Volume 97, Issue 10 - Special Issue: Quantum Science and Technology

In this perspective article, we discuss the link between minimum energy paths and activation parameters for reactions on complex potential energy surfaces involving many possible local minima, as are typically found for enzyme-substrate complexes. Such systems are frequently tackled with hybrid QM/MM methods in order to characterize reactivity. The link between local energy barriers along a minimum energy path, the so-called exponential average of such local energy barriers, and multiconformational transition state theory is discussed. Also, it is shown that in case of positive skewness of the distribution of barrier heights across sets of minimum energy paths, exponential averaging converges relatively quickly with the number of paths used.

This paper reflects the authors’ personal journey in applying quantum chemistry methods to understand one of the most important processes in nature: enzymatic catalysis. The integration of quantum mechanics with biomolecular simulations represents one of the most significant advances in computational enzymology over the past few decades. This approach has revolutionized our understanding of enzyme function and catalytic mechanisms, and has provided powerful tools for enzyme design and optimization. Combined quantum mechanics/molecular mechanics (QM/MM) methods have transformed theoretical studies of enzymatic reactions from qualitative descriptions to quantitative predictions, capable of guiding experimental work with unprecedented accuracy.

Quantum chemistry plays a fundamental role in unraveling the mechanisms by which biological systems sense and use light, driving functions such as light harvesting and energy conversion, and photoreception and signaling. In this Perspective, we first present the fundamental physical principles underlying the light-induced biological functions. We then focus on the key theoretical frameworks and multiscale modeling strategies based on quantum chemistry that enable a detailed, atomistic description of the processes initiating the biological response to light. Special emphasis is placed on three fundamental photophysical and photochemical processes (excitation energy transfer, photochemical reactions, and electron transfer) which form the core of photoactivation mechanisms in biological systems. By highlighting the advances and challenges associated with the quantum chemical modeling, we demonstrate its essential contribution to deepening our understanding of photoinduced biological function and point to future directions for methodological innovation.

In this mini-review I look into the physics underlying the theory of electronic structure of atoms and molecules. Quantum mechanics is needed to understand the structure of the periodic table. Special relativity is indispensable for a correct description of the chemistry of the heavy elements. With increased accuracy of quantum chemical calculations, it is natural to ask if chemistry needs more physics.

The proton motive force (PMF) across the inner mitochondrial membrane delivers approximately 0.2 eV of energy per proton, powering the F o F 1 -ATP synthase molecular motor. Here, we provide a detailed accounting of how this energy is utilized: Approximately 75–83 % is transduced into the chemical free energy of ATP synthesis, while the remaining 17–25 % is dissipated through internal friction, viscous drag, proton leakage, electroviscous effects, elastic deformations, and information-theoretic costs. Each dissipation channel is quantitatively evaluated, revealing that internal friction in the F 1 motor is the dominant loss mechanism. In this work, we did not account for the possible energy contribution due to the intrinsic electrostatic potential of the enzyme itself. In addition to this energy bookkeeping, we also examine the quantum mechanical constraints on the F o unit’s rotation. We find that, as can be expected, the energy spacing between quantized rotational states is several orders of magnitude smaller than thermal energies at physiological temperature, and that the tunneling probability through rotational barriers is practically zero. Furthermore, the biological rotation speed (∼100–650 revolutions per second (rps)) is between one and three orders of magnitude below the quantum limit implied by quantization of angular momentum of the c-ring (which would have been ca . 13,000 to 62,000 rps (depending on the size of the c-ring (17–8 subunits, respectively)) in the first rotational energy level of the c-ring). Nevertheless, experimental estimates of the rotation rates in isolated c-rings suggest rates in the vicinity of 43,000 rps, right within our theoretical quantum estimates. However, ATP synthase as a whole operates firmly within the classical regime, despite its nanoscale dimensions, which highlights its evolutionary optimization for robust and efficient energy conversion at the quantum–classical boundary. This is the result of the rotatory coupling between the F o and the much slower F 1 unit. ATP synthase’s purely classical behavior showcases a remarkable evolutionary optimization of one of life’s most essential rotary motor engineered so as to thrive far from the quantum limit, securing its function against the uncertainties of the quantum world. As Schrödinger stated in What is Life? (1944): “ The submicroscopic world is full of fluctuations. But in large aggregates of atoms, the law of large numbers ensures that these fluctuations become negligible ” a prediction directly confirmed here in the context of the rotational stability of the F o  unit.

