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Rotational Spectrum of FCO2 Molecule with Resolved fs and hfs

This text discusses the rotational spectrum of the FCO2 molecule, including the resolved fine structure (fs) and hyperfine structure (hfs) in its ground vibrational and 2 B2 electronic ground states.

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Rotational Spectrum of FCO2 Molecule with Resolved fs and hfs

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  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

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