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In the case of AtF3, the C2v structure is found to be a local minimum in all RECP calculations without spin−orbit terms, and the D3h structure becom...
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J. Phys. Chem. B 2008, 112, 16021–16029

16021

Spin-Orbit and Electron Correlation Effects on the Structure of EF3 (E ) I, At, and Element 117)† Hyoseok Kim, Yoon Jeong Choi,‡ and Yoon Sup Lee* Department of Chemistry and School of Molecular Science (BK21), KAIST, Daejeon, 305-701, Republic of Korea ReceiVed: June 26, 2008; ReVised Manuscript ReceiVed: August 20, 2008

Structures and vibrational frequencies of group 17 fluorides EF3 (E ) I, At, and element 117) are calculated at the density functional theory (DFT) level of theory using relativistic effective core potentials (RECPs) with and without spin-orbit terms in order to investigate the effects of spin-orbit interactions and electron correlations on the structures and vibrational frequencies of EF3. Various tests imply that spin-orbit and electron correlation effects estimated presently from Hartree-Fock (HF) and DFT calculations with RECPs with and without spin-orbit terms are quite reasonable. Spin-orbit and electron correlation effects generally increase bond lengths and/or angles in both C2V and D3h structures. For IF3, the C2V structure is a global minimum, and the D3h structure is a second-order saddle point in both HF and DFT calculations with and without spin-orbit interactions. Spin-orbit effects for IF3 are negligible in comparison to electron correlation effects. The D3h global minimum is the only minimum structure for (117)F3 in all RECP calculations, and the C2V structure is neither a local minimum nor a saddle point. In the case of AtF3, the C2V structure is found to be a local minimum in all RECP calculations without spin-orbit terms, and the D3h structure becomes a local minimum at the DFT level of theory with and without spin-orbit interactions. In the HF calculation with spin-orbit terms, the D3h structure of AtF3 is a second-order saddle point. AtF3 is a borderline case between the valence-shell-electron-pair-repulsion (VSEPR) structure of IF3 and the non-VSEPR structure of (117)F3. Relativistic effects, including scalar relativistic and spin-orbit effects, and electron correlation effects together or separately stabilize the D3h structures more than the C2V structures. As a result, one may suggest that the VSEPR predictions agree very well with the structures optimized by the nonrelativistic HF level of theory even for heavy-atom molecules but not so well with those from more elaborate theoretical methods. Vibrational frequencies of AtF3 and (117)F3 are modified substantially and nonadditively by spin-orbit and electron correlation contributions. This is one of those rare cases for which vibrational frequencies of the closed-shell molecules are significantly affected by spin-orbit interactions. Spin-orbit interactions decrease all vibrational frequencies of EF3 molecules considered. 1. Introduction Relativistic effects on the structures of molecules that contain heavy elements can be comparable or larger than electron correlation effects, and even make an unexpected structure to be a local or global minimum. It is necessary to consider spin-orbit effects as well as scalar relativistic effects for the accurate theoretical description of the electronic structures of molecules that contain heavy elements. Spin-orbit effects on the bonding of a superheavy-element molecule (118)F4 were reported to be large enough to induce the stability of a Td structure which deviates from the structure predicted by the valence shell electron pair repulsion (VSEPR)1 theory. There are several approaches for those relativistic calculations to include spin-orbit interaction. The spin-orbit density functional theory (SO-DFT or SOREP-DFT) approach,2-4 which uses twocomponent relativistic effective core potentials (RECPs) including spin-orbit terms, has been successfully used to investigate spin-orbit effects on diatomic molecules of halogen elements5-7 and halomethane systems.8-10 We extend the SO-DFT study to the EF3 (E ) I, At, and element 117) molecules for which the † Part of the special issue “Karl Freed Festschrift”. * To whom correspondence should be addressed. E-mail : [email protected]. ‡ Current address: Department of Materials Science and Engineering, KAIST, Daejeon, 305-701, Republic of Korea.

Figure 1. Bent C2V and planar D3h geometries of EF3 (E ) I, At, and element 117).

shape deviates from the VSEPR structure for the heavy ones due to scalar relativistic effects in order to understand the interplay between spin-orbit and electron correlation contributions and also to assess the quality of SO-DFT procedures. The structure of EF3 molecules is well-known to be a bent T-shaped C2V structure (Figure 1) when the central atom is Cl, Br, or I. This can be explained by the VSEPR11-13 theory or the second-order Jahn-Teller (SOJT) effects.14-25 Relativistic effects on the SOJT distortions of group 17 element fluorides have been studied.26,27 Schwerdtfeger26 investigated scalar relativistic effects on the bonding distances and F-E-F

10.1021/jp8056306 CCC: $40.75  2008 American Chemical Society Published on Web 09/27/2008

