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INTERACTION
OF
LITHIUM
WITH
AMMONIA
4803
1
D.
E.
O'Reilly,
J.
Chern. Phys. 41, 3729 (1964).
2 R. Catterall
and
M.
C.
R.
Symons,
J.
Chern. Soc. (London)
1965, 3763.
3D.
E. O'Reilly, J. Chern. Phys. 50, 4743 (1969).
•
K.
Bar-Eli
and
T.
R.
Tuttle,
Jr.,
Bull. Am. Phys. Soc.
8,
352
(1963).
5
K.
Bar-Eli
and
T.
R.
Tuttle,
Jr.,
J.
Chern. Phys. 40, 2508
(1964).
6
K.
Bar-Eli and
T.
R.
Tuttle,
Jr.,
J.
Chern. Phys. 44, 114
(1966).
7 R. Catterall
and
M.
C.
R. Symons,
J.
Chern. Soc. (London)
1965, 6656.
8
R. Catterall, M. C. R. Symons,
and
J.
W. Tipping,
J.
Chern.
Soc. (London) A1966, 1532.
9
R. Catterall, M. C. R. Symons,
and
J.
W. Tipping, Proc.
Conf. Chern. Aspects Electron Spin Resonance, Cardiff (1966)
(cited in Ref. 10).
1
0
R. Catterall, M.
C.
R.
Symons,
and
J.
W.
Tipping,
J.
Chern.
Soc. (London) A1967, 1234.
11
L. R. Dalton, J. D.
Rynbrandt,
E. M. Hansen,
and
J.
L.
Dye,
J.
Chern. Phys. 44,3969 (1966).
12
L.
R.
Dalton, M.
S.
thesis, Michigan
State
University,
East
Lansing, Michigan, 1966.
1
3
J.
L.
Dye
and
L. R. Dalton, ]. Phys. Chern. 71, 184 (1967).
14
K. D. Vos and
J.
L. Dye, J. Chern. Phys.
38,
2033
(1963).
15
K. D. Vos, Ph.D. thesis, Michigan
State
University,
East
Lansing, Mich.,
1963.
THE
JOURNAL
OF
CHEMICAL
PHYSICS
16 R. Catterall, M. C. R. Symons, and ]. W. Tipping, Proc.
Colloque Weyl
II,
Cornell University, 1969 (to be published).
17 E. Becker,
R.
H. Lindquist, and B.
J.
Alder,
J.
Chern. Phys.
25,
971
(1956).
18
W.
E.
Blumberg
and
T.
P. Das,
J.
Chern. Phys. 30,
251
(1959).
1
9
J.
Jortner,
J.
Chern. Phys. 34,
678
(1961).
2o
J.
Jortner,
S.
A. Rice, and E.
G.
Wilson in Metal Ammonia
Solutions, G. Lepoutre and M.
J.
Sienko, Eds. (W.
A.
Benjamin,
Inc., New York, 1964).
2
1
D. E. O'Reilly,
J.
Chern. Phys. 41, 3736 (1964).
22
D.
E.
O'Reilly and T. Tsang,
J.
Chern. Phys. 42, 3333 (1965).
2
3 R.
C.
Douthit
and
J.
L. Dye, J. Am. Chern. Soc. 82, 4472
(1960).
24
M. Gold, W. Jolly, and K. Pitzer,
J.
Am. Chern. Soc. 84,
2264 (1962).
2
•
J.
L. Whitten (private communication).
2
6
J.
L. Whitten,
J.
Chern. Phys. 44, 359 (1966).
27
G. Herzberg, Infrared and Raman Spectra (D. Van Nostrand
Co., Inc., New York, 1945).
'"C.
D. Ritchie and
H.
F. King,
J.
Chern. Phys.
47,564
(1967).
2
9
L.
C.
Snyder and H. Basch,
J.
Am. Chern. Soc. 91, 2189
(1969).
ao
R.
S.
Mulliken,
J.
Chern. Phys. 23, 1833, 1841, 2338,
2343
(1955).
3
1
D. L. Hardcastle,
J.
