Holstein-Herring method

The Holstein- Herring method , also known under the English names surface integral method or Smirnov's method , is an effective method in quantum physics for calculating the exchange energy splitting of asymptotically degenerate energy states in molecular systems. Although the exchange energy splitting is becoming more and more difficult to calculate for increasingly larger internuclear distances , it is of fundamental importance for the theories of binding in molecules and magnetism. ${\displaystyle R}$

theory

The basic idea of ​​the Holstein-Herring method can be illustrated using the example of the hydrogen molecule ion , or more generally, the atom-ion systems or "systems with one active electron" as follows. We consider molecular states, which are described by state functions, which behave even or odd under space inversion. This is identified by the suffixes g and u and is the standard for identifying the electronic states of diatomic molecules (for atomic states, on the other hand, the English expressions "even" and "odd" are used ). The corresponding electronic Schrödinger equation can be written as:

${\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\psi =E\psi ~,}$

where E is the (electronic) energy of a selected quantum mechanical state (eigenstate), with an electronic state function that depends on the position coordinates of the electron, and where is the Coulomb potential of the electron-nucleus interaction. The following applies to the hydrogen molecular ion : ${\displaystyle \psi =\psi (\mathbf {r} )}$ ${\displaystyle V}$

${\displaystyle V=-{\frac {e^{2}}{4\pi \varepsilon _{0}}}\left({\frac {1}{r_{a}}}+{\frac {1}{r_{b}}}\right)}$

For any even state, the electronic Schrödinger equation can be written in atomic units ( ) as: ${\displaystyle \hbar =m=e=4\pi \varepsilon _{0}=1}$

${\displaystyle \left(-{\frac {1}{2}}\nabla ^{2}+V({\textbf {x}})\right)\psi _{+}=E_{+}\psi _{+}}$

For any odd state the corresponding wave equation can be written as:

${\displaystyle \left(-{\frac {1}{2}}\nabla ^{2}+V({\textbf {x}})\right)\psi _{-}=E_{-}\psi _{-}}$

For simplicity we assume real functions (although the end result can be generalized for the case of complex functions). Now we multiply the equation for the even function from the left by , the equation for the odd function from the left by , and get the difference: ${\displaystyle \psi _{-}}$ ${\displaystyle \psi _{+}}$

${\displaystyle \psi _{+}\nabla ^{2}\psi _{-}-\psi _{-}\nabla ^{2}\psi _{+}={}-2\,\Delta E\,\psi _{-}\psi _{+}\;.}$

where is the exchange energy split . In the next step we define, without loss of generality, orthogonal one-particle functions, and , which are localized at the nuclei and write: ${\displaystyle \Delta E=E_{-}-E_{+}}$ ${\displaystyle \phi _{A}^{}}$ ${\displaystyle \phi _{B}^{}}$

${\displaystyle \psi _{+}={\frac {1}{\sqrt {\,2}}}~(\phi _{A}^{}+\phi _{B}^{})\;,\qquad \psi _{-}={\frac {1}{\sqrt {\,2}}}~(\phi _{A}^{}-\phi _{B}^{})\;.}$

This is similar to the LCAO (Linear combination of atomic orbitals molecular orbital method) approach used in quantum chemistry, but we have to emphasize that the functions and are generally “polarized”; i.e., they are not pure eigenfunctions of the angular momentum operators with regard to their respective centers (see also below). However, in the limiting case , the localized functions are reduced to the well-known atomic (hydrogen-like) psi functions . We now denote the plane perpendicular to the line connecting the nucleus in the middle between the two nuclei (see diagram for hydrogen molecular ion for further details), with a unit vector perpendicular to this plane (this vector is parallel to the Cartesian direction), so that the total three-dimensional space is divided into a left ( ) and a right ( ) half-space. From symmetry considerations it follows: ${\displaystyle \phi _{A}^{}}$ ${\displaystyle \phi _{B}^{}}$ ${\displaystyle \phi _{A,B}^{}}$ ${\displaystyle R\rightarrow \infty }$ ${\displaystyle \phi _{A,B}^{0}}$ ${\displaystyle M}$ ${\displaystyle {\mathbf {z} }}$ ${\displaystyle z}$ ${\displaystyle \mathbf {R} ^{3}}$ ${\displaystyle L}$ ${\displaystyle R}$

