Vibrational Frequencies of Fullerenes C60 and C70 under Pressure

Article pubs.acs.org/JPCA

Vibrational Frequencies of Fullerenes C60 and C70 under Pressure Studied with a Quantum Chemical Model Including Spatial Confinement Effects Marco Pagliai,*,† Gianni Cardini,*,‡ and Roberto Cammi*,§ †

Dipartimento di Chimica “Ugo Schiff”, Università degli Studi di Firenze, via della Lastruccia 3, 50019 Sesto Fiorentino, Florence, Italy ‡ Dipartimento di Chimica “Ugo Schiff”, Università degli Studi di Firenze, via della Lastruccia 3, 50019 Sesto Fiorentino, Florence, Italy § Dipartimento di Chimica, Università degli Studi di Parma, Parco Area delle Scienze 17/A, 43124 Parma, Italy S Supporting Information *

ABSTRACT: The equilibrium geometry structural and vibrational spectroscopic properties of fullerenes C60 and C70 under high pressure have been studied with a quantum-chemical computational approach in which ab initio calculations on a single fullerene molecule have been carried out within the polarizable continuum model framework to mimic pressure effects. The adopted approach has been revealed effective to explain the geometry variations and the frequency shifts observed experimentally.

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spectra,13,14 they are usually computationally demanding,15 thus limiting the calculations to few phase space points. In this respect, there is the necessity for a quantum mechanical method which can overcome this problem and allows mimicking on a single molecule the effects of the reduction of the intermolecular distances when an external pressure is applied. The effects on molecular properties of systems that enter the repulsive region of the intermolecular potential dominated by the Pauli repulsion have been analyzed by adopting several approaches, which can be collected in two different categories. In the first one, the effect of the Pauli repulsion is simulated using several types of model potentials.16−23 The basic idea is to use model external effective potential to represent the statistical average description of the intermolecular interaction. This model potential is included in the molecular Hamiltonian in the form of one electron operator, determining a spatial reduction of the electronic distribution of the molecular system. The second category relies on using as a confining environment aggregates of noble gases to model the repulsive potential.24−29 The structural and vibrational properties of C60 and C70 have been computed applying a recently proposed quantum chemical model,30 which takes into account the perturbation effects due to pressure, adopting an approach based on the PCM.31,32 This new model, called PCM-XP (extreme pressure),30,33,34 allows mimicking of the effects of a hydrostatic pressure on a

INTRODUCTION The interest in fullerenes spreads in many scientific and technological fields including high pressure chemistry and physics.1−11 The effects of extreme pressures on the structural and vibrational properties of fullerenes C60 and C70 have been studied, in the present work, by a novel computational approach based on the polarizable continuum model (PCM).31,32 Vibrational spectroscopies, in particular IR and Raman, are usually adopted and allow useful information regarding the structure and the intra- and intermolecular interactions in molecular systems to be obtained. The versatility of these spectroscopic probes relies on the possibility to monitor molecular properties in different thermodynamic conditions, as occurs, for example, in the synthesis of hard- or superhard materials, where high temperatures and pressures are applied to obtain the products. In particular, with the advent of the diamond anvil cell (DAC),12 the IR and Raman spectroscopies are revealed as powerful tools to investigate the phase transitions of bulk molecular materials as a function of the external pressure. In these studies, the phase transitions are inferred from the effect of the pressure on the vibrational frequencies mainly of the intramolecular normal modes. However, for a given phase, no attention is paid to the systematic rationale on the nature of the often linear dependence of the intramolecular vibrational frequencies as a function of the pressure. One of the problems is the unavailability of effective computational methods to easily assist in the interpretation of the behavior of the intramolecular vibrational frequencies as a function of the pressure. In fact, although periodic ab initio calculations are able to accurately reproduce both the harmonic and anharmonic vibrational © 2014 American Chemical Society

Received: April 29, 2014 Revised: June 16, 2014 Published: June 17, 2014 5098

dx.doi.org/10.1021/jp504173k | J. Phys. Chem. A 2014, 118, 5098−5111

The Journal of Physical Chemistry A

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system performing accurate calculations on an isolated molecule without substantial increase in the computational cost. In fact, the principal pressure effect, which is represented by the progressive reduction of intermolecular distances in the repulsive potential region, is taken into account by varying the PCM cavity hosting the quantum system. This allows information to be obtained to correctly interpret the structural and spectroscopic property variations at molecular level30 and to assist the interpretation and assignment of the vibrational modes. The paper is organized as follows: the basic aspects of the PCM-XP method for the description of the properties of molecular systems at extreme-high pressure are summarized in Theoretical Methods; a detailed description on the assessment of the exchange and correlation functionals and basis set for the calculation of the equilibrium geometry and the vibrational frequencies of the C60 and C70 is reported in Computational Details; the computational results of the effect of the pressure on the structural and spectroscopic properties of fullerenes C60 and C70 are presented in Results and Discussion.

dominated by the Pauli repulsion. A detailed description of the two interaction terms is reported in Cammi et al.30 The reference state for Ge−r[Ψ] is given by a hypothetical state composed by the noninteracting electrons and nuclei of the solute, and by the unperturbed medium at the chosen thermodynamic conditions.30 Within the single-determinant approximation expanding the molecular orbitals (MO) over a finite basis set, the stationary conditions of the free energy functional Ge−r[Ψ] lead to the Fock or Kohn−Sham equations for the molecular solute, whose solution gives the MO of the target molecule. Within this QM scheme, first order derivatives of the PCM energy functional are implemented for the evaluation of the stationary point of the potential energy surface (PES) corresponding to the equilibrium geometries, and for the determination of the first-order properties of the solute under the pressure p. In the canonical ensemble,40,41 the pressure can be connected to QM methods within the PCM-XP model,33,33,34 by eq 2:

THEORETICAL METHODS In the PCM-XP model,30 the medium is represented as a continuum distribution of matter which hosts, into a molecular shaped cavity, the target system, which is described at the quantum mechanical (QM) level. The continuum is characterized in terms of its dielectric permittivity and of its averaged electronic density, equal to the valence electronic distribution of the bulk environment at the given pressure condition. The QM description of the molecular solute is based on an effective Hamiltonian which includes terms representing the electrostatic, the Pauli repulsion, and the solute−solvent interactions. The electrostatic interaction is described by the component of the solute−solvent interaction in the PCM model for standard condition of pressure,31,32 by using the integral equation formalism (IEF).35−38 The Pauli repulsion between the molecule and the external medium is represented as the overlap between the charge distribution of the solute and that of the solvent in the region outside the molecular cavity, according to the expression given by Amovilli and Mennucci,39 and the effect of the pressure is therefore modeled by shrinking the volume of the cavity with respect to its reference value corresponding to the standard conditions of pressure (p = 1 bar). For each given volume of the cavity, the pressure is computed in terms of the basic energy functional of the PCMXP model.30 The basic energy functional to be minimized in the QM procedure39 has the thermodynamic status of a free energy for the whole system: molecule and environment. This functional is shown in eq 1:

(2)

⎛ ∂G ⎞ p = −⎜ e − r ⎟ ⎝ ∂Vc ⎠

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Ge − r[Ψ] =

1 Ψ Ĥ ° + Q̅ (Ψ; ϵ) ·V̂ + Vr̂ (V0) Ψ 2

where Vc is volume of the cavity hosting the molecule. The PCM-XP model accounts the effect of the pressure by shrinking the volume of the cavity. In fact, the effect of high pressure (several GPa) on a condensed phase system consists, at the macroscopic level, of a reduction of the volume of the entire system, to which corresponds, at the microscopic level, a reduction of the intermolecular distances below the van der Waals equilibrium distances, where the molecules experience a strong Pauli repulsion interaction.12,42−50 The shrinking of the cavity determines an increase of the amount of electronic charge density of the target molecule that lies outside of the cavity, with an increase of overlap between the electronic charge distribution of the solute with the charge distribution of the solvent, that corresponds to an increase of the Pauli repulsion of the solute with its surroundings. The PCM-XP model takes also into account the variation of the properties of the external medium, as is described in Results and Discussion.

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COMPUTATIONAL DETAILS A series of DFT calculations51−53 have been performed to correctly study the pressure dependence of structural and spectroscopic properties of fullerene C60 and C70 by checking which exchange and correlation functional provides results in better agreement with the experimental measurements at ambient conditions, with particular regard to the vibrational frequencies. The B3LYP exchange and correlation functional in conjunction with 6-31G(d) basis set has been adopted with success in the complete assignment of the vibrational spectra of C60 and C70 fullerenes.54,55 Structural and spectroscopic properties of isolated molecule or solid state fullerenes C60 and C70 have been studied by means of DFT calculations.56−64 However, a series of different exchange and correlation functionals have been recently developed and have been revealed capable of faithfully reproducing structural and spectroscopic properties.65−67 Geometry optimizations and vibrational frequency calculations have been performed with the Gaussian 09 suite of programs,68 using very tight criteria for convergence of both energies and forces and an ultrafine grid for the evaluation of the integrals.

(1)

where Ĥ ° is the Hamiltonian of the isolated molecule and Q̅ (Ψ;ϵ)·V̂ and V̂ r(V0) are the PCM electrostatic interaction term and the nonelectrostatic Pauli repulsion contribution, respectively. The physical meaning of the effective interaction operator of eq 1 is to describes a statistical average interaction between the target molecule and the environment. We remark that the repulsion interaction operator, V̂ r(V0), is essential to describe the properties of the molecule at extreme high pressure, where the intermolecular distances fall within the region 5099



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Table 2. Vibrational Frequencies (in cm−1) of the Raman (Ag and Hg) and IR (T1u) Active Modes for C60a

Table 1. Bond Lengths (in Å) in C60 and C70 Fullerenes Computed at A = B3LYP/6-31G(d), B = PBE0/6-31G(d), C = M06-2X/6-31G(d), D = wB97-xD/6-31G(d), E = TPSSh/ 6-31G(d), and ED = Electron Diffraction C60

A

B

C

D

E

ED70

5,6 6,6 C70

1.4534 1.3955 A

1.4473 1.3910 B

1.4514 1.3874 C

1.4495 1.3860 D

1.4522 1.3979 E

1.458 1.401 ED71

a−a a−b b−c c−c c−d d−d d−e e−e

1.4518 1.3968 1.4482 1.3888 1.4493 1.4345 1.4209 1.4709

1.4458 1.3924 1.4422 1.3844 1.4439 1.4285 1.4159 1.4639

1.4497 1.3892 1.4457 1.3792 1.4514 1.4256 1.4153 1.4690

1.4478 1.3881 1.4435 1.3766 1.4512 1.4221 1.4134 1.4678

1.4507 1.3991 1.4472 1.3921 1.4472 1.4359 1.4218 1.4692

1.461 1.388 1.453 1.386 1.468 1.425 1.405 1.538

symm

assign.54 B3LYP 496 1468 264 430 709 773 1101 1251 1425 1576 525 578 1182 1433

Ag Hg

T1u

scaling factor MAE

The bond lengths of C60 and C70 molecules computed with B3LYP and other functionals, using the 6-31G(d) basis set, have been collected in Table 1. The bond lengths have been labeled in agreement with the nomenclature usually employed and arising by the different symmetry of the two molecules.69 In fact, C60 has a truncated icosahedron shape, characterized by an Ih point group symmetry, in which are present only two different bond lengths (“5,6” and “6,6”), whereas the presence in the C70 of an equatorial belt which separates two hemispheres lowers the point group symmetry to D5h and increases the number of different bond lengths (eight), as shown in Figure 1.

a

PBE0

M06-2X

wB97XD

TPSSh

483 1478 256 418 689 766 1095 1256 1430 1578 514 573 1188 1436 0.955 8.07

484 1471 256 422 696 766 1096 1254 1425 1592 519 573 1185 1428 0.954 6.71

480 1472 257 424 700 764 1094 1250 1429 1594 523 570 1180 1430 0.953 6.86

485 1477 257 421 689 769 1098 1253 1428 1574 514 575 1189 1436 0.979 6.71

486 1470 260 427 704 771 1102 1249 1423 1581 525 575 1187 1428 0.978 3.50

The basis set adopted in the calculations is 6-31G(d).

exchange and correlation functional. The effect of the basis set on the vibrational frequencies has been verified adopting this exchange and correlation functional, as reported in Table 3. Table 3. Vibrational Frequencies (in cm−1) of the Raman (Ag and Hg) and IR (T1u) Active Modes for C60 Using B3LYP Exchange and Correlation Functional symm Ag Hg

T1u

Figure 1. Molecular structure of fullerene C60 (left) and C70 (right). The 5,6 and 6,6 bonds in C60 molecule are represented in red and blue, respectively. The color schema adopted to represents the eight bond lengths in C70 is a−a = magenta, a−b = red, b−c = orange, c−c = yellow, c−d = lime, d−d = green, d−e = blue, e−e = violet.

