Up to now, a system with a finite number of hydrogen atoms
and a finite number of energy level
has been considered. In this section, a hydrogen chain/polymer
will be studied; the system is infinite, with an
infinite number of atoms and an infinite number of energy levels.
Since the chain is periodic, a cell with only a hydrogen atom
that is repeated by translation in one direction
is the way adopted to tackle this problem.
The cell is described by a lattice vector (or lattice parameter)
The infinite number of electronic states, that form a continuum in energy, are grouped together in a band. The equivalent of the energy level diagram of a finite system is the band structure. Each level is labelled by a continuous variable k: each value of k corresponds to an energy level. The quantity k belongs to a space called, the reciprocal space; in this space a cell can be also defined, the reciprocal cell, and it is useful and convenient, as the system is also periodic in the reciprocal space. The reciprocal lattice parameter a* is related to the lattice parameter of the cell in direct space according to the following formula:
In this section you will calculate the wavefunction and the band structure(that is the energy level diagram for a infinite system) for the hydrogen polymer.
Exercise 1: Start DLVisualize, run a CRYSTAL calculation for the H polymer: H_polymer.inp
Exercise 2: Run a CRYSTAL properties calculation
of the band structure for the H polymer,
as explained in Exercise 3
In other to visualize all the infinite energy levels it is necessary to consider a path in the reciprocal cell that has the same length of a*, for instance:
-a*/2 < k < a*/2
In the CRYSTAL Bands panel, type the coordinates of the
(note that the coordinates are expressed in unit of a*):
Compare the band structure obtained with a second band structure calculation,
with the path indicated below,
in order to show the periodicity in the reciprocal lattice:
However, the following path is generally adopted due to the symmetry
between k and -k:
Exercise 3: Run a CRYSTAL properties calculation
of the density of states
for the H polymer, as explained in Exercise 3 for H4.
Exercise 4: Run a CRYSTAL properties calculation
of the band structure + the density of states
for the H polymer, as explained in Exercise 4 for H4. In the CRYSTAL Bands + DOS panel, select the following path: