Evaluating frontier orbital energy and HOMO/LUMO gap with descriptors from density functional reactivity theory
Abstract
Wave function theory (WFT) and density functional theory (DFT)—the two most popular solutions to electronic structure problems of atoms and molecules—share the same origin, dealing with the same subject yet using distinct methodologies. For example, molecular orbitals are artifacts in WFT, whereas in DFT, electron density plays the dominant role. One question that needs to be addressed when using these approaches to appreciate properties related to molecular structure and reactivity is if there is any link between the two. In this work, we present a piece of strong evidence addressing that very question. Using five polymeric systems as illustrative examples, we reveal that using quantities from DFT such as Shannon entropy, Fisher information, Ghosh-Berkowitz-Parr entropy, Onicescu information energy, Rényi entropy, etc., one is able to accurately evaluate orbital-related properties in WFT like frontier orbital energies and the HOMO (highest occupied molecular orbital)/LUMO (lowest unoccupied molecular orbital) gap. We verified these results at both the whole molecule level and the atoms-in-molecules level. These results provide compelling evidence suggesting that WFT and DFT are complementary to each other, both trying to comprehend the same properties of the electronic structure and molecular reactivity from different perspectives using their own characteristic vocabulary. Hence, there should be a bridge or bridges between the two approaches.
Keywords
Shannon entropy Fisher information Frontier orbitals HOMO/LUMO gap Density functional reactivity theoryIntroduction
Density functional theory (DFT) [1] has enjoyed tremendous success in the past few decades as a popular approach to bypass solving the Schrodinger equation. Applications of DFT to understand structural properties and chemical reactivity in a conceptual framework have also seen considerable interest in the literature [2, 3, 4]. From the historical viewpoint, there are two separate undertakings in the literature to appreciate and entertain molecular properties related to stability and reactivity: one using frontier molecular orbitals, and the other through density-related quantities. Well-known examples of the former approach are Fukui’s frontier molecular orbital theory [5, 6] and Woodward–Hoffmann rules, [7] whereas for the latter method, examples are conceptual DFT [2, 3, 4] and its recent developments, which is often called density functional reactivity theory (DFRT) [8]. The molecular orbital theory of chemical reactivity has been the mainstream for over 50 years. The main issue with the orbital method, however, is the fact that orbitals are human inventions so their general applicability is often questionable. In fact, the more accurate the computational results are, the more ambiguous the orbital’s physical meanings will be. As quoted by Mullikan [9], “with old-fashioned chemical concepts, which at first seemed to have their counterparts in Molecular Quantum Mechanics, the more accurate the calculations became the more the concepts tended to vanish into thin air. So we have to ask, should we try to keep these concepts—do they still have a place—or should they be relegated to chemical history. Among such concepts are electronegativity… hybridization, population analysis, charges on atoms, even the idea of orbitals…”
On the other hand, the approach of chemical reactivity theory using density related quantities is still in the development phase [1]. Much has been accomplished in the past few decades [2, 3, 4]. More recent developments have been summarized in a review by one of the present authors [8]. It originates from the chemical application of DFT. Its rigor and validity have been widely outlined and demonstrated. According to DFT, the electron density of a system should determine everything in the ground state, which includes properties related to stability and reactivity. Nevertheless, how to implement this idea and how to make it happen are subject to debate. Different approaches are possible. Conceptual DFT [1, 2, 3, 4] is one such effort, where first, second and third orders of partial derivatives of the total energy with respect to system parameters such as the total number of electrons and external potential are conceptualized to quantify chemical reactivity. The introduction of conceptual DFT and quantification of concepts like electronegativity, hardness, Fukui function, and dual descriptors have evidently witnessed the validity and effectiveness of such alternative tools [1, 2, 3, 4], even though there have been many challenges and controversies reported in the literature for their general validity and applicability for quite some time [10, 11, 12, 13].
