2.3. Local Quantum Chemical Descriptors
Fukui Functions and Dual Descriptor
The Fukui function f(r) represents the change in electron density at point r with respect to the change in the number of electrons N at a fixed external potential ν(r)  :
Parr expanded the use of the Fukui function to show localized areas of nucleophilicity ( ) and electrophilicity ( ) called the condensed Fukui functions   .
where , , and are the atomic charges in the neutral, anionic and cationic species, respectively.
It has been discovered that combining the two functions provides the most useful description in what is known as the dual descriptor    . Where represents an atom k of a molecule that is nucleophilic and an atom k that is electrophilic.
A condensed form is represented by:
3. Results and Discussion
Full unconstrained molecular geometry optimizations of theacrine and caffeine were carried out using Wavefunction Spartan ‘16 software  using the exchange correlation functional by Becke  with gradient-corrected correlation provided by Lee, Yang and Parr  with the 6-31G* basis set. This DFT method was chosen since it is fast, less computationally intensive, takes electron correlation into account, and provides accuracy in reproducing experimental data  . Figure 1 depicts the optimized molecular structures of the most stable forms of theacrine and caffeine. Optimized geometries were confirmed by vibrational analysis calculations and characterized as minima in their potential energy surface via the absence of an imaginary frequency.
Visualization of the contour plots of the Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) are shown in Figure 2. Analysis of the frontier molecular orbitals provides valuable information regarding nodal patterns and individual atom contributions. Extended conjugation in the HOMO of caffeine with C1, C2, C4 and N4 is observed compared to the more localized conjugation of the HOMO of theacrine with C1, C2 and C4. This is rationalized by resonance of N4 with the adjacent carbonyl group. This localization may be a factor in the Fukui dual descriptor results which predict the reactive sites of each molecule. The HOMO-LUMO energy gap for theacrine is 4.73 eV compared to 5.12 eV for caffeine.
Table 1 shows the global chemical reactivity descriptors. Theacrine has an electronic chemical potential (μ) of −3.10 eV compared to −3.41 eV in caffeine which shows that electrons will have a greater escaping tendency in theacrine, thus indicating theacrine to be a more reactive and less stable molecule compared to caffeine.
Table 1. Global chemical reactivity indices of theacrine and caffeine.
Figure 1. The 2D and 3D structures of theacrine (1) and caffeine (2).
Figure 2. Frontier Molecular Orbitals of theacrine and caffeine.
Chemical hardness (η) is also associated with the stability and reactivity of a molecule since it measures the resistance to change in the electron distribution or charge transfer. Theacrine is less hard and thus more reactive than caffeine. Chemical softness (S) is the measure of how readily the molecule exchanges electron density with the environment. Since theacrine is softer than caffeine, it has more ability to react. But another factor to consider is the higher electrophilicity (ω) value for caffeine compared to theacrine which means that caffeine has a greater propensity to accept electrons and is thus a stronger electrophile.
The partition coefficient of a compound is a widely used metric for determining lipophilicity, which is useful for identifying compounds which may have similar likeness to drugs and possible interaction with biological macromolecules such as receptors and enzymes  . Lipophilicity is an important physicochemical property of a potential drug since it affects solubility, absorption, membrane penetration, distribution, and excretion. The partition coefficient is the quotient of the concentration of the compound in a lipophilic solvent (typically 1-octanol) and the concentration of the compound in water. The logarithm of the partition coefficient is referred to as logP. A positive value for logP means the compound is more lipophilic, whereas a negative value means the compound is more hydrophilic. The data shows that while both compounds are hydrophilic, theacrine hydrophilicity is greater than that of caffeine. This can be rationalized by the extra carbonyl group of theacrine.
Table 2 and Table 3 show the natural bond orbital charge, f+ value, f− value, and the dual descriptor ( ) value calculated for every atom in theacrine and caffeine except for the hydrogens. The natural bond orbital charges were used in Equations (10) and (11) to determine the and Fukui indices respectively. The combined values according to Equation (13) yields the dual descriptor, where is a function that represents the reactivity of the atoms in each molecule; positive values represent sites of electrophilic attack and negative values represent sites of nucleophilic attack.
Table 2. Fukui indices for theacrine.
Table 3. Fukui indices for caffeine.
For theacrine, the calculations show positive values of the at C4, O3, N3, N1, O2, and N4. Electrophilic attack has the highest likelihood on these atoms. It is interesting to see the highest value on C4 and not one of the heteroatoms. That could provide insight into preferential reactivity. The two notable negative values of the are at C1 and C2 where nucleophilic attack could occur. Since C1 and C2 are part of the conjugated system, the calculated values make sense. For caffeine, notable positive values of are found on C4, N1, and O1. C4 is again the most likely site for electrophilic attack. Notable negative values of the are found at C1, C5, and C2. All three of these atoms are involved in the conjugated system with the C1-O1 carbonyl which would be predicted to be susceptible to nucleophilic attack. A major difference in reactivity between theacrine and caffeine is seen at C5, where in caffeine it is predicted to be a site for nucleophilic attack but that is not the case for C5 in theacrine. This is rationalized by the resonance stabilization of the carbonyl carbon in the urea moiety of theacrine.
Theacrine is more reactive than caffeine as judged by electronic chemical potential and chemical hardness and softness. Caffeine, however, was found to be more electrophilic. Theacrine has a more negative logP value which means that it is more hydrophilic. This reduced lipophilicity could result in more difficulty diffusing through cellular membranes and the blood-brain barrier. Fukui dual descriptor analysis shows C4 of both theacrine and caffeine to be the most likely site of electrophilic attack and C1 and C2 of both molecules to be sites of nucleophilic attack. The goal of the study was achieved by uncovering the electronic properties and reactivity metrics of theacrine in comparison to caffeine. Using the data gathered from this study, further analyses, such as molecular docking, can be undertaken to further understand how theacrine interacts with cellular systems.
The authors would like to thank Leeward Community College for providing the computer software and hardware necessary to conduct this computational analysis.
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