The Holstein–Herring method, also called the surface integral method, or Smirnov's method is an effective means of getting the 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). The basic idea was pioneered by Theodore Holstein, Conyers Herring and Boris M. Smirnov in the 1950-1960.
Theory The method can be illustrated for the hydrogen molecular ion or more generally, atom-ion systems or 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 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 time-independent Schrödinger equation can be written as:
( − ℏ 2 2 m ∇ 2 + V ) ψ = E ψ , {\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\psi =E\psi ~,}
where E is the (electronic) energy of a given quantum mechanical state (eigenstate), with the electronic state function ψ = ψ ( r ) {\displaystyle \psi =\psi (\mathbf {r} )} depending on the spatial coordinates of the electron and where V {\displaystyle V} is the electron-nuclear Coulomb potential energy function. For the hydrogen molecular ion, this is:
V = − e 2 4 π ε 0 ( 1 r a + 1 r b ) {\displaystyle V=-{\frac {e^{2}}{4\pi \varepsilon _{0}}}\left({\frac {1}{r_{a}}}+{\frac {1}{r_{b}}}\right)}
For any gerade (or even) state, the electronic Schrödinger wave equation can be written in atomic units ( ℏ = m = e = 4 π ε 0 = 1 {\displaystyle \hbar =m=e=4\pi \varepsilon _{0}=1} ) as:
( − 1 2 ∇ 2 + V ( x ) ) ψ + = E + ψ + {\displaystyle \left(-{\frac {1}{2}}\nabla ^{2}+V({\textbf {x}})\right)\psi _{+}=E_{+}\psi _{+}}
For any ungerade (or odd) state, the corresponding wave equation can be written as:
( − 1 2 ∇ 2 + V ( x ) ) ψ − = E − ψ − {\displaystyle \left(-{\frac {1}{2}}\nabla ^{2}+V({\textbf {x}})\right)\psi _{-}=E_{-}\psi _{-}}
For simplicity, we assume real functions (although the result can be generalized to the complex case). We then multiply the gerade wave equation by ψ − {\displaystyle \psi _{-}} on the left and the ungerade wave equation on the left by ψ + {\displaystyle \psi _{+}} and subtract to obtain:
ψ + ∇ 2 ψ − − ψ − ∇ 2 ψ + =
− 2 Δ E ψ − ψ + . {\displaystyle \psi _{+}\nabla ^{2}\psi _{-}-\psi _{-}\nabla ^{2}\psi _{+}={}-2\,\Delta E\,\psi _{-}\psi _{+}\;.}
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