Nonnuclear Attractors in Heteronuclear Diatomic Systems.

Subscriber access provided by UNIV OSNABRUECK

Article

Non-nuclear Attractors in Heteronuclear Diatomic Systems Luiz Alberto Terrabuio, Tiago Quevedo Teodoro, Cherif F. Matta, and Roberto Luiz Andrade Haiduke J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b10888 • Publication Date (Web): 03 Feb 2016 Downloaded from http://pubs.acs.org on February 7, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Non-Nuclear Attractors in Heteronuclear Diatomic Systems

Luiz Alberto Terrabuio,a Tiago Quevedo Teodoro,a Chérif F. Matta,b and Roberto Luiz Andrade Haiduke a,* * [email protected] a

Departamento de Química e Física Molecular, Instituto de Química de São Carlos,

Universidade de São Paulo, Av. Trabalhador São-Carlense, 400 – CP 780, 13560-970, São Carlos, SP, Brazil b

Department of Chemistry and Physics, Mount Saint Vincent University, Halifax, Nova

Scotia, Canada B3M 2J6; Dalhousie University, Halifax, Nova Scotia, Canada B3H 4J3; and Saint Mary's University, Halifax, Nova Scotia, Canada B3H 3C3.

ACS Paragon Plus Environment

1


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 23

Non-Nuclear Attractors in Heteronuclear Diatomic Systems Abstract Non-nuclear attractors (NNAs) are observed in the electron density of a variety of systems but the factors governing their appearance and their contribution to the system’s properties remain a mystery. The NNA occurring in homo- and hetero-nuclear diatomics of main group elements with atomic numbers up to Z=38 is investigated computationally (at the UCCSD/cc-pVQZ level of theory) by varying internuclear separations. This was done to determine the NNA occurrence window along with the evolution of the respective pseudo-atomic basin properties. Two distinct categories of NNAs were detected in the data analyzed by means of catastrophe theory. Type “a” implies electronic charge transfer between atoms mediated by a pseudo-atom. Type “b” shows an initial relocation of some electronic charge to a pseudo-atom, which posteriorly returns to the same atom that donated this charge in the first place. A small difference of polarizability between the atoms that compose these heteronuclear diatomics seems to favor NNA formation. We also show that the NNA arising tends to result in some perceptible effects on molecular dipole and/or quadrupole moment curves against internuclear distance. Finally, successive cationic ionization results in the fast disappearance of the NNA in Li2 indicating that its formation is mainly governed by the field generated by the quantum mechanical electronic density and only depends parametrically on the bare nuclear field/potential at a given molecular geometry.


2

Page 3 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Introduction

A non-nuclear attractor (NNA), also known as non-nuclear maximum (NNM), corresponds to a critical point that accounts for a local maximum in the electron density of molecular systems, which occurs at a position without a nucleus. Despite this difference, a NNM is topologically identical to a (3, -3) nuclear attractor. Hence, the NNA attracts its own density gradient vector field forming a basin that is analogous to the regular atomic basins of the Quantum Theory of Atoms in Molecules (QTAIM),1- 3 2

which are bound by zero-flux surfaces. NNA basins are proper open quantum systems with well-defined properties such as electron population, energy components, electric moments, volume, etc., in virtue of which they are often called “pseudo-atoms”. Pseudo-atoms can share a common interatomic zero-flux surface, a bond critical point, and a bond path to atoms and other pseudo-atoms in a molecule. Thus, it is important to emphasize that this pseudo-atom definition used in the context of QTAIM should not be confused with that one proposed by Stewart, which was born out in studies concerning the refinement of X-ray diffraction structure factors.4 The first theoretical clue of NNMs dates back to 1955 (Li2).5 Since then, they have been observed in several physical systems such as metals,6- 8 semiconductors,9,10 7

electrides,11 crystals defects and color F-centers,12 and associated with solvated electrons.13 Hence, NNAs are of considerable theoretical interest. In addition, their study has gained renewed attention in recent years due to the first experimental evidence, which was obtained in Mg-Mg bonds of dimeric magnesium (I) compounds.14,15 However, the exact role of NNMs in imparting properties such as electrical conductivity or strength of materials remains to be worked out.


