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The quasiclassical approximation in Delos—Thorson close-coupled equations for inelastic scattering
Volume 207, number 2,3
CHEMlCAL PHYSICSLETTERS
21 May 1993
The quasiclassical approximation in Delos-Thorson close-coupled equations for inelastic scattering D.V. Shalashilin ’ Znstiluteof ChemicalPhysics,RussianAcademy of Sciences, Kosygina4, 117334 Moscow,RussianFederation
A.V. Michtchenko ’ and F. Lara Znstituteof Chemistry, National Universityof Mexico, Coyoacctn.04510, Mexico D.F., Mexico Received 1.5 February 1993
For the description of inelastic scattering a new form of the close-coupled equations of Delos and Thorson is considered. The equations proposed involve wavefunctions of elastic scattering in the quasiclassical approximation of Wentzel, Kramers and Brillouin ( WKB ). Under quasiclassical conditions the number of coupled equations for quantum amplitudes is reduced to the number of semiclassical equations. The quasiclassical equations are tested on the two-state exponential model. High accuracy is achieved if the wavefunctions used take into account the complicated structure of the Stokes phenomena. The structure of the Stokes jump is obtained numerically and visualized.
1. Introduclion
The most general description of inelastic scattering is provided by the close-coupled method (CCM) (see ref. [ 11, for example). If n intrinsic states of the colliding partners are involved, quantum CCM is reduced to a system of n coupled second-order or 2n first-order differential equations [ 2,3 1. The wellknown semiclassical approximation is introduced, when the conditions A& *: & t Ei
(1)
and ~;,~,<
(2)
are valid. Here deify is the energy mismatch of transitions under consideration, E is a collisional energy, Ai and & are De Broglie wavelengths corresponding ’ To whom correspondence should be addressed. z Permanent address: Institute of Chemical Physics, Russian Academy of Sciences, Kosygina 4, 117334 Moscow, Russian Federation. 250
to the motion on i and f potential energy curves, respectively; I is a characteristic range of the interaction. Semiclassical approximation allows us to neglect the difference of the translational energy in the initial and final states and introduce a trajectory of classical motion which is supposed to be translational motion everywhere below. After that the scattering problem is simplified and reduced to n coupled first-order equations for amplitudes a, of n coupled quantum states [ 41, To satisfy the detailed balancing, symmetrization of the collisional energy E or relative velocity v is needed in the semiclassical method. This procedure is apparently ambiguous. In the two state problem, for example, E and u could be respectively taken as E= f (E, +&) or E= (_FL,&)‘/~and v=~(u, +v2) or v=(u,v~)“~ [5]. An interval of the collisional energy E typically exists when the translational motion is quasiclassical and condition (2) is satisfied but ~if_Ei, Ef (see fig. 1). An attractive problem to solve is whether some simplifications can be obtained in the quasiclassical case, when AEif should be neglected with respect to Ei, Ef and no common trajectory could be Elsevier Science Publishers B.V.
Volume 207, number 2,3
21 May 1993
CHEMICAL PHYSICS LETTERS
R Fig. 1. Potential energy curves and turning points.
involved. The Landau formula for the quasiclassical matrix element [6] solves this problem within the first-order perturbation theory. If the amplitude and probability of the transition are not small, the Landau method cannot be applied. Generalization of semiclassical CCM for the quasiclassical situation has been considered in a number of studies. In particular, Billing has proposed [ 7,8] the method of selfconsistent trajectory of classical motion, which takes into account a change of the classical energy in av-
erage and an effect of the quantum transitions on classical motion. This approach, applied for a wide number of concrete systems [ 91, brings down the lower energy limit of the semiclassical method but condition ( 1) is still important. Coupled equations for the amplitudes and a manifold of classical trajectories describing motion from initial to final states have been proposed in refs. [ 10,111. These equations can be solved only by an iterative procedure, because the evolution of a system is determined in this multi-trajectory approach not only by its past but also by its future. In particular, eqs. [ IO, 111 for the classical trajectories contain the finite amplitudes of transitions u.(t=co). A starting point for the present study was the quantum approach of Delos and Thorson [2,3]. The goal of this work is to modify the equations [2,3] and make them convenient for numerical solution. The quantum scattering equations are transformed in order to incorporate into them the wavefunctions
of elastic scattering. In quasiclassical conditions the analytical WKB expressions for the wavefunctions are used. Trajectories of the classical motion are not introduced, but it is shown that the number of equations for the amplitudes is reduced. It becomes two times lower than the initial number of equations [2,3] and is equal to the number of semiclassical equations. Bearing in mind the subsequent application to realistic multilevel scattering problems, we only test here the approximate equations obtained on the twostate exponential model. High accuracy may be achieved if the wavefunctions used take into account the complicated structure of wavefunctions in the vicinity of a turning point (Stokes phenomena). The structure of the Stokes jump is obtained numerically and visualized.
