##
1Details and consequences of the nonadiabatic coupling in the Cl(^{2}P) + H_{2} reaction
Type: Object |
Advantage: None |
Novelty: New |
ConceptID: Obj1

1

Details and consequences of the nonadiabatic coupling in the Cl(

^{2}P) + H_{2}reaction
Type: Object |
Advantage: None |
Novelty: New |
ConceptID: Obj1

2

In an investigation of nonadiabaticity in the Cl + H

_{2}reaction we examine the various coupling terms responsible.
Type: Goal |
Advantage: None |
Novelty: None |
ConceptID: Goa1

3

By neglecting various of these terms, we show that the spin–orbit coupling in the Cl atom is primarily responsible for non Born–Oppenheimer effects in this reaction, and that the anisotropies in the contributing electronic potential energy surfaces as well as Coriolis terms, are considerably weaker in their effect.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res1

4

Neglect of all coupling except spin–orbit leads to little change in the calculated reaction probabilities both for the Born–Oppenheimer allowed [Cl(

^{2}P_{3/2}) + H_{2}] and Born–Oppenheimer forbidden [Cl(^{2}P_{1/2}) + H_{2}] reactions.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res2

5

This implies that one can safely neglect the coupling between different values of the electronic projection quantum number

*ω*.
Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con1

6

Consequently, the computational effort involved in an accurate quantum scattering treatment of this (and, hopefully, other similar reactions) can be substantially reduced.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con1

## Introduction

7

Because of their experimental accessibility, the reactions of F and Cl with H

_{2}and its isotopomers have become paradigms for triatomic abstraction reactions.^{1,2}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac1

8

Experimental interest in the Cl + H

_{2}reaction goes back more than 100 years.^{1}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac1

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac1

10

Approach of the H

_{2}molecule to a Cl atom splits the degeneracy of the^{2}P state.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac2

11

As shown schematically in Fig. 1, two electronic states (1

^{2}A′ and 1^{2}A″;^{2}Σ_{1/2}^{+}and^{2}Π_{3/2}in linear geometry) correlate adiabatically with the ground-state atomic reactant (^{2}P_{3/2}) while a third state (2^{2}A′;^{2}Π_{1/2}in linear geometry) correlates adiabatically with the excited-state atomic reactant (^{2}P_{1/2}).^{10}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac2

12

Of these, only the 1

^{2}A′ electronic state correlates with the electronic ground state of the products [HCl(X^{1}Σ^{+}) + H(^{2}S)].
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac2

13

The two other electronic states correlate with electronically excited states of the products [HCl(a

^{3}Π) + H(^{2}S)] which are considerably higher in energy^{11}and, consequently, energetically inaccessible at low to moderate collision energies.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac2

14

It is common to assume that the motion of the atoms in a chemical reaction occurs on a single potential energy surface (PES).

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac3

15

This description is a consequence of the Born–Oppenheimer (BO) approximation.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac3

16

For bimolecular reactive collisions this assumption is supported by the majority of past experimental work.

^{12}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac3

17

In recent molecular beam experiments, Liu and co-workers

^{13}used two different Cl atom sources to separate the reactivity of the two SO states of the Cl atom.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac4

18

Strikingly, except at the lowest collision energies, they conclude that the excited SO state has a significantly larger reactive cross section.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac4

19

As can be seen in Fig. 1, reaction of the spin–orbit excited Cl atom (

^{2}P_{1/2}) can occur only by a non-Born–Oppenheimer transition from the non-reactive^{2}Π_{1/2}PES to the reactive^{2}Σ+1/2 PES.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac5

20

When the two PESs are widely separated, there is little likelihood that this transition can occur.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac5

21

Hence, non-adiabaticity in the Cl + H

_{2}reaction, possibly to the extent suggested by the experiments of Liu and co-workers, must be a result of couplings between the various PESs in the entrance channel of the reaction.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac5

22

Theoretical investigations of the role of excited electronic surfaces in the F + H

