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Tuning the HOMO-LUMO energy gap of small diamondoids using inverse molecular design Jos L. Teunissen, Frank De Proft, and Freija De Vleeschouwer J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b01074 • Publication Date (Web): 20 Feb 2017 Downloaded from http://pubs.acs.org on February 23, 2017

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Journal of Chemical Theory and Computation

Tuning the HOMO-LUMO energy gap of small diamondoids using inverse molecular design Jos L. Teunissen, Frank De Proft, Freija De Vleeschouwer* [email protected] Research Group of General Chemistry, Vrije Universiteit Brussel (VUB) Pleinlaan 2, 1050 Brussels (Belgium)

Abstract Functionalized diamondoids show great potential as building blocks for various new optoelectronic applications. However, until now, only simple mono- and double substitutions were investigated. In this work, we considered up to 10 and 6 sites for functionalization of the two smallest diamondoids, adamantane and diamantane, respectively, in search for diamondoid derivatives with a minimal and a maximal HOMO-LUMO energy gap. To this end, the energy gap was optimized systematically using an inverse molecular design methodology based on the best-first-search algorithm combined with a Monte Carlo component to escape local optima. Adamantane derivatives were found with HOMO-LUMO gaps ranging from 2.42 eV to 10.63 eV, 9.45 eV being the energy gap of pure adamantane. For diamantane, similar values were obtained. The structures with the lowest HOMO-LUMO gaps showed apparent push-pull character. The push character is mainly formed by sulfur or nitrogen dopants and thiol groups, whereas the pull character is predominantly determined by the presence of electron-withdrawing nitro or carbonyl groups assisted by amino and hydroxyl groups via the formation of intramolecular hydrogen bonds. In contrast, maximal HOMO-LUMO gaps were obtained by introducing numerous electronegative groups.

Introduction Nanoparticles play an increasingly important role in modern technology.1,2 Especially carbon nanoparticles like fullerenes and nanotubes are nowadays widely used. Diamondoids, however, are still only scarcely assessed. Diamondoids or nanodiamonds are a class of carbonnanoparticles that consist of fused diamond cages and are fully sp3-hybridized, very stable3, and perfectly size-selectable. The first members of the series of diamondoids, having respectively one, two and three cages, are adamantane, diamantane and triamantane. Since small diamondoids have been extracted from natural oil only recently,4 there is a growing research interest in using diamondoids. Among others, diamondoids find possible applications in the field of optoelectronics. Functionalized diamondoids can for example be used for bioimaging1,5, when the optical gap would be reduced to the visible region. On the other hand, diamondoids have their value in semi-conductors or other electronic devices, when the electrical conductivity is increased by functionalization.6 Moreover, diamondoids can form monolayers on gold surfaces7 or self-assemble into more complex structures.8 Pure diamondoids have, however, in accordance with pure diamond, a high hardness9 and a very low conductivity.2 Henceforth, for

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useful behavior, one tries to manipulate pure diamondoids to exhibit enriched functionality, for example by introducing other chemical substituents into diamondoids. Several types of functionalization have been applied: diamondoids with fluoroalkyl groups show negative electron affinities and electron-phonon scattering,10 amine-functionalized diamondoids display low enhanced conductivity11, and thionated diamondoids have low optical gaps.1 All these properties are largely determined by the position of the frontier molecular orbitals. Hence, in functionalization studies generally the effect of doping and substitution on electronic properties such as the ionization potential, the electron affinity, and the HOMO-LUMO energy gap is examined,12 in addition to their function as binding sites to make usable nanostructures.2 In particular, some initial studies showed that push-pull doping13,14 is the most effective approach to decrease HOMO-LUMO gaps and that the most electron-withdrawing functional group dominates this reduction.15 Fokin and Schreiner also found that group effects are not additive, meaning that the final property value is not simply an summation of individual contributions from substituents or dopants; therefore, double functionalization sometimes shows rather little effect. Moreover, these authors concluded that the nature of the functional groups seems to be significantly more crucial than the size of the diamondoids itself.15 Until now, most studies focused only on mono- and double functionalization of diamondoids. To explore the world of diamondoids to a larger extend, we thoroughly investigated to what degree the electronic properties of adamantane and diamantane can be tuned by multiple functionalization. Since the number of possible chemical structures rises vastly when more sites for functionalization are taken into account, an inverse molecular design approach was employed to search systematically for structures with ideal properties. We probed the boundaries of the HOMO-LUMO energy gap by searching for their minimal and maximal values in a given chemical space of diamondoid derivatives. In addition, a detailed evaluation of the functionalization effect on the different properties has been carried out.

Computational Methodology In the last decade growing attention has been paid to the design and discovery of valuable molecular structures, using quantum chemistry. A fundamental challenge in molecular design and discovery is the large number of possible structures that are accessible through the systematic variation of the composition of the molecular system.16-18 Direct approaches in molecular design are e.g. the systematic modification of structural motives with known favorable chemical properties, quantitative structure–activity relationship methods, and virtual high-throughput-screening of existing molecular libraries.19-23 In this work, however, an inverse molecular design methodology24-32 is applied to design diamondoid derivatives displaying optimal HOMO-LUMO energy gaps. In this methodology, one searches for an optimal external potential, i.e. the potential due to the nuclei of the system, corresponding to a molecular system with the associated target property. One of the most straightforward search methods is the discrete best-first-search (BFS) algorithm.33 The algorithm evaluates the property value of systematically chemically modified structures and then adopts the most optimal modification. The best first search (BFS) algorithm is known to be quite robust and efficient for single property optimizations, when the search space is of moderate size. In those cases, it generally performs better than the genetic algorithm (GA) method and the dead-end elimination (DEE) method.30 Balamurugan et al.30 have shown that the same performance of GA can be achieved with BFS using 20 times lower computational cost for a system with 12 available sites and a substituent library of 13, which results in a chemical space size of approximately 2.3 1013.


