>=(m./4)cotha. and f.=0,so the present development includes

1 1 1 1

harmonic vibrations as a specific case.

Replacing eq. (2.4.12) into eq. (2.3.13) and rearranging terms,

we obtain the following expression for the APCF.

(aa) 2 -OCi -il.t

FV (,t) =[exp{-6' ai)2 [cosh a. + (e +

i

a. + i1.t

+ e )/2]/(2 .sinha.)}. (2.4.18)

i 1

Using the generating function of the modified Bessel functions of

the first kind (I ) (Abramowitz and Stegun, 1972),

n

exp[x(y + l/y)/2] = y I (x), (2.4.19)

n=-on

we finally arrive at

(aa) (aa) + +

F (Kaa t) = p (K;F)exp (-in. Vt), (2.4.20a)

n n

n

(aa) + -

p (K;F) =n I (X .)exp(-n.a. X cosh a.), (2.4.20b)

n. aii i ai 1

n i

X ai = '[K .(r)] /(2w .sinha.), (2.4.20c)

in which we have introduced the abbreviations n={n.} and

1

V= (Wi.}. Comparing these last equations with the ones corresponding

to harmonic vibrations (eq. (4.16) in Micha, 1979b), we find that the

APCF for harmonic and anharmonic motions coincide in form; however,

for anharmonic molecules, the normal frequencies m. must be replaced

1

by the effective ones w. obtained from eq. (2.4.17).

1

Nonzero intercoordinate anharmonic couplings cause mixing of

the normal coordinates Qi'. which is reflected in the off diagonal

parameters biz of eq. (2.4.4). In this case, the linearized equations

of motion for the DDGF lead to the following matrix equation for

their Fourier transforms.

(w2 1 2 + b) G( (w) = 1 l/(27), (2.4.21)

rb 'V ^ 'I lb

(+) (+)

where G (w) = {G. (w)}, b={b..} and OV ={ 6 .ij.}. Solving for

% r\ ij ru ij i

G (w), shows that the effective frequencies are given by the roots of

2 2

the determinant of (w 1-0 +b). Hence, choosing the b.. to reproduce

'. "'V r" 1J

the exact second derivative of the DDCF at t=0 leads to a matrix

equation for b analogous to eq. (2.4.17). Based on the present de-

velopment, the effects of molecular anharmonicity in hyperthermal col-

lisions may be interpreted in the following way.

5. Effects of Molecular Anharmonicity in Hyperthermal Collisions

In analogy to eq. (2.1.4b), the Fourier transform of the

vibrational APCF,

(aa) (;) dt (aa)

Sr) =exp (-iet/() F-(K,t) = (2.5.la)

V 2 exr V

-00

(aa)

= p (K;r) 6(n. w + e), (2.5.1b)

Sn

n

corresponds to the probability that the molecule will absorb vi-

brational energy and momentum in a collision involving atom a, for

the fixed orientation of the principal axes of the polyatomic in-

dicated by r. This has the form of 6-function peaks, located at the

vibrational energy transfer e =-n V where n. is the change in

quantum number (initial minus final) of the ith effective normal mode;

+ (aa) -

the probability for the transition n is given by p (K;.r)- .In

n

practice, these peaks are broadened into Gaussian functions by the

rotational correlation, as is shown in the following section. In

light of this analysis, the effects of molecular anharmonicity in

hyperthermal atom-polyatomic collisions may be summarized as follows.

1) Anharmonic forces cause a shift in the spectrum of the vi-

brational energy transfer, from the set of harmonic frequencies m.

1

to the effective ones W..

2) Equation (2.4.17) shows that this shift is related to

the work done by the anharmonic forces on the vibrational displacements.

3) The change in the vibrational frequencies is accompanied by

a corresponding change in the probabilities for vibrational excitation,

as indicated in eqs. (2.4.20b,c).

4) In contrast to the harmonic case, the effective frequencies

depend on the temperature of the distribution of initial states.

This is due to the fact that the anharmonic forces affect all the

states that are present in the distribution.

The magnitudes of the effects depend on the specific nature and

strength of the intramolecular forces of the target and require the

evaluation of the averages indicated in eq. (2.4.17). These may be

calculated by standard numerical procedures, or by finite-temperature

perturbation theory, when the temperature of the initial distribution

is sufficiently low. The application of this development to specific

targets will be presented in Chap. IV. We now proceed to consider

the rotational correlation, in order to complete the evaluation of

the APCF.

6. Short-Time Approximation to the Rotational Correlation

The rotational correlation of heavy molecules has been previously

evaluated by means of a Taylor-series expansion of the logarithm of the

APCF (Micha, 1979b). In this section, we present an alternative ap-

proach to the evaluation of rotational correlation, which is based

on the cumulant expansion techniques developed in Chap. I. Although

the two approaches are completely equivalent, we feel that the

present one allows a more unified understanding of the role of

correlation functions in the description of many-body collisions.

Substition of eqs. (2.4.20) into eq. (2.2.3) gives the following

expression for the complete self-correlation function

(aa) + (

F(aa) (,t) = F (,t) exp(-in w t)V (2.6.1a)

+ n

n

(aa) (aa) ^ ex (t

F (K,t) = <>

a K- aR R

n II

(2.6.1b)

where the time dependence of F(aa) is due only to the rotational

n

motion of the molecule. Following the procedure described in Chap. I,

A

Sec. 3, we find that the time evolution of exp(-iK*-d ) is given by

a

exp[i.da (t)R] = exp(itLR/7) exp[i 'a(0)]. (2.6.2)

Here, LR is the Liouville superoperator for rotational motions,

which is defined by its action on an arbitrary operator A, as follows

L R = ([R,)nA]. (2.6.3)