_{2}reaction date back to the pioneering work of Tully.^{14}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac6

23

Several years ago, we presented the framework for the essentially exact quantum determination of cross sections for abstraction reactions involving an atom in a

^{2}P electronic state.^{15}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac7

24

This treatment involves six three-dimensional hypersurfaces, four of which describe the diabatic potential energy functions and two of which describe the coordinate dependence of the spin–orbit coupling.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac7

25

All Coriolis coupling terms are included.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac7

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac8

27

Calculated integral cross sections for the non-Born–Oppenheimer reaction of the spin–orbit excited atom, which are a measure of the degree of non-adiabaticity in the reaction, were found to be markedly smaller than those for the Born–Oppenheimer allowed reaction of the ground spin–orbit state.

^{5}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac8

28

The degree of non-adiabaticity was even less than found in our earlier investigation of the F + H

_{2}reaction.^{15,16}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac8

29

This is not surprising, since the spin–orbit splitting in the Cl atom (882.35 cm

^{–1}= 2.528 kcal mol^{–1}) is more than twice as large as that of the F atom (404 cm^{–1}).^{17}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac8

30

Only at the lowest collision energies, where the ground-state channel is suppressed by the high barrier (Fig. 1), will reaction of the spin–orbit excited atom dominate.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac8

31

These theoretical results stand in direct contrast to the conclusions of Liu and co-workers.

^{13}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac8

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac9

33

Differential cross sections for the adiabatically-forbidden reaction of Cl* (

^{2}P_{1/2}) are backward peaked, similar to those for reaction of the ground spin-obit state, but are much smaller in magnitude.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac9

Type: Motivation |
Advantage: None |
Novelty: None |
ConceptID: Mot1

35

Coupling between the adiabatically allowed and forbidden pathways is due to (

*i*) the spin–orbit Hamiltonian, (*ii*) electronic mixing of the two states of A′ symmetry, and (*iii*) Coriolis coupling between the rotational motion of the nuclei and the spin and electronic-orbital angular momenta of the Cl atom.
Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con2

36

One great advantage of a theoretical simulation, as compared to experiment, is that each of these terms may be altered (or eliminated) in turn to allow their effect to be examined.

Type: Method |
Advantage: Yes |
Novelty: None |
ConceptID: Met1

37

This is the goal of the present article.

Type: Goal |
Advantage: None |
Novelty: None |
ConceptID: Goa2

## Hamiltonian, potential surfaces, and couplings

38

We write the total Hamiltonian for collision of Cl (

^{2}P) with H_{2}(or any other diatomic) molecule as*H*(*R⃑*,*r⃑*,*q⃑*) =*T*_{nuc}(*R⃑*,*r⃑*) +*H*_{el}(*q⃑*;*R⃑*,*r⃑*) +*H*_{so}(*q⃑*;*R⃑*,*r⃑*).Here*q⃑*is a collective notation for the electronic coordinates,*H*_{el}is the electronic Hamiltonian, which, in the Born–Oppenheimer sense, depends parametrically on the positions of the three nuclei, and*H*_{so}is the spin–orbit Hamiltonian, which is not included in*H*_{el}.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod1

39

The nuclear coordinates

*R⃑*and*r⃑*in eqn. (1) designate the Jacobi vectors^{19}in any one of the three chemical arrangements (Cl + HH, H + HCl, or HCl + H).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod1

40

In terms of the mass-scaled Jacobi coordinates

*S*and*s*, one defines Delves hyperspherical coordinates*ρ*and*θ.*^{19}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod1

41

At each value of the hyperradius

*ρ*we expand the total wavefunction in an overcomplete basis of rotational-vibrational-electronic wavefunctions in each arrangement.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod1

42

In the Cl + H

_{2}arrangement there are six electronic states.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod2

43

For a halogen atom with a p

^{5}electron occupancy these states correspond to the three spatial orientations of the*p*hole and the two possible spin-projection quantum numbers.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod2

