Multipole effects to the opacity of hot dense gold plasma

Abstract

The contributions of the multipole transitions to the opacity of hot dense gold plasma are taken into account by using an average-atom model. The influences of the E2, E3 and E4 transitions on the Rosseland opacity are studied, respectively. Comparisons with Miao s calculation have been made. It shows that using the Taylor series to account for the multipole transitions is no longer valid since ikr is not much smaller than the unit when the photon energy goes very high.

1 1 Introduction

Hydrogen has been recognized as a promising candidate for environmentally friendly energy storage, aiming to replace traditional fossil fuels and mitigate their undesirable consequences. Green hydrogen production, a vital component of this shift, relies on utilizing solar, wind, or hydropower to electrolytically split water into hydrogen and oxygen. The electrocatalysts used to efficiently produce hydrogen play a crucial role in the shift. Extensive experimental efforts to synthesize novel electrocatalysts with improved efficiency have been complemented by tremendous progress in computational modeling based on density functional theory (DFT). In addition to creating possibilities for the computer-aided design of new catalysts, the merging theory and experiment enables an understanding of the origin of the reactivity of the electrodes at the microscopic level [1,2]. Nørskov and collaborators [3-5] have formulated a straightforward theory to quantify surface efficiency in catalyzing the hydrogen evolution reaction (HER). They identified the Gibbs free energy of hydrogen adsorption [$\mathrm{\Delta }G$(H)] as a pivotal descriptor for catalyst activity, emphasizing the importance of a value close to zero for optimal performance. State-of-the-art HER catalysts in acidic environments predominantly feature platinum. However, the scarcity and high cost of platinum have prompted extensive research into more cost-effective alternatives. This quest has led to a focus on a novel class of low-dimensional structures, including metal nanoparticles [6,7], alloys [4], single-atom catalysts [18,19], two-dimensional (2D) materials [8], and several types of heterostructures [9-11]. Complex nanostructures and interfaces with liquids were also investigated due to their relevance for green technologies other than environmentally friendly hydrogen production [12,13]. Noteworthy among these 2D materials are MoS₂ nanoparticles supported by graphene oxide [14], metallic MoS₂ nanosheets [15], the edges of MoS₂ [16,17], the MoS₂ doped with metal single atoms [18], and various MoS₂ nanocomposites [20], demonstrating promising catalytic properties.

MXenes, a recently discovered family of two-dimensional transition metal carbides, nitrides, or carbonitrides [21-24], have also emerged as promising candidates for the next generation of HER catalysts [25-31]. Doping MXenes with single transition or noble metals significantly enhances their catalytic properties, transforming them into single-atom catalysts embedded in 2D materials [32-34]. Examples include Pt single atoms in passivated Mo₂TiC₂ [35], Co single atom substitution in 2D molybdenum carbide [36], and Ru single atoms in nitrogen-doped Ti₃C₂ [37]. Moreover, nitrogen [38] and phosphorus doping [39] have been explored to further enhance the performance of MXenes. A recent DFT study focused on experimentally realized MXenes (T₂C, V₂C, Nb₂C, Ti₃C₂, and Nb₄C₃) terminated with oxygen, revealing high structural stability [26]. Among them, Ti₂CO₂ and Nb₂CO₂ displayed calculated $\mathrm{\Delta }G$(H) values very close to zero, indicating prospects for being effective nonprecious HER catalysts. However, V₂CO₂ and Ti₃C₂O₂ exhibited stronger than optimal hydrogen binding [26]. Notably, the Nb₄C₃O₂ showed weaker reactivity than required for good-performance HER catalysis, with DFT-calculated free energy of H adsorption in the range from 0.14 to 0.36 eV [26,27], without an apparent reason for such a significant discrepancy. Yet, there are several advantageous properties of the Nb₄C₃O₂ granting it the status of the promising catalytic material. In addition to demonstrated structural stability, Nb₄C₃O₂ is a good conductor with a hydrophilic surface, which is advantageous in applications where the interaction with water is essential. The primary challenge lies in enhancing its reactivity to achieve close-to-optimal hydrogen binding.