Very recently, we advocated using simple density functionals to quantify reactivity properties [8]. Since this approach deviates so much from the original idea of CDFT [1, 2, 3, 4], we call this new effort DFRT [8]. Steric effects quantified using the Weizsäcker kinetic energy density functional [14] were the first example [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Electrophilicity, nucleophilicity, and regioselectivity in terms of information gain or Kullback–Leibler divergence were other examples [30, 31, 32, 33, 34, 35, 36, 37, 38, 39], which have enabled us to provide a completely novel understanding about the orth/para/meta group directing phenomena for electrophilic aromatic substitution reactions [32, 33]. Three representations [8] of the theoretical framework using the electron density, shape function, and atoms-in-molecules (AIM) densities with three different atomic partition methods such as Bader’s zero-flux [40], Becke’s fuzzy atom [41], and Hirshfeld’s stock-holder [42] schemes have been established. Three principles [8], the principle of extreme physical information, minimum information gain principle, and information conservation principle, have also been proposed. These earlier results form the foundation of the so-called information-theoretic (IT) approach in DFRT [8].
From the perspective of molecular properties themselves, these two vastly different methods deal with exactly the same subject, just from two distinctive angles, one using the wave function as the basic quantity and the other using electron density as the basic quantity, so they should be closely related to each other. Meanwhile, because of their close relationship, quantities from one approach should be comprehensible, and able to be represented by, the other approach. For example, the molecular orbital is often thought of being as one of essential quantities from the wave function theory (WFT). According to above assertion, one should be able to appreciate and quantify properties of molecular orbitals using density-related quantities from the perspective of DFRT.
Is this last statement really valid? How accurate is it, if so? To provide an answer to such questions was the main purpose of this work. To that end, we made use of five conjugate molecular chains, such as trans-ethylene, benzene, naphthalene, di-thiophene, and phenanthrene chains, with from one to ten repeating units, n = 1–10, and calculated their frontier orbital (HOMO, highest occupied molecular orbital, and LUMO, lowest unoccupied molecular orbital) energies as well as the HOMO/LUMO gap. The properties and changing patterns of these quantities were then correlated with the quantities from the IT approach from DFRT [8], such as Shannon entropy [43], Fisher information [44], Ghosh-Berkowitz-Parr entropy [45], Onicescu information energy [46], Rényi entropy [47], etc. As will be revealed by the results from this work, we find that the answer to the above questions is definitely yes. We witness strong linear correlations between the properties from the frontier orbital method and quantities from the IT approach. We confirm this favorable answer with numerical results for these systems at both the whole molecule level and the atoms-in-molecules (AIM) level. What we have accomplished in this work is the establishment of a bridge between two different approaches, wave-function-based theory (frontier orbital theory) and density-based theory (DFRT), to comprehend and appreciate the same properties of the electronic structure for molecular systems from different perspectives.
This work is dedicated to Prof. Henry Chermette [48], who, among his many contributions, published the first ever review on conceptual DFT as early as 1999.
Information-theoretic approach in DFRT
The validity of this relationship has been verified and confirmed numerically [49].
We should keep in mind that the AIM IT magnitudes calculated in Eqs. 17–20 are not, in general, equal to their previously defined counterparts. They will have the same numerical values for non-overlapping atoms in molecules (i.e., Bader atoms), but this will not be true for overlapping atomic densities (i.e., Hirshfeld atoms).
Employing the information gain and Hirshfeld charge from above formulations, we successfully correlated them to electrophilicity, nucleophilicity, and regioselectivity [30, 31]. We also applied these ideas to a number of well-studied organic reactions [34, 35, 36, 37, 38]. In particular, we provided a completely novel understanding about the origin and nature of the ortho/para/meta group directing phenomena for electrophilic aromatic substitution reactions [32, 33]. In addition, our recent efforts have shown that these quantities from the IT approach are able to adequately accurately describe the total electronic energy and its components for both atoms and molecules [39]. Instead of looking for long-elusive formulas to approximate the universal energy density functional and its components such as the exchange-correlation energy density functional, we proposed to simulate them alternatively using a set of simple density functionals from the IT approach [39].