44

For the HCl, the lowest excited state is of Π symmetry, and lies far above the ground state (X

^{1}Σ^{+}) in the region of the molecular minimum.^{11}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod2

45

If we neglect these excited Π states, then, in each H + HX arrangement, we need retain only two states, which correspond to the HX molecule in its ground electronic state (

^{1}Σ^{+}) combined with the two possible spin-projection quantum numbers of the H atom.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod2

46

In each arrangement the basis functions are

^{15,20,21}Here*J*is the total angular momentum, with projection*M*along the space-frame*z*-axis and projection*K*along the Jacobi vector*R⃑*of the relevant arrangement.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod3

47

The quantum number

*j*designates the rotational angular momentum of the diatomic moiety in the same arrangement, with projection*k*along*R⃑*.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod3

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod3

49

The quantities

*λ*and*σ*in eqn. (2) are the projection of the electronic orbital and spin angular momenta along*R⃑*.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod3

50

Since we consider here only doublet electronic states, the spin part of the wavefunction corresponds to

*s*= 1/2 and*σ*= ±1/2.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod3

51

The projection quantum numbers are related as follows

*K*=*k*+*λ*+*σ*.To solve the close-coupled reactive scattering equations, one must first construct surface functions in each sector,^{20,24}by diagonalizing the total Hamiltonian, exclusive of the kinetic operator corresponding to radial motion in the hyperradius*ρ*, in the multiple arrangement basis consisting of the states defined by eqn. (2).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod4

52

In addition to the electrostatic interaction potential and the spin–orbit Hamiltonian, it is also necessary to determine matrix elements of the orbital angular momentum

*L*^{2}= (**–***J***–***l***–***s***)***j*^{2}.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod4

53

In evaluating the matrix elements of the electrostatic interaction potential one first integrates over the electronic coordinates to obtain formally 〈

*λ*′*σ*′|*H*_{el}(*q⃑*;*R**⃑*,*r⃑*)|*λ**σ*〉 = δ_{σσ′}*V*_{λ′λ}(*ρ*,*θ*,*γ*).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod4

54

Similarly, the matrix elements of the spin–orbit interaction potential involve the initial determination of the electronic matrix elements 〈

*λ*′*σ*′|*H*_{so}(*q⃑*;*R⃑*,*r⃑*)|*λ**σ*〉 =*W*_{λ′σ′λσ}(*ρ*,*θ*,*γ*).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod4

55

Once these matrix elements, which are functions of the three internal coordinates, are evaluated, the scattering calculations are no different from that of a triatomic reaction on a single potential energy surface.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod4

56

Capecchi and Werner

^{9}used internally-contracted multi-reference configuration-interaction (IC-MRCI) calculations, based on state-averaged (three-state) complete active space self-consistent field (CASSCF) eference wavefunctions with very large atomic basis sets, to determine the three electronically adiabatic Cl + H_{2}PESs in the reactant arrangement: 1A′, 2A′, and 1A″.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac10

57

These all correlate with Cl(

^{2}P) + H_{2}.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac10

58

These three adiabatic electronic states are the IC-MRCI approximations to the three lowest eigenfunctions of

*H*_{el}, namely*H*_{el}(*R⃑*,*r⃑*;*q⃑*)*ψ*_{i}(*R⃑*,*r⃑*;*q⃑*) =*E*_{i}(*R⃑*,*r⃑*)*ψ*_{i}(*R⃑*,*r⃑*;*q⃑*), where the subscript*i*= 1,2,3 designates the |1A′〉, |2A′〉, and |1A″〉 states.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac10

59

The coordinate dependence of the eigenvalues

*E*_{i}defines the three adiabatic PESs.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac10

60

We define the interaction potential by subtracting the energies of the reactants at infinite Cl–H

_{2}separation, so that*V*_{i}(*R⃑*,*r⃑*) =*E*_{i}(*R⃑*,*r⃑*) –*E*_{Cl}–*E*_{H2}(*r*_{e}).Here*E*_{Cl}designates the electronic energy of the Cl atom in its^{2}P state exclusive of the spin–orbit Hamiltonian.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod5

