Frozen Natural Spinors in Relativistic Quantum Chemistry

• Frozen natural spinors are correlated orbital bases derived by diagonalizing the virtual–virtual block of the one-particle reduced density matrix in relativistic quantum chemistry.
• They reduce computational costs in post-Hartree–Fock methods by truncating low-occupancy orbitals, achieving speedups up to 50× and drastically lowering memory demands.
• Integration with Cholesky decomposition and advanced schemes like FNS++ and SS-FNS enables accurate treatment of excited states and response properties while maintaining near-canonical performance.

Frozen natural spinors (FNS) comprise a class of correlated orbital bases obtained via diagonalization of the one-particle (virtual–virtual) reduced density matrix in relativistic quantum chemistry. In the FNS framework, low-occupancy natural spinors are removed (frozen), providing a highly compact virtual space that accelerates post-Hartree–Fock electron correlation treatments, particularly in two- and four-component relativistic coupled-cluster (CC), equation-of-motion (EOM) CC, and algebraic diagrammatic construction (ADC) methodologies for heavy-element systems. FNS offers a "black-box," single-parameter approach for cost reduction while retaining near-canonical accuracy for ground and core-level calculations and enables new state-specific truncation strategies for excited-state and response properties.

1. Mathematical Basis and Construction of Frozen Natural Spinors

The FNS protocol begins with the formation of the virtual–virtual block of the correlated one-particle reduced density matrix (1RDM). This density matrix, often constructed from an unrelaxed MP2 (or higher-level) reference, is defined as

$$\rho_{ab} = \tfrac{1}{2} \sum_{ijc} \frac{\langle ac||ij \rangle \langle ij||bc \rangle}{(\varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c)(\varepsilon_i+\varepsilon_j-\varepsilon_b-\varepsilon_c)}$$

where $i,j$ run over occupied spinors, $a,b,c$ over virtuals, $\varepsilon_p$ are orbital energies, and $\langle pq||rs \rangle$ are antisymmetrized two-electron integrals (Chamoli et al., 24 Dec 2024, Chamoli et al., 2022, Chamoli et al., 7 Jun 2025).

The diagonalization

$\rho_{ab} V = V n$

yields the "natural spinors" (columns of $V$) and associated "occupation numbers" ($n_i$). After ordering $n_i$ in descending order, a user-specified threshold $\tau$ defines the set of retained (active) virtual spinors: $\text{FNS space} = \{ i \mid n_i \geq \tau \}$ Canonical occupied spinors are unaltered; all electron integrals and excitation amplitudes are subsequently formulated in this truncated (semi-canonicalized) FNS basis (Chamoli et al., 2022, Chamoli et al., 24 Dec 2024).

2. Algorithmic Implementation and Integration with Cholesky Decomposition

The FNS workflow is modular and scalable. Its integration with the Cholesky decomposition (CD) of two-electron integrals is critical for treating large relativistic systems:

1. Solve the reference Hartree–Fock problem (e.g., X2CAMF, 4c-DHF) to obtain canonical spinors and Fock matrix.
2. Form Cholesky-decomposed integrals $$(\mu\nu|\kappa\lambda) \approx \sum_{P} L_{\mu\nu}^P L_{\kappa\lambda}^P$$ (Chamoli et al., 24 Dec 2024, Chakraborty et al., 26 Nov 2025).
3. Construct and diagonalize the MP2 virtual–virtual 1RDM; select FNS via the occupation threshold.
4. Transform integrals and intermediates into the FNS basis, typically via $U = Z \widetilde{V}$, where $Z$ diagonalizes the Fock matrix in the truncated virtual space.
5. Execute all correlated post-HF routines using the compressed FNS/CD basis: CCSD, CCSD(T), EOM-CC variants, and ADC(n) (Chamoli et al., 24 Dec 2024, Chakraborty et al., 26 Nov 2025).

Employing FNS in conjunction with CD avoids the storage and computation bottlenecks of four-index integral tensors. For example, explicit storage of VVVV integrals is eliminated entirely, with only blocks such as OOVV and OOOV needed (Chamoli et al., 24 Dec 2024).

