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On the limits of applicability of the classical trajectory equations in the two-state approximation
Volume 21, number 3
CHEMICAL PHYSICS LETTERS
1 September
1973
ON THE LIMITS OF APPLICABILlTY OF THE CLASSICAL TRAJECTORY EQUATIONS IN THE TWO-STATE APPROXIMATION
Clm?listry
Laboratory
Anne M. WOOLLEY and Svend Erik NIELSEN University of Copenhagerr, DK-2103
III, H.C. Q&red Institute,
Copenhagen
0, Denmark
Received 12 March 1973 Revised manuscript received 18 hlay 1973
The intIoducticjn of an average trajectory is much more powerful than can be appreciated from the derivations of the classical equations. This is discussed and exemplified by systems for which the general conditions derived will not be necessary conditions of applicability.
1. Introduction
It is well known that the “classical trajectory equations” for coupled systems in the two-state approximation
often give results in surprisingly good agreement with exact quantum mechanical calculations, even when the classical picture is not valid. Here V, 1 and V22 are the diabatic potentials describing the uncoupled system and V,, is the potential coupling matrix element. b, and b, are timedependent coefficients, lbi12 being the probability that state i is occupied. Delos et al. [l, 21 have shown that the classical equations are valid whenever the semiclassical criterion holds in the form (A/Q) “’ < 1 . However, as a consequence of their derivation they suggest that a necessary condition on the validity of the classical trajecrory equations is that the classical trajectories of the two states must not be too different. This condition may be written in the alternative forms: I(P2-FJ(P2+P$
e 1 >
I (r2-‘JagI
@1,
or
I (J72--J(F2+F1
>I 4 1 .
@,b ,c>
Here pi and fi are the momentum and position coordinates for motion governed by the potential F/ii, a;ld Fi = -(dViJdr). u is the Bohr radius. Delos et al. are well aware that the necessity of these conditions has not been proven, and they ako spend considerable discussion on the choice of a classical trajectory; however, their final conclusion goes that no accurzie results can be expected if the difference between initial- and fiial-state trzjectories is large in the region of ineIastic
coupling. Although the conditions obtained from the general derivation ensure the validity of these equations, we should like to emphasize that there certainly are systems for which these conditions are not necessary, and that then it will matter what average trajectory is used. We wish here to discuss cases where the comparison of the results, in analytical form, of quantum and classical treatments shows that the introduction of an average trajectory is much more powerful than can be appreciated
from the general derivation of the classical trajectory equations.
491
CHEMICAL
Volume 21, number 3 2. The
Landau-Zener
PHYSICS LE-lTERS
1
September 1973
case
The Landau-Zener case, with its approximations of linear potentials and constant interaction matrix element, is appealing because of its simplicity, but is open to serious criticism [_?, 41 . However, it is extremely important that the L-Z approximations inserted in the quantum mechanical equations result in a set of equations identical to those of the classical approximation, if the classical trajectory is defined by [5] : x(t) = (F/~P) t2 - /.rv2/2F,
(3)
F = (FI F2) I’*, and F, and F2 have the same sign. This is the average trajectory for a linear potential devia the geometric mean force, with the position and energy meas_ired from the crossing point. It should be that the L-Z approximations need not hold everywhere, which is of course unrealistic, but only over the where the transitions take place, in which case the classical equations will be “exact”, as long as F, and Fz have the same sign and the trajectory is defined by eq. (3). We can use tie best forms of the potentials, Vii, in eqs. (I) if the potentials are approximately linear and the matrix element 712 is constant, or at least slowly vary-
where fined noted region
ing, in the transition
region.
