Relativistic MBPT+CI Calculations

• Relativistic MBPT+CI is a rigorous ab initio approach that integrates nonperturbative configuration interaction with many-body perturbation corrections to capture electron correlations in heavy atoms.
• It partitions the atomic Hamiltonian into a model space for valence electrons and a complementary space to systematically include core-valence and higher-order effects via effective energy-dependent operators.
• The method underpins accurate predictions of atomic spectra, isotope shifts, and transition properties, achieving benchmark uncertainties often below 1% in complex open-shell systems.

Relativistic Many-Body Perturbation Theory Plus Configuration Interaction (MBPT+CI) Calculations

Relativistic many-body perturbation theory combined with configuration interaction (MBPT+CI, or CI+MBPT) comprises a rigorous ab initio formalism for treating electron correlation in heavy atoms and ions, particularly for systems with open d, f, or multiple valence electrons where both relativistic and electron-correlation effects are strong and cannot be captured by CI or MBPT in isolation. The approach uniquely integrates the nonperturbative treatment of valence-valence correlations via CI with the systematic inclusion of core-valence and higher-order effects through MBPT. This methodology underpins high-accuracy calculations of atomic spectra, isotope and field shifts, transition properties, sensitivity to fundamental constants, and informs the interpretation of astrophysical and laboratory spectroscopic data.

1. Formulation of the Relativistic Hamiltonian and Model Partitioning

The relativistic atomic Hamiltonian in the no-pair approximation is given as

$$H = \sum_{i=1}^N h_D(i) + \sum_{i<j} \Bigl[\frac{1}{r_{ij}} + B_{ij}\Bigr],$$

where $$h_D(i)=c\,\boldsymbol{\alpha}_i \cdot \mathbf{p}_i + (\beta_i-1) c^2 + V_{\rm nuc}(r_i)$$ is the one-electron Dirac Hamiltonian, $B_{ij}$ is the frequency-independent Breit operator, and $V_{\rm nuc}$ includes finite-nucleus and QED radiative potentials as required (Berengut, 2011, Kahl et al., 2018, Yu et al., 28 Dec 2025).

The Hamiltonian is split as $H=H_0 + V_{\rm res}$, taking $H_0 = \sum_i h_{\rm DF}(i)$ with $h_{\rm DF}$ the Dirac–Fock operator in an average central potential, and $V_{\rm res}$ the residual two-body interactions not absorbed into $H_0$ (Cheung et al., 2024). The model Hilbert space is partitioned into a valence (or “model”) space $P$ (all low-lying configurations for the valence electrons or holes) and its complement $Q$, which contains core and highly excited states (Kozlov et al., 2022).

Single-particle orbitals are constructed by solving the Dirac–Fock or Dirac–Fock–Slater equations, with large B-spline bases used to systematically converge virtual and continuum manifolds (Kahl et al., 2018, Liu et al., 24 Jul 2025, Yu et al., 28 Dec 2025).

2. Effective Hamiltonian via Second-Order MBPT

The essence of MBPT+CI is the derivation of an effective energy-dependent Hamiltonian in the model space $P$,

$H_{\rm eff}(E) = P H_0 P + \Sigma(E),$

where $\Sigma(E)=P V Q (E-QHQ)^{-1} Q V P$ is the self-energy (correlation) operator including core-valence and (optionally) higher-body effects (Berengut, 2011, Kozlov et al., 2022, Berengut, 2016).

To second order in $V_{\rm res}$, $\Sigma(E)$ incorporates:

• One-body terms ($\Sigma^{(1)}$): core polarization and self-energy corrections,
• Two-body terms ($\Sigma^{(2)}$): screening and exchange diagrams, including "box" diagrams for screened effective interactions,
• Three-body terms ($\Sigma^{(3)}$): indispensable for systems with multiple valence electrons to cancel large subtraction contributions (particularly when the initial mean field differs from the actual core potential) (Kahl et al., 2018, Geddes et al., 2018).

The explicit form for second-order MBPT contributions (for valence orbitals $a, b$ and intermediate states $m, n$) includes terms such as

$$(\Sigma^{(2)})_{ab}(E) = \sum_{m n} \frac{\langle a b || V || m n \rangle \langle m n || V || a b \rangle}{E + \epsilon_a + \epsilon_b - \epsilon_m - \epsilon_n}$$

and analogous expressions for other classes of diagrams (Berengut, 2011, Kozlov et al., 2022).

