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The importance of multiple-quantum transitions on the dissociation constant of hydrogen in an inert gas
Volume 14, number 2
IS bfhy 1972
CHEMICAL PHYSICS LETTERS
THE IMPORTANCE OF MULTIPLE-QUANTUM TRANSITIONS ON THE DISSOCIATION CONSTANT OF HYDROGEN IN AN INERT GAS G.B. SQRENSEN Chemistry
Laboratory
Ill, H.C. Qrsted Institute, Universi@ 2!00 Copenhage,; Q. Denmark
of Copenhaen,
Recei\zd 4 January 1972
In a calculation of the dissociation constant of hydrogen in arpn, it is shown that multiplequantum transitions are importint near the dissociation limit. The transition probabilities between the vibrational states are obtained by Nth order semi-classical perturbation theory. Reasonable grcement with experimentally determined values of the dissociation
constant
is obtained.
I. Introductioll
the probability per collision of transition from vibrational state i to vibrational
In recent years, several calculations of the dissociation constant k, of hydrogen in an inert gas have been carried out [l-3] . The so-called “ladder-climbing” model, in which the hydrogen molecule is excited from one vibrational level to another primarily by collision with an inert gas atom, has been used. This process will eventually excite the moiecule to the upper levels, from which dissociation can occur. In previous work the vibrationid transition probabilities have been obtained from SSH theory [4J : but since this is a first-order perturbation theory, it underestimates the multiple-quantum transitions. The reason
why one gets reasonable agreement between the calculated and the experimentally determined values of kd, is that the transition probabilities between neighbouring levels are overestimated, as a ccrlsequence of the high values of the intermolecular parameter used. in the present work some detailed calculations on the hydrogen-argon system are presented. An exponential interaction potential and an RKR oscillatormodel [5,6] for the hydrogen molecule, are used. 2.
Theory
The sem&classicaf head-on collision model, described by Rapp and Sharp [7], is used to find P,(V), 274
state j. This model yields
the following set of coupled equations dajdr = @v2/2ih) sech’($rvf) X
C r1
Uj,ra,l(t) exp@z-l(Ei-E,t)t}
,
(1)
where 11,](-m) = 6i,,, P,,,(Y) = la,{=)12, P is the reduced mass, v the initial velocity, iYYjthe vibrational energy for state j, a the intermolecular parameter and the L& are matrix elements defined by ca
HereYcq is the equilibrium internuclear distance in the molecule, and the c+
f 65..Jfi)
{Ej - W)
P&Y> = 0
(3
by the method given in ref. [8]. mki is the rnas of a hydrogen atom and P’(V) the RKR potential [6] (see fig. 1). In order to obtain the best value of the intermolecular potential parameter a, an exponential fit to the
data from molecular beam measurements [Y, IO] has been made. In table I these values are compared with
Vo’olurn= 14, number 2
1.5 hfay 1972
CHEhIICAL PHYSICS LETTERS
od [ 141, namely Y = +(Y,.f vi), was introduced. Figs. 2, 3 and 4 show the resulting transition probability ratios Pi.+l,XT))IP~o(‘V). (pi+~~i(~W’&Y and cPj+3,~{;3>/cP,,(?-)3,-It was not possible to calculate Pi+l,i or Pi+1 i-1 for iZ 8 without including the continuum states’in the expansion of the total wavefunction. The curves were extrapolated so as to obtain the transition probabilities in this region. In fig. 2 a linear least squares fit was made to the last five points. The behaviour of the last two points of fig. 2 can be traced back to the oscillations of the Pi+r.,{v) curves. These osciliations move to lower velociti& as i increases. At i= 6 the maxwellian velocity distribution, for T = 200%2500”K, weights the p76(v)
103: Fig. 1. The RKR
potential for hydrogen as a function intemucIear distance.