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distortion angles for heavier Br, I, and At. He presented a detailed study on the SOJT distortion from ClF3 to AtF3. Schwerdtfeger also reported large scalar relativistic changes in the bent angle of 5.5° for AtF3 and the energy difference between the symmetric D3h and the distorted C2V structures of AtF3 of only 10 kJ/mol at the coupled cluster single and double (CCSD) level of theory. It was reported that scalar relativistic effects diminish the SOJT term. He also reported that the structure of AtF3 is very sensitive to the basis sets and electron correlations applied. In a subsequent study involving one of us, Bae et al.27 performed the optimization of molecular geometries for EF3 (E ) I, At, and element 117) at the two-component Kramers’ restricted Hartree-Fock (KRHF) level of theory and investigated spin-orbit and scalar relativistic effects on the structures, vibrational frequencies, and energies for group 17 fluorides EF3 (E ) I, At, and element 117) using energy-adjusted RECPs (EAPPs). Bae et al.27 reported that spin-orbit effects, as well as scalar relativistic effects, also diminish the SOJT term at the HF level of theory and the D3h structure of (117)F3 is a global minimum, whereas IF3 and AtF3 retain C2V global minima even with scalar relativistic and spin-orbit effects. The D3h structures of IF3 and AtF3 remain a saddle point at both the HF and KRHF calculations. There is no C2V local minimum structure of (117)F3. In this work, we calculated the optimized geometries and vibrational frequencies of group 17 fluorides EF3 (E ) I, At, and element 117) at the HF and SO-DFT levels of theory by using shape-consistent RECPs (SCPPs). Electron correlations are considered at the DFT and gauged against ab initio molecular orbital (MO) methods such as Mo¨ller-Plesset second order perturbation (MP2) levels of theory. Since the SO-DFT calculations can evaluate spin-orbit effects at the DFT level by using the RECPs including spin-orbit terms, one can estimate spin-orbit effects at the DFT level. We studied the optimized geometries of EF3 (E ) I, At, and element 117) modified by both electron correlations and spin-orbit contributions by using SO-DFT calculations with SCPPs. Especially we employ several RECPs, basis sets, and functionals for AtF3, since the AtF3 structure is known to be dependent upon the basis sets and electron correlations critically.26 Our paper is organized as follow. In section 2, we briefly outline the applied RECPs, basis sets, and DFT methods. In section 3, we present our results and then compare them to available theoretical and experimental data. To sum up, conclusions are stated in section 4. 2. Computational Details Optimized molecular geometries of Group 17 fluorides EF3 (E ) I, At, and element 117) are calculated at the DFT level of theory employing RECPs including spin-orbit terms. The RECP employed for the present calculations is expressed by the following form within the framework originated from Lee et al.28 REP U REP ) ULJ (r) + L-1

l+

1 2

j

∑ ∑ ∑ [UljREP(r) - ULJREP(r)]|ljm 〉 〈ljm| l)0

(1)

1 m)-j j)|l- | 2

where |ljm〉〈ljm|represents a two-component projection operator. Molecular spinors, which are obtained as one-electron wave functions of the Hamiltonian containing the above RECP, have two components. The RECP, U REP, which is referred to REP

TABLE 1: Bond Lengths (re in Å) of AtF3 and (117)F3 with the D3h Symmetry at the HF and MP2 Levels of Theory Using Several Basis Sets RECPs and basis sets molecule AtF3

(117)F3

At and element 117 EAPPa SCPP(1f)b SCPP(2f)c SCPP(d2f)d SCPP(1f)b SCPP(2f)c SCPP(d2f)d

method F

HF

MP2

6-311+G* aug-cc-pVTZ aug-cc-pVTZ aug-cc-pVTZ aug-cc-pVTZ aug-cc-pVTZ aug-cc-pVTZ

2.074 2.008 1.998 1.998 2.090 2.086 2.087

2.123 2.029 2.027 2.105 2.102

a The 7 VE EAPP and the corresponding 5s5p1d valence basis set for At. b The 25 VE SCPP and the corresponding valence basis set augmented by one f function are used for At and element 117. c The 25 VE SCPP and the corresponding valence basis set augmented by two f functions are used for At and element 117. d The 25 VE SCPP and the corresponding valence basis set augmented by one d and two f functions are used for At and element 117.

or SOREP here, can be expressed as the sum of the scalar relativistic effective core potential (AREP), U AREP, and the effective one-electron spin-orbit operator,29 U SO, as

U REP ) U AREP + USO

(2)

Values calculated with AREP include scalar relativistic effects, and those with SOREP contain spin-orbit effects as well as scalar relativistic effects. The spin-orbit effects (∆SO) are defined as the difference between AREP and SOREP values calculated with the same basis set at a given level of theory. The electron correlation effects (∆corr) are defined as the difference between the AREP-HF and the AREP correlated method values calculated with the same basis set. To define the size of basis sets adequate for the accurate description of EF3 molecules, the bond lengths for the D3h symmetry structure were calculated with several basis sets as shown in Table 1. Employed RECPs and basis sets are the sptype 7 valence electrons (VE) shape-consistent pseudopotentials (SCPPs) with the corresponding 4s4p3d2f valence basis set7 used for I30 and the spdsp-type 25 VE SCPPs with corresponding valence basis sets for At and element 117.31,32 The basis sets are used as uncontracted forms, and augmented by two f-polarization functions for At and element 117. The aug-ccpVTZ basis set is used for F.33 Some comparative calculations were performed with energy-adjusted pseudopotentials (EAPPs) and the basis sets used in the previous studies.26,27 In this work, the DFT and SO-DFT methods were selected as the main methods of electron correlations, since the SO-DFT method in NWChem program2-4 can perform the procedure of geometry optimization and numerical vibrational analysis with RECPs including spin-orbit terms. We also performed the AREP-MP2 and AREP-CCSD(T) calculations and compared the results with those of DFT calculations. The applied functionals were B3LYP34 (Becke’s three parameter hybrid functional35 with the Lee-Yang-Parr correlation functional36), ACM35 (Becke’s adiabatic connection method, also known as B3PW91), and PBE037,38 (Perdew-Burke-Ernzerhof parameterfree functional) functionals. It is noted that RECPs used here are all generated using relativistic HF results and may introduce some inconsistency when used with DFT methods. We assume this error to be small enough at least for the purposes of the present study. Four additional sets of DFT calculations were performed for AtF3 since the optimized geometries of AtF3 appear sensitive

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Figure 2. Bending potential curves of IF3 from the AREP- and SOREP-PBE0 calculations. The angle refers to the bond angle Re defined in Figure 1. The curve around the D3h structure (120°) is enlarged in the inset.