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In this study, the imposed structural changes are the placements of various functional groups (substitution or doping) on certain atomic positions, so-called sites, within an initially determined molecular framework. On each site, the different functional groups are ranked according to their corresponding property value and, subsequently, the most optimally modified structure is used as a new starting point for the next site to visit. Convergence is reached when the current optimum cannot be improved anymore by modifying a single site. This procedure only converges to a global optimum when the independent site approximation more or less holds. The independent site approximation assumes that the sites can be optimized individually, meaning that the global property value can be written as a sum of individual contributions from functional groups on each of the sites. More details about the algorithm can be found in literature.30,34,35 However, in many cases the independent site approximation is not perfectly valid and, therefore, one could end up in local optima. The simplest solution is to run the design procedure multiple times, using a different starting structure and/or site order.36 However, this way, costly computer time could be wasted on structures that are far from optimal. In addition, variations on the BFS procedure have been proposed that allow to move away from the local optimum by introducing randomization in the early stages of the optimization process, with the BFS converged solution as the new initial structure.30,37 Hence, based on the work of Hu, Beratan and Yang,38,39 we have adapted their gradientdirected Monte Carlo approach to the discrete best first search algorithm. This approach works as follows. When a complete BFS procedure has been performed and converged, we have available, for a given optimal structure, also all the structures with only one site changed. That is, for a converged optimum, 𝑋opt 𝑏! , 𝑏! , … , 𝑏! , … , 𝑏! , with bi the functional group on site i, all the property values of structures with only one site changed, 𝑋!! 𝑏! , 𝑏! , … , 𝑥! , … , 𝑏! , are known. When the independent site approximation would be valid, one can calculate the property value P of every new structure, 𝑋 ! 𝑥! , 𝑥! , … , 𝑥! , … , 𝑥! with possibly N sites modified with respect to the converged optimum, by the property value for the Xopt structure and a difference term: 𝑃 𝑋 ! = 𝑃(𝑋opt ) + ∆𝑃(𝑋 ! )

(1)

This difference term is the sum of the difference in property value between the converged optimum and the structures with only one site of the optimal configuration changed to the functional group present in XN: ∆𝑃(𝑋 ! ) =

! !!!

𝑃 𝑋!! 𝑏! , 𝑏! , … , 𝑥! , … , 𝑏!

− 𝑃 𝑋opt 𝑏! , 𝑏! , … , 𝑏! , … , 𝑏!

(2)

To escape from possible local traps, after each complete BFS convergence, a random structure XN is generated and accepted with a probability, 𝑝 ∆𝑃 𝑋

!

=𝑒

!

∆! !! !

= 𝑒 !!∆!

(3)

using the Metropolis algorithm. kB is the Boltzmann constant and T is the temperature. The temperature controls the acceptance probability and is increased slowly when no structures are accepted. All calculations were performed within the Kohn-Sham framework using the Gaussian09 software package.40 During the inverse design search, all geometries were optimized at the B3PW91/6-31G(d,p)41,42 level of theory and the same computational methodology was used to obtain the HOMO and LUMO energies, in accordance with the work of Fokin and Schreiner.10 It



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was shown that the B3PW9141 density functional outperforms B3LYP in describing polycyclic fully sp3-hybridized structures,43 while B3PW91 and B3LYP show similar good agreement in describing hydrogen bonds.44-46 In addition, the B3PW91 functional outperforms all other density functionals in computing the band gap for inorganic semiconductors.47 In all calculations, 631G(d,p)42 was used as basis set. No diffuse functions were added, despite the spread-out character of the lowest unoccupied molecular orbital, being delocalized around the surface of the diamondoids. Diamondoids, however, often possess negative electron affinities48-50 and for these cases it has been shown that the LUMO energy shows a much poorer agreement with experimentally established trends when diffuse functions are included in the basis set.51,52 To validate the computational approach, the adiabatic ionization potential was calculated as an energy difference between the neutral diamondoid and the cation. B3PW91/6-31G(d,p) reproduces the experimentally observed adiabatic ionization potential reasonably well: 9.23 eV vs. 8.96 eV for adamantane and 8.80 eV vs. 8.43 eV for diamantane.53 Furthermore, we have tested the B3PW91 functional for an additional 15 substituted small diamondoids. The results are listed in the Supporting Information. On average, B3PW91/6-31G(d,p) computed IP values underestimate the experimental values by 0.39 eV. Using the same functional but a larger basis set, B3PW91/6-311++G(d,p), the IP difference decreases to 0.24 eV. Correlation diagrams between experimental and computed values result in correlation coefficients R2 of 0.87 for the smaller basis set and 0.93 for the larger basis set.

Results and discussion To start the inverse design search, one has to establish the properties of interest and the molecular frameworks, with indication of the sites and functional groups that will be used. In this study, the target property is the HOMO-LUMO energy gap. For the molecular framework, the two smallest possible diamondoids, adamantane and diamantane, are chosen. In adamantane the position of each carbon atom is treated as an adjustable site, with four tertiary and six secondary carbon positions, whereas for diamantane only six out of fourteen carbon atoms are handled as a site for functionalization to keep our chemical compound space and computational cost manageable. Diamantane has three symmetrically different carbon positions: two apical tertiary atoms, six medial tertiary atoms, and six secondary atoms. Of each type, two of them are used as a site for functionalization. The scaffolds with indication of the tunable sites are shown in Figure 1. The set of functional groups consist of three groups for internal doping: (N), (O), (S), and twelve groups for external doping or substitution: (C)-COOH, (C)-NO2, (C)-CN, (C)-OH, (C)SH, (C)-NH2, (C)-CH3 (C)-CF3, (C)-Cl, (C)-F, (C)-H, (C)=O. This means that there are twelve functional groups (11 substituents and 1 dopant) available for tertiary carbon positions in the molecular framework and an additional three for the secondary positions (substituent (C)=O and dopants (O) and (S)), by removing also the other hydrogen atom on that site. All of these functional groups have been already used in the synthesis of mono-substituted diamondoids.54-56 Remark that in principle one could introduce two functional groups on secondary carbon atoms, but they are more difficult to synthesize and only sparse examples are found in literature. In this work, we focused on neutral functional groups, but one could also obtain low HOMO-LUMO gap structures by introducing charged groups.15 Based on the number of sites and functional groups as listed above, the investigated chemical compound spaces of adamantane and diamantane derivatives are respectively 2.4*1011 and 4.6*106 (neglecting symmetrically identical molecules).


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1 5

2

6

5

7 1 8

3

4

10 9

(a)

3 4 6 2

(b)

Figure 1. Molecular framework of (a) adamantane and (b) diamantane, with indication of the adjustable sites considered in our work. In adamantane, there are 4 tertiary (1, 2, 3, 4) and 6 secondary (5, 6, 7, 8, 9, 10) positions, whereas in diamantane, 2 apical tertiary (1, 2), 2 medial tertiary (3, 4), and 2 secondary (5, 6) sites are chosen.

Table 1. The HOMO and LUMO of pure adamantane and diamantane, with indication of their corresponding orbital energies, are depicted (isovalue: 0.02). In addition, the HOMO-LUMO energy gap (HLg) and the vertical and adiabatic ionization potential (IPvert, IPad) are given in eV.

Diamondoid

Structure 2

6 1 8

3

LUMO

HLg

IPvert

IPad

9.45 eV

9.29 eV

8.96 eV

9.00 eV

8.92 eV

8.43 eV

5

7

Adamantane

HOMO

4

10 9

εHOMO =-7.43 eV

εLUMO = 2.02 eV

1 5 3

Diamantane

4 6 2

εHOMO = -7.02 eV

εLUMO = 1.98 eV

A risk of using automated algorithms is to lose the insight of the underlying basic principles. Therefore, in order to gain initial understanding of complicated diamondoid derivatives, we start by considering the frontier orbitals of the unsubstituted and mono-substituted adamantane and diamantane structures. The HOMO and LUMO of pure adamantane and diamantane are delocalized over the whole molecule, with the LUMO showing Rydberg state character.57 The HOMO is mainly associated with the carbon-carbon bonds, whereas the LUMO shows anti-bonding character on the carbon-hydrogen bonds and is more diffuse and delocalized over the surface of the whole diamondoid. The orbitals are depicted in Table 1. The LUMO energies for both diamondoids are positive, indicating that these molecules probably have negative electron affinities. Both the HOMO-LUMO energy gap and the ionization potential are around 0.5 eV smaller for the larger diamantane, compared to adamantane. This is in accordance



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with the trend that the energy gap of increasing diamondoid size converges towards the bandgap value of approximately 5.0 eV for diamond. Functionalization of diamondoids by doping and substitution changes the electronic properties. In most cases this is attributed to the placement of localized electronic states into the HOMO-LUMO gap, either occupied or unoccupied.14 We have computed the HOMO-LUMO gap for all possible mono-substituted adamantane and diamantane structures, with functionalization on the different types of carbon positions, as shown in Figure 2. Sulfur doping (thia) or substitution with a carbonyl, nitro, thiol, or carboxyl group introduces both a new HOMO and a new LUMO level into the HOMO-LUMO gap of the pure diamondoids. These groups therefore result in the most significant decrease of the HOMO-LUMO gap. The largest reduction of 3.5 eV by a single functionalization occurs by incorporating a carbonyl or nitro group. Nitrogen doping (aza) and amino groups only introduce a new HOMO level, whereas a new LUMO level only appears when chloro or cyano groups are included. An example of each case is depicted in Figure 3. When visualizing the orbitals, it is observed that the HOMOs of the amine, alcohol, and thiol compounds are the most strongly localized on the lone pair of the functional group. Aza/thia doping, on the other hand, gives rise to HOMO levels that are largely located on the dopant but also still partly delocalized in the cage. Fluor and trifluoromethyl groups, on the contrary, stabilize the pure diamondoid’s HOMO and thus increase the HOMOLUMO gap.

Figure 2. HOMO-LUMO energy gap, LUMO and HOMO energy level values (in eV) for monosubstituted diamondoids. The values for adamantane (with distinction for secondary and tertiary sites) are given in green whereas for diamantane (with distinction between secondary, apical tertiary, and medial tertiary sites) are blue.

Mono-substituted adamantanes and diamantanes show very similar behavior with the only difference that, when no new HOMO levels are introduced in the gap, the HOMO levels of diamantane are higher in energy.


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Effects of substitution on different positions in the molecule are very small. Most notable is that nitrogen doping on secondary positions results in a lower-lying HOMO level than doping on a tertiary position. This complies with the fact that upon electron removal the positive charge on tertiary nitrogen atoms is more stabilized by hyperconjugation than on secondary nitrogen atoms.

Figure 3. MO-diagram showing the introduction of new HOMO and LUMO levels inside the HOMOLUMO gap of pure adamantane. 2-Aminoadamantane introduces a new nN-HOMO level. 2-Cyanoadamantane introduces two new, almost degenerate π*C ≡N anti-bonding π-type LUMO levels. Adamantanone introduces both a new HOMO and a new LUMO level.

Recently, a few studies on multiple substitutions have shown that the most significantly decreased HOMO-LUMO gap is obtained when the HOMO and LUMO are located on different functional groups.14,15 Remark, however, that group effects are not additive. For example, placing one nitro group on adamantane reduces the HOMO-LUMO gap from 9.44 to 6.15 eV. Introducing another nitro group further reduces the gap to 6.01 eV, while 10 nitro groups finally decrease the HOMO-LUMO gap to 5.64 eV, merely an additional 5% lower compared to the reduction obtained with a single nitro substitution. Hence, to obtain a larger closing of HOMOLUMO gaps, one has to establish diamondoids with different groups that cooperate together. Because these optimal multiple functionalized diamondoids are much less intuitive to find, we turn to the Inverse Molecular Design (IMD) approach.

Inverse molecular design: minimizing the HOMO-LUMO energy gap We have explored the limits of the HOMO-LUMO energy gap of adamantane and diamantane derivatives by a minimization and maximization via the best first search algorithm, using the substituents and sites as mentioned in the previous section.



