Electric properties for HCCH, H2CC, H2CSi and

Brill Academic Publishers P.O. Box 9000, 2300 PA Leiden The Netherlands

Lecture Series on Computer and Computational Sciences Volume 6, 2006, pp. 395-404

Electric properties for HCCH, H2CC, H2CSi and H2CGe *

G.Maroulis , D.Xenides1 and P.Karamanis2 Department of Chemistry, University of Patras, GR-26500 Patras, Greece Recived July 20, 2006; accepted 1 August, 2006 Abstract: We have calculated electric dipole moments, static polarizabilities and hyperpolarizabilities for HCCH, H2CC, H2CSi and H2CGe. All properties have been obtained from finite-field Møller-Plesset perturbation theory and coupled cluster calculations with large, purpose-oriented basis sets. We have examined the effect on the electric properties of the acetylene-vinylidene isomerization and the evolution of the (hyper)polarizability in the H2CX sequence, where X=C, Si, Ge. The mean second dipole hyperpolarizability of vinylidene is γ / e4a04Eh-3 = 5878, slightly higher than the 5794 found for acetylene.

γ (H2CGe) is significantly higher than γ (H2CCe) but comparable to γ (H2CSi) Keywords: Acetylene, hyperpolarizability.

vinylidene,

silylidene,

germylidene,

dipole

moment,

polarizability,

1. Introduction The elegant theory of electric polarizability [1,2] has been the starting point of significant progress in molecular science. It is now routinely associated with molecular characteristics as hardness [3,4], softness [1] and hypersoftness [5], stiffness [6] and compressibility [7]. This basic theory is now the cornerstone of the rational approach to the interpretation of phenomena in fields ranging from intermolecular interactions [9,10] to nonlinear optics [11]. Computational quantum chemistry can now provide reliable estimates of polarizabilities for wide classes of systems of all sizes, from atoms, molecules and clusters to nanoparticles [11, 12]. Particular research fronts that benefit from this expansion include the simulation of fluids [13,14], the analysis of spectroscopic observations [15,16] and systematic studies of molecular architectures with potential use as nonlinear optical materials [17,18]. Electric (hyper)polarizability is also used as a molecular descriptor in pharmacological QSAR studies [19,20]. In this work we report electric properties for acetylene (H-C≡C-H), vinylidene (H2C=C), silylidene (H2C=Si) and germylidene (H2C=Ge). Our goal is to investigate, primo, the effect of the acetylenevinylidene isomerization on the electric (hyper)polarizability and secundo, the effect of substitution for the sequence H2CX, X= C, Si and Ge.

2. Theory Our calculation of the electric properties relies on the finite-field method [21]. The energy and the electric multipole moments of an uncharged molecule perturbed by a weak, static electric field can be expanded in terms of the permanent properties (in bold) of the free molecule and the components of the field, as [1,2,22] * 1

2

Corresponding author. E-mail: [email protected] Present address: Department of Computer Science and Technology, University of the Peloponnese, GR-221 00 Tripolis, Greece. Present address: Laboratoire de Chimie Structurale, UMR 5624, Université de Pau et des Pays de l’Adour, F-64000 Pau, France.

396 _________________________________________________________________G.Maroulis, D.Xenides and P.Karamanis

Ep ≡ Ep(Fα, Fαβ, Fαβγ, Fαβγδ, ...) = E0 - μαFα - (1/3)ΘαβFαβ - (1/15)ΩαβγFαβγ - (1/105)ΦαβγδFαβγδ + ... - (1/2)ααβFαFβ - (1/3)Aα,βγ FαFβγ - (1/6)Cαβ,γδ Fαβ Fγδ - (1/15)Eα,βγδFαFβγδ + ... - (1/6)βαβγFαFβFγ - (1/6)Bαβ,γδFαFβFγδ + ... - (1/24)γαβγδFαFβFγFδ + ...

(1)

μαp = μα + ααβ Fβ + (1/3)Aα,βγ Fβγ + (1/2)βαβγ FβFγ + (1/3)Bαβ,γδ FβFγδ + (1/6)γαβγδ FβFγFδ + ...

(2)

Θαβp = Θαβ + Aγ,αβEγ + Cαβ,γδ Fγδ + (1/2)Bγδ,αβ FγFδ + ...

(3)

Ωαβγp = Ωαβγ + Eδ,αβγ Fδ + ...