In this work, continuing our previous efforts to evaluate electronic properties of molecular systems with the IT quantities, we examined their feasibility and applicability to quantify frontier orbital energies and the HOMO/LUMO gap, which are often believed to be unique to WFT. Employing five polymeric systems as illustrative examples, some of which were found responsible for the visible luminescence from low dimensional carbon materials [60], as will be shown below, we observed strong correlations between these properties and IT quantities, providing favorable answers to the questions defined above.
Computational details
All electronic wave functions were obtained from the Gaussian 09 package, version D01 [61]. For all atoms and molecules investigated in this work, the B3LYP hybrid density functional was employed [62, 63]. For the choice of basis sets, the standard Pople’s 6-311G(d,p) basis set was used [64]. A total of 50 molecular systems was investigated in this work. For each conjugated system chosen, its monomer, dimer, trimer, …, and up to 10 repeating units were examined. Both molecular and AIM values of the Shannon entropy, Fisher information, GBP entropy, and Onicescu information energy of orders 2 and 3 were calculated by the Multiwfn 3.3 package [65], whose reliability and applicability have extensively been verified previously. To consider atomic contributions at the AIM level, three schemes to perform the atomic partition are possible: Becke’s fuzzy atom approach, Bader’s zero-flux AIM criterion, and Hirshfeld’s stockholder approach. As demonstrated earlier by us, these three approaches yield similar results [36]. In this work, we chose the Hirshfeld’s stockholder approach to partition atoms in molecules. To consider atomic contributions at the AIM level and their correlations with quantities like HOMO, LUMO and HOMO/LUMP gap, we made use of the average of the IT quantities on carbon atoms for each of the systems for the purpose. The IT quantities in the shape function representation can be obtained using the relationships in Eqs. (13) to (15). All molecules were fully optimized. Tight self-consistent field convergence criterion and ultrafine integration grids were employed. Checkpoint files obtained from these Gaussian calculations were used as the input file to obtain numerical values for the IT quantities. It is well known that the energy of the LUMO orbital can be modified dramatically when diffuse functions are used in the basis set. We anticipate that the IT quantities would change accordingly. That is to say, we expect that, though both HOMO/LUMO energies and IT quantities changes with respect to the choice of the basis set, the correlation between the two sets of quantities will be similar.
Results and discussion
Numerical results of highest occupied molecular orbital/lowest unoccupied molecular orbital (HOMO/LUMO) energies and their gap values for the five conjugate systems from Scheme 1, with from one to up to ten repeating units, n = 1–10. Units in eV
Mol | n | N | HOMO | LUMO | GAP | Mol | n | N | HOMO | LUMO | GAP |
