Introduction

The nonlinear optic (NLO) research field has been focused on attention since its beginning in 1961 [1]. The wide application of nonlinear optic materials in optical devices and other useful applications such as medical imaging and cancer therapies is the reason for the rapid process of their expansion [2,3,4]. So, it is very interesting to design and introduce new material with the high efficiency of NLO feature.

Recently, organic materials with delocalized electrons introduce potential novel NLO materials [5, 6]. On the other hand, the compounds, atoms or molecules, with excess electron such as alkali metals have presented high performance. Hence, the conjugated organic molecules in interaction with alkali metals lead to an essential charge transfer and enhancing polarizability (α) and or hyperpolarizability (β 0).

Difficulties and the high cost of experimental methods have made clear the importance of computational methods in quantum mechanics as well [7,8,9,10,11]. Due to the essentiality of the decreasing of synthesis cost, the design of simple systems is very important. According to this importance, we motivated to focus on simple π-systems with delocalized electrons in interaction with alkali metals. The other interesting point in the present study is the design and comparison of NLO properties in complexes with diverse π-systems in terms of aromaticity nature for the first time.

Computational methods

Calculations were performed using the Gaussian 09 system of codes [12]. All optimizations, electronic properties, polarizability, and hyperpolarizability were done at the second-order Møller–Plesset perturbation theory (MP2) as one of quantum chemistry post-Hartree–Fock ab initio methods and 6-311++G(2d,2p) basis set. The MP2 method is known as a good method due to rather accurate estimates of geometries and energies for complexes with non-covalent interactions and intermediate computational cost [13,14,15,16]. Frequency calculations were performed at the same computational level in order to confirm that the structures obtained correspond to energy minima. The time-dependent density functional theory (TDDFT) [17] calculations were carried out using CAM-B3LYP/6-311++G(2d,2p) to gain the crucial excited states of all studied systems and optical absorption spectra. It has been shown that TDDFT method and more specifically, CAM-B3LYP is appropriate for calculating the signatures of electronically excited states, and optical (absorption) spectra of molecules [18,19,20,21]. The counterpoise approach of Boys and Bernardi [22] was used to correct the stabilization energy for basis set superposition error (BSSE). The natural bond orbital (NBO) charges analysis was done using NBO [23] program as implemented in the Gaussian program package.

Based on Taylor series expansion, the energy of a system in the weak and homogeneous electric field is described as [24, 25]:

$$ \mathrm{E}={\mathrm{E}}^0-{\upmu}_{\mathrm{i}}{\mathrm{F}}_{\mathrm{i}}-1/2{\alpha}_{ij}{F}_i{F}_j-1/6{\beta}_{ij k}{F}_i{F}_j{F}_k-\dots $$

where E 0, F i, μ i, α ij, and β ijk are the molecular total energy in the absence of the electric field, the electric field component along the i direction, the dipole moment, polarizability, and the first hyperpolarizability, respectively. The average polarizability (α), first hyperpolarizability (β 0), and the electric dipole moments μ 0 values are defined as [26]:

$$ {\displaystyle \begin{array}{l}\alpha =1/3\ \left({\alpha}_{xx}+{\alpha}_{yy}+{\alpha}_{zz}\right)\hfill \\ {}{\beta}_0={\left({\beta_x}^2+{\beta_y}^2+{\beta_z}^2\right)}^{1/2}\hfill \end{array}} $$

in which

$$ {\displaystyle \begin{array}{l}{\beta}_i=3/5\left({\beta}_{iii}+{\beta}_{ijj}+{\beta}_{ikk}\right)\kern0.5em \mathrm{i},\mathrm{j},\mathrm{k}=\mathrm{x},\mathrm{y},\mathrm{z}\hfill \\ {}{\upmu}_0={\left({\upmu^2}_{\mathrm{x}}+{\upmu^2}_{\mathrm{y}}+{\upmu^2}_{\mathrm{z}}\right)}^{1/2}\hfill \end{array}} $$

