Localized intrinsic bond orbitals decode correlated charge migration dynamics

Abstract

For decades, scientists have studied the intricate charge migration dynamics, where, after ionization, a localized charge distribution (“hole”) migrates across the molecule on a femtosecond timescale. This has the potential for controlling electrons in molecules, yet a comprehensive understanding of the many aspects of charge migration is still missing. In this work, we analyze charge migration using an extension of localized intrinsic bond orbitals (IBOs). These orbitals lead to a compact representation of the dynamics and map the complex, correlated many-electron charge migration to chemical concepts such as curly arrows and orbital–orbital interactions. By analyzing multiple challenging scenarios, we show how IBOs enable us to identify key mechanisms in charge migration. For example, we show that different mechanisms are responsible for converting a π-shaped hole to a σ-shaped hole and vice versa. We explain these in terms of hyperconjugation interactions and configurations that couple orbitals with different symmetries. We further demonstrate how IBOs can be used to find molecules with high charge migration efficiency. We carry out all simulations using an efficient setup of the time-dependent density matrix renormalization group (TDDMRG), correlating as many as 45 electrons in 50 orbitals. We believe that our results will be useful to design future experiments. The proposed IBO analysis is applicable to other types of real-time electron dynamics and spectroscopy.

Localized bond orbitals offer a compact, chemically intuitive view of ultrafast, correlated charge migration and reveal its driving interactions. This is demonstrated for nine migration scenarios in five molecules using advanced TDDMRG simulations.

1. Introduction

After the removal of an electron, how do the remaining electrons in a molecule respond, and how can this response be tuned and exploited? Answering this is central to the recently established field of attochemistry,^1–4 and is useful to understand light-induced effects in DNA^5,6 and photofragmentation,^7–13 among many others.^14–20 One particular response after an electron removal is charge migration, in which an often localized “charge cloud” or electron “hole” migrates through a molecule.^8,10,21 Charge migration is initiated within attoseconds^22,23 and leads to the initial hole migrating to the opposite end of small molecules such as amino acids in just a few femtoseconds.^24–28 So far, a plethora of simulations have investigated many properties and effects of charge migration, such as the controllability of the dynamics using ultrashort laser pulses,^17,19 nuclei-induced decoherence effects,^29–35 and molecular symmetry³⁶ or lack thereof,^37–39 among others.^26,40–47

To systematically understand charge migration, we must identify universal patterns and rules that allow us to predict charge migration starting from different orbitals in different molecules.^21,48–52 As the nonequilibrium, many-body, real-time electron dynamics of charge migration can be very complex and require the inclusion of electron-correlation effects,²¹ many tools have been used to analyze charge migration. Among others, these include various time-frequency analyses,^21,48,49,53 density matrices (“hole density,” which we will define later) and their eigenstates (“natural charge orbitals”),⁴⁸ partial charges,^54–56 electron fluxes,^{38,43,57–60} decompositions of the time-dependent wavefunction in terms of hole configurations,⁴⁸ and the time-dependent electron localization function.⁴² Despite these tools and both a large number of simulations and experiments, many features of charge migration still remain elusive.^50,61–67

Next to the aforementioned ways to understand charge migration, quantum chemistry offers many more tools to gain insights by mapping the complicated many-body wavefunction to simple chemical concepts such as curly arrows⁶⁸ and atomic charges. Examples of such tools include the quantum theory of atoms in molecules,^69,70 energy decompositions,^71–76 wave function tilings,^77,78 entanglement patterns,^79–82 and localized orbitals.^82–91 In particular, intrinsic bond orbitals (IBOs)⁹⁰ have been shown to be very useful, e.g., to verify empirical laws,^90,92 to understand oxidation states,^90,93,94 and to gain an understanding of chemical reactions in terms of curly-arrow mechanisms,^92,94–96e.g., for gold catalysts,⁹³ electron transfer in proteins,⁹⁵ and electron flow in chemical reactions.^92,97 However, most of these use cases involve ground state electronic structures, with some exceptions covering excited eigenstates.^98,99

In this work, we extend the IBO formalism to describe real-time, correlated, nonequilibrium charge migration. Using various challenging scenarios in complex molecules, we show that IBOs work well even for complex charge migration where the commonly assumed molecular orbital picture breaks down. In addition, their localized and time-independent character makes IBOs easier to analyze than frequently used tools such as time-dependent natural charge orbitals. We also show that IBOs can aid the interpretation of ionization spectra without requiring real-time dynamics simulations. Importantly, we connect established concepts such as hole configurations to IBO-derived quantities and interpret charge migration using local symmetries, curly-arrow mechanisms based on Lewis structures, as well as intermolecular and through-space orbital interactions based on hyperconjugation.¹⁰⁰ We introduce the latter as a new, important concept for understanding charge migration.

Simulating correlated, nonequilibrium real-time electron dynamics extends electronic structure methods beyond their usual domain.¹⁰¹ We use a recently developed efficient pipeline⁵⁶ for the time-dependent quantum-chemistry density matrix renormalization group (TDDMRG).^102–111 As an inherently multireference method, TDDMRG is well suited to charge migration involving superpositions of highly excited configurations. We simulate fully correlated dynamics of molecules with up to 45 electrons in 50 active orbitals, which is, to our knowledge, the largest TDDMRG simulation to date and prohibitively expensive for many other multireference methods. Together with our IBO analysis framework, the TDDMRG may also support studies of other ionization and excitation spectra,^112–115 nonequilibrium dynamics of excited molecules,^116–119 and core-ionized molecules.¹²⁰

The remainder of this paper is organized as follows. In Section 2, we overview theories and tools employed in the present work, including an extension to the IBO concept. We present the results of our simulations and analyze them in Section 3. We conclude our findings and give our perspective on future use cases in Section 4.

2. Theory

This section is outlined as follows. In Section 2.1, we present an overview of the TDDMRG to simulate real-time electron dynamics. We can gain additional insights into such dynamics from ionization spectra, which we discuss in Section 2.2. In Section 2.3, we review charge migration and common tools to analyze it. As a new tool to analyze charge migration, we give an overview of IBOs and their extensions in Section 2.4. Lastly, in Section 2.5, we discuss global and local symmetries. Throughout, we use atomic units unless specified otherwise.

