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| The '''Holstein–[[Conyers Herring|Herring]] method''',<ref>{{cite journal |last=Holstein |first=T. |title=Mobilities of positive ions in their parent gases |journal=J. Phys. Chem. |volume=56 |issue=7 |pages=832–836 |year=1952 |doi=10.1021/j150499a004 }}</ref><ref>{{cite paper |last=Holstein |first=T. |title=Westinghouse Research Report 60-94698-3-R9 |work=(unpublished) |year=1955 }}</ref><ref name=Herring62>{{cite journal |last=[[Conyers Herring|Herring]] |first=C. |title=Critique of the Heitler-London Method of Calculating Spin Couplings at Large Distances |journal=Rev. Mod. Phys. |volume=34 |issue=4 |pages=631–645 |year=1962 |doi=10.1103/RevModPhys.34.631 |bibcode = 1962RvMP...34..631H }}</ref><ref>{{cite journal |last=Bardsley |first=J. N. |last2=Holstein |first2=T. |last3=Junker |first3=B. R. |last4=Sinha |first4=S. |title=Calculations of ion-atom interactions relating to resonant charge-transfer collisions |journal=Phys. Rev. A |volume=11 |issue=6 |pages=1911–1920 |year=1975 |doi=10.1103/PhysRevA.11.1911 |bibcode = 1975PhRvA..11.1911B }}</ref> also called the '''Surface Integral method''',<ref>{{cite journal |first=T. C. |last=Scott |first2=M. |last2=Aubert-Frécon |first3=D. |last3=Andrae |title=Asymptotics of Quantum Mechanical Atom-Ion Systems |journal=AAECC (Applicable Algebra in Engineering, Communication and Computing) |volume=13 |issue=3 |pages=233–255 |year=2002 |doi=10.1007/s002000200100 }}</ref><ref>{{cite journal |last=Aubert-Frécon |first=M. |last2=Scott |first2=T. C. |last3=Hadinger |first3=G. |last4=Andrae |first4=D. |last5=Grotendorst |first5=J. |last6=Morgan III |first6=J. D. |title=Asymptotically Exact Calculation of the Exchange Energies of One-Active-Electron Diatomic Ions with the Surface Integral Method |journal=[[Journal of Physics B: Atomic, Molecular and Optical Physics|J. Phys. B: At. Mol. Opt. Phys.]] |volume=37 |issue=22 |pages=4451–4469 |year=2004 |doi=10.1088/0953-4075/37/22/005 |bibcode = 2004JPhB...37.4451S }}</ref> also called '''Smirnov's method'''<ref>{{cite journal |last=Smirnov |first=B. M. |last2=Chibisov |first2=M. I. |title=Electron exchange and changes in the hyperfine state of colliding alkaline metal atoms |journal=[[Journal of Experimental and Theoretical Physics|Sov. Phys. JETP]] |volume=21 |issue= |pages=624–628 |year=1965 |doi= |bibcode=1965JETP...21..624S }}</ref> is an effective means of getting the [[Exchange interaction|exchange energy]] splittings of asymptotically degenerate energy states in molecular systems. Although the exchange energy becomes elusive at large internuclear systems, it is of prominent importance in theories of molecular binding and magnetism. This splitting results from the symmetry under exchange of identical nuclei ([[Pauli Exclusion Principle]]).
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Spyderco – I haven’t had quite a lot of expertise with these personally, but I do know that this is the one model that is actually made with self protection because the meant use. I can even say that I do know that the quality of the blade and building has all the time been superb.<br><br>Just like Swiss Army Knives this Stainless Metal Small Folding Pocket Knife is a perfect crusing and camping gear. Featuring a set of 13 helpful tools together with a scissor, file, scaler, metal blade, screwdriver, noticed, bottle opener and a bright LED gentle with inbuilt battery. It’s vital software for mountaineering, trekking or any other outside activities. Local assemblywoman Shannon Grove, who was sitting at the similar desk as Larwood, said her husband jumped into action and immediately gave her the Heimlich maneuver. When that failed, he referred to as for a doctor who also tried and failed to clear her airways with the Heimlich maneuver.<br><br>The title of the blade explains it all. 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| | |
| ==Theory==
| |
