Electronic and Vibrational Hyperpolarizabilities of Lithium Substituted

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A: Spectroscopy, Molecular Structure, and Quantum Chemistry

Electronic and Vibrational Hyperpolarizabilities of Lithium Substituted (Aza-)benzenes and (Aza-)naphthalenes Suélio Marques, Marcos A. Castro, Salviano Araujo Leão, and Tertius Lima Fonseca J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b05612 • Publication Date (Web): 28 Aug 2018 Downloaded from http://pubs.acs.org on September 1, 2018

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The Journal of Physical Chemistry

Electronic and Vibrational Hyperpolarizabilities of Lithium Substituted (Aza-)benzenes and (Aza-)naphthalenes Suélio Marques,†,‡ Marcos A. Castro,∗,† Salviano A. Leão,† and Tertius L. Fonseca† Instituto de Física, Universidade Federal de Goiás, Campus Samambaia, 74690-900 Goiânia, GO, Brazil, and Instituto Federal Goiano, Campus Iporá, Av. Oeste 350, Santa Catarina, 76200-000 Iporá, GO, Brazil E-mail: [email protected]

∗

To whom correspondence should be addressed Instituto de Física, Universidade Federal de Goiás, Campus Samambaia, 74690-900 Goiânia, GO, Brazil ‡ Instituto Federal Goiano, Campus Iporá, Av. Oeste 350, Santa Catarina, 76200-000 Iporá, GO, Brazil †

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Abstract In this article we report results for the electronic and vibrational hyperpolarizabilities of ten molecules: Li@benzene, Li@pyridine, Li@pyrimidine and Li@pyrazine; Li@naphthalene, Li@quinoline, Li@isoquinoline, Li@cinnolin, Li@quinazoline and Li@quinoxaline. An electron correlation study shows that second-order many-body perturbation theory and density functional theory with CAM-B3LYP and M05-2X functionals give results to the electronic hyperpolarizabilities in good agreement with coupled cluster with single and doubles reference values. Static and dynamic vibrational corrections were computed through the perturbation theoretical method of Bishop and Kirtman and using a variational approach. In general are obtained notable discrepancies between the results obtained by the two methods for the pure vibrational corrections because of the deficiency of the perturbation method to properly treat low frequency normal modes present in the investigated systems. On the other hand, both methods give similar results to the zero point vibrational average corrections.

Introduction In recent decades there has been a growing demand for materials with large nonlinear optical responses because of their potential applications. 1–3 Among the most studied materials, those based on π−conjugated organic molecules have emerged as a very promising class, mainly because many of these molecules exhibit large optical nonlinearities. 4–8 Furthermore, organic systems customarily present other attractive features as low cost, ease of handling and small refractive indexes. In the case of molecule based materials the macroscopic nonlinear optical properties originate from the molecular hyperpolarizabilities. An usually effective proceeding to get large hyperpolarizabilities in π−conjugated organic molecules is to attach electron donor and acceptor groups on strategic sites of the molecule, being the charge displacement from the donor to the acceptor through π−conjugated backbone, under influence of the electric field, crucial for enhancing these properties. 9–12 In the class of the conjugated organic 2


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molecules, aromatic hydrocarbons and their derivatives have been exhaustively investigated as nonlinear optical materials. Hyperpolarizabilities of molecules containing benzene, naphthalene and their aza analogues (pyridine, pyrazine, quinoline, quinoxaline, etc.) have been experimentally and theoretically studied. 13–27 Heteroaromatic molecules containing two nitrogen atoms, as pyrazine and quinoxaline, can play the electron acceptor role as result of the electron withdrawing effectiveness of the two unsaturated nitrogens. 28–30 Coordination compounds obtained from combinations of organic molecules with metal complexes have been studied as a promising class of materials with nonlinear optical properties to be explored. 31,32 A strategy to enhance hyperpolarizabilities of organic molecules with increasing acceptance is the substitution/doping by alkali metal atoms. Since early studies based on semiempirical methods 33,34 suggested that the replacement of hydrogen by lithium atoms in organic molecules could lead to extremely large second hyperpolarizabilities, this issue has been subject of several investigations. Correlated ab initio calculations reported by Raptis et al. 35 showed that the electronic second hyperpolarizability of hexalithiobenzene is around three orders of magnitude larger than the second hyperpolarizability of benzene. The effect of alkali atom doping on the longitudinal second hyperpolarizability of polyacetylene chains was investigated by Champagne et al. 36 Studies on the first hyperpolarizability of supershort single-walled nanotubes showed that this property is notably enhanced by lithiation effects. 37–39 The effects of successive lithium substitutions on the second hyperpolarizability of acetylene, ethylene and benzene was recently reported by Mondal et al. 40 Excess electron systems obtained through doping of organic molecules by alkali metal atoms have been recurrently investigated because of their large optical nonlinearities. 41–46 First hyperpolarizabilities of Li@n-acenes salts (Li@naphthalene, Li@anthracene and Li@tetracene) were calculated by Zhang et al. 47 through ab initio and density functional theory methods. These molecules were obtained from the n-acenes replacing one hydrogen atom by lithium. The influence of the lithium atom position on the first hyperpolarizability was investigated. It was achieved that this property is decreasing with n growing