Equation (2.6.1b) can now be rewritten as

(aa) (t) (aa) (aa)) +^

F (t) = P a(K) {<

n n n

x exp[-i-- a(0)]>>R (paa) ; (0))>>R, (2.6.4a)

n

p (). <

>R. (2.6.4b)

n n

Whenever the rotational motion of the molecule is much slower than

its vibrational one, we perform a short-time approximation to the

rotational correlation of each vibrational transition. As discussed

in Chap. I, Secs.4,5, this corresponds to expanding the second fac-

tor of eq. (2.6.4) in terms of the cumulant rotational averages of LR

and retaining only the first two cumulants, so that

(aa) (aa) 2 2 2

Fa (K,t) P (K) exp [it<> /R t <>C/(2 )]. (2.6.5)

n n

The lowest-order cumulants are determined from eq. (1.3.5) to be

(aa) ^ +, 1 (aa)

<> = <

> /P (K),

R CR a R a R

n n

(2.6.6a)

2 (aa) ^ )[H R ((aa)

<> (K) -

R CR 'a RR R

n n

<>2R* (2.6.6b)

Replacing eq. (2.6.5) into eqs. (2.6.1) we obtain the following

expression for the APCF,

(aa) -(aa) 2 2/2

F (K,t) = > P (K)exp[i(<>C-n w V )t/-<
+ n

nI

(2.6.7)

Finally, evaluating its Fourier transform, as indicated in

eq. (2.1.4b), we arrive at

(aa) 1 (aa) >>

S (a ,) Y- P( (K) exp {-[(<
4v' n

n

E)/(2<> (2.6.8)

R CR R CR

Examining this last equation we find that, for a given value of the

momentum transfer K, each atom of the target contributes a group of

normalized Gaussian peaks to the differential cross section. Each

peak corresponds to the vibrational excitation indicated by the set

of changes in quantum numbers n, and is centered at the vibrational

energy transferred in.IV plus the average rotational energy trans-

ferred to this vibrational transition <> ; furthermore, each

R CR

2

peak is rotationally broadened by an amount 2<*> Equations*

R CR"

(2.6.6) show that the rotational energy transfer and the width con-

tributed by each atom are functions of the momentum transfer (hence

of collision energy and scattering angle) and depend on the vi-

brational transition indicated by n. The probability that a collision

4-

involving atom a will excite the vibrational transition n is given

(aa)

by P which is also a function of the momentum transfer.

n

However, if atom a is at the center of mass of the molecule,

so that d =0, then instead of eqs. (2.6.7) and (2.6.8), we have

a

F(aa) 4 (aa) 4

F (K,t) = P (K) exp(-in-wvt), (2.6.9a)

4 n

n

(aa) (aa) 4

S (aaK,) = Pa (K) 6(E+ tn v), (2.6.9b)

+ n

n

which shows that such an atom does not absor-brotational energy

but does contributeto the vibrational excitation.

As discussed in Sec. 1, the correlation functions given by eqs.

(2.6.8,10) together with eqs. (2.4.20b,c) and (2.6.6) completely de-

scribe the response of the target to an impulsive collision, regard-

less of the nature of the projectile. The nature of the projectile

atom determines the two-body cross sections of eq. (2.1.5), in

accordance with the atom-pair potentials. Therefore, before

47

proceeding to the applications of the many-body theory, we must con-

struct the multicenter representation of the intermolecular potential

for the collision systems of interest; this is the subject of the

following chapter.

CHAPTER III

MULTICENTER POTENTIAL ENERGY SURFACES

In this chapter we investigate the multicenter representation

of the intermolecular potential energy surface (PES) for nonreactive

atom-molecule systems. Our aim is to develop a parametrization of the

PES which is appropriate for the study of hyperthermal molecular

collisions. In the first section, we weigh the relative advantages

of the single-center and multicenter representations. In the second

section, we develop a parametrization for ion/linear molecule sur-

faces in terms of atom-pair potentials that are based on the known

general properties of intermolecular forces. Next, we evaluate the

atom-pair potentials that reproduce the quantum chemical PES of the

system Li /CO. In the final section, we study the system Li /CO2,

for which no quantum chemical information is yet available. Using

a simple model of short-range forces plus experimental information

on the molecular properties of CO2, we construct a model PES for this

system. Both surfaces developed here will be used in subsequent

atom-molecule scattering calculations.

1. Single-Center vs. Multicenter Expansion

The standard procedure used to describe the interaction between

an atom A and a molecule B consists of assuming the Born-Oppenheimer

separation of nuclear and electronic motions (Tully, 1976) and then

calculating the lowest PES for a set of conformations, by means of

more or less accurate quantum-chemical methods; for examples see

48

Staemmler,1975, Parker et al., 1976 and Thomas et al., 1978.

Working in the body-fixed (BF) coordinate system determined by the

principal axes of B, one thus obtains a table of the intermolecular

potential V for several values of the relative position of A, denoted

by R, and of the positions of the constituent atoms of B, which we

+B +

indicate with r = {r ,aB}.. In order to facilitate scattering

a

calculations, one must usually construct a functional representation

for the surface. The most common of these is the single-center ex-

pansion, in which the dependence on the orientation of R is described

in terms of a set of basis functions of angles, centered at the center

of mass of the molecule, as follows:

.B -B

V(R,r ) = (R,r )Y(n,). (3.1.1)

n

4->

Here, n,E are the polar and azimuthal angles of R in the B1 system,

respectively, and the Y are the elements of the basis set. The radial

n

coefficients V are fitted to reproduce the values of the PES at the

tabulated configurations.

