On the bond-length dependence of the static electric polarizability and hyperpolarizability of F2

Chemical Physics Letters 442 (2007) 265–269 www.elsevier.com/locate/cplett

On the bond-length dependence of the static electric polarizability and hyperpolarizability of F2 George Maroulis Department of Chemistry, University of Patras, GR-26500 Patras, Greece Received 5 May 2007; in final form 30 May 2007 Available online 8 June 2007

Abstract We have calculated accurate electric (hyper)polarizabilities for F2. Relying on finite-field coupled-cluster calculations with very large, purpose-oriented basis sets we propose the following values (reference near-Hartree–Fock results in parentheses) for the mean 2 2 1 ( a=e2 a20 E1 c=e4 a40 E3 h Þ and the anisotropy (Da=e a0 Eh Þ of the dipole polarizability and the mean dipole hyperpolarizability ( h ):  a ¼
8:4812 (8.5870),
Da = 5.9446 (9.0719) and c ¼ 481 (268). For the derivatives of the dipole polarizability at Re we propose d a ¼ 3:11 and dDa ¼ 5:57. Discrepancies between experimental estimates and theoretical predictions of the mean polarizability dR e dR e are noted and discussed.  2007 Elsevier B.V. All rights reserved.

1. Introduction and theory The physicochemical properties of F2 are of certain interest, as this diatomic is a keynote molecule for fluorine chemistry. Rather little is known of its electric polarizability (aab) and hyperpolarizability (cabcd). The mean dipole polarizability ( a) has been extensively studied experimentally [1–5] but no definite value is known [6] for the static limit of this property. We are not aware of experimental data for the anisotropy (Da) or the mean hyperpolarizability (c). The aim of this study is to report accurate values for the static limit of the electric polarizability of F2. In addition, we have explored the bond-length dependence of the properties around the equilibrium internuclear separation in order to obtain reliable theoretical values for the dipole   daab polarizability derivatives , properties of major dR e importance in Raman spectroscopy [7]. Our approach to the calculation of electric properties relies on the finite-field method [8]. We follow closely Buckingham’s notation and terminology [9]. We will not present our computational approach and methodology here, as this has been done in sufficient detail in previous work [10,11]. E-mail address: [email protected] 0009-2614/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.06.024

We employ a large class of ab initio methods, ranging from self-consistent field (SCF) and Møller–Plesset perturbation theory methods (MP) to coupled-cluster (CC) techniques. In standard notation, the methods are SCF, MP2, SDQMP4, MP4, CCSD and CCSD(T). The highest level of theory used in this work is CCSD(T), singles and doubles coupled-cluster theory (CCSD) with an estimate of connected triple excitations obtained via a perturbational treatment. We lean heavily on the predictive capability of coupledcluster techniques. Elaborate presentations of our methodological arsenal may be found in standard sources [12–14]. 2. Basis set design and computational details The prediction of reliable values of electric (hyper)polarizabilities depends strongly on the choice of highly performance basis sets [15]. The quest for suitable basis sets enriches continuously the literature with new findings [16–19]. Our design of suitable, purpose-oriented basis sets for F2 leans on a computational philosophy developed in previous work and well-tested on difficult cases [20–22]. Thus, we have obtained two large basis sets using a flexible substrate of (12s7p) Gaussian-type functions (GTF) contracted to [7s4p] [23]. These are B1 ” [9s6p4d1f] and

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G. Maroulis / Chemical Physics Letters 442 (2007) 265–269

B2 ” [9s6p5d2f]. Even larger basis sets were built upon uncontracted (13s8p) and (18s13p) substrates constructed by Partridge [24]. These are B3 ” (15s10p5d3f) and B4 ” (20s15p9d5f). 5D and 7F GTF were used on all basis sets. All optimizations were carried out at the experimental bond-length of 2.66816a0 [25]. All calculations have been performed with GAUSSIAN 92 [26], GAUSSIAN 94 [27] and GAUSSIAN 98 [28]. Atomic units are used throughout this Letter. Conversion factors to SI units are: length, 1a0 = 0.529177249 · 1010 m, dipole polarizability, 1e2 a20 E1 h ¼ 1:648778 1041 C2 m2 J1 and hyperpolarizability, 1e4 a40 E3 h ¼ 6:235378  1065 C4 m4 J3 . Property values are given as 4 4 3 pure numbers, i.e. R/a0, aab =e2 a20 E1 h and cabcd =e a0 E h .

