1. a b (+c)(+c) cbacbaz acbbacy bacacbx III l IIlIIl IlIl III r IIrIIr IrIr ~ y~ y ~ x~ x Rotational spectrum of FCO 2 molecule with resolved fs and hfs in its ground vibrational and 2 B 2 electronic ground states ~ ( + z) The choice of the molecule-fixed axes system ? F C OO It is an asymmetric top, which belong to C 2v point group oblate prolate state’s multiplicity ~ M S = 2 S + 1=2 <= S = ½.. electron spin I F = 1/2 moments of inertia I a < I b < I c a b c ? ~ ( + y) ~ z~ z x y z ? J.K.G. Watson, V IBRATIONAL S PECTRA and S TRUCTURE NK a K c..asymmetric rotor levels (two limit cases).. symmetric rotor basis functions N z 2 |KNM> = K |KNM> nuclear spin

2.  (  rve ) B2 B2 B1B1 A1A1 A2A2  (  int ) B2 B2 B1B1 A1A1 A2A2 P.R.Bunker and Per Jensen, M OLECULAR S YMMETRY AND S PECTROSCOPY B1B1 o o B2B2 o e A2A2 e o A1A1 e e  (  rot ) KaKcKaKc 1 1B2B2 1 1B1B1 1 1A2A2 1 1 1 1A1A1 (12) *  bc E *  ab (12) C 2b E GROUP C 2v C 2v (M) MOLECULAR WAVE FUNCTION AND NUCLEAR SPIN STATISTICS SYMMTERY OF VIBRATIONAL LEVEL B2B2 B2B2 A1A1  (  el )   (  vib ) =  (  ev )  =  (  int )  (  rve )  (  ns ) A1A1  int =  el  vib  rot  ns total internal:  el.. electronic  vib..vibrational  rot..rotational  ns..nuclear-spin SYMMETRY OF ROTATIONAL LEVELS NK a K c  (  ev )   (  rot ) =  (  rve )

3. G. Herzberg, M OLECULAR S PECTRA AND M OLECULAR S TRUCTURE II. I NFRARED AND RAMAN S PECTRA OF P OLYATOMIC M OLECULES N K a K c 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 21102110 21102110 32213221 32213221 0.5 1.5 0.5 1.5 2.5 J = | N ± ½ | fine splitting (fs) F = | J ± ½ | hyperfine splitting (hfs) electron spin – nuclear spin electron spin - rotational Interactions: nuclear spin - rotational ROTATIONAL LEVELS OF AN ASSYMETRIC TOP

4. Σ koef |IMI > M J M I  int ~ |v> | SM S > | KNM > |IM I > … uncoupled representation S = ½I = ½ MOLECULAR WAVE FUNCTION IN QUANTUM NUMBER NOTATION electron spin, symmetric rotor and nuclear spin wave functions  int ~ |  KNSJIFM F > … coupled representation For a given K N we have J = | N ± ½ | and F = | J ± ½ | quantum numbers assigned with fine and hyperfine levels eigenfunctions of J 2, J Z with eigenvalues J(J+1), M J  eigenfunctions of J 2, F 2, F Z with eigenvalues J(J+1), F(F+1), M F  coupling of molecular angular momenta N  S  J  F  I  electron spin nuclear spin rotational Σ koef |SM S > |KNM > M S M |KNSJM J > R.N. Zare, A NGULAR M OMENTUM

5. HAMILTONIAN (I r representation ~ prolate, z = a) ROTATIONAL H rot = A N a 2 + B N b 2 + C N c 2 + centrifugal distortion (A-reduction, J.K.G. Watson, V IBRATIONAL S PECTRA and S TRUCTURE ) electron spin – rotational H rot cf = - Δ N N 4 - Δ NK N 2 N a 2 - Δ K N a 4 - δ N N 2 (N + 2 + N - 2 ) - 1/2 δ K { N a 2 (N + 2 + N - 2 ) + (N + 2 + N - 2 ) N a 2 } H sr e =  aa N a S a +  bb N b S b +  cc N c S c electron spin – nuclear spin nuclear spin – rotational H ss en = T aa S a I a + T bb S b I b + T cc S c I c H sr n = C aa N a I a + C bb N b I b + C cc N c I c ~ ~ ~ FINE (fs) AND HYPERFINE (hfs) STRUCTURE TERMS W sr e =  N S  = + a FC S I    W sr n = C N I  = W ss en = T S I  = + a FC S I classical energyHamiltonian (only diagonal terms considered)

6. ELECTRON SPIN – NUCLEAR SPIN INTERACTION H ss en = T aa S a I a + T bb S b I b + T cc S c I c ~ W FC = a FC S I    W ss en = T S I  = H FC = a FC S I T aa + T bb + T cc = 0 The second rank reducible tensor T is symmetric and traceless ! = H FC = a FC S I a FC.. Fermi-contact type term H ss en = 1.5 T aa S a I a + 0.25 (T bb – T cc ) [ S + I + + S – I – ] – 0.5 T aa S I S + = S a + i S b S – = S a – i S b I + = I a + i I b I – = I a – i I b

7. NUMERICAL ANALYSIS OF THE SPECTRA (Pickett’s program) Unitsab-initio study previous study Lille This work Lille + Prague 1. 2. A MHz13772.13752.667 (68)13752.2758 (63)13752.755(167) B —11291.11309.962 (52)11310.2307 (55)11309.853(136) C —6192.6192.8077 (21)6192.80035 (58)6192.8196( 68) ΔNΔN kHz6.9787.088 (124)7.691 (18)6.16( 76) Δ NK —1.221—— — ΔKΔK —13.23121.26 (108)15.682 (156) 29.1( 65) δNδN —2.9933.009 (62)3.3119 (92) 2.54( 38) δKδK —10.3919.492 (237)10.690 (34) 7.67(149) Ф KJ Hz -0.316 (48) Rotational constants (+ centrifugal distortion ~ A-reduction )

8. NUMERICAL ANALYSIS OF THE SPECTRA (Pickett’s program) Unitsprevious study Lille This work Lille + Prague 1. 2. ε aa MHz-80.74 (33)-80.233 (211) -84.88(128) ε bb —-788.67 (47)-789.868 (87)-782.9( 33) ε cc —-44.005 (19)-44.2597 (227)-44.307(112) ΔsNΔsN kHz -0.0923 (297) Δ s NK — -1.64 (45) δsKδsK — -3.80 (38) aF aF MHz-208 (27)-243.7 (79)-165( 98) ½ T aa -95.25 (179)-27.16 (90)-712( 47) ¼ (T bb −T cc ) 8.85 (122)6.008 (131) 86.8( 47) Fine structure constants (+ centrifugal distortion ~ A-reduction ) J.M.Brown and T.J.Sears

9. NUMERICAL ANALYSIS OF THE SPECTRA (Pickett’s program) Unitsprevious study Lille This work Lille + Prague 1. Units 2. C aa MHz8.744 (247)12.990 (159)C aa MHz11.18(107) C aa J — -0.02028 (142) — C aa K — 0.02256 (238)— ¼ (C bb − C cc ) MHz-0.2793 (74)-0.3830 (115)C bb MHz ¼ (C bb − C cc ) J kHz 0.3769 (167)C cc —-0.929( 84) ¼ (C bb − C cc ) K MHz -0.01303 (81)— 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 (+ centrifugal distortion )