Semi-empirical potential surfaces for electron transfer reactions: The Li + FH and Li + F2 reactions

ChemicalPhysics 31 (1978) 217-237 0 North-Holland Publishing Company

SEMI-EMPIRICAL POTENTIAL SURFACES FOR ELECTRON TRANSFER REACTIONS: THE Li + FH AND Li -t F, REACTIONS Y. ZEIRI and M. SHAPIRO

Received 23 September

1977

A new semi-empirical method for calculating potential energy surfaces of triatomic molecules, requiring knowledge of .onIygrourrd state potentials of diatomic molecuies has been developed. The method is formulated for the hl f XY + hIX + Y class of reactions, M being an aIkaIi-atom, X a haIog,en atom and Y either a halosen or a hydrogen atom. The method has been used to obtain excited state potentials of LiF and HF and potential energy surfaces of LiFz and LiHF. The results agree weII with ab-Initio studies. The resulting potentiaI surfaces are analytic and are thus we11 suited for classical trajectory studies. The formula obtained can beued to construct surfaces for the heavier hfXY anzdogues, the results of which arc reported elsewhere.

1. Introduction

The alkali hydrogen-halides reactions, (M + XH), and the alkali-halogen molecules reactions, (M +X2), have played a central role in the development of modem reaction dynamics, partly because of their importance during the “alkali age” of molecular-beam experiments (see for example ref. [I])_ In spite of this, the dynamical interpretation of these experiments has been so far confined to rather simplistic models, mainly of the “harpooning” type. The “harpooning” type models(e.g. refs. [2,3]) assume that these reactions proceed via an electron-jump from the alkali atom to the halogen atom, at internuclear separations, determined by the crossing between covaient and ionic surfaces. After the electron-jump the alkali atom is being attracted to the molecule by coulombic forces, and simultaneously the molecule-ion dissociates. This type of model has been very successful in explaining many features of these reactions, such as the large cross section observed. However, in order to understand the detailed dynamics, for example, the internal (electronic, vibrational, rotational) energy distributions of the products*, the shape of the differential cross sections, and

the effect of reagent excitation$, one must have reasonably accurate potential energy surfaces of these reactions. The aim of the present work is to develop a reliable semi-empirical method for constructing such potential surfaces. The need for a semi-empirical method is quite obvious, as of yet only two “abGnitio” type surfaces of this class, the Li f HF [8,9] and the Li t F, [IO] cases, have been obtained, and no “ab-initio” study of the. chemically more interesting, heavier analogues has been made. In addition, even for the systems studied, the “ab-initio” information is of limited usefulness, since “ab-initio” methods being so expensive, only allow for the construction of the potential surfaces over limited domains of configuration space. Thus when performing classical trajectory calculations one is forced to use complicated interpolation (and extrapolation) schemes which may introduce large errors in the fmal results

PllThe most promising semi-empirical method for constructing potential energy surfaces of reactive systems, is the diatomic in molecules (DIM) method [ 121. This method has been successfully applied to many systems involving the formation or breaking of a covalent type * See, for example, ref. [S] (K + HCKu = 1)); ref. [C] (H f F,);

* See, for example ref. [4] @a + HF).

ref. [ 71 Cy + HX).

7-18

Y. Zeiri, M. Shapiro/Semi-empiricaI potential surfaces for electron transfer reactions

bond [ 131. However an attempt by the authors and others [ 141 to construct DIM type surfaces for the systems above has failed in that it produced surfaces which compared very poorly with the ‘ab-initio” ones. The main reason for the failure of the conventional: DIM method is the inaccurate representation of ioniccovalent coupling at small and intemlediate nuclear separations. This comes about because the conventional DIM method uses information about thefree MX diatomics, in which the covalent-ionic intersections occur at large separations. Since the diatomic curves are sensitive to the covalent-ionic coupling near the crossing point only they are relatively insensitive to the shape of the coupling at shorter internuclear distances. However, the shape of the covalent-ionic coupling at intermediate and short distances becomes increasingly Important in the presence of the third atom, because the crossing point, as discussed below, may then be shifted to smaller diatomic separations. The net result is that the conventional DIM triatomic surfaces become inaccurate near the region where covalent and ionic crossings occur if this happens at relatively short MX separations. This indeed is the case for all M + XH + MX f H reactions. In addition to this failing, the conventional DIM method has a more general draw-back in that it requires information, about excited states of the various diatomics, which is not always available. Moreove:, in the DIM method the VB structure associated with each (experimentally determined), potential curve are chosen purely by chemical intuition This is necessary because in this way we save the effort of calculating integrab. associated with each VB structure. However, the experimental curves, and especially those of the excited states, involve the interactions between several structures not considered, and do not exactly yield Integrals associated with the structures imposed by our chemical intuition. In order to overcome these limitations we have developed a semi-empirical method, of the valence-bond type, in which all the necessary integrals, including the ionic-covalent interactions, are derived solely from the ground state diatomic potentials. This is achieved by calculating integrals associated with two valence elec$ By the term “conventional”

we mean DIM based solely on empirical information regarding excited diatomic potentials. If these states are calculated then the above remarks do not necessarily apply.

trons, from which, by comparison with the experimental ground state potential, it is possible to extract an empirical core-core interaction term. This core--core term may then be used to obtain excited diatomic potentials provided they differ from the ground state only in the assignment of the valence electrons. In addition, this procedure enables one to obtain directly off-diagonal terms between various VB structues, such as the covalent-ionic term discussed above_ The diatomic integrals derived in this manner may, in principle, be used in a generalized DIM type method. We have, however, chosen to specialize our treatment to the M + XH and M + XI systems, in which a simple three-configuration model has been used. In section 2 the three-configuration model is presented and it is shown how to express all triatomic matrix elements as sums of diatomic ones. These diatomic terms are chosen so as to minimize the three-center integrals which are eventually neglected. In section 3 we discuss the method by which the empirical core-core diatomic terms are obtained, from which the diatomic integrals needed for the triatomic matrix elements of section 2 are calculated. We use the method to generate various excited states of the HF and LiF molecules. Finally in section 4 we present a detailed study of the LiFH and LiF, surfaces at many nuclear configurations. The surfaces obtained which are given by a simple closed-form formula are very suitable for classical trajectory studies. Surfaces for other MXH and MX, reactions are presented elsewhere [lS] .