Table 2 Bond-length dependence (DR ” (R  Re)/a0) of the electric (hyper)polarizability of F2 calculated with basis set [9s6p4d1f] at the CCSD(T) level of theory

3. Results and discussion



d a ability derivatives dR and dDa calculated with basis dR e e B1. Last, in Table 4 we compare our findings with those of other authors and the available experimental data. In Fig. 1, we have plotted the R-dependence of the longitudinal component azz for various levels of theory. In Fig. 2, we display the R-dependence of the hyperpolarizability invariants. Last, in Fig. 3 we show the R-dependence of the longitudinal component czzzz.

In Table 1, we show SCF and correlated values calculated with basis sets B1, B2 and B3. We have added in this table SCF results obtained with the large B4 set. In Table 2, we give CCSD(T)/B1 values for the invariants  a, Da and c for a wide range of internuclear separations R/a0 = 1.0, 0.8, 0.6, 0.4, 0.2, 0.1, 0, 0.1, 0.2, 0.3 and 0.4. In Table 3, we show the method dependence of the polariz-

Table 1 Electric (hyper)polarizability

a

b

c

1.0 0.8 0.6 0.4 0.2 0.1 0 0.1 0.2 0.3 0.4

6.55 6.24 6.54 7.13 7.83 8.18 8.51 8.81 9.07 9.29 9.48

1.83 1.71 2.32 3.41 4.72 5.36 5.96 6.49 6.94 7.29 7.57

1438 680 462 423 424 449 463 469 552 572 586

cxxxx

cxxzz

c

D1c

D2c

9.1358 5.0811 5.9987 6.0403 5.9986 5.9643

186 916 651 520 668 580

203 384 385 436 371 395

137 170 189 221 157 171

255 524 487 514 457 463

156 1720 982 480 990 674

434 278 100 372 95 50

3.1715

395

192

34

208

518

384

8.5811 8.2510 8.4376 8.6474 8.3892 8.5088

9.1386 5.0868 6.0044 6.0459 6.0039 5.9694

195 933 661 529 679 604

213 405 396 446 415 459

141 178 185 217 169 189

265 545 491 517 493 517

156 1714 955 454 882 543

439 271 55 328 79 73

0.9841

0.0723

3.1691

409

246

48

251

387

366

14.6489 11.6075 12.4152 12.5949 12.3885 12.4443

5.5615 6.5577 6.4018 6.6127 6.3570 6.4997

8.5906 8.2410 8.4063 8.6068 8.3675 8.4812

9.0874 5.0498 6.0134 5.9822 6.0315 5.9446

202 921 647 531 659 579

212 395 388 441 390 422

141 174 183 214 160 175

267 534 482 513 468 481

182 1705 940 472 895 576

433 273 61 313 88 51

ECC

2.2046

0.9382

0.1094

3.1427

377

210

34

214

394

383

SCF

14.6349

5.5630

8.5870

9.0719

202

214

142

268

175

433

azz

axx

[9s6p4d1f]

SCF MP2 SDQ-MP4 MP4 CCSD CCSD(T)

14.6732 11.6372 12.4359 12.6731 12.3880 12.4848

5.5374 6.5561 6.4372 6.6328 6.3893 6.5205

ECC

2.1884

0.9831

SCF MP2 SDQ-MP4 MP4 CCSD CCSD(T)

14.6734 11.6422 12.4406 12.6780 12.3918 12.4884

5.5349 6.5554 6.4362 6.6321 6.3878 6.5190

ECC

2.1850

SCF MP2 SDQ-MP4 MP4 CCSD CCSD(T)

(20s15p9d5f)

Da

czzzz

Method

(15s10p5d3f)

a

for F2 at the experimental bond-length of 2.66816a0

Basis set

[9s6p5d2f]

DR

a

Da 8.5827 8.2498 8.4368 8.6463 8.3889 8.5086

0.074

The total electron correlation correction is defined as ECC ” CCSD(T)  SCF. a The invariants are defined as: mean, a ¼ ðazz þ 2axx Þ=3 and anisotropy, Da = azz  axx, of the dipole polarizability, mean, c ¼ ð3czzzz þ 8cxxxx þ 12cxxzz Þ=15 and anisotropies D1c = 3czzzz  4cxxxx + 3cxxzz and D2c = czzzz + cxxxx  6cxxzz of the hyperpolarizability. b The innermost MO was kept frozen in all post-Hartree–Fock calculations.