Figure 3. Bending potential curves of (117)F3 from the AREP- and SOREP-PBE0 calculations. The angle refers to the bond angle Re defined in Figure 1. The curve around the D3h structure (120°) is enlarged in the inset.

to the amount of electron correlations and the basis sets applied. Additional DFT calculations use local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA (MGGA), and hybrid meta-GGA (HMGGA) of the exchangecorrelation functionals to investigate the effect of the hybrid functionals. The applied LDA and GGA functionals are SVWN5 (Slater exchange39 and Vosko-Wilk-Nusair correlation functional (V)40), BP86 (Becke exchange41 and Perdew correlation functional42), respectively. The applied MGGA and HMGGA functionals are TPSS43 (nonempirical Tao-Perdew-StaroverovScuseria functional) and M0544 (Minnesota 2005), respectively. We calculated the bending potential curves and obtained highest-occupied molecular ortibal-lowest-unoccupied molecular orbital (HOMO-LUMO) gaps as a function of the F-E-F bond angle (Re) by using HF and PBE0 calculations with and without spin-orbit interactions. This is shown in Figures 2-6. HF and DFT calculations with the PBE0 functional, as shown in Table S1 of Supporting Information, were carried out by using several RECPs and basis sets for AtF3. EAPP1 and EAPP2 refer to the 7 VE EAPP and the corresponding 5s5p1d valence basis set45 for At with the 6-311+G* and aug-cc-pVTZ basis sets, respectively, for F. SCPP1 and SCPP2 denote the 17 VE SCPP and the corresponding 5s5p4d1f valence basis set for At and

the 6-311+G* and aug-cc-pVTZ basis sets, respectively, for F. In the case of SCPP3, the 25 VE SCPP and the corresponding 9s9d6d2f valence basis set for At and the aug-cc-pVTZ basis set for F are used. In the case of SCPP4, the 25 VE SCPP and the corresponding 9s9d6d2f valence basis set for At and the 7 VE SCPP and corresponding 4s4p3d2f valence basis set for F are used. AREP calculations were carried out with the Gaussian46 and NWChem programs.2-4 SOREP calculations were carried out with the NWChem2-4 program. 3. Results and Discussions The optimized molecular geometries of the group 17 fluorides EF3 (E ) I, At, and element 117) at the HF, DFT, MP2, and CCSD(T) levels of theory are shown in Table 2. To assess the extent of electron correlations at the DFT level of theory, the structures of EF3 were also optimized at the MP2 and CCSD(T) level of theory without spin-orbit interactions. Among the various optimized structures of DFT calculations in Table 2, the optimized structures using the PBE0 functional are very close to those from MP2 and CCSD(T) calculations. The VSEPR and SOJT effects predict the C2V structure for EF3, which has five

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Figure 4. Bending potential curves of AtF3 from the AREP- and SOREP-PBE0 calculations. The angle refers to the bond angle Re defined in Figure 1. The curve around the D3h structure (120°) is enlarged in the inset.

Figure 5. Bending potential curves of AtF3 from the SO-DFT calculations with various functionals. The angle refers to the bond angle Re defined in Figure 1.

valence electron pairs on the central group 17 element. We optimized the bent T-shaped C2V and trigonal planar D3h structures. The geometrical parameters for C2V and D3h structures are defined in Figure 1. Symbol i behind the value of bond lengths in Table 2 indicates that the structure has an imaginary vibrational frequency for a doubly degenerate mode. Thus the structure is a second-order saddle point. IF3 and AtF3 have the C2V structures as local minima, and (117)F3 has the D3h structure as the only minimum, which is in good agreement with the previous AREP-HF and SOREP-KRHF results.27 We found that the D3h structure of (117)F3 remains as the only minimum even with electron correlations or both electron correlations and spin-orbit interactions. The D3h structure of IF3 is found to be a second-order saddle point. In the case of AtF3, the D3h structure is found to be a local minimum for all the HF and DFT calculations except SOREP-KRHF calculations. The D3h structure of AtF3 is a saddle point in the previous SOREP-KRHF calculation.27 The D3h structure of AtF3 becomes a local minimum by the use of 25VE SCPPs and large basis set at the HF level and becomes a second-order saddle point by the inclusion of spin-orbit terms at the SOREP-HF level. The D3h structure of AtF3 becomes a saddle point at the nonrelativistic MP2 level and becomes a

local minimum at the scalar relativistic MP2 level in the previous results.26 The bond lengths of the present calculations are somewhat shorter than the previous results. Apart from differences in generating procedure for the pseudopotentials, energyadjusted or shape-consistent, the present study employs smaller core for the RECPs with larger basis set. It is usually observed that the small core RECP results agree better with all-electron ones than the large core ones. The D3h structure of AtF3 becomes a local or global minimum when both relativistic and electron correlation effects are taken into account. Spin-orbit (∆SO) and electron correlation (∆corr) effects on optimized bond lengths and angles can be obtained from Table 2. ∆corr values are evaluated from the AREP calculations. Electron correlation effects of the PBE0 results are very close to those of the MP2 and CCSD(T) results. Since variations among structures from various hybrid functionals are rather small, the main analysis on general trends for EF3 will be based on PBE0 results for clarity. Spin-orbit and electron correlation effects increase the bond lengths of C2V and D3h structures. For C2V structures, ∆corr of IF3 at the PBE0 level are 0.055Å for reeq, 0.045 Å for reax, and 0.8° for Re. Compared with electron correlation effects, spin-orbit effects on IF3 are very small. ∆SO of IF3 at the HF level are 0.003, 0.001 Å, and 0.0°, which are

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Figure 6. HOMO-LUMO gaps of EF3 (E ) I, At, and element 117) at various bending angles from the AREP- and SOREP-PBE0 calculations. The angle refers to the bond angle Re defined in Figure 1.

