The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are two important characteristics for analyzing the electronic properties of molecules [20] , and the gap between them explains molecular stability, chemical reactivity and kinetic stability [21] . Generally, HOMO orbital gives electrons while LUMO one accepts them; in our case, LUMO orbital of complex (Cytosine-Cu-Cytosine) has been attacked by HOMO orbital of adducts (NO₂, NO, CO₂, CO, HCN, H₂S, SO₂) (see Figure 1 and Figure 2 ). Hence, more the energy difference between them is reduced more the attack or the interaction is favored. We observe that the lowest value of the gap ( $\Delta E_{{\text{H}}_{\text{2}}\text{S}}$ = 0.063 eV) was obtained with H₂S molecule, followed by ( $\Delta E_{{\text{SO}}_{\text{2}}}$ = 0.54 eV) of SO₂ molecule and then by ( $\Delta E_{{\text{NO}}_{\text{2}}}$ = 0.82 eV) of NO₂ molecule. So, orbital attack of these molecules (NO₂, SO₂ and H₂S) by Cytosine-Cu-Cytosine complex is more favored. Therefore, the orbital interaction will be favored in the following order: $\Delta E_{{\text{H}}_{\text{2}}\text{S}}$ = 0.063 eV < $\Delta E_{{\text{SO}}_{\text{2}}}$ = 0.54 eV < $\Delta E_{{\text{NO}}_{\text{2}}}$ = 0.82 eV < $\Delta E_{\text{NO}}$ = 1.36 eV < $\Delta E_{\text{HCN}}$ = 2.99 eV < $\Delta E_{{\text{CO}}_{\text{2}}}$ = 3.12 eV < $\Delta E_{\text{CO}}$ = 3.27 Ev.

The decomposition of interaction energy is given as [22] :

$E_{\text{interaction}}=E_{\text{electr}}+E_{\text{orb}}+E_{\text{Pauli}}$ (1)

where E_elec is the electrostatic stabilization energy between Cyt-Cu-Cyt and adducts (NO₂, SO₂, CO₂, CO, H₂S, HCN and NO), E_orb represents orbital energy at relaxed structures and E_Pauli term shows Pauli energy due to repulsion clouds in the molecular geometry. We observe that for all terms, energies are close to each other ( Figure 3 ).

E_Pauli is destabilizing term for each compound and this destabilizing is offset by E_orb and E_elect, therefore interaction energy becomes a stabilizing term and varies little from one system to another: (E_inter = −6.07 a.u for Cyt-Cu-Cyt, E_inter = −6.53 a.u for Cyt-Cu-Cyt-H₂S, E_inter = −6.58 a.u for Cyt-Cu-Cyt-NO, E_inter = −6.69 a.u for Cyt-Cu-Cyt-CO, E_inter = −6.71 a.u for Cyt-Cu-Cyt-SO₂, E_inter = −6.79 a.u for Cyt-Cu-Cyt-NO₂, E_inter = −6.84 a.u for Cyt-Cu-Cyt-HCN and E_inter = −6.95 a.u for Cyt-Cu-Cyt-CO₂. It is more stabilizing for Cyt-Cu-Cyt-CO₂ complex. On the bonding energy level E_bond, this interaction is favored and more stabilizing with CO₂ adduct.

The transition state assumes a special type of chemical equilibrium (quasi-equilibrium) between the initial state (reactants) and the final stable state (products). Our results prove that these interactions occur with transition states since frequency calculations give one imaginary for each case. Figure 4 shows the transition states of the interaction of CO₂, NO, CO, H₂S and SO₂ adducts with Cytosine-Cu-Cytosine complex; these results are performed to form one bond between them (adduct and complex). We noticed that energy barriers vary relatively from one adduct to another and the lowest one is obtained with CO₂ adduct (E_TS = 0.26 kcal/mol); hence, we can consider that the interaction with CO₂ is more favored energetically. This energy barrier is followed by (E_TS SO₂ = 2.6 kcal/mol), (E_TS H₂S = 3.15 kcal/mol), (E_TS CO = 6.02 kcal/mol) and (E_TS NO = 8.16 kcal/mol).

