a

b (+c)

cbacbazacbbacybacacbx

IIIlIIlIlIIIrIIrIr

~ y

~ x

Rotational spectrum of FCO2 molecule with resolved fs and hfs in its ground vibrational and 2 B2 electronic ground states

~ (+z)

The choice of the molecule-fixed axes system ?F

C

O O

It is an asymmetric top, which belong to C2v

point group

oblateprolate

state’s multiplicity ~ MS = 2 S + 1=2 <= S = ½ .. electron spin

IF = 1/2moments of inertia Ia < Ib < Ica b c ?

~ (+y)

~ z

x y z ? J.K.G. Watson, VIBRATIONAL SPECTRA and STRUCTURE

NKaKc ..asymmetric rotor levels

(two limit cases)

.. symmetric rotor basis functionsNz2 |KNM> = K |KNM>

nuclear spin

( rve)

B2

B1

A1

A2

( int)

B2

B1

A1

A2

P.R.Bunker and Per Jensen,MOLECULAR SYMMETRY AND SPECTROSCOPY

B1 o o

B2 o e

A2 e o

A1 e e

( rot)KaKc

-1 1-1 1B2

1-1-1 1B1

-1-1 1 1A2

1 1 1 1A1

(12)*

bc

E*

ab

(12)C2b

E E

GROUP

C2v

C2v (M)

MOLECULAR WAVE FUNCTION AND NUCLEAR SPIN STATISTICS

SYMMTERY OF VIBRATIONAL LEVEL

B2B2 A1

( el ) ( vib ) = ( ev )

= ( int )( rve ) ( ns)A1

int = el vib rot ns

total internal:

el .. electronic vib ..vibrational rot ..rotational ns ..nuclear-spin

SYMMETRY OF ROTATIONALLEVELS NKaKc

( ev ) ( rot) = ( rve)

G. Herzberg, MOLECULAR SPECTRA AND MOLECULAR STRUCTURE II. INFRARED AND RAMAN SPECTRA OF POLYATOMIC MOLECULES

N Ka Kc

2 2 02 2 1

2 1 1

2 1 22 0 21 1 01 1 11 0 1

0 0 0

2110

2110

3221

3221

0.51.5

0.51.5

1.5

1.5

2.5

2.5

J = | N ± ½ |

fine splitting (fs)

F = | J ± ½ |

hyperfine splitting (hfs)

electron spin – nuclear spinelectron spin - rotational

Interactions:

nuclear spin - rotational

ROTATIONAL LEVELS OF AN ASSYMETRIC TOP

Σ koef |IMI >MJ MI

int ~ |v> |SMS> |KNM > |IMI > … uncoupled representation

S = ½ I = ½

MOLECULAR WAVE FUNCTION IN QUANTUM NUMBER NOTATION

electron spin, symmetric rotor and nuclear spin wave functions

int ~ | KNSJIFMF > … 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 J2 , JZ with eigenvalues J(J+1) , MJ

eigenfunctions of J2 , F2 , FZ with eigenvalues J(J+1) , F(F+1) , MF

coupling of molecular angular momenta

N

S

J

F

I electron

spinnuclear spin

rotational

Σ koef |SMS> |KNM>MS M

|KNSJMJ >

R.N. Zare, ANGULAR MOMENTUM

HAMILTONIAN (Ir representation ~ prolate, z = a)ROTATIONAL

Hrot = A Na2 + B Nb

2 + C Nc2

+ centrifugal distortion (A-reduction, J.K.G. Watson, VIBRATIONAL SPECTRA and STRUCTURE)

electron spin – rotational

Hrotcf = - ΔN N4 -ΔNK N2Na

2 - ΔK Na4 - δN N2 (N+

2 + N-2 )

- 1/2 δK { Na2 (N+

2 + N-2 ) + (N+

2 + N-2 ) Na

2 }

Hsre = aa NaSa

+ bb NbSb + cc NcSc

electron spin – nuclear spin

nuclear spin – rotational

Hssen = Taa SaIa

+ Tbb SbIb + Tcc ScIc

Hsrn = Caa NaIa

+ Cbb NbIb + Ccc NcIc

~

~

~

FINE (fs) AND HYPERFINE (hfs) STRUCTURE TERMS

Wsre = N S

=

+ aFC S I

Wsrn = C N I

=

Wssen = T S I

=

+ aFC S I

classical energy Hamiltonian (only diagonal terms considered)

ELECTRON SPIN – NUCLEAR SPIN INTERACTION

Hssen = Taa SaIa

+ Tbb SbIb + Tcc ScIc~

WFC = aFC S I

Wssen = T S I

=

H FC = aFC S I

Taa + Tbb + Tcc = 0

The second rank reducible tensor T is symmetric and traceless !

=

H FC= a FC S I

a FC

.. Fermi-contact type term

Hssen = 1.5 Taa SaIa + 0.25 (Tbb – Tcc ) [ S+I+ + S–I– ]

– 0.5 Taa S I

S+ = Sa + i Sb

S– = Sa – i Sb

I+ = Ia + i Ib

I– = Ia – i Ib

NUMERICAL ANALYSIS OF THE SPECTRA (Pickett’s program)

Units ab-initiostudy

previous study Lille

This work Lille + Prague 1. 2.

A MHz 13772. 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)

ΔNkHz 6.978 7.088 (124) 7.691 (18) 6.16( 76)

ΔNK— 1.221 — — —

ΔK— 13.231 21.26 (108) 15.682 (156) 29.1( 65)

δN— 2.993 3.009 (62) 3.3119 (92) 2.54( 38)

δK— 10.391 9.492 (237) 10.690 (34) 7.67(149)

ФKJ Hz   -0.316 (48)

Rotational constants (+ centrifugal distortion ~ A-reduction )

NUMERICAL ANALYSIS OF THE SPECTRA (Pickett’s program)

Units previous study

Lille

This work Lille + Prague 1. 2.

εaa MHz -80.74 (33) -80.233 (211) -84.88(128)

εbb — -788.67 (47) -789.868 (87) -782.9( 33)

εcc — -44.005 (19) -44.2597 (227) -44.307(112)

ΔsN

kHz   -0.0923 (297)

ΔsNK

—   -1.64 (45)

δsK

—   -3.80 (38)

aF MHz -208 (27) -243.7 (79) -165( 98)

½ Taa -95.25 (179) -27.16 (90) -712( 47)

¼ (Tbb −Tcc ) 8.85 (122) 6.008 (131) 86.8( 47)

Fine structure constants (+ centrifugal distortion ~ A-reduction )

J.M.Brown andT.J.Sears

NUMERICAL ANALYSIS OF THE SPECTRA (Pickett’s program)

Units previous study

Lille

This work Lille + Prague 1.

Units

2.

CaaMHz 8.744 (247) 12.990 (159) Caa MHz 11.18(107)

CaaJ —   -0.02028 (142) —

CaaK —   0.02256 (238) —

¼ (Cbb − Ccc) MHz -0.2793 (74) -0.3830 (115) Cbb MHz

¼ (Cbb − Ccc)J kHz   0.3769 (167) Ccc — -0.929( 84)

¼ (Cbb − Ccc)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 )