On-Top Ratio for Atoms and Molecules | The Journal of Physical

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On-Top Ratio for Atoms and Molecules Rebecca K. Carlson, Donald G. Truhlar, and Laura Gagliardi J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b04259 • Publication Date (Web): 22 Aug 2019 Downloaded from pubs.acs.org on August 25, 2019

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The Journal of Physical Chemistry

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On-Top Ratio for Atoms and Molecules Rebecca K. Carlson, Donald G. Truhlar,* and Laura Gagliardi * Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States Abstract The on-top ratio, R, is the ratio of the on-top pair density to the square of half the total density. Here we explore the on-top ratio as a tool to understand different types of bonds, including covalent, polar covalent, and ionic, and dispersion-dominated weak interactions. We show for several diatomic molecules and for ethylene that the partial derivative of R is a useful indicator of covalent-bond breaking. R also presents a local maximum for each electron shell when all electrons are correlated. Introduction Chemical bonding descriptors are useful to identify shell structure in atoms or molecular regions dominated by lone pairs, bonds, or core electrons, and to differentiate between different types of covalent and noncovalent interactions. Some popular topological descriptors are based on the kinetic energy density, such as the electron localization function (ELF)1,2 or the localized orbital locator (LOL).3 Both ELF and LOL use the uniform electron gas (UEG) as a reference. Other descriptors, including those based on atoms in molecules (AIM)4 and noncovalent interactions,5–7 use the gradient of the density or the reduced density gradient to identify bond critical points in the molecule. Since lowering of the kinetic energy density accompanies the formation of covalent bonds,8–10 it is a useful quantity for understanding chemical bonds. The reduced density gradient, given by 𝑠(𝐫) =

|∇𝜌(𝐫)| 1 2 3

4 3

(1)

2(3𝜋 ) 𝜌(𝐫)

where 𝜌 is the total density, and ∇𝜌 is the gradient of the density, is used to correct deviations from the UEG in density functional theory in generalized gradient functionals (GGAs)11 and

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2 shows minima in the bonding region, where the slopes of the reduced density gradient can be used to differentiate types of bonds.7


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3 Other approaches to characterize bonding are based on energetic quantities rather than topological quantities.12–15 A fundamental component of energetic approaches is the use of the two-body density matrix. Rahm and Hoffmann developed a Q metric that is able to differentiate types of bonds by quantifying the amount of covalency relative to other charge transfer contributions in a bond, where 𝑄 = 1 is considered the signature of a covalent bond.15 One of the challenges mentioned by Rahm and Hoffmann is analyzing the “collective will” of electrons in quantum mechanics, and they show that their 𝜔 metric, which depends on the diagonal elements of the two-electron reduced density matrix, is a useful tool to characterize multielectron interactions.15 The two-body density matrix is essential for characterizing electron correlation, including Fermi correlation by which electrons with parallel spins avoid each other, forming a Fermi hole,16 and Coulomb correlation, by which electrons avoid each other because of their Coulomb repulsion, forming a Coulomb hole.17 A standard definition of correlation energy is the energy lowering of the exact electronic energy when compared to the variational energy of the best single-determinant wave function;18 that part of the correlation energy due to nearly degenerate configurations is called “static”, “nondynamic” or “left-right” correlation.19–22 Previously, we showed that the on-top pair density Π and on-top ratio R can be useful to understand left-right correlation in molecules,23 where Π is defined as

()

𝑁 Π(𝐫) = 2 ∫|Ψ(𝑥1,𝑥2,…,𝑥𝑁)|2𝑑𝜎1𝑑𝜎2…𝑑𝜎𝑁𝑑𝐫3𝑑𝐫4…𝑑𝐫𝑁|𝐫 = 𝐫 = 𝐫 1 2

(2)

where N is the number of electrons, 𝜎 is the spin coordinate, and 𝑥 is the general space-spin coordinate (r, 𝜎), and the on-top ratio R is defined as Π(𝐫)

𝑅(𝐫) = [𝜌(𝐫)/2]2

(3)

where 𝜌 is the total density. The on-top pair density is the probability of finding two electrons in the same place, and the on-top ratio is unity for an uncorrelated singlet. Another property of R is that 𝑅 ≤ 1 for a single determinantal wave function, but for a multideterminantal wave function, R may be greater than unity. Here we extend the study of R to show it can be useful to differentiate types of bonds and to understand the motions of electrons in a more general sense than either the Fermi hole or Coulomb hole. While it is instructive to analyze these types of



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4 holes separately, as has previously been done to understand the behavior of electrons with respect to the Coulomb hole,24–27 we do not differentiate between them in the rest of this paper.