61

The zero of energy will be defined by the equilibrium internuclear separation of the H

_{2}molecule as determined in the IC-MRCI calculations, so that*V*_{i}(|*R⃑*| = ∞, |*r⃑*| =*r*_{e}) = 0,where*r*_{e}is the equilibrium internuclear separation of the H_{2}molecule.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod5

62

By analysis of the coefficients in the CI expansion of the ClH

_{2}electronic wavefunctions, the two PES's of*A*′ reflection symmetry are transformed to an approximate diabatic basis,^{25,26}in which the orientation of the p hole on the Cl atom is fixed with respect to the plane defined by the three atoms.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod6

63

Here we take

*R⃑*to define the*z*-axis and chose the*y*-axis to be perpendicular to the triatomic plane.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod6

64

We shall designate the diabatic states by the projections of the electronic orbital and spin angular momenta along

*R⃑*.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod6

65

We use the compact Cartesian notation |Π

_{x}〉, |Π_{y}〉, and |Σ〉.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod6

66

The two adiabatic states of A′ symmetry correspond to a 2 × 2 rotation of the |Π

_{x}〉 and |Σ〉 diabatic states, namely whereHere the transformation angle*ζ*depends on*R*,*θ*and*γ*(or, equivalently,*ρ*,*s*and*γ*.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod6

67

Since there is no coupling between the |Π

_{y}〉 state, of A″ reflection symmetry, with the |Π_{x}〉 and |Σ〉 states, of*A*′ reflection symmetry, the adiabatic and diabatic states of A″ reflection symmetry are identical.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod6

68

Alternatively, we can define the diabatic states in terms of signed-

*λ*, rather than Cartesian, projections, namely |Π_{1}〉, |Π_{–1}〉, and |Σ〉.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod7

69

These signed –

*λ*states are those that appear above in the expansion of the scattering wavefunction.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod7

70

Note that the state we designate as “Σ” corresponds to

*λ*= 0.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod7

71

As presented in detail in an earlier paper,

^{15}the matrix of the interaction potential, in the 6 × 6 basis defined by the three Cartesian diabatic states and the two possible spin projections (which are also defined with respect to*R⃑*) can be described in terms of three diagonal, electronically diabatic PES's:*V*_{xx},*V*_{yy}, and*V*_{zz},^{27}as well as a fourth PES,*V*_{xz}, which is the coupling between the two states of*A*′ symmetry.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod8

72

Each of these four PES's is a function of the three internal coordinates.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod8

73

Thus, the description of the interaction in the electronically diabatic basis, which involves these four potential energy functions, is equivalent to the description in the electronically adiabatic basis, which involves three potential energy surfaces (

*V*_{1A′},*V*_{2A′}, and*V*_{1A″}) plus the coordinate dependent mixing angle*ζ*.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod8

74

In the basis of the signed-

*λσ*states, which are used in eqn. (2) in the expansion of the scattering wavefunction, the matrix of the interaction potential, which we represented schematically as*V*_{λ′λ}(*ρ*,*θ*,*γ*) in eqn. (4), is Here, a bar over the state label designates a spin projection of –1/2 (*β*spin).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod8

75

The relation between the matrix elements in eqn. (11), and those determined by the

*ab initio*calculations, which are carried out in a Cartesian basis, are*V*_{Σ}=*V*_{zz},*V*_{Π}= (*V*_{yy}+*V*_{xx})/2,*V*_{2}= (*V*_{yy}–*V*_{xx})/2, and*V*_{1}=*V*_{xz}/2^{1/2}.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod9

76

In collinear geometry both

*V*_{1}and*V*_{2}vanish.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs1

77

In perpendicular geometry

*V*_{1}also vanishes while*V*_{2}goes through a relative minimum.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs2

78

Figs. (2) and (3) display contour plots of the four diabatic PES's (

*V*_{Σ},*V*_{Π},*V*_{1}and*V*_{2}).
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs3

