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The localized pair model of electronic structure analysis

Zosia A.M. Zielinski, Jason K. Pearson ⇑Department of Chemistry, University of Prince Edward Island, Charlottetown, PE, Canada C1A 4P3

a r t i c l e i n f o

Article history:Received 20 July 2012Received in revised form 27 August 2012Accepted 27 August 2012Available online 6 September 2012

Keywords:Electronic structure theoryHartree–FockPosition intraculeMomentum intraculeReduced density matrixLocalized molecular orbitals

a b s t r a c t

We introduce the localized pair model of electronic structure analysis and propose the two-electronreduced density matrix as an important interpretive tool in chemistry. Interelectronic probability distri-butions in position and momentum space are calculated for individual localized molecular orbitals cor-responding to intuitive chemical features such as lone pairs and chemical bonds. It is demonstrated thatthese may be interpreted as the distribution of electrons within a chemical bond or lone pair and we referto this model as the localized pair model of electronic structure analysis. Specifically, the Hartree–Focklevel of theory is employed in conjunction with a completely uncontracted 6-311G (d,p) basis set to con-struct our localized orbitals. Spherically averaged position and momentum intracules are calculated foreach orbital and we present results for orbitals of p-block hydrides, saturated main group compounds,fluorinated species, N ? B dative structures, and small cyclic molecules. We find that our analysis gener-ally agrees quite well with intuitive predictions based on bond lengths and electronegativities of thebonded atoms. However the trends in the data cannot be predicted using the bond length or electroneg-ativity alone, which demonstrates the unique features of this model.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

In their influential book, ‘‘Reduced Density Matrices: Coulson’sChallenge’’, John Coleman and Vyacheslav Yukalov eloquently de-scribe the impressive predictive power contained within the sec-ond order reduced density matrix, or 2-matrix as it is commonlyknown, given by [1]

q2ðr1; r2Þ ¼nðn� 1Þ

2

ZjWðx1; . . . ; xnÞj2 ds1 ds2 dx3 . . . dxn ð1Þ

where xi = (ri,si) denotes the combined position and spin coordi-nates of electron i. They discuss the attractive prospect of employ-ing the 2-matrix as the fundamental quantity for the generalprediction of the energy levels of chemical systems. Of course theconcept of an electron pair as a fundamental entity in chemistryis ubiquitous and it is therefore a natural choice for electronic struc-ture prediction. Here we offer a complementary application of theso-called 2-matrix for the analysis of electronic structure. We arguethat the 2-matrix can serve as an important interpretive tool inquantum chemistry and we show briefly that ubiquitous conceptssuch as the chemical bond may be scrutinized in new and intuitiveways with some creative applications of it.

Since the days of Lewis [2], chemists have been enamoured withthe concept of a localized electron pair to represent the now intu-itive features of electronic structure such as the so-called lone pair

and the chemical bond itself. Despite its simplicity, the Lewis mod-el yields an impressive wealth of predictive ability in terms ofmolecular structure [3] and chemical properties and is a universalconcept in the chemical literature. Is there a way to link the es-sence of the Lewis model to the quantum mechanical descriptionof electron pairs? Within the orbital approximation in quantummechanics, the delocalized canonical molecular orbitals (CMOs)of a general chemical system bear little resemblance to the featuresof a Lewis structure. In contrast however, CMOs are amenable to aunitary transformation to afford the less well-known [4–10] local-ized molecular orbitals (LMOs)[11–13]. These are an equivalentdescription of electronic structure and yield intuitive and highlytransferable orbitals that can be said to represent individual bonds,lone pairs, and core electrons (vide infra). It is within the space ofLMOs then, that we should find such a link.

While the Lewis model can be remarkably powerful in its pre-diction of molecular structure and some properties; it yields littleinformation regarding the distribution of the electrons themselves.Are electrons within a particular pair generally close together, orfar apart? Do they move quickly or slowly? How do these distribu-tions change with the chemical environment? Most importantly,do these quantities relate to observable chemical properties? Weseek to address these important questions in the present article.To do this, we calculate interelectronic probability distributions,also known as intracules, for particular LMOs. A position intracule,P(u), is the probability distribution for u = jr2 � r1j, and thereforeyields the likelihood of a pair of electrons being separated by a dis-tance u

2210-271X/$ - see front matter � 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.comptc.2012.08.033

⇑ Corresponding author. Tel.: +1 902 566 0934; fax: 1 902 566 0632.E-mail address: [email protected] (J.K. Pearson).

Computational and Theoretical Chemistry 1003 (2013) 79–90

Contents lists available at SciVerse ScienceDirect

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PðuÞ ¼Z

q2ðr1; r2Þdðr12 � uÞ dr1 dr2 dXu ð2Þ

In the above expression, d is a Dirac delta function and dXu indi-cates integration over the angular components of the u vector. Assuch, P(u) is often referred to as the ‘‘spherically averaged’’ intraculedensity.