Computational Methods All calculations of 𝑅 were done using Molcas version 8.2.28 In the present paper, we generate 𝜌,Π, and 𝑅 from complete active space self-consistent field (CASSCF)29 calculations, and in general we use large active spaces, which are specified in Table 1. A variety of basis sets have been used, and after much testing, we concluded that the results do not depend sensitively on basis set, i.e., the basis set effect on 𝑅 is much smaller than the effect of the choice of active space; therefore we present results for only a single basis set for each case, and these basis sets are specified in Table 1. We chose simple atoms and molecules to understand the nature of R in different regions of space (ie lone pairs, single bonds, double bonds, covalent bonds, ionic bonds, etc). We chose molecules with similar atoms to more easily compare trends. For example, can R differentiate between different types of bond in which F participates, such as a covalent bond in F2, mixed covalent-ionic bond in HF, and an ionic bond in NaF? Does R For LiH, a mixed covalent-ionic bond, does R in Li behave more like Li+ in Li2+ or Li in Li2? We note that is natural to wonder about bond strength of similar molecules, such as Li2+ v. Li2, but R alone does not provide sufficient information to make a statement about bond strength. Experimental equilibrium bond distances are used for diatomics, except for Ne2, which is situated at a 3.1 Å internuclear separation, corresponding to the van der Waals distance. For Li2+, Li2, and Li2-, equilibrium geometries are taken from Ref. 30. For LiH, NaF, HF, and F2, the equilibrium geometry was taken from Ref. 31. The geometry of ethylene was optimized using the M06 density functional32 using the Gaussian 09 program package.33 All plots were made using Mathematica version 10.4.1.34


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5 Table 1. Active spaces and basis sets System He Ne

(n,oa Active orbitals Basis set (2,2) two s orbitals aug-pc-435 (10,10) 1s, 2s, and 2p subshells and two s and one p ” correlating subshells Ar (16,16) 2s, 2p, 3s, and 3p subshells and two s and ” two p correlating subshells Li2 (6,10) On each atom, the 1s and 2s orbitals and one cc-pVTZ36 s and two pπ correlating orbitalsb Li2+ (5,10) ” ” – Li2 (7,10) ” ” LiH (4,4) 1sLi, the σ bonding orbital and antibonding pc-435,37 orbital, and a correlating s orbital NaF (16,16) 2sNa and 2pNa core subshells, 3sNa orbital, ANO-RCC-VTZP38 2sF, and 2pF valence orbitals, 3 correlating p orbitals for Na and F, and a correlating s for F (8,8) 3sNa, 2sF, and 2pF valence subshells and ” three correlating p orbitals. HF (8,8) all valence orbitals and three correlating p pc-435,37 orbitals F2 (14,14) all valence orbitals, and six correlating ” orbitals. All lone pairs are correlated. N2 (10,14) same active orbitals as F2 for comparison “ Ne2 (12,12) 2p subshell on each atom, and six aug-pc-437 correlating p orbitals C2H4 (12,12) all valence orbitals. All bonds are correlated. ANO-RCC-VTZP38 an is the number of active electrons, and o is the number of active orbitals. bThe article considers (6,10), (5,10), and (7,10) active spaces, and the supporting information has results for (6,6), (5,6), and (7,6) active spaces; the R profiles are similar for the large and small active spaces. Results and Discussion Atoms Like the reduced density gradient, the on-top ratio R exhibits atomic shell structure. Figures 1–3 show contour plots and line plots of R for He, Ne, and Ar. All core and valence electrons are correlated in the active space, and there is a maximum of R for each electron shell. For He, R has a maximal value of 1.00 at a distance of 0.32 Å from the nucleus (Fig. 1). For Ne, R has two maxima, both with a value of 1.00, and they are at 0.08 Å and 0.48 Å from the