of the
Table 1 The intermokcutar
parameter
Q (A-‘)
for different
systems
System
Ref. [12]
Ref. [Y]
Ref. [ll]
He-H2 Ar-Hz Ar-N2
3.51
3.85 a) 2.86 2.5
6.25 5.56 5.0
a) Ref. [IO].
values calculated by method ‘5’ of Herzfeld and Litovitz [ 1 I], and a quantum mechanical calculation carried out by Krauss and Mies [12]. The table shows that the method of Herzfeld and Litovitz overestimates CY.In the present calculations the value 2.86 is-1 has been used. Eq. (1) was than integrated numericaliy with lo-30 different values of the relative velocity V, including up to 13 vibrational states in the expansion of the total wavefunction. This gave the curves P,iv), from which the averaged transition probabilities (Pi,{n) were calculated by averagiilg over a maxwellian velocity distribution [ 131. In this calculation, the usual correction to the semiclassical meth-
h E k V 102> I C ._ ‘; h V
/ Y
10’ -
/
,I 0
P 2
4
6
i Fig. 2. Probabilities for singlequantum ~I0U-I).
8
10
12
transitions scaled by
275
Volume
14, number
CHEMICAL
2
PHYSICS
LETTERS
to-71 0
15 May 1972
2
6
L
6
10
i i Fig. 3. Probabilities
for doublequantdm @, DF-P.
Fig. 4. Probabilities
for tripfequantum
nerve in a region where the curve at first decreases
transitions
scaled by
(P,,(T)).
transitions scaled by
from the equation
aqd then increases rapidly. In figs. 3 and 4 the twoaxI t~reequ~~tum transitions are shown. It was not possible to calculate more than the three Iowest threequantum transitions from our data. Figs. 3 and 4 Indicate that muItip1equantun-r transitions cannot be negiected in a caIcuIation of the dissociation constant_ The multipiequantum transitions become more important at higher temperatures, since (P~+2,i(f13,f(Plo(Q> increases by a factor of 1.5 and (Pi+3 XT))l(P1@)) by a factor of 3 when the temperature is increased by 500°K. The masterequation for the process is [ISI :
dnp
= c (nj(Pjj(T)) - n$Ppy] i +~~cP,i(~,
- niGPic(~) ,
.
where ni is the concentration of malecuies on the ith vibrational Ievel, ‘ziA the concentration of hydrogen atoms and (P&J> the transition probability from vibrationd state-i to the continuum_ cP,,(T)) is estimated .276
where D is the dissociation and Zi the collision Zi =
energy, geE a steric factor number for state i:
2nX# (2nkT/,t.#i2
,
where jzX is the concentration of the inert gas> where i > i can be found from figs. 2 and 3, since P&r’)} is known from the experimentaf determination of the vi.brationai if&Da tion times f 17 j
Volume
14, number
2
rtib = [P’19(T))(1 -exp{-(El
CHEhWAL
-E,)/kT}
)I-’
.
PHYSICS LETTERS
1.5 hfay
(3)
This formula is exact only for the harmonic oscilIator, but the error incurred by using it here should not be larger than the experimenta! uncertainty in the determination of 7vib. The values of P,,,(T)> calcuiated by means of eq. (1) are a factor of 5 less than the ones obtained using (3) and the data from ref. [17]. This discrepancy is being investigated at the moment. In the present work we have used the value of ‘PLO(T)) obtained from (3) and: ~7’ wb = 10” * e&p(-0/?/3)/(4.
1 A) set-l ,
where 0 = 100 5 2.6,A = 3.9 + 0.8 and the pressure is 1 atm [17]. For cp,,jc3) we have: (Pii(
= Pii(T)) exp {- (Ej--E$icT}.
The dissociation constant kd is given by hO/rzX where A0 is the lowest eigenvalue of the matrix M [15] . Mti = exp (- i(Ei-
Ej)jkT)
3. Results and discussion In fig. 5 we have shown the calculated values of /cd for A = 3.9 and 8 = 100. Curve I was calculated using single-quantum transitions only, while curve II was obtained with the inclusion of double-quantum transitions. In both calculations dissociation could occur from the three uppermost levels. If dissociation were restricted to occuring from the top level curve I wculd be lowered by a factor of 2-3, while cuive II would only be about 10% lower. This is a result of the inclu-. sion of double-quantum transitions, which facilitate the transfer of molecuies among the upper levels. The curves are sensitive to change in PlO(T). If one used 8 = 102.6 and A = 3.1 in (3) both curves rise by a factor of 1.3 to 1.5. Inclusion of triple,and higherorder quantum transitions raise curve II by a factor which increases with temperature. This gives a better correspondence between the calculated and the experimen tally determined [ 18,191 values of &. The effect of the multiplequantum transitions is that they increase the rate at which the mo!ecules climb the vi-
F&. 5. Calculated values of the dissociation constant for hydrwpn dilured by U-I organ gas (solid lines) compared with x~lues obtained from experimental data of refs. [18,19] (dashed lines).
brational ladder, but more important is their influence near the dissociation limit. Since multiple-quantum transitions are very likely to occur in this region they indirectly lower the limit from which dissociation can occur, i.e., they lower the activation energy. The main difference between the present work and previous works is the inclusion of realistic values of doublequantum transitions. In order to determine these transitions it is necessary to integrate a Iarge set of coupled equations, and future work should therefore be concerned with the following two problems (1) integration of coupled equations with the inclusion of continuum states, in order to check the extrapolations made; (2) other, and easier, methods to evaluate the multiple-quantum transitions, e.g., the perturbed stationary state treatment 1201 .
277
Volume,lld, number 2 .,, :
CHEMICAL PHYSICS LETTERS
Ackno&le,&emei&
:.. ‘@me author wishes to thank Dr. Cari Nyeland for stitiulating to .thank
dis&ssions
and sugg&tiotis.
I also want
Dr.3.E. Nielsen and Dr. Anne M. WooIIey for helpful discussions on preparing the manuscript.
References [lj H.O. Ritchaid, J. Phys. Chem. 65 (1961) 504. [21 ,JN. Rich Bnd I. Classman, AGARD Conf. Proc. 7 (1’966). [31 J.E. DC% and D.G. Jones, 3. Chent. Phys. 55 (1971) 1531. [4] R.N. Schwartz,Z.1.Siawsksand K.F. Iierzfeid, 3. Chem. Phys. 20 (1952) 1591. 151 R.N. are, J. Chem. Phys. 40 (196411934. [6f EA. Mason and L. ~~onchick, Adma. Chem. Fhys. f 2 (1967)‘329. [71 T.E. Sharp and D. Rapp, J. Chem. Phys. 43 (1965) 1233.
238
15 l%y 1972
J.W; Cooley, Math. Camp. 15 /I9611 363. 1. Amdur and J.E. Jordan, Advan. Chem. Phys. IO 0966) 29. i. Amdur atid A.L. Smith, J. Chem. Phys. 48 (1968) 565, K.F. Henfeld and T-A. Litouitz, Absorption of dispersoon of ultrasonic waves (Academic Fess, New York, 1959). M. Kraussand F-H: Mies. J. Chem. Phys. 42 (1965) 2703. r131 D. Rappand T.E. Sharp, J. Chem. Phys. 38 (1963) 2641. 1141 K. Tagayana@, Suppl. Rogr. Theoret. Phys. 25 (1963) 1. WI C.A. Bnu, J. Chem. Phys. 47 (1967) 1153. f161 O.K. Rice. Statistical mechanics, thermodynamics and kineticsIFreaman.SanFrancisco. 19671. fi7f J.H. Kiefer and R.W. Lutz, J. Chem. Phys. 44 (1966) 668. R.W. Patch, J. Chem. P&s. 36 (i962) 1919. f:!f E.A. Sutton, J. Chcm. Phys. 36 (2962) 2923. [ZOJ C.‘NycIand and A. Hunding, Chem. Phys. Letters 5 (1970) 143.
