# Research

**Current Research Directions**

**Chérif F. Matta**

**(Updated: 4 Jan. 2018. © C. F. Matta 2018)**

This research programme ranges from **theoretical chemistry **to** chemical physics **both fundamental and applied. The programme has an important thrust in computational quantum chemistry particularly the topographical analysis of the electron density but is broader than that in scope. Applications of interest stretch from nanoscience and catalysis to theoretical biochemistry and biophysics. There is also a recent emerging interest in the thermodynamic efficiency of molecular machines such as ATP synthase especially from the perspective of information theory. The programme is, thus, diverse and broad, but this horizontal spread does not preclude its characteristic intellectual depth. While the core of the research in “in-house”, synergetic collaborations with experimentalists and other theorists have been and are always welcome with open arms. There are four current somewhat overlapping principal research thrusts in this group, briefly outlined below (the interested reader can consult the cited references for a deeper exposition).

**CONTENTS**

**The invention, development, and uses of the electron localization-delocalization matrices (LDMs).****Molecules under strong external electric fields:***2.1**Molecules in static external electric fields.**2.2**Molecules and reactions in time-varying external fields (intense laser fields)*.

**Development of a fragmentation method for the fast approximation of electron densities and of electrostatic potentials/fields of proteins and nanostructures.****Information theoretic investigations of biological sorting molecular machines****Other interests:***5.1***Accounting for the effect of recoil energy on chemical yield in nuclear disintegration reactions.***5.2***Bioisosterism and the molecular electrostatic potential (ESP).***5.3***Electronic structure of excited states.**

**Main Current Lines of Research**

**1.** **The invention, development, and uses of the electron localization-delocalization matrices (LDMs) **

A localization-delocalization matrix (also called “ζ-matrix”) was proposed in 2014 [1] as a concise mathematical representation of a molecule as a network of electron delocalization pathways [2,3]. The idea is to draw simultaneously on the strengths of quantum chemistry (electronic structure calculations) [4-6], Bader’s topographical analysis of the electron density (known as the Quantum Theory of Atoms in Molecules (or QTAIM) [7-9]), and Chemical Graph Theory (CGT) [10-15] LDMs combine these three branches of theoretical chemistry in a novel, exciting, and synergetic way [1,2,16-21] (see Fig. 1).

**Fig. 1** Localization-delocalization matrices (LDMs) as a bridge between Bader’s Quantum Theory of Atoms in Molecules (QTAIM) and Chemical Group Theory (CGT) (*top*). The figure contrasts the conventional chemical view of a molecule (*bottom, left*) as an *incomplete* and *discrete/sharp* (all-or-none) graph (incomplete in the sense that edges connects only those atoms that chemically-bonded (e.g. either share a bond path and its associated interatomic zero-flux surface or do not)) with the localization-delocalization *complete* and *fuzzy graph* (*bottom right*) where every pair of atoms in a molecule are connected by a *continuously variable* electron delocalization communication channel. The complete and fuzzy molecular graph is captured mathematically in an LDM representative of molecular electron distribution (See: Ref. [2]).

*top*

*bottom, left*

*bottom right*

An LDM is constructed from the calculated numbers of shared electrons between every pair of atoms in the molecule (obtained from a quantum chemical calculation of the molecular wavefunction which is then subjected to a QTAIM analysis). The localization-delocalization matrices are manipulated by the tools of chemical graph theory to extract matrix invariants. (Matrix invariants are matrix properties that do not depend on the simultaneous permutations of rows and columns, *i.e.*, are insensitive to the (arbitrary) atomic numbering/labeling). These invariants are then used as predictive molecular descriptors [22] in quantitative structure/property to activity (QSAR/QSPR) type of modeling and were shown to exhibit high accuracy coupled with an unusually wide range of applicability.

