Model Hamiltonian Approach Research Articles

Robust half-metallicity and topological properties in square-net potassium manganese chalcogenides

Combining ab initio and model Hamiltonian approaches, we investigate the electronic, magnetic, and topological properties of potassium manganese chalcogenides that host square nets of Mn and chalcogen atoms. Our analysis establishes these compounds to be robust half-metallic ferromagnets. The origin of ferromagnetism in these compounds is found to be based on a kinetic energy-driven double-exchange mechanism, first proposed for ${\mathrm{Sr}}_{2}{\mathrm{FeMoO}}_{6}$, a double-perovskite compound. The presence of finite spin-orbit coupling at chalcogen sites triggers nontrivial topology of the chalcogen-derived bands at the conducting channel, dominating the electronic structure close to the Fermi level. This puts the studied compounds in the class of topological half-metals with appreciable values of anomalous Hall conductivity, opening up the application possibility in topological quantum spintronics. Cleavability of these layered compounds makes the situation further promising.

An ab initio method on large sized molecular aggregate system: Predicting absorption spectra of crystalline organic semiconducting films.

Theoretical description of electronically excited states of molecular aggregates at an ab initio level is computationally demanding. To reduce the computational cost, we propose a model Hamiltonian approach that approximates the electronically excited state wavefunction of the molecular aggregate. We benchmark our approach on a thiophene hexamer, as well as calculate the absorption spectra of several crystalline non-fullerene acceptors, including Y6 and ITIC, which are known for their high power conversion efficiency in organic solar cells. The method qualitatively predicts the experimentally measured spectral shape, which can be further linked to the molecular arrangement in the unit cell.

Hybrid Molecular Magnets with Lanthanide- and Countercation-Mediated Interfacial Electron Transfer between Phthalocyanine and Polyoxovanadate.

A series of {V12}-nuclearity polyoxovanadate cages covalently functionalized with one or sandwiched by two phthalocyaninato (Pc) lanthanide (Ln) moieties via V-O-Ln bonds were prepared and fully characterized for paramagnetic Ln = SmIII-ErIII and diamagnetic Ln = LuIII, including YIII. The LnPc-functionalized {V12O32} cages with fully oxidized vanadium centers in the ground state were isolated as (nBu4N)3[HV12O32Cl(LnPc)] and (nBu4N)2[HV12O32Cl(LnPc)2] compounds. As corroborated by a combined experimental (EPR, DC and AC SQUID, laser photolysis transient absorption spectroscopy, and electrochemistry) and computational (DFT, MD, and model Hamiltonian approach) methods, the compounds feature intra- and intermolecular electron transfer that is responsible for a partial reduction at V(3d) centers from VV to VIV in the solid state and at high sample concentrations. The effects are generally Ln dependent and are clearly demonstrated for the (nBu4N)3[HV12O32Cl(LnPc)] representative with Ln = LuIII or DyIII. Intramolecular charge transfer takes place for Ln = LuIII and occurs from a Pc ligand via the Ln center to the {V12O32} core of the same molecule, whereas for Ln = DyIII, only intermolecular charge transfer is allowed, which is realized from Pc in one molecule to the {V12O32} core of another molecule usually via the nBu4N+ countercation. For all Ln but DyIII, two of these phenomena may be present in different proportions. Besides, it is demonstrated that (nBu4N)3[HV12O32Cl(DyPc)] is a field-induced single molecule magnet with a maximal relaxation time of the order 10-3 s. The obtained results open up the way to further exploration and fine-tuning of these three modular molecular nanocomposites regarding tailoring and control of their Ln-dependent charge-separated states (induced by intramolecular transfer) and relaxation dynamics as well as of electron hopping between molecules. This should enable us to realize ultra-sensitive polyoxometalate powered quasi-superconductors, sensors, and data storage/processing materials for quantum technologies and neuromorphic computing.

