Full Empirical Potential
Curves and Improved
Dissociation Energies for
the A1Π and X 1Σ+ States of
+
CH
Young-Sang Cho, Robert J. Le Roy
Department of Chemistry, University of Waterloo,
Waterloo, Ontario, Canada
Why are we interested in CH+ ?
Objectives
To represent all types of experimental data compactly within
uncertainties
1.
•
Should simultaneously treat all types of data (MW, IR, electronic), and
data for all isotopologues and multiple connected electronic states in a
single analysis.
To be able to interpolate reliably for missing observations within
the data range.
To provide realistic predictions in the ‘extrapolation region’ outside
the data range.
2.
3.
•
In effect, this presumes that the analysis provides a realistic global potential energy
curve.
To provide reliable estimates of physically interesting properties.
4.
•
e.g. re, De, force constants, long-range potential coefficients
•
Expectation values, matrix elements and transition intensities.
•
Collisional and dilute (atomic) gas properties (e.g. virial coefficients)
What is the nature of the data?
 Conventional
spectroscopic data
Electronic A - X
Transition:
X-state
" 0  3,J" 0 17
A-state
 '  0  4, J '  1  17
Microwave:
" 0,R(0)  R(5)
/Å
 Photodissociation observation of predissociating
υ(A) = 11-14 and of low υ(A) levels at very high J’
What is the nature of the data?
High-J tunneling
pre-dissociation levels
of the A-state seen by
photofragment
spectroscopya
'  0  9
" 0  5,J" 14  36
Plus kinetic energy of
fragments for selected
levelsa
a H.Helm, P.C. Cosby, M.M. Gaff and J.T. Moseley, Phys.Rev.A25, 304 (1982)
What is the nature of the data?
Low-J, high-v Feshbach
pre-dissociation levels
seen by
photodissociation
spectroscopyb
 '  11 14
" 0,J" 0  9
b U.Hechtfischer, C.J. Williams, M.Lange, J.Linkemann, D.Schwalm, R.Wester, A.Wolf and D.Zajfman,
J.Chem.Phys. 117 8754 (2002)
Direct Potential Fits
Simulate transition energies as numerically determined
eigenvalues, E(υ,J), of some parameterized analytic potential
energy function V (r;{ p j })
obs ( ' , J ' ;" , J " )  E ( ' , J ' )  E (", J " )
Partial derivatives of observables w.r.t. parameters pj required
for fitting are generated readily by the Hellmann-Feynmann
theorem:
V (r;{p j })
E(,J)
  ,J
 ,J
p j
p j
Compare with experiment and iterate the least-squares fit to
convergence.

Direct Potential Fits (cont’)
Advantages
satisfies all four ‘objectives of spectroscopic data
reduction’
full quantum mechanical accuracy
Challenge
to determine flexible potential functions that
are robust and ‘well behaved’ (no spurious behaviour in
interpolation or extrapolation regions)
can incorporate physical constraints and limiting behaviour
Compact and portable – defined by ‘modest’ no. of
parameters

Morse Long-Range Potential (MLR)
 uLR (r)   (r) y eq (r) 2
p

VMLR (r)  De 1 
e
 uLR (re )

   

2D
e
rre

De   e
uLR (r)  De  uLR (r)
 uLR (re ) 
where
    (r  )  ln{ 2De uLR (re )}
theory tells us that u LR (r )  Cm1 r m1  Cm 2 r m 2  ...
N
ref
ref
i
we define (r)  MLR (r)   y ref
(r)
[1

y
(r)]

y
(r)

p
p
i q
r p  rep
eq
in which y p (r)  p
r  rep
y ref
p (r) 
r p  rrefp
r p  rrefp
i0
Analysis (the X1Σ+ state)
What is the best value of p,q?
/Å
/Å
/Å
/Å
/Å
ab-initio
/Å
ab-initio
Two ab-initio points
added to the data set
/Å
ab-initio
/Å
Results (the
1
+
X Σ state)
ULR(r) defined by C4=3.872×104, C6=0.0043×106, C8=1.6×108
Param.
No ab-initio
With ab-initio
dd
1.653
1.655
De
re
34361.6
34361.8
1.12845676
1.1284108
{p,q}
{5,1}
{5,1}
rref
1.54×re
2.54×re
β0
-0.067646
-0.152
β1
12.73823
13.71782
β2
62.05
71.578
β3
197.5739
216.673013
β4
444.94
392.4
β5
644.2
393.7484
β6
440.0
169.0
Comparing with previous work
Source
D0 /cm-1
De /cm-1
Method
34361.8 ± 3.0
10302.4 ± 3.0
Direct fit to MLR
potential function
Present work
(2014)
X
A
32945.8 ± 3.0
9326.0 ± 3.0
Hechtfischer et al.
(2002)
X
A
32946.7 ± 1.1
Experimental
(photodissociation)
Helm et al. (1982)
X
A
32907 ± 23
9351 ± 23
Experimental
(photofragment)
Barinovs and
van Hemert (2004)
X
A
32892.51
9304.6
Theory
Kanzler (1991)
X
A
37586.3
12340.6
Theory
Sarre et al. (1989)
X
A
34323.81
10263.44
Theory
Saxon et al. (1980)
X
A
33392.1
7549.5
Theory
Smith et al. (1973)
X
33842
Theory
Conclusions
Obtained accurate potential function that account for all data
within their uncertainties
This analysis yields improved estimated value of the dissociation
energy and other properties of the A1Π and X 1Σ+ states
This analysis yields the first empirical determination of
Born-Oppenheimer Breakdown corrections for CH+
This analysis yields a good explanation of lambda-doubling
corrections spanning the whole A-state well
These potentials will yield linelists that provide improved
predictions for astrophysics application
Acknowledgement
Dr. Takayoshi Amano for helpful discussions
Research supported by Natural Science and Engineering
Research Council of Canada
Thank you
22