Article pubs.acs.org/JPCB
Magnetism and Bond Order in Diatomic Molecules Described by Semiclassical Electrons Solen Ekesan, Damian Y. Lin, and Judith Herzfeld* Department of Chemistry, Brandeis University, 415 South Street MS#015, Waltham Massachusetts 02453, United States S Supporting Information *
ABSTRACT: The past decade has seen the first attempts at quantifying a semiclassical description of electrons in molecules. The challenge in this endeavor is to find potentials for electron interactions that adequately capture quantum effects. As has been the case for density functionals, the challenge is particularly great for the effects that follow from the requirement for wave function antisymmetry. Here we extend our empirical inquiry into effective potentials, from prior work on the monatomic atoms and ions of nonmetals, to diatomic molecules and ions formed by these elements. Newly adjusted and trained for the longer distances relevant to diatomics, pairwise potentials are able to fit the bond orders and magnetic properties of homonuclear species. These potentials are then found to do an excellent job of predicting the magnetism of heteronuclear species. In these molecules the predicted distribution of electrons also correctly reflects increasing ionic character with increasing difference in the electronegativities of the participating atoms. The distinctive features of the current potential are discussed, along with issues calling for further improvements.
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INTRODUCTION One of the early triumphs of quantum mechanics (QM) was its explanation of periodic trends in the properties of atoms and diatomic molecules. In particular, applications to electrons in diatomic molecules shed light on variations in bond order and the occurrence of paramagnetism, notably in dioxygen. Since then, QM has illuminated many aspects of chemical bonding and reactivity, limited only by the computationally onerous nature of the approach. Over the decades, many approximations have been explored in the interest of achieving economies with minimal impact on consistency and reliability. Recently, density functional tight-binding has shown promise when suitable care is given to the extensive parametrization that is required. For applications to large molecules and collections of molecules, nonorbital approaches have been explored over many years. Among these, molecular mechanics constructs are widely used for studies of molecular structure and dynamics. The common atomistic force fields (e.g., OPLS,1 CHARMM,2 and AMBER3) require a prespecified bond connectivity, while the reactive force fields (e.g., ReaxFF,4 REBO,5,6 EVB7,8) calculate bond order on the fly. In both cases, the electrons are implicit and there is no description of the electron charge or spin distribution. Situated between these two extremes, we inquire to what extent we can have the best of both worlds, with an explicit account of electrons that is radically simpler and more efficient than orbital descriptions. In our first generation effort, we found that independently mobile, semiclassical valence electron pairs could describe the polarizability, amphiproticity, and hydrogen bonding of water in a balanced, intuitive, and highly efficient manner, such that large and long simulations of the solvation and dynamics of aqueous hydroxide and hydronium could © XXXX American Chemical Society
provide results for ion diffusion and surface propensity that challenged DFT results.9−12 In our second-generation effort, we consider the electrons singly to address redox and free-radical phenomena. This construct is able to describe the aufbau, ionization energies, and polarizabilities of atoms of the reactive 2p and 3p elements.13,14 Here we extend this approach to the spin states and bond orders of diatomic species at fixed bond length. In this regime, we discover an essential feature of the interparticle potentials that takes the construct one major and essential step closer to the ultimate goal of turn-key simulations of chemical reactions in systems with many degrees of freedom, for example, those in which solvent composition affects the rates and selectivities of chemical reactions.
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MODEL DESCRIPTION LEWIS• is a subatomistic model comprising fully charged, independently mobile valence electrons (e) and kernels of the various elements (X, Y, each including nucleus and core electrons). Each electron particle has one of two possible spins and a variable cloud diameter, with an associated quantum kinetic energy. As specified in the Supporting Information, all particles are governed by smoothly differentiable pairwise interactions that are Coulombic at long distances and softened at short distances, as required by the diffuse distributions of electrons.13,14 Thus, the contributions to the total energy are
∑ UKE + ∑ Uee(u) + ∑ Uee(l) + ∑ UeX + ∑ UXY
(1)
Special Issue: William M. Gelbart Festschrift Received: March 11, 2016 Revised: May 19, 2016
A
DOI: 10.1021/acs.jpcb.6b02576 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
elements are optimized individually with the Uee fixed). Previously the Uee were trained on H and Cl;13 however, the resulting potentials gave good predictions for Cl2. To challenge the new potentials by one of the failures of the old, we trained the new Uee potentials with H and O, including the specification that the quintet state is the highest energy spin configuration of O2.