Interaction of HCN and NO₂ adducts with Cytosine-Cu-Cytosine leads to the formation of two bonds: Cu-N and Cu-C with HCN and Cu-N, O-H23 with NO₂. Potential energy surfaces (PES) are depicted in Figure 5 . We notice that a formation of saddle point at (Cu-N = 1.96 Å, Cu-C = 2.38 Å, E = −6.83 a.u) during the interaction of HCN with Cytosine-Cu-Cytosine ( Figure 5(b) ). Regarding the formation of Cytosine-Cu-Cytosine-NO₂, it reaches a minimum at O…H23 = 1.902 Å, Cu-N = 2.057 Å at energy E = −6.78 a.u.

QTAIM was applied to optimized complexes to quantify and identify intermolecular interactions and the type and structure of bonds [23] [24] . It is based on electron density analysis [21] , more precisely the electron density in “critical point” (cp) is a point in space at which the first derivatives of the density vanish:

$\nabla \rho =i\frac{\partial \rho }{\partial x}+j\frac{\partial \rho }{\partial y}+k\frac{\partial \rho }{\partial z}\to \{\begin{array}{ll}0\hfill & \left(\text{at}\text{critical}\text{points}\wedge \text{at}\infty \right)\hfill \\ \text{generally}\ne 0\hfill & \left(\text{at}\text{all}\text{other}\text{points}\right)\hfill \end{array}$ (2)

The second derivatives of ρ(r) can be arranged in the so-called “Hessian matrix”, which, when evaluated at a CP located at r_c is written:

$A\left(r_c\right)={\left(\begin{array}{ccc}\frac{{\partial }^2\rho }{\partial x^2}& \frac{{\partial }^2\rho }{\partial x\partial y}& \frac{{\partial }^2\rho }{\partial x\partial z}\\ \frac{{\partial }^2\rho }{\partial y\partial x}& \frac{{\partial }^2\rho }{\partial y^2}& \frac{{\partial }^2\rho }{\partial y\partial z}\\ \frac{{\partial }^2\rho }{\partial z\partial x}& \frac{{\partial }^2\rho }{\partial z\partial y}& \frac{{\partial }^2\rho }{\partial z^2}\end{array}\right)}_{r=r_c}$

The Hessian matrix can be diagonalized and transformed to Λ because it is real and symmetric:

$\Lambda ={\left(\begin{array}{ccc}\frac{{\partial }^2}{\partial {x^{\prime }}^2}& 0& 0\\ 0& \frac{{\partial }^2}{\partial {y^{\prime }}^2}& 0\\ 0& 0& \frac{{\partial }^2}{\partial {z^{\prime }}^2}\end{array}\right)}_{r^{\prime }=r_c}=\left(\begin{array}{ccc}{\lambda }_1& 0& 0\\ 0& {\lambda }_2& 0\\ 0& 0& {\lambda }_3\end{array}\right)$

In which λ₁, λ₂ and λ₃ are the curvatures of the density. The trace of Hessian matrix of the density is invariant to rotations of the coordinate system and it is known as the Laplacian of the density [ ${\nabla }^2\rho \left(r\right)$]:

${\nabla }^2\rho \left(r\right)=\nabla \cdot \nabla \rho \left(r\right)=\frac{{\partial }^2\rho \left(r\right)}{\partial x^2}+\frac{{\partial }^2\rho \left(r\right)}{\partial y^2}+\frac{{\partial }^2\rho \left(r\right)}{\partial z^2}={\lambda }_1+{\lambda }_2+{\lambda }_3$ (3)

