---
title: Relativistic MBPT+CI Calculations
url: https://www.emergentmind.com/topics/relativistic-many-body-perturbation-theory-plus-configuration-interaction-mbpt-ci-calculations
type: topic
---

# Relativistic MBPT+CI Calculations

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 [1110.2292, 1805.11265, 2512.22850].

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 $H_0$ [2410.06680]. 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 [2202.02026].

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 [1805.11265, 2507.18490, 2512.22850].

## 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 [1110.2292, 2202.02026, 1605.08111].

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) [1805.11265, 1805.06615].

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 [1110.2292, 2202.02026].

QED corrections (self-energy, vacuum polarization) enter as additional one-body local potentials, while the Breit interaction enters both the DHF and residual interactions [2202.02026, 1805.11265].

## 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 [1805.06615, 2103.11287, 1612.00747].

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 [1110.2292, 2512.22850].

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 [1605.08111, 1612.00747, 1805.11265].

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 [1805.06615, 2103.11287, 2507.18490].

## 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 [1110.2292, 2512.22850].
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 [1805.06615, 1805.11265].
3. Compute all one-, two-, and (as needed) three-body MBPT integrals for $\Sigma(E)$ up to the required order and desired precision [1805.11265, 2202.02026].
4. Assemble the effective Hamiltonian $H_{\rm eff}$, incorporating MBPT corrections directly into the one- and two-electron integrals [2410.06680].
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) [1805.06615, 2410.06680].
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 [2512.22850, 1805.06615].

Parallel software frameworks (AMBiT, pCI) implement these algorithms with distributed memory and hybrid OpenMP/MPI parallelism, supporting inclusion of Breit, QED, and RPA corrections [1805.11265, 2410.06680].

## 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 [2512.22850, 1612.00747].
- **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 [1110.2292, 1605.08111].
- **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 [2512.22850, 2507.18490].
- **Transition properties:** Electric and magnetic multipole transition rates, $g$-factors, and polarizabilities are computed using CI+MBPT correlated wave functions [2410.06680, 2502.17311].

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 [1110.2292, 2103.11287, 2507.18490].

## 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) [2202.02026, 1805.06615].
- Factorial scaling of CI basis size, even if partially mitigated by “emu CI” or blocked algorithms; practical limits are $N_{\rm valence} \sim 5-7$ [1805.06615].
- Approximate treatment of energy-dependence in $\Sigma(E)$ (Brillouin–Wigner or static approximations); full self-consistency is computationally demanding [1805.11265].

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 [1110.2292].
- Inclusion of all relevant second-order diagrams (one-, two-, and three-body) self-consistently; omitting any leads to uncontrolled cancellations and degraded results [1805.11265].
- Careful convergence checking with respect to orbital basis, CI model size, and MBPT virtual space [2512.22850].
- For open-shell systems, leveraging the particle–hole formalism to nonperturbatively treat holes as valence configurations [1605.08111, 1612.00747].

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 [2202.02026, 2410.06680].

## 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 [1110.2292, 1805.06615].
- **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 $<1\%$–$5\%$ [2507.18490, 1805.06615, 2103.11287].
- **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 [2512.22850].
- **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 [2502.17311].

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 [1110.2292, 1805.11265, 2507.18490, 2512.22850, 2410.06680].

Source: https://www.emergentmind.com/topics/relativistic-many-body-perturbation-theory-plus-configuration-interaction-mbpt-ci-calculations