G. Maroulis / Chemical Physics Letters 442 (2007) 265–269 Table 3 Derivatives of the dipole polarizability of F2 calculated with basis set [9s6p4d1f]

da dDa Method dR e dR e

Method

a

Da

c

Theory SCFa SDQ-MP4a B3LYPb CCSDLRc PBE-DFTd B3LYPe B3PW91e SCFf CCSD(T)f SCFg CCSD(T)g SCFh

8.590 8.485 8.57 8.465 8.61 8.73 8.60 8.5827 8.5086 8.5906 8.4812 8.5870

9.213 5.694 6.12 6.007 5.90 6.42 6.32 9.1358 5.9643 9.0874 5.9446 9.0719

268 512

SCF MP2 MP4 CCSD CCSD(T)

20

18

-1

16 2

Table 4 Theoretical predictions and experimental data for the polarizability of F2 (reference results of this investigation in bold italics)

F2 Basis set [9s6p4d1f]

22

15.61 1.82 5.45 6.29 5.31 5.57

2

6.03 1.97 3.06 3.50 2.94 3.11

αzz / e a0 Eh

SCF MP2 SDQ-MP4 MP4 CCSD CCSD(T)

24

267

14

12

10

535 503 255 463 267 481 268

Experiment 8.40i 8.27 ± 0.17j 8.2 ± 0.6k a

Maroulis and Thakkar [29]. Fuentealba et al. [30]. c Dalskov and Sauer [31]. d Basis set d-aug-ccpVTZ, Van Caillie and Amos [32]. e Maroulis and Makris [33]. f Present investigation, basis set [9s6p4d1f]. g Present investigation, basis set [15s10p5d3f]. h Present investigation, basis set [20s15p9d5f]. i Dielectric constant data extrapolation to zero pressure, Straty and Younglove [3]. j Refractive index data extrapolation to the static limit, see Maroulis and Hohm [6]. k Recommended experimental value, see text. b

The SCF/B1 longitudinal and transversal components of aab at Re are azz = 14.6732 and axx = 5.5374. An analytical calculation gives azz = 14.6732035 and axx = 5.5374030, in perfect agreement with our finite-field values. We expect our SCF/B4 results to be very close to the Hartree–Fock limit for all properties. This basis yields a SCF energy for the free molecule 198.522143865Eh. For the (hyper)polarizability invariants we obtain the sequence a ¼ 8:5827 ðB1Þ; 8:5811 ðB2Þ; 8:5906 ðB3Þ; 8:5870 ðB4Þ, Da = 9.1358 (B1), 9.1386 (B2), 9.0874 (B3), 9.0719 (B4) and c ¼ 255 ðB1Þ; 265 ðB2Þ; 267 ðB3Þ and 268 ðB4Þ. Thus, all basis sets yield values very close to the B4/SCF ones. Turning our attention to the effects of electron correlation, we observe that the longitudinal component reduces consider-

8

6

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

(R-Re)/a0 Fig. 1. Bond-length dependence of the longitudinal polarizability component azz.

ably while the transversal one increases slightly. Overall, the electron correlation correction (ECC) is negative for both the mean and the anisotropy of the dipole polarizability. For B3 = [15s10p5d3f] we obtain ECC = 0.1094 ( a) and 3.1427 (Da), which correspond to a reduction by 1.2% and 34.6% of the respective SCF values. For the hyperpolarizability, we obtain CCSD(T)/B3 values for the Cartesian components czzzz = 579, cxxxx = 422 and cxxzz = 175 which are 186.6%, 99.1% and 24.1% above the respective SCF/B3 values. Thus, electron correlation affects significantly the hyperpolarizability of F2. The content of Table 2 shows clearly the strong Rdependence of the a, Da and c invariants, calculated at the CCSD(T)/B1 level. The effect is not uniform for all properties. The hyperpolarizability seems to increase considerably for very short bond-lengths. In Table 3, we show the variation of the first derivative of a and Da with level of theory. The derivatives have been calculated from bondlengths DR = ±0.2, ±0.1 and 0. It is rather interesting to note that the derivatives obtained at the SCF level are significantly reduced
at higher levels.
At the CCSD(T)/B1 d a level we obtain dR ¼ 3:11 and dDa ¼ 5:57, both values dR e e drastically lower than the respective SCF.