(4)

where Fα, Fαβ, etc. are the field, field gradient, etc. at the origin. E0, μα0, Θαβ0, Ωαβγ0 and Φαβγδ0 are the energy and the dipole, quadrupole, octopole and hexadecapole moment of the free molecule. The second, third and fourth-order properties are the dipole and quadrupole polarizabilities and hyperpolarizabilities ααβ, βαβγ, γαβγδ, Aα,βγ, Cαβ,γδ, Eα,βγδ and Bαβ,γδ. The subscripts denote Cartesian components. A repeated subscript implies summation over x, y and z. The number of independent components needed to specify the non-vanishing tensors is regulated by symmetry [1]. For sufficiently weak fields the expansions of Eqs 1-4 converge rapidly. Thus, the finite-field method offers the possibility of a direct approach to the calculation of electric moments, polarizabilities and hyperpolarizabilities. In the case of a homogeneous electric field, Eq 1 reduces to a much simpler one, E = E0 - μα0Fα - (1/2)ααβFαFβ - (1/6)βαβγFαFβFγ - (1/24)γαβγδFαFβFγFδ + ...

(5)

The properties of interest in this work are the dipole ones μα0, ααβ, βαβγ and γαβγδ. The independent components of the respective tensors have been specified in previous work [23,24]. In addition to the Cartesian components, other properties of interest in this work are the mean ( α ) and the anisotropy (Δα) of the dipole polarizability, the mean of the first (βαβγ) and the second (γαβγδ) hyperpolarizability ( γ ). All the above are directly associated to measurable quantities. For H2CC, H2CSi and H2CGe they are defined (with z as the C2 axis) as [1],

α = (αxx + αyy +αzz )/3 Δα = (1/2)1/2[(αxx-αyy)2 + (αyy-αzz)2 + (αzz-αxx)2]1/2

β = (3/5)(βzxx + βzyy + βzzz)

γ

= (1/5)(γxxxx + γyyyy + γzzzz + 2γxxyy + 2γyyzz + 2γzzxx)

In the case of a linear molecule, as HCCH, these definitions reduce to

α = (2αxx + αzz )/3 Δα = αzz-αxx

β = (3/5)(2βzxx + βzzz)

(6)

Electric properties for HCCH, H2CC, H2CSi and H2CGe

γ

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397

(7)

= (1/15)(3γzzzz + 8γxxxx + 12γzzxx)

Our approach to the calculation of electric properties from Eqs 1-4 has been presented in rich detail in previous work [24-30]. The theoretical methods used in this paper are self-consistent field (SCF), Møller-Plesset perturbation theory (MP) and coupled cluster theory (CC). Essential presentations of these methods are available in standard textbooks [31-34]. Electron correlation effects on the molecular properties are accounted for via MPn (n=2,3,4, second, third and fourth order many-body perturbation theory), CCSD (single and double excitation coupled cluster theory) and CCSD(T) (single and double excitation coupled cluster theory including an estimate of connected triple excitations by a perturbational treatment). We restrict our presentation to the definition of the various orders of MP

MP2 = SCF + D2 MP3 = MP2 + D3 DQ-MP4 = MP3 + D4 + QR4 = MP3 + DQ4 SDQ-MP4 = DQ-MP4 + S4 MP4 = SDQ-MP4 + T4 = SCF + D2 + D3 + S4 + D4 + T4 + QR4

(8)

(where the fourth-order terms S4, D4, T4 and Q4 are contributions from single, double, triple and quadruple substitutions from the reference, zeroth-order wavefunction and R4 is the renormalization term). For the CC levels of theory, CCSD = SCF + ΔCCSD CCSD(T) = CCSD + T

(9)

By virtue of Eqs 6 and 7, analogous decompositions are adopted for the mean polarizability or hyperpolarizability, as α , β and γ are linear in the Cartesian components. The same holds true for the anisotropy Δα in the case of acetylene. For the other three systems the anisotropy Δα at a given level of theory is computed by inserting in Eq 6 the respective quantities for the Cartesian components of ααβ.

3. Computational Details Large, carefully optimized Gaussian basis sets were used for all molecules. The details of their construction and composition are given elsewhere [35]. The basis sets are [9s6p5d2f/6s4p2d] (188 contracted GTF) for acetylene, [9s6p4d1f/6s4p1d] (156 CGTF) for vinylidene, [8s6p3d1f/6s4p3d1f/4s3p1d] (124 CGTF) for silylidene and [8s7p6d2f/6s4p4d2f/4s4p2d] (177 CGTF) for germylidene. 5d and 7f GTF were used in all calculations. All calculations were performed at the molecular geometries selected from the available literature. Our source is Martin et al [36] for acetylene, Chang et al [37] for vinylidene, Hilliard and Grev [38] for silylidene and Stogner and Grev [39] for germylidene. The respective parameters are shown in Figure 1. The molecular axis for acetylene is z. For vinylidene, silylidene and germylidene the molecule is on the xz plane, with z as the C2 axis and the hydrogen nuclei on the negative part of the z axis. The 2 innermost MO were kept froze in all post-Hartree-Fock calculations for acetylene. Similarly, 2 MO were kept frozen for vinylidene, 6 for silylidene and 10 for germylidene. All calculations were performed with GAUSSIAN 94 [40] and GAUSSIAN 98 [41]. Atomic units are used throughout this paper. Conversion factors to SI units are, Energy, 1 Eh = 4.3597482 x 10-18 J, length, 1 a0 = 0.529177249 x 10-10 m, μ, 1 ea0 = 8.478358 x 10-30 Cm, α, 1 e2a02Eh-1 = 1.648778 x 10-41 C2m2J-1, β, 1 e3a03Eh-2 = 3.206361 x 10-53 C3m3J-2, γ, 1 e4a04Eh-3 = 6.235378 x 10-65 C4m4J-3. Property values are in most cases given as pure numbers, i.e. μ/ea0, ααβ/ e2a02Eh-1, βαβγ/e3a03Eh-2 and γαβγδ/e4a04Eh-3


