Fingerprints of Adiabatic versus Diabatic Vibronic Dynamics in the Asymmetry of Photoelectron Momentum Distributions.

When the Born-Oppenheimer approximation is valid, electrons adiabatically follow the nuclear motion in molecules. For strong nonadiabatic coupling between electronic states, one encounters a diabatic motion where the electrons remain local and do not adapt to molecular geometry changes. We show that the mentioned limiting cases are reflected differently in the asymmetry of time-resolved photoelectron momentum distributions. Whereas for adiabatic dynamics, the asymmetry directly maps the time-dependent average nuclear momentum, in the diabatic case, the asymmetry is determined by a nonclassical interference effect arising from the mixing of wave function components in different electronic states, which is present at times nonadiabatic transitions take place.

The Born–Oppenheimer (BO) separation of electronic and nuclear motion,[1] leading to the concept of potential energy surfaces, is one of the most important approximations to understand, e.g., the course of a chemical reaction or the vibronic structure of UV–vis absorption spectra. Within the BO approach, the electrons adiabatically follow geometry changes in molecules. Mathematically, the BO approximation works well when the derivatives of the electronic wave function with respect to the nuclear coordinates are small. Often, however, the BO approximation fails, which is the case when potential energy curves of energetically close lying electronic states show an avoided crossing,[2,3] or, in the more general case, the potential surfaces exhibit conical intersections.[4−6] In such regions of nuclear coordinate space, the character of the electronic wave functions changes abruptly, giving rise to large if not divergent couplings between the electronic states.[7] In such situations, it is favorable to adopt another approach where such divergences do not occur. The electronic states that minimize the above-mentioned couplings are called diabatic states.[8,9] In the diabatic picture, the electronic wave functions remain localized and exhibit only minor (and ideally no) changes if a region of nonadiabatic interaction is passed.

Below, we identify signatures of adiabatic and diabatic dynamics without relying on the separation of electronic and nuclear motion, i.e., by solving the time-dependent Schrödinger equation for the coupled vibronic dynamics. Due to the large computational cost, this is only possible within descriptions of reduced dimensionality. We therefore adopt a model that involves a single electron (coordinate x) and a nucleus (coordinate R), moving in one dimension between two fixed nuclei.[10,11] The electron-nuclei interactions are parametrized with error-functions (erf) involving screening parameters, and the potential energy is (in atomic units)

Our computations use a charge number of Z = 1 for all nuclei and a screening parameter of Rf =1.5 Å. The fixed ions are located at R = ± L/2 with L = 10 Å. The beauty of the model is that, upon adjusting the remaining parameter Rc, one is able to switch from a situation of an adiabatic to a diabatic motion.[12,13] As an observable, we regard the time-dependent photoelectron momentum (p) distribution σ(p,T)[14−16] obtained from ionization with a laser pulse interacting at different times T. Separating the electrons that are emitted with positive/negative momenta leads to spectra σ±(p,T), respectively. Integration over p then yields the spectrum σ±(T) which enters into the definition of the (normalized) asymmetry as

Such asymmetries have recently been investigated experimentally[17−19] and theoretically.[20−24] We calculate A(T) from the wave function ψ(x, R, t, T), which is obtained by numerically solving the time-dependent Schrödinger equation.[25] Briefly, after the pulse interaction, the wave function that has reached the asymptotic region of large electron coordinate (|x| > 100 Å in our calculation) is transformed to electronic momentum space (ψ(p, R, t, T)), and S±(p, T) is calculated asby integration over the nuclear coordinate R; for details, see ref (26).

To characterize the system, we first calculate the adiabatic potentials V(R) and corresponding electronic wave functions φ(x, R) by imaginary time-propagation.[27] The potentials for the two lowest electronic states (n = 0,1) are shown in Figure 1. The two curves in the lower panel (Rc = 1.5 Å) are well separated in energy, suggesting that a nuclear motion in the electronic ground state is described accurately within the BO approximation, i.e., the coupling to other electronic states can be neglected. On the other hand, the curves obtained for a screening parameter of Rc = 2.5 Å exhibit an avoided crossing at the configuration with R = 0, suggesting a strong nonadiabatic coupling. The two situations will be referred to as adiabatic and diabatic, respectively, in what follows. Note that the potential energy curves in Figure 1 are only used for the purpose of interpretation; as we solve the full-coupled system, all electronic (and vibrational) eigenstates are implicitly contained in the dynamics.

Figure 1

Potential curves suggesting an adiabatic motion in the electronic ground state and for the case when a strong coupling between the ground and first excited state is present (diabatic).

In Figure 2 we show the dynamics of the electronic and nuclear densities defined aswhere ψ(x, R, t) is the time-dependent wave function of the unperturbed system. For the adiabatic case (left panels), the wave function at time t = 0 is taken as ψ(x, R, 0) = e–β(φ0(x, R), with β = 7.14 Å and R0 = −0.9 Å. Thus, we start in the electronic ground state. The nuclear density shows a localized wave packet motion. Regarding the ground state potential (Figure 1), it is seen that, due to the initial condition, the motion is restricted to take place in the left-hand well of the potential, which means that we encounter an adiabatic motion in a single electronic state. This is confirmed by inspection of the population in the ground state, which remains constant during the time-interval considered (Figure 3, lower left panel). The electronic density (upper left panel of Figure 2) follows the nuclear density as is expected from adiabatic dynamics.

Figure 2

Time-dependent electronic (upper panels) and nuclear (lower panels) densities are displayed for the case of adiabatic and diabatic motion, as indicated.