3



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for the α-Li@n-acenes and increasing for the β-Li@n-acenes, being somewhat larger for the β-Li@n-acene when n is the same. Nonlinear optical responses of the Na@n-acenes and K@nacenes molecules were studied by Xiao and Lu. 48 Although electronic hyperpolarizabilities are much more studied than the corresponding vibrational hyperpolarizabilities, calculations of vibrational contributions to nonlinear optical properties have been reported for several π-conjugated organic compounds. 35,36,49–53 Furthermore the results show that these contributions are in general important. In this article we report an investigation on the vibrational corrections to the first hyperpolarizabilities of the Li@benzene and Li@naphthalene molecules, in addition to eight molecules where one or two C-H fragments are replaced by nitrogen atoms (Li@aza-benzenes and Li@aza-naphthalenes). We limit our study to the isomers in which the lithium atom is placed in the position for which the first hyperpolarizability is larger, which correspond, for example, to β-Li@benzene and β-Li@naphthalene. 47 The vibrational corrections were computed through the perturbation theoretical method of Bishop and Kirtman 54–56 and using a variational approach proposed in previous works. 57,58 Since the molecules studied here present low frequency vibrational modes, that are difficult to manipulate by perturbation methods, a comprehensive study involving a number of molecules may shed light on limitations of the methods for calculating vibrational corrections to hyperpolarizabilities.

Theory We have adopted the conventional partition of the first hyperpolarizability in electronic contribution (el), zero-point vibrational average (zpva) and pure vibrational (pv) corrections, 59 that is zpva pv el + βijk + βijk . βijk = βijk

4


(1)

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The zpva correction is defined as

zpva eq βijk = h0|βijk |0i − βijk ,

(2)

el where |0i represents the vibrational ground state and βijk is the electronic value of βijk

calculated on the equilibrium geometry. The pv correction is written as 59,60

pv βijk (−ωσ ; ω1 , ω2 ) = [µα] + [µ3 ],

(3)

where [µα] =

X X v h0|µi |vihv|αjk |0i P

2v − ωσ2

v6=0

(4)

and [µ3 ] =

X X h0|µi |vihv|¯ µk |uihu|µj |0i (v − ωσ )(u − ω1 ) P v,u6=0

(5)

In this equations: |vi denotes a excited vibrational state and v its energy relative to the P ground state energy; hv|¯ µk |ui = hv|µk |ui − h0|µk |0iδvu ; P indicates a summation over all distinct terms obtained by permuting the pairs (−ωσ , i), (ω1 , j) and (ω2 , k). In order to calculate the terms [µα] and [µ3 ], we employed the variational methodology (VAR) reported in previous articles. 57,58 The vibrational wave functions |vi were written as combinations of basis functions that are product of single-mode harmonic-oscillator functions, denominated here as |mh i. Thus

|vi =

X

cvm |mh i,

(6)

m

so that hv|µi |ui =

X

cvm cun hmh |µi |nh i,

(7)

m,n

with similar expressions for αij and βijk . In this study we have considered harmonic vibrational wave functions in which the sum of the single-mode vibrational quantum numbers is 5



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no more than 3. The coefficients cvm and the vibrational energies v were calculated by diagonalizing the vibrational Hamiltonian

H=−

1 X ∂2 + V (Q1 , Q2 , · · · ) 2 a ∂Q2a

(8)

in the basis of the harmonic functions |mh i. All vibrational states resulting from the diagonalization were taken into account in the calculations of the terms [µα] and [µ3 ]. The vibrational potential energy was expanded up to third order in the normal coordinates as V (Q1 , Q2 , · · · ) = V eq +

∂ 3V 1X 2 2 1X ωa Qa + Qa Qb Qc , 2 a 6 a,b,c ∂Qa ∂Qb ∂Qc

(9)

where the first two terms are parts of the harmonic vibrational Hamiltonian and the last one is the anharmonicity. The µi component of the dipole moment that appear in the Eq. (7) was expanded up to second order as µi =

µeq i

X ∂µi 1 X ∂ 2 µi Qa + Qa Qb , + ∂Qa 2 a,b ∂Qa ∂Qb a

(10)

X ∂µi 1 X ∂ 2 µi hmh |Qa |nh i + hmh |Qa Qb |nh i ∂Qa 2 a,b ∂Qa ∂Qb a

(11)

so that

hmh |µi |nh i = µeq i δmn +

Since the formulas to calculate hmh |Qa |nh i and hmh |Qa Qb |nh i are well known, 61,62 the matrix elements hmh |µi |nh i (as well as those of αij and βijk ) can be obtained from the computation of the derivatives of the energy and electrical properties in relation to the normal coordinates. These derivatives were numerically computed as in previous papers. 63,64 Another approach employed here to compute vibrational corrections is the perturbation theoretical method of Bishop and Kirtman (BKPT). 54–56 When this methodology is carried

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pv is given by out up to first general (electrical + mechanical) order of perturbation, βijk

pv βijk (−ωσ ; ω1 , ω2 ) = [µα]0,0 + [µ3 ]1,0 + [µ3 ]0,1 ,

(12)

where 1 XX 1 ∂µi ∂αjk , 2 2 2 P a ωa − ωσ ∂Qa ∂Qa

(13)

1 XX 1 ∂µi ∂ 2 µj ∂µk 2 P a,b (ωa2 − ωσ2 )(ωb2 − ω22 ) ∂Qa ∂Qa ∂Qb ∂Qb

(14)

[µα]0,0 =

[µ3 ]1,0 = and

3 0,1

[µ ]

