- Email: [email protected]
The massey parameter expansion for the semiclassical scattering matrix
Volume 55. number 3
CHEMICAL PHYSICS LETTERS
1 hfay 1978
THE MASSEY PARAMETER EXPANSION FOR THE SEMICLASSICAL SCA’JTI$RiNG MATRIX
Institute of Chemical Physics.V-334 Moscow. USSR Received 1 December 1977
The Maszy parameter expansion method for the semiclassical scattering mat& is proposed, using the Mapus formalism. The fmt Massey parameter, a unitary member of this expansion, is found for the interaction hamiltorkm, depending on time, as exp(-7) and T-~. Other possible applications of the proposed method are discussed.
1. Introduction The semiclassical approximation is very useful for the consideration of atomic and molecular inelastic collisions. The main problem of this approximation is to solve the system of coupled time dependent differential equations [ 1]
A method of obtaining such an expansion is proposed in this paper. The first Massey parameter $, a unitary correction to the transition matrix, has been found for some types of V(R)_ The accuracy of the formula obtained has been estimated by comparison with the results for the exactly solvable two state model.
2. Method of calculationand results for the population amplitudes CZ~of the colliding particle states. The classical trajectory R(f) of the particle’s relative motion determines the time dependence of the hamiltonian H[R(t)] . For convenience we divide H into two parts: H(R) = Ho + Y(R), where Ho is the hamiltouiau of the free particles and V(R) is the interaction. The system (1) can be solved analytically only for a few models. Therefore it is necessary to develop approximate methods for its solution_ In slow molecular collisions the system passes through the interaction region along the adiabatic curves of H(R), except for some regions of nonadiabaticity. In the highvelocity limit the transition probabilities between adiabatic states in the particular region of non-adiabaticity can be obtained by the projection of the initial states on the finalstates [2] _This gives the uindependent term in the I/u expansion of the transition probabilities. However, for many applications it is important to know the next few terms of this expansion (the Massey parameter expansion [ 1,3]).
It is not practical to obtain the general formula for all cases. Many processes demand separate consideration. Therefore in order to illustrate the method we shall fmd the transition probabilities in the nonadiabaticity region at large distances, where the molecular terms transform into the free atomic states. The interaction p(R) at far distances usually may be represented iu the form p(R) = fif(R) in which I!? is independent of R , and f(R) usually is exp(-oB) or R-“. In further discussion we shah suppose for definiteness f(R) = exp(-QS) and take the simplest rectilinear trajectory with impact parameter p = 0: R(t) = ut. The semiclassical equations (1) in dimensionless variables T = auf, E = ~/fic~u, in the atomic bases G(C) (Hund’s case c) and in the molecular bases @(a), diagonalising p(R) (Hund’s case a), will have the form
535
Volume55, number 3
CHEMICALPHYSICS LETTERS
where a = a, c, @ = if$/~, iiQ = @a!~ are the dimensionless parts of the full hamiltonian matrix in the bases @(al_ According to the defiiition of @(a) and Q(C), gc and fia are diagonal matrices_ The energy parameter E is chosen so that h;gl,
+I_
(3)
Further we shall consider the case 5 Q 1 and it will be seen later, that for the applicability of the perturbation theory it is important to satisfy the conditions hiig 4 1 and uiig 4 1, coinciding with (?)_ be represented as linear comThe functions $‘)can binations of ~$1 I
ic CJ
xCr =
T
1
f &)dr
(iiae-r
xa,
70
exp [i.+(r
- ~~)]x~ .
(8)
In this representation idxal/dr = .#~(+cal,
idxel/dr = J&$(r)+,
,
(9)
where f &t;=exp
(4) The matrix 3 will play an important role in our theory. The semidassical inelastic scattering problem is formulated in the following manner: if
=exp
xaI
1 May 1978
iE I,
(Gae--T f 6 &a) dr TO
X (6” - 6i;“)exp
I’:=
exp[i#(r
c
-iE
1
T (jjae--r +&=)dr J70
- ro)]Gc expl;-
i#(r
1 ,
(IO)
- ro)]
and Sk” is the circular part of ia. From the definitions (5) and (8)-(10) is the amplitude of the state &al population as r-+--03, one must calculate the amplitude xc- exp(-i$tz’& 7) for r + m. That is, one must find the-matrix f(g): Xc = j%4)Xa -
(5)
The first member of the f(g) expansion in powers of 4 is well known f2J I f(O)=&Q:.
