Carbon monoxide interacting with free-electron-laser pulses

We use rate equations to model the interaction of Gaussian FEL pulses with CO. Every energetically accessible state of CO is denoted by its electronic configuration, ($1{\sigma }^{a},2{\sigma }^{b},3{\sigma }^{c},4{\sigma }^{d},1{\pi }_{x}^{e},1{\pi }_{y}^{f},5{\sigma }^{g}$), where a, b, c, d, e, f and g are the number of electrons occupying a molecular orbital. This occupancy is 0, 1 or 2. These rate equations were discussed in detail in previous work in the context of the homonuclear molecule N₂ ($1{\sigma }_{g}^{a},1{\sigma }_{u}^{b},2{\sigma }_{g}^{c},2{\sigma }_{u}^{d},1{\pi }_{ux}^{e},1{\pi }_{uy}^{f},3{\sigma }_{g}^{g}$) interacting with FEL pulses [23]. In the rate equations describing the interaction of CO with an FEL pulse we employ the single-photon ionisation cross-section and Auger rates for all energetically allowed transitions in CO as well as its atomic fragments. Atomic units are used in this work, unless otherwise stated. To obtain these cross sections and rates we need to compute the bound and the continuum atomic and molecular orbitals for all energetically accessible states. To simplify the computations involved for the photo-ionisation cross-section and Auger rates, we express these orbitals in terms of a single-centre expansion (SCE) [24].

In the rate equations, the photon flux, J(t), is used to compute the photo-ionisation rates. For a monochromatic pulse of photon energy ω, we consider J(t) to have a Gaussian temporal form that is given by

$$\begin{equation}J\left(t\right)=\frac{{I}_{0}}{\omega }\enspace \mathrm{exp}\left\{-4\enspace \mathrm{ln}\enspace 2{\left(\frac{t}{\tau }\right)}^{2}\right\},\end{equation} \tag{ 1 }$$

where I₀ is the peak intensity of the FEL pulse, and τ is the full-width half-maximum (FWHM) of the pulse. In what follows, intensity refers to the peak intensity in equation (1).

We obtain the molecular and atomic bound wavefunctions ψ_i using the Hartree–Fock technique in Molpro [25], a quantum chemistry package. We use the cc-pVTZ basis set to obtain the bound wavefunctions for CO with the nuclei fixed at an equilibrium distance of 1.128 Å [26]. In the single-centre expansion the bound and continuum wavefunctions, ψ_i and ψ respectively, are expressed as [24]

$$\begin{equation}{\psi }_{\left(i/{\epsilon}\right)}\left(\mathbf{r}\right)=\sum _{lm}\frac{{P}_{lm}^{i/{\epsilon}}\left(r\right){Y}_{lm}\left(\theta ,\phi \right)}{r},\end{equation} \tag{ 2 }$$

with r = (r, θ, ϕ) denoting the position of the electron, with respect to the centre of mass of the molecule. We denote by Y_lm , a spherical harmonic with quantum numbers l and m while ${P}_{lm}^{i/{\epsilon}}\left(r\right)$ denotes the single centre expansion coefficients for the orbital i/. The index i refers to the ith bound orbital, while refers to a continuum orbital with energy . Since CO is a linear molecule, it has rotational symmetry and hence only one m value is involved in the summation in equation (2). For a heteronuclear molecule, like CO, the wavefunctions have no gerade or ungerade symmetry and therefore both even and odd values of l have to be included in equation (2). This is unlike a homonuclear molecule, like N₂, where only odd or even values of l are included in the summation in equation (2) depending on whether the wavefunction has gerade or ungerade symmetry.

The continuum wavefunctions, ψ , are calculated by solving the following Hartree–Fock equations [23, 24, 27]:

$$\begin{align}\hfill & \underset{\text{Kinetic}\;\text{energy}}{\underbrace{-\frac{1}{2}{\nabla }^{2}{\psi }_{{\epsilon}}\left(\mathbf{r}\right)}}+\underset{\text{Electron}-\text{nuclei}}{\underbrace{\sum _{n}^{\text{nuc.}}\frac{-{Z}_{n}}{\vert \mathbf{r}-{\mathbf{R}}_{n}\vert }{\psi }_{{\epsilon}}\left(\mathbf{r}\right)}}\hfill \\ \hfill & +\underset{\text{Direct}\;\text{interaction}}{\underbrace{\sum _{i}^{\text{orb.}}{a}_{i}\int d\mathbf{r}p\frac{{\psi }_{i}^{{\ast}}\left(\mathbf{r}p\right){\psi }_{i}\left(\mathbf{r}p\right)}{\vert \mathbf{r}-\mathbf{r}p\vert }{\psi }_{{\epsilon}}\left(\mathbf{r}\right)}}\hfill \\ \hfill & -\underset{\text{Exchange}\;\text{interaction}}{\underbrace{\sum _{i}^{\text{orb.}}{b}_{i}\int d\mathbf{r}p\frac{{\psi }_{i}^{{\ast}}\left(\mathbf{r}p\right){\psi }_{{\epsilon}}\left(\mathbf{r}p\right)}{\vert \mathbf{r}-\mathbf{r}p\vert }{\psi }_{i}\left(\mathbf{r}\right)}}={\epsilon}{\psi }_{{\epsilon}}\left(\mathbf{r}\right),\hfill \end{align} \tag{ 3 }$$

where ψ_i is the wavefunction of the ith bound molecular orbital, a_i is the occupation of orbital i and b_i is a coefficient associated with the ith orbital, whose values are determined by the symmetry of the state. For more details, see our previous work [23]. R_n and Z_n are the position with respect to the centre of mass, and the charge of nucleus n, respectively. In order to compute the continuum orbitals ψ , we substitute ψ and ψ_i using equation (2) and use the non-iterative method [23, 24] to solve for the ${P}_{lm}^{{\epsilon}}$ coefficients.