Spin interaction and magnetism in cobaltate Kitaev candidate materials: An ab initio and model Hamiltonian approach

In the quest for materials hosting Kitaev spin liquids, much of the efforts have been focused on the fourth- and fifth-row transition metal compounds, which are spin-orbit coupling assisted Mott insulators. Here, we study the structural and magnetic properties of 3$d$ transition metal oxides, Na$_2$Co$_2$TeO$_6$ and Na$_3$Co$_2$SbO$_6$. The partial occupancy of sodium in former compound is addressed using a cluster expansion, and a honeycomb lattice of sodiums is found to be energetically favored. Starting from the \textit{ab initio} band structures, a many-body second order perturbation theory leads to a pseudospin-$\frac{1}{2}$ Hamiltonian with estimated magnetic interactions. We show that the experimentally observed zigzag magnetic state is stabilized only when the first neighbor Kitaev coupling dominates over the Heisenberg term, both of which are highly suppressed due to presence of $e_g$ orbitals. A third neighbor Heisenberg interaction is found dominant in both these compounds. We also present a phase diagram for Na$_2$Co$_2$TeO$_6$ by varying the electron-electron and spin-orbit interactions. The computed spin excitation spectra are found to capture essential features of recent experimental magnon spectrum, lending support to our results.

Helium droplet infrared spectroscopy of the butyl radicals.

Butyl radicals (n-, s-, i-, and tert-butyl) are formed from the pyrolysis of stable precursors (1-pentyl nitrite, 2-methyl-1-butyl nitrite, isopentyl nitrite, and azo-tert-butane, respectively). The radicals are doped into a beam of liquid helium droplets and probed with infrared action spectroscopy from 2700 to 3125 cm-1, allowing for a low temperature measurement of the CH stretching region. The presence of anharmonic resonance polyads in the 2800-3000 cm-1 region complicates its interpretation. To facilitate spectral assignment, the anharmonic resonances are modeled with two model Hamiltonian approaches that explicitly couple CH stretch fundamentals to HCH bend overtones and combinations: a VPT2+K normal mode model based on coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)] quartic force fields and a semi-empirical local mode model. Both of these computational methods provide generally good agreement with the experimental spectra.

Unified ab initio description of Fröhlich electron-phonon interactions in two-dimensional and three-dimensional materials

\textit{Ab initio} calculations of electron-phonon interactions including the polar Fr\"ohlich coupling have advanced considerably in recent years. The Fr\"ohlich electron-phonon matrix element is by now well understood in the case of bulk three-dimensional (3D) materials. In the case of two-dimensional (2D) materials, the standard procedure to include Fr\"ohlich coupling is to employ Coulomb truncation, so as to eliminate artificial interactions between periodic images of the 2D layer. While these techniques are well established, the transition of the Fr\"ohlich coupling from three to two dimensions has not been investigated. Furthermore, it remains unclear what error one makes when describing 2D systems using the standard bulk formalism in a periodic supercell geometry. Here, we generalize previous work on the \textit{ab initio} Fr\"ohlich electron-phonon matrix element in bulk materials by investigating the electrostatic potential of atomic dipoles in a periodic supercell consisting of a 2D material and a continuum dielectric slab. We obtain a unified expression for the matrix element, which reduces to the existing formulas for 3D and 2D systems when the interlayer separation tends to zero or infinity, respectively. This new expression enables an accurate description of the Fr\"ohlich matrix element in 2D systems without resorting to Coulomb truncation. We validate our approach by direct \textit{ab initio} density-functional perturbation theory calculations for monolayer BN and MoS$_2$, and we provide a simple expression for the 2D Fr\"ohlich matrix element that can be used in model Hamiltonian approaches. The formalism outlined in this work may find applications in calculations of polarons, quasiparticle renormalization, transport coefficients, and superconductivity, in 2D and quasi-2D materials.

Signatures of Dimensionality and Symmetry in Exciton Band Structure: Consequences for Exciton Dynamics and Transport

Exciton dynamics, lifetimes, and scattering are directly related to the exciton dispersion or band structure. Here, we present a general theory for exciton band structure within both ab initio and model Hamiltonian approaches. We show that contrary to common assumption, the exciton band structure contains nonanalytical discontinuities—a feature which is impossible to obtain from the electronic band structure alone. These discontinuities are purely quantum phenomena, arising from the exchange scattering of electron–hole pairs. We show that the degree of these discontinuities depends on materials’ symmetry and dimensionality, with jump discontinuities occurring in 3D and different orders of removable discontinuities in 2D and 1D, whose details depend on the exciton degeneracy and material thickness. We connect these features to the early stages of exciton dynamics, radiative lifetimes, and diffusion constants, in good correspondence with recent experimental observations, revealing that the discontinuities in the band structure lead to ultrafast ballistic transport and suggesting that measured exciton diffusion and dynamics are influenced by the underlying exciton dispersion.