The previous DFT studies of the HER on the Nb₄C₃O₂, as well as the investigations performed with other 2D MXenes in the pristine form or when doped with transition metals, were primarily focused on finding systems with favorable thermodynamics of the H binding. This study aims to go further and rationalize microscopic mechanisms driving the reactivity of the oxygen-terminated 2D Mixens doped with single-metal atom impurities. Using the Nb₄C₃O₂ as an example, we established a general picture of how to tune the H binding on a broad class of 2D materials featuring surface O atoms as the active catalytic sites. Through the utilization of DFT calculations, this paper demonstrates the potential of substituting 3$d$, 4$d$, and 5$d$ transition and noble metals as impurities to transform the Nb₄C₃O₂ monolayer into a highly efficient catalyst for the HER. The electronic properties of the impurity atom play a crucial role in inducing changes in the local electronic structure of the 2D MXene, thereby significantly influencing hydrogen binding. In some instances, these changes lead to $\mathrm{\Delta }G$(H) values approaching zero, signifying enhanced efficiency in HER catalysis. A comprehensive exploration of the reactivity trends is conducted by comparing a diverse range of transition and noble metals considered as substitutional impurities in Nb₄C₃O₂. The observed trends in reactivity will be systematically rationalized based on the analysis of the electronic properties inherent in the studied MXenes.

The manuscript is structured into four sections. Following the Introduction, Section 2 details the computational methodology employed in this study. Section 3 presents the results along with their discussion, and the paper concludes with Section 4.

2 2 Computational methods

The DFT calculations were performed by GPAW code [40,41], based on the projector augmented waves method (PAW) [42,43]. We used atomic PAW setups where O-2$s$, 2$p$ and Te-4$s$, 4$p$, 5$s$, 5$p$ and 4$d$ electrons were treated as valence electrons [44]. The wavefunctions were represented using the plane wave basis set with the energy cutoff of 600 eV. The Nb₄C₃O₂ monolayer was modeled using 3 × 3 surface cell, employing the periodic boundary conditions in the planes of the monolayer. The simulation cell included the vacuum of at least 16 Å in the direction perpendicular to the slab. The surface Brillouin zone was sampled by 16 special Monkhorst−Pack k-points [45], corresponding to the (4 × 4) grid centered on the $\mathrm{\Gamma }$-point. The electronic exchange and correlation effects were described by employing the GGA-PBE functional [46]. The spin-polarization effects were included for magnetic 3$d$ impurities. The iterative diagonalization of the Kohn−Sham Hamiltonian was carried out by applying the Davidson method [47]. To facilitate the convergency procedure, the electronic states were occupied according to the Fermi−Dirac distribution corresponding to the temperature of 0.05 eV. The geometry optimization was performed through the Broyden−Fletcher−Goldfarb−Shanno algorithm [48], as implemented in the Atomic Simulation Environment (ASE) python package [49,51]. The ASE package is also used to set the atomic structure of studied systems and for postprocessing of data obtained from DFT calculations. The electron charges of the individual atoms were calculated by employing Bader approach [50]. The H adsorption energy is calculated as

(1)$E_{\mathrm{a}\mathrm{d}\mathrm{s}}(\mathrm{H})=E(\mathrm{H}/\mathrm{M}\mathrm{x}\mathrm{e}\mathrm{n}\mathrm{e})-E(\mathrm{M}\mathrm{x}\mathrm{e}\mathrm{n}\mathrm{e})-1/2E({\mathrm{H}}_2),$

where $E(\mathrm{H}/\mathrm{M}\mathrm{x}\mathrm{e}\mathrm{n}\mathrm{e})$, $E(\mathrm{M}\mathrm{x}\mathrm{e}\mathrm{n}\mathrm{e})$, and $E$(H₂) are total energies of the H atom adsorbed on the the surface of MXene, the total energy of the Mxene, and the energy of the H₂ molecule in the gas phase. The free energy of H adsorption $\mathrm{\Delta }G$(H) is directly connected to the H adsorption energy $E_{\mathrm{a}\mathrm{d}\mathrm{s}}$(H) via the expression as follows:

(2)$\mathrm{\Delta }G(\mathrm{H})=E_{\mathrm{a}\mathrm{d}\mathrm{s}}(\mathrm{H})+\mathrm{\Delta }{E}_{\mathrm{Z}\mathrm{P}\mathrm{E}}-T\mathrm{\Delta }{S}_{\mathrm{H}}.$

The $\mathrm{\Delta }{E}_{\mathrm{Z}\mathrm{P}\mathrm{E}}$ and $\mathrm{\Delta }{S}_{\mathrm{H}}$ are differences in the zero-point energy and the entropy of the adsorbed H atom and the hydrogen in the gas phase. Following approach from Ref. [3] the $\mathrm{\Delta }G(\mathrm{H})$ and $E_{\mathrm{a}\mathrm{d}\mathrm{s}}(\mathrm{H})$ are connected by a simple formula: $\mathrm{\Delta }G(\mathrm{H})$ = $E_{\mathrm{a}\mathrm{d}\mathrm{s}}(\mathrm{H})$ + 0.24 eV.

