# Topology of the Electron Density and of Its Laplacian from Periodic LCAO Calculations on f-Electron Materials: The Case of Cesium Uranyl Chloride

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## Abstract

**:**

## 1. Introduction

## 2. Computational Details

**k**-points in the symmetry-irreducible Brillouin zone. Scalar relativistic effects of U must be accounted for [27,28,29,41] and, here, are described by use of small-core effective pseudopotentials, ECP60MDF (with 60 electrons in the core for U) [42,43]. The valence of U is described by a (10s9p7d5f1g)/[10s9p7d5f1g] basis set: we indicate within round brackets the number of Gaussian primitive functions used for the various angular quantum numbers and within square brackets the number of shells in which they are contracted. In this case, we use a fully uncontracted basis. With respect to the original basis set optimized for molecular calculations, some very diffuse exponents have been removed (crucially the most diffuse p-type exponent) that were causing linear dependencies in the periodic calculations. While the program has recently been extended to the treatment of spin-orbit coupling [44,45,46,47], this relativistic effect is disregarded here. This is because, while making the calculations significantly more demanding, it has been previously shown to induce minor changes to chemical bonding for such systems [48]. Oxygen and chlorine are described by molecular def2-TZVP basis sets of (11s6p2d1f)/[5s3p2d1f] and (14s9p3d1f)/[5s5p2d1f] type, respectively [49]. For Cs, Hay–Wadt small-core pseudopotentials are used [50] in combination with a (4s4p1d)/[2$sp$1d] valence.

## 3. The Implementation

#### 3.1. Basis Functions

#### 3.2. The Electron Density and Its Derivatives

#### 3.3. Old Strategy Based on the Expansion in Hermite Gaussian Type Functions

#### 3.4. New Strategy Based on a Direct Evaluation of the Electron Density in the RSSH-GTF Basis

#### 3.5. The Electron Localization Function

## 4. Results and Discussion

**b**crystal lattice vector, (center) the equatorial plane of the four Cl atoms, (right) through the O–U–O axis and a pair of Cl atoms. Upper panels report the experimental deformation density from Ref. [19] while bottom panels report the computed one in this study. The same iso-values have been used for the computed and experimental contour maps. In the panels (A${}_{1}$–A${}_{3}$), DD contour lines are overlaid over the $\nabla \rho $ trajectories of the crystal electron density, so as to show the intersection of the atomic boundaries with the three considered planes. Both quantitative and qualitative differences can be seen in the description of the electron distribution in the bonding region around the U atom. Panels (A${}_{1}$) and (B${}_{1}$) both show the nearly axial symmetry of the U–O interactions. In particular, the DD of present quantum-mechanical calculations in panel (B${}_{1}$) corroborates [11,56,57] the previously suggested “triple bond” nature of the U–O interaction with a $sp$ hybridization of the oxygen and the formation of a $\sigma $ bond along the U–O axis and, supposedly, two $\pi $ bonds with a maximum of charge deformation at about 0.71 Å off the axis. The charge accumulation close to the U atom along the U–O axis in the $\sigma $ bond in panel (B${}_{1}$)—suggestive of a covalent character—is instead missing to a large extent in the experimental DD in panel (A${}_{1}$). Other qualitative differences are observed in (i) a higher degree of localized electron build-up in the region of $\pi $ bonds in the theoretical DD and (ii) a more pronounced charge deformation behind the oxygen atoms in the experimental DD. Large differences are also observed in the equatorial plane of the four Cl atoms in panels (A${}_{2}$) and (B${}_{2}$). In particular, the expected nearly 4-fold symmetry of this plane seems to be lost in the experimental DD of panel (A${}_{2}$) while it is still largely there in the computed DD of panel (B${}_{2}$). The large departure from the expected symmetry was acknowledged in Ref. [19] and tentatively attributed to the different crystal environment at the second and third nearest neighbor level. While present calculations do show some asymmetry in the equatorial plane (for instance, inspection of the DD along the two green lines in panel (B${}_{2}$) reveals that those two directions are not exactly symmetry equivalent), they predict it to be very subtle while preserving the overall symmetric distribution of the density around the U atom in this plane. However, both theory and experiments describe a charge depletion close to U and a charge accumulation close to Cl along the U–Cl bonds, indicative of a higher ionic character of this interaction relative to the U–O one. Panels (A${}_{3}$) and (B${}_{3}$) show the DD in a plane through the O–U–O axis and a pair of Cl atoms and basically confirm all of the discrepancies of those features discussed above on the other two planes.