For atom-molecule scattering in the thermal collision energy

regime, such a single-center expansion has several advantages. Since

low energy scattering is mediated by long-range forces (McDowell and

Coleman, 1970) and these forces vary slowly with the orientation of

4->

R, usually only the first few terms of eq. (3.1.1) need be included

in scattering calculations. On the other hand, the PES can be re-

presented to any arbitrary degree of accuracy by increasing the size

of the angular basis set. Furthermore, if the radial dependence of

the coefficients V is judiciously chosen, these can be related to

molecular properties (electrostatic moments, polarizabilities, etc.)

of the isolated collision partners (Hirschfelder et al., 1967),

this can be used to check the accuracy of the fitting procedure,

or to supplement the quantum-chemical information when only a few

points of the surface are available.

However, as the collision energy increases above a few hundredths

of an electron volt, the single-center expansion, although accurate

in principle, presents rather serious disadvantages. At hyperthermal

energies and for small impact parameters, the projectile probes the

inner region of the PES which is of a multicenter nature, so the

single-center representation loses physical meaning. Furthermore,

as the short-range intermolecular forces depend strongly on the

orientation of R, a very large number of terms are necessary for

eq. (3.1.1) to converge. These drawbacks will be further clarified

by the example presented in the next section.

Alternatively, the multicenter nature of the intermolecular

forces can be better represented by expanding the potential about each

atom of the target, as follows:

S+B +* -

V(R,r ) = v (R,r ), (3.1.2)

a

where the sum runs over the N atoms the polyatomic. Here, v

a

corresponds to the interaction between the (A,a) atom-pair which de-

pends on the electronic distribution of A and on that of atom a in

the valence state of the target. Hence, v is a function of the

a

orientation of R with respect to r and of the magnitude of R-r .

a a

In general, for small A-a separation, the atom-pair potentials are

steeply repulsive due to the overlapping of the electronic clouds

of A and a; in the long-range region, the va fall off as inverse

powers of the A-a distance due to the interactions of the charge

distributions of A and a. For example, the PES corresponding to an

atomic ion with charge qA and a molecule could be modeled, most simply,

by the sum of the pair potentials

va(R,r ) = A exp(-B aR-lr) + Aq a (3.1.3)

in which the fractional atomic charges qa reproduce the experimental

values of the N-lowest electrostatic multiple moments of B. The co-

efficients A and B (both positive) can be obtained by fitting to a

previously calculated PES or from combination rules (Gaydaenko and

Nikulin, 1970).

More, generally, eq. (3.1.2) is the leading term obtained by

expanding the potential in contributions from two atom, three atom,

and so on, atom-clusters as indicated below (Micha, 1979a)

SRB ) Q(2) (,+~ 2 (3) -+ -

V ) = va (R,r) + v (Rr ar ) + (3.1.4)

a a**
Recent attempts have been made at understanding the properties of**

large polyatomic systems in terms of contributions of the constituent

atomic centers (Bader, 1980). We hope that, in the future, such studies

would allow the determination of the atom-cluster potentials from

first principles. However, for our present purposes, we will con-

sider the atom-pair potentials to be functions of r and r with para-

a

meters that will be adjusted to reproduce a previously calculated PES

and/or the known molecular properties of the separated collision

partners. As we shall see in the following section, by judiciously

choosing the form of the v the contributions of the higher-order

cluster potentials can be made negligible small.

cluster potentials can be made negligibly small.

2. Atom-Pair Potentials for Ion/Linear Molecule Interactions

As with any other fitting procedure, the functional form of the

Va greatly influences the accuracy of the representation and the num-

ber of parameters required. In order to allow for a physically mean-

ingful interpretation of the pair potentials and to reduce the number

of parameters to a minimum, the v should be chosen in accordance with

a

the known general properties of intermolecular forces (Hirschfelder

et al., 1967, Margenau and Kestner, 1971). For the interaction be-

tween two closed-shell molecules, these properties may be summarized

as follows.

When the relative distance R is much greater than the lengths

over which the electronic clouds of each species are appreciable, the

intermolecular potential can be written as a sum of the following three

contributions:

1) the electrostatic interaction between the permanent multi-

pole moments of the charge distributions of each molecule,

2) the induction contribution, due to the interaction between

the permanent moments of one species and those induced in the other

one, and

3) the dispersion or Van-der-Waals forces, which may be in-

terpreted as arising from the interactions between the induced

moments of each species. However, when at least one of the molecules

has a net permanent charge, the magnitudesof the dispersion terms are

much smaller than the electrostatic and induction ones, so that

dispersion forces may be neglected.

In order to maintain the notation at the clearest level possible,

from here on we take B to be a linear polyatomic at its equilibrium

configuration (r = d ). Hence, the long-range potential for an atom-

ic ion A and B is given by the following single center expression,

SB qq q (cosB2)/R2 3

V(R,r ) qA /R + q (cosn)R + qAQBP2(cosn)/R

q9[aB/2 + ABP2(cosn)]/R4 + .. (3.2.1)

in which we have explicitly indicated the first three leading terms

of the electrostatic forces, the leading induction contribution (of

-4

order R ), and we have neglected the dispersion forces that are of

order R6 and smaller. Here, P (x) = (3x2 1)/2 is the second order

Legendre polynomial, qA is the charge of the ion, while the molecular

properties of B are the net charge qB, the electrostatic dipole

PB and quadrupole Q moments, and the isotropic a = (a + 2a )/3

SB B zz xx

and anisotropic A = (a a )/3 polarizabilities; a and a

B zz xx zz xx

are the components of the static dipole-polarizability tensor parallel

and perpendicular to the molecular axis, respectively. Equation (3.2.1)

follows from a perturbation-theoretical approach, in which the electro-

static interaction between the charge distributions of each species

is expressed in terms of a multiple expansion (Margenau and Kestner,

1971, Chap. 2). Such an expansion is valid only when the distance

between the centers of charge of each distribution is much larger

than the range of the individual densities (Jackson, 1975, Chap. 4).

Therefore, eq. (3.2.1) loses physical meaning whenever R r ; this

is reflected in the unphysical divergence for small R.

is reflected in the unphysical divergence for small R.