The photo-ionisation cross-section for an electron transitioning from an initial molecular orbital ψ_i to a final continuum molecular orbital ψ is given by [28]

$$\begin{equation}{\sigma }_{i\to {\epsilon}}=\frac{4}{3}\alpha {\pi }^{2}\omega {N}_{i}\sum _{M=-1,0,1}{\left\vert {D}_{i{\epsilon}}^{M}\right\vert }^{2}.\end{equation} \tag{ 4 }$$

The fine-structure constant is denoted by α, the photon energy by ω, the occupation number of orbital i by N_i and the magnetic quantum number of the photon by M. In the length gauge, using equation (2), the matrix element ${D}_{i{\epsilon}}^{M}$ is given by

$$\begin{equation}\begin{aligned}\hfill {D}_{i{\epsilon}}^{M}& =\sqrt{\frac{4\pi }{3}}\sum _{lm,{l}^{\prime }{m}^{\prime }}{\int }_{0}^{\infty }dr{P}_{{l}^{\prime }{m}^{\prime }}^{{{\epsilon}}^{{\ast}}}\left(r\right)r{P}_{lm}^{i}\left(r\right)\hfill \\ \hfill & \quad {\times}\int d{\Omega}{Y}_{{l}^{\prime }{m}^{\prime }}^{{\ast}}\left(\theta ,\phi \right){Y}_{lm}\left(\theta ,\phi \right){Y}_{1M}\left(\theta ,\phi \right)\hfill \\ \hfill & =\sum _{lm,{l}^{\prime }{m}^{\prime }}{\left(-1\right)}^{{m}^{\prime }}\sqrt{\left(2l+1\right)\left(2{l}^{\prime }+1\right)}\hfill \\ \hfill & \quad {\times}\left(\begin{matrix}\hfill {l}^{\prime }\hfill & \hfill l\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{matrix}\right)\left(\begin{matrix}\hfill {l}^{\prime }\hfill & \hfill l\hfill & \hfill 1\hfill \\ \hfill -{m}^{\prime }\hfill & \hfill m\hfill & \hfill M\hfill \end{matrix}\right){\int }_{0}^{\infty }dr{P}_{{l}^{\prime }{m}^{\prime }}^{{{\epsilon}}^{{\ast}}}\left(r\right)r{P}_{lm}^{i}\left(r\right).\hfill \end{aligned}\end{equation} \tag{ 5 }$$

Equation (4) clearly shows that, by adapting the SCE for the bound and continuum wavefunctions, we significantly simplify the computation of the cross-section. Namely, the result of the angular integrals is expressed in terms of the Wigner-3j symbols [29] and we only have to solve a 1D integral numerically, which involves the single-centre expansion coefficients, ${P}_{lm}^{i/{\epsilon}}$. The computation of the matrix element ${D}_{i{\epsilon}}^{M}$ is more intensive for the heteronuclear molecule, CO, than the homonuclear N₂ as it involves both odd and even values for the l and l' numbers.

Auger decay rates have been calculated with a variety of different methods in existing work [30–40]. The general expression for the Auger rate, Γ, is given by [41]:

$$\begin{equation}{\Gamma}=\overline{\sum }2\pi \vert \mathcal{M}{\vert }^{2}\equiv \overline{\sum }2\pi \vert \langle {{\Psi}}_{\text{fin}}\vert {H}_{\text{I}}\vert {{\Psi}}_{\text{init}}\rangle {\vert }^{2},\end{equation} \tag{ 6 }$$

where $\overline{\sum }$ denotes a summation over the final states and an average over the initial states. |Ψ_init⟩ and |Ψ_fin⟩ are the wavefunctions of all electrons in the initial and final molecular state, respectively. H_I is the interaction Hamiltonian. In the m_a , m_b , S, M_S scheme, the Auger rate is given by [23]

$$\begin{equation}{{\Gamma}}_{b,a\to s,\zeta }=\sum _{\begin{subarray}{c}{m}_{a}{m}_{b}{m}_{s}{m}_{\zeta }\\ S{M}_{S}{S}^{\prime }{M}_{S}^{\prime }\end{subarray}}\pi {N}_{ab}{N}_{\text{h}}\sum _{L}\vert \mathcal{M}{\vert }^{2},\end{equation} \tag{ 7 }$$

with N_h being the number of holes in the orbital to be filled. a, b refer to the valence orbitals, s to the core orbital which is filled in and ζ refers to the continuum orbitals. S and M_S are the total spin and its orientation before the transition and S' and M'_S are the total spin and its orientation afterwards. As the CO orbitals have well-defined m, the summations over m will take only a single value. N_ab is the weighting factor related to the occupation of the valence orbitals which fill the hole given by

$$\begin{equation}{N}_{ab}=\begin{cases}\frac{{N}_{a}{N}_{b}}{2{\times}2} \quad \hfill & \mathrm{f}\mathrm{o}\mathrm{r}\enspace \enspace \mathrm{d}\mathrm{i}\mathrm{ff}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\enspace \enspace \mathrm{o}\mathrm{r}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{s}\hfill \\ \frac{{N}_{a}\left({N}_{a}-1\right)}{2{\times}2{\times}1}\quad \hfill & \mathrm{f}\mathrm{o}\mathrm{r}\enspace \enspace \mathrm{s}\mathrm{a}\mathrm{m}\mathrm{e}\enspace \enspace \mathrm{o}\mathrm{r}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{l}.\hfill \end{cases}\end{equation} \tag{ 8 }$$