TABLE 2: Bond Lengths (re, Ångstroms) and Bond Angles (re, Degrees) of EF3 (E ) I, At, and element 117) for the C2W and D3h Structures at the HF and DFT Levels of Theory (Values in Parentheses Are SOREP Results)a C2V

D3h

molecule

method

reeq

reax

Re

re

c

HF MP2 CCSD(T) B3LYP ACM PBE0 HF MP2 CCSD(T) SVWN5 BP86 TPSS B3LYP ACM PBE0 M05 HF MP2 CCSD(T) B3LYP ACM PBE0

1.803 (1.806) 1.855 1.854 1.867 (1.871) 1.864 (1.868) 1.858 (1.862) 1.903 (1.942) 1.945 1.942 1.957 1.992 1.982 1.969 (2.037) 1.957 (2.015) 1.948 (2.000) 1.955

1.892 (1.893) 1.934 1.931 1.949 (1.952) 1.943 (1.946) 1.937 (1.939) 2.004 (2.029) 2.035 2.032 2.039 2.076 2.066 2.057 (2.088) 2.045 (2.074) 2.036 (2.064) 2.047

82.4 (82.4) 83.2 82.8 83.6 (83.8) 83.5 (83.7) 83.2 (83.3) 83.5 (86.7) 85.7 84.6 85.6 88.4 86.7 86.3 (96.8) 85.7 (93.1) 85.0 (90.7) 87.6

1.907i (1.910i)b 1.952i 1.951i 1.965i (1.969i) 1.959i (1.962i) 1.952i (1.956i) 1.998 (2.029i) 2.027 2.026 2.032 (2.062) 2.065 (2.097) 2.058 (2.087) 2.049 (2.080) 2.037 (2.068) 2.029 (2.059) 2.036 (2.074) 2.086 (2.191) 2.102 2.104 2.128 (2.216) 2.117 (2.204) 2.110 (2.199)

IF3

AtF3

(117)F3

a ∆SO is defined as the difference between AREP and SOREP values. b Symbol i behind the value of the bond lengths indicates that the structures is a saddle point. c The experimental values from a single-crystal X-ray structure49 are 1.872, 1.983 Å, and 80.2° for reeq, reax, and Re, respectively.

in very good agreement with the previous HF-KRHF results, 0.004, 0.002 Å, and 0.1°,27 respectively. ∆SO of AtF3 at the

PBE0 level are 0.052 Å for reeq, 0.028 Å for reax, and 5.7° for Re, which are comparable to 0.045 Å for reeq, 0.032 Å for reax,

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TABLE 3: Harmonic Vibrational Frequencies (in cm-1) of EF3 (E ) I, At, and Element 117) with the C2W Symmetry at the HF and DFT Levels of Theory (Values in Parentheses are SOREP Results)a molecule

method

ν1(B1) out of plane

ν2(A1) sym bend

ν3(B2) asym bend

ν4(A1) sym str

ν5(B2) asym str

ν6(A1) sym str

IF3

HF B3LYP ACM PBE0 HF SVWN5 BP86 TPSS B3LYP ACM PBE0 M05

231 (228) 195 (191) 198 (195) 202 (200) 195 (175) 164 163 165 169 (138) 171 (153) 172 (153) 176

248 (246) 200 (198) 201 (198) 204 (200) 166 (127) 110 84 102 113 (37) 121 (53) 129 (74) 104

371 (369) 302 (296) 306 (301) 312 (307) 293 (237) 228 209 220 233 (154) 241 (163) 249 (180) 230

617 (615) 542 (538) 552 (549) 562 (559) 601 (568) 544 510 521 532 (502) 543 (513) 551 (522) 544

619 (617) 559 (558) 573 (571) 583 (581) 560 (537) 536 503 514 519 (500) 528 (512) 536 (520) 519

754 (748) 639 (631) 649 (641) 661 (653) 709 (634) 606 567 580 601 (522) 614 (539) 626 (555) 619

AtF3

a

∆SO is defined as the difference between AREP and SOREP values.

and 1.5° for Re of ∆corr. The bond angle Re of AtF3 increases by 5.726 and 3.9°,27 with the inclusion of scalar relativistic effects26 and spin-orbit effects,27 respectively. The bond angle Re increased by 3.2 and 5.7° with the inclusion of spin-orbit effects from the HF and PBE0 calculations, respectively. Spin-orbit effects of correlated DFT calculations on the C2V structure of AtF3 are larger than those of HF calculations. For D3h structures, ∆SO and ∆corr from the PBE0 calculation are 0.004 and 0.045 Å for IF3, 0.030 and 0.031 Å for AtF3, and 0.089 and 0.024 Å for (117)F3, respectively. ∆SO on the bond lengths re of a D3h structure from the previous KRHF calculations27 are 0.004 Å for IF3, 0.028 Å for AtF3, and 0.109 Å for (117)F3, which are also in agreement with the present HF results (0.003, 0.031, and 0.105Å for IF3, AtF3, and (117)F3, respectively). Spin-orbit effects are comparable to electron correlations in AtF3 and become larger than electron correlation effects in (117)F3. Because of the large radial elongation of 7p3/2 caused by spin-orbit splitting of the 7p shell, molecular properties of (117)F3 are strongly influenced by spin-orbit coupling. It is noted that the bond lengths of the present calculations are all shorter than those of the previous ones by 0.03-0.10 Å at the HF level, which implies some differences in EAPP and SCPP RECPs. Harmonic vibrational frequenicies of C2V and D3h structures of EF3 (E ) I, At, and element 117), which were obtained from numerical Hessian calculations for SOREP calculations are listed in Tables 3 and 4, respectively. Changes in the ν1(E′) bending modes of D3h structures due to electron correlation and spin-orbit effects can be qualitatively described with the bending potential curves for EF3 (E ) I, At, and element 117) as shown in Figures 2-4. The same is true for bending modes of the C2V structures. Spin-orbit effects decrease the vibrational frequencies of both C2V and D3h structures. In many cases the decrease is substantial. One may say that spin-orbit interactions flatten the potential energy curve for every normal coordinate for this molecule. Spin-orbit effects on the vibrational frequencies for IF3 are negligible. Electron correlation effects decrease all vibrational frequencies of C2V and D3h structures except the ν1(E′) bending mode of D3h structures. The gradients of bending potential curves at 120° of Re have negative values as shown in Figure 2. Thus the ν1(E′) bending mode of IF3 with the D3h symmetry has imaginary numbers, indicating that IF3 with the D3h symmetry is a saddle point. Electron correlation effects decrease ∆E of IF3 and reduce the slope of the bending potential curves, which causes the ν1(E′) bending mode of IF3 with a D3h symmetry to increase from -131 at the AREP-HF calculations to -57 at the AREP-PBE0 calculations. Spin-orbit effects on

