Diabatic approach to the dynamics of chemical reactions

Quantum reactive scattering: Biabatic approach to the dynamics of chemical reactions

Seokmin Shit?) and John C. Light Department of Chemistry and The James Fran& Institute, University of Chicago, Chicago, Illinois 60637

(Received 20 January 1994; accepted 29 April 1994)

We present a diabatic approach to the dynamics of electronically adiabatic chemical reactions. A method is proposed for constructing diabatic surfaces from a single adiabatic potential surface. By using diabatic surfaces we can choose natural coordinate systems for both the reactant and product arrangements. The diabatic approach can be easily adapted to exact quantum mechanical calculations based on variational scattering formulations. The square integrable basis functions are obtained by diagonalizing the total Hamiltonian matrix which is constructed from the separate eigenfunctions on the reactant and product diabatic surfaces. Applications to one-dimensional barrier problems and the collinear H+H ,-+H,+H reaction demonstrate the feasibility of the diabatic approach. At low total energies, reaction probabilities for adiabatic reactions can be reproduced by the present method. Moderately accurate results can be obtained at high total energies for the collinear H+H ,-+Hz+H reaction, which may be improved by the optimal construction of diabatic surfaces.

I. INTRODUCTION

The understanding of elementary chemical reactions is an important subject in chemistry. Recent experimental progress has made it possible to study detailed state-to-state dynamics of gas phase chemical reactions. There has also been a lot of effort devoted to developing exact quantum mechanical reactive scattering theory for real chemical reactions.1-4 Recent advances in accurate quantum mechanical calculations on simple few atom systems have demonstrated the possibility of obtaining detailed information con- cerning chemical reactions from. first principles.5-‘3 Yet extension to polyatomic systems still appears to be extremely difficult, and one may need to introduce new conceptual and computational approaches to the quantum reactive process.

“Reactive” scattering is intrinsically more complicated than elastic or inelastic scattering because one set of Goordi- nates does not describe both the reactants and the products of a chemical reaction conveniently. Different formulations of reactive scattering has dealt with this coordinate problem dif- ferently. In order to describe the reactive scattering com- pletely, one may use simultaneously all convenient sets of coordinates for various chemical arrangements involved in reaction, or introduce a rather complicated coordinate system, such as hyperspherical coordinates, which can describe different arrangements at once.

In the past few years, variational methods based on the simultaneous use of convenient coordinates (e.g., mass- scaled Jacobi coordinates) in all the arrangements have been successfully applied to quantum scattering problems of reactive collisions. Three variational principles (the Kohn, Schwinger, and Newton variational principles) are used to obtain scattering information. 14*15 The success of these variational approaches has been demonstrated by recent develop- ments such as S-matrix Kohn method of Zhang, Chu, and

*‘Present address: Department of Chemistry, University of California, Santa Barbara, California 93 106.

Miller,‘,’ the log-derivative Kohn method of Manolopoulos, D’Mello, and Wyatt,io7” and the L2-amplitude density gen- eralized Newton variational principle method of Schwenke, Kouri, and Truhlar.‘2713 These methods involve basis set techniques, in which the wave function is expanded in a set of square-integrable basis functions in the interaction region. The implementation of the variational scattering formulations consists of two major parts, evaluation of matrix elements of the Hamiltonian including multidimensional ex- change integrals between basis functions in different arrangements, and the solution of a large set of algebraic equations at each energy, usually by a matrix inversion. It is, therefore, advantageous to have a set of eigenfunctions of the Hamiltonian in the interaction region as a basis set, which simplifies the evaluation of the Green’s function, especially for many energies.i6~t7

Quasiadiabatic or diabatic descriptions of electronically adiabatic chemical reactions have been introduced before.‘8-22 In these studies the possibility of witching off the reactive part of the Hamiltonian, solving separate nonre- active collision problems (reactantlike and productlike), and then obtaining the reactive transition matrix elements was pursued. In particular, for an adiabatic reaction the reaction probability was evaluated in terms of transition between the two surfaces. The transition matrix was calculated by Franck-Condon-type approximations. In another approach, quantum mechanical perturbation theory was used to obtain reaction cross sections from diabatic solutions.

In the present paper we consider the diabatic approach to the dynamics of chemical reactions. We combine diabatic representations for reactants and products with variational formulations of quantum reactive scattering to produce accurate scattering information about an adiabatic chemical reaction. In Sec. II, the main theoretical framework of the diabatic approach is given, which is explained in more detail for a model one-dimensional case in Sec. III. Section IV is devoted to the discussion of adiabatic-diabatic transformations and nonadiabatic corrections. Application of the method to a

2836 J. Chem. Phys. 101 (4), 15 August 1994 0021-9606/94/101(4)/2836/14/$6.00 Q 1994 American Institute of Physics Downloaded 12 Aug 2003 to 128.135.132.83. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

collinear reaction is given in Sec. V. The difference between adiabatic and diabatic representations, the optimal construction of diabatic surfaces, and possible applications of the method to more general chemical reactions are discussed in

S. Shin and J. C. Light: Quantum reactive scattering 2837

These functions will be orthogonal, independent of the nuclear wave functions. Thus for an expansion of the total wave function of the form

Sec. VI.