2.1. Time evolution using the density matrix renormalization group

We simulate electron dynamics by approximately solving the time-dependent Schrödinger equation using the TDDMRG. In this method, the time-dependent state is expressed as a matrix product state (MPS),

1

where K is the number of active spatial orbitals, |σ_i〉 denotes one of the four possible spin configurations in the i-th orbital, and M^σ_i(t) are matrices of size D_i−1 × D_i. The largest of the D_i is called bond dimension, D. The larger D, the better the approximation to the exact full configuration interaction (CI) state. By using the ansatz of eqn (1) to find the extremum of the time-dependent variational principle, one obtains a set of coupled equations to solve for M^σ_i(t). A Lie–Trotter splitting uncouples these equations, which leads to an algorithm that, essentially, sequentially propagates one matrix M^σ_i(t) at a time. For more information, we refer to ref. 104–106 and 111, the Appendix and our recently developed simulation pipeline.⁵⁶

2.2. Ionization spectra

While charge migration focuses on the time-dependent dynamics after ionization, analyzing it in terms of a time-independent view through ionization spectra provides an additional, helpful perspective, as it allows scientists to identify the relevant eigenstates that contribute to charge migration.^21,48 Through linear response theory, the spectrum can be obtained from the dynamics via the autocorrelation function A(t),^121,122

2

A Fourier transform of A(t) leads to the photo-absorption spectrum P as a function of excitation energy E,

3

To describe the experimental energy resolution, a finite value of η is used, which leads to a convolution of P(E) with a Lorentzian.

As an alternative to the time-dependent approach, the spectrum can also be obtained by computing a Green's function G(E) as

4

with

5

In practice, the time-dependent approach, eqn (3), allows us to obtain the total spectrum with a finite resolution determined by the propagation time, whereas the time-independent Green's-function-based approach, eqn (4), allows us to target particular peaks with a resolution determined numerically through a finite value of η (the smaller η, the higher the resolution, but the more difficult it is to compute eqn (4)). The spectrum is then described as a linear combination of Lorentzians.¹²³

The state Ĝ(E)|Ψ(t = 0)〉 provides further insights into the spectrum. Expanding it in terms of eigenstates {|u_i〉} with energies {E_i},

6

shows that Ĝ(E)|Ψ(t = 0)〉 has dominant contributions from eigenstates close to the target energy E, and, on resonance with E → E_i, it turns into eigenstate |u_i〉 after normalization.

Expressing |Ψ(t = 0)〉 as an MPS, eqn (4) can be solved using DMRG-like procedures that consist of solving linear systems of equations sequentially for each of the matrices M^σ_i in eqn (1). For more details, we refer to ref. 110 and 123–125.

2.3. Charge migration

Charge migration happens when, after ionization, a positively charged region or electron hole moves through the molecule on an ultrafast timescale of attoseconds to a few femtoseconds. To model charge migration, we use the common^{8,12,48–50,126–129} sudden ionization approximation,^130,131 and describe the initial state |Ψ(0)〉 by removing an electron from the ground state |Ψ_GS〉. In second quantization, this is described by

7

where â_i is the annihilation operator of a localized orbital |i〉. Sudden ionization can be achieved, e.g., by using laser pulses with dominant frequencies that overlap with orbital energies of one particular functional group in the molecule. Next to sudden ionization, we assume that the nuclei are frozen during the electron propagation, which is a good assumption for the first few femtoseconds. While this setup is idealized, such simulations still illustrate charge migration, often lead to significant insights, and match experimental observables.^21,27,48,49

A central quantity of charge migration is the hole density h(r, t), which is defined as the difference between the spin-summed ground-state one-particle density ρ_GS(r) and the density ρ(r, t) of the ionized wavepacket at time t,

h(r, t) = ρ_GS(r) − ρ(r, t). 8

For single ionization, h(r, t) describes the motion of one hole particle (it integrates to 1). The contribution q_i(t) of orbital |i〉 to h(r, t) is called hole occupancy and is computed as q_i(t) = 〈i|ĥ(t)|i〉, where ĥ(t) is the corresponding operator of h(r, t). According to eqn (8), h(r, t) < 0 indicates an electron excess relative to the neutral molecule, whereas h(r, t) > 0 indicates a hole. The same sign convention applies to the hole occupancies q_i(t): positive and negative values denote hole occupancy and electron excess, respectively, in orbital |i〉. Analogous to natural orbitals, the time-dependent orbitals that diagonalize ĥ(t) are called natural charge orbitals and can aid the analysis.⁴⁸ However, their time dependence and often complex-valued form make them less convenient than time-independent, real-valued orbitals.

Another useful way to analyze charge migration is to expand the time-dependent cationic state |Ψ(t)〉 in terms of a CI-based expansion,⁴⁸

9

where |Φ_GS〉 is an approximation to the neutral ground state. In many cases, |Φ_GS〉 is a Hartree–Fock-based configuration, but in our case, it will be the DMRG-approximated ground state, |Ψ_GS〉. The configurations or states â_j|Φ_GS〉 are called one-hole (1h) configurations, and â^†_râ_kâ_l|Φ_GS〉 are called two-hole one-particle (2h1p) configurations.† In contrast to 1h configurations, 2h1p and higher-order configurations extend the mean-field ansatz and are required to recover electron correlation. In the ground state electronic structure, 1h (1h and 2h1p) configurations correspond to CIS (CISD). Furthermore, 2h1p and higher-order configurations can involve orbitals of different irreducible representations (irreps), which leads to important consequences in charge migration. Since 2h1p configurations are formed through a removal of two electrons from |Φ_GS〉 followed by an addition of another electron in a different orbital, the occurrence of 2h1p (and higher-order, 3h2p, etc.) configurations can be identified by orbitals with negative hole occupancies. These orbitals are often antibonding, and their occupancy weakens the corresponding bond. We will discuss this below in Section 2.5 and Section 3.2.

2.4. Intrinsic bond orbitals and their extensions

IBOs are convenient to create and provide orbital shapes that match chemical intuition.⁹⁰ The original IBO procedure is for bonding orbitals. Below, we discuss an extension to antibonding orbitals. IBO constructions start with the minimal-basis intrinsic atomic orbitals (IAOs), which are not only localized around atoms but also encode the polarization environment of the molecule. IAOs are computed by projecting n_min free-atom minimal basis set orbitals onto polarized Hartree–Fock (HF) occupied and unoccupied molecular orbitals (MOs) |o〉 that are computed using a larger basis set. More details about the IAO construction are described in SI Section S2.

The IBOs {|i〉} are then constructed by localizing the n_occ occupied HF MOs using a variation of Pipek–Mezey orbital localization,⁸⁵ which minimizes the number of atoms on which an orbital is centered using IAO-based atomic charges. That is, IBOs are a rotation of the occupied MOs,

10

The unitary matrix T is chosen to maximize the electron occupancies n_A(i) of IBO |i〉 at atom A, i.e., . Due to eqn (10), the IBOs span exactly the same space as the occupied MOs and generate the same single-determinantal wavefunction. Fig. 1 displays examples of IBOs.

Fig. 1. Comparison of global and local symmetries. (a and b) Venn-diagram illustrations of the relationship between the global and local symmetries for the IBOs localized around the C O bond in furfural. The a′(2) IBOs are oriented parallel to the molecular plane. (c) The two mirror planes that define the global (M₁) and local (M₂) symmetries illustrated in the Venn diagrams. The local symmetries are labeled as a′(x) and a″(x), respectively, where x arbitrarily enumerates them.