Figure 3

Time-dependence of the populations P(t) in the electronic states (n = 0,1), the expectation values of the nuclear momentum ⟨P⟩ and electron momentum ⟨p⟩ and the asymmetry A(T) are displayed for the case of the adiabatic and diabatic motion, as indicated.

Next, we regard the diabatic dynamics. There, the time-dependence of the nuclear density (starting with a Gaussian nuclear wave packet at R0 = −3.0 Å) resembles the one in the adiabatic case with the difference that here the initial condition is chosen such that the motion extends into regions of positive values of R. The population analysis shows that, upon passing the symmetric configuration (R = 0), the population is transferred from the ground state to the first excited state almost completely (Figure 3, lower right panel). This suggests a dynamics on a diabatic potential, which, for negative values of R, consists of the branch of the adiabatic potential V0(R) and, for positive R, is the potential V1(R). Then, by definition, the electronic density should not change essentially during the dynamics and remain localized. This, indeed, is seen in the time-dependent electron density ρ(x,t), which is shown in Figure 2, right upper panel.

At this point, the question arises whether the character of the quantum dynamics, being adiabatic or diabatic, is reflected in an observable. Therefore, we suggest regarding the time-resolved photoelectron asymmetry A(T), as defined above. Before details of the numerical calculation are presented, let us speculate on what to expect intuitively in the cases of adiabatic and diabatic motion. In the former case, the adiabatic motion of the electron, being attached to the moving nucleus, results in an electron momentum expectation value ⟨p⟩, which is positive/negative if the nucleus moves in the positive/negative direction of R, respectively. Then, upon ionization, electrons are emitted preferentially in the direction determined by the mean momentum in the initial state. From this classical consideration, it is then anticipated that ionization produces electron distributions with an asymmetry determined by the nuclear motion, and thus A(T) should reflect its direction. That this expectation indeed is correct is documented in Figure 3 (left panels), which compares the expectation values of the nuclear and electronic momentum with the asymmetry. In the numerical calculation we use an ionizing XUV-pulse with a wavelength of 62 nm, an intensity of 1013 W/cm2, and a width of the Gaussian envelope function of 4 fs. The times t = T refer to the maximum of the pulse envelope-function.

For the diabatic dynamics, a rather different behavior is anticipated. Because the electronic density remains localized and, to a first approximation, does not change during the nuclear wave packet dynamics, the average electron momentum is zero, and only minor changes in the asymmetry are expected. Figure 3 (right-hand panels) shows that the electron momentum indeed remains small if compared to the adiabatic case besides the fact that the nuclear momentum assumes similar positive and negative values. The asymmetry does not exhibit sign changes upon a change in the direction of the nuclear motion and thus does not reflect the latter. It is small for times when the nuclear density is not located close to the region of the avoided crossing. However, it assumes larger values during the time the nonadiabatic transition occurs.

To gain some insight into the origin of this behavior, we calculate the asymmetry employing first-order time-dependent perturbation theory. Expanding the initial state in terms of the adiabatic electronic eigenfunctions (in the electronic momentum representation) of the two electronic states that participate, the wave function describing the detachment is of the formwhere we explicitely noted the time-dependence of the moving wave packets. The operator B̂T maps, to first order, the initial state onto the final state in the detachment continuum. The photoelectron spectrum, and thus the asymmetry, then consists of three terms representing detachment out of the ground state, out of the excited state, and an interference contribution:

In Figure 4, we show the asymmetry and its decomposition. The figure shows that the interference term contributes most significantly to the asymmetry. Let us therefore discuss this term (σ01(p, T)) and, for simplicity, regard the most significant contribution to the spectrum originating from the fixed nuclear distance R = R where the nuclear density has its maximum at time T. Then, one obtains

Figure 4

Time-dependent asymmetry A(T) and its decomposition into a contribution resulting from the initial electronic ground state (A0(T)), the excited state (A1(T)), and the interference term A01(T).

Because we discuss a diabatic nuclear motion, it is more appropriate to replace the adiabatic components of the two-state nuclear wave function by the diabatic ones using the transformationwhere Θ = Θ(R) is the so-called mixing angle depending on the nuclear coordinate.[7,28] This angle is zero at distances smaller than the coupling region and π/2 for distances larger than the coupling region. In our case, where the wave packet dynamics takes place (almost) exclusively in the diabatic state with n = 0, we have ψ1(R) = 0, so that

Here we encounter a modulation of the spectrum with the function (−cos Θ sin Θ). The same then applies to the asymmetry, and thus the particular time-dependence of the latter during the nonadiabatic transition, which is seen in Figure 3, can be rationalized.

To conclude, we have shown that for a BO adiabatic motion, the photoelectron momentum asymmetry directly correlates with the direction of the vibrational motion, information that is difficult to extract from other time-resolved photoionization signals. On the other hand, for a strong nonadiabatic coupling, the asymmetry maps the population transfer dynamics between the involved electronic states, which correlates with the mixing of the components of the wave packet in these states. Although a rather simple model is employed in the present study, we believe that the findings can be generalized to situations involving several electrons and nuclei. The concepts of adiabatic and diabatic nuclear motion is not restricted to the motion involving a single nuclear degree-of-freedom. Rather, it occurs also in geometry changes involving all of the nuclei bound in a molecule.[4−9] We expect the dynamics in multidimensional systems to induce an asymmetry in the photoelectron distribution as well, which might, however, be more easily visualized in channel-resolved photoelectron angular distributions.[29] Work along these lines is in progress.