3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 23

The origin of these peculiar electron density maxima is ascribed to the overlap of specific electron shells of certain pairs of atoms, mostly those known as nonconvex.16 However, although the interaction of homonuclear atoms can lead to cases theoretically well-studied of NNAs,16,17 the research of NNMs in heteronuclear bonds is not so advanced. Pendás et al. have also discussed that these maxima should be exceptional in heterodiatomic compounds.16 Recently, our research group found one NNA in the equilibrium geometry of a heteronuclear diatomic molecule by means of reliable electronic structure calculations, LiNa.17 Moreover, the BP crystal also was presented as one example of NNMs in heteronuclear bonds.18 This last study has also pointed out that a small electronegativity difference favors the occurrence of NNAs. However, many details of the NNM formation process remain unclear. Hence, this work is devoted to a wider study of NNMs in heteronuclear diatomic systems by means of calculations at the unrestricted coupled cluster level with single and double excitations (UCCSD) and quadruple-ζ basis sets. First, we reinvestigate homonuclear diatomic molecules formed by main group elements with atomic number (Z) between 1 and 38 to a better characterization of occurrence windows and other quantities related to NNAs. Next, a combinatorial study is performed at internuclear distances selected according to the results of these homonuclear counterparts and 15 examples of heteronuclear diatomic systems containing NNMs are discussed. Thus, a classification of these molecules in two categories is presented and the role of the polarizability difference between constituting atoms is raised. Finally, extreme points in molecular dipole and quadrupole moment curves against internuclear distance (maximum or minimum) are investigated to verify if there is some relation between these values and the window for NNA occurrence.


4

Page 5 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Computational details

The electronic structure calculations were carried out within the Gaussian 03 package.19 The UCCSD method is chosen along with cc-pVQZ basis sets,20 - 25 except 2122

2324

for K, Rb and Sr atoms, which were treated by basis sets of similar quality.26 Allelectron excitations have been considered in these calculations. Moreover, the investigated quantities are derived from generalized densities. This treatment was already shown to be a satisfactory choice in terms of the predictive capability for NNM occurrence and the associated demand for computational resources, as verified in Ref. 17 by means of a comparative study with other methods such as Complete Active Space Multiconfiguration Self-Consistent Field (CAS-SCF) and Restricted Active Space (RASSCF) variants. The AIMAll package was selected to perform the QTAIM partitions and to obtain the critical point data.27 The molecules were always positioned along the Cartesian z axis. Results and Discussion A. Homonuclear diatomic systems Figure 1 and Table 1 show the windows for NNA occurrence in homonuclear diatomic molecules as obtained from our UCCSD/cc-pVQZ calculations. First, it is important to realize that the interaction of K shell electrons does not provide nonnuclear maxima for H2 and He2 (K, L, M, N,… atomic shells are labelled according to the customary convention in terms of their principal quantum number value, n=1,2,3,4,…). One can see that the experimental equilibrium bond length28 is inside or very close (up to 0.1 Å) to the distance range presenting NNMs only for Li2, B2, C2,


5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 23

Na2, Al2, Si2 and P2. The shrinking in this window size observed for second-row compounds can be easily rationalized because the valence shell of these elements (L) becomes more compact as one moves from lithium to neon due to increasing effective nuclear charge effects over these electrons. Furthermore, the decrease noticed in the minimum distance value of this range along with atomic number increments (also observed for third-row diatomics) is explained similarly.

a Figure 1: The most important internuclear distance window (Å) for observing NNAs in homonuclear diatomic systems according to UCCSD/cc-pVQZ calculations (circles represent the experimental equilibrium bond lengths).28


6

Page 7 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1: Internuclear distance window (Å) for observing non-nuclear maxima in homonuclear diatomic systems according to UCCSD/cc-pVQZ calculations.a Molecule

Number of non-nuclear attractors

(spin multiplicity) Li2 (1)

1

2

2.08 – 3.41

3.42 – 3.74

Be2 (1)

1.39 – 2.13

2.14 – 2.25

B2 (3)

1.23 – 1.53

0.84 – 1.22

C2 (1)

0.81 – 1.21

1.22 – 1.23

N2 (1)

0.65 – 0.93

O2 (3)

0.56 – 0.80

F2 (1)

0.49 - 0.70

Ne2 (1)

0.44 - 0.62

Na2 (1)

3.18 – 3.79

Mg2 (1)

2.52 - 2.60

Al2 (3)

2.02 - 2.18; 2.50 – 2.51

0.71

2.52

Si2 (3)

1.57 – 2.20

2.21 – 2.23

P2 (1)

1.31 – 1.91

1.92 – 1.94

S2 (3)

1.14 – 1.71;

1.72 – 1.74;

(0.32 – 0.34)

(0.21 – 0.30)

1.02 – 1.54;

1.55 – 1.58;

(0.29 – 0.31)

(0.21 - 0.27)

Ar2 (1)

0.97 – 1.34

1.35 – 1.37

K2 (1)

0.82 – 1.22

1.23 – 1.24

Ca2 (1)

0.77 – 1.11

0.64 - 0.70;

Ga2 (3)

0.44 – 0.58

Ge2 (3)

0.42 – 0.55

Cl2 (1)

3

(0.31)

(0.28)

1.12

As2 (1)

0.41 – 0.53

Se2 (3)

0.40 – 0.52

Br2 (1)

0.39 – 0.50

Kr2 (1)

0.38 – 0.48

Rb2 (1)

1.30

a

The values between parenthesis refer to a secondary inner window for NNA occurrence.

