# Research

## Chérif F. Matta

This research programme is primarily in theoretical and quantum chemistry and can be classified into two broad categories: Research with a biological application and drug-design bent, and research in the continuum between physical chemistry and chemical physics.

My PhD training under Prof. Richard F. W. Bader (1931-2012) (McMaster University) was in characterizing the topology and topography of the electron density and a theory of chemistry based on observables. This theory, developed by Bader, is now known as the Quantum Theory of Atoms in Molecules (and Crystals), or QTAIM.1-3 The theory is used by numerous groups to gain insight into the nature of chemical bonding and reactivity.

An early finding, first reported in my PhD thesis,4 was the characterization of a new type of chemical bonding now known as hydrogen-hydrogen (H-H) bonding.5-7 The paper first reporting this finding has received over 350 citations to date including one in the lead textbook on unusual weak bonding involving hydrogen atoms by V. Bakhmutov.8

The discovery of the locally-stabilizing nature of the H…H bonding sparked an intense initial wave of criticism, controversy, and debates since it contradicted textbook arguments of steric repulsion to explain, for example, the non-planarity of biphenyl. The stabilizing nature of the H-H bonding has been verified by both experiment and theory over time. The instability of planar biphenyl is not the result of steric repulsion between ortho-hydrogens but rather of the destabilization of the two ipso-carbons. The ipso-carbons are less stable in planar biphenyl compared to the twisted global energy minimum due to the elongation of the C–C bond linking the two aromatic rings to accommodate the four coplanar ortho-hydrogen. The principal effect of this elongation is a reduction in the attractive contributions from the electrons of one ipso-carbon to the nucleus of the other, which overwhelms other changes leading to the instability of planar biphenyl with respect to the twisted global minimum structure. In fact, even in planar biphenyl the molecule gains local (but not global) stabilization of a few kcal/mol from the short H…H contact (the energy of the o-hydrogen atoms is lower), a stabilization overwhelmed principally by the destabilization of the ipso-carbons, and the net result is this small observed rise in the energy of planar versus twisted biphenyl.5Despite of the mounting evidence supporting the concept of a stabilizing H-H bonding interaction, to this day, once in a while, the work is still occasionally being criticized on the basis of classical models, models that are inherently incapable, by construction, of describing this non-classical interaction. Recent confirmations include, for example, the ability of prediction of boiling points of hydrocarbons as a function of the number of H…H interactions9 and the direct correlation of dimer stabilization energies with the number of these contacts.10 (Fig. 1).

More recently, in collaboration with Prof. Ignacy Cukrowski (University of Pretoria), the role of weak H…H bonding interactions in the stability of organic Zn2+-complexes has been elucidated in detail,11 as was the question of the protonation sequence of linear aliphatic polyamines.12

QTAIM affords a partitioning of the molecular space (up to an outer isosurface of the electron density) into non-overlapping atomic basins over which any molecular property expressible as a sum of atomic contributions can be defined and calculated by numerical integration. In a series of papers we have studied the atomic contributions to the bond dissociation energies (BDEs) of alkanes, alkenes, and magnesium-complexes of phosphates.13,14 This work led to the idea of locating regions in energy-rich biological molecules such as adenosine 5′-triphosphate (ATP) (often described as the “energy currency of all living cells”) that are responsible for their energy transfer function in the living system.15,16

Another line of research in my group is the effects of intense electric fields on the reactivity, electronic structure, and spectra of molecules. This work started with a collaboration with Prof. André D. Bandrauk (Université de Sherbrooke) in 2004. Our calculations demonstrated that the intensity and phase of a plane polarized, phase-stabilized, beam of two-color laser pulses can be selected such as to invert the shape of the potential energy surface (PES) for co-linear collisions of halogens17,18 or metal ions19 [X• + CH4 -> HX + •CH3, (X = F, Cl, Li)] with methane. As a consequence of the action of this laser radiation, a transition state on the PES can be converted to a radiation-stabilized bound state. The effective potential of the field-molecule system as a function of the reaction coordinate (s) can be approximated:18,19

