- Email: [email protected]
An algebraic approach to vibrational transitions in the forced Morse oscillator
23 December 1983
CHEMICAL PHYSICS LETTERS
Volume 103, number 2
AN ALGEBRAIC APPROACH TO VIBRATIONAL
TRANSITIONS
IN THE FORCED MORSE OSCILLATOR
Taikyue
REE
Department
of Chemistty. Korea Advanced
Institute of Science and Technology,
Seoul, Korea
and Y.H. KIM * and H.K. SHIN Department
of Chemistry.
University of Nevaab, Rena, Nevada 89557. USA
Received 14 September 1983; in foal form 5 October 1983
An expression for the probability of vibrational transitions in the Morse oscillator has been derived by use of the anhar-
monk commutation relations in an approximate solution of the SchrZidmgerequation of motion. The inclusion of the oscillator anharmonicity leads to a complicated probability expression, but in the weak-coupling limit the expression reduces to an anharmonic scaling relation of Pv+v+l /(IJ + 1) (1 - LJX#‘~_,~ = 1, where x0 is the anharmonicity parameter. For Hz + He. this relation appears to hold well up to collision energies of =I..5 vibrational quanta.
1. Introduction
The commutation relations of the harmonic oscillator are known to be useful in developing theoretical approaches for molecular energy transfer problems [l-8] _The usefulness is largely due to the simplicity of solving the Schredinger equation of motion and of calculating scattering operators of the interaction system in terms of the usual Boson creation and annihilation operators. Even the complicated properties of angular momentum eigenvectors have been shown to be readily handled in terms of these commutation relations [9]. The necessary condition for the use of these operators in formulating energy transfer probabilities, however, is that the vibrations of the colliding molecules be harmonic_ While it may be satisfactory for transitions among low-lying ener,v levels, such an assumption can lead to serious errors in studying transitions involving highly excited states due to the importance of the anharmonicity. The anharmonicity can be important even for low-lying levels in polyatomic molecules or in van der Waals molecules which often involve shallow potential energy wells. Recently, Levine [lo] has formulated the commutation relations for the anharmonic oscillator starting with the &son operators in the forms appropriate for energy transfer problems_ Application of these relations to the Morse oscillator state, which is sim-
ilar to the angular momentum eigenvector derived by Schwinger [9], leads to the eigenvalue expression which is accurate to the first anharmonicity correction term. In the present letter, using these commutation relations, we shall present an approximate procedure of developing the perturbed oscillator state and show its use in ener,v transfer studies.
2. Commutation relations For a pair of Boson creation
(~7:. c&) and annihilation
(a,,
a_&) operators,
the commutation
relations
are
* Present address: Department of Chemistry, Inha University, Inchon, Korea.
0 009-2614/83/S
(North-Holland
03.00 0 Elsevier Science Publishers Physics Publishing Division)
B.V.
149
CHEMICAL PHYSICS LETTERS
Volume 103,number2
[iz$fz;t] =[a,-,a;;]= 0
;
23 December 1983
[a,-,ug =6 UN -
(1)
The notations u and N are chosen to represent the vibr&ional quantum number and a~armon~city as shown in section 3. According to Levine [IO], the basic operators needed to describe the Morse oscillator can be formulated in terms of these operators as QO = (2N)-“2(a*x-u N+Qf;oU)* - af;a,)
PO = i(2.N)-‘/z(~~*~ I 0 =.N-‘(2
@I ,
(=I
o- - u+u-) vu
m)
NN
where N is related to-the ~~o~~ity parameter xo asN=x&l. It should be interesting to note the relationship between these and the corresponding harmonic oscillator operators. If we write the hamiltonian of the harmonic oscillator as H = Cp2 + M2u$q2)DM, the commutation_relation connecting the position variable q and the momentump is [q, p] = ifi. When we put q = @/MGJ&~Q and p = (j?fh~~)~I~P,k?= $ittwo(P2 + Q2). The commutation relation, then, is [Q, P] = i. From eqs. (Zb) and (2c), however, we find [Qu, Po] = iZo, which suggests that only in the harmoni? limit does Zu approach the identity operator I. In fact, we can show that there exists the harmonic counterpart to each operator relation obtained from Qo, Po, and lo_ From Q. and Po, we obtain creation (A+) and annihilation (A-) operators for the anharmonic Morse oscillator: A
.+=
2-3/2(Qu - ipo) = N-“%r;Lr;;
A-=2-
IZ2(Qo + iPo) = N-‘Z%;;fX;
,
(3a) *
WI
The commutation relations obtained for the Morse oscillator are then
[lo. PO] = --2ixgQo ,
(44
E&vQOJ= “iq)P()
t4b)
D
[A-. A+] =Z, , [lo&J
= r2xoA*
(4c) .
(44
Note that another set of operators, which can be ~oust~~ted from eq. (1) is
which are the characteristic algebraic relations of angular momentum operators. Utilizing these relations in accordance with the angular momentum properties, Dillon and Stephenson have formulated energy transfer probabilities [4].
3. PerturbedMorseusciUatorstate The most general form of the eigenvalues of the anharmo_mc oscillator referred to the lowest &ate as zero is often given by [ 11, pp_ 92,933 E, = fiwov - F~o~~v~ + fiogov3
+ _.__
(51
To calculate anharmonic eigenvalues, therefore, at least two parameters are needed; namely, the quantum number u and the leading anharmonicity parameter ~0. The latter p-eter is fsed for a given molecular system, but in fo~u~~g a form of the ~~0~~ oscillator state, it can be regarded as a variable. Following Levine’s work, we take these two parameters to be v and N = x-lo , and express the Ghsrrnonic Morse state by IN, VI, in which each parameter can be operated, independently, by the harmonic oscillator creation and antiilation operators, -.
