04 TOPOLOGICAL ANALYSIS AND FREQUENCY DEPENDENT HYPERPOLARIZABILITY CALCULATIONS OF FDDNP : A DFT STUDY

The topological and first-hyperpolarizability properties of 2-(1-{6-[(2-fluoroethyl)(methyl)amino]-2naphthyl}ethylidene)malononitrile (FDDNP) were studied using DFTB3LYP method. The static and dynamic electronic (hyper)polarizabilities of conformers were calculated and a simple two-state model was employed to explain the first hyperpolarizability differences in two conformers. The second harmonic generation property was evaluated at the typical wavelengths of Cr:forsterite, Nd:YAG(neodymium-doped yttrium aluminium garnet) and Ti:sapphire lasers to predict the compound conformers potency for second harmonic generation imaging in biological studies.


Introduction
Optical imaging techniques based on nonlinear optical phenomena have been widely incorporated in microscopy using ultra-fast pulsed lasers.There have been described many applications in cellular and tissue imaging due to the ability to spatially resolve subcellular details with high molecular contrast [1].
The employed near-infrared (NIR) wavelengths reduce scattering and maximize tissue penetration.These characteristics boost nonlinear microscopy as an elective method for imaging cells with micrometer-resolution deep into the living tissues.Both labeled and label-free second harmonic generation (SHG) imaging techniques have been employed in biological studies.For example, the non-centrosymmetric structure of fi brillar collagen is the major source of the SHG signals in different tissues [2,3], and for cellular and subcellular samples.For this purpose, several SHG labels have been used [4,5].
According to the importance of second-harmonic generation, as one of the nonlinear excitation families with deep imaging, 3D capabilities and intrinsic contrast in certain tissues, the study of nonlinear optical properties of biomaterials and their labels plays an essential role in development of medicinal optic devices.So far, only a few electronic structure studies on FDDNP and other dicyanovinyl naphthalene biomarkers, based on ab initio calculations have been performed.Petrič et al. [6] performed a combined experimental and theoretical study on FDDNP and its analogs to explain the chemical modifi cation effects on binding affi nity of biomarkers to amyloid plaques and their fl uorescence Quantum yields.Given the close relationship between the structure and nonlinear activities, and the functional importance of default molecule, it is necessary to develop a deeper vision on electron-topological properties, in order to predict its different optical capabilities.In the present work we consider both structural and nonlinear optical properties of FDDNP to introduce it as a potent material in nonlinear optical imaging of biological samples.
Thereupon in this work, beside the study of conformational effect on structural properties, the static and frequency-dependent electronic (hyper)polarizabilities of the FDDNP conformers were also computed.For this purposes, the nonlinear optical properties related to the excited-state dipole moment were studied using a simple two-state model (TSM) [10].

Computational details
The geometry optimization, frequency and single point energy calculations were carried out for all structures, using the Density Functional Theory method supplied in the GAUSSIAN98W suite of programs [11].
The B3LYP functional [12][13][14] with the 6-31G(d,p) basis set [15] were used.The Time-Dependent Density Functional Theory (TD-DFT) was used to calculate the excited state energies and excited state dipole moments.The 10 lowest excited states were considered.The orbital transition contributions were obtained using GAUSSSUM 2.2 program [16].
To estimate the solvent effect on the studied optical parameters, polarizable continuum model (PCM) calculations were performed [17].The topological analyses were carried out using Multiwfn program [18].

Geometry and Natural Bond Orbital (NBO) studies
Two stable conformations of the FDDNP can theoretically exist [6].The optimized structures are shown in Figure 1.The C2-C6-C9-C10 dihedral angles for two conformers are approximately 31° (Syn form) and 142° (Anti form).The Syn conformer has lower total energy than the Anti form by 4.98 kJ/mol.Intra-molecular hyperconjugative interactions can be used as a measure of the interaction intensity between electron-acceptor orbitals.In fact, the larger the value of the hyperconjugative interaction, the more intensive the interaction between the donor and acceptor, and the greater the extent of conjugation of the whole system is [19].
The role of hyperconjugative interactions in the stabilization of Syn and Anti conformers has been studied using NBO analysis.For a better comparison of the conformers' stability, the same values of the donor-acceptor interactions with large differences (≥0.4 kJ/mol) have been summarized in Table 1.Upon the data, hyperconjugation interactions are more favored for the Syn form.The interactions such as πC8-C9→π*C2-C6 and πC2-C6→π*C8-C9 play an important role in conformational properties of FDDNP.In addition, the sum values of hyperconjugation interactions are in accordance with the total energy values.