The conditions
to be satisfied
are easily
identified
in the distorted-wave
formulation
of the curveucrossing problemapplicablein the caseof smalladiabatictransitionprobabilities, i.e,,smallV,,, For example, the exact fcrm of V12 is unimportant, as long as it is slowly varying compared to the cosine of the action difference, S1 - S2, for motion governed by the two potentials [4] . It may be that the L-Z approximations are not applicable to rrzost real systems. It has, however, been demonstrated that for the HefiNe [6] and K/I [7] systems there is almost ccmplete agreement between the L-Z and qilantum distorted-wave results (applicable because the transition probabilities are so small). For the Ha+/Ne case [6] the complete close-coupled results are also in good agreement. It is therefore important to notice that for Verne systems the conditions (2) will not be necessary conditions, and will not be satisfied for example when F, transition probabiB F2. Furthermore for F2 = C!eqs. (1) can be solved exactly [8], and yield the Landau-Zener lity. Hence we find that the L-Z approximations are valid for some real systems, and therefore the inequalities (2) need not always be satisfied, since the classical equations are exact for such systems ifthe average trajectory is defmed by the geometric mean force (F1 F2)“*.
3. Other special cases Only with ‘he Landau-Zener approximations is it possible to reduce the quantum mechanical coupled equations to the classical equations without any further approximations. Tlte introduction of higher terms in the expansions of the Vii complicates the analysis. It is, however, interesting and instructive to look at a somewhat more realistic coupling potential, namely an exponential, in the first-order distorted-wave approximation. Again the discussion is restricted to linear diabatic potentials since the undistorted wavefunctions then have a simple analytical form. The DW transition probability is given by [9] :
P= 141.rfiL2 7
I2 ,
n(lPIp21)-1’2Ce--~Ai(-pl(x-al))Ai(-P2(~-n2))dr
(4)
-‘L= x=~--C,TC
sing point; Ai
being thecrossingpoint;&=(2@Jh2)L’3; Fr,, = Ce+“, and
= (273-l
j
-0) 192
exp{i(zu+$u3)}du
.
ai = -ElFi,
E being the energy measured
from the cros-
Volume 21, number 3
CHEMICAL PHYSICS LETTERS
Child [9] has shown that the result of the integration
1 September 1973
is
j/c,) I2,
P = I ~~~~~ exp (03Ffi’/3@z)Ai(-(F-eL
(5)
where b = (16C3~,lji2FAF)“* F=&F~F~),
,
EO
F = (.Fl F,)“’
To first order the solution b, ,pr’s
=
~7
(VL#)
(?12F4/2p4F’y3 , ~
AF=F,
-F,,
of the classical trajectory
el = (fi2a2F2/2&F’)
,
F, >F,.
equations
(1) is
(6)
exp{iri’AFJ’x(t)dt)dr
--m for linear Vii. We define an average trajectory Cepar.
p=
The
j&Z’3
result
as before,
consistent
with the linear potentials,
and put VI2 =
is:
exp(cY3FTi’/3~aF’)Ai(-(E-E1)/E0)12
,
(7)
which differs from eq. (5) only in having F in place of Fin the exponential term. Hence if the classical approxirmation is to give good results, using the geometric mean force F, we obtain the condition, &/3/1(Ff’~
+Fy2)I
4 1.
(8)
We see again that the classical approximation is valid also when the conditions (2) of nearly equal elastic trajectories are violated, e.g., for Fj 9 F2, but only if the average trajectory is defined via the geometric mean force F which appears in the Airy functions in eqs. (5) and (7). The correct choice of the average trajectory is therefore important. This example illustrates how the introduction of an average trajectory takes account ofboth classical and non-classical effects in the region of the classical turning points. It is clear that in this example, unlike the Landau-Zener case, both the geometric and arithmetic mean of F, and F, appear in the transition probability, and therefore no one average trajectory can give results in exact analytical agreement with the quantum rest&, but numerically the agreement is expected to be excellent in all but the most exceptional cases. Finally we mention a related problem, that of transitions between vibrational states [IO]. This is not a cumecrossing problem, but takes a similar form. The transition probabilities are small and a first-order treatment is often sufficiently accurate. The first-order transition probability of the classical equations is given by: b, =F’12
= 7
(V,,lili)exp(i~r/h)exp{~-1S(V22-~~~)dt}dt,
(9)
ti being the energy difference between the vibrational states. It is usual to choose all the vij to have an exponential form, in which case we should choose some average trajectory appropriate to an exponential. Mies [I !] has studied this problem and has found that the analytical results of the quantum and classical treatments bear a striking resemblance to each other, and the numerical results are in excellent agreement. This is again interesting because such transitions take place near the classical turning points and the nonclassical region is important in determining the transition probability [12] _We can see once more how the introduction of an average ErajectoT
succeeds in encompassing both classical and non-classical effects. General criteria for optimal choice of an average trajectory are under investigation [ 12, 131 with particular emphasis on the problem of vibrational transitions. Acknowledgement The authors rzd.