More specifically, an LDM is composed of the complete set of localization indices [23], {Λ(Ω* _{i}*)}, along the diagonal, and the complete set of the delocalization indices divided by 2 [23], {d(Ω

*,Ω*

_{i}*)/2 = d(Ω*

_{j}*,Ω*

_{j}*)/2,*

_{i}*i*≠

*j*}, as the off-diagonal elements. These two sets of indices provide a bookkeeping of the whereabouts of all

*N*electrons in the molecule. An LI counts the number of electrons localized within an atomic basin while a DI counts the number of electrons shared between two basins. Accordingly, the molecular LDM is defined [1]:

Eq.(1)

Eq.(1)

where the sum of any column or row yields an atomic population by virtue of Bader’s summation rule (the first equality):

**Eq.(2)**

An interpretation of Eq. (2) that that an atomic population arises from a localized contribution (electrons exchanging with one another within the same atomic basin) and a delocalized contribution from those electrons that are shared, *i.e.* exchange, between a given atomic basin and every other atomic basin in the molecule.

Correspondingly, the molecular electron population *N* can be expressed as the sum of a localized and a delocalizaed electron population as well:

##### **Eq.(3)**

where

**Eq.(4)**

and

**Eq.(5)**

Thus, an LDM carries a considerable amount of information about the molecular electron distribution at an atomic and diatomic resolution. The information coded in an LDM include the *fuzzy molecular graph *when the molecule is viewed as a network of “exchange channels [24]” (with those coinciding with bond paths being dominant “privileged exchange channels” [24]),* molecular geometry, **m**olecular branching, **bond strengths, atomic electron populations, atomic charges and volumes,* newly defined *multidimensional atomic charges**, f**ree valences, and NMR J*_{HH’}* c**oupling c**onstants* [2]. See Fig. 2*.*

**Fig. 2** Information coded in a localization-delocalization matrix (LDM) [2].

Given the considerable information contained tacitly within an LDM, it is unsurprising that LDMs are a powerful predictive molecular descriptor of a wider than usual range of properties in QSAR/QSPR studies [1,3,16-18,20,21,25,26]. Instead of thousands of descriptors tailored for restrict ranges of properties [22], LDMs have been shown to be capable of predicting thermodynamic properties such as boiling points of isomers (which is a property that is sensitive to the degree of molecular branching, which is hard to quantify, see example in Fig. 3), enthalpies of vaporization, heats of combustions, enthalpies of formation, total energies [1], p*K*_{a}‘s [20] (with sufficient reliability to discover a mistaken p*K*_{a} entry in the HRC Handbook) [20], aromaticity of rings-in-molecules (RIMs) [17], UV l_{max}‘s [20], mosquito repellency [16], ribotoxicity of trichothecenes antbiotics, and stress corrosion cracking inhibitors’ activities including the identification of the correct active inhibitors’ species [3,18]. LDMs can also be used to assess the quality basis sets in benchmarking studies [18,25].

LDMs, thus appear to have a wide range of possible applications in the prediction of physicochemical and biological properties and reactivities of series of molecules. The challenge now is to find ways to speed the construction of LDMs to be useful industrially. The LDM philosophy can be extended to interacting quantum atoms (IQAs) energy components developed by the *University of Oviedo’s Quantum Chemistry Group* and to the integrated Bader-Gatti source function the analysis of which is pioneered by Dr. Carlo Gatti’s group at the *Italian National Research Council* [2].

**Fig. 3** Correlation of the boiling points (BPs) of the five isomers of hexane (C_{6}H_{14}) with the Frobenius distance of their LDMs from the LDM of the most branched isomer, 2,2-dimethylbutane, taken as reference [2]. (The self-distance of the reference molecule is zero, obviously).

**2. Molecules under strong external electric fields**

*2.1 Molecules in static external electric fields*

*2.1 Molecules in static external electric fields*

This research group has been interested in elucidating the effects of strong external static and time-varying electric fields on chemical bonds and on reactivity (reaction profiles and energy barriers). My former M.Sc. student **Shahin Sowlati-Hashjin** and I proposed a “*field-perturbed Morse potential*” to predict the effects of external static electric fields on bond lengths and on vibrational spectra (vibrational Stark effect) in diatomics whereby the Stark shift is predicted solely from the *field-free* properties of the molecule without recourse to a full-fledged quantum chemical calculation that includes a field term into the Hamiltonian [27]. Our model has been validated numerically by comparison with the results of brute force calculations including a field term in the Hamiltonian. The importance of this undertaking has been underscored in L. Piela’s definitive book *Ideas in Quantum Chemistry* [6]:

“*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*”.