Likewise, the spherically averaged momentum intracule, M(v), isthe probability distribution for v = jp2 � p1j (where pi describes theposition of electron i in momentum space), and describes the rela-tive momenta of two electrons. We can compute this distributionin an analogous fashion as P(u) by replacing q2(r1,r2) with itsmomentum space analogue, p2(p1,p2)[14,15]

Fig. 1. A comparison of the position (top) and momentum (bottom) intracules forthe water O–H bond. Note the FB (green) result is not visible, as it coincides with theER (red) result. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

Fig. 2. A comparison of the five molecular orbitals of water determined with each ofthe three localization procedures. The contour value of these orbitals is 0.10 a.u.,with the exception of the core orbital on the far left, whose contour value is 0.03 a.u.to make it visible.

Fig. 3. A comparison of the position (top) and momentum (bottom) intracules offirst and second row hydride X–H bonds.

Fig. 4. A plot of the experimental bond dissociation energies (kcal/mol) of thep-block hydrides versus the first inverse moment of the position and momentumbond intracules.

80 Z.A.M. Zielinski, J.K. Pearson / Computational and Theoretical Chemistry 1003 (2013) 79–90


MðvÞ ¼Z

p2ðp1;p2Þdðp12 � vÞ dp1 dp2 dXv ð3Þ

As explicit two-electron quantities, these intracules afford valuableinsights regarding electron–electron interactions that are not prac-tically accessible from the much more common one-electron den-sity alone.

We have been interested in the prediction of electronic pair dis-tributions for reasons including the analysis of electronic structure[16,17], the interpretation of correlation effects [16–18] and thecalculation of the correlation energy [19]. In this work, we describeour efforts to develop a model for the analysis of electronic struc-ture in molecules based on the distribution of electron pairs withinLMOs. Using the framework of LMOs, we have calculated the prob-

ability distribution of the relative separation of electron pairs inboth position and momentum space. Such a procedure was firstproposed by Thakkar and coworkers in 1981[20] and then againin 1984[21]. We will show that these localized intracule densitiesmay be interpreted as the distribution of electrons within a chem-ical bond or lone pair and we refer to this model as thelocalized pairmodel of electronic structure analysis. Such a treatment allows fora unique characterization of electronic structure within localizedregions of space and may find many applications in the character-ization and quantification of inter and intra-molecular interactionsas well as covalent interactions. From the vast expanse of chemicalspace, one may chose an infinite array of species with which toprobe the utility of such a model. In the interests of employing amanageable, yet diverse test set of molecules, we present results

Table 1Properties of first and second row p-block hydrides, relating to the X–H bond. Bond length values (in Å) were determined from the GAMESS calculations done in this work. Bonddissociation energies (BDE) are experimental values from Darwent [28] unless otherwise noted. All values are in atomic units unless otherwise stated.

Bond umax P�1 dP vmax M�1 dM rX–H (Å) BDE (kcal/mol) [28]

B–H 1.675 0.6344 �0.5392 1.100 0.9531 �1.8920 1.189 106.6a [29]C–H 1.516 0.6883 �0.7181 1.283 0.8257 �1.2254 1.084 102.7 ± 2.0C–H (CH3F) 1.488 0.7119 �0.7937 1.317 0.8000 �1.1368 1.082 101.3 [30]C–H (CH2F2) 1.462 0.7339 �0.8683 1.345 0.7787 �1.0520 1.080 103.2 [30]C–H (CHF3) 1.443 0.7504 �0.9281 1.364 0.7648 �0.9820 1.076 106.7 [30]N–H 1.367 0.7614 �0.9415 1.447 0.7215 �0.8101 1.001 104 ± 2O–H 1.236 0.8433 �1.1781 1.618 0.6325 �0.5355 0.941 119.2 ± 0.2F–H 1.113 0.9405 �1.4508 1.808 0.5532 �0.2341 0.896 135.8Al–H 1.938 0.5510 �0.3043 0.904 1.1295 �3.4490 1.579 84.8a [31]Si–H 1.807 0.5893 �0.4121 1.025 1.0195 �2.5671 1.477 80P–H 1.716 0.6101 �0.4950 1.120 0.9445 �1.9735 1.408 88.5 [32]S–H 1.630 0.6381 �0.5820 1.213 0.8713 �1.5163 1.331 91 ± 1Cl–H 1.547 0.6713 �0.6732 1.309 0.8024 �1.1776 1.270 103.24

a ab initio.

Fig. 5. A comparison of the position (top) and momentum (bottom) intracules of the lone pairs of first and second row p-block hydrides. The corresponding LMOs are shownon the right with a contour value of 0.10 a.u. and from top to bottom are for HF, H2O, NH3, HCl, SH2 and PH3.

Z.A.M. Zielinski, J.K. Pearson / Computational and Theoretical Chemistry 1003 (2013) 79–90 81


for p-block hydrides, saturated main group compounds, fluori-nated species, N ? B dative structures, and small cyclic molecules.Atomic units are used throughout, unless otherwise noted.