∂ 3V 1 XX ∂µi ∂µj ∂µk 1 =− . 2 2 2 2 2 2 6 P a,b,c (ωa − ωσ )(ωb − ω1 )(ωc − ω2 ) ∂Qa ∂Qb ∂Qc ∂Qa ∂Qb ∂Qc

(15)

The harmonic vibrational wave functions included here in the framework of the VAR methodology are practically the same necessary to obtain these BKPT formulas. 54 The zpva corrections were calculated by the VAR scheme computing the h0|βijk |0i matrix elements through Eqs. (7) and (11). In this case, the zpva correction can be divided in a term arising from the first order derivatives and another from the second order derivatives. We call these contributions as [βijk ]0 and [βijk ]00 , respectively. According with the BKPT method, the zpva correction is given by 59

zpva βijk = [βijk ]1,0 + [βijk ]0,1 ,

(16)

where [βijk ]1,0 =

1 X 1 ∂ 2 βijk 4 a ωa ∂Q2a

(17)

and [βijk ]0,1 = −

1X 1 ∂ 3 V ∂βijk . 4 a,b ωa2 ωb ∂Qa ∂Q2b ∂Qa

(18)

In the case of the two-ring molecules (exception of Li@naphthalene) we computed the pv 7



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corrections for β through the field-induced coordinates (FIC) methodology. 65 The idea is to calculate the vibrational corrections from the Eqs. (13), (14) and (15) with the 3N-6 normal coordinates replaced by 3 first-order field-induced coordinates, defined as

χi1 = −

X 1 ∂µi Qa , 2 ∂Q ω a a a

(19)

where i stands for x, y and z and the summation is over the normal modes. This approach provide the same static results (and in the infinite frequency limit) with appreciable computational savings in the calculations of the terms [µ3 ]1,0 and [µ3 ]0,1 . The main quantity related to the first hyperpolarizability reported here is the vector component along the dipole moment, given by 1,59 µx βx + µy βy + µz βz β¯ = (µ2x + µ2y + µ2z )1/2

(20)

where βi =

1X (βijj + βjij + βjji ) 5 j

(21)

Other quantities reported are the total first hyperpolarizability β0 = (βx2 + βy2 + βz2 )1/2 , the magnitude of the dipole moment µ = (µ2x + µ2y + µ2z )1/2 and the mean polarizability α ¯ = 13 (αxx + αyy + αzz ). The electronic contributions to the electrical properties were calculated through the selfconsistent field Hartree-Fock (SCF), density functional theory (DFT) and coupled cluster (CC) methodologies. Further details are given in the next section.

Results and Discussion We start by analyzing the basis set effects on the dipole moments, polarizabilities and first hyperpolarizabilities of the systems investigated here. We computed these properties for

8


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the Li@benzene, Li@pyridine, Li@pyrimidine, Li@pyrazine and Li@naphthalene molecules through the Gaussian 09 program 66 at the MP2 (second-order Møller-Plesset perturbation theory) level using three different basis set: aug-cc-pVDZ, d-aug-cc-pVDZ and augcc-pVTZ. 67,68 The dipole moments and polarizabilities were computed analytically while the hyperpolarizabilities was calculated by numerical differentiation of the analytical polarizabilities in relation to the electric field. All calculations were performed on geometries optimized at the MP2 level with the aug-cc-pVDZ basis set. The results for the electrical properties are presented in Table 1 and the optimized structures are depicted in Figures 1 and 2. Figure 2 includes, in addition, five Li@aza-naphthalene molecules that are also investigated in this article, but for which we have not performed a study of basis sets. It is observed from Table 1 that the differences between the results for µ and α ¯ el obtained using the three basis sets are less than 1%. Even for β¯el , the results achieved with aug-cc-pVDZ basis set are at most 3% larger than the corresponding results obtained using d-aug-cc-pVDZ, making clear that the second set of diffuse functions is not necessary. Although the differences between the results obtained with aug-cc-pVDZ and aug-cc-pVTZ are larger, they do not exceed 7%, which indicates that the aug-cc-pVDZ basis set is a good compromise between accuracy and computational cost to calculate the electrical properties of the systems investigated here. Table 1: Dipole moment (au), static electronic polarizability (¯ αel , au) and static el electronic first hyperpolarizability (β¯ , au) of the Li@benzene, Li@pyridine, Li@pyrimidine, Li@pyrazine and Li@naphthalene molecules computed at the MP2 level using aug-cc-pVDZ (aDZ), d-aug-cc-pVDZ (dDZ) and aug-cc-pVTZ (aTZ) basis sets.

Molecule Li@benzene Li@pyridine Li@pyrimidine Li@pyrazine Li@naphthalene

aDZ 2.715 3.238 3.729 2.238 2.984

µ dDZ 2.713 3.234 3.725 2.237 2.981

aTZ 2.735 3.255 3.744 2.250 3.008

α ¯ el aDZ dDZ aTZ 88.04 88.39 87.76 81.06 81.35 80.70 73.85 74.13 73.54 73.69 74.03 73.45 141.80 142.24 141.49

aDZ 1291 1115 946 502 1614

β¯el dDZ aTZ 1259 1217 1084 1048 918 889 498 494 1592 1534

Selected geometric parameters optimized at the MP2/aug-cc-pVDZ level for the systems 9



(a) Li@benzene

(c) Li@pyrimidine

(b) Li@pyridine

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(d) Li@pyrazine

Figure 1: Structures of the (a) Li@benzene, (b) Li@pyridine, (c) Li@pyrimidine and (d) Li@pyrazine molecules optimized at the MP2 level with aug-cc-pVDZ basis set.