An intrinsic difficulty in quantum scattering theory of

~(Q,q,)=~R(Q)~R(a,)+~p(Q)~Piqe), (3)

II. DIABATIC FORMALISM

“reactive” collisions is the problem of different “natural” coordinates for reactants and products. No “ideal” solution exists. We approach the problem in this paper by noting that the adiabatic electronic surface leading from reactants to products can be treated as the lower adiabatic potential energy surface of a pair of diabatic surfaces formed by a linear combination of the diabatic electronic energy surfaces for reactants and products. Whether or not-one knows the “true” diabatic surfaces, one can generate a diabatic representation of the two electronic surfaces with a coupling interaction at each coordinate point which produces exactly the lower adiabatic surface desired upon diagonalization at each coordinate point. We call this the “primitive” diabatic representation.

there is no overlap integral between two terms even though they span overlapping ranges of the nuclear coordinates. Therefore, we can use separate coordinates and basis representations (DVR’s) for the two terms, and the Hamiltonian matrix after formal integration over the electronic coordinates will look like

We note that reasonable functional forms of the diabatic surfaces and the coupling interactions are quite easy to define.“3-25 Let V,.(Q) and V,(Q) be the lower and upper adiabatic surfaces. The adiabatic-diabatic transformation with a coupling potential gives the diabatic surfaces for the reactants and the products as

V,(Q)=V,(Q)+[V,(Q)-V,(Q2)lsin2 0, (14 v,,(Q)=[V,(Q)- V,(Q)ls~ ~'~0s 0, (lb) ~,~Q~=~,(Q)+[~,~Q>-~,~Q,lco~~ 0. (14

Here e(Q) is the coordinate dependent mixing angle. Let S(Q) = 0 be a dividing surface between reactants- and products, h(Q) be a switching function going smoothly from 0 for reactants to 1 for products, and being 0.5 on the dividing surface. Then we can let sin” 8(Q) = h(Q) define the linear combination of diabatic reactant and pro~duct electronic states which produces the lower adiabatic surface. Since at each coordinate point we know the true lower adiabatic surface, V,(Q), and the mixing angle sin 0(Q), we’ need only one more relation at every point to define the diabatic surfaces and the couphng interaction. One simple algorithm is to let

HR vRP

( i> vPR HP (4)

where

HR=($R@‘N+VRIGR’R)~ 64

Y?P=(GRlY?PIePL (5b)

and ?N is the nuclear kinetic energy operator. We label the assumptions in Eqs. (3) and (4) the “primitive” diabatic model which is tested below and improved upon in Sec. IV.

The Hamiltonians for the reactant and product surfaces can each be expressed in the appropriate coordinates, evaluated very simply in DVR’s appropriate to each arrangement, and diagonalized by standard techniques.26-‘8 Note that both the matrices H, and HP will be real and sparse in the DVR, and the overlap matrices will be unity for the separate orthogonal bases and zero for the reactant-product overlap. After diagonalizing HR and HP separately, the diabatic coupling terms due to V,, can be represented in the truncated basis of eigenvectors for reactants and products, and the resulting coupled total Hamiltonian matrix can be diagonalized. This procedure gives a set of square integrable (L2) basis functions which will then be used in various quantum reactive scattering formalisms. Although the evaluation of the matrix elements of the coupling interaction V,, will re- quire the integral between different coordinates, it must be done once, not at every scattering energy.

Ill. “PRIMITIVE” DIABATIC MODEL STUDY: 1 D BARRIER

VL(&)+VU(Q)=~ED=VR(Q)+VP(Q)~ or, perhaps even simpler,

!W

~,(Q)==D, @b) where ED is a constant energy, perhaps the dissociation energy.

Assuming we have the two diabatic surfaces, one of which goes to the proper asymptotic surface for reactants, and the other for products, we can then use the coordinates for each surface which will be appropriate for the corresponding asymptotic channel. Since the two electronic states are orthogonal to each other, we may expand the total wave function in terms of orthonormal nuclear wave functions on each surface times the appropriate electronic wave functions,

A. Diabatic potential surfaces

The Eckart potential of the form

V,(x) = V, sech2( ax) (64

has been used as a 1D model for chemical reaction with a potential barrier.29 For simplicity, the distance can be mea- sured in units of 1 la and the energy can be scaled by V, . We also consider the symmetric double maximum barrier given by the interaction potential

VI,(x)=V0[~+(ax)2]sech2(a2). (W

For the switching function in 1D we use the following form:

h(x)=; [l +tanh(ay)]. u-1

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b) 4

0 -4 -2 0 2 4

X

2838 S. Shin and J. C. Light: Quantum reactive scattering

B. L* basis calculation and S-matrix calculation

Let x0 define the range of strong interaction region ( --xO<x<xO). Then we introduce a DVR basis with equally spaced points (x:} ( ti = 1,2, I.. ,Ns) for me reactant surface in the range of --x,<x<xa and {xi} (/I= 1,2,...,iVp) for the product surface for -.Q<x<x,, where -xL and xL are in the asymptotic regions for the reactant and the product arrangements, respectively. The separate diagonalizations of the Hamiltonian matrices for the reactant and the product surfaces with the corresponding diabatic potentials give a set of eigenvectors for each surface.