Here, N_a and N_b are the occupations of orbitals a and b, respectively. The matrix element, $\mathcal{M}$, is given by

$$\begin{align}\hfill \mathcal{M}& ={\delta }_{S}^{{S}^{\prime }}{\delta }_{{M}_{S}}^{{M}_{S}^{\prime }}\sqrt{\left(2{l}_{s}+1\right)\left(2{l}_{a}+1\right)\left(2{l}_{\zeta }+1\right)\left(2{l}_{b}+1\right)}\hfill \\ \hfill & \quad {\times}\left[\sum _{\begin{subarray}{c}k{l}_{\zeta }{l}_{s}\\ {l}_{b}{l}_{a}\end{subarray}}\sum _{q=-k}^{k}{\left(-1\right)}^{{m}_{s}+q+{m}_{\zeta }}\int d{r}_{1}\right.\hfill \\ \hfill & \quad \left.{\times}\int d{r}_{2}{P}_{{l}_{\zeta }{m}_{\zeta }}^{\zeta {\ast}}\left({r}_{1}\right){P}_{{l}_{s}{m}_{s}}^{s{\ast}}\left({r}_{2}\right)\frac{{r}_{{< }}^{k}}{{r}_{{ >}}^{k+1}}{P}_{{l}_{b}{m}_{b}}^{b}\left({r}_{1}\right){P}_{{l}_{a}{m}_{a}}^{a}\left({r}_{2}\right)\right.\hfill \\ \hfill & \quad \left.{\times}\left(\begin{matrix}\hfill {l}_{s}\hfill & \hfill k\hfill & \hfill {l}_{a}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{matrix}\right)\left(\begin{matrix}\hfill {l}_{s}\hfill & \hfill k\hfill & \hfill {l}_{a}\hfill \\ \hfill -{m}_{s}\hfill & \hfill q\hfill & \hfill {m}_{a}\hfill \end{matrix}\right)\left(\begin{matrix}\hfill k\hfill & \hfill {l}_{\zeta }\hfill & \hfill {l}_{b}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{matrix}\right)\right.\hfill \\ \hfill & \quad \left.{\times}\left(\begin{matrix}\hfill k\hfill & \hfill {l}_{\zeta }\hfill & \hfill {l}_{b}\hfill \\ \hfill -q\hfill & \hfill -{m}_{\zeta }\hfill & \hfill {m}_{b}\hfill \end{matrix}\right)+{\left(-1\right)}^{S}\sum _{\begin{subarray}{c}k{l}_{\zeta }{l}_{s}\\ {l}_{b}{l}_{a}\end{subarray}}\sum _{q=-k}^{k}{\left(-1\right)}^{{m}_{s}+q+{m}_{\zeta }}\right.\hfill \\ \hfill & \quad \left.{\times}\int d{r}_{1}\int d{r}_{2}{P}_{{l}_{\zeta }{m}_{\zeta }}^{\zeta {\ast}}\left({r}_{1}\right){P}_{{l}_{s}{m}_{s}}^{s{\ast}}\left({r}_{2}\right)\frac{{r}_{{< }}^{k}}{{r}_{{ >}}^{k+1}}{P}_{{l}_{a}{m}_{a}}^{a}\left({r}_{1}\right){P}_{{l}_{b}{m}_{b}}^{b}\left({r}_{2}\right)\right.\hfill \\ \hfill & \quad \left.{\times}\left(\begin{matrix}\hfill {l}_{s}\hfill & \hfill k\hfill & \hfill {l}_{b}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{matrix}\right)\left(\begin{matrix}\hfill {l}_{s}\hfill & \hfill k\hfill & \hfill {l}_{b}\hfill \\ \hfill -{m}_{s}\hfill & \hfill q\hfill & \hfill {m}_{b}\hfill \end{matrix}\right)\left(\begin{matrix}\hfill k\hfill & \hfill {l}_{\zeta }\hfill & \hfill {l}_{a}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{matrix}\right)\right.\hfill \\ \hfill & \quad \left.{\times}\left(\begin{matrix}\hfill k\hfill & \hfill {l}_{\zeta }\hfill & \hfill {l}_{a}\hfill \\ \hfill -q\hfill & \hfill -{m}_{\zeta }\hfill & \hfill {m}_{a}\hfill \end{matrix}\right)\right]\hfill \end{align} \tag{ 9 }$$

where r_< = min(r₁, r₂) and r_> = max(r₁, r₂). Due to rotational symmetry, each orbital has a single well-defined m number. This calculation is more computationally intensive in the heteronuclear case than in the homonuclear case, as the wavefunctions do not have gerade or ungerade symmetry and therefore will involve both odd and even values of l.