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We first discuss the results for the HOMO-LUMO gap minimization of adamantane. Two searches have been performed, both converging to a minimum within 4 global iterations (over all sites and substituents). This means that, without considering additional Monte Carlo steps, less than 500 structures were visited in each search, which is much smaller than the size of the investigated chemical space (2.4*1011). A first BFS optimization started from a random start configuration having a HOMO-LUMO gap of 4.7 eV, already well below the energy gap for the unsubstituted adamantane. This can be attributed to the fact that the configuration already contained a carbonyl and a nitro group, as well as a nitrogen dopant (see optimization pathways in Supporting Information). Therefore, from the beginning push-pull character is present. The first BFS procedure resulted in an energy gap minimum of 2.85 eV. Subsequent BFS runs, connected via two Monte Carlo steps, further decreased the HOMO-LUMO gap to 2.75 and 2.67 eV, respectively. The resulting structures are depicted in Table 2, whereas the different chemical modifications along the entire optimization process and plots of the frontier orbitals of the structures in Table 2 can be found in the Supporting Information. In the three optima, the LUMO is almost completely localized on the nitro group, assisted by the neighboring hydroxyl and amino groups through intramolecular hydrogen bonding. In fact, a chain of weaker to stronger hydrogen bonds is present in all optima, and this is confirmed by plotting the non-covalent interactions in the structures using NCIplot58,59 (cf. section 8 in the Supporting Information. The HOMO, on the other hand, is mainly located on the sulfur dopant and to a lesser degree on the nearby lying amino groups (on sites 7 and/or 9) in optimal configuration 1 and 2, whereas in the final configuration, ada_min_gap1, the HOMO is located both on the sulfur dopant on site 8 and the thiol group on site 6. Note that in all cases the HOMO and LUMO are spatially separated. All in all, the three minima look very alike, hence the similar energy gap values. We have also depicted the convergence behavior (Figure 4(a)). In the first global iteration, the gap reduces continuously to approximately 3.2 eV. In the three subsequent iterations, the gap decreases further though very slowly to 2.85 eV. Next, a Monte Carlo step is performed. Within one global iteration, already a slightly better energy gap value is obtained, that is further reduced in the next iterations. After another MC step, it takes another three global iterations to find ada_min_gap1. Note that in 8 of the 10 global iteration sections, the spread in HOMO- LUMO gap is very large when site 5 is changed. Site 5, containing the nitro group in the minimal gap configuration, is by far the most important site in this structure, with energy gap values ranging over almost 4 eV. This is in accordance with the results for the mono-substituted adamantane derivatives. Table 2. The optimal chemical structures obtained from three successive best first search runs, connected via Monte Carlo steps (MC1 and MC2), of adamantane with a minimal HOMO-LUMO gap (HLg). More details can be found in the Supporting Information.

Numbered structure

Optimal configuration 1 OH

2

6

5

7 1 8

3

4

10 9

NH 2

H2N

OH

NH 2

CH 3 OH

H3 C S H3 C

CF3

O N

HO O

CH 3

H2 N

OH

HLg = 2.85 eV

NH 2

HLg = 2.75 eV

O N

HS O

S H3 C NH 2

Final configuration (after MC1) (ada_min_gap1)

O N

O

Optimal configuration 2 (after MC1)

O

F

CH 3

H2 N

OH S H2 N OH

HLg = 2.67 eV 8

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(a)

(b)

Figure 4. Chronological order of the BFS for minimal HOMO-LUMO gap adamantane derivatives: path to ada_min_gap1 (a) and ada_min_gap2 (b). The black lines separate the global iterations. Per separate color the lower limit represents the current structure with the minimal HOMO-LUMO gap. The site order varies in every global iteration. In (a), two MC steps were performed, dividing the full optimization pathway in 3 BFS convergence sections with different number of global iterations (cf. numbering in plot).

Since the initial structure of the above-described BFS optimization already had a quite low energy gap, we performed a second BFS optimization. This time, the random start configuration, as shown in Table 3, had a HOMO-LUMO gap of 6.5 eV, and did not contain nitro, carbonyl or thiol substituents. However, in the first four sites that are visited immediately two nitro groups (sites 7 and 4), a thiol group, and a sulfur dopant were placed (site 6 and 8, respectively). These modifications result in a largely decreased HOMO-LUMO gap of 3.81 eV. At this point, pushpull character is again present: the LUMO is localized on the nitro group on site 4, while the thiol and sulfur yield a high HOMO level. In the following two global iterations, the HOMO-LUMO gap further reduces to 2.42 eV (cf. Figure 4(b)). The different chemical modifications along the complete optimization process are given in the Supporting Information. In this particular case, the use of Monte Carlo steps after the first BFS convergence did not result in a lower gap configuration. As is clear from Figure 4, the energy gap variations within a particular site are far less outspoken for the optimization to ada_min_gap2 than for that to ada_min_gap1, even though both minima hold a nitro group. In fact, 8 out of 10 functionalizations are the same in both structures, albeit on different positions. It seems that in ada_min_gap2 more sites are significantly contributing to the minimization of the energy gap.



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Table 3. Three relevant chemical structures obtained from a best first search for an adamantane derivative with a minimal HOMO-LUMO gap (HLg).

Numbered structure

Random start configuration

Optimum after 1st global iteration O

4 10

9

NH 2

HO

5 3

6

H3 C

7

CF3

O

8

NH 2

HO H2 N

SH CF3

HS S H2 N

CN CN

O N

HN

1 2

O

N

Cl

O

HLg = 6.51 eV

HO HO

O

HLg = 3.62 eV

O

S

H2 N

NH 2 H2 N

N

Cl

Final configuration (ada_min_gap2)

