Frozen Natural Spinors (FNS)

• Frozen natural spinors are a virtual-space truncation scheme that uses eigenfunctions of a correlated one-particle density matrix to reduce the size of the virtual manifold in post-Hartree–Fock calculations.
• They maintain canonical occupied spinors while replacing the full canonical virtual space with a compact, MP2-based natural spinor basis, significantly lowering computational scaling in methods like CC and EOM-CC.
• Extensions such as SS-FNS and FNS++ refine the method for excited-state and response properties by incorporating state-specific and perturbation-sensitive densities, thereby enhancing accuracy.

Frozen natural spinors (FNS) are a relativistic virtual-space truncation scheme used in correlated electronic-structure theory to reduce the cost of four-component and two-component post-Hartree–Fock calculations. In the standard formulation, the occupied spinors remain the canonical SCF or DHF occupied spinors, while the virtual spinor manifold is replaced by natural spinors obtained from a correlated one-particle reduced density matrix and then truncated according to occupation numbers. In the cited relativistic literature, FNS is the relativistic analogue of frozen natural orbitals, is usually generated from an MP2 virtual–virtual density, and serves as a central cost-reduction layer in CC, EOM-CC, ADC, and related methods for heavy-element systems (Chamoli et al., 2022, Mandal et al., 26 Aug 2025, Chamoli et al., 2024).

1. Definition and conceptual domain

Natural spinors are the relativistic analogue of Löwdin natural orbitals: they are eigenfunctions of a correlated one-body reduced density matrix expressed in a spinor basis. In the standard FNS construction used across the cited papers, only the virtual space is transformed and truncated; the occupied space is kept fixed at the DHF, SCF, or X2CAMF-HF level. This is why the method is called frozen natural spinors: low-occupation virtual natural spinors are frozen out, whereas the occupied spinors are not naturalized by the FNS step itself (Chamoli et al., 2022, Surjuse et al., 2022, Mandal et al., 26 Aug 2025).

This definition also fixes two common terminological boundaries. First, FNS is not a frozen-core approximation. Frozen core removes selected occupied core orbitals from correlation treatment; FNS truncates the virtual space on the basis of correlated occupation numbers. Second, standard FNS in its most common relativistic form is a ground-state MP2-based compression, not a target-state-specific density unless an explicit extension such as SS-FNS or FNS++ is introduced (Chamoli et al., 2022, Mandal et al., 26 Aug 2025, Mukhopadhyay et al., 11 May 2025).

The computational motivation is explicit throughout the literature. Relativistic correlated calculations are more expensive because spin symmetry is generally lost, complex arithmetic is required, and relativistic basis sets are often uncontracted; the dominant CC and EOM-CC contractions depend steeply on the number of virtual spinors. For example, the four-component CCSD and CCSD(T) implementations discussed in the cited work scale as $O(N_O^2N_V^4)$ and $O(N_O^3N_V^4)$, respectively, so reducing $N_V$ is immediately consequential (Chamoli et al., 2022). The same virtual-space sensitivity appears in relativistic IP-EOM-CCSD, DIP-ADC(3), DIP-EOMCC, and qUCC-based workflows (Surjuse et al., 2022, Mandal et al., 26 Aug 2025, Mukhopadhyay et al., 18 Sep 2025, Majee et al., 3 Sep 2025).

Variant	Density source	Intended use
Standard FNS	Ground-state MP2 virtual–virtual 1-RDM	CC, IP-EOM-CC, DIP-ADC, DIP-EOMCC, qUCC
SS-FNS	MP2 density augmented by state-specific ADC(2) density	EE/EA ADC and EE-EOM-CCSD
FNS++	Perturbation-sensitive second-order density	LR-CCSD polarizabilities

2. Standard mathematical construction

A representative relativistic FNS construction begins from the virtual–virtual block of the MP2 one-particle reduced density matrix. In the DIP-ADC(3) implementation, this block is written as (Mandal et al., 26 Aug 2025)

$$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$

The natural spinors are then obtained by diagonalizing the virtual-space density,

$D_{ab}V = Vn,$

where $V$ contains the virtual natural spinors and $n$ is the diagonal matrix of occupation numbers. Truncation is controlled by an occupation threshold: virtual natural spinors with occupations below the threshold are discarded, yielding a reduced virtual space $\tilde V$ (Mandal et al., 26 Aug 2025, Chamoli et al., 2024).