FIG. 1. Examples of diabatic potentials constructed as described in Eq. (8) for (a) the Eckart potential and (b) the double maximum potential. The parameters used are cr=3.0a and ED=2.0Vo. The solid lines are for the lower (V,) and the upper (V,) adiabatic surfaces, while the dashed-lines are for the corresponding diabatic surfaces of the reactants (V,) and the products (VP) with the coupling potential ( V,,) represented by the dotted- line.

We truncate the eigenvector basis of the two surfaces to construct the coupled total Hamiltonian matrix as in the suc- cessive diagonalizationftruncation method.26-?8 The trunca- tion is done by retaining the eigenvectors for eigenvalues which satisfy an energy cutoff criterion,

p,sp,,, +Ge;,. (9) The resulting sets of eigenvectors are

btezh m=1,2;...,n,(GN,), (104

{ti}, n= l,&...,np(~N~). (lob)

In order to construct coupled total Hamiltonian matrix, we need to evaluate the matrix elements for the coupling interaction between eigenvectors of the reactant and the product surfaces

wRPL=b14wRPI&>. (11)

Since @m and & use different DVR bases, the evaluation of the overlap integral is not simple. In the Appendix. we have proposed a scheme for calculating the overlap integral.

The total Hamiltonian matrix, in the truncated basis for the reactant and the product surfaces, is of the form

Then the resulting diabatic potentials for the reactants and products, and the coupling interaction are given by

where es and ep are the diagonal matrices of 8, and E:. Note that the order of the matrix is N=n,+np. , ,

The diagonalization of the total Hamiltonian matrix gives the desired L2 eigenvector basis used in scattering calculations. The basis functions (including electronic part) are

dwe)= 2 GTdaX>~R(se> m=l

VRp(x)=[ED-VL(x)][ l-tanh’(ax)]‘“, @b)

VP(~)==D- V,(x), (84 + i Cni&(X)+P($e),

n=nR+l

(13)

when we use the relation V,(x) = 2 E, - V,(x) for the upper adiabatic surface. Note that we have two adjustable.parameters, a and ED, in constructing diabatic potentials and the coupling interaction from the lower adiabatic potential. Fig- ures l(a) and l(b) show examples of the decomposition of the diabatic surfaces from the Eckart potential and the symmetric double maximum barrier. Note that (Y determines the range of the coupling potential and ED determines the sepa- ration between the lower and upper adiabatic surfaces and the magnitude of the coupling interaction.

where Clj=(CJli are the elements of the eigenvectors of the total Hamiltonian matrix.

We have used S-matrix version of the Kohn variational principle to calculate the S-matrix elements’ and to calculate the transmission coefficient for the 1D barrier.

C. Numerical results

We used the barrier height V,, = 0.425 eV and the mass 1060 a.u. for the Eckart barrier given in Eq. (6a), which cor-

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0.8

0.6

0 0.4 0.6 0.8 1 1.2 1.4 1.6

E/V0


FIG. 2. Barrier transmission probability P(E) vs E for the Eckart potential. The solid line is the analytic result for the single lower adiabatic surface. The other lines are for the primitive diabatic results with different values for ru as given in the figure.

responds to a model potential for the collinear H+H, collision. The same parameters are used for the symmetric double maximum potential [Eq. (6b)]. The results for the Eckart barrier are compared with the analytical results, and those for the symmetric double maximum potential with numerical results using the lower adiabatic surface alone.

We have three parameters for the two surface diabatic formulation; Q and ED for the decomposition of the potentials and x0 for the range of the strong interaction region. We will look at the behavior of the results when we change these parameters. In general, if x0 is too small (x,<2) we have poor results, and for x93 the results are insensitive to the change in x0 (note that the range of the Eckart potential itself is about -3(x(3). For 2=GxoG3? we have a weak dependence of the results on x0. The transmission coefficients have rather weak dependence on ED. Large ED would be considered as preferable on the ground that we want to have the upper adiabatic surface well above the scattering energy. The calculations of the transmission coefficients show a strong dependence on cy (Fig. 2). It was found that smaller o! ( CY- 1 .Oa) gives better results compared to the exact results for the lower adiabatic surface alone.

We note, however, that the best results we can get from the primitive two surface approach still show substantial differences from the analytical results for the Eckart barrier. The transmission coefficients obtained from the two surface approach are shifted toward higher energy. This means that

the introduction of the two diabatic surfaces effectiveiy raises the potential barrier for the scattering process.

The results from the two surface calculations on the symmetric double maximum potential show similar behavior (Fig. 3). In the next section we refine the diabatic approach to include appropriate nonadiabatic terms which largely cor- rects these errors.