150
CHEMICAL
Volume 103. number 2
23 December 1983
PHYSICS LETTERS
The a:, ai, a:, and ak, where subscripts u and N represent the vibrational quantum number and anharmonicity. state IfV, u> can be generated from the vacuum state IO) following the scheme that has been used to describe angular momenta in terms of the creation operators of a two-dimensional harmonic oscillator [9,10,12] : IN. u> = [u!(N - u)!]-‘~2(a~)u(a~)N-“10)
_
(6)
We now write the hamiltonian for the Morse oscillator action with an incident particle in the form F = :hwo(fi
+ Q;> +
@/~~o)~‘~~OQ~
or, in terms of the creation
force F(r) due to the inter-
W)
,
and annihilation
operators,
H = fiw&I+/l-
t ;l(J) t (r/ziVClI,)“2F(t)(A+
The SchrBdinger
equation
of motion
forced by a timedependent
t A--) =-Ho t H’(r) .
may then be written
W)
in the form
ih I+(t)> = [ff’ + H’(t)] W(r)) ,
(8)
where l*(t)> describes the Morse oscillator state disturbed by the perturbation linear combination of IN. u>. When we write the state vector in the form
I*(t)) = exp(-iHOt/fi)IQ(t)) the Schrtidinger_equation ilt 6(f)>
= H’(t)l@~(t))
H’(r) and can be obtained
,
as a
(9)
becomes
,
(10)
where H’(t) = exp(iHOf/?t)
H’(t) exp(-&r/h)
= (tl/2kf0~)‘/~F(f)
exp[io&4+d-
+ +lg)](A+
Using eqs. (4~) and (4d) we can convert H’(r) = @/2kk~#~F(t)
this expression
exp(ixow0t)[exp(io&&4*
By use of eq. (11) an approximate
solution
expfioO(IO
t F(t) exp[-iw&, J-cm
- xo)f]
dtA-
This
exp[--io&I+A-
+ ilo)]
_
to .
+ exp(-iw#&4-]
of eq. (10) can be written
F(t)
+
+A-)
+ x,-,)f]
(11)
as
dt A+
11 _
(12)
IQ(-00))
solution is in the form I@(t)> = V(t, -=)I@(-m)), and the exponential part is a time evolution operator transforming the initial state I@(-=)> to the state I@(t)) at a later time; thus it contains all the information needed to describe the interaction process at time t, in particular as t + +m. It does not appear possible, however, to use this form of the evolution operator in deriving transition probabilities because of the appearance of the operator I0 in the exponential part of the exponent. We note that it tends toward the identity operator I in the lit x0 --f 0. Using this limiting form, we approximate eq. (12) as @(t)>
=
exp[-iGl(r)A+
- iGa(t)A-]
where G1,2(t) = (2iE?~o)-~/~J~,F(t) entangled approximately as [13,14] exp (-i G, A+ - iG2A-)
= exp(=iG++)
I@(--)>, exp[kiiw,(l exp(-iG+-)
(13) +x&l
dt. The exponential exp(-$G1G210)
_
operator
in eq. (13) can be dis(14) 151
CHEMICAL
Volume 103. number 2 The evolution of the perturbed I@(t)> = exp(--i@+)
state from the initial state I@(--=))
exp(--iG2A-)exp(-~C1C2Zo)!~(~)) operation
(“fI”
n.1
--xQfl
(1
”
(15)
-
(1 - ZLxo)1Z2) [u!/(u - n)!] f/W
k=v-I
F IiV. u) is then
on IN, u) we obtain
I@(r)> =_exp(-$G,G2)exp(uxoG,G2)
nf-l (;%I”
23 December 1983
1
_
Performing the exponential
+
PHYSICS LETTERS
-lug)l/2)(“~~ijy1 (1 -&xo)l+X
u - n)
u-‘1
+,,I
9
(16)
which is a linear combination of all possible Morse oscillator states produced from the initial state IAT,IA. Here the m-sum contains states which are higher than IA’, II>,while the n-sum contains the lower states. Thus they represent, respectively, the excited and de-excited states of the oscillator due to the collision. In the limit xu + 0, eq. (16) reduces to
I@(r)) = exp(-$G2) +
5
2
m=l
n=l
(
ItA + c
(-iG)m(-iG)n m!n!(u
F
m=l
[(u + m)!/u!]
1121~ + m) + 5 n=l
[u!(u - n - m)!] 1Z21u - n + m>
- n)!
(q
[u!/(u - n)!] t/+
- n)
1
(17)
and, in particular for u = 0, ]$~(r))=exp(-$G~)~~~
‘old,
(18)
. which is a well-known result [2 1. Kere G stands for Gi with x0 = 0.
4. Vibrational
transition probabilities
The probability
of transition from IZV,u) to a final state IZV.f> can be written from eq. (15) as
Pu_,f -Elim IUV,fl~$(f))l~ =.I(N,flexp[-iGl(Qdl -t+O.2-
When eq. 116)s
introduced,
the excitation
probability
exp[-iGp(t)KI is
exp(-fG1G$O)IN,
u>12 -
(1%
Volume
CHEhfICAL
103. number 2
and the de-excitation
probability
Pu.+f = u!f!C, 2(u-n
exp(-GIGZ)
PHYSICS
23 December 1983
LETTERS
exp(2ux0G1G2) I
Because of the presence of the k products containing the a~armonic~ty parameter, express the excitation and de-excitation probabilities by a single expression. We now observe several special cases of the above expressions. (i) If the osciliator is initially iu the ground state, eq. (20) reduces to
it does not appear possibfe to
‘f-1 Cf!)-‘G~fexp(-GlG2)k~o
PO+=
Cl - kxo) .
(22)
For 0 + I the anharmonicity appears only in the functions G because the X-product (ii) For one-quantum excitation processes, eq. (20) becomes
is one.
(-ljn(GIG$”
P u-u+ 1 = (1 - uxo) exp(2uxoCrG2) (u + flPo--r
?I?(?2+ l)!(v
2 - n)! I
’
(23)
wherePo_l = Cl exp(-GIGz). Note that the G -+ 0 as the collision energy approaches zero. That is. only in the weak-coupling limit, does the simple scaling relation PV_V+l = fu f I)( 1 - uxa)Po_ f hold; this relarion has already been proposed by Levine and Wulfman fl5] _The latter relation becomes the harmonic scaling law, P,_,,l = (v + l)Po,l. whenxo = 0. Note that in the weak-coupling limit the scaliig relation for one-quantum de-excitation processes isP,,,_I = u[ I - (u - ~)x~]P~_~_ (ii) In the harmonic limit x0 + 0, eqs. (20) and (21) reduce to the well-known expression [7.16--IS] P
V-f
=
u!f!G 2b-fi
exp(_G’)
where 1 is the lesser of v and
I
x
I
po
2
(_l)i(‘$i j!(lfJ
-fl
+i)!(Z
-j)!
I
’
f_
5. Sample calcuiations In the collinear collision of C-B f A, the a~umption of a purely repulsive exponential interaction between B and A leads to U(z) = D exp(-z/a), where z is the A-.-B distance and D and a potential parameters_ Since z = R - r(d + q), we obtain U(z) = U(R, q) =D exp(yd/a) exp(-R/a) exp(Tq/a); here R is the distance between atom A and the center of mass of BC,x is the displacement of the vibrational coordinate from the equilibrium bond distance d, and r = rQ(m~ + me)_ If exp(~q/a~ is expanded in a power series, we obtain U(R, q) = D’ exp(-R/a) + (@r/a) exp(--R/alq + ._-, withD’=D exp(yd/a). In eq. (7b), the perturbation term is then @‘$a) exp(-R/a)q, in which R is parameterized in time along an appropriate trajectory and q = (?~/Mw~)~~~Q~.An approximate but explicit form of the trajectory for this interaction is known to be 1191 exp(-R(t)/@) = sech~r(~/~~)l/2~/~~, where E is the relative collision energy and p is the reduced mass of the collision system. In the derivation of this trajectory from the classical equation of motion, we have introduced the relation D’ = E to set the energy scale. The-energy dependent quantity Gj 3 G,-(E) which enters in the above probability expressions is rhen G1,#Q=
[27ry(l ~~~)w*a/(7~w~~1/2~
csch[(l
~x&~w~~~~~ZT)~~~] _
WI 153
23
CHEMICAL PHYSICS LEmERS
Volume 103, number 2
December 1983
in the present collinear collision process, following the impulsive modification introduced by Mahan [ZO] , we assume that A and B collide elastically, for which case the reduced mass ~1in the above expression must be replaced by p’ F mAmg/(mA + mg). Furthermore, the collisipn energy E must be replaced by E’, the relative energy of A with respect to B, which is related to E as E’ = tji/kOE. In terms of the dimensionless constants m = mAmc/mB(mA + mg + I?& and Q = U-I [fimc/w@B(nlB + n~&Il’/~, we can write eq. (25) as G1,2(e)= [2l/*nm(l -ixo)/(l + m)]csch[(n/a)(m/2#*(1 +x0)], where E= E/k+,. Since x0 = $(oo/Do), where Do is the dissociation energy above the lowest vibrational level [ 11, p. 1001, we can check the variation of the ratio P;_,f(eq. (20))/P,_,f(eq. (24)) = P>!!/PF!f with m, x0, E, and/or LY.To illustrate this variation, we first simply take 0 + 1 for the fared value of LY= 0.2 at eT = E-,-/ho0 = 4.0. Here the total energy ET is related to E as E1j2 = $ [(ET - Eu,#~2.+ (ET - E&/2], where E,,i and Ev,f are the initial and final vibrational energies. When m = 0.1, the ratio changes from 0.89 to 0.99 as DO/O, increases from 5 to 100 corresponding to x0 of 0.05 to 0.0025. When m ~0.6, it changes from 0.67 to 0.98 over the same range of Do/oo. The ratio approaches one with increasing potential well depth, but its departure from one becomes serious when m increases. Note that for collisions of a “typical” diatomic molecule with a light atom, m is e-1 [21]. For example, for N2 + He. m = 0.125 and for 02 + He, it is 0.110. For N2 + Ne and O2 + Ne, they are 0.417 and 0.384, respectively. For these two molecules, (Y= 0.112 and 0.129. For HCi + He, m is only 0.00282 but becomes as large as 3.47 for CIH + He. For these two configurations the values of (Yare 0.0150 and 0.520, respectively. From the generalization pointed above, then, the anharmonicity is expected to be important in the latter configurations, as well as in H2 + He, but negligible in the former and in N2/02 i He. Exact numerical calculations for the Morse oscillator interacting with an atom carried out by Clark and Dickinson [22] show the same trend as the above result, although their calculations predict smaller values of P&!!l. For example, for H2 + He at eT = 2.0, Pf:l = 5.35 X lOa and PoH-9 = 7.07 X 10S4, whereas the quantum results are 2.46 X 10S4 and 7.20 X 104, respectively. Note that the latter results are obtained for U(R. q) = D’exp[-(R - r&z], while the present calculations are based on lJ(R, q) = D’ exp(-R/u) + @‘y/u) exp(-R/a)q. Here we used we = 4395.24 cm-l and c+xe = 117.99 cm-l [ 11, p_ 532]_ for H2 + He as a function of u for E = 1 .O and 3.0. At both energies, this In fig. 1 we plot Pv,u+l/(u + l)Po_l quantity deviates seriously from the harmonic scaling law, which simply predicts one; the deviation is particularly large when both E and u are large. It is interesting to note that the deviation is linear for the lower energy case, where eq. (23) can be approximated as Pv_,+l/(v + l)Po_l = 1 - uxo, the anharmonic scaling in the weak-
.8-
1.0 .6-
P-1
.8-
_
(v+l’&
R-
.2-
19
OO
I
2
B
Fig. 1. Dependence ofP_,/(u and 3.0 for Hz f He.
154
.
4
.
V
*
6
.
+ I>Po-~
*
6 on u
*
I 10
at E = 1.0
”
E
y
Fig. 2. Dependenck of Pvy~l/(u + 110 - UX~)P~~~ on thereducedcollisionenergyE for Hz + He; 1 - 2,s - 6. and 10 -t 11 transitions &e considered.
CHEMICAL
Volume 103, number 2 coupling higher
+ I)(1 - uxo)Po_t
limit. In fact, Pu_,+tl(u
energy
(E = 3.0),
and ex~(2uxoG1GZ).
however, The k
the dependence
sum contains
23 December
PHYSICS LETTERS
1983
changes from 0.999 for u = 1 to 0.993 for u = 10. At the
is no longer
the anharmonic
linear
due to the increased importance
of the k sum
transitions. For example, for 5 -, 6 the two-state approximation (i.e. including u = 5 and 6 only) givesPg+ = 6G:(l - 5x0) exp(-GtG2) X exp(IOxOGlG2) or P5,J6(l - 5x,-,)P,_, = exp(lOxoG,G,) e 1, but the inclusion of all possible transitions suchas5+4+5+6and5+4+3 + 4 + 5 + 6 in addition to the direct step 5 + 6 leads to
P 5_6 = 6G:(l
- 5x0) exp(-Gt G,)
exp(10xoGIG2)I
- &(I
- 2x0)(1
- 3x0)(1
- ~x~)(G~G~)~
- $(I
-x0)(1
- 2x0)(1
- 3x0)(1
contribution
1 - $(I - 4xo)G,G2
+ &,(I -x0)(1
- 4xo)(G,G#
of multistate
- 2x0)(1
+ $(I - 3x0)(1
- 3x0)(1
- 4xo)(G,G2)’
- ~x,)(G,G~)~
I* t
(76)
which has been used in the present calculation. In fig. 2, following Levine and Wulfman’s work [ 151, we plot the dependence of P,_,,+r/(u + l)(l - UX~)P~_~ bn E for u = 1,5, and 10. Up to the point where E is about 1.5. the = 1 holds well, but above this energy it is no longer reliable, anharmonic scaling Pudv+ t /(u + I)(1 - ux~)Po,~ particularly for large u, in predicting the anbarmonic probability P,_u+ 1 in terms of PO_ 1. Finally it may be stated that although the procedure developed here as deficiencies because of the introduction of assumptions and approximations, it is useful in treating vibrational transitions in the anbarmonic oscillator. We feel that this procedure can be developed into a full, rigorous method when additional knowledge on the algebraic operations of exponential operators become available.
Acknowledgement Acknowledgement is made to the Donors of the Petroleum Chemical Society, for the support of this research_
Research
Fund, administered
by the American
References [l] [2] [3] [4]
J. Wei and E. Norman, J. hlath. Phys. 4 (1963) 575. D. ter Haar. Selected problems in quantum mechanics (Academic Press, New York, 1964) pp. 152-154. P. Pechukas and J.C. Light, J. Chem. Phys. 44 (1966) 3897. T-A. Dillon and J-C. Steuhenson. Phvs. Rev. A6 (1972) 1460. [51 J.D. Kelley, J. Chem. Phys. 56 (i972) 61081 [61 G. Jolicard and.L. Galatry, J. Chem. Phys. 63 (1975) 2787; G. JoIicard. J. Chern. Phys. 63 (1975) 2798; 68 (1978) 3445,3454. [71 H.K. Shim, in: Modern theoretical chemistry, Vol. 1, ed. W-H. Miller (Plenum Press, New York, 1976) p_ 131. [al R.T. Skodje, W.R. Gentry and CF. Giese, Chem. Phys. 74 (1983) 347. 191J. Schwinger, in: Quantum theory of angular momentum, eds. L.C. Biedenharn and H. van Dam (Academic Press, New York. 1965) p_ 229. 1101R.D. Levine, Cheni. Phys. Letters 95 (1983) 87. 1111G. Herzberg, Spectra of diitomic molecules (Van Nostrand, Princeton, 1967). I121 A. Messiah, Quantum mechanics, Vol. 2 (North-Holland, Amsterdam, 1966) pp_ 579-580. [I31 W. Magnus. Commun. Pure Appl. Math. 7 (1954) 649. r141 R.M. Wilcox, J. Math. Phys. 8 (1967) 962. [I51 R.D. Levine and C-E. Wulfman. Chem. Phys. Letters 60 (1979) 372. 1161 E.H. Kerner. Can. J. Phys. 36 (1958) 371. El71 C.E. Treanor. J. Chem. Phys. 43 (1965) 532. H.K. Shin, Chem. Phys. Letters 3 (1969) 125.
t:;; D. Rapp, J. Chem. Phys. 32 (1960) 735. 1201 B.H. Mahan, J. Chem. Phys. 52 (1970) 5221. [21]
D. Secrest &nd B.R. Johnson,
J. Chem. Phys. 45 (1966) 4556. 164.
1221 A.P. Clark and AS. Dickinson, J. Phys. B6 (1973)
155
CHEMICAL PHYSICS LETTERS
Volume 103, number 2
AN ALGEBRAIC APPROACH TO VIBRATIONAL
TRANSITIONS
IN THE FORCED MORSE OSCILLATOR
Taikyue
REE
Department
of Chemistty. Korea Advanced
Institute of Science and Technology,
Seoul, Korea
and Y.H. KIM * and H.K. SHIN Department
of Chemistry.
University of Nevaab, Rena, Nevada 89557. USA
Received 14 September 1983; in foal form 5 October 1983
An expression for the probability of vibrational transitions in the Morse oscillator has been derived by use of the anhar-
monk commutation relations in an approximate solution of the SchrZidmgerequation of motion. The inclusion of the oscillator anharmonicity leads to a complicated probability expression, but in the weak-coupling limit the expression reduces to an anharmonic scaling relation of Pv+v+l /(IJ + 1) (1 - LJX#‘~_,~ = 1, where x0 is the anharmonicity parameter. For Hz + He. this relation appears to hold well up to collision energies of =I..5 vibrational quanta.
1. Introduction
The commutation relations of the harmonic oscillator are known to be useful in developing theoretical approaches for molecular energy transfer problems [l-8] _The usefulness is largely due to the simplicity of solving the Schredinger equation of motion and of calculating scattering operators of the interaction system in terms of the usual Boson creation and annihilation operators. Even the complicated properties of angular momentum eigenvectors have been shown to be readily handled in terms of these commutation relations [9]. The necessary condition for the use of these operators in formulating energy transfer probabilities, however, is that the vibrations of the colliding molecules be harmonic_ While it may be satisfactory for transitions among low-lying ener,v levels, such an assumption can lead to serious errors in studying transitions involving highly excited states due to the importance of the anharmonicity. The anharmonicity can be important even for low-lying levels in polyatomic molecules or in van der Waals molecules which often involve shallow potential energy wells. Recently, Levine [lo] has formulated the commutation relations for the anharmonic oscillator starting with the &son operators in the forms appropriate for energy transfer problems_ Application of these relations to the Morse oscillator state, which is sim-
ilar to the angular momentum eigenvector derived by Schwinger [9], leads to the eigenvalue expression which is accurate to the first anharmonicity correction term. In the present letter, using these commutation relations, we shall present an approximate procedure of developing the perturbed oscillator state and show its use in ener,v transfer studies.
2. Commutation relations For a pair of Boson creation
(~7:. c&) and annihilation
(a,,
a_&) operators,
the commutation
relations
are
* Present address: Department of Chemistry, Inha University, Inchon, Korea.
0 009-2614/83/S
(North-Holland
03.00 0 Elsevier Science Publishers Physics Publishing Division)
B.V.
149
CHEMICAL PHYSICS LETTERS
Volume 103,number2
[iz$fz;t] =[a,-,a;;]= 0
;
23 December 1983
[a,-,ug =6 UN -
(1)
The notations u and N are chosen to represent the vibr&ional quantum number and a~armon~city as shown in section 3. According to Levine [IO], the basic operators needed to describe the Morse oscillator can be formulated in terms of these operators as QO = (2N)-“2(a*x-u N+Qf;oU)* - af;a,)
PO = i(2.N)-‘/z(~~*~ I 0 =.N-‘(2
@I ,
(=I
o- - u+u-) vu
m)
NN
where N is related to-the ~~o~~ity parameter xo asN=x&l. It should be interesting to note the relationship between these and the corresponding harmonic oscillator operators. If we write the hamiltonian of the harmonic oscillator as H = Cp2 + M2u$q2)DM, the commutation_relation connecting the position variable q and the momentump is [q, p] = ifi. When we put q = @/MGJ&~Q and p = (j?fh~~)~I~P,k?= $ittwo(P2 + Q2). The commutation relation, then, is [Q, P] = i. From eqs. (Zb) and (2c), however, we find [Qu, Po] = iZo, which suggests that only in the harmoni? limit does Zu approach the identity operator I. In fact, we can show that there exists the harmonic counterpart to each operator relation obtained from Qo, Po, and lo_ From Q. and Po, we obtain creation (A+) and annihilation (A-) operators for the anharmonic Morse oscillator: A
.+=
2-3/2(Qu - ipo) = N-“%r;Lr;;
A-=2-
IZ2(Qo + iPo) = N-‘Z%;;fX;
,
(3a) *
WI
The commutation relations obtained for the Morse oscillator are then
[lo. PO] = --2ixgQo ,
(44
E&vQOJ= “iq)P()
t4b)
D
[A-. A+] =Z, , [lo&J
= r2xoA*
(4c) .
(44
Note that another set of operators, which can be ~oust~~ted from eq. (1) is
which are the characteristic algebraic relations of angular momentum operators. Utilizing these relations in accordance with the angular momentum properties, Dillon and Stephenson have formulated energy transfer probabilities [4].
3. PerturbedMorseusciUatorstate The most general form of the eigenvalues of the anharmo_mc oscillator referred to the lowest &ate as zero is often given by [ 11, pp_ 92,933 E, = fiwov - F~o~~v~ + fiogov3
+ _.__
(51
To calculate anharmonic eigenvalues, therefore, at least two parameters are needed; namely, the quantum number u and the leading anharmonicity parameter ~0. The latter p-eter is fsed for a given molecular system, but in fo~u~~g a form of the ~~0~~ oscillator state, it can be regarded as a variable. Following Levine’s work, we take these two parameters to be v and N = x-lo , and express the Ghsrrnonic Morse state by IN, VI, in which each parameter can be operated, independently, by the harmonic oscillator creation and antiilation operators, -.
