Single-electron wavefunctions, or orbitals, are the mathematical constructs used to describe the multi-electron wavefunction of molecules. Because the highest-lying orbitals are responsible for chemical properties, they are of particular interest. To observe these orbitals change as bonds are formed and broken is to observe the essence of chemistry. Yet single orbitals are difficult to observe experimentally, and until now, this has been impossible on the timescale of chemical reactions. Here we demonstrate that the full three-dimensional structure of a single orbital can be imaged by a seemingly unlikely technique, using high harmonics generated from intense femtosecond laser pulses focused on aligned molecules. Applying this approach to a series of molecular alignments, we accomplish a tomographic reconstruction of the highest occupied molecular orbital of N
2. The method also allows us to follow the attosecond dynamics of an electron wave packet.
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At a glance
: Illustration of the tunnel ionization process from an aligned molecule.
a, The orange line on the potential surface is an isopotential contour slightly above the energy level of the bound state at the peak of the field amplitude. The opening of the contour shows the saddle point region where the bound-state electron wavefunction will tunnel through to the continuum. The lateral spread of the electron wave packet ψ c is determined by the width of the saddle point region Δ r that depends on the molecular alignment. The wave packet expands in the lateral direction during propagation in the continuum because of the initial momentum spread Δ p given by the uncertainty principle, Δ p ≈ h/Δ r. b, Illustration of the re-colliding wave packet seen by a molecule (the real part of the wave packet Re[ ψ c( x = 0, t)] is shown). The kinetic energy at the time of re-collision is taken from the classical trajectory, and determines the instantaneous frequency of the wave packet as seen by the molecule. The lateral spread is calculated by the free expansion of a gaussian wave packet with an initial 1/ e full width of 1 Å.
: Illustration of a dipole induced by the superposition of a ground-state wavefunction
ψ g and a re-colliding plane wave packet ψ c.
a, Bound-state wavefunction (for example, H atom, 1 s). b, The real part of the superposition of the bound-state wavefunction ψ g and the continuum plane wave ψ c. c, The total electron density distribution | ψ g + ψ c | 2. The superposition of the two wavefunctions induces a dipole d( t) as shown by the red arrow. As the wave packet propagates, the induced dipole oscillates back and forth and leads to the emission of harmonics.
: High harmonic spectra were recorded for N
2 molecules aligned at 19 different angles between 0 and 90° relative to the polarization axis of the laser.
For clarity, only some of the angles have been plotted above. The high harmonic spectrum from argon is also shown; argon is used as the reference atom. Clearly the spectra depend on both the alignment angle and shape of the molecular orbital.
: Molecular orbital wavefunction of N
a, Reconstructed wavefunction of the HOMO of N 2. The reconstruction is from a tomographic inversion of the high harmonic spectra taken at 19 projection angles. Both positive and negative values are present, so this is a wavefunction, not the square of the wavefunction, up to an arbitrary phase. b, The shape of the N 2 2 p σ orbital from an g ab initio calculation. The colour scales are the same for both images. c, Cuts along the internuclear axis for the reconstructed (dashed) and ab initio (solid) wavefunctions.
: A one-dimensional Schrödinger calculation shows that attosecond electronic wave-packet motion is resolved in the high harmonic spectra.
The bottom curve shows a spectrum at a particular pump–probe delay time; the minima are due to interference caused by the wave-packet motion. The top picture shows the spectra at a range of time delays, showing that the minima move with pump–probe time delay. The simulation populated the ground and first excited state (9.5
eV above ground) of a model atom to create an electronic wave packet.