For excited states and non-equilibrium dynamics such as those occurring in charge migration, excited configurations with occupied antibonding orbitals play a crucial role. However, by construction, IBOs are all bonding. To generate antibonding orbitals, we here extend the IBO procedure in a straightforward way using split localization. As we have n_occ IBOs, we generate n_min − n_occ localized, antibonding orbitals that we dub complementary IBOs (cIBOs). These cIBOs are related to, e.g., the valence-virtual orbitals from ref. 91 and 132 and have previously been generated by similar procedures.^98,133,134

We generate cIBOs by first projecting the n_min IAOs {|ν〉} into the space orthogonal to the IBOs {|i〉}, yielding . We then remove linearly dependent vectors from the set {|c̃〉} using singular value decomposition, C = U diag(σ) V^†, with C_κc = 〈κ|c̃〉, where {|κ〉} are orthogonalized atomic orbitals. Since |c̃〉 are constructed by projecting the n_min IAOs to the space orthogonal to the n_occ IBOs, there are n_min − n_occ nonvanishing singular values σ. Thus, we use the first n_min − n_occ columns of U to construct the orthogonal orbitals |c〉 = ∑_κU_κc|κ〉. Finally, we obtain the cIBOs by localizing {|c〉} in the same way as the IBOs, as described around eqn (10). For the cIBO localization, we use the same IAO charges as for the IBOs, for consistency. Different charges can produce nearly identical results.⁹⁸ By construction, the cIBOs form a set of localized orbitals that are orthogonal to the IBOs, and the union of IBOs and cIBOs exactly spans the same space as that of the IAOs. Hence, in the following, when no context is needed, we will collectively refer to IBOs or cIBOs as just IBOs.

IBOs and cIBOs together form a valence-internal space⁹¹ and in most cases will suffice for a comprehensive qualitative analysis of our electron dynamics. The remaining orbitals correspond to what is called hard virtual space⁸⁸ or valence-external space.⁹¹ Here, we dub these remaining orbitals oIBOs, where “o” refers to the fact that these orbitals are orthogonal to the IBO and cIBO space. We construct the oIBOs in the same way we construct the cIBOs.

2.5. Global and local symmetries

To characterize hole motion in our electron-dynamics analysis, we distinguish global and local symmetries. Global symmetry refers to the irrep of the molecular point group. Local symmetry is an additional classification within a spatially restricted region spanning only a subset of atoms. The role of global symmetry in charge-migration dynamics has been explored previously,^36–39 and mainly leads to the appearance of orbitals whose symmetry differs from that of the total wavefunction, which is often due to the 2h1p configurations defined in eqn (9).

As shown below, restricting symmetry to local molecular regions—rather than the molecule as a whole—provides a useful framework for interpreting key features of charge migration. Such local symmetry is meaningful only for localized orbitals. This is not a restriction, as the wavefunction is invariant under a unitary transformation that localizes the orbitals. Unlike other uses of local symmetry in different contexts,^135,136 we apply local symmetry here as an approximate classification and neglect small IBO distortions that violate the chosen local symmetry.

Fig. 1 highlights the interplay of global and local symmetries in selected IBOs of the C_s-symmetric furfural. The orbitals in panel (a) transform as a′ because they are even under reflection in the molecular plane M₁ in panel (c), whereas the orbital in panel (b) transforms as a″ due to its odd M₁ parity. Focusing on the formyl group, the a′ orbitals can be further classified by their parity with respect to a second reflection plane M₂ defined in panel (c), which contains the C O bond and is perpendicular to M₁. This leads to a local symmetry with irreps a′(1) and a′(2) that denote even and odd M₂ parity, respectively. Orbitals from different global irreps necessarily fall into different local symmetries, but some orbitals lack a particular local symmetry. These can be decomposed into orbitals with different symmetries, as indicated by the overlap of the Venn diagrams for a′(1) and a′(2). Next to the local symmetries shown in Fig. 1, other local symmetries are possible for furfural.

3. Results and discussion

In the following, we present our investigation of charge migration dynamics using IBO hole occupancies as the main tool of analysis. We start by validating and introducing the IBO-based analysis in the simple scenario of chloroacetylene in Section 3.1. We then apply it to study changes in hole symmetries in the charge migration of phenylacetaldehyde and furfural in Section 3.2, followed by revealing different charge migration efficiencies in conformers in Section 3.3. Details on the computational setup are given in the Appendix and in the SI.

3.1. Introduction of the IBO analysis in chloroacetylene

One of the first experimental observations of charge migration was made in iodoacetylene.²⁷ Since iodoacetylene is small and linear, its dynamics provide a simple setting, which we use to introduce the IBO analysis workflow before turning to more complex scenarios. In this molecule, a p hole initially localized on iodine migrates to the opposite end and returns to iodine in about 1.9 fs. The initial state is approximately a 50–50 superposition of the ground state |u₀〉 and first excited state |u₁〉 of the cation. In general, for such superpositions of the cationic state, |Ψ(t)〉 = c₀|u₀〉 + c₁|u₁〉, the hole density is given by

11

where ρ₀ and ρ₁ are the electron densities of |u₀〉 and |u₁〉, respectively, ρ₀₁ is the transition density between these two eigenstates, and ΔE is their energy difference. Such a situation has been dubbed hole mixing.⁴⁸

Here, we model the charge migration in iodoacetylene using chloroacetylene, which has fewer electrons and weaker relativistic effects and hence is easier to simulate. Our TDDMRG simulation uses a bond dimension of D = 1000 and includes all 41 non-core orbitals of the def2-SV(P) basis; see the Appendix for more details. We construct the initial p hole state by removing an electron from the neutral in an orbital that is a superposition of the HF HOMO and HOMO−1.

The snapshots of the evolving hole density in chloroacetylene are shown in Fig. 2a. Given that only the cationic ground state and the first excited state should dominate the dynamics, we expect that also only two IBOs will be dominant.‡ Indeed, only two IBOs contribute to the hole density. To gauge how well the two dominant IBOs reproduce the dynamics, we recompute the hole density using just these two dominant IBOs, ignoring all other orbitals, and show it in Fig. 2b. The main hole lobes (positive, green surfaces) in Fig. 2a are faithfully reproduced by the two IBOs. The missing features in the IBO-projected hole density are mostly electron lobes (negative, purple surfaces), which are mainly located near the molecular axis. These differences are due to additional correlation effects not accounted for when using the two dominant IBOs only. Importantly, by using the full IBO, cIBO, and oIBO spaces, the dynamics can be exactly reproduced.

