Hermite correction method in hyperspherical coordinates: Application to chemical reactions

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Abstract

The rotational product state distribution for the D+H2(v=1,j=1) HD(v′=1,j′)+H reaction, has been calculated using the Hermite correction method in hyperspherical coordinates. We have also considered the geometric phase effect in our calculations either by introducing a vector potential in the system Hamiltonian or by multiplying the wavefunction by a phase factor. In the present application, we have investigated the performance of the simplest approach where only one time-dependent Gauss–Hermite basis function in the hyperradius is included.

Introduction

Semiclassical treatment of three-center reactions using hyperspherical coordinates has been suggested in earlier publications 1, 2, 3, 4. In this approach the hyperradius, ρ, and its conjugate momentum, Pρ, are just treated classically instead of quantally. In a previous article [6], we have derived a semiclassical method in hyperspherical coordinates by introducing the so-called Hermite correction method to the semiclassical theory. This method is basically a multiconfiguration time-dependent self-consistent field (MTDSCF) method [5] in which we can identify certain degrees of freedom as being more classical than others. If only one time-dependent basis function is introduced then the lowest-order correction to the semiclassical (or classical path) theory is obtained 7, 8, 9, 10. With the introduction of the Hermite basis in the ρ coordinate, the equations of motion for ρ(t) along with the width parameters for the Gaussian wavepacket are obtained [6]. The use of this method in Ref. [6] for 3D reactive scattering with J=0, has clearly indicated that we need to introduce only a few basis functions for selected degrees of freedom. Hence, the classical limit for these degrees of freedom can be explored. Actually this appears to be the most convenient and consistent way of approaching the classical limit in parts of a multidimensional system.

In this Letter, we have performed 3D reactive scattering calculations on the D+H2 system for non-zero values of the angular momentum J, treating however γ, and its conjugate momentum, Pγ, classically. As we wish to explore the classical path limit for the hyperradius ρ we use the Hermite correction method, and introduce only one Hermite basis function in the ρ coordinate. Here we should mention that the opposite limit, the exact limit, can indeed be obtained also with this basis set but is more conveniently obtained using standard techniques like grid-expansions [11]. Earlier, we obtained reasonable results [6] using one wavepacket only for the D+H2 reaction at J=0. Since reasonably good results were obtained in this case we expect that total cross-sections which involve J>0 would be even better. The reason is that the ρ motion is more quantum mechanical for J=0 due to the smaller turning point than for larger values of J. This expectation motivates us to perform the same calculation with J≠0 so as to obtain the rotational state distributions with or without inclusion of the geometric phase effect. Furthermore the considerable improvement of the agreement between experimental 15, 16, 17 and theoretical 18, 19, 20, 21, 22, 23 product rotational state distributions for the D+H2(v=1,j=1)DH(v′=1,j′)+H reaction obtained in the presence of the geometric phase effect has justified additional and alternative ways of treating reactive scattering approximately but in such a way that the geometric phase effect can be incorporated.

As the conical intersection of the ground and excited states of the H3 system occurs at the symmetric triangular configuration, one can incorporate the effect into the basis functions 18, 19, 20 such that the nuclear Schrödinger equation itself does not need a vector potential 12, 13, 14. In order to include the geometric phase effect in more complicated molecules (where the conical intersection does not occur at a special symmetry point of the coordinate system) the introduction of the vector potential becomes more convenient. Alternatively the system should be treated as a genuine two-surface problem rather than a pseudo-one-surface problem.

We have formulated the appropriate vector potential in hyperspherical coordinates 21, 22, 23 for the general case, where the position of the conical intersection is arbitrary. In this present case, as we wish to see the effect either by introducing the vector potential in the Hamiltonian or by multiplying the wavefunction with a phase factor, the expression for the vector potential has been approximated [21] with θ0=0 instead of 11.5° [22] for the D+H2 system.

In the case of the finite total angular momentum vector J, the initial values of the proposed classical variables γ and the conjugate momentum Pγ have to be chosen randomly and each set of initial values propagated separately. The transition probabilities for each propagation, at a given energy can be obtained by projecting the scattered wavefunction on outgoing plane waves, exp(ikFρ), and the initial wavefunction on incoming plane waves, exp(−ikIρ), and then taking the ratio of the outgoing and incoming fluxes. Finally, the rotational state distributions are calculated by averaging over series of trajectories.

In Section 2, we discuss the Hermite correction method in hyperspherical coordinates while in Section 3, the schemes to handle the geometric phase problems are addressed. Section Section 4covers the details of initialization, propagation and projection of the wavefunction. Finally, in Section 5, the rotational product state distributions for the D+H2(v=1,J=1) DH(v′=1,j′)+H reaction are calculated using the Hermite correction method with or without considering the geometric phase effect.

Section snippets

The Hermite correction method in hyperspherical coordinates

If the Euler angles are considered as classical variables and the choice of the orientation [24] of the space-fixed coordinate system is such that the z axis is aligned with the total angular momentum vector J, the quantum Hamiltonian for a three-particle system can be expressed in a modified form of Johnson's hyperspherical coordinates [25] as

Ĥm=1P̂ρ2+4ρ2L̂2(θ,φ)+Pγ[Pγ−4cosθP̂φ]2μρ2sin2θ+J2−Pγ2μρ2cos2θ(1+sinθcos2γ)−22μρ214+1sin2+V̂(ρ,θ,φ),where ρ is the hyperradius, θ and φ the

The geometric phase effect

At high energies, the rotational state distributions of DH products formed in the reaction D+H2DH+H were found to be in discrepancy with experimental predictions. Kuppermann and Wu 18, 19, 20 demonstrated that the consideration of a non-trivial phase in the theory can give improved agreement. The necessity of a non-trivial phase arises when the system has a conical intersection between the two lowest electronic potential surfaces. As a consequence of that, a real electronic wavefunction

Initialization, propagation and projecton

Though in this present calculation we have used only one Hermite basis function, i.e. a Gaussian wavepacket, still at t=t0, the expansion in Eq. (3)having an arbitrary number of Hermite basis functions, will be of the form

Ψ(ρ,θ,φ,t0)=ψ0(θ,φ,t0)Φ(ρ,t0)I(θ,φ,t0GWP(ρ,t0).

The Gaussian wavepacket is

ΦGWP(ρ,t0)=expi[γ(t0)+Pρ(ρ−ρ(t0)]+A(t0)[ρ−ρ(t0)]2),and the expansion coefficient [27]

ψI(θ,φ,t0)=ρsin(−η)/ζgv(ζ)Pj(cosη),where gv and Pj are the Morse vibrational wavefunction and normalized Legendre

Results and discussion

The Hermite correction method using one wavepacket only (GWP) has been applied for the reaction, D+H2(v=1,j=1)DH(v′=1,j′)+H, using the LSTH potential energy surface 31, 32 where we have considered non-zero values of the total angular momentum vector J. We have performed all calculations with total energy 1.80 eV (Etr=1.0 eV) with or without introducing the geometric phase effect where the transition probabilities as a function of total energy can be obtained by Eq. (16)for each trajectory, and

Acknowledgements

This research was supported by the Danish Natural Research Council and the EU TMR-grant contract No. ERBFMRXCT960088.

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