IS bfhy 1972
CHEMICAL PHYSICS LETTERS
THE IMPORTANCE OF MULTIPLE-QUANTUM TRANSITIONS ON THE DISSOCIATION CONSTANT OF HYDROGEN IN AN INERT GAS G.B. SQRENSEN Chemistry
Laboratory
Ill, H.C. Qrsted Institute, Universi@ 2!00 Copenhage,; Q. Denmark
of Copenhaen,
Recei\zd 4 January 1972
In a calculation of the dissociation constant of hydrogen in arpn, it is shown that multiplequantum transitions are importint near the dissociation limit. The transition probabilities between the vibrational states are obtained by Nth order semi-classical perturbation theory. Reasonable grcement with experimentally determined values of the dissociation
constant
is obtained.
I. Introductioll
the probability per collision of transition from vibrational state i to vibrational
In recent years, several calculations of the dissociation constant k, of hydrogen in an inert gas have been carried out [l-3] . The so-called “ladder-climbing” model, in which the hydrogen molecule is excited from one vibrational level to another primarily by collision with an inert gas atom, has been used. This process will eventually excite the moiecule to the upper levels, from which dissociation can occur. In previous work the vibrationid transition probabilities have been obtained from SSH theory [4J : but since this is a first-order perturbation theory, it underestimates the multiple-quantum transitions. The reason
why one gets reasonable agreement between the calculated and the experimentally determined values of kd, is that the transition probabilities between neighbouring levels are overestimated, as a ccrlsequence of the high values of the intermolecular parameter used. in the present work some detailed calculations on the hydrogen-argon system are presented. An exponential interaction potential and an RKR oscillatormodel [5,6] for the hydrogen molecule, are used. 2.
Theory
The sem&classicaf head-on collision model, described by Rapp and Sharp [7], is used to find P,(V), 274
state j. This model yields
the following set of coupled equations dajdr = @v2/2ih) sech’($rvf) X
C r1
Uj,ra,l(t) exp@z-l(Ei-E,t)t}
,
(1)
where 11,](-m) = 6i,,, P,,,(Y) = la,{=)12, P is the reduced mass, v the initial velocity, iYYjthe vibrational energy for state j, a the intermolecular parameter and the L& are matrix elements defined by ca
HereYcq is the equilibrium internuclear distance in the molecule, and the c+
f 65..Jfi)
{Ej - W)
P&Y> = 0
(3
by the method given in ref. [8]. mki is the rnas of a hydrogen atom and P’(V) the RKR potential [6] (see fig. 1). In order to obtain the best value of the intermolecular potential parameter a, an exponential fit to the
data from molecular beam measurements [Y, IO] has been made. In table I these values are compared with
Vo’olurn= 14, number 2
1.5 hfay 1972
CHEhIICAL PHYSICS LETTERS
od [ 141, namely Y = +(Y,.f vi), was introduced. Figs. 2, 3 and 4 show the resulting transition probability ratios Pi.+l,XT))IP~o(‘V). (pi+~~i(~W’&Y and cPj+3,~{;3>/cP,,(?-)3,-It was not possible to calculate Pi+l,i or Pi+1 i-1 for iZ 8 without including the continuum states’in the expansion of the total wavefunction. The curves were extrapolated so as to obtain the transition probabilities in this region. In fig. 2 a linear least squares fit was made to the last five points. The behaviour of the last two points of fig. 2 can be traced back to the oscillations of the Pi+r.,{v) curves. These osciliations move to lower velociti& as i increases. At i= 6 the maxwellian velocity distribution, for T = 200%2500”K, weights the p76(v)
103: Fig. 1. The RKR
potential for hydrogen as a function intemucIear distance.