The binding energies of the metal impurity M in the MXene with an Nb vacancy are calculated according to the standard expression:

(3)$E_{\mathrm{B}}(\mathrm{M})=E(\mathrm{M})+E(\mathrm{v}\mathrm{a}\mathrm{c}-\mathrm{N}{\mathrm{b}}_4{\mathrm{C}}_3{\mathrm{O}}_2)-E(\mathrm{M}-\mathrm{N}{\mathrm{b}}_4{\mathrm{C}}_3{\mathrm{O}}_2),$

where the terms in the formula above are the total energies of free atom M, the Mxene with the Nb vacancy, and the Nb₄C₃O₂ with an Nb surface atom substituted by metal M.

3 3 Results and discussion

3.1 3.1 Substitutional metal impurities

To assess the stability of the doped Nb₄C₃O₂ monolayer, we first calculated the binding of metal atom $E_B(M)$ at the Nb vacancies created on the surface of the Nb₄C₃O₂ monolayer (Fig.1), using as the reference energies those of free metal atoms. The calculated energies are listed in Tab.1.

Fig.1 Top (a) and side (b) view of the atomic structure of substitutional metal impurity in Nb4C3O2. The impurity atoms are depicted as blue spheres. Nb, C, and O atoms are presented as green, light gray, and red spheres. The dz in panel (b) marks impurity displacement out of the surface Nb plane.

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Tab.1 Binding energies ($E_{\mathrm{B}}$) of substitutional metal impurities in Nb4C3O2 monolayer and their displacements (dz) from the surface Nb atomic plane.

Group	Impurity	$E_{\mathrm{B}}$ (eV)	dz (Å)	Impurity	$E_{\mathrm{B}}$ (eV)	dz (Å)	Impurity	$E_{\mathrm{B}}$ (eV)	dz (Å)
IV	Ti	9.60	0.11	Zr	11.02	0.31	Hf	11.73	0.27
V	V	7.92	−0.19	Nb	10.11	0.00	Ta	11.77	0.08
VI	Cr	5.34	−0.06	Mo	7.80	−0.20	W	9.90	−0.18
VII	Mn	4.51	−0.07	Tc	−	−	Re	8.15	−0.28
VIII	Fe	4.98	−0.37	Ru	6.40	−0.31	Os	7.52	−0.33
IX	Co	5.80	−0.68	Rh	5.66	−0.53	Ir	5.79	−0.29
X	Ni	5.05	−0.66	Pd	2.83	−0.44	Pt	4.05	−0.56
XI	Cu	3.09	−0.46	Ag	1.60	2.06	Au	0.35	−0.29
XII	Zn	1.56	1.31	Cd	0.47	1.80	Hg	unstable	−

The set of inspected impurities expands over 3$d$, 4$d$, and 5$d$ elements from Groups IV−XII. The 4$d$ element Tc is not included in the study as a radioactive element unsuitable for green hydrogen production. The binding energy of Nb in Tab.1 is the energy cost of removing an Nb atom from the surface layer to the gas phase. As expected, the calculated binding energies show a clear trend of a monotonic decrease when moving from early to late transition and noble metals. Also, the binding is weaker for 3$d$ than 4$d$ metals and reaches the highest values for 5$d$ elements. The exceptions are Groups XI and XII elements, where displacements of impurities from the plane formed by Nb atoms could be larger than 2 Å, and the metal adsorption geometries differ considerably. Furthermore, these metals bind significantly weaker than others; thus, the structures they form are less stable than those of other metal impurities.

3.2 3.2 Hydrogen adsorption

Our tests and previous studies [52] show that the H atoms at Nb₄C₃O₂ prefer binding on oxygens. We also found that the impurity effect on the H binding is well-localized to the O atoms next to the impurity, and hence, all numbers in Tab.2 correspond to these adsorption sites. An example of the H adsorption geometry is depicted in Fig.2.

Tab.2 Free energy of H adsorption ($\mathrm{\Delta }G$) and the effective charge transfer on an O atom in the vicinity of the impurity ($\mathrm{\Delta }Q$) at Nb4C3O2 monolayer with substitutional metal impurities.