IV. THE ADIABATIC-DIABATIC TRANSFORMATION AND NONADIABATIC CORRECTIONS (REFS. 23-25)

We now look more carefully at the relations between electronic states and nuclear motion. The adiabatic representation can be defined by expanding the total wave function as follows:

~(Q,qe)=xl(Q,4,)Jl,(Q>+xz(Q,4e)~z(Q)~ (14)

where ,yi( Q,q,> are the electronic wave functions and $&Q> are the nuclear wave functions, and we now recognize the implicit dependence of the electronic wavefunctions on the nuclear coordinates. The total Hamiltonian is written as

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0.8

0.6

0.4

0.2

0

Adiabatic - Alphad. ----- Alghax3 - 0 . . . . . . . . . . .

FIG. 3. Barrier transmission probability P(E) vs E for the double maximum potential. Lines as in Fig. 2.

where TN(Q) is the nuclear kinetic energy operator. The electronic wave functions are assumed to be eigenfunctions of the electronic Hamiltonian, parametrically dependent on the nuclear coordinates

~XQ,Se)Xi(Q,q,)=Wi(Q)Xi(Q,q,). &)

Here W,(Q) are the adiabatic potential energy surfaces. The nuclear wave functions now must satisfy the following coupled equations:

i -~ V2+ Wi-E @ i(Q)=2 {71-i”.V~j-t- 7~~‘~j).

I j 07)

..I where the gradients are with respect to the nuclear coordinates and

(18)

When we neglect the electronic nonadiabatic coupling terms, the equation for the uncoupled adiabatic representation reads in matrix notation as ”

09)

where W is a diagonal matrix of the adiabatic potentials and @ I= ( $r, &)‘.~The lower energy surface defines the adiabatic equations which we wish to solve in the diabatic representation.

The diabatic representations are not unique but are obtained by using different electronic wave functions,

~‘(Q.qe)=51(Qo,qe)cpt(Qj+52(Qo,q,)cPz(Qj, (20) where Q. is a fixed nuclear configuration. The equations satisfied by the nuclear wave functions are given in matrix form as

i -g V-E IcpfVp=O, 1 @ I)

where

vij(Q)=(~i(qz)IHetI5j(qe))p; ~~’ 122)

Note that the diabatic potential matrix V is not diagonal. The two different electronic bases can be related by a

transformation matrix

5~‘“(q,)=A(Q)xad(Q,qc), (23) leading to the following relationships for the nuclear wave functions and the potential matrices:

cp dia= Wad, (24) wad= AtVdiaA (25)


For the two electronic state model, the transformation matrix can be written in terms of a mixing angle B=p [VS(Q)] 0 1

i i -1 0’

