static second hyperpolarizability of lamda shapped alkaline earth metal complexes

Static second hyperpolarizability of ¤ shaped

alkaline earth metal complexes

Kaushik Hatua and Prasanta K. Nandi*

Department of ChemistryIndian Institute of Engineering Science and Technology

Shibpur, Howrah 711 103, India*[email protected]

Received 21 April 2014

Accepted 11 June 2014Published

A number of � shaped complexes of alkaline earth metals Be, Mg and Ca with varying terminal

groups have been considered for the theoretical study of their second hyperpolarizability.

The chosen complexes are found to be su±ciently stable and for a chosen ligand the stabilitydecreases in the order: Be-complex > Ca-complex > Mg-complex. The calculated results of

second hyperpolarizability obtained at di®erent DFT functionals for the 6-311þþG(d,p) basis

set are found to be fairly consistent. The � shaped ligands upon complex formation with metals

lead to strong enhancement of second hyperpolarizability. The highest magnitude of cubicpolarizability has been predicted for the metal complex having >C(C2H5)2 group. For a chosen

ligand, the magnitude of second hyperpolarizability increases in the order Be-complex <

Mg-complex < Ca-complex which is the order of increasing size and electropositive character ofthe metal. The variation of second hyperpolarizability among the investigated metal complexes

has been explained in terms of the transition energy and transition moment associated with the

most intense electronic transition.

Keywords: � shaped complexes; ground state structure and stability; second hyperpolariz-

ability; two state model.

1. Introduction

Increasing demand for electro-optic materials in photonic application, optical data

transmission, data processing leads to tremendous growth of research interest in the

¯eld of nonlinear optics.1–4 There have been several attempts to optimize the NLO

response of materials ranging from the isolated gas phase molecules to the bulk

phase. Most of the theoretical investigations have been carried out by considering

isolated molecules for the purpose of designing new kind of NLO phores having

de¯nite structure-property correlation. In the very early days, investigation on

nonlinear optical materials was mostly con¯ned to various kinds of donor–acceptor

*Corresponding author.

June 26, 2014 5:01:16pm WSPC/178-JTCC 1450039 ISSN: 0219-63361stReading

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Journal of Theoretical and Computational Chemistry

Vol. 13, No. 4 (2014) 1450039 (12 pages)

#.c World Scienti¯c Publishing CompanyDOI: 10.1142/S0219633614500394

1450039-1

http://dx.doi.org/10.1142/S0219633614500394

substituted � conjugated systems5–14 for optimizing NLO responses. During the

last few years, with the aid of modern quantum chemical approach, other kind of

novel materials have emerged which have interesting electronic structure and

superior electric response property. Amongst these organometallic complexes

where suitable manipulation of electronic environment by metal atom may result

in larger NLO responses that deserve special attention. Di®use electron sys-

tem,15,16 Li–Calixpyrrole,17 highly deformed halofullernes,18,19 nitrogen edge-

doped of single walled CNT,20 lithiated silicon cage,21 alkali–alkaline earth metal

cluster and alkalides,22 mixed �-conjugated bridge in carbon nanotube-based

X

Z

Y

>CH2 >CMe2

MeN< >CO

>SO2

Fig. 1. Complexes of alkaline earth metals (Be, Mg and Ca) with � shaped ligands.


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K. Hatua & P. K. Nandi

1450039-2

system,23 alkali metal atom-aromatic ring,24 alkaline earth-based alkalides,25 �

and W shaped sandwich metallocarborane-containing chromophores26 are impor-

tant examples to be worth mentioned in the context of modern approach for NLO

enhancement. Interestingly most of these studies involve the enhancement and

optimization of the ¯rst hyperpolarizability (�) while the second hyperpolariz-

ability (�) has been less explored. In our previous work, we intended to optimize

