Electronic structure and second hyperpolarizability of M(NA2)2 (M = Be, Mg, Ca; A = H, Li, Na) complexes

Chemical Physics Letters 637 (2015) 164–171

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Electronic structure and second hyperpolarizability of M(NA2 )2 (M = Be, Mg, Ca; A = H, Li, Na) complexes Paramita Banerjee, Prasanta K. Nandi ∗ Department of Chemistry, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India

a r t i c l e

i n f o

Article history: Received 22 May 2015 In final form 5 August 2015 Available online 13 August 2015

a b s t r a c t The ground state structure and NLO properties of hydrazine molecule, metal diamine complex M(NH2 )2 and its derivatives M(NA2 )2 (M = Be, Mg, Ca and A = Li, Na) are calculated by using different DFT functionals and basis sets. The chosen species are sufficiently stable. The M(NA2 )2 complexes (A = Li, Na) have rather larger magnitude of third-order response property compared to the hydrazine molecule and M(NH2 )2 complexes. The sum-over-state (SOS) calculated one-photon and two-photon part of secondhyperpolarizability showed almost identical pattern of variation as obtained in the DFT calculated results. The largest second-hyperpolarizability is predicted for the complex Ca(NNa2 )2 . © 2015 Elsevier B.V. All rights reserved.

1. Introduction The increasing demand in photonic device, optical data storage, data processing and data transmission results in the significant advancement of the design of potentially active non linear optical (NLO) materials. Many theoretical and experimental efforts during the past few decades lead to significant growth in the development of many interesting NLO materials which finds various applications. The organic chromophores with extended ␲-delocalization have attracted much attention both theoretically and experimentally since they possess many attractive NLO characteristics, i.e., ultrafast response times, lower dielectric constants, large susceptibility, ease of modification, high laser damage thresholds and relatively low cost. Some of the deserving materials having significant NLO properties are polyyne and polyene chains [1–5], radical ion-pair salt [6], open shell intermediate diradical species [7–10], graphene nanohybrid materials covalently functionalized with porphyrin and fullerene [11] and halofullerenes [12,13]. Apart from the easy fabrication and ease of modeling with organic molecules researchers in this field are interested with organometallic systems [14–28] primarily due to the greater scope of modifying the electronic structure to provide significant thermal stability and to enhance the electric response property to a greater extent. By suitably incorporating metal atom in the

∗ Corresponding author. E-mail address: nandi [email protected] (P.K. Nandi). http://dx.doi.org/10.1016/j.cplett.2015.08.008 0009-2614/© 2015 Elsevier B.V. All rights reserved.

hydrocarbon backbone the third-order NLO property can be tuned to a desirable extent. The notable design of potential third-order NLO active materials includes hexalithiobenzene [27], metal incorporated organic framework [28] and lithiated silicone cage [29] etc. In particular, the study of diffuse electron system is noteworthy [30,31]. Many other examples include lithium atom decorated graphene nanoribbon [20], push–pull electronic effect of Li atom doped smolecule having electride character [32], alkaline earthbased alkalides [33], etc. which have been considered as the newly designed NLO-materials. The main objective of the present theoretical study is to design efficient organometallic molecules which have sufficient thermal stability and attractive second-hyperpolarizability. Most of the metal hydrocarbon interactions arise from the participation of atomic orbitals of the outermost subshell of metal atom and the ␲ electrons of hydrocarbon. The diffuse ␲ electron density of later can bind one or more light metal atom(s) forming stable molecular complexes which find various optoelectronic applications. In the present work, we have considered hydrazine molecule (N2 H4 ) which is structurally different from the ethylene (C2 H4 ) molecule. Each nitrogen atom unlike the C atom of ethylene is sp3 hybridized. As a result the interaction between the metal and hydrazine involves participation of ␴-bonded electrons of the later. Therefore, it will be interesting to study the structural characteristics of metal diamine complexes, M(NA2 )2 (M = Be, Mg, Ca and A = H, Li, Na) (Scheme 1) and their nonlinear optical properties. The precise role of different metals on the modulation of electric response properties of the investigated molecular complexes will be examined to explore new structure property correlation.

