The rate constant for radiative association of HF: Comparing quantum and classical dynamicsa) Magnus Gustafsson, M. Monge-Palacios, and Gunnar Nyman Citation: The Journal of Chemical Physics 140, 184301 (2014); doi: 10.1063/1.4874271 View online: http://dx.doi.org/10.1063/1.4874271 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The reaction rate constant in a system with localized trajectories in the transition region: Classical and quantum dynamics J. Chem. Phys. 107, 7787 (1997); 10.1063/1.475092 Centroiddensity quantum rate theory: Dynamical treatment of classical recrossing J. Chem. Phys. 99, 1674 (1993); 10.1063/1.465284 A combined quantumclassical dynamics method for calculating thermal rate constants of chemical reactions in solution J. Chem. Phys. 96, 8136 (1992); 10.1063/1.462316 Time−dependent dynamical symmetries, associated constants of motion, and symmetry deformations of the Hamiltonian in classical particle systems J. Math. Phys. 16, 548 (1975); 10.1063/1.522553 Dielectric constant and association in liquid HF J. Chem. Phys. 59, 1545 (1973); 10.1063/1.1680219
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THE JOURNAL OF CHEMICAL PHYSICS 140, 184301 (2014)
The rate constant for radiative association of HF: Comparing quantum and classical dynamicsa) Magnus Gustafsson,b) M. Monge-Palacios,c) and Gunnar Nyman Department of Chemistry and Molecular Biology, University of Gothenburg, 41296 Gothenburg, Sweden
(Received 23 August 2013; accepted 21 April 2014; published online 8 May 2014) Radiative association for the formation of hydrogen fluoride through the A1 → X1 + and X1 + → X1 + transitions is studied using quantum and classical dynamics. The total thermal rate constant is obtained for temperatures from 10 K to 20 000 K. Agreement between semiclassical and quantum approaches is observed for the A1 → X1 + rate constant above 2000 K. The agreement is explained by the fact that the corresponding cross section is free of resonances for this system. At temperatures below 2000 K we improve the agreement by implementing a simplified semiclassical expression for the rate constant, which includes a quantum corrected pair distribution. The rate coefficient for the X1 + → X1 + transition is calculated using Breit–Wigner theory and a classical formula for the resonance and direct contributions, respectively. In comparison with quantum calculations the classical formula appears to overestimate the direct contribution to the rate constant by about 12% for this transition. Below about 450 K the resonance contribution is larger than the direct, and above that temperature the opposite holds. The biggest contribution from resonances is at the lowest temperature in the study, 10 K, where it is more than four times larger than the direct. Below 1800 K the radiative association rate constant due to X1 + → X1 + transitions dominates over A1 → X1 + , while above that temperature the situation is the opposite. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4874271] H(2 S) + F(2 P ) → HF(X1 + ) → HF(X1 + ) + ¯ω
I. INTRODUCTION
Molecular formation is an integral part of chemistry. For the formation of a molecule from two colliding fragments it is required that energy is expelled so that a bound state can be reached. Under normal conditions the most common process is that a third body carries away the excess energy. In low density and dust-poor regions of interstellar space, however, the presence of a third body is unlikely and alternative processes need to be taken into account.1 One of these is radiative association, in which a photon is emitted during the collision. This paper concerns calculations of cross sections and thermal rate constants for radiative association of the HF diatomic molecule from colliding H(2 S) and F(2 P) atoms. Neglecting spin-orbit coupling the system presents four electronic molecular states: X1 + , 13 + , A1 , and a3 , where only X1 + supports bound vibrational states.2 Triplet states are dipole forbidden to radiatively form the singlet ground state when the spin-orbit coupling is weak, like in the present case.3 Under this assumption radiative association of HF can evolve toward the X1 + ground electronic state from the A1 excited state, or simply follow the X1 + ground state without any electronic transition, i.e., either of the two reactions H(2 S) + F(2 P ) → HF(A1 ) → HF(X1 + ) + ¯ω,
(1)
a) This work is dedicated to the late Professor Alan S. Dickinson for his
contributions to the theory of radiative association. b) Electronic mail: [email protected] c) Current address: Departamento de Química Física, Universidad de