the second hyperpolarizability by considering small Be based organometallic27–30

compounds. Molecular design of NLO phores becomes a state of art today. It was

noted from our previous study27 that acceptor hydrocarbon complex of beryllium

metal has more ground state charge polarization compared to that of the ligand

itself. The second hyperpolarizability has been found to be considerably enhanced

when charge transfer from metal to ligand is delocalized rather than accumulated

in the vicinity of the metal. When Be atom has been replaced by the more elec-

tropositive metal like Mg and Ca, the larger extent of ground state polarization

occurs which in turn results larger magnitude of the longitudinal component of

second hyperpolarizability of Mg and Ca complexes. With this idea in mind, we

have designed few �-shaped molecular complexes of alkaline earth metals (Be, Mg

and Ca). The molecular structure consists of di®erent donor/acceptor group X

(>CH2, >C(CH3)2 (>CMe2), >C(C2H5)2 (>CEt2), >NMe2, >CO and >SO2)

which are covalently bonded with two adjacent –C�C–H units (Fig. 1). The larger

component of second hyperpolarizability is associated with the larger ground state

polarization which leads to strong electronic coupling between the ground and the

low lying excited state. We have rationalized the variation of NLO property by

the calculated NPA charges and the excited state electronic parameters obtained

by using the time dependent density functional theory (TDDFT) formalism.

2. Computational Methods

Geometry of the chosen � shaped ligands and complexes have been fully optimized

using the B3LYP31 functional. The basis set employed consists of triple zeta quality

(GTO) with two sets of polarization functions augmented by di®use functions on

heavy atom as well as over hydrogen atom i.e. 6-311þþG(d,p). Vibrational analysis

under harmonic approximation has been carried out at the same level of theory

which con¯rmed the optimized geometry as a stationary point in the respective

potential energy hypersurface. The atomic charges have been calculated by using

the natural population analysis (NPA) scheme at the B3LYP/6-311þþG(d,p)

level. Interaction energy (�E) of the complexes has been calculated by subtracting

the electronic energy of the ligand and metal from that of the complex ð�E ¼Ecomplex � Eligand � EmetalÞ. The basis set superposition error (BSSE) has been

eliminated by employing the counterpoise method. Also the zero point energy (ZPE)

correction has been incorporated in the calculation of interaction energy.

During the interaction of matter with the static radiation ¯eld (F0), an induce

dipole moment is generated which can be expanded in Taylor series. The


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Static second hyperpolarizability of � shaped alkaline earth metal complexes

1450039-3

hyperpolarizabilities have been calculated in the static limit F0.

� ¼ �0 þ �F0 þ1

2!�F 2

0 þ 1

3!�F 3

0 � � � ð1Þ

�0 is the permanent dipole moment and the coe±cients �, � and � are called

polarizability, ¯rst hyperpolarizability and second hyperpolarizability, respectively.

The average second hyperpolarizability has been calculated by using the following

expression.

�av ¼1

5½�xxxx þ �yyyy þ �zzzz þ 2ð�xxyy þ �yyzz þ �xxzzÞ�: ð2Þ

The Cartesian components of � have been calculated by the numerical di®erentiation

of the analytically evaluated ¯rst-hyperpolarizability obtained at di®erent DFT

functionals: B3LYP, BHHLYP32 and long range corrected CAM-B3LYP.33 Recently

developed double hybrid functional B2PLYP34 which has MP2 like electron corre-

lation has also been used for the sake of comparison with the conventional DFT

functionals.

