- Generated on Tue May 12 2026 01:52:49 for ampsci by
1.9.8
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ampsci
High-precision calculations for one- and two-valence atomic systems
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Atomic Potentials: Hartree-Fock, Nuclear, Breit and QED.
This assumes you already have ampsci compiled and have a basic understanding of how to run and use it.
The full atomic Hamiltonian for an \(N\)-electron atom:
\[ H = \sum_i^N h_0(r_i) + \sum_{i < j}\frac{1}{r_i - r_j} \]
\[ h_0(\vec{r_i}) = c \vec\alpha_i\cdot\vec{p_i} + c^2 (\beta_i-1) + V_{\rm nuc}. \]
The starting approximation is the Hartree-Fock method:
\[ H \approx \sum_i^N [h_0(\vec r_i) + v^{\rm HF} ]. \]
We focus on case of single-valence systems, and start with so-called \(V^{N-1}\) approximation, in which Hartree-Fock potential is due to the \(N-1\) core electrons:
\[ \hat v^{\rm HF}\phi_a(\vec{r_1}) = \sum_{i\neq a}^{N_c}\Bigg( \int \frac{\phi_i^\dagger(\vec{r_2})\phi_i(\vec{r_2})}{|r_{12}|}d^3\vec{r_2}\,\phi_a(\vec{r_1}) -\int \frac{\phi_i^\dagger(\vec{r_2})\phi_a(\vec{r_2})}{|r_{12}|}d^3\vec{r_2}\,\phi_i(\vec{r_1}) \Bigg), \]
First, Hartree-Fock equations are solved self-consistently for all core electrons, then the valence states are found in the Frozen Hartree-Fock potential due to the core.
As always, begin by checking the available options:
For example, to run Hartree-Fock for Cs with a \(V^{N-1}\) potential (i.e. Xe-like core),
If you're unsure of core/valence states, use ampsci's in-built periodic table
or
which will give:
Besides Hartree-Fock, other methods are available:
e.g.,
For a point-like nucleus, \( V_{\rm nuc} = -Z/r \) ; in reality, the nuclear charge is distributed across the finite-size nucleus. By default, to form \( V_{\rm nuc}, \) we assume the nuclear charge follows a Fermi distribution,
\[ \rho(r) = \frac{\rho_0}{1+\exp[(r-c)/a]}, \]
where \(\rho_0\) is a normalisation factor ( \(\int\rho\,{\rm d} V = Z\)), c is the half-density radius, and a is defined via the 90 - 10% density fall-off \({t\equiv 4a\ln3}\) (known as the `‘skin thickness’'), which we take to be \(t=2.3\,{\rm fm}\) by default for all heavy isotopes.
If no inputs are given, the code will assume a Fermi-like distribution for the nuclear charge, and look up default values for the nuclear parameters (charge radius and skin thickness). The default values are chosen from the specified isotope in the Atom{} block. These may also be specified specifically:
Fermi, spherical, point-like, and Gaussian, with Fermi being the default.input_file allows a numerically determined nuclear potential to be read of from a file. note:Default rms values are taken from:
The Breit Hamiltonian accounts for magnetic interactions between electrons (also known as the Gaunt interaction), and retardation effects. It leads to a correction to the electron-electron Coulomb term in the many-body Hamiltonian:
\[ \sum_{ij}\frac{1}{r_{ij}} \to \sum_{ij}\left( \frac{1}{r_{ij}} + \hat h^B_{ij}\right), \]
where, in the limit of zero frequency, the two-particle Breit Hamiltonian is
\[ h^B_{ij} = - \frac{\vec{\alpha_i}\cdot\vec{\alpha_j} + (\vec{\alpha_i}\cdot\hat{n_{ij}})(\vec{\alpha_j}\cdot\hat{n_{ij}})}{2\, r_{ij}}. \]
This can be included at the Hartree-Fock level with the HartreeFock{Breit = true;} setting.
This setting is a scaling factor; i.e., setting Breit = 0.5; will add effective factor of 0.5 in front of \(h^B\). Typically, only 0 or 1 is set; other values are useful for checking for non-physical non-linear-in-Breit effects. Setting to true is equivilant to setting to 1.0.
Radiative QED corrections can be included into the wavefunctions using the Flambaum-Ginges radiative potential method. An effective potential, \(V_{\rm rad}\), is added to the Hamiltonian before the equations are solved. The potential can be written as the sum of the Uehling (vacuum polarisation), Wichmann-Kroll (higher-order vacuum polarisation) and self-energy potentials; the self-energy potential itself is written as the sum of the high- and low-frequency electric contributions, and the magnetic contribution:
\[ V_{\rm rad}(\vec{r}) = V_{\rm Ueh}(r) + V_{\rm WK}(r) + V_{\rm SE}^{h}(r) + V_{\rm SE}^{l}(r) + i (\vec{\gamma}\cdot\hat{n}) V^{\rm mag}(r). \]
This can be included by setting QED=true; in the Hartree Fock block:
QED=valence;You can all set detailed QED options within the RadPot{} Block. If QED=true; in HartreeFock, and the RadPot{} block is not explicitely included, then the QED radiative potential will be included with the default parameters. This is equivilant to
Ueh, SE_h, SE_l, SE_m, WK) options are scaling factors for their corresponding terms in the potential; typically 0 or 1, but can be tuned (for testing).scale_l takes a list of scaling factors for each l; these will rescale \(V_{\rm rad}\) for each partial wave. e.g., scale_l = 0,1,0; will include \(V_{\rm rad}\) for p states, but not s or d states.scale_rN; Re-scales the effective nuclear radius; used to test finite-nuclear size effects on \(V_{\rm rad}\). =1 means normal, =0 means assume point-like nucleus (when calculating \(V_{\rm rad}\)). 1.9.8