${\displaystyle \left.\psi _{-}\right|_{M}=\mathbf {z} \cdot \left.\mathbf {\nabla } \psi _{+}\right|_{M}=0\;.}$

This implies that:

${\displaystyle \left.\phi _{A}^{}\right|_{M}=\left.\phi _{B}^{}\right|_{M}\;,\qquad {\mathbf {z} }\cdot \left.\mathbf {\nabla } \phi _{A}^{}\right|_{M}={}-\mathbf {z} \cdot \left.\mathbf {\nabla } \phi _{B}^{}\right|_{M}\;.}$

The localized functions are normalized so that the following must apply:

${\displaystyle \int _{L}\phi _{A}^{2}~dV=\int _{R}\phi _{B}^{2}~dV}$

and vice versa. Integration of this result over the entire space to the left of the level gives: ${\displaystyle M}$

${\displaystyle 2\int _{L}\psi _{+}\psi _{-}~dV=\int _{L}(\phi _{A}^{2}-\phi _{B}^{2})~dV=1-2\int _{R}\phi _{A}^{2}~dV}$

and

${\displaystyle \int _{L}(\psi _{+}\nabla ^{2}\psi _{-}-\psi _{-}\nabla ^{2}\psi _{+})~dV=\int _{L}(\phi _{B}^{}\nabla ^{2}\phi _{A}^{}-\phi _{A}^{}\nabla ^{2}\phi _{B}^{})~dV}$

Energy (E) of the two lowest bound states of the hydrogen molecular ion , as a function of the core-core distance (R) in atomic units. ${\displaystyle H_{2}^{+}}$

Applying a variant of the Gaussian integral theorem to this result finally leads to the Holstein-Herring formula:

${\displaystyle \Delta E={}-2\,{\frac {\int _{M}\phi _{A}^{}\mathbf {\nabla } \phi _{A}^{}\bullet d{\mathbf {S} }}{1-2\int _{R}\phi _{A}^{2}~dV}}}$

where is a differential surface element of the median plane . With this formula, Herring was able to show for the first time that the leading term of the asymptotic development of the energy difference between the two lowest states of the hydrogen molecular ion, i.e. the first excited state and the ground state (denoted by molecular notation - see above figure for the energy curves ), has the following mathematical form: ${\displaystyle d{\mathbf {S} }}$ ${\displaystyle M}$ ${\displaystyle 2p\sigma _{u}}$ ${\displaystyle 1s\sigma _{g}}$

${\displaystyle \Delta E=E_{-}-E_{+}={\frac {4}{e}}\,R\,e^{-R}}$

Previous calculations based on the LCAO approximation for the atomic orbitals had incorrectly resulted in the prefactor instead . ${\displaystyle 4/3}$ ${\displaystyle 4/e}$

Applications

The Holstein-Herring formula had limited relevance for applications until around 1990 when Tang, Toennies , and Yiu showed that a polarized function can be, i.e. H. an atomic wave function localized at one of the two nucleus locations, which is distorted by the influence of the other nucleus and therefore no longer has a clear symmetry (even or odd). Nevertheless, the Holstein-Herring formula given above can be used and provides the correct asymptotic series expansion for the exchange energy split. In this way, an original two-center problem has also been successfully transformed into an effective one-center problem. Subsequently, this formula was extended for two-center problems with one active electron (e.g. alkali dimer cations). By Scott et al. the understanding of this initially surprising result was deepened, which required the clarification of subtle but important questions about the convergence of the polarized functions. The result of this analysis means that in principle any order of the asymptotic series expansion of the exchange energy splitting can be calculated. The Holstein-Herring method has also been extended to the case of two active electrons, i.e. H. for the two lowest bound states of the hydrogen molecule and more general diatomic systems. ${\displaystyle \phi _{A}^{}}$ ${\displaystyle H_{2}}$