SH

O NH 2

HLg = 2.42 eV

The final configuration, as can be seen in Table 3, has no symmetry. Nevertheless, some local symmetry can be found around the thiol group as the two adjacent sites (1 and 2) are substituted with amino groups and around the nitro group as two of the neighboring sites hold hydroxyl groups. Notably, in this optimum the nitro group is placed on a tertiary site, whereas for ada_min_gap1 NO2 is positioned on a secondary site. The frontier orbitals are shown in Table 4. The HOMO, with an energy level of -5.51 eV, is mainly formed by the non-bonding electron pairs of the thiol group and to a lower extent of the amino group on site 7. These thiol and amino groups are electron-donating and hence responsible for the pushing character by increasing the electron density of the structure. Also the amino groups on sites 1 and 2, seemingly not participating in forming the HOMO, have a considerable impact on increasing the HOMO’s energy level. The LUMO of ada_min_gap2 (-3.09 eV) is a π*-MO located on the nitro group. The nitro group is electron-withdrawing and hence removes electron density from the cage. The π*-MO LUMO of nitroadamantane lies at -1.68 eV. By introducing the neighboring hydroxyl groups, the LUMO energy is further decreased to -2.43 eV. However, the LUMO of ada_min_gap is not located on those neighboring groups. The hydroxyl groups seem to assist the nitro group through intramolecular hydrogen bonding, as we also observed for the ada_min_gap1 structure. Again, we have computed the non-covalent interactions using NCIplot.58,59 In Figure 5, we clearly see two strong hydrogen bonds between the nitro group and the adjacent hydroxyl groups, as well as a weaker attractive interaction between the sulfur dopant and the nitro group. Many other weak attractive interactions are present, as is displayed in an additional NCI plot in the Supporting Information. We have also looked into the site vs. functionalization dependency of the HOMO-LUMO energy gap w.r.t. the optimal structure. One can appreciate the importance of each site or the functionalization on that site by looking at how the property value changes when only that particular site is changed, while the rest of the optimum is kept fixed. The last cycle in Figure 4 shows the variation in energy gap values for the last global iteration, in which no more chemical modifications to the optimum configuration take place. In the Supporting information more detailed plots are included. We will briefly discuss for ada_min_gap2. Site 4, containing the nitro group, is the most important site in this structure, because modifying this group while keeping the other sites fixed induces a minimal increase in HOMO-LUMO gaps of 1.5 eV. Most other sites display more or less the same range in energy gaps, though with a different functionalization preference. Only site 3 seems to be less influential, with many modifications hardly affecting the


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HOMO-LUMO gap. Note that, if the independent site approximation (ISA) would be valid, similar sites would show a similar property vs. substituent trend. In this case, however, the behavior of all the functionalizations per site is very different. This deviation from ISA can be attributed to the fact that in ada_min_gap2 the frontier orbitals become quite localized on the dopants and/or functional groups. Table 4. Optimal structures for the HOMO-LUMO gaps of adamantane and diamantane derivatives.

Framework

Structure

HOMO

LUMO

HLg (eV)

ada_min_gap2 O

O

N

4 10

9 5

1

HO HO

S

2.42

3 8

2 6

H2 N

NH 2

7

O

H2 N

NH 2

SH

εHOMO = -5.51 eV

εLUMO = -3.08 eV

dia_min_gap H

1

O

O

5 N O

3

2.94

H

4

O N

6 2 N

H3 C

εHOMO = 5.57 eV

εLUMO = -2.64 eV

ada_max_gap F F

2

6 7 1 8

3

F

5

10.63

4

10

F

9

F

O O

O F

εHOMO = -8.70 eV

εLUMO = 1.93 eV

dia_max_gap CF3

1

F

5 3

10.08

CF3

CF3

4 6 2

F CF3

εHOMO = -8.45 eV

εLUMO = 1.63 eV



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Figure 5. Non-covalent interactions (with the electron density ρ ranging from 0.020 to 0.050 a.u.; the reduced density gradient s cutoff = 0.5 a.u. and NCI color scale = -0.03 < ρ < 0.03 a.u.) for the ada_min_gap2 structure using NCIplot.59 Blue surfaces indicate strong attractive interactions such as hydrogen bonds.

Next, we turn to the energy gap minimization of diamantane. Our lowest HOMO-LUMOgap diamantane derivative was found after 6 global iterations with one Monte Carlo step in between, the summary of the procedure being depicted in Figure 6. Subsequently, two other Monte Carlo steps and BFSs were performed, one yielding a minimum of 3.0 eV and the other again our lowest minimum structure. The minimal HOMO-LUMO gap structure (denoted as dia_min_gap) is depicted in Table 4 and has an energy gap of 2.94 eV. This is considerably lower than the 9.00 eV found for the unsubstituted diamantane and closer to the minimal gap of the adamantane derivative (2.42 eV), with the former having only 6 functional sites. In fact, the dia_min_gap structure looks quite similar to the adamantane case, with a nitro group assisted by hydrogen-bonded hydroxyl groups on the one side and electron-rich dopants on the other side. Again, the different optimal structures along the BFS optimization are given in the Supporting Information. During the optimization process, the variation in HOMO-LUMO gap values over all possible functionalizations becomes larger for each of the sites. However, for most sites there exist several functionalization possibilities to obtain a similar result.

Figure 6. Chronological order of the BFS for a minimal HOMO-LUMO gap diamantane derivative; the black lines separate the global iterations. Per separate color the lower limit represents the current structure with the minimal HOMO-LUMO gap. After two global iterations, the first BFS converged to a value of 4.09 eV. Subsequently, a Monte Carlo step was undertaken. The second BFS converged in four global iterations to the final value of 2.94 eV.


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Again, we observe a spatial separation between the frontier molecular orbitals and a similar trend in HOMO and LUMO. The high-lying HOMO level is localized in the cage, mainly on the free electron pairs of the two nitrogen atoms. The presence of the methyl group in between can be explained by the fact that it lifts the HOMO energy level by stabilization of possible positive charge on the nitrogen atoms by hyperconjugation. As in the case of adamantane, a low-lying LUMO is observed, located mainly outside the cage, on the nitro group, assisted through strong intramolecular hydrogen bonding by the two neighboring hydroxyl groups. The partially positively charged hydrogen atoms from the two hydroxyl groups stabilize partial negative charge on the oxygen atoms of the nitro group. The confirming NCI plot is given in the Supporting information. The strength of the hydrogen bonds was determined by the energy difference of the conformations with the hydrogens of the hydroxyl groups pointing towards and away from the nitro group. The individual hydrogen bond strength was calculated to be approximately 5 kcal mol-1. Because the result is so largely influenced by the presence of the nitro group, the final result will depend on which site is changed first in the optimization process. The probability of the first chemical modification being the introduction of a nitro group is rather high, because including this group gives by far the lowest-lying LUMO (cf. Figure 2). Therefore, searches were performed using the three symmetrically distinctive sites (1,3,5) as the starting site. This indeed resulted in three different optima with the nitro group present on the starting site. The structures are depicted in Table 5. The minimal gaps are 3.13, 3.51, and 2.94 eV, respectively; the last one corresponds to dia_min_gap presented above. The resulting minima look very similar. The HOMO part is identical in the three cases. It can be seen that the nitro group always participates in intramolecular hydrogen bonding with its neighbor(s). When the nitro group is put on a tertiary carbon atom, the hydrogen bonding is evidently limited to one neighbor, due to our choice of adjustable sites. This, however, confirms the importance of intramolecular hydrogen bonding in lowering the LUMO energy. Table 5. Three structures resulting from a BFS run to a diamantane derivative with the lowest HOMOLUMO gap with starting site 1, 3 and 5, respectively.

