100 Years of the Lennard-Jones Potential.

It is now 100 years since Lennard-Jones published his first paper introducing the now famous potential that bears his name. It is therefore timely to reflect on the many achievements, as well as the limitations, of this potential in the theory of atomic and molecular interactions, where applications range from descriptions of intermolecular forces to molecules, clusters, and condensed matter.

Reduction of CO2 in the presence of light via excited-state hydride transfer reaction in a NADPH-inspired derivative.

The photo-catalytic reduction of CO2 into chemical feedstocks using solar energy has attracted vast interest in environmental science because of global warming. Based on our previous study on the CO2 complex with one of the benzimidazoline (BI) derivatives, we explore the photochemical reduction of CO2 in one of the benzimidazoline derivatives (1,3-dimethyl-5,6-diol-2,3-dihydro-1H-benzimidazole) by quantum-chemical methods. Our results reveal that carbon dioxide can be reduced to formate (HCOO-) by a hydride transfer reaction in the excited state of this complex of benzimidazoline derivative and CO2. While the ground-state hydride transfer reaction in this complex exhibits a substantial barrier, a charge-transfer can occur in the first singlet excited state of the complex in the UV-A region (326 nm), and after overcoming a moderate barrier (∼0.4 eV) the system can have access to the products. The interaction with a polar solvent decreases further the barrier such that the reaction in dimethyl sulfoxide can proceed with a negligibly small barrier (∼0.1 eV) or in a nearly barrierless manner. Our results show that this benzimidazoline derivative may act as a catalyst in the photoreduction of CO2.

Tipping the Balance Between the bcc and fcc Phase Within the Alkali and Coinage Metal Groups.

Why the Group 1 elements crystallize in the body-centered cubic (bcc) structure, and the iso-electronic Group 11 elements in the face-centered cubic (fcc) structure, remains a mystery. Here we show that a delicate interplay between many-body effects, vibrational contributions and dispersion interactions obtained from relativistic density functional theory offers an answer to this long-standing controversy. It also sheds light on the Periodic Table of Crystal Structures. A smooth diffusionless transition through cuboidal lattices gives a detailed insight into the bcc→fcc phase transition for the Groups 1 and 11 elements.

Large vibrationally induced parity violation effects in CHDBrI.

The isotopically chiral molecular ion CHDBrI+ is identified as an exceptionally promising candidate for the detection of parity violation in vibrational transitions. The largest predicted parity-violating frequency shift reaches 1.8 Hz for the hydrogen wagging mode which has a sub-Hz natural line width and its vibrational frequency auspiciously lies in the available laser range. In stark contrast to this result, the parent neutral molecule is two orders of magnitude less sensitive to parity violation. The origin of this effect is analyzed and explained. Precision vibrational spectroscopy of CHDBrI+ is feasible as it is amenable to preparation at internally low temperatures and resistant to predissociation, promoting long interrogation times (Landau et al., J. Chem. Phys., 2023, 159, 114307). The intersection of these properties in this molecular ion places the first observation of parity violation in chiral molecules within reach.

Lattice sum for a hexagonal close-packed structure and its dependence on the c/a ratio of the hexagonal cell parameters.

We continue the work by Lennard-Jones and Ingham, and later by Kane and Goeppert-Mayer, and present a general lattice sum formula for the hexagonal close packed (hcp) structure with different c/a ratios for the two lattice parameters a and c of the hexagonal unit cell. The lattice sum is expressed in terms of fast converging series of Bessel functions. This allows us to analytically examine the behavior of a Lennard-Jones potential as a function of the c/a ratio. In contrast to the hard-sphere model, where we have the ideal ratio of c/a=sqrt[8/3] with 12 kissing spheres around a central atom, we observe the occurrence of a slight symmetry-breaking effect and the appearance of a second metastable minimum for the (12,6) Lennard-Jones potential around the ratio c/a=2/3. We also show that the analytical continuation of the (n,m) Lennard-Jones potential to the domain n,m<3 such as the Kratzer potential (n=2,m=1) gives unphysical results.

Nuclear Quadrupole Splittings in Rotational Spectra for the Possible Detection of Fine Structure Constant Variations: The Diatomic Gold Halides and Gold Hydride as Case Studies.