Fig. 2. Charge migration analysis of chloroacetylene. (a) Hole density snapshots. Regions with positive (negative) density are shown as green (purple) lobes. (b) Hole density snapshots obtained by projecting the density from (a) to the space of the two dominant IBOs. (c) Hole occupancies of the two IBOs that dominate the hole dynamics. The IBOs are shown on the graph. The times at which the snapshots in (a) are taken are indicated with dashed vertical lines. (d) Absolute value of the autocorrelation function (dark green curve) in comparison to a two-state model (cyan curve). (e)–(i) Orbital densities of the three most dominant (largest absolute eigenvalues) natural charge orbitals at the times of the snapshots in (a). (j) Mapping of IBOs to cationic Lewis structures. (k) Curly-arrow representation of the periodic hole oscillations.

The shapes and the time-dependent hole occupancies of the two dominant IBOs are shown in Fig. 2c. They are close to the initial p hole state at the Cl atom (IBO-1) and a π hole state at the C C bond (IBO-2). In agreement with eqn (11), the hole occupancies exhibit a clear periodic oscillation with a period of ∼1.29 fs. To further confirm this, Fig. 2d compares the absolute value of the autocorrelation function, |A(t)|, with a reproduction from the 2-state model in eqn (11) (see SI Section S3 for more details). As the hole density reproduced with two IBOs only, the two-state autocorrelation function closely resembles the TDDMRG reference.

As discussed in Section 2.3, a common way to analyze charge migration is to use time-dependent natural charge orbitals, obtained by diagonalizing the hole density at different time steps. The densities of the three most dominant natural charge orbitals at different times are shown in Fig. 2e–i. For all times shown, the most dominant natural charge orbital reproduces the hole density almost as good as the two IBOs. Note that some small details are better captured by the two IBOs than by the dominant natural charge orbital, e.g., a small contribution to the hole density at the Cl atom at 0.648 fs. Like the excluded IBOs, the additional natural charge orbitals lead to a full reproduction of the hole density. However, unlike IBOs, the natural charge orbitals do not directly reveal the two-state characteristic of the dynamics. The time dependence of the natural charge orbitals and, in most scenarios, their complex-valued nature further complicate an analysis based on them.

Previously, IBOs have been used to draw curly-arrow mechanisms of chemical reactions by analyzing IBOs along reaction paths.^92,94–96 For the simple charge migration scenario of chloroacetylene, we follow a similar procedure but analyze the IBOs not along a reaction path but along the real simulation time for a fixed nuclear structure. We first map each of the two IBOs to Lewis structures of the cation; see Fig. 2j. Following the hole occupancy analysis in Fig. 2c, we then identify the main Lewis structures at times when only one of them dominates and connect them through curly arrows as shown in Fig. 2k. As the single-headed curly arrows (harpoons) correspond to the “motion” of a single electron, their direction is opposite to the motion of the electron hole.§ For this simple case, the Lewis-structure mechanism fully agrees with eqn (11).

The presented concepts used for the charge migration in chloroacetylene can be straightforwardly extended to larger linear molecules. We demonstrate this in SI Section S4 for chlorobutadiyne, which contains one additional conjugated C C pair and hence requires one more π IBO and hole state for the analysis. Therein, we also discuss additional convergence criteria for the IBO analysis.

3.2. Effects of the initial hole symmetry

The dynamics of chloroacetylene in the preceding section are relatively simple; for instance, the dominant IBOs all share the same global symmetry as the initial hole. In the remainder, we will analyze charge migration in more complicated scenarios—phenylacetaldehyde in Section 3.2.1 and furfural in Section 3.2.2—which reveal both local and global symmetry effects.

3.2.1. Phenylacetaldehyde

Our first more complex charge migration scenario is that of a C_s-symmetric conformer of phenylacetaldehyde that is eclipsed between the carbonyl group and the phenyl group; see Fig. 3a. This raises the question of how an initial hole at the carbonyl group migrates into the phenyl ring and how this depends on the hole's symmetry with respect to the phenyl-ring plane. To answer this, we perform three TDDMRG simulations with different initial holes, each localized at the carbonyl group: (1) a σ hole, (2) a π hole, and (3) a hole in the O atom's p lone-pair orbital. We refer to the resulting dynamics as σ, π, and p dynamics, respectively. See the leftmost panels in Fig. 3a and c for the shapes of the initial holes.

Fig. 3. Charge migration analysis of phenylacetaldehyde. (a–c) Hole density snapshots for the σ, π, and p dynamics, respectively. The color code is the same as that in Fig. 2a. (d) Hole occupancies of IBOs localized in the 2-oxoethyl group during the three dynamics. The IBO associated with each hole occupancy and its irrep label are shown in each panel. (e) Hole occupancies of selected σ and π IBOs localized at the phenyl group. Note that the ordinate range differs from that in (d). (f and g) Summed hole occupancies of IBOs within the phenyl ring having σ and π local symmetries, respectively. This includes IBOs localized at the H atoms and those shown in (e).

Snapshots of the hole densities during these three dynamics are shown in Fig. 3a and c. The hole occupancies of dominant IBOs localized at the 2-oxoethyl group are shown in Fig. 3d, while the occupancies of selected IBOs localized at the phenyl ring are shown in Fig. 3e (the other phenyl IBO occupancies are shown in SI Section S5.1). Furthermore, we display summed hole occupancies of all phenyl IBOs in Fig. 3e and f for σ and π IBOs, respectively. Note that, here, σ and π refer to the local symmetry with respect to the phenyl plane.

The three dynamics have various features that can be explained by different physical mechanisms. In the following, we analyze and discuss three particular mechanisms and the resulting dynamics: (a) hyperconjugation-based through-space orbital interactions that are responsible for different charge migration efficiencies and that lead to a change of hole shape when migrating from the carbonyl group into the phenyl ring; (b) excited 2h1p configurations that lead to the appearance of orbitals with different global symmetries; and (c) the “breakdown of the molecular orbital picture” that leads to quasiexponential decays of IBO hole occupancies for the σ and p dynamics. We summarize these effects in Table 1.

Changes of the hole symmetry in phenylacetaldehyde. For each dynamics, the hole conversion is shown next to the relevant IBOs at the 2-oxoethyl group and at the entry of the phenyl group (c.f.Fig. 3), and a qualitative explanation. For the change of local symmetry, the initial hole symmetry corresponds to that of the orbital at the CO group.

Local symmetry
Dynamics/initial hole symmetry	Hole symmetry in the Ph ring	Relevant IBOs	Hyperconjugations around Ph entry
σ	σ + π	4–8, 12	σ − σ and σ − π
π	σ	3, 8, 9	σ − σ
p	π	2, 5–9, 12	σ − π

Global symmetry
Dynamics/initial hole irrep	Final hole irreps	Most relevant IBOs	Explanation
σ (a′)	σ (a′) + π(a″)	3, 4, cIBO-1	Coupling through 2h1p configurations; “orbital breakdown”

3.2.1.1. Hyperconjugation modulates charge migration efficiency and converts the local hole symmetry

A striking difference between the three dynamics is the amount of charge migration into the phenyl ring. For the time range studied, the p (π) dynamics exhibits the smallest (largest) migration, and the maximum total hole occupancy in the ring is 0.34 (0.53). Compared to the π dynamics, the σ dynamics features a similar amount of migration into the ring, whose origin we will discuss later in the context of 2h1p configurations and global symmetries.