As the atomic theory of matter became more established and accepted by the majority of the scientific community, chemists began to dream about the possibility of understanding the physical and chemical properties of different substances in terms of their microscopic composition. Identifying the constituent elements of a substance and their respective proportions allowed chemists to characterize it by a “ chemical formula .” It did not take long for chemists to realize that substances could have different properties even though they had the same “chemical formula” and to conclude that it was not enough to identify the constituent elements of a substance. It was essential to determine how the atoms of those elements were arranged spatially. Thus, chemists began to realize that the physical and chemical properties of a substance were closely related to its “ molecular structure .” The dream now seemed less impossible and more fascinating: how to arrange a certain group of atoms spatially in order to obtain a molecule with specific physical and chemical properties? However, in addition to knowing the spatial arrangement of the atoms in a molecule, it was necessary to know how these atoms are connected, that is, what the “ chemical structure ” of the molecule was and what was the origin of the “ chemical bond ”. The end of the century before last and the first quarter of the last century were the scenery of a fantastic revolution in our knowledge of the atomic structure of matter. Benefiting from this development, chemists began to have at their disposal powerful instruments for the microscopic analysis of the macroscopic properties of chemical systems: quantum mechanics as well as classical and quantum statistical mechanics. Finally, in the last 40 years, with the availability of personal computers and sophisticated multifunctional “software”, the dream began to become reality, and the development of what we now call Quantum Chemistry had an enormous impact on chemistry. However, in order to have a clear measure of the extent of this impact, it is essential to reconstruct the environment in which Chemistry found itself at the end of the 19th century, with all its difficulties and uncertainties. Only then will the reader be able to understand the incredible change brought about by the application of quantum mechanics to Chemistry. This reconstruction will be done by following the evolution of three concepts that form the pillars of modern chemistry: molecular structure, chemical structure and chemical bonding.

The International Year of Quantum Science and Technology (IYQ) publicizes and celebrates the centenary of quantum mechanics (QM). In this essay, I demonstrate that the assemblage of 17 Nobel Prizes in Chemistry awarded since 1960 that are relevant to theoretical and computational chemistry corroborate the literature conclusion that there was a 20th century Quantum Chemistry Revolution. A similar analysis is made for the Instrumental Revolution. These analyses are atypical for assessments in the history of science in that rely on Nobel Prize data. It is the congregation of Nobel Prizes that is a telling pointer to a revolution in science, not the examination of any one Nobel Prize.

A series of valence bond charge-transfer (VB- n CT) models ( n  ∈ [1, 4]) are enacted to assess the influence of molecular symmetry on the second-order nonlinear optical (NLO) response, the first hyperpolarizability, β . This is accomplished for different amplitudes of the ground-state charge-transfer character, quantified by the VB-CT mixing parameter, m CT . The analysis focuses on quantities accessible from the second harmonic scattering (SHS) phenomenon and measurements, β SHS and its depolarization ratio. Among the symmetries considered, D 3 h and C 3 v (VB-3CT, with large θ , i.e. for almost planar tripod-shape molecules) yield the largest β SHS values provided m CT is large, followed by T d (VB-4CT). These cases correspond to dominant octupolar responses and depolarization ratios close to 3/2. Other symmetries result in comparatively weaker responses, and a variety of depolarization ratios. Maximizing the charge-transfer dipole moment μ CT and minimizing the ratio ξ  =  T / t , are also identified as critical factors for enhancing the second-order NLO responses.