L. Gammel,
and
R. Keown,
J.
Chern.
Phys. 49,
1358
(1968), and references therein.
VOLUME
52,
NUMBER
9
1
MAY
1970
High-Order Time-Dependent Perturbation Theory for Classical Mechanics and for Other
Systems of First-Order Ordinary Differential Equations*
R.
A.
MARCUS
Noyes Chemical Laboratory, University
of
Illinois, Urbana, Illinois 61801
(Received 24 November 1969)
A time-dependent perturbation solution
is
derived for a system of first-order nonlinear or linear ordinary
differential equations.
By
means of an ansatz, justified a posteriori, the
latter
equations can be converted to
an
operator equation which is solvable by several methods.
The
solution is subsequently specialized to
the case of classical mechanics. For the particular case of autonomous equations
the
solution reduces to
a well-known one in the literature. However, when collision phenomena are treated and described in a
classical "interaction representation" the differential equations are typically nonautonomous, and the
more general solution is required.
The
perturbation expression
is
related to a quantum mechanical one and
will be applied subsequently
to
semiclassical and classical treatments
of
collisions.
INTRODUCTION
Classical mechanics has been used extensively
to
treat
experimental
data
on reactive collisions,
1
in
part
because exact calculations can be made.
The
corre-
sponding exact
quantum
calculations in three dimen-
sions are absent
at
present.
Exact
classical calculations
of rotational-vibrational-translational energy transfer
have also been useful.
2
The
latter, in conjunction with
classical approximations, provide insight into
quantum
approximations,
3
as do
the
several exact
quantum
results.
4
In
the present
paper
a "high-order" perturbation
theory is developed in a form which gives final
state
properties in terms of integrals over initial
state
ones
and
so
is suited
to
collision phenomena.
5
It
permits
systematic development of certain approximations in
classical mechanics and, in conjunction with a cor-
respondence principle for collisions,
6
permits
an
ap-
proximation
of
semiclassical matrix elements which
occur in some collision problems.
In
the
present paper a perturbation theory is derived
first for a system
of
ordinary differential equations
more general
than
those characterizing classical
mechanics and is
later
specialized to classical mechanics.
In
passing
it
may
be noted
that
in a variation of con-
stants
treatment
(e.g.,
of
collisions in classical me-
chanics7)
the
differential equations for the
"constants"
are typically nonautonomous.
The
first step in the over-all derivation is a conversion
of
the
system
of
equations to an operator equation.
8
To
this end an ansatz
[Eq.
(6)]
will be introduced, one
which is then justified a posteriori.
The
resulting opera-
tor equation can then be solved
by
one
of
several
methods. Using one
of
these
9
we
obtain Eqs. (6)
and
(15),
and
for classical mechanics Eqs. (24)
and
(26).
[Using another method (iteration) one obtains instead
(15'), while a time-ordered method yields instead
(15").]
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4804
R.
A.
MARCUS
Several applications of the present equations are
given in subsequent articles.
PERTURBATION THEORY
The
initial value problem for
the
system
(i=
1,
· ·
·,
n)
of differential equations, nonlinear or linear,
dx;/ dt= h;(x,
t)
=h.o(x,
t)
+h;
1
(x,
t)
(t?_ t
0
),
x;=x;
0
(t=t
0
),
(1)
is considered, where
h;
0
and
h;
1
are the
unperturbed
and
perturbing terms; x denotes
the
totality
of x;'s.
We shall
be
mainly interested in applications where
h;
1
vanishes as
t~/
0
and
as
t~oo.
This
vanishing
may
occur either because
of
an
explicit dependence
of
h;
1
on
t or, even when this dependence is absent, because of
the range
of
x
of
interest
10
in the neighborhood of t
0
and
t=+oo.
Solution
of
the
unperturbed
problem yields constants
of
the
motion
i;.
In
the
perturbed
problem, when the
above vanishing of
h;
1
occurs, these
x/s
vary
from their
initial values
at
t= t
0
to final constant values
at
t"'oo.