## 5. Conclusions and Perspectives

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Structure of the cesium uranyl chloride, Cs${}_{2}$UO${}_{2}$Cl${}_{4}$, crystal viewed (

**A**) in the $ab$ crystallographic plane, and (

**B**) down the b axis. Colors are as follows: O red, Cl, green, U grey, Cs, turquoise.

**Figure 2.**Deformation density, $\Delta \rho \left(\mathbf{r}\right)$, contour maps of the cesium uranyl chloride crystal around the U atom in three different planes: (

**left**) through the O–U–O axis and the

**b**crystal lattice vector, (

**center**) the equatorial plane of the four Cl atoms, (

**right**) through the O–U–O axis and a pair of Cl atoms. Upper panels report the experimental deformation density from Ref. [19] while bottom panels report the computed one in this study. Contour values are ± 0.05, 0.15, 0.25, 0.4, 0.7, 1.0, 1.5, 2.0 $e/$Å${}^{-3}$. Red and blue lines correspond to positive and negative values, respectively. In the experimental upper panels, DD contour lines are overlaid over the $\nabla \rho $ trajectories of the crystal electron density.

**Figure 3.**(

**A**${}_{\mathbf{1}}$) Laplacian function $\mathbf{L}\left(\mathbf{r}\right)$ (upper panel), deformation Laplacian $\Delta \mathbf{L}\left(\mathbf{r}\right)$ (middle panel) and deformation density $\Delta \mathbf{\rho}\left(\mathbf{r}\right)$ (bottom panel) along the U–Cl axis. (

**A**${}_{\mathbf{2}}$) Same as in (

**A**${}_{\mathbf{1}}$) but along the U–O axis. Vertical dashed black lines mark the distance of Cl and O from U, vertical dashed green lines mark the position of the bond critical point and vertical dashed red lines mark the position of (3,+3) CPs of the Laplacian. (

**B**) Spatial distribution of (3,+3) CPs of the Laplacian in the [UO${}_{\mathbf{2}}$Cl${}_{\mathbf{4}}$]${}^{\mathbf{2}-}$ molecular fragment (yellow spheres). (

**C**${}_{\mathbf{1}}$) Electron localization function (ELF) in the region of the O–U–O bonds from a superposition of non-interacting atomic densities. (

**C**${}_{\mathbf{2}}$) Same as in (

**C**${}_{\mathbf{1}}$) but from the computed density of the crystal.

**Figure 4.**Laplacian function $L\left(\mathbf{r}\right)$ (upper panel), deformation Laplacian $\Delta L\left(\mathbf{r}\right)$ (middle panel) and deformation density $\Delta \rho \left(\mathbf{r}\right)$ (bottom panel) along the U–Cl and U–O axes in the vicinity of the U atom. Vertical dashed red lines mark the position of U VSCC CPs of the Laplacian.

**Figure 5.**Positions in space of (3,+3) CPs of the Laplacian of the electron density of the cesium uranyl chloride (yellow spheres), as obtained from our quantum-mechanical periodic calculations. These CPs correspond to maxima of valence charge concentrations of U, O and Cl atoms.

**Table 1.**Atomic charges computed on the actual periodic system cry-Cs${}_{2}$UO${}_{2}$Cl${}_{4}$ and on the extracted molecular fragment [UO${}_{2}$Cl${}_{4}$]${}^{2-}$. Both Mulliken (M) and QTAIMAC Bader (B) charges are reported. Experimental values derived from a QTAIMAC analysis of the reconstructed electron density of the crystal are also reported for comparison [19].