2. Computational methods

For the purposes of our initial exploration into the localized pairmodel we have chosen to study relatively small (2–12 atoms)closed-shell molecules with a single determinant Hartree–Fock(HF) wave function expanded in a basis of one-electron Gaussianfunctions. In particular, we chose a completely uncontracted 6-311G (d,p) Pople basis set for our Gaussian primitives, which iswell suited to our test set of molecules [19,22]. All molecular struc-tures were optimized using this level of theory and subsequentlythe resultant canonical molecular orbitals were localized usingthe GAMESS [23] software package. We chose to employ three ofthe most commonly applied localization procedures including theFoster–Boys [24,11] (FB), the Edmiston–Ruedenberg [12] (ER),and the Pipek–Mezey [13] (PM) routines, all of which are imple-mented in the GAMESS package. The FB method creates localizedmolecular orbitals by maximizing the distance between centersof charge. The ER method involves maximizing the self-repulsionenergy within each orbital while the PM method spatially localizesthe molecular orbitals by having the orbitals span as few atoms aspossible. More simply, FB deals with charge, ER deals with energy,and PM deals with space. The efficacy of these three methods as itpertains to this work is also of key interest here.

If we produce q2(r1, r2) from a Hartree–Fock (HF) single deter-minant wave function expanded in a basis of one-electron Gauss-ian functions, then Eq. (2) may be reduced to

PðuÞ ¼Xlmkr

ClmkrðlmkrÞP ð4Þ

where Clmkr is the localized HF two-particle density matrix and(lmkr)P are the resultant integrals over the Gaussian primitivesindexed by l, m, k and r. An analogous procedure in momentumspace will reduce (3) to

MðvÞ ¼Xlmkr

ClmkrðlmkrÞM ð5Þ

where the (lmkr)M now represent momentum space integrals. Both(lmkr)P and (lmkr)M have been solved analytically with Gaussianprimitives of general angular momentum and have been previouslyreported [25]. The study of P(u) and M(v) for particular LMOs isafforded by restricting the summation in (4) and (5) to be over onlythose primitives belonging to the particular molecular orbital ofinterest. Eqs. (4) and (5) dictate that our analysis formally scalesas K4, where K is the number of basis functions. However, given thatwe work in a space of LMOs, increasing the size of the moleculedoes not significantly increase the computational cost.

In addition to the qualitative shape of the electron pair distribu-tion functions we also attempt to extract quantitive informationfrom P(u) and M(v). In all of our cases, both P(u) and M(v) exhibita unimodal distribution with a global maximum near 1–2 a.u.,monotonically decaying in all directions from this maximum. Gi-ven the fact that they are probability distributions for a singlelocalized electron pair, we should not generally expect this trendto differ, with the possible exception of such exotic species asthree-center-two-electron bonds whereby the number of potentialwells outnumber the electrons. As such, we have chosen severalquantitative metrics, including the positions of the maxima, the

Fig. 6. Position (top) and momentum (bottom) intracules for methane, fluoromethane, difluoromethane and trifluoromethane. The corresponding orbitals are indicated onthe right with a contour value of 0.10 a.u.



Laplacian of the function at the maximum, and the first inversemoments of the functions. The positions of the maxima are anobvious choice as they indicate the most probable interelectronicdistance for a given orbital. The degree to which the electron pairis likely to deviate from this most probable distance is given bythe curvature of the intracule about its maximum, which is mea-sured by the value of the Laplacian of the intracule at that point

dI ¼ r2IðxmaxÞ ð6Þ

where I refers to the intracule (position or momentum) and x refersto the independent variable (u or v). A relatively high value of dI

(since we are investigating a maximum, dI will always be negativeand therefore a value close to 0 is considered high) indicates a highdegree of fluctuation in the interelectronic distance between theelectrons. Conversely, a relatively low value indicates a low degreeof fluctuation in the interelectronic distance.

For our purposes, the first inverse moment of the position intra-cule (P�1) is a natural choice as it is directly equated to the Cou-lomb repulsion energy of an electron pair [19,22]

P�1 ¼Z 1

0PðuÞ u�1du ¼ EJ : ð7Þ

This quantity is therefore of obvious utility. For the purposes of con-sistency we chose the first inverse moment of M(v) as well, which issimilarly defined as

M�1 ¼Z 1

0MðvÞ v�1dv : ð8Þ

While this does not represent a physical quantity like EJ, it is relatedto the kinetic energy of the electron pair [15].

Our code takes LMO coefficients as input (in addition to otherstandard information such as the basis set and molecular geome-try) and uses those to generate Clmkr and contract it with inter-nally calculated (lmkr)P or (lmkr)M integrals according to (4) and(5). The portion of our code that calculates the integrals (lmkr)P

and (lmkr)M was modified from code written by Hollet and Gill[25]. The output is P(u) or M(v) calculated over a series of u or vpoints on a radial grid defined by Mura and Knowles [26]. With thisgrid, a total of imax grid points are chosen such that

ðu or vÞi ¼ �R log 1� iimax þ 1

� �3 !

ð9Þ

where R is a scaling parameter that allows the user to control therange of the variable. We then interpolate the resultant data for fur-ther analysis using the Mathematica package [27]. R and imax musttherefore be chosen such that the interpolated P(u) and M(v) haveconverged. For position intracules, we find that R = 7.0 andimax = 250 affords a suitable grid density. Momentum intraculesgenerally required a denser grid that extended farther out in v spaceand we find that R = 25.0 and imax = 500 are appropriate.

