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On the reflection method and the impulse approximation
Volume 42,
CHEMICAL
number I
PHYSICS
LETTERS
1.5August 1976
ON THE REFLECTION METIiOI.3 AND THE IMPULSE APPROXIMATION Jean Pierre LAPLANTE” Dt!partement
de Chimie,
and Andre D. BANDRAUK
UniJersitC de Sherbmake.
Sherbrooke,
Qutbec.
Canada
Received 20 April 1976 Reused manuscript zecelved 17 May 1976
Usmg the momentum representation of the Airy function, a new derivation of the reflection method is presented and its Ielatlon to the impulse approxlmatlon of scattenng theory is demonstrated.
I _ Introduction Transitions to or from continua involve calculations of overlap integrals between bound state wavefunctions and rapidly oscillating continuum functions. The calculation is facihtated by replacing the continuum functiorr by 2 8 functlor. as for instance in resonant Raman scattering [ 11, line broadening [2] and calculations of Franck-Condon factors, [3,4]. Recentiy, Gislason [4] has examined the validity of the reflection method and extended the method by expanang the bound state wavefunction around the classical turning point Xc of the continuum state in position space (fig. 1). We wiil show that his resulr can be simply obtained by working in momentum space. Furthermore, this latter representation’permits us to connect the reflection method to the well known impulse approxunation of scattering theory [S] as for instance applied by Dub6 and Herzenberg in e--N20 scattering [6]. We will show that the reflection methcd of spectroscopy correspond to neglect of the kinetic energy of the nuclei as does the impulse approximation in electronmolecule scattering. The corrections to these methods are easily derived from an infinite expansion of the Airy functions in momentum space.
* Present address- Service de Chlmie Physique 2, Universirt5 Llbre. Bruxelles, Belgium
184
I ’V, I
= FCC X-
Xc1
Fig. I. Potentials and functions IXIX-space.
2.
Derivation
We consider the potential system illustrated in fig. 1 where we have a potential V,(x), supporting bound states with functions Q,(X), intersected by a linear continuum potential V,(X) with function @c(X). This intersection occurs usually in predissociation [7] whereas in electronic transitions the two potentials are displaced in energy so that the intersection does not occur. It is to be noticed that Franck-Condon factors are therefore the same for the two processes as these factors involve overlap of d,(x) and #,(X) in X or position space, where X is the internuclear coordi-
CHEMICAL PHYSICS LETTERS
Volume 42, number 1
nate for a particular vibration linear potential wavefunctions Airy functions [8,9],
15 August 1076
In fact, the momentum Airy functions Ai(g) are solutions of the momentum SchrGdinger equatron [8 3 ,
in a molecule. The are the well known
(7)
m
&c(x) =
J exp[3p3 a 25rlF,P2 -03
- i&X
-Xc)]
dg, 0)
and Iri(p) = (~z~F,I)-~/~
exp[ip3/6@ - iEpl(FJ 1.
@)
where Xc corresponds to the classical turning point, the parameter a equals (2flIF,])1/3 with p the reduced mass of the system. A general Franck-Condon overlap integral is denoted by 4,
Neglect of the kinetic energy corresponds to an application of the impulse approximation [6]. Hence in our model, this corresponds to setting p2 = 0 in eq. (7) or
4 = (@,(x)1&(x))
exp(ip3
= @,(P)l@&))
s
1
=
-
2rrjFcjl”
_m
#r(p) exp[$(pln)3
(2) + *Xc]
dP- (3)
Eq. (3) corresponds to the momentum representation of the overlap and is obtained by integrating out the coordinate X after introducing the following integrai representation, @r(x) = (2~)-~1~
7 &CD) exp(ipX)dp. -01
(4)
The momentum form of this integral is very transparent as one immediately discerns the origin of the reflection method from eqs. (1) and (3) Thus expanding exp [&$z)3] in a Taylor series, one obtains, 4=
1
5
iPn ax:
(--lY
OJ I$&) elPXc dp. s -m
Hence the overlap integral 4 may be written 4 = IF’@2
5
n=O
(-‘r (3Q3)+2!
