Microscopic Modelling of Correlated Low-dimensional Systems

Chapter 2: Method 7
through the Coulomb interaction. Formally, the Hamiltonian of such a system is written in
the following general form [63] :
ˆH = − �
i
� 2
2me
∇ 2 i − �
I
� 2
2MI
∇ 2 I + e2
2
�
j�=i
1
|ri − rj|
+ e2
2
�
J�=I
ZIZJ
� ZI
− e2
|RI − RJ| |RI − ri|
I,i
(2.1)
where r = ri, i = 1..N, is a set of N electronic coordinates and R = RI, I = 1... Ñ, is a set of
Ñ nuclear coordinates. me, ZI and MI are the electron mass, the nuclear charges and masses
respectively. The first two terms in Eq. (2.1) are the electronic and nuclear kinetic energies,
the next two terms are the electron-electron and nuclei-nuclei Coulomb interactions, the
last term is the electrostatic Coulomb interaction between nuclei and electrons. Here we
neglect relativistic effects which would add extra terms in this Hamiltonian like the spin-
orbit coupling. Electrons are fermions, therefore the total electronic wave function must
be antisymmetric with respect to exchange of two electrons. Strictly speaking, the atomic
nuclei, composed by protons and neutrons, can be of bosonic or fermionic nature depending
on whether the number of protons plus the number of neutrons is an even or odd number 1 .
All the ingredients are perfectly known and, in principle, the properties of the many-body
problem can be derived by solving the many-body stationary Schrödinger equation:
ˆHΨi(r, R) = EiΨi(r, R) (2.2)
Where Ψ is the wave function of the many body system. In practice, this problem is
almost impossible to treat in a full quantum mechanical framework. Only in few cases a
complete analytic solution is available, and numerical solutions are also limited to a very
small number of particles. There are several features that contribute to this difficulty. First,
this is a multicomponent many-body system, where each component obeys a particular
statistics. Second, the complete wave function cannot be easily factorized because of the
Coulomb interaction, therefore, in general, one has to deal with (3Ñ+3N) spatial coupled
degrees of freedom. The usual way of minimizing the degrees of freedom in the many body
problem, is neglecting the kinetic energy of the nuclei in Eq. (2.1). This approximation is
justified by the fact that the ratio of mass me/MI ≈ 1/2000 ≪ 1, therefore the movement
of the nuclei is much slower than that of the electrons. Thus, we can consider the electronic
1 In solid state is common to define the nuclei as the atomic nuclei of protons and neutrons plus the core
electrons in filled shells