of the
Table 1 The intermokcutar
parameter
Q (A-‘)
for different
systems
System
Ref. [12]
Ref. [Y]
Ref. [ll]
He-H2 Ar-Hz Ar-N2
3.51
3.85 a) 2.86 2.5
6.25 5.56 5.0
a) Ref. [IO].
values calculated by method ‘5’ of Herzfeld and Litovitz [ 1 I], and a quantum mechanical calculation carried out by Krauss and Mies [12]. The table shows that the method of Herzfeld and Litovitz overestimates CY.In the present calculations the value 2.86 is-1 has been used. Eq. (1) was than integrated numericaliy with lo-30 different values of the relative velocity V, including up to 13 vibrational states in the expansion of the total wavefunction. This gave the curves P,iv), from which the averaged transition probabilities (Pi,{n) were calculated by averagiilg over a maxwellian velocity distribution [ 131. In this calculation, the usual correction to the semiclassical meth-
h E k V 102> I C ._ ‘; h V
/ Y
10’ -
/
,I 0
P 2
4
6
i Fig. 2. Probabilities for singlequantum ~I0U-I).
8
10
12
transitions scaled by
275
Volume
14, number
CHEMICAL
2
PHYSICS
LETTERS
to-71 0
15 May 1972
2
6
L
6
10
i i Fig. 3. Probabilities
for doublequantdm @, DF-P.
Fig. 4. Probabilities
for tripfequantum
nerve in a region where the curve at first decreases
transitions
scaled by
(P,,(T)).
transitions scaled by
from the equation
aqd then increases rapidly. In figs. 3 and 4 the twoaxI t~reequ~~tum transitions are shown. It was not possible to calculate more than the three Iowest threequantum transitions from our data. Figs. 3 and 4 Indicate that muItip1equantun-r transitions cannot be negiected in a caIcuIation of the dissociation constant_ The multipiequantum transitions become more important at higher temperatures, since (P~+2,i(f13,f(Plo(Q> increases by a factor of 1.5 and (Pi+3 XT))l(P1@)) by a factor of 3 when the temperature is increased by 500°K. The masterequation for the process is [ISI :
dnp
= c (nj(Pjj(T)) - n$Ppy] i +~~cP,i(~,
- niGPic(~) ,
.
where ni is the concentration of malecuies on the ith vibrational Ievel, ‘ziA the concentration of hydrogen atoms and (P&J> the transition probability from vibrationd state-i to the continuum_ cP,,(T)) is estimated .276
where D is the dissociation and Zi the collision Zi =
energy, geE a steric factor number for state i:
2nX# (2nkT/,t.#i2
,
where jzX is the concentration of the inert gas
Volume
14, number
2
rtib = [P’19(T))(1 -exp{-(El
CHEhWAL
-E,)/kT}
)I-’
.
PHYSICS LETTERS
1.5 hfay
(3)
This formula is exact only for the harmonic oscilIator, but the error incurred by using it here should not be larger than the experimenta! uncertainty in the determination of 7vib. The values of P,,,(T)> calcuiated by means of eq. (1) are a factor of 5 less than the ones obtained using (3) and the data from ref. [17]. This discrepancy is being investigated at the moment. In the present work we have used the value of ‘PLO(T)) obtained from (3) and: ~7’ wb = 10” * e&p(-0/?/3)/(4.
1 A) set-l ,
where 0 = 100 5 2.6,A = 3.9 + 0.8 and the pressure is 1 atm [17]. For cp,,jc3) we have: (Pii(
= Pii(T)) exp {- (Ej--E$icT}.