We treat the dissociation of the molecule phenomenologically, with rates based on FEL experiments with N₂ [20, 42]. This approximation is justified by the similarity between CO and N₂, particularly with respect to their dissociative transitions [43]. We assume that, as for N₂, CO dissociates at its equilibrium distance. The molecular ion CO²⁺ is treated as dissociating with a lifetime of 100 fs [42]. The final products of this dissociation, according to the experimental work in reference [44], are C⁺ and O⁺ with 85% probability, C²⁺ and O with 9% probability, and C and O²⁺ with 6% probability. States of CO³⁺ without core holes are treated as instantaneously dissociating to C⁺ and O²⁺ or C²⁺ and O⁺ with equal probability, as is the case for N₂ [20, 23]. All states of CO⁴⁺ are treated as dissociating instantaneously to C²⁺ and O²⁺ [20, 23]. To determine which atomic orbitals are unoccupied after dissociation, we determine how the molecular orbitals with missing electrons map to atomic orbitals by calculating overlaps between molecular and atomic orbitals. Specifically, we calculate the overlap $\langle {\psi }_{a}^{\text{CO}}\vert {\psi }_{\alpha }^{\mathrm{C}/\mathrm{O}}\rangle$ of each molecular orbital of neutral CO with each atomic orbital of neutral C and O, where a corresponds to a molecular orbital and α to an atomic orbital. We find that $\langle {\psi }_{1\sigma }^{\text{CO}}\vert {\psi }_{1s}^{\mathrm{O}}\rangle \approx 1$, which means that the 1σ CO orbital with energy 542 eV corresponds to a 1s O orbital with energy 544 eV. Moreover, we find $\langle {\psi }_{2\sigma }^{\text{CO}}\vert {\psi }_{1s}^{\mathrm{C}}\rangle \approx 1$, which means that the 2σ CO orbital with energy 296 eV corresponds to a 1s C orbital with energy 297 eV. The other molecular orbitals have overlaps with atomic orbitals on both atomic sites. In what follows, we use these overlaps to determine the possible dissociation products and the dissociation rates to different sets of atomic states. For example, for a CO²⁺ state with missing electrons in molecular orbitals a and b, the dissociation rate to atomic states missing electrons in atomic orbitals α and β is given by:

$$\begin{equation}{{\Gamma}}_{a,b\to \alpha ,\beta }^{{\text{C}}^{2+}+\text{O}}=0.09{\times}{{\Gamma}}_{a,b}^{{\text{CO}}^{2+}}\frac{{\left\vert \langle {\psi }_{a}^{\text{CO}}\vert {\psi }_{\alpha }^{\mathrm{C}}\rangle \right\vert }^{2}{\left\vert \langle {\psi }_{b}^{\text{CO}}\vert {\psi }_{\beta }^{\mathrm{C}}\rangle \right\vert }^{2}}{\sum _{i,j}{\left\vert \langle {\psi }_{a}^{\text{CO}}\vert {\psi }_{i}^{\mathrm{C}}\rangle \right\vert }^{2}{\left\vert \langle {\psi }_{b}^{\text{CO}}\vert {\psi }_{j}^{\mathrm{C}}\rangle \right\vert }^{2}}\end{equation} \tag{ 10 }$$

$$\begin{equation}{{\Gamma}}_{a,b\to \alpha ,\beta }^{{\text{C}}^{+}+{\text{O}}^{+}}=0.85{\times}{{\Gamma}}_{a,b}^{{\text{CO}}^{2+}}\frac{{\left\vert \langle {\psi }_{a}^{\text{CO}}\vert {\psi }_{\alpha }^{\mathrm{C}}\rangle \right\vert }^{2}{\left\vert \langle {\psi }_{b}^{\text{CO}}\vert {\psi }_{\beta }^{\mathrm{O}}\rangle \right\vert }^{2}}{\sum _{i,j}{\left\vert \langle {\psi }_{a}^{\text{CO}}\vert {\psi }_{i}^{\mathrm{C}}\rangle \right\vert }^{2}{\left\vert \langle {\psi }_{b}^{\text{CO}}\vert {\psi }_{j}^{\mathrm{O}}\rangle \right\vert }^{2}}\end{equation} \tag{ 11 }$$

$$\begin{equation}{{\Gamma}}_{a,b\to \alpha ,\beta }^{\text{C}+{\text{O}}^{2+}}=0.06{\times}{{\Gamma}}_{a,b}^{{\text{CO}}^{2+}}\frac{{\left\vert \langle {\psi }_{a}^{\text{CO}}\vert {\psi }_{\alpha }^{\mathrm{O}}\rangle \right\vert }^{2}{\left\vert \langle {\psi }_{b}^{\text{CO}}\vert {\psi }_{\beta }^{\mathrm{O}}\rangle \right\vert }^{2}}{\sum _{i,j}{\left\vert \langle {\psi }_{a}^{\text{CO}}\vert {\psi }_{i}^{\mathrm{O}}\rangle \right\vert }^{2}{\left\vert \langle {\psi }_{b}^{\text{CO}}\vert {\psi }_{j}^{\mathrm{O}}\rangle \right\vert }^{2}},\end{equation} \tag{ 12 }$$

where ${{\Gamma}}_{a,b}^{{\text{CO}}^{2+}}$ is the rate of dissociation of molecular ion CO²⁺ with electrons missing from molecular orbitals a and b. This rate corresponds to a lifetime of 100 fs [42]. Note, in equations (10)–(12), we only consider transitions to atomic ions with electrons missing from orbitals α and β if the relevant overlaps have values greater than 0.02.