QED corrections (self-energy, vacuum polarization) enter as additional one-body local potentials, while the Breit interaction enters both the DHF and residual interactions (Kozlov et al., 2022, Kahl et al., 2018).

3. Configuration Interaction in the Valence Space

Within the model space $P$, the CI approach is used to capture strong valence-valence electron correlations nonperturbatively. The many-electron wavefunction is expanded in Slater determinants or configuration state functions (CSFs) built from the single-particle basis: $|\Psi\rangle = \sum_I C_I |\Phi_I\rangle,$ where $|\Phi_I\rangle$ ranges over all determinant or CSF configurations generated by distributing the valence electrons (or particle-hole pairs) over a selected orbital set, typically including single and double excitations from reference configurations (Geddes et al., 2018, Kahl et al., 2021, Torretti et al., 2016).

The effective Hamiltonian matrix to be diagonalized is

$$(H_{\rm eff})_{IJ} = \langle \Phi_I| H_{\rm CI} | \Phi_J \rangle + \Sigma_{IJ}(E),$$

where $H_{\rm CI}$ is the frozen-core CI Hamiltonian (includes relativistic one- and two-body interactions), and $\Sigma_{IJ}(E)$ adds the MBPT corrections (Berengut, 2011, Yu et al., 28 Dec 2025).

For open-shell and near half-filled shell systems—common in transition metals and heavy elements—the particle–hole CI variant is often superior, nonperturbatively including configurations with holes below the nominal Fermi level (Berengut, 2016, Torretti et al., 2016, Kahl et al., 2018).

The secular equation $H_{\rm eff} C = E C$ is solved for the lowest eigenpairs using iterative eigensolvers (Davidson, Lanczos), with “emu CI” compression or analogous truncation schemes employed to make the CI tractable for $N > 5$ valence electrons (Geddes et al., 2018, Kahl et al., 2021, Liu et al., 24 Jul 2025).

4. Computational Implementation and Convergence Control

Numerical implementation proceeds as follows:

1. Solve the Dirac–Fock equations for the core (and valence, as appropriate), constructing a large orthonormal set of single-particle orbitals via B-splines (Berengut, 2011, Yu et al., 28 Dec 2025).
2. Build the model-space CI basis (Slater determinants or CSFs; generate all single/double excitations from selected references), controlling for $n_{\max}, \ell_{\max}$, excitation level, and basis saturation (Geddes et al., 2018, Kahl et al., 2018).
3. Compute all one-, two-, and (as needed) three-body MBPT integrals for $\Sigma(E)$ up to the required order and desired precision (Kahl et al., 2018, Kozlov et al., 2022).
4. Assemble the effective Hamiltonian $H_{\rm eff}$, incorporating MBPT corrections directly into the one- and two-electron integrals (Cheung et al., 2024).
5. Diagonalize $H_{\rm eff}$, typically using parallelized block-Davidson methods capable of handling CI spaces $\gtrsim 10^6$ configurations (using “emu CI”, partitioned matrix, or chunked construction) (Geddes et al., 2018, Cheung et al., 2024).
6. Control convergence by systematic increases in $n_{\max}$, $\ell_{\max}$, active reference set, and MBPT basis, with residual uncertainties estimated by extrapolation or by comparison between different partitionings and starting potentials (Yu et al., 28 Dec 2025, Geddes et al., 2018).

Parallel software frameworks (AMBiT, pCI) implement these algorithms with distributed memory and hybrid OpenMP/MPI parallelism, supporting inclusion of Breit, QED, and RPA corrections (Kahl et al., 2018, Cheung et al., 2024).

5. Physical Applications: Spectra, Sensitivity Coefficients, and Isotope Shifts

The MBPT+CI methodology enables ab initio calculations of

• Low-lying spectra of multi-valence-electron atoms and ions: achieving mean deviations from experiment at the $<0.1\%$ level for heavy ions and complex open-shell systems (Yu et al., 28 Dec 2025, Torretti et al., 2016).
• Relativistic sensitivity coefficients ($q$ values): The response of atomic levels to variation in the fine-structure constant $\alpha$, given by $q = dE/d(\alpha^{2})|_{\alpha=\alpha_{0}}$, is extracted by finite-difference calculations at shifted $\alpha$ values (Berengut, 2011, Berengut, 2016).
• Isotope and field shifts: MBPT+CI treats both the specific mass shift ($k_{\rm SMS}$) and field shift operator within the CI+MBPT framework, allowing high-accuracy isotope-shift predictions for interpretation of precision King plots and sensitivity to nuclear effects (Yu et al., 28 Dec 2025, Liu et al., 24 Jul 2025).
• Transition properties: Electric and magnetic multipole transition rates, $g$-factors, and polarizabilities are computed using CI+MBPT correlated wave functions (Cheung et al., 2024, Wei et al., 24 Feb 2025).