Our field-perturbed Morse potential has been inspired by (but differs *significantly* from) Delley’s earlier work [28]. This potential generalizes the Morse potential by the inclusion of field-terms to be defined [27]:

**Eq.(6)**

where *E* is the electric field strength, *μ*_{0} and *a*_{//0} are the field-free molecular dipole moment and the longitudinal element of the polarizability tensor.

The field-perturbed Morse potential yields the following equations that predict bond lengths and vibrational frequencies in the presence of an external static electric field from their *field-free* values, respectively [27]:

**Eq.(7)**

**Eq.(8)**

where *R*_{0} and *ν*_{0} are the field-free equilibrium bond length and harmonic frequency, *R*** _{E}** and

*ν*

**are the corresponding values under an external electric field**

_{E}**, and**

*E**r*is the inter-nuclear separation. The effect of external fields on some homonuclear and heteronuclear diatomics is displayed in Fig. 4.

The negative or positive signs of the dipole terms in Eqs. (7) and (8) correspond, respectively, to parallel and antiparallel orientations of the permanent molecular dipole moment (if it exists) with respect to the external field. For molecules without a permanent dipole, the dipolar term vanishes and the response to the field is totally determined by the polarizability term. These equations reproduce accurately the results of direct calculations with explicit treatment of the field-effects [27].

**Fig. 4 **Plots of the change in the equilibrium bond lengths (BLs), D*R*, as a function of the strength of externally-applied static uniform electric fields (*E*) for homo-nuclear diatomics (*top, left*), and as a function of the field strength and direction for hetero-nuclear diatiomics (*bottom, left*) (in Å). Plots of the change in harmonic frequencies in cm^{–1} (D*ν*, *right* plots) and as functions of the field strength for the homonuclear diatomics (*top, right*) and as functions of the field strength and direction for the heteronuclear diatomics (*bottom, right*).(The directions of the permanent molecular dipole moments (if they exists) follow the physicist convention, *i.e.* the dipole originates at the negative pole and the arrowhead points to the positive pole so that the parallel orientation in an external field is the most stable [29,30].

*E*

*top, left*

*bottom, left*

*top, right*

*bottom, right*

Our equation (8) has recently been used by experimentalists to estimate the local electric field to which a probe molecule (CO) is subjected from a neighbouring molecule [31]. Our equation has been used in conjunction with the experimental and calculated IR or Raman shifts of CO (See Table 1 of Ref. [31]).

With my former PhD student, now **Professor Alya A. Arabi **(*Zayed University*), we have developed a kinetics model that include tunneling corrections describing proton transfer reactions [32]. (See Fig. 5). This frequently cited paper is arguably considered as a standard protocol in studies of the effects of external fields, solvents, and electronic excitations on the rate of tautomerization of hydrogen bonded systems involving hydrogen transfer reactions. These reactions include the especially important class of double proton transfers in DNA base pairs, the so-called Löwdin mechanism of spontaneous and induced mutation. Our protocol has been used by other research groups to examine, for example, **(a)** the kinetics of Löwdin mechanism of spontaneous and induced mutation in the presence of external uniform electric fields of intensities believed to exist in the typical microenvironment of DNA (|** E**| ~10

^{8}– 10

^{9}V/m) [33-36]

**(b)**the double proton transfer in non-natural nucleic acid base pairs [37-40],

**(c)**the kinetics of enzyme-catalyzed reactions under the influence of the natural electric fields generated within enzyme active sites [41-43], and

**(d)**together with our above-mentioned work on the effect of external fields on the chemical bonds [27] this research appears to have inspired at least one important study making use of STM electric fields to manipulate nanoscale supramolecular assembly on surfaces [44]. At the present, my MSc student

**Youji Cheng**and I are studying the effect of intrinsic electric fields generated by large proteins on biocatalysis.

**Fig. 5 **(*Left*) Acceleration of the double proton transfer reaction in the formic acid dimer by an external uniform electric field measured by the rate constant (after quantum tunneling correction) as a function of field strength at room temperature [32]. (The field is parallel to the *z*-direction). (*Right*) An example of the modeled Löwdin mechanism of spontaneous and induced mutations under external field illustrated for an adenine-thymine base pair (See also [45]).