Conversely, it is obvious that the NNAs are not a result of valence shell overlap for systems from K2 to Rb2. One can notice that the ranges seen in Figure 1 for K2 to Kr2 are characteristic of M shell interactions since there is an obvious tendency that


7


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 23

extends from Na2 to these systems. Moreover, the single point observed for Rb2 is more likely associated to the overlap of N shell electrons. These affirmations can be justified by considering the research performed by Boyd in 1977, who presented the minimum radius values of electron shells in atoms as obtained by finding the position of the minima in respective radial electron densities.29 This researcher also discovered that M, N and even O shells are indistinguishable from each other in atoms from potassium to yttrium. In addition, studies based on the Laplacian of the electron density, which is another quantity used to resolve the atomic shell structure,30,31 also pointed out that the N shell is not discernible in heavier elements such as those from Sc to Ge.32 Thus, the absence of clear shell separations or a more subtle shell resolution may be somehow related to the absence of NNAs from the overlap of valence electrons in diatomic homonuclear molecules of such elements. Next, we can use the prediction of minimum shell radii given by Boyd (multiplied by two), as obtained from radial electron densities, to find an estimate for the smallest distance to NNM occurrence. This is seen in Figure 2. We observe that sulfur and chlorine can exhibit NNAs due to overlap of L and M shells, which are discernible in Table 1 because two clearly distinct distance ranges containing these electron density maxima are separated by an intermediate region presenting only an ordinary bond critical point (BCP). Finally, the data really support overlap of N shell electrons for rubidium and the minimum value predicted by radial densities is extrapolated from heavier elements.29


8

Page 9 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2: Minimum internuclear distance for occurrence of NNAs according to our UCCSD/cc-pVQZ calculations and values predicted by parameters from Boyd29 against atomic number.

A last point that deserves some comments is related to the characteristics of NNAs detected. As one can see in Table 1, the most common situation is related to the presence of only one NNM, although molecules constituted by elements with Z = 3-6, 9 and 13-20 can exhibit bond length ranges with two NNAs, that is, there are transition points between electron densities containing one and two of these attractors. S2 and Cl2 may also show three of these maxima in some internuclear distances characteristic of L shell overlap. These NNMs are always disposed along the internuclear axis between the nuclei except for the few cases with 3 of them. However, more advanced calculations are probably required to confirm that these attractors found out of the molecular axis are not mere calculation artifacts because of the very small internuclear distance associated to their occurrence (0.3 Å). Figure 3 presents another feature of NNAs, which is related the largest difference between the electron density at this point and at the adjacent BCP obtained along the entire window associated with overlap of the valence shell in terms of atomic numbers. One can see that these differences increase as one moves along the period and the largest values are found for noble gases. This is surely expected if one


9


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 23

takes into account the compactness and the number of electrons in the valence shell of such elements, that is, the NNM tends to become more distinguishable as its formation occurs from overlap of shells with larger electron density values.

Figure 3: The largest electron density difference observed between the NNA formed by valence shell overlap and the adjacent BCP as obtained from UCCSD/cc-pVQZ calculations against the atomic number.

B. Heteronuclear diatomic systems A combinatorial study was conducted searching for NNAs in heteronuclear diatomic molecules constituted by the previous elements. Thus, 15 cases showed evidence of one NNM in certain internuclear distance ranges. LiNa was already wellcharacterized.17 Moreover, two of the remaining cases, BeAl and ClAr, exhibit clear signs of an unsatisfactory treatment of electron structure to a definitive answer and will be left apart for future investigation. Hence, the diatomic molecules that will be further discussed here (spin multiplicities) are: AsP (1), BrCl (1), ArKr (1) ClKr (2), PS (2), AsS (2), BS (2), ClS (2), CCl (2), SeS (3), BP (3) and CAr (3). As seen in Table 2, the NNA window of these new cases is always located at smaller internuclear distances than the respective equilibrium bond lengths.