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6 nucleus (Fig. 2). The Ar atom has three maxima of R, one for each electron shell (Fig. 3), where the maximum for the outermost shell occurs at a distance of 0.86 Å from the nucleus, with an R value of 0.99. Although the ground states of the noble gases are well described by a single determinant, R can be significantly different from unity. For He, R has the smallest value of 0.71 at the nucleus, for Ne, R has a local minimum of 0.90 between the 𝑛 = 1 and 𝑛 = 2 shells, and for Ar, R has its deeper local minimum, 0.93, between the 𝑛 = 2 and 𝑛 = 3 shells. Table 2 shows that the maxima of R, Rval, in the outermost shells correspond well to the atomic covalent radii in the CRC Handbook of Chemistry and Physics.39

Figure 1. Contour plot of R (top) and line plot of R (bottom) for He atom. In all plots containing a contour plot and a line plot, the horizontal axis applies to both, i.e., they are lined up.


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7

Figure 2. Contour plot of R (top) and line plot of R (bottom) for Ne atom.



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Figure 3. Contour plot of R (top) and line plot of R (bottom), for Ar atom. Table 2. Atomic radii of noble gases and distances of maximum R values for the valence shell, with the aug-pc-4 basis set. Atomic Radii (Å)a

Rval Distance (Å)

He

0.37

0.32

Ne

0.62

0.48

Ar

1.01

0.86

aValues from Ref. 39 with ~0.1 Å uncertainty

For Ne and Ar, if smaller active spaces are used in which the core orbitals are inactive and only valence orbitals are correlated, 𝑅 is 1 in the core, but it is similar in the outer shell to what was obtained with the larger active spaces (Figs. S1-S2).


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Molecules Li2 Figure 4 shows R for Li2. One sees similar atomic shell structure in R for each Li atom in Li2, with each Li atom showing 1s and 2s maxima. The lower plot, along the internuclear axis, shows that outside the bond, the 2s peak has a value 𝑅 = 0.65, which is much smaller than at the bond midpoint where 𝑅 > 1; we had previously23 seen R > 1 at the bond midpoint in H2. The behavior of R for Li2 at the bond midpoint is similar to that for H2, where 𝑅 > 1. Two electrons have a higher probability of being in the same place at the midpoint, as compared to anywhere else in the molecule (Fig 4).

Figure 4. Contour plot of R (top) and line plot of R (bottom) for Li2. Dominant orbital contributions are labeled on the line plot for each atom in the molecule. Li2+ Li2+ is an interesting case to study because it has only one valence electron, so the on-top ratio should be nearly zero in the valence region, for example in the bond. This is clearly seen in Fig. 5, where the value of R at the midpoint is essentially zero because Π is very small and is about three orders of magnitude smaller than

𝜌2 2

at this point. In the case of strictly one or zero



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10 electrons, the on-top pair density in eq. 2 is zero, so one might expect that R outside the bond, in the valence orbital, would also be close to zero, similarly to the bond midpoint. While 𝑅 is less than 1 in the region outside the bond in the 2s region, it is larger outside the bond than inside it, which is opposite to the case in the neutral molecule.

Figure 5. Contour plot of R (top) and line plot of R (bottom) for Li2+. Dominant orbital contributions are labeled on line plot for each atom in the molecule. 𝑳𝒊𝟐― 2 u The ground state of the lithium dimer anion is also bound, and, as for the neutral molecule, the ground state of the anion has a maximum of R in the bonding region. The natural orbital occupation number for the σ bonding orbital is 1.82, which is similar to that for the neutral, which is 1.81. However, the anion also has one electron in the antibonding orbital. As compared to the neutral, R for the anion has a more constant value in the bonding region (𝑅⩳1.05), it has higher values outside the bond for the 2s electrons (𝑅 = 0.75 at the local maximum), and 𝑅 decreases more slowly away from the molecule. In the core region, labeled 1s in Fig. 6, R is similar to that in the neutral.