Extremadura, 06071 Badajoz, Spain. 0021-9606/2014/140(18)/184301/10/$30.00
(2)
may occur. Radiative association cross sections can be calculated using different methods, e.g., quantum mechanical perturbation theory4 and the semiclassical5, 6 and classical7 approaches. The semiclassical and classical methods are based on classical trajectories and do not include the resonance contribution to the cross section. This contribution arises from (i) quantum mechanical tunneling through the barrier in the effective potential, giving rise to quasi-bound states whereby the system can tunnel back through the barrier to reactants and (ii) over-barrier resonances, i.e., broad structures stemming from the proximity to the peak of the effective potential barrier. The resonance contribution is accounted for in perturbation theory whose cross section generally shows sharp resonance peaks superimposed on a smooth baseline produced by direct (nonresonant) radiative association. Quantum mechanical perturbation theory as it is described below in Sec. II A, however, tends to overestimate the resonance contribution to the cross section, see e.g., Ref. 8, and references therein, and Refs. 6, 9, and 10. For calculations of rate constants when resonances play a significant role Breit–Wigner theory11, 12 may be applied to the perturbation theory cross section.9, 13, 14 Alternatively, if the over-barrier resonance contribution can be considered to be small, we proposed in our 2009 paper15 that classical trajectory methods may be employed for the direct radiative association and that the resonances are accounted for separately with Breit–Wigner theory. In this work we apply some of the methods mentioned above to obtain thermal rate constants for reactions (1) and (2). In addition, we implement two simplified
140, 184301-1
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184301-2
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semiclassical expressions to calculate rate constants for reaction (1) for comparison with those obtained using the conventional semiclassical approach5, 6 and perturbation theory.4 The first simplified expression, which contains no quantum dynamics, dates back to the 1946 work by Kramers and ter Haar.16 The second is based on the Kramers and ter Haar formula, but we have accounted for some quantum effects through the Wigner–Kirkwood corrected pair distribution function.17, 18 The formation of HF through radiative association has been studied to some extent before. First, Puy et al.19 calculated rate constants for reaction (1) by applying the principle of detailed balance to the (X1 + , v = 0) → A1 photodissociation cross section, which was previously obtained by Brown and Balint-Kurti.3 Second, Talbi and BacchusMontabonel20 estimated rate constants for reaction (2) in the temperature range 400–2000 K through a scheme based on the ideas of Bates’s semiclassical approach.5 They found that the rate constants are about 10−20 –10−19 cm3 s−1 , depending on the temperature and the HF rotational level considered. In those two studies the rate coefficients were obtained for particular vibrational and rotational states of the HF molecule, and so they cannot be directly compared with those obtained in the present work, which cover all rovibrational product states. Finally, in recent work, Gustafsson7 studied the direct contribution to reaction (2) in order to test the classical approach for processes without electronic transitions. The aim of that study was not to obtain an accurate rate constant, since for this purpose resonances should also be taken into account.
II. THEORY
The thermal rate constant for the formation of HF by radiative association through reaction (1) or (2) can be written as 1 3/2 8 1/2 k→ (T ) = μπ kB T ∞ × Eσ→ (E)e−E/kB T dE, (3) 0
where μ is the reduced mass, kB is the Boltzmann constant, and E is the collision energy. and indicate the orbital electronic angular momentum, projected on the molecular axis, of the initial state (of approach) and the final (bound) state, respectively. The cross section σ→ can be evaluated with different methodologies. In Secs. II A–II D we summarise four of those. In Sec. II E we present the two simplified semiclassical expressions for calculating the A1 → X1 + rate constant, which bypass the explicit computation of the cross section. Finally, in order to aid the analysis of our results, in Sec. II F we derive a quantum dynamical equivalent of the rate constants presented in Sec. II E.