2. Theory 2.1. Modified Delos- Thorson close-coupled equations In CCM the wavefunction Y(r, R) is expanded over the basis set of the intrinsic quantum states #Jr) y(r,R)=
C Cn(r)Xn(R). n
(3)
Here R is a distance between the colliding partners and r denotes a set of the intrinsic quantum coordinates. For the sake of simplicity the translational motion is supposed to be one dimensional everywhere below but this restriction is not in principle. The Schrijdinger equation is reduced to a system of coupled equations for the wavefunctions xn(R) of the translational motion on the potential energy curve (PES) V,,, correlating with a state n,
wherek,=p,/fi= [ (2m/fi2)(E-~~,J]1’2is a wavenumber, p,, is the momentum of classical motion on the potential energy curve, and V,, and V,,,,,are the nondiagonal potential energy matrix elements. In this approach [2,3] the wavefunctions x,,(R) are ex-
pressed as follows: 251
Volume 207, number 2,3
CHEMICAL PHYSICS LETTERS
=Lexp[-i($+i)] Xn(R) (pi)*,* + &exp[i(+
(5)
+ a)].
In (5) S,, =J$pn dR is the classical action and Rj, is the turning point on PES V,,(R), where V,,(R’,) = E. At R-t +co in the classical region the first term in (5) describes an incident and the second a reflected wave. Two functions a: (R) and a;(R) were introduced in refs. [ 3,4] instead of one function xn(R). Hence, one arbitrary relationship between the coefficients ought to be involved. This relationship is as follows [ 3,4,12]:
+
f)]=O.
(6)
- ; 5 (p~)l~;~,)l/2
-co),
a-(R=
-co).
+co)=h-(R=
(8)
(9)
where the asterisk means complex conjugation, To prove it one has to cut the complex plane in the upper semiplane and solve eqs. (7) by running around the turning points in the lower semiplane with zero initial conditions for amplitudes d- (R = -co) = 0 and nonzero initial conditions for d + (R = - co ) , The equations for amplitudes d+ in the lower semiplane are equivalent to the equations for - (a- ) * in the upper semiplane. The S matrix, containing all the information about the collision, is expressed as follows: &s‘+(F)
exp[ -ire)]
+a,‘. exp[ir*
b+(R=+co)d+6-(R=
s-=--(s+)*,
Substitution of (5) into (4) together with condition (6) leads to a system of 2n coupled linear equations for the amplitudes a,’ and a;,
x{u;
semiplane and, by solving the equations, we run around the turning points in the upper semiplane (see fig. 2), the exponent exp( - I&l /fi) converts to the incident wave exp ( -iS,Jfr ) and the exponent exp ( IS, I/ii ) to the reflected wave exp( iS,Jfi). The boundary condition for the wavefunctions x,,(R = -co) = 0 leads to the zero initial conditions foramplitudesa,+(R=-co)=O.AtR=+a,thesolution for vectors of amplitudes d+(R = +a~) and 6- (R = + co) are expressed through the initial nonzero amplitudes d- (R= + co) by matrix equations,
The matrices S- and S’ are related as
&(&)w[-i(++j)] +&(&)exp[i(+
21 May 1993
-I.
(10)
The squared absolute values of S+ and S- matrix elements are not to be treated as the transition probabilities. This can be done only with the elements of the unitary S matrix. We consider now the useful transformation of eqs. (7 ). When the wavefunction xn (R) is represented as
+is>D,
da,+ -$-su;exp[-%@+:)I dR
(7) This system describes the quantum scattering problem and it is equivalent to the initial standard CCM system (4). If the complex plane is cut in the lower 252
Fig. 2. Turning points and cuts (dashed lines) in the complex plane R =X+ iY.
Volume 207, number 2,3
xn=b,+(x,+ixn+)+b,(xn-ixn+),
(11)
where 1; and xi are, respectively, exponentially decreasing and increasing at R+ -a~ real solutions of the elastic scattering Schriidinger equation (12) the substitution into (4) with the condition analogous to (6) ~(X;+iy.f)+~(Xn-iX:)=O
(13)
leads to the following system of coupled equations for the amplitudes
db, dR
-
W dR-
m fr~i~z [(x~+ix.fP;(x~--ixn+l)
tained for the unitary 9 matrix, the real symmetric Rmatrix (s=(itiR)(I^-&)-I) [13] andtheelementsof matrices Pand T’ [ 14,151 (pand P’ are the notations from ref. [ 15 ] ). Eqs. ( 14) and equations equivalent to them [ 13-151 can be used only if the expressions for ,y; and xz are known. In particular, in refs. [ 13- 15 ] the functions of free motion (Riccati-Bessel functions) are used. We propose in this work to use the quasiclassical WKB expressions for the wavefunctions. That is the main idea of this article. In contrast to the approach [ 13-l 5 ] our eqs. ( 14) take into account the inelastic scattering only. All the information about the elastic scattering on potential energy curve V,,,,,,including information about the Stokes phenomena, is contained in the WKB wavefunctions. 2.2. Quasiclassicalexpressionsfor the
wavefunctionsand the structureof the Stokes phenomena
m h~i~l!j[(XL-ixn+P;(xn(-ix:)
+(xn-ixn+)b~(x~t+x~)lV,,,,
21 May 1993
CHEMICAL PHYSICS LETTERS
Quasiclassical solutions of eq. ( 12) far from the turning points are well known, (14)
where w=x;‘x.+ -x,x.” is the Wronskian, which is equal to a constant depending on the wavefunction normalization. The amplitudes b: and b; are complex conjugated, and the number of eqs. ( 14) is actually two times lower than the initial number of eqs. ( 7 ) . It is equal to the number of semiclassical equations. This is an advantage of the system proposed. If R--co the amplitudes b:(-co)=b;(-m) are real. At R+tcx, [I; -i;rZ]aexp(i&/A) and [xi +ix:] aexp( -iS,/fi). The matrices 3,’ and & , determined by
x_= R
w(-l~nlJ~),
R
21Pnl”2
x- = sin(S,lfi+x/4), n
n,
R,R,
(Pn)1’2
!I’
R
xn’=
cos(&Jfi+~J4), (Pn)"2
R>R,
“.