TABLE 4: Harmonic Vibrational Frequencies (in cm-1) of EF3 (E ) I, At, and Element 117) with the D3h Symmetry at the HF and DFT Levels of Theory (Values in Parentheses Are SOREP Results)a molecule method IF3

AtF3

(117)F3

ν1(E′)

HF -131 (-130) B3LYP -49 (-52) ACM -51 (-50) PBE0 -57 (-62) HF 27 (-36) SVWN5 86 (61) BP86 84 (66) TPSS 85 (64) B3LYP 81 (57) ACM 82 (61) PBE0 79 (55) M05 78 (51) HF 108 (85) B3LYP 107 (81) ACM 110 (83) PBE0 110 (81)

ν2(A2′′) ν3(E2′) out of plane asym str 323 (318) 270 (264) 273 (267) 277 (271) 253 (214) 211 (174) 204 (168) 208 (174) 215 (176) 219 (181) 222 (184) 221 (177) 198 (104) 173 (95) 175 (97) 177 (105)

574 (570) 523 (520) 535 (533) 545 (542) 568 (536) 537 (512) 505 (481) 513 (491) 522 (497) 532 (506) 539 (514) 531 (497) 565 (500) 522 (475) 530 (481) 537 (486)

ν4(A1′) sym str 623 (618) 545 (540) 555 (551) 565 (561) 618 (574) 553 (518) 521 (492) 530 (495) 545 (508) 556 (520) 564 (528) 560 (511) 608 (529) 544 (488) 553 (496) 559 (502)

a ∆SO is defined as the difference between AREP and SOREP values.

the vibrational frequencies of AtF3 are different for different modes as electron correlations are applied. In the case of AtF3 with the C2V symmetry, spin-orbit effects on the vibrational frequencies are -19, -55, -69, -29, -16, and -71 cm-1 for the PBE0 functional for ν1(B1) out of plane, ν2(A1) symmetric, ν3(B2) asymmetric bending modes, ν4(A1) symmetric, ν5(B2) asymmetric, and ν6(A1) symmetric stretching modes, respectively. The ν4(A1), ν5(B2), and ν6(A1) stretching frequencies of C2V structures are probably more affected directly by the elongation of p3/2 due to spin-orbit effects than the increased ionic characters facilitated by the spin-orbit splitting. For AtF3 with the D3h symmetry, the ν1(E′) bending mode is 27 cm-1 at the HF level of theory and changed to -36 cm-1 by spin-orbit effects, which causes the D3h symmetry at the HF level of theory to become a saddle-point. Electron correlations increase the ν1(E′) bending frequency. This bending frequency becomes 55 and 79 cm-1 for the PBE0 functional with and without spin-orbit terms, respectively, which clearly indicates that spin-orbit and electron correlation effects are not additive for the frequencies. The gradients of bending potential curves at 120° of Re have positive values at the AREP-HF, AREP-PBE0, and SOREP-PBE0 levels and yet have negative values at the SOREP-KRHF level as shown in Figure 4. Thus, the ν1(E′) bending modes of AtF3 with the D3h symmetry from the HF and PBE0 calculations except the SOREP-KRHF calculation have real numbers, indicating that the D3h structures of AtF3

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TABLE 5: Energy Difference between C2W and D3h Structures (∆E in kJ/mol) of AtF3 Using the Several Levels of Theory and Functionals ∆E method

NREP

AREP

HFb HFc HF MP2b MP2 CCSD(T)b CCSD(T) SVWN5 BP86 TPSS B3LYP ACM PBE0 M05

124.7

35.9 35.34 42.07 5.6 4.26 10.4 13.27 -0.34 -1.51 -3.23 2.67 4.46 8.47 0.38

66.5 72.4 LDAd GGA MGGA HGGA HGGA HGGA HMGGA

SOREP 18.26 21.49

∆E < 0e ∆E < 0 ∆E < 0 -2.73 -2.16 0.15 ∆E < 0

global minimuma C2V C2V C2V C2V C2V C2V C2V D3h D3h D3h D3h D3h C2V D3h

a The global minimum at the SOREP level. If there are no SOREP results, the global minimum is estimates from structures at the AREP level. b Reference 26. c Reference 27. d LDA, GGA, HGGA, MGGA, and HMGGA indicate local density approximation, generalized gradient approximation, meta-GGA, and hybrid metaGGA, respectively. e There is no C2V local minimum.

