Numerically exact nonadiabatic eigenfunctions are computed for a two-dimensional conical intersection with circular symmetry, for which a pseudorotation quantum number is conserved and all eigenstates are doubly degenerate. In the calculations reported here, the conical intersection is submerged, with energy below the zero point level. The complete real-valued vibrational-electronic eigenfunctions are visualized using Hunter’s exact factorization for the total vibrational amplitude factor and color for the electronic factor. The zero-point levels have nonzero amplitude at the conical intersection. Nodes in the degenerate nonadiabatic eigenfunctions are classified as accidental if they can be moved or removed by a change in degenerate basis and as essential if they cannot. An integer electronic index defines the order of the nodes for nonadiabatic eigenfunctions by simple closed counterclockwise line integrals. Higher eigenstates can have accidental conical nodes around the conical intersection and essential nodes of varying circular orders at the conical intersection. The signs of the essential nodes are all opposite the sign of the conical intersection and the signed node orders obey sum rules. Even for submerged conical intersections, the appearance of the exact eigenstates motivates use of signed, half-odd-integral, pseudorotation quantum numbers j. Essential nodes of absolute order (|j| – 1/2) are located on the conical intersection for |j| greater than or equal to 3/2. The eigenfunctions around essential first order nodes are right circular cones with their vertex at the conical intersection.

Femtosecond two-dimensional Fourier transform spectroscopy is used to determine the static bandgap inhomogeneity of a colloidal quantum dot ensemble. The excited states of quantum dots absorb light, so their absorptive two-dimensional (2D) spectra will typically have positive and negative peaks. It is shown that the absorption bandgap inhomogeneity is robustly determined by the slope of the nodal line separating positive and negative peaks in the 2D spectrum around the bandgap transition; this nodal line slope is independent of excited state parameters not known from the absorption and emission spectra. The absorption bandgap inhomogeneity is compared to a size and shape distribution determined by electron microscopy. The electron microscopy images are analyzed using new 2D histograms that correlate major and minor image projections to reveal elongated nanocrystals, a conclusion supported by grazing incidence small-angle X-ray scattering and high-resolution transmission electron microscopy. The absorption bandgap inhomogeneity quantitatively agrees with the bandgap variations calculated from the size and shape distribution, placing upper bounds on any surface contributions.Keywords: 2D spectroscopy; inhomogeneity; line width; Quantum dots; shape dispersion; size dispersion;

•Nonadiabatic eigenfunctions can have low dimensionality vibrational nodes.•In 2D, a 0D nonadiabatic node lies at the vertex of a right elliptical cone.•Node dimensionality depends on the number of coupled electronic states.•Node dimensionality depends on the dimension of the vibrational space.•Diffraction or Coulomb explosion might detect low dimensionality nodes.It has been argued the total vibrational probability amplitude for a molecular nonadiabatic eigenfunction should not have nodes unless required by symmetry. For a model with two nonadiabatically coupled electronic states, exact factorization of nonadiabatic eigenfunctions into a total vibrational probability amplitude and a normalized electronic factor reveals zero-dimensional nodes in nonadiabatic eigenfunctions over a two-dimensional vibrational space. These conical nodes have the shape of a right elliptical cone with the node at the vertex. Low dimensionality nodes are likely when the number of nonadiabatically coupled electronic states is less than or equal to the dimensionality of the vibrational space.Download high-res image (123KB)Download full-size image