Results and discussion

Geometries and energy analysis

The optimized geometries of all structures have been depicted in Fig. 1. The binding distances between alkali metals and carbons are presented in Table 1. As can be seen, the metal stands in the central location of the carbon chain or carbon cycle expect in Li···hexatriene complex. In Li···hexatriene, lithium leads to the middle position of the carbon number 1 and 2. It is worth mentioning that a lot of conformers of each complex were generated and optimized in several directions, but many of these configurations were abandoned due to three reasons: (1) The metal atom was displaced to the same structure as reported, (2) some structures had negative frequencies so they were not related to the real structures and not considered for more analysis, and (3) some structures have lower stability. So all presented structures are related to the global minimum in the potential energy surface of the studied complexes.

Fig. 1
figure 1

The optimized geometries of studied complexes. a Li···Butadiene. b Li···Cyclobutadiene. c Li···Hexatriene. d Li···Benzene. e Na···Butadiene. f Na···Cyclobutadiene. g Na···Hexatriene. h Na···Benzene. i K···Butadiene. j K···Cyclobutadiene. k K···Hexatriene. l K···Benzene

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Table 1 The obtained interaction distances (metal-carbon) for the obtained structures (all values in Å)
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All stabilization energies of studied complexes are negative, so the M···π complexes are more stable than separate isolated alkali metals and studied π-systems. As can be seen in Table 2, the stability of complexes with cyclic π-systems are more than chain and in order to M···Cyclobutadiene > M···Butadiene and M···Benzene > M ··· Hexatriene. The second comparison is in terms of the type of π-conjugated system. The values reveal that M···Cyclobutadiene are more stable than M···Benzene. The reason for this observation lies in Hückel’s rule. Based on this rule, a cyclic ring molecule with its π-electrons equal to 4n + 2 and 4n where “n” is zero or any positive integer is aromatic and anti-aromatic, respectively [27]. As is well-known, aromatic molecules are more stable than anti-aromatic ones. So, it is expected that benzene as an aromatic system is more stable than cyclobutadiene as an anti-aromatic system. As will be discussed in the next section, during complex formation, substantial charge transfer from alkali metal to cyclobutadiene happens. Indeed, this charge transfer happens for approximately the number of π-electrons to aromatic π-electron (4n + 2) number, and so the amount of stability is increased. While the charge transfer in M···Benzene leads to the number of π-electrons getting distance from 4n + 2, the amount of stability is decreased. So, it can be understood why the absolute charge transfer amount in benzene complexes are very low in comparison with the cyclobutadiene case. These two phenomena lead to the greater stability of M···cyclobutadiene complex than M···benzene complex under the same condition.

Table 2 Values of the uncorrected stabilization energy (kJ mol−1) as well as the corrected ones (SEcorr) with inclusion of the BSSE amounts
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The third evaluation is in terms of the type of alkali metal. The values show that the lithium complexes are more stable among the other ones.

Natural bond orbital (NBO) analysis

The aim of NBO analysis in this work is the investigation of charge direction between metal and π-conjugated hydrocarbons in different categories. From the results in Table 3, the absolute charge of alkali metals increases in order to “Li < Na < K, excepting the complex M-benzene.”

Table 3 NBO charge of alkali metal atoms in studied complexes (all values in a.u)
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The obtained results show that the alkali metal atoms (Li, Na, and K) have negative charge in interaction with chain π-conjugated hydrocarbons (butadiene and hexatriene) and benzene systems that it exhibits the charge transfer from π-conjugated hydrocarbons to alkali metal. Alkali metals display positive charge in interaction with cyclobutadiene. This means charge transfer from the alkali metal atom to the π-conjugated hydrocarbons. The amount of charge transfer in M···Cyclobutadiene is rather large and about 0.70, 0.81 and 0.84 a.u. for Li, Na, and K, respectively. This could be indicative of the effort of the system to increase the number of π-electrons to aromatic π-electron (4n + 2) number for an achievement to more stability. The values show that the amount of charge transfer depends on the type of π-system and changes in the following order: Cyclobutadiene ≫ Benzene > Hexatriene > Butadiene. So, the nature of π-system is the determinant factor, too. The dual role of metal, charge donor, or acceptor could greatly increase the NLO properties [28,29,30,31,32].

