Comparative Study of Green ’ s Function Matrix Elements and Charge Transfers Obtained from Different Partitioning Schemes of Molecular Charge in Hydrogen-Bonded Complexes

Cálculos de orbitais moleculares ab initio RHF e MP2 usando os conjuntos de base 431G**, 6-311G** e cc-pVTZ têm revelado que os valores de G D,A mostram uma boa correlação com transferências de cargas atômicas intermoleculares obtidas a partir de diferentes esquemas de partição para os complexos de hidrogênio CNH...CNH, NCH...CNH, CNH...NCH e NCH...NCH. Isto é especialmente evidente quando a distância da ligação de hidrogênio é progressivamente aumentada até 4,5 Å. Entretanto, valores de G D,A mostram uma melhor correlação linear com valores de ∆Q usando as cargas Mülliken corrigidas, que são obtidas do modelo carga-fluxo de carga-recobrimento (CCFO) para intensidades no infravermelho. Neste caso, ambos G D,A e ∆Q formam duas curvas exponenciais praticamente superpostas. Por outro lado, valores de G D,A mostram uma menor concordância com valores de ∆Q obtidos das cargas atômicas derivadas dos orbitais de ligação naturais. Isso é claramente verificado quando é considerada a taxa de decaimento exponencial de primeira ordem de G D,A versus ∆Q obtida de diferentes esquemas de partição de carga.


Introduction
When a hydrogen-bonded complex is formed, an intermolecular charge transfer occurs, and its magnitude is sometimes directly associated with the hydrogen bond strength. 1Currently, this charge transfer is obtained from atomic charges derived from ab initio molecular orbital calculations with different basis sets.These charges can be computed from different partitioning schemes of molecular charge.This procedure may suggest a qualitative idea of the magnitude of intramolecular and intermolecular interactions and of chemical reactivity at various sites within the molecule.Often quantitative predictions can be made on the same basis.In particular, atomic charges derived from the charge-charge flux-overlap (CCFO) model 2 for infrared intensities have been useful to predict such interactions in hydrogen bonded complexes. 3,4Indeed, the so-called corrected Mulliken atomic charges 5 are calculated by adding a specific element of the overlap tensor of the CCFO model to the standard Mulliken charges.More recently, we have shown 6 that charge transfers obtained from corrected Mulliken charges (∆Qcorr) are linearly correlated with a Green's function matrix elements J. Braz.Chem.Soc.
(G D,A ) 7 in hydrogen-bonded complexes of the C n NH…C n NH and C n NH…NC n H (n = 1 and 3) type.The Green's function formalism was introduced as a tool for the calculation of bridge mediated donoracceptor interactions V da = V dD G DA V Aa 7 that appears in the Fermi's Golden Rule expression of a diabatic electron transfer rate, 8 and molecular wire conductance. 9Then, a correlation is observed among the charge transfer probability and the magnitude of the charge transferred upon the formation of the H-bond.
There we have verified that the larger is the intermolecular charge transfer for a given complex, the larger its G D,A value appears to be.We have also noted that G D,A and ∆Qcorr are associated with the binding energies of these complexes.Furthermore, both G D,A and ∆Qcorr show the same behavior when the hydrogen bonding distance is progressively increased from the equilibrium position until 4.5 Å, forming two practically superposed exponential curves.
It is also interesting to point out that hydrogen bonding is believed to play an important hole in electron transfer.An electron tunneling pathway model, described as a sequence of covalent, hydrogen bonding and through space interactions, was developed and applied for the reproduction and prediction of electron transfer rates in modified metalloproteins. 10Some molecular model systems with hydrogen bonding on the pathway from electron donor to electron acceptor have been designed for experimental studies. 11The behavior of G D,A through this hydrogen bonding bridge was compared with covalent saturated and unsaturated bridges and indeed it shows a faster decay with the separation distance. 12n this paper we intend to compare G D,A values with intermolecular charge transfers obtained from different partitioning schemes of molecular charge.Besides the corrected Mulliken charges 5 , we also consider the standard Mulliken charges 13 , atomic charges derived from electrostatic potentials 14 and those obtained from natural bonding orbitals. 15This comparative study will be performed on the CNH…CNH, NCH…CNH, CNH…NCH and NCH…NCH complexes employing RHF 16 and MP2 17 ab initio molecular orbital calculations with the 4-31G** 18 , 6-311G** 19 and cc-pVTZ 20 basis sets.

Calculations
The procedures to obtain Green's function matrix elements and the corrected Mulliken charges are given in References 6 and 5, respectively.The ab initio calculations were performed with the GAUSSIAN 94 program. 21In this procedure, the molecular geometry of the isolated molecules and their hydrogen-bonded complexes at the equilibrium distance were fully optimized and no imaginary frequency was observed.The MP2 calculations were performed using the electron frozen core approximation.It is also important to call attention that the G D,A values were calculated using natural bonding orbitals while s and p atomic orbitals or hybrids were used as intermediate steps on the construction of these natural orbitals.These later have been shown to be more adequate in analyzing electron transfer interaction propagation. 22he MP2/4-31G** optimized geometries of CNH… CNH, NCH…CNH, CNH…NCH and NCH…NCH complexes and their monomers are given in Figure 1.The internal default criteria of Gaussian 94 were used in all calculations.