When the relative distance decreases, the overlap of the

electronic clouds of each species causes the intermolecular forces

to become steeply repulsive, Hence, the small R behavior of the

potential depends very strongly on the details of the electronic dis-

tributions. However, when the molecular orbitals of each species

are described in terms of Slater-type orbitals centered on each

nucleus, the overlap will be an approximately exponential function

of the A-a internuclear distance. Slater-type bases are most commonly

used in quantum-chemical calculations for atoms and for linear mole-

cules (Pople and Beveridge, 1970); in these cases, the short-range

behaviour of the potential is of the form

+ +B I

V(R,r ) A aexp(-B R-r ). (3.2.2)

R r

a

Here A and B are positive coefficients that are independent of the

a a

magnitude of R, but they may be functions of the orientation of R with

+

respect to r.

a

We thus find that the single-center representation of eq. (3.2.1)

must be smoothly transformed into a multicenter one, as the relative

distance decreases. The transformation must avoid the unphysical

divergences of the multiple expansion and must accurately reproduce

the intermediate region of the surface which includes the potential

wells. One might attempt to carry out this transformation by

multiplying the right-hand side of eq. (3.2.1) by a switching

function that tends to one for large R, and to zero whenever R r

a

Unfortunately, the second requirement would force the switching

function to have an extremely complicated dependence on the orientation

of R. However, this difficulty may be readily avoided by extending

the multicenter representation to large R, as follows.

Assuming that the molecular properties of B can be expressed in

terms of the individual contributions of each atom, we expand the

long-range region of the potential about the atomic centers and write

+ +B +

V(R,r ) = v (R,r ), (3.2.3a)

a a

a

+ (SR) + + f (LR) r (.

(R,r) = a (R ) + (R,r (R,r ), (3.2.3b)

aa a a a a

(LR) + 3

vaLR) (R,r ) = C 0a/R + C (cosn )/R + C P (cosn )/R +

a a 10a a 21a a(Ca) a 32aP2 a a

+ [C40a + C42aP2(cosna)]/Ra, (3.2.3c)

-4 -> ->

where R = R-r is the position of the ion with respect to target

a a

atom a and na= arcos[R *r /(R r )] is the angle between and r .

a a a aa a a

The terms of v LR) represent the interaction of the charge of A with

a

the first three electrostatic moments and polarization of the

electronic distribution about center a, respectively. In

(SR)

eq. (3.2.3b), v corresponds to the short-range repulsive forces

a

and fa is a switching function that eliminates the singularity of

v at R = r The detailed form of f is not crucial, pro-

a a a

vided it satisfies the following requirements: 1) f and its

a

gradient with respect to R must be continuous, 2) f a 0

a.- +

R-r

and 3) f 1 One such function is, for example, a

R-t

f (R,r ) = O(R c )exp[-Y (R c ) ], (3.2.4)

a a a a a a ar

where 6 is the Heavyside unit step-function. Here, c is a radius

a

about atom a within which the long-range pair potential is zero, while

Y determines how fast v LR) is turned off as the A-a distance de-

a a

creases. We may complete the representation by taking the repulsive

potentials to be

(SR) + +

v (R,r ) = A exp(-B R ), (3.2.5)

a a a aa

with A and B positive and independent of R. If necessary for a

a a

specific system, the v(SR) may be rendered more flexible by letting

a

(SR)

A and B depend on n or by expanding v is a basis-set of

a a a a

functions of the orientation of R .

a

Expressing IR-r in a Taylor series in r about R, we find

that eqs. (3.2.3) and (3.2.1) coincide as R + m, whenever the co-

efficients C satisfy

nZa

A B = C10a (3.2.6a)

a

qA;B = l (C10ara C 2laa/ra), (3.2.6b)

QB = (ClOarl + 2C21ara C2a) (3.2.6c)

qB = + /r (3.2.6d)

AB 10a + 21a ra a

a

qQ = (C r + 2C r + C ), (3.2.6c)

2

qa 2= C4a. (3.2.6e)

a

If the Cn a are fit to a previously calculated PES, these equations

provide us with an independent check of the fit, as the Cna should

yield the correct values of the molecular properties of B. On the

other hand, if a quantum-chemical PES is not available, eqs. (3.2.6)

together with experimental values of the molecular properties may be

used to partly determine the atom-pair potentials.

In the multicenter representation defined by eqs. (3.2.3-5),

the set of parameters that must be adjusted to reproduce the tabulated

PES is {p.} = {A ,B ,c ,C ,C ,C ,C C for a=l to N .

i a a a a 10a 21a 32a 40a' 42a

Although at first sight this set seems rather large, the above analysis

shows that this is the smallest possible set that is required by the

physical nature of ion/linear molecule forces. Furthermore, the pi

generally depend on the internal configuration of the polyatomic, and

hence on r. However, for a specific molecule, some of the long-range

parameters CnZa may be negligibly small. In particular, the electro-

static interaction between the charge of A and a permanent multiple

moment of B may be represented, at large R, as the sum of the forces

between the charge of A and the lower-order multipoles of the target

atoms, as shown in eqs. (3.2.6a-c); this usually involves cancellations

between the terms contributed by each center. As the A-a distance

decreases, the atomic contributions become large, due to the 1/R

dependence, and may not cancel to yield a correct description of the

intermediate region; this indicates that such atomic multipoles

should be made vanishingly small. We will encounter examples of this

phenomenon in the following sections.