Transformation from x;'s to x;'s
("variation
of con-
stants")
yields
(I"?.
to),
(t=to),
(2)
where
h;
vanishes when h;
1
does
and
where x denotes the
totality
of
x;'s.
Even
when (1) is autonomous (i.e.,
h;
depends on x alone),
Eq.
(2)
is
generally nonautono-
mous in collision phenomena, as noted earlier.
In
the
case
of
classical mechanics
Eq.
(2) would form a
classical
counterpart
of
an
"interaction representation"
in
quantum
mechanics.
6
The
ensuing results will apply
to
Eq.
(2) and, when
the
barred
symbols are replaced
by
unbarred
ones, to
Eq.
(1) as well.
A first-order
partial
differential operator D(t),
acting on the space of functions of x
and
containing t as a
parameter, can be defined
8
D(t)=
L,ii;(x,t)(ajax;).
i
The
system (2) can be rewritten as
df(x) I dt= D(t)J(x)
(t?_
to),
(t=to),
(3)
(4)
where f(:i) is
an
arbitrary
differentiable function of
x,
since
dj(x)jdt
equals L,;aj(x)/axi(dx;/dt).
If
the system (2) were autonomous,
and
D(t)
then
written as D, the solution
of
(2) or (4) would be
8
f(x)
=[lexp[(t-to)D]lf(x)]x~x'·
(5)
To
treat
the
more general system (2) we seek instead a
generalization of
(5),
and
shall assume
that
j(x)
can
be
written
as
j(x)
= [{
exp0(t)
}.f(x)
]x~x•,
(6)
where 0(1), like
D,
is
to be a first-order partial differ-
ential operator. 0(1) contains
t as a parameter.
Later,
in Eq. ( 15), an explicit expression for this operator
is
given.
For
brevity
exp0(t)
will be denoted
by
D(t):
00
D(t)
=
exp0(t)
=
L,
0n(t)/n!.
(7)
n=O
Equations
(4),
(6)
and
(7) yield
dj(D(t)x)/dt=D(t)j(D(t)x)
at
x=x
0
,
(8)
where
d/
dt
acts only on the t in
1>
( t), because
of
the
restriction
x=xo.
Because of the properties of the exponential of a
first-order
partial
differential operator
11
one
may
write,
for
any
function g(x,
t),
g(D(t)x,
t)
=D(t)g(x,
t),
at
x=XO.
(9)
Thus,
(8) can be rewritten as
[dD(t)/dt]j(x)
=D(t)D(t)j(x),
at
x=x
0
•
(10)
Omission
of
the
arbitrary
initial point
x=x
0
and
the
arbitrary
functionf(x)
yields the operator equation
dD(t)
/dt=D(t)D(t).
(
11)
One form of solution
9
to
a differential equation for
an
operator U (
t,
to),
dU/dt=A(t)U,
U
(to,
to)=
1,
(12)
lS
U(t, t
0
)
=expCB(t,
to),
(13a)
where
12
(13b)
At, denotes the operator A (I;),
and
[ , ] denotes the
commutator. Comparison of (11)
and
(12) shows
that
instead
of
( 12) we need the equation satisfied
by
u-r,
the inverse of U:
dU-
1
/dt=
-U-
1
A,
U-
1
(1o,
to)=
1,
(14)
obtained
by
differentiation of uu-r = 1
and
introduction
of (12). Comparison of (14) with
(11) shows
that
they
are of the same form,
but
with
U-
1
and A replaced
by
~
and
- D, respectively. Since the inverse of exp0 is
exp(
-0)
one finds from (13)
that
the 0 in (6) is
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PERTURBATION
THEORY
FOR
CLASSICAL
MECHANICS
4805
given
by
e(t)
= r Dt,dt1-
~
ft
[Dt,
!
12
Dt
1
dt1]
dt2
to 2 to to
(15)
where each
D
1
,
denotes
D(t;).
Since the commuta-
tors of this
D
1
,
at
various times t; are, like D
1
,
itself,
first-order partial differential operators, e(t) is also, thus
justifying a posteriori the ansatz (
6).
[Thus,
by
reversing the sequence
of
steps in the derivation one
can verify
that
the solution given
by
(6)
and
(15)
satisfies (2)
.]