150
CHEMICAL
Volume 103. number 2
23 December 1983
PHYSICS LETTERS
The a:, ai, a:, and ak, where subscripts u and N represent the vibrational quantum number and anharmonicity. state IfV, u> can be generated from the vacuum state IO) following the scheme that has been used to describe angular momenta in terms of the creation operators of a two-dimensional harmonic oscillator [9,10,12] : IN. u> = [u!(N - u)!]-‘~2(a~)u(a~)N-“10)
_
(6)
We now write the hamiltonian for the Morse oscillator action with an incident particle in the form F = :hwo(fi
+ Q;> +
@/~~o)~‘~~OQ~
or, in terms of the creation
force F(r) due to the inter-
W)
,
and annihilation
operators,
H = fiw&I+/l-
t ;l(J) t (r/ziVClI,)“2F(t)(A+
The SchrBdinger
equation
of motion
forced by a timedependent
t A--) =-Ho t H’(r) .
may then be written
W)
in the form
ih I+(t)> = [ff’ + H’(t)] W(r)) ,
(8)
where l*(t)> describes the Morse oscillator state disturbed by the perturbation linear combination of IN. u>. When we write the state vector in the form
I*(t)) = exp(-iHOt/fi)IQ(t)) the Schrtidinger_equation ilt 6(f)>
= H’(t)l@~(t))
H’(r) and can be obtained
,
as a
(9)
becomes
,
(10)
where H’(t) = exp(iHOf/?t)
H’(t) exp(-&r/h)
= (tl/2kf0~)‘/~F(f)
exp[io&4+d-
+ +lg)](A+
Using eqs. (4~) and (4d) we can convert H’(r) = @/2kk~#~F(t)
this expression
exp(ixow0t)[exp(io&&4*
By use of eq. (11) an approximate
solution
expfioO(IO
t F(t) exp[-iw&, J-cm
- xo)f]
dtA-
This
exp[--io&I+A-
+ ilo)]
_
to .
+ exp(-iw#&4-]
of eq. (10) can be written
F(t)
+
+A-)
+ x,-,)f]
(11)
as
dt A+
11 _
(12)
IQ(-00))
solution is in the form I@(t)> = V(t, -=)I@(-m)), and the exponential part is a time evolution operator transforming the initial state I@(-=)> to the state I@(t)) at a later time; thus it contains all the information needed to describe the interaction process at time t, in particular as t + +m. It does not appear possible, however, to use this form of the evolution operator in deriving transition probabilities because of the appearance of the operator I0 in the exponential part of the exponent. We note that it tends toward the identity operator I in the lit x0 --f 0. Using this limiting form, we approximate eq. (12) as @(t)>
=
exp[-iGl(r)A+
- iGa(t)A-]
where G1,2(t) = (2iE?~o)-~/~J~,F(t) entangled approximately as [13,14] exp (-i G, A+ - iG2A-)
= exp(=iG++)
I@(--)>, exp[kiiw,(l exp(-iG+-)
(13) +x&l
dt. The exponential exp(-$G1G210)
_
operator
in eq. (13) can be dis(14) 151
CHEMICAL
Volume 103. number 2 The evolution of the perturbed I@(t)> = exp(--i@+)
state from the initial state I@(--=))
exp(--iG2A-)exp(-~C1C2Zo)!~(~)) operation
(“fI”
n.1
--xQfl
(1
”
(15)
-
(1 - ZLxo)1Z2) [u!/(u - n)!] f/W
k=v-I
F IiV. u) is then
on IN, u) we obtain
I@(r)> =_exp(-$G,G2)exp(uxoG,G2)
nf-l (;%I”
23 December 1983
1
_
Performing the exponential
+
PHYSICS LETTERS
-lug)l/2)(“~~ijy1 (1 -&xo)l+X
u - n)
u-‘1
+,,I
9
(16)
which is a linear combination of all possible Morse oscillator states produced from the initial state IAT,IA. Here the m-sum contains states which are higher than IA’, II>,while the n-sum contains the lower states. Thus they represent, respectively, the excited and de-excited states of the oscillator due to the collision. In the limit xu + 0, eq. (16) reduces to
I@(r)) = exp(-$G2) +
5
2
m=l
n=l
(
ItA + c
(-iG)m(-iG)n m!n!(u
F
m=l
[(u + m)!/u!]
1121~ + m) + 5 n=l
[u!(u - n - m)!] 1Z21u - n + m>
- n)!
(q
[u!/(u - n)!] t/+
- n)
1
(17)
and, in particular for u = 0, ]$~(r))=exp(-$G~)~~~
‘old,
(18)
. which is a well-known result [2 1. Kere G stands for Gi with x0 = 0.
4. Vibrational
transition probabilities
The probability
of transition from IZV,u) to a final state IZV.f> can be written from eq. (15) as
Pu_,f -Elim IUV,fl~$(f))l~ =.I(N,flexp[-iGl(Qdl -t+O.2-
When eq. 116)s
introduced,
the excitation
probability
exp[-iGp(t)KI is
exp(-fG1G$O)IN,
u>12 -
(1%
Volume
CHEhfICAL
103. number 2
and the de-excitation
probability
Pu.+f = u!f!C, 2(u-n
exp(-GIGZ)
PHYSICS
23 December 1983
LETTERS
exp(2ux0G1G2) I
Because of the presence of the k products containing the a~armonic~ty parameter, express the excitation and de-excitation probabilities by a single expression. We now observe several special cases of the above expressions. (i) If the osciliator is initially iu the ground state, eq. (20) reduces to
it does not appear possibfe to
‘f-1 Cf!)-‘G~fexp(-GlG2)k~o
PO+=
Cl - kxo) .