The dichotomy whether in Quantum Chemistry insight and numbers are to be placed on equal footing is situated in a historical perspective starting from Coulson’s famous quote “Give us insight, not numbers” to Neese’s recent adaptation “Give us insights and numbers”. In parallel, the problem of the chemical interpretation of complex computational results in terms of classical chemical concepts is addressed starting from Mulliken’s quote that “the more accurate the calculations became the more concepts tend to vanish in the air”. Conceptual Density Functional Theory is one of the techniques which avoids the latter issue by its density based approach which can be applied to computational results of any level of sophistication. The crucial role of response functions of the energy E with respect to perturbations in the number of electrons N and/or external potential v ( r ) is thereby highlighted. However a new confrontation “insight versus numbers” appears. When gradually refining the evaluation of these descriptors, in order to pass from a qualitative to a quantitative level, it turns out from our previous studies that either minor influences show up, or that also fundamental issues may arise hidden in the definition and the physical background of the response function. The derivative discontinuity of the E vs. N curve hereby plays a fundamental role. This issue is documented scrutinizing a series of recent studies on the analytical evaluation of the three second order response functions: the linear response function, the Fukui function and the chemical hardness. They are shown to behave in a fundamentally different way under refinement. For the linear response function, the pure second order v functional derivative of the energy, no fundamental problems arise: when passing to a full analytical evaluation an increasing level of complexity of the equations is observed leading however to a smooth convergence. In the Fukui function case, involving a mixed N and v energy derivative, the issue with the E = E ( N ) curve and N derivative can be circumvented by directly deriving the electronic chemical potential with respect to v using a Maxwell type relation. Finally for the chemical hardness involving the pure second order N derivative, a fundamental problem arises due to the derivative discontinuity when refining the venerable Parr-Pearson parabolic E = E ( N ) curve. It forces us to stick to its result identifying the hardness as the (band) gap. On the other hand the analytical expression yields a “condition” that Density Functional Approximations should obey. It is shown how its implementation leads to a straightforward estimate of their delocalization error, on the road for further improvement of DFAs. The inclusion of temperature may be a way out for further refining the chemical hardness and all other response functions involving second or higher order N derivatives, the simplest case being the dual descriptor. Overall this evolution reflects the basic characteristics Löwdin’s accuracy vs. refinement graph.

While limited coupled cluster theory is formally nonvariational, it is not broadly appreciated whether this is a major issue in practice . We carried out a detailed comparison with de facto full CI energies for a relatively large and diverse set of molecules near equilibrium geometries. Fully iterative limited CC methods such as CCSDT, CCSDTQ, and CCSDTQ5 do represent practical upper bounds to the FCI energy, as do CCSDT-3, CCSDTQ-3, and CCSD(cT). While quasiperturbative approaches such as CCSD(T) and especially CCSDT(Q) may significantly over-correlate molecules if there is significant static correlation, this is much less of an issue with Lambda approaches such as CCSDT(Q) Λ .

The calculation of accurate excitation energies and potential energy surfaces of photochemical reactions is a major challenge of current quantum chemistry, especially when one wants to keep the cost low enough to make detailed dynamical simulations affordable. Methods that might be affordable for large complex molecules can be tested against benchmark results for smaller molecules, where accurate benchmarks can be available. Here we report such testing for the excitation and dissociation of the methyl radical, yielding both singlet and triplet CH 2 . The emphasis is on multistate pair density functional theory using compressed-multistate (CMS-PDFT) and linearized (L-PDFT) formulations. We also consider the less affordable XMS-CASPT2 method with the same state-averaged-complete-active-space (SA-CASSCF) reference wave functions, which has seven active electrons in 10 active orbitals. The calculations use state averaging over seven states and a model space that spans the seven lowest SA-CASSCF eigenvectors. We study three on-top density functionals: tPBE, tPBE0, and MC23. Vertical excitation energies, adiabatic excitation energies, and dissociation energies, along with cuts through the potential surfaces along the dissociation coordinate, were computed with the (7, 10) active space. XMS-CASPT2 and L-PDFT with the MC23 functional show consistent and reliable performance for excitation energies, closely reproducing benchmark values, and producing smooth, physically reasonable potential energy surfaces essential for nonadiabatic dynamics simulations, but they are less accurate for bond energies. The L-PDFT calculations with the tPBE functional are more accurate for dissociation energies, but less accurate for excitation energies.