3. Multicenter Potential for the System Li /CO

We now develop the multicenter representation of the intermolecular

potential for the Li /CO system. This is one of the few many-electron

systems for which a detailed PES has been calculated quantum chemi-

cally, at the configuration-interaction level of accuracy

(Thomas et al., 1978). This surface has been tabulated for a wide

range ofvalues of I, at the equilibrium configuration of CO

(Ira-rb1=1.224 X); this will allow us to judge the applicability

of the multicenter expansion for small and large R. Unfortunately,

only a few points have been calculated for nonequilibrium con-

figurations of the molecule; therefore, throughout this section,

we fix the target atoms at their equilibrium positionsand take the

-B

potential parameters to be independent of r

Due to the nonlinear dependence of the potential on several

of the parameters, the usual least-squares optimization (Doren and

McCracken, 1972, Chap. 7) would lead to a system of sixteen coupled

transcendental equations in the p. whose solution represents the best

values of the parameters. However, such a system can only be solved

by iterative methods (Ostrowski, 1966, Sec. 25) whose convergence

is impracticably slow due to the large dimensionality of the problem.

In order to avoid this inconvenience, we optimize the linear (C a)

and nonlinear (B ca, ya) parameters in two separate steps, as

follows.

In the first step, the Cna are obtained by a linear least-

squares fit to the long-range region of the PES. In order to test

the correctness of the functional representation chosen for the

v (LR), we do not a priori constrain the C to reproduce the known

a n~a

molecular properties of CO. Hence, we allow the Ca to vary freely

in order to minimize the root-mean-squared relative error of the

representation; for large R, this error is given by

-2 +- +B B -* +B 2

e = {[VT(Rr ) VRr )]/[N VT(,r )]} (3.3.1)

->F

Here, the sum runs over the N points of the surface for which

4-

R (LR)

R >3.7 X, V is the tabulated value and VF = vR). The

a

optimum values of the Cna are those that satisfy the system of

linear equations

2

3e / C = 0, (3.3.2)

na

2 2 2

together with the conditions @ e /DC >0.

nla

Solving eqs (3.3.1) for all the Cnla indicated in eq. (3.2.3c)

gave C0a, C21, and C32 of the order of 6 e2 n, n=0,1,2,

respectively, with opposite signs for each atom. These values, although

of unphysically large magnitudes, lead the v(LR) to cancel each other

a

in order to reproduce the tabluated PES. Examining the resulting fit,

we concluded that, for the Li /CO system, the long-range potential

could be reproduced with fewer parameters than those indicated in

eq. (3.2.3c). Repeating the fitting procedure while omitting each

time one pair of the C10a C21a, and C32a, we found that the C21a

were redundant. The optimum values of the parameters are shown in

table 3.1. These reproduce the long range region of the PES with an

r. m. s. error e = 0.85% over the 25 points available (four equally-

spaced values of R from 3.7 1 to 5.3 1 and one at R = 7.9 X, each

for the five orientations n= 0, 45, 90, 135 and 180 deg). Throughout

the remainder of the optimization procedure, the Cna were kept fixed.

nta

In the second step, using the full multicenter potential of

eqs. (3.2.3-5), for a given set of values of the nonlinear parameters

Ba, ca, and y a, we least-squares fitted the Aa to the short-range

region of the tabluated surface for which VT > 0. This fit was

repeated for several sets of B c and y each time obtaining

a new A Finally, we chose the B c y and the corresponding A

a a a a a

that yielded the smallest r. m. s. error in the short and intermediate

ranges; these values are also shown in the table below.

Table 3.1.

Ion/atom potential parameters for Li /CO

Li /C

d ()

a

A (eV)

a

B(A -1)

c (A)

a

ya -1)

C 0a(eVA)

C21a(eV 2)

C32a(eVA 3)

C 40(eVA 4)

C 42(eV 4)

42a

0.6447z

1086.33

4.25

1.38

1.38

-0.189892

0

-5.70326

-11.4071

-1.07765

-0.4835i

1191.92

4.63

1.48

2.27

0.193190

0

-0.951450

-2.54812

-4.90754

Substituting the CnZa into eqs. (3.2.6) yields the molecular

properties of CO that correspond to the present representation;

these are in good agreement with the experimental ones (Thomas et al.,

1978), as is seen in table 3.2. This comparison further confirms the

validity of the multicenter expansion for large R.

Li+ /

Table 3.2 Molecular properties of CO2

Present Fit Exact

qB(e) 2.3 x 104 0

B (eX) -0.0150 -0.0236

QB(eA2) -0.4646 -0.4666

o 3

aB (A) 1.939 1.947

AB( 3) 0.415 0.325

Figures 3.1 and 3.2 show the full multicenter potential (eqs.

(3.2.3-5) together with the parameters of table 3.1) and the quantum-

chemical values. We find that the multicenter representation re-

produces the tabulated values with an r. m. s. error e = 4.4% over

the 55 points available. The largest errors occur near the zeroes

of the potential where its absolute value is less than 0.03eV. We

also find a small discrepancy (15%) near the bottom of the potential

wells at the two orientations n=0 and 180 deg. This discrepancy is

most probably due to the simple angular dependence isotropicc) that

(SR)

we have chosen for the v( The errors could be greatly reduced

a

(SR)

by expanding v in a basis of functions of n However, we

a a

feel that the increase in computational effort resulting from a

larger set of parameters is not justified for the description of

hyperthermal collisions.

Figure 3.1 Comparison of the multicenter representation (curves)

of the Li+/CO potential with the quantum chemical values

(points).

10-

8-

6-

W 4

2-

0-

-0.4-

I

I

I

1

I

I

I

I

SI

rI I

I

Li+/CO

-ml'--

-0

=45

=90

R(A)

N X

^

Figure 3.2 Comparison of the multicenter representation (curves)

of the Li+/CO potential with the quantum chemical values

(points).

Li+/CO

10-

j-

8-

6-

=18 0

? = 1350

R (A)

4 5

3

I

1

I

I

I

ci,

4-

2-

Excepting the two regions just mentioned, the multicenter re-

presentation gives a very accurate description of the Li /CO sur-

face over the four-decade range of magnitudes of the potential, with

a minimum of adjustable parameters. The single-center expansion de-

veloped by Thomas et al., which is of comparable accuracy, required

fifty parameters, in contrast to the sixteen used in the present one.