3. Accuracy, Cost Scaling, and Performance Benchmarks

The principal benefit of FNS is a dramatic reduction in the scaling pre-factor associated with the virtual space, with negligible impact on accuracy for ground-state energies, valence and core ionization potentials (IPs), and certain excited-state phenomena:

• CCSD and CCSD(T) scaling drops from $\mathcal{O}(N_o^2 N_v^4)$ to $\mathcal{O}(N_o^2 N_\text{kept}^4)$, where $N_\text{kept}$ is the number of FNS-retained virtuals (typically $30$–$50\%$ of the canonical space for thresholds $\tau=10^{-4.5}$–$10^{-5}$) (Chamoli et al., 2022, Chamoli et al., 24 Dec 2024).
• FNS truncation leads to wall-time reductions of $10$–$50\times$ and memory reductions from TB to GB scale for large systems, with mean absolute errors (MAEs) in energies and IPs below "chemical accuracy" ($<$0.2 kcal/mol for CCSD(T), $<$0.07 eV for IP-EOM-CCSD) (Chamoli et al., 24 Dec 2024, Chamoli et al., 7 Jun 2025).
• For double ionization potential (DIP) EOM-CC, FNS truncation with thresholds $n_\text{thresh}=10^{-4.5}$ yields errors $<0.05$ eV for DIP energies across noble gas and halide molecules, while lowering virtual space indices by factors of 2–3 and cost scaling to $\mathcal{O}(N^7)$ (Mukhopadhyay et al., 18 Sep 2025).

A representative summary of FNS thresholds and efficiency is provided below:

FNS Tier	Threshold $\tau$	Speedup vs canonical	Typical Energy Error
LOOSEFNS	$10^{-4}$	$10$–$40\times$	$\sim0.07$ eV (IP), $<0.2$ kcal/mol (CCSD(T))
NORMALFNS	$10^{-4.5}$	$15$–$50\times$	$\sim0.03$ eV (IP), $<0.15$ kcal/mol (CCSD(T))
TIGHTFNS	$10^{-5}$	$2$–$3\times$ less vs LOOSEFNS	$\sim0.02$ eV (IP)

FNS-based approaches maintain near-canonical accuracy for ground- and core-state calculations in both two- and four-component frameworks (Chamoli et al., 24 Dec 2024, Surjuse et al., 2022).

4. Limitations, Correction Strategies, and Property Dependence

While FNS is robust for total energies and IPs, it is less straightforward for certain differential or response properties:

• Ground-state energies and valence/core IPs show rapid and monotonic convergence with respect to FNS threshold; for most systems, $\tau=10^{-5}$ yields sub-0.05 eV IP error (Surjuse et al., 2022, Chamoli et al., 7 Jun 2025).
• For first-order properties (dipole moments, bond lengths), convergence can be oscillatory; a perturbative correction technique is established: $$E^{\text{corr}}_{\mathrm{CCSD(T)}} = E^{\mathrm{FNS}}_{\mathrm{CCSD(T)}} - \Delta E_{\mathrm{MP2}}$$ where $$\Delta E_{\mathrm{MP2}} = E^{\mathrm{FNS}}_{\mathrm{MP2}} - E^{\text{canonical}}_{\mathrm{MP2}}$$ (Chamoli et al., 2022, Chamoli et al., 24 Dec 2024).
• Linear response (polarizabilities, dynamic response) is poorly described by ground-state FNS because low-occupancy, diffuse spinors are essential for accuracy; to obtain $<1\%$ error in $\alpha$, standard FNS requires retention of $80-90\%$ of canonical virtuals (Chakraborty et al., 29 Mar 2025). A perturbation-sensitive scheme (FNS++), based on the perturbed density matrix, compacts the basis while achieving $<0.5\%$ error with only $30-40\%$ of spinors (Chakraborty et al., 29 Mar 2025).

For excited-state and attachment-type problems, especially those with substantial Rydberg or charge-transfer character, state-specific extensions to FNS are necessary for systematic convergence (Mukhopadhyay et al., 11 May 2025, Chakraborty et al., 26 Nov 2025).