2. Triatomic surfaces We wish to construct potential surfaces for the M+XY+MX+Y

(I)

reactions, where M is an alkali atom, X is a halogen atom and Y is either H or another halogen atom. The ground state potential energy of the three atoms is assumed to result from mixing between three VB configurations, two ionic, $5, = kl4+x-Yl=

1xX Yl,

$J,=l&xY-I=IXPYl, and one covalent,

(2a) (2b)

Y. Zeiri, M. S~iapiro~lse,,elnpin~aI $3 = 2-1’2

(IMZ

F-I -

IMX FI).

potentiat mrfaces for electron transfer reactions

(2c)

Each configuration, describes the inner electrons and three valence electrons. The notation used in eqs. (2) is explained by the following example, Pfx

YI C4(l%P x+ PM(l)

CY(l)X(2)B(2) Y(3) o(3)), (3)

A being the normalized antisymmetrizer for all electrons, M’ being the wavefunction of the inner electrons of the M atom, with similar definitions for x’ and I’+. The atomic orbitals of the valence electrons are chosen as follows, M(l)=ns(l),

H&2,3) =H[(XY)-]

219

(1,2,3) +H[AfX] (223)

+ H[MtY] (1) - H[X-]

(2,3) -H[Y]

(l),

and a similar decomposition with the negative charge on Y. The third decomposition involves only neutral terms, H(1,2,3)=H[MX]

(l,‘)+H[XY]

(2,3)

(I)-H[X]

+H[MY] (1,3)-H[M]

(2)-H[Y](3)

(7b) where the diatomic hamiltonians of eqs. (7) are defined as follows,

(44

where ns is a Slater-type s orbital, II being the principal quantum number appropriate to the M alkali atom,

riY1 3

X(2) =mp(2),

(4b)

tc

mp being a SIater-type p orbital whose principal quan-

tum number is VI. Similar definitions hold for the Y orbital if Y is a halogen atom. For the Y =Hcase we simply write, Y(3) = Is(3).

‘+’ ‘ii

i>i

(84

'

HIMXI=Hcc[MX]&v; II +-+‘12

H[MYl

= 0,

R MX

+& +$)

=f$@‘l

(8b) ’

-i

(+

v;

+-$

+-$

(5)

where H is the hamiltonian matrix +‘+-,

‘12

(6)

and S

Rx,

(4c)

The chaise of direction of the mp orbital is discussed below. The secular equation for the above three configurations is, det(H -ES)

(7a)

is the 3 X 3 overlap matrix. We now turn our attention to the evaluation of the hamiltonian matrix. We write the triatomic matrix elements in terms of diatomic and atomic matrix elements. This is achieved by writing the triatomic harniltonian as a sum of diatomic and atomic terms. Three decompositions are possible, two involving ionic terms,

1

R

(8c)

MY

JWYI =ffJXYl - i(+

v; +r; L)

(84

(se)

(80

= @, IH- E&) = C(MXIM+X-)

HI3 -ES,, f 2-l’”

[Ec(XY) -JqMX)

- E(M) - E(Y) -E(X)

H[(MX)+] =H,e[MX]

-*T&+~-&

+&

x [%x

MX

W where H, [A] and If,, [AB] are the core and core-core hamiltonians respectively. They include the hamiltonians of the inner electrons and interaction terms with the electron. In eqs. (8) riA is the distance between electron iand nucleus A; rii is the distance between electron i and electronj, andRAB is the distance between nucleus A and nucleus Et. In the appendix it is shown that the H -Es matrix elements assume the following form,

-

2s,.sxY

+qMY)

- E] (94

+%xs;Yl’

and

Hq3 -ESt3 =(@2~-El@3)= C(MYlM+Y-) E;(MY)+E;(MX)

+2-“2[Ec(XY)-

VdenCe

H,, -ES,, =@,lH-E14J1) = [E(X-Y) + E(hf+X-) - E(M+) -E(Y)

f E(ti+Y) - E(X-)

- E] (1 - S&),

(9a)

Hz2 - ES22 = (9, IH - E13,>

-E(X)

-E]

(1 -S&),

(9b)

+ E;(MX)

-I-E;(MY)

-E(M)

-E(X)

(%I

+S&y]-

Thedefinition of the diatomic and atomic integrals appearing in eqs. (9) are as follows, S xy = (.m%

(104

S*ix = @an

(lob)

s hly = VMIY),

WC)

= (lX%‘l

/H[(XY)-

] IIXXYI),

(1la)

I IXrnl>,

Olb)

E(XY_) = tLnYl

W[(XY)_]

E(M’Y-)

lH[MY] l lyYl,,

=
E(M*X-) =
Hj3 - ESJ3 = G$ IH - El@,> = [l?(XY)

' [S,,, -=&Xy

E(X-Y)

= [E(XY-) t E(M+Y-) + E(hl*X) - E(Y-) -E(M+)

-E(M)- E(X)-E(Y)-E]

(I Ic) (IId)

EC(XY)=+(IXYI - IxYlm[XY] l lXvl - l;uYl>, (I Ie)

-E(Y)-El [l-ShrYSblXSXY-%S;fy+Six)l + E;(hlX)

[.Siy - SixI

+E;(MY)

E;(MX) = cos2(oY)Ec(MX) + sin2(ol,)E;(MX), (llf)

[Sgy - S&I, (9c)

where, E’(MX) is analogous to EC(XY), ay is the (MXY) bond angle and

H,2 -ES12 =<@,IH-El$)=C(X-YIXY-) E;(MX) = + tW%J - @X,1

+ [E(M+X-) + E(M+Y) - E(X-) - E(MY) - E] X&y[I -S&J,

@d)

WWXII WZ??J - @X11), (I Ig)

X, being a Px orbital perpendicular to the MX axis. The definition of E#dY) is analogous when Y is a hydrogen atom.

Y. Z&i, M. ShapirofSemi-empiricalpotentid surfacesfor electron transferreactions

E(M+Y) = (YlkZ[(MY)+] IY), E(M+X) = (XW[(MX)+] E(X_)=

(I lh)

IX),

(l_FXl IH[X_] I Ixxl),

E(M)+E(X)+E(Y)=O

(13)

(Iii)

we obtain the much simplitied matrix elements,

(1 U)

HI1 -ES,,

= E(X-Y)+ E(M*X-)

f EA(X) - IP(M)

and E(Y_) = (1PYl l/T[Y_] I IFYI).

221

flW

+ E(M+Y)

- E,

(144

H22 -ES22=E(XY-)+ E(M+Y-)+E(M'X) + EA(Y) - IP(M) - E,

E(X), E(Y), E(M) are the atomic energies. The diatomic coupling terms appearing in eqs. (9) are defined as,

(lab)

H33 - ESs3=E=(XY)+E;(MX)+ E;;(MY)-

E, (I4c)

C(X-YlXY-)

=
C(MXIM+X-)

= 2-1’2(lkfFl

IIXPYl>,(12a)

- hl?Xl IH[MX] : 1Xx1,,

HI2 -

ES,,=C(X-YIXY-),

IfI3-ES,, = C(MXIM+X-),

(I4e)

ES,3= C(MYlM+Y-),

(I4f)

(12b) C(MYlM+Y-)

= 2- 1/2 &%fYl-

l%Yl W[MY]l

I@l>. (I2c)

Having reduced the triatomic integrals to sums of diatomic ones, we further simplify eqs. (9) by neglecting aI1 terms proportional to Sib, .S~u where SAB is a diatomic overlap integral In addition we note that the three terms appearing in the off-diagonal matrix elements, SXY [E(M+X-)

f E(M+Y) - E(X-)

-E(Y)

- E],

H23 -

where IP(M) is the ionization

potential

of M and EA(X)

is the electron affinity of X. Solving for eq. (5) we express the first three adiabatic potential surfaces as the solutions of a cubic equation,

El =S1 cos+/3 - a213

(154

E2 = -2 S, sin [($r + @)/3] E3 = -2 S, sin[(&-

2-112 ShiX [EC(XY) - E;(MX)

W-U

$)/3]

II, /3,

- a?/3,

(15b)

WC)

f E;(MY) where

- E(M) - E(X) - E(Y) -E] s, = (2r2 f $)‘I6

,

W4

and

@= tan-I[(-q3/r2 2- 1’2 SMY [E’XY)

-E;(MY)

- 1)1”],

12 4=+Ql -gQ2> - E(M) -E(Y)

- E(X) - E] , t-=+(QlQ2

are roughly proportional

3 -3Q0)-&a?,

WC) Wd)

to S&I and are also neglected

for simplicity_ If we choose to measure E relative to the energy of the separated atoms, i.e. our zero point of energy is

fixed by the equation

Wb)

+E;(MX)

and U7a)

222

Y. Zein; M. S.kapiro/Semi-empirical

“1 =%lH22 -(q2

+H22H33

3. Diatomic potentials

+H33H~l

+ HZ3 + $3),

a,, =HllH&

potential surfaces for electron transfer reactions

Wb)

+H.,,HT3 +Hx3H& __ U7c)

- m12H23H13 -H11H22H33-

The eigenvalues of eq. (15) are not necessarily ordered.

Thus the ground state is given as Ea = min(Er , E2, E3).

0 7d)

Eqs.(14)-(17) togetherwiththedeftitionsofthe diatomic integrals [eqs. (ll)-(12)] constitute the essence of our model for the MXY potential surfaces. Physically eqs. (14) describe resonance between two ionic and one covalent configurations. The approximations introduced while deriving the model were meant to reduce the hamiltonian matrix elements to a sum of pairwise interactions. This is partly justified as the three body terms are proportional to S2 and higher orders of S. Pictorially eqs. (14) can be represented by the following diagram n?

* *

/ x-* * * * y

n+ w

** * \ x****y-

In the previous section we have shown how to reduce the triatomic matrix elements to a form containing diatomic terms only. Diatomic terms of two and three-electron hamiltonians can in principle be computed quite accurateiy. However, in the present case, as in most chemical systems, one needs to estimate the effect of the inner (core), electrons on the diatomic matrix elements of interest. These type of effects are grouped here under the name “core-core interactions”, although they contain, in addition to the interactions between the core electrons, the effect of the core electrons on the outer (valence) electrons. The main assumption we make is that the “corecore interactions” are the szzzzze for all configurations which only differ in the respective assignments of the outer electrons. We thus propose to evaluate the ‘&corecore” term from the (experimental) ground-state potential and the (calculated) outer-electrons terms. The “core-core” term can then be used for all diatomic configurations in which the outer electrons occupy different orbitals. In addition, this procedure will automatically correct for inaccuracies in approximations used for the outer-electrons terms which are common to all the diatomic integrals. We proceed in the following way; we first factorize all the diatomic hamihonians of eq. (S),

(18) n .

;

. . X.........

(194 where, (A, B) = (X, Y) or (M, X) or (M, Y),

.

/

. _

{z(i)= -$v;

- I/Q* - l/QB,

Wb)

‘Y

where the following notation is used: is an ionic interaction term, *** is an ion-neutral term, is a covalent interaction term, .._ ++ is a covalent-ionic coupling term, and 0 is an (X-Y-XT) coupling. This type of diagram allows for an easy interpretation of the resulting surfaces as discussed below.

and the range of variance of i is either 1
l/r&(l))

= --p(A),

-IP(A),

(20e) (20b)

p(A) being the screening constant of the outer electron of A [17],

Y. Zeiri, M. ShapirofSemi-empirical potential surfaces for electron transfet reactions

(204

Or(l)1- l/rlBl&l)~ = -l/R,, Ol(~)l~(l)l~(1)) = q&)

EC(MX) = Hec [MX] - IP(M) - IP(X) - ShrX [IP(M) + IP(X) + l/R,,

X {-II’(A) - IP(B) - l/R,,

-*b(A)

223

+ !G(M) +$(X)1, (23d)

+p(B)]}. (20d)

E(M+X) = Rc, [MX] - IP(x) - [or(M+)+ o(X)] I@$,,), (?3c)

Similarly for the two-electron integrals we write tbat, ol(l)A(2)ll/r,21~(1)A(2)~

= IP(A) - EA(A),

(21a)

++[-l/RXY

oi~l~~(2)ll/r~~l~(1~~~2)~ =&b/2) oi(l)B(2)1

(A(1)B(2)ll/rl,lB(l)A(2))

[XY] - [P(X) - IP(Y)

+$&~I 1,

+ EA(X) + EA(Y) +&~o()

(231)

[IP(A) - EA(A) + I/q., IL40)W))

C(X_Y IXY--) = -Sxy{-Hec

1/R,,&

@lb)

= l/RAB,

(214

= (5B/2)2

C(MXIM+X-) = 2*‘2 S&H,

[MXJ - IP(X)

- + [IP(M) f EA(X) ++ p(M) +&J(X)] ],

(23g)

Eqs. (20d), (21b) and (21d) implicitly incorporate the M&ken approximation [ 181,

where the o’s are the polarizabilities of the various atoms or ions. Similar expressions hold for E(Y-X), E(M’Y-), f?(MX), EC(MY) and C(MY]M+Y-). The E(MfY) term is treated according to eq. (23e) if Y is a halogen. For the Y = H case an explicit expression for H,,[MH] can be used. The resulting expression for the M+H potential is,

A(1)8(2) =(S,J2)

W+Hl = ev(-~&

X [IP(A) - EA(A) + IF’(B) - EA(B) + 2/R&. t2ld)

[A’(l) 4(2)].

(22)

(1 f&J&

-

IP(H). (23h)

Using eqs. (20) and (21) we can express all the diatomic integrals appearing in eqs. (11) and (12), in terms of SAB, IP(A), IP(B), EA(A), EA(B), p(A) and p(B). The results after neglect of terms proportional to SiB and addition of the appropriate polarization terms are, E(x-Y)

=Hcc [XY] - IP(X) - II’(Y) - EA(X), (23a)

E(M*X-) = Hcc [MX] - l/R,X - [e&‘)

+ 4X-N

EC(XY) =HJXY] - S,,[IP(X)

- IP(X) - EA(X) (23b)

&!R”,,), - IP(X) - IP(Y)

+ IP(y) + ‘IR,

++PS

+h’(Y)k

(23~)

The diatomic overlaps, SA,, can be readily computed using analytic formulae given by MuUiken et al. [ 191. AlI the diatomic integrals of eqs. (23) are seen to depend on three core-core energies, Hc,_[XY] , If,, [MX] and H,[MY]. All the other terms appearing in eqs. (23) are readily evaluated. These same terms appear also in all twelve diatomic integrals of eqs. (10). Moreover, the same Rcc term appears in our expressions for all of the excited states in which only the assignment of the yalence electron has been changed. The H,, terms can be obtained, as discussed below from the ground state potentials_ Since our expressions predict simple relations between various diatomic potentiaIs we can use potentials available from ab-iitio calculations or experiments to check the accuracy of eqs. (23). We thus, as a first step towards obtaining the triatomic surfaces, compute various excited states of the diatomic pairs

Y. Zeiri, M. ShapirolSemi-empinkalpotential surfacesfor electron transferreactions

224

which appear in the triatomic systems of interest. Once this is done we shall have established a very useful method for obtaining excited states potentials, which are essential, for example, for DIM type of surfaces [12,13]. The various H,, terms are extracted from the ground state diatomic potentials. The procedure used varies slightly according to the available experimental or abinitio ground state energies. In this paper we discuss the diatomic potentials important for the Li + FH and Li + F2 reactions.

If we now demand that E, - the experimental ground state energy of the X’ C+ state, be one of the roots of the secular determinant,

f(ELl=O,

(27)

we obtain from eq. (27) that H,,, expressed in terms of El, is H,, = (2 - 2S;,)-’

{2El(1 -‘S&)

-h,,

- Zz22

3.1. Grortud aud excited state of HF - 4S&./r 12)2 - 4$( 1 - 2S&)] “2). In order to obtain the H,,[HF] term we assume that the ground state of HF may be written as a linear combination of ionic and covalent configurations*, tix”+(HF)=

C,(IHFI - j@1)/21” -I-C,jFFI.

(24)

The notation for the orbitals follows eq. (3), F being the 2p, orbital of F, z being directed along the H-F axis. The corresponding secular determinant can be written, using eqs. (23b), (23~) and (23g), as

The H, thus obtained is used to compute all the desired HF integrals. We first evaluate the other root of eq. (27) which we identify with the energy of the A’Zf state. The results for the X’ Zf and A’Z” states are shown in figs. (la) and (b). The X’ Zf curve is taken from the work of Murreli and Sorbie [20] who use the form E”‘(HF)

f(E) =

+ C&R

hll = -[P(F) - IP(H) - SH,{IP(F) + U’(H) f l/R,, h,, = -1/R,,

++[p(H) + p(F)1 1,

(26a)

- IP(F) - EA(F) - c$F’)/2&,

(28)

=A,,

exp[-PHF(R

-R,)]

- Re)2 +DHFtR - RePI.

[l +BHF(R -R,) (2%

(The constants used are given in tables 1 and 2.) This form accounts better than a Morse function for the spectroscopically determined parameters. As shown in fig. lb our result for the A’Z? state compare extremely well with the multi-structure VB calculations of Yardley and Baliit-Kurti 121,221. The agreement is better than expected as our&, was extracted from the experimental X’ 2+ curve which is not identical with the VB one as shown in fig. la. We can further utilize our formulation to calculate other excited states of HF. Assuming that the 3Z’ state can be described as a single configuration,

(26’~) G3’+(HF) = WFI, %2 =

-IP(F) - +(IP(H) -t EA(F) +&I(H)

(26~) * In principle the ground state of HF has contribution

from the 3 Z state. This is necessary to ensure that the ground state dissociates to the H(*S) F(~P,,,) limit. The triplet state is implicitly included in ffcc, as shown below, since our two electron terms do not in&de any spin-orbit coupling.

l

(3Oa)

f p(F)] }, we obtain from eqs. (20) and (21) that, E

3”+wF) = H,, [HF] - IP(H) - IP(F).

(3Ob)

Thus, up to a constant, the energy of the 3Z+ state is simply H, [HF] _As shown in fig. lc, the result we ob-

Y. Zeiri, M. Shapb-o/Semi-empiricat potential surfaces for electron transfer reactions Table 1 Parameters

for diatomic

potentials

225

a)

HF - Murrell and Sorbie parameters b, = -0.225 11 BHF = 2.15536 CHF = 0.93848

;”

HF = 0.45257

PHF= 2.15536

R,= 1.7325

LiF - Rittner parameters c) ALiF = 28.17 pLiF = 1.947 LiH - Morse parameters d, De = 0.09246

p = 0.597734

R, = 3.0147

F2 - Morse parameters e, De = 0.0606 p = 1.60655

R, =

Fg - Morse parameters f) e = 1.10433 De = 0.0864

R, = 3.3

2.683

a) All values are in au. b) Ref .&_& 701 12c9),cef. [74]. d)Ref. [33]. e, Ref. [34]. .

Table 2

Atomic parameters

R(HF)m

A

WA)

EA(A)

cr(A)

P(A)

F PH HLi Li+

0.6402

0.1257

2.6

0.49982

0.02773

4.04917 7.0862 4.5 93.22

0.1981

0.02206

au

Fig. 1. HF potential energy curves: -this work; o multistructure VB calculation - refs. 121,221. (a) Ground state; -fit to experimental data of ref. [20]; @)ionic-covalent coupling; (c)-(e) potentials for electronically excited states.

tain agrees very well with the multi-structure VB calculation. A slightly modified H,,[HF] is used to compute the energies of the ‘lI and 311 states. The moditications to H,, [HF] are necessary because the II states involve a different assignment of the core electrons_ Assuming that the ‘ll and 311 states can be described a%

(31)

I$~"(HF)= jfE"I,

(32)

by the Hf nucleus. The second difference between the Z znd lI states is due to the vanishing of SHF Thus I we write, qc [HFI = HE WI

Combining

where F, is a 2p orbital of Fperpendicular to the H-F axis. We note two sources of differences between the Z and iI states. The Hg [HF] is slightly less repulsive than the Hg[HF], because in the II state an extra electron occupies the 2pZ orbital of F which is polarized

0.65 0.1930

- (u(F,)/$,,

Pa) (33b)

SHF, = ‘-

$"(HF)=(IHFJ - ~~x])/2"z

1.0

eqs. (33) with eqs. (20) and (2 1) we obtain

that, E’“(HF)

= _&HF)

= H;[HF]

- [P(H) - [P(F). (34)

We have computed the energies of both Ifi and 311 states. The results as shown in figs. Id and le agree very well with the multistructures VB results [21,22].