A. Perturbation theory
The quantum mechanical perturbation treatment for radiative association of a diatom leads to the Fermi golden rule
J. Chem. Phys. 140, 184301 (2014) TABLE I. Hönl–London factors, SJ → J , from Ref. 24 and statistical weights, P , for HF. The Hönl–London factors are parity averaged for the case A1 → X1 + . A1 → X1 +
X1 + → X1 +
(J + 1)/2 (2J + 1)/2 J/2
J 0 J+1
2/12
1/12
SJ → J −1 SJ → J SJ → J +1 P
cross section4, 21 σ→ (E) =
σ→ v J (E),
(4)
v J
where the partial cross section is defined as σ→ v J (E) =
h2 1 2 P 3 3 (4π 0 )c 2μE 3 2 × ωE v J SJ → J MEJ, v J
(5)
J
and c is the speed of light, ωE v J is the angular frequency of the emitted photon, and SJ → J are the Hönl–London factors given in Table I. The initial rotational quantum number is labeled with J and the final vibrational and rotational quantum numbers with v and J , respectively. The statistical weight factor for approach in the molecular electronic state, , is given by (see, e.g., Ref. 22) (2S + 1)(2 − δ0, ) , (6) P = (2LH + 1)(2SH + 1)(2LF + 1)(2SF + 1) where LH = 0, SH = 1/2, LF = 1, SF = 1/2 are the electronic orbital angular momentum and spin quantum numbers of the atoms and = 0 and 1 for the + and molecular states, respectively. We consider only singlet states of HF, so that the total electronic spin is S = 0. The values of P are given in Table I. Finally, the dipole matrix element is defined as ∞ FEJ (R) D (R) (7) MEJ, v J = v J (R) dR, 0
where D (R) is the matrix element of the dipole moment operator between the two electronic states as a function of the (R) is the continuum wave funcinternuclear distance R. FEJ tion of the initial state for the partial wave J, normalized to the collision energy E (see, e.g., Landau and Lifshitz23 and Sec. II F) and v J (R) is a bound state wave function, normalized to unity. In Eqs. (4) and (7) the approximation for the total angular momentum and spin that J − S ≈ J has been used (see, e.g., Ref. 4). B. Semiclassical approach for transitions to a lower electronic state
In the semiclassical theory, which is applicable to the formation of HF through reaction (1), the cross section is6 μ 1/2 σ→ (E) = 4π P 2E ∞ ∞ AEb → (R) dR db × b , (1 − V (R)/E − b2 /R 2 )1/2 0 Rc (b,E) (8)
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184301-3
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J. Chem. Phys. 140, 184301 (2014)
where b is the impact parameter, for which Rc (b, E) is the outer classical turning point on the effective potential V + Eb2 /R 2 . The transition probability AEb → (R) (with dimension time−1 ) is defined as ⎧ if E < V (R) − V (R) and ⎪ ⎪ ⎨ A→ (R) 2 0
⎩ √2
π
−y √ 0
xe−x dx
(18) otherwise
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J. Chem. Phys. 140, 184301 (2014)
is the incomplete gamma function. The term fb (V /kB T ) in the brackets serves to include radiative transitions to bound states only (i.e., it expresses the energy condition E < V (R) − V (R) in Eq. (9)) while fb (V /kB T ) excludes transitions from bound states on the upper potential. A version of Eq. (17) appropriate for radiative transitions to both bound and continuum states was later derived and implemented by Julienne30 and Dickinson.31 The formula for the rate constant in Eq. (17) was corrected in the work by Bates5 for the case when there is a centrifugal barrier on the upper potential curve. Calculations using Eq. (17) include radiative association from quasibound states in a classical manner, and it should be used with caution if resonances play a big role. Here we have ignored Bates’s corrections since, as we will see below, the upper state (A1 ) potential energy curve is purely repulsive, apart from a shallow well with a depth of about 1.3 meV. If this well produces any resonances they ought to contribute noticeably only at rather low temperatures. So, for the system we study here we expect the formula in Eq. (17) to work well over some range of temperatures. From the derivation of the quantum dynamical equivalent of Eq. (17) below in Sec. II F it becomes apparent that the function exp( −VkBT(R) ) in Eq. (17) is the classical pair distribution function, g(R, T). At low temperatures there may be a significant difference between the classical and quantum pair distributions. To include this effect in an approximate manner we have implemented the lowest-order Wigner–Kirkwood corrected pair distribution17, 18
gW K (R, T ) = e
−V (R) kB T
d 2 V (R) ¯2 1− 12μ(kB T )2 dR 2
1 2 dV (R) − + R dR 2kB T
dV (R) dR
2 (19)
and computed the corresponding quantum corrected rate constant
∞
k→ (T ) = 4π P
R 2 A→ (R) gW K (R, T )
0
V (R) V (R) − fb dR. (20) × fb kB T kB T In addition to the lowest-order Wigner-Kirkwood correction introduced in Eq. (19) additional corrections (O(¯4 )) may be implemented for improved low temperature behaviour.18 For the purposes of our study, however, the quantum correction in Eq. (19) is sufficient.