Had
( 1) been used instead
of
(2),
a solution identical
with (6) and (15) would have been obtained
but
having
x;, x;
0
,
and
h/s
instead
of
x;,
:X;
0
,
and
h;'s.
(The
restric-
tion
of
h;
1
need not be imposed then, except for possible
convergence questions
at
large
t.
This point
is
returned
to in a later section.) Solving the operator Eq. (
11)
could also have been based on time ordering or on simple
iteration [cf. Refs. 13-15, or Eqs. (15') and (15")
below].
When the system
of
differential equations (2) is
autonomous, the commutators in (15) vanish since the
D
1
,'s
no longer depend on t;. Equation (15) then reduces
to the well-known result (5).
The
ansa tz (
6)
, leading from ( 4) to (
11)
, could
undoubtedly be replaced
by
a more basic Lie-algebraic
argument, a point to which
we
shall
return
in a later
communication,
D(t)
being an infinitesimal generator
of
a Lie algebra. However, the argument given earlier
suffices for the immediate purpose. Again, the results
(6)
and
(15), together with (24)
and
(26),
may
be
known to workers in
that
field,
but
have
not
to our
knowledge been published explicitly.
An iterative solution of (11) or (5) leads to a known
15
result for
J(i):
f(i)
=
[D(t)f(x)
]x=.t",
where
a result consistent (after some manipulation) with
(7)
and
(15).
A time-ordered solution of (11) yields instead
f(i)
=[D(t)f(i)]x=x',
D(t)
=P
exp
ft
Dt
1
dt~,
(15")
to
where P denotes the time-ordering operator.
16
"
APPLICATION TO CLASSICAL MECHANICS
The
preceding solutions can be specialized now to the
case where the independent variables occur in canoni-
cally conjugate pairs.
The
classical mechanical equa-
tions
of
motion for a system with generalized co-
ordinates
q;,
canonically conjugate momenta
p;,
and
Hamiltonian H(q,
p,
t)
are
dqJdt=aHjap;, dp/dt= -aHjaq;. (16)
(Throughout,
q
and
p will denote the totality
of
q/s
and
p;'s, respectively.)
The
Hamiltonian is the sum
of
unperturbed
and
perturbed terms, H
0
and
H1:
H(q,
p,
t)
=Ho(q,
p,
t)
+H1(q,
p,
t). (17)
Transformation to new variables
ij;
and
p;
which are
constants
of
the motion
of
the unperturbed problem
is
conveniently made by means of a generating function
W(q,
p,
t) satisfying the Hamilton-Jacobi1
7
equation
for the unperturbed problem,
Ho(q,
p,
t)+aW(q,
p,
t)jat=O. (18)
The
transformation equations are
q;=aw jap;,
p;=aw
;aq;.
(19)
The
generating function W transforms H to a new
Hamiltonian
H(ij,
p,
t),
H(ij,
p,
t)
=H(q,
p,
t)+aW(q,
p,
t)jat, (20a)
which in virtue
of
( 17) and ( 18) becomes
H(ij,
p,
t)
=Hl(q,
p,
t). (20b)
[Thus,
to
obtainfl,
the solutions
q(ij,
p,
t)
and p(ij,
p,
t)
of
the unperturbed problem are introduced into
H1(q,
p,
t).]
The
new equations
of
motion are
dij;/
dt
= afl.
q,
p,
t)
1
ap;,
dpJdt=
-afl(q,
p,
l)/aq,. (21)
The operator
D(t), defined
by
(3), thus becomes
16
h
D(t)=-IH(t),
l (22)
where { , } denotes the Poisson-Bracket:
{X,
Y}=
I:
(axjaq;aYjap;-aXjap;aYjaq;). (23)
i
The
solution to (21) is
f(ij;, p;) = [ ( exp0)f(ij, p) ]ii=q'.P=P' (24)
for
any
function f
of
the ij;'s
and
p/s,
where 0 is given
by
(
15)
and
(22).
