arXiv subjects

Henryk A. Witek

Publications and source records attributed to Henryk A. Witek.

7 recordsLinked to original sources

arXiv2026

Elimination of angular dependency in the quantum three-body problem made easy

We present a systematic account of the separation of the angular degrees of freedom from the nonrelativistic Schrödinger equation for a three-body quantum system with arbitrary masses, charges, total angular momentum, and parity. The resulting reduced Schrödinger equation (RSE) for the partial-wave components, expressed as functions solely of the interparticle distances, is reported in a compact matrix operator form. The remnants of the angular dependence, essential for the hermiticity of the RSE and consequently the stability of variational computations, appear in the RSE formalism as additional angular factors, derived by expanding minimal bipolar harmonics into a basis of Wigner functions \texorpdfstring{$\mathcal{D}$}{D}. We validate the final form of the RSE by computing accurate energy levels for helium states using an explicitly correlated Hylleraas-type basis. This work serves as a self-contained reference for the RSE formulation, consolidating elements previously scattered throughout the literature, thereby offering a convenient foundation for further analytical and numerical studies of general three-body quantum systems.

physics.atom-ph↗

arXiv2025

An Algorithm for Automated Extraction of Resonance Parameters from the Stabilization Method

The application of the stabilization method [A.~U.\ Hazi and H.~S.\ Taylor, Phys.~Rev.~A {\bf 1}, 1109 (1970)]) to extract accurate energy and lifetimes of resonance states is challenging: The process requires labor-intensive numerical manipulation of a large number of eigenvalues of a parameter-dependent Hamiltonian matrix, followed by a fitting procedure. In this article, we present \dosmax, an efficient algorithm implemented as an open-access \texttt{Python} code, which offers full automation of the stabilization diagram analysis in a user-friendly environment while maintaining high numerical precision of the computed resonance characteristics. As a test case, we use \dosmax to analyze the natural parity doubly-excited resonance states (${}^{1}\textnormal{S}^{\textnormal{e}}$, ${}^{3}\textnormal{S}^{\textnormal{e}}$, ${}^{1}\textnormal{P}^{\textnormal{o}}$, and ${}^{3}\textnormal{P}^{\textnormal{o}}$) of helium, demonstrating the accuracy and efficiency of the developed methodology. The presented algorithm is applicable to a wide range of resonances in atomic, molecular, and nuclear systems.

physics.comp-ph↗

arXiv2025

SpectraMatcher: A Python Program for Interactive Analysis and Peak Assignment of Vibronic Spectra

SpectraMatcher is a cross-platform desktop application for interactive comparison of experimental and computed vibronic spectra, designed to assist in the recognition and assignment of spectral patterns. It provides an intuitive graphical interface -- with no coding or scripting required -- for importing experimental spectra, visualizing them alongside the corresponding theoretical spectra constructed from Gaussian frequency calculations, and adjusting key parameters such as peak width, intensity scaling factors, and vibration-type-specific anharmonic corrections. SpectraMatcher features an automated peak-matching algorithm that assigns experimental and computed peaks based on their intensity ratio and proximity. Assignments and spectra can be exported in multiple formats for publication or for further analysis. The software remains responsive even for large datasets, and supports efficient and reproducible interpretation of vibronic spectra.

physics.chem-ph↗

arXiv2023

ZZPolyCalc: An open-source code with fragment caching for determination of Zhang-Zhang polynomials of carbon nanostructures

Determination of topological invariants of graphene flakes, nanotubes, and fullerenes constitutes a challenging task due to its time-intensive nature and exponential scaling. The invariants can be organized in a form of a combinatorial polynomial commonly known as the Zhang-Zhang (ZZ) polynomial or the Clar covering polynomial. We report here a computer program, ZZPolyCalc, specifically designed to compute ZZ polynomials of large carbon nanostructures. The curse of exponential scaling is avoided for a broad class of nanostructures by employing a sophisticated bookkeeping algorithm, in which each fragment appearing in the recursive decomposition is stored in the cache repository of molecular fragments indexed by a hash of the corresponding adjacency matrix. Although exponential scaling persists for the remaining nanostructures, the computational time is reduced by a few orders of magnitude owing to efficient use of hash-based fragment bookkeeping. The provided benchmark timings show that ZZPolyCalc allows for treating much larger carbon nanostructures than previously envisioned.

physics.comp-ph↗

arXiv2021

ZZ Polynomials of Regular $m$-tier Benzenoid Strips as Extended Strict Order Polynomials of Associated Posets -- Part 1. Proof of Equivalence