Strong influence of nonmagnetic ligands on the momentum-dependent spin splitting in antiferromagnets

Recent studies have shown that the non-relativistic antiferromagnetic ordering could generate momentum-dependent spin splitting analogous to the Rashba effect, but free from the requirement of relativistic spin-orbit coupling. Whereas the classification of such compounds can be illustrated by different spin-splitting prototypes (SSTs) from symmetry analysis and density functional theory calculations, the significant variation in bonding and structure of these diverse compounds representing different SSTs clouds the issue of how much of the variation in spin splitting can be traced back to the symmetry-defined characteristics, rather to the underlining chemical and structural diversity. The alternative model Hamiltonian approaches do not confront the issues of chemical and structural complexity, but often consider only the magnetic sublattice, dealing with the all-important effects of the non-magnetic ligands via renormalizing the interactions between the magnetic sites. To this end we constructed a 'DFT model Hamiltonian' that allows us to study SSTs at approximate 'constant chemistry', while retaining the realistic atomic scale structure including ligands. This is accomplished by using a single, universal magnetic skeletal lattice (Ni2+ ions in Rocksalt NiO) and designing small displacements of the non-magnetic (oxygen) sublattice which produce, by design, the different SSTs magnetic symmetries. We show that (i) even similar crystal structures having very similar band structures can lead to contrasting behavior of spin splitting vs. momentum, and (ii) even subtle deformations of the non-magnetic ligand sublattice could cause a giant spin splitting in AFM-induced SST. This is a paradigm shift relative to the convention of modeling magnets without considering the non-magnetic ligand that mediate indirect magnetic interaction (e.g., super exchange).

Emergence of strong tunable linear Rashba spin-orbit coupling in two-dimensional hole gases in semiconductor quantum wells

Two-dimensional hole gases (2DHGs) in semiconductor quantum wells are promising platforms for spintronics and quantum computation, but suffer from the lack of the $\mathbf{k}$-linear term in the Rashba spin-orbit coupling (SOC), which is essential for spin manipulations without magnetism. The Rashba SOC in 2DHGs is commonly believed to be a $\mathbf{k}$-cubic term as the lowest order. Here, contrary to conventional wisdom, we taking Ge/Si system as an example uncover a strong and tunable $\mathbf{k}$-linear Rashba SOC in 2DHGs of semiconductor quantum wells (QWs) by performing atomistic pseudopotential calculations in conjunction with theoretical analysis based on the effective model Hamiltonian approach. We illustrate that this emergent $\mathbf{k}$-linear Rashba SOC is a first-order direct Rashba effect, originating from a combination of heavy-hole-light-hole mixing and direct dipolar coupling to the external electric field. The enhanced interband mixing renders [110]-oriented Ge/Si QWs a much stronger linear Rashba SOC than [001]-oriented counterpart with the maximal strength exceeding 120 meV\AA{}, comparable to the highest values reported in two-dimensional electron gases made by narrow band-gap III-V semiconductors, which suffers from short spin lifetime due to the presence of nuclear spin. These findings confirm Ge-based 2DHGs to be an excellent platform for large-scale quantum computation.

A quantum chemical model for a series of self-assembled nanocages: the origin of stability behind the coordination-driven formation of transition metal complexes up to [M12L24]24.

Herein, we present a systematic computational model to study the electronic states and free energies of a self-assembled multi-metal complex series. By combining the previously developed model Hamiltonian approach for transition-metal complexes and the generalized Born model, the thermodynamics, optimized geometries, and electronic states of the [Pd12L24]24+ nanocage are revealed, together with [PdnLm]2n+ complex series. The effective model Hamiltonian is a theoretical method to obtain the d-electron wavefunction and potential energy including interaction energy between the transition-metal and ligands. In the present improvement, the electronic state on each transition-metal center is focused as a building unit and solved under the whole electronic field of the assembling system. We realize a reliable and systematic treatment of multi-transition-metal complexes with different sizes and charges. Consequently, our model could reproduce binding energies of the [PdnLm]2n+ complex series quantitatively as compared to density functional theory (DFT). Regarding free energy, we revealed that the assembling solute becomes unstable due to the electrostatic interaction, and effects of the solvent and counter anions mainly compensated it. Optimized geometries were also analysed. The local square-planar coordination structures around the palladium centres were characterized in the complex series. The relationships between the entire symmetrical geometries and the local coordination structures are also discussed. Finally, electronic structures of the [Pd12L24]24+ nanocage were well characterized as a single-determinant, where only dx2-y2 is unoccupied due to the ligand-field effect. We also found that the solvent polarized the electronic states of the Pd ions, whereas the counter anion suppressed the polarization. The present method realizes size-independent reliable and rapid computations, and therefore can be expected to further application studies on self-assembly dynamics.