P. Banerjee, P.K. Nandi / Chemical Physics Letters 637 (2015) 164–171

Scheme 1. B3LYP/6-311++G(d,p) calculated optimized ground state structures.

165

166

P. Banerjee, P.K. Nandi / Chemical Physics Letters 637 (2015) 164–171

2. Computational methods

All calculations are performed using Gaussian 09 quantum chemical program package [41].

Geometry optimizations of the chosen molecules (Scheme 1) have been carried out by using the B3LYP method and MP2 methods for the 6-311++G(d,p) basis set. The stability of geometry of each molecule on the potential energy hypersurface has been checked by calculating harmonic vibrational frequencies at each level which are found to be real. The optimized geometry has subsequently been taken in the calculation of nonlinear optical properties. The Gibb’s free energy corresponding to the formation binding energy (GB0 ) of the complex H2 N M NH2 (M = Be/Mg/Ca) in the isolated gas phase has been calculated based on the following reaction scheme. N2 H4 + M → H2 N M NH2 (M = Be/Mg/Ca) The Gibb’s free energy change for the following reaction.

(G0 )

(1)

has also been calculated

H2 N M NH2 + 4A → A2 N M NA2 + 2H2 (A = Li/Na; M = Be/Mg/Ca)

(2)

(G0 )

of overall reaction leading to the formaThe free energy tion of the complex A2 N M NA2 (A = Li, Na; M = Be, Mg, Ca), N2 H4 + M + 4A → A2 N M NA2 + 2H2 (A = Li/Na; M = Be/Mg/Ca)

(GB0

(3)

+ GR0 ).

The necessary thermal can be obtained as the sum energy and zero point energy correction have been included in the calculation of reaction free energies. The mean polarizability also called the isotropic polarizability (˛iso ) has been calculated as one third of the trace of linear polarizability tensor. aiso =

(˛xx + ˛yy + ˛zz ) 3

(4)

The orientationally averaged second-hyperpolarizability ( av ) has been calculated by using the following expression [34]. av =

1 [xxxx + yyyy + zzzz + 2(xxyy + xxzz + yyzz )] 5

(5)

The Cartesian components of polarizability and firsthyperpolarizability of the chosen metal complexes are calculated analytically while the dipole second-hyperpolarizability are evaluated by numerical differentiation of the analytically obtained dipole first-hyperpolarizability by using the restricted DFT method consisting of BHHLYP [35], long range corrected CAM-B3LYP [36], wb97XD [37] and B2PLYP (hybrid density functional with perturbative second-order correlation) [38] functionals. Among the chosen methods the CAM-B3LYP functional has been found [39,40] to be more reliable for (hyper)polarizability calculation since it can reproduce the results obtained with other correlated ab initio levels. For the present hyperpolarizability calculations default electric field strength (F) equal to 0.00033 au has been used. However, the numerical accuracy of the second hyperpolarizability of the studied complexes has been checked by comparing the results of  xxxx and <> obtained at the CAM-B3LYP/6-311++G(3df,3pd) level for different electric field strengths varied within 0.00003–0.0006 au (Table 1s, Supplementary Section). For molecules 1a, 2a, 2b, 2c, 3a, 3b, 3c, 4a and 4b the calculated second hyperpolarizability obtained at the default field strength remains within 0 to ±1% with respect to that obtained at F = 0.00003 au. However, for molecule 4c the magnitude of  xxxx obtained at default field strength is found to be smaller (by about 4.2%) than that obtained for the lowest field strength. Since no experimental or theoretical results of second-hyperpolarizability for the investigated molecules are available, a comparison of the calculated results obtained by the chosen DFT methods may provide the reliability of calculations a least qualitatively. The different basis sets 6-311++G(d,p), 6311++G(3df,3pd), aug-cc-pVTZ and aug-pc-2 have been used in the calculation of second-hyperpolarizability.