(6)
but it is not sufficient for many practical problems. Here we shah calculate the next correction in the framework of the Magnus formalism [4]. First, however, we divide the whoIe interval r(-00, -) into two parts a - (-=, ru) and c - (ru, -)_ The point r. will be defined later. Then let us find for each part, the evohrtion matrices in the @(a) and G?(C)bases, respectively
p= &,(-,
r&Wa&ru.
-)
*
01)
Now let the perturbation theory be appiicabie in both parts a and c. This gives the condition defmhrg i-ao: eb70 = 1 7. = 1. Using the Magnus expansion for c/, [4] we can write * where 2,
and A, to first order in 5 are =0
>a=-ig
s @r)dr, -DJ
A, = -iE
i
i;@)dr
.
(13)
=0
next members of the expansion are expressed by the commutators of&r) and cl(r) at the different times and for r. * 0 are of higher order (g* in& ~$2). The simple calculations give, for the linear parts of _.& and A-, The
70
Xc(Q)
= &(r2 2q$+-)~
Aakj -
= -i@$
(7)
It will be convenient to do further caiculations in the interaction representation [S] :
exp{2i<[dh&(r s --oD
- Lkcki(e-’
-e-To)])dr
=Z-iz$hij[c
-
ln(2$e-To)
- i ni sign(Au&)], 535
- rr,)
- lnAu,,.
k f j;
(14a)
CHEMICAL PHYSICS LETTERS
Vofume 55. number 3
Ackj
= [email protected]
J
e-T&-
= -i~u~je-~O
.
(IS)
fb
Analogously the matrices A1 , a2 and cc, ea can be calculated for a I/R@*‘) potential. It is worthwhile replacing simuIr~eousIy ,$by n.$ Then after simple cakulations we find
Here AP _= P ha. Au& = U& - 4. In or& to go; &a t the pro~ab~tres are independent of 70 let us write the following equations ;=n;r=&ja,
&“fi = $h”
(16)
Xfz(l A(“)
2kj
and also
-+- ~2ss&$2,,&32,4, -eB1
-(W
with an accuracy e*, it easy to show that the contributions of (15) and the two first members of (14) give the phase shifts of the transition amplitudes. These shifts can be, reduced to tyo additional diagonal matrices exp(iC,.) and exp(iCa) in the fmal expression for 2^with Cc@=Pz~(ln2E
- ~0 -c).
= f@h?.l~(2~e-~~o+~)) v
%j
- 1~n~~os~~~~it~
- u?.e-70 1 $1
(tide infra). The anaIogons shifts have been discussed earlier in the two state modeIs [6]_ These additional phases are small (of order @G$ if pertyrbation theory is applicable. The final expression for T may be written in the form (1%
(22)
)
= -5(24)‘9A4ji’h
X itfT(I -
remembe~g that rla and fi” are diagonal matrices. Using (17) and the fact that for two matrices & and & of order 5
c
1 May 1978
1.fn)
sin(rr/2.n) sIgn(Ar@
_
(23)
From (22) and (23) we have lism,,, ApI = A I, I.imm_, Ay) = AZ. This means, that (as in the adiabatic limit g %- I) l/;i” potentials with II % 1 (really n > 3) can be approximated by the exp(-r) potential, Similar to the adiabatic limit not o&y the values but aIso the slopes of the potential l/P and the approximating potentiai are to coincide in the nonad~abatIcity region (near T& The slope of 1j&dependence at the point r() a 1 (this is the nonadiabaticity region for the ha~toni~ in the right hand side of@ and the potential 1/tiz therein) equals ( l/rn)‘If=rO * --II. That is why we have replaced g by n$ earlier. The proposed method of transition matrix expansion in powers of 2 may be used in more complicated cases than that considered above. Firstly, the case of a rectitinear trajectory with p f 0 can be considered. Secondly, the matrix & may have elements, depeud~g on t, such that the integrals (13) converge. Thirdly, the 5 expansion may be calculated for the models with fmite term splitting at both ends, T = fm. The main problem of the analysis by the proposed method is to show that the probability is independent of ro. There are no @fficuIties in numericai calcuIations of the mamrkes 4 I and &2 even For a large number of coupfed states.