(22)
For 0 + I the anharmonicity appears only in the functions G because the X-product (ii) For one-quantum excitation processes, eq. (20) becomes
is one.
(-ljn(GIG$”
P u-u+ 1 = (1 - uxo) exp(2uxoCrG2) (u + flPo--r
?I?(?2+ l)!(v
2 - n)! I
’
(23)
wherePo_l = Cl exp(-GIGz). Note that the G -+ 0 as the collision energy approaches zero. That is. only in the weak-coupling limit, does the simple scaling relation PV_V+l = fu f I)( 1 - uxa)Po_ f hold; this relarion has already been proposed by Levine and Wulfman fl5] _The latter relation becomes the harmonic scaling law, P,_,,l = (v + l)Po,l. whenxo = 0. Note that in the weak-coupling limit the scaliig relation for one-quantum de-excitation processes isP,,,_I = u[ I - (u - ~)x~]P~_~_ (ii) In the harmonic limit x0 + 0, eqs. (20) and (21) reduce to the well-known expression [7.16--IS] P
V-f
=
u!f!G 2b-fi
exp(_G’)
where 1 is the lesser of v and
I
x
I
po
2
(_l)i(‘$i j!(lfJ
-fl
+i)!(Z
-j)!
I
’
f_
5. Sample calcuiations In the collinear collision of C-B f A, the a~umption of a purely repulsive exponential interaction between B and A leads to U(z) = D exp(-z/a), where z is the A-.-B distance and D and a potential parameters_ Since z = R - r(d + q), we obtain U(z) = U(R, q) =D exp(yd/a) exp(-R/a) exp(Tq/a); here R is the distance between atom A and the center of mass of BC,x is the displacement of the vibrational coordinate from the equilibrium bond distance d, and r = rQ(m~ + me)_ If exp(~q/a~ is expanded in a power series, we obtain U(R, q) = D’ exp(-R/a) + (@r/a) exp(--R/alq + ._-, withD’=D exp(yd/a). In eq. (7b), the perturbation term is then @‘$a) exp(-R/a)q, in which R is parameterized in time along an appropriate trajectory and q = (?~/Mw~)~~~Q~.An approximate but explicit form of the trajectory for this interaction is known to be 1191 exp(-R(t)/@) = sech~r(~/~~)l/2~/~~, where E is the relative collision energy and p is the reduced mass of the collision system. In the derivation of this trajectory from the classical equation of motion, we have introduced the relation D’ = E to set the energy scale. The-energy dependent quantity Gj 3 G,-(E) which enters in the above probability expressions is rhen G1,#Q=
[27ry(l ~~~)w*a/(7~w~~1/2~
csch[(l
~x&~w~~~~~ZT)~~~] _
WI 153
23
CHEMICAL PHYSICS LEmERS
Volume 103, number 2
December 1983
in the present collinear collision process, following the impulsive modification introduced by Mahan [ZO] , we assume that A and B collide elastically, for which case the reduced mass ~1in the above expression must be replaced by p’ F mAmg/(mA + mg). Furthermore, the collisipn energy E must be replaced by E’, the relative energy of A with respect to B, which is related to E as E’ = tji/kOE. In terms of the dimensionless constants m = mAmc/mB(mA + mg + I?& and Q = U-I [fimc/w@B(nlB + n~&Il’/~, we can write eq. (25) as G1,2(e)= [2l/*nm(l -ixo)/(l + m)]csch[(n/a)(m/2#*(1 +x0)], where E= E/k+,. Since x0 = $(oo/Do), where Do is the dissociation energy above the lowest vibrational level [ 11, p. 1001, we can check the variation of the ratio P;_,f(eq. (20))/P,_,f(eq. (24)) = P>!!/PF!f with m, x0, E, and/or LY.To illustrate this variation, we first simply take 0 + 1 for the fared value of LY= 0.2 at eT = E-,-/ho0 = 4.0. Here the total energy ET is related to E as E1j2 = $ [(ET - Eu,#~2.+ (ET - E&/2], where E,,i and Ev,f are the initial and final vibrational energies. When m = 0.1, the ratio changes from 0.89 to 0.99 as DO/O, increases from 5 to 100 corresponding to x0 of 0.05 to 0.0025. When m ~0.6, it changes from 0.67 to 0.98 over the same range of Do/oo. The ratio approaches one with increasing potential well depth, but its departure from one becomes serious when m increases. Note that for collisions of a “typical” diatomic molecule with a light atom, m is e-1 [21]. For example, for N2 + He. m = 0.125 and for 02 + He, it is 0.110. For N2 + Ne and O2 + Ne, they are 0.417 and 0.384, respectively. For these two molecules, (Y= 0.112 and 0.129. For HCi + He, m is only 0.00282 but becomes as large as 3.47 for CIH + He. For these two configurations the values of (Yare 0.0150 and 0.520, respectively. From the generalization pointed above, then, the anharmonicity is expected to be important in the latter configurations, as well as in H2 + He, but negligible in the former and in N2/02 i He. Exact numerical calculations for the Morse oscillator interacting with an atom carried out by Clark and Dickinson [22] show the same trend as the above result, although their calculations predict smaller values of P&!!l. For example, for H2 + He at eT = 2.0, Pf:l = 5.35 X lOa and PoH-9 = 7.07 X 10S4, whereas the quantum results are 2.46 X 10S4 and 7.20 X 104, respectively. Note that the latter results are obtained for U(R. q) = D’exp[-(R - r&z], while the present calculations are based on lJ(R, q) = D’ exp(-R/u) + @‘y/u) exp(-R/a)q. Here we used we = 4395.24 cm-l and c+xe = 117.99 cm-l [ 11, p_ 532]_ for H2 + He as a function of u for E = 1 .O and 3.0. At both energies, this In fig. 1 we plot Pv,u+l/(u + l)Po_l quantity deviates seriously from the harmonic scaling law, which simply predicts one; the deviation is particularly large when both E and u are large. It is interesting to note that the deviation is linear for the lower energy case, where eq. (23) can be approximated as Pv_,+l/(v + l)Po_l = 1 - uxo, the anharmonic scaling in the weak-
.8-
1.0 .6-
P-1
.8-
_
(v+l’&
R-
.2-
19
OO
I
2
B
Fig. 1. Dependence ofP_,/(u and 3.0 for Hz f He.