5. State-Specific and Perturbation-Sensitive Extensions

Limitations of ground-state FNS for excited states and response properties motivate two key generalizations:

• FNS++ (Perturbation-Sensitive FNS): Constructs the truncation basis from the dipole-perturbed MP2 density matrix, producing a compact virtual space for accurate polarizabilities and linear responses. At an occupation threshold $n_{\text{cut}}=10^{-5}$, FNS++ can freeze $\sim70\%$ of virtuals while retaining $<0.5\%$ error in $\alpha$ (Chakraborty et al., 29 Mar 2025).
• State-Specific FNS (SS-FNS): For EE-EOM-CCSD, excitations, and attachment processes, a state-specific virtual–virtual density is constructed by augmenting the MP2 density with ADC(2)-based difference densities for each target state. Diagonalization yields a state-optimized FNS basis. SS-FNS truncates $\sim50\%$ of virtuals at thresholds $\lambda=10^{-4.5}$–$10^{-5}$, achieving sub-$0.02$ eV errors for excited-state energies (Mukhopadhyay et al., 11 May 2025, Chakraborty et al., 26 Nov 2025). Perturbative ADC(2)-based corrections further smooth convergence. For linear response and double ionization potentials, state-specific approaches are essential to avoid large errors or convergence irregularities.

6. Applications and Large-Scale Performance

FNS and its variants are now implemented for routine use in large-scale, relativistic electronic structure codes (e.g., DIRAC ExaCorr, custom X2CAMF/ADC implementations) enabling:

• Ionization and double-ionization energy calculations for systems with $>1000$ virtual spinors and basis sets up to 2600 functions (Chakraborty et al., 26 Nov 2025, Chamoli et al., 24 Dec 2024).
• Correlated EOM-CC and ADC(3) applications, with state-specific FNS producing balanced excitation energies, transition moments, and electron attachment energies at a fraction of the computational cost of canonical treatments (Mukhopadhyay et al., 18 Sep 2025, Chakraborty et al., 26 Nov 2025).
• Wall-time reductions from several days to hours for complex open-shell or heavy-element calculations (e.g., [I(H$_2$O)$_{12}$]$^-$, [UO$_2$(NO$_3)_3$]$^-$) (Chamoli et al., 24 Dec 2024, Chakraborty et al., 26 Nov 2025).

A typical medium-sized calculation (HI with aug-cc-pVTZ/dyall.acv3z) reduces from 434 to 174 virtuals after FNS, eliminates 2.1 TB of four-index integral storage, and achieves a $38\times$ speedup versus full 4c CCSD (Chamoli et al., 24 Dec 2024).

7. Practical Recommendations and Best Practices

• For black-box CCSD(T) and EOM-CCSD calculations, NORMALFNS ($\tau\sim10^{-4.5}$) is generally robust, balancing accuracy ($<0.2$ kcal/mol for CCSD(T), $<0.07$ eV for IPs) with order-of-magnitude cost reductions.
• For response properties, use FNS++ or SS-FNS, formed from the appropriate perturbed or state-specific density; standard FNS is inadequate for polarizabilities, excited-state, or dynamically delocalized properties (Chakraborty et al., 29 Mar 2025, Mukhopadhyay et al., 11 May 2025).
• Always monitor convergence with respect to threshold, especially for diffuse or near-degenerate target states.
• For high accuracy, a perturbative correction ($\Delta E_{\mathrm{MP2}}$ or analogous) is recommended; this typically restores sub-$\mathrm{kcal/mol}$ or sub-$0.01$ eV accuracy (Chamoli et al., 2022, Chamoli et al., 24 Dec 2024).
• Large-basis or heavy-element systems may require lower thresholds and/or tailored state-specific FNS construction to avoid missing essential virtual correlation (Mukhopadhyay et al., 18 Sep 2025, Chakraborty et al., 26 Nov 2025).

Frozen natural spinors, augmented with Cholesky integral decompositions and state-specific generalizations, define a state-of-the-art paradigm for scalable, controllably accurate relativistic electron-correlation methods in modern computational chemistry.