If
the iterative solution ( 15') or
time-ordered one (15") were employed, expe would
be given
by
D in (15') or
(15"),
respectively, where
now (22) is introduced. When (15) is used instead,
(26) is obtained, as follows.
Since
D is now a Poisson-Bracket, Eq. (15) can
first be simplified: One readily verifies
that
Jacobi's
identity
18
for Poisson-Brackets can be rewritten in
operator form as
where
[X,
Y]=
II
X,
Y},
},
X=
{X, }
and
Y=
IY,
}.
(25a)
(25b)
Downloaded 08 Mar 2006 to 131.215.225.174. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
4806
R.
A.
MARCUS
Consequently, (15) becomes
e={B,
},
where
+
~
{ [
Hts,
f:l
Ht.,
f:'Rr!dtt]
dt2]
dl3
+ :
2
~:
[[
Hr"
~:
3
Rt,d12],
1:
3
Hr
1
rllt]
dl3+ • · •.
(26a)
(26b)
We note
that
the
symbol exp{B, } arising from (24)
and
(26a) represents
exp{B,
}=1+{B,
}+(1/2!){B,
{B,
}}
+(1/3!)
{B, {B, {B, } }
}+·
· •. (27)
The
solution
to
(21) is given
by
(24)
and
(26).
Had
(1) been used instead
of
(2) as a starting point,
(26) would again have been obtained,
but
with ij,
p,
ij
0
,
p
0
,
and
fi
replaced
by
q,
p,
q
0
,
p
0
,
and
H.
In
some problems interest lies in
the
perturbation
of
variables
c;
(e.g.,
the
orbital elements in celestial
mechanics
19
) which are functions
of
the canonically
conjugate pairs,
rather
than
the
pairs themselves.
Equations (24)
and
(26) can still be applied.
20
An alternative derivation
of
(24) and (26),
but
not
of
(6) and (15), can be given
6
using a
quantum
mechanical
expression
and
the correspondence between classical
and
quantum
mechanics.
COMPARISON
WITH
PREVIOUS
WORK
Grobner
8
has employed an operator formalism
("solution
by
Lie series"), particularly for
the
case
that
the system (1) is autonomous.
21
Solutions were
made iteratively or
by
other
13
methods, though
not
employing Magnus' result. When the system
of
equa-
tions (1) is autonomous, D (with x's and h's instead
of
x's and fi's) becomes
the
operator which enters into
Lie's theory
of
ordinary differential equations.
22
An
iterative solution for nonautonomous systems was
noted previously.
1
5
Operator methods were introduced into classical
mechanics
by
Koopman.
23
A rather different operator
formalism has been employed
by
Garrido
14
in a per-
turbation theory for classical mechanics.
He
noted
that
the operator
!J
defined in
dF
(aH
a
aH
a)
&
=SJF=
2"1
ap;
aq;-
ap;
aq;
F
(28)
is a linear differential operator and
that,
for
that
reason,
the
equation for
the
evolution
of
a function
of
phase space, F(q, p), is equivalent
to
an operator
equation
24
dF
I
dl
= [!J,
F],
(29)
if
F in (29) is reinterpreted in
an
operator acting on
the
space
of
functions
of
phase space. An automorphism
was next tacitly assumed,
25
by
analogy with a known
quantum
mechanical result, and a solution was obtained
both
in terms
of
a time-ordered product and iteratively
(Magnus' method was
not
employed). Appealing to
another analogy
26
between an ordinary and an operator
equation, he obtained an expression for the time
evolution of
the
junction F.
The
present method
of
derivation
of
( 15')
and
( 15") can be regarded as
providing a more rigorous derivation
of
his final results.
27
Garrido's final equation has been applied to rota-
tional-translational energy transfer in a plane.
28
Similarly, the present results can be applied to collision
phenomena, either using the solutions (24) and (26) or
using ( 15') or (
15").
In
the
perturbation
treatment
of
(1) or (2)
we
were
particularly interested in the case where
h;YO
as
I---"
oo
.
In
problems such as forced harmonic oscillator
q=
p,
p=w
0
2
q+a
sinwt,
where clearly h;
1
+0
as
I---'>
oo
,
the
series for
(.')
