State-Specific Frozen Natural Spinor (SS-FNS)

• State-Specific Frozen Natural Spinor (SS-FNS) is a method that constructs a tailored virtual basis using state-specific density matrices, overcoming limitations of ground-state MP2 approaches.
• The technique integrates density construction, threshold-based eigenvector truncation, and semi-canonicalization within relativistic frameworks such as EOM-CCSD and ADC(3) using Dirac-Coulomb or X2CAMF Hamiltonians.
• Benchmark studies show SS-FNS can reduce virtual-space dimensions by up to 70% while keeping excitation energy errors within 0.02 eV, significantly enhancing computational efficiency and accuracy.

State-Specific Frozen Natural Spinor (SS-FNS) denotes a state-targeted virtual-space truncation scheme for relativistic correlated electronic-structure theory in which the virtual spinor space is generated separately for each target state from a state-specific density matrix, rather than once from a ground-state MP2 density. In the available relativistic formulations, SS-FNS is developed as an excited-state and electron-attachment adaptation of the frozen natural spinor idea for relativistic EOM-CCSD and ADC(3), using either the four-component Dirac-Coulomb Hamiltonian or the exact two-component atomic mean-field (X2CAMF) Hamiltonian, and is motivated by the observation that ground-state MP2-based frozen natural spinors can be unreliable for excited states because they “hardly have any information about the excited states” (Mukhopadhyay et al., 11 May 2025, Chakraborty et al., 26 Nov 2025).

1. Emergence from the frozen natural spinor framework

The immediate background to SS-FNS is the earlier relativistic frozen natural spinor (FNS) literature. A lower-scaling four-component relativistic coupled-cluster formulation introduced a ground-state, property-agnostic FNS-CCSD / FNS-CCSD(T) framework in which the virtual space is reduced using natural spinors derived from a relativistic MP2 density matrix, while the occupied spinors are kept as the original Dirac-Hartree-Fock occupied spinors (Chamoli et al., 2022). A subsequent four-component IP-EOM-CCSD implementation likewise used ground-state MP2 natural spinors to build a state-universal frozen-natural-spinor basis for ionized states, with core-ionized states treated by core-valence separation (CVS), and explicitly noted that extensions to excited- and electron-attached states would require state-specific FNS generated from a second-order approximate method (Surjuse et al., 2022).

This distinction is essential to the meaning of SS-FNS. Standard relativistic FNS compresses the virtual space once from the reference-state MP2 density and then reuses that truncated basis across the target manifold. SS-FNS instead modifies the density so that it reflects the electronic distribution of a chosen target root. A plausible implication is that SS-FNS should be understood not as a different parent many-body theory, but as a state-resolved basis-construction layer that sits on top of relativistic EOM-CCSD or ADC(3), replacing a one-size-fits-all MP2-derived virtual space by a target-adapted one (Mukhopadhyay et al., 11 May 2025).

2. State-specific density construction and spinor truncation

The formal construction begins from the standard FNS density. From the MP2 virtual-virtual density block,

$$D_{ab}=\frac{1}{2}\sum_{cij}\frac{\langle ac||ij\rangle \langle ij||bc\rangle}{\varepsilon_{ij}^{ac}\,\varepsilon_{ij}^{bc}},$$

one solves

$D_{ab}V = V\eta,$

retains only eigenvectors with occupation numbers above a threshold $\eta_{\text{crit}}$, projects the virtual Fock block,

$\tilde F=\tilde V^\dagger F\tilde V,$

semi-canonicalizes the retained space through

$\tilde F\tilde Z=\tilde Z\tilde\epsilon,$

and defines the final transformation as

$B=\tilde V\tilde Z.$

This is the basic frozen-natural-spinor machinery inherited by SS-FNS (Mukhopadhyay et al., 11 May 2025).

SS-FNS changes the density itself. For relativistic EE-EOM-CCSD, the defining relation is

$$D_{ab}^{\text{SS}(k)} = D_{ab}^{\text{MP2}} + D_{ab}^{\text{EE-ADC(2)}(k)},$$

and for closed-shell systems the data further specify

$D^{SS}_{ab} = D^{MP2}_{ab} + D^{EE-ADC(2)}_{ab},$

with

$$D^{MP2}_{ab} = \frac{1}{2}\left(D_{vv}+D_{vv}^{\dagger}\right), \qquad D_{vv} = \frac{1}{2}t_{in}^{ea*(1)}t_{in}^{eb(1)},$$

and

$$D^{EE-ADC(2)}_{ab} = r_i^{a*}r_i^b + \frac{1}{2}r_{ij}^{ca*}r_{ij}^{cb}.$$

Here $D_{ab}V = V\eta,$0 and $D_{ab}V = V\eta,$1 are the ADC(2) excited-state eigenvector amplitudes in the ISR formulation. The crucial consequence is that each excited state gets its own tailored truncated basis (Mukhopadhyay et al., 11 May 2025).