Physical interpretation

The Holstein-Herring formula given above can be interpreted physically as follows: The electron tunnels back and forth between the two nuclei, thereby generating a current whose flux density through the central plane allows the exchange energy split to be determined. Regarding the tunnel effect, a supplementary interpretation of Sidney Coleman 's "Aspects of Symmetry" has an " instanton " journey to the vicinity and via the classic path within the path integral . So this energy is shared by both nuclei, i.e. H. exchanged . It should also be noted that the volume integral is subdominant in the denominator of the Holstein-Herring formula, so that for sufficiently large core-core distances the denominator can simply be set to one and only the surface integral needs to be calculated in the numerator. ${\displaystyle M}$ ${\displaystyle R}$

Individual evidence

1. ↑ Holstein T., "Mobilities of positive ions in their parent gases," J. Phys. Chem. 56 , 832-836 (1952).
2. ↑ Holstein T., Westinghouse Research Report 60-94698-3-R9, (unpublished), (1955).
3. ^ ^{A b} Herring C. , "Critique of the Heitler-London Method of Calculating Spin Couplings at Large Distances", Rev. Mod. Phys. 34 : 631-645 (1962).
4. ↑ Bardsley JN, Holstein T., Junker BR, and Sinha S., "Calculations of ion-atom interactions relating to resonant charge-transfer collisions", Phys. Rev. A 11 , 1911-1920 (1975).
5. ^ TC Scott, M. Aubert-Frécon, D. Andrae, “Asymptotics of Quantum Mechanical Atom-Ion Systems”, AAECC (Applicable Algebra in Engineering, Communication and Computing) 13 , 233-255 (2002).
6. ↑ Aubert- Frécon M., Scott TC, Hadinger G., D. Andrae, Grote Dorst J., and Morgan III JD, "Asymptotically Exact Calculation of the Exchange Energies of One-Active Electron Diatomic ion with the Surface Integral Method" J. Phys. B: At. Mol. Opt. Phys., 37 , pp. 4451-4469 (2004). Asymptotically exact calculation of the exchange energies of one-active-electron diatomic ions with the surface integral method Abstract in: Journal of Physics B: Atomic, Molecular and Optical Physics - IOPscience
7. ↑ Smirnov BM and Chibisov MI, "Electron exchange and changes in the hyperfine state of colliding alkaline metal atoms" Sov. Phys. JETP 21 , 624-628 (1965).
8. ↑ Tang KT, Toennies JP , and Yiu CL, “The exchange energy of H2 + calculated from polarization perturbation theory”, J. Chem. Phys. 94 : 7266-7277 (1991).
9. ^ Scott TC, Dalgarno A. and Morgan III JD (1991). "Exchange Energy of H2 + Calculated from Polarization Perturbation Theory and the Holstein-Herring Method", Phys. Rev. Lett. 67 : 1419-1422. Phys. Rev. Lett. 67, 1419 (1991): Exchange energy of H_ {2} ^ {+} calculated from polarization perturbation theory and the Holstein-Herring method
10. ↑ Scott TC, Babb JF, Dalgarno A. and Morgan III JD, "Resolution of a Paradox in the Calculation of Exchange Forces for H2 +", Chem. Phys. Lett. 203: 175-183 (1993). Abstract
11. ↑ Scott TC, Babb JF, Dalgarno A. and Morgan III JD, "The Calculation of Exchange Forces: General Results and Specific Models", J. Chem. Phys., 99 , 2841-2854, (1993). bibcode : 1993JChPh..99.2841S
12. ^ Herring C., and Flicker M., "Asymptotic Exchange Coupling of Two Hydrogen Atoms," Phys. Rev. A 134 , 362-366 (1964).
13. ^ Scott TC, Aubert-Frécon M., Andrae D., Grotendorst J., Morgan III JD and Glasser ML, "Exchange Energy for Two-Active-Electron Diatomic Systems Within the Surface Integral Method", AAECC, 15 , 101-128 (2004). doi : 10.1007 / s00200-004-0156-6