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Advanced Mathematics for Physical Chemistry and Chemical Physics
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Advanced Mathematics for Physical Chemistry and Chemical Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 April 2020) | Viewed by 12251

Special Issue Editor

Prof. Dr. Darin J. Ulness

E-Mail Website
Guest Editor
Department of Chemistry, Concordia College, Moorhead, MN 56562, USA
Interests: physical chemistry; spectroscopy; liquids; physical neuroscience; physical chemistry
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to collect current work in mathematics that has applicability to any area of physical chemistry and chemical physics. In the spirit of the well-known quote by G. N. Lewis, “physical Chemistry is anything that is interesting,” the scope of the project is broad, with all areas of mathematics invited to participate. Work can be (i) fundamental by contributing to the base mathematical foundation of physical chemistry and chemical physics, (ii) applied by contributing a solution to a problem of chemical interest, or (iii) pedagogical by providing analysis of “toy models” that deepen our understanding of nature. Authors should provide an exposition on how their work contributes to the molecular sciences. This should include a discussion of the range of applicability and any physical insights gained from the underlying mathematics.

Prof. Dr. Darin J. Ulness
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Quantum mechanics
  • Statistical mechanics
  • Chemical kinetics
  • Symmetry
  • Time dynamics
  • Spectroscopy
  • Light–matter interaction
  • Condensed matter
  • Analytic techniques

Published Papers (5 papers)