The same logic was generalized in relativistic ADC to electron attachment and excitation through

$D_{ab}V = V\eta,$2

The target-state ADC(2) density contribution is obtained from the zeroth-order ISR/ADC(2) eigenvectors for that root, and the diagonalization, threshold truncation, projection, and semi-canonicalization steps are then carried out separately for each target state. This state-by-state construction is the formal reason the method is “state-specific” (Chakraborty et al., 26 Nov 2025).

3. Embedding in relativistic EOM-CCSD and ADC(3)

In the relativistic EOM-CCSD realization, the coupled-cluster reference is

$D_{ab}V = V\eta,$3

with similarity-transformed Hamiltonian

$D_{ab}V = V\eta,$4

Excited states are represented as

$D_{ab}V = V\eta,$5

and obtained from

$D_{ab}V = V\eta,$6

Transition properties are evaluated in the standard biorthogonal EOM-CC framework,

$D_{ab}V = V\eta,$7

with

$D_{ab}V = V\eta,$8

and

$D_{ab}V = V\eta,$9

The paper emphasizes that because each excited state has its own tailored truncated basis, the state-specific excited-state wave function is naturally biorthogonal to the ground state in that basis, simplifying transition-property evaluation (Mukhopadhyay et al., 11 May 2025).

The relativistic Hamiltonian can be either the four-component Dirac-Coulomb Hamiltonian or X2CAMF. In the EOM-CCSD implementation, the X2CAMF Hamiltonian is introduced to avoid explicit construction of relativistic two-electron integrals and is combined with Cholesky decomposition to reduce memory and integral cost. The same Hamiltonian strategy appears in relativistic ADC up to third order, where the ISR/ADC secular problem,

$\eta_{\text{crit}}$0

is expanded as

$\eta_{\text{crit}}$1

with ADC(3) corresponding to truncation at third order, and the scaled-matrix variant

$\eta_{\text{crit}}$2

using $\eta_{\text{crit}}$3 as the default semiemperical choice. In the SS-FNS ADC realization, the reduced virtual basis is root-specific, so a root-specific Davidson solver and state-homing by overlap with canonical ADC(2) guesses are used (Chakraborty et al., 26 Nov 2025).

4. Convergence behavior, benchmarks, and threshold dependence

Benchmark data for relativistic SS-FNS-EE-EOM-CCSD show a consistent improvement over conventional MP2-based FNS. For the Zn atom, standard MP2-FNS converges only when about 70% of the virtual space is retained, whereas SS-FNS converges around 40% virtual space, and with perturbative correction around 30%. At threshold $\eta_{\text{crit}}$4, SS-FNS errors are below 0.02 eV for all four Zn states. For Ga$\eta_{\text{crit}}$5, the contrast is sharper: at threshold $\eta_{\text{crit}}$6, standard FNS gives 7.622 eV, while SS-FNS gives 13.121 eV, essentially matching the canonical value 13.112 eV with only 0.009 eV error; the perturbative correction changes it to 13.105 eV. For AuH, standard FNS converges slowly and needs very tight threshold ($\eta_{\text{crit}}$7), whereas SS-FNS converges much faster, around $\eta_{\text{crit}}$8, and with correction convergence is good already near $\eta_{\text{crit}}$9. At threshold $\tilde F=\tilde V^\dagger F\tilde V,$0, SS-FNS uses 132 of 382 virtual spinors and yields a much smaller error than standard FNS, which selects 126 virtual spinors but still has a much larger error of about 0.54 eV (Mukhopadhyay et al., 11 May 2025).

The same study reports that for I$\tilde F=\tilde V^\dagger F\tilde V,$1, SS-FNS excitation energies show typical deviations of about 0.02–0.04 eV from canonical 4c-DC EOM-CCSD results, with perturbative correction improving the mean absolute error from 0.03 to 0.02 eV. For the first four bright Rydberg states of Xe, SS-FNS truncates about 70% of virtuals in nearly all cases while keeping excitation energies within 0.02 eV of canonical values and preserving the ordering of the brightest and weakest transitions. For Ga$\tilde F=\tilde V^\dagger F\tilde V,$2, In$\tilde F=\tilde V^\dagger F\tilde V,$3, and Tl$\tilde F=\tilde V^\dagger F\tilde V,$4, SS-FNS excitation energies agree with canonical values within about 0.01 eV even with less than 40% of virtual space retained, and the X2CAMF-SS-FNS-EE-EOM-CCSD results typically differ from the 4c-DC version by 0.00–0.01 eV (Mukhopadhyay et al., 11 May 2025).