(a) Li@naphthalene

(b) Li@quinoline

(c) Li@isoquinoline

(d) Li@cinnolin

(e) Li@quinazoline

(f) Li@quinoxaline

Figure 2: Structures of the (a) Li@naphthalene, (b) Li@quinoline, (c) Li@isoquinoline, (d) Li@cinnolin, (e) Li@quinazoline and (f) Li@quinoxaline molecules optimized at the MP2 level with aug-cc-pVDZ basis set. studied here are displayed in Table 2. The parameters obtained for the corresponding nonlithiated molecules (optimized at the same level) are also exhibited for comparison. It is noticeable that the Li-C distances are approximately twice the corresponding H-C bond lengths. Comparing the C-C bond lengths involving the carbon atom that is bonded to the lithium with the corresponding C-C bond lengths in the non-lithiated molecules, one can observe increases around 0.02 Å. The N-N distances are little affected by lithiation, except for the Li@pyrazine. Regarding the LiCC angles, one can see small modifications in relation 10


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to the corresponding HCC angles. In the aza-molecules the modifications are larger for the angles trans to the nitrogen atoms (outside the parentheses). The Li-C distances optimized in this work for Li@benzene and Li@naphthalene (2.005 and 2.006 Å) are somewhat larger than those obtained by Zhang et al. 47 at the B3LYP level using 6-31+G(d) basis sets for C and H and 6-311+G(3df) for Li (1.958 Å for both molecules). The Li-C bond lengths for the Li@pyridine and Li@pyrimidine (2.009 and 2.013 Å) are slightly larger than for Li@benzene. Regarding the two-ring molecules, the Li-C bond lengths are also slightly larger for Li@isoquinoline, Li@cinnolin, Li@quinazoline and Li@quinoxaline (around 2.010 Å), while for Li@quinoline this distance is practically equal to that of the Li@naphthalene. In the case of Li@pyrazine, the lithium atom form two shorter bonds, being the Li-C and Li-N distances equal to 1.944 and 1.909 Å, respectively. A consequence of these two bonds is a much smaller value of the first hyperpolarizability for this molecule, as can be seen in Table 1. After defining aug-cc-pVDZ as the basis set, we calculated the first hyperpolarizabilities of the same molecules of Table 1 at various levels of electron correlation in order to define the most convenient method to compute the vibrational corrections. The results are presented in Table 3. At the SCF and DFT (with CAM-B3LYP and M05-2X functionals) levels the hyperpolarizabilities were computed analytically using the Gaussian 09 package. 66 We included only these two functionals advised by the work of Zhang et al. 47 that considered also B3LYP and BHandHLYP methods and achieved that CAM-B3LYP and M05-2X lead to values closer to the MP2 results for Li@n-acenes. The coupled cluster calculations were performed analytically through quadratic response theory 69 using the Dalton 2016 program 70 at the following levels: CCS (coupled cluster with single excitations); CCSD (coupled cluster with single and double excitations); CC2 (an approximation to CCSD 71 ). It can be seen that the SCF results for β¯el correspond to little more than half of the CCSD reference values and that CCS practically does not improve these results. In its turn, the CC2 results are much (34 to 89%) larger than the corresponding CCSD values. In general MP2 is the method that

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Table 2: Selected geometric parameters optimized at MP2/aug-cc-pVDZ level (bond lengths in angstrom and angles in degree). H-C and HCC refer the hydrogen that is replaced by the lithium atom while C-C refers to the bond lengths in which one of the carbons is bonded to the lithium atom (or the hydrogen that is replaced by it). The C-C distance and the LiCC and HCC angles in parentheses are those above in figures (in general cis to the nitrogens) Molecule Benzene Li@benzene Pyridine Li@pyridine Pyrimidine Li@pyrimidine Pyrazine Li@pyrazine Naphthalene Li@naphthalene Quinoline Li@quinoline Isoquinoline Li@isoquinoline Cinnolin Li@cinnolin Quinazoline Li@quinazoline Quinoxaline Li@quinoxaline

H-C 1.094

Li-C 2.005

1.094 2.009 1.093 2.013 1.095 1.944 1.094 2.006 1.094 2.005 1.094 2.011 1.094 2.011 1.094 2.010 1.094 2.010

C-C 1.408 (1.408) 1.428 (1.428) 1.405 (1.407) 1.424 (1.426) 1.403 (1.403) 1.421 (1.421) 1.407 1.423 1.425 (1.392) 1.445 (1.411) 1.425 (1.392) 1.444 (1.411) 1.426 (1.391) 1.446 (1.411) 1.428 (1.390) 1.447 (1.410) 1.426 (1.392) 1.446 (1.411) 1.426 (1.391) 1.446 (1.410)

N-N

LiCC

HCC 120 (120)

123 (123) 121 (120) 127 (121) 2.420 2.414 124 (124) 2.856 2.816 178

122 (122) 121 120 (120)