where two types of electron−electron repulsions are required to distinguish between the case of like and unlike spins (Uee(l) and Uee(u), respectively), with the former including an additional exchange term that reflects the required antisymmetry of the underlying wave function Uee(l) = Uee(u) + Uexchange
(2)
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15−18
As expected, the exchange term needs to be strongly positive at short distances and vanish at long distances; however, we also found that a sign change is necessary for a correct ordering of the energies of different spin configurations in the atoms and ions of 2p and 3p elements.13,14
RESULTS New Uexchange. With the new training set, we could focus on the inadequacies of Uexchange at distances relevant to diatomic species. Previously we had shown that a sign change was necessary to describe the spin orders of monatomic species,13 and it now appeared that a second sign change was essential for the diatomic species, at a distance inconsistent with the previous form of Uexchange. These suspicions were confirmed when we could fit our training set only with exchange potentials that had flexibility in the location of the second sign change. Of the different forms we have tried, we have gotten the best results with χ Uexchange = 2ee exp(− (τee1reff )2 ) deff
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FROM SINGLE ATOMS TO MOLECULES Initial work on LEWIS• considered only monatomic species (for which UXY is not relevant) so that attention could be focused on developing forms and parameters for Uee(l), Uee(u), and UeX. In principle, these terms also suffice to describe the electronic structure of molecules with fixed nuclear coordinates. In particular, minimization of the energy
∑ UKE + ∑ Uee(u) + ∑ Uee(l) + ∑ UeX
(3)
should predict the correct bond orders and the correct sequences of spin states. An example is the case of dioxygen, O2, where the triplet (paramagnetic) species is the doublebonded ground state, while the singlet (diamagnetic) species is an excited state. Using the published potentials trained on monatomic species, we performed Monte Carlo (MC) simulations for the singlet (s) and triplet (t) states of the homonuclear diatomics of all 2p and 3p elements previously addressed by LEWIS•.14 We fixed the bond lengths in both spin states to the experimental value for the ground state with the understanding that the higher energy spin state would be at a disadvantage at that bond length and therefore still be the more energetic species. Comparing the results with experiment, it became clear that potentials trained for the distances relevant to monatomic species are not adequate over the distances relevant to diatomic species. One problem was that most of the homonuclear diatomic molecules were predicted to hold only two electrons in the bonding region between the kernels, effectively forming only single bonds. In addition, almost half of the diatomics had the wrong orders of the spin states. While both properties were predicted correctly for B2, C2, F2, and Cl2, both were predicted incorrectly for O2 and Si2. These failures indicated that the potentials, Uee(l), Uee(u), and UeX needed to be revisited, but in a conservative fashion: our strategy is to consider potential functions that retain the successful short-range features while innovating at longer range.
× [exp(− (τee2λee1)2 ) − exp(− (τee2reff )2 )] × [exp(− (τee3λee2)2 ) − exp(− (τee3reff )2 )]
(4)
where χee controls the amplitude, deff is the effective cloud diameter (as specified in the Supporting Information), reff is the distance between the electrons in units of deff, τee1 controls the overall range of the potential, τee2 and τee3 control the shapes of the negative and positive regions, while λee1 and λee2 control the distances of the two sign changes. With this modification of the exchange potential, no changes were required in the forms of Uee(u) or UeX. The resulting electron−electron potentials are plotted in Figure 1 (for the optimized potential parameters shown in Table S3).
Figure 1. Charge-normalized potentials versus effective distance for electron−electron repulsions with the new exchange form. Uee(u) (solid line) and Uee(l) (dashed line) based on the parameters in Table S3 are compared with a simple Coulomb potential (dotted line). (Although Uee(l) appears to diverge, it levels off to a finite value outside the range of this graph.)
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NEW TRAINING SET The monatomic training set continued to comprise the energies of the ground and stable spin-excited state(s) of the atoms in various degrees of ionization, with extra weights given to the electron affinities (EAs) and the first ionization energies (IE1s). For longer distances, we included the number of bonding electrons and the order of the triplet and singlet configurations for the homonuclear diatomics (fixed at their ground-state bond length). To make the optimization manageable, the Uee potentials are optimized with just two elements (before the UeX for the other
For electrons of unlike-spin (solid line), the softening of Coulomb repulsion (dotted line) at short distances reflects the diffuse nature of the electron cloud. The dramatic difference for electrons of like-spin (dashed line) is due to the exchange term. Although the crossover region between the sign changes in the exchange term might seem small, with variations in the cloud diameter it is highly populated. We are not certain whether the bump following the crossover region is necessary or a consequence of this particular exchange form. B
DOI: 10.1021/acs.jpcb.6b02576 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
Figure 2. Ground-state LEWIS• structures of the homonuclear diatomic molecules of the reactive 2p and 3p elements. Valence electrons with spins α and β are rendered in magenta and light pink, respectively, with diameters scaled to the corresponding cloud diameter. Where α and β pair up, they are shown as a single particle that is half magenta and half light pink.