| The basic idea pioneered by Holstein and Herring approach can be illustrated for the [[hydrogen molecular ion]] or more generally, [[atom-ion]] systems or ''[[One-electron_bond#One-_and_three-electron_bonds|one-active electron]]'' systems, as follows. We consider states that are represented by even or odd functions with respect to behavior under space inversion. This is denoted with the suffixes [[Molecular term symbol|g and u]] from the German ''gerade'' and ''ungerade'' and are standard practice for the designation of electronic states of diatomic molecules, whereas for atomic states the terms ''even'' and ''odd'' are used. The electronic [[Schrödinger equation]] can be written as: | |
| :<math>
| |
| \left( -\frac{\hbar^2}{2m} \nabla^2 + V \right) \psi = E \psi~,
| |
| </math>
| |
| where ''E'' is the (electronic) energy of a given quantum mechanical state (eigenstate), with the electronic state function <math> \psi=\psi(\mathbf{r}) </math> depending on the spatial coordinates of the electron and where <math> V </math> is the electron-nuclear Coulomb potential energy function. For the [[hydrogen molecular ion]], this is:
| |
| :<math>
| |
| V = - \frac{e^{2}}{4 \pi \varepsilon_0 } \left( \frac{1}{r_a} + \frac{1}{r_b} \right)
| |
| </math>
| |
| For any gerade (or even) state, the electronic Schrödinger wave equation can be written in [[atomic units]] (<math>\hbar=m=e=4 \pi \varepsilon_0 =1 </math>) as:
| |
| :<math>
| |
| \left( -\frac{1}{2} \nabla^2 + V(\textbf{x}) \right) \psi_{+} = E_{+} \psi_{+}
| |
| </math>
| |
| For any ungerade (or odd) state, the corresponding wave equation can be written as:
| |
| :<math>
| |
| \left( -\frac{1}{2} \nabla^2 + V(\textbf{x}) \right) \psi_{-} = E_{-} \psi_{-}
| |
| </math>
| |
| For simplicity, we assume real functions (although the result can be generalized to the complex case). We then multiply the gerade wave equation by <math>\psi_{-}</math>on the left and the ungerade wave equation on the left by <math>\psi_{+}</math>and subtract to obtain:
| |
| :<math>
| |
| \psi_{+} \nabla^2 \psi_{-} - \psi_{-} \nabla^2 \psi_{+} = {} - 2 \, \Delta E \, \psi_{-} \psi_{+} \; .
| |
| </math>
| |
| where <math>\Delta E = E_{-} - E_{+} </math> is the ''exchange energy splitting''. Next, without loss of generality, we define orthogonal single-particle functions, <math>\phi_A^{}</math> and <math>\phi_B^{}</math>, located at the nuclei and write:
| |
| :<math>
| |
| \psi_{+} = \frac{1}{\sqrt{\,2}} ~ ( \phi_A^{} + \phi_B^{}) \; ,\qquad \psi_{-} = \frac{1}{\sqrt{\,2}} ~ ( \phi_A^{} - \phi_B^{}) \; .
| |
| </math>
| |
| This is similar to the LCAO [[Linear combination of atomic orbitals molecular orbital method|(Linear Combination of Atomic Orbitals)]] ansatz used in quantum chemistry, but we emphasize that the functions <math>\phi_A^{}</math> and <math>\phi_B^{}</math> are in general ''polarized'' i.e. they are not pure eigenfunctions of angular momentum with respect to their nuclear center, see
| |
| also below). Note, however, that in the limit as <math>R \rightarrow \infty</math>, these localized functions <math>\phi_{A,B}^{}</math> collapse into the well-known atomic (hydrogenic) psi functions <math>\phi_{A,B}^{0}</math>. We denote <math>M</math> as the mid-plane located exactly between the two nuclei (see diagram for [[hydrogen molecular ion]] for more details), with <math>{\mathbf{z}}</math> representing the unit normal vector of this plane (which is parallel to the Cartesian <math>z</math>-direction), so that the full <math>\mathbf{R}^3</math> space is divided into left (<math>L</math>) and right (<math>R</math>) halves. By considerations of symmetry:
| |
| :<math>
| |
| \left. \psi_{-} \right|_M = \mathbf{z} \cdot \left. \mathbf{\nabla} \psi_{+} \right|_M = 0 \; .
| |
| </math>
| |
| This implies that:
| |
| :<math>
| |
| \left. \phi_{A}^{} \right|_M = \left. \phi_{B}^{} \right|_M \; , \qquad
| |
| {\mathbf{z}} \cdot \left. \mathbf{\nabla} \phi_{A}^{} \right|_M = {} - \mathbf{z} \cdot \left. \mathbf{\nabla} \phi_{B}^{} \right|_M \; .
