1. P. JANSEN, W. UBACHS, H. L. BETHLEM
SENSITIVITY OF TRANSITIONS IN INTERNAL ROTOR MOLECULES TO A POSSIBLE VARIATION OF THE PROTON-TO-ELECTRON MASS RATIO
P. JANSEN, W. UBACHS, H. L. BETHLEM
Institute for Lasers, Life and Biophotonics, VU University, Amsterdam, The Netherlands
I. KLEINER
Laboratoire Interuniversitaire des Systèmes Atmosphériques, CNRS et Universités Paris Diderot et Paris Est, Créteil, France
L-H. XU
Department of Physics and Centre for Laser, Atomic and Molecular Sciences, University of New Brunswick, Saint John, Canada
a=e2/hc4pe0

2. General Objectives:
Predict spatio- temporal variations of the constants of nature
detecting a possible drift of a fundamental constant on a cosmological time scale :
possible variation of the fine-structure a constant by Webb et al (PRL 1999)
possible variation of the proton-to-electron mass ratio μ (Ubachs,Rheinhold et al (PRL 2006)
variation of μ > shift in the position of a spectral line
----> change in the spectrum of atoms and molecules
. Mu= (75). Not all lines will shift in the same amount or direction. The response
of a transition to a variation of α or μ is characterized by
its sensitivity coefficient, Kμ or Kα , which is defined as the proportionality constant between the fractional frequency shift of the transition, Dν/ν, and the fractional shift in α or μ
number with a value of about 1/137, the constant dictates the strength of the electromagnetic force and, hence, determines the exact wavelengths of light an atom will absorb 2

3. Sensitivity coefficient Km
The response of a transition to a variation of μ is characterized by its sensitivity coefficient, Km : proportionality constant between the fractional frequency shift of the transition, Dν/ν, and the fractional shift in μ
Varshalovich and
Levshakov (1993)
Laboratory measurements
Astronomical measurements
m = (75).

4. Previous Cosmological observations:
1) comparing optical transitions of molecular hydrogen (H2) in high-redshifted objects (QUASARS) with accurate laboratory measurements 
limit of Dμ/μ < 10−5 for look-back times of 12 billion years,
Km = −0.05 to (Rheinhold et al PRL 2006)
2) Transitions between inversion levels of NH3 (Km = −4.2 )
Astronomical observations of NH3 in microwave 
stringent constraints around (1.0 ± 4.7) × 10−7 (Flambaum and Kozlov PRL 2007)
3) Torsional-rotational transitions in CH3OH (Km = −88 to 330) :
Jensen et al , Levshakov,Koslov and Reimers 2011
4 transitions used to constrain Dμ/μ at (11.0 ± 6.8) × 10−8
at a look-back time of 7 billion years
(Bagdonaite, jensen, Bethleem, Ubachs, Henkel, Menten Science 2013)

5. The spectroscopic point of view: which molecules (and which type of transitions) have to highest Km ?
Sensitivity of Internal rotation transition
To calculate K, the energy levels and their dependance in m
(the mass) has to be known :
Numerical calculations (precise)
« Toy » model (physically simple)
Sensitivity is increased for transitions between nearly degenerate energy levels showing different dependances in m : internal rotation can fulfill this

6. What is internal rotation ?
Ts symmetry
s = 0 : A species
s= ±1, E species
vt: torsional quantum number
J = |K| = 1
± K, s= +1
K, s= -1
Xu et al Methanol, JMS 2008

7. HOW TO MODEL INTERNAL ROTATION?
For one C3v top, and a frame with a plane of symetry Cs
HRAM = Htor + Hrot + Hd.c + Hint
1) Diagonalization of the torsional part of the Hamiltonian in an axis system where torsion-rotation coupling is minimal (Rho Axis Method, RAM), Kirtman et al, Lees and Baker , Herbst et al:
Htor= F (pa - r.Jz)2 + V(a) F: internal rotation constant
r depends on Itop/Imolecule
Eigenvalues = torsional energies
2) Eigenvectors are used to set up the matrix of the rest of the Hamiltonian:
Hrot = ARAMJa2 + BRAMJb2 +CRAMJc2 + Dab(JaJb + JbJa)
Hd.c usual centrifugal distorsion terms
Hint higher order torsional-rotational interactions terms : cos3a et pa and global rotational operators like Ja, Jb , Jc

