Noniterative Finite Amplitude Method (FAM)

• Noniterative FAM is a formulation that constructs QRPA matrices explicitly (matrix FAM) from finite-amplitude probes, bypassing the iterative frequency-by-frequency solution.
• It relies on finite-difference evaluations or analytic linearization of the induced fields to generate residual interactions within nuclear density functional theory.
• Applications in giant resonances, spin-flip M1 transitions, and inelastic neutron scattering demonstrate its balance of direct matrix construction and computational efficiency.

The noniterative finite amplitude method (FAM) denotes a family of linear-response formulations in nuclear density functional theory in which the residual interaction is generated from infinitesimal variations of the self-consistent mean field, but the unknown amplitudes are not obtained by the standard frequency-by-frequency iterative FAM loop. The terminology is not fully uniform across the literature. In some papers, the noniterative route is identified with matrix FAM (m-FAM), where explicit RPA or QRPA matrices are reconstructed from finite-amplitude probes and then solved by matrix methods; in related work, contour integration is used to extract discrete QRPA poles without explicit matrix construction, although the FAM equations at sampled complex frequencies remain iterative (Nakatsukasa, 2014, Avogadro et al., 2013, Hinohara et al., 2013, Sasaki et al., 2022).

1. Terminology, scope, and historical setting

FAM was introduced as a way to perform self-consistent linear-response calculations without explicitly constructing the full residual-interaction matrix. In the superfluid case, it allowed an HFB code to be transformed into a QRPA code with simple modifications, while avoiding explicit construction of the full QRPA matrix $A,B$ (Avogadro et al., 2011). A later review organized the field around three distinct formulations: the standard iterative FAM (i-FAM) for response functions, matrix FAM (m-FAM) for explicit QRPA-matrix construction, and a contour-integral method for extracting discrete modes from complex-frequency response (Nakatsukasa, 2014).

In that review, m-FAM is described as the closest method to a noniterative FAM, because it does not iteratively solve the FAM response equation at each frequency; instead, it builds the QRPA matrices and then diagonalizes them. In covariant density functional theory, the same distinction appears explicitly as iterative FAM versus matrix FAM, and the latter is the noniterative counterpart in the sense that the response amplitudes are not obtained through iterative linear solves (Liang et al., 2013). Later Skyrme-based papers adopted the phrase non-iterative FAM directly for explicit matrix formulations of RPA and QRPA (Sasaki et al., 2022, Sasaki et al., 2022).

Formulation	Literature description	Operational feature
i-FAM	standard iterative FAM	solves $({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$ at chosen $\omega$
m-FAM	matrix FAM; closest thing to noniterative FAM	builds explicit $A$ and $B$ from finite-amplitude probes, then solves the matrix problem
contour-integral FAM	eigenmode-extraction extension of i-FAM	integrates $S(F;\omega)$, $X(\omega)$, and $Y(\omega)$ around poles

This taxonomy is essential because a substantial part of the FAM literature is matrix-free but still iterative. A method can avoid explicit QRPA-matrix construction without being noniterative in the solver sense.

2. Standard iterative FAM as the baseline

The reference point for any noniterative formulation is the standard FAM response equation. In QRPA form, the unknown forward and backward amplitudes satisfy

$$(E_\mu+E_\nu-\omega)X_{\mu\nu}(\omega)+\delta H^{20}_{\mu\nu}(\omega)=-F^{20}_{\mu\nu},$$

$$(E_\mu+E_\nu+\omega)Y_{\mu\nu}(\omega)+\delta H^{02}_{\mu\nu}(\omega)=-F^{02}_{\mu\nu},$$

and the induced residual field is generated from perturbed densities rather than from explicitly stored QRPA matrices (Avogadro et al., 2011, Nakatsukasa, 2014).

In the original FAM logic, the induced field is evaluated numerically by a finite difference of the self-consistent Hamiltonian,

$({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$0

or, in the commonly used schematic form,

$({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$1

This is the defining matrix-free step: the code needs only the action of the residual kernel on a trial response, not the four-index tensors $({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$2 and $({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$3 (Nakatsukasa, 2014).

Standard i-FAM then solves the linear system

$({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$4

iteratively at each chosen complex frequency $({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$5, and constructs the response function

$({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$6

That structure is explicit in the review literature and in later large-scale applications (Nakatsukasa, 2014).