(I) ter-
minates after the second term, and no difficulty arises.
However, in problems such as the anharmonic oscillator
q=
p,
p
=w2q+aq2,
secular terms develop.
They
can be avoided
by
re-
sorting
to
other methods, such as Lindsted's proce-
dure29
or canonical perturbation theory.
30
In
the
latter
theory, some old variables appear as a perturbation
series in terms
of
the new.
In
collision problems on the
other hand, one is much more interested in an expression
for the new variables (i.e.,
the
new constants
of
the
motion)
at
I'""
oo
in terms
of
the
old, as for example
in the solution given
by
(24) and (26).
* Acknowledgment is made to
the
donors
of
the
Petroleum
Research Fund, administered
by
the
American Chemical Society,
for
partial
support of this research. This research was also sup-
ported by a
grant
from
the
National Science Foundation
at
the
University
of
Illinois.
1
See, e.g., N. C. Blais
and
D.
L. Bunker, J. Chern. Phys. 41,
2377 (1964)
and
references cited therein;
M.
Karplus
and
L.
M.
Raff, ibid. 41,
1267
(1964)
and
subsequent articles in this journal;
P.
J.
Kuntz,
E.
M.
Nemeth,].
C. Polanyi,
S.D.
Rosner,
and
C.
E.
Young, ibid. 44, 1168 (1966), and subsequent articles
of
this series.
2
R. ]. Cross,
Jr.
and
D.
R. Herschbach,
J.
Chern. Phys., 43,
3530 (1965);
S.
W. Benson
and
G.
C.
Berend, ibid. 44, 4247
(1966), references cited therein; 47, 4199 (1967);
J.
D.
Kelley
and
M. Wo!fsberg, ibid. 44, 324 (1966); R. G. Gordon, ibid. 44,
3083 (1966);
D.
Secrest, ibid. 51,
421
(1969).
3
See, e.g., M. Attermeyer
and
R.
A.
Marcus, J. Chern. Phys.
52,
393
(1970);
A.
0.
Cohen
and
R.
A.
Marcus, ibid. 52, 3140
(1970); see also comparison
of
classical
and
quantum
results in
C. C.
Rankin
and
J.
C.
Light, ibid. 51,
1701
(1969)
and
R.
Russell and
J.
C.
Light, ibid. 1720 (1969).
4
Exact numerical
quantum
calculations for inelastic energy
transfer using smooth potential energy surfaces
and
several
open channels
(e.g.,j=O,
±2)
are given
by
B. R. Johnson
and
D.
Secrest,
J.
Math.
Phys. 7,
2187
(1966);
A.
C. Allison
and
A.
Dalgarno, Proc. Phys. Soc. (London) 90, 609 (1967): W. A.
Lester,
Jr.
and
R. B. Bernstein, Chern. Phys. Letters 1, 207,
347
(1967); B. R. Johnson,
D.
Secrest, W.
A.
Lester
and
R. B.
Bernstein,
ibid. 1, 396 (1967); B.
R.
Johnson
and
D. Secrest,
J.
Chern. Phys. 48, 4682 (1968); W. Erlewein, M. von Seggern,
and
J.
P. Toennies,
Z.
Physik 211,
35
(1968). For a recent rapid
exact
quantum
mechanical method for this energy transfer,
competitive
in
time
(at
low
quantum
numbers) with the classical
one, see R.
G.
Gordon,
J.
Chern. Phys. 51,
14
(1969);
A.
S.
Cheung
and
D.
J.
Wilson, ibid. 51, 3448, 4733 (1969).
5
This method may be contrasted with others. Surveys
of
other methods are given in (a) R. Bellman, Perturbation
Downloaded 08 Mar 2006 to 131.215.225.174. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
PERTURBATION
THEORY
FOR
CLASSICAL
MECHANICS
4807
Techniques
in
Mathematics, Physics and Engineering (Holt,
Rinehart
and
Winston, Inc., New York, 1964); (b) W. F. Ames,
Nonlinear Ordinary Differential Equations
in
Transport Processes
(Academic Press Inc., New York, 1968); (c) Differential Equa-
tions and Dynamical Systems,
J.
