eight-valence electron species

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1 DOI: 0.038/NCHEM.263 Quadruple bonding in C 2 and analogous eight-valence electron species Sason Shaik*, David Danovich, Wei Wu 2, Peifeng Su 2, Henry Rzepa 3, Philippe C. Hiberty 4. Institute of Chemistry and The Lise Meitner-Minerva Center for Computational Quantum Chemistry, Hebrew University of Jerusalem, 9904, Jerusalem, Israel 2. The State Key Laboratory of Physical Chemistry of Solid Surfaces, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry, and College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, Fujian 36005, China 3. Department of Chemistry, Imperial College London, South Kensington Campus, SW7 2AZ, United Kingdom. 4. Laboratoire de Chimie Physique, UMR CNRS 8000, Université de Paris Sud, 9405 Orsay Cédex, France Table of Contents I. VB methods and results 2 I.. VB structures 2 I.2. The 4 th bond: VB calculations of the QC state and D in and ΔE ST 4 II. FCI results 7 II.. FCI/6-3G* data 7 II.2. FCI wave functions 8 II.3. GVB Transformation of the FCI/6-3G* Wave functions 2 III. Wiberg bond order indices 6 III. The " + g and " + states 6 IV. Relaxed Force Constants 6 NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

2 DOI: 0.038/NCHEM.263 I. VB methods and results I.. VB structures The VB calculations for C 2 and for Σ + in CN +, CB, and BN include all the 92 structure-set used recently. The dominant Group set of structures for C 2 is shown in Scheme S.. These structures account for all the covalent structures and their corresponding ionic components of the four potential bonds in the molecule.. Su, P., Wu, J., Gu, J., Wu, W. Shaik, S. & Hiberty, P. C. Bonding Conundrums in the C 2 Molecule: A Valence Bond Study. J. Chem. Theor. Comput. 7, 2-30 (20). Note that structure 73 was misrepresented in Ref.. The correct representation is the following: Scheme S.: VB structures for Group. The numbering of the structures follows Ref NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

3 DOI: 0.038/NCHEM Table S. below shows that Group dominates the total VB wave function of the 92 structure-set Table S.. VBSCF/6-3G* weights of VB structures gathered by groups which take into account all the 92 structures at the equilibrium distances obtained by FCI/6-3G* Groups/ Weights groups Lowdin Inverse C 2 group group group group group group group CN + group group group group group group group BN group group group group group group group CB - group group group group group group group NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

4 DOI: 0.038/NCHEM.263 I.2. The 4 th bond: VB calculations of the QC state and D in and ΔEST The D in values were calculated as follows: (a) We uncoupled the electrons in the hybrid atomic orbitals (HAOs) of the 4 th bond in all the structures which contain this bond (structures, 2, 3, 6, 7, 8, 9, 4, 5, 8, 9, 20, and 2 in Scheme S.), by taking only one of the two spin arrangement patterns, e.g. spin-up/spin-down. The energy of the resulting set of 3 structures is E(QC), where QC is the quasi-classical state (see Text). The energy difference between E(QC) and the full VB energy of the 2 set of structures set gave D in. The ΔEST values were calculated as follows: The electrons in the hybrid atomic orbitals (HAOs) of the 4 th bond in all the structures which contain this bond (structures, 2, 3, 6, 7, 8, 9, 4, 5, 8, 9, 20, and 2 in Scheme S.) were given parallel spins. The resulting set of 3 structures was used to calculate the energy E(TS) triplet state of the 4 th bond. The energy difference between E(TS) and the full VB energy of the 2 set of structures set gave ΔEST. Tables S.2-S.5 collect the VB weights for the four molecules in their singlet states, as calculated using the 2-structure set and the 3-structure set, and in the triplet state using the 3-structure set where the electrons of the fourth bond have parallel spins: Table S.2. Löwdin and Inverse weights of the structures in Group for C 2 calculated at VBSCF/6-3G* level. Structure 2-structure singlet 3-structure singlet a 3-structure triplet a Löwdin Inverse Löwdin Inverse Löwdin Inverse E(au) b a. The set of the 3 structures involves those structures, which pair the electrons of fourth (i.e. structures are, 2, 3, 6, 7, 8, 9, 4, 5, 8, 9, 20, 2). b. Total energies of the corresponding wave functions. The QC state energy obtained from the 3-structure calculation, by un-pairing the 4 th bond, is au NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved. 4

