Born-Oppenheimer approximation

This article deals with the approximation of the many-particle Schrödinger equation. See Born's approximation for the approximation in scattering theory, also named after M. Born.

Calculations of the properties of molecules, e.g. B. binding energies or vibrational states are usually carried out on the basis of the Born-Oppenheimer approximation.

The Born-Oppenheimer approximation or Born-Oppenheimer approximation (according to Max Born and J. Robert Oppenheimer ) or adiabatic approximation is an approximation for simplifying the Schrödinger equation of systems made up of several particles . It takes advantage of the fact that heavy and light particles in a system change their direction of movement on very different time scales, and that the equations of motion of the fast, light particles can therefore be solved meaningfully without taking into account the movement of the slow, heavy ones.

The Born-Oppenheimer approximation is used in the quantum mechanical treatment of molecules and solids , as these consist of at least two atomic nuclei and a large number of much lighter electrons . The approximation is also widely used in physical chemistry , since here only for the simplest systems, e.g. B. the hydrogen atom , an analytically exact solution of the Schrödinger equation is known. The Born-Oppenheimer approximation was first published in 1927 in the Annalen der Physik .

The Born-Oppenheimer approximation leads to good results for molecules in the ground state , especially for those with heavy nuclei. However, it can lead to very poor results for excited molecules and cations , which is particularly important for photoelectron spectroscopy .

motivation

The quantum mechanical state function of a molecule or solid is a function of the degrees of freedom of all electrons and atomic nuclei. In the following, the spin degrees of freedom of the particles are dispensed with; then the positions of all electrons in the vector are summarized, the positions of all atoms in the vector . ${\displaystyle \Psi _{\mathrm {mol} }({\vec {r}},{\vec {R}})}$ ${\displaystyle n}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle N}$ ${\displaystyle {\vec {R}}}$

As usual, the state function is determined from the associated Schrödinger equation:

${\displaystyle {\hat {H}}_{\mathrm {mol} }\cdot \Psi _{\mathrm {mol} }({\vec {r}},{\vec {R}})=E\cdot \Psi _{\mathrm {mol} }({\vec {r}},{\vec {R}})}$

The molecular Hamilton operator

${\displaystyle {\hat {H}}_{\mathrm {mol} }={\hat {T}}_{\mathrm {e} }+{\hat {T}}_{\mathrm {N} }+{\hat {V}}_{\mathrm {ee} }+{\hat {V}}_{\mathrm {NN} }+{\hat {V}}_{\mathrm {eN} }}$

contains the kinetic operators

${\displaystyle {\hat {T}}_{\mathrm {e} }=-\sum \limits _{i=1}^{n}{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{i}^{2}}$( kinetic energy of the electrons)

and

${\displaystyle {\hat {T}}_{\mathrm {N} }=-\sum \limits _{I=1}^{N}{\frac {\hbar ^{2}}{2M_{I}}}\nabla _{I}^{2}}$ (kinetic energy of the nuclei)

also the repulsion between the electrons

${\displaystyle {\hat {V}}_{\mathrm {ee} }={\frac {q_{0}}{4\pi \varepsilon _{0}}}\sum \limits _{i<j}{\frac {1}{|{\vec {r}}_{i}-{\vec {r}}_{j}|}}}$,

the repulsion between the nuclei

${\displaystyle {\hat {V}}_{\mathrm {NN} }={\frac {q_{0}}{4\pi \varepsilon _{0}}}\sum \limits _{I<J}{\frac {Z_{I}Z_{J}}{|{\vec {R}}_{I}-{\vec {R}}_{J}|}}}$,

and the attraction between nuclei and electrons

${\displaystyle {\hat {V}}_{\mathrm {eN} }=-{\frac {q_{0}}{4\pi \varepsilon _{0}}}\sum \limits _{i,I}{\frac {Z_{I}}{|{\vec {r}}_{i}-{\vec {R}}_{I}|}}}$.