154
.
4
.
V
*
6
.
+ I>Po-~
*
6 on u
*
I 10
at E = 1.0
”
E
y
Fig. 2. Dependenck of Pvy~l/(u + 110 - UX~)P~~~ on thereducedcollisionenergyE for Hz + He; 1 - 2,s - 6. and 10 -t 11 transitions &e considered.
CHEMICAL
Volume 103, number 2 coupling higher
+ I)(1 - uxo)Po_t
limit. In fact, Pu_,+tl(u
energy
(E = 3.0),
and ex~(2uxoG1GZ).
however, The k
the dependence
sum contains
23 December
PHYSICS LETTERS
1983
changes from 0.999 for u = 1 to 0.993 for u = 10. At the
is no longer
the anharmonic
linear
due to the increased importance
of the k sum
transitions. For example, for 5 -, 6 the two-state approximation (i.e. including u = 5 and 6 only) givesPg+ = 6G:(l - 5x0) exp(-GtG2) X exp(IOxOGlG2) or P5,J6(l - 5x,-,)P,_, = exp(lOxoG,G,) e 1, but the inclusion of all possible transitions suchas5+4+5+6and5+4+3 + 4 + 5 + 6 in addition to the direct step 5 + 6 leads to
P 5_6 = 6G:(l
- 5x0) exp(-Gt G,)
exp(10xoGIG2)I
- &(I
- 2x0)(1
- 3x0)(1
- ~x~)(G~G~)~
- $(I
-x0)(1
- 2x0)(1
- 3x0)(1
contribution
1 - $(I - 4xo)G,G2
+ &,(I -x0)(1
- 4xo)(G,G#
of multistate
- 2x0)(1
+ $(I - 3x0)(1
- 3x0)(1
- 4xo)(G,G2)’
- ~x,)(G,G~)~
I* t
(76)
which has been used in the present calculation. In fig. 2, following Levine and Wulfman’s work [ 151, we plot the dependence of P,_,,+r/(u + l)(l - UX~)P~_~ bn E for u = 1,5, and 10. Up to the point where E is about 1.5. the = 1 holds well, but above this energy it is no longer reliable, anharmonic scaling Pudv+ t /(u + I)(1 - ux~)Po,~ particularly for large u, in predicting the anbarmonic probability P,_u+ 1 in terms of PO_ 1. Finally it may be stated that although the procedure developed here as deficiencies because of the introduction of assumptions and approximations, it is useful in treating vibrational transitions in the anbarmonic oscillator. We feel that this procedure can be developed into a full, rigorous method when additional knowledge on the algebraic operations of exponential operators become available.
Acknowledgement Acknowledgement is made to the Donors of the Petroleum Chemical Society, for the support of this research_
Research
Fund, administered
by the American
References [l] [2] [3] [4]
J. Wei and E. Norman, J. hlath. Phys. 4 (1963) 575. D. ter Haar. Selected problems in quantum mechanics (Academic Press, New York, 1964) pp. 152-154. P. Pechukas and J.C. Light, J. Chem. Phys. 44 (1966) 3897. T-A. Dillon and J-C. Steuhenson. Phvs. Rev. A6 (1972) 1460. [51 J.D. Kelley, J. Chem. Phys. 56 (i972) 61081 [61 G. Jolicard and.L. Galatry, J. Chem. Phys. 63 (1975) 2787; G. JoIicard. J. Chern. Phys. 63 (1975) 2798; 68 (1978) 3445,3454. [71 H.K. Shim, in: Modern theoretical chemistry, Vol. 1, ed. W-H. Miller (Plenum Press, New York, 1976) p_ 131. [al R.T. Skodje, W.R. Gentry and CF. Giese, Chem. Phys. 74 (1983) 347. 191J. Schwinger, in: Quantum theory of angular momentum, eds. L.C. Biedenharn and H. van Dam (Academic Press, New York. 1965) p_ 229. 1101R.D. Levine, Cheni. Phys. Letters 95 (1983) 87. 1111G. Herzberg, Spectra of diitomic molecules (Van Nostrand, Princeton, 1967). I121 A. Messiah, Quantum mechanics, Vol. 2 (North-Holland, Amsterdam, 1966) pp_ 579-580. [I31 W. Magnus. Commun. Pure Appl. Math. 7 (1954) 649. r141 R.M. Wilcox, J. Math. Phys. 8 (1967) 962. [I51 R.D. Levine and C-E. Wulfman. Chem. Phys. Letters 60 (1979) 372. 1161 E.H. Kerner. Can. J. Phys. 36 (1958) 371. El71 C.E. Treanor. J. Chem. Phys. 43 (1965) 532. H.K. Shin, Chem. Phys. Letters 3 (1969) 125.
t:;; D. Rapp, J. Chem. Phys. 32 (1960) 735. 1201 B.H. Mahan, J. Chem. Phys. 52 (1970) 5221. [21]
D. Secrest &nd B.R. Johnson,
J. Chem. Phys. 45 (1966) 4556. 164.
1221 A.P. Clark and AS. Dickinson, J. Phys. B6 (1973)
155