Theoretical analysis of pressure induced spin crossover phenomenon in a di-nuclear Fe(II) molecular complex

We have studied a Fe-based di-nuclear molecular complex having the chemical formula [{Fe(bpp)(NCS)2}2(’-bipy)]·2MeOH (where bpp = -bis(pyrazol-3-yl) pyridine and ’-bipy = ’-bipyridine, 1) using density functional theory and model Hamiltonian approach. Our study provides insight to the pressure driven spin-crossover (SCO) phenomena observed experimentally in these systems. Upon increasing the pressure, the spin state of Fe(II) cation gradually changes from a high spin state (S =2) to a low spin (LS) state (S =0) accompanied by volume contraction. The gradual increase in pressure shrinks Fe–N bond length and also causes angular deviation of the FeN6 octahedron leading to full conversion to the LS state without global structural phase transition. We have carried out exact diagonalization study of an effective single site Hamiltonian and confirmed the importance of intramolecular interaction for SCO phenomena. We have investigated the cooperativity of the observed SCO phenomena. We have also studied the effect of Co doping on the spin state of Fe and find that the spin state of Fe has a subtle dependency on the concentration of dopant atoms. Excess Co doping pave the way towards the possibility of an intermediate spin state for Fe and can give rise to a bistable spin transition process.

NiRh2O4 : A spin-orbit entangled diamond-lattice paramagnet

Motivated by the interest in topological quantum paramagnets in candidate spin-$1$ magnets, we investigate the diamond lattice compound NiRh$_2$O$_4$ using {\it ab initio} theory and model Hamiltonian approaches. Our density functional study, taking into account the unquenched orbital degrees of freedom, shows stabilization of $S=1$ and $L=1$ state. We highlight the importance of spin-orbit coupling, in addition to Coulomb correlations, in driving the insulating gap, and uncover frustrating large second-neighbor exchange mediated by Ni-Rh covalency. A single-site model Hamiltonian incorporating the large tetragonal distortion is shown to give rise to a spin-orbit entangled non-magnetic ground state, largely accounting for the entropy, magnetic susceptibility, and inelastic neutron scattering results. Incorporating inter-site exchange within a slave-boson theory, we show that exchange frustration can suppress exciton condensation. We capture the dispersive gapped magnetic modes, uncover `dark states' invisible to neutrons, and make predictions for future experiments.

Multiple band crossings and Fermi surface topology: Role of double nonsymmorphic symmetries in MnP-type crystal structures

We use relativistic ab initio methods combined with model Hamiltonian approaches to analyze the normal-phase electronic and structural properties of the recently discovered WP superconductor. Remarkably, the outcomes of such study can be employed to set fundamental connections among WP and the CrAs and MnP superconductors belonging to the same space group. One of the key features of the resulting electronic structure is represented by the occurrence of multiple band crossings along specific high-symmetry lines of the Brillouin zone. In particular, we demonstrate that the eightfold band degeneracy obtained along the $\mathit{SR}$ path at $({k}_{x},{k}_{y})=(\ensuremath{\pi},\ensuremath{\pi})$ is due to inversion time-reversal invariance and a pair of nonsymmorphic symmetries. The presence of multiple degenerate Fermi points along the $\mathit{SR}$ direction constrains the topology of the Fermi surface, which manifests distinctive marks when considering its evolution upon band filling variation. If the Fermi level crosses the bands along the $\mathit{SR}$ line as it happens at the nominal filling of the MnP, these Fermi surfaces are open or closed Fermi pockets. Moving the relative position of the Fermi level away from the eightfold degenerate bands as for the WP and CrAs compounds, the electronic changeover exhibits a simultaneous modification of the Fermi-surface dimensionality and topology. Four two-dimensional (2D) Fermi-surface sheets are centered around the $\mathit{SR}$ line with a corrugated profile along the ${k}_{z}$ direction. Moreover, we show that the spin-orbit interaction determines a selective removal of the band degeneracy and, consequently, a splitting of the quasi-2D Fermi sheets, as it happens in WP. Finally, we comment on the connections between our results and recent experimental and theoretical proposals about the triplet superconductivity in this class of compounds.