In conjunction with different charge migration efficiencies observed above, for all three dynamics, the initial hole changes its local symmetry when migrating from the carbonyl group into the phenyl ring as follows: (1) σ → σ + π, (2) π → σ, and (3) p → π. This is most noticeable from the phenyl IBO hole occupancies in Fig. 3e and from the summed phenyl IBO occupancies in panels (f) and (g). Changes in local hole symmetry are also evident in the time-dependent natural charge orbitals (SI Section S5.2), but the time-independent, localized IBOs dramatically speed up the analysis and thus are shown here.

This change of local symmetry is most obvious for the π dynamics, as shown by the hole density at t = 0.968 fs in Fig. 3b and by the selected phenyl σ IBO occupancies in panel (e). What leads to these phenyl ring σ holes? Initially, the C O π hole (IBO-3) migrates into the two mirror-symmetric C–H σ bonds of 2-oxoethyl; compare the initial hole density with that at t = 0.242 fs in Fig. 3b and the occupancies of IBO-8 in panel (d). Note that, aside from IBOs 3 and 8, no other 2-oxoethyl IBOs have large occupancies. After arriving at the C–H IBO-8, the hole migrates through the nearby phenyl C–C σ IBO-9 in the ring, compare the snapshots at t = 0.484 fs and t = 0.726 fs in panel (b), and the σ-ring IBO occupancies in panel (e).¶ Comparing the occupancies of IBOs 9 and 10 shows that migration from one end of the ring to the other requires only 0.5 fs.

Qualitatively, we explain both the different charge migration efficiencies and the change of local symmetries in terms of hyperconjugations/through-space orbital interactions¹⁰⁰ that occur as the hole migrates down the 2-oxoethyl group and arrives at the entry point into the ring.||Fig. 4a illustrates this mechanism. These interactions first occur between the initial C O π IBO-3 and the C–H σ IBO-8. Once the hole reaches the latter IBO, additional interactions occur with the nearby phenyl C–C σ IBO-9, leading to the migration into the ring.

Fig. 4. Main IBOs taking part in the hyperconjugation interactions in phenylacetaldehyde. (a) π, (b) p, and (c) σ dynamics. The interactions lead to ring σ holes in the first and ring π holes in the last two dynamics, respectively. The arrows indicate the interactions.

In contrast to the π dynamics, the p dynamics leads to π character in the ring. Note that there is no π-type conjugation visible in phenyl in Fig. 3c; instead, we see p-orbital-shaped lobes on the C atoms. However, plots at a different isosurface value reveal conjugation; see SI Section S5.4. We confirm this from the π IBO occupancies in panel (e) and from the summed IBO hole occupancies in panels (f) and (g), where, for the shown time range, the summed π ring IBO occupancy of the p-dynamics reaches a maximum value of 0.22, whereas that of the σ ring IBOs is only 0.14.

The change of local symmetry in the p dynamics is due to the 2-oxoethyl C–C σ IBO 6, which interacts through hyperconjugation with the phenyl π IBO-12, as shown in Fig. 4b. IBO 6 is not active in the π dynamics, as its symmetry is different to that of the initial hole. A similar analysis holds for IBO 5. The limited interactions between the initial p hole (IBO-2) and the C–H σ-bonding IBOs 7 and 8 lead to a reduced migration into the ring through these IBOs and consequently almost no σ holes in the ring for the p dynamics.

The σ dynamics lead to hole densities in the ring of both σ and π character. Because the initial IBO-4 is mirror-symmetric, like the p-type IBO-2, the same orbitals active in the p dynamics give rise to the π-ring hole occupancies; see Fig. 3e for the occupancies and Fig. 4c for the mechanism. Through another mechanism that we will explain in the next paragraph, in the σ dynamics, the π IBO-3 becomes active and hence also leads to active σ ring IBOs.

3.2.1.2. 2h1p interactions lead to a change of global hole symmetry

In the preceding paragraph, we have analyzed how the hole shape and thereby its local symmetry changes when migrating from the carbonyl group into the phenyl ring. Now, we shift our analysis to a change of the global symmetry, i.e., the occurrence of dominant hole orbitals with irreps different from that of the initial hole. This can occur through an excitation mechanism that contains orbitals with different irreps.⁴⁸ Note that this does not affect the symmetry of the total wavefunction, which is conserved.

This global symmetry effect is very pronounced in the σ dynamics, where within the first 0.8 fs the occupancy of the initial hole in the σ-type C O IBO-4 (a′) rapidly decreases while other IBOs become dominant, particularly the positive occupancy of the π-type C O IBO-3 (a″) and the negative occupancy of the π* cIBO-1 (a″); c.f.Fig. 3d. These orbitals of different global symmetries are coupled through excited configurations as shown in eqn (9): the initial state corresponds to a one-hole (1h) configuration, â_IBO-4|Ψ_GS〉, and it couples with 2h1p configurations of type â^†_cIBO-1â_IBO-3â_IBO−X|Ψ_GS〉, where X is one of the orbitals of a′ symmetry whose hole occupancy is increasing, e.g., X = 5, while that of cIBO-1 is decreasing. The 2h1p configurations with electrons in the π* C O bond weaken this bond and hence facilitate the charge migration into other regions of the molecule. Through this mechanism, the π-bonding C O IBO-3 becomes active during the σ dynamics and enables a similar mechanism and charge migration efficiency as that of the π dynamics.

When starting from a 1h state, electron-correlation-induced couplings to 2h1p states and higher-order states (3h2p etc.) are easily recognizable by the appearance of negative occupancies in the antibonding cIBOs (see Section 2.3). For example, common for all three dynamics is a negative p-type lobe in the hole density of the phenyl ring at 0.242 fs in Fig. 3a. This leads to very similar π and π* ring IBO occupancies at the onset of the dynamics; c.f., Fig. 3e and SI Fig. S2, where negative occupancies of π* IBOs are shown. As another, related example, at later times, once the ring becomes populated, the σ dynamics exhibit large negative occupancies in the antibonding π orbitals of the phenyl ring; see Fig. 3g. However, the occupancies of the π and π* IBOs almost cancel each other and the maximum value of the combined π and π* occupancy is 0.11, which explains why only a few π-like features are visible in the hole density in Fig. 3a. The large negative occupancies in the σ dynamics that are due to many 2h1p configurations indicates that there are many more states in the σ dynamics that are excited than in the other two dynamics. We will analyze this in more detail in the next paragraph.