(17)
differ from the matrices s: and 9; (8) determined above by unimportant phase factors. Note that in contrast to (9) SC = (S,’)* not asymptotically at R= +cx but for any R. The s matrix is expressed through s,+ and s,- in the same manner as ( lo),
Only the physical WKB solution ,y; is commonly considered in the theory of quasiclassical wavefunctions [ 61. But if we are focused on the inelastic scattering, the solution xz is necessary as well. To describe the wavefunctions in the vicinity of the turning points the uniform approximation is used. This approach reduces eq. ( 12) to the standard Airy equation. After the substitution <= (+$f$,k,dR)2/3 for the function X=x,,/(RR’,)‘12 we can obtain [ 151
$&@a’)-’
x"tt;x=f(r).
~+(R)=S‘,+(R)~+(R=-W), 6-(R)=&-(~)b-(R=-co)
,
(15)
(16)
Similarly to (7) and ( 14) the equations can be ob-
(18)
If the quasiclassical condition ( 1) is valid the right 253
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CHEMICALPHYSICSLETTERS
side f( 5) can be neglected and eq. ( 18) is reduced to the Airy equation xll+yx=o.
(19)
21 May 1993
_.” Stokes
4.0
line
3.0 2.0
This equation has two standard solutions Ai and Bi, which lead to the following expressions for x; and +. :,=ck,1,2~~~:qdR)116Ai(_i), x.’ =Ck,L’2(1~1nk,dR)1’6Bi(.-5).
See also refs. [ 16, I7 ] and references therein. normalization constant. The accuracy of x; and x,’ determines curacy of the inelastic scattering equations is important to analyze the wavefunctions
(20) C is the -3.0
the ac( 14). It (20) in
the vicinity of the turning point and to investigate the solutions of the Airy equations in order to compare the solutions ( 17 ) and (20). In this region the properties of the wavefunctions are closely connected with the Stokes phenomena. We visualize below the Stokes jump when crossing the Stokes line in the complex plane. The Stokes phenomena play an important part in many problems of modem physics (see, for example, the recent work [ 181). Representing X as X=$exp(-i$~312)+-$exp(i~<3’z)
(21)
and using the condition analogous to (5)
(22)
-2.5
-2.0
-1.5
phase=-
-1.0
p
-0.5
0.0
0
Fig. 3. Real and imaginaryparts of amplitudes (I+ and CI-along the contour running around the turning point in the upper semiplane. The structure of the Stokesjump on the Stokesline.
fig. 3 demonstrates the coefficients in expansion (21) of this function. The coefficient a - remains constant while the amplitude a+ changes sharply when crossing the Stokes line, where the absolute value of exp( -2i3<3i2) reaches its maximum. Fig. 3 shows the complicated and symmetric structure of the Stokes phenomena. If we integrate eqs. (23 ) on the real axis, the Stokes jump takes place around the turning point. It is noteworthy that the uniform approximation (20) for the wavefunctions describes adequately the structure of the Stokes phenomena in this region. The simplest “step” approximation for the Stokes jump, corresponding to the solutions ( 17 ) , does not describe xand xf near the turning point. As will be shown in section 3, the quality of the wavefunctions in this region is critical for the accuracy of our approach.
we obtain da-= dt da+ -=de
3. Modeling example and discussion
Lexp(2i$<3fz)a+, 4< . _L exp( - 2i$<3/1)a45
(23)
The solutions of similar equations were constructed in ref. [ 121 in the form of integral series. The numerical solution, obtained on the contour 1Cl = 3 around the turning point C=O with initial condition a - = 1, is shown in fig. 3. The abovementioned initial condition leads to the function X=2 Ai( -0 and 254
To test an accuracy of the quasiclassical approximation in CCM let us apply eqs. (14) for the twostate exponential model I’,, =A exp( -olR), V,, = AEfA exp( -aR), with the exponential coupling V,, = VA exp( - crR) (see fig. 1). The parameters used (AE=2000 cm-‘, cuc4.0 A-’ and reduced mass m = 10 au) are typical of the conditions of vibrational-translational energy exchange. The diminishing of the coupling parameter v provides the ap-
Volume207,number2,3
CHEMICALPHYSICSLETTERS
plicability of perturbation theory. The results of the transition probability ( P,2 = )s,, j2) calculations with the solutions of eqs. ( 14) are presented in figs. 4 and 5 and in table 1. The simplest wavefunctions ( 17 ) and the uniform approximation wavefunctions
Fig. 4. The transition probability (P,*) dependence on energy E,. ( 1) Quantum calculation, (2) first-order perturbation theory, (3 ) semiclassicalcalculation, (4 ) quasiclassicalcalculation with the wavefunctions (17), (5) quasiclassicalcalculationwith the wavefunctions (20). v=O.O5.
