# Continuum Electronic States: The Tiresia Code

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Tiresia Program

#### 2.1. The Basis Set

#### 2.2. The Galerkin Approach

#### 2.3. The Many Electron Wavefunction

#### 2.3.1. The Static Exchange Approach

#### 2.3.2. The TDDFT Approach

#### 2.3.3. The Correlated Single Channel Approach

## 3. Multiphoton and Strong Field Processes

- 1.
- Calculation of ${\Phi}_{DI}$, which is ${L}^{2}$ and depends on the particular excitation mechanism considered.
- 2.
- Calculation of ${\Psi}_{F\overrightarrow{k}}^{(-)}$ or equivalently ${\Psi}_{FElm}^{(-)}$, which is independent of the former, as eigenvector of the molecular hamiltonian already considered.

- 1.
- In the multiphoton domain, generally the lowest order perturbation theory is employed$${\Phi}_{DI}^{Nph}=D{(H-{E}_{N-1})}^{-1}D{(H-{E}_{N-2})}^{-1}D\cdots D{\Phi}_{I}=D{(H-{E}_{N-1})}^{-1}{\Phi}_{DI}^{Nph-1}$$$${E}_{m}={E}_{i}+m\hslash \omega $$
- 2.
- For nonperturbative fields a standard approach is now the solution of the time dependent Schrödinger equation (TDSE)$${\Phi}_{DI}\left(t\right)=U\left(t\right){\Psi}_{I}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}-i\frac{d}{dt}{\Phi}_{DI}\left(t\right)=H\left(t\right){\Phi}_{DI}\left(t\right)$$

## 4. Ab Initio Developments

## 5. Applications

#### 5.1. Molecular Photoionization

_{3}H

_{5})

_{2}[19], shows the large changes in $\sigma $ and $\beta $ parameters from those afforded by HF or DFT orbitals (Figure 3). Moreover, it shows the value of photoionization observables concerning aiding in the difficult problem of assigning the spectrum in strongly correlated systems, e.g., the appearance of the strong $3p\to 3d$ autoionization resonance in ionizations involving metal 3d participation.

#### 5.2. Strong Field and Ultrafast Processes

## 6. Conclusions

- A dense set within a finite range (a sphere) of arbitrary length. It can approach completeness and therefore converge to the required solutions.
- A complete control of the overlap matrix, hence numerical linear independence and stability.
- Accurate solutions of homogeneous and inhomogeneous equations within the range, with proper boundary conditions, which are easily implemented.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## References

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**Figure 2.**C${}_{60}$ photoionization. Upper left: Total DFT and TDDFT cross section, from Ref. [36] with permission. Upper right: High resolution DFT core cross section [37]. Lower left: HOMO and HOMO-1 cross sections; lower right HOMO/HOMO-1 cross section ratio (circles are experimental data), from Ref. [38] with permission.

**Figure 3.**Nickel bis-allyl cross section and $\beta $ parameter at the DFT, HF and Dyson-DFT level. From Ref. [20] with permission.

**Figure 4.**Dichroic parameter ${\beta}_{1}$ for Camphor HOMO ionization. Results for different potential choices. Left upper panel from ref. [49] with permission.

**Figure 5.**Valence photoelectron spectrum of cobalt tris-acetylacetonate, with LB94 eigenvalues (black) and OVGF IEs (red). Experimental spectrum from [51] with permission.

**Figure 6.**${\beta}_{1}$ parameter computed for the valence levels of Cobalt tris-acetylacetonate (from [51] with permission). Right panel in the region of autoionization resonance.

**Figure 7.**Time resolved photoelectron spectra of pyrazine. Cross section (

**upper**panels), $\beta $ parameter (

**lower**panel). (

**a**,

**c**) experimental; (

**b**,

**d**) calculated. From [52] with permission.

**Figure 8.**Strong field ionization yield in water (from [56] with permission).

**Figure 9.**Photoelectron spectrum (ionization probability) relative to HOMO ionization in water at $I=1.0\times {10}^{14}$, $3.0\times {10}^{14}$, $5.0\times {10}^{14}$ and $7.0\times {10}^{14}$ (blue) W/cm${}^{2}$.

**Figure 10.**HHG spectra of $C{O}_{2}$ at $I=0.6\times {10}^{14}$ (red), $0.85\times {10}^{14}$ (green) and $1.4\times {10}^{14}$ (blue) W/cm${}^{2}$.

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**MDPI and ACS Style**

Decleva, P.; Stener, M.; Toffoli, D.
Continuum Electronic States: The Tiresia Code. *Molecules* **2022**, *27*, 2026.
https://doi.org/10.3390/molecules27062026

**AMA Style**

Decleva P, Stener M, Toffoli D.
Continuum Electronic States: The Tiresia Code. *Molecules*. 2022; 27(6):2026.
https://doi.org/10.3390/molecules27062026

**Chicago/Turabian Style**

Decleva, Piero, Mauro Stener, and Daniele Toffoli.
2022. "Continuum Electronic States: The Tiresia Code" *Molecules* 27, no. 6: 2026.
https://doi.org/10.3390/molecules27062026