3. Results and discussions 3.1. Optimized structure and energetics The ground state optimized structures of hydrazine and its metal derivatives are displayed in Scheme 1. The B3LYP calculated geometrical parameters of the chosen molecules are reported in Table 1. The MP2 calculated results are given in Table 2s (Supplementary Section). The optimized geometry of hydrazine is in the gauche form which is consistent with the previous theoretical results [42] obtained at the HF and MP2 levels for the 6-31G* basis set. The B3LYP/6-311++G(d,p) calculated N H bond ˇ ˚ length (1.01 A)/N N bond length (1.43A)/ 2b > 2c. When H atoms of hydrazine are replaced by alkali metal (M = Li/Na) atoms the value of MNM (versus HNH angle) becomes larger but follows an identical pattern of variation 3a > 3b > 3c/4a > 4b > 4c, i.e., with a given alkali metal the increasing size of alkaline earth metal significantly lowers the MNM angle. These methods, however, predict substantially different NCaN angles for molecules Ca(NH2 )2 (2c) and Ca(NLi2 )2 (3c). The MP2 method invariably predicts the linear NCaN structure while the B3LYP method produces the bent structure for molecules 2c and 3c. On the other hand, the N Be N angle in molecule 4a and N Mg N angle in molecule 4b obtained at both levels are found to be identical. In order to assess the reliability of MP2 and B3LYP methods in predicting the correct electronic structure of molecules 2c and 3c, geometry optimization of two molecules has also been carried out at the CASSCF (4,4)/6311++G(d,p) level. This method gives the linear N Ca N structure of molecules 2c and 3c which is similar to the corresponding MP2 structures. Thus for the chosen diamine complexes of calcium the NCaN angle is linear. The calculated distance between the nitrogen atom and the alkaline earth metal atom increases with increasing size of the metal atom. The metal-nitrogen bond lengths are found ˚ at the MP2 level compared to that somewhat larger (by about 03 A) obtained at the B3LYP level. However, the replacement of terminal H atoms with alkali metal atoms (Li/Na) does not have a noticeable effect on the central metal–nitrogen bond length. At equilibrium the electrostatic repulsion between two adjacent nitrogen lone pairs of hydrazine rotates one NH2 group with respect to other by about 60◦ from the ideal perpendicular orientation. As a results one pair of hydrogen atoms attached to adjacent nitrogen atoms come in shorter distance with a torsion angle of 29.4◦ (B3LYP)/28.2◦ (MP2) while the other hydrogen pair go farther away from each other making a torsion angle of 90.8◦ (B3LYP)/90.4◦ (MP2). Interestingly, when an alkaline earth metal atom is inserted between two bonded nitrogen atoms of hydrazine the mutual repulsion of nitrogen lone pairs increases due to greater accumulation of electron density on N atoms arising from the charge transfer from Be/Mg/Ca atom. As a result H atoms at one terminal site are mutually orthogonal to the other side (MP2). Indeed, the A2 N M atoms of the complex, A2 N M NA2 remain in a plane when the N M N angle is 180◦ . In this case, the two A2 N planes are orthogonal to each other which is consistent with the

Table 1 Optimized structural parameters (distances in A˚ and angles in degree) of molecules in Scheme 1 obtained at the B3LYP level and free energies (kcal/mol) of reactions (1) (GB0 ), (2) (GR0 ) and their sum (G0 ) calculated at the MP2 geometry for the 6-311++G(d,p) basis set. Optimized structures (Scheme 1)

Molecules

M

A

1a

˚ r2 (A)

1.01

1.43

N M N



Dihedral angle ( 2174 / 5471 )

108.6

90.8/29.4a

GR0

G0

−128.2

−37.5

−165.8

GB0

−128.2

2a

Be

1.01

1.51

180.0

109.4

70.8/19.2

2b 2c

Mg Ca

1.01 1.02

1.90 2.19

180.0 141.4

107.2 104.0

70.8/19.2 158.5/−19.0

3a

Be

1.72

1.53

180.0

136.0

51.1/38.9

3b 3c

Mg Ca

1.72 1.73

1.93 2.19

180.0 157.0

123.9 113.6

51.2/38.8 142.2/−35.4

−60.8 −64.2

−35.3 −26.6

−96.1 −90.8

4a

Be

2.12

1.54

168.4

121.2

169.1/−54.7

−128.2

47.0

−81.2

4b 4c

Mg Ca

2.13 2.14

1.95 2.16

174.2 179.6

116.9 105.8

159.2/−62.7 45.0/45.0

−60.8 −64.2

47.9 49.4

−12.9 −14.9

−60.8 −64.2

P. Banerjee, P.K. Nandi / Chemical Physics Letters 637 (2015) 164–171

a

˚ r1 (A)