79

Note that the ordering of the states is here grouped by values of

*ω*=*λ*+*σ*, and is consequently different form that chosen in our earlier paper.^{15}
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs3

80

As seen in Fig. 1, four states correlate with the Cl(

^{2}P_{3/2}) + H_{2}reactant (*ω*= ±1/2 and*ω*= ±3/2).
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs4

81

Only two states correlate with the Cl(

^{2}P_{1/2}) + H_{2}reactant (*ω*= ±1/2).
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs4

82

The matrix of the spin–orbit Hamiltonian is determined fully by two components,

^{26}*A*(*R*,*r*,*θ*) ≡*i*〈Π_{y}|*H*_{so}|Π_{x}〉 and*B*(*R*,*r*,*θ*) ≡ 〈_{x}|*H*_{so}|Σ〉, where The spin–orbit functions*A*and*B*can be determined in the*ab initio*calculations, along with the PESs.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res3

83

The dependence of these on the reactant arrangement Jacobi coordinates are shown in Fig. 4.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs5

84

In terms of

*A*and*B*, the matrix of the spin–orbit Hamiltonian in the |*λσ*〉 basis is If the*V*_{1}and*V*_{2}terms are neglected, then the electrostatic Hamiltonian [eqn.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs6

85

(11)] is diagonal in the

*λσ*basis while the matrix of the spin–orbit Hamiltonian [eqn.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs6

86

(15)] is blocked into two identical 2 × 2 matrices and two single channels.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs6

87

Consequently, the matrix of

*H*_{el}+*H*_{so}in the*ω*= 1/2 (or equivalently,*ω*= –1/2) states, is At long, range, where*V*_{Σ}and*V*_{Π}vanish, this can be diagonalized by transforming to a coupled (*j*_{a}*ω*) basis (where*j*_{a}is the total electronic angular momentum of the molecule), with energies –*A*/2 for the*j*_{a}= 3/2 states and +*A*for the*j*_{a}= 1/2 states.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res4

88

As short range, where the splitting between

*V*_{Π}and*V*_{Σ}is much larger than the spin–orbit constant, the Hamiltonian is approximately diagonal in the (uncoupled)*λσ*basis.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res5

89

Early work on inelastic collisions of atoms in

^{2}P states has shown^{28}that non-adiabatic transitions will take place predominately at the value of*R*where the difference between the diagonal terms in eqn. (16), which increases as the collision partners approach, becomes equal to the off-diagonal term.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac11

90

At this point a frame transformation takes place.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs7

91

This point (or two-dimensional seam in the three-dimensional ClH

_{2}PES) is marked by a heavy line in Fig. 2.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs7

92

It is in the neighborhood of this seam that the nonadiabatic transitions will occur, mixing the

*ω*= ±1/2 states which correlate with Cl(^{2}P_{1/2}) and the*ω*= ±1/2 states which correlate with Cl(^{2}P_{3/2}).
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res6

93

We observe that this transition point occurs well outside the barrier to reaction.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs8

94

Further, we observe from Fig. 3 that in the region of this seam both the

*V*_{1}and*V*_{2}coupling potentials are considerably smaller than the off-diagonal spin–orbit coupling (2^{1/2}*B*≅ 3.57 kcal mol^{–1}).
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs9

95

Fig. 5 compares the off-diagonal coupling in eqn. (16), the splitting between the diagonal terms, (

*V*_{Π}+*A*–*V*_{Σ}), and*V*_{1}, as a function of*R*for the hydrogen molecule held at its equilibrium distance (1.4 bohr) and a fixed Jacobi angle of*γ*= 45°.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs10

96

Finally, we observe, by comparing Figs. 2 and 4, that in the region of the seam in Fig. 2, where, presumably, nonadiabatic transitions are most likely to occur, the two spin–orbit terms are little changed from their asymptotic value.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs11