$V_{{\rm{eff}}{\rm{.}}} (s) = V_{{\rm{field-free}}} (s)-\mu _z (s)E_0 \cos (\phi )-{1 \over 2}\alpha _{zz} (s)E_{\rm{0}}^{\rm{2}} \cos ^2 (\phi ), (1) where $\mu _z$ and $\alpha _{zz}$ are the components of the dipole moment and of the polarizability tensor element parallel to the C3 axis, E0 is the amplitude and $\phi$ the phase of the electric component of the electromagnetic field. The parallel (z) components of the dipole moment and zz polarizability tensor element of the reacting system is found to exhibit remarkably sharp peaks near the transition state (TS) region. Therefore, and through Eq. (1), one can anticipate a maximal effect of the field on the reaction near the TS region where the field-molecule interaction is maximal. The phase and the intensity of the external field can be adjusted to eliminate or, as mentioned above, even invert the potential energy barrier converting a TS into a bound state. (See Fig. 2 for an example). My former student Shahin Sowlati-Hashjin and I, in collaboration with Prof. Bandrauk, examined atomically-decomposed dipole moment surfaces to pinpoint those atoms responsible of the sharp dipole moment peak near the transition state.17 The calculations show that it the halogen is the principal contributor and hence it is the nature of this atom that most affect the controllability of this reacting system by an external laser field.17 In 2011, my student Alya A. Arabi and I developed a protocol to predict the change in the rate of double proton transfer reactions under an intense external electric field.20 Our protocol has been used by several research groups that examine (a) the kinetics of Löwdin mechanism of spontaneous and induced mutation in electric fields of intensities encountered in the microenvironment of DNA (~108 – 109 V/m),21-23 (b) the double proton transfer in non-natural base pairs,24 and (c) the kinetics of enzyme-catalyzed reactions under electric field in active sites.25 In 2013 we have developed a model of the vibrational Stark effect in diatomics where it is predicted from the properties of the field-free system and only the characteristics of the field without the explicit solution of the Hamiltonian that contains the term for the field.26 The vibrational frequencies and bond lengths obtained from the direct brute force calculations (including a field term in the Hamiltonian) agree well with the results obtained from our simple model. The importance of such an undertaking has been underscored in L. Piela’s definitive text Ideas in Quantum Chemistry:27 The problem is how to compute the molecular properties in the electric field from the properties of the isolated molecule and the characteristics of the applied field”, which is exactly what we have accomplished for diatomic molecules in this work, work that will be extended to larger more complex molecules in the future. The model we propose, following the lead (but differing in significant ways from the work) of Delley,28 is a generalization of the Morse potential to include field effects. Thus, we define a field-perturbed Morse potential:26$V(r) = D_{\rm{e}} \left( {1-\zeta } \right)^2 \mp \mu _0{} (r)E-{1 \over 2}\alpha _{//0} (r)E^2 $, (2) where E is the electric field strength, μ0 and a//0 are the field-free molecular dipole moment and the longitudinal element of the polarizability tensor, and the remaining symbols in the first term have their usual meaning. This field-perturbed Morse potential leads to the following equations that predict bond lengths and vibrational frequencies in a field, respectively, from the field-free values:26$R_E = R_0-{1 \over a}\ln {1 \over 2}\left( {1 + \sqrt {1 \mp {2 \over {aD_e }}{{\partial \mu _0^{} } \over {\partial r}}E-{1 \over {aD_e }}{{\partial \alpha _{//0} } \over {\partial r}}E^2 } } \right)$, (3) $$\nu _E = \nu _0 \sqrt {{1 \over 2}\left[ {1 + 4\left( { \mp {2 \over {aD_e }}{{\partial \mu _0^{} } \over {\partial r}}E – {1 \over {aD_e }}{{\partial \alpha _{//0} } \over {\partial r}}E^2 } \right)} \right]^{1/2} \left\{ {1 + \left[ {1 + 4\left( { \mp {2 \over {aD_e }}{{\partial \mu _0^{} } \over {\partial r}}E – {1 \over {aD_e }}{{\partial \alpha _{//0} } \over {\partial r}}E^2 } \right)} \right]^{1/2} \right\}}$$, (4) where R0 and ν0 are the field-free equilibrium bond length and harmonic frequency, RE and νE are the corresponding values under an external electric field E, and r is the inter-nuclear separation. The negative or positive signs of the dipole term are for a parallel or an antiparallel orientation of the permanent molecular dipole moment (if it exists) with respect to the external field, respectively. For molecules without a permanent dipole, the dipolar term vanishes and the response to the field is totally determined by the second (polarizability) term. These equations reproduce accurately the results of direct calculations with explicit treatment of the field-effects.26 We have recently reported a generalization of a linear scaling fragmentation scheme called the Kernel Energy Method (KEM),29-31 known to yield extremely accurate approximations of total energies, binding energies, and energies of interaction of large biomolecules at a small fraction of the computational cost. However, so far the KEM has never been used for predicting other than energies and their changes. In our recent work,32 KEM is used to predict (very accurately) response properties induced by external fields in a challenging case of a delocalized system such as a hydrogen-terminated zigzag graphene nanoribbon. The studied properties include the change in the energy (ΔE) and the change in the dipole moment components (Δμi, i = x,y,z) of a finite hydrogen terminated armchair graphene nanoribbon.32 The errors in predicted values are practically zero. By reproducing field-responses so accurately and fast, this study opens the door for non-periodic quantum mechanical (cluster) calculations on large systems of nanotechnological interest (Fig. 3). Since the dipole moment is the second (and a lead) term in an infinite sum of multipolar terms, and given that several other properties are well reproduced by KEM, we are currently investigating the reconstruction of the total electron density scalar field itself from fragments according to an equation similar in form to the constitutive KEM equation:32,33$\rho _{{\rm{KEM}}} ({\bf{r}}) = \sum\limits_{i = 1}^n {\sum\limits_{i < j}^n {\rho _{ij} ({\bf{r}})-(n-2)} } \sum\limits_{k = 1}^n {\rho _k } ({\bf{r}})$, (5) where $\rho$KEM is the approximate KEM electron density of the full molecule, $\rho$ij and $\rho$k are the electron densities of the ijth double kernel and of the kth single kernel, respectively, and r is a position vector. With an approximate KEM density in hand, the molecular electrostatic potential can readily be reconstructed using the known charges Zi (in atomic units) and positions (Ri) of the nuclei:34 $$V_{{\rm{KEM}}} ({\bf{r}}) = \sum\limits_{\scriptstyle \,\,\,i = 1 \hfill \atop \scriptstyle {\bf{R}}_i \ne {\bf{r}} \hfill} ^{\rm{M}} {{{Z_i } \over {\left| {{\bf{R}}_i-{\bf{r}}} \right|}}}-\int {{{\rho _{{\rm{KEM}}} ({\bf{r’}})} \over {\left| {{\bf{r’}} – {\bf{r}}} \right|}}} d{\bf{r}}’$$ , (6) where the ith term in the first sum must be eliminated when Ri = r. Alternatively, the VKEM(r) can be reconstructed from the MESP calculated for the kernel fragments on a point-by-point basis using an equation of the same form of Eq. 5. Further, given an approximation to the density [Eq. (5)], all one-electron properties represented by multiplicative operators can be calculated in addition to several derived topological properties of the density such as those obtained from Bader’s QTAIM. This work is currently being undertaken in collaboration with the CUNY research group of Prof. Lou Massa and the NRL research group of Dr. Lulu Huang. With Prof. Massa and Dr. Huang, we are presently extending KEM to calculate energy derivatives and hence opening the possibility of performing geometry optimizations (whether in presence or absence of external fields). This will enable the calculation of vibrational frequencies of large molecules, not at the molecular mechanics levels but rather at high ab initio levels of theory, at a fraction of the cost. Prof. Massa, Prof. Ada Yonath (Weizmann Institute of Science), Dr. Jerome Karle (1919-2013) (NRL), and I described the sequence of steps that lead to the formation of the peptide bond in the active site of the ribosome. The mechanism of peptide bond formation inside the ribosome remained a subject of intense debate (and mystery) as a C&EN article in 2007 explains.35 Some light have been shed on the mechanism of this step by our joint study whereby we examined the evolution of the electron density and the associated bond paths-dependent molecular graph as a function of the reaction coordinate in a model of the ribosome active site. The principal result is a simpler direct mechanism proposed as a replacement of the more elaborate shuttle mechanism of the peptide bond formation. The preliminary results of this work appeared in the form of a chapter in the book Quantum Biochemistry.36 Earlier, with Prof. Russell J. Boyd (Dalhousie University), we have calculated and analyzed electron density distributions of a number of other biological molecules. Our paper on the chemical bonding in $\pi$-stacked DNA base pair37 has received over 130 citations. We have also examined the non-nuclear electron density attractor associated with the solvated electron that results from the photoionization of water under radiation and which can initiate damage to biological macromolecules.38 Electron density descriptors were also shown to be powerful predictors of a range of physicochemical and biological properties of molecules with an accuracy often at least comparable to traditional descriptors such as Hammett $\sigma$-substituent constants15,39-48 Another more specific contribution to medicinal chemistry is our resolution of the puzzle of the existence of structurally and chemically very different bioisosteric moieties, tetrazole and carboxylate anions, interchangeable in a drug with little effect on its pharmacology? We found that these bioisosteres exhibit an almost identical topology of their electrostatic potential irrespective of the capping group.42 (Fig. 4). A curious side finding reported in the same paper42 is that, despite of the totally different chemical nature of these two bioisosteres, the average electron densities of these bioisosteric groups are almost identical. A novel integration of chemical graph theory and QTAIM has just been proposed,39 and followed-up in an article with Prof. Paul W. Ayers (McMaster University) and our joint M.Sc. student Ismat Sumar (Saint Mary’s University) where these ideas are used to predict the pKa‘s and UV $\lambda$max of substituted benzoic acids.49 In Ref. 39, it is demonstrated that QTAIM localization and delocalization indices (LIs and DIs), cast in matrix format, provide a molecular fingerprinting measure of molecular similarity useful in the empirical modeling of physicochemical properties in ground and excited states. The newly defined matrix, termed the localization-delocalization matrix (LDM, or ζ-matrix) is a compact and efficient numerical representation of the electronic structure introduced and used for the first time in Ref. 39. The ζ-matrix lists the complete set of localization indices, {Λ(Ωi)}, along the diagonal and the complete set of half of the delocalization indices, {δ(Ωij)/2 = δ(Ωji)/2, ij}, as the off-diagonal element (where the LIs and DIs are defined in Refs. 1-3. LIs and DIs provide a bookkeeping of the electron population of the molecule (N).50 The LI counts the number of electrons localized within an atomic basin while the DI counts the number of electrons shared between two basins. One defines the LDM in terms of LIs and DIs/2:39 (7) where the sum of any column or row yields the corresponding atomic population, as:$N(\Omega _i ) = \Lambda (\Omega _i ) + {1 \over 2}\sum\limits_{j \ne i}^n {{\rm{\delta }}(\Omega _i ,\Omega _j )} $. (8) The total molecular electron population is then given by the sum of the column or row sums and can be expressed as the sum of two sub-populations: $$N = \sum\limits_{i = 1}^n {N(\Omega _i )} = \sum\limits_{i = 1}^n {\Lambda (\Omega _i )} + {1 \over 2}\sum\limits_{i = 1}^n {\sum\limits_{j \ne i}^n {{\rm{\delta }}(\Omega _i ,\Omega _j )} } = N_{{\rm{loc}}} + N_{{\rm{deloc}}}$$ , (9) where$N_{{\rm{loc}}} \equiv \sum\limits_{i = 1}^n {\Lambda (\Omega _i )} $, (10) and$N_{{\rm{deloc}}} \equiv {1 \over 2}\sum\limits_{i = 1}^n {\sum\limits_{j \ne i}^n {{\rm{\delta }}(\Omega _i ,\Omega _j )} } = N-tr({\bf{\zeta }}) = N-N_{{\rm{loc}}} \$
.