Fits to the Training Set. Our strategy of minimizing change in the short-range interactions proved successful in preserving the success with monatomics,13,14 including the energetics (see Tables S4−S6) and the correspondence of electron distributions with the major lobes of hybrid orbitals. For all homonuclear diatomics, we obtain the correct energy order for the singlet versus the triplet and the correct number of bonding electrons for the ground state. The energyminimized structures of homonuclear diatomics are illustrated in Figure 2. Geometries are almost identical for states with the same spin configuration (i.e., within a group). While diatomics within a group generally have the same experimental total spin in the ground state, this is not the case in group 4A, and it is noteworthy that LEWIS• correctly captures the difference in ground-state spin between C2 and Si2. As was the case for monatomics, electron cloud diameters do not vary significantly within the molecules and shrink going across a row and up a column, consistent with periodic trends of atomic radii. The symmetries of the molecules are in accord with Linnett’s description19 of these bonds, such that electrons of like spin avoid each other by forming tetrahedra. This can be clearly seen in all of the ground-state molecules except for B2 and Al2, in which the small number of electrons precludes the suggested symmetries. Validation Set. To test the validity of the potentials, we gathered all diatomic species with available experimental bond lengths that can be constructed with the elements for which we have trained potentials, a total of 85 molecules (XX: 10 neutral and 7 ions shown in Table 1, XH: 10 neutral and 13 ions shown in Table 2, XY: 37 neutral and 8 ions shown in Tables 3 and 4). For each species, we modeled the three lowest spin states at the experimental bond length of the ground state and performed MC simulations to check the relative energies of the spin states as well as the bond orders. For the neutral homonuclear diatomic molecules we modeled the third spin state that had not been included in the training set. LEWIS• produced stable structures for all diatomic species except for PO− and O2−, each of which lost one electron, resulting in the neutral molecule. Of the remaining molecules, LEWIS• successfully predicts the correct energy orders for a total of 64 molecules among the validation set hydrides and heteronuclear diatomic molecules, a success rate of 85%. The
Table 1. Ground- and Excited-State Multiplicities for Homonuclear Diatomic Molecules of the Reactive 2p and 3p Elementsa,b experimental20 RXY (Å)
E0
E1
LEWIS•
E2
E0
E1
E2
c
c
B2 C2 N2 O2 F2
1.59 1.24 1.10 1.21 1.41
t s s t s
t t s t
qnt -
t sc sc tc sc
s tc tc sc tc
qnt qnt qnt qntc qnt
Al2 Si2 P2 S2 Cl2
2.70 2.25 1.89 1.89 1.99
t t s t s
t s t
-
tc tc sc tc tc
sc sc tc sc sc
qnt qnt qnt qnt qnt
C2− O2+ O2− F2+ P2+ S2+ Cl2+
1.27 1.12 1.35 1.32 1.99 1.83 1.89
d d d d d d d
q q q q -
-
d d d d q d
q q q q d q
sxt sxt sxt sxt sxt sxt
error
-
x
a
Groups are the neutral homonuclear diatomics of the 2nd row (top), the neutral homonuclear diatomics of the 3rd row (middle), and the stable ions (bottom). bs: singlet, t: triplet, qnt: quintet, d: doublet, q: quartet, sxt: sextet, x: order could not be reproduced, -: unstable. c Order of these states is included in the training set.