After truncation, the retained virtual space is semi-canonicalized by projecting the virtual–virtual Fock block into the truncated natural-spinor space,

$\tilde F_{VV}=\tilde V^\dagger F_{VV}\tilde V,$

and diagonalizing

$\tilde F_{VV}\tilde Z=\tilde Z\tilde\varepsilon.$

One explicit form of the final transformation used in relativistic ADC is

$O(N_O^3N_V^4)$0

The working basis is therefore composed of canonical occupied spinors plus semi-canonical truncated virtual natural spinors (Mandal et al., 26 Aug 2025). Closely related four-component and X2CAMF implementations use the same sequence: build the MP2 virtual density, diagonalize, threshold by occupation, and semi-canonicalize the retained virtuals (Chamoli et al., 2022, Chamoli et al., 2024, Thapa et al., 25 Feb 2026).

The methodological significance of this construction is that it ranks virtual spinors by correlated importance rather than by canonical energy. Several papers emphasize that naive removal of high-energy canonical virtuals is inefficient, especially for ionized and doubly ionized states where orbital relaxation and higher excitation sectors such as $O(N_O^3N_V^4)$1 or $O(N_O^3N_V^4)$2 are important (Mandal et al., 26 Aug 2025, Mukhopadhyay et al., 18 Sep 2025).

3. Embedding in relativistic many-body workflows

FNS has been implemented in both four-component no-pair Dirac–Coulomb frameworks and exact-two-component frameworks such as X2CAMF and X2CMP. In four-component CC and EOM-CC, the reference spinors are positive-energy DHF spinors under the no-pair approximation; in the X2CAMF implementations, the spinors come from X2CAMF-HF, after which FNS acts as an orbital-space reduction layer on top of the relativistic spinor basis (Chamoli et al., 2022, Surjuse et al., 2022, Chamoli et al., 2024).

The integration of FNS with Cholesky decomposition (CD) is a recurring implementation pattern. In the relativistic DIP-ADC(3) workflow, X2CAMF-HF spinors are generated first, then LOO and LOV Cholesky vectors are generated prior to the construction of the FNS basis and stored, the FNS basis is constructed from ground-state MP2 amplitudes, and LVV-type Cholesky vectors are generated only in the FNS basis. The same section states that generating LVV vectors directly in the reduced FNS space both speeds up the integral transformation step and greatly reduces memory usage (Mandal et al., 26 Aug 2025). The X2CAMF-CC implementation describes an analogous design: only LOO and LOV three-center quantities are built in the canonical basis, while virtual-heavy quantities are formed directly in the FNS basis, and $O(N_O^3N_V^4)$3 and $O(N_O^3N_V^4)$4 are generated on the fly rather than stored (Chamoli et al., 2024).

This leads to a characteristic three-part efficiency strategy. X2CAMF reduces the cost of the relativistic Hamiltonian by replacing explicit molecular relativistic two-electron integrals with an effective one-electron spin–orbit term plus standard nonrelativistic two-electron integrals; CD factorizes the latter and reduces integral storage; FNS reduces the size of the virtual spinor manifold entering the correlated treatment. The same combined philosophy is used in relativistic CC, EOM-IP-CC, DIP-ADC(3), qUCCSD[T], and triples-corrected EOM-IP-CC work (Mandal et al., 26 Aug 2025, Chamoli et al., 2024, Majee et al., 3 Sep 2025, Thapa et al., 25 Feb 2026).

Operationally, FNS is applied before the expensive correlated steps. In the cited implementations, the reduced space is used for CCSD amplitudes, perturbative triples intermediates, Davidson diagonalization of EOM or ADC matrices, and response equations. This does not generally change the formal exponent of the asymptotic scaling, but it reduces the effective virtual dimension that enters the dominant $O(N_O^3N_V^4)$5- or $O(N_O^3N_V^4)$6-dependent contractions (Chamoli et al., 2022, Surjuse et al., 2022, Mukhopadhyay et al., 18 Sep 2025).

4. Thresholds, convergence, and quantitative performance

A notable practical feature of FNS is threshold-based accuracy control. Several papers describe the method as controlled by a single occupation threshold, although the recommended value is method- and observable-dependent (Chamoli et al., 2022, Surjuse et al., 2022). In four-component IP-EOM-CCSD, the recommended thresholds are $O(N_O^3N_V^4)$7 for valence ionization potentials and $O(N_O^3N_V^4)$8 for core ionization energies (Surjuse et al., 2022). In relativistic DIP-ADC(3), the production threshold is $O(N_O^3N_V^4)$9, chosen from a Cl$N_V$0 benchmark where the FNS-DIP-ADC(3) results converge closely to the canonical values at that threshold (Mandal et al., 26 Aug 2025). In four-component DIP-EOMCC with $N_V$1 sectors, $N_V$2 is likewise adopted as the default production threshold; in the Cl$N_V$3 test, $N_V$4 gives an error below $N_V$5 eV, $N_V$6 gives about $N_V$7 eV error, and $N_V$8 is virtually identical to the untruncated calculation (Mukhopadhyay et al., 18 Sep 2025). In triples-corrected relativistic EOM-IP-CC, the benchmark on HCl/dyall.v3z leads to a recommended FNS threshold of $N_V$9 (Thapa et al., 25 Feb 2026).

