DIRECT POTENTIAL FITTING FOR THE A31u and X1+g STATES

DIRECT POTENTIAL FITTING FOR THE A 3Π1u andX 1Σ+g STATES OF Br2

T.YUKIYA∗, N. NISHIMIYA∗, M.SUZUKI∗, and R. J. LE ROY∗∗

*Dept. of Electronics and Information Technology, Tokyo Polytechnic University, Iiyama 1583,Atsugi City, Kanagawa 243-0297, Japan

** Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

June 21, 2012

T.YUKIYA et al. (Tokyo Polytechnic University) Potential Fitting for Br2 June 21, 2012 1 / 17

Our ResearchThe A − X system of ICl(1),(2), IBr(3),(4), I(5)

2 and Br(6),(7)2 have been

studied in the 0.8µm region.

The line positions of these spectra were determined in order toestablish a frequency standard.

Potential energy curves and spectroscopic constants of the Aand X state are determined.

(1) N. Nishimiya et al., “Nuclear Quadrupole Coupling Effect in the A-X System and Spectroscopic Constants for v=2′-4′′ ,3′-5′′

and 4′-5′′ Progressions of ICl”, JMS, 163, 43(1994).(2) T. Yukiya et al., “ Doppler-limited spectroscopy of the A3Π1–X1Σ+ system of ICl in the 0.8µm region using a Ti:sapphire ring

laser”, JMS 269, 193(2011).(3) N. Nishimiya et al., “Laser Spectroscopy of the A3Π1–X1Σ+ System of IBr”, JMS, 173,8(1995).(4) T. Yukiya et al., “ High-Resolution Laser Spectroscopy of the A3Π1–X1Σ+ System of IBr with a Titanium: Sapphire Ring Laser”,

JMS 214, 132( 2002).(5) T. Yukiya et al., “High-Resolution Laser Spectroscopy of the A3Π1u–X1Σ+g System of I2 with Ti: Sapphire Ring Laser”, JMS,

182, 271(1997).(6) N. Nishimiya et al., “High-Resolution Laser Spectroscopy of the A3Π1–X1Σ+ of Br2”, OHIO Columnbus Meeting (2005).(7) N. Nishimiya et al., “Improvement of Spectroscopic Constants for the A3Π1–X1Σ+ System of Br2”, OHIO Columnbus Meeting

(2011).


Previous WorkM.A.A.Clyne and J.A.Coxon,“The Emission Spectra of Br2 and IBr Formedin Atomic Recombination Processes”, J.Mol.Spectroscopy, 23, 58 (1967).

P.Venkateswarlu, V.N.Sarma, and Y.V.Rao, “Resonance series of Br2 in thevacuum ultraviolet“, J.Mol.Spectrosc., 96, 247 (1982).

M. Saute,et. al., ”Coefficients d’interaction à grande distance C5 et C6 pourles 23 états moléculaires de Cl2 et de Br2”, Mol. Phys., 51, 1459 (1984).

S. Gerstenkorn and P. Luc, ”Analysis of the long range potential of 79Br2 inthe B3Π0+u state and molecular constants of the three isotopic brominespecies 79Br2, 79Br81Br2 and 81Br2”, J. Phys. (France), 50, 1417 (1989).

C. D. Boone, ” Magnetic rotation study of the A3Π1u − X1Σ+g system of Br2”,Thesis (PhD), British Columbia (1999).

C. Focsa, H. Li, and P. F. Bernath, “Characterization of the Ground State ofBr2 by Laser-Induced Fluorescence Fourier Transform Spectroscopy of theB3ΠO+u − X1Σ+g system”,J.Mol.Spectroscopy, 200, 104 (2000).

D.J.Postell,“Laser Induced Fluorescence Spectroscopy of Br2:Observations of Emission to High Vibrational Levels in the Ground X(1Σ+g )State”, Thesis(Master of Science), Wright State University, 2005.


Block diagram of Ti:sapphire ring laser spectrometer.

91MHz

100kHz

Br2

Local Bus

PC

Function

Generator

Sweep Sig.

D/A

Etalon Drive Sig.

Servo Amp.

Laser

Cntrl.

Ti:Al2O3 Laser Ar+ LaserM

BS

BS

Chopper

F.P.I

ref.

Det.

Wavelength

Meter

Lock-in

Amp.

Chopper

Cntrl.

Function Gen.

RF Gen.

Balanced

MixerBurst Signal

Det.

E.O.M.

ref.

Lock-inAmp.

GP-IB

Bus

To BRF

White type CellM

amp.