10

Page 11 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2 also presents the data of electronegativity and polarizability differences for each pair of elements composing these heteronuclear systems.33,34 As expected,18 one can see that small electronegativity differences are generally observed in all these heteronuclear molecules. However, since some large values are indeed found depending on the scale used and others are missing, the best electronegativity formalism to be used in such an investigation seems to be a controversial question. We have found that the well-defined atomic polarizability is also able to provide satisfactory insights in the prediction of NNAs. The largest atomic polarizability difference in these molecules is only 10 a.u. Thus, these data reinforce the notion that NNAs are favored as little perturbation of the electron clouds of constituent atoms are observed by the formation of the chemical bond. However, if some disturbances take place, as probably occurs in the interaction of highly polarizable elements (Li and Na, for instance), both atoms should respond similarly to this perturbation in order to result in NNM formation, as evidenced by the small polarizability differences found. Hence, this result can guide the studies about the formation mechanism of NNAs occurring, for instance, in diatomic homonuclear systems composed by convex atoms.16 However, it is important to notice that this criterion alone is not sufficient to guarantee the NNM occurrence since there are diatomics that present small or null polarizability differences but do not exhibit NNAs, as is the case in some homonuclear molecules.


11


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 23

Table 2: Spin multiplicity (2S+1), equilibrium bond length (re) and window range for NNA occurrence as obtained in UCCSD/cc-pVQZ calculations along with differences of electronegativity (ε) and polarizability (Pol) between the two atoms constituting each heteronuclear diatomic molecule. Differencesa Molecule 2S+1 re (Å) Distance range (Å) ∆ε(MJ) ∆ε(P) LiNa 1 2.813 2.65 – 3.64c 0.10 0.05 AsP 1 1.987 1.87 – 1.91 0.25 0.01 BrCl 1 2.132 1.30 – 1.49 0.33 0.20 d ArKr 1 4.073 1.14 – 1.25 0.19 ClKr 2 3.616 1.26 – 1.53 0.05 0.26 PS 2 1.887 1.40 – 1.54 1.37 0.39 AsS 2 2.008 1.48 – 1.50 1.62 0.40 BS 2 1.604 1.08 – 1.43 1.40 0.54 ClS 2 1.982 1.19 – 1.43 0.26 0.58 CCl 2 1.647 1.05 – 1.36 0.34 0.61 SeS 3 2.024 1.52 – 1.69 1.03 0.03 BP 3 1.732 1.27 – 1.70 0.03 0.15 d CAr 3 3.282 0.91 – 1.14 0.10

∆ε(S) ∆ε(AR) ∆ε(A) ∆Pol (a.u.)b 0.01 0.04 0.04 1 0.37 0.14 0.04 5 0.32 0.09 0.18 6 0.75 0.20 0.28 6 0.11 0.27 0.10 2 0.50 0.38 0.34 5 0.13 0.24 0.38 10 0.78 0.43 0.54 1 0.62 0.39 0.28 5 0.81 0.33 0.33 3 0.10 0.04 0.17 7 0.28 0.05 0.20 4 1.45 0.80 0.70 0

a

The electronegativity scales used were Mulliken-Jaffe (MJ), Pauling (P), Sanderson (S), Allred-Rochow (AR) and Allen (A);33 b Reference 34; c Reference 17; d Not available.

Further, the heteronuclear systems with NNMs can be divided into two categories with respect to critical points characteristics. The first group (type “a”), which is associated with electronic charge transfer between atoms mediated by a pseudo-atom, is composed by LiNa, AsP, BrCl, ClKr, AsS, BS, CCl, SeS and CAr. The second category (type “b”), in which the electronic charge associated with a pseudoatom leaves from and posteriorly returns to the same atom, includes ArKr, PS, ClS and BP. Some details of both groups become clear as one uses the Catastrophe Theory formalism to investigate NNAs, an approach introduced by Cioslowski in 1990 for Li2.35


12

Page 13 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figures 4 and 5 illustrate an example of the first group, AsP (see the Supplementary Material for other type “a” molecules). Figure 4 shows that this heteronuclear system with NNA exhibits a characteristic crossing of curves representing the electron density ascribed to both BCPs, which lie between the NNM and each nucleus. This occurs because the non-nuclear maximum splits from one and finally merges to the other BCP in the limits of the respective distance range. Moreover, the curve in Figure 5 shows a remarkable regular “S” like shape for the same reason (an inverted “S” would also be possible here). The data in Table 3 point out that the molecules belonging to this group are also associated with a polarity inversion according to QTAIM charges, except by BS (the charges for boron and sulfur are almost zero in the largest distance before the NNA window). The polarity observed in little larger bond lengths than those containing NNMs are always in agreement with the majority of these electronegativity scales. Furthermore, as the internuclear distance is increased along the NNA window, this non-nuclear maximum basin separates from the most polarizable atom and migrates towards a merging with the least polarizable one. The exception is CCl, while the polarizability differences in CAr are almost negligible to allow any reliable characterization.