97

We see from matrices (11) and (16) that both the spin–orbit and electrostatic Hamiltonians can cause transitions between the spin–orbit excited reactant channel to the

*ω*= ±1/2 and ±3/2 states which correlate with the ground spin–orbit reactant channel.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res7

98

In addition, off-diagonal Coriolis coupling, which arises from the expansion of the orbital (end-over-end) angular momentum operator for the relative (end-over-end) motion of Cl with respect to H

_{2}, can also contribute to spin–orbit changing transitions.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res7

99

Since

**= (***L***–***J***–***l***–***s***), one can show, that the effect of***j**L*^{2}on the |*JMK**v**jkλσ*〉 states of eqn. (2) is where, for simplicity, we have suppressed the*J*,*M*,*v*,*j*quantum numbers.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

100

Here, we define

*a*_{±}≡ 〈*λ*± 1|*l*_{±}|*λ*〉.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

101

Premultiplying by various 〈

*JMK*′*v*′*j*′*k*′*λ*′*σ*′| states will give the matrix of the rotational Hamiltonian.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

102

The first term in eqn. (17) is responsible for the diagonal matrix elements.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

103

The second, third and fourth terms, which all vary roughly linearly with

*J*, will make a more significant contribution than the other terms, which are independent of*J*.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

104

The former terms give rise to the matrix element This term is to be multiplied, of course, by 1/(2

*µ**R*^{2}), where*µ*is the Cl–H_{2}reduced mass (1.96 u).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

105

The third term in eqn. (18) is identical to the off-diagonal Coriolis term which appears in the standard centrifugally decoupled (coupled-states) treatment of atom-molecule collisions without electronic degrees of freedom.

^{29,30}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod10

106

Because of the δ

_{λ′λ}δ_{σ′σ}factor, this term will mix the two*ω*= 1/2 states which correlate with the two spin–orbit manifolds.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs12

107

In the region of the non-adiabatic mixing seam (Fig. 1), the magnitude of this term will be approximately 2.5 × 10

^{–3}*jJ*kcal mol^{–1}.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs12

108

The other two terms will have magnitudes approximately 2.5 × 10

^{–3}*J*.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs12

109

Consequently, since the rotational angular momentum of the H

_{2}molecule (*j*) will never be larger than 4 or 5, we predict that the contribution of the off-diagonal Coriolis coupling to nonadiabaticity in the Cl + H_{2}reaction will be important only for total angular momentum*J*greater than about 25.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res8

110

As discussed in the introduction, theoretical simulation allows us to vary (or eliminate) one (or more) of these terms which contribute to the nonadiabaticity in the reaction and hence investigate the relative importance of these various terms.

Type: Method |
Advantage: Yes |
Novelty: None |
ConceptID: Met1

111

This will be explored in the next Section.

Type: Goal |
Advantage: None |
Novelty: None |
ConceptID: Goa2

## Calculations

112

The time-independent, close-coupling calculations we use to investigate the scattering (both reactive and inelastic) of Cl by H

_{2}have been described in detail in several earlier publications.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac12

113

For each value of the hyperradius

*ρ*, we expand the total scattering wavefunction as outlined by eqn. (2).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod11

114

A log-derivative method

^{31}is used to propagate the solution numerically, from small*ρ*to large*ρ*.
Type: Method |
Advantage: None |
Novelty: Old |
ConceptID: Met2

115

The parameters which control the accuracy of the integration are increased until the desired quantities (transition probabilities, integral and/or differential cross sections) have converged to within a reasonable limit.

Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met2

116

For abstraction reactions dominated by linear (or near linear) barriers, converged results can be obtained with only a few values of

*K*, which greatly reduces the necessary computer time.
Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met2

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod12

118

To reduce further the number of basis functions which must be simultaneously considered, we use definite-parity linear combinations of the signed-

*K*rotation–vibration–electronic basis functions (eqn. (2)).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod12

119

Further reduction in the size of the basis can be made by using eigenfunctions of the H ↔ H interchange symmetry.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod12

120

The propagation of the solutions to the CC equations are carried out entirely in the fully-uncoupled, body-frame basis of eqn. (2).