The most explicit demonstration of the superiority of FNS over canonical virtual truncation in a relativistic double-ionization problem comes from the X2CAMF-DIP-ADC(3) study. For the lowest vertical DIP of Cl$$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$0 in dyall.av4z, retaining only $$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$1 of the frozen virtual spinors in the FNS basis yields an absolute error of less than $$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$2 eV, whereas the error at the same $$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$3 retention in the canonical basis is approximately $$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$4 eV. The same study states that convergence is achieved with only $$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$5 of the virtual spinors retained in the FNS basis, while canonical truncation requires $$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$6 (Mandal et al., 26 Aug 2025). In the corresponding four-component DIP-EOMCC study, the FNS basis reaches canonical-quality accuracy with about $$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$7 of the virtual space, whereas canonical truncation requires about $$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$8 (Mukhopadhyay et al., 18 Sep 2025).

Production calculations using FNS also preserve high-level relativistic accuracy. In X2CAMF-DIP-ADC(3), the efficient FNS-CD-X2CAMF implementation reproduces 4c-FNS reference DIPs for Kr, Xe, Cl$$D_{ab} = \frac{1}{2} \sum_{ijc} \frac{\langle ac || ij \rangle \langle ij || bc \rangle} {\varepsilon^{ij}_{ac}\,\varepsilon^{ij}_{bc}}, \qquad \varepsilon^{ij}_{ac} = \varepsilon_i+\varepsilon_j-\varepsilon_a-\varepsilon_c.$$9, and HBr with error not exceeding $D_{ab}V = Vn,$0 eV (Mandal et al., 26 Aug 2025). In the four-component valence IP benchmark, a threshold of $D_{ab}V = Vn,$1 gives outer-valence IPs for HF, HCl, HBr, HI, and HAt in excellent agreement with experiment, with maximum reported error below $D_{ab}V = Vn,$2 (Surjuse et al., 2022).

The cost reduction can be large. One timing example for HI reports a canonical 4c calculation with 18 occupied and 474 virtual spinors taking 7 days, 3 hours, 47 minutes, and 15 seconds; the corresponding 4c-FNS calculation retains 140 virtual spinors and takes 7 hours, 15 minutes, and 51 seconds; the CD-X2CAMF-FNS version takes 1 hour and 55 seconds (Thapa et al., 25 Feb 2026). Another HI IP-EOM-CCSD example reduces the virtual space from 462 to 256 and yields a nearly sixfold wall-time speedup (Surjuse et al., 2022). In X2CAMF-CC for HI, the FNS-CD implementation is reported as about 11 times faster than four-component FNS-CCSD and about 38 times faster than canonical four-component CCSD (Chamoli et al., 2024).

5. Extensions beyond standard MP2-FNS

The standard MP2-based FNS construction is explicitly ground-state-based, and several later papers show that this is not always sufficient. For excited states and electron-attachment problems, the relevant virtual manifold can differ strongly from the one favored by the ground-state MP2 density. This observation led to state-specific frozen natural spinors (SS-FNS) in relativistic EE-EOM-CCSD and ADC, where the virtual density is augmented by a target-state ADC(2) contribution. One defining expression is

$D_{ab}V = Vn,$3

The purpose is to generate a separate compact virtual basis for each target root rather than reuse a single ground-state MP2-FNS space (Mukhopadhyay et al., 11 May 2025, Chakraborty et al., 26 Nov 2025).

The quantitative effect can be decisive. In Ga$D_{ab}V = Vn,$4, for the $D_{ab}V = Vn,$5 excited state at threshold $D_{ab}V = Vn,$6, canonical 4c-DC-EE-EOM-CCSD gives $D_{ab}V = Vn,$7 eV, standard 4c-DC-FNS-EE-EOM-CCSD gives $D_{ab}V = Vn,$8 eV, and 4c-DC-SS-FNS-EE-EOM-CCSD gives $D_{ab}V = Vn,$9 eV; the corrected SS-FNS value is $V$0 eV (Mukhopadhyay et al., 11 May 2025). In the relativistic ADC(3) work, standard MP2-FNS is described as not well suited for excited states and electron attachment, whereas SS-FNS reduces AuH excitation-energy errors from $V$1 eV to $V$2 eV at threshold $V$3 while keeping a similar number of retained virtuals, and gives electron-affinity errors below $V$4 eV already at $V$5 for IBr (Chakraborty et al., 26 Nov 2025).