5 DOI: 0.038/NCHEM.263 Table S.3. Löwdin and Inverse weights of the structures in Group for BN calculated at VBSCF/6-3G* level. a Structure 2-structure singlet 3-structure singlet 3-structure triplet Löwdin Inverse Löwdin Inverse Löwdin Inverse E(au) b a. See Table S.2. b. Total energies of the corresponding wave functions. The energy of the QC state obtained from the 3-structure calculations is au. Table S.4. Löwdin and Inverse weights of the structures in Group for CN + calculated at VBSCF/6-3G* level. a Structure 2-structure singlet 3-structure singlet 3-structure triplet Löwdin Inverse Löwdin Inverse Löwdin Inverse NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved. 5

6 DOI: 0.038/NCHEM E(au) b a. See Table S.2. b. Total energies of the corresponding wave functions. The energy of the QC state obtained from the 3-structure calculations is au. Table S.5. Löwdin and Inverse weights of the structures in Group for CB - calculated at VBSCF/6-3G* level. a Structure 2-structure singlet 3-structure singlet 3-structure triplet Löwdin Inverse Löwdin Inverse Löwdin Inverse E(au) a a. See Table S.2. b. Total energies of the corresponding wave functions. The energy of the QC state obtained from the 3-structure calculations is au. Table S.6. D in values calculated by three methods Method for D in calculation C 2 BN CN+ CB- D in,vb (QC) a 4.30(9.64) 6.97(9.04) 7.38(0.32) 4.6(9.45) NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

7 DOI: 0.038/NCHEM.263 D in,vb (/ ΔE ST ) b D in,fci (/2 ΔE ST ) c a. D in was calculated as a difference in energy between the VB energy of the 2 singlet coupled structures in group and the QC state of the 3 covalent bonded structures E(E(2st-singlet state)- E(3st-quasiclassical state)). In parentheses are shown the covalent contributions to D in, i.e., E(E(3st-singlet state)-e(3st-quasiclassical state)). b ) D in,vb was calculated as half of the difference between total energy of the 2 singlet coupled structures on Group and the energy of the 3 singly couple structures in the triplet state, E(E(2st-singlet state)-e(3st-triplet state))/2. c ) D in,fci was calculated half of the difference between the FCI/6-3G* energies of the X! g + and c 3! u + states. II. FCI results II.. FCI/6-3G* data Table S.7. Total energies in atomic units (au) for different states of C 2, Si 2 and Ge 2, and equilibrium bond lengths (R eq, in Å) of the corresponding X! + g state calculated by FCI/6-3G*. C 2 a Si 2 b Ge 2 c X! g c 3 +! u " u # " g ΔE( X! g + - c 3! u + ) d R eq ( X! + g ) Å a) -75. au. b) au. c) au. d) kcal/mol Table S.8. Total energies in atomic units (au) of the different states of CN +, BN, and CB -, and equilibrium bond lengths (R eq, in Å) in the " + state calculated by FCI/6-3G*. CN +a BN b CB - c " " " " # ΔE( " " + ) d R eq ( " + ) Å a) - 9. au. b) au. c) au. d) kcal/mol. NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

8 DOI: 0.038/NCHEM.263 II.2. FCI wave functions II.2.. C 2 The X! + g state (FCI/6-3G* optimized bond length is.260 Å) Coefficients larger than 0.05 are listed: 2" 2 g 2" 2 u # 2 2 ux # uy " 2 g # 2 ux # 2 2 uy 3" g 2" 2 g 2" 2 u # 2 2 ux # gy 2" 2 g 2" 2 u # 2 2 uy # gx " 2 g # 2 ux 3" g # uy 2" u # gy 2" 2 g # 2 uy 3" g # ux 2" u # gx 2" 2 g # 2 ux 2" u # gy 3" g # uy 2" 2 g # 2 uy 2" u # gx 3" g # ux 2" 2 g # 2 ux 2" u 3" g # gy # uy 2" 2 g # 2 uy 2" u 3" g # ux # gx 2" 2 g # 2 ux # uy # gy 2" u 3" g 2" 2 g # 2 uy # ux # gx 2" u 3" g 2" 2 u # 2 ux # 2 uy 2" g 3" g 2" 2 u # 2 ux # 2 uy 3" g 2" g 2" 2 g 3" 2 g # 2 2 uy # gx 2" 2 g 3" 2 g # 2 2 ux # gy 2" 2 g 2" 2 u # uy # gx # ux # gy 2" 2 g 2" 2 u # 2 2 ux 3" g 2" 2 g 2" 2 u # 2 2 uy 3" g The c 3! u + state Coefficients larger than 0. are listed 2" 2 g # 2 ux # 2 uy 2" u 3" g NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved. 8