y. zeiri, bi. Shapiro/Semi-empirical

226

We have thus established

that the procedure

powttial

we

have proposed can be used for obtaining accurate excited states HF potentials. We proceed to use our expressions

to obtain ihe HF integrals of eqs. (14) and

(23) needed for constructing the LiHF potential surface. As an example we show (fig. If) the normalized coupling between ionic and c&dent configuration, (1 +s:,,)-“~C(HFIHfF-), where C(HFIH+F-) is analogous to C(MXIMcX-) of eq. (23g). We have aIso computed the HF- and H-F potentials using eq. (23a) and our H,,[HF]. As evident from eq. (23a) and the

surfaces for electron transfer reactions

Rai [25] - To test eq. (37), we proceed, in a manner identical to the HF case, to calcuIate the potential cruves for the X12+ A ‘EC, 3Z+,111 and 311 states of LiF. We use the fouo’wing VB configurations, $ ’ “‘(LiF) $A++

= ClllFFl

f CIZ(ILiFI - Izi PI), (38a)

(LiF) = CzL IFFI + C,,(lLiFI

$3z+(LiF) = ILiFt, = (ILiF x I - I&F x 1)/2’D I

- Izi fi),

(38b) (38~)

shape of the H,,[HF] curve, (fig. Ic), the HF- and H-F curves are very repulsive. This fact has a significant bearing on the triatomic potential surfaces. As discussed below, the bent LitH-F arrangement al-

$I”(LiF)

for the stabilization resonance,

where the mixing Coefficients of eqs. (38a) and (38b) are obtained by solving the 2 X 2 secular equations. The results for the first four states and the nonnalized ionic covalent coupling, (1 + S&)-‘1’ X C(LiFILi+F-), are shown in fig. 2. These results compare well with the multi-structure VB ones [21,22] as shown in fig. 2. The agreement between the two cal-

lows

Li+

*Li+

* *

/

of the (HF) molecule via the

(35)

W*

* H-t * * * F

* \ H * * * i

F-

The collinear Li’F-H configuration prohibits this resonance because Li* lowers the energy of the F-H configuration. Hence we expect the Li + HF minimum energy path to occur in the bent configuration.

$3”(LiF)=

ILiFxl,

W-U (3ge)

culations is not however as close as that of the HF case. One possible reason for this has to do with the use of the Rittner model as a source for the HJLiF] term.

In the ground state Rittner potential the -4 exp(-R/p) term compensates in part for the errors introduced by the --1/R term at short distances. It is precisely for this

3.2. Ground and twited states of LiF The procedure we use to generate the various LiF integrals and LiF excited states is very similar to the one used for HF. except that the Hc,[LiF] was taken from a simplified Rittner potential to which the energy of LifF- is, ERit(LiF)

[23,24],

according

= A LF exp(-N~,J

- [dLi’)

+ a(F-)]

/2R4 - I/R + Ip(Li) - EA(F), (36)

where ALr and pLiF are Rittner.parameters given in table 1_ This form suggests identifying the H,, [ LiF] a% H,,PFl

=ALF

exp(--R/pLiF).

A similar procedure

(37)

has been employed by Rai and

Fig. 2. LiF potential energy curve: -this work; o refs. 121, 221. (a)-(c) Potentials for electrorrically excited states.

(d) Ionic-covalent coupling.

Y. Zeiri. M Shapiro/Semi-empirical potential surfacesfor electron transfer reactions by invoking for by adopting a procedure similar to HF case. This has not been done because the Rittner potential dissociates to the ionic Emit and cannot be considered as represent-

227

reason that some error is introduced

Table 3

eq. (37). In principle this can be corrected

Dissociation energies of XT and X2 and equilibrium distances ofx; X

ing the full ground state potential at large separations. A more accurate Hc, [LiF] can also be obtained from .the electrongas model of Gordon and Kim [26]. The procedure used for other diatomics follows the two cases outlined above. We briefly discuss a few of the other diatomics needed for the Li f HF or Li + F, surfaces.

D&G) ON

F

De(&) (au)

R&G 1 (4 3.125 23 3.30 b)

0.068 a)

0.0876 b) 0.0474 c) Cl

0.0606 d) 4.875 3 5.1496 e,

0.076 1.5al 0.04603 e, 0.09244 d)

Br

3.3. F; and F2

5.50 a)

0.0658 3) 0.04603 e,

5.6012e)

0.07246 d,

The ground state of the F, moIecuIe is assumed to result from a resonance between FF- and F-F. We thus write that,

The ‘2,

energy can be computed using eqs. (23a) and

(230. The HcJFz] term can be extracted from the experimental F2 potential. As an example we present in table 3 the dissociation energies and equilibrium separation of a series of XT

molecules. These results are compared to the pseudopotential calculations of Tasker et al. [27], the SCF resuIts of Gilbert and Wahl [28J and the multi-structure VB calculation of Balint-Kurti and Karplus [29]. The dissociation energies we obtain for the XT molecules, are generally larger than the “ab-initio” c$es (except for the Fz case, where our De is smaller than the “abinitio” D, but closer to the experimental value [30]). The equilibrium distances are in a much better agreement with other calculations [28, -291. In particular our results for the FT-case, and those of Balint-Kurti and Karplus [29], violate the bond-order bond-length rule, since the equilibrium distance of F, is longer than that of F2 [R,(Fz) = 2.683 au], while the dissociation energy of F7 is larger than that of F-,. The errors in the dyssociation energies of thi XT series probably come from the inaccuracy of the H,, [X,] obtained from the neutral diatomic when used as the “core-core” interactions of the rzegative diatomics. Thus in order to compute the E(F-F) and C(F-FIFF-) integrals [eqs. (23a) and (2301, which enter into the triatomic matrix elements, eqs. (14a)

I

6.5 a) 6.1511 e,

0.0602 a) 0.04103 e) 0.05668 d)

a) b) cl d)

This work. Multi-structure VB calculation - ref. [29]. Esperimental value - ref. [30]. Experimental vahes from ref. [34] _ e) Pseudo-potential calculation - ref. 1271.

and (14d) we chose to extract theH,_,[F2] term from the more exact F2 potenti& of BaIint-Kurti and Karplus [29] _The procedure used to do that follows closely that of the HF case (eq. (28)), where as input we use the computed curve [29]. The important

consequence

of this calculation,

as

shown in fig. 3, is the repulsive nature of the E(F-F) curve. The stability of the F, results from the resonance between FF- and F-F (eq.139)). Similarly to the HFcase, the possibility for efficient resonance greatly depends on the position of the Li’ ion, and consequently

IO

5.0

3.0

70

90

RCF- F) in a u.

Pig. 3. Single VB configuration

potential

curve for FF-.