*Left*

*Right*

*2.2 Molecules and reactions in time-varying external fields (intense laser fields)*

*2.2 Molecules and reactions in time-varying external fields (intense laser fields)*

In a collaboration with **Professor André D. Bandrauk** (*Université de Sherbrooke*) we demonstrated that the *intensity* and *phase* of a plane polarized, phase-stabilized, beam of two-color laser pulses can be tuned to *invert* the potential energy surface (PES) for co-linear collisions of methane with either halogens [46,47] or metal ions [48] [X• + CH_{4} –> HX + •CH_{3}, (X = F, Cl, Li)]. In this way, *a transition state on the PES can be converted to a radiation-stabilized bound state*. (See Fig. 6).

The effective potential of the field-molecule system as a function of the reaction coordinate (*s*) can be approximated as [47,48]:

**Eq.(9)**

and are the components of the dipole moment and of the polarizability tensor element parallel to the *C*_{3} axis, *E*_{0} is the amplitude and 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 sharp peaks near the transition state (TS). Thus, and through Eq. (9), one can anticipate a maximal effect of the field on the reaction near the TS where the field-molecule interaction is maximal. The phase and the intensity of the external field can be adjusted to eliminate or invert the potential energy barrier converting a TS into a bound state. (See Fig. 6).

**Fig. 6** Energy profile along the reaction coordinate, for a 3.0 x 10^{13 }W.cm^{-2} laser pulse along the long axis of a radiation-aligned Cl+CH_{4} system plus the dipole moment and polarizability contributions. Field-induced bound states are shown to have a deep minimum at *Φ* = π at this field intensity.

With **Professor Bandrauk** and my former student **Shahin Sowlati-Hashjin**, we examined atomically-decomposed dipole moment surfaces to pinpoint those atoms responsible of the sharp dipole moment peak near the transition state [46]. The calculations show that 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 [46].

**3. Development of a fragmentation method for the fast approximation of electron densities and of electrostatic potentials/fields of proteins and nanostructures**

The kernel energy method (KEM) has originally been developed by **Prof. Lou Massa** (*City University of New York (CUNY)*), and **Dr. Lulu Huang** and **Dr. Jerome Karle ( 1985 Chemistry Nobel Laureate)** (

*US Naval Research Laboratory*

*(NRL)*) [49-56]. This fragmentation scheme allows one to estimate the properties of a large molecules from single and double fragments (kernels). The double fragments account for all interaction between all different parts of the molecule, and then through the following summation rule the properties of the full molecule is estimated after accounting for the double counting associated with representing the single kernels again in the double kernels (see Fig. 7).

**Fig. 7** An illustrative example of a large molecule broken into three single kernels and three double kernels.

The constitutive equation of the KEM method is totally general, and while written below for the total energy and the electron density it applies equally well to approximate QTAIM atomic charges [57], the molecular dipole moment components (with one KEM equation for each individual component) [58], changes in properties induced by applied external fields [58], the molecular electrostatic potential, and, most recently, it has been proposed as the basis for reconstructing one-body and two-body reduced density matrices [57,59-61]. An interesting application has recently demonstrated that one can also obtain accurate electron localization-delocalization matrices (LDMs) of a delocalized electronic system such as a graphene nanoribbon [21,62]. (See Fig. 7).

The KEM delivers an accurate approximation to the total molecular energy *E* (whether in vacuum or perturbed by a solvent reaction field or by an externally-applied electric field) as:

**Eq. (10)**

where *E*_{KEM} is the KEM approximation to the total energy of the full system, *E _{ij}* is the energy of the

*ij*

^{th}double kernel,

*E*is the energy of the

_{k}*k*

^{th}single kernel,

*i*,

*j*, and

*k*are running integer indices, and

*n*is the number of single kernels.