125 (121) 120 (120) 128 (118) 120 (120) 125 (121) 1.328 119 (120) 1.331 128 (117) 2.429 119 (120) 2.430 127 (119) 2.879 120 (120) 2.881 127 (119)

produces results closer to CCSD, with differences varying from 4% for Li@naphthalene up to 11% for Li@pyrimidine. The performances of the DFT methods are similar to MP2, a conclusion that was previously obtained by Zhang et al. 47 for Li@n-acenes. The differences between CAM-B3LYP and CCSD are in the range of 7 to 14% while the differences between M05-2X and CCSD vary from 3 to 16%. Considering that to calculate the pure vibrational corrections for the first hyperpolarizability it is necessary to compute the derivatives of the dipole moment and polarizability in relation to the normal coordinates and that these properties can be computed analytically at the MP2 level (using Gaussian 09 program), we adopted this method to calculate these corrections for all molecules investigated here. On the

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other hand, to compute the zpva corrections for the first hyperpolarizability it is necessary to calculate their own derivatives. Since the numerical MP2 calculations for this property are computationally much more expensive, we limit the calculations of this correction to the one-ring molecules. Table 3: Static electronic first hyperpolarizability (β¯el , au) of the Li@benzene, Li@pyridine, Li@pyrimidine, Li@pyrazine and Li@naphthalene molecules computed at different levels using the aug-cc-pVDZ basis set. Molecule SCF MP2 CAM-B3LYP M05-2X CCS CC2 CCSD Li@benzene 713 1291 1269 1266 773 2056 1413 Li@pyridine 639 1115 1127 1088 700 1993 1053 Li@pyrimidine 557 946 949 920 612 1582 1057 Li@pyrazine 315 502 464 454 346 723 540 Li@naphthalene 903 1614 1552 1515 961 2515 1683

Electronic contributions to the hyperpolarizabilities are related to the electronic spectrum. A two-level model 72,73 states that β0el ∝ ∆µf /∆E 3 , where ∆E is the transition energy, f the oscillator strength and ∆µ the difference between the dipole moment in the crucial excited state and the dipole moment in the ground state. To rationalize the lithiation effects on the first hyperpolarizabilities of the systems investigated here, we present in Table 4 the β0el values together with the three first electronic transition energies and their corresponding oscillator strengths for the lithiated and non-lithiated molecules. The electronic spectrum was computed through time-dependent density functional theory (TDDFT) with the CAM-B3LYP functional. To maintain the uniformity of the data, the quoted values of β0el were also obtained at the CAM-B3LYP level. It can be observed that in general the values of ∆E are smaller for the lithiated molecules, in line with the larger values of the hyperpolarizabilities of these systems. However, only this factor can not quantitatively explain the remarkable increases of β0el . Large differences between the ground state dipole moment and that of the excited state of interest are also very important. The values of ∆µ for the crucial transitions (largest oscillator strengths) are also displayed in Table 4, being much higher for the lithiated molecules. The factor ∆µf /∆E 3 is 350 times greater for Li@pyridine than 13



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for pyridine and 670 times greater for Li@pyrimidine than for pyrimidine. Considering the two-ring molecules (with ∆µ 6= 0), this ratio varies from 22 to 190, which is in agreement with the noticeable enhancing of the first hyperpolarizability in the lithiated molecules. The large values of ∆µ for the lithiated molecules are related to the electron transfers from the ring to the lithium atom. With the exception of Li@pyrazine, the lithium atom has positive charge in the ground state and negative charge in the crucial excited state, giving rise to large modification in the dipole moment. To illustrate this issue, the values of the CHELPG charges (CHarges from ELectrostatic Potentials using a Grid-based method) for the lithium atoms are also displayed in Table 4. Table 4: Static electronic first hyperpolarizability (β0el , au), transition energy (∆E, eV), oscillator strength (f ), dipole moment difference in the crucial transition (∆µ, au) and CHELPG charge in the Li atom (in electron charge unit) computed at the CAM-B3LYP/aug-cc-pVDZ level. For the Li charge, g indicates ground state and e crucial excited state. 1st Molecule Benzene Li@benzene Pyridine Li@pyridine Pyrimidine Li@pyrimidine Pyrazine Li@pyrazine Naphthalene Li@naphthalene Quinoline Li@quinoline Isoquinoline Li@isoquinoline Cinnolin Li@cinnolin Quinazoline Li@quinazoline Quinoxaline Li@quinoxaline

β0el 0 1269 9 1153 6 949 0 640 0 1455 62 1313 12 1411 50 1159 56 1092 112 1168

∆E 5.36 3.38 5.00 3.52 4.50 3.77 4.14 3.20 4.47 3.46 4.52 3.54 4.40 3.57 3.32 3.41 4.10 3.68 3.79 3.63

f 0.000 0.078 0.004 0.079 0.005 0.080 0.006 0.004 0.000 0.103 0.030 0.102 0.058 0.102 0.003 0.004 0.003 0.098 0.003 0.109

2nd ∆E 6.04 4.01 5.42 4.22 4.87 4.41 5.00 3.81 4.50 3.58 4.56 3.90 4.65 3.78 4.30 3.68 4.48 3.97 4.36 3.75

14

f 0.000 0.011 0.000 0.016 0.000 0.000 0.000 0.032 0.071 0.005 0.055 0.006 0.002 0.006 0.059 0.098 0.050 0.003 0.036 0.000


3rd ∆E 6.40 4.44 5.43 4.34 5.69 4.51 5.27 4.21 5.57 4.31 4.57 4.16 4.72 4.19 4.63 3.96 4.80 4.19 4.43 4.04

f 0.000 0.000 0.045 0.015 0.045 0.007 0.111 0.013 0.000 0.045 0.002 0.008 0.028 0.009 0.015 0.005 0.016 0.004 0.098 0.031