21 May 1993
(20) were used with two values of the coupling parameter, v =O.OS and v=O.4. The results are compared with the results of solution of the exact quantum equation (7) and semiclassical equations with the simplest energy symmetrization 4 (E, +&). The first-order perturbation theory results obtained by the quantum Djekson-Mott formula are also demonstrated. The transition probability oscillations and sharp minima take place in the points, where amplitude s,, alters sign. In these points plZ=O. The semiclassical and both of the quasiclassical approximations are consistent with quantum results at the limit of high energies E ze AE. Table 1 demonstrates that for low energies Es AE the perturbation theory reproduces not absolute value but only exponential energy dependence of the transition probability even for v=O.OS. The quasiclassical approach with the simplest wavefunctions (17) is in bad agreement with the quantum results. This can be explained by the fact that in the exponential model the region in the vicinity of turning points determines the transition probability. The functions ( 17) are not accurate just in the turning point. Under the quasiclassical conditions aJ, a 1 and LX&a 1 the use of the uniform approximation wavefunctions (20) leads to the accuracy of about 5% even for the energies EC AE and for both of the values of cou-
Fig. 5. The transition probability (PJ dependence on energy E,. ( 1) Quantum calculation, (2) first-order perturbation theory, (3) semiclassicalcalculation, (4) quasiclassicalcalculation with the wavefunctions ( I7), (5) quasiclassicalcalculation with the wavefunctions (20). kO.4. 255
Volume 207, number 2,3
CHEMICALPHYSICSLEl-fERS
21 May 1993
Table I Transition probabilities. PG - first-order perturbation, Pft - semiclassical,P@ - quasiclassicalwith the wavefunctions (17), Pfq* quasiclassicalwith the wavefunctions (20), PR -exact quantum calculation
0.05
1000 2000 3000
0.16 0.115 0.094
0.094 0.082 0.073
0.5 1.0 1.5
3.4x lo-‘0 9.3x 10-s 2.1 x 10-s
1.6x lO-22 2.3x IO-” 6.9x IO-lo
2.9x lo-’ 4.9x 10-l 8.0x lo-)
I.zxlo-‘e 2.8~ lo-’ 5.4x 10-7
1.2x lo-‘0 2.8x LO-* 5.4x lo-’
0.4
1000 2000 3000
0.16 0.115 0.094
0.094 0.082 0.073
0.5 1.0 1.5
2.5x 10-s 5.9x 10-e 1.4x 1o-4
9.1x lo-‘7 2.5x lO-9 9.3x lo-’
7.6x lo-* 7.5x 10-Z 4.3 x 1o-2
9.4x 1o-9 2.5x 1O-6 5.4x 1o-5
9.7x 10-g 2.6x lO-6 6.0x lO-5
pling parameter Y.This result is well known in the perturbation theory (see ref. [ 171, for example). Figs. 4 and 5 demonstrate that approximation (20) for the diabatic wavefunctions is accurate out of the perturbation theory limits as well. This approximation does not reproduce only the short period oscillations of the quantum solution at v=O.4 shown in the square in tig. 5. The considered scattering equations, involving the quasiclassicalWKB wavefunctions, seem thus to be effective for the numerical analyses. High accuracy of this approach is achieved if the uniform approximation is used for the elastic scattering wavefunctions. This approximation describes adequately the complicated structure of the Stokes phenomena.
Acknowledgement
We highly acknowledge Professor M.Ya. Ovchinikova and Professor S.Ya.Umanskii for their interest in the work and fruitful discussions.
256
References [ 1] A.J. Voronon and V.I. Osherov, Dynamics of molecular reactions (Nauka, Moscow, 1990) (in Russian). [2] J.B. Delos and W.R. Thorson, Phys. Rev. A 6 (1972) 709. [ 31J.B. Delos and W.R. Thorson, Phys. Rev. A 6 (1972) 720. [4] E.E. Nikitin and S.Ya. Umanskii, Theory of slow atomic collisions (Springer, Berlin, 1984). [ 51G.D. Billing, Chem. Phys. Letters 30 (1975) 391; Chem. Phys. 9 (1975) 359. [6] L.D. Landau and E.M. Lifshitx, Quantum mechanics, 3rd Ed. (Pergamon Press, Oxford, 1977). [7] G.D. Billing,Chem. Phys. Letters 100 (1983) 535. [ 81 N.-H. Kwong,J. Phys. B 20 ( 1987) L647. [9] G.D. Billing, Comput. Phys. Commun. 32 (1984) 45; Comput. Phys. Rept. 1 (1984) 237. [ lo] P. Pechukas,Phys. Rev. 181 ( 1969) 166,174. [ 111P. Pechukasand J.C. Laight,J. Chem. Phys. 44 ( 1966)3897. [ 121N. Friiman and P.O. Froman, JWKB approximation (North-Holland, Amsterdam, 1965). [ 131V.V. Babikov, Method of phase function in quantum mechanics (Nat&a, Moscow, 1988) (in Russian). [14]E.F.HayesandD.J.Kouri,J.Chem.Phys.54(1971)878. [ 151D. Secrest, in: Methods of computational physics, Vol. 10. Atomic and molecularscattering, eds. B. Alder, S. Fembach and M. Rote&erg (Academic Press, New York, 1971). [ 161J. Heading, An introduction to phase-integral methods (Wiley, New York, 1962), [ 171T. UserandMXChild, Mol. Phys. 41 (1980) 1177. [ 181C. Zhu, H. Nakamura, N. Re and V. Aquilanti, J. Chem. Phys. 97 (1992) 1892.