(11)

As can be seen from Eqs. (7) – (11), the LDM contains information on electron localization and delocalization, atomic populations (and hence atomic charges since q(Ω) = ZΩN(Ω), ZΩ being the atomic number), and implicitly on inter-atomic distances since the DI generally decays with distance.11

The predictive value of the LDI matrices, when mathematically treated with the tools of Chemical Graph Theory (CGT),51-53 is demonstrated through the modeling of the pKa‘s of a series of 14 para-substituted benzoic acids (r2 = 0.986) and the UV $\lambda$max‘s of a subset of 8 of those compounds (r2 = 0.972).

Several solutions have been proposed to circumvent the dimensional inequality and atomic labeling ambiguity inherent in all matrix representations of a molecule.39 An appealing solution inspired by the work of Pye and Poirier54,55 led us to develop the concept of “super-atoms” defined by “pruning” the branches of the molecular graph of substituents bringing to equality the matrix representatives of all compared benzoic acids.49

The disagreement of one data value obtained from the CRC Handbook of Chemistry and Physics56 for p-dimethylaminobenzoic acid (p-DMABA) led us to consult the original literature and we to find that the Handbook has erroneously entered its pKa as 6.03, a value that has propagated in many other references and web pages. The primary literature (e.g. Ref. 57) give a value of 5.03 for p-DMABA coinciding to the second decimal with the ζ-matrix prediction.