In our earlier investigations,27–29 it was noted that the 6-311þþG(d,p) basis set

can reproduce the result obtained with the aug-cc-pVXZ (X¼D,T) basis set with

reasonable computational cost. Furthermore, the double augmentation by another

set of di®use function i.e. d-aug-cc-pVDZ does not enhance the longitudinal com-

ponent of second hyperpolarizability appreciably.30 Thus the basis set dependence of

second hyperpolarizability is expected to be insigni¯cant and has not been considered

in the present study. All calculations have been carried out by using the G09

quantum chemistry program package.35

3. Results and Discussion

3.1. Ground state equilibrium structure

Important equilibrium structural parameters, natural atomic charges and the metal

ligand interaction energy have been presented in Table 1. An inspection of Table 1

reveals that C�C bond distance in ligands is nearly constant (� 1.20�A) but in

Table 1. B3LYP/6-311þþG(d,p) calculated equilibrium C�C triple bond length (rCC , �A) in the ligand and in the

complex, the nearest metal–carbon bond distance (rC-M , �A), natural atomic charges (q, a.u.) and the metal ligand

interaction energy (�E, Kcal/mol).

rCC rC-M q �E

�X Ligand Be Mg Ca Be Mg Ca Be Mg Ca Be Mg Ca

>CH2 1.201 1.298 1.255 1.264 1.647 2.254 2.462 0.842 0.671 0.910 �84.194 �36.114 �43.822

>CMe2 1.202 1.298 1.256 1.267 1.646 2.236 2.437 0.832 0.695 0.966 �123.050 �73.734 �82.220

>CEt2 1.202 1.298 1.256 1.266 1.645 2.246 2.446 0.817 0.667 0.927 �158.139 �108.855 �117.002

>NMe 1.203 1.315 1.274 1.284 1.649 2.175 2.406 0.898 0.840 1.073 �103.268 �52.283 �60.927

>CO 1.203 1.328 1.337 1.347 1.682 2.100 2.332 0.889 1.407 1.670 �38.343 �18.558 �48.316

>SO2 1.202 1.291 1.271 1.281 1.663 2.152 2.336 0.848 1.063 1.516 �90.089 �41.390 �63.321


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1450039-4

complexes, it varies signi¯cantly which may be due to the di®erent extent of charge

transfer (CT) interaction. From Table 2, it can be seen that with the exception of

>SO2, the <CXC angle of ligands decreases signi¯cantly upon complex formation

with metals. The amount of CT from a chosen metal depends on the nature of the

end group of the ligand. Electron transfer from a metal is smaller with electron

donating group. On the other hand, the extent of CT enhances when electron

withdrawing groups (>CO and >SO2) are present. Owing to the greater electro-

positive character Mg and Ca prefer to form ionic complexes. The greatest amount

of charge transfer from metal occurs when the ligand contains the >CO group

which is re°ected in the signi¯cant polarization (elongation) of C�C bond length

the extent of which increases with the amount of CT varying in the order

Be < Mg < Ca. As can be seen from Table 2 that this exceptionally larger extent of

charge transfer from alkaline earth metals is also accompanied by substantial re-

duction of the <CXC angle. It should be noted that C–Be distance remains almost

same (� 1.65�A) except for the complex with >CO group. The possible reason

behind the smallest interaction energy obtained for the chosen metal complexes

having >CO group may be rather signi¯cant increase of angular strain due to the

shortening of <CXC angle with respect to the free ligand. The metal–carbon dis-

tance increases with increasing size of the metal atom. The calculated interaction

energy has been found to be negative which suggest that the metal–ligand CT

interaction stabilizes the complexes with respect to free metal and ligand. The

order of stability of Be and Mg complexes decreases in the order >CEt2, >CMe2,

>NMe, >SO2, >CH2 and >CO. This order slightly di®ers from that obtained for

Ca complexes.

The relative order of stability of metal complexes with a particular end

group (>X) can be qualitatively understood from the second-order stabilization

energy (see Table 1A in the Supplementary Material section) associated with

metal–ligand CT interactions. The Be metal forms stronger covalent interactions

with \C�C" moieties while Mg and Ca uses valence s and p orbitals in the

weaker � interaction with acetylene moieties. This accounts for substantially

Table 2. B3LYP/6-311þþG(d,p) calculated

angles (deg.).

