Selected Configuration Interaction Methods

• Selected configuration interaction (SCI) is a family of electronic structure methods that approximates full CI by variationally optimizing over an adaptively chosen subset of determinants.
• SCI methods iteratively select determinants based on their energetic relevance using criteria like Epstein–Nesbet contributions and Hamiltonian couplings to recover correlation energy efficiently.
• Modern SCI approaches integrate perturbative corrections, symmetry adaptation, and emerging quantum-classical and machine-learning techniques to achieve near-FCI accuracy while reducing computational resources.

Selected configuration interaction (SCI) denotes a family of electronic-structure methods that approximate full configuration interaction (FCI) by variationally optimizing the wave function in an adaptively chosen subset of Slater determinants, typically augmented by a second-order perturbative correction. In contrast to truncated CI by excitation rank, SCI uses an arbitrary determinant set chosen for energetic relevance, and modern variants such as CIPSI, ASCI, HCI, and SHCI can deliver near-FCI energies with only a small fraction of the determinants in the FCI space (Chilkuri et al., 2018, Holmes et al., 2016, Loos et al., 2024).

1. Formal structure of SCI

In CI, the many-electron wave function is expanded as

$|\Psi\rangle = \sum_I c_I |D_I\rangle,$

and the electronic Schrödinger equation becomes the matrix eigenvalue problem

$$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$

FCI includes all determinants compatible with the chosen basis, electron number, spin, and symmetry, and is exact within that one-particle basis, but its dimension grows combinatorially with system size (Zeni et al., 23 Jun 2026, Loos et al., 2024).

SCI replaces the full determinant space by a selected variational space, often denoted $\mathcal V$ or an internal space $I$, together with an external space of connected but unselected determinants. The SCI wave function is therefore a CI wave function in a truncated determinant basis that is not restricted to fixed excitation ranks. As the selected space grows, the variational energy approaches the exact eigenvalue in the chosen CI Hamiltonian, and the perturbative correction tends to zero (Chilkuri et al., 2018, Burton et al., 2023).

A defining feature of SCI is that determinant selection is iterative. Starting from a small reference space, one diagonalizes the Hamiltonian in the current variational space, generates connected external determinants, scores their importance, admits a subset, and repeats. This logic underlies determinant-driven CIPSI, heat-bath CI, adaptive sampling CI, and related schemes, even though their importance measures and search strategies differ (Loos et al., 2024, Holmes et al., 2016).

2. Determinant importance and selection criteria

Classical SCI methods differ primarily in how they quantify determinant importance. In CIPSI, external determinants are ranked by their individual second-order Epstein–Nesbet contributions,

$$\delta E_\mu^{(2)} = \frac{\left|\langle \Psi_0 | \hat{H} | D_\mu \rangle\right|^2}{E_\text{var} - E_\mu},$$

and those with the largest contributions are promoted into the variational space (Loos et al., 2024). This ties selection directly to perturbative recovery of the missing correlation energy.

Heat-bath configuration interaction (HCI) introduced a simpler criterion based directly on the product of Hamiltonian couplings and current CI coefficients. A determinant $D_k$ is added whenever

$|H_{ki} c_i| > \epsilon_1$

for at least one determinant $D_i$ already present in the variational space (Holmes et al., 2016, Zeni et al., 23 Jun 2026). HCI also uses a separate matrix-element screen to avoid enumerating connected excitations with negligible $|H_{ki}|$, which is central to its efficiency. In the original HCI benchmarks, this strategy recovered the FCI energies of the carbon dimer in the cc-pVTZ basis and the strongly correlated chromium dimer in the Ahlrichs VDZ basis to better than $1$ mHa in just a few minutes on a single core, despite full variational spaces of $$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$0 and $$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$1 determinants, respectively (Holmes et al., 2016).

A recurrent misconception is that SCI is merely truncated CI with a smarter cutoff. The literature instead treats selection as an importance-ordering problem over determinant space. This perspective is explicit in recent machine-learning formulations: classifier-assisted SCI recasts determinant importance as binary classification, whereas ranking configuration interaction (RCI) treats selection as a pairwise learning-to-rank problem in which only the relative ordering of determinants matters (Zeni et al., 23 Jun 2026, Nie et al., 11 May 2026).

3. Perturbative correction, extrapolation, and dressed Hamiltonians

Most modern SCI methods combine the variational selected-space energy $$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$2 with an Epstein–Nesbet second-order perturbative correction,

$$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$3

so that

$$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$4

This correction estimates the residual contribution of determinants outside the variational space and is also used as a convergence and extrapolation variable (Loos et al., 2024, Chilkuri et al., 2018).

A large part of SCI practice relies on extrapolation to the FCI limit. Burton and Loos gave a rationale for the widely used near-linear relation between $$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$5 and $$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$6, showing from the structure of the exact electronic energy landscape that the energy gap and the coupling between internal and external spaces determine when the linear regime is reached (Burton et al., 2023). In the asymptotic regime one uses

$$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$7

and extrapolates to $$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$8 (Loos et al., 2024, Burton et al., 2023). The same work also derived a non-linear extrapolation formula designed to improve post-processing when the SCI points are not yet deep in the linear regime (Burton et al., 2023).