For hydrazine molecule (1a) the dihedral angles are  2146 = 90.8,  2145 = 29.4, respectively. 167

0.51 1.27 107.53 1229.50 2.84 125.58 627.02 4.33 223.88 1438.19 0.52 0.901 55.45 1936.69 1.922 149.03 930.17 2.37 392.19 3497.00 0.22 0.34 1.30 3.32 0.43 1.68 3.55 0.52 2.13 4.86 0.25 0.41 1.34 3.71 0.48 1.90 3.89 0.60 2.51 7.09 0.50 1.07 94.26 1007.83 2.74 114.59 526.87 4.11 206.30 1166.93 0.52 0.90 51.16 1557.88 1.92 144.23 794.56 2.38 379.18 3067.03 0.22 0.34 1.27 3.16 0.43 1.65 3.41 0.52 2.09 4.67 0.25 0.41 1.32 3.53 0.48 1.87 3.79 0.60 2.48 6.78 0.62 1.38 100.75 869.57 3.55 118.48 480.14 5.24 189.99 977.31 0.61 1.00 58.60 1237.99 2.26 148.80 579.11 2.58 332.75 2478.89 aug-pc-2 basis set for Ca atom and aug-cc-pVTZ basis set for other atoms. a

0.23 0.36 1.30 2.93 0.45 1.69 3.29 0.54 2.13 4.32

0.61 0.98 51.43 927.14 3.40 142.79 533.90 2.65 351.85 2314.99

0.60 1.30 82.07 611.72 3.09 102.55 335.09 4.61 178.25 813.86

0.26 0.42 1.48 3.55 0.50 2.09 4.16 0.65 2.34 6.58 0.26 0.43 1.39 3.26 0.51 1.96 3.77 0.64 2.60 6.10 1a 2a 3a 4a 2b 3b 4b 2c 3c 4c

0.23 0.36 1.41 3.12 0.46 1.85 3.60 0.55 2.78 4.79

 xxxx ˛iso ˛xx

B2PLYP

 av  xxxx ˛iso ˛xx

BHHLYP

 av  xxxx ˛xx

˛iso wb97XD

˛iso

 xxxx

 av ˛xx

The predominant charge transfer interaction of the investigated metal complexes takes place along the molecular x-axis passing through the N M N moiety. Thus the axial x-component of  ( xxxx ) has been taken as the longitudinal component. The results of the longitudinal component of linear polarizability (˛xx ) and the second hyperpolarizability ( xxxx ) along with their mean obtained for the metal complexes (Scheme 1) calculated at different methods are compared in Tables 2 and 3 for the aug-cc-pVTZ and aug-pc-2 basis sets, respectively. To examine the effect of higher angular momentum polarization functions on the electric response properties of the investigated molecular species the 6311++G(3df,3pd) calculated results obtained for the chosen DFT methods are also reported in Table 3s (Supplementary Section). Due to the centrosymmetric structure the dipole moment and first-hyperpolarizability of the chosen species are found to be zero. The results of polarizability obtained by different methods and basis sets do not show noticeable variation. The lowest and the highest values of polarizability differing by an order of magnitude are obtained for hydrazine (1a) and its derivative Na2 N Ca NNa2 (4c), respectively. However, the results of secondhyperpolarizability obtained by different methods show significant variation. The magnitude of cubic polarizability obtained for a given method is predicted significantly larger in aug-cc-pVTZ and aug-pc-2 basis sets compared to the 6-311++G(3df,3pd) basis set. At a given level of calculation, the longitudinal component of second-hyperpolarizability increases by about an order when an alkaline earth metal is placed between two N atoms of hydrazine (1a). However, a dramatic enhancement of  xxxx (106 –107 a.u.) occurs when alkali metal atoms replace H atoms of hydrazine. The substitution of hydrogens with sodium enhances the secondhyperpolarizability by an order of magnitude compared to that with replacement with lithium atoms. The largest magnitude of secondhyperpolarizability has been predicted for molecule 4c which is about four-order of magnitude larger than that of hydrazine (1a). The results of  obtained at CAM-B3LYP and wb97XD levels are comparable, in general. Also the BHHLYP and B2PLYP results are comparable but are predicted much larger compared to the CAMB3LYP and wb97XD calculated results (Figure 1) and Figs. 1s and 2s