Group	Impurity	$\mathrm{\Delta }G$ (eV)	$\mathrm{\Delta }Q$ (e)	Impurity	$\mathrm{\Delta }G$ (eV)	$\mathrm{\Delta }Q$ (e)	Impurity	$\mathrm{\Delta }G$ (eV)	$\mathrm{\Delta }Q$ (e)
IV	Ti	0.14	−0.003	Zr	0.36	0.038	Hf	0.36	0.045
V	V	0.05	−0.030	Nb	0.20	0.000	Ta	0.32	0.016
VI	Cr	−0.01	−0.046	Mo	0.14	−0.034	W	0.17	−0.015
VII	Mn	−0.09	−0.055	Tc	−	−	Re	0.10	−0.041
VIII	Fe	−0.01	−0.089	Ru	−0.11	−0.088	Os	−0.11	−0.062
IX	Co	−0.15	−0.106	Rh	−0.19	−0.125	Ir	−0.17	−0.088
X	Ni	−0.18	−0.107	Pd	−0.07	−0.129	Pt	0.01	−0.136
XI	Cu	0.00	−0.111	Ag	0.13	−0.058	Au	−0.02	−0.139
XII	Zn	0.28	0.014	Cd	0.30	−0.011	Hg	−	−

Fig.2 The top view of the H adsorption geometry at Nb4C3O2 monolayer with Mn impurity. The side view is shown in the inset. Blue and yellow spheres represent Mn and H atoms. The color coding of other atoms is the same as in Fig.1.

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We calculated the free adsorption energy of the H atom at pristine Nb₄C₃O₂ monolayer of 0.20 eV (Tab.2), not far from the value in Ref. [26]. Since this MXene is not among those with the optimal catalytic properties regarding the HER, further modifications are required to improve its efficiency. A convenient and efficient approach to tailor Nb₄C₃O₂ reactivity relies on doping with transition and noble metal atoms. Our systematic study of 3$d$, 4$d$, and 5$d$ transition and noble metals incorporated into Nb₄C₃O₂ demonstrates that these single-atom impurities can be used for an effective adjustment of the free energy of H adsorption at nearby O atoms over the energy range of ~0.6 eV (Fig.3). The plot in Fig.3 shows an increase in the reactivity of the O adsorption sites with an increase in the occupation of the impurity $d$ shell. Once it gets fully occupied, the binding of H weakens. The highest H binding (the lowest $\mathrm{\Delta }G$(H) values) occurs for the Group IX or X impurites. In Fig.3 we highlighted the impurities that produce $\mathrm{\Delta }G$(H) values in the range from −0.1 to 0.1 eV, and hence convert the Nb₄C₃O₂ monolayer into a highly efficient catalyst for HER. The list of the impurities includes 3$d$ metals V, Cr, Fe, Mn, Fe, and Cu, Pd as a 4$d$ metal, as well as 5$d$ elements Re, Pt, and Au. The Ru, and Os are marginally (by 0.01 eV) outside this energy window and should also be considered promising candidates to tune the Nb₄C₃O₂ reactivity in the desired direction.

Fig.3 The variation in $\mathrm{\Delta }G$(H) with the Group number of substitutional impurity in Nb4C3O2 monolayer. The points corresponding to Nb (representing the pristine Nb4C3O2) and impurities giving rise to the $|\mathrm{\Delta }G(\mathrm{H})|\le 0.1$ eV are labeled.