3.1.1. Ion yield dependence on FEL pulse parameters

In what follows, we identify how the C and O ion yields depend on different FEL pulses interacting with CO. In figure 1, we show the atomic ion yields of C and O produced by FEL pulses interacting with CO. These ion yields are obtained as a function of intensity for FEL pulses of photon energy 1100 eV and pulse duration 4 fs figures 1(a) and 80 fs figure 1(b) as well as of photon energy 350 eV and pulse duration 4 fs figures 1(c) and 80 fs figure 1(d). The photon energy 1100 eV is sufficient to ionise an electron from the 1σ orbital, the innermost orbital of CO, while 350 eV allows for an electron to be removed from the 2σ orbital. Therefore, the photon energy of 350 eV does not allow for an electron to be removed from the O site of CO. An 80 fs duration FEL pulse allows for more photo-ionisations transitions to take place compared to a 4 fs duration pulse.

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At a given intensity, for 1100 eV photon energy, comparing figure 1(a) with figure 1(b) and for 350 eV photon energy, comparing figure 1(c) with figure 1(d), we find that as expected the longer 80 fs pulse results in larger yields for the higher-charged states compared to the shorter 4 fs pulse. This is due to a larger number of photons being absorbed during the longer duration pulse. Moreover, we identify the effect of the photon energy, for a given pulse duration, by comparing the ion yields of the 350 eV pulses with the ion yields of the 1100 eV pulses. That is, we compare figure 1(a) with figure 1(c) and figure 1(b) with figure 1(d). We find that, at a given intensity, the ion yields for the higher-charged states are larger for the smaller 350 eV photon energy pulse. This is attributed to two factors. For a given intensity, a lower photon energy corresponds to a higher photon flux. In addition, the photo-ionisation cross-sections are higher for the lower 350 eV photon energy compared to the higher 1100 eV photon energy, resulting in more photo-ionisation processes.

3.1.2. C versus O atomic ion yields

For a given charge state, the C and O atomic ion yields as a function of intensity are generally different, see figure 1. For both 1100 eV and 350 eV photon energies and 4 fs and 80 fs pulse durations, we compare C with O, C⁺ with O⁺ and C²⁺ with O²⁺. We find these ion yields to be similar, since they are created mostly by the dissociation of CO²⁺. Specifically, C⁺ and O⁺ are created in equal amounts, see equation (11). For small intensities, the atomic ion C²⁺ has a slightly higher ion yield than O²⁺, since C²⁺ is produced with a dissociation rate of $0.09{\times}{{\Gamma}}_{a,b}^{{\text{CO}}^{2+}}$ (equation (10)), while O²⁺ is produced with a rate of $0.06{\times}{{\Gamma}}_{a,b}^{{\text{CO}}^{2+}}$ (equation (12)). For higher intensities, the biggest difference between C²⁺ and O²⁺ ion yields, occurs for pulse parameters 350 eV photon energy and 80 fs pulse duration. The reason is that significantly more atomic photo-ionisation transitions after dissociation lead to the formation of O²⁺ compared to C²⁺. This is shown in figure 2(d). Note, in figures 2(a)–(d), we plot, as a function of intensity, the average number of atomic single-photon ionisation transitions that lead to the formation of each atomic ion state.

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We now focus on the higher-charged states Cⁿ⁺ and Oⁿ⁺, where n = 3, 4, 5. At a given intensity, we find that for the 1100 eV photon energy, both for the 4 fs and 80 fs pulses, the yield of the Oⁿ⁺ ion is higher than the yield of the Cⁿ⁺ ion. The reason is that the single-photon ionisation cross-section to remove an electron from the 1σ molecular orbital or from the 1s orbital in oxygen is higher than the cross-section to remove an electron from the 2σ molecular orbital or from the 1s orbital in carbon. Figures 2(a) and (b) show that atomic photo-ionisation transitions play a more important role for the formation of C⁵⁺ and O⁵⁺, roughly two transitions, compared to one or less atomic transitions leading to the formation of Cⁿ⁺ and Oⁿ⁺ with n = 3, 4. At a given intensity, we find that for the 350 eV photon energy, both for the 4 fs and 80 fs pulses, the yield of the Cⁿ⁺ ion is higher than the yield of the Oⁿ⁺ ion where n = 3 or 4. The reason is that a photon energy of 350 eV is insufficient to ionise a core electron corresponding to the oxygen atomic site in CO or to ionise a 1s electron from an oxygen atomic ion. Hence, while C⁴⁺ is formed by an inner-shell atomic photo-ionisation from C²⁺ followed by an Auger process, O⁴⁺ is formed by two valence atomic photo-ionisation transitions from O²⁺. Indeed, comparing figures 2(c) and (d) with figures 2(a) and (b), we find that the number of atomic transitions to form C⁴⁺ remains roughly equal to one both for 1100 eV and 350 eV. However, for O⁴⁺ the number of photo-ionisations increases to two in the 350 eV case compared to one in the 1100 eV case. Finally, for the 350 eV case, C⁵⁺ and O⁵⁺ are both created by three valence atomic photo-ionisation transitions, resulting in similar ion yields.