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Figure 6. Contour plot of R (top) and line plot of R (bottom) for 𝐿𝑖2― 2 u . Dominant orbital contributions are labeled on line plot for each atom in the molecule. LiH LiH has a mixture of covalent and ionic character. Figure 7 shows that in LiH the R contours present two interacting ions, H–, with a very large radius, and Li+, with a smaller radius. The contours for H– are similar to those in the He atom, and the local maximum due to H– overlapping with the 2s orbital of Li+ occurs at a distance of 0.74 Å from the Li nucleus, whereas the geometric center of the bond is 0.80 Å from the Li nucleus. As compared to the neutral and charged Li dimers, LiH has a higher R value in the 2s region outside the bond; in particular, Fig. 7 shows this maximum is close to 1 at 1.32 Å from the Li nucleus.



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Figure 7. Contour plot of R (top) and line plot of R (bottom) for LiH. Dominant orbital contributions are labeled on the line plot for each atom. Cartesian coordinates have Li at the origin. From left to right, the first arrow indicates a peak corresponding to the overlap of the H 1s and Li 2s orbitals. The next two arrows correspond to the Li 1s orbital on either side of the nucleus. NaF We studied NaF with two active space sizes (see Table 1). Both active spaces include the valence orbitals; the larger active space also includes the outer core (2s and 2p subshells) of Na. NaF has an ionic bond, where, similarly to LiH, the contours show two interacting atoms. Figure 8, with the larger active space, shows that, as for Ar, R has the lowest minimum value going from the 𝑛 = 2 to 𝑛 = 3 shell of Na, just inside of the local maximum outside the bond. Figure 9, though, shows the results for the smaller space, and, as one might expect, when the 𝑛 = 2 shell of Na is inactive, 𝑅 = 1 on the Na side, but the R profile on the F side is unchanged. Since there is a large amount of charge transfer, almost all the valence electrons are on the F atom, and the largest difference between the two active spaces occurs for Na between


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13 the 𝑛 = 2 and 𝑛 = 3 electron shells, where R decreases with the larger active space due to correlating more electrons. Does the size of the active space matter in understanding the nature of the bond? The inflection point of the (8,8) curve at z near -1 Å corresponds to a local maximum of the (16,16) curve in Fig 9. This region is where the reduced density gradient 𝑠(𝐫) has a critical point, as described by Boto et al,7 where at the bond critical point (CP) one has ∇𝜌(𝐫) = 0. The reduced density gradient 𝑠(𝐫) has similar features to those of R and |∇𝑅(𝐫)|, where a plot of both 𝑠(𝐫) and R is shown in Fig. S9b, and we focus on comparing 𝑠(𝐫) and |∇𝑅(𝐫)| in the main text. Figure 10 shows that |∇𝑅(𝐫)| has similar features to 𝑠(𝐫), where one can see that despite the different active space sizes, the critical point of 𝑠(𝐫), which are labeled CP in Fig. 10, occurs very close to where, for the (16,16) active space,

the (8,8) active space, at -1.05 Å (Fig. 10d),

2 R z 2

R  0 (at -1.06 Å in Fig. 10d) and where for z  0.

Moving from the nuclei, towards the middle of the bond, in Fig. 10a, the first minimum in 𝑠(𝐫) after the nuclei corresponds to the zero in |∇𝑅(𝐫)| for both active spaces (Fig. 10b,c), which is the transition region between the core and n = 2 shell for both atoms. The inflection point in 𝑠(𝐫) corresponds closely to the zero in |∇𝑅(𝐫)| at -0.58 Å, which in turn coincides with the n = 2 maximum in R. The maximum in 𝑠(𝐫) near the bond midpoint occurs closely to the zero in |∇𝑅(𝐫)| at -0.95 Å. Generally, the comparison between |∇𝑅(𝐫)| and 𝑠(𝐫) for the other molecules follows the same trend, where the minima and maxima of 𝑠(𝐫) correspond closely to the zeros and shoulders of |∇𝑅(𝐫)| , respectively, and the inflection point in 𝑠(𝐫) near the bond midpoint corresponds to the maxima in R in the valence orbital. While the CP for 𝑠(𝐫) is close to where

R  0 for most molecules presented in this paper, we note that this was not the case z

for all molecules tested, such as HCN, where

R  0 occurred almost 0.2 Å closer to the N z

atom. In the paper of Boto et al.,7 the authors note that for CO, which has a similar bond to



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14 HCN, the CP in 𝑠(𝐫) occurs closer to C, rather than O, which results in incorrect charges calculated using AIM methods. Perhaps, R and |∇𝑅(𝐫)| may be useful in understanding such cases.

