Nonadiabatic coupling and vector correlation in dissociation of triatomic hydrogen

We determine experimentally the vector correlation among the three neutral ground state hydrogen atoms which appear in dissociation of neutral H3* molecules.

The sum of the kinetic energies of the three H-atoms is fixed by selecting the energy of the H3* molecule by laser excitation in the range between 0.85 and 3.60 eV.

The highly structured maps of correlation in the motion of the three atoms provide a direct view of the internal molecular couplings which initiate dissociation.

We discuss this feature in a model calculation and in terms of a new quantum chemical calculations of the potential energy surfaces of H3*.

A basis of quantum chemistry is the Born–Oppenheimer (BO) approximation, according to which nuclei move on single adiabatic potential energy surfaces created by the much faster moving electrons.

This approximation permits the definition of isolated electronic molecular states and their energy levels.

Many interesting aspects of molecular dynamics such as molecule formation and dissociation arise from the breakdown of this approximation.

This is due to small terms in the molecular Hamiltonian which originate from the finite response time of electron motion to changing nuclear position.

A direct access to the dependence of these couplings on molecular coordinates has eluded experimental observation to date, except for diatomics.1

In our experiment we monitor the reaction H3* → H(1s) + H(1s) + H(1s).for individual, state-selected H3* molecules and determine separately for each molecule the three atomic momentum vectors in coincidence using technologies described previously.2–4

We prepare metastable triatomic hydrogen molecules in a fast (3 keV) beam by charge transfer neutralization of H3+.

The rotationless 2p 2A2″ state of H3 is immune against rapid predissociation.5

This state is present in the neutral beam of H3 molecules in a range of vibrational levels {v1,v2}, where v1 and v2 describe the symmetric stretch and degenerate bending mode vibrational quantum numbers.

Photoexcitation spectra of H3 and D3 have been studied in detail6 and firm electronic, vibrational, and rotational assignments of the spectra have emerged, backed by MQDT theory based solely on ab-initio parameters.7

In the photodissociation experiment described here the molecules are photoexcited inside the cavity of a narrowband dye laser which is tuned to a specific absorption transition to a molecular Rydberg state below the ionization threshold.

In this way the total energy of the molecule, W, defined relative to three separated hydrogen atoms, H(1s) + H(1s) + H(1s), is precisely fixed in the experiment.

This situation is indicated in Fig. 1, which gives a cut through the potential energy surfaces of H3 along the symmetric stretch coordinate.

Following photoexcitation the molecules typically predissociate on time scales of 1–10 ns,8 their center of mass propagating at an energy of 3 keV.

The photofragments separate spatially from this direction according to their transverse momentum and they are detected in coincidence using position- and time-sensitive multihit-detectors3 after a free-flight of 1510 mm.

Over this flight distance the fragments separate in space by as much as 100 mm.

High speed time-to-digital converters permit to measure the spatial coordinates of the impact positions of the neutral atoms with a resolution of <100 µm and the arrival time differences between the three atoms with a resolution of <100 ps.

A triple-coincidence logic-routine10 examines the positions and arrival-times to distinguish process (1) from the fragmentation channel H3* → H2(1Σ+g) + H(1s).For each triple coincidence event, the momentum vectors {k⃑1,k⃑2,k⃑3} in the center-of-mass frame are evaluated from the time and position information.

After recording the dissociation of ≈104 molecules, we obtain a map of preferred momentum correlation, under which the selected H3* state escapes into the three-particle continuum.

These data are coded into a Dalitz plot (see below) after accounting for the geometric detection efficiency.4

The exit channel in reaction (1) is well defined in terms of three plane waves with center-of-mass momenta k⃑i.

Neglecting spin- and orbital angular momenta it may be written as the product wavefunction of the three hydrogen atoms ψc = Φ1(k⃑1)Φ2(k⃑2)Φ3(k⃑3).Since the molecular state under study is selected in the laser-excitation step the energy condition holds (m being the hydrogen mass), in addition to momentum conservation k⃑1 + k⃑2 + k⃑3 = 0.