cos e(Q)’ sin O(Q) -sin S(Q) cos e(Q) 1

and the following relations hold between the adiabatic and the diabatic potential elements:

~~~(Q)=~~(Q)+~~~~Q>-~~~Q>lsi~* 4 (274

V,,(Q)=[W~(Q)-W~(Q)IS~~ 0 ~0s 0, W’b)

V,,(Q)=W,(Q>+[W?(Q)-W~(Q)ICOS~ 0. (274

Now consider the transformation of the diabatic representation given by Eq. (21) into the corresponding adiabatic representation. Using Eq. (24) and Eq. (25), Eq. (21) can be rewritten as

+A’(V’A)@}. 128)

Thus the diabatic representation of Eq. (20) leads to a “coupled” adiabatic representation. In order to find a diabatic representation corresponding to the desired “uncoupled” adiabatic representation we need to eliminate the coupling terms in Eq. (28). As a first approximation, the second term on the right-hand side of Eq. (28) can be eliminated by can- celling it with an effective potential matrix in the original diabatic representation such as

Ai i2 Veff’~ (V’A)A’=% [V20(Q)]

(2%

The first term in Eq. (29) is an off-diagonal, non-Hermitian term which we neglect. The diagonal nonadiabatic correction terms can be considered as effective potentials added to the diagonal elements of the diabatic potential matrix,

vtot= v-k v&= i

Vll-UCZff VI2

VI2 i V22--7Jeff : (30)

where v,~=(~~~/~,u)(VLY)~. A more exact expression for the diabatic representation

equivalent to the uncoupled adiabatic representation includes both the first derivative of the nuclear wave function term and the effective potential term in the diabatic equations. They are

(

where

Irp+B.Vcp+(V+?eT,ff)cp=O, (31)


(324

Wb)

Since the above equations are complicated by the presence of the first derivative terms, we do not use them in the present study, but rather use only the effective potential corrections of Eq. (30); after all, the purpose of the approach is to sim- plify the calculations.

In this formulation, the coupling term can be written in terms of the derivatives of the switching function used in constructing diabatic surfaces. In order to find the same result as for the lower “uncoupled” adiabatic surface alone, we may construct diabatic surfaces with smaller coupling terms or include the nonadiabatic correction terms of Eq. (30). Both corrections are implemented and compared below.

We first consider modification of the switching function alone. The switching function given in Eq. (7) has its first derivative largest around the barrier region. Since the nonadiabatic coupling term involves the first derivative of the switching function, we may reduce its magnitude by reduc- ing the slope of the switching function (governed by the parameter cu). The fact that smaller (Y shows less deviation from the adiabatic results (see Figs. 2 and 3) is consistent with this idea. However, the parameter cz cannot be too small, because very small a makes the range of the coupling potential undesirably large. We can improve the situation by introducing a coordinate dependent a(x) such as

h(x)=; {I +tanh[cr(x)xl}, (33a)

a(x)=as[l-exp(-yx’)]. CW

Note that n%, determines the overall range of the coupling potential region while y determines the range of the slowing down region around the barrier. The transmission coefficients for the Eckart barrier calculated using the above switching function in the primitive two surface diabatic approach are shown in Fig. 4(a). The results are greatly improved with deviations from the adiabatic results now very small. Similar improvements are shown for the symmetric double maximum potential [Fig. 4(b)].

Alternatively, as shown above, we can calculate the nonadiabatic correction terms using the simpler switching functions. As a first approximation, we have the correction terms as diagonal effective potentials added to the diabatic potentials. In effect, the diabatic potential surfaces are low- ered by (fi’/2,~)[ @‘(x)]~ with sin2 e(x) =h(x). The rest&s for the two model ID barriers after including the nonadiabatic effective potential corrections are shown in Figs. 5(a) and 5(b), which reproduce the adiabatic results very closely. It is also found that with the nonadiabatic corrections the results are not very sensitive to changes in CY.



(a) 1

0.8

0.6

G PI

0.4

0.2

0 0.4 0.6 0.8 1 1.2 1.4 1.6

E/V0

(b)

0.8

0.6

z PC

0.4

0.2

0 0.4 0.5 0.6 0.7 0.8 0.9 1

E/V0

FIG. 4. Barrier transmissidn probability P(E) vs E for (a) the Eckxt potential and (b) the double maximum potential. The solid line is the result for the single lower adiabatic surface. The other Iines are for the primitive diabatic results with modified switching function given in Eq. (33).

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0.8

0.6

G PC

0.4

0.4 0.6 0.8 1 1.2 1.4 1.6

.S. Shin and J. C. Light: Quantum reactive scattering 2843

0.8

0.6

: I

. 0 0.4 0’,5 .. . . .,. 0.6 0.7 0 . 8’ 0.9 i

E/V0 ,.-

E/Vo

(b) r-: *

FIG. 5. Barrier transmission probability P(E) vs E for (a) the E&art potential and (b) the double maximum potential.,The solid line is the result for the single lower adiabatic surface..‘& dotted-lines are for the diabatic results witQ.the diago@ effective po[ential nonadiabatic correction as described in the te.$ [q. (3011.

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(a) V tb) V L R

0 12 3 4 5 6 7

Rtau) R(au)

(c) v (d) V P RP

6

0 12 3 4 5 6 7 12 3 4 5 6 :

R(au) Rtau)

FIG. 6. The LSTH potential for the collinear H+H, -+H,+H reaction as a function of mass-weighted Jacobi coordinates {RJ}. (a) Lower adiabatic surface. (b) Reactant diabatic surface. (c) Product diabatic surface. (d) CoupIing potential. The switching function of Eq. (36) is used with parameters of a=20 (a.u.)-’ and E,=5.0 eV.