CHEMlCAL PHYSICSLETTERS
21 May 1993
The quasiclassical approximation in Delos-Thorson close-coupled equations for inelastic scattering D.V. Shalashilin ’ Znstiluteof ChemicalPhysics,RussianAcademy of Sciences, Kosygina4, 117334 Moscow,RussianFederation
A.V. Michtchenko ’ and F. Lara Znstituteof Chemistry, National Universityof Mexico, Coyoacctn.04510, Mexico D.F., Mexico Received 1.5 February 1993
For the description of inelastic scattering a new form of the close-coupled equations of Delos and Thorson is considered. The equations proposed involve wavefunctions of elastic scattering in the quasiclassical approximation of Wentzel, Kramers and Brillouin ( WKB ). Under quasiclassical conditions the number of coupled equations for quantum amplitudes is reduced to the number of semiclassical equations. The quasiclassical equations are tested on the two-state exponential model. High accuracy is achieved if the wavefunctions used take into account the complicated structure of the Stokes phenomena. The structure of the Stokes jump is obtained numerically and visualized.
1. Introduclion
The most general description of inelastic scattering is provided by the close-coupled method (CCM) (see ref. [ 11, for example). If n intrinsic states of the colliding partners are involved, quantum CCM is reduced to a system of n coupled second-order or 2n first-order differential equations [ 2,3 1. The wellknown semiclassical approximation is introduced, when the conditions A& *: & t Ei
(1)
and ~;,~,<
(2)
are valid. Here deify is the energy mismatch of transitions under consideration, E is a collisional energy, Ai and & are De Broglie wavelengths corresponding ’ To whom correspondence should be addressed. z Permanent address: Institute of Chemical Physics, Russian Academy of Sciences, Kosygina 4, 117334 Moscow, Russian Federation. 250
to the motion on i and f potential energy curves, respectively; I is a characteristic range of the interaction. Semiclassical approximation allows us to neglect the difference of the translational energy in the initial and final states and introduce a trajectory of classical motion which is supposed to be translational motion everywhere below. After that the scattering problem is simplified and reduced to n coupled first-order equations for amplitudes a, of n coupled quantum states [ 41, To satisfy the detailed balancing, symmetrization of the collisional energy E or relative velocity v is needed in the semiclassical method. This procedure is apparently ambiguous. In the two state problem, for example, E and u could be respectively taken as E= f (E, +&) or E= (_FL,&)‘/~and v=~(u, +v2) or v=(u,v~)“~ [5]. An interval of the collisional energy E typically exists when the translational motion is quasiclassical and condition (2) is satisfied but ~if_Ei, Ef (see fig. 1). An attractive problem to solve is whether some simplifications can be obtained in the quasiclassical case, when AEif should be neglected with respect to Ei, Ef and no common trajectory could be Elsevier Science Publishers B.V.
Volume 207, number 2,3
21 May 1993
CHEMICAL PHYSICS LETTERS
R Fig. 1. Potential energy curves and turning points.
involved. The Landau formula for the quasiclassical matrix element [6] solves this problem within the first-order perturbation theory. If the amplitude and probability of the transition are not small, the Landau method cannot be applied. Generalization of semiclassical CCM for the quasiclassical situation has been considered in a number of studies. In particular, Billing has proposed [ 7,8] the method of selfconsistent trajectory of classical motion, which takes into account a change of the classical energy in av-
erage and an effect of the quantum transitions on classical motion. This approach, applied for a wide number of concrete systems [ 91, brings down the lower energy limit of the semiclassical method but condition ( 1) is still important. Coupled equations for the amplitudes and a manifold of classical trajectories describing motion from initial to final states have been proposed in refs. [ 10,111. These equations can be solved only by an iterative procedure, because the evolution of a system is determined in this multi-trajectory approach not only by its past but also by its future. In particular, eqs. [ IO, 111 for the classical trajectories contain the finite amplitudes of transitions u.(t=co). A starting point for the present study was the quantum approach of Delos and Thorson [2,3]. The goal of this work is to modify the equations [2,3] and make them convenient for numerical solution. The quantum scattering equations are transformed in order to incorporate into them the wavefunctions
of elastic scattering. In quasiclassical conditions the analytical WKB expressions for the wavefunctions are used. Trajectories of the classical motion are not introduced, but it is shown that the number of equations for the amplitudes is reduced. It becomes two times lower than the initial number of equations [2,3] and is equal to the number of semiclassical equations. Bearing in mind the subsequent application to realistic multilevel scattering problems, we only test here the approximate equations obtained on the twostate exponential model. High accuracy may be achieved if the wavefunctions used take into account the complicated structure of wavefunctions in the vicinity of a turning point (Stokes phenomena). The structure of the Stokes jump is obtained numerically and visualized.