3.2.1.3. The “breakdown of the molecular orbital picture” leads to damped oscillations

Both the σ and p dynamics lead to quasiexponentially decaying oscillations of IBO occupancies; compare the hole occupancies in Fig. 3d of IBOs 1, 4, 5, and 6 for the σ dynamics and those of IBOs 2, 5, and 6 for the p dynamics, respectively. This quasiexponential decay has previously been analyzed by Breidbach and Cederbaum,^48,49 and was identified as a signature of a highly correlated initial state produced by ionization of a core orbital or, as in our case, an inner valence orbital. A signature of such a scenario is contributions to the initial state from a quasicontinuum of states in the ionization spectrum. Consequently, this scenario is dubbed “breakdown of the molecular orbital (MO) picture.”¹³¹

In the following, we will analyze this behavior for the σ dynamics, where it is most dominant. Indeed, the initial state in the σ dynamics corresponds to inner valence ionization with an ionization potential of 34 eV, which is a typical energy for an orbital breakdown. The exponentially damped oscillation is not only visible in the hole occupancies, but also in the autocorrelation function in Fig. 5a, dark green curve. Remarkably, the autocorrelation closely corresponds to the hole occupancy of IBO-4 (red) for all shown times. We used this IBO to create the initial state.

Fig. 5. Autocorrelation function and ionization spectrum of the σ dynamics of phenylacetaldehyde. (a) TDDMRG autocorrelation function (dark green) and IBO-4 hole occupancy from Fig. 3d (red). (b) DMRG ionization spectrum of the σ dynamics (light purple).

The ionization spectrum provides further insights (see Section 2.2), and it is shown in Fig. 5b. While these spectra are difficult to converge, with the used spectral broadening of η = 0.08, we clearly see two peaks at 14.5 eV and 35.5 eV, and a shoulder near 27.4 eV. The shoulder is due to another peak, which we confirmed through unconverged computations with smaller broadening (not shown). Since the broadening used does not allow for resolution at the eigenstate level, we assume that multiple eigenstates contribute to the peaks. The spectrum in Fig. 5b can be fully described by three Lorentzians, and the Fourier transform qualitatively matches the TDDMRG autocorrelation function; see SI Section S5.3. The energy difference of the two main peaks leads to the oscillations, whereas the Lorentzian shape leads to the exponential decay. These observations are in full agreement with the breakdown of the MO picture.⁴⁹

We gain additional insights by analyzing the response wavefunctions that contribute to the peaks (see eqn (6) and SI Section S5.3). The hole densities for these wavefunctions are all localized at the 2-oxoethyl group (shown in SI Fig. S6). The wavefunction at 14.5 eV has large contributions from IBOs 1, 3, 4, and cIBO-1, indicating that this peak is due to both 1h and the 2h1p states discussed in the previous paragraph. In contrast, the wavefunction at 35.5 eV has large contributions from IBOs 1 and 4 only; hence no 2h1p configurations contribute to this peak. This discussion not only shows that IBOs can be used to analyze ionization spectra, but also that, remarkably, only the few IBOs shown in Fig. 3 are necessary to analyze a breakdown of the molecular orbital picture.

3.2.2. Furfural

Following a setting similar to that in phenylacetaldehyde, here we compare charge migration dynamics in furfural, where the initial hole is either a π C O orbital (dubbed π dynamics) or a σ C O orbital (dubbed σ dynamics). Unlike phenylacetaldehyde, furfural is planar, and the carbonyl group is π-conjugated with the furyl group. We have already studied furfural in ref. 56 in terms of natural orbitals and dipole moments. Here, we use IBOs, which provide a much more straightforward analysis and more insights.

Snapshots of the hole density for the π dynamics are shown in Fig. 6a. As the molecule is fully π conjugated, the initial π hole rapidly migrates into the ring, which starts within the first 0.2 fs. This is more clearly visible through the IBO occupancies shown in Fig. 6c. The π-conjugated IBOs 2–4 in the furyl ring quickly become populated, which is due to conjugation of these orbitals with the initial hole. Note that these orbitals have the same a″ symmetry and no σ-type orbitals with a′ symmetry become populated during the π dynamics. A closer inspection of Fig. 6c shows that the hole occupancy first increases rapidly for the furyl C C π IBO-2, followed by a similar but slightly delayed increase for IBO-3, which is the farthest away from the initial hole. At later times, the furyl oxygen p orbital also gains hole occupancy. To complete this analysis, next to the IBO occupancies, we need to include the negative occupancies of the antibonding cIBOs localized at the same bond as the bonding IBOs, which correspond to active 2h1p configurations and weaken a chemical bond. Adding the IBO and cIBO occupancies leads to an effective bond occupancy. These occupancies are shown in Fig. 6d and reveal a sequential migration to the different orbitals. We have seen a similar type of migration in the π dynamics of phenylacetaldehyde.

Fig. 6. Charge migration analysis of furfural. (a and b) Hole density snapshots for the π and σ dynamics, respectively. The color code is the same as that in Fig. 2a. (c) Dominant IBO hole occupancies. The IBO associated with each hole occupancy and its irrep label are shown in each panel. (d) Summed hole occupancies of the dominant IBOs and the corresponding cIBOs within the first 1 fs during the π dynamics. The red arrow suggests the movement of the hole. (e) Curly-arrow mechanism demonstrating the initial hole migration; compare with the red arrow in panel (d).

Like our analysis of chloroacetylene in Section 3.1, the π dynamics can be understood in terms of Lewis structures and curly arrows; see Fig. 6e. While helpful, such a curly-arrow mechanism of course only provides a rough qualitative understanding of the dynamics. It is difficult to extend curly-arrow mechanisms to more complex migrations that display symmetry effects, strong entanglement and/or wavepacket bifurcations. For such situations, methods that explicitly analyze the electron flux may be useful in determining the curly arrows.^43,137

Snapshots of the hole density for the σ dynamics are shown in Fig. 6b. Surprisingly, despite the vastly different initial states of the σ and π dynamics, within 0.484 fs, parts of the σ dynamics resemble the π dynamics. This is more clearly seen in the IBO analysis in panel (c); compare the occupancies of the π IBOs 2–4, and that of the cIBOs 1–3. All of these have a″ symmetry whereas the initial σ hole has a′ symmetry. Hence, while the hole shape in the π dynamics remains unchanged, the hole in the σ dynamics changes to that of a π type as the hole enters the furyl group. Note the absence of active σ orbitals within the ring in Fig. 6c. As the migration into the π furyl orbitals is the main mechanism, the σ dynamics is less efficient than the π dynamics and the total sum of the furyl IBO occupancies reaches a maximum value of only 0.60 for the σ dynamics, compared to 0.89 for the π dynamics (c.f. SI Section S6).