scaling factor MAE

assign.54

6-31G(d)

6-311G(d)

cc-pvdz

cc-pvtz

496 1468 264 430 709 773 1101 1251 1425 1576 525 578 1182 1433

486 1470 260 427 704 771 1102 1249 1423 1581 525 575 1187 1428 0.978 3.50

488 1468 261 433 716 773 1105 1244 1418 1582 535 577 1188 1423 0.983 5.14

486 1474 259 426 704 770 1101 1248 1424 1579 523 576 1187 1432 0.976 3.57

489 1466 263 436 728 774 1107 1243 1417 1581 542 579 1188 1422 0.984 7.00

Although the vibrational frequencies are less affected by basis set than exchange and correlation functional as confirmed by the value of MAE obtained by calculations, the most accurate results have been achieved with the B3LYP/6-31G(d) level of theory. The application of the computed scaling factor has been extended to the C70 fullerene. Because C70 belongs to D5h symmetry group, the A′1, E′2, and E″1 modes are active in Raman, while those E′1 and A″1 are active in IR. The vibrational frequencies computed for fullerene C70 are summarized in Table 4. It is important to note that the results in better agreement with experiments have been obtained adopting the B3LYP/ 6-31G(d) level of theory. As reported by Schettino et al.,55 the largest discrepancy between calculations and measurement is related to the normal mode assigned to one with symmetry E′1

Starting from the equilibrium geometries, the vibrational frequencies of C60 fullerene have been calculated and uniformly scaled to obtain better agreement with the available experimental measurements at ambient conditions. The scaling factor has been determined by taking as reference the assignment of the vibrational frequencies of the active IR and Raman normal modes of C60,54 adopting a least-squares procedure. The scaled frequencies (ordered by symmetry), the scaling factor, and the mean absolute error (MAE) are summarized in Table 2. Since the C60 molecule belongs to the Ih point group, the normal modes with symmetry Ag and Hg are active in Raman while those with T1u symmetry are active in IR. The better agreement between experimental and computed vibrational frequencies has been obtained by using the B3LYP 5100



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Table 4. Vibrational Frequencies of the Raman (A1′, E2′, and E1″) and IR (E1′ and A2″) Active Modes for Fullerene C70a symm A′1

E1′

E″1

a

assign.55 B3LYP 260 396 455 568 697 701 1060 1182 1222 1459 1468 1576 225 303 430 505 535 668 688 721 738 743 760 768 948 1055 1196 1258 1332 1349 1374 1500 1520 1580 250 410 480 520 548 675 712 733 741

252 392 448 564 700 705 1060 1183 1227 1448 1469 1571 220 299 425 502 533 664 688 722 734 739 749 766 944 1057 1184 1255 1327 1348 1371 1499 1521 1571 245 408 477 514 545 674 712 734 738

PBE0

M06-2X

249 385 445 552 690 696 1052 1190 1232 1454 1475 1567 216 293 416 492 521 650 672 700 717 725 734 748 938 1056 1178 1255 1332 1346 1372 1496 1519 1568 241 399 468 503 534 658 696 717 730

249 387 446 557 696 699 1050 1189 1220 1454 1478 1586 216 293 418 495 521 649 679 710 728 732 743 748 938 1042 1179 1246 1328 1341 1371 1504 1525 1584 240 401 471 503 537 657 705 729 731

wB97XD TPSSh 252 389 442 559 696 702 1049 1188 1217 1455 1483 1591 215 294 419 498 526 656 682 709 729 734 747 754 938 1038 1179 1243 1328 1340 1371 1506 1530 1588 242 401 474 506 538 665 708 727 735

symm

251 387 446 552 690 699 1056 1187 1235 1449 1473 1562 217 296 420 494 527 655 674 704 718 725 736 753 941 1062 1181 1258 1331 1348 1372 1495 1518 1563 243 402 459 507 536 662 696 718 732

A2″

E1′

scaling factor MAE

assign.55 B3LYP 800 1050 1172 1227 1296 1311 1367 1445 1511 1545 320 458 564 707 901 1133 1203 1320 1460 1557 328 361 418 509 534 578 642 674 728 795 835 904 1087 1176 1251 1291 1321 1414 1430 1489 1563

793 1052 1167 1223 1294 1311 1364 1430 1514 1571 318 458 563 700 895 1141 1204 1319 1461 1564 325 358 415 507 532 571 639 665 727 750 826 904 1086 1176 1253 1288 1316 1413 1429 1488 1566 0.978 4.38

PBE0

M06-2X

776 1048 1163 1224 1293 1308 1370 1437 1513 1567 311 448 560 685 890 1137 1209 1327 1467 1560 319 352 407 499 528 560 624 650 709 725 809 898 1083 1172 1254 1287 1312 1413 1435 1487 1563 0.955 10.76

773 1039 1156 1221 1282 1298 1357 1431 1521 1584 310 450 561 695 876 1140 1177 1307 1459 1577 320 352 409 501 529 567 623 655 721 739 805 898 1082 1168 1252 1284 1307 1411 1428 1495 1582 0.954 11.32

wB97XD TPSSh 781 1035 1152 1225 1284 1298 1356 1435 1526 1587 311 452 558 699 870 1140 1179 1307 1458 1578 322 355 409 501 528 570 629 658 725 742 817 898 1081 1168 1249 1284 1308 1411 1430 1501 1587 0.953 10.73

782 1053 1167 1224 1295 1313 1373 1436 1511 1562 314 449 562 684 898 1137 1215 1332 1468 1557 321 355 410 502 530 560 628 654 708 727 816 901 1084 1175 1255 1288 1317 1414 1435 1484 1559 0.979 9.46

The basis set adopted in the calculations is 6-31G(d).

at 795 cm−1, which is computed in the frequency range between 725 and 750 cm−1 with the different exchange and correlation functional. The complete list of the calculated frequencies for C60 and C70 is reported in Tables S1 (frequencies for C60 with different exchange and correlation functionals), S2 (frequencies for C60 with different basis set), and S3 (frequencies for C70 with different exchange and correlation functionals) of the Supporting Information.

terms of a set of spheres each centered on a carbon atom with radii Ri equal to the corresponding atomic van der Waals radii72 (RvdW = 1.75 Å) times a scaling factor f, i.e., Ri = RvdW f. To determine the SES cavity, a solvent molecular probe modeled as a hard sphere of given radius (Rsolv) is used.30 A further sphere has been added in the center of C60 (RvdW(dummy) = 2.00 Å) and two in C70 (RvdW(dummy) = 2.95 Å) centered symmetrically along the C5 axis at 0.2 Å from the origin. These parameters have been chosen to not allow the solvent molecules to be inserted inside the fullerenes. The bond lengths and the vibrational frequencies of the fullerenes outside the PCM-XP protocol have been considered at the pressure of 0 GPa, whereas values of pressure until ∼20 GPa have been reached shrinking the cavity with the progressive reduction of the scaling factor f, from f = 1.2 to f = 1.0.