for those calculations are local minima. The D3h structure of AtF3 from the SOREP-KRHF calculation is a saddle point. This is quite interesting since the spin-orbit effects reduce the energy differences between the C2V and the D3h structures by ∼20 kJ/ mol in Table 5. Electron correlation effects decrease ∆E of AtF3 and reduce the slope of the bending potential curves. As a result of incorporating both spin-orbit and electron correlation effects, all vibrational frequencies of AtF3 are positive, and the D3h structure becomes a local minimum. For (117)F3 with the D3h symmetry, ∆SO on ν1(E′) bending modes and ν2(A2′) out-of-plane bending modes are -29 and -72 cm-1 for the PBE0 functional, respectively. Electron correlations hardly affect the ν1(E′) bending mode of (117)F3 in Table 4. As shown in Figure 3, the gradients of bending potential curves at 120° of Re have positive values for both the AREP and the SOREP calculations. Thus ν1(E′) bending modes of (117)F3 with the D3h symmetry from the AREP and SOREP calculations have real frequencies and (117)F3 with the D3h symmetry is the global minimum. There is no C2V local minima or saddle points of (117)F3 since there are no stationary points except at 120° of Re, which is the bond angle of the D3h minimum. Electron correlations only slightly affect the slope of the bending potential curves. Spin-orbit effects reduce the slope of the bending potential curves, which can be seen in the ν1(E′) bending modes of (117)F3 with the D3h symmetry, 108 (85) and 110 (81) cm-1 at the AREP-HF (SOREP-KRHF) and AREP-PBE0 (SOREP-PBE0) levels, respectively. Four sets of additional DFT and SO-DFT calculations for the optimized C2V and D3h structures of AtF3 were performed to determine whether spin-orbit and electron correlation effects are caused by the use of hybrid functionals or not, employing LDA (SVWN5), GGA (BP86), MGGA (TPSS), and HMGGA (M05) functionals. The results are also summarized in Tables 2-5. ∆SO on the bond length re of LDA, GGA, MGGA, and HMGGA functionals are 0.030, 0.032, 0.029, and 0.038 Å, respectively. These are very similar to those of hybrid functionals (0.031 Å for B3LYP, 0.031 Å for ACM, and 0.030 Å for PBE0 functionals). Vibrational frequencies of the D3h structure using LDA, GGA, MGGA, and HMGGA functionals are all real positive in both the AREP and the SOREP

calculations. ∆SO values on the vibrational frequencies of LDA, GGA, MGGA, and HMGGA functionals are also in good agreement with those of hybrid functionals. Although there are some minor differences depending on functional employed, the electron correlation effects on the D3h and C2V structures are consistent among all DFT calculations. We performed the HF and PBE0 calculations for AtF3 using six cases of RECPs and basis sets listed in Table S1 of Supporting Information in order to investigate the effects of RECPs and basis sets applied. Optimized geometries for the C2V and D3h structures of AtF3 are listed in Tables S2 and S3 of Supporting Information, respectively. In the case of EAPP1, the AREP-HF bond lengths are 1.981 and 2.079 Å for reeq and reax, respectively, and the AREP-HF Re is 84.8°, which is identical to the previous KRHF results.27 As noted earlier, the bond lengths differ somewhat among different RECPs and/or basis sets. It appears that smaller core for RECPs and larger basis sets for F yield shorter bond lengths as shown in Table 1. In the absence of results from any all-electron or experiments, shorter bond lengths of SCPP3 and SCPP4 cases are probably closer to the exact ones among the present ones. Electron correlations and spin-orbit interactions increase bond lengths and angles in all calculations. The use of large basis sets decreases bond lengths and angles in all HF and PBE0 calculations. For the C2V structure of AtF3, the magnitude of spin-orbit effects on reax and Re increase slightly in PBE0 calculations and that of electron correlation effects on reax and Re increases slightly in SOREP calculations. There is a small nonadditivity between electron correlation effects and spin-orbit effects on reax and Re at the given RECPs and basis set. ∆SO on reeq and reax of the C2V structure at the HF level are on average 0.042 and 0.025 Å, respectively, and ∆SO at the PBE0 calculations are on average about 0.055 and 0.026 Å, respectively. ∆SO on Re of a C2V structure are 3.4 and 6.5° at the HF and PBE0 levels, respectively. The average ∆SO on re of the D3h structure are about 0.031 and 0.029 Å for HF and PBE0 calculations, respectively. Table S4 of Supporting Information shows the vibrational frequencies of AtF3 with the D3h symmetry. Spin-orbit effects decrease the vibrational frequencies in all modes. Electron correlations also decrease the vibrational frequencies in all modes except the ν1(E′) bending mode. Electron correlations increase the ν1(E′) mode by an average of about 76 cm-1 in AREP calculations. The D3h structure of AtF3 is a second-order saddle-point at the HF level when the smaller basis sets are used. With the larger basis set, the D3h structure becomes a local minimum at the HF level of theory without spin-orbit terms and becomes a saddle-point at the HF level with spin-orbit terms. In all the case of AREP- and SOREP-PBE0 calculations, the D3h structure is found to be a local minimum. The energy difference (∆E) between the C2V and the D3h structures is also dependent upon both relativistic and the level of electron correlation effects as shown in Table 5. ∆E values of AtF3 are 21 and 42 kJ/mol at the HF level with and without spin-orbit terms, respectively. ∆E values of AtF3 become 4 and 13 kJ/mol from the AREP-MP2 and the AREP-CCSD(T) calculations, respectively. For the DFT calculations with spin-orbit terms, the D3h structure becomes comparably stable to or slightly more stable than the C2V structure. Relativistic effects, including scalar relativistic and spin-orbit effects, and electron correlation effects decrease ∆E of AtF3 as shown in Table 5. ∆E values of AtF3 from DFT calculations with spin-orbit terms are negative. When there are no C2V minima as in LDA and GGA functionals with SOREP, we can estimate

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Kim et al.

