Evaluation of the even wave functions of the hydrogen atom in the path-integral formalism

Goovaerts, Marc; Broeckx, F.; Camp, Piet Van

doi:10.1016/0031-8914(73)90113-4

Cited by 27 publications

(22 citation statements)

References 5 publications

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“…When the HOMO-LUMO schemes of the other molecules in Table 3, which are formed by substitutions to the parent compound, are examined, it is seen that this distinction becomes more evident, as a result, it separates and is clearly dispersed. "Descriptors that depend on the electronic order and derive from the electronic structure of the compound are called electronic structure descriptors" (Koopmans, 1933). Some of them are "highest occupied molecular orbital energy (EHOMO), lowest unoccupied molecular orbital energy (ELUMO), LUMO-HOMO energy gap (ΔE), absolute hardness (η), absolute softness (σ), electronegativity (χ), chemical potential (μ), electrophilicity index (ω), nucleophilicity index (ε) and dipole moment (DM)".…”

Section: Frontier Orbitalsmentioning

confidence: 99%

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Bazı Sentezlenmiş 3-Arilamino-5-[2-(Sübstitüe Edilmiş 1-İmidazol) Etil 1]-1,2,4-Triazol Türevlerinin Teorik İncelenmesi

Yiğit

Turhan

2023

Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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Bu çalışmada, literatürde çalışmalara konu olan sekiz adet 1,2,4-triazol türevinin kimyasal, doğrusal olmayan optik özellikleri ve antibakteriyel aktivite parametrelerini elde etmek için kuantum kimyasal hesaplamaları yapılmış ve yorumlanmıştır. Bu kapsamda çalışma kapsamında, bilimsel araştırmalara konu olan sekiz bileşiğin yoğunluk fonksiyonel teorisi kullanılarak optimize edilerek en kararlı yapı B3LYP/6-311++G(d,p) yönteminde bulunmuştur. literatürde çalışmalar. Optimize edilmiş moleküler sonuçlardan elde edilen HOMO-LUMO enerjileri ve bu enerji değerlerinden hesaplanan kimyasal parametreler (I.iyonizasyon potansiyeli, A: elektron ilgisi, ΔE enerji aralığı, χ: elektronegatiflik, σ: moleküler yumuşaklık, η: moleküler sertlik, ω: elektrofilik indeks, ε: nükleofilik indeks, μ: kimyasal potansiyel) değerleri belirlendi.

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Section: Frontier Orbitalsmentioning

confidence: 99%

“…Other descriptors can be calculated with the help of the equations found in the literature. According to Koopman's theorem, as seen from equations ( 1) and ( 2), the EHOMO and ELUMO values of any chemical species are related to its ionization energy and electron affinity values (Koopmans, 1933;Sastri & Perumareddi, 1997).…”

Section: Frontier Orbitalsmentioning

confidence: 99%

Bazı Sentezlenmiş 3-Arilamino-5-[2-(Sübstitüe Edilmiş 1-İmidazol) Etil 1]-1,2,4-Triazol Türevlerinin Teorik İncelenmesi

Yiğit

Turhan

2023

Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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“…The obtained data in Now, with the identi cation of the ground state structures and their related structural properties and Based on the lowest-energy structures, it is possible to investigate their energetic, and electronic properties. In order to evaluation of the stability and size-dependent properties of gold clusters, the total binding energies (E b ), binding energies per atom (E b/n , dissociation energies), HOMO-LUMO gaps energies (E gap ), vertical ionization potentials (VIP), fragmentation energies (E f , D (n,n−1) ), and second differences in total cluster energies (Δ2E n ), calculated and de ned as the following formulas Based on the Koopmans theorem [2,17,31]:…”

Section: Vacuum Environmentmentioning

confidence: 99%

A DFT-Based Investigation of the properties of gold nanoclusters up to Au20

Kargar

Mohammadnejad

2023

Preprint

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The structures and stabilities of gold clusters with up to 20 atoms have been studied by Density Functional Theory (DFT) and compared in two vacuum and COSMO environment here for the first time. The structure optimizations and frequency analysis are performed with the Generalized Gradient Approximation (GGA) BLYP (Becke, 1988; Lee et al., 1988) functional in Dmol3 combined with the all-electron relativistic core treatment and corresponding Double Numerical plus Polarization (DNP) basis set. The transition point from two-dimensional to three-dimensional geometry for gold clusters occurs at Au17 in both the vacuum and COSMO environment. The structural, energetic, and electronic properties of the small gold clusters strongly depend on sizes and structures, which are in good agreement with other theoretical and experimental calculations. It was also found that the effect of relativistic properties has a great influence on the properties of gold clusters. The even-odd oscillation (saw-tooth patterns) observed in the different properties of the clusters predicts the stability of the clusters and their electronic properties. The result shows that the clusters with even numbers of atoms were more stable than the neighboring clusters with odd numbers of atoms due to their closed-shell orbital structure. Also, the properties of gold nanoclusters in COSMO seem to be different from the vacuum environment.

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“…A comprehensive study of a semi-classical approximation of the path integral for the Green's function at a given energy has been performed by Gutzwiller [2]; in particular, he found an exact Green's function for negative energy in the polar coordinates of momentum space by a semiclassical approximation. It is also known that a path integral of the s-wave propagator ( or evolution operator ) is obtained by summing a perturbative series [3]. In 1979, Duru and Kleinert [4] showed an elegant path integral method to evaluate the Green's function for the hydrogen atom exactly.…”

Section: Introductionmentioning

confidence: 99%

“…The path integral treatment of the hydrogen atom is interesting not only for a methodological interest but also for a pedagogical purpose. After several early attempts [1,2], Duru and Kleinert [3] showed an elegant path integral method to evaluate the Green's function for the hydrogen atom exactly. Two basic ingredients in their method are the use of a re-scaled time variable and the so called Kustaanheimo-Stiefel transformation [4]which reveals the O(4) symmetry explicitly in the coordinate space.…”

Section: Introductionmentioning

confidence: 99%

Path integral of the hydrogen atom, Jacobi's principle of least action and one-dimensional quantum gravity

Fujikawa¹

1997

Nuclear Physics B

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The general treatment of a separable Hamiltonian of Liouville-type is well-known in operator formalism. A path integral counterpart is formulated if one starts with the Jacobi's principle of least action, and a path integral evaluation of the Green's function for the hydrogen atom by Duru and Kleinert is recognized as a special case. The Jacobi's principle of least action for given energy is reparametrization invariant, and the separation of variables in operator formalism corresponds to a choice of gauge in path integral. The Green's function is shown to be gauge independent,if the operator ordering is properly taken into account. These properties are illustrated by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates.

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Evaluation of the even wave functions of the hydrogen atom in the path-integral formalism

Cited by 27 publications

References 5 publications

Bazı Sentezlenmiş 3-Arilamino-5-[2-(Sübstitüe Edilmiş 1-İmidazol) Etil 1]-1,2,4-Triazol Türevlerinin Teorik İncelenmesi

Bazı Sentezlenmiş 3-Arilamino-5-[2-(Sübstitüe Edilmiş 1-İmidazol) Etil 1]-1,2,4-Triazol Türevlerinin Teorik İncelenmesi

A DFT-Based Investigation of the properties of gold nanoclusters up to Au20

Path integral of the hydrogen atom, Jacobi's principle of least action and one-dimensional quantum gravity

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