A,,=AZkk=O; A,
3. Two state model
_= i$!&tj a Wu;+,
k!
Azkj
= -$rf
sign(A~~~), k #j _
cw
The matrix ?, defined by (IS), is unitary, therefore
this formula may have a larger range of applicability than the ordinary nonunitary expression: ~=eisc&[i+~l
+iz]eica
_
(211
In this section, in order to illustrate and test the accuracy of (19), we shalt consider the two state exponential model [ 1.71. The transition probability in this model can be calculated analytically. The ham& ton&n of this made1 is d~fmed by the eq~t~~ns 41
- ff22 =cos@
+e-T,
42
=&le
*
(24) 537
CHEMICAL PHYSICS LE’I-IIERS
Volume 55, number 3
1 May 1978
4. Conclusion In this paper we have proposed a method of expanding the transition matrix over a smaII Massey parameter E_The first term of the expansion, found by the
I
.
.
-
-
-
’
_
’
.
.
1.0
0.5
5
Fig. 1. Comparison of the Masseyparameter E dependence of the exact (25) (dashed line) and approximate (26) (solid line) transition probability in the two state exponential model: (1) 6 = 2~13, (2) e = lr!2, (3) e = sr/3. Thct
exact transition probability known [I,73 pex = exp[-&@(I X sh[$rE(I
for this model is well
f cos 0)]
- cost??f/shn$j .
in order to derive an approximate
cm expression
Magnus method, has a unitary form. The comparison with the results of the two state solvable model shows that this approximate formula gives the probability with an error of less than 0.05 for f up to OS. It is easy to calculate the probability according to (19) even for many state models. The main problem is to diagonalize the matrices, In following papers we shall use the proposed meth-
od for the consideration of more interesting systems of a practical nature. In particular, the influence of the error of the projection procedure on the accuracy of different mechanisms in predictions for the Na+Ne system will be found by making use of the relative reaction cross section oscillation amplitudes and by using the relative average cross sections [89]_ The reinterpretation of the experimktal results on NeNe+, A&e+, KrNe will be discussed_ In the first interpretation [IO] the projection approximation is used at very low energies (comparatively large Massey parameters).
for the
probability, note that in the same manner, as when obtaining (15), one may prove that the matrix 4~ for
References
the two state model is a phase factor and can be omitted.
Simple calculations give the approximate without a phase factor
F-matrix
0% with q
1
=
z-22 = co&e
T12 = -T21
= six@
-
$sr4;~~ e) , - $r&ixl6).
The comparison of (25) and (261, given in fig_ I for several B (t? = 2x13, x/2, n/3), shows good agreement
between IT,, 12and Pa for Massey parameters $2 0.6. It is difficult to expect better agreement for such a large c in all models, but it seems to us that using (19) one may obtain the transition probabilities with an accuracy of = 0.05 for Es 0.5.
538
I1 j E-E. Nikitin, Advan. Quantum Chem. 5 (1970) 195. [2] E-E. Nikitin. inr Chemisehe Elementarprozesse (Spriqer, BerIin, 1968). [3] MS. ChSd, Molecular collision theory (Academic Press, New York, 1974). [4) hf.L. Goldberger and K.M. Watson, Collision theory (Wiley, New York, 1964). [S] Ph. Pechukas and J.C. Light, J. Chem. Phys. 44 (1966) 3897_ [6] J.E. Bayfield, E.E. Nikitin and A-1. Reznikov, Chem. Phys- Letters 19 (1973) 471. (71 E-E. Nil&in. Discussfons Faraday Sot. 33 (1962) 14. [S] E-E. Nikitin, M.Ya. &wbinn&ova and ki. Soviet Phys- JJ5TP43 (1976) 646.
Shushin,
[9] N-H. Talk, J-C. Tu&g, C.W. White, J. Krause, A.A. Mange,
[lo]
D-L- Sirmns, M.F. Robbins, S.H. Neff and W. Lichten, Phys. Rev- A 13 (1976) 765. V. Kempter, G. Rieke, F. Veith and L. Zehule, J. Phys. B 9 (1976) 3081.