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod12

121

At the end of the propagation, where all the PESs (

*V*_{Σ},*V*_{Π},*V*_{1}, and*V*_{2}) are negligibly small, but before extraction of the*S*matrix, we transform the log-derivative matrix into a partially-coupled basis which diagonalizes the residual spin–orbit coupling [eqn.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod12

122

(16)].

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod12

123

In this partially-coupled basis, the electronic states, which correspond to the Russell–Saunders spin–orbit states of the Cl atom, are labeled by the total electronic angular momentum of the atom,

*j*_{a}, and the projection of*j*_{a}along*R⃑*, which we designate*k*_{a}.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod12

124

Figs. 6 and 7 display the total probability of reaction (summed over all energetically accessible rotational and vibrational states of the HCl products) as a function of collision energy.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs13

125

These figures refer to reaction of H

_{2}in*v*= 0,*j*= 0 (the lowest state of*para*-H_{2}) at the lowest possible value of the total angular momentum*J*= 1/2.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs13

126

The results shown refer to states of positive overall parity; the results for the states of negative parity are virtually identical.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res9

127

Results are shown for calculations with the full Hamiltonian, including all couplings (spin–orbit, electrostatic, and Coriolis) as well as from calculations in which we (a) eliminated

*V*_{1}(the coupling between the two states of A′ reflection symmetry), (b) eliminated*V*_{1}and*V*_{2}(the coupling between the two Π states), (c) held the two spin–orbit constants to their asymptotic value (*A*=*B*= 2.52 kcal mol^{–1}), or (d) eliminated all off-diagonal Coriolis coupling.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res10

128

The probabilities for reaction out of the ground (

^{2}P_{3/2}) and excited (^{2}P_{1/2}) spin–orbit states are shown separately in Fig. 6, and together in Fig. 7.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs14

129

As we have remarked earlier,

^{5}the stair-case structure in the energy dependence of the probabilities for reaction of Cl(^{2}P_{3/2}) correspond, in a transition-state model, to the opening of successively higher bend-stretch states of the ClH_{2}complex at the transition states.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac13

130

The position of the lower of these states is shown in Fig. 1.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs15

131

We observe that the results change little when the approximations discussed in the preceding paragraph are introduced.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs15

132

Accurate reaction probabilities can be obtained by retaining only the spin–orbit coupling, and, further, without taking into account the small variation of these constants with the approach of the H

_{2}molecule.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res11

133

We might have anticipated this result from the discussion in the preceding Section.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res11

134

In the region where significant non-adiabatic transitions will occur, at the point where the off-diagonal spin–orbit coupling (2

^{1/2}*B*) is equal to the splitting between the two*ω*= 1/2 states, the*V*_{1}and*V*_{2}coupling potentials are only a small fraction of this off-diagonal coupling.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res12

135

In reality, this relative size difference will be greater than is seen in the comparison of Figs. 3 and 5, because all Jacobi angles

*γ*are sampled during the collision.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res12

136

Both

*V*_{1}and*V*_{2}reach the maximum values shown in Fig. 3 at*γ*= 45° and decline sharply as*γ*goes to zero (where they both vanish) or to 90° (where*V*_{1}vanishes and where*V*_{2}goes through a local minimum).
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs16

137

We also observe that the off-diagonal Coriolis coupling contributes little to the nonadiabaticity.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs16

138

However, since three of the off-diagonal Coriolis terms increase roughly linearly with

*J*, it may be worthwhile to investigate the effect of neglecting the Coriolis coupling at higher values of the angular momentum.
Type: Goal |
Advantage: None |
Novelty: None |
ConceptID: Goa3

139

Fig. 8 shows the dependence on the collision energy of the total reaction probabilities for several values of the total angular momentum.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs17

140

The total reaction probability decreases with increasing

*J*.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs17

141

This is a consequence of the the increasingly large centrifugal barrier which shifts the threshold to higher energy.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res13