| |
| </math>
| |
| Also, these localized functions are normalized, which leads to:
| |
| :<math>
| |
| \int_{L} \phi_A^2 ~dV = \int_{R} \phi_B^2 ~dV
| |
| </math>
| |
| and conversely. Integration of the above in the whole space left to the mid-plane yields: | |
| :<math>
| |
| 2 \int_{L} \psi_{+} \psi_{-} ~ dV = \int_{L} ( \phi_A^2 - \phi_B^2 ) ~ dV = 1 - 2 \int_R \phi_A^2 ~ dV
| |
| </math> | |
| and
| |
| :<math>
| |
| \int_{L} ( \psi_{+} \nabla^2 \psi_{-} - \psi_{-} \nabla^2 \psi_{+} ) ~dV = \int_{L} ( \phi_{B}^{} \nabla^2 \phi_{A}^{} - \phi_{A}^{} \nabla^2 \phi_{B}^{} ) ~dV
| |
| </math>
| |
| [[Image:h2plus figure 1.png|thumb|400px|right| Energies (E) of the two lowest discrete states of the Hydrogen Molecular Ion <math>H_2^{+}</math>, as a function of inter-nuclear distance (R) in atomic units.]] From a variation of the [[divergence theorem]] on the above, we finally obtain:
| |
| :<math>
| |
| \Delta E = {} - 2 \, \frac{\int_M \phi_A^{} \mathbf{\nabla} \phi_A^{} \bullet d{\mathbf{S}} }{1-2 \int_R \phi_A^2 ~dV}
| |
| </math>
| |
| where <math> d {\mathbf{S}}</math> is a differential surface element of the mid-plane. This is the Holstein-Herring formula. From the latter, Herring was the first to show<ref name=Herring62/> that the lead term for the asymptotic expansion of the energy difference between the two lowest states of the hydrogen molecular ion, namely the first excited state <math>2 p \sigma_{\mu}</math> and the ground state <math> 1 s \sigma_g </math> (as expressed in [[Molecular term symbol|molecular notation]]—see graph for energy curves), was found to be:
| |
| :<math>
| |
| \Delta E = E_{-} - E_{+} = \frac{4}{e} \, R \, e^{-R}
| |
| </math> | |
| Previous calculations based on the LCAO of atomic orbitals had erroneously given a lead coefficient of <math>4/3</math> instead of <math> 4/e </math>. While it is true that for the Hydrogen molecular ion, the eigenenergies can be mathematically expressed in terms of a generalization of the [[Lambert W function#Generalizations|Lambert W function]], these asymptotic formulae are more useful in the long range and the Holstein–Herring method has a much wider range of applications than this particular molecule.
| |
| | |
| ==Applications==
| |
| The Holstein-Herring formula had limited applications until around 1990 when Tang, [[Jan Peter Toennies|Toennies]], and Yiu<ref>{{cite journal |last=Tang |first=K. T. |authorlink2=Jan Peter Toennies |last2=Toennies |first2=J. P. |last3=Yiu |first3=C. L. |title=The exchange energy of H<sub>2</sub><sup>+</sup> calculated from polarization perturbation theory |journal=[[Journal of Chemical Physics|J. Chem. Phys.]] |volume=94 |issue=11 |pages=7266–7277 |year=1991 |doi=10.1063/1.460211 |bibcode = 1991JChPh..94.7266T }}</ref> demonstrated that <math>\phi_A^{}</math> can be a ''polarized'' wave function, i.e. an atomic wave function localized at a particular nucleus but perturbed by the other nuclear center, and consequently without apparent gerade or ungerade symmetry, and nonetheless the Holstein-Herring formula above can be used to generate the correct asymptotic series expansions for the exchange energies. In this way, one has successfully recast a two-center formulation into an effective one-center formulation. Subsequently, it has been applied with success to one-active electron systems. Later, Scott ''et al.'' explained and clarified their results while sorting out subtle but important issues concerning the true convergence of the polarized wave function.<ref>{{cite journal |last=Scott |first=T. C. |authorlink2=Alexander Dalgarno |last2=Dalgarno |first2=A. |last3=Morgan III |first3=J. D. |year=1991 |title=Exchange energy of H<sub>2</sub><sup>+</sup> calculated from polarization perturbation theory and the Holstein-Herring method |journal=[[Physical Review Letters|Phys. Rev. Lett.]] |volume=67 |issue=11 |pages=1419–1422 |doi=10.1103/PhysRevLett.67.1419 |bibcode = 1991PhRvL..67.1419S }}</ref><ref>{{cite journal |last=Scott |first=T. C. |last2=Babb |first2=J. F. |last3=Dalgarno |first3=A. |last4=Morgan III |first4=J. D. |title=Resolution of a Paradox in the Calculation of Exchange Forces for H<sub>2</sub><sup>+</sup> |journal=[[Chemical Physics Letters|Chem. Phys. Lett.]] |volume=203 |issue=2–3 |pages=175–183 |year=1993 |doi=10.1016/0009-2614(93)85383-Y |bibcode = 1993CPL...203..175S }}</ref><ref>{{cite journal |last=Scott |first=T. C. |last2=Babb |first2=J. F. |authorlink3=Alexander Dalgarno |last3=Dalgarno |first3=A. |last4=Morgan III |first4=J. D. |title=The Calculation of Exchange Forces: General Results and Specific Models |journal=[[Journal of Chemical Physics|J. Chem. Phys.]] |volume=99 |issue=4 |pages=2841–2854 |year=1993 |bibcode=1993JChPh..99.2841S |doi=10.1063/1.465193 }}</ref>