K.
Hale
and
J.
P. LaSalle, Eds.
(Academic Press Inc., New York, 1967).
6
R. A. Marcus,
J.
Chern. Phys. (to be published).
7 Compare A.
0.
Cohen
and
R. A. Marcus,
J.
Chern. Phys.
49, 4509 (1968).
8 E.g., W. Grobner, Die Lie-Reihen und Ihre Anwendungen
(VEE Deutscher Verlag der Wissenschaften, Berlin, 1967), 2nd ed.
(autonomous systems);
cf. K.
T.
Chen, Arch.
Rat.
Mech. Anal.
13, 348 (1963), which came to our attention after submission of
the
present article.
It
contains a quite different derivation of (11),
lengthier
than
the
present though self-contained.
9
W. Magnus, Commun. Pure Appl.
Math.
7, 649 (1954).
1
° For example when (1) represents a system of equations
of
classical mechanical motion for a collision
of
two particles,
h;
1
reflects their interaction.
In
a conservative system neither
h;
0
nor h;
1
depend explicitly on time
t,
but
nevertheless
h;
1
vanishes
as
t--+±
oo, because in
the
usual collision problems the particles
are
then
far
apart.
(1
0
is chosen to be
any
arbitrary
time in
the
region where
the
initial interaction of
the
particles is negligible.)
11 W. Grobner, Ref. 8, p.
17.
12
In
a related problem, coefficients of the
Baker-Campbell-
Hausdorff series
have
been calculated to a high order by computer
[R.
D.
Richtmyer
and
S. Greenspan, Commun.
Pure
Appl.
Math.
18, 107 (1965)
.]
Applications
of
Magnus' solution are
given in
D.
W. Robinson, Helv. Phys. Acta 36, 140 (1963); P.
Pechukas
and
J.
C. Light,
J.
Chern. Phys. 44, 3897 (1966);
S.
Chan,
J.
C. Light
and
J.
Lin, ibid. 49, 86 (1968); E.
H.
Wich-
mann,
J.
Math.
Phys. 2, 876 (1961); R. M. Wilcox, ibid.
8,
962 (1967)
and
references cited therein; M. Lutzky, ibid.
9,
1125 (1968). Sometimes
the
third
term
of
the
series is written
more symmetrically,
but
the
present form, due
to
Magnus,
emphasizes
that
the series terminates if
Au
and
commute.
13
E.
H.
Abate
and
F. Hofelich,
Z.
Physik 209,
13
(1968).
14
(a) L.
M.
Garrido, Proc. Phys. Soc. (London) 76,
33
(1960);
J.
Math.
Anal. Appl.
3,
295
(1961); (b) L.
M.
Garrido and
F.
Gascon, Proc. Roy. Soc. (London) 81, 1115 (1963).
15
See, e.g.,
K.
T.
Chen, Ref. 8;
J.
Diff. Equations 2, 438 (1966);
for classical mechanics see L.
M.
Garrido
and
F. Gascon, Ref.
14,
Eq.
12, a result discussed later.
THE
JOURNAL
OF
CHEMICAL
PHYSICS
16
(a) For example
E.
Merzbacher, Quantum Mechanics (John
Wiley
& Sons, Inc., New York, 1961), p. 464. (b) Thus, as seen
from (21)
D now becomes for
the
system (20),
the
"Liouville
operator," defined, for example,
in
R. W. Zwanzig, in Lectures
in
Theoretical Physics, W. E. Brittin, B. W. Downs
and
J.
Downs,
Eds. (Interscience Publishers, Inc., New York, 1961), p. 107.
1
7
(a) See, e.g.,
H.
C.
Corben
and
P. Stehle, Classical Mechanics
(John Wiley & Sons, Inc., New York, 1960), 2nd ed., pp. 178,
184; (b) E. W. Brown
and
C.
A.
Shook, Planetary Theory (Cam-
bridge University Press, London, 1933), p. 125.
18 Reference
17
(a), p. 221.