The DIs have been also shown to provide a measure of aromaticity,58 work in collaboration with Prof. Jesús Hernández-Trujillo (UNAM). We proposed a “local aromaticity index”, q, as a measure of the degree of alternation of the delocalization index d between atoms within a ring of n atoms:58

$$\theta = 1-{c \over n}\sqrt {\sum\limits_{i = 1}^n {\left( {\delta _0-\delta _i } \right)^2 } }$$,

(12)

where c is a constant such that θ = 0 for cyclohexane, and δ0 is a reference delocalization index. For example, for benzenoid polycyclic aromatic hydrocarbons (PAHs), δ0 is the total number of delocalized electrons of a carbon atom in benzene with the remaining five carbons while δi is the total electron delocalization of a carbon atom in the given 6-membered ring with the other carbons in the same ring.

Thermochemical properties are important in predicting reaction kinetics and energetics. The determination of reliable heats of formations of large molecules remains a difficult problem for both quantum chemistry and experiment. Our θ-aromaticity index has crossed from the domain of theory to the laboratory where it has been used to predict heats of formations (ΔHf°298K) of a series of PAHs by Sivaramakrishnan et al.59 These authors use the θ-index to define “ring conserved isodesmic reactions (RCIR)” in PAHs as reactions in which the total aromaticity is conserved. This approach predicts heats of formations of large PAH molecules with an accuracy that significantly exceeds the traditional bond-separation and bond-centred group additivity schemes.59

The paper where the θ-aromaticity index is first proposed58 has been cited in excess of 120 times including prominent citations such as in Ref. 60, and in five reviews61-65 in a special issue of Chemical Reviews (Issue of October 2005, Vol. 105, Number 10, pp. 3433-3947) dedicated to aromaticity, edited by Prof. Paul von Ragué Schleyer. A review by Prof. Schleyer lists our index among “Some Important Aromaticity Criteria and Key Developments” in a chronological tabulation that goes back to the nineteenth century (Table 2 of Ref. 64).

Earlier, Prof. Nick Werstiuk (McMaster University) and I elucidated the significance and the numerical behavior of the QTAIM localization and delocalization indices obtained from several correlated levels of theory including post-Hartree-Fock as well as DFT,66 using AIMDELOC, a program that I wrote that calculated the delocalization indices from wavefunction files.67

Ab initio methods scale rapidly with the size of the system, and as a result they often cannot be applied to large biological molecules. As a graduate student at McMaster University, I proposed a fragmentation solution to this problem based on QTAIM (developed by my supervisor Prof. Bader). This method is termed “buffered fragments” since the properties of the large system are obtained from calculations on small fragments embedded in an appropriate electronic environment which are extracted from this environment and combined to reconstruct the target molecule. The method has been illustrated using morphine analogues (opioids) as the test.68 The work has been highlighted in Chemical and Engineering News (C&EN) as part of the cover story “Computational Chemistry, Scenes from the Cutting Edge” in a section entitled with the question “Opioid Breakthrough?69

A definitive and positive answer to the 2001 question of the C&EN reporter came later, in 2005, from high-resolution X-ray diffraction experiments coupled with non-spherical multipolar refinement.70 The buffered fragments method68 has been closely replicated experimentally by the X-ray crystallography group of Prof. Peter Luger (Free University of Berlin).70 These authors show how to reconstruct an approximation to the experimentally-derived electron density of morphine from experimental buffered fragments,70 in a procedure that parallel the one proposed on the basis of our theoretical calculations.68

The goal of the group of Luger extends beyond the particular chosen system (opioids) to a broader “proof of principle” that the buffer fragments methods can be used to obtain the electron density distributions of large molecules. The motivation is to reconstruct good approximations of the electron densities of large molecules that are (often) difficult to crystallize, especially when they contain floppy moieties, from crystallographic densities of smaller more readily crystallizable molecules.

Recently, my Honors thesis student Matthew Timm and I have undertaken a study of the evolution of the electron density immediately after a nuclear radioactive transmutation.71 We developed what appears to be the first mathematical model in the literature that predicts the primary retention yield, that is, the fraction of molecules that survive the nuclear recoil following a β-decay event. We have shown that the primary retention yield can be obtained by the fraction of nuclear transmutation events that will result in an angle equal to or less than a critical value θcritical between the ejected β-electron and the accompanying antineutrino as a function of a maximal allowed recoil energy Erecoil (set for example as the bond dissociation energy (BDE) of the weakest bond in the compound):71

$$\theta _{critical} = \arccos \left\{ {{{\left( {{{M \times E_{{\rm{recoil}}} } \over {5.36 \times 10^2 }}} \right)-E_e^2-1.02E_e-\left[ {\max (E_e )-E_e } \right]^2 } \over {2\left[ {\max (E_e )-E_e } \right]\sqrt {E_e^2 + 1.02E_e } }}} \right\}$$
,

(13)

where the imposed upper bounds on the recoil energy, Erecoil, is converted into maximum allowable angular deviations from π between the directions of the ejected β-electron and the accompanying antineutrino that will result in an Erecoil that does not exceed the imposed upper bound, and where the max(Ee) is the highest energy of the ejected β-electrons. The angular bounds can be converted into fractions of ejected β-electrons that would leave the residual nucleus with a recoil energy less (or, at most, equal to) the chosen upper bound (See example in Fig. 5).

In collaboration with Dr. Seyedabdolreza Sadjadi (University of Hong Kong) and Prof. Ian Hamilton (Wilfrid Laurier University) we have undertaken a study of metal clusters whereby the cores of heavy metal atoms, missing due to the use of effective core potentials (ECP) in electronic structure calculations, have been augmented from pre-calculated relativistic cores to generate approximate continuous electron densities with relativistic core densities72 without the need for an explicit relativistic treatment possible only for the smallest systems.73 These continuous densities do not exhibit the discontinuous behaviors at the cores boundaries and hence are amenable to a complete QTAIM analysis.

