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Rotational excitation in molecular collisions: A strong coupling approximation
Volume 6. number
1
CWEMCAL
PHYSlCS LKTTERS
.
; suly-~97o
_ .’
-.
..
ROTATIOiAL
EXCIiATICN
A STRONG
COUPLING
iN
MOLECULAR
+
.
COLLIiIONS:
AP?ROXIMATfON
*
THOMA# P. TSXEN 2nd RUSSEL T. PACK
Depatiment
of Chemisiry . Sm’gttarn Young Unive,ersity, Prom, Utah 84601, USA f Eeccived 21 May 1970
A genera&&on of Takayanagils fully auantummechanical strong coupling approximation for rotationally inelastic collisior,s. valid for small energy exchange,. is discussed and tested numericaILy on ft.model problem. The m&hoc? is co~~u~~on~tly very simple, and the results are most encouraging.
In this letter we present and test a strong coupling approximation, valid far Small energy exchange, for rotational excitation in molecAr collisions. The approximation was suggested by Takayanagi [II] and is a generalization of the “exact resonance” method [2,3], It is a type of quantum mechanical sudden f4j approximation very similar to that used by Curtiss [!3]. But. it . does not seem to have been tested numerically, In molecular scattering prublems involving rotational excitation of heavy diatomic molecules, one must solve sets of strongly coupled eauations
k& = (2ghi2~ (E- &>, E is the total energy, c5Tm is the energy of interi& state m, @ is the reduced mass of the collldfng molecules, Im labels the relative orbital ang&ar momentum, and the Y = cWf2) (m Iu bL)are matrix elements of ~~~aterrnole~~l~ potential over the internal states, The 22~ Fe eiementa of the scattering matrix, and jSmi12 is the probability of a tr;G7sition from state The numerical
i to state m. solution of eqs.
To obtain simple approximate
(3) is tedious. solutions we first
write the set of coupled eqs. (1) in matrix form as [K+ E --V]G = 0, where G = (Gm) is a column vector, Xrz
(3) =
rr,, = &,&kg - 2,(1,+1)/i?+, and Vmn was defined above. Now let 0 = U where F! is an orthogonal transformation matr% _. in the adiabatic a proximation one chooses U = C 02) 8uch that ep(E - V) C is diagonal for all 12. in the 3,,d%‘dR2,
subject to the boundary conditions $#l,
= 0,
uncoupled and usual distorted wave approxima-
and
tions one.chooses U to be the unit matrix, U= .f - G j-), which makes K + E-diagonal,
- %i
Mere f is
the
&&[i(kJ2 - $-2m~lJ)*
initial (incomingf state.
Here
* &search supported in part by &ants from Re---, search Corporatick and the S3righamYoung Univer- .. sity E&ear& Fund. ** See ref. [6] for the coupled equations of cur example problem..
54
Here we choos_e the other ‘lix#t f , 0 = c (0); which xn@es U(K- V) Udiagohat (at least for smaiS R). Sicce this U is independent of R, kg. (3) becomes
Volume 6, number 1
CHEMICAL PHYSICS LETTERS
Neglecting off-diagonal terms, we obtain zerothorder solutions gA”) (R) which satisfy the boundary conditions,
g(O)(O) =0 , n
(54
probabilities for the S-wave 0 =O) scattering of an atom by a homonuclear rigid rotor. The model potential chosen was a Lennard-Jones (12,6) potential with a short range fi(cos6) anisotropy, so that vm, = 4E[ (@I2
g”)(R)
=
?2
K-~‘~ Sin(K
A
R-~J
n
‘2n
R-+X
n’
-r+
%I’
(5b)
where K~ and xtt come from (6 E iJ), and represent wavenumbers and angulpr momenta, respectively. However, to have G (0) = Ug(O) satisfy (2b) requires that Kn = Ki and X, = Xi for all ?z. This requires a further approximation which ca? be made either by replacing k, by ki and 5 by Zi in E for all states n_coupled to the jzitial state i, or by replacing (UE U), by (fjE U)ii in (4) and also k- by Ki and 4 by Ai in (2b). Then from g(O) = GGfo) and the known amplitude of the incoming A,
wave one easily
shows
= -2iUin exp(io,J.
This determines the zero&-order
that
3
(V
and
This approximation thus gives a simple, exactly unitary scattering matrix directly *. Computations are particularly simple: U is independent of R, so that E - V must on1.y be diagonalized once. For energies high enough that the approsimation is justified, the phase shift differences are easily calculated.using the WTGl approximatiOll. If only energy-averaged cross sections are needed, it may often be possible to make a random phase approximation in (8) to obtain
This result is cross sections inant coupling To test eq.
