Rhorix: An interface between quantum chemical topology and the 3D graphics program blender

Chemical research is assisted by the creation of visual representations that map concepts (such as atoms and bonds) to 3D objects. These concepts are rooted in chemical theory that predates routine solution of the Schrödinger equation for systems of interesting size. The method of Quantum Chemical Topology (QCT) provides an alternative, parameter‐free means to understand chemical phenomena directly from quantum mechanical principles. Representation of the topological elements of QCT has lagged behind the best tools available. Here, we describe a general abstraction (and corresponding file format) that permits the definition of mappings between topological objects and their 3D representations. Possible mappings are discussed and a canonical example is suggested, which has been implemented as a Python “Add‐On” named Rhorix for the state‐of‐the‐art 3D modeling program Blender. This allows chemists to use modern drawing tools and artists to access QCT data in a familiar context. A number of examples are discussed. © 2017 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Steric repulsions, rotation barriers, and stereoelectronic effects: a real space perspective

Widely used chemical concepts like Pauli repulsion or hyperconjugation, and their role in determining rotation barriers or stereoelectronic effects, are analyzed from the real space perspective of the interacting quantum atoms approach (IQA). IQA emerges from the quantum theory of atoms in molecules (QTAIM), but is free from the equilibrium geometry constraint of the former. A framework with both electronically unrelaxed and relaxed wavefunctions is presented that leads to an approximate correspondence between the IQA concepts and those used in the EDA (energy decomposition analysis) or NBO (natural bond orbital) procedures. We show that no net force acts upon the electrons in an electronically relaxed system, so that any reasonable definition of Pauli repulsion must involve unrelaxed state functions. Using antisymmetrized fragments clarifies that Pauli repulsions are energetically connected to the IQA deformation energies, leaving footprints in the finally relaxed states. Similarly, EDA or NBO hyperconjugative stabilizations are found to be naturally related to the IQA electron delocalization patterns. Applications to the rotation barrier of ethane and other simple systems are presented, and the very often forgotten role of electrostatic contributions in determining preferred conformations is highlighted.     

Long range behavior of high-rank topological multipole moments

The construction of a high-rank multipolar force field (for peptides) is a complex task, leading to several intermediate questions in need of a clear answer. Here we focus on the convergence of the (electrostatic) multipolar expansion at medium and long range. Using molecular electron densities, quantum chemical topology (QCT) defines the atoms as finite volumes, each endowed with multipole moments. The terms in the multipole expansion are grouped according to powers of the internuclear distance, R(-L). Given two atom types at a given distance, we determine which rank (L) is necessary for the electrostatic energy to converge to the exact interaction energy within a certain error. With this information, the rank of the expansion for each interaction can be adapted to the required accuracy and the available computing power.     

A vector‐based representation of the chemical bond for the normal modes of benzene

We introduce a vector‐based interpretation of the chemical bond within the quantum theory of atoms in molecules (QTAIM), the bond‐path framework set B = {p, q, r}, to follow variations in the 3D morphology of all bonds for the four infrared active normal modes of benzene. The bond‐path framework set B comprises three unique paths p, q, and r where r is the familiar QTAIM bond concept of bond‐path (r) while the two new paths p and q are formulated from the least and most preferred directions of electron density accumulation, respectively. We find 3D distortions including bond stretching/compression, torsion, and curving. We introduce two fractional measures to quantify these variations away from linearity of the bond.     

Atomic Properties of Amino Acids: Computed Atom Types as a Guide for Future Force‐Field Design

The quantum chemical topology (QCT) is able to propose atom types by direct computation rather than by chemical intuition. In previous work, molecular electron densities of 20 amino acids and smaller derived molecules were partitioned into a set of 760 topological atoms. Each atom was characterised by seven atomic properties and subjected to cluster analysis element by element, that is, C, H, O, N, and S. From the respective dendrograms, 21 carbon atom types were distinguished, 7 hydrogen, 2 nitrogen, 6 oxygen, and 6 sulfur atom types. Herein, we contrast the QCT atom types with those of the assisted model building with energy refinement (AMBER) force field. We conclude that in spite of fair agreement between QCT and AMBER atom types, the latter are sometimes underdifferentiated and sometimes overdifferentiated. In summary, we suggest that QCT is a useful guide in designing new force fields or improving existing ones. The computational origin of QCT atom types makes their determination unbiased compared to atom type determination by chemical intuition and a priori assumptions. We provide a list of specific recommendations.     