<CXC

>X Ligand Be Mg Ca

>CH2 113.82 105.54 107.27 110.26>CMe2 109.52 101.98 104.09 107.18

>CEt2 108.69 100.03 102.93 105.27

>NMe 120.41 111.78 115.29 117.96

>CO 115.14 70.36 63.55 65.40>SO2 100.81 101.98 103.78 114.81


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1450039-5

higher stabilization energy of beryllium complexes compared to the magnesium

and calcium complexes.

3.2. Second hyperpolarizability and TDDFT analysis

The dominant component of the ground state dipole moment of the chosen ligands

and metal complexes lies along the Z axis. The calculated results of all the axial

components of � along with the mean second hyperpolarizability obtained at the

B3LYP/6-311þþG(d,p) level are presented in Table 3. The results obtained with

other DFT functional are reported in Table 2A (Supplementary Materials sec-

tion). The z-component of second hyperpolarizability (�zzzz) is found to be the

major axial component. It is evident that �zzzz value of Mg and Ca complexes has

been predicted larger than that of the ligands by an order of magnitude. This can

be attributed to the enhanced CT interaction taking place in these metal com-

plexes. For a chosen ligand, the magnitude of second hyperpolarizability of the

complexes increases in the order Be < Mg < Ca. This trend follows a nice corre-

lation with the <CXC angle of the complexes which increases gradually in the

order Be < Mg < Ca and approaches the angle of free ligands. It can be noted that

the B3LYP and B2PLYP calculated �zzzz values obtained for Be-complexes are

found to lie within a narrow margin. Likewise, the BHHLYP and CAM-B3LYP

calculated result of �zzzz di®ers within a close margin. However, in the case of Mg

and Ca complexes B3LYP and BHHLYP calculated �zzzz values obtained for the

¯rst three ligands lies within a small range while the BHHLYP and CAM-B3LYP

values of Mg-complexes having >CO and >SO2 groups show an excellent agree-

ment. For a chosen alkaline earth metal, the maximum value of �zzzz has been

predicted for complexes having >CEt2 group. However, the magnitude of �zzzz and

�av obtained for complexes having >CEt2 and >CMe2 groups di®ers by a narrow

margin. In the case of Be-complexes, the �zzzz values obtained with >CMe2 and

>NMe groups are almost identical. Longitudinal component of second hyperpo-

larizability obtained for ligands with di®erent >X varies in the order >CEt2>>

CMe2>>NMe>>CH2>> SO2>>CO for magnesium and calcium complexes, and

in the order >CEt2>NMe �>CMe2>>CH2>>SO2>>CO for beryllium com-

plexes. It should be noted that at the B3LYP level, the �zzzz value of complexes

with >CMe2 ð>CEt2Þ group has been predicted larger by 0.22% (7.12%) for Be,

26.13% (28.03%) for Mg and 28.37% (32.80%) for Ca than the corresponding

complexes with >NMe group. It is interesting to note that replacement of two H

atoms of >CH2 group with two methyl (ethyl) groups enhances the magnitude of

�zzzz by 64.38% (75.68%) in Be, 51.92% (59.94%) in Mg, and 60.80% (67.28%) in

Ca at the B3LYP level. On the other hand, complexes containing electron with-

drawing groups >CO and >SO2 involve greater amount of charge transfer from

the metal atom (Table 1) but have smaller values of second hyperpolarizability

which may be due to the higher angular strain arising from the substantial

shortening and widening of <CXC angle, respectively.


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1450039-6

Tab

le3.

Axialcom

pon

entsof

secondhyperpolarizab

ility(�

iiii,104a.u.)an

dav

erag

esecondhyperpolarizab

ility(�

av,104a.u.)of

thechosen

metal

complexes

obtained

attheB3L

YP/6-311þþ

G(d,p)level.