As in phenylacetaldehyde in Section 3.2.1, the furyl π holes that appear in the σ dynamics are accompanied by strongly negative occupancies of antibonding cIBOs and a quasiexponential decay of key IBOs, e.g. IBO-7 in Fig. 6c. Thus, also the furfural σ dynamics exhibits a breakdown of the MO picture and 2h1p configurations transfer the dominant hole character to orbitals of different symmetry. However, in contrast to phenylacetaldehyde, the migration into the ring is mediated through π conjugation rather than hyperconjugation.

3.3. Conformer effects

In Section 3.2, we have shown that both global and local symmetries can play a pivotal role in determining how the hole shape changes as it migrates through the molecule. So far, all molecules studied have global symmetry. What if the molecule has local symmetries but no global symmetry? To address this for a particular scenario, we study two C₁-symmetric conformers of 3-fluoro-2-methylpropanal, namely the synclinal one (sc-3-fluoro-2-methylpropanal, dubbed SC), and the synperiplanar one (sp-3-fluoro-2-methylpropanal, dubbed SP); see Fig. 7a and b for their geometries. SP is the global minimum, and SC is about 3.1 kJ mol⁻¹ higher in energy. The conformers primarily differ in the placement of fluorine, whose lone pairs and high electronegativity can significantly affect charge migration. We will show below that the placement of fluorine can be crucial for efficient charge migration and thus suggest design principles for optimal charge migration.

Fig. 7. Charge migration analysis of sc-3-fluoro-2-methylpropanal. (a and b) Hole density snapshots of sc-3-fluoro-2-methylpropanal (SC) and sp-3-fluoro-2-methylpropanal (SP), respectively. Note that only the densities projected into the dominant IBO space are shown. (c) Dominant IBO hole occupancies. The IBOs of the two conformers associated with each hole occupancy are shown in each panel. (d) Structure of SP highlighting the six atoms whose IBOs are in a nearly planar arrangement and therefore facilitate charge migration. (e) Same as (d) but viewed at an angle that reveals the relative alignment of the initial hole with IBOs 2 and 8.

For both conformers, our simulations start with a hole in the π bond of C O. In Fig. 7a and b, we show snapshots of the hole density for SC and SP, respectively. Unlike in our previous studies, we here only show the hole densities projected into the dominant IBO space, which facilitate the analysis.* As both conformers have the same initial hole, the conformer effect is not large during the initial dynamics but becomes larger at later times. In particular, the hole density snapshots indicate that dominant changes in the charge migration start to appear at ∼0.7 fs. After ∼1 fs, the initial hole in SP becomes almost completely drained out. Hence, the placement of fluorine in SP leads to more efficient charge migration than in SC. The IBO hole occupancies provide more details and are shown in Fig. 7c. The charge migration efficiency is clearly seen from IBO-1, whose occupancy starts close to 1 and for SP reaches a minimum of 0.090, whereas for SC it only reaches 0.26 for the time range studied.

The enhanced charge migration in SP arises from the molecular geometry and orbital alignment highlighted in Fig. 7d and e, in particular a nearly planar arrangement of the shown atoms. This leads to a favorable mutual orientation of IBOs 1, 2, and 8, which then enables sequential hyperconjugative coupling, first from IBO-1 to IBO-2 and then from IBO-2 to IBO-8. As shown in panels (b) and (c), the hole therefore migrates from the C O π IBO-1 to the C₃–C₆ σ IBO-2 and finally reaches the F atom's p lone-pair IBO-8.

The local symmetry of the initial hole further determines which orbitals participate in the dynamics. For example, the C₂–C₃ σ IBO (not shown) remains inactive because its local symmetry is incompatible with that of IBO-1. The same restriction applies to IBOs 3 and 7 (see their SP occupancies in Fig. 7c).

This mechanism is less effective in SC because its geometry does not yield a similarly favorable alignment involving the occupied fluorine orbitals. IBO-1 nevertheless couples weakly to SC IBO-7, which explains the IBO-7 occupancy in Fig. 7c; however, their incomplete alignment limits the strength of this coupling.

While the migration toward the F atom differs markedly between the two conformers, we observe some similarities in other migration paths: the occupancy curves of cIBO-1, IBOs 4–6, and that of SC/SP IBOs 9/10 are very similar for both conformers. These IBOs are associated to migrations to other parts of the molecule, e.g. the hydrogen atoms. The similarity between the two conformers is due to the same type of initial dynamics, which is not affected by the placement of the F atom at the opposite end of the molecule.

Our results on SC and SP have demonstrated an interplay between initial hole symmetry, the location of a functional group with lone pairs, and geometry in enhancing charge migration, and thus will be useful in the search for molecules that exhibit a clear charge migration signal. As the dynamics involve both σ, π, and p orbitals, they also present an example of the underlying similarity between conjugation and hyperconjugation¹⁰⁰ in the context of electron dynamics.

4. Conclusions

In this work, we introduced a new way of analyzing charge migration dynamics that is based on mapping the hole density and related quantities to an extension of intrinsic bond orbitals (IBOs). This extension includes antibonding orbitals and leads to a very compact representation of the essentials of the real-time many-body quantum dynamics, even in challenging cases such as correlated inner-valence ionized wavefunctions. The real-valued and time-independent nature of IBOs makes them particularly convenient to use for time-dependent simulations. Our simulations were based on an efficient setup of the time-dependent density matrix renormalization group (TDDMRG), which allowed us to simulate charge migration in molecules as large as phenylacetaldehyde with 45 fully correlated electrons in 50 orbitals.

IBOs bridge quantum mechanics with chemical concepts, and, in our case, enabled us to not only connect IBOs to established quantities in charge migration, such as hole configurations, but also to understand complex charge migration in terms of through-bond/-space interactions such as hyperconjugation and Lewis structures. We demonstrated this capability for various molecules, with a focus on how ionization from orbitals of various shapes affects the charge migration. For example, we showed that in phenylacetaldehyde a sudden ionization at the carbonyl group leads to a migration into the σ bonds of the phenyl ring regardless of whether the initial hole has a π or a σ shape. We found that through-bond and through-space hyperconjugation interactions between different IBOs of similar local symmetry are responsible for this surprising effect. We also observed a rapid change of character from a σ hole to a π hole at the carbonyl bond, which we explained by connecting antibonding extensions of IBOs to excited two-hole one-particle (2h1p) configurations that couple orbitals with π and σ symmetries. Surprisingly, a small set of IBOs can provide a semiquantitative description of the correlated dynamics even in the so-called “breakdown of the molecular orbital picture” regime.

In contrast to phenylacetaldehyde, in furfural, we found that charge at the carbonyl group migrates into the π bonds of the formyl group, regardless of whether the ionization created a π or σ hole at the carbonyl group. The π-hole-initiated migration is driven by a full conjugation of the π orbitals throughout the molecule, while the σ-hole-initiated migration is mediated by excited 2h1p configurations. Taking advantage of the straightforward mapping of IBOs to Lewis structures, we also demonstrated how an IBO analysis leads to a qualitative mechanism in terms of curly arrows for charge migrations without strong wavefunction bifurcations and interference effects.