Figure 6 shows critical points (CP) in Cyt-Cu-Cyt complexes. Those in red represent BCP (bond critical points) while those in green are (RCP) ring critical points. Generally, $\rho \left(r_c\right)<0.07$, ${\nabla }^2\rho \left(r_c\right)>0$ we have ionic bond, $\rho \left(r_c\right)>0.15$, ${\nabla }^2\rho \left(r_c\right)<0$ it’s about covalent bond and $0.07<\rho \left(r_c\right)<0.15$, ${\nabla }^2\rho \left(r_c\right)<0$ it’s about an intermediate bond. In the interaction regions, BCP1 was formed during the interaction between the adducts and the Cyt-Cu-Cyt complex, which had low density and positive Laplacian. For that formed between the oxygen atom of CO₂ and N1 of complex BCP1(O-N1) $\left(\rho \left(r_c\right),{\nabla }^2\rho \left(r_c\right)\right)=\left(0.056,0.173\right)$ and between the sulfur atom of SO₂ and copper atom BCP1(S-Cu) $\left(\rho \left(r_c\right),{\nabla }^2\rho \left(r_c\right)\right)=\left(0.038,0.062\right)$ these criteria are assigned for intermediate closed-shell interaction [21] . About the interaction of CO adduct with Cyt-Cu-Cyt complex BCP1 has ρ(r) = 0.192 a.u with negative Laplacian ${\nabla }^2\rho \left(r\right)=-0.159\text{a}\text{.u}$ and hence BCP1(C_CO-Cu) is assigned as a covalent bond. Low density and positive Laplacian are assigned for the interaction of NO and H₂S adducts with Cyt-Cu-Cyt complex; BCP1(N_NO-Cu) have $\left(\rho \left(r_c\right),{\nabla }^2\rho \left(r_c\right)\right)=\left(0.106\text{a}\text{.u},0.096\text{a}\text{.u}\right)$, BCP1( ${\text{S}}_{{\text{H}}_{\text{2}}\text{S}}\text{-Cu}$) have $\left(\rho \left(r_c\right),{\nabla }^2\rho \left(r_c\right)\right)=\left(0.077\text{a}\text{.u},0.109\text{a}\text{.u}\right)$, these parameters correspond to electrostatic interactions. For N-Cu bonds, electronic density ranges from [0.065 a.u] to [0.173 a.u] and Laplacian varies in the interval [0.05 a.u, 0.271 a.u]. We can scribe them as electrostatic interaction. Exception for N4-Cu bond in Cyt-Cu-Cyt-CO₂ complex which has negative Laplacian ${\nabla }^2\rho \left(r_c\right)=-0.11\text{a}\text{.u}$, so it is considered as a covalent bond. In the interaction region, it was formed pseudo cycle characterized by a ring critical point having low electronic density from $\rho \left(r_c\right)=0.007\text{a}\text{.u}$ to $\rho \left(r_c\right)=0.01\text{a}\text{.u}$. QTAIM performs hydrogen bonds with very low values of electronic density from $\rho \left(r_c\right)=0.028\text{a}\text{.u}$ to $\rho \left(r_c\right)=0.034\text{a}\text{.u}$ with positive Laplacian.

Figure 7 exposes topological properties of Cyt-Cu-Cyt-NO₂ and Cyt-Cu-Cyt-HCN complexes. One bond was formed during the interaction of NO₂ with Cyt-Cu-Cyt complex, more precisely between ${\text{N}}_{{\text{NO}}_{\text{2}}}$ and Cu whose critical point atoms BCP1 having $\left(\rho \left(r_c\right),{\nabla }^2\rho \left(r_c\right)\right)=\left(0.065\text{a}\text{.u},0.117\text{a}\text{.u}\right)$ According to these criteria (low electronic density and positive Laplacian), we can assign it as electrostatic interaction. The interaction of HCN adduct with Cyt-Cu-Cyt complex leads to the form of two bonds: C_HCN-Cu having BCP1 characterized by $\left(\rho \left(r_c\right),{\nabla }^2\rho \left(r_c\right)\right)=\left(0.102\text{a}\text{.u},0.248\text{a}\text{.u}\right)$, the second one BCP2, was formed between N_HCN-Cu having ( $\left(\rho \left(r_c\right),{\nabla }^2\rho \left(r_c\right)\right)=\left(0.106\text{a}\text{.u},0.286\text{a}\text{.u}\right)$ it is also assigned as electrostatic interaction. In this way, it formed a ring with three atoms (Cu-C-N) characterized by RCP1 ${\nabla }^2\rho \left(r_c\right)=0.092\text{a}\text{.u}$. Likewise, Cu-N bonds have low values of electronic density ranging from [0.071 a.u] to [0.143 a.u] and positive Laplacian in the range [0.023 a.u - 0.199 a.u], so we can assign them as intermediate closed-shell interaction. Ring critical points have a very low electronic density from [0.008 a.u] to [0.092 a.u]. It was formed a new hydrogen bond between ${\text{O}}_{{\text{NO}}_{\text{2}}}$….H20 having BCP2 with ( $\left(\rho \left(r_c\right),{\nabla }^2\rho \left(r_c\right)\right)=\left(0.033\text{a}\text{.u},0.119\text{a}\text{.u}\right)$. The other hydrogen bonds also have a low electronic density from [0.031 a.u] to [0.111 a.u] and positive Laplacian in the range [0.031 a.u] to [0.391 a.u] and hence they are assigned as intermediate closed-shell interaction.