The results discussed so far have been obtained by using Eq. (2a)~for the upper adiabatic surface. In order to study the effect of the form of the upper adiabatic surface we have done calculations using a constant upper potential as in Eq. (2b). It is found that the two different choices for the upper adiabatic surfaces give the same result unless the upper surface is too close to the lower surface (E,< 1 .O).

V. COLLINEAR H+H, REACTION

We now turn to a somewhat more realistic and intuitive test for the diabatic approach to adiabatic reactions, collinear reactive scattering. We use the collinear HS H, reaction with LSTH potential energy surface.

A. Construction of diabatic potential surfaces

For the collinear reaction, we use mass weighted Jacobi coordinates, (R,r). The LSTH potential for H+H,+H,+H

collinear reaction is shown in Fig. 6(a). We define a dividing surface (dividing line for the 2D collinear case) by the equation S(R,r) = 0. The simplest choice is a straight line through saddle point

S(R,r)=r-tcR, (34)

where K= r,lR, with (R, , r,) the coordinate of the saddle point.

For the 2D switching function h( R,r), we can use a 1D switching function perpendicular to the dividing line and a constant parallel to it. Let p be the perpendicular distance from the dividing line, defined such that p<O for the reactant side and p=O on the dividing line

r---R P’y7irT

Then the switching function can be defined as

(35)



0.8

0.6 0 h I 0

z

0.4

Adiabatic - Diabatic ----.

With Correction ..........'

0.2 0.4 0.6 0.8 1 1.2 1-L 1.6 E (eV)

FIG. 7. u = 0-u ’ = 0 reaction probabilities for the LSD3 potential surfake. The solid line is the result for the lower adiabatic surface. The dashed-line shows the primitive diabatic results with the same parameters as in Fig. 6. The dotted-line shows the same diabatic result with the diagonal nonadiabatic correction included.

h(R,r)=h[p(R,r)]=~ [l +tanh(cup)]. (36).

To construct diabatic surfaces, we assume an upper adiabatic surface of the form

V&,r)=&, for V,(R,;)GC,

= V&b-), for VL(R,r)>ED. (37)

Diabatic potentials for the reactant and the product surfaces and the coupling potential are given by Eqs. (la)-(lc). Fig- ures 6(b)-6(d) show these potential surfaces.

6. Results

We have two sets of mass weighted Jacobi coordinates appropriate to the reactant, (R, , ra), and the product, (R, ,I-,), surfaces respectively. We introduce direct product (Chebychev) DVR bases for these coordinates {R~}@{r~} for the reactant surface and {R$} @{rf,} for the product surface. The Hamiltonian matrices for the reactant and the product diabatic surfaces are diagonalized separately by standard methods to give a set of eigenvectors for each surface. The matrix elements for the coupling potential (V,,) between truncated eigenvectors of the reactant and the product surfaces are evaluated as in the 1D case (see aIso Appendix). The resulting total Hamiltonian matrix is diagonalized to

. give an efficient L2 basis for the problem. The S-matrix is calculated by using the Kohn variational principle.

We have calculated the reactive transition’probabilities P(udu’) for the collinear H+H,-+H,-tH reaction where% and u ’ are the initial and final vibrational states. We compare the results from the two surface diabatic approach with those using the lower adiabatic surface alone. The single adiabatic surface calculation was done using hyperspherical coordinates and the Kohn variational method.17

Figure 7 shows P(O--+O) for the two approaches. As in the 1D problem, the transition probabilities are very similar but have shifted energies for the primitive diabatic approach. When we include the diagonal nonadiabatic corrections for the diabatic potentials (see Sec. IV), the two surface diabatic results are much closer to the adiabatic results, particularly in the threshold region. However, this simple nonadiabatic correction does not fully correct the diabatic results to the adiabatic results at higher energies.

When we use a modified switching function as in Eq. (33), the difference between the double surface diabatic approach and the single adiabatic surface approach is found to be rather small (Fig. 8). The results are somewhat sensitive to the parameter y, which determines’the range of the slowly varying switching function around the dividing surface, and the shape of the energy dependence of the probability tends to be slightly distorted for y too small.

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2848 S. Shin and J. C. Light; Quantum reactive scattering

0.8

- 0.6 0 A I

1 8 Adiabatic -

Diabatic . . . . . . . . . . .

0.6 0.8 1 1.2 1.4 1.6 E (eV)

FIG. 8. v =O-+IJ’ = 0 reaction probabilities for the LSTH potential surface. The solid line shows the result for the lower adiabatic surface. The dotted-line shows the diabatic results when the modified switching function [Eq. (33)] is used with parameters of cU=2.0 1a.u.j”. Ed=S.O eV, and y=O.25 (a.u.)-‘.

The switching function used above depends only on the normal distance from the dividing line. This 1D character of the switching function results in a rather broad coupling potential extending well into the asymptotic region. We prefer to restrict the coupling potential to the interaction region around saddle point. This can be. done by making the curva- ture parameter (Y dependent on the distance from-the saddle point along the dividing line. This 2D switching function js &fined as _y ._ -::

h[p(R,r),s(R.r).]=i (1 +tanh[a(s)p]}. (38) I .7”

We choose two different forms for a(s),

(i) a(s)=ao exp[ys(s-sij2], ” . &9a)

(4 4s)=a0(1 ~~~C!-sech[y,(s_~o?l))~ (39k)