The molecular Schrödinger equation can only be solved analytically for the simplest systems. A numerical solution of the complete system is also not feasible due to the high dimensionality . In order to make the molecular Schrödinger equation solvable, an approximation is necessary.

principle

The Born-Oppenheimer approximation separates the molecular Schrödinger equation into an equation for the electrons and one for the nuclei. The two sub-problems can then be solved much more easily by using symmetries . The separation of the electronic and nuclear degrees of freedom is based on the large difference in mass, which leads to greater inertia of the nuclei. Since all particles with each other mainly by Coulomb forces interact that are equally, essentially, the light electrons are much more accelerated than the cores.

The essence of the Born-Oppenheimer approximation can be represented as follows:

• From the perspective of the electrons, the nuclei practically stand still. At first the kinetic operator of the nuclei is neglected . This results in a Schrödinger equation for the electrons, in which the position of the nuclei is included as a parameter in the attractive potential and in the repulsive potential . This results in electronic eigenstates and associated eigenenergies that depend parametrically on the positions of the nuclei. ${\displaystyle ({\hat {T}}_{\mathrm {N} }\approx 0)}$ ${\displaystyle {\hat {V}}_{\mathrm {eN} }}$ ${\displaystyle {\hat {V}}_{\mathrm {NN} }}$ ${\displaystyle \phi _{h}({\vec {r}},{\vec {R}})}$ ${\displaystyle E_{h}}$
• In contrast to this, the movement of the nuclei is almost unaffected by the instantaneous position of the electrons. However, the nuclei feel the inherent energy of the electronic state. Every electronic state creates its own potential, in which the nuclei then move.

Mathematical formulation

The precondition for the Born-Oppenheimer approximation is that the movement of the electrons and that of the nuclei can be separated. This assumption leads to a molecular wave function , which consists of a product of the wave function of the electrons and the wave function of the nuclei: ${\displaystyle \Psi _{\mathrm {mol} }}$ ${\displaystyle \phi }$ ${\displaystyle \eta }$

${\displaystyle \Psi _{\mathrm {mol} }=\phi \cdot \eta }$

It is also assumed that:

• the electron wave function except positions of the electrons and of the positions dependent upon the nuclei, but not of their speeds: . This means that the nuclear motion is so much smaller than the electron motion that it can be assumed to be fixed and only flows in as a parameter. ${\displaystyle \phi }$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {R}}}$ ${\displaystyle \phi =\phi ({\vec {r}},{\vec {R}})}$
• the core wave function only of the core coordinate dependent: . ${\displaystyle \eta }$ ${\displaystyle {\vec {R}}}$ ${\displaystyle \eta =\eta ({\vec {R}})}$

If you now apply the Hamilton operator to the entire wave function, you get two separate expressions: ${\displaystyle {\hat {H}}_{\mathrm {mol} }}$

• a Schrödinger equation for the motion of the electrons:

${\displaystyle {\hat {H}}_{\mathrm {e} }\cdot \phi _{h}({\vec {r}},{\vec {R}})=E_{h}({\vec {R}})\cdot \phi _{h}({\vec {r}},{\vec {R}})}$

With

${\displaystyle {\hat {H}}_{\mathrm {e} }={\hat {T}}_{\mathrm {e} }+{\hat {V}}_{\mathrm {ee} }+{\hat {V}}_{\mathrm {eN} }}$

• and a Schrödinger equation for the motion of the nuclei:

${\displaystyle ({\hat {T}}_{\mathrm {n} }+{\hat {V}}_{\mathrm {NN} }+E_{h}({\vec {R}}))\cdot \eta _{hk}({\vec {R}})=E_{hk}\cdot \eta _{hk}({\vec {R}})}$

method

The electronic Schrödinger equation is solved successively for different core distances . Finally, one obtains a relationship between bond length or equilibrium distance and the energy of the molecule or the dissociation energy of the bond . This relationship is shown in the potential curve or potential hypersurface, in which the electronic energy of the molecule is shown as a function of one or more parameters of the nuclear geometry.

See also

• Franck-Condon principle

Individual evidence

1. ^ JC Slater : Quantum Theory of Molecules and Solids, Vol. 1: Electronic Structure of Molecules . In: American Journal of Physics . tape 32 , 1964, pp. 65 , doi : 10.1119 / 1.1970097 .
2. ↑ M. Born, R. Oppenheimer: To the quantum theory of molecules . In: Annals of Physics . tape 389 , no. 20 , 1927, pp. 457-484 , doi : 10.1002 / andp.19273892002 .