228

Y- Zeiti, M_ S~tapirofSemi4r~~piricai potentid

the J-S, “molecule” is expected to be most stable for the iSOSC&S triangular configuration, as indeed found by Balint-Kurti [lo]. Contrary to the (HF)- case, in which the HF- configuration is dominant, the resonance here is more efficient and leads to a formation of a well in the LiF2 surface. 3.4. LiHatzd LiiH Integrals associated with LM, Li*H appear in the expressions for triatomic matrix elements eqs. (14a)(14~) and we evaluate them using eqs. (23b), (23d) and (23g) where M is now Li and Y is H. The magnitude of the C(LiH/Li+H-) coupling at the LiH/Li+Hcrossing point is in reasonable agreement with the semiempirical values of Olson et al. [3 I] _However, our procedure cannot be used to generate excited states of LiH due to the diffuse nature of the H- ion, which makes point-charge like appro&ations, [eqs. (20)(22)], unrealistic. In the Li’H case the explicit form @ven in eq. (23h) is used.

4. Potential surfaces of LiFH and LiF, We now use the diatomic integrals (section 3), and the formulae for the triatomic surfaces (section 2), to calculate the potential surfaces for the M f XH and hl +X2 type reactions. In this section we report our results for the Li f FH and Li + F, systems, for which ab-initio results [9,10] at various geometries are available. The results for other, M f Xl and M + XH (M = Li, Na, K, F = F, Cl, Br, I) are presented elsewhere [ 15]_ 4.1. LiFH The basicshape of the Li + FH + LiF + H, and all other M t XH reactions, is determined by the position of the covalent-ionic crossing line. In fig_4 we show the covalent surface [Hz3, eq. (14c)], one of the ionic surfaces [HII, eq. (14a)]. and a surface, defined as min(H1 1, HJ3) in which the crossing line is clearly seen as a cusp. It is evident from fig. 4c that an electron can ‘f.ump” from the Li to the F atom only when the Li is reasonably close to the HF molecule. Harpooning, is therefore, not an appropriate term to use here, since the Li is already very close when ‘Sarpooning” may be said to occur. The basic reason for the

surfaces for electron transfer reactions

“non-harpooning” behaviour is the strong repulsion felt by HF- at close range (section 3). The Li atom has got to approach the F atom to almost the Li+Fequilibrium distance, and simultaneously the HF bond has got to stretch considerably to allow for the electron transfer to occur. This ‘line of crossing” type analysis shows that the Ii + FH -+ LiF + H reaction is expected to be governed by a barrier between the entrance and exit channels. In figs. 5 and 6 the full LiFH ground state surface, obtained using eqs. (14)-( 17), is shown at five L&F-H bond angles. Two features are clearly evident: (1) The barrier is indeed located between the entrance and exit channels; (2) The barrier diminishes and moves towards the entrance channel as the system deviates from collinearity- These two-findings are in good agreement with the OM results of Balint-Kurti and Yardley [9]. The OM surface, shown for the co!linear configuration in fig. 7, is similar in shape to ours (fig. 5a). The agreement in barrier height is excellent (21.3 * 0.5 k&/mole for the OM surface versus 20 + 1 kcal/mole for our surface). The difference in the absolute magnitude between the two surfaces comes from the fact that we use the experherztal diatomic potentials, while the OM calculations give only =90% of the diatomic dissociation energies. On the other hand our use of the Rittner model for LiF leads, as discussed in section 3, to a slightly too repulsive EC(LiF). As a result, the saddle point of our surface shifts to a slightly greater LiF separation. The reduction of the barrier height as the system moves away from collinearity is a result of the emergence of the third ionic configuration, that of Li+H-F [f& of eq. (ldb)] . As depicted in (18) the extra possib%y of resonance serves to reduce the repulsion between the HF- pair. This allows for the electron jump to occur at larger LiF separations and causes Li+Fionic attraction to be felt more in the entrance channel. Both effects serve to reduce the barrier height which reaches a minimum for LLiFH 1: 70”. The shift of the barrier to the entrance channel is a result of the./?(LiH) repulsion, which is felt more strongly for the bent configuration- The reduction of barrier height with bending is in complete agreement with the OM calculation of Balint-Kurti and Yardley [9] _ The anisotropy of the LiFH surface is most readily seen from fig. 8 where polar plots of the surface at various HF separations are shown. The change in the

Y. Zeiri. M. SJ~apiro/Sef?~i-empi

5

PO’9

potential surfaces for electron transfer reactions

‘10’1

LOG

'ki-F

f.00

Rt_i-F

Fig. 4. Ionic and covalent potential surfaces for collinear LiFH. Energy values are in au. (A) Ionic contigurntions (resonance between configurations 1 and 2 - eqs. (Za), (%I. (14a) and (14b)j. (B) Covalent configuration - eqs. (3) and (14~); (C) Potential surface composed of the min (A, B). The cusp corresponds to the ionic covalent crossing line (A = 8).

character of the surface due to the possibility of reaction is clearly demonstrated. In fig. 8a where the HF molecule is constrained to its equilibrium separation, the Li feels a repulsive, almost isotropic, potential. This

potential is due to the dominance of the covalent configuration (H&, at small HF separation, in which no electron jump can occur due to the large HF- repulsion. As the H and F separate the emergence of the

y- Zeiri, M. SltlipiroiSemi-enri~cal potfit~id siwjiicesfor electron trnnsfer reactions

LI

. F H.THElR-135

RLI-F

L[

lllrlltlI I I i

. F “.TkETR.OCo

i

RL,-F

R,, -F

Fig_ 5. Ground electronic state for LiFH ns a function OfRLiF and RFH_ Energy values are in au. (A) LLiFH = 180” ;(B) LLiPH = 135” ; (C) LLiFH = 90” ;(D) LLiFH = 50”.

ionic surfaces shows up at noncollinear configurations. At 2.5 au separation (fig. Sb), a small well is formed around LLiFH = 70’. At this configuration the resonance depicted in eq. (18) stabilizes the molecule. At

3.5 au separation (fig. 8c) the valley formed by the contribution of ionic configurations extends to the LLiFH = 180” domain, corresponding to rhe formation of a Li’F- molecule. The small well at an almost

Y. Zeiri, M. Srlapiro/Semi-empi

potmtrid

surfacesfor electrm hmsfer reactioNs

231

Lt . Ii F.THETR.MO

2.0

3.0

4.0

5.0

6.0

7.0

RLi-F RLi-H

Fig. 6. Ground electronic state for LiFH at the LiHF collinear configuration IS a function OfRLiH and RHF. isosceles triangle remains. Finally at 4.5 au HF separation, the formation of a stable LiH and LiF can be seen. In addition, the configuration of Li between the H and the F with LLiFH = 40’ is seen to be quite stable. 4.2. Conmwnts OIIthe djvramics of the Li f FH-t LiF +HarldBa+FH+Ba+H

The analysis above leads us to believe that there is a strong possibility of rotational excitation of the LiF product. The bending motion induced by the existence of a well, fig. Sb, can easily become a free rotation upon departure of the H atom. This effe’ct, induced by the potential, may be just as dominant as the HHL kinematic effect [32] which tends to transform the orbital angular momentum in the reactants’ channel to the products’ internal angular momentum. The energy available for the products which, at room temperature, is mainly the HF zero point vibration, may to a large extent be channeled into the rotational motion of LiF, as well as vibrational motion. This effect may be more significant than the existence of a large barrier in the collinear arrangement as the system has a way of surmounting the barrier by avoiding the collinear path. A trajectory study, which we are

Fig. 7. OBl calculation (LLiFH = 180”).