ModulationMethod

:Tone Burst

EOM :191 MHz

Cell Temp. :150C

Path length :13.6 m

Gas Pressure:10 Torr

WavelengthMeter

:BurleighWA-1500


Example of Br2 Spectrum

12711.0 12711.5 12712.0 12712.5 12713.0 12713.5

14 18

39

68

31

66

69

23

89

15 13 17

64

3427

32

14 12

68

91

30

38

13

56


Assigned Bands for the A State

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8

A s

tate

X state

Q

PR

Boone Q

Boone PR


Assigned Ro-vibrational Levels in the A State

0

10

20

30

40

50

60

70

80

90

100

0 5 10 15 20 25 30 35 40

J'

v'

This Work

Boone

Boone(QB)


Bυ′ and the First Differences for the A State

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0.055

0.060

0 10 20 30 40

B' v

cm

-1

v'

79,79Br2

79,81Br2

81,81Br2

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

0 10 20 30 40

B

' vx1

03cm

-1

v'

79,79Br2

79,81Br2

81,81Br2

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 5 10 15 20 25 30

qv'x105cm-1

v'

79,79Br2

79,81Br2

81,81Br2


Schrödinger equation and Potential Function.Schrödinger equation..

......

− ~

2

2µα

d2

dr2 +[V1ad(r) + ∆V (α)

ad (r)] +~2[J(J + 1) − Λ2]

2µαr2 [1 + g(α)(r)]

+ sgΛ(e/ f )∆V (α)Ω

(r)[J(J + 1)]Λψv,J(r) = Ev,Jψv,J(r)

.Potential Function(Morse Long Range)..

......

VMLR(r) = De

1 − µLR(r)

µLR(re)e−β(r)·yp(r;re)

2

µLR(r) =last∑i=1

Dmi (r)Cmi

rmi, Dm(r) =

(1 − exp

−b · (ρr)

m− c · (ρr)2

√m

)m

β(r) = βMLR(yp(r; rref)) = yp(r; rref)β∞ + [1 − yp(r; rref)]NS ,NL∑

i=0

βi · yq(r; rref)i

yp(r; rref) =rp − rref

p

rp + rrefp , yq(r; rref) =

rq − rrefq

rq + rrefq

Robert J. Le Roy et al. “Long-range damping functions improve the short-range behaviour of ’MLR’ potential energy functions”, Mol.

Phys. 109, p435(2011)


Born–Oppenheimer Breakdown Correction and Λ–Doubling Functions.Born–Oppenheimer Breakdown Correction..

......

∆V (α)ad (r) =

∆M(α)A

M(α)A

S Aad(r) +

∆M(α)B

MαB

S Bad(r), g(α)(r) =

∆M(1)A

MαA

RAna(r) +

∆M(1)B

M(α)B

RBna(r)

S Aad(r) = [1 − ypad (r; re)]

NAad∑

i=0

uAi yqad (r; re)i + uA

∞ypad (r; re)

RAna(r) = [1 − ypna (r; re)]

NAna∑

i=0

tAi yqna (r; re)i + tA

∞ypna (r; re)

∆M(α)A = M(α)

A − M(1)A

.Λ–Doubling Functions..

......

∆V (α)Ω

(r) =(~2

2µαr2

)2Ω

fΩ(r)

fΩ(r) =NΩ∑i=0

ωΩi yqΩ(r; re)i


Dimensionless Root-Mean-Square Deviation(dd)

dd =

1N

N∑i=1

ycalc(i) − yobs(i)

unc(i)

2

1/2

yobs(i) : measured valueycalc(i) : calculated valueunc(i) : The estimated uncertainty of datumN : The total number of data


Choosing of a Potential Model minimizing DRMS andNumber of Parameters.

0.914

0.916

0.918

0.920

0.922

0.924

0.926

0.928

0.930

3.0 3.2 3.4 3.6 3.8 4.0

dd

rref

N12 p=5 q=4N12 p=5 q=5N13 p=5 q=4N13 p=5 q=5N14 p=5 q=4N14 p=5 q=5

0.910

0.915

0.920

0.925

0.930

0.935

0.940

0.945

0.950

0.955

0.960

2.8 3.0 3.2 3.4 3.6

dd

rref

N10 p5 q6N11 p5 q6N12 p5 q6N10 p5 q5N11 p5 q5N12 p5 q5

(a) A State (b)X State

Number of β terms , p and q values of Morse Long Range(MLR) potential functionin the A and X state were changed in this calculation. C5 ,C6 and C8 of the X stateand C6 and C8 of the A state are constrained at values reported by M. Saute. Teand C5 of the A state, De , Re and β were computed by DPotFita . In the A state,Born – Oppenheimer Breakdown term t1 ∼ t6 and Λ doubling term ω0 ∼ ω3 wereused. The lines of the v′=27 in A state were not included in this calculation.

