Born-oppenheimer approximation

The Born- Oppenheimer (BO) guess is the most popular numerical estimate of atomic elements in quantum science and sub-atomic physical science. In particular, it is the presumption that the wave elements of nuclear cores and electrons in a particle can be dealt with independently because of the way that the cores are a lot heavier than the electrons. Because of the enormous relative mass of a core contrasted with an electron, the directions of the cores in a framework are approximated as fixed, while the directions of the electrons are dynamic. The methodology is named after Max Conceived and J. Robert Oppenheimer, who proposed it in 1927, in the early time of quantum mechanics.

In quantum chemistry, an approximation is frequently employed to expedite the computation of big molecules’ molecular wavefunctions and other features.

The approximation can lose validity in some situations where the presumption of separable motion is broken down, but even in these situations, it is frequently used as a jumping-off point for more sophisticated techniques.

Molecules have N nuclei (µ, v,….) and n electrons (I, j,…..) the complete Hamiltonian operator, neglecting the spin-orbit interaction and other magnetic interactions are,

\[\displaystyle \hat{H}=-\frac{{{{\hbar }^{2}}}}{2}\sum\limits_{\mu }^{N}{{\frac{1}{{m\mu }}}}\nabla _{\mu }^{2}-\frac{{{{\hbar }^{2}}}}{{2{{m}_{e}}}}\sum\limits_{i}^{n}{{\nabla _{i}^{2}}}+\sum\limits_{{\mu ,v}}{{\frac{{{{z}_{\mu }}{{z}_{v}}{{e}^{2}}}}{{{{{\left( {4\pi {{\varepsilon }_{0}}} \right)}}_{{{{r}_{{\mu v}}}}}}}}}}-\sum\limits_{{i,\mu }}{{\frac{{{{z}_{\mu }}{{e}^{2}}}}{{{{{\left( {4\pi {{\varepsilon }_{0}}} \right)}}_{{{{r}_{{iu}}}}}}}}}}+\sum\limits_{{i,j}}{{\frac{{{{e}^{2}}}}{{{{{\left( {4\pi {{\varepsilon }_{0}}} \right)}}_{{{{r}_{{ij}}}}}}}}}}-----(1)\]

Above equation (1) there is a five-term is described as a different indication, first term act as nuclear kinetic energy, the second electronic kinetic energy; third, nuclear potential energy; the fourth, nuclear – electron attraction potential energy, and the fifth, electric repulsion potential energy.

Schrodinger equation for the molecule is as follows,

\[\displaystyle \hat{H}\psi =E\psi -----(2)\]

where E = Total energy,

ψ = Total wave function

Compared to electrons, the proton is heavy about 1840 times, which is why compared to electronic motion fast than nuclear motion.

Suppose two motions are independent of each other, as a product of the electronic wave function ψ_eand nuclear function Ψ_n is the total wave function written as

\[\displaystyle \psi ={{\psi }_{e}}{{\psi }_{n}}-----(3)\]

The above equation (1) and (2) write a Schrodinger equation for Born-approximation for electronic motion, the nuclear kinetic energy is eliminated,

\[\displaystyle \left[ {-\frac{{{{\hbar }^{2}}}}{{2{{m}_{e}}}}\sum\limits_{i}^{n}{{\nabla _{i}^{2}}}+\sum\limits_{{\mu ,v}}{{\frac{{{{z}_{\mu }}{{z}_{v}}{{e}^{2}}}}{{{{{\left( {4\pi {{\varepsilon }_{0}}} \right)}}_{{{{r}_{{\mu v}}}}}}}}}}-\sum\limits_{{i,\mu }}{{\frac{{{{z}_{\mu }}{{e}^{2}}}}{{{{{\left( {4\pi {{\varepsilon }_{0}}} \right)}}_{{{{r}_{{iu}}}}}}}}}}+\sum\limits_{{i,j}}{{\frac{{{{e}^{2}}}}{{{{{\left( {4\pi {{\varepsilon }_{0}}} \right)}}_{{{{r}_{{ij}}}}}}}}}}} \right]{{\psi }_{e}}=E{{\psi }_{e}}----(4)\]

See that in equation (4), for a particular fixed nuclear framework, the electronic wave function ψ_e depends only on the coordinates of the electrons, and the nuclear coordinates do not affect ψ_e and E_e.

But if the nuclear configuration is fixed, the second term of the nuclear repulsion potential energy is considered constant. The electronic energy E_e wrote as,

\[\displaystyle {{E}_{e}}=E-\sum\limits_{{\mu ,v}}{{\frac{{{{z}_{\mu }}{{z}_{v}}{{e}^{2}}}}{{{{{\left( {4\pi {{\varepsilon }_{0}}} \right)}}_{{{{r}_{{\mu v}}}}}}}}}}-----(5)\]

And equation 4 becomes,

\[\displaystyle \left[ {\sum\limits_{i}{{\left( {-\frac{{{{\hbar }^{2}}}}{{2{{m}_{e}}}}\nabla _{i}^{2}-\sum\limits_{{i,\mu }}{{\frac{{{{z}_{\mu }}{{e}^{2}}}}{{{{{\left( {4\pi {{\varepsilon }_{0}}} \right)}}_{{{{r}_{{iu}}}}}}}}}}} \right)+\sum\limits_{{i,j}}{{\frac{{{{e}^{2}}}}{{{{{\left( {4\pi {{\varepsilon }_{0}}} \right)}}_{{{{r}_{{ij}}}}}}}}}}}}} \right]{{\psi }_{e}}={{E}_{e}}{{\psi }_{e}}----(6)\]

The Schrodinger equation for nuclear motion is written as,

\[\displaystyle \left[ {-\frac{{{{\hbar }^{2}}}}{2}\sum\limits_{\mu }^{N}{{\frac{1}{{m\mu }}}}\nabla _{\mu }^{2}+\sum\nolimits_{e}{{\left( {{{r}_{{\mu v}}}} \right)}}} \right]{{\psi }_{n}}={{E}_{n}}{{\psi }_{n}}-----(7)\]

Nuclear wave function ψ_n = Rotational and vibrational motions in the potential field supplied by electron

Therefore equation (7) represents the nuclear motion using an effective Hamiltonian in which the potential energy is the one the fixed nuclei approximation gives it.

The Born – Oppenheimer approximation of the electronic and nuclear motions has been separated is depicts equations (6) and (7)

\[\displaystyle {{E}_{e}}={{E}_{{e\left( R \right)}}}\]

Here, obtaining E_e, a function of R in equation (1) is solved for a fixed value of internuclear separation, and r_µ_v is denoted as R.

For a diatomic molecule plot, a graph E_{e (R)} vs R gives the potential energy curve shown in the graph;

Energy vs. Internuclear distance

The most stable nuclear configuration for a given electronic state corresponds to the value of R to a minimum in energy. And the value of E_e obtained from equation (6) is used in equation (7) as potential energy.

The molecule’s electronic and nuclear motions are the total energy, E.

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Bhoomika Sheladiya

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