The dependence of nuclear quadrupole coupling constants CNQC(α) on the fine-structure constant α for various diatomic gold molecules AuX (X = H, F, Cl, Br, and I) at the density functional level of theory is investigated. While the electric field gradient at gold is very sensitive to the density functional applied, the derivative with respect to α is less sensitive. From this, we can estimate the upper limit for the α variation in time, ∂CNQC/∂t, for the 197Au nuclear quadrupole coupling constant, which is on the order of 10-9 Hz/year. This is currently beyond the limit of high-precision spectroscopy. I demonstrate that α∂CNQC∂α can be estimated from relativistic effects in CNQC, which will be useful in further investigations.

Solving a problem with a single parameter: a smooth bcc to fcc phase transition for metallic lithium.

Density functional calculations for metallic lithium along a cuboidal bcc-to-fcc transformation path demonstrate that the bcc phase is quasi-degenerate with the fcc phase with a very small activiation barrier of 0.1 kJ mol-1, but becomes the dominant phase at higher temperatures in accordance with Landau theory. This resolves the long-standing controversy about the two phases for lithium.

Toward Detection of the Molecular Parity Violation in Chiral Ru(acac)3 and Os(acac)3.

We present a theory-experiment investigation of the helically chiral compounds Ru(acac)3 and Os(acac)3 as candidates for next-generation experiments for detection of molecular parity violation (PV) in vibrational spectra. We used relativistic density functional theory calculations to identify optimal vibrational modes with expected PV effects exceeding by up to 2 orders of magnitude the projected instrumental sensitivity of the ultrahigh resolution experiment under construction at the Laboratoire de Physique des Lasers in Paris. Preliminary measurements of the vibrational spectrum of Ru(acac)3 carried out as the first steps toward the planned experiment are presented.

Light-driven reduction of CO2: thermodynamics and kinetics of hydride transfer reactions in benzimidazoline derivatives.

CO2 capture, conversion and storage belong to the holy grail of environmental science. We therefore explore an important photochemical hydride transfer reaction of benzimidazoline derivatives with CO2 in a polar solvent (dimethylsulfoxide) by quantum-chemical methods. While the excited electronic state undergoing hydride transfer to formate (HCOO-) shows a higher reaction path barrier compared to the ground state, a charge-transfer can occur in the near-UV region with nearly barrierless access to the products involving a conical intersection between both electronic states. Such radiationless decay through the hydride transfer reaction and formation of HCCO-via excited electronic states in suitable organic compounds opens the way for future photochemical CO2 reduction. We provide a detailed analysis for the chemical CO2 reduction to the formate anion for 15 different benzimidazoline derivatives in terms of thermodynamic hydricities (ΔGH-), activation free energies (ΔG‡HT), and reaction free energies (ΔGrxn) for the chosen solvent dimethylsulfoxide at the level of density functional theory. The calculated hydricities are in the range from 35.0 to 42.0 kcal mol-1i.e. the species possess strong hydride donor abilities required for the CO2 reduction to formate, characterized by relatively low activation free energies between 18.5 and 22.2 kcal mol-1. The regeneration of the benzimidazoline can be achieved electrochemically.

From the gas phase to the solid state: The chemical bonding in the superheavy element flerovium.

As early as 1975, Pitzer suggested that copernicium, flerovium, and oganesson are volatile substances behaving like noble gas because of their closed-shell configurations and accompanying relativistic effects. It is, however, precarious to predict the chemical bonding and physical behavior of a solid by knowledge of its atomic or molecular properties only. Copernicium and oganesson have been analyzed very recently by our group. Both are predicted to be semiconductors and volatile substances with rather low melting and boiling points, which may justify a comparison with the noble gas elements. Here, we study closed-shell flerovium in detail to predict its solid-state properties, including the melting point, by decomposing the total energy into many-body forces derived from relativistic coupled-cluster theory and from density functional theory. The convergence of such a decomposition for flerovium is critically analyzed, and the problem of using density functional theory is highlighted. We predict that flerovium in many ways does not behave like a typical noble gas element despite its closed-shell 7p1/2 2 configuration and resulting weak interactions. Unlike the case of noble gases, the many-body expansion in terms of the interaction energy does not converge smoothly. This makes the accurate prediction of phase transitions very difficult. Nevertheless, a first prediction by Monte Carlo simulation estimates the melting point at 284 ± 50 K. Furthermore, calculations for the electronic bandgap suggests that flerovium is a semiconductor similar to copernicium.

Relativistic Effects Stabilize Unusual Gold(II) Sulfate Structure via Aurophilic Interactions.