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Research

17 pages, 2147 KiB  
Article
A Dynamic Study of Biochemical Self-Replication
by Desire T. Gijima and Enrique Peacock-López
Mathematics 2020, 8(6), 1042; https://doi.org/10.3390/math8061042 - 26 Jun 2020
Cited by 3 | Viewed by 1849
Abstract
As it is well understood, in biological systems, small regulatory motifs are present at all scales, thus looking at simple negative feedback loops give us some information of how autocatalytic systems may be affected by regulation. For a single template self-replication, we consider [...] Read more.
As it is well understood, in biological systems, small regulatory motifs are present at all scales, thus looking at simple negative feedback loops give us some information of how autocatalytic systems may be affected by regulation. For a single template self-replication, we consider a plausible mechanism, which we reduce to a 2-variable dimensionless set of ordinary differential equations, (ODE). The stability analysis of the steady states allows us to obtain exact relations to describe two-parameter bifurcation diagrams. We include a negative feedback to the reactants input to study the effect of regulation in biochemical self-replication. Surprisingly, the simpler regulation has the largest impact on the parameter space. Full article
(This article belongs to the Special Issue Advanced Mathematics for Physical Chemistry and Chemical Physics)
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14 pages, 3130 KiB  
Article
Interference among Multiple Vibronic Modes in Two-Dimensional Electronic Spectroscopy
by Camille A. Farfan and Daniel B. Turner
Mathematics 2020, 8(2), 157; https://doi.org/10.3390/math8020157 - 22 Jan 2020
Cited by 11 | Viewed by 3464
Abstract
Vibronic coupling between electronic and vibrational states in molecules plays a critical role in most photo-induced phenomena. Many key details about a molecule’s vibronic coupling are hidden in linear spectroscopic measurements, and therefore nonlinear optical spectroscopy methods such as two-dimensional electronic spectroscopy (2D [...] Read more.
Vibronic coupling between electronic and vibrational states in molecules plays a critical role in most photo-induced phenomena. Many key details about a molecule’s vibronic coupling are hidden in linear spectroscopic measurements, and therefore nonlinear optical spectroscopy methods such as two-dimensional electronic spectroscopy (2D ES) have become more broadly adopted. A single vibrational mode of a molecule leads to a Franck–Condon progression of peaks in a 2D spectrum. Each peak oscillates as a function of the waiting time, and Fourier transformation can produce a spectral slice known as a ‘beating map’ at the oscillation frequency. The single vibrational mode produces a characteristic peak structure in the beating map. Studies of single modes have limited utility, however, because most molecules have numerous vibrational modes that couple to the electronic transition. Interactions or interference among the modes may lead to complicated peak patterns in each beating map. Here, we use lineshape-function theory to simulate 2D ES arising from a system having multiple vibrational modes. The simulations reveal that the peaks in each beating map are affected by all of the vibrational modes and therefore do not isolate a single mode, which was anticipated. Full article
(This article belongs to the Special Issue Advanced Mathematics for Physical Chemistry and Chemical Physics)
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12 pages, 914 KiB  
Article
Proof and Use of the Method of Combination Differences for Analyzing High-Resolution Coherent Multidimensional Spectra
by Peter C. Chen and Jeffrey Ehme
Mathematics 2020, 8(1), 44; https://doi.org/10.3390/math8010044 - 01 Jan 2020
Cited by 2 | Viewed by 1959
Abstract
High-resolution coherent multidimensional spectroscopy is a technique that automatically sorts rotationally resolved peaks by quantum number in 2D or 3D space. The resulting ability to obtain a set of peaks whose J values are sequentially ordered but not known raises the question of [...] Read more.
High-resolution coherent multidimensional spectroscopy is a technique that automatically sorts rotationally resolved peaks by quantum number in 2D or 3D space. The resulting ability to obtain a set of peaks whose J values are sequentially ordered but not known raises the question of whether a method can be developed that yields a single unique solution that is correct. This paper includes a proof based upon the method of combined differences that shows that the solution would be unique because of the special form of the rotational energy function. Several simulated tests using a least squares analysis of simulated data were carried out, and the results indicate that this method is able to accurately determine the rotational quantum number, as well as the corresponding Dunham coefficients. Tests that include simulated random error were also carried out to illustrate how error can affect the accuracy of higher-order Dunham coefficients, and how increasing the number of points in the set can be used to help address that. Full article
(This article belongs to the Special Issue Advanced Mathematics for Physical Chemistry and Chemical Physics)
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26 pages, 18975 KiB  
Article
Centered Polygonal Lacunary Graphs: A Graph Theoretic Approach to p-Sequences of Centered Polygonal Lacunary Functions
by Keith Sullivan, Drew Rutherford and Darin J. Ulness
Mathematics 2019, 7(11), 1021; https://doi.org/10.3390/math7111021 - 28 Oct 2019
Cited by 2 | Viewed by 2268
Abstract
This work is on the nature and properties of graphs which arise in the study of centered polygonal lacunary functions. Such graphs carry both graph-theoretic properties and properties related to the so-called p-sequences found in the study of centered polygonal lacunary functions. [...] Read more.
This work is on the nature and properties of graphs which arise in the study of centered polygonal lacunary functions. Such graphs carry both graph-theoretic properties and properties related to the so-called p-sequences found in the study of centered polygonal lacunary functions. p-sequences are special bounded, cyclic sequences that occur at the natural boundary of centered polygonal lacunary functions at integer fractions of the primary symmetry angle. Here, these graphs are studied for their inherent properties. A ground-up set of planar graph construction schemes can be used to build the numerical values in p-sequences. Further, an associated three-dimensional graph is developed to provide a complementary viewpoint of the p-sequences. Polynomials can be assigned to these graphs, which characterize several important features. A natural reduction of the graphs original to the study of centered polygonal lacunary functions are called antipodal condensed graphs. This type of graph provides much additional insight into p-sequences, especially in regard to the special role of primes. The new concept of sprays is introduced, which enables a clear view of the scaling properties of the underling centered polygonal lacunary systems that the graphs represent. Two complementary scaling schemes are discussed. Full article
(This article belongs to the Special Issue Advanced Mathematics for Physical Chemistry and Chemical Physics)
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34 pages, 31826 KiB  
Article
Centered Polygonal Lacunary Sequences
by Keith Sullivan, Drew Rutherford and Darin J. Ulness
Mathematics 2019, 7(10), 943; https://doi.org/10.3390/math7100943 - 11 Oct 2019
Cited by 5 | Viewed by 2164
Abstract
Lacunary functions based on centered polygonal numbers have interesting features which are distinct from general lacunary functions. These features include rotational symmetry of the modulus of the functions and a notion of polished level sets. The behavior and characteristics of the natural boundary [...] Read more.
Lacunary functions based on centered polygonal numbers have interesting features which are distinct from general lacunary functions. These features include rotational symmetry of the modulus of the functions and a notion of polished level sets. The behavior and characteristics of the natural boundary for centered polygonal lacunary sequences are discussed. These systems are complicated but, nonetheless, well organized because of their inherent rotational symmetry. This is particularly apparent at the so-called symmetry angles at which the values of the sequence at the natural boundary follow a relatively simple 4 p -cycle. This work examines special limit sequences at the natural boundary of centered polygonal lacunary sequences. These sequences arise by considering the sequence of values along integer fractions of the symmetry angle for centered polygonal lacunary functions. These sequences are referred to here as p-sequences. Several properties of the p-sequences are explored to give insight in the centered polygonal lacunary functions. Fibered spaces can organize these cycles into equivalence classes. This then provides a natural way to approach the infinite sum of the actual lacunary function. It is also seen that the inherent organization of the centered polygonal lacunary sequences gives rise to fractal-like self-similarity scaling features. These features scale in simple ways. Full article
(This article belongs to the Special Issue Advanced Mathematics for Physical Chemistry and Chemical Physics)
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