The electron density of the full molecule, in its turn, can be approximated as [21,58]

**Eq. (11)**

where *r _{ij}* and

*r*are the electron densities of the

_{k}*ij*

^{th}double kernel and of the

*k*

^{th}single kernel, respectively, and

**r**is a position vector. With an approximate KEM density, the molecular electrostatic potential can readily be reconstructed using the known charges

*Z*(in atomic units) and positions (

_{i}**R**

*) of the nuclei [63]:*

_{i}**Eq. (12)**

where the *i*^{th} term in the first sum must be eliminated when **R*** _{i}* =

**r**. Alternatively,

*V*

_{KEM}(

**r**) can be reconstructed from the molecular ESP calculated for the kernel fragments on a point-by-point basis using an equation of the same form of Eq. (11).

**Fig. 8 **(*Left*) Relative orientations of a finite graphene nanoribbon under an external uniform electric field reconstructed from KEM fragments. (The little arrow in the middle and antiparallel to the *x*-axis depicts the direction of the permanent dipole moment of this nanoribbon). (*Middle*) Comparison of field-induced changes in the total energy of the nanoribbon calculated directly for the intact molecule and from the KEM approximation as a function of field strength. (*Right*) Comparison of field-induced changes in the principal (*y-*) component of the dipole moment of the nanoribbon calculated directly for the intact molecule and from the KEM approximation as a function of field strength.

*Left*

*Middle*

*Right*

The KEM method is currently being extended by its incorporation to what is now termed “** Quantum Crystallography (QCr)**” [60,61] in collaboration with

**Prof. Lou Massa**(who has started the field of QCr [56,64-72]). QCr aims at generating entire density matrices (not only electron densities), or even of complete wavefunctions [73], of large molecules that are consistent simultaneously with crystallographic X-ray diffraction data and with the underlying quantum mechanical mathematical structure (Fig. 9). By this we mean that QCr in its original definition aims at ensuring the derivability of the observed electron density from an underlying properly antisymmetrized wavefunction (satisfying quantum mechanics) and from the experimental structure factors (satisfying experiment) simultaneously [60,61,74]. There has been a recent revival of interest in QCr as indicated by the renaming of the

*International Union of Crystallography (IUCr)*’s long-standing

*Commission on Charge, Spin, and Momentum Densities*with its new name as

**a name adopted by the IUCr’s General Assembly in Hyderabad (India) during its meeting in 2017.**

*IUCr Commission on Quantum Crystallography***Fig. 9** Main steps of quantum crystallography in conjunction with the kernel energy method (KEM).

## **4. Information theoretic investigations of biological sorting molecular machines**

Following the lead of Johnson and Knudsen’s work on the thermodynamics of the kidney (an entire organ acting as a sorting machine) [75-77], we apply their approach, not to an organ but rather to a constituent of a cell organelle, namely, ATP synthase. Our work led to the finding that the thermodynamic efficiency of mitochondrial ATP synthase may have been underestimated by as much as 30% in standard biochemistry textbook. This underestimation resulted from overlooking the free-energy cost associated with the act of sorting (with a minimal cost of *kT*ln2 per bit of information [78,79]). We argue that the kidney and mitochondrial ATP synthase are two realizations of Maxwell demon [80-89], one is in the form of an entire organ and the latter in the form of a sorting molecular machine [75-77].

Considerations of the quantum mechanical (Heisenberg) uncertainty in this light leads to an expression for the uncertainty in the time of the measurement (or sorting act) which is inversely proportional to the temperature at which the measurement occurs [76,90]. Further, because of Einstein’s famous energy-mass equivalence, an almost imperceptible mass is predicted to accompany every bit of information accumulated. Finally, an additional term is added to Mitchell’s chemiosmotic equation to account for the energy cost of sorting protons by ATP synthase viewed as a molecular machine whose primary job is ions sorting and proton selection. (See Fig. 10).

**Fig. 10 **Three consequences of accounting for the minimal energy cost of information: On Mitchell’s chemiosmotic theory’s central equation (by adding the last information-theoretic term (*top*)), on a minimal quantum uncertainty on the time of affecting a binary selection (*bottom, left*), and on a predicted tiny mass associated with every bit of information (*bottom, right*).

