BO approximation

BO approximation#

What you need to know

Born-Oppenheimer (BO) approximation, Because nuclei are much heavier than electrons, the Schrodinger equation can be approximately separated into the nuclear and the electron parts. Thus the electronic Schrodinger equation for a molecule can be solved separately at each fixed nuclear configuration. This is called the Born-Oppenheimer approximation.
BO approximation allows us to extend multi-electron treatment of atoms to molecules.
Single electron wavefunctions of molecules are called Molecular Orbitals (MO)
Geometry of $H_{2}$ molecule defined by its bond length $R$ is fixed when determining molecular orbitals.
Nuclear geometry is a varyable parameter in the problem: For different values of R one gets different energies and MOs.

Simplest molecule#

In the following, we will consider the simplest molecule $H_{2}^{+}$ , which contains only one electron. This simple system will demonstrate the basic concepts in chemical bonding. The Schr”odinger equation for H $_{2}^{+}$ is:

H ψ ({\vec{r}}_{1}, {\vec{R}}_{A}, {\vec{R}}_{B}) = E ψ ({\vec{r}}_{1}, {\vec{R}}_{A}, {\vec{R}}_{B})

where ${\vec{r}}_{1}$ is the vector locating the (only) electron and ${\vec{R}}_{A}$ and ${\vec{R}}_{B}$ are the positions of the two protons. The Hamiltonian for H $_{2}^{+}$ is:

\hat{H} = - \frac{ℏ^{2}}{2 M} (Δ_{A} + Δ_{B}) - \frac{ℏ^{2}}{2 m_{e}} Δ_{e} + \frac{e^{2}}{4 π ϵ_{0}} (\frac{1}{R} - \frac{1}{r_{1 A}} - \frac{1}{r_{1 B}})

where $M$ is the proton mass, $m_{e}$ is the electron mass, $r_{1 A}$ is the distance between the electron and nucleus A, $r_{1 B}$ is the distance between the electron and nucleus B and $R$ is the A - B distance.

Note that the Hamiltonian includes also the quantum mechanical kinetic energy for the protons. As such the wavefunction depends on ${\vec{r}}_{1}$ , ${\vec{R}}_{A}$ and ${\vec{R}}_{B}$ .

The BO approximation#

Because the nuclear mass $M$ is much larger than the electron mass $m_{e}$ , the wavefunction can be separated (Born-Oppenheimer approximation):

ψ ({\vec{r}}_{1}, {\vec{R}}_{A}, {\vec{R}}_{B}) = ψ_{e} ({\vec{r}}_{1}, R) ψ_{n} ({\vec{R}}_{A}, {\vec{R}}_{B})

where $ψ_{e}$ is the electronic wavefunction that depends on the distance $R$ between the nuclei and $ψ_{n}$ is the nuclear wavefunction depending on ${\vec{R}}_{A}$ and ${\vec{R}}_{B}$ . It can be shown that the nuclear part can be often be separated into vibrational, rotational and translational parts. The electronic Schr”odinger equation can now be written as:

{\hat{H}}_{e} ψ_{e} = E_{e} ψ_{e}

Note that equation depends parametrically on $R$ (``one equation for each value of $R$ ’’). The electronic Hamiltonian is:

{\hat{H}}_{e} = - \frac{ℏ^{2}}{2 m_{e}} Δ_{e} + \frac{e^{2}}{4 π ϵ_{0}} (\frac{1}{R} - \frac{1}{| r_{1} - R_{A} |} - \frac{1}{| r_{1} - R_{B} |})

Because $R$ is a parameter, both $E_{e}$ and $ψ_{e}$ are functions of $R$ .

BO approximation

Contents

BO approximation#

Simplest molecule#

The BO approximation#