Apical site O

Medial site

Secondary site H

O H

N

N

O

H N

O

H

O N O

O N

N

O N

O

N H3 C

H3 C

H

N

N H3 C

N

N

HOMO-LUMO gap = 3.13 eV HOMO-LUMO gap = 3.51 eV HOMO-LUMO gap = 2.94 eV Since the nitro-group is so significantly determining the results, it is interesting to examine which minima can be found when the nitro group is removed from the functionalization library. The start and converged configuration for each BFS procedure, followed by a Monte Carlo step, are given in the Supporting Information. In total, we have performed six subsequent Monte Carlo steps, in order to escape possible local minima. It turns out that the carbonyl group takes over the role as electron-withdrawing group on which the LUMO mainly resides, in accordance with our findings for the mono-substituted diamantanes. Even though the carbonyl group can only be 13 ACS Paragon Plus Environment


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placed at a secondary site, the BFS results still depend a lot on the start structure. Almost every consecutive Monte Carlo step led to a more suitable structure, ultimately lowering the energy gap of the initial minimum by 0.3 eV. This lowest-energy-gap structure was the result of 3 out of 7 Monte Carlo directed BFS searches, indicating that this might be the global minimum in our search space. All minima have the LUMO corresponding to a σ*-orbital located on the carbonyl group. The carbonyl has in all cases a hydroxyl group as a neighbor forming an intramolecular hydrogen bond. The structural part on the other end of the carbonyl group is different each time, containing either a sulfur atom or, one or two nitrogen atoms. Only the lowest minimum structure shows similar ending as dia_min_gap in Table 4.

Inverse molecular design: maximizing the HOMO-LUMO energy gap Although fewer applications are foreseen for large-gap diamondoid derivatives, we are interested in how much the HOMO-LUMO energy gap can be stretched. Based on the monosubstituted diamondoid results in Figure 2, a competition between hydrogen, fluorine, trifluoromethyl, and methyl groups is expected. In addition, we anticipate only a small energy gap increase compared to the pristine diamondoid structures. After five global iterations an adamantane structure is found with a HOMO-LUMO gap of 10.63 eV, an increase of merely 1.18 eV compared to the unsubstituted case. This structure, depicted in Table 4, has C3v symmetry and contains methyl, trifluoromethyl, fluoro groups, and oxygen dopants. For diamantane, also after five global iterations, a structure is found containing only fluoro and fluoromethyl groups, having a HOMO-LUMO gap of 10.08 eV. The two ends of the structure are identical in this case. For both maxima, no new HOMO or LUMO levels are introduced. As illustrated in Figure 7, the increase in energy gap can be attributed to a lowering of the HOMO level. Functionalization with fluorine and trifluoromethyl already lowers the HOMO energy level in the mono-substituted cases. The LUMO is still a surface state of the C-H anti-bonding sigma-bonds. Mono-substitution of a diamondoid site with methyl, alcohol, fluorine, or trifluoromethyl, or doping with oxygen has no significant effect on the LUMO energy level (cf. Figure 2). Hence, the dia_max_gap structure is easy to understand. The ada_max_gap structure contains electron-withdrawing fluorine and trifluoromethyl groups, but also three oxygen atoms. Oxa-adamantane, however, has a HOMO level of -6.3 eV, around 1 eV more than for pure adamantane. So surprisingly, the behavior of the oxygen dopant is changed when positioned between fluorine groups. More detailed information about the maximization of HOMO-LUMO gaps of adamantane and diamantane are gathered in the Supporting Information. The general trend of energy gap vs. substituent is in these cases much more consistent among all sites. Effects of interchanging methyl, fluoromethyl, fluoro, and oxygen are rather small. All give HOMO-LUMO gaps around 10 eV. Because the HOMO and LUMO are now both delocalized in or over the cage and not reside on a single functional group, all sites show similar behavior. Hence, the independent site approximation is largely valid in this subclass of diamondoid chemical space. Note that, like for the mono-substituted diamondoids, carbonyl is predicted to have a larger impact on reducing the HOMO-LUMO energy gap than the nitro group.


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Figure 7. The frontier molecular orbital levels of pure adamantane and diamantane and the minimal and maximal HOMO-LUMO energy gap structures.

The HOMO-LUMO gap: HOMO versus LUMO A HOMO-LUMO energy gap optimization is in essence a linear scalarization of a multipleobjective problem. The actual optimization property is a linear combination of individual optimization parameters. In this case, the HOMO-LUMO energy gap is a linear combination of two optimization parameters, the HOMO and LUMO energy with coefficients -1 and 1, respectively. A more thorough understanding of the HOMO-LUMO energy gap optimization could thus be obtained by understanding how the individual parameters behave under optimization. The HOMO and LUMO energy levels could be subjected to individual maximization and minimization. Because we are mainly interested in minimal HOMO-LUMO gaps, only the HOMO maximization and the LUMO minimization were performed. The results are presented in Table 6. Note that via Koopmans’ theorem the HOMO energy level can be related to the ionization potential (IP). Therefore, HOMO maximization can be conceptually understood as a minimization of the IP. Dopants or functional groups with free electron pairs, such as nitrogen and sulfur, and amino, thiol, and carbonyl groups, having high-lying HOMOs, can introduce a new HOMO level into the HOMO-LUMO gap of the pure diamondoids and as such assist in maximizing the HOMO. Accordingly, the resulting structures of the HOMO maximization contain mainly amino groups and nitrogen substituents. It can be observed, however, that in dia_max_homo, possessing Ci symmetry, the lone pairs of the amino groups do not contribute to the HOMO, although they might influence the energy level. On the other hand, functional groups with free electron pairs, such as nitro, carbonyl, and carboxyl groups, having low-lying LUMOs, can introduce a new LUMO level into the energy gap of the pure diamondoids, which helps to minimize the LUMO. The structures of the LUMO minimization indeed contain many nitro and carbonyl groups. In ada_min_lumo, with Cs symmetry, the LUMO