9 DOI: 0.038/NCHEM.263 2" 2 g # 2 ux 2" 2 u # uy # gy 2" 2 g # 2 uy 2" 2 u # ux # gx 2" 2 g # 2 ux # 2 gy 2" u 3" g 2" 2 g # 2 uy # 2 gx 2" u 3" g II.2.2. The CN + molecule Coefficients larger than 0. are listed The " + state (FCI/6-3G* optimized bond length is.94 Å) 2" 2 s 2" 2 pz # 2 2 x # y 2" 2 s 3" 2 pz # 2 2 x # y ! 2 s! 2 y! 2 x 2! pz 3! pz 2! 2 s! 2 y! 2 x 2! pz 3! pz 2" 2 s 2" 2 pz # 2 2 x 2# y 2" 2 s 2" 2 pz # 2 2 y 2# x The 3 " + state 2" 2 s # 2 x # 2 y 2" pz 3" pz 2" 2 s 2" 2 pz # 2 x # y 2# y 2" 2 s 2" 2 pz # 2 y # x 2# x 2" 2 s # 2 x 2# 2 y 2" pz 3" pz 2" 2 s 2# 2 x # 2 y 2" pz 3" pz II.2.3. The CB - molecule Coefficients larger than 0. are listed The " + state (FCI/6-3G* optimized bond length is.3877 Å) 2" 2 s 2" 2 pz # 2 2 x # y 2" 2 s 3" 2 pz # 2 2 x # y 2" 2 s 2" 2 pz # 2 2 x 2# y 2" 2 s 2" 2 pz # 2 2 y 2# x The 3 " + state NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

10 DOI: 0.038/NCHEM.263 2" 2 s # 2 x # 2 y 2" pz 3" pz 2" 2 s 2" 2 pz # 2 x # y 2# y 2" 2 s 2" 2 pz # 2 y # x 2# x 2" 2 s # 2 x 2# 2 y 2" pz 3" pz 2" 2 s 2# 2 x # 2 y 2" pz 3" pz II.2.4. The BN molecule Coefficients larger than 0. are listed The " + state (FCI/6-3G* optimized bond length is.292 Å) 2" 2 s 2" 2 pz # 2 2 y # x 2" 2 s 3" 2 pz # 2 2 y # x ! 2 s! 2 y! 2 x 2! pz 3! pz 2! 2 s! 2 y! 2 x 2! pz 3! pz The 3 " + state 2" 2 s # 2 y # 2 x 2" pz 3" pz 2" 2 s 2" 2 pz # 2 x # y 2# y 2" 2 s 2" 2 pz # 2 y # x 2# x II.2.5. The Si 2 molecule Coefficients larger than 0. are listed The X! g + state (FCI/6-3G* optimized bond length is 2.26 Å) 3" 2 g 2# 2 ux 2# 2 2 uy 3" u 3" 2 g 4" 2 g 2# 2 2 ux 3" u 3" 2 g 4" 2 g 2# 2 2 uy 3" u 3" 2 g 2# 2 ux 3" 2 2 u 2# gy 3" 2 g 2# 2 uy 3" 2 2 u 2# gx The c 3! + u state 3" 2 g 2# 2 ux 2# 2 uy 3" u 4" g 3" 2 g 2# 2 ux 3" 2 u 2# uy 2# gy 3" 2 g 2# 2 uy 3" 2 u 2# ux 2# gx NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