Topological analysis
The topological study of electron density based on Quantum Theory of Atom In Molecules (QTAIM) as a rigorous interpretative method can unambiguously describe atoms existing in the system.In this theory the chemical bonds can be defi ned as the set of lines known as bond paths (BPs) linking neighboring atoms.In addition, other concepts and quantities are also necessary to characterize a bonding interaction, namely critical point (CP), electron density ρ(r), Laplacian of electron density  2 ρ(r), kinetic energy density [G(r)], potential energy density [V(r)], and total energy density [H(r)] [20].
Hydrogen bonding can be explained and classifi ed as a main factor in determining the preferred conformers according to the Roza's criteria [21]: 1. weak hydrogen bonds:  2 ρ BCP ˃0 and H BCP ˃0; 2. medium hydrogen bonds:  2 ρ BCP ˃0 and H BCP ˂0; 3. strong hydrogen bonds:  2 ρ BCP ˂0 and H BCP ˂0, where  2 ρ BCP and H BCP are the Laplacian of electron density and the total electron energy density at critical point respectively.
The nature of hydrogen bonds can be specifi ed using the following criteria [22]: 1. for 1, the hydrogen bond has non-covalent nature; 2. for 0.5 1, the hydrogen bond is partly covalent.
At the fi rst step, we verifi ed if C3H27 and C3H26 (see Figure 2) were in fact hydrogen bonds.Then, the strength of bonds was studied through the criteria above.Koch and Popelier [23] established some criteria for hydrogen atom that are involved in hydrogen bond as follows: 1. increasing the net positive charge [q(H)]; 2. decreasing the fi rst atomic dipole moment [M(H)]; 3. decreasing the atomic volume [V(H)]; 4. increasing the atomic total energy [E(H)].
To compare the atomic properties with a reference, the C2-C6-C9-C10 dihedral angle was fi xed at 90° and QTAIM calculations were performed to obtain the atomic properties of H26 and H27 when they are not involved in hydrogen bond.The atomic QTAIM parameters related to H26 and H27 for Syn conformer, Anti conformer and the reference state are gathered in Table 2.As can be seen, all Koch's criteria for identifi cation of C3H26 and C3H27 as hydrogen bonds are satisfi ed.For classifi cation of the intra-molecular hydrogen bonds in Syn and Anti conformers, the topological quantities at characteristic critical points (see Figure 2) have been calculated and summarized in Table 3.Both Laplacians of electron density at the bond critical points are positive, which reveals a depletion of electron charge in these bond critical points.[22].
In addition, the ratio of related to critical points of C3H27 and C3H26 are 1.34 and 1.35 respectively.Then according to Roza's criterion, both intra-molecular bonds can be classifi ed as weak hydrogen bonds.Figure 2 shows that the formation of intra-molecular hydrogen bonds leads to a ring critical point (RCP) in each conformer.

Frequency dependence of linear and nonlinear optical properties
In order to investigate the linear and nonlinear optical performances of FDDNP conformers, polarizability (α), polarizability anisotropy (Δα) and fi rst hyperpolarizability (β) have been calculated according to the following relations [24][25][26]: α=1/3(α xx +α yy +α zz ), (1) Polarizability and hyperpolarizability parameters for some common frequencies in functional lasers are gathered in Table 4.As can be seen, the average polarizability, polarizability anisotropy, and Electro-Optic Pockels Effect (EOPE)-hyperpolarizability are increased with increasing frequency.SHG-hyperpolarizability has maximum value in ω=0.051770 au, which is the typical wavelength of the Ti:sapphire laser (880 nm).
p-Nitroaniline (pNA) is prototypical compound used in the study of the nonlinear optical (NLO) properties as a reference one.Hence, we calculated SHG-hyperpolarizability of this molecule at λ= 880 nm.The calculated value is 43.99 times smaller than the same value of FDDNP (Syn form).Thus, we can conclude that the FDDNP has considerable nonlinear optical properties.K. Akhtari et al. / Chem. J. Mold. 2016, 11(2), [84][85][86][87][88][89][90][91][92] To study the medium effect on the optical properties of the FDDNP conformers, the water, DMSO, and benzene solvent were employed to simulate the protic polar, aprotic polar and nonpolar media.Similar method has been adopted to calculate thermodynamic and electronic properties in biological environments [29][30][31].The linear and nonlinear optical parameters of FDDNP conformers have been shown in Table 5.The notable point is a large SHG-hyperpolarizability of title compound in nonpolar medium.

Two-state model
As shown from Table 5, the static fi rst hyperpolarizabilities of Syn form are larger than the similar values in Anti form.A two-state model has been employed to explain these differences.According to this model, the fi rst hyperpolarizability can be explained using the following relation [10]: In the above expression, β is proportional to the difference of dipole moment in the ground state and the crucial excited states (Δμ ge ) and the oscillator strength (f os ), but inversely proportional to the third power of the transition energy (ΔE).Calculated parameters by using TD-DFT method (Table 6) for FDDNP conformers indicate that the largest dipole moment changes and oscillator strengths correspond to the HOMO→LUMO transition.The transition energies and corresponding oscillator strengths are almost the same for both conformers.The difference can be attributed to a larger dipole moment change in Syn form.The transition has p→π * character (Figure 3).As seen in molecular orbital plots (Figure 3), the highest occupied molecular orbitals mainly concentrate on naphthalene group and nitrogen atom in the fi ve-member chain, while for the lowest unoccupied molecular orbitals the population analysis of molecular orbitals indicates that the C2 and C6 atoms on rotating group contribute 13% and 28% for Syn conformer and 13% and 30% for Anti conformer.These orbitals play the crucial role in the intramolecular chargetransfer (ICT).In this way, the electron density transfers from the electron donor moiety to the electron acceptor group.K. Akhtari et al. / Chem. J. Mold. 2016, 11(2), [84][85][86][87][88][89][90][91][92]

Figure 1 .
Figure 1.Structures of two possible minimum energy conformations of FDDNP.

Figure 2 .
Figure 2. Molecular graphs for the electron density topology of (a) Syn and (b) Anti forms, with their bond critical points (BCP) and ring critical points (RCP).

Table 6 Results of the TD-DFT calculations for FDDNP conformers*.
ΔE (eV), the difference of dipole moment between the ground state and the excited state Δμ ge (Debye).