---|---|---|---|---|---|---|---|---|---|---|---|
5 | 1 | 16 | −7.529 | 0.151 | 7.681 | 4 | 1 | 86 | −5.878 | −1.509 | 4.368 |
2 | 30 | −6.495 | −0.896 | 5.599 | 2 | 170 | −5.342 | −2.133 | 3.208 | ||
3 | 44 | −5.956 | −1.469 | 4.487 | 3 | 254 | −5.159 | −2.416 | 2.742 | ||
4 | 58 | −5.623 | −1.822 | 3.801 | 4 | 338 | −5.076 | −2.487 | 2.589 | ||
5 | 72 | −5.396 | −2.062 | 3.334 | 5 | 422 | −5.030 | −2.553 | 2.476 | ||
6 | 86 | −5.232 | −2.235 | 2.997 | 6 | 506 | −4.985 | −2.266 | 2.719 | ||
7 | 100 | −5.107 | −2.366 | 2.741 | 7 | 590 | −4.970 | −2.286 | 2.683 | ||
8 | 114 | −5.009 | −2.468 | 2.541 | 8 | 674 | −4.961 | −2.306 | 2.655 | ||
9 | 128 | −4.930 | −2.549 | 2.380 | 9 | 758 | −4.956 | −2.324 | 2.632 | ||
10 | 142 | −4.865 | −2.616 | 2.249 | 10 | 842 | −4.948 | −2.332 | 2.615 | ||
4 | 1 | 42 | −6.952 | −0.219 | 6.733 | 5 | 1 | 68 | −6.034 | −1.226 | 4.807 |
2 | 82 | −6.313 | −0.934 | 5.378 | 2 | 120 | −5.760 | −1.516 | 4.243 | ||
3 | 122 | −6.037 | −1.244 | 4.792 | 3 | 172 | −5.664 | −1.630 | 4.033 | ||
4 | 162 | −5.893 | −1.414 | 4.478 | 4 | 224 | −5.621 | −1.693 | 3.928 | ||
5 | 202 | −5.813 | −1.513 | 4.299 | 5 | 276 | −5.599 | −1.739 | 3.860 | ||
6 | 242 | −5.762 | −1.578 | 4.184 | 6 | 328 | −5.588 | −1.692 | 3.895 | ||
7 | 282 | −5.730 | −1.621 | 4.109 | 7 | 380 | −5.581 | −1.696 | 3.884 | ||
8 | 322 | −5.705 | −1.655 | 4.049 | 8 | 432 | −5.577 | −1.696 | 3.881 | ||
9 | 362 | −5.688 | −1.678 | 4.010 | 9 | 484 | −5.575 | −1.691 | 3.883 | ||
10 | 402 | −5.675 | −1.698 | 3.976 | 10 | 536 | −5.573 | −1.704 | 3.869 | ||
3 | 1 | 42 | −6.952 | −0.219 | 6.733 | ||||||
2 | 68 | −6.034 | −1.226 | 4.807 | |||||||
3 | 94 | −5.472 | −1.888 | 3.584 | |||||||
4 | 120 | −5.101 | −2.330 | 2.771 | |||||||
5 | 146 | −4.843 | −2.640 | 2.202 | |||||||
6 | 172 | −4.655 | −2.867 | 1.788 | |||||||
7 | 198 | −4.513 | −3.037 | 1.475 | |||||||
8 | 224 | −4.404 | −3.169 | 1.234 | |||||||
9 | 250 | −4.318 | −3.273 | 1.045 | |||||||
10 | 276 | −4.093 | −3.582 | 0.511 |
An illustrative example of information-theoretic (IT) quantities using Molecule 1 in Scheme 1 as an example at the molecular level. Also shown are their correlation coefficients with HOMO/N and GAP/N quantities, with N the total number of electrons in each of the systems. Values in atomic units
N | S_{S} | S_{σ} | I_{F} | I_{σ} | S_{GBP} | I_{G} | R_{2} | R_{2}^{σ} | R_{2}^{r} |
---|---|---|---|---|---|---|---|---|---|
16 | 21.774 | 2.565 | 508.090 | 1.985 | 104.763 | 0.336 | 63.496 | 0.248 | 16.632 |
30 | 36.963 | 2.709 | 1005.813 | 1.118 | 195.954 | 0.552 | 126.798 | 0.141 | 31.046 |
44 | 52.128 | 2.828 | 1503.473 | 0.777 | 287.135 | 0.766 | 190.097 | 0.098 | 45.458 |
58 | 67.288 | 2.924 | 2001.115 | 0.595 | 378.312 | 0.980 | 253.396 | 0.075 | 59.869 |
72 | 82.446 | 3.002 | 2498.750 | 0.482 | 469.488 | 1.194 | 316.696 | 0.061 | 74.279 |
86 | 97.603 | 3.069 | 2996.379 | 0.405 | 560.663 | 1.407 | 379.994 | 0.051 | 88.690 |
100 | 112.759 | 3.128 | 3494.006 | 0.349 | 651.838 | 1.620 | 443.296 | 0.044 | 103.100 |