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RESULTS AND DISCUSSION Pressure effects on the structural and spectroscopic properties of the fullerenes C60 and C70 have been obtained with PCM-XP calculations, using the B3LYP hybrid functional and the 631G(d) basis set. In this approach, the solvent excluding surface (SES) cavity hosting fullerenes C60 and C70 has been defined in 5101



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All the calculations have been performed following the protocol proposed by Cammi et al.,30 with the repulsive step potential V0(f) defined in eq 3: V0(f ) =

⎛ f ⎞9 4π 0 0 n S⎜ ⎟ 0.7E ha03 ⎝ f ⎠

(3)

where n0S is the numeral density of the external medium at standard condition of pressure. The dielectric permittivity, ϵ( f), used for the determination of the polarization charges of the solvent is parametrized as in eq 4: e(f ) = 1 + (e0 − 1)(f0 /f )3

(4)

with ϵ0 the dielectric permittivity of the medium at the standard pressure condition. Since cyclohexane has been revealed an excellent medium in high pressure experiments,73 it has been used as solvent in the PCM-XP scheme (ε0 = 2.0165, n0S = 2.581 × 1023 cm−3, and R0solv = 2.815 Å), with a functional form derived by the Onsager equation74,75 for the dielectric permittivity, by assuming the polarizability of the medium molecules to be pressure independent.30 In Table S4 of the Supporting Information the parameters adopted in the calculations increasing pressure are summarized. The equilibrium geometries of the molecular solutes have been obtained at different pressure values, performing all the optimizations within the fixed cavity approximation, which allows the relaxation of the positions of the nuclei but not that of the centers of the spheres. The geometry optimizations for the various cavity factors have been performed with a step-bystep procedure, in which the optimized structure to a cavity coefficient f has been employed as input in the subsequent calculation with the reduced f, starting from the equilibrium geometry of the isolated fullerene. The harmonic vibrational frequencies have been numerically computed from the analytical gradient30 within the fixed cavity approximation for the scaling cavity coefficient f = 1.2, 1.1, and 1.0 at the corresponding optimized geometries. The Pressure Effects on Structural Properties. In this subsection, the equilibrium structures of C60 and C70 as a function of pressure have been obtained by a shrinking of the molecular cavity enclosing the fullerene system. The pressure p as a function of the cavity volume, Vc, hosting fullerene C60 or C70, is shown in Figure 2. The lines reported in Figure 2 have been obtained by fitting the computed pressure (in GPa) values at each cavity volume (in Å3) with a Murnaghan type equation, eq 5:76 p=

B′ ⎡ ⎤ B0 ⎢⎛ V c0 ⎞ · ⎜ ⎟ − 1⎥ ⎥ B′ ⎢⎣⎝ Vc ⎠ ⎦

Figure 2. Pressure, p, as a function of the cavity volume, Vc for C60 (blue ○) and C70 (red ◇). The data have been fitted with a Murnaghan type equation. The solid blue and dashed red curves represent the fit of the data for C60 and C70, respectively.

between molecular properties computed by the PCM-XP model and the available experimental counterpart. Analyzing the effect of the pressure on the bond lengths of fullerenes C60 and C70, Figure 3 and Figure 4 show that the

Figure 3. Bond length variation as a function of pressure in fullerene C60. The 5,6 and 6,6 bonds in C60 molecule are represented in red and blue, respectively.

compression occurs uniformly in the simulated range of pressure. In fact, the bond length variation exhibits a linear dependence as reported in Table 5 and Table 6. In particular, the data for C60 in Table 5 shows that the 5,6 bond is slightly more compressed than the 6,6, as expected on the basis of its nature. Similarly in the case of C70 a more pronounced slope related to the bond of the equatorial belt (e−e) is obtained, as a consequence of a higher contraction. The coordinates of the optimized geometries for C60 and C70 are reported in Tables S5 and S6 of the Supporting Information, respectively. The results regarding the geometry of C60 and C70 can be rationalized explaining the effect of the pressure on the bond length contraction and its linear variation. These two elements are strongly correlated and can be analyzed considering in first instance the origin of the forces acting on the carbon nuclei, and consequently on the bond length contraction. This aspect is related to the direct effect of the pressure on the electronic

(5)

where B0 is the bulk modulus, B′ is its derivative with respect to pressure, and V0c is the cavity volume for the factor f = 1.2. The fitted values for B0 and B′ are 22.50 GPa and 9.80 for C60 and 25.47 GPa and 9.99 for C70, respectively. The Murnaghan type equation of state76 for fullerenes C60 and C70 is in satisfactory agreement with some of the available experimental equations of state of the pressure p as a function of the molar volume.77 In fact, Duclos et al.78 have determined by X-ray measurement a value of B0 = 18.1 GPa and B′ = 5.7 for C60, whereas Chrystides et al.79 have obtained a value of B0 = 25 GPa and B′ = 10.6 for C70. These results are particularly useful for a correct comparison 5102



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Figure 4. Bond length variation as a function of pressure in fullerene C70. Selected bond lengths of C70 have been reported adopting the following color scheme: aa = red, cc = green, dd = blue, ee = black.