Figure 8. Contour plot of R (top) and line plot of R (bottom) for NaF with a (16,16) active space. Dominant orbital contributions are labeled on the line plot for each atom in the molecule. Cartesian coordinates have F at the origin and Na at -1.93 Å. From left to right, the first two arrows point to maxima due to the Na 2s and 2p valence orbitals. The arrow in the middle indicates the feature due to Na 3s orbital, and the last two arrows indicate peaks for the valence 2s and 2p orbitals of the F atom.


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Figure 9. Line plot of R for NaF for (16,16), solid line, and (8,8) active space, dotted line. Cartesian coordinates in Å, with F at the origin and Na at -1.93 Å.

Figure 10. (a) Line plot of s(r) for NaF from (16,16) active space. (b) Line plot of |∇𝑅(𝐫)| for (16,16) active space. (c) Line plot of |∇𝑅(𝐫)| for (8,8) active space. (d) zoomed in plot of a-c showing behavior at bond critical point (CP) of s(r) defined in Ref. 7. The Cartesian coordinates have F at the origin and Na at -1.93 Å.

HF The contours of R for HF are shown in Fig. 11; they almost look like the contours of F– perturbed by a proton, but unlike NaF, where the contours on F are almost symmetric due to a



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16 nominally full valence shell, the contours on F in the HF molecule are asymmetric. HF is considered to have a polar-covalent bond (roughly 50:50), where there is incomplete charge transfer, and the asymmetric R values through the pz bonding orbital indicate a different interaction with H than with Na.

Figure 11. Contour plot of R (top) and line plot of R (bottom) for HF, with Cartesian coordinates having F at the origin and H at –0.92 Å. Dominant orbital contributions are labeled on line plot for each atom in the molecule.

F2 The bond in F2 can be categorized as a covalent bond,40 as a “charge-shift” bond, due to the covalent–ionic mixing,41 as a bond with correlation that is “truly molecular (namely strongly dependent upon the internuclear separation),”42 as a bond with "appreciable" nondynamic correlation even at the equilibrium internuclear separation,43 or as a nuclearresisted15 bond. Figure 12 shows that the R contours for F2 are qualitatively different than those


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17 for any of the molecules already considered. As in Li2, there is a maximum at the bond midpoint, but whereas the plot for Li2 has the appearance of two interacting atoms, the bonding in F2 is clearly different. The contours in Fig. 12 show that the σ bond has a minimum in R of less than 0.75, and R deviates from unity throughout nearly the entire bonding region, but in the lone pairs, 𝑅 is closer to unity.

Figure 12. Contour plot of R (top) and line plot of R (bottom) for F2 with nuclei at 0.709 Å and –0.709 Å. Regions are labeled on the line plot for each atom. N2 The bond in N2 is a formally a covalent, triple bond, where unlike F2, there are no electrons in the π symmetry antibonding orbitals. Comparing the line plot for N2 (Fig. 13) to that of F2, there are some noticeable similarities. For example, along the z-axis 𝑅 < 1 in the region of the σ bond and 𝑅 > 1 at the bond midpoint. Looking at the contour plot in Fig 13, there is a difference in R for the lone pairs formed from the atomic 2s orbitals that make up the 2σg and 2σu* molecular orbitals. For N2, there is a distinct region (labeled LP in Fig 13) where



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18 R is unity. For F2, this distinct region is absent and R is lower, around 0.9, in the corresponding region. Using Lewis dot diagrams, we assign these lone pairs similarly for N2 and F2, but looking at the on-top ratio, it is clear that these regions are different. It may be useful to consider molecular orbital theory, where the MO diagrams for N2 and F2 can be found in any good quantum mechanics textbook. For F2, the atomic 2s orbitals, and therefore the corresponding molecular orbitals they form are much lower in energy than the 2p atomic orbitals and their corresponding molecular orbitals. In MO theory, there is no s-p mixing in the molecular orbitals for F2, whereas there is for N2. This s-p mixing causes the 2σu* orbital in N2 to be closer in energy to the 1πu orbitals, changing the correlation between these electrons.