�zzzz

�yyyy

�xxxx

�av

>X

Ligan

dBe

Mg

Ca

Ligan

dBe

Mg

Ca

Ligan

dBe

Mg

Ca

Ligan

dBe

Mg

Ca

>CH

22.06

58.99

635

.498

77.781

1.10

13.07

5.30

822

.253

0.92

33.40

520

.521

92.796

1.33

35.23

920

.974

64.420

>CMe 2

2.56

414

.788

53.930

125.779

2.03

44.41

06.25

622

.451

2.35

74.19

720

.076

86.337

2.37

47.65

127

.797

84.058

>CEt 2

2.95

515

.805

56.774

130.115

2.59

41.57

813

.646

20.615

2.34

02.94

84.98

441

.808

2.76

57.37

028

.570

87.091

>NMe

4.10

914

.755

44.343

97.979

2.23

13.37

94.64

214

.960

2.25

83.90

113

.555

56.964

2.97

07.10

320

.774

60.622

>CO

1.60

73.17

011

.161

20.766

1.33

62.66

23.10

05.04

50.96

21.23

52.69

54.15

20.67

12.06

15.21

99.47

9

>SO

22.49

82.67

917

.419

70.928

0.95

01.61

21.65

63.43

20.69

21.10

44.70

96.90

31.29

11.75

37.36

223

.271


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1450039-7

In order to explain the relative variation of second hyperpolarizability, the

following two state model (TSM) has been invoked

� 2Lzzzz �

� 2gn��2

gn

�E 3gn

� �4gn

�E 3gn

� �; ð3Þ

where �Egn, �gn and ��gn are the transition energy, transition dipole moment and

dipole moment di®erence between the ground \g" and the crucial excited state \n".

It should be noted that for the crucial excited state \n", �zzzz depends strongly on the

magnitude of �Egn, �gn and ��gn terms. For a nonpolar or weakly polar molecule,

the ¯rst term (dipolar contribution) is generally insigni¯cant due to very small

magnitude of ��gn but the second term may be signi¯cant. Hence, considering only

the magnitude of the nondipolar contribution, the above equation can be further

approximated to the following expression.

� 2Lzzzz �

�4gn

�E 3gn

��: ð4Þ

Here, it is worthwhile to mention that Eq. (3) is the simpli¯ed SOS expression of

the longitudinal component of second hyperpolarizability and contains the spectro-

scopic properties of the crucial state corresponding to the most intense linear tran-

sition. However, the exact SOS expression of � also involves two and three photon

electronic transitions encompassing many higher lying excited states. Thus the neg-

ative sign of second hyperpolarizability cannot be ascertained even the second term of

Eq. (3) is larger. Nevertheless, the later term may be useful in the evolution of second

hyperpolarizability. In our previous study,29 it has been noted that the overall sign of

SOS � obtained by considering about 150 excited states is positive even though the

one photon contribution (OPC) is largely negative. Also if the sign is discarded, the

larger magnitude of OPC corresponds to the larger magnitude of �. The spectroscopic

parameters in Eq. (4) has been calculated by employing the time dependent DFT

(TDDFT) method for the B3LYP functional and 6-311þþG(d,p) basis set for 50

lowest lying singlet excited states. The most dominant electronic transition has been

determined by the larger magnitude of �gn associated with the signi¯cant oscillator

strength which are given in Table 4. The comparable but larger values of �zzzzobtained for Be complexes having >CMe2, >CEt2 and >NMe groups may arise from

the relatively smaller transition energy and higher transition moment. In the case of

Mg and Ca complexes, the highest value of second hyperpolarizability obtained with

>CEt2 group may be explained for notably smaller transition energy and higher

transition moment. Also for a chosen ligand, the increasing trend of second hyper-

polarizability with increasing electropositive character of metal primarily originates

from the decreasing transition energy gap.

