Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew S. Petit and Anne B. McCoy The Ohio State University

What Are the Challenges??? Harmonic analysis pathologically fails

What Are the Challenges??? Harmonic analysis pathologically fails shared proton stretch H3O2-H3O2-

What Are the Challenges??? Harmonic analysis pathologically fails An accurate and general treatment of these highly fluxional systems requires either a global PES or a grid-based sampling of configuration space

What Are the Challenges??? Harmonic analysis pathologically fails An accurate and general treatment of these highly fluxional systems requires either a global PES or a grid-based sampling of configuration space Large basis set expansions required to fully capture the anharmoncity of these systems

The General Scheme

I I Monte Carlo Sample Configuration Space

Monte Carlo Sample Configuration Space Use importance sampling Monte Carlo

Monte Carlo Sample Configuration Space

Use importance sampling Monte Carlo Normalized probability distribution that is peaked, ideally, where the function of interest, f, is peaked. Monte Carlo Sample Configuration Space

Sampling function based on harmonic ground state Monte Carlo Sample Configuration Space R Relative Probability

Sampling function based on harmonic ground state Monte Carlo Sample Configuration Space R Relative Probability

Sampling function based on harmonic ground state Monte Carlo Sample Configuration Space R Relative Probability

Sampling function based on harmonic ground state Monte Carlo Sample Configuration Space R Relative Probability

Sampling function based on harmonic ground state Monte Carlo Sample Configuration Space R Relative Probability

I I I I The General Scheme Construct Initial Basis Monte Carlo Sample Configuration Space

I I I I I I Build and Diagonalize Hamiltonian Matrix The General Scheme Construct Initial Basis Monte Carlo Sample Configuration Space

I I I I I I I I Build and Diagonalize Hamiltonian Matrix Assign Eigenstates & Build New Basis The General Scheme Construct Initial Basis Monte Carlo Sample Configuration Space

I I I I I I I I Build and Diagonalize Hamiltonian Matrix Assign Eigenstates & Build New Basis The General Scheme Construct Initial Basis Monte Carlo Sample Configuration Space

The Evolving Basis

Harmonic oscillator basis

The Evolving Basis Harmonic oscillator basis

The Evolving Basis Harmonic oscillator basis

Eigenstates used to construct new basis Assignment via: The Evolving Basis Harmonic oscillator basis

Eigenstates used to construct new basis Assignment via: The Evolving Basis Harmonic oscillator basis

Eigenstates used to construct new basis Assignment via: Define an active space: The Evolving Basis Harmonic oscillator basis

Eigenstates used to construct new basis Assignment via: Define an active space: The Evolving Basis Harmonic oscillator basis

Eigenstates used to construct new basis Assignment via: Define an active space: The Evolving Basis Harmonic oscillator basis

The Evolving Basis Eigenstates used to construct new basis Assignment via: Define an active space:

MxM Matrix Construction of 2 nd Iteration Basis

MxM Matrix Suppose active space only contains ground state Construction of 2 nd Iteration Basis

MxM Matrix Suppose active space only contains ground state Basis functions shown prior to orthonormalization Construction of 2 nd Iteration Basis

MxM Matrix Suppose active space only contains ground state Basis functions shown prior to orthonormalization Construction of 2 nd Iteration Basis

MxM Matrix Suppose active space only contains ground state Basis functions shown prior to orthonormalization Construction of 2 nd Iteration Basis

MxM Matrix Suppose active space only contains ground state Basis functions shown prior to orthonormalization Construction of 2 nd Iteration Basis Effective size of basis grows each iteration

I I I I I I I I Build and Diagonalize Hamiltonian Matrix Assign Eigenstates & Build New Basis The General Scheme Construct Initial Basis Monte Carlo Sample Configuration Space

Testing the Method Δr (Å)Energy (cm -1 ) E 0 =237.5 cm -1 E 1 =637.5 cm -1 E 2 =937.5 cm -1 E 3 = cm -1 E 4 = cm -1

Testing the Method E 0 =237.5 cm -1 E 1 =637.5 cm -1 E 2 =937.5 cm -1 E 3 = cm -1 E 4 = cm -1

Testing the Method E 0 =237.5 cm -1 E 1 =637.5 cm -1 E 2 =937.5 cm -1 E 3 = cm -1 E 4 = cm -1

Testing the Method E 0 =237.5 cm -1 E 1 =637.5 cm -1 E 2 =937.5 cm -1 E 3 = cm -1 E 4 = cm -1

Moving on to 2D Horvath et al. J. Phys. Chem. A 2008, 112, (Å) 2D PES and G matrix elements constructed by S. Horvath from a bicubic spline interpolation of a grid of points on which MP2/aug-cc-pVTZ calculations were performed. and Highly anharmonic system with a strong coupling observed between Our goal is to accurately capture all states with up to 3 total quanta of excitation.