wish to express their gratitude
for the support
received from Statens
naturvidenskabelige
Forsknings493
Volume 21, number 3
CHEMICAL PHYSICS LETTERS
References [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo] [ll] [12] (131
J.B. Delos, W.R. Thosson and S.K. Knudson, Phys. Rev. A6 (1972) 709. J.B. Delos and W.R. Thorson, Phys. Rev. A6 (1972) 720. D.R. Bates, PIOC. Roy. Sot. A257 (1960) 22. C.A. Coulson and K. Zalewski. Proc. Roy. Sot. A268 (1962) 437. V.K. Bykhovskii, E.E. Nikitin and M.Ya. Ovchinnikova, Soviet Phys. JETP 20 (1965) 500. R.E. O!son and F.T. Smith, Phys. Rev. A3 (1971) 1607; A6 (1972) 526. A.M. Woolley, unpublished. M.Ya. Ovchinnikova, Opt. i Spektroskopiya 17 (1964) 447. M.S. Child, hlol. Phys. 23 (1972) 469. D. Rapp and T. Kassal, Chem. Rev. 69 (1969) 61. F.H. Mies, J. Chem. Phys. 41 (1964) 903. G.B. Sdrensen, Dissertation, University of Copenhagen (1972). G.B. Sirensen, to be published.
1 September
1973
CHEMICAL PHYSICS LETTERS
1 September
1973
ON THE LIMITS OF APPLICABILlTY OF THE CLASSICAL TRAJECTORY EQUATIONS IN THE TWO-STATE APPROXIMATION
Clm?listry
Laboratory
Anne M. WOOLLEY and Svend Erik NIELSEN University of Copenhagerr, DK-2103
III, H.C. Q&red Institute,
Copenhagen
0, Denmark
Received 12 March 1973 Revised manuscript received 18 hlay 1973
The intIoducticjn of an average trajectory is much more powerful than can be appreciated from the derivations of the classical equations. This is discussed and exemplified by systems for which the general conditions derived will not be necessary conditions of applicability.
1. Introduction
It is well known that the “classical trajectory equations” for coupled systems in the two-state approximation
often give results in surprisingly good agreement with exact quantum mechanical calculations, even when the classical picture is not valid. Here V, 1 and V22 are the diabatic potentials describing the uncoupled system and V,, is the potential coupling matrix element. b, and b, are timedependent coefficients, lbi12 being the probability that state i is occupied. Delos et al. [l, 21 have shown that the classical equations are valid whenever the semiclassical criterion holds in the form (A/Q) “’ < 1 . However, as a consequence of their derivation they suggest that a necessary condition on the validity of the classical trajecrory equations is that the classical trajectories of the two states must not be too different. This condition may be written in the alternative forms: I(P2-FJ(P2+P$
e 1 >
I (r2-‘JagI
@1,
or
I (J72--J(F2+F1
>I 4 1 .