A recurrent source of confusion is that some implementations replace numerical finite differences by explicit linearization of the induced fields, yet remain fully iterative. In the deformed Skyrme application to giant dipole resonances, the induced fields are obtained by explicit linearization rather than a literal finite-difference quotient, and “thanks to this explicit linearization, the infinitesimal parameter $({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$7 is no longer needed”; however, the amplitudes are still solved iteratively, and the paper explicitly states that “the QRPA linear response problem is solved iteratively” and that “for an iterative solution of the FAM amplitudes, the Broyden method was utilized” (Oishi et al., 2015). Explicit linearization of residual fields therefore does not, by itself, define a noniterative FAM.

3. Matrix FAM and direct construction of $({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$8 and $({\cal H}-\omega{\cal N})\vec Z(\omega)=-{\cal F}$9

The canonical noniterative route is matrix FAM. The basic idea is to use the finite-amplitude machinery not to solve the response directly, but to construct the QRPA matrix column by column. In the review formulation, one defines a forward unit vector $\omega$0 by

$\omega$1

and a backward unit vector $\omega$2 by

$\omega$3

Then

$\omega$4

with, for example,

$\omega$5

Once the QRPA matrix is constructed this way, the normal modes are obtained by diagonalization (Nakatsukasa, 2014).

This same idea was developed explicitly for superfluid QRPA as m-FAM. There, one chooses basis unit vectors such as

$\omega$6

or

$\omega$7

so that each FAM call returns one column of $\omega$8 or $\omega$9. After $A$0 such evaluations, the full $A$1 QRPA matrix is assembled and diagonalized (Avogadro et al., 2013). The same paper emphasizes that m-FAM is exact in principle with respect to standard QRPA; only the route to the matrix elements changes.

For covariant density functionals, matrix FAM was formulated in spherical coordinate space as a counterpart to iterative FAM. Its key benefit is that the tedious analytical derivation of the relativistic residual interaction is replaced by finite differences of the mean-field Hamiltonian. In that setting, the rearrangement terms due to density-dependent couplings are implicitly calculated without extra computational costs in both iterative and matrix FAM schemes, whereas the negative-energy Dirac-sea sector must be included explicitly in m-FAM but is taken into account automatically in coordinate-space i-FAM through completeness (Liang et al., 2013).

Matrix FAM is therefore noniterative only in a precise sense: it avoids iterative solution of the response amplitudes at each $A$2, but it does so by reverting to an explicit matrix representation and the associated diagonalization burden.

4. Explicit-linearization noniterative FAM in Skyrme RPA and HF+BCS QRPA

A later line of work adopted the term non-iterative FAM explicitly for Skyrme-like functionals. In the RPA formulation for E1 and M1 giant resonances, the perturbed single-particle states are written as

$A$3

and the induced field is treated as

$A$4

Instead of leaving this as a numerical finite-difference object inside an iterative loop, the method takes the $A$5 limit analytically, evaluates the derivatives $A$6 and $A$7, and arrives at the block RPA equation

$A$8

with

$A$9

$B$0

The single-particle Hamiltonian is split into time-even and time-odd parts, and the explicit matrix is then solved directly rather than through iterative FAM updates (Sasaki et al., 2022).

The QRPA extension of this explicit noniterative program was developed on top of HF+BCS rather than fully general HFB. The quasiparticle amplitudes are

$B$1

and the coherence factor is

$B$2

For a one-body external field,

$B$3

while the induced quasiparticle fields separate into time-even and time-odd pieces,

$B$4

After explicit linearization, the QRPA equation takes the direct matrix form

$B$5

with $B$6 and $B$7 written as derivatives of the time-even and time-odd mean fields multiplied by $B$8 and $B$9, respectively (Sasaki et al., 2022).

In this HF+BCS QRPA formulation, the authors explicitly neglect residual pairing contributions from $S(F;\omega)$0 and $S(F;\omega)$1 (Sasaki et al., 2022). That approximation limits the pairing-sector self-consistency, but it preserves the central noniterative idea: FAM becomes a residual-kernel generator for explicit QRPA matrices.