This technique of relativistic core augmentation in heavy-metal quantum chemical calculations, invented by Dr. Michael Frisch and Dr. Todd A. Keith,74 was also used in calculations that demonstrate the retention of the electron distribution in the catalytic site of a zwitterionic form of a rhodium catalyst, combining both the catalytic power and water-solubility in one compound.75 This work has been undertaken jointly with Prof. Claude Lecomte and Dr. El-Eulmi Bendeif (Université de Lorraine), Prof. Mark Stradiotto (Dalhousie University), and Dr. Pierre Fertey (Synchrotron Soleil).

In collaboration with Prof. Lou Massa, Dr. Lulu Huang, and Prof. Ivan Bernal (University of Huston) we proposed a new series of structures of solvated hydronium ions.76 This work is still progressing at the time of writing.

My former PhD student, Hugo Bohórquez (co-supervised with Prof. Russell J. Boyd (Dalhousie)), has contributed significantly to “local quantum chemistry” characterization of weak chemical bonding interactions and applied it to rationalized the structural stability of a number of biological macromolecules.77,78

In 2010, the dependence of molecular and atomic properties on the level of ab initio quantum chemical theory have been analyzed to establish the sensitivity of QTAIM results used in QSAR-type studies on underlying levels of theory.79 In collaboration with Prof. Paul W. Ayers (McMaster University) and my former undergraduate student, Martin Sichinga, we elucidated the effect of the level of theory on information theoretic properties of atoms in molecules and found that they (statistically) correlate strongly with dynamic electron correlation energy.80

In 2007, with my then summer student Alya A. Arabi, and in collaboration with Dr. Todd A. Keith (SemiChem, Inc.), we have explored the physical meaning of QTAIM atomic energies derived from density functional calculations (DFT), that is from Kohn-Sham orbitals.14 The kinetic energy density derived from KS-orbitals is missing the correlation energy but, fortunately, this missing energy is approximately proportional to the dominant non-interacting kinetic energy leading to trends that are generally physically sound when these energies are obtained and interpreted as a black-box. This work gave the missing physical meaning to the numerous atomic energies reported in the literature using DFT as a black-box using the same procedures used in deriving atomic energies from Hartree-Fock calculations through the application of the atomic virial theorem.14

In the course of a study of non-classical fluorine-fluorine bonding81-83 we found a cage critical point within a single ring of atoms.83 This potentiality has remained only a mathematical possibility as described in several monographs,1,2 never observed before in neither experimental nor calculated electron density distributions. 83 Bader on page 37 of his book states:1

[W]hile it is mathematically possible for a cage to be bounded by only two ring surfaces, the minimum number found in an actual molecule so far is three, as in bicyclo [1.1.1] pentane, for example.”

The first example of “an actual molecular system where a cage is bounded by two ring surfaces” has been reported in our 2005 paper.83

During my first postdoctoral fellowship, in Prof. John C. Polanyi’s group (University of Toronto), we have elucidated the factors governing the observed statistical tendencies of homolytic fission reactions of halogenated hydrocarbons when they react with silicon surfaces (Si111(7×7)) under ultrahigh vacuum observed by scanning tunneling microscopy (STM).84,85 This work contributes to nanolithography by formulating the rules driving reaction dynamics on reactive surfaces. The ultimate goal is to elaborate the principles governing self-assembly on chemically-active surfaces in predictable and reproducible fashion. Self-assembly is a prerequisite to nanolithography since the atom-by-atom STM manipulation cannot be scaled to an industrial level.

One of my earliest PhD projects led to a trilogy of papers, co-authored with Prof. Bader, about the electron densities of the 20 genetically-encoded amino acids.46-48 This work has been selected by The Faculty of 1000 (F1000) and ranked by Professor-Dame Janet Thornton, (Director, European Bioinformatics Institute), as “Exceptional“, a category of papers defined by F1000 as “top 1% of all selected publications“.