In relativistic ADC, the production thresholds are channel-dependent. For IP, convergence in IBr suggests $\tilde F=\tilde V^\dagger F\tilde V,$5 with CD threshold $\tilde F=\tilde V^\dagger F\tilde V,$6, whereas for EA and EE the authors choose $\tilde F=\tilde V^\dagger F\tilde V,$7 after testing AuH and IBr. At that threshold, the SS-FNS basis can remove roughly 65% of the virtual space in AuH while still reproducing canonical excitation energies to within $\tilde F=\tilde V^\dagger F\tilde V,$8 eV after correction, whereas standard FNS at similar truncation gave much larger errors. For I$\tilde F=\tilde V^\dagger F\tilde V,$9, the uncorrected SS-FNS-EE-ADC(3) gives a MAD of 0.0128 eV, improving to 0.0100 eV with correction, while the RMSD drops from 0.0251 to 0.0203 eV. For Xe excited states, SS-FNS-EE-ADC(3) excitation energies deviate from 4c by less than 0.03 eV and transition dipole moments by less than 0.005 a.u.. For Ga$\tilde F\tilde Z=\tilde Z\tilde\epsilon,$0, In$\tilde F\tilde Z=\tilde Z\tilde\epsilon,$1, and Tl$\tilde F\tilde Z=\tilde Z\tilde\epsilon,$2, SS-FNS-EE-ADC(3) differs from 4c by about 0.01–0.02 eV, and fine-structure splittings are reproduced to well below 0.01 eV (Chakraborty et al., 26 Nov 2025).

5. Perturbative correction and computational role

A defining practical feature of SS-FNS is that the occupation threshold $\tilde F\tilde Z=\tilde Z\tilde\epsilon,$3 is the control parameter for the virtual-space reduction, and the truncation error can be reduced by a low-order correction. In the relativistic EE-EOM-CCSD formulation, the correction is

$\tilde F\tilde Z=\tilde Z\tilde\epsilon,$4

which is used as a perturbative estimate of the basis-truncation error. The study describes the threshold as more meaningful than merely fixing a percentage of orbitals and adopts a conservative threshold of $\tilde F\tilde Z=\tilde Z\tilde\epsilon,$5 for subsequent calculations (Mukhopadhyay et al., 11 May 2025).

The same idea is carried into relativistic ADC through

$\tilde F\tilde Z=\tilde Z\tilde\epsilon,$6

This is explicitly a basis-set-truncation correction rather than a higher-order physical correction to ADC(3) itself. The computational gains arise from a smaller truncated virtual space together with reduced cost in integral transformations and tensor contractions, especially when SS-FNS is combined with X2CAMF and Cholesky decomposition. For IBr, SS-FNS/FNS calculations reduce wall time dramatically versus canonical ADC(3): about 6× for IP, 15× for EA, and 10× for EE. The same implementation reports feasibility for larger systems, with the largest system successfully treated comprising more than 2600 basis functions (Chakraborty et al., 26 Nov 2025).

The main tradeoff is that each excited or attached state needs its own ADC(2) density and its own CC/EOM or ADC calculation. The EOM-CCSD paper explicitly notes this extra state-by-state overhead, but also argues that it is more than compensated by the reduction in virtual-space size and the improved state-specific accuracy (Mukhopadhyay et al., 11 May 2025).

6. Scope, related methods, and common misidentifications

SS-FNS should not be conflated with the earlier relativistic FNS literature. The four-component FNS-CCSD / FNS-CCSD(T) method is a ground-state, property-agnostic truncation strategy, and the four-component FNS-IP-EOM-CCSD method is a ground-state-MP2-generated, state-universal basis for ionized states; neither introduces a formal SS-FNS protocol. Likewise, the relativistic DIP-ADC(3) work based on X2CAMF, Cholesky decomposition, and FNS uses a ground-state MP2 density and explicitly states that it does not introduce a separate state-specific FNS protocol for DIP-ADC(3) (Chamoli et al., 2022, Surjuse et al., 2022, Mandal et al., 26 Aug 2025).

SS-FNS is also closely related to state-specific frozen natural orbitals (SS-FNO), but the two are not identical. In the nonrelativistic electron-attachment context, SS-FNO is defined from

$\tilde F\tilde Z=\tilde Z\tilde\epsilon,$7

followed by the same sequence of diagonalization, thresholding, and semi-canonicalization. The relationship is conceptual and methodological: SS-FNO and SS-FNS belong to the same family, differing mainly by whether the one-electron basis is orbital-based or spinor-based and by the underlying Hamiltonian (Mukhopadhyay et al., 8 Nov 2025).

A separate possible misidentification arises from spinor Bose-Einstein-condensate literature, where “frozen” and “spinor” occur in an entirely different sense. The work on spin-glass-like behavior in spin turbulence of spinor BECs introduces a spin-glass order parameter

$\tilde F\tilde Z=\tilde Z\tilde\epsilon,$8

to quantify a spin texture that is spatially disordered but temporally frozen. That paper is relevant only as an analogy for frozen spin structure; it does not discuss frozen natural spinors, many-body electronic structure, or state-specific virtual-space truncation (Tsubota et al., 2013).

SS-FNS therefore occupies a specific methodological position: it is the relativistic spinor analogue of state-specific frozen natural orbital compression, developed to correct the failure mode of ground-state MP2-FNS for excitation and attachment problems by using ADC(2)-derived, root-resolved densities to define the truncated virtual space (Mukhopadhyay et al., 11 May 2025, Chakraborty et al., 26 Nov 2025).