where s = (R + m-)l.dm and SO corresponds’ to the saddle point, In both forms, a(s) has minimum at the saddle point and increases -as the distance from the saddle point increases. The potentials for the 2D switching function (i) are shown in Figs. 9(a)-9(d). The transition probability calculated using the 2D switching function is very close to the adiabatic result (Fig. 10). The two forms of (Y(S) give similar results. The differences in transition probabilities which per- sist at higher energies are probably due to the “true” nondi- agonal nonadiabatic coupling. .‘.’ ~1

Vi. CONCLUDING REMARKS

In this paper we demonstrated the possibility of using a diabatic approach to the dynamics of electronically adiabatic reactions.. A method has been proposed for constructing “reasonable” diabatic surfaces from a single adiabatic potential surface. Using the diabatic surfaces allows us to choose “natural” coordinate systems for both the reactant and.prod- ucf arrangements., thus simplifying .the treatment of the asymptotic regions. The diabatic approach can be easily adapted to variational scattering formulations. The square integrable bases needed in the scattering calculation are ob- tamed by diagonalizing total Hamiltonian using’DVR’s and standard techniques developed by Light and co-workers.“6-28 The overlap integral between different arrangements is needed only for the coupling potential which can have a relatively small range.

The present study is different from the other studies of quasiadiabatic or diabatic descriptions of adiabatic chemical reactions-in that we provide the first exact quantum me&&i- cal calculations of the diabatic description and examine the effects of changing the diabatic surfaces. It-earlier studies, the quasiadiabatic description was used as the basis for Franck-Condon approximations for reaction dynamics.‘8-20 These applications to the collinear AB +C+A+BC reactions showed that the relative internal state distributions are accurately reproduced and are fairly insensitive to the choice of diabatic surfaces. However, the absolute reaction-probabili-



7

6

5

‘3$ lu 5;3

2

1 ~

R(au) R(m)

O4---- R(au)

(b) V R

OL 12 3 4 5 6 7

(d) V RP

6= ."".. "... ...-'.. ".'

5.

1.

oi:. 12 3 4 5'6

R(au)

FIG. 9. The LS’I’H potential for the collinear H+H 2+H2+H reaction as a function of mass-weighted Jacobi coordinates {R,r}. (a) Lower adiabatic surface. (b) Reactant diabatic surface. (c) Product diabatic surface. (d) Coupling potential. The 2D switching function as given in !I$. (38) and FQ. (39a) is used with parameters of q,=OS (a.u.)-‘, E,=5.0 eV, and y,=O.25 (a.u.)-*.

ties were very sensitive to the choice. No systematic attempts were made to choose the quasiadiabatic surfaces so as to maximize the effectiveness of the Franck-Condon model. More recent work by Eno et al. explored the application of quantum mechanical perturbation theory to the determination of reactive transition probabilities based on a quasiadibatic t:QA) description of the dynamics.21,22 The quasiadibatic (QA) curves were constructed in a similar fashion to ours, but they only considered 1D model problems at energies below the barrier height. The QA results were highly sensitive to the variation of the QA potential parameters, and may reflect the perturbation scheme used to evaluate the reaction probabilities.

In this paper accurate numerical applications to 1D model problems and collinear H+H,+H,+H reaction show the feasibility of the diabatic approach to give accurate adiabatic surface reaction probabilities. In both cases, the probabilities from the primitive diabatic method have the same

general shape as those for the single adiabatic surface, but at shifted energies. This shift can be largely corrected by add- ing effective potentials to the diabatic potentials due to diagonal nonadiabatic correction terms. These are determined from the transformation between the adiabatic and diabatic representations. When the total energy is near the threshold region (about the barrier height), the diabatic results with the nonadiabatic correction can reproduce the adiabatic results essentially exactly. For higher total energies, as shown in the collinear reaction, the differences between diabatic and adiabatic results are made small but are not fully resolved by the diagonal nonadiabatic correction terms, presumably due to the neglected off-diagonal nonadiabatic correction terms. Some of these terms [Eq. .128)] are approximately propor- tional to &-& where Ei is the internal energy, and can be important at higher energies. Introducing a modified switching function as in Eq. (33) or 2D switching function [Eq.



0.8

Adiabatic - ,,iabatic . . . . . . . . . .

0.6 0.8 1 1.2- 1.4 1.6 E (eW

FIG. 10. II = O--to ’ =0 reaction probabilities for the LSTH potential surface. The solid line shows the result for the lower adiabatic surface. The dotted-line shows the diabatic results when the 2D switching functioniis~nsed with the same parameters as in Fig. 9.

,.

(38)] allows us to have a small coupling region and to reproduce adiabatic results quitecloseljz

One interesting feature of 2D switching function defined in Eq. (38) is that the coupling potential V,, is rather flat and stretched around the saddle point, while the coupling potential from the original switching function has a peak at the saddle point [compare Fig. 9(d)’ with Fig. 6(d)]. Sensitivity analysis of the reaction probability has been used to deter- mine that the relative importance of different regions of the potential depends on the total energy of the system.30 Usu- ally, a higher total energy such as the first resonance region around E-O. 89 eV has a broader region of the potential contributing to the dynamics than the lower threshold region. Perhaps an optimal choice of-the diabatic surfaces for higher total energies should%be able to describe a rather broad region of the potential’surface correctly.