| |
| | |
| The outcome meant that it was possible to solve for the asymptotic exchange energy splittings to any order. The Holstein–Herring method has been extended to the ''two-active electron'' case i.e. the hydrogen molecule for the two lowest discrete states of <math>H_2</math><ref>{{cite journal |last=Herring |first=C. |last2=Flicker |first2=M. |title=Asymptotic Exchange Coupling of Two Hydrogen Atoms |journal=[[Physical Review|Phys. Rev. A]] |volume=134 |issue=2A |pages=362–366 |year=1964 |doi=10.1103/PhysRev.134.A362 |bibcode = 1964PhRv..134..362H }}</ref> and also for general atom-atom systems.<ref>{{cite journal |last=Scott |first=T. C. |last2=Aubert-Frécon |first2=M. |last3=Andrae |first3=D. |last4=Grotendorst |first4=J. |last5=Morgan III |first5=J. D. |last6=Glasser |first6=M. L. |title=Exchange Energy for Two-Active-Electron Diatomic Systems Within the Surface Integral Method |journal=AAECC |volume=15 |issue=2 |pages=101–128 |year=2004 |doi=10.1007/s00200-004-0156-6 }}</ref>
| |
| | |
| ==Physical Interpretation==
| |
| The Holstein–Herring formula can be physically interpreted as the electron undergoing "[[quantum tunnelling]]" between both nuclei, thus creating a current whose flux through the mid-plane allows us to isolate the exchange energy. The energy is thus shared, i.e. ''exchanged'', between the two nuclear centers. Related to the tunnelling effect, a complementary interpretation from [[Sidney Coleman]]'s ''"Aspects of Symmetry"'' (1985) has an "[[instanton]]" travelling near and about the classical paths within [[path integral formulation]]. Note that the volume integral in the denominator of the Holstein-Herring formula is sub-dominant in <math>R</math>. Consequently this denominator is almost unity for sufficiently large internuclear distances <math>R</math> and only the surface integral of the numerator need be considered.
| |
| | |
| ==See also==
| |
| * [[Delta_potential#Double_Delta_Potential|Dirac Delta function model]] ( 1-D version of H<sub>2</sub><sup>+</sup>)
| |
| * [[Exchange interaction]]
| |
| * [[Exchange symmetry]]
| |
| * [[Conyers Herring]]
| |
| * [[Hydrogen molecular ion]]
| |
| * [[Lambert_W_function#Generalizations|Lambert W function]]
| |
| * [[Quantum tunneling]]
| |
| | |
| ==References==
| |
| {{Reflist|30em}}
| |
| | |
| {{DEFAULTSORT:Holstein-Herring Method}}
| |
| [[Category:Quantum chemistry]]
| |
I have others, some of which I’ve stored since I was a boy—two Barlows, which will probably be passed along to my grandsons at some point; several lock-backs I picked up somewhere or another; and a fairly giant Schrade that I keep in my fishing bag in case I’m attacked by a bear on the river. I have a number of Swiss Military knives and 4 or five multi-instruments, which I contemplate more instrument than knife. I have a combination knife/cash clip—two ways to guard your money in one tool. Horváth, László; Kovács, Tibor; Szabó, Miklós (1987). Transdanubia I (Corpus of Celtic Finds in Hungary) (v. 1) Akademiai Kiado. pp. 38–forty one. ISBN 978-963-05-3807-7
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A pocket knife is a must-have, nifty instrument for everyone. You never know when the circumstances demand a pocket knife and that too in a fix! So, it is best to be prepared and carry the most prime-rated, sharpest knife around. A cool pocket-knife may even garner lots of envy! Do you need a pocket knife? Spyderco – I haven’t had quite a lot of expertise with these personally, but I do know that this is the one model that is actually made with self protection because the meant use. I can even say that I do know that the quality of the blade and building has all the time been superb.
Just like Swiss Army Knives this Stainless Metal Small Folding Pocket Knife is a perfect crusing and camping gear. Featuring a set of 13 helpful tools together with a scissor, file, scaler, metal blade, screwdriver, noticed, bottle opener and a bright LED gentle with inbuilt battery. It’s vital software for mountaineering, trekking or any other outside activities. Local assemblywoman Shannon Grove, who was sitting at the similar desk as Larwood, said her husband jumped into action and immediately gave her the Heimlich maneuver. When that failed, he referred to as for a doctor who also tried and failed to clear her airways with the Heimlich maneuver.
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