Adds calculating second derivative of the on-top kernel with respect 1- and 2-RDM as well as computing the vector-hessian product on the grid. Numerically these terms are the one- and two- body on-top gradient response terms defined as
where $\mathbf{f}^\mathrm{ot}$ is the Hessian of the on-top potential kernel with respect to the density variables. Since on-top kernels are translated from the corresponding Kohn-Sham functionals via a non-linear mapping, some care is taken.
Tests are added that to numerically check that the analytical equations agree. See a discussion here of what is being tested.
Adds calculating second derivative of the on-top kernel with respect 1- and 2-RDM as well as computing the vector-hessian product on the grid. Numerically these terms are the one- and two- body on-top gradient response terms defined as
$$ [\gamma \cdot\mathbf{F}]^qp = \int \vec{\rho\gamma}^T\cdot\mathbf{f}^\mathrm{ot}\cdot \frac{\partial\vec{\rho}}{\partial\gamma^p_q} $$
$$ [\gamma \cdot\mathbf{F}]_{pr}^{qs} = \int \vec{\rho\gamma}^T\cdot\mathbf{f}^\mathrm{ot}\cdot \frac{\partial\vec{\rho}}{\partial\gamma\{qs}^{pr}} $$
where $\mathbf{f}^\mathrm{ot}$ is the Hessian of the on-top potential kernel with respect to the density variables. Since on-top kernels are translated from the corresponding Kohn-Sham functionals via a non-linear mapping, some care is taken.
Tests are added that to numerically check that the analytical equations agree. See a discussion here of what is being tested.