2.7 Hamilton Operator of the Solid
33
where r i and R i are the position operators and p i and P i are the momentum operators of the electrons
and nuclei, respectively. The first term is the kinetic energy of the electrons, the second term is the
kinetic energy of the nuclei. The third term is the electrostatic interaction of the nuclei, the fourth term
is the electrostatic interaction of the electrons. In the third and fourth terms the summation over the
same indices is left out. The fifth term is the electrostatic interactions of electrons and nuclei.
In the following, the usual approximations in order to treat (2.19) are discussed. First, the nuclei
and the electrons tightly bound to the nuclei (inner shells) are united with ion cores. The remaining
electrons are the valence electrons.
The next approximation is the Born–Oppenheimer (or adiabatic) approximation. Since the ion cores
are much heavier than the electrons (factor ≈ 10
3 ) they move much slower. The frequencies of the
ion vibrations are typically in the region of several tens of meV (phonons, cf. Sect. 5.2), the energy
to excite an electron is typically 1 eV. Thus, the electrons always ‘see’ the momentary position of the
ions, the ions, however, ‘see’ the electron motions averaged over many periods. Thus, the Hamiltonian
(2.19) is split into three parts:
H = H ions (R j ) + H e (r i , R j 0 ) + H e−ion (r i , δR j ) .
(2.20)
The first term contains the ion cores with their potential and the time-averaged contribution of the
electrons. The second term is the electron motion around the ion cores at their averaged positions
R j 0 . The third term is the Hamiltonian of the electron–phonon interaction that depends on the electron
positions and the deviation of the ions from their average position δR j = R j − R j 0 . The electron–
phonon interaction is responsible for such effects as electrical resistance and superconductivity.
33
where r i and R i are the position operators and p i and P i are the momentum operators of the electrons
and nuclei, respectively. The first term is the kinetic energy of the electrons, the second term is the
kinetic energy of the nuclei. The third term is the electrostatic interaction of the nuclei, the fourth term
is the electrostatic interaction of the electrons. In the third and fourth terms the summation over the
same indices is left out. The fifth term is the electrostatic interactions of electrons and nuclei.
In the following, the usual approximations in order to treat (2.19) are discussed. First, the nuclei
and the electrons tightly bound to the nuclei (inner shells) are united with ion cores. The remaining
electrons are the valence electrons.
The next approximation is the Born–Oppenheimer (or adiabatic) approximation. Since the ion cores
are much heavier than the electrons (factor ≈ 10
3 ) they move much slower. The frequencies of the
ion vibrations are typically in the region of several tens of meV (phonons, cf. Sect. 5.2), the energy
to excite an electron is typically 1 eV. Thus, the electrons always ‘see’ the momentary position of the
ions, the ions, however, ‘see’ the electron motions averaged over many periods. Thus, the Hamiltonian
(2.19) is split into three parts:
H = H ions (R j ) + H e (r i , R j 0 ) + H e−ion (r i , δR j ) .
(2.20)
The first term contains the ion cores with their potential and the time-averaged contribution of the
electrons. The second term is the electron motion around the ion cores at their averaged positions
R j 0 . The third term is the Hamiltonian of the electron–phonon interaction that depends on the electron
positions and the deviation of the ions from their average position δR j = R j − R j 0 . The electron–
phonon interaction is responsible for such effects as electrical resistance and superconductivity.