Figure 13. Contour plot of R (top) and line plot of R (bottom) for N2 with nuclei at 0.549 Å and –0.549 Å. Regions are labeled on the line plot for each atom.

Ne2 Unlike the other dimers considered in this article, Ne2 is bound only through damped dispersion forces, so one might expect a difference in R compared to other types of bonds, and


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19 Fig. 14 shows a qualitative difference from the other diatomics. The noncovalent interaction leads to a much lower R value between the atoms (corresponding to the electrons avoiding other electrons), but there is a small region where there is a local maximum in R.

Figure 14. Contour plot of R (top) and line plot of R (bottom) for Ne2. Features are labeled on line plot for each atom in the molecule, with atoms centered at ±1.505 Å. Ethylene Figure 15 shows that the R profile for ethylene is more similar to that of F2 than to that of the other diatomics. Looking at the contour plot in the molecular plane, one sees similarities to AIM gradient paths, where the critical points in AIM are very similar to the maxima in the 𝑅 contours.4 R has a lower value for the C-H σ bonds than the C-C σ bond (0.75 versus 0.9), and the lowest value of R occurs outside the C-H bond, behind the H atoms, similarly to what occurs in HF. Like the σ bonds, the π bond has a maximum in the plane through the midpoint ( 𝑅 = 1.02), although the maximum R value is about 0.02 larger for the C-C σ bonds than the σ C-H bonds. The R contours through the π system resemble the shape of the orbitals.



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Figure 15. Ethylene. (top, left) Contour plot of R in the molecular plane, which is the xz plane. (bottom, left) Line plot of R along the C-C axis. (top,right) Contour plot of R in yz plane (carbon atoms oriented vertically). (bottom, right) Line plot of R for ethylene through the π orbitals on one nucleus. Bond Breaking The effective bond order (EBO)44 is defined as the difference between the sum of the natural orbital occupation numbers of the bonding orbitals and the antibonding orbitals, divided by two. In a previous paper, 23 we showed analytically for H2 that Rmid, defined as R at the bond midpoint, may be another useful descriptor for bond breaking; for H2 the internuclear separation r at which there is an inflection point of Rmid as a function of r, is close to the internuclear separation where the EBO has an inflection point. For H2, the inflection point in the Rmid curve corresponds to an internuclear distance where Rmid begins to be greater than unity, a distinguishing feature of multideterminantal wave functions. This is the point where the wave function is no longer well described as a single determinantal wave function. By drawing comparisons between familiar bonding metrics, such as the EBO, and something less familiar,


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21 the on-top ratio, we hope to show that the on-top pair density, through R, can be a useful quantity in understanding wave functions and bonding. Figure 16 shows that for F2 there is an inflection point in Rmid at r = 2.3 Å. This corresponds to the point in the dissociation process where there is a fundamental change in the character of the wave function. For comparison we note that the inflection point of the EBO curve for F2 occurs at 2.0 Å, at an EBO of value of 0.5, similarly to H2. At equilibrium, the EBO for F2 is nearly 1, indicating a single bond. A bond order of 0.5 can be considered the point where the covalent bond is broken, although there still is an interaction between the atoms.

Figure 16. Plot of R at bond midpoint for F2, where r is the internuclear distance.

Figure 17 compares the R profile for F2 to that for F atom at three values of the internuclear distance r, showing the deviation of R from the atomic values as the bond forms. At the van der Waals distance, the R profile for the molecule overlaps the atomic values closely, until about 1 Å away from the nucleus, inside the bond where R tends towards its maximum value for the molecule. For r = 2.3 Å, corresponding to the inflection point for the EBO, the R curves for the molecule are intermediate between those for the atom and those for even smaller r.



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22

Figure 17. Line plots of R for F2 (colored, with one atom at z = 0) and of F atom (black, with the atom at z = 0) at (a) r = 2.94 Å (the van der Waals distance), (b) r = 2.3 Å, and (c) r = 1.418 Å (the equilibrium distance). Figure 18 shows the behavior of

∂𝑅 ∂𝑧

as the bond breaks in F2. Monitoring a position 0.61

Å away from the right nucleus only, indicated by arrows in Fig 18, we find that behavior as a function of r. We chose the point 0.61 Å, because this is where

∂𝑅 ∂𝑧

R 2 R  0 . z z 2

This corresponds well to the covalent radius of F. At the van der Waals distance,


changes

∂𝑅 ∂𝑧

> 0 (Fig.