1
9
See, e.g.,
T.
E. Sterne,
An
Introduction
to
Celestial Mechanics
(Interscience Publishers, Inc., New York, 1960), p. 100ff;
S.
W.
Groesberg,
Advanced Mechanics (John Wiley & Sons, Inc., New
York, 1968), p. 306ff.
2
0
Since
the
c;
0
are functions of the q
0
's
and
p
0
's, (24)
and
(26)
yield
c;
equal to exp
!B,
)c;
0
•
Finally,
the
chain rule for dif-
ferentiation converts
lB, l to
~k.ilck
0
,
c;
0
)(aBjack
0
)(a/c;
0
).
21
Nonautonomous systems were considered by adjoining t to
the
set
of dependent variables. [See, however,
G.
R.
Sell, in Ref.
5(c),
p. 531.]
In
our case
the
use of this device would have
destroyed
the
similarity
of
classical
and
(the
customary)
quantum
equations used elsewhere in applications.
22
See, e.g., R. Hermann, Differential Geometry and
the
Calculus
of
Variations (Academic Press Inc., New York, 1968), Chap.
6; E. L. Ince,
Ordinary Differential Equations (Dover Pub-
lications, Inc., New York, 1956), Chap. IV.
2
3
B.
0.
Koopman, Proc.
Nat.
Acad. Sci. 17,
315
(1931); cf.
J.
von Neumann, Ann.
Math.
33,
587
(1932);
E.
H.~
Wichmann,
Ref. 12.
"Reference
14(b),
Eq.
(5).
25
Compare assumption
of
Eqs. (6)
and
(7) in Ref. 14(b)
whose analog in our case would be our ansatz (6), leading to
the automorphism represented by our (9).
26
Analogy was made by comparison
of
Eq.
(3) of Ref. 14(b)
with the "operator equation" there,
Eq.
(2).
27
See also Ref. 6, which treats
what
we
have termed there
the
"interaction"
and
"mixed-interaction" pictures (or representa-
tions) in classical mechanics.
In
Ref.
14
the first is used, while
in
the
present paper
the
second is employed,
it
being
the
standard
one in classical mechanics.
Both
pictures are treated
and
compared
in Ref.
6.
2
8
F.
J.
Zeleznik,
J.
Chern. Phys. 47, 3410 (1967).
2
9 Reference 5 (a), p.
57.
30
See, e.g.,
D.
Ter
Haar,
Elements of Hamiltonian Mechanics
(North-Holland Publishing Company, Amsterdam, 1961), p. 153;
Ref.
17
(a), p.
251ff.
VOLUME
52,
NUMBER
9 1
MAY
1970
Calculation of Transition Probabilities for Collinear Atom-Diatom and Diatom-Diatom
Collisions with Lennard-] ones Interaction
VINCENT
P.
GuTsCHICK*
A)ID
VINCENT
McKoY
Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, t Pasadena, California 91109
AND
DENNIS
J.
DIESTLER
Department of Chemistry, University
of
Missouri,t St. Louis, Missouri 63121
(Received
10
March
1969)
Numerical integration of
the
close coupled scattering equations is performed
to
obtain vibrational transi-
tion probabilities for three models of
the
electronically adiabatic H
2
-H
2
collision. All three models use a
Leonard-Jones interaction potential between the nearest atoms in the collision partners.
The
results are
analyzed for some insight
into
the
vibrational excitation process, including the effects of anharmonicities
in
the
molecular vibration
and
of
the
internal structure (or lack
of
it) in one
of
the
molecules. Conclusions
are drawn on
the
value
of
similar model calculations. Among them is
the
conclusion
that
the
replacement
of earlier and simpler models
of
the
interaction potential
by
the
Leonard-Jones potential adds very little
realism for all
the
complication
it
introduces.
INTRODUCTION
There is current interest in quantum-mechanical
treatments of molecular collisions involving excitation
of
internal degrees
of
freedom and possibly reaction.
The
collision systems pose a multichannel scattering
problem, commonly solved
by
the coupled channels
(CC) method.
The
CC equations are coupled differen-