8. Theoretical Model: the global approach
RAM = Rho Axis Method (axis system) for a Cs (plane) frame
HRAM = Hrot + Htor + Hint + Hd.c.
Torsional operators and potential function V(a)
Constants 1
1-cos3a
p2a
Japa
1-cos6a
p4a
Jap3a
V3/2 F r
V6/2 k4 k3 J2
(B+C)/2* Fv Gv Lv Nv Mv
k3J
Ja2
A-(B+C)/2* k5 k2 k1 K2 K1
k3K
Jb2 - Jc2
(B-C)/2* c2 c1 c4
c11 c3
c12
JaJb+JbJa
Dab or Eab
dab
Dab
dab6
DDab
ddab
Rotational Operators
Rotational constants give the structure.
Using the diagonalization of the inertial tensor, one can get the orientation angles of the CH3 group relative to the PAS
= angle of torsion, r = couples internal rotation and global rotation, ratio of the moment of inertia of the top and the moment of inertia of the whole molecule
Kirtman et al 1962
Lees and Baker, 1968
Herbst et al 1986
Hougen, Kleiner, Godefroid JMS 1994

9. How to scale internal rotation contants in m
How to scale internal rotation contants in m ? we assume that the neutron mass has a similar time variation as the proton mass
V3, V6, .. : no mass dependance (B.O approx.): m0
A, B, C, Dab, F, direct dependance in the mass: scale as 1/m
r depends on Itop/Imolecule: scales as m0
M=mp/me if m changes the mass(neutron)/me changes the same way !
we assume that the neutron mass has a similar timevariation
as the proton mass.

10. Generating the molecular constants, using the scale relation in m
calling Belgi as a « slave » program …
3) Calculated the energy levels in terms of m
Xu et al JMS 2008
Methanol energy levels, Dm =0

11. some
Transitions of CH3OH
51  60 A MHz
Km = -42.1
32  31 E MHz
Km =18.
22  21 E MHz
Km =18.
20 ->3-1 E MHz
Km = -32.8

12. What about other internal rotors present in interstellar objects?
Molecules V F s r Jmax Kmax vtmax Nlines Range Npar Std
cm cm =4V3/9F
Xu et al JMS 2008
Kleiner et al JMS 2009
Ilyushin et al, JMS 2009
Ilyushin et al JPCRD 2008
Ilyushin et al , JMS 2004
CH3SH methyl mercaptan MW-THz-FIR
Xu et al JCP 2012

13. The « toy » model
The m dependance of a state:(neglecting vibration-rotation-torsion interactions)
Where
Is the sensitivity of a vibrational level
Is the sensitivity of a rotational level
Neglecting the vibrational part:
With DERot = E’rot- E’’rot
DEtor= E’tor - E’’tor

14. How to estimate DErot and DEtor ?
Rotational energy (symmetric top)
A, B, C are inversionaly proportional to moments of inertia (mass) ,
scale like 1/m
Torsional energy (Lin and Swalen 1959)
F internal rotation
constant, scales like
1/m
For a DK =+1 transition:
s = 4V3/9F
A1, B1, C1 : coefficients

15. Difference in energy DErot and DEtor for the A J,K J+1, K-1 transitions in CH3OH
The highest sensitivity coefficients occur when DEtor ≈ -DErot

16. hn (exp. Data) = DErot + DEtor
n, F, g assumed 1 GHz x Km

17. Maximum Km Km
Km max when n = 1GHz
DEtors max when ?

19. Thank you for your attention

20. Sensitivity Constants
Although implicit in previous work, Varshalovich and Levshakov (1993) explicitly developed the sensitivity constant which for a line i is defined as
The rest frame wavelengths are related to the observed wavelengths by
where m0 is the proton-to-electron mass ratio in the present epoch, at zero redshift, and mz the mass ratio for the absorbing cloud at high redshift.
Each line has a unique sensitivity constant Ki which can be slightly negative, zero or positive.
The higher the vibrational quantum number the larger the sensitivity constant.

21. The spectroscopic point of view: which molecules (and which type of transitions) have to highest Km ?
1) Vibrational energy
Sensitivity of an vib. Level:
w is the vibrational frequency,
m is the reduced mass
The rest frame wavelengths are related to the observed wavelengths by
2) Rotational energy

22. How to estimate DErot and DEtor ?
Rotational energy (symmetric top)
A, B, C are inversionaly proportional to moments of inertia (mass) ,
scale like 1/m
Torsional energy (Lin and Swalen 1959)
F rotation constant
Interne , scale like
1/m
For a DK =+1 transition:
s = 4V3/9F

23. How to estimate DErot and DEtor ?
Torsional energy (Lin and Swalen 1959) 1)
2) Fit
a0,, a1
r=0.8
s = 4V3/9F
3) Fit
r=0.3
r=0.07

24. F scales as 1/m

25. Coefficients a1 in function of s
Lin and Swalen (1959)

26. Calcul of Km using the toy model
hn (exp. Data) = DErot + DEtor