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G. Maroulis / Chemical Physics Letters 442 (2007) 265–269 5000

4000

F2 Basis set [9s6p4d1f]

F2 CCSD(T) [9s6p4d1f]

3000

4000 SCF MP2 MP4 CCSD CCSD(T)

2000

-3

1000 2000

4

4

4

γzzzz / e a0 Eh

4

γαβγδ / e a0 Eh

-3

3000

0

1000

-1000

γ Δ 1γ Δ 2γ

-2000

0

-3000

-1000

-1.2

-0.8

-0.4

0.0

0.4

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2

(R-Re)/a0 Fig. 2. Bond-length dependence of the hyperpolarizability invariants.

Let us now turn our attention to Figs. 1–3. Fig. 1 shows that the R-dependence of azz is very smooth for jDRj < 0.6. Electron correlation modifies the shape of this curve. The MP2 curve is most dissimilar to all others for bond-lengths R > Re. The behaviour of the Møller–Plesset methods in calculations far from Re has been systematically studied and commented upon (see Ref. [14]). The R-dependence of the mean and the anisotropies of cabcd at the CCSD(T)/ B1 shows a rather unusual behaviour for DR < 0. The displayed curves demonstrate clearly that the relative size of the Cartesian components czzzz, cxxxx and cxxzz varies largely with the internuclear separation. In the last figure, we observe that the R-dependence of the longitudinal component has much in common with that of azz. The MP2 and MP4 methods show extremely strong R-dependence for DR > 0, while the most accurate CCSD(T) curve varies smoothly in the same range of bond-lengths. Our findings are in good agreement with previous theoretical values for the dipole polarizability by Maroulis and Thakkar [29], Fuentealba [30], Dalskov and Sauer [31], Van Caillie and Amos [32]. Some more attention is due the B3LYP and B3PW91 values of the hyperpolarizability reported recently by Maroulis and Makris [33]. The respective values c ¼ 535 (B3LYP) and 503 (B3PW91) are a few percent above our best CCSD(T)/B3 results. To compare our theoretical values of  a to the available experimental data we have evaluated the zero-point vibra-

0.0

0.2

0.4

0.6

(R-Re)/a0 Fig. 3. Bond-length dependence of the longitudinal hyperpolarizability component czzzz.

tional correction (ZPVC). We rely on the simple scheme of Schlier [34] and experimental spectroscopic constants [26] to obtain the dependence of the mean polarizability on the vibrational number t as aðtÞ  aðRe Þ ¼ 0:0697ðtþ 1=2Þ. The ZPVC ðaÞ ¼ 0:0348 is very small. Experimentalists striving to determine the polarizability of fluorine have to face problems arising from the high reactivity and relatively low purity of the samples [35]. An overall examination of the experimental estimates [35] suggests a static value of a ¼ 8:2  0:6. Relying on our CCSD(T)/B3 results and the calculated ZPVC, we advance an estimate of the mean polarizability aðt ¼ 0Þ  8:52. This is in very good agreement with the above estimate. 4. Conclusions We have calculated accurate values for the electric (hyper)polarizability of F2. Reference values have been obtained for the invariants at the CCSD(T)/[15s10p5d3f] level of theory: a ¼ 8:4812, Da = 5.9446 and c ¼ 481. We have also obtained for the first time a clear picture of the R-dependence of the (hyper)polarizability of F2. Our final estimate for the static limit of the mean polarizability aðt ¼ 0Þ  8:52 compares well with the experimental 8.2 ± 0.6, suggesting agreement with experiment but more

G. Maroulis / Chemical Physics Letters 442 (2007) 265–269

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