Unified theory of plasmon-induced resonance energy transfer and hot electron injection processes for enhanced photocurrent efficiency.

Plasmons in metal nanoparticles (MNPs) promise to enhance solar energy conversion in semiconductors. Two essential mechanisms of enhancement in the near-field regime are hot electron injection (HEI) and plasmon-induced resonance energy transfer (PIRET). Individual studies of both mechanisms indicate that the PIRET efficiency is limited by the short lifetime of the plasmon, whereas the hot electrons result from the plasmon decay. The development of a unified theory of the coupled HEI and PIRET processes is fundamentally interesting and necessary for making reliable predictions but is complicated by the multiple interactions between various components that participate in the enhancement process. In this paper, we use the model-Hamiltonian approach to develop a combined theoretical framework including both PIRET and HEI. The coupled dynamics as well as the time evolution of hot electron energy distribution are studied. The theory further predicts an interference-induced asymmetry in the spectral dependence of PIRET, which can be used to distinguish it from HEI. As the relative contributions of PIRET and HEI strongly depend on the size of the MNPs, this presents itself as a simple route to control the strength of their contributions. The results presented here can further guide future applications of plasmonic solar energy harvesting.

Enhanced Magnetoresistance in Chiral Molecular Junctions.

Chirality-induced spin selectivity (CISS) is a recently discovered effect, whose precise microscopic origin has not yet been fully elucidated; it seems, however, clear that spin-orbit interaction plays a pivotal role. Various model Hamiltonian approaches have been proposed, suggesting a close connection between spin selectivity and filtering and helical symmetry. However, first-principles studies revealing the influence of chirality on the spin polarization are missing. To clearly demonstrate the influence of the helical conformation on the spin polarization properties, we have carried out spin-dependent Density-Functional Theory (DFT) based transport calculations for a model molecular system. It consists of α-helix and β-strand conformations of an oligo-glycine peptide, which is bonded to a nickel electrode and to a gold electrode in a two-terminal setup, similar to a molecular junction or a local probe, for example, in STM or AFM configurations. We have found that the α-helix conformation displays a spin polarization, calculated through the intrinsic magneto-resistance of the junction, about 100-1000 times larger than the linear β-strand, clearly demonstrating the crucial role played by the molecular helical geometry on the enhancement of spin polarization associated with the CISS effect.

Emergence of Landauer transport from quantum dynamics: A model Hamiltonian approach.

The Landauer expression for computing current-voltage characteristics in nanoscale devices is efficient but not suited to transient phenomena and a time-dependent current because it is applicable only when the charge carriers transition into a steady flux after an external perturbation. In this article, we construct a very general expression for time-dependent current in an electrode-molecule-electrode arrangement. Utilizing a model Hamiltonian (consisting of the subsystem energy levels and their electronic coupling terms), we propagate the Schrödinger wave function equation to numerically compute the time-dependent population in the individual subsystems. The current in each electrode (defined in terms of the rate of change of the corresponding population) has two components, one due to the charges originating from the same electrode and the other due to the charges initially residing at the other electrode. We derive an analytical expression for the first component and illustrate that it agrees reasonably with its numerical counterpart at early times. Exploiting the unitary evolution of a wavefunction, we construct a more general Landauer style formula and illustrate the emergence of Landauer transport from our simulations without the assumption of time-independent charge flow. Our generalized Landauer formula is valid at all times for models beyond the wide-band limit, non-uniform electrode density of states and for time and energy-dependent electronic coupling between the subsystems. Subsequently, we investigate the ingredients in our model that regulate the onset time scale of this steady state. We compare the performance of our general current expression with the Landauer current for time-dependent electronic coupling. Finally, we comment on the applicability of the Landauer formula to compute hot-electron current arising upon plasmon decoherence.

Chirality-Dependent Electron Spin Filtering by Molecular Monolayers of Helicenes.

