vibrational states, exp(—OHv), where O=(kBTv)1. In these cases Wis calculated from ~.

W=

Tr[pb(O)U(6)(QQT T I

—

fa\11(i~\1

Qf~,,h)]

Pll(O) = exp(—6H~),

(3.4.17a, b)

354

F. Vulallonga and D.A. Micha, The calculation of time-correlation function.c for molecular collision,,

U(O) = 1— Jdo’ ~(O’)+ V(O)

=

Ph(0)VaflhPh(0)

f

J

do’ ~(O’) dO” ~(O”)

(3.4.17c)

+~,

.

(3.4.17d)

where Hh is the purely harmonic component of H,,, and the traces are taken with respect to the vibrational degrees of freedom. For the above operators, matrix elements between eigenstates of Hh are available analytically [671,so this procedure can also be implemented via symbolic programming. In our studies of CO, N2, CO,. N20 and OCS, we have found that the first two terms of the above series suffice to determine the elements of W to within ±0.1%for 100 K ~ T,,. ~ 600 K. It is important to realize that perturbation theory is here used only for the initial density, i.e. a static property, and not for the collisional dynamics. The resulting F~(t, ic; F) will still contain the dynamical effects of Vdlh to infinite order, because the correlation functions were evaluated algebraically without perturbation expansions. Once the_ average-work matrix W has been calculated, it is diagonalized to yield the matrix C. Physically, C describes the degree to which V,,,~,mixes the normal modes Q1 of Vh while also shifting their frequencies away from the harmonic values. The diagonalization can be done analytically for small molecules, and a number of standard subroutine packages can be employed for large polyatomics, e.g., IMSL [80]. The eigenvalues w of Wsatisfy w, = ~(/ith1coth ci,), so this equation is then solved for the ~i1by Newton’s method. This completes the additional calculations required for anharmonic vibrations. Comparing eqs. (3.4.7) and (3.4.14) with (3.3.4), one sees that the structure of the present DCFs is very similar to the exponent of the ~ for purely harmonic vibration. Therefore, from here onwards one takes advantage of the procedures already described in section 3.3 by replacing the harmonic to and C with the anharmonic effective w and C’ = CC, respectively. -

3.5. Slow rotational motion: short-time expansions Since the rotational periods of most polyatomics are usually much longer than vibrational periods and longer than the collision times, the correlation due to molecular rotation can be evaluated by means of the short-time expansions discussed in section 2.2. For this purpose, the overall TCF is written F~(t,ic)

=

1(t,K)

=

~ exp(inw~t)f~(t,ic) ((exp[iK

f~’

d

(3.5.la)

,

1(K,F) exp[—iic 6(t)] P~

d 6(0)])),

(3.5.lb)

N5,

P~(ic,F)

=

fJ p~1(ic,F),

(3.5.lc)

1=1

where n and to,,, denote the row and column matrices composed of elements n~and w,, respectively, and the average is taken with respect to the distribution of initial rotational states. The logarithm of f~(t.K) is then expanded in a Taylor series to second order in t and centered about t = 0; the necessary time derivatives are evaluated from =

((([HR,)tm exp(iic d 6)] P~(ic,F) exp(—iic d6))) .

.

(3.5.2)

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

355

The resulting TCF becomes a sum of Gaussian functions of t, and its Fourier transform is evaluated analytically, giving Sbb(K, w)

f(b)(o K)

—

=

)‘f~(0, ic) ~n (~“2h~ ((P~(K,F)))

~[(

w + flw~+ g~(K) h~(ic)

(3.5.3a)

)2]

(3.5.3b)

,

g~(K)

=

(([HR,exp(iK.db)]P~(K,F)exp(—iK~db))) [kf~

h~(ic)

— -

(2(([HR, [HR, exp(iicdb)]]P~(K,F) [h2f~(0, ic)] - [g~(K)]2 exp(—iic ~db)))511/2

ic)]’

,

.

(3.5.3c)

(..

The above commutators are obtained by means of elementary operator algebra, as outlined below. The required averages are calculated most conveniently by working in the coordinate representation, so that in general,

~

Ida Jsin/3d/3 Jdy

~

(a/3y~L)~2(a/3yj(~a/3y),

(3.5.4)

where L denotes the set of rotational quantum numbers of the molecule and the WL are the populations of the initial rotational eigenstates (af3y L). The upper limits of the integrations will usually be smaller for a specific molecule, depending on its point-group symmetry. As an illustration of the procedure, we work out the details for the case of a linear molecule. One lets K define the z axis of the SF reference frame, so that Kdb=Kdbcosf3,

HR=(—h2121)(0/Ocosf3)sin2f3(8/Ocosf3),

where I is the moment of inertia. Repeated differentiations with respect to cos /3 lead to ([HR,)’ exp(iicdb)] =[(hwbsin2I3)’ +

. .

~

Wb

and the omitted terms are of order (Kdb)” with Kdb ~ 100 typically for impulsive collisions; hence

j’ ~j

—

~(~b)/21,

2. The latter terms can be neglected since

((([HR,)’ exp(iic db)]P~(K,F) exp(—iic db))) .

Wirn

J

daJsin

/3

d/3

~im(/3’

a)~2sin2~/3 P~(ic,F),

(3.5.5)

where the ‘~‘1mare spherical harmonics and F = (a, /3, 0). In the more common experimental situations, the initial population of rotational states is not polarized, i.e., the Wirn are independent of m. One can then take advantage of the addition theorem of spherical harmonics [81] to further simplify the right-hand member of the above equation into

356

F. Vulallonga and D.A. Micha. The calculation of time-correlation functions for molecular collisions

(hw,,)’

f

sin /3 d/3 sin2~/3fda P~(ic,F).

which is independent of the w,. Finally, the integrations are performed numerically by standard quadratures. In our calculations, we have found that 20-point Gauss—Legendre quadrature typically yields 0.01% accuracy when ~ n 1~~ 10 and K ~ 150 A [32, 33, 70, 77]. More quadrature points will probably be necessary for larger values of n1~because the P~ [1~[ic D61(a, /3, y)1H as was discussed in section 3.3.1. .

3.6. Applications to scattering by polyatomics 3.6.1. Atom—molecule interactions and atom-pair cross sections For impulsive collisions, we have found that the intermolecular forces can be adequately represented by means of spherical atom-pair potentials v6(R, r6) U6(~R r6~).where —

U6(r)

=

A6 exp(—B6r)

t’1(r)= 1

+

~ f~(r)C6,,/r”,

(3.6.la)

~ (B

-

f~

6r)k exp(—B6r)/k!.

(3.6.lb)

k

The first term of U6 represents the steeply repulsive forces arising from the overlap of electronic clouds at small separations. The parameters A,,, and B6 are determined by fitting the short-range region of the ab initio atom—molecule potential surface, if available; otherwise, they can be estimated by means of semiempirical combination rules based on the electronic density of the separated collision partners [82]. The remaining terms of U6 model the long-range intermolecular forces. The C6,~are chosen such that the overall potential reproduces the known asymptotic behavior for R ~ rh, which in turn is determined from well-known expansions in inverse powers of R [56—59].The damping functions f~(r) serve to remove the unphysical singularities of the r” when R r6~—~0 [831. For the superthermal impact energies characteristic of impulsive collisions, the wavelength of relative motion is usually much shorter than the range of the v6. Accordingly, the atom-pair cross sections dff6/dQ are evaluated by means of uniform semiclassical treatments [84—86].Here we outline the calculation for potentials (]6(r) exhibiting only one attractive well at intermediate distances and a long-range tail such that limr~[rU6(r)]_* 0; the following procedure can be systematically extended to multi-well potentials and Coulombic forces. Given the impact energy Ek, for each atom pair (A, b) one first calculates the classical deflection function Xh ( p) and the semiclassical (JWKB) phase shift ij6 ( ii). —

x6(p)= ~—2p

J

2g dr[r

6(r)]~,

(3.6.2a)

r05

776(p)

=

K~(~p~r06 + —

2/r2

g6(r) = [1— p

—

f

dr [g~(r)

U

2, 6(r)/r~]”

—

n).

(3.6.2b) (3.6.2c)

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

357

where p denotes the (A, b) impact parameter, and rOb is the classical turning point for U,,, at the relative energy r~,cf. eqs. (3.2.6). The turning point is defined as usual by g~(r)= 0, which is easily solved for rOb by Newton’s method. To efficiently calculate the Xb’ the singularity at r—~rOb must be subtracted out of the integrand. This is done by expanding Ub in a Taylor series in r centered about rOb; the numerical convergence of the integral is also improved by changing the integration variable from r to x = rObir, so that Xb(p)=2JdxL(1 —x2)112—g~1(r0b/x)].

(3.6.3)

The integrations are then performed numerically by repeated application of Simpson quadrature, progressively doubling the number of points until the integrals converge to the desired relative error. In our calculations we have found that 128 points (or less) usually yield the Xb(P) converged to ±0.10. The Xb(P) and flb(P) are thus tabulated over a range of impact parameters sufficiently wide to span the desired scattering angles; the typical step size is ~ 0.1 A. This calculation is performed only once for a given value of Ek and the resulting tables Xb and q,, and versus are stored for later parameters use. After 0b = ofmin~ Xb(P)~ the pcorresponding impact tabulation, one locates the rainbow angles Pb, which will be required by the subsequent steps. Next, for a given scattering angle 0, one evaluates the (A, b) relative scattering angle 0b; from eqs. (3.2.6) it follows that 0b = arcos{~[1 — (k’/k)2 + 2(k’Ik) cos 0]}. Onç then calculates the impact parameters ~ 1=1, 2,..., that correspond to °b’ i.e., the roots of lXb(P)~ = 0b~ This equation is solved for the ~ by inverse interpolation based on the previously c4lculated table of Xb versus p; typically, inverse-quadratic interpolation suffices for 1 eV ~ ~ 10 eV when the Xb(P) are tabulated in steps z~p 0.1 A. The smallest root p~corresponds mostly to a direct rebound from the repulsive core of Ub; if Ub exhibits a single well and 0b
=

sin Ob])

~ —

exp{i[2-q~(p~11) K°bP~,1~Ob2~]} —

(2’rrK~/sinOb)”2[p Ai(s)

iq Ai’(s)] exp[i(/3

—

—

+

~n)]

,

(3.6.4)

where Ai denotes the Airy function, the primes in the right-hand member indicate derivatives, and the parameters p, q, and /3 are defined as follows from each atom pair: .~

/ /

=

\1/2

‘‘2r /

(2)

) L(

~o 2Kb

q

,

)

Pb (2) Xb(Pb

r

=

1/2

(3)

fb

(3)

Xb(Pb S

— —

Pb

+ ~

) )I

~ 1/2

/ —

~

~ib(Pb

(3) )

+ ~Jb(Pb

1 )

I)

+ ~Kb(Pb

(2)

\ 1/21

(3)

Xb(Pb

(2) ,

) j~ ) ]~ \ 1/21

(2)

(3)

(3.6.5b)

)~

12/3 Pb(3)~~ )1mb J

+ Pb )Ob

(3.6.5a)

)~

Xb(Pb

i3 / (2)~ ( (3)\ 1 Oj (2) ~2 37b~,Pb ) — TIb~Pb ) + ~Kb~,Pb — (2)

/3 =

(3)

)~

/

I2(_s)1/2K~]1/2[(

/

3 6 5c .