114 | 127.915 | 3.179 | 3991.631 | 0.307 | 743.012 | 1.833 | 506.601 | 0.039 | 117.509 |
128 | 143.070 | 3.225 | 4489.254 | 0.274 | 834.186 | 2.046 | 569.896 | 0.035 | 131.919 |
142 | 158.225 | 3.267 | 4986.877 | 0.247 | 925.359 | 2.260 | 633.202 | 0.031 | 146.329 |
R^{2}(HOMO/N) | 0.622 | 0.788 | 0.622 | 0.991 | 0.622 | 0.623 | 0.622 | 0.990 | 0.622 |
R^{2}(GAP/N) | 0.565 | 0.737 | 0.565 | 0.975 | 0.565 | 0.567 | 0.565 | 0.974 | 0.565 |
An illustrative example of IT quantities on the carbon atoms using Molecule 1 in Scheme 1 as an example at the atoms-in-molecules (AIM) level using Hirshfeld’s share-holder partition scheme. Also shown are their correlation coefficients with HOMO/Nc, and GAP/Nc quantities, with Nc representing the average of the total number of electron populations on the carbon atoms in each of the systems, and n as the number of repeating units. Values in atomic units
N | S_{S} | S_{σ} | I_{F} | I_{σ} | S_{GBP} | I_{G} | R_{2} | R_{2}^{σ} | R_{2}^{r} | Hirshfeld |
---|---|---|---|---|---|---|---|---|---|---|
1 | 5.236 | 2.666 | 245.446 | 40.371 | 39.555 | 0.153 | 31.525 | 0.853 | 6.304 | −0.080 |
2 | 5.021 | 2.630 | 245.073 | 40.441 | 39.379 | 0.128 | 31.532 | 0.859 | 6.255 | −0.060 |
3 | 4.946 | 2.618 | 244.938 | 40.462 | 39.319 | 0.120 | 31.533 | 0.861 | 6.238 | −0.053 |
4 | 4.909 | 2.611 | 244.868 | 40.473 | 39.289 | 0.116 | 31.534 | 0.861 | 6.230 | −0.050 |
5 | 4.886 | 2.608 | 244.826 | 40.479 | 39.271 | 0.113 | 31.535 | 0.862 | 6.225 | −0.048 |
6 | 4.870 | 2.605 | 244.797 | 40.483 | 39.258 | 0.111 | 31.535 | 0.862 | 6.221 | −0.047 |
7 | 4.859 | 2.603 | 244.776 | 40.486 | 39.250 | 0.110 | 31.536 | 0.863 | 6.219 | −0.046 |
8 | 4.851 | 2.602 | 244.761 | 40.488 | 39.243 | 0.109 | 31.536 | 0.863 | 6.217 | −0.045 |
9 | 4.845 | 2.601 | 244.749 | 40.490 | 39.238 | 0.108 | 31.536 | 0.863 | 6.215 | −0.045 |
10 | 4.840 | 2.600 | 244.739 | 40.491 | 39.234 | 0.108 | 31.536 | 0.863 | 6.214 | −0.044 |
R^{2}(HOMO/Nc) | 0.998 | 0.998 | 0.996 | 1.000 | 0.998 | 0.998 | 0.995 | 0.999 | 0.998 | 0.999 |
R^{2}(GAP/Nc) | 0.988 | 0.988 | 0.984 | 0.996 | 0.989 | 0.988 | 0.990 | 0.991 | 0.988 | 0.991 |
Correlation coefficients R^{2} of HOMO energy and HOMO/LUMO gap with quantities from density functional reactivity theory (DFRT) at both molecular and AIM levels
Molecular level | AIM level | ||||||||
---|---|---|---|---|---|---|---|---|---|
Mol | S_{σ} | S_{GBP}/N | R_{2}^{σ} | Hirshfeld | S_{σ} | S_{GBP}/N_{C} | R_{2}^{σ} | ||
1 | HOMO/N | 0.788 | 0.993 | 0.990 | HOMO/N_{C} | 0.999 | 0.998 | 0.995 | 0.999 |
GAP/N | 0.737 | 0.979 | 0.974 | GAP/N_{C} | 0.991 | 0.988 | 0.983 | 0.991 | |
2 | HOMO/N | 0.848 | 0.998 | 0.997 | HOMO/N_{C} | 0.999 | 0.999 | 0.999 | 0.999 |
GAP/N | 0.808 | 0.990 | 0.988 | GAP/N_{C} | 0.992 | 0.992 | 0.993 | 0.992 | |
3 | HOMO/N | 0.848 | 0.997 | 0.991 | HOMO/N_{C} | 0.995 | 0.998 | 0.998 | 0.995 |
GAP/N | 0.791 | 0.984 | 0.971 | GAP/N_{C} | 0.978 | 0.986 | 0.996 | 0.979 | |
4 | HOMO/N | 0.784 | 0.997 | 0.997 | HOMO/N_{C} | 0.987 | 0.991 | 0.998 | 0.996 |
GAP/N | 0.856 | 0.976 | 0.978 | GAP/N_{C} | 0.989 | 0.964 | 0.984 | 0.983 | |