CHEMICAL PHYSICS LETTERS
1 hfay 1978
THE MASSEY PARAMETER EXPANSION FOR THE SEMICLASSICAL SCA’JTI$RiNG MATRIX
Institute of Chemical Physics.V-334 Moscow. USSR Received 1 December 1977
The Maszy parameter expansion method for the semiclassical scattering mat& is proposed, using the Mapus formalism. The fmt Massey parameter, a unitary member of this expansion, is found for the interaction hamiltorkm, depending on time, as exp(-7) and T-~. Other possible applications of the proposed method are discussed.
1. Introduction The semiclassical approximation is very useful for the consideration of atomic and molecular inelastic collisions. The main problem of this approximation is to solve the system of coupled time dependent differential equations [ 1]
A method of obtaining such an expansion is proposed in this paper. The first Massey parameter $, a unitary correction to the transition matrix, has been found for some types of V(R)_ The accuracy of the formula obtained has been estimated by comparison with the results for the exactly solvable two state model.
2. Method of calculationand results for the population amplitudes CZ~of the colliding particle states. The classical trajectory R(f) of the particle’s relative motion determines the time dependence of the hamiltonian H[R(t)] . For convenience we divide H into two parts: H(R) = Ho + Y(R), where Ho is the hamiltouiau of the free particles and V(R) is the interaction. The system (1) can be solved analytically only for a few models. Therefore it is necessary to develop approximate methods for its solution_ In slow molecular collisions the system passes through the interaction region along the adiabatic curves of H(R), except for some regions of nonadiabaticity. In the highvelocity limit the transition probabilities between adiabatic states in the particular region of non-adiabaticity can be obtained by the projection of the initial states on the finalstates [2] _This gives the uindependent term in the I/u expansion of the transition probabilities. However, for many applications it is important to know the next few terms of this expansion (the Massey parameter expansion [ 1,3]).
It is not practical to obtain the general formula for all cases. Many processes demand separate consideration. Therefore in order to illustrate the method we shall fmd the transition probabilities in the nonadiabaticity region at large distances, where the molecular terms transform into the free atomic states. The interaction p(R) at far distances usually may be represented iu the form p(R) = fif(R) in which I!? is independent of R , and f(R) usually is exp(-oB) or R-“. In further discussion we shah suppose for definiteness f(R) = exp(-QS) and take the simplest rectilinear trajectory with impact parameter p = 0: R(t) = ut. The semiclassical equations (1) in dimensionless variables T = auf, E = ~/fic~u, in the atomic bases G(C) (Hund’s case c) and in the molecular bases @(a), diagonalising p(R) (Hund’s case a), will have the form
535
Volume55, number 3
CHEMICALPHYSICS LETTERS
where a = a, c, @ = if$/~, iiQ = @a!~ are the dimensionless parts of the full hamiltonian matrix in the bases @(al_ According to the defiiition of @(a) and Q(C), gc and fia are diagonal matrices_ The energy parameter E is chosen so that h;gl,
+I_
(3)
Further we shall consider the case 5 Q 1 and it will be seen later, that for the applicability of the perturbation theory it is important to satisfy the conditions hiig 4 1 and uiig 4 1, coinciding with (?)_ be represented as linear comThe functions $‘)can binations of ~$1 I
ic CJ
xCr =
T
1
f &)dr
(iiae-r
xa,
70
exp [i.+(r
- ~~)]x~ .
(8)
In this representation idxal/dr = .#~(+cal,
idxel/dr = J&$(r)+,
,
(9)
where f &t;=exp
(4) The matrix 3 will play an important role in our theory. The semidassical inelastic scattering problem is formulated in the following manner: if
=exp
xaI
1 May 1978
iE I,
(Gae--T f 6 &a) dr TO
X (6” - 6i;“)exp
I’:=
exp[i#(r
c
-iE
1
T (jjae--r +&=)dr J70
- ro)]Gc expl;-
i#(r
1 ,
(IO)
- ro)]
and Sk” is the circular part of ia. From the definitions (5) and (8)-(10) is the amplitude of the state &al population as r-+--03, one must calculate the amplitude xc- exp(-i$tz’& 7) for r + m. That is, one must find the-matrix f(g): Xc = j%4)Xa -
(5)
The first member of the f(g) expansion in powers of 4 is well known f2J I f(O)=&Q:.