the approximate values of ∆E of AtF3 from the inflection point in Figure 5. ∆E values for AtF3 from SO-DFT calculations range from -2 to -8 kJ/mol except for PBE0 calculations. The decrease in ∆E due to electron correlation effects diminishes as descending the group in the periodic table,26 which is also evident for At and 117 here. Although the global minimum of AtF3 has the D3h symmetry in SO-DFT calculations with many functionals, it is hard to decide that the global minimum of AtF3 is whether the C2V or D3h structure. The leading SOJT distortion term is given by47

| | | | 〈 〉 ∑

2

∂Η

q2

ψi

j*i

ψj

∂q Ei0 - Ej0

(3)

which contributes to the energy stabilization of a molecular system by mixing the jth excited state |ψj〉 into the ground state |ψi〉 through distortion coordinate q. The energy gap, E0i - E0j , appears in the denominator; thus the term with an excited state |ψj〉 lying close to the ground state |ψi〉 dominates the SOJT expression. The magnitude of the stabilization energy due to this term may be qualitatively estimated from HOMO-LUMO gaps (∆E(HOMO-LUMO)) when compared within the same molecular system. The SOJT terms of EF3 (E ) Cl, Br, and I), for which spin-orbit effects are negligible, increase for heavier members of the group in the periodic table since the bending angle Re of the Jahn-Teller distorted C2V structures decreases with increasing atomic number.26 In the case of IF3, the curvature and the value of ∆E(HOMO-LUMO)48 as a function of Re, as shown in Figure 6, are similar in both the AREP-PBE0 and SOREP-PBE0 calculations since the spin-orbit effects on the ∆E(HOMO-LUMO) are negligible. The curvature of AREP-PBE0 calculations at ∼120° of Re (D3h structure) shows the cusp typical of symmetry swithching, indicating that the symmetry of the LUMO of the AREPPBE0 calculations changes at that point. When the bending angle Re is lower than 120°, LUMO and LUMO+1 have a1 and b2 symmetries, respectively. LUMO and LUMO+1are degenerate and have the e′ symmetry at 120°. When the Re is larger than 120°, LUMO and LUMO+1 have b2 and a1 symmetries, respectively. However, at the SOREP-PBE0 level of theory, LUMO is defined by the two-component spinor which has the same symmetry across the whole range of the bending angle due to the spin-orbit mixing at the orbital level, and the gap shows the pattern of avoided crossing rather than the cusp. In the case of AtF3, the anomaly of decreasing in the Re of the C2V structure compared with that of IF3 is a result of the reduced SOJT term due to the scalar relativistic effects.26 This reduction of SOJT is further amplified by spin-orbit effects. The ∆E(HOMO-LUMO) in Figure 6 increases due to spin-orbit effects which stabilize HOMO more than LUMO, reducing the SOJT distortion and increasing the bond angle Re of the C2V structure for AtF3. The reduction of the SOJT term also decreases the energy difference between the C2V and D3h structures by destabilizing the former. The curvature of the ∆E(HOMO-LUMO) around 120° of the bending angle Re in AREP-PBE0 calculations is steeper than that in SOREP-PBE0 calculations mainly because of the symmetry properties mentioned above. This may be responsible for the reduction of bending vibrational frequencies due to spin-orbit terms even though the D3h structure is stabilized more than the C2V one by the spin-orbit interactions as can be estimated from the larger value of the ∆E(HOMO-LUMO) from SOREP-PBE0 than AREP-PBE0.

In the case of (117)F3, the global minimum in AREP and SOREP calculations is the D3h symmetry and that in NREP calculations is the C2V symmetry. For more comprehensive interpretation, the ∆E(HOMO-LUMO) from NREP-PBE0 calculations is also plotted with those from AREP- and SOREPPBE0 calculations in Figure 6. The curvature of the ∆E(HOMO-LUMO) around 120° in NREP-PBE0 and AREPPBE0 calculations is more acute than that in SOREP-PBE0 calculations, leading to the reduction of vibrational frequencies due to spin-orbit effects. The value of the ∆E(HOMO-LUMO) from SOREP-PBE0 calculations is larger than that from AREPPBE0 calculations which is in turn larger than those from NREPPBE0 calculations, explaining why the global minimum structure of (117)F3 from relativistic AREP-PBE0 and SOREP-PBE0 calculations is different from the nonrelativistic NREP-PBE0 calculation as noted previously.27 4. Conclusions We have calculated the optimized geometries of EF3 (E ) I, At, and element 117) at the HF and DFT levels of theory with and without spin-orbit terms and evaluated spin-orbit and electron correlation effects for the optimized C2V and D3h structures. Various tests imply that spin-orbit and electron correlation effects estimated presently from HF and DFT calculations with RECPs with and without spin-orbit terms are quite reasonable. Spin-orbit and electron correlation effects increase bond lengths and/or angles in both C2V and D3h structures. For IF3, the C2V structure is a global minimum, and the D3h structure is a second-order saddle point in both HF and DFT calculations with and without spin-orbit interactions. Spin-orbit effects for IF3 are negligible compared to electron correlation effects. (117)F3 has the D3h global minimum in all AREP and SOREP calculations and the C2V structure is neither a local minimum nor a saddle point. With spin-orbit and electron correlation effects, the D3h structure of (117)F3 becomes a more stable global minimum. In the case of AtF3, the C2V structure is found to be a local minimum in all AREP and SOREP calculations and the D3h structure becomes a local minimum at the DFT level of theory with and without spin-orbit interactions. Only in the SOREP-HF calculation, the D3h structure of AtF3 is a second-order saddle point. AtF3 is a borderline case between the VSEPR structure of IF3 and the non-VSEPR structure of (117)F3. Relativistic effects, including scalar relativistic and spin-orbit effects, and electron correlation effects together or separately decrease the SOJT term thus decreasing the energy difference between C2V and D3h structures. Although the global minimum of AtF3 has the D3h symmetry from SO-DFT calculations with most functionals, it is not fully supported by the CCSD(T) result. The global minimum of AtF3 is probably D3h with C2V being a local minimum but more sophisticated calculations with spin-orbit terms by using larger basis sets will be needed for clarification. One may suggest that the VSEPR predictions agree very well with nonrelativistic HF models even for heavy-atom molecules but not so for more sophisticated and thus more realistic theoretical methods. Vibrational frequencies of AtF3 and (117)F3 are modified substantially and nonadditively by spin-orbit and electron correlation, suggesting that SO-DFT and other RECP methods with spin-orbit terms could be essential to correctly place the vibrational frequencies of some molecules. Spin-orbit interactions decrease all vibrational frequencies of EF3 molecules considered. Acknowledgment. This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the