Results and Discussion
Table 1 shows G D,A and ∆Q values using the MP2 and RHF levels of calculation with the 4-31G**, 6-311G** and cc-pVTZ basis sets for the CNH…CNH, NCH…CNH, CNH…NCH and NCH…NCH complexes.Here ∆Q is the amount of transferred atomic charge from a molecule to another when the hydrogen bond is formed.This quantity is obtained from four different partitioning schemes of molecular charge: (i) ∆Qcorr is obtained from the corrected Mulliken charges using the charge-charge flux-overlap (CCFO) model for infrared intensities; (ii) ∆Q M is calculated from the traditional Mulliken charges; (iii) ∆Q CHELPG is obtained from atomic charges derived from electrostatic potentials and, finally, (iv) ∆Q NBO is obtained from atomic charges using natural bonding orbitals.Initially, we can note from Table 1 that the G D,A values show a good correlation with the ∆Q values for these four different partitioning schemes of molecular charge.In general, we can verify that the larger is the Green's function matrix elements (G D,A ) for a given complex, the larger its corresponding charge transfer appears to be.This can be visualized in Figure 2 considering G D,A and ∆Q CHELPG values for the four hydrogen-bonded complexes here employed using the RHF/4-31G**, MP2/4-31G**, MP2/6-311G** and MP2/ cc-pVTZ calculations.
For example, G , respectively, whereas their corresponding values for ∆Q CHELPG are 0.222 e, 0.145 e, 0.134 e and 0.075 e, respectively.We have also investigated a possible correlation between lnG D,A values and r H...X hydrogen bond lengths involving the hydrogenbonded complexes here studied.However, our results show that these parameters are not correlated.While lnG D,A values follow the order: CNH...CNH > NCH...CNH > CNH...NCH > NCH...NCH, as can be seen in Table 1, the hydrogen bond lengths, in turn, follow the order: NCH...CNH > NCH...NCH > CNH...CNH > CNH...NCH.This later can be seen in considering the r H...X values obtained from the MP2/4-31G** calculations shown in Figure 1.
G D,A values always follow for a given calculation the order: CNH…CNH > NCH…CNH > CNH…NCH > NCH…NCH.However, some inversions are found for the values of the intermolecular charge transfers involving the NCH…CNH and CNH…NCH complexes.For example, G D,A values for NCH…CNH and CNH…NCH are 0.265 E h -1 and 0.204 E h -1 , respectively, using the MP2/ 4-31G** calculation whereas their corresponding ∆Qcorr values are 0.048 e and 0.065 e, respectively.Although the trend is always consistent for the CNH…CNH and NCH…NCH complexes, some inversions occur for the NCH…CNH and CNH…NCH complexes.It is also important to point out that RHF/4-31G** calculations show G D,A and ∆Q values smaller than its MP2/4-31G** corresponding values.For example, the G D,A values using The larger values for both G D,A and ∆Q are verified for the CNH…CNH complex, whereas their smaller values are verified in NCH…NCH.This can be understood in considering the greater acid character of CNH with respect to NCH.In this sense, it is also interesting to take into account their HOMO and LUMO energies.The values of ε HOMO and ε LUMO , for CNH are -13.363eV and 3.608 eV, respectively, using the MP2/cc-pVTZ calculation level whereas its corresponding values for NCH are -13.374eV and 3.848 eV, respectively.The energy gap for CNH is 16.970 eV, whereas for NCH is 17.223 eV.Therefore, the charge transfer in CNH…CNH is larger than in NCH…CNH.Furthermore, this transfer is larger in CNH…NCH than in NCH…NCH.
We have already verified 6 that both G D,A and ∆Qcorr follow an exponential behavior when the hydrogen bonding distance is progressively increased from the equilibrium position until 4.5 Å.Indeed, their exponential curves are practically superposed for both MP2 and RHF calculation levels.Figure 3 clearly shows this behavior for the CNH…CNH complex employing the MP2 level of calculation with the 4-31G**, 6-311G** and cc-pVTZ basis sets, analogously to what was verified for the other complexes.
Furthermore, we can note in Figure 4 that this exponential behavior is also verified for the other charge partitioning schemes.
However, a still better agreement is verified when corrected Mulliken charges are used to obtain charge transfer values.This can be better understood in considering that the overlap term, which is added to standard Mulliken charge to obtain the corrected Mulliken charge of the α atom, contains electronic contributions from atomic dipoles and lone pairs. 6As a consequence, it is expected that the corrected Mulliken charges can give a better description of the distribution of molecular atomic charge.On the other hand, it is less accentuated when NBO atomic charges are employed.This can be also visualized in Table 2 considering the first order exponential decay rate (t) of these parameters using the MP2 H-bond distance at equilibrium position with the 4-31G**, 6-311G** and cc-pVTZ basis sets.Here t is given by (1)   where r is the H-bond equilibrium distance, Γ is G D,A or ∆Q obtained at distance r and A is the pre-exponential factor of the fitting curve.From this Table we can observe that the exponential decay rate obtained from ∆Qcorr is in better agreement with that obtained from the G D,A than its corresponding use of ∆Q M , ∆Q CHELPG and ∆Q NBO values for the CNH…CNH, NCH…CNH, CNH…NCH and NCH…NCH complexes.The major difference has been verified for ∆Q NBO .

Figure 3 .
Figure 3. Graphs of G D,A and ∆Qcorr as function of the hydrogen bonding distance in the CNH…CNH complex using the MP2 calculation level with the (a) 4-31G**, (b) 6-311G** and (c) cc-pVTZ basis sets.