Open-Shell Calculations

Transcription

1 Open-Shell Calculations Unpaired Electrons and Electron Spin Resonance Spectroscopy Video IV.vi

2 Winter School in Physical Organic Chemistry (WISPOC) Bressanone, January 27-31, 28 Thomas Bally University of Fribourg Switzerland Lecture 2: open-shell species WISPOC 28, Prof. Thomas Bally 2

3 open-shell species: atoms or molecules that contain one or more unpaired electrons one must deal with the issue of the electron s spin electrons have a magnetic moment, called spin S, associated with a spin quantum number S for the electron S is equal to 1/2 in the presence of a magnetic field B, the spin precesses rapidly around the axis of the field (which defines the z-direction) B M S = -1/2 S=1/2 M S = +1/2 S z S z z S S µ e µ e,z µ e µ e,z depending on its magnetic quantum number M S (±1/2, corresponding to and β-electrons, respectively), the z-component of the spin, S z, is oriented parallel or antiparallel to B the magnetic moment µ e is proportional but antiparallel to S the energy of interaction of the spin with the magnetic field is E = µ e,z B = g e µ B M S B WISPOC 28, Prof. Thomas Bally 3

4 Due to the interaction with the magnetic field, the energy levels of and β electrons are different (Zeeman-splitting). Transitions between these levels can be induced by electromagnetic radiation ( ESR spectrosocpy) E M S = +1/2 hν for a free electron: ν = B 28 Mz/mT resonance M S = -1/2 B B If this would be all there is to ESR spectroscopy, it would not be a very interesting experiment. What makes it interesting are the nuclear magnetic moments and their interaction with the magnetic moments of the electrons Like electrons, some important nuclei ( 1, 13 C) have a spin I of 1/2 which can be parallel or antiparallel to a magnetic field (M I =±1/2). As for electrons, the energy levels of opposite nuclear spins undergo Zeeman-splitting in a magnetic field, and transitions between the levels can be incuded by electromagnetic radiation (NMR-spectroscopy) WISPOC 28, Prof. Thomas Bally 4

5 Interaction of electron and nuclear magnetic moments (spins) dominant anisotropic contribution: the Fermi contact term E Fc = C ρ s () M S M I ρ s = ρ -ρ β spin density at the nucleus the contribution of this interaction to the energy is much smaller than that of the interaction with the external field hyperfine splitting E M S = +1/2 I=1/2 X hν M I = +1/2 M I = -1/2 ESR selection rules: ΔM S = ±1; ΔM I = M S = -1/2 M I +1/2-1/2 M I = -1/2 M I = +1/2 ESR-lines: a x hyperfine splitting a X : a X = K X ρ s () [T] B WISPOC 28, Prof. Thomas Bally 5

6 Interaction of electron and nuclear magnetic moments (spins) two equivalent nuclei two non-equivalent nuclei E M S = +1/2 M I (1) M I (2) +1/2 +1/2 +1/2-1/2-1/2 +1/2-1/2-1/2 ΣM I +1-1 E M S = +1/2 M I (A) +1/2 M I (B) +1/2-1/2 +1/2 +1/2-1/2-1/2-1/2 ΣM I +1-1 hν hν M S = -1/2 ΣM I /2-1/2-1 +1/2-1/2-1/2 +1/2 +1/2 +1/2 +1 M S = -1/2 ΣM I +1-1 a A -1/2-1/2 +1/2-1/2-1/2 +1/2 +1/2-1/ a x B a B B WISPOC 28, Prof. Thomas Bally 6

7 1 ESR-spectrum of the allyl radical (Fessenden & Schuler 1963) a = +.46 mt a = mt nodal plane a = mt singly occupied MO (SOMO) of the allyl radical? 1) why is there any electron "spin on the -atoms? 2) "why is there any electron "spin on the central -atom? 3) why is there negative spin "density on the outer s? WISPOC 28, Prof. Thomas Bally 7

8 spin polarization (purely statistical) probability of finding e B at distance r AB of e A β Pauli principle: two electrons of the same spin can never be at the same place at the same time e B Fermi-hole e A r AB for the same distribution of a pair of electrons, two electrons of the same spin suffer less repulsion than two electrons of opposite spin ( exchange interaction ) C planar π-systems β C β electrons of opposite spin have a higher propensity to avoid being in similar regions of space than two electrons of the same spin do. more favorable situation less favorable situation excess negative (β) spin density on π-σ spin polarization WISPOC 28, Prof. Thomas Bally 8

9 π-spin polarization excess spin positive a β β excess β spin π-π spin polarization negative a π-σ spin polarization? - and β-electrons in different obitals? no more paired - and β-electrons? WISPOC 28, Prof. Thomas Bally 9

10 Open-Shell Calculations andling Unpaired Electrons Restricted Open-shell vs Unrestricted Video IV.vii

11 π-spin polarization excess spin positive a β β excess β spin π-π spin polarization negative a π-σ spin polarization? - and β-electrons in different obitals? no more paired - and β-electrons? WISPOC 28, Prof. Thomas Bally 11