@,b ,c>
Here pi and fi are the momentum and position coordinates for motion governed by the potential F/ii, a;ld Fi = -(dViJdr). u is the Bohr radius. Delos et al. are well aware that the necessity of these conditions has not been proven, and they ako spend considerable discussion on the choice of a classical trajectory; however, their final conclusion goes that no accurzie results can be expected if the difference between initial- and fiial-state trzjectories is large in the region of ineIastic
coupling. Although the conditions obtained from the general derivation ensure the validity of these equations, we should like to emphasize that there certainly are systems for which these conditions are not necessary, and that then it will matter what average trajectory is used. We wish here to discuss cases where the comparison of the results, in analytical form, of quantum and classical treatments shows that the introduction of an average trajectory is much more powerful than can be appreciated
from the general derivation of the classical trajectory equations.
491
CHEMICAL
Volume 21, number 3 2. The
Landau-Zener
PHYSICS LE-lTERS
1
September 1973
case
The Landau-Zener case, with its approximations of linear potentials and constant interaction matrix element, is appealing because of its simplicity, but is open to serious criticism [_?, 41 . However, it is extremely important that the L-Z approximations inserted in the quantum mechanical equations result in a set of equations identical to those of the classical approximation, if the classical trajectory is defined by [5] : x(t) = (F/~P) t2 - /.rv2/2F,
(3)
F = (FI F2) I’*, and F, and F2 have the same sign. This is the average trajectory for a linear potential devia the geometric mean force, with the position and energy meas_ired from the crossing point. It should be that the L-Z approximations need not hold everywhere, which is of course unrealistic, but only over the where the transitions take place, in which case the classical equations will be “exact”, as long as F, and Fz have the same sign and the trajectory is defined by eq. (3). We can use tie best forms of the potentials, Vii, in eqs. (I) if the potentials are approximately linear and the matrix element 712 is constant, or at least slowly vary-
where fined noted region
ing, in the transition
region.
The conditions
to be satisfied
are easily
identified
in the distorted-wave
formulation
of the curveucrossing problemapplicablein the caseof smalladiabatictransitionprobabilities, i.e,,smallV,,, For example, the exact fcrm of V12 is unimportant, as long as it is slowly varying compared to the cosine of the action difference, S1 - S2, for motion governed by the two potentials [4] . It may be that the L-Z approximations are not applicable to rrzost real systems. It has, however, been demonstrated that for the HefiNe [6] and K/I [7] systems there is almost ccmplete agreement between the L-Z and qilantum distorted-wave results (applicable because the transition probabilities are so small). For the Ha+/Ne case [6] the complete close-coupled results are also in good agreement. It is therefore important to notice that for Verne systems the conditions (2) will not be necessary conditions, and will not be satisfied for example when F, transition probabiB F2. Furthermore for F2 = C!eqs. (1) can be solved exactly [8], and yield the Landau-Zener lity. Hence we find that the L-Z approximations are valid for some real systems, and therefore the inequalities (2) need not always be satisfied, since the classical equations are exact for such systems ifthe average trajectory is defmed by the geometric mean force (F1 F2)“*.
3. Other special cases Only with ‘he Landau-Zener approximations is it possible to reduce the quantum mechanical coupled equations to the classical equations without any further approximations. Tlte introduction of higher terms in the expansions of the Vii complicates the analysis. It is, however, interesting and instructive to look at a somewhat more realistic coupling potential, namely an exponential, in the first-order distorted-wave approximation. Again the discussion is restricted to linear diabatic potentials since the undistorted wavefunctions then have a simple analytical form. The DW transition probability is given by [9] :
P= 141.rfiL2 7
I2 ,
n(lPIp21)-1’2Ce--~Ai(-pl(x-al))Ai(-P2(~-n2))dr
(4)
-‘L= x=~--C,TC
sing point; Ai
being thecrossingpoint;&=(2@Jh2)L’3; Fr,, = Ce+“, and
= (273-l
j
-0) 192
exp{i(zu+$u3)}du
.