(ref.

191)of the collinear LiFH surface

currently performing ought to provide us with definite answeres concerning this problem. It is questionable whether similar mechanism is operative in the Ba f FH + BaF + H reaction, in which surprisingly little amount of energy has been detected as the BaF vibration [4]. Preliminary studies we have made indicate that the Ba + FH is more distinctly repulsive. This is a direct result of the change in the crossing line between covalent and ionic BaFH surfaces. As a result, even departure from collinearity will not reduce the Ba + FH barrier height by nearly as much as in Li f FH system. As reported elsewhere [15] similar trends are seen to occur in the Na + FH and K + FH reactions_ 4.3. LiFF The basic difference between the M + X7 reactions and the M + XH reactions is due to the difference petween the XT and (t-lx)- molecule-ions. While the resonance in the X5 system [eq. (39)] leads to a formation of a stable molecule-ion, the resonance in the @IX)- system is not enough to compensate for the HX- repulsion. As a result the two ionic surfaces M+X-X and MtXX- [H, 1 and Iiz2, eqs. (14a), (14b)] become energetically lower than the covalent one [HS3, eq. (14c)] at large MX separation. In fig. 9 a surface defined as min( I/;, V,), where B, is the covalent [H 33, eq. (14c)] and V, is a resonance

Y. Zeiri. M. SJmpiro/Semi.empiricaI potential surfaces for electron transfer reactions

232

LI

l

FH.A~HF1.1.7300

LI

R.u.

LI

l

FH.AeiF)=2.5000

- FB.R~JW-~.SOOO

A.U.

Ad.

Fig. 8. Polar plots of the LiFH surface. The midpoint between H and F is fried at the origin, and the surface isplotted as a function of the X and Y displacemen t of Li from the origin for a fvcd H and F positions. (A) RHF = 1.73 au (=== equilibrium distance of free HF); (B)RHp = 2.5 911;(C) RHF = 3.5 au; (D)RHF = 4.5 au.

Y. Zeiri, M. ShpirofSemi-empirical

potential surfaces for electron transfer reactions

Pig.. 9. Potential surface composed of the min(Li+& LiFa) ionic and cowlcnt surfaces. The cusp corresponds to the ionic cov3lent

crossins

line.

between the ionic structures [HII and H22, eqs. (14a) and (14b)], is shown for the collinear LiFF system. The crossing is clearly marked as the cusp in the surface. The FF potential (section 3), is more repulsive than the F, potential at short distances. As a result, at short F-F separations the ionic surfaces are energetically higher than the covalent one. As the Li approaches the F2, the ionic configurations become energetically lower and the crossing line moves to shorter and shorter F-F separations. For F-F separations in the vicinity of the F2 equilibrium distance (R, = 2.683 au) the crossing can occur at large LiF sepa arations. The LiFF surface has a peculiarity associated with the fact that the dissociation energy of F2 is larger than that of F2 (0.0864 au versus 0.0606 au) and, as discussed in section 3, its equilibrium distance is also

233

the reaction, because of the large velocity of the nu-. clei at the region of crossing. Thus the separate covalent and ionic surfaces, together with the coupling between them [eqs. (14a)-( 14e)], may be used directly as a starting point for a dynamical calculation. They have the advantage of a simpler analytic representation, and the near absence of the dynamic coupling differential-operators which appear in the BomOppenheimer representation. The ground state (‘C+) Born-Oppenheimer surface at three different J_.iFF bond angles is shown in figs. lOa-10~. In figs. 1 la and 1 lb we show the first ?Z+ excited state at two LiFF bond angles. Also shown (fig. 12). is the OM surface of &lint-Kurti [lo] at the collinear Li-F-F configuration. As shown in figs. 10a and 12 there is an excellent agreement between our semi-empirical ground state and that of the OM calculation. Those sections of our excited state surface which can be compared with the published OM surface also show good agreement. Thus, in good agreement with our results (fig. 1 la) an RF-T: ~3.3 au cut of the OM surface [lo] at LLiFH = 180” is essentially repulsive. Moreover, a cut along the RLil-‘ = 3.0 au line of the OM surface, shows the first excited %+ state to have a minimum of about -0.04 au again in good agreement with our surface (fig. 1 la). The dissociation limit shown in fig. 1 la is the F(‘P’) + LiF(3n), corresponding [IO] to the dissociation limit of the seco,zd excited ‘Z’ state, while thefirst excited ‘Z+ state dissociates [7] to the F- + iiF+(‘Z+) limit, which is not considered by our simple three configurational model. Good agreement is also obtained for the bent configurations, both in the ground and excited states. This is most readily seen by examining the polar plots of fig. 13 for the ground state and fig. i lb for the excited state. In both states the well deepens as LLiFF decreases, until a minimum is reached at the isosceles triangular configuration.

larger (3.3 au [29] versus 2.683 au). As a result for

R F_F= 3.3 au the ionic configurations become ener-

Acknowledgement

getically lower than the covalent one at Li-F separations as large as 44 au. Thus as discussed by BalintKurti [lo], harpooning may take place at large LiF separations. Under such conditions it was pointed out by Herschbach [2] that the Born-Oppenheimer (adiabatic) surfaces need not be the best representation of

Many helpful discussions with Dr. Balint-Kurti are gratefully acknowledged. One of us (MS.) wishes to i_ thank the Department of Theoretical Chemistry, Bristol University, for their kind hospitality during the writing of this paper.

Y. Zeiri. M. Shapiro/Semi-empirical potential surfaces for electron transfer reactibns

234

0

3.04

5.00

7.00

9.00

11.00

IS;00

13.00

RI_,-F LI.F

F.TETR-090.

~~i5--7T-

gi.90

.

.

.

.

Il.00

.

I

RLi-F Fig. 10. Ground electronic state for LiFl as a function of the RLiF and RFF. (A) LLiFF = 180”: (B) LLiFF = 1350; (0 LLiFF =90”.

Appendix In this appendix we indicate how eqs. (9) are obtained. As an example, let us evaluate (#Q p - E~$J~> of eq. (9a). Using eqs. (2) we have that,

(0, IH - Elr#~,)= CKLYlfi - Eldet(XzY)) = (XXYIH - ElXXY - YXX), where “det” denotes the unnormalized

(A-1) determinant.