- ($)6]6,n
similar to but should give better than the usual statistical or domapproximation 17-J. (8) we calculated some transition
* In the adiabatic ,and distorted wave appro&ations bne must go to first order and evaluate another in.tegral to obtain a nolr-r,erb approximation to the transition probability. .-
c afintpE($12.
(10)
This model has been used recently by several authors, who have explained the parameters involved [8,9]. The parameters (see, e.g., ref. [iO]) were chosen to simulate a-room temperature He-N2 collision: u = 3.164A, E/A = 25.11°K and E/k = 300°K. This gave as reduced parameters B= 2/.&/1$
= 36.29,
E’ = 21;~2E/fi2 = 433.6, and E$ P 21102&j/2i2 = (~$/i)j(jtl)
0%
the outgoing waves and gives as scattering matrix elements,
s mi = $J umnuin exp@iVJ
1 July 1970
= 4.16lj(jtl).
To slmpiify calculation of the exact results Gsed for comparison, all the computxtions were restricted to the two-state approximation. Many more than two channels are open here and strongly ccupled, but comparison of the two-state strong coupling results with the exact two-stare results should show whether this approximation handles the coupled differential equations properly. The two states chosen are 1 = u, I) = (0,O) and 2 = (i’, I’) = (2,2). In this special case one has [7] fll = 0, f22 = 2/7, andf21 =fl2 = 5-l12. In this approximation eq. (8) becomes
w Transition probabilities lS2I I2 calculated in several approximations are plotted V~PSU.Sthe asymmetry parameter cLin fig. 1. It is seen drat the usual distorted-wave (DW) approximation [3j gives good results for small ICZIbut becomes ridiculously large Sor large i ~1. The adiabatic (-40) approximation [2,1X] **, which works well at low (near threshold) energies does rather poorly at all a values in this case. But the strongcoupling (SC) approximation, which is computationaly about ten times faster than either the DW or the AD approximation, gives surprisingly good results for this problem for all values cf c.
** See particularlyeq. @8) of ref. [Ill. This is a better approximation thsn &at of ref. [9]. -In these computatLons.theadiabatic corrections to the potential, (d8/dR)L, change the results negligibly. 55
.:. Volume. 6, number 1
CHEMICALPHYSICSLETTERS
,’
13uly 1970
%k thank Dr. Richard i, Snow for helpfut discussioiLs and Mr.-Wesley D. Smith for prep@+ _the figures. REFERENCES
Fig. 1. Dependenco of the transition probability fS2j ’ on the asymmetry parametw Q for a room temperature He-NZ collMon. The solid tinegives the exact two&ate results: tie dotted tine is the present strong couoIiug (SC) approximation; the dashed iine is the usuat digtortedwave (Dm approximation; and the dot-dash tine is the adfahatcc (AD) approximation. . (it will not work. so well wt?cnkrg& energy exchange is involved.) Preliminary results af many-state strong owpling ctilculaticns curently in process are encotiraging nnd wi!! be reported later.
;5s
[! ] K. Takayaaagi, Fwgr. Theoret, Phy& (Kyoto) %ppl. 25 (1963) 40. ;2J F. London, 2. Physik 74 (l932) 143. [3J N. F. Mott and H. 5, W. Massey, The theory of ~&XXic collieions, 3rd ed. (Oxford Univ. Press, London, 1965) p. 349. [4J R. D. Levine, Chem. Phys, Letters 4 (1969) 211. [5j C. F. Curtiss, J. Chem.Phys. 49 (1968) 1952; 52 (1970) 1078; University of Wisconsin Theoretical Chemistry KnstituteReport WIS-TCI-370 (1969)[SJ A. M. Arthurs and A. Dalgarno, Prcc. Roy. Sot. A256 (l960) 546. [?] 5~.B. Bernstein, A. Dalgarno, Ei.S. W.Massey and 1. C. Percival, Pro& Roy. Sac. A274 (1963) 427. [Sj R. D. Levine, 3. R.JoEmson, J.T. Mucke~an and R. B. Bernstein, Chem. Phys. Letters 1 g9SE) 617; J. Chem. Phys. 49 (lSS8) 56. JQjR. D. Levine, M. Shapiro, 3. T. Muckerman and B.R. Johnson, Chem. Phys. Letters 2 (l968) 545: B. R.Johnson am R. D. Levine, J. Chem. Phys, 50 (1969) 1902. [ZOJJ.0. Hirschfetder, C. F. Curtisa and R. B. Bird. MoIecuhr theory of gases and Uquids (Wi!ey, New York. 1954). [LX] R. D. Levine, M. Shapiro-and B. 11.Johnson, 3. Chem. Phys. 52 (l970) 1755.