2. Theory 2.1. Modified Delos- Thorson close-coupled equations In CCM the wavefunction Y(r, R) is expanded over the basis set of the intrinsic quantum states #Jr) y(r,R)=
C Cn(r)Xn(R). n
(3)
Here R is a distance between the colliding partners and r denotes a set of the intrinsic quantum coordinates. For the sake of simplicity the translational motion is supposed to be one dimensional everywhere below but this restriction is not in principle. The Schrijdinger equation is reduced to a system of coupled equations for the wavefunctions xn(R) of the translational motion on the potential energy curve (PES) V,,, correlating with a state n,
wherek,=p,/fi= [ (2m/fi2)(E-~~,J]1’2is a wavenumber, p,, is the momentum of classical motion on the potential energy curve, and V,, and V,,,,,are the nondiagonal potential energy matrix elements. In this approach [2,3] the wavefunctions x,,(R) are ex-
pressed as follows: 251
Volume 207, number 2,3
CHEMICAL PHYSICS LETTERS
=Lexp[-i($+i)] Xn(R) (pi)*,* + &exp[i(+
(5)
+ a)].
In (5) S,, =J$pn dR is the classical action and Rj, is the turning point on PES V,,(R), where V,,(R’,) = E. At R-t +co in the classical region the first term in (5) describes an incident and the second a reflected wave. Two functions a: (R) and a;(R) were introduced in refs. [ 3,4] instead of one function xn(R). Hence, one arbitrary relationship between the coefficients ought to be involved. This relationship is as follows [ 3,4,12]:
+
f)]=O.
(6)
- ; 5 (p~)l~;~,)l/2
-co),
a-(R=
-co).
+co)=h-(R=
(8)
(9)
where the asterisk means complex conjugation, To prove it one has to cut the complex plane in the upper semiplane and solve eqs. (7) by running around the turning points in the lower semiplane with zero initial conditions for amplitudes d- (R = -co) = 0 and nonzero initial conditions for d + (R = - co ) , The equations for amplitudes d+ in the lower semiplane are equivalent to the equations for - (a- ) * in the upper semiplane. The S matrix, containing all the information about the collision, is expressed as follows: &s‘+(F)
exp[ -ire)]
+a,‘. exp[ir*
b+(R=+co)d+6-(R=
s-=--(s+)*,
Substitution of (5) into (4) together with condition (6) leads to a system of 2n coupled linear equations for the amplitudes a,’ and a;,
x{u;
semiplane and, by solving the equations, we run around the turning points in the upper semiplane (see fig. 2), the exponent exp( - I&l /fi) converts to the incident wave exp ( -iS,Jfr ) and the exponent exp ( IS, I/ii ) to the reflected wave exp( iS,Jfi). The boundary condition for the wavefunctions x,,(R = -co) = 0 leads to the zero initial conditions foramplitudesa,+(R=-co)=O.AtR=+a,thesolution for vectors of amplitudes d+(R = +a~) and 6- (R = + co) are expressed through the initial nonzero amplitudes d- (R= + co) by matrix equations,
The matrices S- and S’ are related as
&(&)w[-i(++j)] +&(&)exp[i(+
21 May 1993
-I.
(10)
The squared absolute values of S+ and S- matrix elements are not to be treated as the transition probabilities. This can be done only with the elements of the unitary S matrix. We consider now the useful transformation of eqs. (7 ). When the wavefunction xn (R) is represented as
+is>D,
da,+ -$-su;exp[-%@+:)I dR
(7) This system describes the quantum scattering problem and it is equivalent to the initial standard CCM system (4). If the complex plane is cut in the lower 252
Fig. 2. Turning points and cuts (dashed lines) in the complex plane R =X+ iY.
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xn=b,+(x,+ixn+)+b,(xn-ixn+),
(11)
where 1; and xi are, respectively, exponentially decreasing and increasing at R+ -a~ real solutions of the elastic scattering Schriidinger equation (12) the substitution into (4) with the condition analogous to (6) ~(X;+iy.f)+~(Xn-iX:)=O
(13)
leads to the following system of coupled equations for the amplitudes
db, dR
-
W dR-
m fr~i~z [(x~+ix.fP;(x~--ixn+l)
tained for the unitary 9 matrix, the real symmetric Rmatrix (s=(itiR)(I^-&)-I) [13] andtheelementsof matrices Pand T’ [ 14,151 (pand P’ are the notations from ref. [ 15 ] ). Eqs. ( 14) and equations equivalent to them [ 13-151 can be used only if the expressions for ,y; and xz are known. In particular, in refs. [ 13- 15 ] the functions of free motion (Riccati-Bessel functions) are used. We propose in this work to use the quasiclassical WKB expressions for the wavefunctions. That is the main idea of this article. In contrast to the approach [ 13-l 5 ] our eqs. ( 14) take into account the inelastic scattering only. All the information about the elastic scattering on potential energy curve V,,,,,,including information about the Stokes phenomena, is contained in the WKB wavefunctions. 2.2. Quasiclassicalexpressionsfor the
wavefunctionsand the structureof the Stokes phenomena
m h~i~l!j[(XL-ixn+P;(xn(-ix:)
+(xn-ixn+)b~(x~t+x~)lV,,,,
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CHEMICAL PHYSICS LETTERS
Quasiclassical solutions of eq. ( 12) far from the turning points are well known, (14)
where w=x;‘x.+ -x,x.” is the Wronskian, which is equal to a constant depending on the wavefunction normalization. The amplitudes b: and b; are complex conjugated, and the number of eqs. ( 14) is actually two times lower than the initial number of eqs. ( 7 ) . It is equal to the number of semiclassical equations. This is an advantage of the system proposed. If R--co the amplitudes b:(-co)=b;(-m) are real. At R+tcx, [I; -i;rZ]aexp(i&/A) and [xi +ix:] aexp( -iS,/fi). The matrices 3,’ and & , determined by
x_= R
w(-l~nlJ~),
R
21Pnl”2
x- = sin(S,lfi+x/4), n
n,
R,R,
(Pn)1’2
!I’
R
xn’=
cos(&Jfi+~J4), (Pn)"2
R>R,
“.