3.1.3. Pathways for the formation of C and O atomic ions

Next, we discuss the prevalent pathways leading to the formation of Cⁿ⁺ and Oⁿ⁺, with n = 1, 2, 3, 4, when CO interacts with an FEL pulse of 10¹⁶ W cm⁻² intensity, and 1100 eV and 350 eV photon energies and for FWHM equal to 4 fs and 80 fs. For the FEL pulse of photon energy 1100 eV and FWHM 4 fs and 80 fs we find that the main pathways leading to the formation of ions up to C²⁺ and O²⁺ do not involve any atomic processes, see tables 1 and 2. Moreover, we find that the main pathway involves a single-photon absorption from a 1σ orbital and an Auger decay filling in the 1σ hole. For both pulses, we find that the main pathway leading to the formation of C³⁺ and O³⁺ involves a single-photon ionisation from the 1σ molecular orbital and an Auger decay filling in the 1σ hole, while it involves an atomic single-photon ionisation of a core electron and an Auger decay. Regarding C⁴⁺ and O⁴⁺, one of the main pathway leading to their formation involves two single-photon ionisation processes from the 1σ orbital and two Auger decays filling in the 1σ core hole as well as an atomic single-photon ionisation of a core electron and an Auger decay. For the FEL pulse of photon energy 350 eV and FWHM 4 fs and 80 fs, we find that the main pathways leading to the formation of ions up to C²⁺ and O²⁺ do not involve any atomic processes, see tables 3 and 4. This is similar to the respective pathways for the FEL pulse of photon energy 1100 eV. The difference between the 350 eV and the 1100 eV FEL pulses, is that for the former (latter) the main pathways involve a single-photon absorption from a 2σ (1σ) orbital and an Auger decay filling in the 2σ (1σ) core hole. Another difference between the 350 eV and 1100 eV FEL pulses is that the formation of O³⁺ and O⁴⁺ involves single-photon absorptions of valence electrons. This is consistent with our finding, discussed in the previous section, that, for the 350 eV photon energy, the ion yields of C³⁺ and C⁴⁺ are larger than the yields of O³⁺ and O⁴⁺, respectively.

Table 1. The % contributions to the atomic ion yields up to C⁴⁺/O⁴⁺ of the dominant pathways of ionisation for an FEL pulse of 4 fs FWHM, 1100 eV photon energy and an intensity of 10¹⁶ W cm⁻². P_1σ and P_2σ refers to a photo-ionisation either from the 1σ or 2σ orbital, respectively. A_1σ and A_2σ refers to an Auger transition in which either the 1σ or the 2σ orbital is filled, D refers to a dissociation. Where transitions are in braces, the different orders of transitions are also included.

Ion	Mol pathway	Atom pathway	Proportion (%)
C	P1σ A1σ D		69
C+	P1σ A1σ D		69
C2+	P1σ A1σ D		62
C3+	$\left\{{P}_{1\sigma }{P}_{1\sigma }{A}_{1\sigma }{A}_{1\sigma }\right\}D$	A	27
C3+	P1σ A1σ D	PC A	20
C4+	$\left\{{P}_{1\sigma }{P}_{2\sigma }{P}_{2\sigma }{A}_{1\sigma }\right\}D$	AA	27
C4+	$\left\{{P}_{1\sigma }{P}_{1\sigma }{A}_{1\sigma }{A}_{1\sigma }\right\}D$	PC A	25
O	P1σ A1σ D		69
O	P2σ A2σ D		29
O+	P1σ A1σ D		69
O+	P2σ A2σ D		29
O2+	P1σ A1σ D		59
O2+	P2σ A2σ D		25
O3+	P1σ A1σ D	PC A	48
O3+	P2σ A2σ D	PC A	13
O4+	$\left\{{P}_{1\sigma }{P}_{1\sigma }{P}_{1\sigma }{A}_{1\sigma }\right\}D$	AA	30
O4+	$\left\{{P}_{1\sigma }{P}_{1\sigma }{A}_{1\sigma }{A}_{1\sigma }\right\}D$	PC A	23

Table 2. Same as for table 1 for an FEL pulse of 80 fs FWHM, 1100 eV photon energy and an intensity of 10¹⁶ W cm⁻².