The Problem of Assignment Eigenstates used to construct new basis

The Problem of Assignment Eigenstates used to construct new basis DVR

The Problem of Assignment Eigenstates used to construct new basis DVR Iteration 2

The Problem of Assignment Eigenstates used to construct new basis DVR Iteration 2 Iteration 3

The Problem of Assignment DVR Iteration 2 Iteration 3 Identity of problem states dependent on parameters of algorithm AND exact distribution of Monte Carlo sampling points

The Hunt for an Effective Metric Basis function Un-assigned eigenstate

Basis function Un-assigned eigenstate The Hunt for an Effective Metric

Basis function Un-assigned eigenstate The Hunt for an Effective Metric

Basis function Un-assigned eigenstate Compare moments of the two sets of wavefunctions: The Hunt for an Effective Metric

Gruebele et al. Int. Rev. Phys. Chem 1998, 17, The Hunt for an Effective Metric

Perform calculation using an assignment scheme based on

Gruebele et al. Int. Rev. Phys. Chem 1998, 17, The Hunt for an Effective Metric Perform calculation using an assignment scheme based on Significantly contaminated eigenstates are rebuilt

The Hunt for an Effective Metric Perform calculation using an assignment scheme based on Significantly contaminated eigenstates are rebuilt (Å) DVR

The Hunt for an Effective Metric Perform calculation using an assignment scheme based on Significantly contaminated eigenstates are rebuilt Iteration 23 (Å) DVR

The Hunt for an Effective Metric Perform calculation using an assignment scheme based on Significantly contaminated eigenstates are rebuilt Iteration (Å) DVR

The Hunt for an Effective Metric Iteration (Å) DVR Contamination occurs under the radar of overlaps, transition moments, moment distributions, etc.

The Hunt for an Effective Metric Iteration (Å) DVR Contamination occurs under the radar of overlaps, transition moments, moment distributions, etc.

The Hunt for an Effective Metric Iteration Number F 0,2;m,n Largest F 0,2;m,n 2 nd Largest F 0,2;m,n

The Hunt for an Effective Metric Iteration Number F 0,2;m,n Largest F 0,2;m,n 2 nd Largest F 0,2;m,n (Å) DVR

Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample configuration space and minimize size of basis required Conclusions and Future Work

Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample configuration space and minimize size of basis required The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator Conclusions and Future Work

Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample configuration space and minimize size of basis required The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator Promising developments in tackling the problem of multi- dimensional eigenstate assigment Conclusions and Future Work

Complete the optimization of the multi-dimensional algorithm’s treatment of spurious resonances Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample configuration space and minimize size of basis required The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator Promising developments in tackling the problem of multi- dimensional eigenstate assigment Conclusions and Future Work

Complete the optimization of the multi-dimensional algorithm’s treatment of spurious resonances Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample configuration space and minimize size of basis required The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator Promising developments in tackling the problem of multi- dimensional eigenstate assigment Conclusions and Future Work Further multi-dimensional benchmarking

Complete the optimization of the multi-dimensional algorithm’s treatment of spurious resonances Couple algorithm with ab initio electronic structure calculations Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample configuration space and minimize size of basis required The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator Promising developments in tackling the problem of multi- dimensional eigenstate assigment Conclusions and Future Work Further multi-dimensional benchmarking

Acknowledgements Dr. Anne B. McCoy Dr. Sara E. Ray Bethany A. Wellen Andrew S. Petit Annie L. Lesiak Dr. Charlotte E. Hinkle Dr. Samantha Horvath

The Initial Basis (2D Example)

Basis functions change each iteration

The Initial Basis (2D Example) Basis functions change each iteration

The Initial Basis (2D Example) Basis functions change each iteration Phase factors

The Initial Basis (2D Example) Basis functions change each iteration Phase factors Will be made orthogonal to all previously built states and normalized

The Initial Basis (2D Example)