F. Distorted wave optical potential approach for transitions to a lower electronic state
In order to obtain a quantum dynamical version of the rate constant presented in Sec. II E we will exploit the optical potential method.6, 14, 26 In that approach, in one dimension,
the Schrödinger equation ¯2 (J (J + 1) − 2 ) ¯2 d 2 + + V (R) − 2 2μ dR 2μR 2 i¯ EJ → − A→ (R) − E FEJ (R) = 0 2
(21)
is solved so that the radiative deactivation cross section σ→ (E) =
π ¯2 → (2J + 1)(1 − e−4ηEJ ) P 2μE J
(22)
→ is the imaginary part of the phase can be computed. ηEJ → shift of FEJ (R) and the transition probability in Eq. (21) is defined as6 ⎧ if E < V (R) − V (R) and ⎪ ⎪ ⎨ A→ (R) 2 J (J +1) V (R) + ¯ 2μR
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THE JOURNAL OF CHEMICAL PHYSICS 140, 184301 (2014)
The rate constant for radiative association of HF: Comparing quantum and classical dynamicsa) Magnus Gustafsson,b) M. Monge-Palacios,c) and Gunnar Nyman Department of Chemistry and Molecular Biology, University of Gothenburg, 41296 Gothenburg, Sweden
(Received 23 August 2013; accepted 21 April 2014; published online 8 May 2014) Radiative association for the formation of hydrogen fluoride through the A1 → X1 + and X1 + → X1 + transitions is studied using quantum and classical dynamics. The total thermal rate constant is obtained for temperatures from 10 K to 20 000 K. Agreement between semiclassical and quantum approaches is observed for the A1 → X1 + rate constant above 2000 K. The agreement is explained by the fact that the corresponding cross section is free of resonances for this system. At temperatures below 2000 K we improve the agreement by implementing a simplified semiclassical expression for the rate constant, which includes a quantum corrected pair distribution. The rate coefficient for the X1 + → X1 + transition is calculated using Breit–Wigner theory and a classical formula for the resonance and direct contributions, respectively. In comparison with quantum calculations the classical formula appears to overestimate the direct contribution to the rate constant by about 12% for this transition. Below about 450 K the resonance contribution is larger than the direct, and above that temperature the opposite holds. The biggest contribution from resonances is at the lowest temperature in the study, 10 K, where it is more than four times larger than the direct. Below 1800 K the radiative association rate constant due to X1 + → X1 + transitions dominates over A1 → X1 + , while above that temperature the situation is the opposite. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4874271] H(2 S) + F(2 P ) → HF(X1 + ) → HF(X1 + ) + ¯ω
I. INTRODUCTION
Molecular formation is an integral part of chemistry. For the formation of a molecule from two colliding fragments it is required that energy is expelled so that a bound state can be reached. Under normal conditions the most common process is that a third body carries away the excess energy. In low density and dust-poor regions of interstellar space, however, the presence of a third body is unlikely and alternative processes need to be taken into account.1 One of these is radiative association, in which a photon is emitted during the collision. This paper concerns calculations of cross sections and thermal rate constants for radiative association of the HF diatomic molecule from colliding H(2 S) and F(2 P) atoms. Neglecting spin-orbit coupling the system presents four electronic molecular states: X1 + , 13 + , A1 , and a3 , where only X1 + supports bound vibrational states.2 Triplet states are dipole forbidden to radiatively form the singlet ground state when the spin-orbit coupling is weak, like in the present case.3 Under this assumption radiative association of HF can evolve toward the X1 + ground electronic state from the A1 excited state, or simply follow the X1 + ground state without any electronic transition, i.e., either of the two reactions H(2 S) + F(2 P ) → HF(A1 ) → HF(X1 + ) + ¯ω,
(1)
a) This work is dedicated to the late Professor Alan S. Dickinson for his
contributions to the theory of radiative association. b) Electronic mail: [email protected] c) Current address: Departamento de Química Física, Universidad de
Extremadura, 06071 Badajoz, Spain. 0021-9606/2014/140(18)/184301/10/$30.00
(2)