∆µ 0 5.29 0.10 5.43 0.05 5.49 0 0.52 0 5.76 0.91 5.77 0.10 5.75 0.69 5.72 0.52 5.76 0.40 5.69

q(Li) g

e

0.76

−0.37

0.78

−0.36

0.80

−0.36

0.67

0.68

0.76

−0.37

0.77

−0.37

0.76

−0.36

0.77

−0.36

0.77

−0.37

0.77

−0.35

Page 15 of 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The results for the static pure vibrational corrections to the hyperpolarizabilities of the Li@aza-benzene and Li@aza-naphthalene molecules investigated in this article are presented in Table 5 (the quoted values correspond to β¯pv ). For the Li@benzene, Li@pyridine, Li@pyrimidine and Li@pyrazine one-ring molecules and for Li@naphthalene we computed these properties using the BKPT and VAR methodologies while for the other two-ring molecules we only calculated them using the FIC method. Regarding the relative importance of the term [µα], a similar pattern can be observed for all molecules with exception of Li@pyrazine, the only system in which the lithium atom forms two bonds (see Figures 1 and 2). Considering the other systems, this term has magnitude similar to the corresponding electronic contribution when calculated by BKPT. Nevertheless it is noticeable that the VAR results for the one-ring molecules (excluding Li@pyrazine) correspond to no more than half of the BKPT values. For Li@pyrazine, the term [µα]0,0 has roughly the same magnitude when calculated by the BKPT and VAR methods, while for Li@naphthalene the VAR result represents 75% of the BKPT value. In order to understand the differences between the BKPT and VAR results for [µα], we performed two test calculations. First we calculated this term by means of Eq. (4) using matrix elements computed with the harmonic vibrational wave functions, but including first and second order derivatives of µ and α. The results for Li@benzene, Li@pyridine, Li@pyrimidine, Li@pyrazine and Li@naphthalene molecules are −1375, −1139, −870, 197 and −1537 au, that are very close to the BKPT values, indicating that the second order derivatives are not important. Then we performed calculations using the corrected wave functions and including only first order derivatives in the calculation of the matrix elements. In this time the results are −499, −498, −421, 207 and −1043 au, that are close to the VAR results with first and second order derivatives included. These results confirm that the contrast in the calculated values of the term [µα] came not from the second order derivatives, but from the differences between the BKPT and VAR methodologies. To illustrate the lithiation effect, the individual contributions of normal modes to the term [µα]0,0 for the pyridine and Li@pyridine molecules are exhibited in Figure 3. It is

15



Page 16 of 34

Table 5: Pure vibrational correction to the static first hyperpolarizability (β¯pv , au) computed at the MP2/aug-cc-pVDZ level. For Li@benzene, Li@pyridine, Li@pyrimidine, Li@pyrazine and Li@naphthalene the results were obtained by BKPT and VAR methodologies. For the other two-ring molecules the pv corrections were computed through FIC method. The electronic contribution (β¯el ) computed at the same level is also quoted for comparison.

Molecule Li@benzene Li@pyridine Li@pyrimidine Li@pyrazine Li@naphthalene Li@quinoline Li@isoquinoline Li@cinnolin Li@quinazoline Li@quinoxaline

β¯el 1291 1115 946 502 1614 1360 1407 1095 1142 1221

0,0

[µα] −1357 −1125 −859 195 −1509 −1476 −1321 −1240 −1169 −1469

BKPT [µ3 ]1,0 −5870 −4048 −2605 552 −8227 −7301 −6330 −5377 −5804 −6943

VAR 3 0,1

[µ ] −8554 −7711 −5009 −647 −17701 −25549 −16527 −24115 −15222 −17564

[µα] −518 −519 −433 207 −1129

[µ3 ] −3595 −3932 −2069 143 −16380

observed that for the Li@pyridine molecule, the major contributions (in magnitude) come from the low-frequency normal modes. These modes correspond to vibrations involving the lithium atom. The corresponding normal modes in pyridine have higher frequencies and give much smaller contributions. In this case, the total value of the term [µα]0,0 when calculated by BKPT is only −0.90 au. Similar conclusions would be obtained for the other molecules, which were not included in the graph for easier visualization. In general, the low-frequency normal modes involving the lithium atom (in the region of 100 cm−1 ) present strong anharmonicities and their contributions for [µα] are overestimated when calculated by BKPT. The discrepancies between the BKPT and VAR results for the term [µ3 ] are even larger than for [µα]. Aiming to rationalize this disagreement, we calculated the term [µ3 ] by means of Eq. (5) using matrix elements computed with the harmonic vibrational wave functions and including first and second order derivatives of µ. As no mechanical anharmonicity is taken into account, these numbers should be compared with [µ3 ]1,0 , with the difference that the VAR results include additional terms arising from products of more 16