142

To investigate the effect of increasing

*J*on the validity of the approximations made earlier, we present in Fig. 9 plots of the total reaction probabilities out of H_{2}(*v*= 0,*j*= 0), positive parity states at*J*= 17.5.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs18

143

Although the spread in calculated transitions probabilities, as a function of the various simplifying approximations, is a bit larger than seen in Fig. 6 at

*J*= 0.5, nevertheless, we see that the conclusion made earlier for*J*= 0.5 is still valid, namely that the degree of inelasticity in the Cl + H_{2}reaction can be determined to a high degree of accuracy by neglecting all off-diagonal couplings other than the spin–orbit coupling.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res14

## Discussion

144

As we have seen here, the overall degree of inelasticity in the Cl + H

_{2}reaction is governed, almost exclusively, by the spin–orbit coupling.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res1

145

If the smaller

*V*_{1}and*V*_{2}PESs, as well as Coriolis coupling, can be safely neglected, then the computational problem becomes considerably reduced.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res2

146

We see from eqns. (11), (15), and (16), that neglect of

*V*_{1}and*V*_{2}as well as the off-diagonal Coriolis coupling block-diagonalizes the coupling matrix into four blocks corresponding to the four possible values of*ω*(±1/2 and ±3/2).
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res15

147

For

*ω*= ±1/2, the blocks, of dimension 2, are identical.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res15

148

For

*ω*= ±3/2 the blocks, also identical, are of dimension 1.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res15

149

One consequence is that the two

*ω*= ±3/2 states, which correlate with the^{2}Π_{3/2}PES, will not contribute to reaction.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res16

150

This is consistent with the validity of the approximation, used earlier by several groups,

^{4,33}that cross sections and rate constants for halogen-H_{2}abstraction reactions determined from calculations on a single PES should be reduced by a factor of 1/2 to account for the flux which, in reality, is associated with the*ω*= ±3/2 states and hence follows the nonreactive^{2}Π_{3/2}PES.
Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con3

151

A second consequence is that the dimensionality of the quantum scattering calculation is reduced by a factor of 3, at least in the reactant arrangement, since one need include only two electronic states in the expansion of the wavefunction in eqn. (2), namely the state with

*ω*= ±1/2.
Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con4

152

Since the difficulty of any close coupling calculation goes as the cube of the number of states (channels) included in the expansion of the wavefunction, this implies that the computational effort will be reduced by at least an order of magnitude.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con1

153

This is not insubstantial.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con1

154

Because the calculation of the overall reactivity of the excited spin–orbit state of Cl is insensitive to the neglect of Coriolis coupling, calculations of this can be carried out within the computationally faster centrifugal-decoupling (coupled states) approximation.

^{30}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac14

155

This will further decrease the computational effort, by block-diagonalizing the channel coupling into blocks of one value of

*K*, the body-frame projection of the total angular momentum.
Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con5

156

Finally, we see from the discussion in Section II, that nonadiabaticity in the Cl + H

_{2}reaction will be determined by the relative size of the spin–orbit constant (which can be considered a fixed number), and the splitting between the reactive and repulsive, nonreactive, PESs, particularly in the region where these two terms are equal in magnitude.
Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con6

157

For the ClH

_{2}system, this occurs (Fig. 2) at a Cl–H_{2}separation of roughly 6*a*_{0}, very near the minimum in the van der Waals well for this system.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res17

158

The equilibrium geometry of the Cl–H

_{2}negative ion occurs in linear geometry at a Cl–H_{2}separation of 5.7*a*_{0}.^{34}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac15

159

Consequently, the recent photodetachment experiments of Neumark and co-workers

^{35}will probe the ClH_{2}PESs in exactly the region responsible for the nonadiabatic transitions which will contribute to the reactivity of Cl in its excited spin–orbit state.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac15

160

As a result, a theoretical simulation of these experiments will provide an invaluable assessment of the accuracy of the CW PESs in this region.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con7

161

Work along these lines is already in progress.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con7