NEW LINE LISTS WITH INTENSITIES FOR THE C2 SWAN SYSTEM AND ROVIBRATIONAL TRANSITIONS WITHIN THE NH GROUND STATE

James S.A. Brookea*, Peter F. Bernathb, Colin M. Westernc, Timothy W. Schmidtd, George B. Bacskayd, Marc C. van Hermerte & Gerrit C. Groenenboomf

a: Department of Chemistry, University of York, York, UK. b: Department of Chemistry & Biochemistry, Old Dominion University, Norfolk, VA, USA. c: School of Chemistry, University of Bristol, Bristol, UK. d: School of Chemistry, The University of Sydney, New South Wales, Australia. e: Department of Chemistry, Gorlaeus Laboratories, Leiden University, The Netherlands. f: Theoretical Chemistry, Institute for Molecules and Materials (IMM), Radboud University Nijmegen, Nijmegen, The Netherlands.

Funded by:

Line lists

• Quantum number assignments, line positions and intensities

• Positions– Recorded directly from laboratory spectra

• Intensities– Obtained with a combination of experimental and

theoretical methods– Require potential energy curve and (transition)

dipole moment function

Potential Energy Curves – Experimental (1)

• Spectrum is obtained from lab observations

• Lines are assigned and a fit of line positions provides molecular constants for each vibrational level.

• Equilibrium constants are obtained from fitting to and molecular constants:

• RKR procedure generates potential energy curve – Bob Le Roy’s RKR1 program

Spectrum

Molecular constants

Line assignment

and fit

RKR1

Potential energy curve

fit

Equilibrium constants

Potential Energy Curves – Experimental (2)

C2 Swan system (d3Πg-a3Πu) potential energy curves and TDM function

d

(Transition) Dipole Moment Function - Theoretical

• Line intensities cannot be accurately obtained from the experimental spectra that we use.

• Obtained from ab initio methods• Usually using good level of theory, e.g. MRCI,

and basis set such as aug-cc-PV6Z (used for NH), as they’re only diatomic molecules

Ab initio

Electronic (transition) dipole moment function

LEVEL and PGOPHER to Final Line List

• Bob Le Roy’s LEVEL calculates vibrational wavefunctions by solving the one-dimensional Schrödinger equation.

Vibrational wavefunctions and transition dipole moment matrix elements

LEVEL

Potential energy curve+

Electronic (transition) dipole moment function

Transition Dipole Moment Matrix Elements

• It then overlaps them and multiplies the result by the electronic (transition) dipole moment function, which is then integrated to give the transition dipole moment matrix element (TDMME).

LEVEL and PGOPHER to Final Line List

• Bob Le Roy’s LEVEL calculates vibrational wavefunctions by solving the one-dimensional Schrödinger equation.

Vibrational wavefunctions and transition dipole moment matrix elements

Einstein As and f-values – line list with positions and intensities

LEVEL

PGOPHER

Potential energy curve+

Electronic (transition) dipole moment function

• It then overlaps them and multiplies the result by the electronic (transition) dipole moment function, which is then integrated to give the TDMME.

• Colin Western’s PGOPHER calculates Hönl-London factors, and combines them with the TDMMEs and line positions to calculate Einstein A values and then f-values.

Line Strength Equations• PGOPHER calculates Hönl-London factors, and combines them

with the TDMMEs and line positions to calculate Einstein A values:3

• Where is the Hönl-London factor and is the TDMME.• The Einstein A values are also converted to oscillator strengths (f-

values):

𝑓𝐽 ′← 𝐽′ ′

¿=1.499

1~ν2

(2 𝐽 ′+1 )(2 𝐽 ′ ′+1 )

𝐴 𝐽 ′→ 𝐽 ′ ′❑¿

LEVEL and PGOPHER to Final Line List

• Bob Le Roy’s LEVEL calculates vibrational wavefunctions by solving the one-dimensional Schrödinger equation.

Vibrational wavefunctions and transition dipole moment matrix elements

Einstein As and f-values – line list with positions and intensities

LEVEL

PGOPHER

Potential energy curve+

Electronic (transition) dipole moment function

• It then overlaps them and multiplies the result by the electronic (transition) dipole moment function, which is then integrated to give the TDMME.