Table 5. Pressure Effects on the Bond Lengths in Fullerene C60a pressure/GPa

5,6/Å

6,6/Å

0.0 0.9 1.7 5.4 8.6 12.8 19.3 (dr/dp)/Å·GPa−1

1.4534 1.4528 1.4521 1.4508 1.4481 1.4462 1.4433 −0.00053

1.3955 1.3949 1.3943 1.3931 1.3907 1.3890 1.3864 −0.00048

Figure 5. Isosurfaces representing the decrease (green) and increase (orange) of electron density ρ with pressure in C60 (top) and C70 (bottom), respectively. The pressure values adopted to compute ρhp and ρlp are 19.3 and 0.9 GPa for C60 and 19.6 and 0.7 GPa for C70 respectively. In both cases the molecular geometry corresponds to the equilibrium structure at 0 GPa. The cutoff of isodensity surfaces is 0.000165 au.

where ρhp and ρlp are the densities computed at the higher and lower values of the cavity volume, respectively. Figure 5 shows that the electron density partially moves from the peripheral intermolecular region toward the atoms and bonds of the fullerene, as a consequence of the Pauli repulsion between the molecule and the external medium due to the increase of the external pressure. This result provides a first explanation on the bond length contraction discussed in Table 5 and Table 6. However, while in the case of C60 the pressure effect induces an almost uniform increase of the electron density on the bond length, the situation related to the C70 molecule reflects the structural variations. In fact, Figure 5 shows a more pronounced variation of the electron density in the hemisphere region than in the equatorial belt, where however an increase in the density occurs on the e−e bond. This result not only allows elucidation of the structural variations induced by pressure but also gives a first explanation

The angular coefficients dr/dp (Å·GPa−1) obtained by a linear regression procedure of the variation of the bond length with pressure are reported. a

charge distribution. This relation has its roots on the force concept based on the Hellmann−Feynman theorem, which provides the necessary link between the changes in the molecular structure and electronic distribution, as described in Cammi et al.30 The effect of the pressure can be qualitatively discussed by analyzing the deformation of the electronic density of fullerene C60 and C70 reported Figure 5. Performing the calculation on the equilibrium structure of C60 and C70, the variation of the electronic density has been obtained by eq 6: Δρ = ρhp − ρlp

(6)

Table 6. Pressure Effects on the Bond Lengths in Fullerene C70a

a

pressure/GPa

a−a/Å

a−b/Å

b−c/Å

c−c/Å

c−d/Å

d−d/Å

d−e/Å

e−e/Å

0.0 0.7 2.0 4.4 6.0 9.8 14.1 19.6 (dp/dp)/Å·GPa−1

1.4518 1.4511 1.4505 1.4493 1.4486 1.4472 1.4452 1.4424 −0.00046

1.3968 1.3961 1.3956 1.3944 1.3938 1.3925 1.3908 1.3881 −0.00042

1.4482 1.4475 1.4469 1.4456 1.4449 1.4434 1.4414 1.4386 −0.00047

1.3888 1.3882 1.3876 1.3864 1.3858 1.3844 1.3826 1.3798 −0.00044

1.4493 1.4486 1.4480 1.4468 1.4460 1.4444 1.4424 1.4396 −0.00048

1.4345 1.4338 1.4331 1.4317 1.4309 1.4294 1.4275 1.4245 −0.00049

1.4209 1.4201 1.4196 1.4182 1.4174 1.4159 1.4139 1.4110 −0.00049

1.4709 1.4701 1.4692 1.4675 1.4665 1.4645 1.4619 1.4580 −0.00063

The angular coefficients dr/dp (Å·GPa−1) obtained by a linear regression procedure of the variation of the bond length with pressure are reported. 5103



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Figure 6. Left: radial mode with symmetry Ag(1) for C60 (breathing mode). Center: tangential mode with symmetry Ag(2) for C60. Right: radial mode with symmetry A1′ (2) for C70 (breathing mode).

on the structural reactivity of C70 under pressure, which occurs essentially involving the two hemispheres. Although the variation of the electron density allows the explanation of the bond length contraction, the linear dependence with pressure requires an analysis in terms of symmetry considerations. In fact, the pressure acting on the molecular solute within the PCM-XP model is spatially isotropic (i.e., hydrostatic), and as a consequence the symmetry point group of the molecule is invariant under pressure. It has to be noted that this description of the pressure can be justified in the case of fullerenes by considering that the experimental shear modulus anisotropy of C60 is very close to the one of an isotropic system such as solid argon.12 In this respect, fullerenes C60 and C70 maintain, respectively, the Ih and the D5h point groups, although some small deviations are due to the intrinsic limit of the numerical procedures used by the PCM-XP model to treat the cavity surface elements. The effect of the pressure on equilibrium geometries of the two studied systems under pressure corresponds to bond length contractions and shifts along the totally symmetric normal modes of the isolated fullerenes. In fact, fullerene C60 has two totally symmetric (TS) vibrational normal modes, with Ag(1) corresponding to in phase radial displacements of the nuclei (i.e., breathing), and Ag(2) corresponding to a displacement of each carbon atom in the plane perpendicular to its radial axis (tangential vibrational motion), as reported in Figure 6. Fullerene C70, instead, presents 12 TS vibrational normal modes, with the A1′ (2) mode corresponding to in phase radial displacements of the nuclei, and the other corresponding to more complex radial or tangential displacements of the carbon nuclei. The breathing mode of fullerene C70 is graphically represented in Figure 6. The variation of the molecular geometry of fullerenes C60 and C70 under pressure has been also quantified by projecting the atomic coordinates along the corresponding totally symmetric normal modes. The pressure induces a shift of the geometry along a single totally symmetric normal mode, the Ag(1) mode for C60 and the A′1(2) mode for C70 as shown in Figure 7. According to Cammi et al.,30 the slope of the corresponding linear correlations of the TS mode as a function of the pressure can be interpreted in terms of suitable response properties of the molecular solute evaluated at the equilibrium geometry at 0 GPa. The shift of totally symmetric normal coordinates Qi under pressure p is expressed by eq 7: Γ Q i(p)eq = − i p ki

i ∈ TS

Figure 7. Shift of the Ag(1) and A′1(2) TS modes at different values of pressure (Qeq i (p)) in C60 (blue ○) and C70 (red ◇), respectively. The result of the linear regression (last column of Table 7) is represented by solid blue and dashed red lines for C60 and C70, respectively.

of the isolated molecules. The coupling parameter Γi is defined as a mixed second derivative of the basic energy functional Ge−r, with respect to the normal coordinate Qi and to the pressure, as reported in eq 8: ⎛ ∂ 2G ⎞ e−r ⎟⎟ Γi = ⎜⎜ ∂ ∂ p Qi ⎠ ⎝ Q =0 i

(8)

being the derivative evaluated at the equilibrium geometry in vacuo. The pressure coefficient Γi has been formally introduced by Fujimura et al.80 in a series of pioneer studies on the effect of the pressure on the potential energy surfaces of molecules under pressure and has the physical meaning of a molecular response function describing the effect of the external pressure on the component of the gradient of the energy along the normal coordinate Qi. We note that, for symmetry reasons, Γi is different from zero only for TS normal modes. The computed values of the force constant ki and the pressure coupling constant Γi for fullerenes C60 and C70 have been reported in Table 7. Their ratio −Γi/ki is then compared with the pressure coefficient (dQi/dp) of the molecular compression along the normal mode i, showing that eq 7 is verified quantitatively for both the fullerenes C60 and C70. Pressure Effects on Vibrational Properties. In this subsection the pressure effects on the calculated vibrational frequencies are discussed and the results are compared with the available experimental measurements.