ahttp://scienide2.uwaterloo.ca/∼rleroy/dpotfit/

X-State C5 cm−1Å5 : -0.0033×105

C6 cm−1Å6 : 6.274 ×105

C8 cm−1Å8 : 1.55 ×107

A-State C5 cm−1Å5 : 0.32 ×105

C6 cm−1Å6 : 6.37 ×105

C8 cm−1Å8 : 1.55 ×107

M. Saute (Mol. Phys., 51, 1459(1984)).


Long-Range Extrapolation behavior of the Potential

0.0x100

5.0x104

1.0x105

1.5x105

2.0x105

2.5x105

3.0x105

0.00 0.05 0.10 0.15 0.20

-(V

(r)-

D)*

r5=

C5+

C6/r

(A

Sta

te)

1/r [Ang.-1

]

Data


Number of Λ-Type Doubling Constants and CentrifugalBorn–Oppenheimer Breakdown Terms

1.0

2.0

3.0

4.0

5.0

0 1 2 3 4 5 6

dd

N

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

0.0 5.0 10.0 15.0 20.0

V

/r**

4 x

10

4

Å

N

=0123456

(a) Number of Lambda-Type Doubling terms (b) ∆VΩ(r) function

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

0 1 2 3 4 5 6 7 8 9

dd

Nna

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 5.0 10.0 15.0 20.0

Rnax 1

04

Å

Nna=12

3

4

5

6

7

8

9

(a) Number of Centrifugal BOB terms (b) Centrifugal BOB function


Potential Function Parameters for the A and X state.For the A-Sate: MLR N=13, p=5, q=4 , Rre f =3.3..

......

Values Error Values Error Values ErrorTe 13909.01274 0.023 De 2147.9239 0.014 Re 2.7046951 2.1×10−5

C5 3.72212×104 720 C6 6.37×105 – C8 1.55×107 –β0 1.1604 6.2×10−4 β1 0.60688 0.0028 β2 -3.12787 0.005β3 -3.4697 0.012 β4 10.3542 0.025 β5 14.758 0.081β6 -42.133 0.31 β7 -47.605 0.54 β8 142.91 1.9β9 64.24 2.2 β10 -295.0 5.6 β11 42.8 3.8β12 260.0 7.0 β13 -147.0 4.7

t0 0.0 – t1 0.00171 6.4×10−4 t2 -0.0174 0.00545t3 -0.036 0.025 t4 1.205 0.24 t5 -5.99 0.93t6 12.84 1.7 t7 -12.9 1.6 t8 5.0 0.55ω0 0.00386 5.0×10−5 ω1 -0.0122 0.0042 ω2 0.037 0.011ω3 0.312 0.0084

The lines of the v′=27 in A state were not included in this calculation.

.For the X-Sate: MLR N=11, p=5, q=6, Rre f =3.12..

......

Values Error Values Error Values ErrorDe 16056.93664 0.0088 Re 2.28102682 6.9×10−7

C5 -3.3×102 – C6 6.274×105 – C8 1.55×107 –β0 1.0418061 3.8×10−5 β1 0.860947 1.7×10−4 β2 1.24355 7.4×10−4

β3 0.16568 0.0035 β4 1.48566 0.0076 β5 -1.9265 0.025β6 0.5507 0.043 β7 3.806 0.061 β8 2.972 0.12β9 -1.523 0.048 β10 -3.38 0.11 β11 -1.43 0.057u0 0.323 0.0068

.Overall..

......

No. of Data: 16916v′=27 is not included

No. of Param.:45

dd: 0.896

vmin(A): 2

vmax(A): 37

D-vmax(A): 2.0 cm−1

vmin(X): 0

vmax(X): 76

D-vmax(X): 116 cm−1


Band Constant Calculated by Parameter Fits and by Direct-Potential

Fits

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 10 20 30

Bv'/

cm

-1

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

0 10 20 30

B

v' x 1

03/

cm

-1

79,79Br2(BOB)

79,79Br2(No C-BOB)

ParameterFitting Coxon(J.M.S.1972)

v ' v '


Conclusion

Spectra in the new bands weremeasured and assigned in 0.71 –0.83 µm region.Irregular behavior of Bv′ for the Astate were confirmed.Electronic isotope shift andcentrifugal Born – Oppenheimerbreakdown terms were determinedfor the first time.Accurate MLR potential function forthe A and X state were determined.

-15

-10

-5

0

5

2.0 3.0 4.0 5.0

R [Å]x1

00

0 c

m-1


DIRECT POTENTIAL FITTING FOR THE A31u and X1+g STATES

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