Lastly, we used our extension of IBOs to explain different charge migration efficiencies of two conformers of 3-fluoro-2-methylpropanal. The localized character of the IBOs dramatically sped up this analysis. We found that, for one conformer, the charge migration is facilitated by through-space interactions of three key IBOs that are favorably aligned. This leads to an almost complete drain of the initial hole, in contrast to the other conformer, where the IBOs are aligned less favorably. This illustrates a potential way to find molecules and functional groups with efficient, long-lasting, coherent charge migration, which is an ongoing quest and crucial for clear experiments.^35,53,138 As such, we believe that the present work might help in realizing future experiments, and by extension, ultrafast control of chemical reactions.

While we focused on charge migration, the proposed IBO analysis can be extended to understand not only other types of nonequilibrium real-time molecular dynamics but also time-independent quantities such as response functions in ionization spectra, which we briefly demonstrated. Importantly, IBOs are not restricted to DMRG wavefunctions but can be combined with other quantum chemistry methods, including density functional theory.

Author contributions

Imam S. Wahyutama: conceptualization (equal); data curation (lead); formal analysis (equal); investigation (lead); methodology (equal); software (lead); supervision (supporting); validation (lead); visualization (lead); writing – original draft (equal); writing – review & editing (equal); Madhumita Rano: formal analysis (supporting); visualization (supporting); writing – review & editing (supporting); Henrik R. Larsson: conceptualization (equal); formal analysis (equal); funding acquisition (lead); investigation (supporting); methodology (equal); project administration (lead); resources (lead); software (supporting); supervision (lead); validation (supporting); writing – original draft (equal); writing – review & editing (equal).

Conflicts of interest

There are no conflicts to declare.

5. Appendix computational details

Our TDDMRG implementation is based on the block2 ^110,139 and pyscf^140–143 codes. We follow our recently established guidelines for efficient TDDMRG charge migration simulations.⁵⁶ To enforce the spin SU(2) symmetry of open-shell wavefunctions resulting from ionization, we use a spin-adapted MPS. This leads to a slightly different basis than that given in eqn (1), which only supports particle-number U(1) symmetry. Based on the findings in our previous work,⁵⁶ we use singlet-embedding for enforcing the nonzero total spin within the spin-adapted MPS, that is, the nonzero spin state is embedded in a state that has a total spin of zero.^144,145 To propagate the complex-valued wavefunction, we use a fully complex MPS representation, i.e., all matrices in eqn (1) are complex-valued. The individual MPS matrices are propagated using the short iterative Lanczos algorithm¹⁴⁶ with a relative convergence tolerance of 5 × 10⁻⁶. The main TDDMRG propagation parameters are outlined in Table 2. See SI Section S8 for convergence studies.

Table 2. TDDMRG propagation parameters. D_prelim and D are the bond dimensions for the preliminary and main TDDMRG propagations, respectively. The former is used for selecting the CAS orbitals.

System/dynamics	D prelim	D	Active space	Δt/as
Chloroacetylene	N/A	1000	CAS(15e, 41o)	0.968
Phenylacetaldehyde	100	500	CAS(45e, 50o)	0.484
Furfural (π)	200	500	CAS(35e, 40o)	0.968
Furfural (σ)	200	500	CAS(35e, 40o)	0.484
3-Fluoro-2-methylpropanal	150	500	CAS(35e, 40o)	0.726

We use 6-31G basis set^147,148 for all molecules except for chloroacetylene, for which we use the def2-SV(P) basis set.¹⁴⁹ All simulations use the frozen core approximation. The geometries were optimized using density functional theory (see SI Section S1). For phenylacetaldehyde, the particular conformer is not the global minimum (other conformers that are 2.7 kJ mol⁻¹ lower in energy exist), but it is accessible at room temperature.

To select the active orbitals for the CAS, we use our procedures developed in ref. 56. In short, we select the active orbitals for the dynamics from the dominant orbitals of both the reduced density matrix (RDM) of the neutral molecule's ground state and, in addition, from the time-averaged hole density matrix of a preliminary TDDMRG simulation using all orbitals and a small bond dimension D_prelim shown in Table 2. To improve the DMRG efficiency, the selected orbitals are then split-localized using Pipek–Mezey localization.⁸⁵ We employ these hole-DM-selected orbitals for all molecules except for chloroacetylene, for which we use split-localized Hartree–Fock orbitals and the full orbital space. Note that these orbitals used for the TDDMRG simulations are not IBOs. The IBO analysis is performed through a change of basis.

For all molecules except chloroacetylene, we use the annihilation operator â_k of a particular IBO |k〉 to create a localized initial hole. For chloroacetylene, we create the initial hole from an equal–weight superposition of Hartree–Fock HOMO and HOMO−1, which leads to a p lone-pair orbital on Cl. We compute restricted-Hartree–Fock-based IBOs using the implementation in pyscf, which differs slightly from the original IBO procedure outlined in ref. 90. We compute the IBOs by localizing the MOs without directly enforcing symmetry, except for phenylacetaldehyde, where we localize the MOs within each irrep of the C_s point group. This makes the IBOs less localized but facilitates the charge migration analysis.

For the hole density and natural orbital density plots, we use isosurface values of ±0.006, while for the IBOs, this value is ±0.06 for all molecules except for furfural, for which it is ±0.08.

The ionization potentials of the states in the σ, π, and p dynamics of phenylacetaldehyde in Section 3.2.1 are 34.0 eV, 15.6 eV, and 14.9 eV, respectively. For furfural, the ionization potentials are 15.6 eV and 33.6 eV for the π dynamics and σ dynamics, respectively. We note that the energy of the σ dynamics for both molecules is higher than the second ionization potential, which means that some influence from Auger–Meitner decay to this dynamics should be expected at times longer than our simulation time.¹⁵⁰

The Green's function computations for phenylacetaldehyde in Section 3.2.1 are based on a bond dimension of D = 550, use a final relative convergence tolerance of 10⁻⁸, a maximal sweep number of 8, and a linear solver tolerance of 10⁻⁶. To steer the optimization problem out of local minima, we use noise on the order of 10⁻⁵ for the first 4 sweeps.

Data availability

The data that support the findings of this study are available within the article and the supplementary information (SI). Supplementary information: used geometries; description of the intrinsic atomic orbital procedure; fitting procedure of the n-state model; simulation and analysis of chlorobutadiyne and conformers; natural charge orbital analysis of the σ, π, and p dynamics; additional hole density snapshots of the p dynamics; Green's function analysis of the σ dynamics in phenylacetaldehyde; total hole occupancies in the phenyl and furyl rings; snapshots of the unprojected hole densities of the 3-fluoro-2-methylpropanal conformers; convergence studies for all dynamics. See DOI: https://doi.org/10.1039/d6sc02580c.