In Part 1 of the current series of papers, we demonstrate the equivalence between the Zhang-Zhang polynomial $\text{ZZ}(\boldsymbol{S},x)$ of a Kekuléan regular $m$-tier strip $\boldsymbol{S}$ of length $n$ and the extended strict order polynomial $\text{E}_{\mathcal{S}}^{\circ}(n,x+1)$ of a certain partially ordered set (poset) $\mathcal{S}$ associated with $\boldsymbol{S}$. The discovered equivalence is a consequence of the one-to-one correspondence between the set $\left\{ K\right\}$ of Kekulé structures of $\boldsymbol{S}$ and the set $\left\{ μ:\mathcal{S}\supset\mathcal{A}\rightarrow\left[\,n\,\right]\right\}$ of strictly order-preserving maps from the induced subposets of $\mathcal{S}$ to the interval $\left[\thinspace n\thinspace\right]$. As a result, the problems of determining the Zhang-Zhang polynomial of $\boldsymbol{S}$ and of generating the complete set of Clar covers of $\boldsymbol{S}$ reduce to the problem of constructing the set $\mathcal{L}(\mathcal{S})$ of linear extensions of the corresponding poset $\mathcal{S}$ and studying their basic properties. In particular, the Zhang-Zhang polynomial of $\boldsymbol{S}$ can be written in a compact form as $\text{ZZ}(\boldsymbol{S},x)=\sum_{k=0}^{\left|\mathcal{S}\right|}\sum_{w\in\mathcal{L}(\mathcal{S})}\binom{\left|\mathcal{S}\right|-\text{fix}_{\mathcal{S}}(w)}{\,\,k\,\,\hspace{1pt}-\text{fix}_{\mathcal{S}}(w)}\binom{n+\text{des}(w)}{k}\left(1+x\right)^{k}$, where $\text{des}(w)$ and $\text{fix}_{\mathcal{S}}(w)$ denote the number of descents and the number of fixed labels, respectively, in the linear extension $w\in\mathcal{L}(\mathcal{S})$.

math.CO↗

arXiv2020

Zhang-Zhang Polynomials of Ribbons

We report a closed-form formula for the Zhang-Zhang polynomial (aka ZZ polynomial or Clar covering polynomial) of an important class of elementary pericondensed benzenoids $Rb\left(n_{1},n_{2},m_{1},m_{2}\right)$ usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem. 84, 143--176 (2020)]. The discovered formula provides compact expressions for various topological invariants of $Rb\left(n_{1},n_{2},m_{1},m_{2}\right)$: the number of Kekulé structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes $O\left(k,m,n\right)$ and oblate rectangles $Or\left(m,n\right)$.

math.CO↗

arXiv2020

In how many distinct ways can flocks be formed? A problem in sheep combinatorics

In this short paper, we extend the concept of the strict order polynomial $Ω_{P}^{\circ}(n)$, which enumerates the number of strict order-preserving maps $ϕ:P\rightarrow\boldsymbol{n}$ for a poset $P$, to the extended strict order polynomial $\text{E}_{P}^{\circ}(n,z)$, which enumerates analogous maps for the elements of the power set $\mathcal{P}(P)$. The problem at hand immediately reduces to the problem of enumeration of linear extensions for the subposets of $P$. We show that for every $Q\subset P$ a given linear extension $v$ of $Q$ can be associated with a unique linear extension $w$ of $P$. The number of such linear extensions $v$ (of length $k$) associated with a given linear extension $w$ of $P$ can be expressed compactly as $\binom{\text{del}_{P}(w)}{k}$, where $\text{del}_{P}(w)$ is the number of deletable elements of $w$ defined in the text. Consequently the extended strict order polynomial $\text{E}_{P}^{\circ}(n,z)$ can be represented as $ \text{E}_{P}^{\circ}(n,z)=\sum_{w\in\mathcal{L}(P)}\sum_{k=0}^{p}\binom{\text{del}_{P}(w)}{p-k}\binom{n+\text{des}(w)}{k}z^{k}$. The derived equation can be used for example for solving the following combinatorial problem: Consider a community of $p$ shepherds, some of whom are connected by a master-apprentice relation (expressed as a poset $P$). Every morning, $k$ of the shepherds go out and each of them herds a flock of sheep. Community tradition stipulates that each of these $k$ shepherds will herd at least one and at most $n$ sheep, and an apprentice will always herd fewer sheep than his master (or his master's master, etc). In how many ways can the flocks be formed? The strict order polynomial answers this question for the case in which all $p$ shepherds go to work, and the extended strict order polynomial considers also all the situations in which some of the shepherds decide to take a day off.

math.CO↗