(17)
differ from the matrices s: and 9; (8) determined above by unimportant phase factors. Note that in contrast to (9) SC = (S,’)* not asymptotically at R= +cx but for any R. The s matrix is expressed through s,+ and s,- in the same manner as ( lo),
Only the physical WKB solution ,y; is commonly considered in the theory of quasiclassical wavefunctions [ 61. But if we are focused on the inelastic scattering, the solution xz is necessary as well. To describe the wavefunctions in the vicinity of the turning points the uniform approximation is used. This approach reduces eq. ( 12) to the standard Airy equation. After the substitution <= (+$f$,k,dR)2/3 for the function X=x,,/(RR’,)‘12 we can obtain [ 151
$&@a’)-’
x"tt;x=f(r).
~+(R)=S‘,+(R)~+(R=-W), 6-(R)=&-(~)b-(R=-co)
,
(15)
(16)
Similarly to (7) and ( 14) the equations can be ob-
(18)
If the quasiclassical condition ( 1) is valid the right 253
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side f( 5) can be neglected and eq. ( 18) is reduced to the Airy equation xll+yx=o.
(19)
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_.” Stokes
4.0
line
3.0 2.0
This equation has two standard solutions Ai and Bi, which lead to the following expressions for x; and +. :,=ck,1,2~~~:qdR)116Ai(_i), x.’ =Ck,L’2(1~1nk,dR)1’6Bi(.-5).
See also refs. [ 16, I7 ] and references therein. normalization constant. The accuracy of x; and x,’ determines curacy of the inelastic scattering equations is important to analyze the wavefunctions
(20) C is the -3.0
the ac( 14). It (20) in
the vicinity of the turning point and to investigate the solutions of the Airy equations in order to compare the solutions ( 17 ) and (20). In this region the properties of the wavefunctions are closely connected with the Stokes phenomena. We visualize below the Stokes jump when crossing the Stokes line in the complex plane. The Stokes phenomena play an important part in many problems of modem physics (see, for example, the recent work [ 181). Representing X as X=$exp(-i$~312)+-$exp(i~<3’z)
(21)
and using the condition analogous to (5)
(22)
-2.5
-2.0
-1.5
phase=-
-1.0
p
-0.5
0.0
0
Fig. 3. Real and imaginaryparts of amplitudes (I+ and CI-along the contour running around the turning point in the upper semiplane. The structure of the Stokesjump on the Stokesline.
fig. 3 demonstrates the coefficients in expansion (21) of this function. The coefficient a - remains constant while the amplitude a+ changes sharply when crossing the Stokes line, where the absolute value of exp( -2i3<3i2) reaches its maximum. Fig. 3 shows the complicated and symmetric structure of the Stokes phenomena. If we integrate eqs. (23 ) on the real axis, the Stokes jump takes place around the turning point. It is noteworthy that the uniform approximation (20) for the wavefunctions describes adequately the structure of the Stokes phenomena in this region. The simplest “step” approximation for the Stokes jump, corresponding to the solutions ( 17 ) , does not describe xand xf near the turning point. As will be shown in section 3, the quality of the wavefunctions in this region is critical for the accuracy of our approach.
we obtain da-= dt da+ -=de
3. Modeling example and discussion
Lexp(2i$<3fz)a+, 4< . _L exp( - 2i$<3/1)a45
(23)
The solutions of similar equations were constructed in ref. [ 121 in the form of integral series. The numerical solution, obtained on the contour 1Cl = 3 around the turning point C=O with initial condition a - = 1, is shown in fig. 3. The abovementioned initial condition leads to the function X=2 Ai( -0 and 254
To test an accuracy of the quasiclassical approximation in CCM let us apply eqs. (14) for the twostate exponential model I’,, =A exp( -olR), V,, = AEfA exp( -aR), with the exponential coupling V,, = VA exp( - crR) (see fig. 1). The parameters used (AE=2000 cm-‘, cuc4.0 A-’ and reduced mass m = 10 au) are typical of the conditions of vibrational-translational energy exchange. The diminishing of the coupling parameter v provides the ap-
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plicability of perturbation theory. The results of the transition probability ( P,2 = )s,, j2) calculations with the solutions of eqs. ( 14) are presented in figs. 4 and 5 and in table 1. The simplest wavefunctions ( 17 ) and the uniform approximation wavefunctions
Fig. 4. The transition probability (P,*) dependence on energy E,. ( 1) Quantum calculation, (2) first-order perturbation theory, (3 ) semiclassicalcalculation, (4 ) quasiclassicalcalculation with the wavefunctions (17), (5) quasiclassicalcalculationwith the wavefunctions (20). v=O.O5.