(6)
but it is not sufficient for many practical problems. Here we shah calculate the next correction in the framework of the Magnus formalism [4]. First, however, we divide the whoIe interval r(-00, -) into two parts a - (-=, ru) and c - (ru, -)_ The point r. will be defined later. Then let us find for each part, the evohrtion matrices in the @(a) and G?(C)bases, respectively
p= &,(-,
r&Wa&ru.
-)
*
01)
Now let the perturbation theory be appiicabie in both parts a and c. This gives the condition defmhrg i-ao: eb70 = 1 7. = 1. Using the Magnus expansion for c/, [4] we can write * where 2,
and A, to first order in 5 are =0
>a=-ig
s @r)dr, -DJ
A, = -iE
i
i;@)dr
.
(13)
=0
next members of the expansion are expressed by the commutators of&r) and cl(r) at the different times and for r. * 0 are of higher order (g* in& ~$2). The simple calculations give, for the linear parts of _.& and A-, The
70
Xc(Q)
= &(r2 2q$+-)~
Aakj -
= -i@$
(7)
It will be convenient to do further caiculations in the interaction representation [S] :
exp{2i<[dh&(r s --oD
- Lkcki(e-’
-e-To)])dr
=Z-iz$hij[c
-
ln(2$e-To)
- i ni sign(Au&)], 535
- rr,)
- lnAu,,.
k f j;
(14a)
CHEMICAL PHYSICS LETTERS
Vofume 55. number 3
Ackj
= [email protected]
J
e-T&-
= -i~u~je-~O
.
(IS)
fb
Analogously the matrices A1 , a2 and cc, ea can be calculated for a I/R@*‘) potential. It is worthwhile replacing simuIr~eousIy ,$by n.$ Then after simple cakulations we find
Here AP _= P ha. Au& = U& - 4. In or& to go; &a t the pro~ab~tres are independent of 70 let us write the following equations ;=n;r=&ja,
&“fi = $h”
(16)
Xfz(l A(“)
2kj
and also
-+- ~2ss&$2,,&32,4, -eB1
-(W
with an accuracy e*, it easy to show that the contributions of (15) and the two first members of (14) give the phase shifts of the transition amplitudes. These shifts can be, reduced to tyo additional diagonal matrices exp(iC,.) and exp(iCa) in the fmal expression for 2^with Cc@=Pz~(ln2E
- ~0 -c).
= f@h?.l~(2~e-~~o+~)) v
%j
- 1~n~~os~~~~it~
- u?.e-70 1 $1
(tide infra). The anaIogons shifts have been discussed earlier in the two state modeIs [6]_ These additional phases are small (of order @G$ if pertyrbation theory is applicable. The final expression for T may be written in the form (1%
(22)
)
= -5(24)‘9A4ji’h
X itfT(I -
remembe~g that rla and fi” are diagonal matrices. Using (17) and the fact that for two matrices & and & of order 5
c
1 May 1978
1.fn)
sin(rr/2.n) sIgn(Ar@
_
(23)
From (22) and (23) we have lism,,, ApI = A I, I.imm_, Ay) = AZ. This means, that (as in the adiabatic limit g %- I) l/;i” potentials with II % 1 (really n > 3) can be approximated by the exp(-r) potential, Similar to the adiabatic limit not o&y the values but aIso the slopes of the potential l/P and the approximating potentiai are to coincide in the nonad~abatIcity region (near T& The slope of 1j&dependence at the point r() a 1 (this is the nonadiabaticity region for the ha~toni~ in the right hand side of@ and the potential 1/tiz therein) equals ( l/rn)‘If=rO * --II. That is why we have replaced g by n$ earlier. The proposed method of transition matrix expansion in powers of 2 may be used in more complicated cases than that considered above. Firstly, the case of a rectitinear trajectory with p f 0 can be considered. Secondly, the matrix & may have elements, depeud~g on t, such that the integrals (13) converge. Thirdly, the 5 expansion may be calculated for the models with fmite term splitting at both ends, T = fm. The main problem of the analysis by the proposed method is to show that the probability is independent of ro. There are no @fficuIties in numericai calcuIations of the mamrkes 4 I and &2 even For a large number of coupfed states.