Chemical reactivity or electronic properties

Electronic properties of the studied complexes were investigated in terms of the energy gap (E g) [33], the chemical potential (μ) [34] and the global hardness (η) [34]. The values are measured and gathered in Table 4. The E g values are measured by using the following equation:

$$ {\mathrm{E}}_{\mathrm{g}}={\mathrm{E}}_{\mathrm{LUMO}}\hbox{--} {\mathrm{E}}_{\mathrm{HOMO}} $$
Table 4 The electronic properties of different π-systems and studied complexes
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Which EHOMO and ELUMO are the energies of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), respectively.

According to atomic physics, the chemical potential (μ) of the electrons in a molecule could be defined as the negative of the atom’s electronegativity and can be obtained with Mulliken equation as follow:

$$ \upmu =\hbox{-} {\upchi}_{\mathrm{Mulliken}}=\hbox{-} \left(I+A\right)/2 $$

where I and A are the ionization potential and electron affinity of the atom, respectively.

On the other hand, based on the Koopman’s theorem [35], (HOMO) and (LUMO) energies are related to the ionization potential (I) and the electron affinity (A), respectively, and can be stated as I = − EHOMO and A = − ELUMO. By combining Mulliken equation and Koopman’s theorem, chemical potential can be calculated using the following equation:

$$ \upmu =\left({\mathrm{E}}_{\mathrm{HOMO}}+{\mathrm{E}}_{\mathrm{LUMO}}\right)/2 $$

Chemical hardness (η) is one of the important properties of atoms, ions, or molecules and described as the resistance towards the deformation or polarization of electron cloud [36, 37]. Chemical hardness is formulated as [34]:

$$ \upeta =\left({\mathrm{E}}_{\mathrm{LUMO}}\hbox{-} {\mathrm{E}}_{\mathrm{HOMO}}\right)/2 $$

The global electrophilic index (ω) was established by Parr et al. stated formula [33]:

$$ \upomega ={\upmu}^2/2\upeta . $$

From the values in Table 4, following results were obtained in studied complexes in comparison with π-conjugated systems. In all cases, the energy gap (E g) has decreased. The obtained chemical potential (μ) was negative in the all-designed complexes. Due to the fact that the escaping tendency of electrons from a stable system can be described by μ; negative values demonstrate stable systems [38]. So, all systems are stable in our work. The chemical hardness (η) was decreased at least 25 and up to 63%, so the chemical softness was increased. The electrophilic index (ω) values are relatively small and have not changed much upon complex formation.

NLO response and optical absorption spectra

The characteristics of optical properties are reported in Table 5. The resulted data illustrate that the average polarizability (α), hyperpolarizability (β 0), and the electric dipole moment (μ 0) values have been larger for studied complexes in comparison with pure π-systems and urea. As we know, urea has been used as the standard reference for investigation of nonlinear optical (NLO) properties. The results reveal following trend in polarizability: Hexatriene > Benzene > Butadiene > Cyclobutadiene in interaction with the same metal. On the other hand, with the change of alkali metal, α value is increased in the following order: K > Na > Li. Also, a very satisfactory result was obtained from hyperpolarizability data. Interaction of π-systems with alkali metals results to about 5000–271,000 times enhancement of β 0 values in the M···π-molecule adducts. Cyclobutadiene, as an anti-aromatic system has maximum enhancement of β 0 after interaction with alkali metals among of the rest of the types of studied π-molecules. So, cyclobutadiene is obtained 129,847, 271,595, and 212,287 times enhancement of β 0 values in interaction with Li, Na, and K, respectively. Also, atomic number is one affective factor on the first hyperpolarizability so that increasing of atomic number induces greater β 0 [39]. Herein, heavier alkali metals lead to greater enhancement in β 0. Therefore, the nature of π-system and metal is the determinant factor.