CAM-B3LYP

3.2. Static electronic second hyperpolarizability

Molecule

sum of the twisted angles  2174 and  5471 (Table 1 and Table 2s in Supplementary Section). The planarity of A2 N M is, however, lost in diamine complexes 4a, 4b and 4c having bent N M N angle. The NBO population analysis showed that each nitrogen atom of A2 N M NA2 complex is sp3 hybridized. However, the nature of hybridization of the central alkaline earth metal atom depends on the size of atom ‘A’ of the NA2 moiety. In H2 N M NH2 (M = Be and Mg) complexes the central metal atom is sp hybridized while in A2 N M NA2 (A = Li and Na) the alkaline earth metal atom has poor mixing of s and p orbitals and is accompanied by significant population of higher lying s and p orbitals of Be and Mg, and 3d orbitals of Ca. As can be seen from Table 1 the stability of the chosen metal complexes depends largely on the size of the metal atom. The highest stability has been predicted for the complex H2 N Be NH2 the binding energy of which is almost twice of that of the corresponding Mg and Ca complexes. The displacement of hydrogen atoms in H2 N M NH2 (M = Be/Mg/Ca) with lithium atoms forming Li2 N M NLi2 complexes is energetically favorable. The direct displacement with sodium is, however, not feasible. The formation of Na2 N M Na2 (M = Be/Mg/Ca) complexes is possible when coupled with the first reaction (overall Gibb’s free energy of reaction being negative). The preferential binding with beryllium and lithium atoms may be due to their smaller size which favors greater covalent interaction.

 av

P. Banerjee, P.K. Nandi / Chemical Physics Letters 637 (2015) 164–171 Table 2 Longitudinal component of polarizability (˛xx ,102 a.u.), second-hyperpolarizability ( xxxx , 104 a.u.), average polarizability (˛iso , 102 a.u.) and second-hyperpolarizability ( av , 104 a.u.) obtained for molecules (Scheme 1) at different DFT functionals for the aug-cc-pVTZ basis set.a

168

6

(γXXXX, 10 a.u.)

0.59 1.22 104.82 1159.03 2.84 124.39 605.07 3.80 224.21 1288.45 0.60 0.94 57.35 1809.43 2.06 147.85 884.24 2.70 391.54 3415.84 0.22 0.34 1.34 3.24 0.43 1.68 3.49 0.52 2.13 4.78 0.26 0.40 1.30 3.63 0.48 1.89 3.84 0.61 2.51 6.93

169

40 CAM-B3LYP Wb97xd BHHLYP B2PLYP

20

0 1a 2a 3a 4a 2b 3b 4b 2c 3c 4c Molecules

Figure 1. Plot of  xxxx of the chosen molecules obtained at different DFT functional for the aug-pc-2 basis set.