11 DOI: 0.038/NCHEM.263 3" 2 g 4" 2 g 2# 2 ux 2# uy 2# gy 3" 2 g 4" 2 g 2# 2 uy 2# ux 2# gx 3" 2 g 3" 2 u 2# 2 gx 2# uy 2# gy 3" 2 g 3" 2 u 2# 2 gy 2# ux 2# gx 3" 2 g 2# 2 ux 2# 2 gy 4" g 3" u 3" 2 g 2# 2 uy 2# 2 gx 4" g 3" u For 3 " u state (ground state according to FCI/6-3G*) 3" 2 g 2# 2 uy 3" 2 u 2# ux 4" g " 2 g 3" 2 u 2# 2 gy 2# ux 4" g 3" 2 g 4" 2 g 2# 2 uy 3" u 2# gx 3" 2 g 3" 2 u 4" g 2# uy 2# gx 2# gy II.2.6. The Ge 2 molecule Coefficients larger than 0. are listed The X! g + state (FCI/6-3G* optimized bond length is Å) 4" 2 g 4" 2 u 5" 2 2 g 3# uy 4" 2 g 4" 2 u 5" 2 2 g 3# ux 4" 2 g 4" 2 u 5" 2 2 g 3# gx 4" 2 g 4" 2 u 5" 2 2 g 3# gy The c 3! u + state 4" 2 g 5" 2 g 4" 2 u 3# uy 3# gx 4" 2 g 5" 2 g 4" 2 u 3# ux 3# gy 4" 2 g 3# 2 uy 4" 2 u 3# ux 3# gy 4" 2 g 3# 2 ux 4" 2 u 3# uy 3# gx The 3 " g # state (ground state according to FCI/6-3G*) 4" 2 g 5" 2 g 4" 2 u 3# ux 3# uy 4" 2 g 5" 2 g 4" 2 u 3# gx 3# gy NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

12 DOI: 0.038/NCHEM.263 II.3. GVB Transformation of the FCI/6-3G* Wave functions II.3.. The 4 th s-bond in the C 2 Molecule The FCI wave function is dominated by two configurations, which cover about 80% of the full FCI wave function: " FCI # g 2 $ u 4 2# u 2 % # g 2 $ u 4 3# g 2 + There are smaller contributions taking care of correlating the π- bonds and dynamic correlation of all pairs. If we take just the two major configurations and renormalize the two- configuration (TC) wave function (N ), we obtain: " TCSCF # 0 $ # D " TCSCF [# 0 $ # D ] λ , λ " TCSCF (2# g 2 $ u 4 )(2# u + %3# g )(2# u & %3# g ) & (2# g 2 $ u 4 )(2# u + %3# g )(2# u & %3# g ) " TCSCF (2# 2 g $ 4 u )% R % L & (2# 2 g $ 4 u )% R % L GVB- pair " L + # (2$ + #3$ ) + 2 u g " " R + # (2$ % #3$ ) u g + " 2 φl φr S " L " R # II.3.2. The π x -bond in C 2. Similarly, the fundamental configuration and the one correlating the π bonds constitute about 70% of the full FCI wave function:! FCI ! 2 g 2! 2 u! 2 2 uy! ux " ! 2 g 2! 2 u! 2 2 uy! gx + Renormalizing the two dominant configuration (N ) yields: " 0 # " D NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

13 DOI: 0.038/NCHEM [" 0 # " D ] λ , λ (2! 2 g 2! 2 u! 2 uy )(! ux + "! gx )(! ux " "! gx ) " (2! 2 g 2! 2 u! 2 uy )(! ux + "! gx )(! ux " "! gx ) (2! 2 g 2! 2 u! 2 uy )! R! L " (2! 2 g 2! 2 u! 2 uy )! R! L GVB- pair! L! R + " (# 2 ux + "# gx ) + " (# 2 ux! "# gx ) +! φl φr S! L! R ! II.3.3. The π y - bond in C 2. By complete analogy to II.3.2 we get here the following GVB pair: φl φr S " L " R # II.3.4. The 4 th σ- bond in the CN + Molecule NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