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23

18b), but for r = 2.21 Å,

∂𝑅 R 2 R   0 . For the equilibrium, ∂𝑧 < 0 at the point 0.61 Å away z z 2

from the nucleus. In the previous paragraphs, we discussed looking at the change in EBO and Rmid as the bond breaks, finding that 2.3 Å was the internuclear separation where the bond may be considered broken. Similarly, we find that the internuclear distance where

R 2 R  0 z z 2

(2.21 Å) is close to 2.3 Å, where Rmid has its maximal change, indicating that an analysis of the change in R at a point near the covalent radius may be another useful indicator of bond breaking. This point also indicates a fundamental change in the wave function, as the R profiles (Fig. 17) no longer share large similarities between the individual atomic profile and that of atom in the molecule.



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24

Figure 18. F2. Line plot of R (a, c, e) and

∂𝑅 ∂𝑧

(b, d, f) for (a,b) r = 2.94 Å, (c,d), r = 2.21 Å, and

(e-f), r = 1.418 Å Arrows in 18e indicate a point 0.61 Å from the nucleus. Comparing the behavior of

∂𝑅 ∂𝑧

for the dissociation of another covalent bond, namely the

C=C bond in ethylene, we see similar behavior to that of F2 , where Figs. 19 and 20 show two

R 2 R   0 , one for the breaking of the π bond, and one for the breaking of the regions where z z 2 σ bond. We only consider the change in C=C bond, and do not dissociate the C-H bonds, where ∂𝑅

r is the C=C distance. For the π bond, ∂𝑦 = 0 at an internuclear distance of 1.90 Å at a point 0.8 Å away from the carbon nuclei, approximately the length of the covalent radius (Fig. 19d). Like


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25 the π bond, the σ bond has

∂𝑅 ∂𝑧

= 0 r = 3.30 Å at a point 0.8 Å away from the carbon nucleus,

also approximately the value of the covalent radius (Fig. 20).

∂𝑅

Figure 19. Line plot of R (a, c, e) and ∂𝑦 (b, d, f) for r = 2.52 Å (a-b), r = 1.90 Å (c-d), and r = 1.32 Å (e-f), showing the behavior of

∂𝑅 ∂𝑦

in the π bond is stretched in ethylene. Arrows indicate

a point 0.8 Å away from the nucleus, about the length of the covalent radius.



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26

∂𝑅 ∂𝑧 (b, d, f) for r = 3.30 Å (a-b), r = 2.52 Å (c-d), and ∂𝑅 of ∂𝑧 as the σ bond is stretched in ethylene. Arrows in

Figure 20. Line plot of R (a, c, e) and

r=

1.32 Å (e-f), showing the behavior

20c

indicate a point 0.8 Å from the nucleus. No arrows are indicated in 19e and f because the two atoms are only 1.32 Å apart. The EBO for ethylene at a C–C internuclear separation of 1.90 Å is 1.65 and the natural orbital occupation number of the bonding π orbital is 1.73, while at r = 3.30 Å, the EBO is 0.30 and the natural orbital occupation of the σ orbital is 1.25. Interestingly, the average of these two distances (2.60 Å) and EBO value (0.98) correspond to where the EBO versus distance curve has an inflection point (Fig. S14). For F2, we found that the internuclear distance where


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27

R 2 R   0 and where the Rmid and EBO versus r curves had an inflection point occurred z z 2 close to the same r. However, for ethylene, we find that the internuclear separation where the EBO has an inflection point is the average of the distances where

R 2 R   0 for the π orbital z z 2

and σ orbital. Since the EBO is an average measure of bond order, accounting for all the configurations in the wave function, the EBO of 0.98 at 2.60 Å is an average bond order including contributions from both the σ and π orbital. While 2.60 Å is the distance where the bond order is half its equilibrium value, there is no information about the individual orbitals, although looking at the natural occupation numbers can be useful. As a multiple bond breaks, it may be useful to understand more concretely how the individual orbitals that contribute to the bond are changing and not just look at their average behavior.