=
1.
(9)
Thus the term n = 0 in eq. (6) is nothing but the impulse approximation from a scattering vrew pomt and equivalently the reflection method in spectroscopy. It is to be noticed that relation (9) indeed transforms the continuum wavefunction r&(X), eq. (I), 6(X- Xc), i.e. into localized nuclei. Furthermore, since a = p1j3, we notice that this approximation is excellent for heavy or classical particles. The formal justification of our impulse approximatlon* crimes from the expansion of the Green’s operator (_Kf V- E‘) in powers of the kinetic energy K = p2/2P, i.e., (K+ V-E)--1 f
(V-q-1
= (V--E)-1
K(V---E)-1 + .. .
This operator is important for resonant processes such as in electron-molecule scattering [6]. Resonance Raman scattering, i.e., photon-molecule scattering, is formally the same process. For this latter, the reflection method has been used to calculate cross sections [l] . We have shown above that the impulse approximation used in scattering theory gives the same result for Franck-Condon factors as does the reflection method. Hence for resonant prclcesses, the first term in the expansion (10) of the Green’s function, I.e.,
2njF,I U2 n=O (3a3)%! x
/3Q3)
as (6)
which is the result obtained by Gislason [4] who expanded the bound state wavefunction $+(X) about the turning point X, and did all integrals in X space. Our method demonstrates clearly that only 3nth derivatives contribute to corrections as they come from the exp(ip/@)3 factor in the momentum representation of the Airy function.
G = [E-
V(X)]--’
(11)
corresponds to the impulse approximation and tberefore also the reflection method for such processes. it is to be observed that again this is an excellent approxithank the referee for emphasizing of this approximation [S ,6]. However the same.
* We
the drfferent versions the physical idea IS
Volurn~ 42, number 1
CHEMICAL PHYSICS LETTERS
mation for systems with large reduced masses, since eq. (10) is an expansion ifl terms of y-l. Corrections to both the impulse and reflection method are there= fore the same as they arise from the neglect of the kinetic energy as in nonadiabatic corrections. The latter have been succenftiy treated in the momentum representation [7,10]. The present work illustrates once more the advantage of wor_tig in the momentum representation when one is dealing with nuclear kinetic energy effects. As a final remark, we must emphasize that the impulse approximation is based on the physicaI idea that the scattering process is short compared to the periods of internal motion of the target [S]. The neglect of nuclear motion whrch follows from this approximation, or equivalentIy the reflection method, should be acceptable for resonant continuous states (zero frequency) but may not be adequate for resonant bound states with high vibrational frequencies. in this latter case, the exact Franck-Condon factors will have to be used.
15 August
1976
References [l] M. Mingardi and W. Siebrand, Chem. Phys. Letters 23 (1973) 1; J. Chem. Phys. 62 (1975) 1074. [21 F-H Mies and AL. Smith, J. Chem. Phys. 45 (1966) 994. [3] I. Riess, I. Chem. Phys. 56 (1972) 1613. [41 E-A. Gis%son. 3. Chem. Phys. 58 (1973) 3702. [S] C F. Chew and M.L. Goldberger, Phys. Rev. 87 (1952) 718; M.L. Goldberger and KM. Watson. Collision theory (wrley, New York, 1964). [61 L. Dub6 and A Herzenberg, Phys. Rev. 11A (1975) 1314. 171 A D. Bandrauk and JP. Laplante, to be published; A D. Bandrauk and KS. Child, Mol. Phys. 19 (1970) 95. 181 L.D Landau and E.M. Lifshitz, Non-relativistic quantum mechamcs (Addaon-Wesley, Reading, 1958), sectron 87. [9] M. Abramowrtz and LA. Stegun, Handbook of mathematical functrons (Dover, New York, 1965). [ 101 E-E. Nlkitin. ix Chemische Etementarprozesse, ed. H. Hartmann (Springer, Berlin, 1968).