A solution to Maxwell’s equations in the three-dimensional frequency domain is used to calculate rephasing two-dimensional Fourier transform (2DFT) spectra of the D2 line of atomic rubidium vapor in argon buffer gas. Experimental distortions from the spatial propagation of pulses through the sample are simulated in 2DFT spectra calculated for the homogeneous Bloch line shape model. Spectral features that appear at optical densities of up to 3 are investigated. As optical density increases, absorptive and dispersive distortions start with peak shape broadening, progress to peak splitting, and ultimately result in a previously unexplored coherent transient twisting of the split peaks. In contrast to the low optical density limit, where the 2D peak shape for the Bloch model depends only on the total dephasing time, these distortions of the 2D peak shape at finite optical density vary with the waiting time and the excited state lifetime through coherent transient effects. Experiment-specific conditions are explored, demonstrating the effects of varying beam overlap within the sample and of pseudo-time domain filtering. For beam overlap starting at the sample entrance, decreasing the length of beam overlap reduces the line width along the ωτ axis but also reduces signal intensity. A pseudo–time domain filter, where signal prior to the center of the last excitation pulse is excluded from the FID-referenced 2D signal, reduces propagation distortions along the ωt axis. It is demonstrated that 2DFT rephasing spectra cannot take advantage of an excitation–detection transformation that can eliminate propagation distortions in 2DFT relaxation spectra. Finally, the high optical density experimental 2DFT spectrum of rubidium vapor in argon buffer gas [ J. Phys. Chem. A 2013, 117, 6279−6287] is quantitatively compared, in line width, in depth of peak splitting, and in coherent transient peak twisting, to a simulation with optical density higher than that reported.

Absolute molecular number concentration and extinction coefficient are simultaneously determined from linear and nonlinear spectroscopic measurements. This method is based on measurements of absolute femtosecond pump–probe signals. Accounting for pulse propagation, we present a closed form expression for molecular number concentration in terms of absorbance, fluorescence, absolute pump–probe signal, and laser pulse parameters (pulse energy, spectrum, and spatial intensity profile); all quantities are measured optically. As in gravimetric and coulometric determinations of concentration, no standard samples are needed for calibration. The extinction coefficient can then be determined from the absorbance spectrum and the concentration. For fluorescein in basic methanol, the optically determined molar concentrations and extinction coefficients match gravimetric determinations to within 10% for concentrations from 0.032 to 0.540 mM, corresponding to absorbance from 0.06 to 1. In principle, this photonumeric method is extensible to transient chemical species for which other methods are not available.

The absolute femtosecond pump–probe signal strength of deprotonated fluorescein in basic methanol is measured. Calculations of the absolute pump–probe signal based on the steady-state absorption and emission spectrum that use only independently measured experimental parameters are carried out. The calculation of the pump–probe signal strength assumes the pump and probe fields are both weak and includes the following factors: the transverse spatial profile of the laser beams; the pulse spectra; attenuation of the propagating pulses with depth in the sample; the anisotropic transition probability for polarized light; and time-dependent electronic population relaxation. After vibrational and solvent relaxation are complete, the calculation matches the measurement to within 10% error without any adjustable parameters. This demonstrates quantitative measurement of absolute excited state population.

The delocalized, anticorrelated component of pigment vibrations can drive nonadiabatic electronic energy transfer in photosynthetic light-harvesting antennas. In femtosecond experiments, this energy transfer mechanism leads to excitation of delocalized, anticorrelated vibrational wavepackets on the ground electronic state that exhibit not only 2D spectroscopic signatures attributed to electronic coherence and oscillatory quantum energy transport but also a cross-peak asymmetry not previously explained by theory. A number of antennas have electronic energy gaps matching a pigment vibrational frequency with a small vibrational coordinate change on electronic excitation. Such photosynthetic energy transfer steps resemble molecular internal conversion through a nested intermolecular funnel.