Table 1 displays covalent and non-covalent percentage of various bonds as well as their bond energies. From results, it can be observed that Cu-C_CO, ${\text{Cu-O}}_{{\text{CO}}_{\text{2}}}$ and Cu-C_HCN bonds possess a high non-covalent character which is equal to 97.8% (74.8 kcal/mol) and 89.1% (72.73 kcal/mol) 98.6% (9.5 kcal/mol) respectively, and it is more stabilizing with CO. We also notice that Cu-N_NO, ${\text{Cu-N}}_{{\text{NO}}_{\text{2}}}$, ${\text{Cu-O}}_{{\text{NO}}_{\text{2}}}$, ${\text{Cu-N}}_{{\text{NO}}_{\text{2}}}$ and Cu-N_HCN bonds hybrid character between covalent and non-covalent interactions having the following values: for covalent part 58.3% (539.28 kcal/mol), 98.1% (36.99), 94%, 97.4% and 43.5% (86.21 kcal/mol) respectively and it is more stabilizing with Cu-N_NO. For non-covalent parts, 41.7% (748.06 kcal/mol), 1.9% (5.89 kcal/mol), 6% (61.74 kcal/mol), 2.6% (65.14 kcal/mol) and 56.5% (111.87 kcal/mol) respectively and it is more stabilizing with Cu-N_NO. Hybrid character is also dominant in Cu-N1 and Cu-N4 bonds, giving a mixture of covalent and non-covalent interaction for the majority of complexes, except for Cyt-Cu-Cyt-NO which are purely covalent bonds. ${\text{Cu-S}}_{{\text{SO}}_{\text{2}}}$ bond is purely non-covalent, having (130.82 kcal/mol) whereas ${\text{Cu-S}}_{{\text{H}}_{\text{2}}\text{S}}$ bond is mostly covalent 95.7% (83.10 kcal/mol). Hydrogen bonds are described as non-covalent interactions with very low covalent character.

Values between parenthesis are bond energies in (kcal/mol).

Reduced density gradient (RDG) approach used to reveal non-covalent intermolecular and intramolecular interactions [25] . It is a visualization index based on electronic density and its derivatives using RDG results at low densities [26] [27] :

$\text{RDG}\left(r\right)=\frac{1}{2{\left(3{\pi }^2\right)}^{\frac{1}{3}}}\frac{\left|\Delta \rho \left(r\right)\right|}{\rho {\left(r\right)}^{\frac{4}{3}}}$ (4)

According to the sign λ₂ ρ (the second eigenvalue of the electronic density Hessian matrix), we can identify the nature of interaction: sign λ₂ ρ < 0 signifies a strong attractive interaction such as a hydrogen bond, sign λ₂ ρ > 0 denotes a steric interaction. If sign λ₂ ρ ≈ 0, that implies weak attractive interaction such as Van der Waals interactions.

Figure 10 and Figure 11 show reduced density gradient (RDG) and Non-covalent interaction (NCI) plots respectively of Cyt-Cu-Cyt-HCN, Cyt-Cu-Cyt-H₂S complexes. Likewise, sign λ₂ < 0 indicates strong interaction or hydrogen bond, more precisely from (−0.05 a.u to −0.02 a.u) for Cyt-Cu-Cyt-HCN and which is represented by blue color between NH…O atoms of thymine bases. Hydrogen bond is absent in Cyt-Cu-cyt-H₂S complex, but there exists a strong interaction between sulfur atom and copper ion. Weak interactions are designated at sign λ₂ ≈ 0 it belongs to the interval [−0.01 a.u, −0.01 a.u] for the two complexes denoting Van der Waals interactions that are visualized by green color and that are very marked in Cyt-Cu-Cyt-HCN complex. While, positive sign λ₂ ρ(r) in the range [0.02 a.u, 0.05 a.u] specifies steric effects that are visualized by red color for the two complexes.