12 how to model open-shell systems? paired orbitals: restricted open-shell (ROF or RODFT) - physically incorrect (prevents spin polarization) - technically cumbersome (multiple operators, MP2) - leads often to artefactual symmetry breaking different orbitals for different spins (DODS, unrestricted F or DFT) - allows (in principle) to model spin polarizazion - technically easy to implement, including MP2 - gives lower electronic energy than ROF/RODFT owever: unrestricted wavefunctions show spin contamination! S (or S 2 ) and S z are molecular properties that can be computed as expectation values from wavefunctions using corresponding operators S 2 and ^ S z ^ Ψ S^ 2 Ψ = S 2 Ψ S^ z Ψ = S z the correct values for S 2 is S(S+1), i.e..75 for radicals (S=1/2), 2 for triplets (S=1) S 2 for restricted open-shell wavefunctions correspond to these (correct) values S 2 for unestricted open-shell wavefunctions are invariably higher than these values WISPOC 28, Prof. Thomas Bally 12

13 unrstricted wavefunctions are not eigenfunctions of the S 2 operator, ^ because they contain admixtures from (they are contaminated by) higher spin states this is demonstrated below for the allyl radical: the bigger λ, the higher is spin contamination linear combination of a doublet configuration and the S z =.5 component of a quartet state! Ψ ROF = π 1 π 1 β π 2 π 1a = π 1 λ π 3 π 1b = π 1 + λ π 3 Ψ UF = π 1a π β 1b π 2 = ( π 1 + λ π 3 ) β ( π 1 λ π 3 ) π 2 = π 1 π 1 β π 2 λ 2 π 3 π 3 β π 2 +λ π β 1 π 2 π 3 π 1 π β ( 2 π 3 ) Ψ ROF ** Ψ ROF provides for some dynamic correlation WISPOC 28, Prof. Thomas Bally 13

14 This spin contamination can become quite a nuisance, especially in highly delocalized systems where the and β-electrons in subjacent MOs are easily polarized polyenyl radicals soliton in polyacetylene 2 C!! C! C!!! C! n-3" C 2! As a consequence of spin contamination, UF overestimates spin polarization WISPOC 28, Prof. Thomas Bally 14

15 Spin contamination causes also problems in post-f methods to recover dynamic correlation that are based on many-body perturbation theory (MP2, CCSD), because the perturbation through high-spin states is too big to be handled by these methods This can lead to quite absurd results, as shown below for the benzyl radical E rel [kcal/mol] (correct answer) UF UMP2 S 2 S 2 /h spin polulation º bond lengths ROF " " ω " 9 º" º" º" 3 º" º" 5 º" º" 7 º" º" 9 º" ω ω WISPOC 28, Prof. Thomas Bally 15

16 Note that for well localized radicals (alkyl, oxo- aminyl- or nitroxy radicals) these problems are usually less severe. But: be watchful of S 2 in UF-based calculations! ow about DFT? within the KS model, DFT can be formulated in an unrestricted way, just like F, by optimizing individual spin densities ρ (r) and ρ β (r) instead of the total densityy ρ(r) = ρ (r) + ρ β (r). Of course unrestricted KS wavefunctions (for a fictional system of noninteracting electrons) will also contain terms due to higher spin states, but it is not quite clear whether spin contamination of a KS wavefunction means that the true wavefunction is bad (which is what it means in UF!) Nevertheless it is comforting to note that spin contamination in KS wavefunctions is usually much less severe than in F wavefunctions (the more F exchange density is admixed in hybrid functionals, the worse spin contamination becomes). In spite of this UDFT is quite good at modelling spin polarization (better than UF which overestimates it), and the energetics and properties of open-shell systems seem to be predicted just as well as those of closed-shell systems. WISPOC 28, Prof. Thomas Bally 16

17 modelling ESR spectra: does this work? remember: the dominant anisotropic contribution to hyperfine coupling is the Fermi contact term E Fc = C ρ s () M S M I ρ s = ρ -ρ β but in calculations AOs are usually composed of Gaussian functions Ψ 2 = A exp(-2ξ (r/r ) 2 ) spin density at the nucleus Ψ 2 (r=) 1s-AO of : Ψ 2 = 1/(π r 3 ) exp(-2r/r ), r = m Ψ 2 (r=) = m -3 no cusp! cusp r ow can this ever work? use very compact Gaussians (large ξ) to compose your AOs, and thus mimick a cusp (ESR specific basis sets). suprisingly, with DFT, one can make pretty good predictions with normal basis sets such as 6-31G*, probably due to a fortuitious cancellation of errors. WISPOC 28, Prof. Thomas Bally 17

18 some ESR hyperfine coupling constants BLYP B3LYP exp <.2 (Batra et. al. J. Phys. Chem. 1, (1996) BLYP or B3LYP/6-31G* C 3 P C 3 C F F F F F F N N WISPOC 28, Prof. Thomas Bally 18

19 an IR-spectrum of a radical cation -e + hν * + aselbach et al. elv. Chim. Acta. 84, 167 (21) WISPOC 28, Prof. Thomas Bally 19

20 some energetics of radical cations E rel [kcal/mol] activation energies B3LYP/6-31G* B3LYP/cc-pVTZ CCSD(T)/cc-pVDZ 2 1 +" " +" " +" radical cation of [1.1.1]propellane " " +" hν" +" " " -3 +" " +" " +" " (first observed species!) WISPOC 28, Prof. Thomas Bally 2

21 ow about the pathological benzyl radical? S 2 /h 2 planar (best answer) B(3)LYP " " ω " spin population perpendicular C-C bond lengths S 2 (correct value) 9 º" º" 3 º" 5 º" 7 º" 9 º" ω ω DFT ist quite well-behaved WISPOC 28, Prof. Thomas Bally 21

22 TANK YOU FROM CRIS TO TOMAS! Fribourg a friendly little city in the heart of Switzerland Chemistry" WISPOC 28, Prof. Thomas Bally 22