14 DOI: 0.038/NCHEM.263 The following two configurations cover about 76% of the full FCI wave function. There are small contributions taking care of correlating the p- bonds and dynamic correlation of all pairs. " FCI # 2 $ x 2 $ y 2 4# 2 % # 2 $ x 2 $ y 2 5# 2 + Using a TC wave function and renormalizing (N ), we get: " 0 # " D [" 0 # " D ] λ , λ " TCSCF (3# 2 $ x 2 $ y 2 )(4# + %5# )(4# & %5# ) & (3# 2 $ x 2 $ y 2 )(4# + %5# )(4# & %5# ) " TCSCF (3# 2 $ 2 x $ 2 y )% R % L & (3# 2 $ 2 x $ 2 y )% R % L GVB- pair " L (4$ + #5$) 2 + # " R + # (4$ % #5$) 2 + " 2 φl φr S " L " R # II.3.5. The 4 th σ- bond in the CB - Molecule Two configurations cover about 79% of the full FCI wave function: " FCI # 2 $ 2 x $ 2 y 4# 2 % # 2 $ 2 x $ 2 y 5# 2 + There are small contributions taking care of correlating the π- bonds and dynamic correlation of all pairs. Renormalizing the TC wave function (N ) yields, NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

15 DOI: 0.038/NCHEM " 0 # " D [" 0 # " D ] λ , λ (3! 2 " x 2! y 2 )(4! + "5! )(4! " "5! ) " (3! 2 # x 2! y 2 )(4! + "5! )(4! " "5! ) (3! 2! x 2! y 2 )! R! L " (3! 2! x 2! y 2 )! R! L GVB- pair! L! R (4# + "5# ) 2 + " + " (4#! "5# ) 2 +! φl φr S " L " R # II.3.6. The 4 th σ- bond in the BN Molecule Two configurations cover about 77% of the full FCI wave function: " FCI # 2 $ x 2 $ y 2 4# 2 % # 2 $ x 2 $ y 2 5# 2 + There are small contributions taking care of correlating the π- bonds and dynamic correlation of all pairs. Renormalizing the TC wave function (N ), we get: " TCSCF # 0 $ # D " TCSCF [# 0 $ # D ] λ , λ " TCSCF (3# 2 $ 2 x $ 2 y )(4# + %5#)(4# & %5# ) & (3# 2 $ 2 x $ 2 y )(4# + %5#)(4# & %5# ) " TCSCF (3# 2 $ 2 x $ 2 y )% R % L & (3# 2 $ 2 x $ 2 y )% R % L GVB- pair " L (4$ + #5$) 2 + # " R + # (4$ % #5$) 2 + " 2 NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.

16 DOI: 0.038/NCHEM.263 φl φr S " L " R # III. Wiberg bond order indices III. The " g + and " + states The calculations were done using B2PLYP/aug- cc- pvtz level of theory using NBO3. Table S.9. Wiberg bond order indices for the " + g and " + states calculated using B2PLYP/aug-cc-pVTZ level of theory a SCF density MP2 density C N " C H 2 C BN CN BC a) using NBO 3. program from Gaussian 09 package. IV. Relaxed Force Constants Table S.. Force constant (FC, in N/cm), Reciprocal compliance force constant (RCFC, in cm/n), equilibrium distance and frequencies of C2 ( X! g + and c 3! u + states) and H2C2 ( " g + state) using 6-3G* basis set. RCFC a FC b v(cm- ) c Req (Å) d " + g - C2- CCSD(T) e (2.24) f (858.6) f " + g - C2- MRCI NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved. 6

17 DOI: 0.038/NCHEM.263 " g + - C2- FCI.2603 " + g - H2C2- CCSD(T) e (5.68) f ( ) f.070 " + g - H2C2- MRCI a) RCPC (relaxed force constant) were calculated using Compliance 3.02 program and output from CCSD(T)/6-3G* calculations using G03 program. b) Force constant were obtained with the formula presented in the method section of the paper. c) Frequencies (v, in cm- ) obtained from corresponding Molpro calculations. d) Optimized parameters from corresponding calculations using Molpro. e) CCSD(T)/6-3G* geometry used in the frequency calculation corresponds to MRCI/6-3G* optimized geometry. f) Frequencies and FC correspond to CCSD(T)/6-3G* calculations using G03 program. NATURE CHEMISTRY Macmillan Publishers Limited. All rights reserved.