Atomic properties of selected biomolecules: quantum topological atom types of hydrogen,oxygen, nitrogen and sulfur occurring in natural amino acids and their derivatives

Molecular electron densities are generated at B3LYP/6-311+G(2d,p)//HF/6-31G(d) level for 57 molecules, including one conformation of each naturally occurring amino acid and smaller derived molecules. The electron densities are partitioned into atomic fragments according to the approach of quantum chemical topology (QCT). A set of 547 unique topological atoms is obtained, containing 421 hydrogens, 63 oxygens, 57 nitrogens and 6 sulfurs. Each atom is described by seven properties: volume, kinetic energy, monopole, dipole, quadrupole, octupole and hexadecapole moment. Cluster analysis groups atoms into atom types based on their similarity expressed in the discrete 7D space of atomic properties. Using a separation criterion we distinguish seven hydrogen, six oxygen, two nitrogen and six sulfur atom types.     

The quantum divided basins: A new class of quantum subsystems

The rigorous theory of the quantum divided basins (QDB), the quantum subsystems emerging from the net zero‐flux equation, is considered in this article. This framework, the quantum theory of proper open subsystems, is derived from the extension of the quantum theory of atoms in molecules to encompass the new class of quantum subsystems. It is demonstrated that the regional hypervirial theorem and the associated regional observables as well as the subsystem variational procedure are all expressible for the QDB. The history of QDB is briefly reviewed and the bundles, which were introduced by other researchers, are offered as typical examples whereas new examples of QDB (not yet mentioned in literature) are also presented. Based on some model systems as well as the nitrogen molecule, the regional properties and the morphologies of typical QDB are scrutinized. It is also demonstrated that the QDB may be used to study the fine structure of the electron localization and delocalization. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Interatomic exchange-correlation interaction energy from a measure of quantum theory of atoms in molecules topological bonding: A diatomic case

The potential relations between the measure of topological interatomic bonding—integrals of electron density with respect to internuclear axis over the corresponding quantum theory of atoms in molecules (QTAIM)-defined interatomic surface (IAS)—and interatomic exchange-correlation contributions from the interacting quantum atoms approach are discussed. The quantum chemical computations of 38 equilibrium diatomic systems at different levels of theory (HF, MP2, MP4SDQ, and CCSD) are invoked to support abstract considerations. Parameters of excellent correlations between IAS integrals and interatomic exchange-correlation energy are found by the optimization. The performance of these trends depends on the accuracy of the electronic correlation treatment. The resulting trends are a unique feature of equilibrium states, whereas more complicated dependencies are explored for several systems at non-equilibrium conditions. The relations of established trends with other IAS-based estimations of strength of bonding interactions between topological atoms and issues explored for multiatomic systems are briefly discussed.     

The quantum theory of atoms in positronic molecules: The subsystem variational procedure

This contribution deals with the subsystem variational procedure within the context of the quantum theory of atoms in positronic molecules (QTAIPM). Before introducing the subsystem energy functional termed as joint subsystem energy functional, a novel notation and the combination strategy are disclosed in detail by restating the positronic subsystem hypervirial theorem. They are employed in proposing the proper subsystem energy functional, the validity of which is checked by various criteria. The zero flux surfaces of the joint density distribution are used to define the topological atoms in the positronic molecules, and they are incorporated into the subsystem variational procedure as proper real space boundary conditions. The variational procedure finally yields the flux of the joint current property density that also appears in the positronic subsystem hypervirial theorem. At every stage, the corresponding equations for the purely electronic systems within the context of the quantum theory of atoms in molecules (QTAIM) are presented to clearly reveal the analogy between these two formalisms and to emphasize the importance of combining the property density distributions in the QTAIPM. The presented material demonstrates the internal consistency of the whole framework and discloses the fact that the QTAIM must be regarded as a variant of the QTAIPM. Furthermore, this formalism promises an extended QTAIM, which is hoped to resolve the issue of molecular structure beyond the clamp nuclei approximation. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Revisiting the foundations of the quantum theory of atoms in molecules: Toward a rigorous definition of topological atoms