Figure 4: Electron density at critical points against interatomic distances as obtained in UCCSD/cc-pVQZ calculations for AsP.


13


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 23

Figure 5: Diagram of distances between critical points and the phosphorus nucleus against interatomic distances as obtained in UCCSD/cc-pVQZ calculations for AsP. Figures 6 and 7 present a molecule that belongs to the second category of heteronuclear systems, BP (the Supplementary Material contains similar illustrations for other members of this group). In contrast to type “a”, Figure 6 does not show any crossing of the electron density curves displayed because the NNM splits from and eventually merges again to the same BCP. Moreover, Figure 7 also illustrates that an ellipsoidal like shape now appears. All the molecules within this group do not show any QTAIM charge inversion as found by comparing the results right before and after the NNA window (see Table 3). Curiously, this finding is in disagreement with the BP crystal data from Density Functional Theory calculations, which pointed to polarity inversion.18 One can also observe that the most electronegative atom according to the majority of the scales used here is negatively charged at distances little outside the NNM window, except by BP. Finally, the NNA in type “b” molecules is always formed nearby the least polarizable atom.


14

Page 15 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 3: QTAIM charges at internuclear distances lying immediately before and after the window for NNA occurrence from UCCSD/cc-pVQZ calculations.

a

Molecule Charge before (e) Charge after (e) LiNaa qLi = 0.48 qLi = -0.27 AsP qP = 0.59 qP = -0.12 BrCl qCl = 0.88 qCl = -0.50 ArKr qKr = 0.81 qKr = 0.61 ClKr qCl = 1.16 qCl = -0.43 PS qP = 1.43 qP = 1.29 AsS qS = 0.27 qS = -0.52 BS qB = 0.05 qB = 1.28 ClS qS = 1.32 qS = 0.95 CCl qCl = 1.93 qCl = -0.51 SeS qS = 0.91 qS = -0.37 BP qP = 1.37 qP = 0.56 CAr qAr = 2.21 qAr = -0.52

Reference 17.

Figure 6: Electron density at critical points against interatomic distances as obtained in UCCSD/cc-pVQZ calculations for BP.

Figure 7: Diagram of distances between critical points and the boron nucleus against interatomic distances as obtained in UCCSD/cc-pVQZ calculations for BP.


15


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 23

C. Effects over molecular multipoles Another important aspect to be discussed here is related to the possible effects of NNA formation over molecular dipole (μ) and quadrupole (Θ) moments. For instance, LiNa shows a maximum in the zz component of Θ at a bond length of 3.7 Å, almost at the same point that is associated with the upper limit for NNM occurrence (3.64 Å).17 In addition, Li2 exhibits a similar maximum nearly where one NNA splits in two.36 The other homonuclear diatomic molecules studied here customarily present a maximum or minimum in Θzz that occur approximately at the same points where NNMs appear / disappear or also when the number of these non-nuclear maxima changes. For instance, this is apparently noticed for every second-row diatomics except perhaps for Be2, in which case the electronic structure treatment should be improved for a definitive answer. The interaction between valence shell electrons also tends to result in similar situations for Na2, Al2, Si2, Cl2 and Ar2. Other examples are likewise observed for overlap of inner shell electrons. The heteronuclear systems discussed here also present some kind of electrical signature of this nature inside or nearby the distance limits for NNA occurrence, except for AsP and AsS. Hence, local minima and/or maxima in μz (internuclear distance) are found for BrCl (1.07 Å), ArKr (1.26 Å), ClKr (1.06 and 1.37 Å), ClS (1.18 Å), CCl (1.10 Å), SeS (1.18 Å) and CAr (1.31 Å). Moreover, a minimum in Θzz is detected in ClKr (1.44 Å) while maxima of this multipole are seen in ClKr (1.08 Å), PS (1.54 Å), BS (1.11 Å), CCl (1.30 Å) and BP (1.71 Å). The agreement between the positions of such extreme points and the NNM window limits for molecules such as ArKr, ClS, CCl, PS, BS and BP is certainly remarkable.