.

(3.6.5d)

358

F. Vulallonga and D. A. Micha. The calculation of time-correlation functions for molecular collision.,

whenever =

0h ~ 06;

otherwise,

~i:2(

)‘

-

~,

q

=

(2~’2)

(

(i

2

-

(3.6.6a, h)

)2~3

K 6X6(p6)

-

K6X6(p6)

)

‘K6X6(p6)

K~(o6—o6),

(3.6.6c,d)

~

The Airy function and its derivative are calculated by means of commercial subroutine packages [801. The do-,,,/dQ are then obtained from the (K ~ as is indicated by eq. (3.2.llb).

3.6.2. Li”—CO and Li ‘—N2 collisions Impulsive TCFs were first applied to describe the scattering of Li’ ions from N. and CO at impact energies ranging from 4 to 7 eV [32]. A detailed comparison with experimental results [87] showed that the calculations tended to overestimate the rotational excitation. This was not wholly unexpected. since the impulsive model neglects the dynamical effects of long-range torques. which are more significant for ionic projectiles than for neutrals. Such torques tend to lock molecular rotation onto the projectile orbit leading to more grazing collisions and an apparent decrease in the momentum into rotation. 1 transferred in the rotational energy This effect is readily accounted for by replacing K with an effective Keff = KC transfer kwh [see eq. (3.5.5)1, or equivalently by replacing the moment of inertia I of the target with an effective moment i~= ci, where c> 1. The single adjustment c = 3 for both CO and N 2 led to good agreement between TCF and experimental results over all the impact energies and scattering angles measured by the experiments. For example, fig. 1 compares the impulsive (full line) and experimental (dashed line) distributions of final energy for Li” scattering from CO for Ek = 7.07 eV and 0 = 43.2°.Here, the distributions of energy transferred into rotation are also displayed separately for vibrationally elastic and inelastic scattering (dotted lines), since they readily explain the structure of the experimental results. The calculations typically required just a few seconds of CPU time on a VAX-780 equivalent. This unique capability for

~

Lit: ~7O

60

62

,//‘

eV

dcM~,~”

\ ~

‘F

72

Ej leVI Fig. I. Final-energy distribution of Li’ ions due to collisions with CO at the impact energy F = 7.07eV and scattering angle ~ = 43.2”. Solid line: impulsive TCF calculations: dashed line: experimental results,

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

359

incorporating large numbers of rotational states at low computational cost is very useful for analyzing experimental results, particularly when the rotational transitions are so numerous that they cannot be resolved experimentally. 3.6.3. Li’—C02 and Li~—N2Ocollisions When comparing with unresolved experimental results, it is often convenient to integrate the differential cross section for a given vibrational transition over the energy ER transferred into rotation. One thus avoids the ambiguities that may arise from the procedure used to deconvolve experimental results from the laboratory to the center-of-mass reference frame. The partially integrated cross sections, which we denote by d&~/du1,are obtained by taking advantage of the factorization of slow rotation from fast vibration, as follows [70]: d&

~

du I JI dER~-~(k,k’) j

dt ~—~exp(iERt/h)

x ((exp[iic db(t)] P~(ic,F) exp[—iK d,,(0)])) .

,

(3.6.7)

where we have decomposed the total transferred energy into rotational and vibrational components by writing e = ER + ~ kn1w1 hence the inner integral is the distribution of rotational energy transferred concurrently with the vibrational transition i.’. In principle, the outer integral would have to be evaluated numerically because k’ and K are functions of ER. In practice, the above expression can be simplified because the rotational distributions often are peaked sharply about the average transferred energies (r~), cf. fig. 1. In those cases we have 1

1/2

k’—k~=~[2M(Ek+~knJwI+(E~))]

,

KK~kk~,

so that k’ and ic become approximately independent of ER, and the dob/dQ can be factored outside the integral. Exchanging the order of the integrations then yields a s-function of t, which leads to cancellation of the exponential operators after integrating over t, so that do~,(k,k~) ((p(b)(KF)))

(3.6.8)

For example, fig. 2 compares the calculated [70] and experimental [88] values of d&~(0,Ek)! d&0(O, Ek) versus scattering angle 0 for the first few vibrational excitations of CO2 in collisions with Li~ at Ek = 2.81 eV. Here the results have been normalized with respect to the vibrationally-elastic cross section, d&01d12, because these experiments were not calibrated to yield absolute cross sections; moreover, the experiments could not resolve the pairs of transitions (020, 100) nor (030, 110) because they are quasidegenerate in vibrational energy. The TCF and experimental results agree well, even in the value of 0 where the relative cross section of the 010 transition intersects that of the pair (020, 100). Similar calculations for Li~/N2Ocan be found in ref. [70], where the TCFs are exploited further to obtain much physical insight into the structure of the cross sections. The present TCF treatment has also been employed to assess the role of vibrational anharmonicity in diatomic targets (N2, CO) [76]as well as linear triatomics (CO2, N2O and OCS) [77].For instance, fig.

F. Vilallonga and D. A. Micha. The calculation of time-correlation functions for molecular collisions

36(1

0.10

.~

+

Li 1C0 10

Li~/C02 E~2.8I eV

000 020

~.

020 and 100

:°~

CM scattering angle (deg)

Fig. 2. Ratio of the vibrationally inelastic to the elastic differential cross section, do,,(O. E)!do-,(O, E). for the transitions n = (n1n~mt,) versus scattering angle 0 in collisions of Lu’ with CO, at the impact energy F = 2.81 eV; the initial vibrational temperature us T, = 300 K. Here mu, mu, and n, respectively indicate transitions of the symmetric stretch, bending and asymmetric stretch modes of C0. Full lines: results of impulsive TCF calculations; crosses: experimental measurements; dashed lines have been drawn through the experimental points for clarity. Experiments do not resolve the pairs (020) and (100) nor (030) and (110) because these arc quasidegenerate in energy.

— ..

~111111~o~lb0 15

2 E~4.72eV

C005

.

anharmoruic harmonic

~0l0

30 CM scattering angle (deg)

90

cross section. dir,)6. F) du~(~, F). for the vibrational transitions n = (nn,n,) versus scattering angle 0 in collisions of Li’ with CO. at the impact energy F = 2.81 es’. Full lines: TCF results for anharmonic (cubic plus quartic) vibrational force held of CO,: dotted lines: similar. hut for the harmonic CO. force field. Fig. 3. Relative vibrational differential ~,,

3 shows the relative vibrational differential cross sections dó-~(0,Ek)I~fldo~(0,Ek) versus 0 calculated for Li f—CO, at Ek = 4.72 eV. Here we compare the TCF results obtained from a spectroscopically determined anharmonic vibrational potential (quadratic plus cubic plus quartic) [89] with those of the purely harmonic force field. In these systems, we observed that the shapes of the d&~/~~ d&~versus 0 for anharmonic forces would usually follow those of the harmonic case: nevertheless, anharmonicity tended to significantly increase the amount of vibrational excitation. Since anharmonicity was incorporated via the quasi-algebraic procedure outlined in section 3.4, the calculations required only a few more seconds of CPU than those for purely harmonic vibration. A detailed analysis of vibrational excitation in terms of intra- and inter-mode anharmonic couplings can be found in ref. [771. 3.6.4. H—CO and H—CO. collisions Collision experiments with fast H atoms have been carried out using target molecules initially at thermal equilibrium, under experimental conditions suitable for application of the equations derived for short collision times [90]. The experiments can be analyzed for specific electronic—vibrational transitions, for which the final rotational distributions are presented as functions of the final rotational energy E~or quantum number J’ rather than as a function of the amount of energy transfer F. Therefore, one should modify the Gaussian shaped double differential cross section to obtain an expression in terms of

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

361

E1~,for a cross section which has been averaged over the initial rotational state distribution. Treating the initial rotational quantum number J as a continuous variable, one obtains 2

J

dE~du2=

2

dJ dE w

(3.6.9)

WR(J).

Here unprimed and primed quantities refer to the initial and final states, respectively, and WR refers to the distribution of initial rotational quantum numbers J. We separate the total energy E~0~ into electronic—vibrational plus rotational terms, and obtain the following relations E=

~

E~v=

+ E1~)— E,~v

wR(J)

—

=

(Eev + ER)

Eev,

ER

=

=

(3.6.lOa)

Fey + ER,

(3.6. lOb, c)

E~ER,

(2J + 1) exp(—ERIkBT),

ER(J)

=

Bhc[J(J + 1)],

(3.6.lOd, e)

where B is the rotational constant, h is the Planck constant and c the speed of light. For hyperthermal collisions for which the kinetic energy is in the range of a few eV, most of the transferred energy goes into electronic and vibrational excitation and only a small fraction goes into rotational excitation. Hence the collisions are rotationally quasielastic so that F(0) and k’ are only very weakly dependent upon the rotational energy. Letting 2)=((z~ER)2), (3.6.lla,b) E—(E)=ER—(ER),

((~E)

the new double differential cross section becomes

dE~dQ=

~ev

(~~R

2fr2J+

1)exp(-

(2~2

-

khT)’

(3.6.12)

2)”

where ER depends on J. Therefore, the double differential cross section as a function of the final rotational energy, E~,is given by d2o -t dO~ey j’~ (ER)—E~ dE~du2= (Bhc) exp~,~a +2 kBT t’2[a x erfc{(2a2) 2 + ((ER) — E~)IkBT]}, 2(a = A exp(at + ~a2— x’) erfc[(2a2)’~ 1 + a2 x’)], —~-~-

—

=

(3.6.13a) (3.6.13b) (3.6.13c)

Af(x’),

where A is defined by the second member of the equation, and a1, a2 and x’ are dimensionless quantities defined as 2)I(kBT)2 (3.6.14a, b) a1 = (ER)I(kBT), a2 = ((,~ER) ,

362

F. Vulallonga and D. A. Micha, ihe calculation of time-correlation functions for mmtolecular collusio,t,s

=

Et~/kBT°” J’(J’ + 1)Bhc/k~T.

