^{1}and David F. Coker

^{1,a)}

### Abstract

We describe a new method for constructing empirical valence bond potential energy surfaces using a parallel multilevel genetic program (PMLGP). Genetic programs can be used to perform an efficient search through function space and parameter space to find the best functions and sets of parameters that fit energies obtained by *ab initio* electronic structure calculations. Building on the traditional genetic program approach, the PMLGP utilizes a hierarchy of genetic programming on two different levels. The lower level genetic programs are used to optimize coevolving populations in parallel while the higher level genetic program (HLGP) is used to optimize the genetic operator probabilities of the lower level genetic programs. The HLGP allows the algorithm to dynamically learn the mutation or combination of mutations that most effectively increase the fitness of the populations, causing a significant increase in the algorithm's accuracy and efficiency. The algorithm's accuracy and efficiency is tested against a standard parallel genetic program with a variety of one-dimensional test cases. Subsequently, the PMLGP is utilized to obtain an accurate empirical valence bond model for proton transfer in 3-hydroxy-gamma-pyrone in gas phase and protic solvent.

We gratefully acknowledge support for this research from the National Science Foundation under Grant No. CHE-0911635. D.F.C. acknowledges support provided by his Stokes Professorship in Nano Biophysics from Science Foundation Ireland. M.A.B. acknowledges the support provided by his IGERT fellowship from the National Science Foundation. We also acknowledge a grant of supercomputer time from Boston University's Office of Information Technology and Scientific Computing and Visualization. The authors thank Dr. Yann Tambouret and Dr. Daniel Montemayor for useful discussions.

I. INTRODUCTION

II. METHODS

A. Lower level genetic program

1. Gene representation: Tree data structures

2. Creating an initial population of random functions

3. Fitness function

4. Natural selection

5. Genetic operators

6. Directed search strategy

B. Higher level genetic program

1. Gene representation

2. Fitness function

3. Natural selection and genetic operators

4. Migration

III. RESULTS AND DISCUSSION

A. One-dimensional test cases

1. Quality of fits

2. Efficiency of searches

3. Statistical significance and effect size

B. Empirical valence bond model

C. Proton transfer

1. Simulation details

2. Mean proton transfer time

3. Solvent mediated back proton transfer

IV. CONCLUSION

### Key Topics

- Reaction mechanisms
- 45.0
- Potential energy surfaces
- 30.0
- Hydrogen bonding
- 29.0
- Ab initio calculations
- 25.0
- Solvents
- 24.0

## Figures

Tree representation of various functions.

Tree representation of various functions.

General overview of the PMLGP algorithm.

General overview of the PMLGP algorithm.

(a) Average relative errors of Morse functions. (b) Average relative errors of Gaussian functions. (c) Average relative errors of double well functions.

(a) Average relative errors of Morse functions. (b) Average relative errors of Gaussian functions. (c) Average relative errors of double well functions.

(a) Number of evolutionary cycles to reach an average relative error of 0.05 for Morse functions. (b) Number of evolutionary cycles to reach an average relative error of 0.05 for Gaussian functions. (c) Number of evolutionary cycles to reach an average relative error of 0.05 for double well functions.

(a) Number of evolutionary cycles to reach an average relative error of 0.05 for Morse functions. (b) Number of evolutionary cycles to reach an average relative error of 0.05 for Gaussian functions. (c) Number of evolutionary cycles to reach an average relative error of 0.05 for double well functions.

Molecular structures of the normal (enol) reactant state (left) and the tautomer (keto) product state (right) of 3-hydroxy-gamma-pyrone.

Molecular structures of the normal (enol) reactant state (left) and the tautomer (keto) product state (right) of 3-hydroxy-gamma-pyrone.

(a) Contour plot of the ground state DFT data of 3-hydroxy-gamma-pyrone with the intrinsic reaction path overlaid (white-light dots). (b) Contour plot of the EVB ground state of 3-hydroxy-gamma-pyrone with the DFT intrinsic reaction path (white-light dots) and EVB intrinsic reaction path (blue-dark dots) overlaid.

(a) Contour plot of the ground state DFT data of 3-hydroxy-gamma-pyrone with the intrinsic reaction path overlaid (white-light dots). (b) Contour plot of the EVB ground state of 3-hydroxy-gamma-pyrone with the DFT intrinsic reaction path (white-light dots) and EVB intrinsic reaction path (blue-dark dots) overlaid.

Contour plot of the readjusted EVB ground state of 3-hydroxy-gamma-pyrone with the DFT intrinsic reaction path (white-light dots) and EVB intrinsic reaction path (blue-dark dots) overlaid.

Contour plot of the readjusted EVB ground state of 3-hydroxy-gamma-pyrone with the DFT intrinsic reaction path (white-light dots) and EVB intrinsic reaction path (blue-dark dots) overlaid.

Probability, *P*(*t*) for back proton transfer to the ground state normal form as a function of time after fluorescence from the equilibrated excited state tautomer minimum. The red-rapidly decaying curve gives the gas phase results at *T* = 300 K and green-slowly decaying curve gives results for back proton transfer in methanol solution at *T* = 300 K.

Probability, *P*(*t*) for back proton transfer to the ground state normal form as a function of time after fluorescence from the equilibrated excited state tautomer minimum. The red-rapidly decaying curve gives the gas phase results at *T* = 300 K and green-slowly decaying curve gives results for back proton transfer in methanol solution at *T* = 300 K.

Trajectories of various bond distances between the transferring proton on 3HGP and (1) the O-atom on the blocking methanol to which it is initially intermolecularly H-bonded (red curve that jumps up at 5 ps); (2) the O-atom on the mediator methanol molecule that approaches from solution and disrupts the blocking H-bond (green curve that fluctuates around a mean value of 4 Angstrom throughout); and (3) the normal form intramolecular O-atom on 3HGP to which the transferring proton first H-bonds (at 5 ps) and then transfers to (beyond 6 ps) (blue curve that jumps down at 5 ps).

Trajectories of various bond distances between the transferring proton on 3HGP and (1) the O-atom on the blocking methanol to which it is initially intermolecularly H-bonded (red curve that jumps up at 5 ps); (2) the O-atom on the mediator methanol molecule that approaches from solution and disrupts the blocking H-bond (green curve that fluctuates around a mean value of 4 Angstrom throughout); and (3) the normal form intramolecular O-atom on 3HGP to which the transferring proton first H-bonds (at 5 ps) and then transfers to (beyond 6 ps) (blue curve that jumps down at 5 ps).

(a) One-dimensional best fit function for *O* _{2} Morse function. (b) One-dimensional best fit function for Gaussian test case 5. (c) One-dimensional best fit function for double well test case 5.

(a) One-dimensional best fit function for *O* _{2} Morse function. (b) One-dimensional best fit function for Gaussian test case 5. (c) One-dimensional best fit function for double well test case 5.

## Tables

Group mean, standard deviation, *t* value, confidence level, and effect size of the average relative errors and number of evolutionary cycles to obtain an average relative error of 0.05.

Group mean, standard deviation, *t* value, confidence level, and effect size of the average relative errors and number of evolutionary cycles to obtain an average relative error of 0.05.

Relative barrier heights of the DFT ground state and EVB ground state of 3-hydroxy-gamma-pyrone.

Relative barrier heights of the DFT ground state and EVB ground state of 3-hydroxy-gamma-pyrone.

Parameters used for (a) one-dimensional Gaussian test functions and (b) one-dimensional double well test functions.

Parameters used for (a) one-dimensional Gaussian test functions and (b) one-dimensional double well test functions.

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