The interaction of low-energy photoelectrons with well-ordered monolayers of enantiopure helical heptahelicene molecules adsorbed on metal surfaces leads to a preferential transmission of one longitudinally polarized spin component, which is strongly coupled to the helical sense of the molecules. Heptahelicene, composed of only carbon and hydrogen atoms, exhibits only a single helical turn but shows excess in longitudinal spin polarization of about P Z = 6 to 8% after transmission of initially balanced left- and right-handed spin polarized electrons. Insight into the electronic structure, that is, the projected density of states, and the spin-dependent electron scattering in the helicene molecule is gained by using spin-resolved density functional theory calculations and a model Hamiltonian approach, respectively. Our results support the semiclassical picture of electronic transport along a helical pathway under the influence of spin-orbit coupling induced by the electrostatic molecular potential.

The magnetic effects on the energetic landscape of Fe-Cu alloy: A model Hamiltonian approach

Using ab initio calculations and the magnetic cluster expansion formula, a model with the magnetic effects on the energetic landscape of Fe–Cu was developed. The (partially) disordered local moment picture and Monte Carlo simulation method were combined to study the magnetically disordering effects on the thermodynamic properties in Fe–Cu alloys. The simulation results show nonlinear enhancements of Fe magnetism with increasing Cu concentration under varying magnetic ordering/disordering states, and the nonlinearity of the enhancement is more significant under the ferromagnetic state. It is also found that magnetism has subtle effects on the clustering of Cu because both the substitutional and binding energies could be affected but in different ways depending on the magnetic states. The results suggest that the alloying-induced enhancement of the Fe magnetism is a fundamental feature of Fe–Cu, which introduces difference on the concentration dependence of mixing, depending on the magnetic ordering/disordering states. In conjunction with the concentration dependence of short-range ordering, the enhancement of Fe magnetism also strongly affects the Curie temperatures in Fe–Cu.

Origin of Plasmon Lineshape and Enhanced Hot Electron Generation in Metal Nanoparticles

Plasmon-generated hot carriers are currently being studied intensively for their role in enhancing the efficiency of photovoltaic and photocatalytic processes. Theoretical studies of the hot electrons subsystem have generated insight, but we show that a unified quantum-mechanical treatment of the plasmon and hot electrons reveals new physical phenomena. Instead of a unidirectional energy transfer process in Landau damping, back energy transfer is predicted in small metal nanoparticles (MNPs) within a model-Hamiltonian approach. As a result, the single Lorentzian plasmonic line shape is modulated by a multipeak structure, whose individual line width provides a direct way to probe the electronic dephasing. More importantly, the hot electron generation can be enhanced greatly by matching the incident energy to the peaks of the modulated line shape.

First-Principles Models for Biological Light-Harvesting: Phycobiliprotein Complexes from Cryptophyte Algae.

There have been numerous efforts, both experimental and theoretical, that have attempted to parametrize model Hamiltonians to describe excited state energy transfer in photosynthetic light harvesting systems. The Frenkel exciton model, with its set of electronically coupled two level chromophores that are each linearly coupled to dissipative baths of harmonic oscillators, has become the workhorse of this field. The challenges to parametrizing such Hamiltonians have been their uniqueness, and physical interpretation. Here we present a computational approach that uses accurate first-principles electronic structure methods to compute unique model parameters for a collection of local minima that are sampled with molecular dynamics and QM geometry optimization enabling the construction of an ensemble of local models that captures fluctuations as these systems move between local basins of inherent structure. The accuracy, robustness, and reliability of the approach is demonstrated in an application to the phycobiliprotein light harvesting complexes from cryptophyte algae. Our computed Hamiltonian ensemble provides a first-principles description of inhomogeneous broadening processes, and a standard approximate non-Markovian reduced density matrix dynamics description is used to estimate lifetime broadening contributions to the spectral line shape arising from electronic-vibrational coupling. Despite the overbroadening arising from this approximate line shape theory, we demonstrate that our model Hamiltonian ensemble approach is able to provide a reliable fully first-principles method for computation of spectra and can distinguish the influence of different chromophore protonation states in experimental results. A key feature in the dynamics of these systems is the excitation of intrachromophore vibrations upon electronic excitation and energy transfer. We demonstrate that the Hamiltonian ensemble approach provides a reliable first-principles description of these contributions that have been detailed in recent broad-band pump-probe and two-dimensional electronic spectroscopy experiments.