Beyond additive PT2 corrections, SCI has also been combined with effective-Hamiltonian dressing. Garniron and co-workers revised the internal space under the effect of its interaction with the outer space using a shifted-Bk effective Hamiltonian, and in the multi-state case removed the storage bottleneck of the dressing matrix through a low-rank factorization (Lyu et al., 2018). This establishes a second major branch of SCI post-processing: instead of only correcting the energy, one renormalizes the variational problem itself.

4. Symmetry adaptation, determinant representations, and large-scale implementations

Determinant-based SCI does not automatically guarantee spin purity. In particular, selection performed only over Slater determinants does not ensure that the resulting SCI wave function is an eigenfunction of $$\mathbf{H}\mathbf{c} = E\mathbf{c}, \qquad H_{IJ} = \langle D_I|\hat H|D_J\rangle.$$9, which can become quantitatively problematic in bond breaking, magnetic systems, and excited states (Chilkuri et al., 2018). To address this, an efficient algorithm was introduced that, given an arbitrary determinant space, generates all missing Slater determinants needed to obtain spin-adapted wave functions while avoiding manipulations involving configuration state functions during selection (Chilkuri et al., 2018). The same work showed that generating all determinants with 6 spin-up and 6 spin-down electrons in 12 open shells takes 21 CPU cycles per generated Slater determinant, and that storing Davidson vectors in a CSF basis can reduce the Davidson memory footprint by about a factor 4 for dissociated N$\mathcal V$0 (Chilkuri et al., 2018).

Representation choices also affect scalability. Determinant-driven SCI implementations, such as CIPSI in Quantum Package, avoid storing the full Hamiltonian and instead generate connected determinants on the fly using bitstring representations and semistochastic PT2 evaluation (Loos et al., 2024). At a different scale, tensor-product SCI (TPSCI) separates selected $\mathcal V$1- and $\mathcal V$2-spin bitstrings and forms determinant spaces through their tensor product, a structure motivated in part by QC+HPC workflows (Xu et al., 13 Mar 2025).

Large-scale parallelization has recently become a central implementation theme. A distributed-storage TPSCI implementation reported the first SCI framework with distributed storage of the CI vector and demonstrated massively parallel FCI-mode calculations on Fugaku. Its largest case, N$\mathcal V$3 in cc-pVTZ under $\mathcal V$4 symmetry with frozen core orbitals, involved $\mathcal V$5 determinants, surpassing the previous SCI record of $\mathcal V$6 determinants by over three orders of magnitude and exceeding the largest earlier FCI calculation of $\mathcal V$7 determinants (Xu et al., 13 Mar 2025). This suggests that the practical boundary between “selected” and “full” determinant-space algorithms is increasingly shaped by storage and communication architecture as much as by formal selection logic.

5. Extensions beyond ground-state Hermitian energies

SCI has been extended from ground-state energies to response theory, resonances, and non-Hermitian transcorrelated Hamiltonians. Linear-response SCI (LR-SCI) introduced determinant selection criteria specifically motivated by response theory, because ground-state-optimized selection alone can omit determinants important for wave-function response. Using these criteria, LR-SCI was shown to converge systematically toward the FCI limit for static polarizabilities, to compute the water K-edge X-ray absorption spectrum in active spaces up to $\mathcal V$8, and to deliver NMR spin-spin coupling constants for water with accuracy beyond CCSDT (Reinholdt et al., 3 Oct 2025).

Perturbatively corrected LR-SCI further combined variational LR-SCI with Epstein–Nesbet perturbation theory through second order. In this formulation, the finite-order perturbative treatment preserves the pole structure of the parent variational LR-SCI theory, so the method is useful for static properties but not suitable for frequency-dependent molecular response properties; numerically, second-order corrections substantially improved static polarizabilities of water, ethene, boron hydride, and hydrogen chloride and reduced the oscillatory convergence behavior of the parent variational theory (Reinholdt et al., 25 Jun 2026).

SCI has also entered non-Hermitian problems. CAP-SCI generalized selected CI to electronic resonances by combining SCI with complex absorbing potentials, producing resonance positions and widths of FCI quality for the shape resonances of $\mathcal V$9 and $I$0; high-order correlation shifted CAP-EOM-EA-CCSD values by more than $I$1 eV in the reported examples (Damour et al., 2024). In the transcorrelated setting, SCI was reformulated around a bilinear left-right functional for the non-Hermitian TC Hamiltonian, and numerical tests showed that taking the non-Hermitian character into account in the selection criterion is mandatory for fast convergence of the TC energy (Ammar et al., 2022). A notable consequence is that criteria based on first-order coefficients and second-order energies, usually similar in Hermitian SCI, can lead to significantly different convergence rates in TC-SCI (Ammar et al., 2022).