The results above also suggest mat ‘a single adiabatic surface description of a real chemical reaction may be insuf- ficient whenever there are other adiabatic surfaces nearby. The present diabatic formulation can be easily applied to “real” nonadiabatic reactions in which actual lowerand up’ per adiabatic surfaces- am-known. Even when the accurate coupling potential is not available in that case, the switching function defined in this paper can be used to study the relative irirportance of the strength’of the coupling between adiabatic surfaces on the dynamics of a reaction. ,I

The present study has showed that the diabatic formalism provides a reasonable’ alternative description of reactive

scattering processes. The diabatic approach allows one to describe the reactant -and product asymptotic regions simple and correctly. At low total energies, the state to state reaction probabilities for adiabatic reactions can be reproduced very accurately by the relatively simple diabatic approach pre- sented in this paper. These results are relatively insensitive to the choice of diabatic surfaces. At higher energies, the deviations of the diabatic results from the adiabatic ones are still relatively small. Further studies are needed if one wants to solve the diabatic representation which is formally equivalent to the adiabatic representation (Sec. IV).

ACKNOWLEDGMENT

This research was supported in part by Department of Energy Grant No. DE-FG02-87ER13679.

APPENDIX: DOUBLE QUADRATURE EVALUATION FOR THE MATRIX ELEMENTS OF THE COUPLING POTENTIAL

We will consider the 1D case for simplicity. Let (C&X)} and {$(x)} be the sets of orthogonal polynomials (FBR) corresponding to the DVR bases {x:} and {xs} for the reactant and the product surfaces, respectively. The FBR and DVR bases are related to each other by transformation matrices given by

(TR),,= l/i&$(& fz,i= 1,2 ,...) Nn, (Ala)



(TP)jp= I&&$(X;), ,f?,j- 1,2 ,..., NP . (Al@

The matrix elements of the coupling potential in the FBR is defined as

Note that the FBR bases are real functions. The evaluation of the overlap integral given above can be done by using both the reactant and the product DVR quadrature points as follows:

NR fc4 d-b~b",> hdx:) Pp(-c>

+ 2 [ 1 -f(x;>]o;&x;) .’ /?=I

’ W. H. Miller, Annu. Rev; Phys. Chem. 41, 245 (1990). ’ W. H. miller, in Methods in Comp&tational Molecular Physics, edited by

S. Wilson an& G: ii%~die&sen (P&urn, New York, 1992). fi. 519. ‘A. Kuppermann and P. G. Hipes, J. Chem. Phy!, 84, 5962 (1986). 6R. T Pack and G. A. Parker, J. Chem. Phys. 87, 3888 (1987); 89, 3511

(1989). 7E Webster and J.-c. Light, J, Chem. Phys. 90, 265, 300 (1989). s W. H. Miller and B. M. D. D. Jansen op de Haar, J. Chem. Phys. 86.6213

(1987). ‘J. Z. H. Zhang S-I. Chu, and W. H. Miller, J. Chem. Phys. 88, 6233

(1988); 89, 4454 (i988). ?‘I “D. E. Manolopoulos and R. E. Wyatt, Chem. Phys. Lett. 152, 23 (1988). ‘ID. E. Manolopoulos, M. D’Mello, and R. E. Wyatt, J. Chem. Phys. 91,

6096 (1989). l2 J. Z. H. Zhang, p. J. Koqi,.K. Haig, D. W. Schwenke, Y. Shima, and D.

G. Truhlar, J. Chem. Phys. 88, 2492 (1988). 13D’ G. Truhlar, D. W. Schwenke;a&l D. J. Kouri, J. Phys. Chem. 94,7346

(1490). “R. K. Nesbet, Variational Methods in Electron-Atom Scattering Theory .-(Plenum, New York, 1980). 15R. G. Newton, Scattering Theory of Particles and Waves (Springer, New x &&;> 4ojpb$. * (A3)

Here f(x) is a switching function which g&s from 0 at the asymptotic region of the reactant surface to 1 at the asynip- totic region of the product surface. We can use the same switching function as for the construction of the diabatic surfaces, h(x).

The matrix elements in the FBR can then be transformed into the DVR,

Finally, in the truncated eigenbasis for the reactant and the product surfaces the matrix elements become

vRp= (CR)fV~~cp=(Vp~)f, CA9

where C” and CP are the matrices formed by the truncated eigenvectors. The resulting matrices V,, and VP, are used in constructing the total Hamiltonian matrix.

‘Tlteory of Chemical Reaction Dynamics, edited by M. Baer (Chemical Rubber, Boca Raton, 1985).

*D. E. Manolopoulos and D. C. Clary, Annu. Rep. C (R. Sot. Chem.) 86, 9.5 (1989).

- -- _ - York, 1982).

16D. Brown and J. C. Light, J. Chem. Phys. 97, 5465 (1992). 17D. Brown, Ph.D. thesis;The University of Chicago, 1993. f*G. C. Schatz and J. Ross, J. Chem. Phys. 66, 1021, 1037 (1977); C. L.

Vi& J. L. Kinsey, J. Ross, and G. C. Schatz, ibid. 70, 2414 (1979). “S. Mukamel and J. Ross, J. Chem. Phys. 66, 3759 (1977). ‘OK. H. Fung and K. F. Freed, Chem. Phys. Lett. 30, 249 (1978). “I. P. Dillon md L. Eno, J. Chem. Phys. 83.5696 (1985); T. R. Horn and L.

Eno, ibid. 85, 2631 (1986). =J. C. Peploski and L. Eno, J. Chem. Phys. 95, 895 (1991). “M. Baer, Chem. Phys. Lett. 35, 112 (1975). 24M. S. Child, in Atom-Molecule Collision Theory, edited by R. B. Bem-

stein (Plenum, New York, 1979), p. 427. zM. Baer, in The Theory of Chemical Reaction Dynamics, edited by M.

Baer (Chemical Rubber, Qca Raton, 198$), Vol. 2, Chap. 4. 26Z Bacic R. M. Whitnell, D. Brown, and J. C. Light, Comput. Phys.

C&mud 51, 35 (1988). ’ “J: C. Light, R. M. Whitnell, T. J. Park, and S. E. Choi, in Supercomputer

Algorithms foi Reactivity, Dynamics and Kinetics of Small Molecules, NATO ASI Series C, edited by A. Lagana (Kluwer, Dordrecht, 1989). Vol. 277, p, 187.

‘ss. E. Choi and J. C Light, J. Chem. Phys. 92, 2129 (1990). 2gH. S. Johnston, GaYs Phase Reaction Rate Theory (Ronald, New York,

.1966), pp. 37-47. 3oJ. Chang, N. J. Brown, M. D’Mello, R. E. Wyatt, and H. Rabitz, J. Chem.

Phys. 97, 6226, 6240 (1992).

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Diabatic approach to the dynamics of chemical reactions