nine molecules with erroneous energy orders are S2+, CH−, SiH−, BC, BN, NO−, PCl, AlN, and PF. There does not seem to be a clear pattern for categorizing these errors. In the validation set, LEWIS• also extends the description of electronic structure from the pure covalent bonds in the neutral homonuclear diatomics (Figure 2) to the bonds in species with strong ionic character (Figure 3). The elements with lowest electronegativity in our set are B, Al, and Si, and in all of their bonds with highly electronegative elements (F and Cl) we see clear ionic nature. An example of this is shown in Figure 3 for the molecule AlCl, where chlorine has clearly adopted an electron from aluminum and has become chloride, while aluminum carries the positive charge. C
DOI: 10.1021/acs.jpcb.6b02576 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B Table 2. Ground- and Excited-State Multiplicities for Hydrides of the Reactive 2p and 3p Elementsa,b experimental20
Table 3. Ground- and Excited-State Multiplicities for SameRow Heteronuclear Diatomic Molecules of Reactive 2p and 3p Elementsa,b
LEWIS•
RXY (Å)
E0
E1
E2
E0
E1
E2
BH CH NH OH HF
1.23 1.12 1.04 0.97 0.92
s d t d s
t q s t
-
s d t d s
t q s q t
qnt sxt qnt sxt qnt
AlH SiH PH SH HCl
1.65 1.52 1.42 1.34 1.27
s d t d s
t s t
-
s d t d s
t q s q t
qnt sxt qnt sxt qnt
BH+ CH− NH+ OH+ OH− HF+ AlH+ SiH+ SiH− PH+ PH− SH+ HCl+
1.22 1.09 1.07 1.03 0.96 1.01 1.60 1.50 1.47 1.44 1.41 1.37 1.32
d t d t s d d s t d d t d
s q s t s -
-
d s d t s d d s s d d t d
q t q s t q q t t q q s q
sxt qnt sxt qnt qnt sxt qnt qnt sxt sxt qnt sxt
experimental20
error
x
x
LEWIS•
RXY (Å)
E0
E1
E2
E0
E1
E2
error
BC BN BO BF CN CO CF NO NF FO
1.49 1.33 1.20 1.27 1.17 1.13 1.27 1.15 1.32 1.35
q t d s d s d d t d
d t q t q q s -
-
d s d s d s d d t d
q t q t q t q q s q
sxt qnt sxt qnt sxt qnt sxt sxt qnt sxt
x x
AlS AlCl SiS SiCl PS PCl SCl
2.03 2.13 1.93 2.06 1.90 2.02 1.98
d s s d d t d
t t s -
-
d s s d d s d
q t t q q t q
sxt qnt qnt sxt sxt qnt sxt
CN− CO+ NO+ NO−
1.18 1.12 1.07 1.26
s d s t
t s
-
s d s s
t q t t
qnt sxt qnt qnt
x
x
a
Groups are the neutral heteronuclear diatomics of the 2nd row, the neutral heteronuclear diatomics of the 3rd row, and the stable ions of these species. bs: singlet, t: triplet, qnt: quintet, d: doublet, q: quartet, sxt: sextet, x: order could not be reproduced.
a
Groups are the neutral hydrides of the 2nd row, neutral hydrides of the 3rd row, and the stable ions of these species. bs: singlet, t: triplet, qnt: quintet, d: doublet, q: quartet, sxt: sextet, x: order could not be reproduced.
molecule, which is followed by the spin-down electrons, except for the single one at the sulfur kernel. In PS• spin-up electrons are distributed as 2−1−3, while spin-down electrons hold a reciprocal distribution of 1−3−1. Regardless of the distributions, they both yield a double bond. Overall we observed that the electron distributions were least satisfactory in the diatomics of elements B, Al, and Si. We suspect that this is because these elements are farthest from oxygen, which is the element that was used, along with hydrogen, to train the potentials for the electron−electron repulsions. Prospects. Although the potential that we report here generally does an impressive job of describing the electronic structure of a wide range of diatomic molecules, we have noted some instances of incorrect spin orders and irregular electron distributions. In addition, we have found that even where the potential does give good spin orders and electron distributions, the global energy minima are accompanied by many local minima separated by enormous barriers. To have confidence in finding the global minimum in a given system, it was necessary to begin MC calculations from many different initial distributions of the electrons. Such an extremely rugged surface seems to be a poor candidate for good simulations of reactions, and we suspect that, to give a good description close to the global minimum, the present potential functions are compensating elsewhere in unfortunate ways. Clearly there is a need for further exploration of alternative potential forms. A clue as to how to proceed is suggested by the sign changes in the present Uexchange. In previous attempts to describe