(7)

where Γi is a pressure coupling constant for the force acting on the nuclei, and ki is the force constant of the ith normal mode 5104



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−1 Table 7. Pressure Coupling Constant Γi (Eha−1 0 GPa ), −1 Harmonic Force Constants in Gas Phase ki (Eha−1 ·Å ), 0 −1 Ratio Γi/ki (Å·GPa ), and Pressure Coefficients dQi/dp (Å·GPa−1) of C60 and C70a

Γi −1 (Eha−1 0 GPa )

ki −1 (Eha−1 0 ·Å )

−Γi/ki (Å·GPa−1)

dQeq i /dp (Å·GPa−1)

0.00194 0.00249

0.21178 0.17975

−0.0091 −0.0115

−0.0097 −0.0105

C60 C70 a

The reported values refer to the PCM/B3LYP/6-31G(d) level of calculation, and regard the totally symmetric normal coordinates Ag(1) and A′1(2). The pressure coupling Γi has been evaluated from linear fitting of the nuclear gradients of fullerene GQe−ri as a function of the pressure, and at the equilibrium energy in gas phase. The pressure −1 coefficients dQeq i /dp (Å·GPa ) have been evaluated by linear fitting of the normal mode coordinate Ag(1) and A′1(2), respectively. Figure 8. Correlation between the relaxation pressure coefficient obtained as a difference between pressure coefficient and the curvature contribution and though the anharmonic vibrational constant. The blue and red data refer to C60 and C70, respectively.

A selection of computed vibrational frequencies of fullerenes C60 and C70, for which is present the experimental measurement of the pressure coefficient (dv/dp), are summarized in Table 8 and Table 9. The calculated values of dv/dp have been determined by a linear regression of the ν(p) reported in Table 8 and Table 9; the linear trend is confirmed by the value of the coefficient of determination, R2, always close to 1. All the computed frequencies for each value of pressure are reported in Tables S7 and S8 of the Supporting Information for C60 and C70, respectively, while in Tables S9 and S10 of the Supporting Information are reported the IR intensities for selected normal modes. To explain the observed linear behavior of the harmonic vibrational frequencies as a function of the pressure, the vibrational frequency shift of fullerenes C60 and C70 has been analyzed by partitioning the pressure coefficient for the harmonic vibrational frequencies (∂v(p)/∂p) obtained within the PCM-XP procedure in the curvature and relaxation contribution,30 as reported in eq 9: ⎛ ∂ν(p) ⎞ ⎛ ∂ν(p) ⎞ ⎛ ∂ν(p) ⎞ =⎜ ⎜ ⎟ ⎟ +⎜ ⎟ ⎝ ∂p ⎠Q(p) ⎝ ∂p ⎠cur ⎝ ∂p ⎠rel

confinement effect of the repulsive intermolecular interactions, due to the variation of the second derivative (i.e., curvature) of the potential energy surfaces, evaluated at the zero-pressure equilibrium geometry. Instead, the relaxation contribution is related, throughout the anharmonic force constant to the relaxation of the equilibrium geometry of the molecule under the effect of the pressure. Although a similar analysis of the vibrational frequency shift as a consequence of external pressure has been previously undertaken with a semiclassical approach,81 PCM-XP overcomes the incapability to accurately estimate, for general molecular systems, the single effects. In this respect, the PCM-XP partition of the pressure coefficient for the harmonic vibrational frequencies can be formulated in the following equations, for the curvature (eq 10) and relaxation (eq 11) contribution:

(9)

where Q(p) denotes the vibrational frequencies evaluated at the equilibrium geometries. Such a treatment is important to rationalize the reasons for the shift observed in the experimental measurements of the system under consideration. In particular, the curvature contribution is related to the

⎛ ∂ν(p) ⎞ ⎛ ∂ν(p) ⎞ ⎜ ⎟ =⎜ ⎟ ⎝ ∂p ⎠cur ⎝ ∂p ⎠Q(0)

(10)

⎞ ⎛ TS ⎛ ∂ν(p) ⎞ Γ ⎜ ⎟ = ⎜⎜ −∑ giij̃ (0) i ⎟⎟ ki(0) ⎠ ⎝ ∂p ⎠rel ⎝ j Q(0)

(11)

where Qi(0) denotes the equilibrium geometry of the isolated molecule, g̃iij(0) is the cubic anharmonic coupling constant

Table 8. Computed Vibrational Frequencies for the Raman and IR Active Modes of C60 at the Simulated Pressure Valuesa

a

symm

assign. cm−1

0 GPa cm−1

0.9 GPa cm−1

5.4 GPa cm−1

19.3 GPa cm−1

dv/dp cm−1 GPa−1

Ag(1) Ag(2) Hg(1) Hg(2) Hg(3) Hg(4) Hg(7) Hg(8) T1u(1) T1u(2) T1u(3) T1u(4)

496 1468 264 430 709 773 1425 1576 525 578 1182 1433

486 1470 260 428 704 770 1423 1582 525 575 1188 1428

488 1473 262 429 705 772 1425 1584 526 577 1189 1431

494 1482 266 431 707 776 1434 1591 528 582 1195 1440

516 1520 283 438 713 792 1469 1620 531 600 1218 1474

1.54 2.6 1.17 0.5 0.43 1.12 2.38 1.96 0.29 1.28 1.57 2.39

A scaling factor of 0.978 has been applied. 5105



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Table 9. Computed Vibrational Frequencies for the Raman and IR Active Modes of C70 at the Simulated Pressure Valuesa

a

symm

assign. cm−1

0.0 GPa cm−1

0.7 GPa cm−1

4.4 GPa cm−1

19.6 GPa cm−1


A1′ (4) A1′ (7) A′1(8) A′1(9) A1′ (11) A1′ (12) E′2(1) E′2(16) E2′ (19) E1″(4) E″1 (9) E1″(16) E1″(17) E1″(18) A″2 (2) A2″(3) A2″(6) A2″(9) E′1(4) E1′ (5) E1′ (6) E′1(7) E′1(8) E1′ (10) E1′ (15) E′1(16) E′1(19) E1′ (20) E1′ (21)

