Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
ATZ-Fachkongress

Chassis.tech plus 2024


4 Kongresse in einer Veranstaltung

Treffen Sie in München die Fachleute von Chassis, Lenkung, Bremsen sowie Reifen/Räder.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Particles and Waves Kapitel lesen Erstes Kapitel lesen

2024 | OriginalPaper | Buchkapitel

7. Molecules

verfasst von : Giulio Armando Ottonello

Erschienen in: Quantum Geochemistry

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In these short notes we encounter first the conceptually simplest molecule, the H2+ ion, i.e. a single electron rotating about two nuclear centers, and we move then toward a more complex case: the H2 molecule.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Many-Electron Atoms
Nächstes Kapitel 3D and 2D Periodic Structures
Fußnoten
1
Note that in the following I will change sign to the original data.
 
2
It is even for the states σg, πu, δg and odd for the states σu, πg, δu.
 
3
To get the normalization constant we must make the square of the wave function and integrate over all space. A quick account of the procedure is given elsewhere.
 
4
See Appendix 2 in Slater, (1963); for a proof of this theorem.
 
5
l changes by ±1 units or 0 and this limits greatly the number of observed lines.
 
6
Note that in Pekeris(1934) the development is directly on \( {\mathcal{R}} \)(R); moreover Eq. (7.2.56.4) is replaced by \({\mathcal{R}}(\mathcal y)=\text N \exp (-z / 2) z^{{\text b} / 2} \text F(\overline{z})\).
 
7
H 2 + dissociates into H and H+; the theoretical electronic energy of H is −0.5 Hartree while H+ has no electrons, hence no electron energy. The dissociation energy of H 2 + is therefore the minimum energy attained by the molecule with the nuclear centers placed at equilibrium distance (i.e. bottom of potential well), plus the Zero Point Correction, less the theoretical (or calculated) energy of H.
 
8
There are several methods of estimating the initial ci,μ coefficients. The most common is using the eigenvectors from a truncated form of the actual Hamiltonian.
 
9
The Hartree energy is often denoted as Eh and corresponds to 4.359748×10−18 J; it corresponds also to 27.2114 eV and to 2 Ry.
 
10
More precisely, a product of a spatial function and a spin function with a spin-free Hamiltonian (cf. Meckler, 1953).
 
11
I recall here again that Σ is the MO counterpart symbolic representation of the σ AO symbolic representation; the-at apex denotes change of sign under reflection through an axial plane. The 1,3 at apex preceding the symbols denote the multiplicity, i.e. “singlet” or “triplet”). A Σ state arises whenever ML = 0, a π state whenever ML = + 1 or −1 and a ∆ state whenever ML = +2 or −2.
 
12
D0 is, except for the zero-point correction, the energy difference of the stable molecule and the dissociated atoms. It is this quantity which is experimentally determined by analysis of band spectra.
 
13
The notation adopted in the following treatment by Mulligan assigns o, s, z, x, y, in the place of 1s, 2s, 2pz, 2px,2py to the first oxygen atom; then o’, s’, z’, x’, y’, for the corresponding orbitals of the second oxygen atom. The carbon orbitals are designated by c, sc, zc, xc, yc.
 
14
A lone pair is an electron pair in the outermost shell of an atom that is not shared or bonded to another atom. It is also called “unshared pair”. Lone pairs affect substantially the polarity of charge of a molecule, reinforcing or reducing its dipole and quadrupole moment.
 
15
For spherical molecules, the ratio μ*/μ increases from unity to 3/2 with the increase of the dielectric constant from 1 to ∞.
 
16
The bold character of s implies a vectorial notation with a charge distribution which attains zero only at the cavity surface.
 
17
Calculations can be performed also with Pauling radii, Bondi radii (Bondi, 1964), Universal Force Field radii (UFF; Rappè et al., 1992).
 
18
In Mennucci et al. (1997) mi is replaced by ci the concentration of the ith ionic species in moles per cubic meter.
 
19
Green’s function are used as a convenient method for solving more complicated differential equations. They are numerical solutions to an inhomogenous differential equation with a “driving term” given by a delta function. In the case of an electromagnetic field they describe how the field behaves according to the laws of quantum mechanics. In this case the Green’s functions are quantum-mechanical operators having no longer simple numerical values. See Qin (2014) for an extended treatment. Few notes are also available in the Mathematics section of this book (Par. A.20).
 
20
The Calderón projector is a pseudo-differential operator used widely in applied mathematics to solve linear partial differential equations which have been formulated as integral equations (i.e. Boundary Element Method, BEM). See Steinbach (2008) for an extended treatment.
 
21
The subscript n specifies which root λn of the secular equation has been used.
 
22
Assuming the first and higher excited states to be inaccessible at any T, the electronic contribution to the partition function (qe) corresponds to the electronic spin multiplicity of the molecule, which gives no contribution to the internal energy of the molecule.
 
23
For a single atom the rotational partition function is identically 1 at all T, so that there are no contributions or rotational terms to internal energy, heat capacity and entropy; for a linear molecule the rotational contribution varies linearly with T so that the partial derivative of the natural logarithm of the rotational term reduces to T−1.
 
24
Examples of practical application of the procedure to crystalline compounds are given later on.
 
Metadaten
Titel
Molecules
verfasst von
Giulio Armando Ottonello
Copyright-Jahr
2024
Buch
Quantum Geochemistry

Print ISBN: 978-3-031-21836-1

Electronic ISBN: 978-3-031-21837-8

Copyright-Jahr: 2024

DOI
https://doi.org/10.1007/978-3-031-21837-8_7

Premium Partner

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

    Zur Marktübersicht
    Bildnachweise