(3.6.14c)

This double differential cross section is no longer a Gaussian distribution, but instead a distribution of E~values that peaks at an intermediate value x’ and tapers off at both small and large x’. A similar expression can be adopted integrating over the solid angle (2 and assuming that the dependence of a~and a, on scattering angles is weak for hyperthermal collisions, to obtain

=

f

dQ dE~df1=

dfI A) f(x’)

=

A’f(x’),

(3.6.15)

where A’ is another constant independent of E1~. Finally, the scattering cross section for the final rotational state J’ can he obtained by numerical integration, ‘2

~r(J’) = BhcJ

dJ’ (2J’

+

1)

=(2J’ + l)A”f[x’(J’)].

(3.6.16)

Therefore, o-(J’) can be expressed as a function of the final rotational quantum number J’ and of three parameters a1, a. and a.~ as follows: 2(a

~(J’)=(2J’+ 1)a1exp(a1

+

~a,—x’)erfc[(2a,)

“

1 +a2—x’)].

(3.6.17)

The essential features of the final rotational distribution are determined solely by the parameters a1 and a2. The third parameter a-, is a scaling factor which is necessary to fIt the experimental distribution usually expressed in an arbitrary, relative scale. O’Neill et al. determined the nascent rotational distribution of the product CO,(OO°l)by timeresolved diode laser spectroscopy [91]. Their experimental data together with the initial rotational distribution at 298 K are shown in fig. 4. Also shown in the same figure is the theoretical distribution curve obtained by fitting the theoretical u(J’) to the experimental points. The best fit parameters are a1 = 0.52, a. = 1.44, and a., = 0.029, respectively. As is evident from the figure. the fit itself seems to be

~

Rotational Quontum Number(J)

Fig. 4. Final rotational state distribution of C0,(00’l I in collisions with H. Circles: experimental results: curve a: initial Boltzmann distribution at 298 K: curve h: theoretical fit using eq. (3.6.17).

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

363

very good and within the experimental range over the whole region for which experimental results are reported. This fact also bears out from the average rotational energy and its dispersion for the final 1/2 calculated from the theoretical distribution are 1.93k 32kB T state. The (E~)and ((~E~)2) 8 T and 1 . (T = 298 K), respectively, while the corresponding quantities deduced from the experimental distribution are l.88kBT and l.1OkBT, respectively. Chawla et al. have studied collisional T—~V, R energy transfer from H (Ek = 1.58 eV) into CO(v = 0, J = 2), corresponding to a rotational temperature TR —30 K, and determined the rotational state distribution of the product CO(v = 0) and CO(v = 1). Since the temperature is so low, the fitting parameters a~and a 2 are quite large in magnitude, causing computational overflow in the exponent range. Therefore, a least-squares fit was based instead on the expression 212a 2I(a o~(J’)= (2J’ + 1)a3 exp[—(a1 — x’) 2] (2a2)~ 1+ a2 x’) (3.6.18) —

,

which is equivalent to the previous one in the limit of large arguments. Figure 5 shows the experimental rotational distribution for CO(v = 1) and the theoretical best fit curve to the experimental data. The best fit parameters are a1 = —239, 2) a2 = 16 384, and a., = 17.9. A measure of goodness of fit, provided by t/2 gives an indication that the theoretical fit is very good. In effect, the theoretical values of (E1~)and ((~E~)average rotational energies (E~)are SO/CB T and 53k the and experimental 8 T, respectively, 2) 1/2 values are 34kB T and 4OkB T (T = 30 K) respectively, in and theoretical and experimental ((~E~) close proximity of each other. Average rotational quantum number (J’) for both distributions also agree very well; they equal 21 for the theoretical distribution and 22 for the experimental one. Unlike the C0 2(OO°l)and CO(v = 1) cases, the agreement between the experimental data and the best fit curve is unsatisfactory for the CO(v = 0) final state. This signals the breakdown of the assumption of short duration, impulsive collisions invoked in deriving the equations. Based on an extensive trajectory study, Chawla et al. argued that the production of low J’ excitation comes from short-duration collisions of a purely direct and impulsive nature, while collisions leading to high J’ are more likely to proceed through a collision complex with a statistical distribution with an effective temperature as suggested by surprisal theory [92].Consequently, a general procedure for the CO(v = 0) final state consists of using the distribution [90]

1.00

I

____________________

0.00

‘

0

10

20

30

40

50

Rotational Quantum Number(J)

Fig. 5. Final rotational state distribution of CO(v = 1) in collisions with H. Circles, triangles and squares: experimental results; curve: theoretical fit using eq. (3.6.18).

F. Vilallonga and 1). A .Micha, The calculation of tirne-correlatuon functions for molecular collusuo,u.c

364

u(J’)

=

Ojmp(J’) + o~,

(3.6.19)

1~(J’) ,

where the first term is the impulsive collision distribution and the second term is a surprisal distribution. This procedure has provided an excellent fit to the experimental results with five parameters related to physical observables.

4. The calculation of cross sections in the short-wavelength limit The impulsive treatment outlined above is most appropriate to scattering into large final angles, since collisions are then mediated by short-range forces. We next outline an alternative treatment that fully incorporates long-range interactions and therefore spans small through large scattering angles. Here the transition operator is evaluated in a semiclassical limit appropriate to polyatomic molecules whenever the impact energy is larger than a few meV [16, 60—621. In these cases, the translational wavelength is much smaller than bond lengths and smaller than the range of the intermolecular potential, so the relative motion can be described semiclassically, in terms of trajectories and associated phases. while internal motions are treated quantum mechanically. This dual description is achieved via the pathintegral formulation of quantum mechanics [93,94] followed by stationary-phase approximations for the translational degrees of freedom. Other classical-path treatments have been formulated for atom—molecule [43.95—1011 and molecule— molecule [102—107]collisions; however, they were mostly based on internal-state expansions that become computationally impractical as the collision energy increases. Semiclassical calculations have also been implemented by means of time-dependent wavepackets [39],whose propagation can become expensive for motions with widely differing time scales. The following TCF—semiclassical approach encompasses very high densities of internal states as well as fast and slow motions. 4.1. Separation of relative from internal motions We first summarize the conventional approach to the short-wavelength limit and then outline the modifications necessary to achieve a computationally practical treatment. One starts from the relation G(E) = G,,(E) + G1,(E)T(E)G(E), where G and G are the propagators corresponding to the interacting and noninteracting collision partners, respectively. From the known form of G,1 in the R. a) representation, one has 2/h]6~,~6(k,,k) exp(ik,,R) /R lirn (R. a’~G,,(E)Ik,a) = ~~[M(2.,r)l‘ with k,~= k~R/R,which leads to

(4.1.1)

,

(k~,aj Tlk~,a~)= (k4/2’irM2) lim{R

a~)

1R~exp[—i(k~R~ + k~R~)] (R~.a~~(GG,)~R., —

}

,

(4.1.2)

where the limit is taken for R and R1—s with R = —Rk~!k~, R1 = RfkfIkf; the second term tn the right-hand member vanishes asymptotically, possibly excepting the special case of extreme forward scattering. Next, G(E) is evaluated from its Fourier transform, exp(—iHt/h), and the matrix elements (R~,c~exp(—iHtIk)IRj,a) are expressed in terms of the path integral for relative motion [93,94], so that

E. Vilallonga and D.A- Micha, The calculation of time-correlation functions for molecular collisions

(R1, af~G(E~)~R1, a~)=

—~

f

365

drexp(iEWr/h)JDRexp(iSr[R]!h) (af~U’[R]~a1), (4.1.3a)

11

U’[R] H’

=

=

~

(Rf~exp(—iH’rINh)~R~)... (R1~exp(—iH’rINti)~R1) + V,

(4.1.3c)

where Sr is the action for free relative motion. Here

~

(4.1.3b)

,

.f

DR(...) is used as an abbreviation for

[(-~) f dR1 J dR2~ •f dRN(...)] .

which corresponds integratingmolecules, over all possible start fromoscillates R1 at t~=veryr12rapidly, and endsoatthe Rf 12. Fortopolyatomic Sr P1,paths and Rthethatintegrand when t~ = T integrations are performed by well known stationary-phase methods [95,108]. One writes —

~

(a~~U’[R]~a 1)(a~~U’[R]~a1)~ exp(iflfj),

=

so the total phase of the integrands is P~1([R],r) = (S, + h~+ Er)/h, where we use the abbreviations = (R, a1, ti), f= (Rf, af, tf) and indicate that ck~~ is a functional of the path [R] and an explicit function of r. The stationary-phase condition, 3t~I1~Iar=O, defines the transit time r~(E)for given E and endpoints (R1, t1), (Rf, ti); here a/ar is evaluated as the first difference along the fixed path [R]. The stationary-phase approximation to the integral over r gives (Ri, a1~G(E~)~R1, a1) = —! P1

J

DR exp[i~~1([R],r~~)]~ ( a~U’[R]~a1)

(

2~i ~1/2 ~.([R] r.) fi ‘

(4.1.4)

where fI~f is the second difference in r along the path. To perform the functional integration, one defines Si., = Sr + h~j11and introduces the trajectory Q(t) for relative motion that satisfies ~S1~ = 0. This requires that Q(t) be calculated from the Lagrange-like equation

[a/aQ— (d/dt) a/aQ]L~— (~)f/aR)Q= 0,

(4.1.5)

where Lr is the Lagrangian for free relative motion. Here, (~i7Jf1/~R)Q plays the role of a nonlocal force, as is seen by writing ~n~~1/~R explicitly in terms of ~V/~R [95]; the computational implications of (~n)f1I~R)Q will be addressed shortly. The functional integral becomes approximately (R

1, £XfIG(E(+) )1R1, a~)=

_~

21T1

if~\~(rR1 fiyI.

f M ~3/2 ~

32

r~1R~

tiJ

(doCl/dQ) ~ exp{i[I~1([Q], r11) 2nfilT]} (4.1.6) 2),n~ Q(r~1/ 1is the number of zeros encountered in the Jacobian aRf/aQl~R. X

where R1 = Q(—r13/2), Rf =

J’

‘~ ~t/2

T.’I”

—

,

366

F. Vilallonga and LI. A . Mic/ua, The calculation of tu,ne-correlation function.s for molecular collision.,

along Q(t), and da-~~/dQ is the classical cross section defined by Q(t) for each pair (f. i). The semiclassical limit of the (kr, a~~T~k a,)1,is then obtained by substituting the above result into eq. (4.1.2).