16

Page 17 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


D. How Strongly does the Appearance of the NNA Depend on the Bare Nuclear Potential? Esquivel and coworkers proposed in 1993 that the bare nuclear potential is the main factor responsible for the occurrence of nonconvex regions in atomic electron densities, which eventually leads to NNM formation.37 Thus, as a numerical test of such hypothesis, the Li2 molecule, at its fixed ground-state equilibrium geometry, has been subjected to successive ionization to determine the survivability of the NNM under these artificial conditions. Results are displayed in Table 4. Calculations show that, in fact, the NNA only persists up to and including Li2+ with a charge q(NNA) of only 0.724 e instead of the -1.242 e observed for Li2. Any further stripping of electrons that leaves only the former K shell electrons results in the disappearance of the NNM as can be seen from Table 4. Interestingly enough, anionic species preserve the NNA with significant basin electron populations. The density in all these systems is rather flat, though, as can be seen from the difference in the electron density at the BCP and at the NNM (last column of Table 4). These results strongly suggest that the appearance of the NNA is a quantum phenomenon that depends mostly on the field generated by the system of electrons rather than on the classical Coulombic bare nuclear field.


17


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 23

Table 4: QTAIM charges (e) of Li2 in the anionic, neutral, and cationic states (at the frozen optimized geometry of the neutral ground-state species) along with electron densities (a.u.) at the bond and non-nuclear critical points from UCCSD/cc-pVQZ calculations. qLi

qNNA

3-

-0.560

Li2 2-

Li2

Li2

-

ρBCP[Li,Li]

ρBCP[NNA,Li]

ρNNA

ρNNA-ρBCP[NNA,Li]

-1.649

0.01113

0.01163

0.00050

-0.409

-1.067

0.01155

0.01214

0.00058

-0.030

-0.929

0.01136

0.01197

0.00061

0.622

-1.242

0.01328

0.01416

0.00088

Li2

+

0.862

-0.724

0.01579

0.01701

0.00122

Li2

2+

1.000

0.00009

Li2 3+

1.500

0.00008

Li2

4+

2.000

0.00001

Li2

5+

2.500

0.00001

Li2

Conclusions This work provides evidence that a small polarizability difference between the atoms that compose a heteronuclear diatomic molecule is an important feature that favors the occurrence of NNMs. This is somehow related to the previously proposed requirement of similar electronegativities18 and reinforces the view that the perturbation of electronic clouds associated to the atomic species approaching must be small to result in the appearance of such non-nuclear maxima. However, this empirical rule also indicates that NNAs can occur even in systems composed by highly polarizable atoms provided that their susceptibilities to electron cloud deformations are nearly the same. Hence, this finding can help to understand the NNM formation mechanism in molecules composed by convex atoms. The appearance of the NNA is also shown to be a quantum mechanical phenomenon that depends mainly on the field generated by the system of electrons and only depends parametrically on the bare nuclear potential.


18

Page 19 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


In addition, two types of heteronuclear systems with NNAs have been characterized. Type “a” is composed by diatomic molecules that exhibit electronic charge exchange between the atoms by means of a pseudo-atom. Type “b” includes molecules in which the NNM basin separates from an atom at the lowest limit of the window for NNA occurrence but eventually merges again to the same atom at the final point of this distance range. These two categories seem to be the only possible alternatives for heteronuclear diatomics containing a single NNM along the internuclear axis. However, new categories can also be defined if, for instance, new heteronuclear diatomic systems containing more than one NNA were discovered, especially if these cases could present NNMs out of the internuclear axis. Finally, our findings also show that the NNAs usually result in extreme values (maximum or minimum) in molecular electrical properties (dipole and quadrupole moments) that occur nearly at the same internuclear distances of the NNM window limits or at intermediary points associated with variations in the number of such nonnuclear maxima. The large number of cases detected and the astonishing concordance observed for some of them practically excludes a mere coincidence hypothesis and provides support to cogitate that NNAs may have effects on other physical chemical properties of these molecules as well.

Acknowledgements The authors thank FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) (grant number 2010/18743-1) for financial support. T.Q.T. acknowledges FAPESP for a doctoral fellowship (2012/22143-5). L.A.T. also thanks CAPES for a doctoral fellowship and CNPq (Science without Borders scholarship program –


19


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 23

205445/2014-4). C. F. M. acknowledges the funding of the Natural Sciences and Engineering Research Council of Canada (NSERC) and of Canada Foundation for Innovation (CFI).

Supporting Information. Figures presenting electron density values at critical points and diagrams of distances between critical points and selected nuclei against interatomic distances for the twelve heteronuclear systems described here.