• Colin Western’s PGOPHER calculates Hönl-London factors, and combines them with the TDMMEs and line positions to calculate Einstein A values and then f-values.

• A final line list is created including positions and intensities for observed and non-observed transitions.

Line strengths

Ab initio

Electronic (transition) dipole moment function

Vibrational wavefunctions and transition dipole moment matrix elements

Einstein As and f-values – line list with positions and intensities

+LEVEL

Start

Start

PGOPHER

Spectrum

Molecular constants

Line assignment

and fit

RKR1

Potential energy curve

fit

Equilibrium constants

End

C2 Swan System - Introduction

• C2 is found in interstellar clouds, comets, cool stars and flames.

• Available line strengths based on assignments from 1968 by Phillips and Davis [1]

• Tanabashi and Amano in 2002 [2] disagreed with Phillips and Davis.

• Tanabashi et al. in 2007 [3] confirmed the disagreement.• A new line list would be very beneficial.

[1] Phillips JG, Davis SP. The Swan system of the C2 molecule, The spectrum of the HgH molecule. University of California Press; 1968. [2] Tanabashi A, Amano T. J Mol Spectrosc 2002;215:285–94. [3] Tanabashi A, Hirao T, Amano T, Bernath PF. Astrophys J Suppl Ser 2007;169:472–84.

• Swan system is the most prominent and therefore heavily studied

C2 Swan System – Perturbations (1)• Most vibrational levels of the d3Πg

state are perturbed.• Worst affected are the v=4 and 6

levels.– v=16 perturbs d3Πg v=4

– v=19 perturbs d3Πg v=6

• Perturbations quantified by Bornhauser et al. in 2010 and 2011 [4,5] – a 5Πg state also discovered and identified to perturb d3Πg v=6

• We refitted the available line positions and included these perturbations, and new data from Yeung et al..

[4] Bornhauser P, Knopp G, Gerber T, Radi P. J Mol Spectrosc 2010;262:69–74. http://dx.doi.org/10.1016/j.jms.2010.05.008. [5] Bornhauser P, Sych Y, Knopp G, Gerber T, Radi PP. J Chem Phys 2011;134:044302. http://dx.doi.org/10.1063/1.3526747. [6] Yeung SH, Chan MC, Wang N, Cheung. Chem Phys Lett 2013;557:31–6 doi: http://dx.doi.org/10.1016/j.cplett.2012.11.092.

u

C2 Swan System – Perturbations (2)Change in perturbation constants in fit:Parameter Bornhauser et al. value Value from fit

-0.6401(86) -0.6147(59)0.24737(61) 0.24869(21)0.7855(110) 0.7417(82)0.31192(37) 0.31123(12)4.6220(88) 4.6150(94)

Change in line errors with inclusion of perturbation constants:

Involving upper vibrational level

Average line position error (cm-1)Without perturbation

constantsWith perturbation

constants4 0.210 0.069

6 0.571 0.038

all 0.071 0.025

C2 Swan System – Intensities

• Electronic transition dipole moment calculated by Tim Schmidt and George Bacskay (University of Sydney) [8]

• Earlier mentioned procedure performed

v’Our value(ns)

Theor. [8](ns)

Expt.(ns)

Expt.(ns)

98.0 95.1 101.8 ± 4.2 106 ± 1599.8 96.7 96.7 ± 5.2 105 ± 152 102.4 99.1 104.0 ± 17 106.0 102 110.9 107 5 118.2 113

Vibrational lifetime comparison

[8] Schmidt TW, Bacskay GB. J Chem Phys 2007;127(23).