(Supplementary Section). In the case of Mg compounds the substitution of H atoms with Li/Na atoms enhances  xxxx up to 106 orders. On replacing Li atoms with Na second-hyperpolarizability of Mg complexes increases by about 6 times at the B2PLYP level. It may be noted that although the magnitude of  xxxx and  av increases appreciably on inclusion of higher angular momentum diffuse functions in the basis set the relative variation of  xxxx obtained for a given method and basis set is identical (see Figure 1, 1s and 2s). A similar pattern of variation of  av has been obtained for the chosen molecules. The notably larger magnitude of ␥ (∼107 a.u.) has been predicted for molecules Be(NNa2 )2 (4a), Mg(NNa2 )2 (4b) and Ca(NNa2 )2 (4c). The ratio of  xxxx of molecules 4a, 4b and 4c obtained at each DFT level for aug-cc-pVTZ/aug-pc-2 basis sets are: 1.74: 1.00: 4.34/1.70: 1.00: 4.34 (CAM-B3LYP), 2.14: 1.00: 4.28/2.06: 1.00: 3.86 (wb97XD), 1.96: 1.00: 3.86/1.92: 1.00: 4.06 (BHHLYP) and 2.08: 1.00: 3.76/2.05: 1.00: 3.86 (B2PLYP), respectively. Thus CAM-B3LYP and B2PLYP methods give rather more consistent results which are less sensitive to the change of basis set. It should be worth mentioning that the almost identical ratio of  xxxx (1.68: 1.00: 4.19 (CAM-B3LYP), 1.98: 1.00: 3.90 (wb97XD), 1.89: 1.00: 4.08 (BHHLYP) and 2.02: 1.00: 3.80 (B2PLYP)) is obtained for molecules 4a, 4b and 4c at the respective level for the 6311++G(3df,3pd) basis set when compared with that predicted with the aug-pc-2 basis set.

12

6

,10 a.u.)

 av  xxxx ˛iso ˛xx

0.58 1.18 90.59 952.21 2.73 114.42 506.04 5.04 203.43 1131.93 0.59 0.94 49.10 1465.75 1.92 143.65 763.55 2.69 380.19 3096.74 0.73 1.49 106.77 787.85 3.69 124.66 451.22 5.78 205.75 885.35 0.70 1.06 52.85 1128.26 2.55 154.33 547.33 3.05 362.93 2115.86

 xxxx

0.25 0.40 1.32 3.45 0.48 1.87 3.74 0.61 2.48 6.64 0.23 0.36 1.40 3.04 0.47 1.82 3.52 0.56 2.32 4.71 0.26 0.42 1.46 3.48 0.50 2.05 4.08 0.65 2.75 6.43 0.70 1.34 81.05 579.17 3.05 102.98 323.82 7.46 183.99 799.23

0.22 0.34 1.27 3.10 0.43 1.64 3.37 0.52 2.08 4.59

B2PLYP

 av  xxxx ˛xx

˛iso BHHLYP

˛xx

 av ˛iso wb97XD

 av

(γ

TPC

xxxx

6 0

0.69 1.04 50.62 874.36 2.31 143.05 516.98 3.04 352.74 2242.88

1a 2a 3a 4a 2b 3b 4b 2c 3c 4c Molecules 6

xxxx

(γ

0.26 0.43 1.38 3.20 0.51 1.95 3.72 0.65 2.60 5.99

OPC

0.23 0.36 1.30 2.87 0.45 1.69 3.24 0.55 2.13 4.26

6

,10 a.u.)

 xxxx ˛iso

3 0 1a 2a 3a 4a 2b 3b 4b 2c 3c 4c Molecules

1a 2a 3a 4a 2b 3b 4b 2c 3c 4c

CAM-B3LYP

˛xx Molecule

Table 3 Longitudinal component of polarizability (˛xx , 102 a.u.), second-hyperpolarizability ( xxxx , 104 a.u.), average polarizability (˛iso , 102 a.u.) and second-hyperpolarizability ( av , 104 a.u.) obtained for molecules (Scheme 1) at different DFT functionals for the aug-pc-2 basis set.

P. Banerjee, P.K. Nandi / Chemical Physics Letters 637 (2015) 164–171

Figure 2. Plot of one photon and two photon part of  xxxx of molecules of Scheme 1 obtained at the CIS/6-311++G(d,p) level for about 50 lower lying singlet excited states.

