---
title: State-Specific Frozen Natural Spinor (SS-FNS)
url: https://www.emergentmind.com/topics/state-specific-frozen-natural-spinor-ss-fns
type: topic
---

# State-Specific Frozen Natural Spinor (SS-FNS)

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” [2505.07080; 2511.21640].

## 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 [2201.07752]. 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 [2207.08627].

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 [2505.07080].

## 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 [2505.07080].

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 $r_i^a$ and $r_{ij}^{ab}$ 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 [2505.07080].

The same logic was generalized in relativistic ADC to electron attachment and excitation through
$$
D_{ab}^{\text{SS}(k)} = D_{ab}^{\text{MP2}} + D_{ab}^{\text{EA/EE-ADC(2)}(k)}.
$$
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” [2511.21640].

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

In the relativistic EOM-CCSD realization, the coupled-cluster reference is
$$
\left| \psi_{\text{CC}} \right\rangle = e^{\hat T}\left| \phi_0 \right\rangle, \qquad \hat T = \hat T_1+\hat T_2,
$$
with similarity-transformed Hamiltonian
$$
\bar{H} = e^{-\hat T}\hat H e^{\hat T}.
$$
Excited states are represented as
$$
\left| \psi_k \right\rangle = \hat R_k e^{\hat T}\left| \phi_0 \right\rangle,
$$
and obtained from
$$
[\bar H, \hat R_k]\left| \phi_0 \right\rangle = \omega_k \hat R_k \left| \phi_0 \right\rangle, \qquad \omega_k = E_k - E_0.
$$
Transition properties are evaluated in the standard biorthogonal EOM-CC framework,
$$
\langle \phi_0| \hat L_i \hat R_j |\phi_0\rangle = \delta_{ij},
$$
with
$$
\left| \mu_{0\to k} \right|^2 = \langle \phi_0 | (1+\hat\Lambda)\bar{\mu}\hat R_k|\phi_0\rangle \langle \phi_0 | \hat L_k \bar{\mu}|\phi_0\rangle,
$$
and
$$
f_{0\to k} = \frac{2}{3}\Delta\omega_k \left|\mu_{0\to k}\right|^2.
$$
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 [2505.07080].

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,
$$
\mathbf M\mathbf X = \mathbf X\boldsymbol\Omega, \qquad \mathbf X^\dagger\mathbf X=\mathbf 1,
$$
is expanded as
$$
\mathbf M = \mathbf M^{(0)}+\mathbf M^{(1)}+\mathbf M^{(2)}+\mathbf M^{(3)}+\cdots,
$$
with ADC(3) corresponding to truncation at third order, and the scaled-matrix variant
$$
\mathbf M_{\mathrm{sm-ADC}[(2)+x(3)]} = \mathbf M^{(0)}+\mathbf M^{(1)}+\mathbf M^{(2)}+x\,\mathbf M^{(3)},
$$
using $x=0.5$ 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 [2511.21640].

## 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 $10^{-4}$, SS-FNS errors are below **0.02 eV** for all four Zn states. For Ga$^+$, the contrast is sharper: at threshold $10^{-4.5}$, 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 ($10^{-7}$), whereas SS-FNS converges much faster, around $10^{-4}$, and with correction convergence is good already near $10^{-3}$. At threshold $10^{-4.5}$, 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** [2505.07080].

The same study reports that for I$_3^-$, 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$^+$, In$^+$, and Tl$^+$, 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** [2505.07080].

In relativistic ADC, the production thresholds are channel-dependent. For IP, convergence in IBr suggests $\eta_{\text{crit}}=10^{-4}$ with CD threshold $10^{-3}$, whereas for EA and EE the authors choose $\eta_{\text{crit}}=10^{-4.5}$ 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 $\sim 0.01$ eV after correction, whereas standard FNS at similar truncation gave much larger errors. For I$_3^-$, 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$^+$, In$^+$, and Tl$^+$, 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** [2511.21640].

## 5. Perturbative correction and computational role

A defining practical feature of SS-FNS is that the occupation threshold $\eta_{\text{crit}}$ 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
$$
\omega_{\mathrm{SS\text{-}FNS\text{-}EOM}}^{\mathrm{corrected}(k)}
=
\omega_{\mathrm{SS\text{-}FNS\text{-}EOM}}^{\mathrm{uncorrected}(k)}
+
\omega_{\mathrm{EE\text{-}ADC(2)}}^{\mathrm{canonical}(k)}
-
\omega_{\mathrm{EE\text{-}ADC(2)}}^{\mathrm{SS\text{-}FNS}(k)},
$$
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 $10^{-4.5}$ for subsequent calculations [2505.07080].

The same idea is carried into relativistic ADC through
$$
\omega_{\mathrm{SS\text{-}FNS\text{-}ADC(3)}}^{\mathrm{corrected}(k)}
=
\omega_{\mathrm{SS\text{-}FNS\text{-}ADC(3)}}^{\mathrm{uncorrected}(k)}
+
\omega_{\mathrm{ADC(2)}}^{\mathrm{canonical}(k)}
-
\omega_{\mathrm{ADC(2)}}^{\mathrm{SS\text{-}FNS}(k)}.
$$
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** [2511.21640].

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 [2505.07080].

## 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) [2201.07752; 2207.08627; 2508.19012].

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
$$
\mathbf{D}_{ab}^{SS}(k)=\mathbf{D}_{ab}^{MP2}+\mathbf{D}_{ab}^{EA\text{-}ADC(2)}(k),
$$
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 [2511.05897].

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
$$
q(t)=[\langle\hat{ \bm{s} (\bm{r},t)\rangle_T^2]
$$
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 [1304.4674].

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 [2505.07080; 2511.21640].

Source: https://www.emergentmind.com/topics/state-specific-frozen-natural-spinor-ss-fns