C2 Swan System – Data Included

• Final line list with positions and intensities produced:v' v'' J' J'' F' F'' p' p'' Observed Calculated Residual Perturbation E'' A f Description Paper Quality Weight 0 0 2 3 1 1 e e 19371.76200 19371.7621 -0.00010 0.00000 -1.4481 2.621927E+6 7.481903E-3 pP1e(3) T7 a 0.00500 0 0 3 4 1 1 f f 19369.31700 19369.3187 -0.00170 0.00000 9.8325 3.363432E+6 1.045363E-2 pP1f(4) T7 a 0.00500 0 0 4 5 1 1 e e 19367.02600 19367.0221 0.00390 0.00000 24.0522 3.652000E+6 1.194297E-2 pP1e(5) T7 a 0.00500 0 0 5 6 1 1 f f 19364.84600 19364.8809 -0.03490 0.00000 41.3152 3.780763E+6 1.278964E-2 pP1f(6) T7 c 0.05000 0 0 6 7 1 1 e e 19362.97400 19362.9435 0.03050 0.00000 61.5972 3.844201E+6 1.332207E-2 pP1e(7) T7 c 0.05000 0 0 7 8 1 1 f f 19361.19500 19361.1967 -0.00170 0.00000 85.0354 3.874655E+6 1.367316E-2 pP1f(8) T7 a 0.00500 0 0 8 9 1 1 e e 19359.69200 19359.6924 -0.00040 0.00000 111.5372 3.890445E+6 1.392373E-2 pP1e(9) T7 a 0.00500 0 0 9 10 1 1 f f 19358.39500 19358.3892 0.00580 0.00000 141.2663 3.896652E+6 1.410409E-2 pP1f(10) T7 a 0.00500 0 0 10 11 1 1 e e 19357.36400 19357.3560 0.00800 0.00000 174.0863 3.899270E+6 1.424427E-2 pP1e(11) T7 a 0.00500 0 0 11 12 1 1 f f 19356.51400 19356.5168 -0.00280 0.00000 210.1675 3.898503E+6 1.435123E-2 pP1f(12) T7 a 0.00500 0 0 12 13 1 1 e e 19355.95800 19355.9729 -0.01490 0.00000 249.3643 3.897172E+6 1.443957E-2 pP1e(13) T7 a 0.00500 0 0 13 14 1 1 f f 19355.61300 19355.6049 0.00810 0.00000 291.8268 3.894456E+6 1.450963E-2 pP1f(14) T7 a 0.00500 0 0 14 15 1 1 e e 19355.61300 19355.5592 0.05380 0.00000 337.4353 3.892098E+6 1.457015E-2 pP1e(15) T7 y 99999.00000 0 0 15 16 1 1 f f 19355.61300 19355.6644 -0.05140 0.00000 386.2900 3.889092E+6 1.461975E-2 pP1f(16) T7 c 0.05000 0 0 16 17 1 1 e e 19356.15900 19356.1219 0.03710 0.00000 438.3304 3.886728E+6 1.466390E-2 pP1e(17) T7 y 99999.00000 0 0 17 18 1 1 f f 19356.69700 19356.6999 -0.00290 0.00000 493.5777 3.884009E+6 1.470083E-2 pP1f(18) T7 a 0.00500 0 0 18 19 1 1 e e 19357.64500 19357.6639 -0.01890 0.00000 552.0606 3.882009E+6 1.473487E-2 pP1e(19) T7 c 0.05000 0 0 19 20 1 1 f f 19358.71500 19358.7135 0.00150 0.00000 613.6943 3.879775E+6 1.476359E-2 pP1f(20) T7 a 0.00500 0 0 20 21 1 1 e e 19360.19400 19360.1861 0.00790 0.00000 678.6244 3.878267E+6 1.479079E-2 pP1e(21) T7 a 0.00500 0 0 21 22 1 1 f f 19361.70600 19361.7057 0.00030 0.00000 746.6338 3.876572E+6 1.481407E-2 pP1f(22) T7 a 0.00500

• Published in JQSRT [9]. Line list available from article website, http://uk.arxiv.org/abs/1212.2102, and http://bernath.uwaterloo.ca/download/AutoIndex.php?dir=/C2/

[9] Brooke JSA, Bernath PF, Schmidt TW, Bacskay GB. JQSRT 2013;124(0):11-20. doi:10.1016/j.jqsrt.2013.02.025.

NH state vibration-rotation transitions

• NH is present in cool stars, comets, diffuse interstellar clouds, the Sun, and probably the upper atmospheres of extrasolar planets.