4.2. Separation of time scales ,fbr intramolecular motions A rigorous numerical implementation of the above procedure would be prohibitive for polyatomics, because the force (~i~~1/~R)0 is not only nonlocal but also dependent on and a~. Expensive root-search procedures would be required to self-consistently calculate each of the Q(t) that connect the numerous transitions (k, (k1, a1) [105]. However, (~ii~~1/~iR)0 in general can be written as the sum of a local force plus a residual nonlocal term. In molecular collisions at energies above a few meV. the forces along paths vary slowly compared to the local de Broglie wavelength. It is then sufficient to use approximate trajectories calculated only from the local force, which in turn is derived from an effective potential V. A realistic V is constructed based on the relative strengths of the couplings between the various degrees of freedom, as follows [61]. Since the unperturbed evolution of the internal states a) is known in principle, it is convenient to write:

a

a)—*

U’=exp(~iHjnutv/~) U”[R]exp(—iJdtV[RI/~)exp(iH11it~/h).

(4.2.1)

where U” is implicitly defined by comparing against eq. (4.1.3) once V has been specified. For this purpose we refer back to section 2.4 and let X and x denote the internal coordinates corresponding to slow and fast internal motions, e.g., molecular rotations and vibrations, respectively. A change of representation from the a) to Xx) basis sets then gives

(a~~ U’{R]~a~)

=

exp[i(E,,,t~ E,,.t~) /h] —

J

dX1 dx~ dX dx (af~XIxl

x (X~x~~ U”[R] X~x~) exp[i~([R], X~x~)] (Xx1 I at).

~([R], X~x)=

—~

J

dt(X1x~IV[R1IX~x1).

(4.2.2a)

(4.2.2b)

By examining the comparative strength of internal-relative couplings as well as the time scales of the various motions, one constructs V so as to render ~ X1x1)I. In particular. the slow coordinates X strongly influence the relative motion, due to the large anisotropy of polyatomics. In contrast, the relative motion is quite insensitive to the fast coordinates x, because vibrational amplitudes are usually much smaller than equilibrium bond lengths in nonreactive collisions; hence it is sufficient to take into account only their average value (x)~.Accordingly, we define the effective potential by (X1xIV[R]IX1x) = V(R, X, (x)), from which follows 1~~)IR a1)1,

(Ri, a~IG(E

dX~dx1JdXj dx~(a,IXfxf)I(Xfxf, X~x1)(X~xja~),

(4.2.3a)

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

I(Xfxf, X~x1)= (—i/h)

J

dr exp{i[E~ — ~ (E~— Ea)]T/h}

x JDRexp[iS([R], X1)/ti] (X~x~IU”[R]~X1x1),

~([R], X1)

=

Sr

—

J

367

dtV(R, X~,(x)),

(4.2.3b)

(4.2.3c)

where S can be interpreted as the effective action. A practical semiclassical approximation follows from performing the integrations in eq. (4.2.3b) via stationary-phase methods for each value of X, while neglecting the phase of (X~x~~U”[R]~X1x1) in comparison with S([R], X~)/h.As a consequence, the stationary-phase quantities depend parametrically on the initial values of the slow variables. The stationary condition with respect _to r is now approximately satisfied by Kr + V(R, X~,(x)) = E ~(E,~1+ Ea), and the condition FiS = 0 leads to trajectories Q(t; XL) for relative motion determined from the N~wton-likeequation —

MQ(t) =

—(aV/aQ)~Q(~)X.(X~,

(4.2.4)

.

This equation is solved numerically for Q(t; X~)while keeping constant the X1. One thus neglects the changes of the slow variables during the course of the relative motion. Nevertheless, the Q(t; X1) fully incorporate the anisotropy of the intermolecular potential. Therefore the present treatment is more general than previous classical-path approximations that have employed straight-line trajectories [99] or, alternatively, a single trajectory calculated from the spherical average of V [98, 102, 104]. Depending on the shape of the potential surface, one may find that several trajectories Q~’~(t; X1), j = 1, 2,. satisfy ~iS= 0 for fixed E and X1. As is well known from the special case of central potentials, such trajectories correspond to branches of the classical deflection function and lead to rainbow interference phenomena. In the more general case of anisotropic potentials, the resultant I(Xfxf, X.x1) is similarly given by the sum of the contributions from each Q~’~(t; X). Substituting the stationary-phase result for I(X~x1,X1x~)into eq. (4.2.3a), the latter into eq. (4.1.2), and taking advantage of the completeness relation of the XfXf), one arrives at a semiclassical limit for the (k1, a~ I T1k1, a~)that is computationally practical [61], .

1T1k (k1, a~ bkk

=

WIX1)

1VVbkk a 1, a~) ~

(

a~

1) ~

(4.2.5a)

,

f

e2xp[i~(X~)](X dX1 X1)(d~~/dfl)” 11,

=

—

=

~

+ Jdt [V(Q, X1, (x))

lim U’[Q(t; X1)]IX~)

R1,R1—~o

,

—

~

Q]/h,

(4.2.5b)

(4.2.5c) (4.2.5d)

F. Vilallonga and LI. A. Micha, The calculation of ti,ne-correlatio,u function.s for molecular collusion.,

368

where du~~/dfl is the elastic differential cross section defined by Q(t; X). The relative motion can now be calculated separately from the internal transitions, since bkk is independent of a~and af. The operator bkk corresponds to the probability amplitude that the relative motion will undergo the transition k1 k~for fixed X in the limit of short translational wavelengths. Accordingly bkk

is

—*

determined by do~~~Id11 and the associated phases These quantities are implicit functions of the X. as is appropriate to the strong coupling between the relative motion and the large-amplitude (i.e., slow) internal variables. Once the Q(t; X~)have been calculated, the operator W determines the corresponding transition amplitudes for the internal degrees of freedom. Here, W the asymptotic limit of the evolution operator, which obeys the well known equation ill aU’/at = H’U’ with H’ evaluated along Q(t; X). ~.

is

4.3. Collisional TCFs in the short-wavelength limit Collisional TCFs for short translational wavelengths are obtained from the above (kf, a~ITIk..,a1) following the general procedure outlined in section 2.1. Presently, however, the sum over branches gives rise to interference terms in4~’1(X~)> the transition rate, i.e. so coherent terms, which involve 1(X)]}. Usually ir for polyatomics, the interference terms average exp{i[~’~(X~) ~ out to negligible values upon integrating over X, and X~.It then suffices to consider only the incoherent contribution to the detailed transition rate. —

R(k~,a

( a~WakkIa )(J)12

1 —~k~,a~)= (27r/h) ~

ak.k

=

J

2(X,I.

(4.3.

la)

(4.3. lb)

dX1 X)(d~/d12) ‘

Following the steps indicated by eqs. (2.1.4) through (2.1.7), one finds that the total transition rate is given by the Fourier transform of the TCF F(t; kf, k) W(t)

=

=

((a~k(t)1”(t)W(0)akk(0)))1’1

~

.

(4.3.2a)

exp(iH 1~1t/h)W exp(—iH1~1t/h)

(4.3.2b)

.

with a similar expression for ak1k(t). The calculation of F(t; k~,k) can be simplified by taking further advantage of the separation of time scales discussed in section 2.4. Whenever slow and fast internal motions are weakly coupled, one may write H5~ H~+ H1 and {Ia)} = {IL)}® {In)}, where IL) and In) are the eigenstates of H, and Hf. respectively. Upon taking V= V~+ V~for the internal-relative coupling, one has W—~WtrWsr, where

W~~IX1) ~exp((-i/h)f

dt~r)

(4.3.3)

IX),

a

with the integrand evaluated along Q(t; X); similar expression defines Wtr in terms of Hf and V~..As a consequence, the TCF takes on the nested form

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

F(t; k~,k) — kk()

=

~

(akk(t)Wsr(t)((Wfr(t)lVfr(O)))x1’VsI(O)aktk.(O)))

369

(4.3.4a)

,

exp(iH,t/h) akk exp(—iH~t/h),

(4.3.4b)

W~~(t) = exp(iH~t/h)Wsr exp(—iH~t/h),

(4.3.4c)

exp(iH~t/h)Wtr exp(—iH~t/h),

(4.3.4d)

Wtr(t)

=

where the inner average is evaluated for fixed X1 and with respect to the distribution of the the outer average pertains to the distribution of the IL1).

In), while

4.4. Classical relative motion 4.4.1. Trajectories for fixed target orientations Letting R(t; 1~)hereafter indicate the stationary-phase trajectories Q(t; X,), the Hamilton equations for the relative motion,

aR/at=P(t)/M,

aP/at= —aV(R, 1)/aR,

(4.4.la,b)

are solved numerically by one of several procedures, e.g., Runge—Kutta or Adams—Moulton algorithms [109,110], that employ variable step size z~tin order to efficiently handle the slowly varying long-range forces as well as the steep repulsion at short range. The trajectories are calculated in the SF reference frame for fixed orientations [ = {a1, ~ y~}of the target molecule. Conventionally, the z axis of the SF frame is defined by the direction of the initial momentum, so the initial conditions are R(t,; 1) = b(cos ~ x + sin x y) + Z.z, (4.4.2a) 112z, (4.4.2b) P(t1 1) = (2E/M) where b is the impact parameter, x is the angle between the original scattering plane and the (x, z)-plane, and Z. (<0) is chosen such that IV(RI, ~~)I <~Ewith 0< ij 1; typically i~is of the order of ~ The integration is stopped when IV(R, 1)I again becomes less than ~‘jEbut with aR/at >0; typically, R(t 1) R(tf) = 20 A in collisions of atomic ions with diatomic and triatomic molecules at E — 5 eV. Due to the symmetry of the initial-value problem, R(t; 1) depends on x and on a~only through the difference a~— x, so that x can be conveniently fixed at x = 0 while a1 is sampled as follows. The computational procedure is divided into two steps based on the characteristic times of motion, in order to take full advantage of the separation of translational and internal motions. Inspecting eqs. (4.3.4) one sees that the TCFs will usually require considerable computational effort because the vibrational displacements oscillate rapidly. Instead, R(t) can be calculated quite inexpensively because V by construction is independent of u(t). Therefore, much CPU time can be saved by calculating TCFs only for those trajectories that reach the specific scattering angles that are of interest. Accordingly, in the first step trajectories are calculated at a given collision energy for a large sample of initial conditions, in order to generate a table of final scattering angles U and 4i versus b, a~,f~ and y~. Interpolating from this table, one can later identify which initial conditions, and therefore which ~

