Rotational Spectrum of FCO2 Molecule with Resolved fs and hfs

1. moments of inertia Ia < Ib < Ic a b c? x y z? J.K.G. Watson, VIBRATIONALSPECTRAand STRUCTURE prolate oblate Ir IIr IIIr Il IIl IIIl x b c a c a b y c a b b c a Nz2 |KNM> = K|KNM> .. symmetric rotor basis functions z a b c a b c Rotational spectrum of FCO2 molecule with resolved fs and hfs in its ground vibrational and 2 B2 electronic ground states a ~y ~z state’s multiplicity ~ MS = 2 S + 1=2 <= S = ½ .. electron spin The choice of the molecule-fixed axes system ? IF = 1/2 F nuclear spin (+c) ~ (+y) ~ (+z) b ~x C O O It is an asymmetric top, which belong to C2v pointgroup NKaKc ..asymmetric rotor levels (two limit cases)

2. SYMMTERY OF VIBRATIONAL LEVEL ( el )  ( vib) = ( ev) B2 A1 B2 ( ns) A1 SYMMETRY OF ROTATIONAL LEVELS NKaKc ( ev)  ( rot) = ( rve) KaKc (rot) e e A1 GROUP C2v(M) E E (12) C2b E* ab (12)* bc e o A2 C2v o e B2 A1 1 1 1 1 A2 1 1 -1 -1 o o B1  = ( int) ( rve) B1 1 -1 -1 1 B2 1 -1 1 -1 MOLECULAR WAVE FUNCTION AND NUCLEAR SPIN STATISTICS total internal:  int =  el vib rot ns  el .. electronic  vib ..vibrational  rot ..rotational  ns ..nuclear-spin P.R.Bunker and Per Jensen, MOLECULAR SYMMETRYAND SPECTROSCOPY

3. 3 2 2 1 2.5 1.5 2.5 3 2 2 1 1.5 2 1 1 0 1.5 0.5 1.5 2 1 1 0 0.5 J =| N ± ½ | F =| J± ½ | fine splitting (fs) hyperfine splitting (hfs) Interactions: electron spin - rotational electron spin – nuclear spin nuclear spin - rotational ROTATIONAL LEVELS OF AN ASSYMETRIC TOP 2 2 0 2 2 1 2 1 1 2 1 2 2 0 2 1 1 0 1 1 1 1 0 1 0 0 0 N Ka Kc G. Herzberg, MOLECULAR SPECTRAAND MOLECULAR STRUCTURE II. INFRARED ANDRAMAN SPECTRAOF POLYATOMIC MOLECULES

4. coupling of molecular angular momenta  S Σkoef|SMS>|KNM> MS M   J I   nuclear spin electron spin eigenfunctions of J2 ,JZ with eigenvalues J(J+1) , MJ  N |KNSJMJ > Σ koef |IMI > MJ MI  F    rotational eigenfunctions of J2 , F2 ,FZ with eigenvalues J(J+1) , F(F+1) , MF andF =| J± ½ | For a givenKNwe have J =| N ±½ | quantum numbers assigned with fine and hyperfine levels MOLECULAR WAVE FUNCTION IN QUANTUM NUMBER NOTATION • int ~ |v> |SMS> |KNM > |IMI > … uncoupled representation S = ½ I = ½ electron spin, symmetric rotor and nuclear spin wave functions • int ~ |KNSJIFMF >… coupled representation R.N. Zare, ANGULAR MOMENTUM

5. =  Wsrn= CN I HAMILTONIAN (Ir representation ~ prolate, z = a) ROTATIONAL Hrot= A Na2+B Nb2 + C Nc2 + centrifugal distortion (A-reduction, J.K.G. Watson, VIBRATIONALSPECTRAand STRUCTURE) Hrotcf=-ΔN N4 -ΔNKN2Na2 - ΔKNa4-δNN2(N+2+ N-2) - 1/2 δK { Na2 (N+2 + N-2 ) + (N+2 + N-2 )Na2 } FINE (fs) AND HYPERFINE (hfs) STRUCTURE TERMS =  ~ Hsre=aaNaSa+bbNbSb + cc NcSc electron spin – rotational Wsre= N S =  Wssen= TS I Hssen=TaaSaIa+TbbSbIb + Tcc ScIc electron spin – nuclear spin ~  +aFC S I + aFCS I nuclear spin – rotational ~ Hsrn=CaaNaIa+CbbNbIb + Ccc NcIc classical energy Hamiltonian (only diagonal terms considered)

6. The second rank reducible tensor T is symmetric and traceless ! = S+= Sa+iSb I+= Ia+iIb S–= Sa–iSb I–= Ia–iIb Taa+ Tbb+ Tcc= 0 Hssen=1.5TaaSaIa + 0.25(Tbb–Tcc ) [S+I++ S–I–] – 0.5TaaS I a FC ELECTRON SPIN – NUCLEAR SPIN INTERACTION =  Wssen= TS I Hssen=TaaSaIa+TbbSbIb + Tcc ScIc ~  WFC = aFC S I HFC= aFCS I .. Fermi-contact type term HFC= a FCS I

7. NUMERICAL ANALYSIS OF THE SPECTRA (Pickett’s program) (+ centrifugal distortion ~ A-reduction ) Rotational constants

8. NUMERICAL ANALYSIS OF THE SPECTRA (Pickett’s program) Fine structure constants (+ centrifugal distortion ~ A-reduction ) J.M.Brown and T.J.Sears

9. NUMERICAL ANALYSIS OF THE SPECTRA (Pickett’s program) (+ centrifugal distortion ) Hyperfine structure constants -14.9(37) MICROWAVE AVG = 0.039987 MHz, IR AVG = 0.00000 MICROWAVE RMS = 3.955783 MHz, IR RMS = 0.00000 END OF ITERATION 5 OLD, NEW RMS ERROR= 1.16633 1.16633