CHEMICAL
number I
PHYSICS
LETTERS
1.5August 1976
ON THE REFLECTION METIiOI.3 AND THE IMPULSE APPROXIMATION Jean Pierre LAPLANTE” Dt!partement
de Chimie,
and Andre D. BANDRAUK
UniJersitC de Sherbmake.
Sherbrooke,
Qutbec.
Canada
Received 20 April 1976 Reused manuscript zecelved 17 May 1976
Usmg the momentum representation of the Airy function, a new derivation of the reflection method is presented and its Ielatlon to the impulse approxlmatlon of scattenng theory is demonstrated.
I _ Introduction Transitions to or from continua involve calculations of overlap integrals between bound state wavefunctions and rapidly oscillating continuum functions. The calculation is facihtated by replacing the continuum functiorr by 2 8 functlor. as for instance in resonant Raman scattering [ 11, line broadening [2] and calculations of Franck-Condon factors, [3,4]. Recentiy, Gislason [4] has examined the validity of the reflection method and extended the method by expanang the bound state wavefunction around the classical turning point Xc of the continuum state in position space (fig. 1). We wiil show that his resulr can be simply obtained by working in momentum space. Furthermore, this latter representation’permits us to connect the reflection method to the well known impulse approxunation of scattering theory [S] as for instance applied by Dub6 and Herzenberg in e--N20 scattering [6]. We will show that the reflection methcd of spectroscopy correspond to neglect of the kinetic energy of the nuclei as does the impulse approximation in electronmolecule scattering. The corrections to these methods are easily derived from an infinite expansion of the Airy functions in momentum space.
* Present address- Service de Chlmie Physique 2, Universirt5 Llbre. Bruxelles, Belgium
184
I ’V, I
= FCC X-
Xc1
Fig. I. Potentials and functions IXIX-space.
2.
Derivation
We consider the potential system illustrated in fig. 1 where we have a potential V,(x), supporting bound states with functions Q,(X), intersected by a linear continuum potential V,(X) with function @c(X). This intersection occurs usually in predissociation [7] whereas in electronic transitions the two potentials are displaced in energy so that the intersection does not occur. It is to be noticed that Franck-Condon factors are therefore the same for the two processes as these factors involve overlap of d,(x) and #,(X) in X or position space, where X is the internuclear coordi-
CHEMICAL PHYSICS LETTERS
Volume 42, number 1
nate for a particular vibration linear potential wavefunctions Airy functions [8,9],
15 August 1076
In fact, the momentum Airy functions Ai(g) are solutions of the momentum SchrGdinger equatron [8 3 ,
in a molecule. The are the well known
(7)
m
&c(x) =
J exp[3p3 a 25rlF,P2 -03
- i&X
-Xc)]
dg, 0)
and Iri(p) = (~z~F,I)-~/~
exp[ip3/6@ - iEpl(FJ 1.
@)
where Xc corresponds to the classical turning point, the parameter a equals (2flIF,])1/3 with p the reduced mass of the system. A general Franck-Condon overlap integral is denoted by 4,
Neglect of the kinetic energy corresponds to an application of the impulse approximation [6]. Hence in our model, this corresponds to setting p2 = 0 in eq. (7) or
4 = (@,(x)1&(x))
exp(ip3
= @,(P)l@&))
s
1
=
-
2rrjFcjl”
_m
#r(p) exp[$(pln)3
(2) + *Xc]
dP- (3)
Eq. (3) corresponds to the momentum representation of the overlap and is obtained by integrating out the coordinate X after introducing the following integrai representation, @r(x) = (2~)-~1~
7 &CD) exp(ipX)dp. -01
(4)
The momentum form of this integral is very transparent as one immediately discerns the origin of the reflection method from eqs. (1) and (3) Thus expanding exp [&$z)3] in a Taylor series, one obtains, 4=
1
5
iPn ax:
(--lY
OJ I$&) elPXc dp. s -m
Hence the overlap integral 4 may be written 4 = IF’@2
5
n=O
(-‘r (3Q3)+2!