[UO${}_{2}$Cl${}_{4}$]${}^{2-}$ | cry-Cs${}_{2}$UO${}_{2}$Cl${}_{4}$ | ||||
---|---|---|---|---|---|

Q${}_{\mathbf{M}}$ | Q${}_{\mathbf{B}}$ | Q${}_{\mathbf{M}}$ | Q${}_{\mathbf{B}}$ | Q${}_{\mathbf{B},\mathbf{exp}}$ | |

U | +2.18 | +2.90 | +2.14 | +2.94 | +2.75 |

O | −0.61 | −0.93 | −0.60 | −1.02 | −0.92 |

Cl | −0.74 | −0.76 | −0.74 | −0.74 | −0.60 |

Cs | - | - | +1.00 | +0.98 | +0.77 |

**Table 2.**Descriptors of chemical bonding of cesium uranyl chloride from the QTAIMAC: bond length l, distance between first atom and bond critical point ${d}_{\mathrm{CP}}$, value of several local quantities at the bond critical point such as the electron density $\rho $, the Laplacian of the density ${\nabla}^{2}\rho $, the ratio between the potential energy density and kinetic energy density $\left|V\right|/G$, and the bond degree $H/\rho $ (i.e., ratio between total energy density and electron density). Calculations are performed at the experimental geometry. Experimental data are taken from Ref. [19]. Calculations are performed both on the actual periodic extended lattice (Cry) and on the extracted molecular fragment [UO${}_{2}$Cl${}_{4}$]${}^{2-}$ (Mol).

O–U | Cl–U | O–Cs | Cl–Cs | Cl–Cs | Cl–Cs | Cl–Cs | ||
---|---|---|---|---|---|---|---|---|

l (Å) | Exp | 1.776 | 2.670 | 3.259 | 3.502 | 3.518 | 3.544 | 3.624 |

Mol | 0.797 | 1.298 | - | - | - | - | - | |

${d}_{\mathrm{CP}}$ (Å) | Cry | 0.797 | 1.298 | 1.458 | 1.693 | 1.715 | 1.734 | 1.773 |

Exp | 0.818 | 1.279 | 1.494 | 1.729 | 1.703 | 1.757 | 1.771 | |

Mol | 2.059 | 0.421 | - | - | - | - | - | |

$\rho $ (e/Å${}^{3}$) | Cry | 2.058 | 0.425 | 0.057 | 0.070 | 0.069 | 0.066 | 0.058 |

Exp | 1.695 | 0.486 | 0.053 | 0.067 | 0.071 | 0.076 | 0.068 | |

Mol | 7.47 | 3.44 | - | - | - | - | - | |

${\nabla}^{2}\rho $ (e/Å${}^{5}$) | Cry | 7.55 | 3.42 | 0.88 | 0.81 | 0.79 | 0.76 | 0.65 |

Exp | 15.77 | 3.28 | 0.72 | 0.65 | 0.73 | 0.64 | 0.63 | |

Mol | 1.780 | 1.256 | - | - | - | - | - | |

$\left|V\right|/G$ | Cry | 1.778 | 1.264 | 0.740 | 0.799 | 0.825 | 0.823 | 0.806 |

Exp | 1.587 | 1.417 | 0.724 | 0.845 | 0.831 | 0.902 | 0.860 | |

Mol | −0.903 | −0.196 | - | - | - | - | - | |

$H/\rho $ (a.u.) | Cry | −0.902 | −0.202 | 0.224 | 0.141 | 0.119 | 0.121 | 0.128 |

Exp | −0.926 | −0.340 | 0.204 | 0.091 | 0.105 | 0.053 | 0.079 |

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**MDPI and ACS Style**

Cossard, A.; Casassa, S.; Gatti, C.; Desmarais, J.K.; Erba, A.
Topology of the Electron Density and of Its Laplacian from Periodic LCAO Calculations on *f*-Electron Materials: The Case of Cesium Uranyl Chloride. *Molecules* **2021**, *26*, 4227.
https://doi.org/10.3390/molecules26144227

**AMA Style**

Cossard A, Casassa S, Gatti C, Desmarais JK, Erba A.
Topology of the Electron Density and of Its Laplacian from Periodic LCAO Calculations on *f*-Electron Materials: The Case of Cesium Uranyl Chloride. *Molecules*. 2021; 26(14):4227.
https://doi.org/10.3390/molecules26144227

**Chicago/Turabian Style**

Cossard, Alessandro, Silvia Casassa, Carlo Gatti, Jacques K. Desmarais, and Alessandro Erba.
2021. "Topology of the Electron Density and of Its Laplacian from Periodic LCAO Calculations on *f*-Electron Materials: The Case of Cesium Uranyl Chloride" *Molecules* 26, no. 14: 4227.
https://doi.org/10.3390/molecules26144227