37(1

F. Vilallonga and LI. A. Micha, The calculation of time-correlation functions for molecular collisions

trajectories, correspond to the desired scattering angle(s). In general, for an asymmetric target molecule, the initial Euler angles must be sampled within the ranges [0, 2ir], [—1,1] and [0. 2’rr], respectively for a 1, cos ~ and ‘y~.Depending on the point-group symmetry of the target, these ranges can often be reduced by symmetry considerations, e.g.. = 0 and 0 ~ a ~ i~ for linear molecules, 0~y,~2i~/n for C,, symmetry, etc. It is convenient to choose an equally-spaced grid instead of the more usual random sampling [41]so that the average over the initial rotational distribution can later be done by numerical quadratures. For instance, in our studies of Lit—N, collisions at impact energies E = 2eV through 10eV, a and cos were each sampled in 33 equally-spaced steps, for a total of 1024 distinct orientations; the impact parameter b was sampled in steps of 0.1 A in the interval [0,6.0 A], which spanned 180° U ~2° [62]. The second step of the calculation is performed for each scattering angle of interest. At each 1~1(O;a orientation in the grid of Euler angles one evaluates the quantities b 1. /3~, of the impact parameter corresponding to the chosen 0 by means of inverse interpolation based on the table of (0, q~) ‘y versus (b, a1, 1); here the index j indicates that more than one impact parameter may lead to the same value of 0 near rainbow angles for given a~.f~land Our experience has indicated that quadratic t’1. one calculates the interpolation suffices for the above mentioned grid steps. Next for each b R1’t(t; 17) necessary for the subsequent evaluation of the collisional TCFs. At this stage one may wish to confirm that the resulting trajectories reach the desired scattering angle. In our calculations the final directions of the relative momentum fell within ±0.5° of the desired value of 0. As an example of the above tabulation, fig. 6 shows the polar scattering angle ,0 as a function of the impact parameter b calculated for several initial orientations of the target in collisions of Li with N 2 at E = 4.72 eV; these results are based on a realistic potential surface previously evaluated by ab initio techniques [111,112]. The figure illustrates that the relative motion can be very sensitive to 17 at superthermal impact energies. Therefore, it would be inappropriate to obtain the trajectories from the spherical average of the potential, as was proposed in other classical-path treatments [96, 98, 102, 1041. y1)

3j,

/

‘~.

20

\\

lb)

80

st

HI

5 Ci~1

ill

~1i

Ill

iJ\2

LI~~N

St~\~60t

!t35 .E 90

to,~ot

to ,i 201

190,601\

2/~\3 ~

\

N.

U

\~~l45,60t

.5 a-

~45 190,601

0

0-

-I.’—, 2

3

4.

Impact parameter (A)

5

0

I

•

2

Impact parameter (A)

Fig. 6. Polar scattering angle 0 versus impact parameter b calculated for several initial orientations (a. ~) of the target in collisions of Li’ with N at the impact energy E = 4.23 eV. The points labeled 1 through 3 correspond to the trajectories shown in fig. 7.

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

371

It is also evident that several trajectories starting from different initial conditions may reach the same scattering angle, giving rise to rainbow phenomena that can be broadly classified as follows. At small scattering angles (OR 5° to 100 at this energy, cf. fig. 6a) and for fixed a1 and f3~one finds rainbows similar to those of central potentials [84].They are due to the interplay between attractive long-range forces and short-range repulsion, as can be seen in fig. 7a which shows the three trajectories that reach 0 = 7° for a1 = 0 and f31 = ir/3. 2 in fig. large scattering angles: OR> 6b. For Thecertain latterorientations, are due to another the facttype thatof Urainbow and/or occurs 4) are atdiscontinuous functions of (b,Tn17). The discontinuity is caused by the strong anisotropy of the short-range forces, as is illustrated by fig. 7b which shows the two trajectories that reach 0 = 160°after being sharply deflected by the repulsive core for a~= 0 and I3~= 2Tn/3. In addition, the anisotropy of the potential leads to rotational rainbows, i.e., structure in the distribution of final rotational quantum numbers, because trajectories corresponding to different values of a 1 and f3~can also interfere. The rainbow structure of anisotropic potentials has been extensively studied in the literature [113—119],so we shall not dwell on it here. In general, rainbows lead to singularities in the classical elastic cross section dUc/dLI when one varies a1, f3~and y, for fixed 0.

4

;::. (

0

• —-‘

5\~_

—2

-

—6

—4

(a)

e=’r

E=4.23eV

—2

0

Z

(R)