1
CWEMCAL
PHYSlCS LKTTERS
.
; suly-~97o
_ .’
-.
..
ROTATIOiAL
EXCIiATICN
A STRONG
COUPLING
iN
MOLECULAR
+
.
COLLIiIONS:
AP?ROXIMATfON
*
THOMA# P. TSXEN 2nd RUSSEL T. PACK
Depatiment
of Chemisiry . Sm’gttarn Young Unive,ersity, Prom, Utah 84601, USA f Eeccived 21 May 1970
A genera&&on of Takayanagils fully auantummechanical strong coupling approximation for rotationally inelastic collisior,s. valid for small energy exchange,. is discussed and tested numericaILy on ft.model problem. The m&hoc? is co~~u~~on~tly very simple, and the results are most encouraging.
In this letter we present and test a strong coupling approximation, valid far Small energy exchange, for rotational excitation in molecAr collisions. The approximation was suggested by Takayanagi [II] and is a generalization of the “exact resonance” method [2,3], It is a type of quantum mechanical sudden f4j approximation very similar to that used by Curtiss [!3]. But. it . does not seem to have been tested numerically, In molecular scattering prublems involving rotational excitation of heavy diatomic molecules, one must solve sets of strongly coupled eauations
k& = (2ghi2~ (E- &>, E is the total energy, c5Tm is the energy of interi& state m, @ is the reduced mass of the collldfng molecules, Im labels the relative orbital ang&ar momentum, and the Y = cWf2) (m Iu bL)are matrix elements of ~~~aterrnole~~l~ potential over the internal states, The 22~ Fe eiementa of the scattering matrix, and jSmi12 is the probability of a tr;G7sition from state The numerical
i to state m. solution of eqs.
To obtain simple approximate
(3) is tedious. solutions we first
write the set of coupled eqs. (1) in matrix form as [K+ E --V]G = 0, where G = (Gm) is a column vector, Xrz
(3) =
rr,, = &,&kg - 2,(1,+1)/i?+, and Vmn was defined above. Now let 0 = U where F! is an orthogonal transformation matr% _. in the adiabatic a proximation one chooses U = C 02) 8uch that ep(E - V) C is diagonal for all 12. in the 3,,d%‘dR2,
subject to the boundary conditions $#l,
= 0,
uncoupled and usual distorted wave approxima-
and
tions one.chooses U to be the unit matrix, U= .f - G j-), which makes K + E-diagonal,
- %i
Mere f is
the
&&[i(kJ2 - $-2m~lJ)*
initial (incomingf state.
Here
* &search supported in part by &ants from Re---, search Corporatick and the S3righamYoung Univer- .. sity E&ear& Fund. ** See ref. [6] for the coupled equations of cur example problem..
54
Here we choos_e the other ‘lix#t f , 0 = c (0); which xn@es U(K- V) Udiagohat (at least for smaiS R). Sicce this U is independent of R, kg. (3) becomes
Volume 6, number 1
CHEMICAL PHYSICS LETTERS
Neglecting off-diagonal terms, we obtain zerothorder solutions gA”) (R) which satisfy the boundary conditions,
g(O)(O) =0 , n
(54
probabilities for the S-wave 0 =O) scattering of an atom by a homonuclear rigid rotor. The model potential chosen was a Lennard-Jones (12,6) potential with a short range fi(cos6) anisotropy, so that vm, = 4E[ (@I2
g”)(R)
=
?2
K-~‘~ Sin(K
A
R-~J
n
‘2n
R-+X
n’
-r+
%I’
(5b)
where K~ and xtt come from (6 E iJ), and represent wavenumbers and angulpr momenta, respectively. However, to have G (0) = Ug(O) satisfy (2b) requires that Kn = Ki and X, = Xi for all ?z. This requires a further approximation which ca? be made either by replacing k, by ki and 5 by Zi in E for all states n_coupled to the jzitial state i, or by replacing (UE U), by (fjE U)ii in (4) and also k- by Ki and 4 by Ai in (2b). Then from g(O) = GGfo) and the known amplitude of the incoming A,
wave one easily
shows
= -2iUin exp(io,J.
This determines the zero&-order
that
3
(V
and
This approximation thus gives a simple, exactly unitary scattering matrix directly *. Computations are particularly simple: U is independent of R, so that E - V must on1.y be diagonalized once. For energies high enough that the approsimation is justified, the phase shift differences are easily calculated.using the WTGl approximatiOll. If only energy-averaged cross sections are needed, it may often be possible to make a random phase approximation in (8) to obtain
This result is cross sections inant coupling To test eq.