The crystal structure of gold(II) sulfate is strikingly different from other coinage metal(II) sulfates. Central to the unsual AuSO4 bulk structure is the Au24+ ion with a very close Au-Au contact, which is a structural feature that does not appear in CuSO4 and AgSO4. To shed some light on this unusual behavior, we decided to investigate the relative stabilities of the coinage metal(II) sulfates utilizing periodic Density Functional Theory. By computing relative energies of the hypothetical nonrelativistic gold(II) sulfate (AuNRSO4) in different structural arrangements and performing chemical bonding analyses employing the Electron Localization Function as well as the Quantum Theory of Atoms in Molecules method, we show that the stability of the unsual AuSO4 bulk structure can be related to aurophilic interactions enabled by relativistic effects. From the relative stabilities and UV-vis spectra computed via GW methodology, we predict that AuNRSO4 would assume the structure of either copper(II) sulfate or silver(II) sulfate with almost equal likelihood and appear as bright-violet or deep-blue substances, respectively.

It's Complicated: On Relativistic Effects and Periodic Trends in the Melting and Boiling Points of the Group 11 Coinage Metals.

While the color of metallic gold is a prominent and well-investigated example for the impact of relativistic effects, much less is known regarding the influence on its melting and boiling point (MP/BP). To remedy this situation, this work takes on the challenging task of exploring the phase transitions of the Group 11 coinage metals Cu, Ag, and Au through nonrelativistic (NR) and scalar/spin-orbit relativistic (SR/SOR) Gibbs energy calculations with λ-scaled density-functional theory (λDFT). At the SOR level, the calculations provide BPs in excellent agreement with experimental values (1%), while MPs exhibit more significant deviations (2-10%). Comparing SOR calculations to those conducted in the NR limit reveals some remarkably large and, at the same time, some surprisingly small relativistic shifts. Most notably, the BP of Au increases by about 800 K due to relativity, which is in line with the strong relativistic increase of the cohesive energy, whereas the MP of Au is very similar at the SOR and NR levels, defying the typically robust correlation between MP and cohesive energy. Eventually, an inspection of thermodynamic quantities traces the trend-breaking behavior of Au back to phase-specific effects in liquid Au, which render NR Au more similar to SOR Ag, in line with a half-a-century-old hypothesis of Pyykkö.

Instability of the body-centered cubic lattice within the sticky hard sphere and Lennard-Jones model obtained from exact lattice summations.

A smooth path of rearrangement from the body-centered cubic (bcc) to the face-centered cubic (fcc) lattice is obtained by introducing a single parameter to lattice vectors of a cuboidal unit cell. As a result, we obtain analytical expressions in terms of lattice sums for the cohesive energy where the interaction is described by a Lennard-Jones (LJ) interaction potential or a sticky hard-sphere (SHS) model with a r^{-n} long-range attractive term. These lattice sums are evaluated to computer precision by expansions in terms of a fast converging Bessel function series. Applying the whole range of lattice parameters for the SHS and LJ potentials we prove that the bcc phase is unstable (or, at best, metastable) toward distortion into the fcc phase in the low temperature and pressure limit. Even if more accurate potentials are used, such as the extended LJ potential for argon or chromium, the bcc phase remains unstable. This strongly indicates that the appearance of a low temperature bcc phase for several elements in the periodic table is due to higher than two-body forces in atomic interactions.

The Lennard-Jones Potential Revisited: Analytical Expressions for Vibrational Effects in Cubic and Hexagonal Close-Packed Lattices.

Analytical formulas are derived for the zero-point vibrational energy and anharmonicity corrections of the cohesive energy and the mode Grüneisen parameter within the Einstein model for the cubic lattices (sc, bcc, and fcc) and for the hexagonal close-packed structure. This extends the work done by Lennard-Jones and Ingham in 1924, Corner in 1939, and Wallace in 1965. The formulas are based on the description of two-body energy contributions by an inverse power expansion (extended Lennard-Jones potential). These make use of three-dimensional lattice sums, which can be transformed to fast converging series and accurately determined by various expansion techniques. We apply these new lattice sum expressions to the rare gas solids and discuss associated critical points. The derived formulas give qualitative but nevertheless deep insight into vibrational effects in solids from the lightest (helium) to the heaviest rare gas element (oganesson), both presenting special cases because of strong quantum effects for the former and strong relativistic effects for the latter.