References
[1]   Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Oxford University Press: Oxford, U.K., 1990.
[2]   Popelier, P. L. A. Atoms in Molecules: An Introduction; Prentice Hall: London, 2000.
[3]   Matta, C. F.; Boyd, R. J. (Eds.) The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design; Wiley-VCH: Weinheim, 2007.
[4]   Matta, C. F. Applications of the Quantum Theory of Atoms in Molecules to Chemical and Biochemical Problems; Ph.D. Thesis; McMaster University: Hamilton, Canada, 2002.
[5]   Matta, C. F.; Hernández-Trujillo, J.; Tang, T. H.; Bader, R. F. W. Hydrogen-hydrogen bonding: a stabilizing interaction in molecules and crystals. Chem. Eur. J. 2003, 9, 1940-1951.
[6]   Hernández-Trujillo, J.; Matta, C. F. Hydrogen-hydrogen bonding in biphenyl revisited. Struct. Chem. 2007, 18, 849-857.
[7]   Matta C. F. Chapt. 9 In: Hydrogen Bonding – New Insight, (Challenges and Advances in Computational Chemistry and Physics Series); Grabowski, S. (Ed.); Springer: 2006.
[8]   Bakhmutov, V. I. Dihydrogen Bonds: Principles, Experiments, and Applications; Wiley-Interscience: New Jersey, 2008.
[9]   Monteiro, N. K. V. Firme, C. L. Hydrogen-hydrogen bonds in highly branched alkanes and in alkane complexes: A DFT, ab initio, QTAIM, and ELF study. J. Phys. Chem. A 2014, 118, 1730-1740.
[10]   Echeverría, J.; Aullón, G.; Danovich, D.; Shaik, S.; Alvarez, S. Dihydrogen contacts in alkanes are subtle but not faint. Nature Chemistry 2011, 3, 323-330.
[11]   Cukrowski, I.; Matta, C. F. Hydrogen-hydrogen bonding: A stabilizing interaction in strained chelating rings of metal complexes in aqueous phase. Chem. Phys. Lett. 2010, 499, 66-69.
[12]   Cukrowski, I.; Matta, C. F. Protonation sequence of linear aliphatic polyamines from intramolecular atomic energies and charges. Comput. Theor. Chem. 2011, 966, 213-219.
[13]   Matta, C. F.; Castillo, N.; Boyd, R. J. Atomic contributions to bond dissociation energies in aliphatic hydrocarbons. J. Chem. Phys. 2006, 125, 204103.
[14]   Matta, C. F.; Arabi, A. A.; Keith, T. A. Atomic Partitioning of the Dissociation Energy of the PO−(H) Bond in Hydrogen Phosphate Anion (HPO42−): Disentangling the Effect of Mg2+. J. Phys. Chem. A 2007, 111, 8864-8872.
[15]   Matta C. F.; Arabi, A. A. Chapt. 15 In: Quantum Biochemistry: Electronic Structure and Biological Activity; Matta, C. F. (Ed.), Wiley-VCH: Weinheim, 2010.
[16]   Arabi, A. A.; Matta, C. F. Where is energy stored in adenosine triphosphate? J. Phys. Chem. A 2009, 113, 3360-3368.
[17]   Matta, C. F.; Sowlati-Hashjin, S.; Bandrauk, A. D. Dipole moment surfaces of the CH4 + •X -> CH3• + HX (X = F, Cl) reactions from atomic dipole moment surfaces, and the origins of the sharp extrema of the dipole moments near the transition states. J. Phys. Chem. A 2013, 117, 7468-7483.
[18]   Bandrauk, A. D.; Sedik, E. S.; Matta, C. F. Effect of absolute laser phase on reaction paths in laser-induced chemical reactions. J. Chem. Phys. 2004, 121, 7764-7775.
[19]   Bandrauk, A. D.; Sedik, E. S.; Matta, C. F. Laser control of reaction paths in ion-molecule reactions. Mol. Phys. 2006, 104, 95-102.
[20]   Arabi, A. A.; Matta, C. F. Effects of external electric fields on double proton transfer kinetics in the formic acid dimer. Phys. Chem. Chem. Phys. (PCCP) 2011, 13, 13738-13748.
[21]   Cerón-Carrasco, J. P.; Cerezoa, J.; Jacquemin, D. How DNA is damaged by external electric fields: Selective mutation vs. random degradation. Phys. Chem. Chem. Phys. (PCCP) 2014, 16, 8243-8246.
[22]   Cerón-Carrasco, J. P.; Jacquemin, D. Electric field induced DNA damage: An open door for selective mutations. Chem. Commun. 2013, 49, 7578-7580.
[23]   Cerón-Carrasco, J. P.; Jacquemin, D. Electric-field induced mutation of DNA: A theoretical investigation of the GC base pair. Phys. Chem. Chem. Phys. (PCCP) 2013, 15, 4548-4553.
[24]   Brovarets, O. O.; Zhurakivsky, R. O.; Hovorun, D. M. The physico-chemical mechanism of the tautomerisation via the DPT of the long Hyp…Hyp WatsonCrick base pair containing rare tautomer: A QM and QTAIM detailed look. Chem. Phys. Lett. 2013, 578, 126-132.
[25]   Timerghazin, Q. K.; Talipov, M. R. Unprecedented external electric field effects on s-nitrosothiols: Possible mechanism of biological regulation? J. Phys. Chem. Lett. 2013, 4, 1034-1038.
[26]   Sowlati-Hashjin, S.; Matta, C. F. The chemical bond is external electric fields: Energies, geometries, and vibrational Stark shifts of diatomic molecules. J. Chem. Phys. 2013, 139, 144101 (Erratum: J. Chem. Phys. 141, 039902, 2014).
[27]   Piela, L. Ideas of Quantum Chemistry; Elsevier: Amsterdam, 2007.
[28]   Delley, B. Vibrations and dissociation of molecules in strong electric fields: N2, NaCl, H2O and SF6. J. Mol. Struct. (THEOCHEM) 1998, 434, 229-237.
[29]   Huang, L.; Bohorquez, H.; Matta, C. F.; Massa, L. The Kernel Energy Method: Application to Graphene and Extended Aromatics. Int. J. Quantum Chem. 2011, 111, 4150-4157.
[30]   Huang L.; Massa, L.; Karle, J. Chapt 1 In: Quantum Biochemistry: Electronic Structure and Biological Activity; Matta, C. F. (Ed.), Wiley-VCH: Weinheim, 2010.
[31]   Huang, L. ; Massa, L.; Karle, J. Kernel energy method applied to vesicular stomatitis virus nucleoprotein. Proc. Natl. Acad. Sci. USA 2009, 106, 1731-1736.