6. Quantum-classical and machine-learning reformulations

Quantum-selected configuration interaction (QSCI) uses a quantum device to generate configurations and a classical computer to diagonalize the Hamiltonian in the selected subspace. In this paradigm, measured bitstrings define the SCI space, and the quantum device replaces the classical search over determinant space (Erhart et al., 20 Jun 2025, Vaquero-Sabater et al., 22 May 2026). A time-evolved-state variant, TE-QSCI, proposed using $I$2 as the input state instead of an optimized variational ansatz, thereby providing a systematic and optimization-free method to prepare the input state of QSCI (Mikkelsen et al., 2024). Related QSCI studies have shown that noiseless ansatz sampling can produce compact and biased configurational spaces, whereas sampling noise combined with configuration recovery can enlarge the effective Hilbert-space exploration and systematically improve the resulting CI energies (Vaquero-Sabater et al., 22 May 2026).

QSCI has also been embedded into post-CI frameworks. QSCI-TCC tailors coupled-cluster theory with a QSCI wave function, mapping active-space CI coefficients obtained from quantum sampling onto fixed cluster amplitudes and then optimizing the remaining amplitudes classically. In N$I$3 with a $I$4 active space, the shot-count analysis showed that, with the $I$5 correction, chemically sufficient precision $I$6 is achieved with only $I$7 shots in the strongly correlated regime at $I$8 Å, an order of magnitude fewer than required by an earlier matchgate-shadows implementation (Erhart et al., 20 Jun 2025).

Machine learning has recently shifted SCI from hand-crafted heuristics toward learned selection rules. A neural-network-assisted framework integrated a binary classifier into both classical heat-bath CI and quantum-classical sample-based quantum diagonalization, yielding roughly a $I$9 reduction in memory and per-iteration cost for the classical cHCI variant and markedly fewer iterations for the quantum-classical cSQD variant on N$$\delta E_\mu^{(2)} = \frac{\left|\langle \Psi_0 | \hat{H} | D_\mu \rangle\right|^2}{E_\text{var} - E_\mu},$$0 (Zeni et al., 23 Jun 2026). RCI then argued that SCI selection is intrinsically a ranking problem rather than a regression or classification problem, and reported overall computational time reductions of 23% to over 50% depending on the system, only 55% of the determinant count in representative N$$\delta E_\mu^{(2)} = \frac{\left|\langle \Psi_0 | \hat{H} | D_\mu \rangle\right|^2}{E_\text{var} - E_\mu},$$1 and CO cases, and chemical accuracy on a highly challenging iron-sulfur system using only 12% of the full CI space (Nie et al., 11 May 2026). This suggests that learned determinant ordering may become a generic plugin for existing supervised SCI workflows.

7. Benchmark role, sustainability, and contemporary significance

SCI now occupies a dual role: it is both an approximation strategy for strongly correlated systems and a benchmark generator for small and medium molecules. A particularly explicit benchmark came from propane in STO-3G, where the FCI Hilbert space contained $$\delta E_\mu^{(2)} = \frac{\left|\langle \Psi_0 | \hat{H} | D_\mu \rangle\right|^2}{E_\text{var} - E_\mu},$$2 determinants. Determinant-driven CIPSI in Quantum Package reached microhartree accuracy at a fraction of the computational and memory cost: with $$\delta E_\mu^{(2)} = \frac{\left|\langle \Psi_0 | \hat{H} | D_\mu \rangle\right|^2}{E_\text{var} - E_\mu},$$3 determinants, the SCI+PT2 energy was within $$\delta E_\mu^{(2)} = \frac{\left|\langle \Psi_0 | \hat{H} | D_\mu \rangle\right|^2}{E_\text{var} - E_\mu},$$4Ha of FCI, and the extrapolated exFCI energy differed by $$\delta E_\mu^{(2)} = \frac{\left|\langle \Psi_0 | \hat{H} | D_\mu \rangle\right|^2}{E_\text{var} - E_\mu},$$5Ha (Loos et al., 2024). On a single node, the corresponding run used 47 GB of memory, 13:16 hours of wall time, and 4.1 kWh, whereas the reported trillion-determinant FCI calculation required 256 nodes, 113.6 hours, and 19 TB of memory (Loos et al., 2024).

This resource profile has led to an explicit “go green” interpretation of SCI. The propane study argued that SCI can achieve microhartree accuracy using more reasonable and sustainable computational resources, reducing the carbon footprint relative to brute-force FCI (Loos et al., 2024). At the same time, the distributed-storage TPSCI results show that SCI implementations are now entering the trillion-determinant regime on modern supercomputers (Xu et al., 13 Mar 2025).

Selected configuration interaction is therefore best understood not as a single algorithm but as a general determinant-space strategy. It encompasses perturbatively selected variational expansions, heat-bath screening, dressed effective Hamiltonians, spin-adapted determinant completion, non-Hermitian and resonance extensions, linear-response theories, and quantum- or learning-assisted selection mechanisms. Across these variants, the invariant principle is the same: exploit the sparsity of the exact wave function by identifying and optimizing over only the determinants that matter most, while controlling the residual correlation by perturbation, extrapolation, or both (Burton et al., 2023, Loos et al., 2024).