may occur. Radiative association cross sections can be calculated using different methods, e.g., quantum mechanical perturbation theory4 and the semiclassical5, 6 and classical7 approaches. The semiclassical and classical methods are based on classical trajectories and do not include the resonance contribution to the cross section. This contribution arises from (i) quantum mechanical tunneling through the barrier in the effective potential, giving rise to quasi-bound states whereby the system can tunnel back through the barrier to reactants and (ii) over-barrier resonances, i.e., broad structures stemming from the proximity to the peak of the effective potential barrier. The resonance contribution is accounted for in perturbation theory whose cross section generally shows sharp resonance peaks superimposed on a smooth baseline produced by direct (nonresonant) radiative association. Quantum mechanical perturbation theory as it is described below in Sec. II A, however, tends to overestimate the resonance contribution to the cross section, see e.g., Ref. 8, and references therein, and Refs. 6, 9, and 10. For calculations of rate constants when resonances play a significant role Breit–Wigner theory11, 12 may be applied to the perturbation theory cross section.9, 13, 14 Alternatively, if the over-barrier resonance contribution can be considered to be small, we proposed in our 2009 paper15 that classical trajectory methods may be employed for the direct radiative association and that the resonances are accounted for separately with Breit–Wigner theory. In this work we apply some of the methods mentioned above to obtain thermal rate constants for reactions (1) and (2). In addition, we implement two simplified
140, 184301-1
© 2014 AIP Publishing LLC
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184301-2
Gustafsson, Monge-Palacios, and Nyman
semiclassical expressions to calculate rate constants for reaction (1) for comparison with those obtained using the conventional semiclassical approach5, 6 and perturbation theory.4 The first simplified expression, which contains no quantum dynamics, dates back to the 1946 work by Kramers and ter Haar.16 The second is based on the Kramers and ter Haar formula, but we have accounted for some quantum effects through the Wigner–Kirkwood corrected pair distribution function.17, 18 The formation of HF through radiative association has been studied to some extent before. First, Puy et al.19 calculated rate constants for reaction (1) by applying the principle of detailed balance to the (X1 + , v = 0) → A1 photodissociation cross section, which was previously obtained by Brown and Balint-Kurti.3 Second, Talbi and BacchusMontabonel20 estimated rate constants for reaction (2) in the temperature range 400–2000 K through a scheme based on the ideas of Bates’s semiclassical approach.5 They found that the rate constants are about 10−20 –10−19 cm3 s−1 , depending on the temperature and the HF rotational level considered. In those two studies the rate coefficients were obtained for particular vibrational and rotational states of the HF molecule, and so they cannot be directly compared with those obtained in the present work, which cover all rovibrational product states. Finally, in recent work, Gustafsson7 studied the direct contribution to reaction (2) in order to test the classical approach for processes without electronic transitions. The aim of that study was not to obtain an accurate rate constant, since for this purpose resonances should also be taken into account.
II. THEORY
The thermal rate constant for the formation of HF by radiative association through reaction (1) or (2) can be written as 1 3/2 8 1/2 k→ (T ) = μπ kB T ∞ × Eσ→ (E)e−E/kB T dE, (3) 0
where μ is the reduced mass, kB is the Boltzmann constant, and E is the collision energy. and indicate the orbital electronic angular momentum, projected on the molecular axis, of the initial state (of approach) and the final (bound) state, respectively. The cross section σ→ can be evaluated with different methodologies. In Secs. II A–II D we summarise four of those. In Sec. II E we present the two simplified semiclassical expressions for calculating the A1 → X1 + rate constant, which bypass the explicit computation of the cross section. Finally, in order to aid the analysis of our results, in Sec. II F we derive a quantum dynamical equivalent of the rate constants presented in Sec. II E.
A. Perturbation theory
The quantum mechanical perturbation treatment for radiative association of a diatom leads to the Fermi golden rule
J. Chem. Phys. 140, 184301 (2014) TABLE I. Hönl–London factors, SJ → J , from Ref. 24 and statistical weights, P , for HF. The Hönl–London factors are parity averaged for the case A1 → X1 + . A1 → X1 +
X1 + → X1 +
(J + 1)/2 (2J + 1)/2 J/2
J 0 J+1
2/12
1/12
SJ → J −1 SJ → J SJ → J +1 P
cross section4, 21 σ→ (E) =
σ→ v J (E),
(4)
v J
where the partial cross section is defined as σ→ v J (E) =
h2 1 2 P 3 3 (4π 0 )c 2μE 3 2 × ωE v J SJ → J MEJ, v J
(5)
J