[32]   Huang, L.; Massa, L.; Matta, C. F. A graphene flake under external electric fields reconstructed from field-perturbed kernels. Carbon 2014, 76, 310-320.
[33]   Timm, M. J.; Matta, C. F.; Massa, L.; Huang, L. The localization-delocalization matrix and the electron density-weighted connectivity matrix of a finite graphene flake reconstructed from kernel fragments. J. Phys. Chem. A 2014, in press (DOI: 10.1021/jp508490p).
[34]   Bonaccorsi, R.; Scrocco, E.; Tomasi, J. Molecular SCF calculations for the ground state of some three-membered ring molecules: (CH2)3, (CH2)2NH, (CH2)2NH2+, (CH2)2O, (CH2)2S, (CH)2CH2, and N2CH2. J. Chem. Phys. 1970, 52, 5270-5284.
[35]   Borman, S. Protein factory reveal its secrets: Researchers picture and poke the ribosome to learn how it works. Chem. Eng. News 2007, 85(8), 13-16.
[36]   Massa L.; Matta, C. F.; Yonath, A.; Karle, J. Chapt. 16 In: Quantum Biochemistry: Electronic Structure and Biological Activity; Matta C. F. (Ed.), Wiley-VCH: Weinheim, 2010.
[37]   Matta, C. F.; Castillo, N.; Boyd, R. J. Extended weak bonding interactions in DNA: π-Stacking (base-base), base-backbone, and backbone-backbone interactions. J. Phys. Chem. B 2006, 110, 563-578.
[38]   Taylor, A.; Matta, C. F.; Boyd, R. J. The hydrated electron as a pseudo-atom in cavity-bound water clusters. J. Chem. Theor. Comput. 2007, 3, 1054-1063.
[39]   Matta, C. F. Modeling biophysical and biological properties from the characteristics of the molecular electron density, electron localization and delocalization matrices, and the electrostatic potential. J. Comput. Chem. 2014, 35, 1165-1198.
[40]   Matta, C. F.; Arabi, A. A. Electron-density descriptors as predictors in quantitative structure-activity/property relationships and drug design. Future Med. Chem. 2011, 3, 969-994.
[41]   Bensasson, R. V.; Sowlati-Hashjin, S.; Zoote, V.; Dauzonne, D.; Matta, C. F. Physicochemical properties of exogenous molecules correlated with their biological efficacy as protectors against carcinogenesis and inflammation. Int. Rev. Phys. Chem. 2013, 32, 393-434.
[42]   Matta, C. F.; Arabi, A. A.; Weaver, D. F. The bioisosteric similarity of the tetrazole and carboxylate anions: Clues from the topologies of the electrostatic potential and of the electron density. Eur. J. Med. Chem. 2010, 45, 1868-1872.
[43]   Matta, C. F.; Massa, L. Subsystem quantum mechanics and in-silico medicinal and biological chemistry. Future Med. Chem. 2011, 3, 1971-1974.
[44]   Matta C. F. (Ed.) Quantum Biochemistry: Electronic Structure and Biological Activity; Wiley-VCH: Weinheim, 2010.
[45]   Zhurova, E. A.; Matta, C. F.; Wu, N.; Zhurov, V. V.; Pinkerton, A. A. Experimental and theoretical electron density study of estrone. J. Am. Chem. Soc. 2006, 128, 8849-8861.
[46]   Matta, C. F.; Bader, R. F. W. Atoms-in-molecules study of the genetically-encoded amino acids. III. Bond and atomic properties and their correlations with experiment including mutation-induced changes in protein stability and genetic coding. Proteins: Struct. Funct. Genet. 2003, 52, 360-399.
[47]   Matta, C. F.; Bader, R. F. W. Atoms-in-molecules study of the genetically-encoded amino acids. II. Computational study of molecular geometries. Proteins: Struct. Funct. Genet. 2002, 48, 519-538.
[48]   Matta, C. F.; Bader, R. F. W. An atoms-in-molecules study of the genetically-encoded amino acids. I. Effects of conformation and of tautomerization on geometric, atomic, and bond properties. Proteins: Struct. Funct. Genet. 2000, 40, 310-329.
[49]   Sumar, I.; Ayers, P. W.; Matta, C. F. Electron localization and delocalization matrices in the prediction of pKa‘s and UV-wavelengths of maximum absorbance of p-benzoic acids and the definition of super-atoms in molecules. Chem. Phys. Lett. 2014, 612, 190-197.
[50]   Fradera, X.; Austen, M. A.; Bader, R. F. W. The Lewis model and beyond. J. Phys. Chem. A 1999, 103, 304-314.
[51]   Janezic, D.; Milicevic, A.; Nikolic, S.; Trinajstic, N. Graph Theoretical Matrices in Chemistry (Mathematical Chemistry Monographs, Vol. 3); University of Kragujevac: Kragujevac, 2007.
[52]   Hall, L. H.; Kier, L. B. Molecular Connectivity in Chemistry and Drug Research; Academic Press: Boston, 1976.
[53]   Dmitriev, I. S. Molecules without Chemical Bonds (English Translation); Mir Publishers: Moscow, 1981.
[54]   Pye, C. C.; Poirier, R. A. Graphical approach for defining natural internal coordinates. J. Comput. Chem. 1998, 19, 504-511.
[55]   Pye, C. C. Applications of Optimization to Quantum Chemistry, PhD Thesis; Memorial University of Newfoundland: Saint John’s (NF), Canada, 1997.
[56]   Lide, D. R. CRC Handbook of Chemistry and Physics 88th Edition; CRC Press: 2007-2008.
[57]   Jover, J.; Bosque, R.; Sales, J. QSPR prediction of pKa for benzoic acids in different solvents. QSAR Combin. Sci. 2008, 27, 563-581.
[58]   Matta, C. F.; Hernández-Trujillo, J. Bonding in polycyclic aromatic hydrocarbons in terms of the the electron density and of electron delocalization. J. Phys. Chem. A 2003, 107, 7496-7504 (Correction: J. Phys. Chem A, 2005, 109, 10798).
[59]   Sivaramakrishnan, R.; Tranter, R. S.; Brezinsky, K. Ring conserved isodesmic reactions: a new method for estimating the heats of formation of aromatics and PAHs. J. Phys. Chem. A 2005, 109, 1621-1628.