Furthermore, the multicenter representation allows us to understand

the physical origin of the parameters, even in the short-range regions.

On the other hand, the ion/atom potentials just obtained may be

utilized to gain insight into the physical structure of the CO mole-

cule. Recognizing the dangers of extrapolating excessively from in-

direct results, we propose the following interpretation to the

differences between the Li /C and Li /O potentials.

A comparison of the short-range regions of the v shown in

a

fig. 3.3, suggests that the electronic density about carbon extends

farther from its nucleus than does that of oxygen. This observation

is consistent with the signs of the C10a which indicate a slight ex-

cess of electrons on the C-center and a corresponding deficit on the

O-one. This causes the Li /0 interaction to be repulsive at large R,

ag

while the Li /C potential is attractive for large distances, as is

seen if figs. 3.4 and 3.5, respectively. Comparing the magnitudes

of the C32a, we conclude that the charge distribution about C is more

anisotropic than the one about 0, and that both distributions are

prolate spheroidal. Finally, the C40 indicate that the electrons

around C are more polarizable than those about 0; however, the

anisotropy of the polarizability of 0 is quite large, so that, at

Figure 3.3 Short-range region of the ion/atom potentials of Li /CO.

--- Li /C

8-

6

4-

2-

' I I

1.00

1.25

Ra (A)

Li /O--

1.50

1.50

C)

I 1 1 1

- 0 0d

(A)D A

0

II

P--

L0n -

0

Cr

0

II

F`

I I I

-- O -DA

o 5 o;

(A)D^

0

-J

0

0

0)

II

\

d

I

C\J

0

I i

an=90 deg. the Li /O potential is repulsive at all distances. We

expect that these conclusions would be confirmed by the configuration-

interaction charge density of the isolated CO molecule; unfortunately,

the wave-function of CO is not available to us at the present time.

4. Multicenter Potential for the System Li+/C02

Due to the large computational effort required by the number of

electrons in CO2, a quantum chemical PES for the Li+/C02 system has

not been calculated. Until such a surface becomes available, we

propose a multicenter potential which is based on the electron-gas

model of short-range forces (Gordon and Kim,1972) and on the long-

range interaction between the charge of Li and the known electro-

static multipoles and polarization of CO2.

Briefly, the uniform-electron-gas model of the interaction be-

tween two closed-shell atoms and/or molecules A and B is based on

the following assumptions. Firstly, the potential is written as the

sum of four terms that represent the kinetic energy of the electrons,

the electrostatic interactions between the charge distributions of A

and B, the electron exchange and correlation effects, respectively.

Each term is separately calculated as an integral over all space of

the corresponding energy density, which is a functional of the

electronic density of the system. The electrostatic energy-density

functional may be readily obtained by considering the Coulombic inter-

actions between the charge distribtuions. However, the kinetic,

exchange and correlation density functionals are taken to be those of a

uniform gas of electrons (Slater, 1968, Carr et al., 1961,

Carr and Maradudin, 1964). Furthermore, the electronic density of

the A-B system is approximated by the sum of the densities of the

isolated partners, regardless of their separation. Hence, if the

density of each species is known, the calculation of the PES is

reduced to the numerical evaluation of the integrals of the energy-

density functionals.

This model has been found to provide a reasonably accurate de-

scription of the short-range repulsive forces in atom-atom (Kim and

Gordon, 1974a, b) atom-molecule (Green et al., 1975, Parker et al.,

1976, Davies et al., 1979) and molecule-molecule (Parker et al., 1975)

systems, at a small fraction of the computational cost of even the

most approximate quantum-chemical methods. Unfortunately, the

model does not correctly describe the long-range forces, for the

following reasons. Due to the additive nature of the approximate

total density, the correlation effects depend on the overlap between

the individual densities. Hence, for large R, the correlation con-

tribution decreases exponentially and does not reproduce the

-2n

Van-der-Waals potential which is of order Rn, n 3. This de-

ficiency may be remedied by smoothly switching from the electron-

gas correlation potential to the dispersion one, as R increases

(Parker et al., 1976). In the Li /CO system, we expect that

the dispersion forces are much smaller than the induction ones, due

to the net charge of the ion. However, by assuming that the

electronic densities of each species remain unchanged for all dis-

tances, the model excludes the rearrangement of the charge distribution

of CO2 caused by the charge of Li and hence the induction forces.

Nevertheless, we may obtain a realistic model of the Li +/C02 PES

by combining the short-range results of the electron-gas method with

the long-range multicenter expansion studied in the previous section,

as shown below.

Electron-gas calculations were performed for the collinear

(n= 0 deg) and perpendicular (n = 90 deg) configurations of Li +/C02,

using the program developed by Green and Gordon, 1974. These calcu-

lations incorporated the correction due to Rae, 1973, that avoids

overcounting of the electrons in the density functionals. The

electronic density of CO2 was approximated by that of the self-

consistent-field (SCF) molecular wave-function calculated by McClean

and Yoshimine, 1967, p. 192. The equilibrium C-0 bond distance was

fixed at the value that corresponds to the SCF wave-function

(1.1348 X) instead of the experimental one (1.161 X, in Eastes et al.,

1977), because the experimental bond-length represents a slightly

stretched CO2 molecule in the SCF approximation. The Li ion was also

described by an SFC wave-function (Clementi, 1965, table 03-01).

The numerical integration of the energy density functionals was

carried out by means of three-dimensional Gaussian quadratures, the

number of quadrature points was gradually increased until the cal-

culated potential converged to 1%, at all the values of 1. Due to

the approximate nature of the model, we felt that the increase in

computational cost required for a greater accuracy was not justified.