Response properties motivate a different extension, FNS++, because conventional FNS is reported to be unsuitable for linear-response properties such as static and dynamic polarizabilities. FNS++ replaces the ground-state MP2 density with a perturbation-sensitive second-order density built from approximate first-order perturbed amplitudes, for example

$V$6

This change is explicitly motivated by the observation that diffuse virtuals can have very small ground-state occupations and still be crucial for response (Chakraborty et al., 29 Mar 2025, Chakraborty et al., 14 Apr 2026).

The response benchmarks are correspondingly sharp. In Zn and HBr, standard FNS converges very poorly for polarizabilities, whereas FNS++ converges rapidly. One paper states that standard FNS can still have errors exceeding 20 a.u. for Zn at threshold $V$7, while FNS++ at the same threshold yields error below 2.5 a.u.; standard FNS may require a threshold as tight as $V$8, whereas FNS++ works well already around $V$9 (Chakraborty et al., 14 Apr 2026). The same work reports that, with the adopted averaged-density FNS++ construction, about $n$0 of the virtual spinor space is removed on average while preserving high accuracy for static and dynamic polarizabilities (Chakraborty et al., 14 Apr 2026). In the four-component LRCCSD implementation, the abstract states that nearly $n$1 of the total virtual spinors can be truncated while maintaining excellent accuracy (Chakraborty et al., 29 Mar 2025).

These extensions suggest a useful classification. Standard MP2-FNS is well matched to ground-state correlation and many ionization problems; SS-FNS is introduced when the target state itself should influence the virtual-space ranking; FNS++ is introduced when the property depends on a perturbation-sensitive response density rather than the unperturbed ground-state density.

6. Limitations, caveats, and terminological boundaries

The cited literature is explicit that FNS is not harmless at arbitrary cutoffs. Threshold choice matters, and different problems require different settings. The four-component CC paper recommends $n$2 for uncorrected FNS-CCSD/FNS-CCSD(T) correlation energies but $n$3 with MP2 correction as a practical default; the qUCCSD[T] paper recommends LOOSEFNS ($n$4), NORMALFNS ($n$5), and TIGHTFNS ($n$6), with TIGHTFNS specifically recommended for finite-field properties (Chamoli et al., 2022, Majee et al., 3 Sep 2025). This suggests that FNS should be viewed as systematically improvable but not universally threshold-independent.

A second caveat is observable dependence. The ground-state MP2 ranking that works well for energies and many ionization problems can be suboptimal for properties, particularly response properties and states with diffuse character. This is why uncorrected FNS may show non-smooth convergence for dipoles, bond lengths, and frequencies in four-component CC, and why FNS++ or SS-FNS were developed for polarizabilities and excited-state problems, respectively (Chamoli et al., 2022, Chakraborty et al., 29 Mar 2025, Chakraborty et al., 14 Apr 2026, Mukhopadhyay et al., 11 May 2025).

A third boundary concerns terminology outside relativistic quantum chemistry. The phrase frozen natural spinor in the arXiv literature summarized here refers to a virtual-space truncation scheme for electronic-structure calculations. It does not refer to the ultracold-atom regime where “magnetization is dynamically frozen” in spinor Bose gases, nor to spin-turbulent states in which spin density vectors are “spatially random but temporally frozen.” Those are distinct concepts from spinor-condensate physics rather than orbital-compression methods in relativistic many-body theory (Pasquiou et al., 2011, Tsubota et al., 2013).

In present arXiv usage within relativistic electronic structure, FNS therefore denotes a family of controlled virtual-space compression strategies built around correlated spinor densities. The baseline form is MP2-based and ground-state-oriented; the later SS-FNS and FNS++ variants modify the density source when excited-state or response-specific virtual spaces are needed. Across CC, EOM-CC, ADC, and qUCC implementations, the common invariant is the same: retain canonical occupied spinors, replace the full canonical virtual manifold by a more compact correlated natural-spinor basis, and freeze the low-occupation part of that virtual space (Chamoli et al., 2022, Mandal et al., 26 Aug 2025, Mukhopadhyay et al., 11 May 2025, Chakraborty et al., 14 Apr 2026).