21 May 1993
(20) were used with two values of the coupling parameter, v =O.OS and v=O.4. The results are compared with the results of solution of the exact quantum equation (7) and semiclassical equations with the simplest energy symmetrization 4 (E, +&). The first-order perturbation theory results obtained by the quantum Djekson-Mott formula are also demonstrated. The transition probability oscillations and sharp minima take place in the points, where amplitude s,, alters sign. In these points plZ=O. The semiclassical and both of the quasiclassical approximations are consistent with quantum results at the limit of high energies E ze AE. Table 1 demonstrates that for low energies Es AE the perturbation theory reproduces not absolute value but only exponential energy dependence of the transition probability even for v=O.OS. The quasiclassical approach with the simplest wavefunctions (17) is in bad agreement with the quantum results. This can be explained by the fact that in the exponential model the region in the vicinity of turning points determines the transition probability. The functions ( 17) are not accurate just in the turning point. Under the quasiclassical conditions aJ, a 1 and LX&a 1 the use of the uniform approximation wavefunctions (20) leads to the accuracy of about 5% even for the energies EC AE and for both of the values of cou-
Fig. 5. The transition probability (PJ dependence on energy E,. ( 1) Quantum calculation, (2) first-order perturbation theory, (3) semiclassicalcalculation, (4) quasiclassicalcalculation with the wavefunctions ( I7), (5) quasiclassicalcalculation with the wavefunctions (20). kO.4. 255
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21 May 1993
Table I Transition probabilities. PG - first-order perturbation, Pft - semiclassical,P@ - quasiclassicalwith the wavefunctions (17), Pfq* quasiclassicalwith the wavefunctions (20), PR -exact quantum calculation
0.05
1000 2000 3000
0.16 0.115 0.094
0.094 0.082 0.073
0.5 1.0 1.5
3.4x lo-‘0 9.3x 10-s 2.1 x 10-s
1.6x lO-22 2.3x IO-” 6.9x IO-lo
2.9x lo-’ 4.9x 10-l 8.0x lo-)
I.zxlo-‘e 2.8~ lo-’ 5.4x 10-7
1.2x lo-‘0 2.8x LO-* 5.4x lo-’
0.4
1000 2000 3000
0.16 0.115 0.094
0.094 0.082 0.073
0.5 1.0 1.5
2.5x 10-s 5.9x 10-e 1.4x 1o-4
9.1x lo-‘7 2.5x lO-9 9.3x lo-’
7.6x lo-* 7.5x 10-Z 4.3 x 1o-2
9.4x 1o-9 2.5x 1O-6 5.4x 1o-5
9.7x 10-g 2.6x lO-6 6.0x lO-5
pling parameter Y.This result is well known in the perturbation theory (see ref. [ 171, for example). Figs. 4 and 5 demonstrate that approximation (20) for the diabatic wavefunctions is accurate out of the perturbation theory limits as well. This approximation does not reproduce only the short period oscillations of the quantum solution at v=O.4 shown in the square in tig. 5. The considered scattering equations, involving the quasiclassicalWKB wavefunctions, seem thus to be effective for the numerical analyses. High accuracy of this approach is achieved if the uniform approximation is used for the elastic scattering wavefunctions. This approximation describes adequately the complicated structure of the Stokes phenomena.
Acknowledgement
We highly acknowledge Professor M.Ya. Ovchinikova and Professor S.Ya.Umanskii for their interest in the work and fruitful discussions.
256
References [ 1] A.J. Voronon and V.I. Osherov, Dynamics of molecular reactions (Nauka, Moscow, 1990) (in Russian). [2] J.B. Delos and W.R. Thorson, Phys. Rev. A 6 (1972) 709. [ 31J.B. Delos and W.R. Thorson, Phys. Rev. A 6 (1972) 720. [4] E.E. Nikitin and S.Ya. Umanskii, Theory of slow atomic collisions (Springer, Berlin, 1984). [ 51G.D. Billing, Chem. Phys. Letters 30 (1975) 391; Chem. Phys. 9 (1975) 359. [6] L.D. Landau and E.M. Lifshitx, Quantum mechanics, 3rd Ed. (Pergamon Press, Oxford, 1977). [7] G.D. Billing,Chem. Phys. Letters 100 (1983) 535. [ 81 N.-H. Kwong,J. Phys. B 20 ( 1987) L647. [9] G.D. Billing, Comput. Phys. Commun. 32 (1984) 45; Comput. Phys. Rept. 1 (1984) 237. [ lo] P. Pechukas,Phys. Rev. 181 ( 1969) 166,174. [ 111P. Pechukasand J.C. Laight,J. Chem. Phys. 44 ( 1966)3897. [ 121N. Friiman and P.O. Froman, JWKB approximation (North-Holland, Amsterdam, 1965). [ 131V.V. Babikov, Method of phase function in quantum mechanics (Nat&a, Moscow, 1988) (in Russian). [14]E.F.HayesandD.J.Kouri,J.Chem.Phys.54(1971)878. [ 151D. Secrest, in: Methods of computational physics, Vol. 10. Atomic and molecularscattering, eds. B. Alder, S. Fembach and M. Rote&erg (Academic Press, New York, 1971). [ 161J. Heading, An introduction to phase-integral methods (Wiley, New York, 1962), [ 171T. UserandMXChild, Mol. Phys. 41 (1980) 1177. [ 181C. Zhu, H. Nakamura, N. Re and V. Aquilanti, J. Chem. Phys. 97 (1992) 1892.