20

Page 21 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References 1

Bader, R.F.W. Atoms in Molecules: A Quantum Theory; Clarendon Press: Oxford, U.K.,1990. Bader, R.F.W. Atoms in Molecules. Acc. Chem. Res. 1985, 18, 9-15. 3 Matta, C.F.; Boyd, R.J. The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, Wiley-VCH, Weinheim, 2007. 4 Stewart, R.F. Electron Population Analysis with Rigid Pseudoatoms. Acta Cryst. 1976, A32, 565-574. 5 Besnainou, S.; Roux, M.; Daudel, R. Retour Sur Leffet de la Liaison Chimique sur la Densite Electronique. Compt. Rend. Acad. Sci. 1955, 241, 311-313. 6 Gatti, C.; Fantucci, P.; Pacchioni, G. Charge-Density Topological Study of Bonding in Lithium Clusters. 1. Planar LiN Clusters (N=4,5,6). Theor. Chim. Acta. 1987, 72, 433-458. 7 Cao, W.L.; Gatti, C.; MacDougall, P.J.; Bader, R.F.W. On the Presence of Nonnuclear Attractors in the Charge-Distributions of Li and Na Clusters. Chem. Phys. Lett. 1987, 141, 380385. 8 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. 9 Sakata, M.; Sato, M. Accurate Structure Analysis by the Maximum-Entropy Method. Acta Cryst. A 1990, 46, 263-270. 10 de Vries, R.Y.; Briels, W.J.; Fell, D.; te Velde, G.; Baerends, E.J. Charge Density Study with the Maximum Entropy Method on Model Data of Silicon. A Search for Non-Nuclear Attractors. Can. J. Chem. 1996, 74, 1054-1058. 11 Dale, S.G.; Otero-de-la-Roza, A.; Johnson, E.R. Density-Functional Description of Electrides. Phys. Chem. Chem. Phys. 2014, 16, 14584-14593. 12 Bader, R.F.W.; Platts, J.A. Characterization of an F-Center in an Alkali Halide Cluster. J. Chem. Phys. 1997, 20, 8545-8553. 13 Taylor, A.; Matta, C.F.; Boyd, R.J. The Hydrated Electron as a Pseudo-Atom in CavityBound Water Clusters. J. Chem. Theor. Comput. 2007, 3, 1054-1063. 14 Platts, J.A.; Overgaard, J.; Jones, C.; Iversen, B.B.; Stasch, A. First Experimental Characterization of a Non-Nuclear Attractor in a Dimeric Magnesium(I) Compound. J. Chem. Phys A. 2011, 115, 194-200. 15 Wu, L.C.; Jones, C.; Stash, A.; Platts, J.A.; Overgaard, J. Non-Nuclear Attractor in a Molecular Compound Under External Pressure. Eur. J. Inorg. Chem. 2014, 5536-5540. 16 Pendás, A.M.; Blanco, M.A.; Costales, A.; Mori-Sánchez, P.; Luaña, V. Non-Nuclear Maxima of the Electron Density. Phys. Rev. Lett. 1999, 83, 1930-1933. 17 Terrabuio, L.A.; Teodoro, T.Q.; Rachid, M.G.; Haiduke, R.L.A. A Systematic Theoretical Study of Non-Nuclear Electron Density Maxima in Some Diatomic Molecules. J. Phys. Chem. A 2013, 117, 10489-10496. 18 Mori-Sánchez, P.; Pendás, A.M.; Luaña, V. Polarity Inversion in the Electron Density of BP Crystal. Phys. Rev. B 2001, 63, 125103. 19 Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, Jr., J. A.; Vreven, T.; Kudin, K. N.; Burant, J. C et al. Gaussian 03, Revision D.02; Gaussian, Inc.: Wallingford CT, 2004. 20 Dunning, Jr., T.H. Gaussian-Basis Sets for Use in Correlated Molecular Calculations. 1. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007-1023. 21 Woon, D.E.; Dunning, Jr., T.H. Gaussian-Basis Sets for Use in Correlated Molecular Calculations. 4. Calculation of Static Electric Response Properties. J. Chem. Phys. 1994, 100, 2975-2988. 22 Prascher, B.P.; Woon, D.E.; Peterson, K.A.; Dunning, Jr., T.H.; Wilson, A.K. Gaussian Basis Sets for Correlated Molecular Calculations. VII. Valence, Core-Valence, and Scalar Relativistic Basis Sets for Li, Be, Na, and Mg. Theor. Chem. Acc. 2011, 128, 69-82. 23 Woon, D.E.; Dunning, Jr., T.H. Gaussian-Basis Sets for Use in Correlated Molecular Calculations. 3. The Atoms Aluminum through Argon. J. Chem. Phys. 1993, 98, 1358-1371. 2