N

V=E 2

4

6

(bt 2

~~~4.23ev9=i60

0

—2

a=0° fl=120~ —6

—4

—2 Z(A)

0

2

Fig. 7. Trajectories that reach the same valueof the scattering angle 8 for given initial orientation (a,, 13,) of the target in collisions of Li~with N2 at the impact energy E = 4.23 eV. The dashed curve indicates the equipotential contour V(R, d,) = E. (a) 8 = 70 for (a,, 13,) = (0, irI3); (b) 0 = i60°for (a, 13,) = (0, Tu,-/3).

372

L’. Vilallonga and LI. A. Micha, The calculation of time-correlation fulnctuon.c for molecular cOllisiOfl5

Such singularities require the following treatment in order to ensure reliable numerical implementa-

tions. 4.4.2. Calculation of angular distributions A nonsingular approximation to do-~/dQ can be readily obtained by simulating experimental measurements. A physical detector has a nonzero aperture, so that it integrates the flux of particles scattered over a small solid angle ~Q centered about (0, 4)). For simplicity one may assume that the polar and azimuthal angles subtended by the detector are equal, so that ~XQ= 2~i~ sin where 4, is the angular resolution of the experiment. The detected particles originate from a cross sectional area ~ of the incident beam, so that the du~/d(1corresponding to an actual measurement is given by ~,

du~/dI1~du~ff/du1 = ~~7(2~i~ sin 0).

(4.4.3)

Physically, rainbows lead to rapid but finite increases of

~

so du~1~/dI2 remains finite for all

17.

since ~ is nonzero in experiments; typically ~i5 1°[87, 120]. Numerically, ~ is obtained by following 1’>(O: a

a narrow bundle of four trajectories initially centered around h 1, -y1). The solid angle finally subtended by the bundle is fitted by a polynomial quadratic in b and in x. and inversion of the fit then gives the value of corresponding to the aperture of the detector. The above calculation of doCff/duI. together with that of the corresponding TCFs (see below), is repeated similarly for all the tabulated values of a~. and ‘y1. In accordance with eq. (4.3.4), the contributions from all branches j are accumulated, and then the average over the initial rotational distribution is performed by standard quadratures. e.g., Simpson and/or Gauss—Legendre depending on the tabulation previously chosen for a,, /3~and y1. In selected cases, the integrals over ./7 are calculated using one half as many values for each of the Euler angles, in order to check the accuracy of the quadratures. In our studies, relatively smallsignificant grids for 2u/ds dQ converged to three or more each angle (20 to 40 points) yielded an overall d figures. This accuracy is significantly better than one could practically achieve by the more customary Monte Carlo procedures [41]. There, initial conditions would be sampled at random and differential cross sections would be estimated from histograms. Although the accuracy of the Monte Carlo method can be systematically improved by enlarging the initial sample, computational costs limit its resolution in 0 to about 5°with numerical uncertainties of ±20%to ±50%for the histogrammed cross sections [41, 111, 121—123]. Instead, the above procedure yields better angular resolutions (±0.5°)and smaller numerical errors (±0.1%)[62] by taking advantage of the decoupling of relative from internal motions and by carefully sampling the initial conditions. ~.

~.

~,

4.5. Quantal internal motions 4.5. 1. Algebraic methods for driven harmonic oscillators In accordance with the separation of fast vibrational from slow rotational motions, the unperturbed vibration is described by the Hamiltonian operator H~= K,, + Vh + ti’1Inh and it is driven by the translational—vibrational coupling VTV[R(t; 17), u; 17], where K~ is the vibrational kinetic energy operator, and we have separated harmonic (h) and anharmonic (anh) terms in the intramolecular potential. This defines the Hamiltonian to be used in the description of the driven molecular oscillators while working in the molecular BF frame. The treatment can be done at several levels of accuracy. To begin with, we consider the case of linearly driven harmonic oscillators, since this is sufficient for many

E.

Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

373

problems and has been extensively implemented in applications [16, 61, 62]. We then review recent developments that allow for anharmonic internal motions, and for nonlinear driving terms. For linearly driven harmonic oscillators, we make as before a transformation from the atomic displacements u = {ub; b = 1, N) to the internal bond variables S = {s,; i = 1, f~,,},and from the latter to the mass-weighted normal coordinates 0 = {Q1 j = 1,. fv} with frequencies w1. The . .

.

. .

,

.

. ,

. ,

standard quantization by means of creation and annihilation operators {a~,a.; j = 1, leads to H,,,

hw1(aa1 +

~.

=

VTV

~),

v0(t) = VTV[R(t; 17), 0; 17],

=

v0(t) + ~ v1(t)(a~+ a.),

v.(t) =

(i-)

1/2

.

. .

,

f,~,}then

(4.5.la, b)

V ~ D~,1(17) ~

,

(4.5.lc, d)

where the ö i/TV/ar,, are evaluated at R = R(t; 17) and u = 0 for fixed 17, and the coefficients D,,1(17) are the matrix elements of the transformation from vibrational displacements u in the SF frame to normal coordinates in the BF frame, as defined in section 3.3.1. It is convenient to proceed in the interaction picture generated by the Hamiltonian H~.The time evolution operator U1(t, t) from the initial time t1, in this picture, satisfies the equation Q

ih 8U1/at

V~(t,t)

=

=

V1(t, t)U1(t, t1)

exp[i(t

—

t1)H~/h]VTv exp[—i(t

where V1 is derived from a.(t)

=

a~exp[—i(t

(4.5.2a)

,

—

VTV

—

t~)H~/h},

(4.5.2b)

replacing in the latter a. with

t,)w1]

(4.5.3)

.

The operator U1, in its time-ordered exponential form, readily follows from the Magnus expansion [161

U1(t, t~)= exp(_i ~ Z~(t,ti)), Z1(t, t1)

Z2(t, t1)

=

f

(4.5.4a)

dt’ V1(t’, t1)/h,

dt’

J

(4.5.4b)

2,

dt”

(4.5.4c)

[V1(t’,ti), V1(t”, t~)]/2h

where the higher order terms in the exponent are null, due to the commutators of the creation and annihilation operators, [a Using this expansion, one also finds the form of the wave operator 1,a~]= Wfr = W,, in the present notation, 6jk~

W~= U~, —~)=

fl

~.

exp(—iz 1),

~

=

exp[—i(A1a1

+

A~a~)],

(4.5.5a, b)

374

L. Vulallonga and LI. A. Micha. The calculation of time-correlation functions for molecular collisions

A,

=

f

dtv~(t)exp[—i(w1t+ 61)]/h,

f f

z1 =

dt

dt’ v1(t)v1(t’) sin[w1(t

—

(4.5.5c)

t’ )1 /~,

(4.5.5d)

where is the initial phase of the jth oscillator. The vibrational correlation function finally takes the form F~(t;17) 17(t; .17)

=

=

~W~(t)W,,(0)))~=FJIj(t; 17), KKWj(t)W’,.(0))~,.

W,(t)

=

exp{—i[A1a1(t)

(4.5.6a) +

A~a,(t)]}-

(4.5.6h.

c)

This expression can be evaluated using the algebra of operators as done earlier, and again results in an expansion of the TCF in terms of modified Bessel functions 1,,, [16] F~(t;17) =

exp[n(ihwt

~

A*A ,

a

=

a)

—

hw/2kBl

\,

,

—

v coth a] l,,(v/sinh a),

(4.5.7a) (4.5.7h, c)

where v implicitly depends on 17 and we have suppressed the subindex j. The calculation of the Bessel functions is implemented as was indicated by eqs. (3.3.7). in order to avoid overflows for large i.’ and/or small a. We next consider the case where a collection of harmonic oscillators are driven by nonlinear potentials, keeping however only terms quadratic in the s, within the expansion of the driving potentials. The time-dependent Hamiltonian operator is then written 7(t) + V H(t)

=

H,,, + 1

2(t) ,

(4.5.8)

where V~(t)and V2(t) are linear and quadratic potential energy functions, respectively, in the s,, and the time dependence arises from the translational trajectory. The total vibrational Hamiltonian is then a bilinear function of the operators {Q1, P,}, where P. = —ill ~I3Q,. In what follows we show how one can calculate time evolution operators for these time-dependent Hamiltonians [124], and we mention how one can obtain TCFs describing translational—vibrational energy transfer [125, 126]. To simplify matters, we first consider a single normal mode, and then generalize the treatment to several normal modes. We further do this including all the contributions coming from the quadratic terms in the driving potentials; some of these contributions have frequently been neglected in the literature to simplify the algebras. For a single normal mode, the total Hamiltonian can be written as a combination of the operators X,, given by 2, X,=KA(QP+PQ), X 2, X X=(KQ) 5=(AP) 4=KQ, X5=AP, X,~=U. (4.5.9)

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

375

where I is the identity operator, K has dimensions of (length)~and A dimensions of (momentum)~,so that ~ = hKA is dimensionless. These are found to form a set closed under the commutation operation, and hence constitute a Lie algebra. Using this closure property, it is relatively straightforward to construct first the Hamiltonian H1(t, t) in the interaction picture, and then its time-evolution operator. We write H1(t, t)

=

f~(t,t~)X~,

(4.5.10)

where the functions f~ are easily derived [125]. The time evolution operator is written in the product form [127] U1(t, t~)=

exp[—ia~(t,tI)Xfl].

(4.5.11)

Replacing in the equation for U1 one can derive the coupled differential equations for the a~,which are first transformed into linear coupled differential equations for two new functions q(t) and p(t), and are then solved [124]. The functions a~(t,t~) are given in terms of two independent solutions

{q~(t),pm(t);m1,2}. The collisional TCFs involve the asymptotic form Wi,, of the time-evolution operator,

w~=

fl w~.

(4.5.12)

Transition amplitudes between harmonic oscillator states, and the averages present in the TCFs can be conveniently evaluated in the coordinate representation using the algebra of the X,, operators and some properties of harmonic oscillator eigenfunctions [126]. Compact expressions have been derived in this way for thermally averaged TCFs in terms of { q,,, (°°),Pm (oo), m = 1, 2}. The algebraic approach has been the basis for several approximate treatments of oscillations in systems with several degrees of freedom, coupled by bilinear potentials. They are briefly reviewed in what follows because they are suitable for derivations of TCFs. An approach introduced for a single degree of freedom [103], and using creation and annihilation operators, was extended to several degrees of freedom [43,101, 128] and extensively applied to cases where direct two-quanta transitions, mediated by operators aa~,a1a1, can be neglected. The importance of these neglected terms was investigated some time ago using the algebra of the X,, operators and a sequence of unitary transformations [129,130], and was applied to collisional energy transfer in H2 + HF, N-, + CO, N2 + 02 and H2 + H2. Anharmonic1”anh oscillators have also with was expressed in been terms described of creation anda similar algebraic approach. The anharmonic potential annihilation operators, and its terms regrouped into a term given by a combination of operators within a Lie algebra, and a residual term outside the algebra. The internal Hamiltonian within the algebra generates a transformation to an intermediate picture and the new Hamiltonian in this picture is regrouped in the same way. A sequence of such transformations rapidly leads to results for He + H 2 including anharmonicity effects [131]. These approximate treatments have been followed more recently by exact treatments for multiple degrees of freedom coupled by bilinear potentials, using a recursive algebraic procedure [132,133], and

F. Vilallonga and D.A. Micha. ihe calculation of time-correlation fu?uctuons for molecular collusto,u.u

376

representations of the dynamical groups of the coupled oscillators [134]. The recursive procedure provides a simple and computationally convenient way for calculating transition probabilities and TCFs. For f~ degrees of freedom, it starts with the f,,, x I matrices P1. Q1 of momenta and position operators in the interaction picture. It derives expressions for the matrices of creation and annihilation operators a+(t)

=

(~ U)[P1(t) ±iQ~(t)],

(4.5.13)

of the form a

(t)

=

G,,(t)

a,(t)

=

G~(t)+ G~(t)a.j0)+ G’~(t)aJO)

+

G.~(t)a_(0)+ G5 (t)a...(0)

(4.5. l4a) (4.5. l4b)

.

where G,, is an f,,, x I matrix and the G. are f,,.. X ~ matrices obtained from linear differential equations. Transition probabilities and TCFs between states IN) = In n,) can be obtained from simple recursion relations for the matrix elements (N’Ia. UIIN) [132, 133]. ~.

4.5.2. Asymptotic expansions for rotational motion Given the separation of slow rotational from fast vibrational motions outlined in section 4.2, unperturbed rotations are described by the Hamiltonian HR. and the translational—rotational coupling is VTR(R, F) = V(R, (u), d). Using again the Magnus expansion one finds that the wave operator W.,,~WR,where [16]

WR

=

exp(—iAR),

AR

=

h’

I

dtVfR[R(t~ 17), 17].

(4.5.l5a. h)

and we have neglected the slow evolution of the molecular Euler angles in comparison with the much faster relative motion. By comparing against vibrational motions, one is similarly justified in setting ak,k(t) akk(O) within the overall collisional TCF of eq. (4.3.4a), so that F(t; k,, k,)

~

exp(inw,,,t)

~ ~a~,(0) exp[iA

P~(k,,k1 17) exp[ —iA R(~~)l akk(O)))

R(t)]

(4.5.lôa) Pfl(kf, k; 17) =

fl exp(—n1a,

—

a1).

i-’~ coth a~)I,,(i~’,/sinh

AR(t) = exp(iHRt/h) AR exp(—iHRt/h) ,

(4.5.l6h) (4.5. 16c)