Understanding how dynamical behaviour and structural features influence protein function and stability is crucial. While extensive experimental data exist, studying real-time protein dynamics and enzyme catalysis remains challenging. Computational advances have been instrumental in overcoming experimental limitations, enabling molecular-level insights into biological macromolecules. The integration of experimental and computational approaches has proven to be very valuable in protein studies. Here, we demonstrate this synergy by investigating the conformational stability of the urethanase UMG-SP3, which exhibited a lower optimum temperature than expected and rapid loss of activity. Molecular dynamics simulations of the UMG-SP3-substrate complex at various temperatures revealed structural rearrangements outside the optimum temperature range (25–35 °C), leading to loss of the native protein fold and impaired substrate binding. Even at the optimum temperature for activity, the enzyme struggled to maintain a catalytically favourable orientation, aligning with experimental findings. Unfolding profiles were determined through differential scanning fluorimetry. Notably, the computational results provided a rationaly for the structural instability observed experimentally, emphasizing the strength of computational methods in elucidating protein behaviour at the atomic level. This study highlights the importance of combining experimental and computational approaches to deepen our understanding of protein stability and function.

Hydrogen bonded complexes and their lithium bonded analogues are investigated using DFT CAM-B3LYP quantum chemistry methods. The accuracy of DFT is verified by CCSD(T) calculations. Structures, binding energies, and Hirshfeld charge and spin densities for several charged closed shell and neutral open shell oxygen-containing complexes with O–H⋯O and O–Li⋯O bonds are compared. We also studied H-bonded and Li-bonded C–H⋯C and C–Li⋯C complexes in which the hydrogen and lithium bond donor and bond acceptor act through carbon-containing groups. Attention is paid to the bonding character in the model (H 3 C) 2 –CH–Li–CH–(CH 3 ) 2 (diisopropyllithium) doublet. Its binding energy with respect to the isopropyllithium and the isopropyl radical is −45 kJ/mol. The potential energy curve for the transfer of the lithium atom between the two carbon atoms shows double minima with a barrier of 11 kJ/mol. The Hirshfeld charge and spin density analysis shows that the charge transfer from the isopropyl radical to the isopropyllithium molecule occurs, and in combination with electrostatic interaction and (about 25 %) dispersion contribution are responsible for the formation of the Li-bonded (H 3 C) 2 –CH–Li–CH–(CH 3 ) 2 complex.

Metal–ligand bonding and noncovalent interactions (NCIs), such as hydrogen bonding or π – π interactions, play a crucial role in determining the structure, function, and selectivity of both biological and artificial metalloproteins. In this study, we employed a hybrid quantum mechanics/molecular mechanics (QM/MM) approach to investigate the ligation of water or cyanide in a mutated myoglobin system, in which the native heme scaffold was replaced with M-salophen or M-salen Schiff base complexes (M = Cr, Mn, Fe). Using our local vibrational mode analysis, particularly local vibrational mode force constants as intrinsic bond strength parameters, complemented with electron density and natural orbital analyses we explored the role of metal–ligand bonding and NCIs in different environments within the myoglobin pocket. Our analysis revealed that metal–ligand bonding, for both water and cyanide ligands, is strongest in the delta form of distal histidine and favors salophen prosthetic groups, as indicated by an overall increase in metal–ligand bond strength. Hydrogen bonding between the distal histidine and ligand also exhibited greater strength in the delta form; however, this effect was more pronounced with salen prosthetic groups. Additionally, the NCIs within the active pocket of the protein were found to be variable, highlighting the adaptability of local force constants. In summary, our data underscore the potential of computational methodologies in guiding the rational design of artificial metalloproteins for tailored applications, with local vibrational mode analysis serving as a powerful tool for bond strength assessment.