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is localized both over the entire cage and on the substituents’ lone pairs. This is not the case for dia_min_lumo, for which the LUMO resides outside the cage, on three out of four nitro groups. The cyano groups help to stabilize the LUMO, but not through their lone pairs. Also note that the LUMO of ada_min_lumo is even lower than the HOMO of ada_max_homo. When we compare these orbital energy values with the corresponding values in the minimal HOMO-LUMO gap structures, we observe a further increase in the HOMO level of around 0.80.9 eV for ada_max_homo and dia_max_homo, and a lowering in the LUMO level of approximately 0.5 eV for dia_min_lumo and a significant 1.8 (!) eV for ada_min_lumo. Nevertheless, the same functionalization patterns appear in both sets of structures. This confirms that, instead of selecting substituents or dopants that perform well on average by both lowering the LUMO and increasing the HOMO via stabilization/destabilization effects or the introduction of new relevant orbital levels, push-pull doping is the most effective. Incorporating electronwithdrawing functional groups on the one end and electron-donating groups on the other end results in the largest energy gap reduction, spatially separating the two frontier molecular orbitals. Table 6. Diamondoid derivatives found by searching for maximizations of the HOMO energy level of adamantane and diamantane and for minimizations of the LUMO energy level of adamantane and diamantane

ada_max_homo

dia_max_homo

NH 2

ada_min_lumo O

NH 2

N

N

N

O

O

H2 N H2 N

N

N NH H2 N

-4.68 eV

O

O

O

N

N

N

O

N

N

O O

O

O

N O

-4.65 eV

O O

O

H2 N

N

O

O

N

N

dia_min_lumo

O

O O

-4.90 eV

N N O

O

-3.16 eV

The optimal structures from the diamantane derivatives were rather symmetrical and intuitive, while the ada_min_gap structure is much more complex. We therefore propose some adamantane derivatives with fewer substituted sites that still have significantly lowered HOMOLUMO gap. These structures should all have some push-pull character. The push component is formed primarily by sulfur (on secondary site) or nitrogen dopants. Nitrogen dopants give higher HOMO levels when placed on tertiary rather than on secondary sites. Multiple nitrogen dopants can be included on different tertiary sites that are next nearest neighbors, as is the case in the dia_min_gap structure. Furthermore, the pulling component should consist of an electronwithdrawing group, a nitro or a carbonyl group. Placing neighboring electronegative groups lowers the LUMO energy levels even further, as is done here with two hydroxyl groups that form hydrogen bonds with the nitro group. Doubly substituted adamantane with a nitro-group and a 16 ACS Paragon Plus Environment

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secondary sulfur or tertiary nitrogen dopant lowers the gap to 4.2 eV or 4.1 eV, respectively (Table 7). Triply and quaternary substitution of adamantane by adding hydroxyl groups besides the nitro group lowers the gap to 3.9 and 3.5 eV, respectively. Subsequent addition of two amine groups on the next nearest neighbors of the sulfur atom results in a symmetrical structure with a HOMO-LUMO energy gap of 3.0 eV. Interestingly, when adamantane is substituted in a similar pattern as the dia_min_gap structure, the HOMO-LUMO gap is also 3.0 eV. Table 7. Two proposed pathways with increasing number of functionalizations leading to structures with small HOMO-LUMO energy gaps, namely ada_min_gap_red1 and ada_min_gap_red2. The upper part follows a path towards structures similar to ada_min_gap while the lower part follows the path of increasing functionalizations towards the same substitution pattern of the dia_min_gap structure. OH NO 2

OH

NO 2

H2 N

NO 2

NO 2

OH

OH

NO 2 S

S

4.23 eV

S

OH S

3.91 eV

NH 2

3.54 eV

3.02 eV OH

OH

9.45 eV

NO 2

NO 2

6.14 eV

NO 2

NO 2

N

N

N

N

4.00 eV

3.88 eV

N

3.05 eV

N

OH H3 C

N

OH

2.99 eV

The suggested structures were based on ada_min_gap1 (and earlier minima in that BFS) and dia_min_gap. We also considered reducing the number of substituents of ada_min_gap2, the structure with the lowest energy gap. However, it turned out that nearly all functionalizations in that structure have a non-negligible impact on the energy gap reduction. To support this idea, we built up the final structure, starting from pure adamantane and adding step by step the functionalization having the largest impact at this particular step. Since a carbonyl group in a mono-substituted adamantane results in a larger energy gap decrease than a nitro group, two different pathways were considered: one starting with the introduction of a carbonyl group and one introducing firstly a nitro group. We will briefly discuss the NO2-pathway, as shown in Figure 8. You can consult the Supporting Information for more details. The largest energy gap reduction happens in the first two steps, with NO2 causing a significant lowering of the LUMO level and S causing a large increase in the HOMO level, inducing already push-pull character. The next three steps further decrease the LUMO level, especially via the inclusion of hydrogenbonded hydroxyl groups. The functionalizations added in steps 6 to 10 are then used to further push up the HOMO level. In this pathway, it is clear that lowering the LUMO energy level is more beneficial than increasing the HOMO level. However, in order to obtain a similar energy gap value as the proposed reduced structures in Table 7, we would have to include 8 of the 10 functionalizations. This clearly shows the synergetic effect of substituents and dopants in ada_min_gap2 to attain a large HOMO-LUMO gap reduction. What is very interesting to see is that throughout the build-up, the HOMO-LUMO spatial separation is manifesting more distinctly, with the LUMO residing on the nitro-group and the HOMO shifting from the sulfur dopant, over CO and S, then to NH2 on site 1, to finally being located on SH. The structure is thus being divided in a part that helps to lower the LUMO and a part that aids in increasing the HOMO.



4

location of HOMO

6.16

HOMO energy

NO 2

NH 2

H 2N

4.50

4.02

3.83

3.70

3.33

3.07

2.68

2.42

CNHH (1)

CSH (6)

CNHH (2)

CNHH (7)

CNHH (3)

NH 2

CO (8)

COH (10)

S (9)

CNOO (4)

-1

4.21

O

COH (5)

H 2N SH

1

energy gap

S

HO

2

0

LUMO energy

HO

3

reduction [eV]

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-2 Figure 8. The reduction in HOMO and LUMO energy level, and the resulting energy gap, when adding the functionalization with the largest impact (the site number is included in parenthesis). The numbers above the bars indicate the energy gap value at that stage. In addition, the increasing HOMO-LUMO spatial separation is shown by the dashed lines.