where n = {n1: j = I A~} is a row matrix of integers and t.~ is the column matrix of harmonic vibrational frequencies w1. Since AR is of the order of the transferred angular momentum (in units of 11), and the latter is of the order of 10011 at superthermal impact energies, one usually has IIARII ~ 1 Hence exp[iAR(t)] oscillates rapidly, even though AR(t) varies slowly with time, so that it would be inappropriate to take exp[iAR(t)]zo~exp[iAR(O)]. Nevertheless, the rotational correlation can be readily

F. Vilallonga and D.A. Micha. The calculation of time-correlation functions for molecular collisions

377

evaluated by means of an asymptotic expansion specifically developed for TCFs of exponential operators with large exponents, as is shown next [16]. To outline the procedure in its most general way while abbreviating the notation, consider the TCF

f(t) = ((Z(t)Z’(O))) Y(t)

=

,

Z(t)

=

exp[iAY(t)],

(4.5.17a, b)

exp(iH1~1t/1l)Yexp(—iH~~1t/h) ,

(4.5.17c)

where A is a real number, V is an operator involving the internal degrees of freedom, and the internal Hamiltonian Hint is time-independent. In order to keep track of the relative magnitudes of the terms involved, it is convenient to assume that II I’ll is of order unity while A 1. A Taylor expansion of Z(t) in powers of t centered at t = 0 would give ~‘

Z(t)

=

(t’/n!)(a~Z/at~)I~.,,

~

(4.5.18)

.

n=

Obviously, this series cannot be truncated because the resulting TCF would not have a definite Fourier transform. Instead of truncating, one develops an approximation to the time derivatives that is valid for all n. Starting with n = 1, one identifies the highest power of A in az/at by writing Z(ty’ 8Z/at= ~ bm(t)A”’,

(4.5.19)

m=))

where the bm(t) are operators to be determined. Expanding the left-hand member in a power series of the exponent of z and inspecting the coefficients of A’~in the resulting equality, one finds that b,,(t) = 0 and that

b~,(t)= _[(_iA)0u/m!]{[Y(t),}m~a V/at] ,

m >0 ,

(4.5.20)

where {[y,)~~ a V/at] indicates the m-tuple commutator of V with aY/at. An important simplification is possible in the cases of vibrational—rotational motions. Throughout this review we have seen that the relevant operators V are functions of position operators (i.e., vibrational displacements and molecular Euler angles), and that they do not explicitly involve the conjugate momenta. Presently, Hinu is quadratic in the conjugate momenta, and as a consequence 2 aY/at] = i[Y,[Y,[HI~~, Yj]]/h = 0. {[Y,} Hence the series in eq. (4.5.19) terminates after m = 2, and

az/at

=

Z(t){iA2[Y(t), ~

Y(t)]] /211

—

A[I1~

1, Y(t)] /11)

.

(4.5.21)

The term linear in A has an obvious classical correspondence, while quantal effects are represented by the term quadratic in A, which is much more significant whenever A 3’ 1. In these cases one may retain only the highest power of A and write

378

F. Vilallotuga and LI. A. Micha. The calculation of time—correlatio,u fi~nctio,u.sfor ouolecu~larcollisions

az/at

Z(t){iA2[ Y(t), [H,,, Y(t)]] /211 + 0(A))

(4.5.22)

.

Repeated differentiations with respect to time then give (a”Z/at” )I,

,

Z[(iA2[ Y, I H

0, Y]] /2h )‘

+

O( A’ -

(4.5.23)

)1.

for A ~ I, where all the operators in the right-hand member are evaluated at

I

=

0. This result allows

one to re-sum the Taylor seriesY(4.5.19). 2[Y.IH,,,~, I1t/2h)~obtaining (4.5.24) f(t) — ~exp(iA asymptotically for A ~ I. As an example, we outline the calculation for a symmetric-top molecule by taking AY = AR and H,,, = HR.

HR = (112/21, )J2 + [h(i

1/)/21,1/]J~.

—

(4.5.25a)

J2BOtPdO2P(d+d)+ 2tPPr)

(4.5.25h)

.

(Ia

cia ely

ely -

J~= —ia/a~,

(4.5.25c)

where I~,i = 1, 2, 3 are the moments of inertia about the x, y and z BF axes, respectively, with I, = for spherical-top symmetry. Here J is the rotational angular momentum (in units of 11) with respect to the SF frame, while J~is the angular momentum about the BF z axis [81]. The above expressions include as special cases spherically symmetric molecules (I, = I. = I,) and linear rotors (I., = 0): in the latter case, the second term of eq. (4.5.25a) is not present. To proceed further, one writes J in terms of its Cartesian components =

i(cos a cot /3 a/aa

+

sin a ~!ciJ3 cos a cosec /3

=

i(sin a cot /3 8!8a

—

cos a aiai~ sin a cosec /3 alay) , —

and one introduces the operators J [f , J]

=

I,

±if,,.

ci/ciy)

(4.5.26a)

,

J

=

—ia/an

.

(4.5.26b, c)

Taking advantage of the commutation relation

2J, the rotational Hamiltonian takes the more convenient form 2/21,)(J J + J7 J.) + [112(1, I~)/211 HR = (h 5]J~. =

,

—

—

(4.5.27)

The commutator [AR, [HR. ARI] can then be calculated explicitly be repeated applications of the relations [A, BC]=[A, B]C+ B[A. C].

[AB, C]= A[B, C]+[A, C]B.

(4.5.28a, b)

valid for any three operators A, B and C. The average over the distribution wR(L; TR) of initial

E. Vilallonga and D.A. Micha. The calculation of time-correlation functions for molecular collisions

379

rotational states IL) is then evaluated for the temperature TR in the coordinate representation, and one obtains for the overall TCF F(t; kf, k1) — ~ exp(inw~t)~ wR(L; TR)

J

d17 ~

x [(do~/d1l) exp(i7)~t/h)P~(k1,k1 ~7R =

(4.5.29b)

(17~[AR, [HR, AR]]~17),

112 1/BA =

(4.5.29a)

F)](i) ,

~/

\~2

1~~)L~05p

~

+

~

~ R

\2

R) + ~R)

—

2

~

\2]

j(sin f3)~ (4.5.29c)

.

ll)/21,IS](aAR/ay) Finally, taking the Fourier transform of F(t; k + [h2(1~

—

1, k) yields the doubly differential cross section,

dEdQ

=

k ~ wR(L, TR)

J

d17 ~

(~‘KF~L)I2~

P~(k 1,k~17)6(r

-

hnw~ -

(4.5.30) 2u/dr dQ consists of a continuous Examiningof the above expression fromatright to left, one sees equal that dto the vibrational (hno~)plus the distribution 6-function peaks located transferred energies rotational energy (mi) going into the internal molecular motions for initial orientations 17. The intensity of this distribution is determined by the P,,, which correspond to the probabilities for exciting vibrational transitions, and by the absolute value squared of the rotational eigenfunctions which determine the distribution of energy ~R transferred into rotation. This product is next multiplied by the classical cross section for the relative motion and summed over all the trajectories j that arrive at the given scattering angles (0, 4)), thus yielding the overall transition probability for given initial orientation. The overall probabilities are then integrated over the initial orientations by numerical quadratures based on the grid of 17 chosen in section 4.4.1, and finally averaged over the distribution of initial rotational states.

(17I~~),

4.5.3. Collisional action integrals

The expression for ij~ indicates that the energy transferred into rotation is determined by the intermolecular torques integrated along the trajectory of relative motion. In fact,

aA

aV

, I

=

11” j

dt

~

R(r;l~)

(4.5.31)

for each of the Euler angles ~ = a~,/3 and y, so the integrands are the torques around the BF axes of the target. To calculate the aAR/a~,and also the vibrational action integrals A. defined by eqs. (4.5.5c) and (4.5.ld), one first takes a derivative with respect to the upper limit of integration. The resultant time-differential equations are then numerically integrated in parallel with the Hamilton equations for the classical trajectories, as was outlined in section 4.4.1.

381)

F. Vulallonga and LI. A. Micha. I’hr calculation of time-correlatuon functions for molecular collu,suo,ts

4.6. Applications to molecular collisions 4.6.1. Li —N, collisions The above treatment of linearly driven vibrations has been thoroughly tested by calculations of Li’ ions scattering from N. [62], for which there is available a wealth of experimental data [87, 120] as well as independent theoretical studies. For instance. Secrest et al. did a fully-quantal study using the infinite-order sudden approximation to simplify the rotational problem [135]; Billing classically treated the relative and rotational motions while describing vibrations quantum mechanically by means of eigenfunction expansions with time-dependent coefficients [Ill]; in addition, purely classical simulations have also been performed for Li ‘—N2 [121, 122]. We first summarize the comparison between our TCF and Billing’s semiclassical results, and then illustrate a TCF analysis of experimental measurements. Semiclassical results were presented in the form of state-to-state differential sections 2u/dE d[l that are cross measured by du,~,,1/dI2,experiments instead of and the calculated distributions of transferred dand j,- denote the final vibrational— time-of-flight by collisional TCFs;energy here n~ rotational quantum numbers of N,. For the purpose of comparison, do~,,,,,/dul were derived from do/dr do as follows. Since the population of excited internal states was initially low in these cases, the change in vibrational quantum number was set to n — n,. and the energy transferred into rotation (ER) was represented by a final rotational index j’ defined by the relation t~= nhw,, + nfhw\,. + h2’j’(j’ + 1)121, where j’ is a continuous variable, / is the moment of inertia of N. and the initial internal energy was assumed to be negligibly small. Accordingly, do~,.

~f 11

d2odj’ (2j’ + I) dr dOI’

(4.6.1)

for integer values of], and with e evaluated at n~hw~ +h2j’( j’ + 1)121. Figure 8 compares the du,, 2,,,11d02 versus j, with Billing’s semiclassical results, both calculated for E = 7.07eV and O~= 49°using the same potential surface [112]. The main features of the cross sections agree very well, particularly in the maximum values of do~,,,.,,,/df2and in their position on the j, axis. Differences in the details of the distributions were expected since each calculation corresponded to somewhat different treatments of the initial conditions. Specifically, Billing considered only the case of = 2 and even values of j~in contrast, the initial experimental distributions (and therefore our calculations) included several rotational states. Typically TR = 15 K in experiments [120], so the TCFs allowed transitions into even and odd values of j,.. Nevertheless, both theories agreed quantitatively in the location of the maxima, which defines the most probable value j, of the final rotational angular momentum, and also agreed in the maximum value du,,,,~,,,1/dQof the cross sections for all the calculated values of E, 6 and n, [62]. As an example of the distributions of transferred energy resulting from collisional TCFs, fig. 9 shows duld(2 dr calculated for Li ‘—N. at the impact energy E = 8.4eV and scattering angle 0 = 73°[62].The TCFs yield a very dense distribution of transitions, due to the large number of final vibrational— rotational states that are accessible at this collision energy. Also shown in fig. 9 is the experimental distribution measured by time-of-flight techniques [120], and at first sight there appears to be little agreement. In fact, this figure also serves to illustrate the pitfalls of comparing theoretical and experimental results in a naïve way, i.e., without taking into account the averaging properties of the

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

10

E=7M7eV

1.0

J

0)

~

E=8.4eV

)‘.

~~./ \

: 0

0

381

0.5

~‘

0

2040

60

0.0

e=73°

•

0

Jf

1

2

Energy transfer (eV)

Fig. 8. Differential cross sections du,,,,,,,,,/dQ in collisions of Li~ with N. at the impact energy E = 7.07 eV and scattering angle 0 43°. Full line: semiclassical TCF results; dashed line: results of the semiclassical vibrational basis-set expansion of ref. [lllj; the error bars indicate the numerical uncertainties of the latter.

Fig. 9. Doubly-differential cross section d’o/d(2 dr versus transferred energy r in collisions of Li’ with N, at the impact energy E = 8.4 eV and scattering angle 0 = 73°. The experimental intensity is indicated by circles.

experimental setup. Since the intensity was very low at large scattering angles, the experiments sacrificed resolution of final energy in order to improve the signal to noise ratio. The measured spectra corresponded to an effective resolution in r of 0.2 eV, which is three orders of magnitude larger than the rotational constant of N2 (112/21 = 2.5 x i0’ eV) [136]. Hence the low experimental resolution represented an additional average over the final energy, which “washed out” the fine rotational structure and instead gave rise to the broad peak measured around r 1.2 eV. It then becomes necessary to average the theoretical cross sections over the resolution profile g(e) of the experimental detector I(r; 0, E)

=

J

ds’g(r

—

e’) dQ

(4.6.2)

where 1(r; U, E) is the measured intensity. Typically g(r) is well approximated by a normalized Gaussian function, so that from eq. (4.5.30) it follows that

2

1(e; 0, E)E wR(L, TR)JdI7~ [dUct (17~L)j —

x ~ (i —

~

(1)

1/2

c exp[—b(r — hnw~—

~R)2]

P~(k,,k~,17)]

,

(4.6.3a)

382

F. Vilallonga and LI .4. Mic/ua, The calculatuon of’ time-correlation fd,uction.s for molecular u’olli.sio,u.c

h

=

c

4(ln 2)I~.

=

(7/~)[(In 2)/~J

(4.6.3h, c)

2

