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

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

General approaches to solving Variational Perturbative Converge numerically exact energies and wavefunctions Non-rigidity handled naturally BUT Very expensive (ca. 103+ 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?

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) 1198.14 1213.94 RMS error 14.80 *with rectilinear quartic force field Soft, anharmonic bending mode N. Reilly, et al, J. Chem. Phys. 142, 231101 (2015) M. McCarthy, et al, J. Phys. Chem. Lett. 6, 2107 (2015) PBC & J. Baraban, J. Chem. Phys. 145, 174106 (2016)

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

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, 124102 (2011) L.S. Norris et al, J. Chem. Phys. 105, 11261 (1996); O. Christiansen, J. Chem. Phys. 119, 5773 (2003)

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) 1198.14 1213.94 1198.17 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, 174106 (2016)

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, 174106 (2016). After a bunch of machinery, we get A/B/C rotational constants and quartic centrifugal distortion constants.

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

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!!!

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, 174106 (2016)

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. 57 301 (1975); G. Sørensen et al J. Mol. Struct. 97, 77 (1983). PBC & J. Baraban, J. Chem. Phys. 145, 174106 (2016)

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!!!

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) 1198.14 1213.94 1198.17 --- 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, 231101 (2015) M. McCarthy, et al, J. Phys. Chem. Lett. 6, 2107 (2015) J. Cernicharo, et al, ApJL, 806, L3 (2015)