Fig 1. Molecular geometry specification and atomic charges from a NBO population analysis.


_________________________________________

Figure 2. HOMO and LUMO for HCCH, H2CC, H2CSi and H2CGe .

399


4. Results and Discussion In Figure 1, in addition to the geometrical parameters of the molecules, we show also atomic charges obtained from a NBO analysis at the MP2/cc-pvdz level of theory for acetylene, vinylidene and silylidene and MP2/[8s7p4d/6s4p2d/4s2p] for germylidene. We observe that there is a negative charge on the C atom both for acetylene and H2CX. In Figure 2 we show the evolution of the HOMO-LUMO in the sequence HCCH→H2CC→H2CSi→H2CGe. We give in Tables 1 and 2 a detailed analysis of electron correlation effects on the electric properties of acetylene and vinylidene. Table 1. Electron correlation effects on the dipole (hyper)polarizability of acetylene.

α

Method SCF D2 D3 S4 D4 T4 QR ΔCCSD T MP2 SDQ-MP4 MP4 CCSD CCSD(T) ECC

23.41 -0.58 -0.03 0.01 -0.12 0.35 -0.12 -0.86 0.24 22.83 22.80 22.92 22.55 22.78 -0.62

Δα 12.04 -0.66 0.21 0.12 -0.05 0.06 0.08 -0.44 0.06 11.38 11.59 11.80 11.59 11.65 -0.39

γ 5299 833 -489 -51 9 670 -311 113 382 6133 5644 5960 5413 5794 495

Table 2. Electron correlation effects on the dipole moment and (hyper)polarizability of vinylidene. Method SCF D2 D3 S4 D4 T4 QR ΔCCSD T MP2 SDQ-MP4 MP4 CCSD CCSD(T) ECC

μ 0.9318 0.0232 -0.0311 0.0134 -0.0069 0.0081 0.0002 -0.0100 0.0041 0.9550 0.9239 0.9386 0.9218 0.9259 -0.0059

α 23.86 0.25 -0.32 0.15 -0.05 0.27 -0.06 0.09 0.27 24.12 23.79 24.10 23.95 24.23 0.36

Δα

β

9.15

38.7 3.8 -1.4 1.7 -0.8 2.4 -0.3 4.0 2.7 42.5 41.2 44.1 42.7 45.4 6.7

0.19 8.07 8.20 8.41 8.92 8.95 -0.20

γ 4481 1224 -485 137 56 470 -209 933 465 5705 5220 5674 5414 5878 1397

Electron correlation has a small effect on the dipole polarizability of acetylene. The magnitude of both invariants is reduced by electron correlation at all levels of theory. The electron correlation correction (ECC), defined as ECC = CCSD(T) – SCF is -0.62 and -0.39 for α and Δα, respectively. This corresponds to a reduction of the SCF value by 2.7 and 3.2%, respectively. The post-Hartree-Fock values of the mean hyperpolarizability vary significantly. The presumably most accurate CCSD(T) methods yields γ = 5794, or 9.3% above the SCF value of 5299. Electron correlation has a small effect for the dipole moment and polarizability of vinylidene. The dipole moment is predicted to be μ = 0.9318 at the SCF level but the CCSD(T) is lower at 0.9259. The


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401

mean polarizability α increases from 23.86 (SCF) to 24.23 (CCSD(T)), a change of only 1.5%. A reduction of the magnitude of the anisotropy is predicted by all post-Hartree-Fock methods but the overall effect is rather small. The CCSD(T) value of the mean first hyperpolarizability is β = 45.43. This represents an strong increase of 17.3% over the SCF result. The effect is quite strong for the second dipole hyperpolarizability, as the CCSD(T) value is γ = 5878, or 31.2% above the respective SCF. Overall, vinylidene is characterized by a mean dipole polarizability slightly higher than that of acetylene. This is reversed for the anisotropy. At the SCF level of theory γ (HCCH) > γ (H2CC). Electron correlation inverses this inequality. Thus, taking into account the effect on α and γ , vinylidene is more (hyper)polarizable than acetylene. The calculated electric properties for silylidene and germylidene are shown in Tables 3 an 4, respectively. Table 3. Electric properties for silylidene. Method SCF MP2 SDQ-MP4 MP4 CCSD CCSD(T) ECC