Ion	Mol pathway	Atom pathway	Proportion (%)
C	P1σ A1σ D		69
C+	P1σ A1σ D		68
C2+	$\left\{{P}_{1\sigma }{P}_{1\sigma }{A}_{1\sigma }{A}_{1\sigma }\right\}D$		28
C2+	P1σ A1σ D		27
C2+	$\left\{{P}_{1\sigma }{P}_{2\sigma }{A}_{1\sigma }{A}_{2\sigma }\right\}D$		23
C3+	P1σ A1σ D	PC A	63
C3+	P2σ A2σ D	PC A	26
C4+	$\left\{{P}_{1\sigma }{P}_{1\sigma }{A}_{1\sigma }{A}_{1\sigma }\right\}D$	PC A	37
C4+	$\left\{{P}_{1\sigma }{P}_{2\sigma }{A}_{1\sigma }{A}_{2\sigma }\right\}D$	PC A	28
O	P1σ A1σ D		69
O	P2σ A2σ D		29
O+	P1σ A1σ D		68
O+	P2σ A2σ D		29
O2+	$\left\{{P}_{1\sigma }{P}_{1\sigma }{A}_{1\sigma }{A}_{1\sigma }\right\}D$		29
O2+	$\left\{{P}_{1\sigma }{P}_{2\sigma }{A}_{1\sigma }{A}_{2\sigma }\right\}D$		24
O2+	P1σ A1σ D		22
O3+	P1σ A1σ D	PC A	66
O3+	P2σ A2σ D	PC A	26
O4+	$\left\{{P}_{1\sigma }{P}_{1\sigma }{A}_{1\sigma }{A}_{1\sigma }\right\}D$	PC A	40
O4+	$\left\{{P}_{1\sigma }{P}_{2\sigma }{A}_{1\sigma }{A}_{2\sigma }\right\}D$	PC A	29

Table 3. Same as for table 1 for an FEL pulse of 4 fs FWHM, 350 eV photon energy and an intensity of 10¹⁶ W cm⁻².

Ion	Mol pathway	Atom pathway	Proportion (%)
C	P2σ A2σ D		91
C+	P2σ A2σ D		90
C2+	P2σ A2σ D		57
C2+	P2σ A2σ P2σ A2σ D		21
C3+	$\left\{{P}_{2\sigma }{A}_{2\sigma }{P}_{V}\right\}D$	PC A	37
C3+	P2σ A2σ D	PC A	24
C4+	P2σ A2σ P2σ A2σ D	PC A	38
C4+	$\left\{{P}_{2\sigma }{A}_{2\sigma }{P}_{V}\right\}D$	PC A	37
O	P2σ A2σ D		92
O+	P2σ A2σ D		90
O2+	P2σ A2σ D		37
O2+	P2σ A2σ P2σ A2σ D		30
O3+	$\left\{{P}_{2\sigma }{A}_{2\sigma }{P}_{V}\right\}D$	PV	42
O3+	P2σ A2σ P2σ A2σ D	PV	35
O4+	$\left\{{P}_{2\sigma }{A}_{2\sigma }{P}_{V}\right\}D$	PV PV	40
O4+	P2σ A2σ P2σ A2σ D	PV PV	38

Table 4. Same as for table 1 for an FEL pulse of 80 fs FWHM, 350 eV photon energy and an intensity of 10¹⁶ W cm⁻².

Ion	Mol pathway	Atom pathway	Proportion (%)
C	P2σ A2σ D		90
C+	P2σ A2σ D		85
C2+	$\left\{{P}_{2\sigma }{A}_{2\sigma }{P}_{V}\right\}D$		73
C3+	P2σ A2σ D	PC A	40
C3+	$\left\{{P}_{2\sigma }{A}_{2\sigma }{P}_{V}\right\}D$	PC A	32
C4+	P2σ A2σ P2σ A2σ D	PC A	64
O	P2σ A2σ D		87
O+	P2σ A2σ D		55
O+	$\left\{{P}_{2\sigma }{A}_{2\sigma }{P}_{V}\right\}D$		30
O2+	P2σ A2σ D		53
O2+	$\left\{{P}_{2\sigma }{A}_{2\sigma }{P}_{V}\right\}D$		21
O3+	P2σ A2σ P2σ A2σ D	PV	61
O3+	$\left\{{P}_{2\sigma }{A}_{2\sigma }{P}_{V}\right\}D$	PV	12
O4+	P2σ A2σ P2σ A2σ D	PV PV	68

3.1.4. Comparison of C, O and N atomic ion yields

Next, we compare the N atomic ion yields produced by the interaction of N₂ with an FEL pulse with the C and O atomic ion yields produced when the same FEL pulse interacts with CO. Specifically, we compare the N, C and O atomic ion yields for an FEL pulse of 1100 eV photon energy and duration 4 fs figure 3(a) and 80 fs figure 3(b). Figures 3(a) and (b) clearly show that, at a given intensity, each N ion yield is larger than the respective C and O ion yields. This is due to the cross-sections of photo-ionisation transitions in N₂ being higher than the cross-section of photo-ionisation transitions in CO. Indeed, in N₂ the core orbitals 1σ_g and 1σ_u have very similar ionisation energies, roughly equal to 420 eV, and large photo-ionisation cross-sections approximately equal to 0.0023 a.u. for the pulse with photon energy of 1100 eV. However, in CO, the ionisation energy of the core orbital 1σ, 544 eV, is significantly higher than the ionisation energy of the 2σ, 296 eV. As a result, the photo-ionisation cross-section from the 1σ orbital (0.0018 a.u.) in CO is significantly higher than the photo-ionisation cross-section from the 2σ orbital (0.000 77 a.u.).