568 1060 1182 1222 1468 1576 225 1258 1374 520 741 1367 1445 1511 458 564 1133 1460 509 534 578 642 674 795 1251 1291 1430 1489 1563

564 1060 1183 1227 1469 1571 220 1255 1371 514 738 1364 1430 1514 458 563 1142 1461 507 532 572 638 665 751 1253 1288 1429 1487 1566

564 1061 1186 1229 1472 1573 221 1257 1373 515 739 1367 1433 1516 458 564 1143 1464 507 534 572 638 666 751 1255 1290 1432 1490 1568

566 1066 1194 1236 1481 1580 226 1263 1380 516 741 1375 1441 1524 460 569 1148 1472 510 539 574 640 667 753 1261 1297 1440 1497 1576

573 1085 1224 1262 1512 1606 244 1287 1406 522 748 1405 1471 1551 469 589 1168 1502 520 556 581 645 674 758 1284 1322 1469 1524 1602

0.44 1.26 2.03 1.75 2.14 1.78 1.24 1.61 1.77 0.40 0.46 2.04 2.02 1.86 0.57 1.33 1.35 2.08 0.69 1.18 0.50 0.35 0.46 0.39 1.56 1.68 1.99 1.84 1.82

A scaling factor of 0.978 has been applied.

Table 10. Experimental and Computed Pressure Coefficients and Curvature and Relaxation Contributions for a Series of Normal Modes of Fullerene C60a

a

symm

assign. cm−1


(dv/dp)cur cm−1 GPa−1

(dv/dp)rel cm−1 GPa−1

exptl cm−1 GPa−1

type

Ag(1) Ag(2) Hg(1) Hg(2) Hg(3) Hg(4) Hg(7) Hg(8) T1u(1) T1u(2) T1u(3) T1u(4)

496 1468 264 430 709 773 1425 1576 525 578 1182 1433

1.54 2.6 1.17 0.5 0.43 1.12 2.38 1.96 0.29 1.28 1.57 2.39

0.92 −0.36 1.14 0.97 0.67 0.25 −0.30 −0.19 0.88 0.62 −0.04 −0.36

0.60 2.95 0.01 −0.46 −0.24 0.85 2.67 2.14 −0.61 0.65 1.61 2.74

0.5 5.5 3.3 0.5 −0.8 0.1 3.9 4.8 −0.5 2.3 3.8 4.3

radial tangential radial radial radial radial tangential tangential radial radial tangential tangential

For each mode has been indicated the radial or tangential character. The experimental data have been taken from ref 77.

between the ith and the jth normal modes, and ki(0) and Γi are the harmonic force constant for the isolated molecule and the pressure coupling constant (eq 7). The curvature contribution has been evaluated by the calculation of the vibrational frequencies as a function of the pressure at the fixed equilibrium geometry in vacuo. Instead, the evaluation of the relaxation contribution is, in general, more complex and computationally more expensive, because it requires the calculation of the anharmonic vibrational constants, g̃iij(0), for the isolated molecules. However, this contribution can be easily obtained as the difference between eq 9 and eq 10.

The evaluation of the anharmonic vibrational constants g̃iij(0) becomes feasible, limiting the analysis to the totally symmetric normal modes j, Ag(1) and A1′ (2) for fullerene C60 and C70, respectively. The anharmonic vibrational constants are reported in Tables S11 and S12 of the Supporting Information for C60 and C70, respectively. In this respect, the methods for the determination of the relaxation contribution can be evaluated and compared for consistency with those obtained from the difference, as shown in Figure 8. In fact, by the linear regression of the data reported has been obtained a coefficient of determination for C60 and C70 of 0.996 and 0.982, respectively. 5106



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Table 11. Experimental and Computed Pressure Coefficients and Curvature and Relaxation Contributions for a Series of Normal Modes of Fullerene C70a symm

assign. cm−1


A1′ (4) A1′ (7) A′1(8) A1′ (9) A1′ (11) A′1(12) E′2(1) E2′ (16) E2′ (19) E″2 (4) E″2 (9) E2″(16) E2″(17) E″2 (18) A2″(2) A2″(3) A2″(6) A″2 (9) E1′ (4) E1′ (5) E′1(6) E′1(7) E1′ (8) E1′ (10) E′1(15) E′1(16) E1′ (19) E1′ (20) E′1(21)

568 1060 1182 1222 1468 1576 225 1258 1374 520 741 1367 1445 1511 458 564 1133 1460 509 534 578 642 674 795 1251 1291 1430 1489 1563

0.44 1.26 2.03 1.75 2.14 1.78 1.24 1.61 1.77 0.40 0.46 2.04 2.02 1.86 0.57 1.33 1.35 2.08 0.69 1.18 0.50 0.35 0.46 0.39 1.56 1.68 1.99 1.84 1.82

(dv/dp)

cur

cm−1 GPa−1

(dv/dp)

rel

cm−1 GPa−1

−0.44 1.29 2.23 1.93 2.43 2.36 −0.18 1.78 1.96 0.01 −0.21 2.31 2.35 2.06 −0.42 0.69 1.36 2.43 −0.16 0.52 −0.32 0.08 −0.17 −0.22 1.68 1.80 2.31 2.02 2.02

0.87 −0.05 −0.22 −0.21 −0.32 −0.59 1.41 −0.20 −0.22 0.39 0.67 −0.30 −0.36 −0.23 0.99 0.65 −0.03 −0.38 0.86 0.63 0.81 0.27 0.62 0.60 −0.14 −0.14 −0.36 −0.19 −0.22

exptl cm−1 GPa−1

type

−0.06a 2.24b 6.98b 3.88b 4.17b 2.2b −0.3a 2.08b 1.1a 0.38a 0.12a −1b 2.49b 2.2b 0.6c 2.11c 2.51c 1.3a 0.47c 1.95c −0.30c 0.21c −0.45c −0.38c 2.30c 10.2a 5.44d 5.38d 3.01d

radial, eq tangential tangential, eq tangential tangential tangential radial tangential tangential tangential, eq radial tangential tangential tangential radial radial tangential tangential radial radial radial radial radial radial, eq tangential tangential, eq tangential tangential tangential

a

For each mode has been indicated the radial or tangential character, and if is involved in the motion of the equatorial belt (eq). The experimental data have been taken from a = ref 77, b = ref 82, c = ref 84, d = ref 84.