3. Results and discussion

In Section 3.1 we compare results from each of the three local-ization procedures. Section 3.2 describes some initial applicationsto simple p-block hydrides and also begin investigating effects ofinduction by electron withdrawing substituents. Then, in Section3.3 we expand to include molecules with two first row p-blockatoms and consider dative bonds and strained moleculargeometries.

Fig. 7. A comparison of the position (top) and momentum (bottom) intracules of C–C, C–N, C–O and C–F bonds. Orbitals are depicted to the right with a contour value of0.10 a.u.



3.1. Localization methods

As mentioned previously, three localization methods wereexamined: FB ‘‘charge’’ localized orbitals, ER ‘‘energy’’ localizedorbitals, and PM ‘‘spatially’’ localized orbitals. Each of these iswell known and described in the literature [11–13] so we willnot dwell on them here except to report the extent to whichour results depend upon the choice of orbitals. We use a proto-typical water molecule to illustrate our conclusions here,although they are consistent across our complete set of mole-cules. Intracules of the O–H bond LMOs from H2O are shown inFig. 1 for each type of orbital. The figure illustrates that the pairof electrons described by this particular LMO will generally notbe found coincident with each other (i.e. P(0) = 0) nor will theybe found separated by large distances (i.e. P(>8) � 0). Their pref-erence would instead appear to be a separation of about 1.2 a.u.,where P(u) exhibits a maximum. In momentum space, the mostprobable relative momentum between the electron pair is about1.6 a.u. Qualitatively, there is very little difference between eachtype (the FB and ER LMO intracules are indistinguishable in thefigure) in position or momentum space with the exception of aslightly broader P(u) distribution from the PM LMO and hencea slightly lower maximum in P(u) to retain normalization. Quan-titatively, we can measure this difference in several ways. Themaximum difference between PFB(u) and PER(u) is 0.00076 a.u.In comparison, the maximum difference between PFB(u) andPPM(u) is 0.014 a.u., which still only equates to a 2.6% differencein the functions at that particular point. The analogous quantitiesin momentum space are smaller still at max(jMFB(v) �MER

(v)j) = 0.00057 a.u. and max(jMFB(v) �MPM(v)j) = 0.0097 a.u.

We may also calculate the normalized absolute difference con-tent (NADC) [19] as

DI ¼Z 1

0jDIðxÞj dx ð10Þ

where I refers to the intracule type and x is the associated indepen-dent variable. In comparing the ER intracules to either of the othertwo, we find that DP = 0.029 and DM = 0.024, which indicatesexcellent agreement between the results for all three orbital typesconsidering that these distributions are all normalized to unity.While the analysis is shown here for the water molecule only, theseresults were consistent for all molecules under investigation. Inother words, the FB, ER and PM bond LMOs gave almost identicalintracules in both position and momentum space for all moleculesstudied. These results suggest, thankfully, that our analysis is gener-ally independent of the localization scheme.

A significant difference is observed though in the LMOs repre-senting lone pairs due to the fact that PM LMOs do not enforcethe same symmetry constraints as the FB and ER analogues. PMlone pair LMOs do not posses degenerate symmetry as in the caseof FB and ER LMOs and thus are not comparable (see Fig. 2).

If one insists upon the chemically intuitive picture of havingequivalent lone pairs on a central atom (where appropriate) as ina standard Lewis structure, then FB or ER LMOs are clearly prefer-able. As with the case of the intracules from O–H bond LMOs, theintracules from lone pair LMOs employing either FB or ER localiza-tion are practically equivalent.

One final consideration is the ease with which these LMOs areconverged. Generally, FB orbitals are more difficult to convergeand have more problematic cases than the other types. Since intra-

Fig. 8. A comparison of the position (top) and momentum (bottom) intracules of N–C, N–N, N–O and N–F bonds. Orbitals are depicted to the right with a contour value of0.10 a.u.



cules from ER LMOs are essentially identical to those calculatedfrom FB LMOs, ER localization is a more robust choice for our anal-ysis and consequently all of the subsequent results reported in thispaper are generated with ER LMOs.

3.2. X–H bonds in p-block hydrides

Our first foray into the localized pair model of electronic struc-ture began with the X–H bond in simple p-block hydrides; the ser-ies of BH3, CH4, NH3, H2O and HF, and the series of AlH3, SiH4, PH3,SH2 and HCl. The intracules for the X–H bond LMOs in these mol-ecules are shown in Fig. 3. Additionally, some properties of thesedistributions are provided in Table 1, including the values of uand v at the maximum, the first inverse moments, the Laplacianat the maximum and the optimized bond lengths.