Hence, the number of points ranged from (56, 80, 1), at n=0deg, to

(24, 40, 24),at n=90 deg, for the (X, y, @) spheroidal coordinates,

respectively.

Table 3.3 Electron-gas Li /C02 potential.

R V(R,0 deg) R V(R,90 deg)

(Bohr) (Hartree) (Bohr) (Hartree)

3.7444 0.81826 1.60 0.87678

3.8444 0.63961 1.80 0.62150

3.9444 0.52927 2.00 0.44061

4.0444 0.40132 2.20 0.31309

4.1444 0.32335 2.40 0.22474

4.2444 0.25550 2.60 0.16382

4.3444 0.20132 2.80 0.12175

4.4444 0.15878 3.00 0.09251

4.5444 0.12499 3.20 0.07347

4.6444 0.09931 3.40 0.05994

4.7444 0.07718 3.60 0.04937

4.8444 0.01634 3.80 0.04059

4.00 0.03335

The results of the above calculations are shown in table 3.3

(the odd-looking scale chosen for R at n= 0 is due to the face that

the coordinate most appropriate to short-range interactions is the

separation between the ion and the nearest atom of CO2). These re-

sults correspond to the so called Hartree-Fock model potential, which

is the sum of the kinetic, electrostatic and exchange contributions.

The electron-gas correlation is not presented here, due to the in-

accuracies of the model previously discussed. However, the correlation

effects were calculated to be of the order of 1% of the "Hartree-Fock"

potential, and hence comparable to the numerical errors of the quad-

ratures. In order to facilitate future scattering calculations, the

short-range potential was fitted to a multicenter expansion,as follows.

A plot of the logarithm of the potential versus the distance

between Li and the nearest molecular atom yielded an approximately

straight line, for the collinear configuration. This indicated that

the Li /O potentials, v1 and v3, could be represented by the single

exponential of eq. (3.2.5). However, an analogous plot for the per-

pendicular configuration showed a steep straight line for small R,

that smoothly changed to a shallower slope for larger distances. This

graph suggested that two exponential terms were required for the Li /C

potential. The parameters of the ion/atom potentials were adjusted

to reproduced the tabulated electron-gas surface, by a procedure

analogous to the second step described in Sec. 3.3. Briefly, for

fixed values of the exponential coefficients B1 = B3, B2 and B ,

the preexponential factors A =A3, A2 and A2, were simultaneously

least-squares fitted to all the points of the tabulated surface.

This fitting was repeated for several sets of values of

B1 = B3, B2 and B finally selecting the exponential coefficients

and corresponding preexponential ones that yielded the lowest r. m. s.

relative error. These are shown in table 3.4 (below), and they re-

produced the surface with an r. m. s. error e = 0.43% over the 25

tabulated points; the largest error (3.6%) occurred at

R = (4.8444 a ro= 0 deg). These short range ion/atom potentials may

now be combined with the known long-range forces of the Li+/CO2 system,

as follows.

Without further knowledge of the properties of the exact charge

distribution of CO2, there is some ambiguity in the determination of

(LR) (LR)

the va because only lim v may be determined from experimental

aR a

R- coa

measurements of CO2. In particular, the Li -charge/CO2-quadrupole

interaction, may be represented in several different ways that are in

accordance with eq. (3.2.6c). For example, Eastes et al., 1977,

assumed fractional charges qa located on the atomic centers of the

molecule. The values q = q3 = -0.331 e and q2 = +0.662 e reproduce

the experimental quadrupole moment QB = -0.90 eL 2, at large dis-

tances. However, for small R, these charges yield values for the

perpendicular Li /C02 interaction that are much larger than those

calculated above, because the Coulomb potentials q q /R do not

cancel in the appropriate way. Similarly, one may set the qa equal

to zero, and consider fractional atomic dipoles pa (note that =

a P2

due to the symmetry of oxygen); however, such a model is also in dis-

agreement with the above short-range results. Therefore, we presently

assume that q = 0 and P = 0, for all a, and take nonzero atomic

a a

quadrupoles Qa, such that Qa = QB We must emphasize that this choice

a

is not unique; a combination of small atomic charges, dipoles and

quadrupoles may not be excluded as a possibility. On the other hand,

the electronic clouds about each of the atomic centers are polarized

by the net charge of the ion; hence, we represent the isotropic and

and anisotropic polarizabilities of CO2 by the sum of the atomic

polarizabilities a and A respectively. Furthermore, since the num-

a a

ber of electrons in C and in 0 are comparable, we let Qa = Q/3,

a = a /3 and A = A /3, with aB = 2.602 3 and AB = 0.6931 A3

(Buckingham and Orr, 1967). Finally, the resulting v(LR) are switched

a

off at small A-a distances as in eqs. (3.2.3b,4) using the switching

parameters c and ya of C and of 0 determined in the previous section.

a a

The parameters for the long range ion/atom potentials of Li /CO2 are

also included in the table below.

Table 3.4 Ion/atom potential parameters for Li /CO2.

Li+/C Li+/O

A (eV) 650.697 1340.16

a

A2(eV) 17.5648

-1

B ( -) 4.346 4.856

a

B2( -1 1.455

2

c (A) 1.38 1.48

a

-1

c ( -1) 1.38 2.27

a

C0a(eVA) 0 0

C21a(eV2) 0 0

C32a(eVA3) -4.32 -4.32

o4

C 40a(eVA ) -6.246 -6.246

C 42a(eVA4) -3.327 -3.327

42a

The complete model potential (eqs. (3.2.3-5) together with

table 3.4) is presented in figs. 3.6,7. Examination of the short-

range region shows the large anisotropy of the repulsive forces,

which is due to the many-body nature of the target. In this region,

the accuracy of the model is determined only by that of the electron-

gas approximation because the v(LR) are switched off. By design,

a

the present model reproduces the exact long-range forces, as deter-

mined by the properties of CO2. A judgement of the validity of this

surface in the intermediate regions must be postponed until a quantum-

chemical PES becomes available. The latter region includes the

potential wells and is thus sensitive to the atomic quardupoles and

polarizabilities, which are not uniquely defined by QB, aB and AB.

