## The H polymer

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) a.
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:

.
There is a periodic reciprocal direction for each of the periodic direction in the direct space, the one where the cell with an hydrogen atom can be visualised.
In the reciprocal space, there are particular points that are important for their symmetry; in fact, when the energy levels are plotted as a faction of k, the degeneracy can occur at these point. The origin of the reciprocal space is called GAMMA, .

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 for H4.
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 points (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:

Questions:

1. How many atoms are there in the cell? Look also the the output from the CRYSTAL calculation (the LogFile).
2. How many bands are visualised in the band structure?
3. What are the differences between the band structure and the energy levels diagrams obtained in the previous sections, H100 and H100 II?
4. Where are the bonding and anti-bonding states?

Optional questions::

1. Is there any non-bonding state?
2. Consider the band structure of the hydrogen chain, can you predict the band structure for a chain of Li atoms?

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