Page 17 of 34

4 Contribution to [µα]0,0 (102 au)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 0 −2 −4 −6 −8 −10

Pyridine Li@pyridine

−12 0

5

10

15

20

25

30

35

Frequency (102 cm−1) Figure 3: Contributions of normal modes to the term [µα]0,0 of the pure vibrational correction to the static first hyperpolarizability of the pyridine and Li@pyridine molecules. than one second order derivative, that do not appear in [µ3 ]1,0 . The results −5871, −4052, −2607, 553 and −8236 au for the Li@benzene, Li@pyridine, Li@pyrimidine, Li@pyrazine and Li@naphthalene molecules are in very good agreement with the [µ3 ]1,0 values, indicating that these additional terms are not relevant. Thus, the disagreement between the BKPT and VAR results arise from the differences between these methodologies. We also calculated dynamic vibrational corrections to the hyperpolarizabilities of the one-ring molecules and for Li@naphthalene. Attention is given to the term [µα] of the first hyperpolarizability β (−ω; ω, 0), that is related to the dc-Pockels effect, because it does not vanish in the limit ω → ∞ (in fact, it tends to 1/3 of the static value 74 ). We included the dumping in a phenomenological way replacing v → v − iγv in Eq. (4) (or ωa → ωa − iγa in Eq. 13), where γv (or γa ) represents the inverse of the lifetime of the excited vibrational

17



state. 75–77 We adopted the approximation in which γv (or γa ) is equal to 10−4 hartree for all excited states. This value was chosen to prevent the first hyperpolarizability go to infinite in the poles without significantly affecting the calculated values away from them. A consequence of the introduction of dumping is that very near poles overlap. In special, at high-frequencies where there are many poles nearby, no pole structure is observed. 75–77 The dynamic results to the term [µα] obtained by BKPT and VAR methods for the Li@benzene, Li@pyridine, Li@pyrimidine, Li@pyrazine and Li@naphthalene molecules are displayed in Figure 4 (values correspond to β¯pv ). The graphs present similar patterns, with larger differences for Li@pyrazine and Li@naphthalene. The VAR curves are right shifted in relation to the BKPT curves, which is mainly consequence of the increase of the energy differences between the excited vibrational states and the ground state resulting from the interaction between normal modes. Consider, for instance, the case of Li@pyridine. The computation of the term [µα] by the VAR method reveals that the excited states (obtained from diagonalizing of the vibrational Hamiltonian) that provide the largest contributions to this term are 1st, 2nd, 14th, 16th, 19th and 23rd (the states are ordered in increasing energy). The energies of these states in relation to the vibrational ground state are 0.00109, 0.00110, 0.00260, 0.00268, 0.00330 and 0.00358 hartree, which correspond to the first four resonances of the VAR (blue) curve of Figure 4(b) (1 hartree = 219474.63 cm−1 ). Notice that due to the proximity of the energies, the resonances related to 1st and 2nd states, as well as those associated with 14th and 16th, overlap. The identification of these states in terms of harmonic vibrational wave functions, as in Eq. 6, shows that |1i = 0.98|11 i − 0.12|11 121 i − 0.10|11 131 i + · · · |2i = 0.98|21 i − 0.13|21 101 i − 0.11|21 111 i + · · · |14i = 0.43|51 i + 0.87|11 31 i + · · · |16i = 0.86|51 i − 0.44|11 31 i − 0.10|41 251 i − 0.10|41 261 i + · · · |19i = 0.98|61 i − 0.13|51 191 i − 0.12|51 201 i + · · · |23i = 0.94|71 i − 0.20|21 31 i − 0.13|21 251 i − 0.12|61 121 i + · · ·

18


Page 18 of 34

Page 19 of 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


where |1i, |2i, · · · indicate 1st, 2nd, · · · excited states and |ai bj i indicates the harmonic vibrational wave function in which i and j are the excitation numbers of modes a and b, respectively. Notice that in general the states that give the most relevant contributions in the VAR method predominantly correspond to modes 1, 2, 5, 6 and 7 in the first excited state. These modes are those giving the largest contributions to the term [µα]0,0 in BKPT and their harmonic vibrational energies are ω1 = 0.00059, ω2 = 0.00060, ω5 = 0.00216, ω6 = 0.00281 and ω7 = 0.00310 hartree (see Figure 3). These values are much smaller than the excitation energies obtained by diagonalization of the vibrational Hamiltonian and are in agreement with the first four resonances of the BKPT (red) curve of Figure 4(b). The zpva corrections to the first hyperpolarizabilities of Li@benzene, Li@pyridine, Li@pyrimidine and Li@pyrazine one-ring molecules, computed through BKPT and VAR methodologies, are displayed in Table 6 (the quoted values correspond to β¯zpva ). It is observed that the results obtained by the two methods are in very good agreement. In addition, it is notable that the term [β]1,0 (or its corresponding [β]00 ) provides most of the total zpva correction. The term [β]0,1 ([β]0 ) is slightly higher for Li@pyridine, contributing with 28% (23%) of the total zpva value when computed by BKPT (VAR). The comparison with the electronic contribution shows that the total zpva correction have magnitude around 14% of β el for Li@pyridine and around 5% for the other molecules. Torrent-Sucarrat et al. 78–80 have proposed a way to evaluate the convergence of the BKPT series that is based on the examination of two sequences: (A) β e , [β zpva ]I , [β zpva ]III , · · · (B) β nr , β (c−zpva)(I) , β (c−zpva)(III) , · · · where [β zpva ]I = [β zpva ]1,0 +[β zpva ]0,1 and β nr = [µα]0,0 +[µ3 ]1,0 +[µ3 ]0,1 . The roman superscript indicates the total (electrical + mechanical) order of anharmonicity. When the [β zpva ]I /β e and β (c−zpva)(I) /β nr ratios are much smaller than unity the series tend to present good convergence. The small [β zpva ]I /β e ratios obtained here for the one-ring molecules (from 0.05 to 0.14) indicate the appropriate convergence of the series (A). Although the term β (c−zpva)(I) is not