The electronic transitions in metal complexes having >CMe2 group are displayed

in Fig. 2. As can be seen from the orbital picture, overlapping interaction is signif-

icant in both HOMO and LUMOþ3 of Be complex. In the case of Mg complex, the


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1450039-8

orbital overlap in HOMO is similar to that of Be complex but the overlap of metal

orbitals in LUMO is rather poor resulting in spreading of electron density which

accounts for the notably larger transition moment. For the more electropositive

alkaline earth metal Ca, the charge distribution in both transition orbitals are

comparable showing no orbital interaction between the metal atom and the ligand

which is consistent with the electrostatic interaction. Owing to the greater electron

withdrawing power of the polar functional groups >CO and >SO2, the corre-

sponding metal complexes have highly polar ground state which may result poor

orbital overlap with the metal atom (having higher positive charge) (Table 1) and

result in larger transition energy gap which in turn substantially lowers the

Mg-CMe2 Ca-CMe2Be-CMe2

Fig. 2. Major electronic transition of complexes of Be, Mg and Ca containing the >CMe2 group.

Table 4. Transition energy (�Egn, eV), transition moment (�gn, a.u.), oscillator strength (f0, a.u.) and the major

transitiona involved in the crucial excitation for the chosen complexes obtained at TD-B3LYP/6-311þþG(d,p) level.

Be-complexes Mg-complexes Ca-complexes

X �Egn �gn f0

Major

transition �Egn �gn f0

Major

transition �Egn �gn f0

Major

transition

>CH2 4.499 1.051 0.121 H ! Lþ 3 3.431 1.875 0.296 H ! Lþ 1 1.673 1.976 0.160 H ! L

>CMe2 4.464 1.214 0.161 H ! Lþ 3 3.043 2.527 0.476 H ! L 1.587 2.136 0.177 H ! L

>CEt2 4.416 0.921 0.092 H ! Lþ 3 3.028 2.541 0.479 H ! L 1.581 2.177 0.183 H ! L

>NMe 4.385 1.352 0.196 H ! Lþ 1 3.085 2.169 0.355 H ! L 1.828 2.269 0.230 H ! L

>CO 5.827 1.148 0.188 H� 4 ! L 3.929 1.502 0.217 H� 1 ! Lþ 2 3.042 0.853 0.054 H� 1 ! Lþ 2

>SO2 6.051 0.946 0.133 H ! Lþ 3 3.328 2.063 0.347 H ! L 1.922 0.819 0.031 H ! L

aH and L stand for highest occupied MO and lowest unoccupied MO, respectively.


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1450039-9

magnitude of cubic polarizability. Thus it emerges that apart from the ground state

polarization, the energy gap between the ground and crucial excited states plays a

key role in the modulation of second hyperpolarizability of the chosen � shaped

molecular complexes.

4. Conclusion

In the present investigation, a number of complexes between the alkaline earth

metals Be, Mg and Ca and � shaped ligands with varying functional groups have

been considered for the theoretical study of the ground state structure and second

hyperpolarizability by employing di®erent DFT functionals and 6-311þþG(d,p)

basis set. The � shaped complexes are found to be thermally stable. For a particular

choice of ligand having a functional group X, the order of stability of complexes with

metal varies in the order Be > Ca > Mg except for the >CO group. For a chosen

metal, the maximum stability has been found for the ligand having>C(C2H5)2 group

and also for each metal, the maximum value of second hyperpolarizability is also

predicted for the same ligand. One of the most interesting ¯nding of the present work

is the preference of bulky functional group rather than the polar electron with-

drawing group while optimizing the second hyperpolarizability. The gradual opening

of <CXC angle on substitution with larger alkaline earth metals plays an important

role in enhancing the CT interaction and second hyperpolarizability of the chosen �

shaped complexes. The variation of NLO property of the chosen � shaped metal

complexes have been explained satisfactorily in terms of the transition energy and

transition moment corresponding to the most intense electronic transition.

Acknowledgments

(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 the valuable comments of the Reviewer to improve the quality

of the manuscript.

References

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static second hyperpolarizability of lamda shapped alkaline earth metal complexes

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