ai = -ElFi,
E being the energy measured
from the cros-
Volume 21, number 3
CHEMICAL PHYSICS LETTERS
Child [9] has shown that the result of the integration
1 September 1973
is
j/c,) I2,
P = I ~~~~~ exp (03Ffi’/3@z)Ai(-(F-eL
(5)
where b = (16C3~,lji2FAF)“* F=&F~F~),
,
EO
F = (.Fl F,)“’
To first order the solution b, ,pr’s
=
~7
(VL#)
(?12F4/2p4F’y3 , ~
AF=F,
-F,,
of the classical trajectory
el = (fi2a2F2/2&F’)
,
F, >F,.
equations
(1) is
(6)
exp{iri’AFJ’x(t)dt)dr
--m for linear Vii. We define an average trajectory Cepar.
p=
The
j&Z’3
result
as before,
consistent
with the linear potentials,
and put VI2 =
is:
exp(cY3FTi’/3~aF’)Ai(-(E-E1)/E0)12
,
(7)
which differs from eq. (5) only in having F in place of Fin the exponential term. Hence if the classical approxirmation is to give good results, using the geometric mean force F, we obtain the condition, &/3/1(Ff’~
+Fy2)I
4 1.
(8)
We see again that the classical approximation is valid also when the conditions (2) of nearly equal elastic trajectories are violated, e.g., for Fj 9 F2, but only if the average trajectory is defined via the geometric mean force F which appears in the Airy functions in eqs. (5) and (7). The correct choice of the average trajectory is therefore important. This example illustrates how the introduction of an average trajectory takes account ofboth classical and non-classical effects in the region of the classical turning points. It is clear that in this example, unlike the Landau-Zener case, both the geometric and arithmetic mean of F, and F, appear in the transition probability, and therefore no one average trajectory can give results in exact analytical agreement with the quantum rest&, but numerically the agreement is expected to be excellent in all but the most exceptional cases. Finally we mention a related problem, that of transitions between vibrational states [IO]. This is not a cumecrossing problem, but takes a similar form. The transition probabilities are small and a first-order treatment is often sufficiently accurate. The first-order transition probability of the classical equations is given by: b, =F’12
= 7
(V,,lili)exp(i~r/h)exp{~-1S(V22-~~~)dt}dt,
(9)
ti being the energy difference between the vibrational states. It is usual to choose all the vij to have an exponential form, in which case we should choose some average trajectory appropriate to an exponential. Mies [I !] has studied this problem and has found that the analytical results of the quantum and classical treatments bear a striking resemblance to each other, and the numerical results are in excellent agreement. This is again interesting because such transitions take place near the classical turning points and the nonclassical region is important in determining the transition probability [12] _We can see once more how the introduction of an average ErajectoT
succeeds in encompassing both classical and non-classical effects. General criteria for optimal choice of an average trajectory are under investigation [ 12, 131 with particular emphasis on the problem of vibrational transitions. Acknowledgement The authors rzd.
wish to express their gratitude
for the support
received from Statens
naturvidenskabelige
Forsknings493
Volume 21, number 3
CHEMICAL PHYSICS LETTERS
References [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo] [ll] [12] (131
J.B. Delos, W.R. Thosson and S.K. Knudson, Phys. Rev. A6 (1972) 709. J.B. Delos and W.R. Thorson, Phys. Rev. A6 (1972) 720. D.R. Bates, PIOC. Roy. Sot. A257 (1960) 22. C.A. Coulson and K. Zalewski. Proc. Roy. Sot. A268 (1962) 437. V.K. Bykhovskii, E.E. Nikitin and M.Ya. Ovchinnikova, Soviet Phys. JETP 20 (1965) 500. R.E. O!son and F.T. Smith, Phys. Rev. A3 (1971) 1607; A6 (1972) 526. A.M. Woolley, unpublished. M.Ya. Ovchinnikova, Opt. i Spektroskopiya 17 (1964) 447. M.S. Child, hlol. Phys. 23 (1972) 469. D. Rapp and T. Kassal, Chem. Rev. 69 (1969) 61. F.H. Mies, J. Chem. Phys. 41 (1964) 903. G.B. Sdrensen, Dissertation, University of Copenhagen (1972). G.B. Sirensen, to be published.
1 September
1973