A unique map of three correlated momentum vectors can be represented in a Dalitz11 plot.

This probability-density plot gives the vector correlation in terms of the reduced energies of the three atoms where εi = |k⃑i|2/(2mW).

Fig. 2 illustrates the meaning of the position of an event in the Dalitz plot in terms of the orientation and magnitude of the fragment momenta.

Thus islands of preferrred population in a Dalitz plot refer to specific orientations of final-state fragment momenta.

In this work we determined the correlation map for about twenty electronic and vibrational molecular states of H3 and D3 in the energy range between 0.85 and 3.60 eV above the three-particle asymptote H(1s) + H(1s) + H(1s).

Typical examples are shown in Fig. 3.

Pronounced islands of high probability appear, their location depending on the type of electronic excitation, on the vibrational state, and on nuclear mass.

The initial electronic states have nearly identical nuclear geometry because the adiabatic potential energy surfaces of the excited states are all close to the geometry of the parent ion core H3+.12

The tightly bound ion is of D3h geometry with a proton separation of 1.64 a0 at the potential minimum.

The attached Rydberg electron adds small modifications to the binding, leading to small (<5%) variations in the proton separation at equilibrium.

However the electronic, rovibrational, and nuclear symmetries of the excited states differ and thereby control the coupling matrix elements for predissociation into the ground state continuum.

This feature dictates the appearance of the Dalitz plots as we will discuss below.

At present, no rigorous theory has treated a molecular three-body problem such as (1), however a formal discussion can be given.

The initially bound H3* molecule is prepared at time t0.

At low vibrational excitation its heavy particle wavefunction ψR(t0) is restricted by locally quadratic potentials.

It is characterized by products of harmonic oscillator wavefunctions χ in the symmetric stretch and bending normal mode coordinates, Qs and Qb, with {i,j} quanta of vibrational excitation ψRi,j(t0) = χi(Qs)χj(Qb).In a time-dependent approach we may formally view process (1) as a sequence of two steps.

In a first phase the bound molecular state accesses the repulsive ground state surface of H3ψm(t1) = ψRi,j(t0)where ψm(t1) is a continuum wavepacket at molecular distances and the operator  describes the modification of the normal mode wave functions due to the coupling.

This phase is followed by the evolution of the wavepacket on the continuum energy surface13 where  is the Hamiltonian describing the motion of the three atoms.

In the limit of t → ∞ the function ψc(t) approaches eqn. (3).

Significant pieces of information required to carry out the propagation in eqns. (7) and (8) have been developed recently.

Among these are studies of some of the nonadiabatic14,15 and Jahn–Teller induced couplings16 as well as time-dependent dynamics simulations.13–15,17

A simple model based on a geometry- and state-independent coupling operator  fails to explain the finer details in the measured correlation maps.18

The projection of the wavepacket in eqn. (7) depends sensitively on the overlap of the initial H3* state with the continuum and restrictions imposed on the projection by symmetry considerations.

On the basis of symmetry arguments we know that s A′1 states are predissociated by the degenerate bending mode, d E″ states by rotational coupling,19 and p E′ states by Jahn–Teller coupling.16

This difference implies that for each electronic state different areas of phase space play the leading role in step (7) and hence dictate a specific form of .

We therefore conclude that our experiment provides a direct image of the action of the operator , which couples excited molecular states with the final state continuum.

The dependence of  on electronic and nuclear coordinates and symmetries is embedded in an inverse problem of relating probability density in the Dalitz plot to phase-space density of the molecular level.

Since final and initial states are precisely defined from our experiment, the challenge is to perform a quantum calculation with proper account of electronic and nuclear degrees of freedom.

The complexity of such a task is apparent from Fig. 4 which gives two selected cuts through the potential energy surfaces of H3.

As H3 is the primary example of a polyatomic system an ab initio treatment of the nonadiabatic couplings in this system will significantly extend our microscopic understanding of molecular dynamics.

Certainly the experiment has reached a level of sophistication which warrants an in-depth confrontation with this fundamental system of three protons and three electrons.