All of our intracules show a unimodal distribution with a globalmaximum near 1–2 a.u., monotonically decaying in all directionsfrom this maximum. The intracules show a consistent trend, wherethe electronegativity of the heavy atom seems to play a significantrole in determining the shape of the distribution. The P(u) plots(top) show the hydrogen–fluorine bond peaking sharpest and fur-thest to the left with dP = �1.45 and umax = 1.113. This indicatesthat the electrons in the H–F bond are not only the closest togetheron average, but that they are the least likely to fluctuate signifi-cantly from their most probable distance and it can be said thatthis is a ‘‘tight’’ bond in terms of the electron pair distribution.The intracules generally are broader and exhibit a higher dP andumax as one moves left across the periodic table, where the Al–Hbond has the broadest distribution. This trend correlates stronglywith the well known electronegativity of the atoms. Since fluorine

Fig. 9. A comparison of the position (top) and momentum (bottom) intracules of O–C, O–N, O–O and O–F bonds. Orbitals are depicted to the right with a contour value of0.10 a.u.

Table 2Properties of selected bonds in a set of small molecules. Bond length values (in Å) were determined from the GAMESS calculations done in this work. Bond dissociation energies(BDE) are experimental values from Darwent [28] unless otherwise noted. All values are in atomic units unless otherwise stated.

Molecule Bond umax P�1 dP vmax M�1 dM rX–Y (Å) BDE (kcal/mol) [28]

C2H6 (C–C) 1.516 0.6878 �0.7890 1.612 0.7404 �0.6989 1.527 88 ± 2 [33]CH3NH2 (C–N) 1.408 0.7440 �0.9346 1.757 0.6755 �0.5601 1.455 79 ± 3CH3OH (C–O) 1.294 0.8134 �1.0553 1.917 0.6079 �0.4229 1.400 90 ± 3CH3F (C–F) 1.168 0.9015 �1.1782 2.093 0.5408 �0.3092 1.364 108 ± 5N2H4 (N–N) 1.365 0.7688 �1.0336 1.882 0.6407 �0.5119 1.413 59 ± 3NH2OH (N–O) 1.312 0.8024 �1.0530 2.029 0.5964 �0.4422 1.390 61.3 [34]NH3F (N–F) 1.231 0.8588 �1.0253 2.195 0.5423 �0.3629 1.376 73.5a [35]H2O2 (O–O) 1.319 0.7980 �0.9823 2.169 0.5715 �0.4328 1.385 51.1 ± 0.5HOF (O–F) 1.277 0.8324 �0.9401 2.323 0.5304 �0.3814 1.361 54 [35]

a ab initio



is the most electronegative atom, it has the ability to pull the elec-trons closer than the less electronegative aluminum atom. For M(v)(bottom), this trend is reversed. In the momentum case, the H–Fbond shows the broadest distribution (dM = �0.23), whereas theAl–H bond peaks higher and furthest to the left of the others(dM = �3.45 and vmax = 0.904). This is to be expected, since elec-trons that are very close together (as in the H–F bond) will alsomove more quickly relative to one another. Consequently, they willexhibit a broad distribution in momentum space [15]. Conversely,electrons that are further apart will tend to move more slowlyrelative to one another.

It would appear that although these results suggest an intuitiverelationship between the shape of P(u) or M(v) and the disparitybetween the electronegativity of the bonded atoms, this is not auniversal trend. The overlap of the first and second row X–H bondintracules is offset from what would be expected based on electro-negativity values alone (e.g. Cl is more electronegative than C andyet their intracules in Fig. 3 are nearly coincident). One might alsoexpect that the average distance between electrons within a cova-lent bond would be related to the bond length. However, this isalso not observed. Returning again to the example of the C–H bondin methane as compared to the Cl–H bond in hydrogen chloride,despite the fact that the intracular distributions of their X–H bondsis essentially identical, they differ significantly in bond length. Alikely interpretation is that the increased electronegativity of theCl atom draws the electron pair closer to it (and consequently clo-ser to each other) and thus P(u) for this system effectively matchesthat of the C–H bond despite having a significantly longer bond.

Interestingly, the overlapping X–H bond position intracules ofthe BH3 and SH2 molecules are not so similar in momentum space.This indicates that despite these electrons having a practically

identical P(u), the bond is fundamentally different and the elec-trons exhibit differing relative momenta in each of these bonds.Conversely, both the P(u) and M(v) distributions are nearly identi-cal for the X–H bonds in methane and hydrogen chloride. Thesefacts suggest that both P(u) and M(v), in the context of LMOs, offernovel information about the electronic structure and properties ofelectrons in these systems and may readily serve as a bond charac-terization tool. Our results are both intuitive and unique.

Given that the bond intracules for methane and hydrogen chlo-ride seem to be so remarkably identical despite obvious differencesin the molecules themselves, we chose to consider these furtherand discovered that these molecules also have identical (to withinexperimental uncertainty [28]) bond dissociation energies (BDEs)for the X–H bond. Table 1 lists the bond types studied here alongwith their experimentally determined BDEs (where available; the-oretical data supplied in the absence of experiment) as well as P�1,dP, M�1 and dM.