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We are interested in the proportion of each atomic ion yield that is reached by accessing various types of DCH states. We calculate these proportions by using an expanded set of rate equations. That is, for each electronic configuration, we consider four equations, which keep track of the population that reaches this state after accessing a TSDCH state, after accessing an SSDCH state, on the C or on the O site respectively, as well as the population which does not access a DCH state. The proportions of each atomic ion yield that are formed via accessing different DCH states are shown in figure 4 for a 4 fs pulse duration with intensity of 10¹⁸ W cm⁻². We choose these pulse parameters as they favour the production of DCH states. Indeed, FEL pulses of short duration and high intensity favour more single-photon ionisation transitions occurring in a certain time interval compared to longer and lower intensity FEL pulses. Moreover, in addition to the 1100 eV photon energy used in our calculations in the previous sections, we also consider a photon energy of 700 eV. The reason is that 700 eV is sufficient to photo-ionise both the 1σ and 2σ molecular orbitals in CO, as well as the 1σ_g and 1σ_u orbitals in N₂ and at the same time these cross-sections are higher than the 1100 eV case.

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Examining figure 4, we see that the proportion of the ion yields which accesses a DCH generally increases at higher-charged ion yields. This is expected as transition pathways which involve a pair of two core photo-ionisations typically lead to higher-charged ions. For this high-intensity and short-duration FEL pulse the C and O ions with charges 2, 3 and 4 are mainly produced by the dissociation of CO⁴⁺. We find that almost all of the C³⁺ and O³⁺ ion yields are produced via pathways involving TSDCH states. The reason is that when CO⁴⁺ breaks to C²⁺ and O²⁺, each doubly-charged ion is created with a core hole. Each core hole is then filled in by an Auger decay leading to the production of C³⁺ and O³⁺. A large proportion of the O⁴⁺ yield accesses an SSDCH state with the core holes localised on the oxygen. Regarding the O⁴⁺ ion, it is formed by dissociation of CO⁴⁺ to O²⁺ with two 1s core holes, which are then filled in by two Auger transitions. Hence, O⁴⁺ is mostly formed by accessing an SSDCH state of CO. In addition, we find that C²⁺ and O²⁺ are formed after the dissociation of CO⁴⁺ with no core holes. However, CO⁴⁺ before dissociation, accesses mostly SSDCH states on the oxygen side. This is due to the much higher photo-ionisation cross-section to transition from the 1σ orbital compared to transitioning from the 2σ orbital.

Comparing figure 4(d) with figure 4(a) and figure 4(e) with figure 4(b), we find that, for the 700 eV case, higher-charged ion states are produced with a higher proportion of these ion states accessing a TSDCH state. This is in accord with higher photo-ionisation cross-sections from both the 1σ and 2σ molecular orbitals for 700 eV photon energy, compared to 1100 eV. Finally, comparing the N ion yields in figure 4(c) with the C and O ion yields in figures 4(a) and (b) for 1100 eV and the N ion yields in figure 4(f) with the C and O ion yields in figures 4(d) and (e) for 700 eV, we find that N₂ dissociates into higher-charged ion states. Moreover, we find that a higher proportion of each of these N higher-charged ion states accesses a TSDCH state. This is due to the higher photo-ionisation cross-sections of the N₂ core orbitals, particularly the 1σ_u cross-section, which is much higher than the 2σ cross-section of CO.

To gain further understanding into the proportion of the ion states of C, O and N, that is formed by accessing a TSDCH state, we plot in figure 5 these proportions as a function of time. In what follows, for simplicity, we refer to these states as C-TSDCH, O-TSDCH and N-TSDCH. We find that the proportion of an ion state that is formed via a TSDCH state can significantly change with time depending on the atom and on the pulse parameters. For instance, the yield of C²⁺-TSDCH in figure 5(d) reaches a high value in a short time interval, while for large times this yield becomes very small. This means that, in the first few femtoseconds C²⁺-TSDCH is formed, it has core holes. As time increases, atomic transitions take place leading to higher carbon charged states. As a result, the yield of C²⁺-TSDCH at large times is significantly smaller than the one at small times. For an FEL pulse of photon energy of 1100 eV, FWHM of 4 fs and intensity of 10¹⁸ W cm⁻² interacting with CO, we find that in the first few femtoseconds C²⁺-TSDCH and C³⁺-TSDCH (see figure 5(a)) and O²⁺-TSDCH and O³⁺-TSDCH (see figure 5(b)) are created mostly without core holes. This is the reason why the yields of C⁴⁺-TSDCH and C⁵⁺-TSDCH (see figure 5(a)) and O⁴⁺-TSDCH and O⁵⁺-TSDCH (see figure 5(b)) are small. However, for N₂, we find that the proportion of N²⁺-TSDCH and N³⁺-TSDCH that has core holes is larger compared to the respective proportions for the ion states of C and O. As a result, the yields of N⁴⁺-TSDCH and N⁵⁺-TSDCH are larger compared to the respective yields for C and O. In contrast, for an FEL pulse of photon energy of 700 eV, FWHM of 4 fs and intensity of 10¹⁸ W cm⁻² interacting with CO and N₂, we find that in the first few femtoseconds C²⁺-TSDCH and C³⁺-TSDCH (see figure 5(d)) and O²⁺-TSDCH and O³⁺-TSDCH (see figure 5(e)) and N²⁺-TSDCH and N³⁺-TSDCH (see figure 5(f)) are created mostly with core holes. As a result, the yields of C⁴⁺-TSDCH and C⁵⁺-TSDCH (see figure 5(d)) and O⁴⁺-TSDCH and O⁵⁺-TSDCH (see figure 5(e)) and N⁴⁺-TSDCH and N⁵⁺-TSDCH (see figure 5(f)) are significantly larger compared to the respective yields for the case of the 1100 eV FEL pulse.

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