Figure 9. Pressure coefficient dν/dp for C60: the blue and yellow bars represent the curvature and relaxation contributions, respectively.

Figure 10. Pressure coefficient dν/dp for C70: the blue and yellow bars represent the curvature and relaxation contributions, respectively.

Moreover, the angular coefficient calculated in the linear regression analysis allows determination of Γi/ki. The computed Γi/ki for C60 and C70 are 0.0116 and 0.0127 Å· GPa−1, respectively, further validating the computational procedure adopted in the present work. On the basis of these results, the pressure coefficients along with the curvature and relaxation contributions are summarized in Table 10 and Table 11.

Considering the vibrational pressure coefficients of fullerene C60, it is possible to observe that • (dν/dp)cur is positive for the radial normal modes Ag(1), Hg(1), Hg(2), Hg(3), Hg(4), T1u(1), and T1u(2); • (dν/dp)cur is negative for the tangential normal modes Ag(2), Hg(7), Hg(8), T1u(3), and T1u(4); 5107



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Figure 11. Comparison between calculated (green bars) and experimental (gray bars) pressure coefficients in C60.

Figure 12. Comparison between calculated (green bars) and experimental (gray bars) pressure coefficients in C70.

• (dν/dp)rel shows a different behavior and it is generally positive, with the exception of the three radial normal modes Hg(2), Hg(3), and T1u(1). It is to be noted that the radial modes with a positive (dν/dp) cur are more affected by confinement effects, while it is more difficult to describe the pressure effect on the tangential modes with a negative trend in (dν/dp) cur and further studies are required. Instead, the negative value of (dν/dp) rel can be ascribable to the negative value of the anharmonic constant, and it is important to note that the experimental frequencies characterized by negative vibrational pressure coefficients present also a negative relaxation contribution. The result of the curvature and relaxation contribution with increasing of the computed vibrational frequencies for C60 is graphically represented in Figure 9.

The partition of the pressure coefficient in the curvature and relaxation contribution has been extend also to the fullerene C70: • (dν/dp)cur is positive for the A′1(4), E′2(1), E″1 (4), E″1 (9), A″2 (2), A″2 (3), E′1(4), E′1(5), E′1(6), E′1(7), E′1(8), and E′1(10) radial modes; • (dν/dp)cur is negative for the A′1(7), A′1(8), A′1(9), A′1(11), A′1(12), E′1(16), E′1(19), E″1 (16), E″1 (17), E″1 (18), A″2 (6), A″2 (9) E′1(15), E′1(16), E′1(19), E′1(20), and E′1(21) tangential modes; • (dν/dp)rel is negative for the A′1(4), E′1(1), E″1 (9), A″2 (2), E1′ (4), E1′ (6), E1′ (8), and E1′ (10) vibrational modes. Considerations similar to those for the C60 can be adopted also in the discussion of the trend of the C70 pressure coefficients. In fact, the effect of confinement determines a 5108



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vibrational frequencies of C60 and C70; parameters adopted in the PCM-XP protocol; optimized coordinates of C60 and C70 at selected values of pressure; vibrational frequencies of C60 and C70 computed for selected optimized geometries at different values of pressure; IR intensities for selected normal modes of C60 and C70; anharmonic vibrational constants of C60 and C70. This material is available free of charge via the Internet at http://pubs.acs.org.

positive (dν/dp)cur, while to a negative anharmonic force constant corresponds a negative (dν/dp)rel. Analogously to what occurs in C60, if the experimental pressure coefficient presents a negative trend, a similar behavior is observed for the relaxation contribution. Figure 10 shows the results of this analysis. On the basis of this analysis it is possible to compare the calculated pressure coefficient with those measured. To make it easier to compare the results in Figure 11 and Figure 12, the partitioning of the normal modes in radial and tangential normal modes has been maintained. The correlation between experimental and computed values is satisfactory for tangential modes both in sign and in values for C60 and C70. This allows to consider the frequency shift in the tangential modes as due essentially to the relaxation contribution. In the case of radial modes the agreement is less satisfactory, but it is possible to observe that, when the positive pressure coefficient dominates, the most important contribution is represented by the curvature, whereas if they are negative, the dominating contribution is the relaxation. If, experimentally, a pressure coefficient shows a negative trend, its computed value is usually low, with the only exception of the low frequency mode at 225 cm−1 for C70, and in any case, the analysis in terms of curvature and relaxation contribution shows that the latter, related to the anharmonicity, is negative.

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*E-mail: marco.pagliai@unifi.it. *E-mail: gianni.cardini@unifi.it. Phone +39 055 4573072. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) under the Contract FIRB - Futuro in Ricerca 2010 No. RBFR109ZHQ. The authors wish to thank Prof. Vincenzo Schettino for critically reading the manuscript and helpful discussion.

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CONCLUSIONS For the first time, the effects of pressure on the structural and spectroscopic properties of fullerenes C60 and C70 have been computed adopting the PCM-XP method. Although in the PCM-XP approach the ab initio calculations are performed on single molecule, i.e., without explicitly considering the surrounding molecules, the method has been revealed particularly effective in the description of the perturbation induced by pressure on the geometry and vibrational spectra of the studied systems. In particular, the PCM-XP approach allows the main effects of pressure on a series of molecular properties to be obtained, representing a complementary method in the study of solid state systems. Within the PCM-XP method the comprehension of hydrostatic pressure effects on the studied system is achieved, while the complete behavior in the crystal requires periodic calculations, where the molecules and the interactions in the sample are treated explicitly. In fact, ab initio periodic calculations allow useful information on the variations of the intermolecular interactions to be obtained not only for the vibrational frequencies but also for the IR and Raman intensities. In fact, a series of different approaches13,14,85−94 have been adopted for the determination of IR and Raman spectra of condensed phase systems allowing a complete comparison with measurements. The PCM-XP method has been applied with success in the analysis of the vibrational properties of fullerenes C60 and C70, allowing the partition of the pressure effects in the curvature and relaxation contribution overcoming the problems encountered in semiclassical analysis,81 essentially due to the accuracy in the determination of the force constant increasing the pressure. This type of analysis could represent a starting point in the developing of analogous approaches in periodic calculations.

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AUTHOR INFORMATION

Corresponding Authors

REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Effect of the exchange and correlation functional on the vibrational frequencies of C60; effect of the basis set on the 5109



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Vibrational Frequencies of Fullerenes C60 and C70 under Pressure

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