5 | HOMO/N | 0.850 | 0.999 | 0.999 | HOMO/N_{C} | 0.999 | 1.000 | 1.000 | 1.000 |
GAP/N | 0.879 | 0.995 | 0.993 | GAP/N_{C} | 0.995 | 0.996 | 0.997 | 0.996 | |
Avg. | 0.819 | 0.991 | 0.988 | 0.992 | 0.991 | 0.994 | 0.993 |
Put together, the results of this work clearly demonstrate that one is able to accurately evaluate frontier orbital properties such as the HOMO energy and HOMO/LUMO gap with IT quantities such as Shannon entropy, Fisher information, GBP entropy, Rényi entropy, Onicescu information energy, etc. This work is consistent with what we have recently proposed in our previous work [39], where we showed that these same IT quantities can be employed to evaluate the electronic properties of a Coulombic system such as the total electronic energy as well as its components like the exchange-correlation energy. Where this work differs from our previous study is in the fact that, as one of the main products in WFT, orbitals are often believed to be unique to that theory. What we have accomplished in this work is not only to present a convincing piece of evidence that orbital properties can be accurately simulated and appreciated by IT quantities, where only the electron density and its associated quantities are involved, but also to establish a bridge between the two vastly different approaches of wave-function-based theory (frontier orbital theory) and density-based theory (DFRT), to comprehend and entertain the same properties of the electronic structure for molecular systems from different perspectives. WFT and DFT share the same origin, deal with the same objective, yet use distinct methodologies and different vocabularies in their language of understanding and appreciating molecular structure and reactivity properties.
Conclusions
As we continue to look into the possibility of employing IT quantities to evaluate and quantify properties related to molecular structure and chemical reactivity, our current work presents numerical results from a different perspective. Using IT quantities such as Shannon entropy, Fisher information, GBP entropy, Onicescu information energy, Rényi entropy, etc., we evaluate and then correlate these quantities with properties such as frontier orbital levels and the HOMO/LUMO gap, which are often believed to be unique to wave function theory. With five polymeric systems as illustrative examples, we have unambiguously demonstrated that these orbital-related properties can be simulated accurately by IT quantities. The wave-function-based theory like molecular orbital theory and density-based theory such as DFRT are complementary to each other, both trying to comprehend and appreciate the same properties of the electronic structure for molecular systems from different perspectives with their own language.
Notes
Acknowledgements
S.B.L. and C.Y.R. acknowledge support from the National Natural Science Foundation of China (No.21503076) and RQZ is supported in part by a grant from Environmental Conservation Fund (No. 921100 (29/2015)).
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