Applications range from benchmarking atomic clocks, probing $\alpha$-variation (in quasar absorption spectra and laboratory clocks), to modeling astrophysical spectra for heavy and superheavy elements (Berengut, 2011, Kahl et al., 2021, Liu et al., 24 Jul 2025).

6. Limitations, Extensions, and Best Practices

The formalism’s limitations primarily arise from:

• Truncation of MBPT at second or third order (neglecting some diagrams relevant for $N_{\rm valence}>5$, deep-core correlation, or highly correlated open d/f shell systems) (Kozlov et al., 2022, Geddes et al., 2018).
• Factorial scaling of CI basis size, even if partially mitigated by “emu CI” or blocked algorithms; practical limits are $N_{\rm valence} \sim 5-7$ (Geddes et al., 2018).
• Approximate treatment of energy-dependence in $\Sigma(E)$ (Brillouin–Wigner or static approximations); full self-consistency is computationally demanding (Kahl et al., 2018).

Best practices for achieving optimal accuracy include

• Using a starting potential (often $V^{N-1}$) that closely matches the actual electron configuration to minimize large subtraction diagrams and enhance cancellation of MBPT terms (Berengut, 2011).
• Inclusion of all relevant second-order diagrams (one-, two-, and three-body) self-consistently; omitting any leads to uncontrolled cancellations and degraded results (Kahl et al., 2018).
• Careful convergence checking with respect to orbital basis, CI model size, and MBPT virtual space (Yu et al., 28 Dec 2025).
• For open-shell systems, leveraging the particle–hole formalism to nonperturbatively treat holes as valence configurations (Berengut, 2016, Torretti et al., 2016).

Extensions include

• Determinant-based perturbation theory (VPT) for treating high-excitation spaces,
• Hybridization with relativistic coupled-cluster for deep-core correlation (RCC+CI),
• Inclusion of QED corrections via local radiative potentials (Kozlov et al., 2022, Cheung et al., 2024).

7. Illustrative Results and Benchmarks

Recent large-scale benchmarks using MBPT+CI include:

• Cr II and five-valence-electron systems: Converged spectra and $q$-values with $<5\%$ mean deviation; all second-order diagrams are imperative (Berengut, 2011, Geddes et al., 2018).
• Heavy ions and superheavy elements (Ta, Db, Lu, Lr, Hg, Cn): Uncertainties in calculated energies at the $\sim 0.1\%$–$1\%$ level; isotope shift constants and $g$-factors accurate to $<1\%$–$5\%$ (Liu et al., 24 Jul 2025, Geddes et al., 2018, Kahl et al., 2021).
• Highly charged ions (Ni$^{12+}$): Excitation energies within $<10\,\mathrm{cm}^{-1}$ of experiment, with relative uncertainties below 0.2%, mass and field shift constants with $<1\%$ uncertainty (Yu et al., 28 Dec 2025).
• Multipole transition properties (Al II): Transition rates and oscillator strengths computed for hundreds of lines, 1–5% agreement with experiment, E1-M1-E2-M2 multiplet coverage (Wei et al., 24 Feb 2025).

A summary of representative accuracy achieved across systems is given below.

System	States/Properties	Energy Uncertainty	Comments
Cr II	Low-lying levels	$\lesssim 5\%$	Full second-order MBPT required
Ta, Db	Extensive spectra	$\lesssim 10\%$	Emu CI for $N=5$ valence
Lr, Lu	Multiple levels	$\sim 0.1$–$0.2\%$	Benchmarked to experiment
Hg, Cn	IP, field shifts	$50$–$500\,\mathrm{cm}^{-1}$	QED & Breit included
Ni$^{12+}$	4 excited states	$<0.2\%$	Isotope shifts at $<1\%$
Al II	400 transitions	$\sim 1\%$	Multipole lines, E1–M2

This systematic, highly parallelizable approach is now standard for precision modeling of heavy-atom structure, atomic clocks, and astrophysical spectra (Berengut, 2011, Kahl et al., 2018, Liu et al., 24 Jul 2025, Yu et al., 28 Dec 2025, Cheung et al., 2024).