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| The '''Born–Huang approximation'''<ref name=BHCrystalBook>{{cite book|last=Born|first=Max|title=Dynamical Theory of Crystal Lattices|year=1954|publisher=Oxford University Press|location=Oxford|coauthors=Kun, Huang}}</ref> (named after [[Max Born]] and [[Huang Kun]]) is an approximation closely related to the [[Born–Oppenheimer approximation]]. It takes into account diagonal nonadiabatic effects in the electronic Hamiltonian than the Born–Oppenheimer approximation.<ref>[http://www.chemistry.mcmaster.ca/courses/3bb3/z_HW/Set%201.pdf Mathematical Methods and the Born-Oppenheimer Approximation]</ref> Despite the addition of correction terms, the electronic states remain uncoupled under Born–Huang approximation, making it an adiabatic approximation.
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| ==Mathematical Formula==
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| The Born–Huang approximation asserts that the representation matrix of nuclear kinetic energy operator in the basis of Born-Oppenheimer electronic wavefunctions is diagonal:
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| :<math>
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| \langle\chi_{k'}(\mathbf{r};\mathbf{R}) |
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| T_\mathrm{n}|\chi_k(\mathbf{r};\mathbf{R})\rangle_{(\mathbf{r})}= \mathcal{T}_\mathrm{k}(\mathbf{R})\delta_{k'k}
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| </math>
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| ==Consequences==
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| The Born–Huang approximation loosens the Born-Oppenheimer approximation by including some electronic matrix elements, while at the same time maintains its diagonal structure in nuclear equations of motion. As a result, the nuclei still move on isolated surfaces, obtained by addition of a small correction to the [[potential energy surface]].
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| Under Born–Huang approximation, the Schrödinger equation of the molecular system simplifies to
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| :<math>
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| \left[ T_\mathrm{n} +E_k(\mathbf{R})+\mathcal{T}_\mathrm{k}(\mathbf{R})\right] \; \phi_k(\mathbf{R}) =
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| E \phi_k(\mathbf{R})
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| \quad\mathrm{for}\quad k=1,\ldots, K,
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| </math>
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| The quantity <math>\left[E_k(\mathbf{R})+\mathcal{T}_\mathrm{k}(\mathbf{R})\right]</math>serves as the corrected potential energy surface.
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| ==Born-Huang approximation as a method to evaluate non-adiabatic correction==
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| The evaluation of correction term <math>\mathcal{T}_\mathrm{k}(\mathbf{R})</math> is straightforward and can be obtained with little additional cost when the electronic wavefunction is known. However, the Born–Huang approximation is rarely used as a way to obtain non-adiabatic correction.
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| The Born–Huang approximation shares the same applicability as the Born-Oppenheimer approximation and requires that the electronic states involved are well separated. However, the diagonal correction from Born–Huang approximation is only large when potential energy surfaces comes close to each other, on which occasion the approximation itself breaks down. In fact, with a small additional cost, the accuracy of energy under Born–Huang approximation is very similar to that obtained under Born-Oppenheimer approximation. These factors prevented Born–Huang approximation from being used as a way to evaluate non-adiabatic corrections to energies.
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| ==Upper-Bound property==
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| The value of Born–Huang approximation is that it provides the upper-bound for the ground state energy.<ref name=BHCrystalBook/> The Born-Oppenheimer approximation, on the other hand, provides the lower-bound for this value.<ref>{{cite journal|last=Epstein|first=Saul T.|title=Ground-State Energy of a Molecule in the Adiabatic Approximation|journal=The Journal of Chemical Physics|date=1 January 1966|volume=44|issue=2|pages=836|doi=10.1063/1.1726771|url=http://dx.doi.org/10.1063/1.1726771|bibcode = 1966JChPh..44..836E }}</ref>
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| As a result, the Born–Huang approximation can be used as a very efficient way to validate the Born-Oppenheimer approximation, avoiding the need for explicit treatment of multiple electronic states and [[vibronic coupling]]s, which is very complicated and computationally demanding. When the energy obtained from Born-Huang and Born-Oppenheimer approximations provide nearly identical values, adiabatic approximations can be safely used for the process. When these two values differ by a significant value, explicit non-adiabatic methods have to be employed to yield correct results.
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| ==See also==
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| * [[Vibronic coupling]]
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| * [[Born-Oppenheimer Approximation]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Born-Huang Approximation}}
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| [[Category:Quantum chemistry]]
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| [[Category:Approximations]]
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Nice to satisfy you, my name is Refugia. To do aerobics is a thing that I'm completely addicted to. For many years he's been operating as a receptionist. California is exactly where I've usually been living and I love each working day residing here.
my web site; cs.sch.ac.kr