[60]   Mandado, M.; Gonzalez Moa, M. J.; Mosquera, R. A. Aromaticity: Exploring Basic Chemical Concepts with the Quantum Theory of Atoms in Molecules; Nova Science Publishers, Inc.: New York, 2008.
[61]   Krygowski, T. M.; Stepien, B. T. π- and σ-electron delocalization: Focus on substituent effects. Chem. Rev. 2005, 105, 3482-3512.
[62]   Cyranski, M. K. Energetic aspects of cyclic π-electron delocalization: Evaluation of the methods of estimating aromatic stabilization energies. Chem. Rev. 2005, 105, 3773-3811.
[63]   Merino, G.; Vela, A.; Heine, T. Description of electron delocalization via the analysis of molecular fields. Chem. Rev. 2005, 105, 3812-3841.
[64]   Chen, Z.; Wannere, C. S.; Corminboeuf, C.; Puchta, R.; Schleyer, P. v. R. Nucleus-independent chemical shifts (NICS) as an aromaticity criterion. Chem. Rev. 2005, 105, 3842-3888.
[65]   Poater, J.; Duran, M.; Solà, M.; Silvi, B. Theoretical evaluation of electron delocalization in aromatic molecules by means of atoms in molecules (AIM) and electron localization function (ELF) topological approaches. Chem. Rev. 2005, 105, 3911-3947.
[66]   Wang, Y.-G.; Matta, C. F.; Werstiuk, N. H. Comparison of localization and delocalization indices obtained with Hartree-Fock and conventional correlated methods: Effect of Coulomb correlation. J. Comput. Chem. 2003, 24, 1720-1729.
[67]   Matta, C. F., AIMDELOC: Program to calculate AIM localization and delocalization indices (QCPE0802) (Quantum Chemistry Program Exchange, Indiana University, IN, 2001).
[68]   Matta, C. F. Theoretical reconstruction of the electron density of large molecules from fragments determined as proper open quantum systems: the properties of the oripavine PEO, enkephalins, and morphine. J. Phys. Chem. A 2001, 105, 11088-11101.
[69]   Wilson, E. K. Cover Story. Computers and chemistry: Symposium speakers showcase new developments in computational chemistry – (p.44: Opioid breakthrough?). Chem. Eng. News 2001, 79, 39-44.
[70]   Scheins, S.; Messerschmidt, M.; Luger, P. Submolecular partitioning of morphine hydrate based on its experimental charge density at 25 K. Acta Cryst. B 2005, 61, 443-448.
[71]   Timm, M.; Matta, C. F. Primary retention following nuclear recoil in β-decay: Proposed Synthesis of a metastable rare gas oxide (38ArO4) from (38ClO4) and the evolution of chemical bonding over the nuclear transmutation reaction path. Appl. Rad. Isotopes 2014, 94, 206-215.
[72]   Sadjadi, S.; Matta, C. F.; Lemke, K. H.; Hamilton, I. P. Relativistic-consistent electron densities of the coinage metal clusters M2, M4, M42-, and M4Na2 (M = Cu, Ag, Au): A QTAIM study. J. Phys. Chem. A 2011, 115, 13024-13035.
[73]   Sadjadi, S.; Matta, C. F.; Hamilton, I. P. Chemical bonding in groups 10, 11, and 12 transition metal homodimers: An electron density study. Can. J. Chem. 2013, 91, 583-590.
[74]   Keith, T. A.; Frisch, M. J. Subshell fitting of relativistic atomic core electron densities for use in quantum theory of atoms in molecules analyses of effective core potential-based wave functions. J. Phys. Chem. A 2011, 115, 12879-12894.
[75]   Bendeif, E.-E.; Matta, C. F.; Stradiotto, M.; Fertey, P.; Lecomte, C. Can a formally zwitterionic rhodium(I) complex emulate the charge density of a cationic rhodium(I) complex? A combined synchrotron X-ray and theoretical charge density study. Inorg. Chem. 2012, 51, 3754-3769.
[76]   Wallace, S.; Huang, L.; Matta, C. F.; Massa, L.; Bernal, I. New structures of hydronium cation clusters. C. R. Chim. 2012, 15, 700-707.
[77]   Bohórquez, H.; Boyd, R. J.; Matta, C. F. Molecular model with quantum mechanical bonding information. J. Phys. Chem. A 2011, 115, 12991-12997.
[78]   Bohórquez, H. J.; Matta, C. F.; Boyd, R. J. The localized electron detector as an ab initio representation of molecular structures. Int. J. Quantum Chem. 2010, 110, 2418-2425.
[79]   Matta, C. F. How dependent are molecular and atomic properties on the electronic structure method? Comparison of Hartree-Fock, DFT, and MP2 on a biologically relevant set of molecules. J. Comput. Chem. 2010, 31, 1297-1311.
[80]   Matta, C. F.; Sichinga, M.; Ayers, P. W. Information theoretic properties from the quantum theory of atoms in molecules. Chem. Phys. Lett. 2011, 514, 379-383.
[81]   Matta, C. F.; Castillo, N.; Boyd, R. J. The characterization of a closed-shell fluorine-fluorine bonding interaction in aromatic compounds on the basis of the electron density. J. Phys. Chem. A 2005, 109, 3669-3681.
[82]   Castillo, N.; Matta, C. F.; Boyd, R. J. Fluorine-Fluorine spin-spin coupling constants: Correlations with the delocalization index and with the internuclear separation. J. Chem. Inf. Mod. 2005, 45, 354-359.
[83]   Castillo, N.; Matta, C. F.; Boyd, R. J. The first example of a cage critical point in a single ring: A novel twisted α-helical topology. Chem. Phys. Lett. 2005, 409, 265-269.
[84]   Dobrin, S.; Harikumar, K. R.; Matta, C. F.; Polanyi, J. C. An STM study of the localized atomic reaction of 1,2 and 1,4-dibromoxylene at Si(111)7×7. Surf. Sci. 2005, 580, 39-50.
[85]   Matta, C. F.; Polanyi, J. C. Chemistry on a peg-board: The effect of adatom-adatom separation on the reactivity of dihalobenzenes at Si(111)7×7 surfaces. Phil. Trans. R. Soc. Lond. A 2004, 362, 1185-1194.