• Vibrational transitions have been used to calculate the actual nitrogen abundance in cool stars and the Sun.

• Boudjaadar et al. 1986 - The v=1 sequence up to v =5 [10]′• Ram et al. 1999 - more transitions in the same sequence in

1999 [11]• Ram and Bernath 2010 - (6,5) band [12]• Robinson et al. 2007 – rotational transitions in in v=1 and 2

[13][10] Boudjaadar D, Brion J, Chollet P, Guelachvili G, Vervloet M. J Mol Spectrosc 1986;119(2):352-66. doi:10.1016/0022-2852(86)90030-5.[11] Ram RS, Bernath PF, Hinkle KH. J Chem Phys 1999;110:5557-63. doi:10.1063/1.478453.[12] Ram RS, Bernath PF. J Mol Spectrosc 2010;260:115-9. doi:10.1016/j.jms.2010.01.006.[13] Robinson A, Brown J, Flores-Mijangos J, Zink L, Jackson M. Mol Phys 2007;105:639-62. doi:10.1080/00268970601162085.

NH state calculations• No new line

position fit - used Ram and Bernath 2010 [12]

• Mostly same procedure as before

• Dipole moment function calculated by Gerrit Groenenboom – not published [14]

[14] Campbell WC, Groenenboom GC, Lu HI, Tsikata E, Doyle JM. Phys Rev Lett 2008;100(8). doi:10.1103/PhysRevLett.100.083003.

NH - The Herman-Wallis Effect (1)

• Rotation - centrifugal force - causes change in bond length

• Results in a change in the vibrational wavefunctions

• TDMMEs are changed• Effect is greater in NH than C2 due to

light H atom

NH - The Herman-Wallis Effect (2)

• The noticeable effect for NH is an increase in the strength of the R branch and a decrease in the strength of the P branch

• To the right is the calculated spectrum of the NH (1,0) band.

• The quantum number N was used in the Herman-Wallis calculations

NH spectra

CN A2П-X2Σ+ and B2Σ+-X2Σ+ systems and theCP A2П-X2Σ+ system

• Will be discussed tomorrow by Ram• Herman-Wallis effect is not as strong in these

systems as the atoms are heavier• Maximum Herman-Wallis effect for observed

bands is:– 70% for CN A2П-X2Σ+

– 40% for CN B2Σ+-X2Σ+

– 2% for CP A2П-X2Σ+ (up to 30% for unobserved bands)

Conclusions

• New line list with positions and intensities produced for C2 Swan system – up to v =10 and v =9.′ ′

• Similar lists will soon be finished for NH, CN and CP.• The combination of experimental and theoretical

results is very effective and will be used for more molecules in the future.

• The Herman-Wallis effect is very important for NH, as expected, but also affects heavier diatomic molecules.

Thanks for listening!

NH - The Herman-Wallis Effect (3)

𝐹 𝑇𝐷𝑀 (𝑚𝑁 )=¿𝜓𝑣𝑖𝑏(𝑣¿¿ ′ ,𝑁 ′ )|𝑅𝑒 (𝑟 )|𝜓𝑣𝑖𝑏(𝑣¿¿ ′ ′ ,𝑁 ′ ′ )> ¿¿𝜓𝑣𝑖𝑏(𝑣¿¿ ′ ,𝑁¿ 0

′ )|𝑅𝑒 (𝑟 )|𝜓𝑣𝑖𝑏(𝑣¿¿ ′ ′ ,𝑁 ¿0′ ′ )>¿¿ ¿¿

¿ ¿¿

Four of these polynomials are used, one for each .

One of these polynomials is used – for both P and R branches.

for an R branch, for a Q branch for a P branch

for for for for

Standard definition Used for NH

Possible values of ΔN

657

546

435

JN

657

546

435

6

5

4

6

5

4

768

324

7

3

ΔJ -1ΔN

0 +1 -1 +1

-1 0 +1 0

-1 0 +1 -1 +1

ΔJΔN

ΔJΔN

-1 -1 +1 +1

-3 -1 -1 -1 +1

-1 +1 +1 +1 +3