Table 5 The average polarizability α (a.u.), hyperpolarizability β 0 (a.u.), and the molecular electric dipole moment μ 0 (Debye) values for urea as the standard molecule, π-systems, and studied complexes
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The molecular electric dipole moment (μ 0) parameter indicates molecular charge distribution in three directions. All the studied molecular π-systems have no dipole moments while after interaction with the alkali metal; the resulted complexes were obtained as polar nature. It is clear that the decreasing of symmetry as the result of interaction with a moiety, for example, an alkali metal in this work, is the main reason for increasing in μ 0 values.

Generally, every factor that can be affected on α, β 0, and μ 0 parameters can change NLO properties.

Furthermore, the UV-visible absorption spectra of the all studied complexes are characterized in Table 6 and exhibited in Fig. 2. From the results, strong absorption peaks according to a rather high oscillator strength values in the M···π-complexes have occurred in the UV and visible regions. While, π-conjugated systems, show a few number of absorption peaks that located in the UV region. Therefore, appearance of much number of strong absorption peaks shifted to larger wavelengths after complex formation, and this can be attributed to the interaction between π-conjugated systems and alkali metals.

Table 6 Selected excitation energies (ΔE), wavelength (λ), oscillator strength (f 0), and the most strongly contributing transitions between single-electron states and providing their percentages
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Fig. 2
figure 2

UV-Vis spectrum

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The interesting point inferred from the comparison of the transitions given in Table 6 is shift effect in the major peaks of optical spectrums and explained as follow:

In the optical spectrum of Li···Butadiene, the peak with the largest oscillator strength is located in 215.8 nm and when Li atom is replaced with Na and K this peak is shifted to 213.4 and 211.8 nm, respectively, and shows blue shift about 2.4 and 1.6 nm. Hence, in the case of M···Butadiene, excitation energies are increased in order to K > Na > Li. By comparing the same way, Li···Cyclobutadiene shows 30.8 and 68.4 nm red shift with replacing of Li atom with Na and K and so in case of M··· Cyclobutadiene, excitation energies are increased in order to Li > Na > K. Similarly, Li···Hexatriene reveals red shift in Na···Hexatriene and K···Hexatriene and finally Li···Benzene shows blue and red shift after metal changing with Na and K, respectively.

Conclusion

The most important achievement of this research work is the introduction of simple molecules with high optical and nonlinear optical (NLO) properties. For the attainment of this aim, we have investigated the effect of interaction with alkali metals (Li, Na, and K) on the average polarizability (α), hyperpolarizability (β 0), and the molecular electric dipole moment (μ 0) of π-conjugated systems (π = butadiene, cyclobutadiene, hexatriene, and benzene). The M···π-conjugated interaction leads to about 5000–271,000 times enhancement of β 0 values in the π-systems and the nature of π-system and metal is the determinant factor in this enhancement.

The NBO analysis represents that the studied π-conjugated systems act as charge donors as well as acceptors in interaction with alkali metals and lead to charge transfer between the alkali metal and π-conjugated system.

The energy gap (E g) of studied π-systems is decreased; α and β 0 are increased dramatically and μ 0 finds the notable values during complex formation. The UV-Vis spectra revealed strong absorption after complex formation that it is attributed to the interaction between π-conjugated systems and alkali metals. The major peak (the absorption peak with the largest oscillator strength) of Li···Butadiene reveals blue shift, while Li···Cyclobutadiene and Li···Hexatriene show red shift effects with the change of Li with Na and K, respectively. Interestingly, LiBenzene shows blue and red shift after metal replacing with Na and K, respectively.