Even lower HOMO-LUMO gaps might be obtained by using e.g. charged functional groups, diamondoids with more cages like triamantane, tetramantane etc., or by using other functionalizations such as boron or phosphor dopants, or large conjugated functional groups as phenyl and sulfonyl. To conclude, we would like to shed some light on how the HOMO-LUMO gap using the Kohn-Sham (KS) frontier orbitals from DFT can be interpreted. For molecular systems, the KS gap in fact becomes a good approximation to the molecular optical gap.60 In addition, we have performed extra calculations to more accurately describe the molecular “band gap” for a selection of diamondoids. In bulk materials, the band gap is a well-defined quantity. The equivalent in molecular systems is the so-called fundamental gap, being the energy difference between the vertical electron affinity (EA) and the vertical ionization potential (IP). It is important to note, however, that the band gap for solids is typically considerably smaller in energy than the fundamental gap in molecules.61 There is thus a clear distinction between the optical gap and the fundamental gap with the difference being a measure of the electron–hole pair binding energy.61 For a detailed discussion on the difference between HOMO-LUMO energy gap and IP-EA in DFT, we refer to Baerends et al.,60 Tozer et al.,62 and Teale et al. 63. It is therefore no surprise that there exist large differences between the two sets of quantities. In Table 8, we have collected the IP-EA results for pure adamantane and diamantane, the reduced structures ada_min_gap_red1 and ada_min_gap_red2 (Table 7), and dia_min_gap using the following levels of theory: B3PW91/6-31G(d,p), MP2/cc-pVDZ, MP4(SDQ)/cc-pVDZ, and CCSD(T)/cc-pVDZ. The results for other computational methods are reported in the Supporting Information. Moreover, to obtain more accurate EA and IP values, extrapolations were performed toward the CCSD(T) level of theory in the limit of an asymptotically complete basis set 18 ACS Paragon Plus Environment

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(CCSD(T)/cc-pV∞Z) using the principles of a focal point analysis (FPA)64-71. In such an approach, the faster convergence of the higher order correlation corrections to the calculated energy differences is exploited in well-suited extrapolations of results obtained using CCSD(T) theory. More specifically, the so-called FPA-QZ method 72-75 was used, already proven to be able to reproduce IPs and positive EAs much better than 0.1 eV, and negative EAs better than 0.2 eV accuracy. Remark that for the structures with a positive EA, namely ada_min_gap_red1, ada_min_gap_red2, and dia_min_gap, the DFT functional B3PW91 with basis set 6-31G(d,p) is deviating less than 0.3 eV from our benchmark data. Table 8. Computed IP-EA values (in eV) with IP the vertical ionization potential and EA the vertical electron affinity, using different levels of theory. CCSD(T)/cc-pV∞Z using the FPA-QZ method serves as benchmark.

B3PW91 MP2 MP4(SDQ) CCSD(T) FPA-QZ 6-31(d,p) cc-pVDZ cc-pVDZ cc-pVDZ adamantane 12.77 12.57 13.36 13.02 11.43 diamantane 12.27 12.14 12.92 12.30 10.78 ada_min_gap_red1 7.39 8.34 8.19 8.13 7.64 ada_min_gap_red2 7.44 8.61 8.17 8.25 7.72 dia_min_gap 7.22 8.15 7.96 7.93 7.42 structure

Conclusions We used an Inverse Molecular Design approach to search for diamondoid derivatives with minimal and maximal HOMO-LUMO gaps. Using Monte Carlo steps within the design algorithm effectively helped to escape local optima towards more optimal structures and also decreased the number of calculations that are far from optimal. These combined techniques resulted in the discovery of several adamantane and diamantane derivatives with HOMO-LUMO gaps even smaller than 3.0 eV, and without incorporating charged functional groups. The minimal gap structures clearly show push-pull character with the HOMO and LUMO located on respectively electron-donating (n-doping) and electron-withdrawing (p-doping) groups. The structural element showing the lowest-lying LUMO was identified to be an electron withdrawing nitro group that is assisted by neighboring hydroxyl or amino groups forming intramolecular hydrogen bonds. A high-lying HOMO was mostly obtained by electron-donating dopants (S, N) in the diamondoid cages. Sulfur and nitrogen dopants were assigned to respectively secondary and tertiary sites. Groups that stabilize positive charge were placed close to these dopants, mostly stabilized by hyperconjugation. In all cases, we observed a spatial separation of the frontier molecular orbitals. Maximization of the HOMO-LUMO gap led to diamondoid derivatives with only electronegative groups as trifluoromethyl and fluorine groups, sometimes accompanied by oxygen dopants. The HOMO and LUMO were, as in pure diamondoids, located on or over the whole cage of the molecule and only the HOMO energy level was decreased. No push-pull doping was present. Optionally, larger diamondoids or diamondoids with charged groups could bring down the HOMO-LUMO gap even further, but similar trends and cooperation of functional groups are expected.



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Acknowledgments F.D.V. acknowledges the Research Foundation-Flanders (FWO) for a postdoctoral fellowship (1227014N). F.D.P. acknowledges the Fund for Scientific Research-Flanders (FWO) and the Free University of Brussels (VUB) for continuous support to his research group, in particular the VUB for awarding a Strategic Research Program (SRP) to the ALGC research group started on January 1, 2013. J.L.T also wishes to acknowledge the Free University of Brussels (VUB) for support via the SRP.

Supporting Information The Supporting Information contains computed adiabatic ionization potential values and their comparison with experimental values, the HOMO-LUMO gap data for the mono-substituted diamondoids, the optimization pathways for all BFS runs, plots of the BFS convergence and the substituent vs. energy gap correlation for each site, plotted HOMO and LUMO of the first adamantane energy gap minimization, a table showing the Monte Carlo steps of the diamantane gap minimization without NO2 in the functionalization library, Noncovalent Interaction index plots for all minima, data on the build-up of structure ada_min_gap2, and energy data used for the molecular energy gap calculations.

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Table of Contents / Abstract graphic: CF3

F CF3

F3C F CF3

MAX O

H

PURE

O N+ O

N

O– H

N

MIN


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