Y. Zeiri, M. .Qapiro/Semi-empirical potential mrfaees for electrorl transfer reactions

;

gl.00

3.00

5.00

7.00

3.00

If.CO

Fig. 12.OLl culcuiation face (LLilT = 180”).

13.00

RLi-F

-H[Y]

(Ref.

(3) - H[M+]

235

[ 101 of the collinear LiF2 sue-

- EIXXY - YXX)

(A-2)

which, using the definitions of eq. (1 I), becomes eq. (9a). Similarly we can derive eq. (9b). The derivation of eq. (9c) requires making some approximations in evaluating the three-center integrals. Using eqs. (2) we have that, @I, III - EIQ,) = +(IMXYl= + [OlrxYlH~lxY

+Gm'

fbci .

-5’ .00

7’ .00

9.00

11.00

Fig. 11. 2X+e~cifed state of LiFz. (A) LLiFF = 180”; (6) LLiFF = 90”.

The orthononnality of the LYand /3 spin functions used to obtain the second equality_ Decomposing Has in eq. (7a) we have that,

is

(1*2,3)

+H[MX] (1,2) +H[(MY)+] (3) -H[X-1

- YXIII

B 15.00 I-

Ii .co

RLi-F

<~,I~-Ei~,)=(XXYIH[(XY)-l

YriXYl /HI WXYI - WXPI)

(132)

If we now decompose H according to eq. (7b) and further approximate the three-center integrals as in the following examples,

(MX[H[MX] IYX) = EC(MX) &,

64.4)

vMXlH[XY] IYX) =EC(XY)Sh*y,

(A-5)

VMXIHWI IYM) = EC(MX&S,,,,

(A.61

(MXIH[XY] IXY) = J%XY)S,,~ S,,,

(A-7)

236

Y. zeiri, M. S~:epiro/Setni-etnpirieol

potential surfaces for electron transfer reactions

U+FF.P@F)=~.?OOI).u.

ag;r- -ws-

WE-

DO’,-

00’1

OO'E

or+

01

A

X LL+F_F.R(FFJ=taoo

OKL- 00’5 b -I/

OWE-

OO'I-

R.U. 00'1

-

OD'E C

OD'S

0o.G i

Fig. 13. Polar plots of the LiFF surface. The coocdinates are, similarly to fig. 8, the separation of Li from the midpoint between the two F atmos which are held at fixed points. (A) RFF = 2.7 au (-- equilibrium distance of free Fz); (B) RFF = 4.0 au; (C) RFF = 6.0 au.

Y. Zeiri, M. SkpiroJSemi-empirical

potential surfacesfor electron transfer reactions

we obtain eq. (9c) after some rearranging of terms. Eqs. (S&of) are derived in a similar way, using the covalent decomposition of H [eq. (7b)] and the definitions of the diatomic integrals as given in eqs. (11).

References

111J.L.

Kinsey, in: MTP international review of science, Vol. 9, ed. J.C. Polanyi (Buttenvorths, London, 1972) p_ 173, and references therein.

I21 D.R. Herschbach, Advan. Chem. Phys. 10 (1966) 3 19. r31 J.L. Magee, J. Chem. Phys. 8 (1940) 687; J.C. Polanyi,

Appl. Opt. Suppl. 2 (1965)

109.

[41 J. Gary Pruelt and R.N. Zare, J. Chem. Phys. 64 (1976) 1774. [51 T.J. Odrione, P.R. Brooks and J.V.V. Kasper, J. Chem. Phys. 55 (1971)

1980.

[‘31J.C. Polanyi, J.J. Sloan and J. Wanner, Chem. Phys. 13 (1976) 1. 171 D.J. Douglas, J.C. Polanyi and J.J. Sloan, Chem. Phys. 13 (1976) 15. [81 W.A. Lester and M. Krauss, J. Chem. Phys. 52 (1970) 4775. [91 G.G. Bali&Kurti and R.N. Yardley, Faraday Discussions Chem. Sot. 62 (1977) 77. I101 G.C. Balint-Kurti, Mol. Phys. 25 (1973) 393. [111 M.H. Alexander, Chem. Phys. 8 (1975) 86. 117-l F.O. Ellison, J. Am. Chem. Sot. 85 (1963) 3540. 1131 F-0. Ellison, N.T. Huff and J.C. Patel, J. Am. Chem. Sot. 85 (1963) 3544; R.K. Preston and J.C. TuUy, J. Chem. Phys. 54 (1971) 4297; P.J. KunQ and A.C. Roach, 3. Chem. Sot. Faraday II 68 (1972) 259;Erratum J. Chem. Sot. Faraday II 69 (1973) 926.

1141 G.G. Balint-Kurti,

237

private com&nication.

[W M. Shapiro and Y. Z&i, to be published. tw H.A. Pohl, R. Rein and K. Appel, J. Chem.

Phys. 41

(1964) 3385. 1171 R. Pariser, J. Chem. Phys. 21 (1953) 568. [W R.S. Mulliien, I. Chem. Phys. 46 (1949)497, 675. --. 1191 R.S. hlulliken, C.A. Rieke, D. Orloff and H. Orloff, J. Chem. Phys. 17 (1949) 1248. f201 J.N. MurreU and K.S. Sorbie, J. Chem. Sot. Faraday 11 70 (1974) 1552. WI R.N. Yardley and C.G. Balint-Kurti, ;Ilol. Phys. 31 (1976) 921. WI R.N. Yardley, Ph.D. Thesis Bristol University (1976). I231 E.S. Rittner, J. Chem. Phys. 19 (1951) 1030. 1241 P. Brumer and M. Karplus, J. Chem. Phys. 58 (1973) 3903. VI S-B. Rai 2nd D.K. Rai, Chem. Phys. Letters 30 (1975) 326. WI R.G. Gordon and S.Y. Kim, J. Chem. Phys. 56 (1972) 3122; R.G. Gordon, private communication. t271 P-IV. Tasker, G.G. Balint-Kurti and R.N. Dixon, MO!. Phys. 32 (1976) 1651.

v31 T.L. Gilbert and A.C. Wahl, J. C&em. Phys. 55 (1971) 5247.

1291 G.G. Balint-Kurti and hl. Karplus, J. Chem. Phys. 50 (1969) 478. [301 W.A. Chupka, J. Berkowitz and D. Cutman, J. Chem. Phys. 5.5 (1971) 2724.

I311 R.E. Olson, F.T. Smith and E. Bauer, Appl. Optics 10 (1970)

1848.

1321 P.J. KunQ, EM. Nemeth and J.C. Polanyi, J. Chem. Phys. 50 (1969) 4607.

1331 R. Velasco, Can. J. Phys. 35 (1957) 1204. Tables, 2nd Ed. (NSRDS-NBS 1341 JANAF Thermochemical 37, 1971).