Exclusively Relativistic: Periodic Trends in the Melting and Boiling Points of Group†12.

First-principles simulations can advance our understanding of phase transitions but are often too costly for the heavier elements, which require a relativistic treatment. Addressing this challenge, we recently composed an indirect approach: A precise incremental calculation of absolute Gibbs energies for the solid and liquid with a relativistic Hamiltonian that enables an accurate determination of melting and boiling points (MPs and BPs). Here, we apply this approach to the Group 12 elements Zn, Cd, Hg, and Cn, whose MPs and BPs we calculate with a mean absolute deviation of only 5 % and 1 %, respectively, while we confirm the previously predicted liquid aggregate state of Cn. At a non-relativistic level of theory, we obtain surprisingly similar MPs and BPs of 650±30 K and 1250±20 K, suggesting that periodic trends in this group are exclusively relativistic in nature. Ultimately, we discuss these results and their implication for Groups 11 and 14.

Oganesson: A Noble Gas Element That Is Neither Noble Nor a Gas.

Oganesson (Og) is the last entry into the Periodic Table completing the seventh period of elements and group 18 of the noble gases. Only five atoms of Og have been successfully produced in nuclear collision experiments, with an estimate half-life for 294 118 Og of 0. 69 + 0 . 64 - 0 . 22  ms.[1] With such a short lifetime, chemical and physical properties inevitably have to come from accurate relativistic quantum theory. Here, we employ two complementary computational approaches, namely parallel tempering Monte-Carlo (PTMC) simulations and first-principles thermodynamic integration (TI), both calibrated against a highly accurate coupled-cluster reference to pin-down the melting and boiling points of this super-heavy element. In excellent agreement, these approaches show Og to be a solid at ambient conditions with a melting point of ≈325 K. In contrast, calculations in the nonrelativistic limit reveal a melting point for Og of 220 K, suggesting a gaseous state as expected for a typical noble gas element. Accordingly, relativistic effects shift the solid-to-liquid phase transition by about 100 K.

The periodic table and the physics that drives it.

Mendeleev's introduction of the periodic table of elements is one of the most important milestones in the history of chemistry, as it brought order into the known chemical and physical behaviour of the elements. The periodic table can be seen as parallel to the Standard Model in particle physics, in which the elementary particles known today can be ordered according to their intrinsic properties. The underlying fundamental theory to describe the interactions between particles comes from quantum theory or, more specifically, from quantum field theory and its inherent symmetries. In the periodic table, the elements are placed into a certain period and group based on electronic configurations that originate from the Pauli and Aufbau principles for the electrons surrounding a positively charged nucleus. This order enables us to approximately predict the chemical and physical properties of elements. Apparent anomalies can arise from relativistic effects, partial-screening phenomena (of type lanthanide contraction) and the compact size of the first shell of every l-value. Further, ambiguities in electron configurations and the breakdown of assigning a dominant configuration, owing to configuration mixing and dense spectra for the heaviest elements in the periodic table. For the short-lived transactinides, the nuclear stability becomes an important factor in chemical studies. Nuclear stability, decay rates, spectra and reaction cross sections are also important for predicting the astrophysical origin of the elements, including the production of the heavy elements beyond iron in supernova explosions or neutron-star mergers. In this Perspective, we critically analyse the periodic table of elements and the current status of theoretical predictions and origins for the heaviest elements, which combine both quantum chemistry and physics.

Copernicium: A Relativistic Noble Liquid.

The chemical nature and aggregate state of superheavy copernicium (Cn) have been subject of speculation for many years. While strong relativistic effects render Cn chemically inert, which led Pitzer to suggest a noble-gas-like behavior in 1975, Eichler and co-workers in 2008 reported substantial interactions with a gold surface in atom-at-a-time experiments, suggesting a metallic character and a solid aggregate state. Herein, we explore the physicochemical properties of Cn by means of first-principles free-energy calculations, which confirm Pitzer's original hypothesis: With predicted melting and boiling points of 283±11 K and 340±10 K, Cn is indeed a volatile liquid and exhibits a density very similar to that of mercury. However, in stark contrast to mercury and the lighter Group 12 metals, we find bulk Cn to be bound by dispersion and to exhibit a large band gap of 6.4 eV, which is consistent with a noble-gas-like character. This non-group-conforming behavior is eventually traced back to strong scalar-relativistic effects, and in the non-relativistic limit, Cn appears as a common Group 12 metal.