From the table we confirm that bond length does not correlateto the shape of either P(u) or M(v) as any two cases with similarfirst inverse moments or Laplacians exhibit very different bondlengths. What is apparent from the data in the table however, isthat the BDE of a particular bond does seem to correlate well tothe values for P�1 and/or M�1. Fig. 4 illustrates the linear correla-tion obtained between the experimentally determined BDEs andP�1 (R2 correlation coefficient of 0.95) as well as M�1 (R2 correlationcoefficient of 0.94). It is important to note here that we have notincluded the BDE data for BH3 or AlH3 in our linear fits becausethese species do not typically exist in monomeric form and there-fore their X–H BDEs are somewhat spurious.

Our correlation is somewhat remarkable, since our data is ulti-mately from the Hartree–Fock theory 2-matrix and is therefore

Fig. 10. Position (top) and momentum (bottom) intracules for N ? B dative bonds in H3NBH3, F3NBH3 and H3NBF3. The corresponding orbitals are depicted on the right with acontour value of 0.10 a.u.



lacking the necessary physical basis to describe the bond breakingprocess. BDEs determined explicitly from Hartree–Fock theorywould not even be qualitatively correct, let alone accurate to with-in 5 kcal/mol, as our fitted functions are. Therefore, to see a rela-tively strong correlation is encouraging and indicative of adeeper potential within the localized pair model. Of course weshould not expect P�1 or M�1 to be generally related to BDEs dueto the increased complexity of the bond breaking process with lar-ger molecules. It is merely an interesting result that points to anunderlying potential within the model.

We have also explored the distribution of electron pairs withinthe lone pair LMOs of the same p-block hydrides and these areshown in Fig. 5. Again, we note that the electronegativity of thecentral atom has a contracting effect on the distribution of the elec-tron pair within the specific LMO. This is evidenced by the gradualbroadening of the P(u) as one proceeds from right to left across theperiodic table. Similarly, as the electron pairs contract in positionspace, they move more quickly relative to one another and broadentheir momentum space intracular distributions. It would be inter-esting to determine the extent to which these distributions changewith participation in non-covalent interactions.

We were also interested in whether the localized pair modelcould resolve differences in the electronic structure of bonds thatwere more distant from the chemical substitutions and thus providea direct measure of inductive effects. The C–H bond in the series ofmethane, fluoromethane, difluoromethane and trifluoromethanewas examined. The position and momentum intracules for theseC–H bond LMOs are shown in Fig. 6 and we did indeed observe smallbut significant differences between the electron pair distributions in

these bonds (data presented in Table 1). An expected trend emerges;where the more fluorinated compounds exhibit narrower positionintracules and broader momentum intracules. However, this is amore subtle trend than seen in the previous cases due to the relativeproximity of the F substituent to the C–H bond.

3.3. Bonds involving two heavy atoms

To expand upon our test set and increase the complexity of thecovalent bonds under scrutiny we chose to explore simple mole-cules with two heavy atoms. We again turned our attention to C,N, O, and F and studied the series of saturated small molecules con-taining a single bond between the various permutations of pairs ofthese atoms. Our computed intracules for the X–Y bond LMOs(where X and Y refer to the two central heavy atoms in each mole-cule) are shown in Figs. 7–9 and the accompanying data is listed inTable 2. For the purposes of clarity we show our intracules in threeseries of molecules to afford a convenient description of their prop-erties and trends. The first series is that of C2H6, CH3NH2, CH3OH andCH3F, which includes some of the most ubiquitous bond types in or-ganic chemistry, C–C, C–N, C–O and C–F bonds, as shown in Fig. 7.

When one of the carbons is replaced with increasingly electro-negative atoms, both umax and dP decrease, indicating that the elec-trons are closest together in the C–F bond (and presumably alsocloser to the F atom). Conversely, the momentum intracule is morebroadly distributed with the more electronegative atom. This,again, is intuitive as the more electronegative atom will draw theelectrons closer to it, and they will consequently become closer to-gether. Compared with the C–H bond in methane, P(u) for the C–H

Fig. 11. Position (top) and momentum (bottom) intracules for N–H and B–H bonds in H3NBH3, F3NBH3, H3NBF3, NH3, and BH3. The corresponding orbitals are depicted on theright with a contour value of 0.10 a.u.



bond in the fluorinated analogues are very similar despite eachhaving a �50% longer C–H bond. This suggests a very similar spa-tial distribution of electrons. In momentum space however, there isa more marked difference, where M(v) is significantly more broadfor the larger molecules indicating a greater relative momentumfor these electron pairs.

The nitrogen and oxygen containing series are presented inFig. 8 (containing NH2CH3, N2H4, NH2OH and NH3F) and Fig. 9 (con-taining HOCH3, HONH2, H2O2 and HOF), respectively. Again withthese series, the position intracules exhibit a lower umax and dP

with increasing disparity in the electronegativity of the bondedheavy atoms, and the momentum intracules exhibit a higher vmax

and dM (see Table 2). An intriguing trend that emerges is the factthat P(u) becomes increasingly indistinguishable between the ser-ies of X–Y bonds (Y = C, N, O, F) as X grows increasingly electroneg-ative. For the case of the O–Y bonds, it is difficult to resolvedifferences in P(u) between the distributions within Fig. 9 with

the eye. These electron pairs are much more easily distinguishedusing M(v). Therefore, M(v) can effectively resolve differences be-tween bond types even when P(u) distributions are very similar.It is also illustrated in these figures that M(v) appears to be moresensitive to the nodal structure of the particular LMO than P(u),resulting in ‘‘shoulders’’ that appear within M(v).