where ~ is the final-energy resolution of the detector (full width at half maximum). We must emphasize that g(e) does not arise from any limitation inherent in the TCF treatment; it is introduced a posteriori into the theoretical results with the sole purpose of facilitating the comparison with low-resolution experiments. In fact, the TCFs naturally lead to distributions of 6-functions for the quantal transitions, as is clearly seen in eq. (4.5.30) as well as in fig. 9. When the experimental resolution profile was incorporated into the theoretical results, the measured and calculated intensities for Li —N. came into remarkably good agreement. as is demonstrated by fig. 10 for E = 8.4 eV, 0 = 37° and 73°.The semiclassical TCFs gave excellent results throughout the wide ranges E and 6 measured by experiments (4eV ~ E ~ 17eV and 35°~ 6 ~ 105°),and they explained the complicated structures of the measured cross sections [62]. For instance, fig. 11 shows the distributions of transferred rotational energy corresponding to each vibrational transition n. One thus sees that the narrow peak measured near e 0 arises from purely rotational excitation (n = 0). while the broad peak centered near r — 1.2 eV is due to combined vibrational—rotational excitations. 4.6.2. Scattering of atoms from molecules adsorbed on linear lattices The semiclassical TCF treatment can also he applied to collisions with molecules bound to a substrate or to an extended system. We consider here a model substrate given by a linear chain of metal atoms, moving under a harmonic force field. This can be considered a one-dimensional model of an

1.0

E=8.4eV Q.730

/

a

_______ _________ 11.0

~

I

0=37°

:[~‘~\\

0.5

I o.o

~E=6

.ri I..

I’)

n

-‘

/

~0.5

0=73°

1

f

/

~

1

2

0

Energy transfer

1

2

\ ~

/~

~\

4

/ -i/777j1L~’\\

4eV~

I 0

/

3

(eV)

Fig. IP. Intensity of scattered particles l(r; 0. F) versus transferred energy r in collisions of Li with N, at the impact energy F = 8.4eV and scattering angles 0 = 37° and 73°. Curve: results of the semiclassucal TCF theory convolved with a Gaussian detector profile of width = ((.2 eV: circles: experimental results. The intensities have hccn normalized so that their maxima equal unity,

0.0

0.0

-/

~./

1.0

~..

“..

~.

2.0

‘v~

3.0

Energy transfer (eV) Fig. II. intensity of scattered particles 1(r: 0. L’) versus transterred energy r in collisions of Li’ with N. calculated for the inspact energy

F = 8.4 eV. scattering angle 0 = 73° and detector resolution i, = ((.2 eV. The dashed lines indicate the contributions of each vubratuonal transitiou,. The overall intettsitv has been displaced upwards for clarity.

E. Vilallonga and D.A. Micha, The calculation of time-correlation functions for molecular collisions

383

adsorbate on a metal surface. The calculation of cross sections for scattering of atoms by this target illustrates the possibilities of the collisional TCF approach for studies of interactions with solid surfaces and with other extended systems. Figure 12 summarizes results corresponding to one-dimensional models of atom—adsorbate/solid collisions [137,138]. Here we compare the energy-loss spectra of He scattering from N2WN, OCPtN, OCNi,,,,, WN, PtN and NiN, calculated for the impact energy E = 0.5 eV and for N = 80 substrate atoms. The energy-loss spectra are displayed as they would be measured by a one-dimensional “experiment”: they include a Boltzmann distribution of initial vibrational states for the target cooled to T~= 77 K, and the energy resolution of the simulated detector is /.1,. —0.01 x E = 5 meV, as in modern time-of-flight machines. Considering first the clean substrates (dashed lines), one observes a smooth distribution of r due to multiphonon transitions of very high order. There is a preponderance of phonon excitation (r >0), because the impact energy is much larger than the excitation energies of bulk phonons, presently hWph 0.01 eV. Nevertheless, there is a small but nonzero probability for r <0, which indicates that some thermal energy can be transferred out of the solid into projectile translation, particularly in the cases of WN and PtN. When the adsorbates are present (solid lines), one observes additional peaks due to excitation of the adsorbate—substrate bond, !IWAS 0.05 eV in these systems; the internal bond of the adsorbate not 11WAA 0.3 eV.is The significantly excited at this energy, since it has a considerably larger frequency location of each peak is displaced from the expected position, 11flWAS, by the average energy (EPh) transferred into the substrate phonons. Each peak is also broadened by the large number of °~

O.~EO~V

Energy transfer (eV) Fig. 12. One-dimensional simulations of energy-loss spectra for He scattering from diatomic molecules adsorbed on metal substrates: N 2W8,,, OCPt80, OCNi50 (full lines) and from the pure metals w80, Pt80, Ni80 (dashed lines), calculated for the collision energy E = 0.5 eV, target temperature T = 77 K and detector resolution Li, = 5 meV. The intensities have been normalized so that their maxima equal unity.

384

L’. Vilalloouga and 1). A .Micha. ‘flue calculation of tiou’-correlatio,u fiouctiosus for molecular colli,su,ns

simultaneous transitions of bulk phonons: in the case of He—OCNi~.,the phonon broadening is so large that the transitions of the C—Ni stretch cannot be resolved. 4.6.3. Estimated computational cost It is worth remarking that in these semiclassical calculations the largest fraction of CPU time goes into the calculation of classical quantities: the trajectories of relative motion and vibrational—rotational action integrals. As a consequence, the overall computational cost is mostly determined by the grid steps necessary to adequately sample the initial conditions for the relative motion: impact parameters plus molecular orientations. In our three-dimensional studies, we have found that a typical trajectory requires approximately 25 s of CPU time on a VAX-780 equivalent, from which one can estimate the overall cost by multiplying with the number of initial grid points. Moreover, the trajectories are naturally independent of each other, so the procedure is ideally suited for parallelization in multiprocessor systems, should the need arise in more extensive studies. After the classical quantities have been tabulated, the calculation of d2u/dr dO) from TCFs takes a few minutes of CPU time (VAX-780). The distributions of energy transferred into quantal internal states are thus obtained for the cost equivalent to a much simpler classical calculation. Collisional TCFs in the short-wavelength limit therefore provide a highly efficient computational treatment of polyatomic systems with very large numbers of energetically accessible states (~l0” in the above examples).

5. Conclusion 5. 1. Advantages of calculations based on collisional TCFs The relation between collisional TCFs and double-differential cross sections provides not only physical insight, but also alternative ways to calculate cross sections suggested by considerations of the time scales of the degrees of freedom in the interacting molecules. For events of a short duration compared with collisions times, the cumulant expansions have led to expressions showing that energy transfers into slow’ degrees of freedom are described by Gaussian distributions, with their parameters easily calculated from the theory. This was particularly useful in the description of rotational excitation, for example in the collisions of Li with CO and N 2, and of H with CO and CO2. Arguments based on time scales were also important in suggesting the decomposition of TCFs into nested expressions, where one first calculates the correlation of fast degrees of freedom for fixed slow ones, and then one constructs the total TCFs by suitable weighted averages. This was the procedure followed to describe vibrational—rotational transitions in both impulsive and short-wavelength calculations. The two methods emphasized in this review, sequences of encounters for impulsive collisions and driving forces along trajectories for short de Broglie wavelengths of relative motions. are the ones that lead to the simplest computational procedures while at the same time providing numerical results for realistic models, that can be compared with experimental measurements. The physical picture of a sequence of atom—atom encounters mediated by repulsive potentials could be implemented in our calculations starting from the formalism of multiple scattering. The resulting TCFs then factored out into scattering amplitudes relating to relative atom—atom motions, and correlation functions for the atomic distributions within the targets. In this way the calculations could be done separately for both factors. The Poisson distributions of transferred vibrational energies, cx-

E. Vilallonga and D. A. Micha. The calculation of time-correlation functions for molecular collisions

385

perimentally observed, were then found in a natural way from a formulation that could also be used to calculate the distribution parameters. For phenomena characterized by relative motions with short wavelengths, the collisional TCF approach pointed the way to a computational procedure where one first calculates not only the relative trajectories of the centers of mass, but also the action integrals accumulated over these trajectories. These quantities are then used in the construction of the TCFs at relatively small computational expenses. 5.2. Extensions of the theory and prospects for future calculations

The brief discussion of the basic formalism hinted at other procedures which could be followed to calculate collisional TCFs. These involve for example formulations in Liouville’s space [15, 22] and formulations based on time-dependent transition amplitudes [24,25, 139]. Formulations in Liouville space, which introduce superoperators through commutation relations, are very useful in the derivation of compact expressions for double-differential cross sections, for example taking the form of generalized Lorentzian distributions. They can also lead to cumulant expansions as shown here for dynamical form factors, but for more general situations. They also provide a way of treating thermal averages over initial distributions in a consistent way, where all the initially populated states are incorporated simultaneously in the calculations. Formulations based on time-dependent transition amplitudes were found very useful in the calculations of TCFs where the projectiles are photons, and the cross sections give line shapes. The procedure can be applied to any type of interaction, and incorporates from the outset the properties of the coupling to the probe particle, such as a transition dipole or a collisional operator. It requires the solutions of partial differential equations for the amplitude functions, where the variables are the time and the degrees of freedom active in the collision. These two formulations and other aspects of the formalism of collisional time-correlation functions are further described in a review chapter in preparation [17]. A particularly useful aspect of collisional TCFs is the natural way in which they can describe extended systems where the boundaries are kept under certain thermodynamic constraints, such as fixed temperature and volume. This can be easily incorporated in calculations through the statistical densities mentioned in the section on the basic formalism. An important application of this feature is to scattering by solid surfaces. We have already mentioned in the introduction some of the developments by us and by other groups in this area. Collisional TCFs can be used to describe energy transfer in collisions of atoms and molecules with clean solid surfaces and with adsorbates on solids. Present studies using models of low dimensionality or cluster models are being generalized to more realistic models. Three-dimensional calculations are well under way and their results will be published in forthcoming articles [140]. The formalism is also being developed by us and by other groups to cover a great variety of phenomena including also overlayers and disordered systems. We are working on a review of work in this area by us and others.

6. Acknowledgements Our comparisons with experimental results originated with an extended visit of one of us (D.A.M.) to the Max-Planck Institut für Strömungsforschung (Gottingen, Germany) some time ago. We thank Dr. Peter Toennies for very helpful discussions at that time and later on. Research in this subject has

386

F. Vilallonga and LI. A .Micha. The calculation of’ time-correlatiotu fuinction.s for molecular collu,vuotu,s

been supported by grants from the U.S. National Science Foundation to one of us (D.A.M.) over several years, most recently with the Grant CHE-8918925. The work reviewed in section 4.6.2 on energy transfer into adsorbates was partly supported by a grant from the U.S. Air Force Office for Scientific Research (F 49620-87-C-0045) to Herschel Rabitz and E.V. This review was partly written while D.A.M. was visiting the Institute for Theoretical Atomic and Molecular Physics at Harvard University and the Smithsonian Astrophysical Observatory, with support by the NSF, and the Chemistry Department and the Supercomputer Computations Research Institute of Florida State University, with support by the contract DE-FG-5-87ER60517 of the U.S. Department of Energy. While writing this review, E.V. was partially supported by grants from ONR (N 00014-91-i- 1209) and DOE (DE FG-02-86ER13480).

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F. Vulallonga amid LI.A. Mucha, ‘The calculation of time-correlation functions for molecular colhisuotus

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