The pump−probe polarization anisotropy is computed for molecules with a nondegenerate ground state, two degenerate or nearly degenerate excited states with perpendicular transition dipoles, and no resonant excited-state absorption. Including finite pulse effects, the initial polarization anisotropy at zero pump−probe delay is predicted to be r(0) = 3/10 with coherent excitation. During pulse overlap, it is shown that the four-wave mixing classification of signal pathways as ground or excited state is not useful for pump−probe signals. Therefore, a reclassification useful for pump−probe experiments is proposed, and the coherent anisotropy is discussed in terms of a more general transition dipole and molecular axis alignment instead of experiment-dependent ground- versus excited-state pathways. Although coherent excitation enhances alignment of the transition dipole, the molecular axes are less aligned than for a single dipole transition, lowering the initial anisotropy. As the splitting between excited states increases beyond the laser bandwidth and absorption line width, the initial anisotropy increases from 3/10 to 4/10. Asymmetric vibrational coordinates that lift the degeneracy control the electronic energy gap and off-diagonal coupling between electronic states. These vibrations dephase coherence and equilibrate the populations of the (nearly) degenerate states, causing the anisotropy to decay (possibly with oscillations) to 1/10. Small amounts of asymmetric inhomogeneity (2 cm−1) cause rapid (130 fs) suppression of both vibrational and electronic anisotropy beats on the excited state, but not vibrational beats on the ground electronic state. Recent measurements of conical intersection dynamics in a silicon napthalocyanine revealed anisotropic quantum beats that had to be assigned to asymmetric vibrations on the ground electronic state only [Farrow, D. A.; J. Chem. Phys. 2008, 128, 144510]. Small environmental asymmetries likely explain the observed absence of excited-state asymmetric vibrations in those experiments.

Hot electronic dynamics in lead sulfide nanocrystals is interrogated by degenerate pump−probe spectroscopy with 20−25 fs pulses over a broad frequency range around three times the nanocrystal band gap. For each nanocrystal diameter, an initial reduction in absorption is seen only at the peak of the quantum confined E1 transition, while increased absorption is seen at all other wavelengths. The signals from the nanocrystals are ∼300 times weaker than expected for a two-level system with the same absorbance and molar extinction coefficient and are weaker near time zero. These results appear to be inconsistent with quantum confinement of the initially excited high energy states. Arguments based on carrier scattering length, the wave packet size supported by the band structure, and effective mass are advanced to support the hypothesis that, for many direct-gap semiconductor quantum dots, the carrier dynamics at three times the band gap is localized on the 1−2 nm length scale and essentially bulklike except for frequent collisions with the surface.

Four-level two-dimensional (2D) Fourier transform relaxation spectra are simulated with response functions for a chromophore pair in the exponential relaxation (optical Bloch model) limit. The parameters in this study are chosen to model coupled carbonyl stretching vibrations. As long as coherence persists, every peak in the real 2D spectra has a partially mixed absorptive/dispersive (“phase-twisted”) shape because the nonlinear signals are not symmetric with respect to interchange of the first two pulses. This asymmetry in 2D relaxation spectra arises from coherence between singly excited states and a red shift of the doubly excited state. Coherence between the singly excited states causes oscillation of the 2D spectra and the associated spectrally resolved pump−probe (SRPP) transients at the quantum beat frequency. Projecting the phase-twisted nature of the 2D peaks onto the detection frequency axis, the SRPP peaks are also asymmetric about their maximum when not at maximum or minimum amplitude. Three-dimensional Fourier transform (3DFT) methods are used to simulate absorption/dispersion and beam geometry distortions of the multilevel 2D spectra with cross peaks. The distortions can be understood by consideration of their effects on individual coherence pathways that contribute to peaks in the 2D spectra. The beam geometry distortion explains some unequal cross peak amplitudes previously observed experimentally by Khalil et al. (J. Chem. Phys. 2004, 121, 362). A representation of 2D spectra that reduces beam geometry distortion is presented. If the transformation to correct for beam geometry distortion is combined with the transformations that correct absorptive/dispersive propagation distortions (J. Chem. Phys. 2007, 126, 044511), the recovered 2D spectrum matches the ideal 2D spectrum after all coherence is destroyed. In the presence of coherence, the new representation reduces the error in the distorted 2D spectrum by a factor of 4 for practical 2D-IR experimental conditions.