Various Effects on the Structure of EF3 Center for Space-Time Molecular Dynamics and Grants R112007-012-03001-0 and R01-2007-000-11015-0 and Grant 06K1401-01010 from CNMM. Computational resources were provided by the supercomputing center of the Korea Institute of Science Technology Information. Supporting Information Available: Tables of six cases of RECPs and basis sets applied at the HF and DFT levels for AtF3 (Table S1), C2V bond lengths (reeq and reax in Å), angles (Re in degrees) of AtF3 with the C2V symmetry from the HF and PBE0 calculations using several RECPs and basis sets, bond lengths (re in Å) of AtF3 with the D3h symmetry from the HF and PBE0 calculations using several RECPs and basis sets, harmonic vibrational frequencies (in cm-1) of AtF3 with the D3h symmetry from the HF and DFT calculations using several RECPs and basis sets. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Han, Y.-K.; Lee, Y. S. J. Phys. Chem. A 1999, 103, 1104. (2) Kendall, R. A.; Apra, E.; Bernholdt, D. E.; Bylaska, E. J.; Dupuis, M.; Fann, G. I.; Harrison, R. J.; Ju, J.; Nichols, J. A.; Nieplocha, J.; Straatsma, T. P.; Windus, T. L.; Wong, A. T. Comput. Phys. Commun. 2000, 128, 260. (3) Harrison, R. J.; Nichols, J. A.; Straatsma, T. P.; Dupuis, M.; Bylaska, E. J.; Fann, G. I.; Windus, T. L.; Apra, E.; Jong, W. d.; Hirata, S.; Hackler, M. T.; Anchell, J.; Bernholdt, D.; Borowski, P.; Clark, T.; Clerc, D.; Dachsel, H.; Deegan, M.; Dyall, K.; Elwood, D.; Fruchtl, H.; Glendening, E.; Gutowski, M.; Hirao, K.; Hess, A.; Jaffe, J.; Johnson, B.; Ju, J.; Kendall, R.; Kobayashi, R.; Kutteh, R.; Lin, Z.; Littlefield, R.; Long, X.; Meng, B.; Nakajima, T.; Nieplocha, J.; Niu, S.; Rosing, M.; Sandrone, G.; Stave, M.; Taylor, H.; Thomas, G.; Lenthe, J. v.; Wolinski, K.; Wong, A.; Zhang, Z. NWChem, A Computational Chemistry Package for Parallel Computers, Version 4.1; Pacific Northwest National Laboratory: Richland, WA, 2002. (4) Bylaska, E. J.; Jong, W. A. d.; Kowalski, K.; Straatsma, T. P.; Valiev, M.; Wang, D.; Apra`, E.; Windus, T. L.; Hirata, S.; Hackler, M. T.; Zhao, Y.; Fan, P.-D.; Harrison, R. J.; Dupuis, M.; Smith, D. M. A.; Nieplocha, J.; Tipparaju, V.; Krishnan, M.; Auer, A. A.; Nooijen, M.; Brown, E.; Cisneros, G.; Fann, G. I.; Fru¨chtl, H.; Garza, J.; Hirao, K.; Kendall, R.; Nichols, J. A.; Tsemekhman, K.; Wolinski, K.; Anchell, J.; Bernholdt, D.; Borowski, P.; Clark, T.; Clerc, D.; Dachsel, H.; Deegan, M.; Dyall, K.; Elwood, D.; Glendening, E.; Gutowski, M.; Hess, A.; Jaffe, J.; Johnson, B.; Ju, J.; Kobayashi, R.; Kutteh, R.; Lin, Z.; Littlefield, R.; Long, X.; Meng, B.; Nakajima, T.; Niu, S.; Pollack, L.; Rosing, M.; Sandrone, G.; Stave, M.; Taylor, H.; Thomas, G.; Lenthe, J. v.; Wong, A.; Zhang, Z. NWChem, A Computational Chemistry Package for Parallel Computers, Version 5.0; Pacific Northwest National Laboratory: Richland, WA, 2006. (5) Choi, Y. J.; Lee, Y. S. J. Chem. Phys. 2003, 115, 3448. (6) Cho, W. K.; Choi, Y. J.; Lee, Y. S. Mol. Phys. 2005, 103, 2117. (7) Lee, H. S.; Cho, W. K.; Choi, Y. J.; Lee, Y. S. Chem. Phys. 2005, 311, 121. (8) Lee, M.; Kim, H.; Lee, Y. S.; Kim, M. S. Angew. Chem., Int. Ed. 2005, 44, 2929. (9) Lee, M.; Kim, H.; Lee, Y. S.; Kim, M. S. J. Chem. Phys. 2005, 122, 244319. (10) Lee, M.; Kim, H.; Lee, Y. S.; Kim, M. S. J. Chem. Phys. 2005, 123, 024310. (11) Gillespie, R. J.; Hargittai, I. The VSEPR Model of Molecular Geometry; Allyn and Bacon: Boston, 1991. (12) Gillespie, R. J.; Nyholm, R. S. Q. ReV. Chem. Soc. 1957, 11, 339. (13) Gillespie, R. J.; Robinson, E. A. Angew. Chem., Int. Ed. 1996, 35, 495. (14) Bersuker, I. B. Chem. Phys. 2001, 101, 1067.

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