^{1}, Michael S. Schuurman

^{1}and David R. Yarkony

^{1,a)}

### Abstract

A self-consistent procedure for constructing a quasidiabatic Hamiltonian representing coupled electronic states in the vicinity of an arbitrary point in nuclear coordinate space is described. The matrix elements of the Hamiltonian are polynomials of arbitrary order. Employing a crude adiabatic basis, the coefficients of the linear terms are determined exactly using analytic gradient techniques. The remaining polynomial coefficients are determined from the normal form of a system of pseudolinear equations, which uses energy gradient and derivative coupling information obtained from reliable multireference configuration interactionwave functions. In a previous implementation energy gradient and derivative coupling information were employed to limit the number of nuclear configurations at which *ab initio* data were required to determine the unknown coefficients. Conversely, the key aspect of the current approach is the use of *ab initio* data over an extended range of nuclear configurations. The normal form of the system of pseudolinear equations is introduced here to obtain a least-squares fit to what would have been an (intractable) overcomplete set of data in the previous approach. This method provides a quasidiabatic representation that minimizes the residual derivative coupling in a least-squares sense, a means to extend the domain of accuracy of the diabatic Hamiltonian or refine its accuracy within a given domain, and a way to impose point group symmetry and hermiticity. These attributes are illustrated using the and states of the 1-propynyl radical, .

This work was supported by NSF Grant No. CHE0513952 to D.R.Y.

I. INTRODUCTION

II. THEORETICAL APPROACH

A. The form of the quasidiabatic Hamiltonian

1. First-order terms

2. Higher-order terms

B. Determining the higher-order terms

1. Pseudolinear equations

2. Independent equation enumeration

3. Least-squares solution: The normal equations method

4. Quasidiabaticity of

III. NUMERICAL RESULTS

A. The and states of 1-propynyl

B. Electronic structure treatment

C. Nuclear configurations and the form of the normal equations

D. Discussion of the solutions to the normal equations

1. symmetry constraint

2. Domain of accuracy of

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Ab initio calculations
- 29.0
- Polynomials
- 8.0
- Wave functions
- 6.0
- Matrix equations
- 4.0
- Potential energy surfaces
- 4.0

## Figures

Symmetry behavior of the (a) and (b) defined surfaces as characterized by the relations derived in the Appendix. The plot shows the of actual vs symmetry predicted values for coefficients. The black circles denote the block, the blue squares denote the blocks, and the red triangles denote the blocks. The dashed lines bound a region of error on values.

Symmetry behavior of the (a) and (b) defined surfaces as characterized by the relations derived in the Appendix. The plot shows the of actual vs symmetry predicted values for coefficients. The black circles denote the block, the blue squares denote the blocks, and the red triangles denote the blocks. The dashed lines bound a region of error on values.

Fit characteristics for the surface. The plot shows the of predicted vs *ab initio* values for energy, energy gradients, and coupling gradients. The blue circles denote the set of points and the red triangles denote the (*min*) set. (a), (b), and (c) present the , , and state energies, respectively. (d), (e), and (f) present the , , and energy gradients, respectively. (g), (h), and (i) present the , , and coupling gradients, respectively.

Fit characteristics for the surface. The plot shows the of predicted vs *ab initio* values for energy, energy gradients, and coupling gradients. The blue circles denote the set of points and the red triangles denote the (*min*) set. (a), (b), and (c) present the , , and state energies, respectively. (d), (e), and (f) present the , , and energy gradients, respectively. (g), (h), and (i) present the , , and coupling gradients, respectively.

Fit characteristics for the surface. The plot shows the of predicted vs *ab initio* values for energy, energy gradients, and coupling gradients. The blue circles denote the set of points and the red triangles denote the (*min*) set. (a), (b), and (c) present the , , and state energies, respectively. (d), (e), and (f) present the , , and energy gradients, respectively. (g), (h), and (i) present the , , and coupling gradients, respectively.

*ab initio* values for energy, energy gradients, and coupling gradients. The blue circles denote the set of points and the red triangles denote the (*min*) set. (a), (b), and (c) present the , , and state energies, respectively. (d), (e), and (f) present the , , and energy gradients, respectively. (g), (h), and (i) present the , , and coupling gradients, respectively.

Fit characteristics for the (*all*) surface. The plot shows the of predicted vs *ab initio* values for energy, energy gradients, and coupling gradients. The blue circles denote the set of points and the red triangles denote the (*min*) set. (a), (b), and (c) present the , , and state energies, respectively. (d), (e), and (f) present the , , and energy gradients, respectively. (g), (h), and (i) present the , , and coupling gradients, respectively.

Fit characteristics for the (*all*) surface. The plot shows the of predicted vs *ab initio* values for energy, energy gradients, and coupling gradients. The blue circles denote the set of points and the red triangles denote the (*min*) set. (a), (b), and (c) present the , , and state energies, respectively. (d), (e), and (f) present the , , and energy gradients, respectively. (g), (h), and (i) present the , , and coupling gradients, respectively.

## Tables

Sets of nuclear displacements.

Sets of nuclear displacements.

Nonzero vibronic matrix elements. If symmetry unique term is omitted the nonzero term is symmetry unique.

Nonzero vibronic matrix elements. If symmetry unique term is omitted the nonzero term is symmetry unique.

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