We present a self-consistent implementation of the correlation-factor model for exchange and correlation within the generalized Kohn--Sham (gKS) framework. In the correlation-factor approach, the exchange-correlation hole is written as the product of a model exchange hole and a correlation factor,

\[ \rho_{xc}(\mathbf r,u)=f_c(\mathbf r,u)\rho_x(\mathbf r,u),\]

where \(\mathbf r\) is the reference position and \(u=|\mathbf r'-\mathbf r|\) is the electron--electron separation[1,2]. In the present implementation, the exchange hole is represented with a generalized Becke--Roussel form [2,3], and the resulting hole model is used to construct the exchange-correlation energy,

\[ E_{xc}=\int d^3r\,\rho(\mathbf r)\int d^3u\,\frac{\rho_{xc}(\mathbf r,u)}{2u}.\]

A central difficulty in the self-consistent use of such functionals is that the energy depends explicitly on orbital quantities and nonstandard intermediate ingredients. Accordingly, the effective gKS contribution is obtained from derivatives with respect to the orbitals rather than from a purely multiplicative density derivative,

\[ \hat v^{\mathrm{gKS}}_{xc,\sigma}\phi_{i\sigma}(\mathbf r)=\frac{\delta E_{xc}}{\delta \phi_{i\sigma}^*(\mathbf r)}.\]

To evaluate these derivatives efficiently and robustly, we implement the functional in Python using JAX and employ automatic differentiation to generate the orbital-dependent terms entering the self-consistent-field cycle [4]. This avoids lengthy manual derivations and fragile finite-difference procedures.

The implementation is assessed through self-consistent calculations for representative atoms and molecules. The results show that automatic differentiation provides a practical and stable route to fully self-consistent gKS calculations with correlation-factor-based functionals. More generally, this work demonstrates that physically motivated, hole-based exchange-correlation models can be brought to the self-consistent level in a flexible computational framework suitable for further functional development.

[1] Roy, P.-O., Cuierrier, É., & Ernzerhof, M. (2020). The correlation factor approach: Combining density functional and wave function theory. The Journal of Chemical Physics, 152(21), 211101.

[2] Wang, R., Zhou, Y., & Ernzerhof, M. (2019). The correlation factor model for the exchange-correlation energy and its application to transition metal compounds. The Journal of Chemical Physics, 150(8), 084107.

[3] Becke, A. D., & Roussel, M. R. (1989). Exchange holes in inhomogeneous systems: A coordinate-space model. Physical Review A, 39(8), 3761–3767.

[4] Bradbury, J., Frostig, R., Hawkins, P., Johnson, M. J., Katariya, Y., Leary, C., Maclaurin, D., Necula, G., Paszke, A., VanderPlas, J., Wanderman-Milne, S., & Zhang, Q. (2018). JAX: composable transformations of Python+NumPy programs (Version 0.3.13) [Computer software].