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To rationalize the impurity effect on the H binding and trends across the periodic table, in Fig.4, we plotted $\mathrm{\Delta }G$(H) dependence on the center of the 2$p$ states of O adsorption site prior to H binding. The plot is inspired by the well-known $d$-band model of Hammer and Nørskov [53] successfully used to quantify the reactivity of metal surfaces. The correlation between the position of O-2$p$ center and the $\mathrm{\Delta }G$(H) is apparent for 3$d$ metal impurities and less evident for 4$d$ and 5$d$ elements. We attribute the dispersion of the point in the plot from a straight line, which would correspond to the perfect correlation, to the difference in the adsorption geometries of metal impurities embedded into the Nb₄C₃O₂. As seen in Tab.1, the dz meassuring displacement of single-atom impurities from the Nb atomic plane varies from −0.68 to 2.06 Å. Thus, the intrinsic assumptions of the theories like the $d$-band model that we compare only very similar adsorption geometries is fulfilled only aproximately. Still, the plot reveals that the major trend in reactivity can be understtod as the electronic effect. In Fig.4, we highlighted as examples Ta and Ir the impurities with the lowest $\mathrm{\Delta }G$(H) and the highest O-2$p$ center positions. At the 2D MXene with Ta impurity, the $\mathrm{\Delta }G$(H) is 0.32 eV, and the corresponding value for Ir is −0.17 eV. It is evident that the upshift of O-2$p$ center by ~0.4 eV leads to an increase of H binding by 0.49 eV. For pristine Nb₄C₃O₂ monolayer, we calculated $\mathrm{\Delta }G$(H) of 0.20 eV. The correlation between the O-2$p$ center and the $\mathrm{\Delta }G$(H) raises the question of the origin of the changes in the position of the O-2$p$ center. To get insight into this issue, we further compared Nb₄C₃O₂ monolayer with Ta and Ir, focusing on the impurity atoms and three O atoms nearby. In addition to Ta, and Ir, in further analysis of the electronic properties of doped Nb₄C₃O₂ we also included Cu, the metal with theoretically optimal $\mathrm{\Delta }G$(H) value of zero. For these three transition metals, in Fig.5(a)−(c), we plotted the impurity-induced charge density $\delta \rho$, defined as the difference in the charge densities of the Nb₄C₃O₂ with the impurity and the pristine monolayer. The isocontour plot of $\delta \rho$ for the Nb₄C₃O₂ monolayer with a Ta impurity [Fig.5(a)] illustrates that alterations in electron density induced by Ta are concentrated around the impurity, with minimal and invisible effects on the three closest O atoms, as depicted in the figure. This observation aligns seamlessly with small differences in $\mathrm{\Delta }G$(H) observed between the pristine and Ta-doped Nb₄C₃O₂ monolayer. A distinctive contrast emerges in the $\delta \rho$ plot when examining Cu or Ir impurities in Nb₄C₃O₂ [Fig.5(b) and (c)]. Beyond the electron density perturbation at the impurity site, noticeable changes manifest at the neighboring O atoms. This is exemplified by an apparent depletion of the O-2$p$ orbital oriented toward the impurity atom. The electron depletion on the O atoms signifies the elevated electronegativity of Cu or Ir relative to Nb. Fig.5(d) illustrates the impact of three different impurities on the energy of the 2$p$ states of the adjacent O atoms. The evident energy shift of the O states in the MXene with an Ir impurity, compared to that doped with a Ta atom, contributes to higher reactivity of the former, resulting in significantly stronger H binding. Notably, the Cu impurity induces changes in local electronic properties, representing a perfect compromise between Ta and Ir and hence providing an environment for optimal H binding. The $\delta \rho$ and density of states (DOS) projected onto nearby O-2$p$ orbitals, as examined for other metals considered in this study, are consistent with the calculated $\mathrm{\Delta }G$(H) values. To quantify the effective charge transfer on O atoms near the metal impurity, the $\mathrm{\Delta }Q$, we employed the Bader analysis [50]. The values presented in Tab.2 are calculated as the difference in the Bader charges of the O atom on the Nb₄C₃O₂ monolayer with the impurity compared to the value found for pristine Nb₄C₃O₂. The Bader analysis confirms that the free energies of the H adsorption can be correlated to the effective charge transfer on O atoms near the metal impurity.

Fig.4 The correlation between $\mathrm{\Delta }G$(H) and the 2$p$ center of the O atoms where H binds, for Nb4C3O2 monolayer with 3$d$, 4$d$ and 5$d$ transition and noble metal impurities.

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Fig.5 Impurity-induced charge density perturbation $\delta \rho$ in Nb4C3O2 monolayer with (a) Ta, (b) Cu, and (c) Ir substitutional impurities; $\delta \rho$ is defined as the difference in the charge densities of the MXene with the impurity and the pristine one; the orange and yellow isocontours are plotted at the values of 0.05 e/Å3 and −0.05 e/Å3, respectively. (d) DOS projected on the 2$p$ states of an O atom next to the impurity. The color scheme: Nb − green; O − red; C − gray. The impurities atoms are enclosed in orange and yellow isocontoures and thus not visible.

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4 4 Conclusions

Based on extensive DFT studies investigating metal impurities within Nb₄C₃O₂, our findings highlight several transition and noble metals capable of finely tuning the free energy of H adsorption. These tunable values match or surpass those observed on the Pt(111) surface. Consequently, we identify the corresponding 2D structures as highly promising candidates for efficiently catalyzing the hydrogen evolution reaction. The list of the single-atom metals eble to tune H binding on the Nb₄C₃O₂ monolayer to theoretically optimal value includes cheap and widely available elements Fe, Cr, and Cu. Our study reveals that enhanced hydrogen binding occurs in the presence of impurities exhibiting higher electronegativity than Nb. This effect stems from the effective electron depletion these impurities induce in the oxygen sites where hydrogen binds. We anticipate that this alteration in local reactivity, driven by oxygen charging, is a general phenomenon extending beyond Nb₄C₃O₂ to other 2D MXenes. This broadens the scope of our findings, suggesting the broader applicability of substitutional impurities for tuning reactivity in diverse 2D materials.