170

P. Banerjee, P.K. Nandi / Chemical Physics Letters 637 (2015) 164–171

Table 4 TD-CAM-B3LYP/6-311++G(d,p) calculated results of transition energy (Egn , eV), transition moment (gn , a.u.), oscillator strength (fng , a.u.) and the major electronic transitiona obtained for molecules in Scheme 1. Molecule 1a 2a 3a 4a 2b 3b 4b 2c 3c 4c a

Egn

gn

fng

Major transition

S0 → Sn

7.562 8.212 3.111 1.462 7.357 2.382 1.633 3.138 1.641 1.158

0.589 1.468 1.338 1.804 1.608 1.626 1.746 0.854 1.510 2.342

0.064 0.434 0.136 0.116 0.466 0.154 0.122 0.056 0.092 0.155

H-1 → L + 4 H→L+9 H-4 → L H→L+2 H-2 → L H-2 → L H-1 → L + 2 H→L H-2 → L H→L

S0 → S8 S0 → S13 S0 → S12 S0 → S5 S0 → S19 S0 → S7 S0 → S6 S0 → S1 S0 → S3 S0 → S3

H and L refer to the HOMO (highest occupied MO) and LUMO (lowest unoccupied MO), respectively.

Since the ground state geometry of molecules Ca(NH2 )2 (2c) and Ca(NLi2 )2 (3c) optimized at the B3LYP, MP2 and CASSCF levels for the 6-311++G(d,p) basis set differ substantially it will be important to find whether it has any significant effect on the calculated second-hyperpolarizability. For this purpose, the NLO calculations at the CAM-B3LYP level have been carried out using the 6-311++G(3df,3pd) and aug-pc-2 basis sets taking the B3LYP geometry (bent structure), MP2 and CASSCF geometries (linear structures), and the calculated results are listed in Table 4s (Supplementary Section). The change of geometry has insignificant effect on  xxxx of molecule 2c for both basis sets. However,  xxxx of molecule 3c increases by about 39% (at the MP2 geometry) and 55% (at the CASSCF geometry) with respect to the bent structure (at the B3LYP geometry) for both basis sets. For molecule 4c,  xxxx (2548.40 × 104 a.u.) and  av (867.09 × 104 a.u.) increase by about 14% at the MP2 geometry with respect to the B3LYP optimized structure. To find the spectroscopic origin of second hyperpolarizability of the investigated molecules the standard sum-over-state (SOS) scheme of Orr–Ward–Bishop (OWB) [45,46] has been split into the two-level, three-level and four-level terms which can be related to one photon, two photon and three photon transitions, respectively [21],



xxxx = 24

 = 24

   gm (mn − x ımn )(np − x ınp )pg Emg Eng Epg

m= / gn = / gp = / g

  gm ¯ 2 mg mm 2

2

3 Emg

m= / g

  gm mn nm mg m= / gn = / m,g

2 E Emg ng

−

−

4mg

−

= 24

  gm ¯ 2 mg mm

m= / g

2

3 Emg

m= / gn = / m,g



m= / gn = / m,g

2gm 2ng 2 Eng Eng

+

2 Emg Eng

 

4mg

m= / gn= / m,g

3 Emg

(8)

+

  gm ¯ mm mp pg m= / gp= / m,g

2 Emg Eng

−

m= / gn= / m,g

2 Emg Epg



+

(9)

2gm 2ng 2 Eng Eng

Since for the chosen molecules, the ground state dipole moment is zero the dipole moment difference terms in Eqs. (8) and (9) do not contribute to the OPC and TPC of . Therefore, the one photon contribution (OPC) reduces to the second term which is negative while the two photon contribution arises from the third and fourth double summation terms. For molecules involving stronger coupling between the intermediate state and the final TPA state the third term of Eq. (9) should dominate. The evolution of  OPC ,  TPC and  SOS as a function of the number of excited states obtained for each molecule are shown in Fig. 3s (Supplementary Section). The OPC plots compared to TPC exhibit much faster convergence, in general. The variation of calculated OPC and TPC contributions among the chosen molecules are plotted in Figure 2. It is interesting to note that the pattern of variation SOS calculated OPC and TPC contributions is nearly identical to the  xxxx plot obtained by DFT different methods (Figure 1) and Fig. 1s and 2s (Supplementary Section).