The dissociation constant kd is given by hO/rzX where A0 is the lowest eigenvalue of the matrix M [15] . Mti = exp (- i(Ei-
Ej)jkT)
3. Results and discussion In fig. 5 we have shown the calculated values of /cd for A = 3.9 and 8 = 100. Curve I was calculated using single-quantum transitions only, while curve II was obtained with the inclusion of double-quantum transitions. In both calculations dissociation could occur from the three uppermost levels. If dissociation were restricted to occuring from the top level curve I wculd be lowered by a factor of 2-3, while cuive II would only be about 10% lower. This is a result of the inclu-. sion of double-quantum transitions, which facilitate the transfer of molecuies among the upper levels. The curves are sensitive to change in PlO(T). If one used 8 = 102.6 and A = 3.1 in (3) both curves rise by a factor of 1.3 to 1.5. Inclusion of triple,and higherorder quantum transitions raise curve II by a factor which increases with temperature. This gives a better correspondence between the calculated and the experimen tally determined [ 18,191 values of &. The effect of the multiplequantum transitions is that they increase the rate at which the mo!ecules climb the vi-
F&. 5. Calculated values of the dissociation constant for hydrwpn dilured by U-I organ gas (solid lines) compared with x~lues obtained from experimental data of refs. [18,19] (dashed lines).
brational ladder, but more important is their influence near the dissociation limit. Since multiple-quantum transitions are very likely to occur in this region they indirectly lower the limit from which dissociation can occur, i.e., they lower the activation energy. The main difference between the present work and previous works is the inclusion of realistic values of doublequantum transitions. In order to determine these transitions it is necessary to integrate a Iarge set of coupled equations, and future work should therefore be concerned with the following two problems (1) integration of coupled equations with the inclusion of continuum states, in order to check the extrapolations made; (2) other, and easier, methods to evaluate the multiple-quantum transitions, e.g., the perturbed stationary state treatment 1201 .
277
Volume,lld, number 2 .,, :
CHEMICAL PHYSICS LETTERS
Ackno&le,&emei&
:.. ‘@me author wishes to thank Dr. Cari Nyeland for stitiulating to .thank
dis&ssions
and sugg&tiotis.
I also want
Dr.3.E. Nielsen and Dr. Anne M. WooIIey for helpful discussions on preparing the manuscript.
References [lj H.O. Ritchaid, J. Phys. Chem. 65 (1961) 504. [21 ,JN. Rich Bnd I. Classman, AGARD Conf. Proc. 7 (1’966). [31 J.E. DC% and D.G. Jones, 3. Chent. Phys. 55 (1971) 1531. [4] R.N. Schwartz,Z.1.Siawsksand K.F. Iierzfeid, 3. Chem. Phys. 20 (1952) 1591. 151 R.N. are, J. Chem. Phys. 40 (196411934. [6f EA. Mason and L. ~~onchick, Adma. Chem. Fhys. f 2 (1967)‘329. [71 T.E. Sharp and D. Rapp, J. Chem. Phys. 43 (1965) 1233.
238
15 l%y 1972
J.W; Cooley, Math. Camp. 15 /I9611 363. 1. Amdur and J.E. Jordan, Advan. Chem. Phys. IO 0966) 29. i. Amdur atid A.L. Smith, J. Chem. Phys. 48 (1968) 565, K.F. Henfeld and T-A. Litouitz, Absorption of dispersoon of ultrasonic waves (Academic Fess, New York, 1959). M. Kraussand F-H: Mies. J. Chem. Phys. 42 (1965) 2703. r131 D. Rappand T.E. Sharp, J. Chem. Phys. 38 (1963) 2641. 1141 K. Tagayana@, Suppl. Rogr. Theoret. Phys. 25 (1963) 1. WI C.A. Bnu, J. Chem. Phys. 47 (1967) 1153. f161 O.K. Rice. Statistical mechanics, thermodynamics and kineticsIFreaman.SanFrancisco. 19671. fi7f J.H. Kiefer and R.W. Lutz, J. Chem. Phys. 44 (1966) 668. R.W. Patch, J. Chem. P&s. 36 (i962) 1919. f:!f E.A. Sutton, J. Chcm. Phys. 36 (2962) 2923. [ZOJ C.‘NycIand and A. Hunding, Chem. Phys. Letters 5 (1970) 143.