=
1.
(9)
Thus the term n = 0 in eq. (6) is nothing but the impulse approximation from a scattering vrew pomt and equivalently the reflection method in spectroscopy. It is to be noticed that relation (9) indeed transforms the continuum wavefunction r&(X), eq. (I), 6(X- Xc), i.e. into localized nuclei. Furthermore, since a = p1j3, we notice that this approximation is excellent for heavy or classical particles. The formal justification of our impulse approximatlon* crimes from the expansion of the Green’s operator (_Kf V- E‘) in powers of the kinetic energy K = p2/2P, i.e., (K+ V-E)--1 f
(V-q-1
= (V--E)-1
K(V---E)-1 + .. .
This operator is important for resonant processes such as in electron-molecule scattering [6]. Resonance Raman scattering, i.e., photon-molecule scattering, is formally the same process. For this latter, the reflection method has been used to calculate cross sections [l] . We have shown above that the impulse approximation used in scattering theory gives the same result for Franck-Condon factors as does the reflection method. Hence for resonant prclcesses, the first term in the expansion (10) of the Green’s function, I.e.,
2njF,I U2 n=O (3a3)%! x
/3Q3)
as (6)
which is the result obtained by Gislason [4] who expanded the bound state wavefunction $+(X) about the turning point X, and did all integrals in X space. Our method demonstrates clearly that only 3nth derivatives contribute to corrections as they come from the exp(ip/@)3 factor in the momentum representation of the Airy function.
G = [E-
V(X)]--’
(11)
corresponds to the impulse approximation and tberefore also the reflection method for such processes. it is to be observed that again this is an excellent approxithank the referee for emphasizing of this approximation [S ,6]. However the same.
* We
the drfferent versions the physical idea IS
Volurn~ 42, number 1
CHEMICAL PHYSICS LETTERS
mation for systems with large reduced masses, since eq. (10) is an expansion ifl terms of y-l. Corrections to both the impulse and reflection method are there= fore the same as they arise from the neglect of the kinetic energy as in nonadiabatic corrections. The latter have been succenftiy treated in the momentum representation [7,10]. The present work illustrates once more the advantage of wor_tig in the momentum representation when one is dealing with nuclear kinetic energy effects. As a final remark, we must emphasize that the impulse approximation is based on the physicaI idea that the scattering process is short compared to the periods of internal motion of the target [S]. The neglect of nuclear motion whrch follows from this approximation, or equivalentIy the reflection method, should be acceptable for resonant continuous states (zero frequency) but may not be adequate for resonant bound states with high vibrational frequencies. in this latter case, the exact Franck-Condon factors will have to be used.
15 August
1976
References [l] M. Mingardi and W. Siebrand, Chem. Phys. Letters 23 (1973) 1; J. Chem. Phys. 62 (1975) 1074. [21 F-H Mies and AL. Smith, J. Chem. Phys. 45 (1966) 994. [3] I. Riess, I. Chem. Phys. 56 (1972) 1613. [41 E-A. Gis%son. 3. Chem. Phys. 58 (1973) 3702. [S] C F. Chew and M.L. Goldberger, Phys. Rev. 87 (1952) 718; M.L. Goldberger and KM. Watson. Collision theory (wrley, New York, 1964). [61 L. Dub6 and A Herzenberg, Phys. Rev. 11A (1975) 1314. 171 A D. Bandrauk and JP. Laplante, to be published; A D. Bandrauk and KS. Child, Mol. Phys. 19 (1970) 95. 181 L.D Landau and E.M. Lifshitz, Non-relativistic quantum mechamcs (Addaon-Wesley, Reading, 1958), sectron 87. [9] M. Abramowrtz and LA. Stegun, Handbook of mathematical functrons (Dover, New York, 1965). [ 101 E-E. Nlkitin. ix Chemische Etementarprozesse, ed. H. Hartmann (Springer, Berlin, 1968).