 (6)

2 Emg Eng

+

  gm ¯ mm mp pg m= / gp = / m,g

2 E Emg pg

 gm mn np pg

m= / gn = / m,g p = / m,n,g



2 Emg Eng

  gm mn nm mg  

m= / gn = / g

+

 

−

m= / gn= / m,g

   gm mn  ¯ nn ng

3 Emg

= 24

   gm mn ¯ nn ng

  gm mg gn ng

where  ¯ mm is dipole moment difference between the excited state (|m) and ground state (|g), mg is the ground to excited state transition moment, mn refers to the transition moment integral between the excited states |m and |n and Emg is the energy difference between the excited and ground states. In the above equations, the x-component of the dipole moment and transition moments are used. These spectroscopic quantities for each molecule have been calculated at the CIS/6-311++G(d,p) level by considering by about 50 lower lying singlet excited states and the B3LYP optimized geometry. In the present work, the one photon contribution (OPC) corresponding to one photon absorption (OPA) and the two photon contribution (TPC) corresponding to the two photon absorption (TPA) have been calculated by using the following expressions, respectively. 2L xxxx

3L xxxx



+ (7)

Emg Eng Epg

The two photon contribution is substantially larger compared to the one photon contribution. However, for molecule 4c the magnitude of OPC is substantially larger compared to that of molecule 4a although their TPC contributions are rather comparable resulting in an incorrect order of  SOS . 3.3. Charge transfer transition The time dependent DFT calculated results of spectroscopic parameters such as transition energy (Egn ), transition dipole moment (gn ), oscillator strength (fng ) along with the nature of orbital transitions obtained at the CAM-B3LYP/6-311++G(d,p) level are listed in Table 4. The significantly larger magnitude of second-hyperpolarizability obtained for complexes Be(NNa2 )2 (4a), Mg(NNa2 )2 (4b) and Ca(NNa2 )2 (4c) compared to the analogous metal complexes M(NA2 )2 (M = Be/Mg/Ca and A = H/Li) may be

P. Banerjee, P.K. Nandi / Chemical Physics Letters 637 (2015) 164–171

attributed to the larger transition moment and appreciably smaller transition energy. This qualitative agreement is consistent with the two-state model of second-hyperpolarizability. Of the two metal complexes 4a and 4c the rather larger magnitude of  obtained for molecule 4c may be accounted for the largest one photon contribution  OPC since the value of  TPC of molecules 4a and 4c does not differ significantly (see Figure 2). The appreciably smaller E and larger gn of molecule 4c compared to that of 4a reasonably accounts for the larger  OPC of the former. Further the nature of orbital transition obtained for molecules 4a, 4b and 4c are compared in Fig. 4s (Supplementary Section). In each case electron transfer takes place from the nitrogen p-orbital to the vacant orbital ( type) of alkaline earth metal atom but the greater amount of charge accumulates on the Ca atom in molecule 4c. The stronger enhancement of second-hyperpolarizability 4a (versus 3a) and 4b (versus 3b) may arise from the greater extent of charge transfer from the nitrogen p-orbital to the vacant diffuse -type MO of Be/Mg compared to the weaker transitions from the ␴ MOs of 3a and 3b. This different kind of electronic transition is fairly consistent with the calculated spectroscopic quantities (Table 4). 4. Conclusions The present theoretical study demonstrates that the diamine metal complexes formed by the interaction of alkali and alkaline earth metals with hydrazine molecule are sufficiently stable. These new molecular species showed notably large secondhyperpolarizability especially when the size of chosen metal atoms increases. The enhancement of NLO response is rather significant when alkali metal atoms replace the H atoms of hydrazine. The results of NLO property obtained by different DFT functional and basis sets exhibit a consistent trend. The two photon contribution has been found significant and follows identical pattern of variation as obtained in the DFT calculated results of secondhyperpolarizability. The variation of cubic-polarizability has been explained qualitatively in terms of the TD-CAMB3LYP calculated transition energy and transition moment corresponding to the most crucial electronic transition. Acknowledgements The authors are grateful for the valuable comments of the Reviewers. (PKN) acknowledges the grant from UGC, Government of India under the Major Research Project (F. No. 42-339/2013 (SR) for carrying out this research work. The authors are grateful to Professor Sudip Kumar Chattopadhyay for helpful discussions. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.cplett.2015.08.008

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