An explicit classification of consistent variational constraints within the context of the “quantum theory of proper open subsystems” as well as the “quantum theory of atoms in molecules” (QTAIM) it presented. It is demonstrated that the general variational procedure is not sensitive enough to discriminate between different mathematically consistent variational conditions. The uniqueness of the regional kinetic energy is employed to derive the net zero‐flux condition and the regions satisfying this condition are named as quantum divided basins. A modified form of the local zero‐flux is proposed in order to define topological atoms within the context of the orthodox QTAIM. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Bond paths as privileged exchange channels

Evidence that the bond paths of the quantum theory of atoms-in-molecules (QTAIM) signal preferred quantum-mechanical exchange channels is presented. We show how bond paths between an atom A and the atoms B in its environment appear to be determined by competition among the A-B exchange-correlation energies that always contribute to stabilize the A-B interactions. These pairwise additive stabilizations depend neither on the attractive or repulsive nature of the classical electrostatic interaction between the atoms' charge densities, nor on the change in the self energies of the atoms involved. These other terms may well cause an overall molecular-energy increase in spite of a possibly large A-B exchange-correlation stabilization. After our proposal, bond paths, both at and out of equilibrium geometries, are endowed with a specific energetic meaning that should contribute to reconcile the orthodox QTAIM interpretation with other widely accepted views, and to settle recent controversies questioning the meaning of hydrogen-hydrogen bonding and the nature of the so-called "steric interactions", the role of bond paths in endohedral complexes, and the generality of the results provided by the QTAIM. Implications for the nature of more general closed-shell interactions are also briefly discussed.     

The effects of higher orders of perturbation theory on the correlation energy of atoms and bonds in molecules

We examine, for the first time, the effects of higher orders of Møller–Plesset perturbation theory on the individual atoms within a molecule and the bonds between them, via the topological energy partitioning method of interacting quantum atoms. In real terms (i.e., not by absolute value) MP3 decreases the correlation energy of a bond, and MP4SDQ also decreases the energy of the atoms at either end of the bond. In addition, we investigated long‐range through‐space dispersive effects on a H₂ oligomer. Overall, MP3 is the largest correction to the correlation energy, and most of that energy is allocated to chemical bonds, reducing their values in actual terms. The MP4SDQ bond correlation correction, despite being relatively small, tends to have two effects: (i) for small or negative correlation energies MP4SDQ tends to decrease the bond correlation values even more, and (ii) for large (positive) bond correlation energies MP4SDQ tends to restore the bond correlation energies from the MP3 back toward the MP2 values. Furthermore, each individual part of a molecule or complex (atom or bond) has a specific convergence pattern for the MPn series: through‐space interactions converge at MP2 but bonds converge at MP3 level. The atomic correlation energy appears to head toward convergence at the MP4 level.     

An efficient method for computing the QTAIM topology of a scalar field: The electron density case

An efficient method for computing the quantum theory of atoms in molecules (QTAIM) topology of the electron density (or other scalar field) is presented. A modified Newton–Raphson algorithm was implemented for finding the critical points (CP) of the electron density. Bond paths were constructed with the second‐order Runge–Kutta method. Vectorization of the present algorithm makes it to scale linearly with the system size. The parallel efficiency decreases with the number of processors (from 70% to 50%) with an average of 54%. The accuracy and performance of the method are demonstrated by computing the QTAIM topology of the electron density of a series of representative molecules. Our results show that our algorithm might allow to apply QTAIM analysis to large systems (carbon nanotubes, polymers, fullerenes) considered unreachable until now. © 2012 Wiley Periodicals, Inc.     