5. Contour-integration and residue methods: related, but distinct

A second major extension of FAM extracts discrete QRPA modes from the analytic structure of the response in the complex $S(F;\omega)$2-plane. In this formulation, the FAM amplitudes have simple poles at $S(F;\omega)$3,

$S(F;\omega)$4

$S(F;\omega)$5

Contour integration around a pole $S(F;\omega)$6 then yields

$S(F;\omega)$7

and the normalized QRPA amplitudes follow from

$S(F;\omega)$8

with an analogous expression for $S(F;\omega)$9 (Hinohara et al., 2013).

This strategy is highly effective for isolated low-lying collective states in deformed superfluid nuclei, because it avoids explicit QRPA-matrix construction and diagonalization while still recovering discrete amplitudes and transition strengths (Hinohara et al., 2013). It also became the basis of later local-QRPA calculations of collective inertia in spontaneous fission, where contour integrals of $X(\omega)$0 and $X(\omega)$1 were used to extract $X(\omega)$2 and $X(\omega)$3 without explicit QRPA-matrix construction or diagonalization (Washiyama et al., 2020).

However, the contour method is not noniterative in the same sense as m-FAM or explicit matrix construction. In both the deformed low-lying-mode work and the fission-inertia work, the FAM equations at each sampled complex frequency are still solved iteratively, using modified Broyden acceleration (Hinohara et al., 2013, Washiyama et al., 2020). The contour integral is therefore a non-diagonalizational mode-extraction scheme built on top of iterative FAM response.

6. Applications, computational tradeoffs, and recurrent misunderstandings

The computational tradeoff between iterative and noniterative formulations was made explicit in the QRPA matrix-construction study. For $X(\omega)$4Sn with $X(\omega)$5 MeV and matrix dimension $X(\omega)$6, m-FAM was about six times faster than i-FAM for that particular problem, but i-FAM scaled weakly, close to linear in $X(\omega)$7, whereas m-FAM grew between $X(\omega)$8 and $X(\omega)$9, due especially to diagonalization (Avogadro et al., 2013). This suggests that no single formulation dominates across all problem sizes: explicit noniterative matrix methods are attractive when discrete modes or eigenvectors are required, while iterative FAM remains preferable for very large spaces and smooth strength functions.

In direct applications, the explicit non-iterative RPA formulation for Skyrme-like functionals was used to calculate E1 and M1 giant resonances. For E1 transitions in heavy nuclei, the calculations reproduced well the resonance energy of the photoabsorption cross sections. For M1 transitions in double-magic nuclei, the residual interaction did not affect the transition strength, which suggested that the spin terms in the Skyrme force neglected in that computation could improve agreement between FAM and experiment (Sasaki et al., 2022). The HF+BCS noniterative QRPA extension then produced both large spin-flip M1 transitions in the 5 to 10 MeV region and a low-energy orbital transition corresponding to the M1 scissors mode in deformed gadolinium isotopes; when those strengths were used in Hauser-Feshbach calculations, the neutron-capture cross section was enhanced due to the low-energy M1 contribution, although the calculated cross section still underestimated experiment (Sasaki et al., 2022).

A further extension embedded noniterative FAM in a distorted-wave Born approximation framework for neutron-induced reactions. There the noniterative FAM was used to derive QRPA equations for inelastic neutron scattering to both discrete and continuum states in a consistent manner, with the Skyrme force employed as the interaction between the projectile neutron and nucleons inside the target nucleus. In the application to $Y(\omega)$0Pb, the calculated differential inelastic scattering cross sections to low-lying states reproduced available experimental data without phenomenological parameters often introduced in conventional DWBA calculations, and the calculated double differential cross section to the continuum state also agreed with the experimental data in the energy region relevant to the direct and pre-equilibrium processes (Sasaki et al., 25 Jul 2025).

A recurrent misunderstanding is to identify every matrix-free FAM implementation with noniterative FAM. Large-scale applications to giant dipole resonances in heavy rare-earth nuclei, local-QRPA collective inertia in spontaneous fission, the deformed relativistic Hartree-Bogoliubov theory in continuum, and numerical convergence studies of electromagnetic response all remain iterative in the solver sense, even when they avoid explicit QRPA matrices or evaluate induced fields through explicit linearization rather than literal finite differences (Oishi et al., 2015, Washiyama et al., 2020, Sun et al., 2022, Li et al., 2023). The stable conceptual distinction is therefore not between “finite-difference” and “analytic” residual fields, but between formulations that solve the response amplitudes iteratively at each $Y(\omega)$1 and formulations that replace that loop by explicit matrix construction or by residue extraction of discrete modes.