The Hückel model by means of the path integral method

Nagao, Hidemi; Nishikawa, Kiyoshi; Aono, Shigeyuki

doi:10.1016/0009-2614(92)86109-u

Cited by 14 publications

(3 citation statements)

References 13 publications

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“…This further can be correlated with Feynman’s idea of accounting for all possible electron paths . Nagao, Nishikawa, and Aono demonstrated that the transition amplitude in the Feynman path integral method can be obtained by summing contributions from all independent trajectories . Each SRW in a molecular graph would be interpreted as an independent Feynman’s electron trajectory …”

Section: Resultsmentioning

confidence: 99%

From Infection Clusters to Metal Clusters: Significance of the Lowest Occupied Molecular Orbital (LOMO)

Tsuji

Yoshizawa

2021

ACS Omega

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In this paper, the nature of the lowest-energy electrons is detailed. The orbital occupied by such electrons can be termed the lowest occupied molecular orbital (LOMO). There is a good correspondence between the Huckel method in chemistry and graph theory in mathematics; the molecular orbital, which chemists view as the distribution of an electron with a specific energy, is to mathematicians an algebraic entity, an eigenvector. The mathematical counterpart of LOMO is known as eigenvector centrality, a centrality measure characterizing nodes in networks. It may be instrumental in solving some problems in chemistry, and also it has implications for the challenge facing humanity today. This paper starts with a demonstration of the transmission of infectious disease in social networks, although it is unusual for a chemistry paper but may be a suitable example for understanding what the centrality (LOMO) is all about. The converged distribution of infected patients on the network coincides with the distribution of the LOMO of a molecule that shares the same network structure or topology. This is because the mathematical structures behind graph theory and quantum mechanics are common. Furthermore, the LOMO coefficient can be regarded as a manifestation of the centrality of atoms in an atomic assembly, indicating which atom plays the most important role in the assembly or which one has the greatest influence on the network of these atoms. Therefore, it is proposed that one can predict the binding energy of a metal atom to its cluster based on its LOMO coefficient. A possible improvement of the descriptor using a more sophisticated centrality measure is also discussed.

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Section: Resultsmentioning

confidence: 99%

From Infection Clusters to Metal Clusters: Significance of the Lowest Occupied Molecular Orbital (LOMO)

Tsuji

Yoshizawa

2021

ACS Omega

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“…= Pjj+I =fj=fi +fj+i -B-/-B+2 + - (47) Pi) * Pjj+1 = 2(fi -B-/ + B-2 -•• ) (48) Equations 47 and 48 can also be used for calculating bond orders in linear polymers composed of bridged evenmembered rings.…”

Section: Pi]mentioning

confidence: 99%

Topological Atomic Charges, Valencies, and Bond Orders

Bonchev¹,

Gordeeva²

1995

J. Chem. Inf. Comput. Sci.

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Theorems and corollaries are proved and relationships are derived for the atomic and bond moments of energy in molecules and solids or, in terms of graph theory, for the self-returning walks SRWs that begin in graph vertexes and edges, respectively. Topological atomic charges, valencies, and bond orders are introduced by means of the limit distributions of the SRWs. A hierarchical system of topological rules on atomic and bond reactivity is presented proceeding from the alternating positive and negative contributions to the atomic and bond moments. Some light is thus shed on the topological control of chemical reactivity.

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“…Recently, the path integral method has been applied to the Huckel model [4] and the manyelectron problems [51. We have also developed the path integral method without using the Grassmann number to estimate the electron correlation in molecules [6].…”

Section: Introductionmentioning

confidence: 99%

Path integral Monte Carlo method for ab initio calculation

Kawabe

Nagao

Nishikawa

1996

Int. J. Quantum Chem.

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mThe Feynman path integral method is applied to the many-electron problem. We first give new closure relations in terms of ordinary complex and real numbers, which could be derived from an arbitrary complete set of state vectors. Then, in the path integral form, the partition function of the system and the ensemble average of energy are explicitly expressed in terms of these closure relations. It is impossible to evaluate, the path integral by direct numerical integrations because of its huge amount of integration variables. Therefore, we develop an algorithm by the Monte Carlo method with constraints corresponding to the normalization condition of states to calculate the required integral. Finally, the ensemble average of energy for the hydrogen molecule is explicitly evaluated by the quantum Monte Carlo method and results are compared with the result obtained by the ordinary full configuration interaction (CI) method.

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The Hückel model by means of the path integral method

Cited by 14 publications

References 13 publications

From Infection Clusters to Metal Clusters: Significance of the Lowest Occupied Molecular Orbital (LOMO)

From Infection Clusters to Metal Clusters: Significance of the Lowest Occupied Molecular Orbital (LOMO)

Topological Atomic Charges, Valencies, and Bond Orders

Path integral Monte Carlo method for ab initio calculation

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