and c is the speed of light, ωE v J is the angular frequency of the emitted photon, and SJ → J are the Hönl–London factors given in Table I. The initial rotational quantum number is labeled with J and the final vibrational and rotational quantum numbers with v and J , respectively. The statistical weight factor for approach in the molecular electronic state, , is given by (see, e.g., Ref. 22) (2S + 1)(2 − δ0, ) , (6) P = (2LH + 1)(2SH + 1)(2LF + 1)(2SF + 1) where LH = 0, SH = 1/2, LF = 1, SF = 1/2 are the electronic orbital angular momentum and spin quantum numbers of the atoms and = 0 and 1 for the + and molecular states, respectively. We consider only singlet states of HF, so that the total electronic spin is S = 0. The values of P are given in Table I. Finally, the dipole matrix element is defined as ∞ FEJ (R) D (R) (7) MEJ, v J = v J (R) dR, 0
where D (R) is the matrix element of the dipole moment operator between the two electronic states as a function of the (R) is the continuum wave funcinternuclear distance R. FEJ tion of the initial state for the partial wave J, normalized to the collision energy E (see, e.g., Landau and Lifshitz23 and Sec. II F) and v J (R) is a bound state wave function, normalized to unity. In Eqs. (4) and (7) the approximation for the total angular momentum and spin that J − S ≈ J has been used (see, e.g., Ref. 4). B. Semiclassical approach for transitions to a lower electronic state
In the semiclassical theory, which is applicable to the formation of HF through reaction (1), the cross section is6 μ 1/2 σ→ (E) = 4π P 2E ∞ ∞ AEb → (R) dR db × b , (1 − V (R)/E − b2 /R 2 )1/2 0 Rc (b,E) (8)
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J. Chem. Phys. 140, 184301 (2014)
where b is the impact parameter, for which Rc (b, E) is the outer classical turning point on the effective potential V + Eb2 /R 2 . The transition probability AEb → (R) (with dimension time−1 ) is defined as ⎧ if E < V (R) − V (R) and ⎪ ⎪ ⎨ A→ (R) 2 0
⎩ √2
π
−y √ 0
xe−x dx
(18) otherwise
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184301-4
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J. Chem. Phys. 140, 184301 (2014)
is the incomplete gamma function. The term fb (V /kB T ) in the brackets serves to include radiative transitions to bound states only (i.e., it expresses the energy condition E < V (R) − V (R) in Eq. (9)) while fb (V /kB T ) excludes transitions from bound states on the upper potential. A version of Eq. (17) appropriate for radiative transitions to both bound and continuum states was later derived and implemented by Julienne30 and Dickinson.31 The formula for the rate constant in Eq. (17) was corrected in the work by Bates5 for the case when there is a centrifugal barrier on the upper potential curve. Calculations using Eq. (17) include radiative association from quasibound states in a classical manner, and it should be used with caution if resonances play a big role. Here we have ignored Bates’s corrections since, as we will see below, the upper state (A1 ) potential energy curve is purely repulsive, apart from a shallow well with a depth of about 1.3 meV. If this well produces any resonances they ought to contribute noticeably only at rather low temperatures. So, for the system we study here we expect the formula in Eq. (17) to work well over some range of temperatures. From the derivation of the quantum dynamical equivalent of Eq. (17) below in Sec. II F it becomes apparent that the function exp( −VkBT(R) ) in Eq. (17) is the classical pair distribution function, g(R, T). At low temperatures there may be a significant difference between the classical and quantum pair distributions. To include this effect in an approximate manner we have implemented the lowest-order Wigner–Kirkwood corrected pair distribution17, 18
gW K (R, T ) = e
−V (R) kB T
d 2 V (R) ¯2 1− 12μ(kB T )2 dR 2
1 2 dV (R) − + R dR 2kB T
dV (R) dR
2 (19)
and computed the corresponding quantum corrected rate constant
∞
k→ (T ) = 4π P
R 2 A→ (R) gW K (R, T )
0
V (R) V (R) − fb dR. (20) × fb kB T kB T In addition to the lowest-order Wigner-Kirkwood correction introduced in Eq. (19) additional corrections (O(¯4 )) may be implemented for improved low temperature behaviour.18 For the purposes of our study, however, the quantum correction in Eq. (19) is sufficient.
F. Distorted wave optical potential approach for transitions to a lower electronic state
In order to obtain a quantum dynamical version of the rate constant presented in Sec. II E we will exploit the optical potential method.6, 14, 26 In that approach, in one dimension,
the Schrödinger equation ¯2 (J (J + 1) − 2 ) ¯2 d 2 + + V (R) − 2 2μ dR 2μR 2 i¯ EJ → − A→ (R) − E FEJ (R) = 0 2
(21)
is solved so that the radiative deactivation cross section σ→ (E) =
π ¯2 → (2J + 1)(1 − e−4ηEJ ) P 2μE J
(22)
→ is the imaginary part of the phase can be computed. ηEJ → shift of FEJ (R) and the transition probability in Eq. (21) is defined as6 ⎧ if E < V (R) − V (R) and ⎪ ⎪ ⎨ A→ (R) 2 J (J +1) V (R) + ¯ 2μR