Table 3. Bond breaking distances for covalent bond dissociations for selected neutral molecules Re (Å)

Bond Breaking Distance (Å)

F2

1.42

2.2

ethylene π

1.32

1.9

ethylene σ

1.32

3.3

Li2

2.67

3.3

LiH

1.59

2.0

We also find that there are points where

R 2 R   0 for other covalent type bonds z z 2

(Table 3), where the line plots can be found in supporting information. The bond breaking distance in Table 3, is the point where there is a fundamental change in the character of the wave function, as left-right correlation increases when the atoms separate, identified by the internuclear distance where

R 2 R R 2 R   0 . For Li2 and LiH,   0 at a distance close to z z 2 z z 2



Page 28 of 32

28 the covalent radius of the Li atom, 1.3 Å and 1.1 Å, respectively. We note that all the distances listed in Table 3 occur at an internuclear separation that is less than the van der Waals distance for those atoms, bearing in mind that for ethylene, we only have broken the C-C bond. We did find bond breaking point for HF at an internuclear distance of 2.5 Å, which is close to the sum of the van der Waals distances for these atoms, and is much longer than the bond breaking distance for the other covalent molecules. Since HF is described as a mixture of covalent and ionic bonding, other effects may be at play. We also tried to find a similar point for the dissociation of NaF, but

∂𝑅 ∂𝑧

did not behave the same as other covalent molecules. It would be

interesting if future work were to look at bond breaking indicators for other cases, like ionic bonds and mixed-type bonds.

Conclusions The on-top ratio R can differentiate types of bonds and can be an indicator of bond breaking in molecules. The R profiles of molecules have similar features to other topological bond descriptors, like the reduced density gradient s(r), but offer more information about the wave function and general behavior of the electrons. Both s(r) and |∇𝑅(𝐫)| have similar features, and as such, |∇𝑅(𝐫)| and R may also be useful for understanding s(r), a fundamental quantity in DFT. When all electrons are correlated, R has a local maximum value for each electron shell. When the valence electron shell is full for atoms with atomic number greater than two, R is very close to one for the valence shell maximum, as occurs for the noble gas atoms and F in NaF and HF as opposed to when the valence shell is partially filled, such as with Li in Li2. Dynamic correlation added by including correlating orbitals tends to lower R, with the greatest decrease in R occurring between the valence electron shell and next lower electron shell, as seen in the Ar atom. For covalent and noncovalent bonds, R shows different behavior in the bond. Covalent bonds show an overlap-generated feature in R at the bond midpoint for the valence orbitals, whereas for ionic bonds the R profile at equilibrium does not show overlap for the valence


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29 orbitals; rather the profiles are distinct for each atom, but pushed close together, such as what occurs for NaF. A characteristic of bonds to hydrogen is that R tends to be lower in the region around H outside the bond, such as occurs for HF and for the C-H bonds in ethylene. A characteristic feature of covalent bonds is a maximum at the bond midpoint, where 𝑅 tends to be greater than unity, and this feature increases when the bond breaks. It seems that a feature of bond breaking in covalent bonds is that

R 2 R   0 at a distance that is approximately equal z z 2

to the length of the covalent radius from the nucleus. Interesting future work would be to consider molecules with multiple bonds, larger molecules, and higher spin states and to consider dissociation of a greater variety of cases.

 ASSOCIATED

CONTENT Supporting Information Plots corresponding to different active spaces for the lithium dimers and F2 can be found in the SI, as well as plots of s(r) and |∇𝑅(𝐫)| for all dimers and plots of

∂𝑅 ∂𝑧

corresponding to Table 2.

 AUTHOR

INFORMATION Corresponding Authors *E-mail: [email protected], [email protected] Notes

The authors declare no competing financial interest. ORCID Rebecca Carlson: 0000-0002-1710-5816 Laura Gagliardi: 0000-0001-5227-1396 Donald G. Truhlar: 0000-0002-7742-7294

 ACKNOWLEDGMENT

The authors are grateful to Prachi Sharma for helpful discussions. This work was supported in part by the National Science Foundation by grant no. CHE-1464536.



Page 30 of 32

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