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Back | Show only these items: | Details and consequences of the nonadiabatic coupling in the Cl(2P) + H2 reaction

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Details and consequences of the nonadiabatic coupling in the Cl(²P) + H₂ reaction

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

In an investigation of nonadiabaticity in the Cl + H₂ reaction we examine the various coupling terms responsible.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

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

Neglect of all coupling except spin–orbit leads to little change in the calculated reaction probabilities both for the Born–Oppenheimer allowed [Cl(²P_3/2) + H₂] and Born–Oppenheimer forbidden [Cl(²P_1/2) + H₂] reactions.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

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

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

Because of their experimental accessibility, the reactions of F and Cl with H₂ and its isotopomers have become paradigms for triatomic abstraction reactions.^1,2

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

Experimental interest in the Cl + H₂ reaction goes back more than 100 years.¹

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Numerous quasiclassical trajectory and various quantum scattering investigations have been reported^3–6 on several potential energy surfaces (PESs).^7–9

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Approach of the H₂ molecule to a Cl atom splits the degeneracy of the ²P state.

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As shown schematically in Fig. 1, two electronic states (1²A′ and 1²A″; ²Σ_1/2⁺ and ²Π_3/2 in linear geometry) correlate adiabatically with the ground-state atomic reactant (²P_3/2) while a third state (2²A′; ²Π_1/2 in linear geometry) correlates adiabatically with the excited-state atomic reactant (²P_1/2).¹⁰

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

Of these, only the 1²A′ electronic state correlates with the electronic ground state of the products [HCl(X ¹Σ⁺) + H(²S)].

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

The two other electronic states correlate with electronically excited states of the products [HCl(a ³Π) + H(²S)] which are considerably higher in energy¹¹ and, consequently, energetically inaccessible at low to moderate collision energies.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

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

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

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

For bimolecular reactive collisions this assumption is supported by the majority of past experimental work.¹²

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

In recent molecular beam experiments, Liu and co-workers¹³ 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

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

As can be seen in Fig. 1, reaction of the spin–orbit excited Cl atom (²P_1/2) can occur only by a non-Born–Oppenheimer transition from the non-reactive ²Π_1/2 PES to the reactive ²Σ+1/2 PES.

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

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

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

Hence, non-adiabaticity in the Cl + H₂ 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

Theoretical investigations of the role of excited electronic surfaces in the F + H₂ reaction date back to the pioneering work of Tully.¹⁴

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

Several years ago, we presented the framework for the essentially exact quantum determination of cross sections for abstraction reactions involving an atom in a ²P electronic state.¹⁵

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

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

All Coriolis coupling terms are included.

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

More recently, we applied⁵ this methodology to the reaction of Cl with molecular hydrogen, using new ab initio potential energy surfaces developed by Capecchi and Werner (the CW PESs).⁹

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

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

The degree of non-adiabaticity was even less than found in our earlier investigation of the F + H₂ reaction.^15,16

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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

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

These theoretical results stand in direct contrast to the conclusions of Liu and co-workers.¹³

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

This past year¹⁸ we have determined differential cross sections for the Cl + H₂ reaction.⁶

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

Differential cross sections for the adiabatically-forbidden reaction of Cl* (²P_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

Because of the continuing disagreement between experiment¹³ and theory,⁵ it is worthwhile to examine in more detail the factors which control the degree of non-adiabaticity in the Cl + H₂ reaction.

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

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

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

This is the goal of the present article.

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

Hamiltonian, potential surfaces, and couplings

We write the total Hamiltonian for collision of Cl (²P) with H₂ (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

The nuclear coordinates R⃑ and r⃑ in eqn. (1) designate the Jacobi vectors¹⁹ in any one of the three chemical arrangements (Cl + HH, H + HCl, or HCl + H).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

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

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

In the Cl + H₂ arrangement there are six electronic states.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

For a halogen atom with a p⁵ 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

For the HCl, the lowest excited state is of Π symmetry, and lies far above the ground state (X ¹Σ⁺) in the region of the molecular minimum.¹¹

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

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 (¹Σ⁺) combined with the two possible spin-projection quantum numbers of the H atom.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

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

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

Here, also, DJ*MK(Ω) is a Wigner rotation matrix element,²²Y_jk(γ,0) is a spherical harmonic,²² and φ_vj(θ;ρ) is the solution of a “vibrational” reference problem corresponding to motion in the hyperangle θ at the given (fixed) value of ρ.²³

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

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

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

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

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 L² = (J – l – s – j)².

