1. Ab initio effective rovibrational Hamiltonians for non-rigid molecules via curvilinear VMP2
Bryan Changala
JILA & Dept. of Physics, Univ. of Colorado Boulder
Joshua Baraban
Dept. of Chemistry, Univ. of Colorado Boulder
ISMS 2017

2. Rigid vs. non-rigid polyatomic molecules
Well defined, unique equilibrium geometry
Possibly no unique eq. geometry
Structure
Small amplitude, approximately harmonic normal modes
Vibrational
dynamics
Large amplitude, highly anharmonic, tunneling
Total rotations approximately separable from internal motion (e.g. Eckart conditions)
Potentially significant rotation-vibration interaction (e.g. internal rotors)
Rotational
dynamics

3. General approaches to solving
Variational
Perturbative
Converge numerically exact energies and wavefunctions
Non-rigidity handled naturally
BUT
Very expensive (ca per atom)
Computationally economical
Near spectroscopic accuracy in favorable cases
BUT
Standard methods (e.g. VPT2) geared for semi-rigid systems.
Alternative perturbative rovibrational approach for non-rigid molecules?

4. Variational (“exact”) / cm-1
Disilicon carbide, Si2C Si C
Mode
Variational (“exact”) / cm-1
Standard VPT2 */ cm-1
ν2 (Si-C-Si bend)
140.49
148.82
ν1 (sym. stretch)
828.24
846.62
ν3 (asym. stretch)
RMS error
14.80
*with rectilinear quartic force field
Soft, anharmonic bending mode
N. Reilly, et al, J. Chem. Phys. 142, (2015)
M. McCarthy, et al, J. Phys. Chem. Lett. 6, 2107 (2015)
PBC & J. Baraban, J. Chem. Phys. 145, (2016)

5. Perturbative theories are only as good as Ψ0
Standard zeroth order picture:
3N-6 harmonic oscillator normal modes
Rectilinear (Cartesian) normal modes Qi are not natural choices
Q2 bend 1
Use curvilinear internal coordinates (e.g. bond angles & lengths, …)
Vibrational factors are harmonic oscillator wavefunctions
Allow arbitrary, anharmonic 1D vibrational wavefunctions 2

6. Holding all other fixed, vary one to minimize …___……….
Vibrational SCF & MP2
Perturbatively correct for remaining vibrational correlation (VMP2)
How do we choose
the factors??
Guess some initial
Repeat
Holding all other fixed, vary one to minimize …___……….
Do this for each factor.
… until converged
J. Bowman, Acc. Chem. Res. 19, 202 (1986); Gerber & Ratner, Adv. Chem. Phys. 70, 97 (1988)
Strobusch & Scheurer, J. Chem. Phys. 135, (2011)
L.S. Norris et al, J. Chem. Phys. 105, (1996); O. Christiansen, J. Chem. Phys. 119, 5773 (2003)

7. How does VMP2 fare for Si2C?
Mode
Variational (“exact”) / cm-1
Standard VPT2 / cm-1
Curvilinear VMP2*/ cm-1
ν2 (Si-C-Si bend)
140.49
148.82
140.48
ν1 (sym. stretch)
828.24
846.62
828.29
ν3 (asym. stretch)
RMS error
14.80
0.03
Why such an improvement?
*with curvilinear normal coords.
VSCF/VMP2 zeroth order approximation = separability of 3N-6 qi vibrational degrees of freedom (which we choose … critical user input!)
All “diagonal anharmonicity” and “mean-field cross anharmonicity” are accounted for already at zeroth order.
Remaining vibrational correlations tend to be “small”; handled well by VMP2.
PBC & J. Baraban, J. Chem. Phys. 145, (2016)

8. Adding molecular rotation to VMP2
NEW
Typically: so let’s treat them perturbatively as well.
Important: choice of body-fixed frame!
For well defined eq. geometry, use Eckart frame for curvilinear KEO.
If no well defined eq. geometry, more elaborate schemes are used. Details: J. Chem. Phys. 145, (2016).
After a bunch of machinery, we get A/B/C rotational constants and quartic centrifugal distortion constants.

9. Si2C rotational constants
Variational (“exact”)/MHz
VPT2
|rel. error| x 104
VMP2
Av=0
63627
49.3
2.1
Bv=0
4339
2.9
0.3
Cv=0
4051
2.3
0.7
Abend
70668
315.8
5.8 a

10. Unhindered internal rotation in nitromethane
50 cm-1
22 cm-1
So far, we’ve assumed
5.5 cm-1
But in the torsional manifold
V = 2 cm-1
Uh-oh!!!

11. Dealing with near-resonant interactions with rotational-VMP2
Energy
Target state
well isolated
Energy
Target state
not isolated
Standard perturbative correction sum
Account for remaining (weak) interactions perturbatively (via contact/Van Vleck transformation)
Treat subset of states non-perturbatively
PBC & J. Baraban, J. Chem. Phys. 145, (2016)

12. CH3NO2 torsion-rotation effective Hamiltonian
Parameter
Expt. / MHz
VPT2
VMP2 A
13342
12190
13330 B
10544
10464
10507 C
5876
5849
5862 F
166703
---
166896 A’
13283
13249
ΔJK x 103
17.8
953
ΔK x 103
-7.5
-949
-10.7
δK x 103
15.8
-268 a
All the “problem” parameters involve internal or total rotation about the CH3 top axis (a-axis)
F. Rohart, J. Mol. Spec (1975); G. Sørensen et al J. Mol. Struct. 97, 77 (1983).
PBC & J. Baraban, J. Chem. Phys. 145, (2016)

13. Conclusions
Rotational curvilinear VMP2 is a flexible and efficient tool for accurate rovibrational predictions for non-rigid molecules displaying various types of non-trivial nuclear motion dynamics.
Applications to tunneling gauche-1,3,-butadiene (10 atom molecule, see talk TK07 this afternoon) and c-C3H2 (see talk WF08 by H. Gupta)
Future work:
inclusion of rotations in SCF stage (explicit RVSCF)
extensions to vibronic/JT systems???
Thanks for your attention!!!

14. Si2C experimental molecular constants
Mode
Variational (“exact”) / cm-1
Standard VPT2 / cm-1
Curvilinear VMP2 / cm-1
Expt / cm-1
ν2 (Si-C-Si bend)
140.49
148.82
140.48
140(2)
ν1 (sym. stretch)
828.24
846.62
828.29
830(2)
ν3 (asym. stretch)
---
RMS error
14.80
0.03
Rotational constant
Variational (“exact”)/MHz
VPT2
|rel. error| x 104
VMP2
Expt / MHz
Av=0
63627
49.3
2.1
64074
Bv=0
4339
2.9
0.3
4396
Cv=0
4051
2.3
0.7
4102
Abend
70668
315.8
5.8
71230
N. Reilly, et al, J. Chem. Phys. 142, (2015)
M. McCarthy, et al, J. Phys. Chem. Lett. 6, 2107 (2015)
J. Cernicharo, et al, ApJL, 806, L3 (2015)