μ 0.1076 0.1217 0.1125 0.0918 0.1143 0.0907 -0.0169

α 48.19 47.66 47.09 47.51 47.07 47.39 -0.80

Δα 18.70 16.71 18.08 17.94 18.32 18.28 -0.42

β -268.4 -130.1 -167.6 -159.7 -173.5 -171.6 96.8

γ 24124 26945 24793 26591 25048 26639 2515

The SCF dipole moment of silylidene is significantly smaller than that of vinylidene. Electron correlation reduces further this value by 15.7%. In absolute terms this change is rather small. It is worth noticing that MP2, SDQ-MP4 and CCSD predict a positive correlation effect, in disagreement with MP4 and CCSD(T). At the CCSD(T) level we obtain α =47.39 and Δα = 18.28, lower by 1.7 and 2.2% lower than the respective SCF values. Electron correlation effects are very important for the mean first hyperpolarizability but considerably lower for the second one. We observe an overestimation of the correlation effect by MP2 for both properties. Table 4. Electric properties for germylidene. Method SCF MP2 SDQ-MP4 MP4 CCSD CCSD(T) ECC

μ 0.1853 0.1510 0.1095 0.0796 0.1271 0.0981 -0.0872

α 52.58 50.25 50.05 50.32 50.23 50.35 -2.23

Δα 24.56 21.02 22.97 22.66 22.83 22.65 -1.91

β -290.2 -48.2 -97.5 -81.0 -124.5 -120.3 169.9

γ 28949 28597 25055 27189 26824 28186 -763

The dipole moment of germylidene is higher than that of silylidene. Electron correlation more than halves this property. We observe a rather larger electron correlation effect on the dipole polarizability: at the CCSD(T) level of theory α = 50.35 and Δα = 22.65, lower by 4.2 and 8.8% than the respective SCF values. As in the case of silylidene the electron correlation effect on the first hyperpolarizability is very pronounced. The MP methods are in disagreement with both CC ones. Nevertheless, in the case of the second hyperpolarizability all post-Hartree-Fock methods do not change significantly the mean γ . The CCSD(T) value is just 2.6% lower than the SCF one. The mean dipole polarizability increases, as expected, for the three systems: α = 24.23 (H2CC), 47.39 (H2CSi), 50.35 (H2CGe). We observe that although the value of this property nearly doubles from vinylidene to silylidene, the increase is not as large from slilylidene to germylidene. In Figures 3 and 4 we have traced the evolution of the mean first and second hyperpolarizability for the vinylidene, silylidene and germylidene.


H2CX

50

X = C, Si, Ge 0

-100

3

3

β / e a0 Eh

-2

-50

-150

SCF MP2 CCSD(T)

-200

-250

-300

5

10

15

20

25

30

35

Z Figure 3. Evolution of the mean first dipole hyperpolarizability in the sequence H2CX, X=C,Si,Ge with the nuclear charge of X.

30000

H2CX

20000

X = C, Si, Ge

4

4

γ / e a0 Eh

-3

25000

15000

SCF MP2 CCSD(T)

10000

5000

5

10

15

20

25

30

35

Z

Figure 4. Evolution of the mean second dipole hyperpolarizability in the sequence H2CX, X=C,Si,Ge with the nuclear charge of X. At the SCF level of theory the magnitude of the mean first hyperpolarizability increases monotonically. Due to the very strong electron correlation effects, the monotonicity is destroyed at the


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403

post-Hartree-Fock level. This is not the case of the mean second hyperpolarizability, where γ (H2CC) < γ (H2CSi) < γ (H2CGe). As in the case of the mean dipole polarizability, the second hyperpolarizability of silylidene is much larger than that of vinylidene. The change from silylidene to germylidene is less important. We are not aware of previous electric property values for vinylidene, silylidene and germylidene. We have compared our findings for acetylene to those of other workers [42-44] in another paper [35].

5. Conclusions We have calculated electric (hyper)polarizabilities for acetylene, vinylidene, silylidene and germylidene. We have found that vinylidene is more (hyper)polarizable than acetylene. The electric (hyper)polarizability increases in the sequence H2CC→ H2CSi → H2CGe. The mean polarizability and second hyperpolarizability of silylidene is considerably larger than that of vinylidene. The same properties of germylidene are not significantly larger than those of silylidene.

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