However, we expect this model to provide at least a qualitative de-

scription of the eact PES for intermediate R; in particular, the dis-

appearance of the wellatg= 0 deg as n-*90 deg, seen in fig. 3.7,

clearly reflects the change in sign of the Li+ charge/CO2-polarization

interactions. Finally, in figs. 3.8-10 we present the ion/atom

potentials. Examination of their short-range behaviour (fig. 3.8)

indicates that the charge distribution about the C-center extends

farther away from its nucleus than do those of the 0-centers. Con-

sequently, the larger extent of the Li /C repulsion causes the wells

of v2 to be much shallower than thoseof v1 and v3, as is seen in

figs. 3.9,10.

Having determined multicenter potentials for Li /CO and Li+/CO2,

we may now proceed to study energy transferprocesses in these systems

by means of the many-body theory developed in Chap. II.

Figure 3.6 Short-range region of the Li/C model potential.

Figure 3.6 Short-range region of the Li /CO model potential.

2

Li /C02

---o=00

R (A)

=45

I0-

8-

6-

4-

2-

0-

1.5

77=90.

-l

*d

4-J

O

4-

0

0

1r

+

a

CI-

0

0

1-

4-)

ct

0

a)

b0

C\j

0

C.)

0

CD

It1

0

o

II

F`

(A-)

(As) A

0

II

F'

Figure 3.8

Short-range region of the ion/atom potentials of the

Li+/CO2 model surface.

10-

0 Li +/C02

8-

-Li /C

S6-

4-

Li /O 0

2-

0-A^-------- ^

1.00 1.25 1.50

Ra(A)

I-q U

o

0

Ca)

01

4-1 CM

CO 0

(U

4-JC

C)

m

88

LO

O O

0

00

*I-II1

0 Co

SII

0 0^

na~)00

r-4

0

C1

0

+

*-

0

-41

e

10

I0

t0 ca

H *H

,-4 c0

I

cO +

Srj

r4

4-4

J 0

*r-1 *H

'0 l-

a)(

F

LO

o<

0

nr

C\J

0

C)

0

.-

.

_I

0

0

II

C\J

I I l

-0 7

(A9) DA

0

II

0

,-

)

CHAPTER IV

ROTATIONAL AND VIBRATIONAL ENERGY TRANSFER IN HYPERTHERMAL

COLLISIONS OF Li+ WITH N2, CO AND CO2

The many-body theory of atom-molecule collisions that was presented

in Chapter II is applied here to the study of rotational and vibrational

energy transfer in the scattering of Li+ ions from N2, CO and CO2. In

Section 1 we evaluate the quantities that are required for the calcula-

tion of the atom-pair correlation function (APCF) of a diatomic target.

In Section 2 we present the theoretical results for the systems Li /N2

and Li /CO, and compare them with the experimental measurements of

Bottner et al., 1976. In the following section, we briefly outline the

steps of the evaluation of the APCF of a linear triatomic molecule.

Theoretical probabilities of vibrational excitation and average energy

transfers for the system Li /CO2 are presented in Section 4. Comparing

these results with those of experiments (Eastes et al., 1977) and analyzing

the theoretical model, we gain insight into the dynamical processes and

intermolecular forces that dominate atom-molecule scattering in the hyper-

thermal energy regime. Finally, in Section 5 we calculate the effects of

molecular anharmonicity on the vibrational energy transferredto the tar-

gets N2, CO and CO2.

1. The Atom-Pair Correlation Function of a Diatomic Molecule

The evaluation of the APCF of equation (2.6.8) requires knowledge

4->

of the equilibrium positions of the atoms (d ) and of the matrix elements

(C ai) of the transformation between Cartesian vibrational displacements

ai

(ua) and normal coordinates (Qi). For a diatomic molecule, equation

(2.2.1a) shows that there is only one vibrational degree of freedom, and

that d1 = -m2dl2d/mB and d2 = mldl2 d/mB; here m is the mass of the target

atom a, mB is the mass of the diatomic, dl2 is the equilibrium bond-length,

and d is a unit vector in the direction of the molecular axis, from atom 1

to atom 2. From the normal-mode analysis outlined in equations (2.3.8-15)

'-2+ 4 12

we obtain C11 = -[m2/(mlm) ] d and C21 [m/1(m2mB)] d, and for the har-

monic vibrational frequency, w1 = [kllmB/(m m2)] where kll is the quad-

11 12 11

ratic force-constant.

The rotational energy transfers and widths require the commutators

([i ,) exp (-iK *d )], for n = 1 and 2. These are readily evaluated for

a linear polyatomic in the coordinate representation, where the rotational

Hamiltonian is given by (Edmonds, 1974)

2 2

H 2[ ( -cos2 ) I (4.1.1)

R 21 [(cosB) ((cos)B) 2

4->

Here, a and B are the azimuthal and polar angles of d in the space fixed

(SF) reference frame, respectively, and I = m d2 is the moment of in-

B a a

a

ertia of the polyatomic. Taking the z-axis of the SF frame along the

direction of K, so that K-d = Kda cosB, and performing the necessary

a

differentiations, yields

[HR, exp(-iK d)] = (a /2IB) exp(-iKd cosB)x

x{K2d2 (1 cosi2)- 2iKd [cosB (1 cos2 B)3/9(cos 2). (4.1.2)

a a

Whenever Kda >> 1, only the highest power of Kda need be retained in the

above commutator; in these cases, one has that

([,)n exp (-i a