Of course we would not expect such a correlation here betweenP�1 or M�1 and BDEs as observed for the case of X–H bonds in theprevious section, and indeed none is found. However, it is interest-ing to note that P(u) can recover the non-monotonic behavior ofBDEs in these more complicated cases. Note for example the seriesof O–Y bonds (Y = C, N, O, F). In all other cases, our intracules varysmoothly and monotonically (as measured by such properties asumax, vmax, dP, dM, P�1 and M�1) as one progresses across a seriesaccording to the periodic table. For the case of O–Y bonds however,both umax and P�1 predict that the O–F bond will be an outlier inthe series, just as their BDE values confirm.

Table 3Properties of selected bond intracules in H3NBH3, F3NBH3 and H3NBF3, as compared to BH3 and NH3. Bond length values were determined from the GAMESS calculations done inthis work. All values are in atomic units unless otherwise stated.

Molecule Bond umax P�1 dP vmax M�1 dM rX–Y (Å)

H3NBH3 (N ? B) 1.402 0.7304 �0.7154 1.662 0.6946 �0.6636 1.678F3NBH3 (N ? B) 1.252 0.8306 �1.1101 1.631 0.6248 �0.5717 1.786H3NBF3 (N ? B) 1.404 0.7444 �0.7497 1.617 0.6756 �0.4681 1.683NH3 (N–H) 1.367 0.7614 �0.9415 1.447 0.7215 �0.8101 1.001H3NBH3 (N–H) 1.349 0.7762 �0.9941 1.489 0.7015 �0.7559 1.003H3NBF3 (N–H) 1.351 0.7762 �0.9897 1.486 0.7031 �0.7521 1.004BH3 (B–H) 1.675 0.6344 �0.5392 1.100 0.9531 �1.892 1.189H3NBH3 (B–H) 1.700 0.6198 �0.5020 1.097 0.9642 �1.9773 1.210F3NBH3 (B–H) 1.684 0.6284 �0.5254 1.110 0.9413 �1.8989 1.195

Fig. 12. Position (top) and momentum (bottom) intracules for C–C bonds in ethane, cyclopropane and cyclobutane. The orbitals are depicted on the right with a contour valueof 0.10 a.u.



3.3.1. Dative bondsWe have also pursued the analysis of the electronic structure of

dative bonds to complement our analysis of simple covalent bonds.In addition to predicting the electron pair distribution within a da-tive bond, we investigated the effect on the dative bonds whenhydrogen atoms were replaced with electronegative fluorine atoms(again, probing inductive effects). The series of H3NBH3, F3NBH3

and H3NBF3 were studied and we investigated the N ? B dativebonds as well as the covalent N–H and B–H bonds. These resultsare presented in Figs. 10 and 11 as well as Table 3.

One can immediately notice that the N ? B dative bond orbitalssignificantly differ from those of the other covalent bonds consid-ered thus far, due to the primary contribution from the N atom. De-spite the qualitatively different shape of the N ? B bond LMOhowever, the intracular distributions are similar to what wasobserved for N–N bonds. The key difference is in the response tosubstitution by fluorine. The N ? B dative bond intracules are rela-tively insensitive to having electronegative substituents placed onthe boron atom. However, the bond P(u) exhibits a significantlysmaller umax when fluorine atoms are placed on the nitrogen. Thisis intuitive since nitrogen is the electron donor and thus the fluorineatoms directly impact the density that N is able to contribute to thedative bond. Conversely, since boron is the acceptor, it’s electrondeficiency has a nearly negligible effect on the bond. Again, we notethat the localized pair model recovers the qualitatively intuitivechemical differences between these various bond types while offer-ing additional insight in terms of the specific electron distribution.

Both the N–H and B–H bond intracules retain a high degree ofsimilarity to their respective hydrides, NH3 and BH3, regardless ofwhether the N or B participate in a dative bond or the level of Fsubstitution on the neighboring group. In Fig. 11, we see thatalthough the N–H and B–H bond intracules differ markedly fromeach other, they follow closely with their substituted analogues.

3.3.2. Bonds in molecules under geometrical strainFinally we chose to study bond intracules in the series of eth-

ane, cyclopropane and cyclobutane to investigate the effect ofvarying degrees of strain in a molecule. These results are presentedin Fig. 12 and Table 4.

The ethane and cyclobutane intracules seem to correlate well tothe bond lengths. Cyclobutane has a longer C–C bond length and aslightly broader P(u) distribution than ethane, although the M(v)for all three are similar. Cyclopropane, on the other hand, has ashorter bond length than the other two but its position intraculeis broader, not narrower, as might be expected. Of course, the rea-son is due to the proximity of the cyclopropane C–C bond to theother C–C bonds in the system. The electron pair within a particu-lar C–C bond is forced away from the midpoint of its constituentatoms by Coulomb repulsion from the nearby electron pairs andis therefore bowed. Interestingly, this results in a smaller Coulombrepulsion energy (see P�1 in Table 4) between the electrons in theC–C bond in cyclopropane than the analogous bond in cyclobutaneor even ethane (where there is obviously no strain).

