^{1,a)}, Stephen L. Coy

^{1,b)}and Robert W. Field

^{1,c)}

### Abstract

An *ab initio* -matrix method for determining the molecular reaction matrix of scattering theory is introduced. The method makes use of a principal-value Green function to compute the collision channel wave functions for the scattered electron, in combination with the Kohn variational scheme for the evaluation of -matrix eigenvalues on a spherical boundary surface at short range. This technique permits the size of the bounded volume in the variational calculation to be reduced, making the computations fast and efficient. The reaction matrix is determined in a form that minimizes its energy dependence. Thus the procedure does not require modification or an increase in the computational effort to study the electronic structure and dynamics in Rydberg molecules with extremely polar ion cores. The analysis is specialized to examine the bound-state and free-electron scattering properties of nearly one-electron molecular systems, which are characterized by a Rydberg/scattering electron incident on a closed-shell ion core. However, it is shown that the treatment is compatible with all-electron/*ab initio* representations of open-shell and nonlinear polyatomic ion cores, emphasizing its generality. The introduced approach is used to calculate the electronic spectrum of the calcium monofluoride molecule, which has the extremely polar closed-shell ion-core configuration. The calculation utilizes an effective single-electron potential determined by M. Arif, C. Jungen, and A. L. Roche [J. Chem. Phys.106, 4102 (1997)] previously. Close agreement with experimental data is obtained. The results demonstrate the practical utility of this method as a viable alternative to the standard variational approaches.

The computer code for the numerical computations carried out in this work was written in MATHEMATICA 5.1, a product of Wolfram Research. This research was supported by the NSF Grant No. CHE-04050876.

I. INTRODUCTION

II. THEORY

A. Solutions of the zero-order equations

B. Coupled equations

C. Iterative procedure to calculate the matrix

1. Solutions of the zero-order equations

2. Solutions of the coupled equations

3. The reaction matrix

D. The energy dependence of the reaction matrix

1. Alternate pair of basis functions

E. The transformation between and

III. CONCLUSIONS

### Key Topics

- Eigenvalues
- 18.0
- Rydberg states
- 16.0
- Chemical reaction theory
- 12.0
- Green's function methods
- 12.0
- Ab initio calculations
- 9.0

## Figures

(A) The calculated eigenquantum defects of CaF in the electronic symmetry, obtained from the matrix. The eigenquantum defect curves characterize the four penetrating Rydberg series of CaF, which have been observed in the electronic spectrum (Ref. 16). The Rydberg series are labeled by the value where is the effective principal quantum number (Refs. 16 and 44). The filled circles are the experimentally determined eigenquantum defects, whereas the plus signs are the eigenquantum defects determined from a global fit by Field *et al.* (see Ref. 42). (B) The calculated eigenquantum defects of CaF in the electronic symmetry. The eigenquantum defect curves characterize the three penetrating Rydberg series of CaF, which have been observed in the electronic spectrum (see Ref. 16). The Rydberg series are labeled by the value where is the effective principal quantum number (see Refs. 16 and 44). The experimental eigenquantum defects are marked by the filled circles and the plus signs are the eigenquantum defects determined from a global fit by Field *et al.* (see Ref. 42).

(A) The calculated eigenquantum defects of CaF in the electronic symmetry, obtained from the matrix. The eigenquantum defect curves characterize the four penetrating Rydberg series of CaF, which have been observed in the electronic spectrum (Ref. 16). The Rydberg series are labeled by the value where is the effective principal quantum number (Refs. 16 and 44). The filled circles are the experimentally determined eigenquantum defects, whereas the plus signs are the eigenquantum defects determined from a global fit by Field *et al.* (see Ref. 42). (B) The calculated eigenquantum defects of CaF in the electronic symmetry. The eigenquantum defect curves characterize the three penetrating Rydberg series of CaF, which have been observed in the electronic spectrum (see Ref. 16). The Rydberg series are labeled by the value where is the effective principal quantum number (see Refs. 16 and 44). The experimental eigenquantum defects are marked by the filled circles and the plus signs are the eigenquantum defects determined from a global fit by Field *et al.* (see Ref. 42).

The calculated eigenquantum defects of CaF in the electronic symmetry, obtained from the integer- matrix. The calculation of the matrix was carried out for the ionization continuum only and the eigenchannels are labeled using the scheme of Fig. 1. The eigenquantum defects show a prominent energy variation, due to the effects of the long-range dipole field; this strong energy dependence is not present in the behavior of the eigenquantum defects of the matrix shown in Fig. 1. The strong energy dependence of the matrix eigenquantum defects makes it difficult to extrapolate their values to the region of negative energies. The eigenquantum defects for the eigenchannels labeled as and experience an avoided crossing at The eigenquantum defect continues to resonantly rise on the high-energy side of the avoided crossing, intersecting 0.5 at This conspicuous behavior is due to a molecular shape resonance, whose properties are explored in more detail in Part II.

The calculated eigenquantum defects of CaF in the electronic symmetry, obtained from the integer- matrix. The calculation of the matrix was carried out for the ionization continuum only and the eigenchannels are labeled using the scheme of Fig. 1. The eigenquantum defects show a prominent energy variation, due to the effects of the long-range dipole field; this strong energy dependence is not present in the behavior of the eigenquantum defects of the matrix shown in Fig. 1. The strong energy dependence of the matrix eigenquantum defects makes it difficult to extrapolate their values to the region of negative energies. The eigenquantum defects for the eigenchannels labeled as and experience an avoided crossing at The eigenquantum defect continues to resonantly rise on the high-energy side of the avoided crossing, intersecting 0.5 at This conspicuous behavior is due to a molecular shape resonance, whose properties are explored in more detail in Part II.

## Tables

Summary of the terms used in the reaction matrix calculations.

Summary of the terms used in the reaction matrix calculations.

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