- ($)6]6,n
similar to but should give better than the usual statistical or domapproximation 17-J. (8) we calculated some transition
* In the adiabatic ,and distorted wave appro&ations bne must go to first order and evaluate another in.tegral to obtain a nolr-r,erb approximation to the transition probability. .-
c afintpE($12.
(10)
This model has been used recently by several authors, who have explained the parameters involved [8,9]. The parameters (see, e.g., ref. [iO]) were chosen to simulate a-room temperature He-N2 collision: u = 3.164A, E/A = 25.11°K and E/k = 300°K. This gave as reduced parameters B= 2/.&/1$
= 36.29,
E’ = 21;~2E/fi2 = 433.6, and E$ P 21102&j/2i2 = (~$/i)j(jtl)
0%
the outgoing waves and gives as scattering matrix elements,
s mi = $J umnuin exp@iVJ
1 July 1970
= 4.16lj(jtl).
To slmpiify calculation of the exact results Gsed for comparison, all the computxtions were restricted to the two-state approximation. Many more than two channels are open here and strongly ccupled, but comparison of the two-state strong coupling results with the exact two-stare results should show whether this approximation handles the coupled differential equations properly. The two states chosen are 1 = u, I) = (0,O) and 2 = (i’, I’) = (2,2). In this special case one has [7] fll = 0, f22 = 2/7, andf21 =fl2 = 5-l12. In this approximation eq. (8) becomes
w Transition probabilities lS2I I2 calculated in several approximations are plotted V~PSU.Sthe asymmetry parameter cLin fig. 1. It is seen drat the usual distorted-wave (DW) approximation [3j gives good results for small ICZIbut becomes ridiculously large Sor large i ~1. The adiabatic (-40) approximation [2,1X] **, which works well at low (near threshold) energies does rather poorly at all a values in this case. But the strongcoupling (SC) approximation, which is computationaly about ten times faster than either the DW or the AD approximation, gives surprisingly good results for this problem for all values cf c.
** See particularlyeq. @8) of ref. [Ill. This is a better approximation thsn &at of ref. [9]. -In these computatLons.theadiabatic corrections to the potential, (d8/dR)L, change the results negligibly. 55
.:. Volume. 6, number 1
CHEMICALPHYSICSLETTERS
,’
13uly 1970
%k thank Dr. Richard i, Snow for helpfut discussioiLs and Mr.-Wesley D. Smith for prep@+ _the figures. REFERENCES
Fig. 1. Dependenco of the transition probability fS2j ’ on the asymmetry parametw Q for a room temperature He-NZ collMon. The solid tinegives the exact two&ate results: tie dotted tine is the present strong couoIiug (SC) approximation; the dashed iine is the usuat digtortedwave (Dm approximation; and the dot-dash tine is the adfahatcc (AD) approximation. . (it will not work. so well wt?cnkrg& energy exchange is involved.) Preliminary results af many-state strong owpling ctilculaticns curently in process are encotiraging nnd wi!! be reported later.
;5s
[! ] K. Takayaaagi, Fwgr. Theoret, Phy& (Kyoto) %ppl. 25 (1963) 40. ;2J F. London, 2. Physik 74 (l932) 143. [3J N. F. Mott and H. 5, W. Massey, The theory of ~&XXic collieions, 3rd ed. (Oxford Univ. Press, London, 1965) p. 349. [4J R. D. Levine, Chem. Phys, Letters 4 (1969) 211. [5j C. F. Curtiss, J. Chem.Phys. 49 (1968) 1952; 52 (1970) 1078; University of Wisconsin Theoretical Chemistry KnstituteReport WIS-TCI-370 (1969)[SJ A. M. Arthurs and A. Dalgarno, Prcc. Roy. Sot. A256 (l960) 546. [?] 5~.B. Bernstein, A. Dalgarno, Ei.S. W.Massey and 1. C. Percival, Pro& Roy. Sac. A274 (1963) 427. [Sj R. D. Levine, 3. R.JoEmson, J.T. Mucke~an and R. B. Bernstein, Chem. Phys. Letters 1 g9SE) 617; J. Chem. Phys. 49 (lSS8) 56. JQjR. D. Levine, M. Shapiro, 3. T. Muckerman and B.R. Johnson, Chem. Phys. Letters 2 (l968) 545: B. R.Johnson am R. D. Levine, J. Chem. Phys, 50 (1969) 1902. [ZOJJ.0. Hirschfetder, C. F. Curtisa and R. B. Bird. MoIecuhr theory of gases and Uquids (Wi!ey, New York. 1954). [LX] R. D. Levine, M. Shapiro-and B. 11.Johnson, 3. Chem. Phys. 52 (l970) 1755.