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

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;R⃑,r⃑)|λσ〉 = δ_σσ′V_λ′λ(ρ, θ, γ).

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

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)|λσ〉 = W_{λ′σ′λσ}(ρ,θ,γ).

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

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

Capecchi and Werner⁹ 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 + H₂ PESs in the reactant arrangement: 1A′, 2A′, and 1A″.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

These all correlate with Cl(²P) + H₂.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

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

The coordinate dependence of the eigenvalues E_i defines the three adiabatic PESs.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

We define the interaction potential by subtracting the energies of the reactants at infinite Cl–H₂ 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 ²P state exclusive of the spin–orbit Hamiltonian.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

The zero of energy will be defined by the equilibrium internuclear separation of the H₂ 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 H₂ molecule.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

By analysis of the coefficients in the CI expansion of the ClH₂ 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

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

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

We use the compact Cartesian notation |Π_x〉, |Π_y〉, and |Σ〉.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

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

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

Alternatively, we can define the diabatic states in terms of signed-λ, rather than Cartesian, projections, namely |Π₁〉, |Π_–1〉, and |Σ〉.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

These signed –λ states are those that appear above in the expansion of the scattering wavefunction.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

Note that the state we designate as “Σ” corresponds to λ = 0.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

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

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

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

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

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

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, V₂ = (V_yy – V_xx)/2, and V₁ = V_xz/2^1/2.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

In collinear geometry both V₁ and V₂ vanish.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

In perpendicular geometry V₁ also vanishes while V₂ goes through a relative minimum.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

Figs. (2) and (3) display contour plots of the four diabatic PES's (V_Σ, V_Π, V₁ and V₂).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

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

As seen in Fig. 1, four states correlate with the Cl(²P_3/2) + H₂ reactant (ω = ±1/2 and ω = ±3/2).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

Only two states correlate with the Cl(²P_1/2) + H₂ reactant (ω = ±1/2).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

The matrix of the spin–orbit Hamiltonian is determined fully by two components,²⁶A(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

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

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

In terms of A and B, the matrix of the spin–orbit Hamiltonian in the |λσ〉 basis is If the V₁ and V₂ terms are neglected, then the electrostatic Hamiltonian [eqn.

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

(11)] is diagonal in the λσ basis while the matrix of the spin–orbit Hamiltonian [eqn.

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

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

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

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_aω) 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

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

Early work on inelastic collisions of atoms in ²P states has shown²⁸ 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

At this point a frame transformation takes place.

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

This point (or two-dimensional seam in the three-dimensional ClH₂ PES) is marked by a heavy line in Fig. 2.

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

It is in the neighborhood of this seam that the nonadiabatic transitions will occur, mixing the ω = ±1/2 states which correlate with Cl(²P_1/2) and the ω = ±1/2 states which correlate with Cl(²P_3/2).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

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

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

Further, we observe from Fig. 3 that in the region of this seam both the V₁ and V₂ coupling potentials are considerably smaller than the off-diagonal spin–orbit coupling (2^1/2B ≅ 3.57 kcal mol^–1).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

Fig. 5 compares the off-diagonal coupling in eqn. (16), the splitting between the diagonal terms, (V_Π + A – V_Σ), and V₁, 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

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

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

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 H₂, can also contribute to spin–orbit changing transitions.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

Since L = (J – l – s – j), one can show, that the effect of L² on the |JMKvjkλσ〉 states of eqn. (2) is where, for simplicity, we have suppressed the J,M,v,j quantum numbers.