A complex Gaussian representation of continuum wavefunctions respectful of their asymptotic behaviour

TL;DR

The paper develops a complex Gaussian-type orbital (cGTO) framework to represent continuum radial functions $u_{l,k}(r)$, addressing two key challenges: instability in exponent optimization and loss of correct asymptotics due to finite fitting boxes. It introduces dimensionless optimization to grant uniform relative flexibility across exponents and a two-step fitting loop, improving fit quality for Coulomb functions. It then introduces an indirect fitting strategy based on asymptotic factorization with an offset $C$, fitting a distortion factor instead of the raw continuum function to recover the correct oscillatory behavior at infinity. The indirect method preserves closed-form expressions for common one-electron integrals, yields accurate asymptotics, and remains computationally competitive, enabling more reliable continuum-state calculations in molecular contexts and potential extensions to multicentric problems.

Abstract

Complex Gaussian basis sets are optimized to accurately represent continuum radial wavefunctions over the whole space. First, attention is put on the technical ability of the optimization method to get more flexible series of Gaussian exponents, in order to improve the accuracy of the fitting approach. Second, an indirect fitting method is proposed, allowing for the oscillatory behaviour of continuum functions to be conserved up to infinity as a factorized asymptotic function, while the Gaussian representation is applied to some appropriately defined distortion factor with limited spatial extension. As an illustration, the method is applied to radial Coulomb functions with realistic energy parameters. We also show that the indirect fitting approach keeps the advantageous analytical structure of typical one-electron transition integrals occurring in molecular ionization applications.

Figures (7)

• Figure 1: Comparison of complex Gaussian expansions of the Coulomb radial function for $k=0.75$ a.u. and $l = 0$ represented with 30 cGTOs, obtained from three different sets of optimization parameters A, B, C as described in table \ref{['tab:fitting_parametersABC']}. The upper panel shows the function and its different Gaussian fits, the second and third panels show the real and imaginary parts of the error, respectively.
• Figure 2: Unphysical long-range behaviour of complex Gaussian expansions for Coulomb radial wavefunctions ($k=0.75$ a.u., $l=0$) represented with 30 cGTOs, obtained from three different sets of optimization parameters A, B, C as described in table \ref{['tab:fitting_parametersABC']}. The vertical dashed line indicates the radial limit of the fitting box.
• Figure 3: Original Coulomb radial function $u_{l,k}(r)$ for $k=2$ a.u. and $l=1$ (black solid line), together with their associated asymptotic distortion factor $\tilde{u}_{l,k}(r)$ as defined in eq. \ref{['eq:distortion_factor']}, for three selected values of the offset parameter ($C=2$, red dotted line; $C=4$, green dashed line; $C=8$, blue long-dashed line).
• Figure 4: Distortion functions $\tilde{u}_{l,k}(r)$ (for $k=2$ a.u., $l=1$), together with their complex Gaussian representations using 30 cGTOs, obtained with three different values of the offset constant $C$. The upper panel shows the function and its different Gaussian fits, the second and third panels show the real and imaginary parts of the error, respectively.
• Figure 5: Reconstruction of the original radial continuum wavefunction $u_{l,k}(r)$ ($k=2$ a.u., $l=1$) within the fitting box, using eq. \ref{['eq:reconstruction']} with 30 cGTOs. We show the results obtained with three different values of the offset constant $C$ (red dotted line for $C=2$, green dashed line for $C=4$, blue long-dashed line for $C=8$. See table \ref{['tab:par_conv_distortion_function']} for the details). The indirect fit is also compared to an example of direct fit (magenta dotted-dashed line). The upper panel shows the function and its different Gaussian fits, the second and third panels show the real and imaginary parts of the error, respectively.