*top*

*bottom, left*

*bottom, right*

**5. Other recent miscellaneous interests**

**5.1. Accounting for the effect of recoil energy on chemical yield in nuclear disintegration reactions**

**5.1. Accounting for the effect of recoil energy on chemical yield in nuclear disintegration reactions**

In 2014, my (then) student **Matthew Timm** and I have undertaken a study of the evolution of the electron density immediately after a nuclear radioactive transmutation [91]. 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 *θ _{critical}* between the ejected

*β*-electron and the accompanying antineutrino as a function of a maximal allowed recoil energy

*E*

_{recoil}(set for example as the bond dissociation energy (BDE) of the weakest bond in the compound). Thus, from mass and linear momentum conservation arguments, we derived the formula [91]:

**Eq. (13)**

where the imposed upper bounds on the recoil energy, *E*_{recoil}, is converted into maximum allowable angular deviations from *p* between the directions of the ejected *β*-electron and the accompanying antineutrino that will result in an *E*_{recoil} that does not exceed the imposed upper bound, and where the max(*E _{e}*) 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 (Fig. 11).

**Fig 11 (a) **The abscissa represents the deviations of the angle *θ* from *p* (in ^{o}) between the trajectories of the ejected *β*-electron and the antineutrino the three values of the experimentally-observed max(*E*_{e}) for an upper bond rupture threshold *E*_{recoil} = 20 kcal/mol. Each curve delimits a region contained to the left of the curve and which corresponds to energies of recoil £ to 20 kcal/mol and areas to its right that correspond to angular deviations greater than the critical values would result in bond rupture. **(b)** The abscissa represents the percent fraction of nuclei that will survive a decay event with the ejection of a *β*-particle given in the ordinate axis.

_{e}

**5.2. Bioisosterism and the molecular electrostatic potential (ESP)**

**5.2. Bioisosterism and the molecular electrostatic potential (ESP)**

A long standing problem is to explain why there exists structurally and chemically *very different* bioisosteric groups such as tetrazole and carboxylate anions that are interchangeable in a drug with little effect on the drug’s pharmacological profile. At least for this pair of bioisosteres (tetrazole and carboxylate), we found that they exhibit an almost identical topology and topography of their electrostatic potential irrespective of the capping group [92] (Fig. 12). A curious finding is that, despite of the different chemical nature of these two bioisosteres, their *average* electron densities are almost identical [92]. Similar results were also observed for the bioisosteric couple methylsquarate and carboxylic acid [93]. These observations have not yet been generalized to other bioisisteric couples.

**Fig. 12 ** The two bioisosteres carboxylate and tertrazole anions and a model of the region of a receptor complementary to the –COO^{−} and −CN_{4}^{−} regions. Isosurfaces of the electrostatic potential (ESP): Pink for ESP > 0, violet for ESP < 0, and the displayed isosurfaces are (in atomic units) ±0.250 for CH_{3}-COO^{-} and ±0.275 for CH_{3}-CN_{4}^{-}. The distance matrix of the four minima in the ESP of each bioisostere is given at the bottom.

** ***5.3 Electronic structure of excited states*

*5.3 Electronic structure of excited states*

In collaboration with the group of **Roberto L. A. Haiduke** (Universidade de São Paulo) and our joint former PhD students (**Dr. Luiz A. Terrabuio**) we have examined the structures and electron densities of difluorodiazirine in the ground and two low-lying excited states [94,95] and that of carbon monoxide (CO) [96] to explain their unusual and unexpected dipole moments in their ground and low-lying excited states. This is achieved by decomposing the molecular dipole moment into atomic polarization (AP) and inter-atomic charge transfer (CT) contributions according to the quantum theory of atoms in molecules (QTAIM) [7,97-99] (Fig. 13).

As an example, it has been found that large but opposite AP and CT contributions essentially cancel in the ground state (*S*_{0}) resulting in CO’s small dipole moment [96]. In the lowest triplet (*T*_{1}) and singlet (*S*_{1}) excited states this balance is disturbed resulting in significant dipole moments mainly driven by an induced charge transfer from O to C. The reduction in the intensity of the vibrational stretching band is shown to result mainly from changes in the atomic charges as well, although other contributions become more significant in the vibrations of excited states but largely cancel.

**Fig. 13 ** The dipole moment of CO shown to arise from two terms: A charge transfer term and an atomic polarization (deformation) term expressed in an origin independent manner (See Refs: [97,98]).

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