夢で見た論文　　　　Wavelet-Resolved Electron Flux Representation of Configuration Interaction States in Conjugated Polymer Systems

Wavelet-Resolved Electron Flux Representation of Configuration Interaction States in Conjugated Polymer Systems

Abstract

We propose a wavelet-based representation of electronic motion in ab initio Configuration Interaction (CI) calculations for conjugated polymer systems. Conventional molecular orbital representations provide accurate electronic energies but often obscure the local pathways through which electrons propagate. By projecting CI wavefunctions onto a multiresolution wavelet basis, electronic transitions may be interpreted as localized hopping events between spatial cells.

This framework enables the construction of an electron flux network derived from the one-particle transition density matrix. The resulting representation yields a vector field describing electron transport across molecular structures. Unlike conventional molecular orbital pictures, the proposed formalism offers a direct visualization of electronic motion and exciton migration.

Polyethylene-like and polyacetylene-like systems are discussed as model examples. We argue that wavelet-localized CI analysis provides a bridge between quantum chemistry and transport theory, allowing electronic structure calculations to be interpreted in terms of physically intuitive flow processes.

1. Introduction

Electronic structure theory has traditionally focused on the determination of stationary states and total energies. Hartree–Fock and post-Hartree–Fock methods describe molecular systems with remarkable accuracy, yet the resulting wavefunctions are often difficult to interpret in terms of local electronic motion.

In conjugated molecular systems, concepts such as charge transport, exciton diffusion, and electron hopping are frequently invoked. However, these phenomena are typically inferred indirectly from delocalized molecular orbitals.

The present work explores a different viewpoint. Rather than describing electrons through global molecular orbitals, we seek a representation in which electronic motion emerges naturally as a localized flow.

Wavelet bases provide an attractive framework due to their simultaneous localization in space and scale. We investigate how CI wavefunctions may be expressed within a wavelet decomposition and how the resulting coefficients may be interpreted as electron fluxes between neighboring regions.

2. Wavelet Representation of Electronic States

Let the electronic wavefunction be expanded as

Ψ(r) = Σ cjk ψjk(r)

where ψjk(r) denotes a wavelet basis function at scale j and position k.

Unlike Gaussian orbitals, wavelets possess compact spatial support and form a multiresolution hierarchy. This allows electronic density to be decomposed into local spatial contributions.

The electronic density becomes

ρ(r) = |Ψ(r)|²

which may be analyzed at multiple spatial scales.

3. CI Wavefunctions and Transition Density

The CI wavefunction is expressed as

|ΨCI⟩ = ΣI CI |I⟩

where |I⟩ represents Slater determinants.

The one-particle density matrix is

γij = ⟨ΨCI| a†i aj |ΨCI⟩

Projecting γij onto a wavelet basis yields a localized transition matrix between wavelet cells.

This matrix may be interpreted as a hopping network connecting neighboring spatial regions.

4. Electron Flux Network

We define an effective electron flux

Fab

between wavelet cells a and b.

The collection of all Fab generates a directed graph describing electronic transport.

This graph may be visualized as a vector field

J(r)

analogous to probability current density.

Unlike conventional current density representations, the wavelet formulation naturally separates transport contributions according to spatial scale.

Large-scale components describe collective polarization, whereas fine-scale components reveal local bond-to-bond electron motion.

5. Application to Polyethylene-like Systems

Consider a linear carbon chain.

Traditional molecular orbital analysis predicts delocalized bonding and antibonding states extending over the entire chain.

In contrast, the proposed representation decomposes electronic motion into localized transitions

C1 → C2 → C3 → C4 → ...

The resulting flow map provides a direct picture of electron migration pathways.

Excitonic motion may likewise be represented as a sequence of localized flux events.

6. Discussion

The proposed framework shares conceptual similarities with Wannier functions and tight-binding models.

However, unlike Wannier approaches, localization is introduced at the basis level rather than through a post-processing transformation.

Consequently, the representation remains compatible with fully ab initio CI calculations while preserving spatial interpretability.

The method may provide new insight into:

- Charge transport

- Exciton diffusion

- Polaron migration

- Molecular electronics

- Organic semiconductors

7. Conclusion

We propose a wavelet-resolved interpretation of CI wavefunctions in which electronic transitions are represented as localized flux vectors between spatial cells.

This formulation transforms the conventional electronic structure problem into a transport-like picture while maintaining an ab initio foundation.

Future work will focus on numerical implementation and comparison with conventional current-density analyses.