^{1,a)}, Motomichi Tashiro

^{2}, Masahiro Ehara

^{2}and Lorenz S. Cederbaum

^{1}

### Abstract

Core vacancies created on opposite sides of a molecule operate against each other in polarizing the environment between them. Consequently, the relaxation energy associated with the simultaneous creation of these two core holes turns out to be smaller than the sum of the relaxation energies associated with each individual single core vacancy created independently. The corresponding residual, termed interatomic relaxation energy, is sensitive to the environment. In the present paper we explore how the interatomic relaxation energy depends on the length and type of carbon chains bridging two core ionized nitrile groups (–C≡N). We have uncovered several trends and discuss them with the help of simple electrostatic and quantum mechanical models. Namely, the absolute value of the interatomic relaxation energy depends strongly on the orbital hybridization in carbons being noticeably larger in conjugated chains (*sp* and *sp* ^{2} hybridizations) possessing highly mobile electrons in delocalized π-type orbitals than in saturated chains (*sp* ^{3} hybridization) where only σ bonds are available. The interatomic relaxation energy decreases monotonically with increasing chain length. The corresponding descent is determined by the energetics of the molecular bridge, in particular, by the HOMO-LUMO gap. The smallest HOMO-LUMO gap is found in molecules with the *sp* ^{2}-hybridized backbone. Here, the interatomic relaxation energy decreases slowest with the chain length.

N.V.K. and L.S.C. gratefully acknowledge financial support by the DFG. M.T. and M.E. were supported by a Grant-in-Aid for Scientific Research from the Japanese Society for the Promotion of Science (JSPS).

I. INTRODUCTION

II. COMPUTATIONAL DETAILS

III. RESULTS AND DISCUSSION

A. Trends in single and double ionization potentials

B. Interatomic relaxation energy

1. Electrostatic model for interatomic relaxation in saturated systems

2. Quantum mechanical model for interatomic relaxation in systems with conjugation

C. Generalized Wagner plots

IV. SUMMARY

### Key Topics

- Polarizability
- 23.0
- Carbon
- 21.0
- Ionization potentials
- 20.0
- Electron mobility
- 12.0
- Ionization
- 12.0

## Figures

Optimized ground-state geometries of three exemplary dinitrile compounds: N_{2}C_{8}H_{12} (*top*), N_{2}C_{8}H_{6} (*middle*), and N_{2}C_{8} (*bottom*).

Optimized ground-state geometries of three exemplary dinitrile compounds: N_{2}C_{8}H_{12} (*top*), N_{2}C_{8}H_{6} (*middle*), and N_{2}C_{8} (*bottom*).

Computed N1*s* single ionization potentials and their constituting parts (see Eq. (1)) as functions of the molecular length and the carbon bridge type in the bridged dinitriles. *n* refers to the number of the chain units. These units are shown at the left lower corner of panel (c). Filled circles correspond to the *ab initio* results. Dashed lines are fitting functions defined by the polynomials .

Computed N1*s* single ionization potentials and their constituting parts (see Eq. (1)) as functions of the molecular length and the carbon bridge type in the bridged dinitriles. *n* refers to the number of the chain units. These units are shown at the left lower corner of panel (c). Filled circles correspond to the *ab initio* results. Dashed lines are fitting functions defined by the polynomials .

Computed tsDIP(N1*s*, N1*s*) as a function of the molecular length and the carbon bridge type in the bridged dinitriles. *n* refers to the number of chain units. Filled circles correspond to the *ab initio* data. Dashed lines are fitting functions defined by the polynomials .

Computed tsDIP(N1*s*, N1*s*) as a function of the molecular length and the carbon bridge type in the bridged dinitriles. *n* refers to the number of chain units. Filled circles correspond to the *ab initio* data. Dashed lines are fitting functions defined by the polynomials .

Calculated interatomic relaxation (IR, color circles) and correlation (IC, color squares) energies as functions of the molecular length and the carbon bridge type in the bridged dinitriles. The two grey symbols refer to the unbridged dinitrile (cyanogen). The values of the IC energies are joined by solid lines. Dashed lines are fitting functions for the IR energies defined by the polynomials .

Calculated interatomic relaxation (IR, color circles) and correlation (IC, color squares) energies as functions of the molecular length and the carbon bridge type in the bridged dinitriles. The two grey symbols refer to the unbridged dinitrile (cyanogen). The values of the IC energies are joined by solid lines. Dashed lines are fitting functions for the IR energies defined by the polynomials .

The IR energy as calculated using Eq. (19) (open circles). The solid line is a fitting function of the type *a*(ln *N*)^{ b }/*N* ^{ c }. The IR energy is given in η = −1/(β*d* ^{2}) units, where β is the resonance integral and *d* is the distance between neighboring atoms.

The IR energy as calculated using Eq. (19) (open circles). The solid line is a fitting function of the type *a*(ln *N*)^{ b }/*N* ^{ c }. The IR energy is given in η = −1/(β*d* ^{2}) units, where β is the resonance integral and *d* is the distance between neighboring atoms.

Generalized Wagner plots for the dinitriles bridged by different carbon chains drawn as described in Sec. III C. The core level IP of a nitrogen atom is plotted on the horizontal axis. The core level IP of the same nitrogen atom in the presence of a core vacancy on the other nitrogen atom is plotted on the vertical axis. The core holes repulsion energies are subtracted from the latter IPs. *Ab initio* values (filled circles) are joined by straight solid lines to guide the eye. The diagonal lines with slope +1 corresponds to the levels of constant IR energies.

Generalized Wagner plots for the dinitriles bridged by different carbon chains drawn as described in Sec. III C. The core level IP of a nitrogen atom is plotted on the horizontal axis. The core level IP of the same nitrogen atom in the presence of a core vacancy on the other nitrogen atom is plotted on the vertical axis. The core holes repulsion energies are subtracted from the latter IPs. *Ab initio* values (filled circles) are joined by straight solid lines to guide the eye. The diagonal lines with slope +1 corresponds to the levels of constant IR energies.

## Tables

Selected bondlengths and total lengths (in Å), the Mulliken charges on the terminal nitrogen atoms (*Q*, in *e*) and the HOMO-LUMO gaps (H-L, in eV) in the bridged dinitrile molecules studied.

Selected bondlengths and total lengths (in Å), the Mulliken charges on the terminal nitrogen atoms (*Q*, in *e*) and the HOMO-LUMO gaps (H-L, in eV) in the bridged dinitrile molecules studied.

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