19



15

15

[µα]0,0 − BKPT

10

[µα] − VAR

[µα]0,0 − BKPT

10

[µα] − VAR 5

5

[µα] (10 au)

0

0

2

[µα] (102 au)

20

−5 −10

−5 −10

−15 −15

−20 −25

−20 0

1

2

3

4

5

Frequency (10

6 −3

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

9

10

Frequency (10−3 hartree)

hartree)

(a) Li@benzene

(b) Li@pyridine

15

12 [µα]0,0 − BKPT

[µα]0,0 - BKPT

10

10

8

[µα] − VAR

[µα] - VAR

6

5

[µα] (10 au)

4

2

[µα] (102 au)

0 −5

2 0 -2 -4

−10

-6 −15

-8 0

1

2

3

4

5

Frequency (10

6 −3

7

8

9

10

0

1

2

3

4

5

6

7

8

Frequency (10-3 hartree)

hartree)

(c) Li@pyrimidine

(d) Li@pyrazine

15 [µα]0,0 - BKPT

10

[µα] - VAR 5 [µα] (102 au)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 34

0 -5 -10 -15 -20 0

1

2

3

4

5

6

7

8

9

10

Frequency (10-3 hartree)

(e) Li@naphthalene

Figure 4: Terms [µα] of the pure vibrational corrections to β (−ω; ω, 0) of the (a) Li@benzene, (b) Li@pyridine, (c) Li@pyrimidine, (d) Li@pyrazine and (e) Li@naphthalene molecules computed by perturbation and variational methods. directly computed here, the differences between the VAR and BKPT results give some idea of its magnitude. The ratios between the differences obtained from Table 5 and β nr are around 0.7 for Li@benzene, Li@pyridine, Li@pyrimidine, 0.4 for Li@naphthalene and 2.5 for 20


Page 21 of 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Li@pyrazine, which is an indication of the poor convergence of the series (B). Table 6: Zpva correction to the static first hyperpolarizability (β¯zpva , au) of Li@benzene, Li@pyridine, Li@pyrimidine and Li@pyrazine molecules computed throug BKPT and VAR methods at the MP2/aug-cc-pVDZ level.

Molecule Li@benzene Li@pyridine Li@pyrimidine Li@pyrazine

BKPT [β] [β]0,1 63 6 122 34 59 −2 23 5 1,0

VAR 00

[β] 64 123 59 24

[β]0 2 28 −3 4

Conclusion In this article we report results of calculations to the electronic and vibrational hyperpolarizabilities of ten molecules based on(aza-)benzenes and (aza-)naphthalenes in which one hydrogen is replaced by a lithium atom. They are four one-ring molecules (Li@benzene, Li@pyridine, Li@pyrimidine and Li@pyrazine) and six two-ring molecules (Li@naphthalene, Li@quinoline, Li@isoquinoline, Li@cinnolin, Li@quinazoline and Li@quinoxaline). A basis set analysis shows that aug-cc-pVDZ is adequate to compute the dipole moments and the electronic polarizabilities and first hyperpolarizabilities of the studied molecules. An investigation about the electron correlation role in the calculation of the first hyperpolarizability reveals that second-order many-body perturbation theory is able to reproduce the coupled cluster with single and doubles reference values with accuracy varying from 4% to 11%. The results also show that density functional theory with CAM-B3LYP and M05-2X functionals can produce results similar to the CCSD ones. Pure vibrational corrections to the first hyperpolarizabilities of the one-ring molecules and Li@naphthalene were computed at the MP2 level through the perturbation theoretical method of Bishop and Kirtman and using a variational approach. For the other two-ring molecules, the vibrational hyperpolarizabilities were computed through field-induced coordi21



nates method. In general are obtained notable discrepancies between the perturbation and variational results for the terms [µα] and [µ]3 of the pure vibrational corrections. Considering the Li@benzene, Li@pyridine and Li@pyrimidine one-ring molecules, the perturbation results for the term [µα] have similar order of magnitude to the corresponding electronic hyperpolarizabilities, while the variational results correspond to approximately half of the electronic values. We report graphics showing the frequency dependent vibrational corrections to the first hyperpolarizabilities of the one-ring molecules and Li@naphthalene using an phenomenological approximation to the dumping. These graphics emphasize the differences between the two methodologies employed to compute the vibrational hyperpolarizabilities. Remarkable shifts of the variational curves in relation to the perturbation ones are observed, which may be understood by the increase of the energy differences between the excited and ground vibrational states when the anharmonicity is taken into account in the vibrational Hamiltonian. The results of the current investigation give broad support to the conclusion reached in the previous article 57 that the variational method provides an most appropriate option to treat low-frequency normal modes. Results to the zpva correction for the one-ring molecules are also presented. In general they show that both perturbation and variational methods yield similar results and that the contributions coming from second order derivatives of β are dominant.

Acknowledgement The authors gratefully acknowledge the financial support of the Brazilian funding agencies CNPq, CAPES and FAPEG/GO as well as the computational resources provided by LCCUFG.

conflict of interest The authors declare no competing financial interest.

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Graphical TOC Entry 15

10

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0

-5

-10

-15

Li@pyridine

-20 0

1

2

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4

5

34

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8

9

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Electronic and Vibrational Hyperpolarizabilities of Lithium Substituted

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