Molecules HF, HCl, OH−, ClF, CO, and CN− are well known and are often used in textbooks for illustrating molecular orbitals. LEWIS• succeeds in describing all of them accurately. HF (Figure 3), HCl, and OH− have the same spin configuration and result in the same geometry, in which all valence electrons are paired up, with one of the pairs located between H and the heavy atom kernel forming a single bond. A single bond is also observed in ClF (Figure 3), where the lone pairs form staggered triangles around the kernels. CO and CN− are known to form triple bonds, and this is beautifully replicated by LEWIS• (Figure 3) with three electron pairs between the two kernels, in a trigonal planar array suggestive of banana bonds. While the bonding electrons in CN− are centered at the midpoint of the bond, in CO they are slightly shifted toward the more electronegative oxygen. The single electrons in LEWIS• (vs the pairs in LEWIS9−12) enable us to describe free radicals. In Figure 3 we show OH•, SO+•, PS•, and CP• as examples of radicals with different bond orders. These molecules also differ in the location of the unpaired electron. In OH•, SO+•, and PS• the unpaired electron resides as a lone electron on one of the kernels, sustaining an integer bond order; in CP•, the bonding region accommodates the unpaired electron, resulting in a fractional bond order of 2.5. SO+• and PS• also illustrate a case of the same electron configurations, resulting in different geometries. While they both have a 6α5β electron configuration, in SO+• we see a 2− 2−2 distribution of spin-up electrons along the length of the D
DOI: 10.1021/acs.jpcb.6b02576 J. Phys. Chem. B XXXX, XXX, XXX−XXX
The Journal of Physical Chemistry B
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CONCLUSIONS LEWIS• has been extended to diatomic molecules. Via suitable pairwise interactions of semiclassical electrons with each other and with kernels, it can accurately describe not just the valence shells of atoms but also the electronic structures of diatomic molecules, including both magnetism and bond order. A single set of electron−electron repulsions is again shown to be transferable across the reactive 2p and 3p elements. Potentials that have been fit to the electronic structures of homonuclear diatomic molecules are able to predict the electronic structures of heteronuclear diatomic molecules, including hydrides, radicals, and molecules with strong ionic character. Some limitations have been noted, but the present results are encouraging, and unexpected features of the current potentials suggest a way forward.
Table 4. Ground- and Excited-State Multiplicities for Heteronuclear Diatomic Molecules between Reactive 2p Elements and Reactive 3p Elementsa,b experimental20
LEWIS•
RXY (Å)
E0
E1
E2
E0
E1
E2
BS BCl CP CS CCl NS NCl AlN AlO AlF SiN SiO SiF PN PO PF SO SF ClO ClF
1.61 1.72 1.56 1.53 1.65 1.49 1.61 1.79 1.62 1.62 2.13 2.13 2.13 1.49 1.48 1.59 1.48 1.60 1.57 1.63
d s d s d d t t d s d s d s d t t d d s
t t s t t q q s s t
-
d s d s d d t s d s d s d s d s t d d s
q t q t q q s t q t q t q t q t s q q t
sxt qnt sxt qnt sxt sxt qnt qnt sxt qnt sxt qnt sxt qnt sxt qnt qnt sxt sxt qnt
NS+ PO− PF+ SO+
1.44 1.54 1.50 1.42
s t d d
s q
-
s d d
t q q
qnt sxt sxt
Article
error
x
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b02576. Training set, potential forms, optimization techniques, optimized parameters, fit results, Tables S1−S6. (PDF)
x
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AUTHOR INFORMATION
Corresponding Author
*Tel: 781-736-2538. Fax: 781-736-2516. E-mail: herzfeld@ brandeis.edu.
-
Notes
The authors declare no competing financial interest.
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a
Groups are the neutral species and the stable ions. bs: singlet, t: triplet, qnt: quintet, d: doublet, q: quartet, sxt: sextet, x: order could not be reproduced, -: unstable.
ACKNOWLEDGMENTS We thank Francesco Pontiggia for his help with the parallelization of our code. This work was sponsored in part by NSF grant 1305713. Additional computational support was provided by the Brandeis HPC.
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REFERENCES
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Figure 3. Ground-state LEWIS• structures of variously bonded heteronuclear diatomic molecules formed by 2p and 3p elements. Valence electrons with spins α and β are rendered in magenta and light pink, respectively, with diameters scaled to the corresponding cloud diameter. Where α and β pair up they are shown as a single particle that is half magenta and half light pink.
fermions semiclassically,15−18 it has been assumed that Uexchange is dominated by Pauli exclusion; however, the contribution of Pauli exclusion to Uexchange is strictly positive. The need for sign changes calls for a closer look at other contributions to Uexchange, and we are presently undertaking a reexamination of all of the exchange integrals that Uexchange is meant to represent. E
DOI: 10.1021/acs.jpcb.6b02576 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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