A,,=AZkk=O; A,
3. Two state model
_= i$!&tj a Wu;+,
k!
Azkj
= -$rf
sign(A~~~), k #j _
cw
The matrix ?, defined by (IS), is unitary, therefore
this formula may have a larger range of applicability than the ordinary nonunitary expression: ~=eisc&[i+~l
+iz]eica
_
(211
In this section, in order to illustrate and test the accuracy of (19), we shalt consider the two state exponential model [ 1.71. The transition probability in this model can be calculated analytically. The ham& ton&n of this made1 is d~fmed by the eq~t~~ns 41
- ff22 =cos@
+e-T,
42
=&le
*
(24) 537
CHEMICAL PHYSICS LE’I-IIERS
Volume 55, number 3
1 May 1978
4. Conclusion In this paper we have proposed a method of expanding the transition matrix over a smaII Massey parameter E_The first term of the expansion, found by the
I
.
.
-
-
-
’
_
’
.
.
1.0
0.5
5
Fig. 1. Comparison of the Masseyparameter E dependence of the exact (25) (dashed line) and approximate (26) (solid line) transition probability in the two state exponential model: (1) 6 = 2~13, (2) e = lr!2, (3) e = sr/3. Thct
exact transition probability known [I,73 pex = exp[-&@(I X sh[$rE(I
for this model is well
f cos 0)]
- cost??f/shn$j .
in order to derive an approximate
cm expression
Magnus method, has a unitary form. The comparison with the results of the two state solvable model shows that this approximate formula gives the probability with an error of less than 0.05 for f up to OS. It is easy to calculate the probability according to (19) even for many state models. The main problem is to diagonalize the matrices, In following papers we shall use the proposed meth-
od for the consideration of more interesting systems of a practical nature. In particular, the influence of the error of the projection procedure on the accuracy of different mechanisms in predictions for the Na+Ne system will be found by making use of the relative reaction cross section oscillation amplitudes and by using the relative average cross sections [89]_ The reinterpretation of the experimktal results on NeNe+, A&e+, KrNe will be discussed_ In the first interpretation [IO] the projection approximation is used at very low energies (comparatively large Massey parameters).
for the
probability, note that in the same manner, as when obtaining (15), one may prove that the matrix 4~ for
References
the two state model is a phase factor and can be omitted.
Simple calculations give the approximate without a phase factor
F-matrix
0% with q
1
=
z-22 = co&e
T12 = -T21
= six@
-
$sr4;~~ e) , - $r&ixl6).
The comparison of (25) and (261, given in fig_ I for several B (t? = 2x13, x/2, n/3), shows good agreement
between IT,, 12and Pa for Massey parameters $2 0.6. It is difficult to expect better agreement for such a large c in all models, but it seems to us that using (19) one may obtain the transition probabilities with an accuracy of = 0.05 for Es 0.5.
538
I1 j E-E. Nikitin, Advan. Quantum Chem. 5 (1970) 195. [2] E-E. Nikitin. inr Chemisehe Elementarprozesse (Spriqer, BerIin, 1968). [3] MS. ChSd, Molecular collision theory (Academic Press, New York, 1974). [4) hf.L. Goldberger and K.M. Watson, Collision theory (Wiley, New York, 1964). [S] Ph. Pechukas and J.C. Light, J. Chem. Phys. 44 (1966) 3897_ [6] J.E. Bayfield, E.E. Nikitin and A-1. Reznikov, Chem. Phys- Letters 19 (1973) 471. (71 E-E. Nil&in. Discussfons Faraday Sot. 33 (1962) 14. [S] E-E. Nikitin, M.Ya. &wbinn&ova and ki. Soviet Phys- JJ5TP43 (1976) 646.
Shushin,
[9] N-H. Talk, J-C. Tu&g, C.W. White, J. Krause, A.A. Mange,
[lo]
D-L- Sirmns, M.F. Robbins, S.H. Neff and W. Lichten, Phys. Rev- A 13 (1976) 765. V. Kempter, G. Rieke, F. Veith and L. Zehule, J. Phys. B 9 (1976) 3081.