In momentum space there is a less pronounced difference be-tween the bond intracules in this series, although M(v) for C3H6

is somewhat shifted to smaller v as compared to the others.

4. Conclusion

We have developed a novel electronic structure analysis tech-nique whereby we determine the distribution (in both positionand momentum space) of electron pairs within the chemical bond,as well as other entities such as a Lewis lone pair. This is accom-plished by localizing the canonical molecular orbitals and calculat-ing the relative interelectronic probability distributions, hereinreferred to as intracules. In this initial exploratory work, we se-lected a wide range of covalent and dative bonding structures tounderstand the relationship between this localized pair modelinterpretation and known intuitive chemical features.

From a pragmatic perspective, the localization scheme of Edm-iston and Ruedenberg [12] seemed to be the most reasonable forthe application of our model and it was therefore employed on aseries of molecules including p-block hydrides, saturated maingroup compounds, fluorinated species, N ? B dative structures,and small cyclic molecules. The resultant intracules for a wide vari-ety of bond types and lone pairs show a unimodal distribution witha global maximum near 1–2 a.u., monotonically decaying in bothdirections from this maximum. We subsequently examined theseintracules both qualitatively and quantitatively using metrics suchas the positions of maxima, their Laplacians at the maxima, andtheir first inverse moments.

We find that our analysis generally agrees quite well with intu-itive predictions based on bond lengths and electronegativities ofthe bonded atoms. However the trends in the data cannot be pre-dicted using the bond length or electronegativity alone, which dem-onstrates the unique features of this method. In fact, the correlationcoefficient for a linear fit between umax and r for all molecules stud-ied is surprisingly poor (R2 = 0.03). Generally though, a short bondwith a large disparity between the electronegativities of its constit-uent atoms will produce a very narrow P(u) distribution that peaksnearer the origin. Consequently, M(v) for such a bond will be broad,to reflect the fact that electrons which are close together are gener-ally moving more quickly relative to each other. Additionallythough, it is not uncommon to see different bond types with nearlyidentical P(u) or M(v), differ significantly in the other. This suggeststhat chemical bonds may be uniquely characterized by the relativepositions and momenta of the electrons within them, and the local-ized pair model accomplishes exactly this. In the curious rare casewhere we observed nearly identical P(u) and M(v), we found thatthe experimental bond dissociation energy (BDE) of these bondswere also identical. While there is a strong correlation betweenthe inverse moments of P(u) and M(v) and BDE, more work is re-quired to explore the potential of the localized pair model as a pre-dictor of bond strength. In the event that the model is useful in thisregard, one may conceivably pursue the localized pair model as anenergy decomposition scheme, not entirely unlike that of the local-ized methods of Head-Gordon et al. [36].

The localized pair model of electronic structure analysis allowsone to uncover a rich and yet untapped topology of electron–elec-tron interactions in atomic and molecular systems. This differenti-ates the model from many other electron density analysistechniques that rely on the one-electron density alone. Given therelatively high dimensionality of the 2-matrix as compared withthe one-electron density, there are obviously many other pair dis-tributions that may be interesting and relevant, which are relatedto our work. Extracule (center-of-mass) densities and higherdimensional intracules promise to have an even richer topologyand should be fruitful for future research. Also, because we areconcerned only with individual pairs of electrons described by asingle molecular orbital, we may be able to explore the use ofDFT with the localized pair model, as the Coulomb component ofan intracule is completely determined by the one-electron density.

Table 4Properties of C–C bonds in ethane, cyclopropane and cyclobutane. Bond length valueswere determined from the initial geometry optimization. All values are in atomicunits unless otherwise stated.

Molecule umax P�1 dP vmax M�1 dM rC–C (Å)

C2H6 1.516 0.6878 �0.7890 1.612 0.7404 �0.6989 1.527C3H6 1.548 0.6568 �0.6556 1.567 0.7210 �0.7897 1.499C4H8 1.523 0.6826 �0.7662 1.630 0.7240 �0.7161 1.546



This will also allow for the analysis of how the effects of electroncorrelation change the distribution of electron pairs in chemicalbonds and will provide a link between the localized pair modeland other analysis techniques such as Bader’s QTAIM [37].

We never had the great privilege of meeting the late Prof. A.John Coleman in person, though we spent countless hours withhis written words. The wealth of potential he described that lieswithin the reduced density matrix served as great personal inspira-tion for our work and that of many others. We therefore humblydedicate this work to his memory.

Acknowledgements

The authors acknowledge the University of Prince Edward Is-land, the Natural Sciences and Engineering Research Council ofCanada, and the Canadian Foundation for Innovation for financialsupport. The authors thank the Atlantic Computational ExcellenceNetwork for computational resources and Dr. Josh Hollet for pro-viding code which we modified for this work.

We would also like to thank an anonymous referee for pointingout Refs. [20,21] to us in addition to recommending the use of theLaplacian of the intracule density as a quantitative metric.

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