QTAIM charge-charge flux-dipole flux interpretation of electronegativity and potential models of the fluorochloromethane mean dipole moment derivatives

Infrared fundamental vibrational intensities and quantum theory atoms in molecules (QTAIM) charge-charge flux-dipole flux (CCFDF) contributions to the polar tensors of the fluorochloromethanes have been calculated at the QCISD/cc-pVTZ level. A root-mean-square error of 20.0 km mol(-1) has been found compared to an experimental error estimate of 14.4 and 21.1 km mol(-1) for MP2/6-311++G(3d,3p) results. The errors in the QCISD polar tensor elements and mean dipole moment derivatives are 0.059 e when compared with the experimental values. Both theoretical levels provide results showing that the dynamical charge and dipole fluxes provide significant contributions to the mean dipole moment derivatives and tend to be of opposite signs canceling one another. Although the experimental mean dipole moment derivative values suggest that all the fluorochloromethane molecules have electronic structures consistent with a simple electronegativity model with transferable atomic charges for their terminal atoms, the QTAIM/CCFDF models confirm this only for the fluoromethanes. Whereas the fluorine atom does not suffer a saturation effect in its capacity to drain electronic charge from carbon atoms that are attached to other fluorine and chlorine atoms, the zero flux electronic charge of the chlorine atom depends on the number and kind of the other substituent atoms. Both the QTAIM carbon charges (r = 0.990) and mean dipole moment derivatives (r = 0.996) are found to obey Siegbahn's potential model for carbon 1s electron ionization energies at the QCISD/cc-pVTZ level. The latter is a consequence of the carbon mean derivatives obeying the electronegativity model and not necessarily to their similarities with atomic charges. Atomic dipole contributions to the neighboring atom electrostatic potentials of the fluorochloromethanes are found to be of comparable size to the atomic charge contributions and increase the accuracy of Siegbahn's model for the QTAIM charge model results. Substitution effects of the hydrogen, fluorine, and chlorine atoms on the charge and dipole flux QTAIM contributions are found to be additive for the mean dipole derivatives of the fluorochloromethanes.     

Toward a Consistent Interpretation of the QTAIM: Tortuous Link between Chemical Bonds,Interactions, and Bond/Line Paths

Currently, bonding analysis of molecules based on the Quantum Theory of Atoms in Molecules (QTAIM) is popular; however, “misinterpretations” of the QTAIM analysis are also very frequent. In this contribution the chemical relevance of the bond path as one of the key topological entities emerging from the QTAIM’s topological analysis of the one‐electron density is reconsidered. The role of nuclear vibrations on the topological analysis is investigated demonstrating that the bond paths are not indicators of chemical bonds. Also, it is argued that the detection of the bond paths is not necessary for the “interaction” to be present between two atoms in a molecule. The conceptual disentanglement of chemical bonds/interactions from the bonds paths, which are alternatively termed “line paths” in this contribution, dismisses many superficial inconsistencies. Such inconsistencies emerge from the presence/absence of the line paths in places of a molecule in which chemical intuition or alternative bonding analysis does not support the presence/absence of a chemical bond. Moreover, computational QTAIM studies have been performed on some “problematic” molecules, which were considered previously by other authors, and the role of nuclear vibrations on presence/absence of the line paths is studied demonstrating that a bonding pattern consistent with other theoretical schemes appears after a careful QTAIM analysis and a new “interpretation” of data is performed.     

Chemical bonding in view of electron charge density and kinetic energy density descriptors

Stalke's dilemma, stating that different chemical interpretations are obtained when one and the same density is interpreted either by means of natural bond orbital (NBO) and subsequent natural resonance theory (NRT) application or by the quantum theory of atoms in molecules (QTAIM), is reinvestigated. It is shown that within the framework of QTAIM, the question as to whether for a given molecule two atoms are bonded or not is only meaningful in the context of a well‐defined reference geometry. The localized‐orbital‐locator (LOL) is applied to map out patterns in covalent bonding interaction, and produces results that are consistent for a variety of reference geometries. Furthermore, LOL interpretations are in accord with NBO/NRT, and assist in an interpretation in terms of covalent bonding. © 2008 Wiley Periodicals, Inc.J Comput Chem, 2009.     

Relativistic (SR‐ZORA) quantum theory of atoms in molecules properties

The Quantum Theory of Atoms in Molecules (QTAIM) is used to elucidate the effects of relativity on chemical systems. To do this, molecules are studied using density‐functional theory at both the nonrelativistic level and using the scalar relativistic zeroth‐order regular approximation. Relativistic effects on the QTAIM properties and topology of the electron density can be significant for chemical systems with heavy atoms. It is important, therefore, to use the appropriate relativistic treatment of QTAIM (Anderson and Ayers, J. Phys. Chem. 2009, 115, 13001) when treating systems with heavy atoms. © 2016 Wiley Periodicals, Inc.