21


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 23

24

Koput, J.; Peterson, K.A. Ab Initio Potential Energy Surface and Vibrational-Rotational Energy Levels of X2Σ+ CaOH. J. Phys. Chem. A 2002, 106, 9595-9599. 25 Wilson, A.K.; Woon, D.E.; Peterson, K.A.; Dunning, Jr., T.H. Gaussian Basis Sets for Correlated Molecular Calculations. IX. The Atoms Gallium through Krypton. J. Chem. Phys. 1999, 110, 7667-7676. 26 Ceolin, G.A.; de Berrêdo, R.C.; Jorge, F.E. Gaussian Basis Sets of Quadruple Zeta Quality for Potassium Through Xenon: Application in CCSD(T) Atomic and Molecular Property Calculations. Theor. Chem. Acc. 2013, 132, 1339. 27 AIMAll (Version 12.08.21), T. A. Keith, TK Gristmill Software, Overland Park KS, USA, 2012 (aim.tkgristmill.com). 28 Huber, K.P.; Herzberg, G., Constants of Diatomic Molecules (data prepared by J.W. Gallagher and R.D. Johnson, III) in NIST Chemistry WebBook, NIST Standard Reference Database Number 69, Linstrom, P.J.; Mallard, W.G., Eds., National Institute of Standards and Technology, Gaithersburg MD, 20899 (retrieved April 30, 2015). 29 Boyd, R.J. Electron Density Partitioning in Atoms. J. Chem. Phys. 1977, 66, 356-358. 30 Bader, R.F.W.; MacDougall, P.J.; Lau, C.D.H. Bonded and Nonbonded Charge Concentrations and Their Relation to Molecular Geometry and Reactivity. J. Am. Chem. Soc. 1984, 106, 1594-1605. 31 Bader, R.F.W.; Essén, H. The Characterization of Atomic Interactions. J. Chem. Phys. 1984, 80, 1943-1960. 32 Shi, Z.; Boyd, R.J. The Shell Structure of Atoms and the Laplacian of the Charge Density. J. Chem. Phys. 1988, 88, 4375-4377. 33 Huheey, J.E. Inorganic Chemistry: Principles of Structure and Reactivity; Harper & Row Publishers: New York, 1983. 34 Schwerdtfeger, P. Atomic Static Dipole Polarizabilities, in Computational Aspects of Electric Polarizability Calculations: Atoms, Molecules and Clusters, ed. G. Maroulis, IOS Press: Amsterdam, 2006; pg.1-32 (Updated static dipole polarizabilities are available as pdf file from the CTCP website at Massey University). 35 Cioslowski, J. Nonnuclear Attractors in the Lithium Dimeric Molecule. J. Phys. Chem. 1990, 94, 5496-5498. 36 Penotti, F.E. On the Eletronic Structure of Li2 (X1Σg+) and Its Change with Internuclear Distance. Int. J. Quantum Chem. 2000, 78, 378-397. 37 Esquivel, R.O.; Sagar, R.P.; Smith, Jr., V.H.; Chen, J.; Stott, M.J. Pseudoconvexity of the Atomic Electron Density: A Numerical Study. Phys. Rev. A 1993, 47, 4735-4748.


22

Page 23 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table of Contents Image: Two categories of NNA


23

Dynamical systems, attractors, and neural circuits.

Classification of attractors for systems of identical coupled Kuramoto oscillators.

Anderson attractors in active arrays.

A test for a conjecture on the nature of attractors for smooth dynamical systems.

Tumors as chaotic attractors.

Unity of cosmological inflation attractors.

Design of Heteronuclear Metalloenzymes.

Features of chemical bonds based on the overlap polarizabilities: diatomic and solid-state systems with the frozen-density embedding approach.

Heteronuclear transverse and longitudinal relaxation in AX4 spin systems: application to (15)N relaxations in (15)NH4(+).

Spontaneous motion on two-dimensional continuous attractors.

Nonlinear fate of internal wave attractors.

2D Magneto-optical trapping of diatomic molecules.

Pattern Selection in Network of Coupled Multi-Scroll Attractors.

Pure in-phase heteronuclear correlation NMR experiments.

Attractors in Sequence Space: Peptide Morphing by Directed Simulated Evolution.

Polarization attractors in harmonic mode-locked fiber laser.

Tunable dark modes in one-dimensional "diatomic" dielectric gratings.

Relativistic Effects Break Periodicity in Group 6 Diatomic Molecules.

Three-dimensional heteronuclear NMR studies of RNA.

Spectral analysis of two coupled diatomic rotor molecules.

Stabilization of perturbed Boolean network attractors through compensatory interactions.

Site-specific analysis of heteronuclear Overhauser effects in microcrystalline proteins.

Heteronuclear dipolar coupling in spin-1 NQR pulsed spin locking.

Control of asymmetric Hopfield networks and application to cancer attractors.