The function calculateProjectedHamiltonianEnergySpectrum() calculates the energy spectrum of the molecular projected self-consistent Hamiltonian (MPSH). By this, we mean that the self-consistent Hamiltonian is projected onto specified atoms in the central region, and then diagonalized to produce an energy spectrum. This can be used to show how the molecular levels for a molecule (e.g. hydrogen) are affected when placed between between the two electrodes.

The code sample below shows how the energy spectrum, projected onto the hydrogen atoms, is calculated. We export the calculated spectrum to a VNLFile, which can be imported and visualized in VNL.

from ATK.TwoProbe import *

# Restore calculation
scf = restoreSelfConsistentCalculation('lih2li-scf.nc')

# Calculate energy spectrum for the two H atoms
mpsh_spectrum = calculateProjectedHamiltonianEnergySpectrum(
    scf, projection_atoms = [3,4]
    )

# Store energy spectrum in a VNL file
vnl_file = VNLFile("lih2li-mpsh.vnl")
vnl_file.addToSample(mpsh_spectrum,'lih2li-mpsh')

lih2li-mpsh-spectrum.py

The projection_atoms specify which atoms we want to calculate the eigenstates for. The numbers refer to the order of the elements in the central region, as they were specified in the TwoProbeConfiguration used in the self-consistent field calculation. In the above script, we have specified atom 3 and 4 corresponding to the two hydrogen atoms.

If you cannot remember or do not know which atoms that are in the central region, the following script will display all elements in the central region along with their Cartesian coordinates.

def displayCentralRegionAtoms(atomic_configuration):
    '''
    This function will print information about the atoms located
    in the central region of a two-probe system. This information
    involves:
    * Internal order (number)
    * Element type
    * Cartesian coordinates of the atom
    '''
    print 'Central region atomic configuration:'
    print '='*50
    print 'Number\tElement\tCartesian coordinates (Ang)'
    for i in  range(len(atomic_configuration.elements())):
        print i,'\t',atomic_configuration.elements()[i].symbol(),'\t',
        for x in atomic_configuration.cartesianCoordinates()[i]:
            print x.inUnitsOf(Angstrom),'\t',
        print
    print '='*50

displayCentralRegion.py

You can use this function in the following way:

from ATK.TwoProbe import *
from displayCentralRegion import displayCentralRegionAtoms

vnl_file=VNLFile("lih2li.vnl")
atomic_configuration_dict = vnl_file.readAtomicConfigurations()
atomic_configuration = atomic_configuration_dict["lih2li"]
displayCentralRegionAtoms(atomic_configuration)
      

which should print the following to your screen provided you have been using the file lih2li.vnl for storing the atomic configuration used throughout these examples.

Atoms located in the scattering region:
==================================================
Number  Element Cartesian coordinates
0       Li      [0.0 Ang 0.0 Ang -8.568 Ang]
1       Li      [0.0 Ang 0.0 Ang -5.668 Ang]
2       Li      [0.0 Ang 0.0 Ang -2.768 Ang]
3       H       [0.0 Ang 0.0 Ang -0.402 Ang]
4       H       [0.0 Ang 0.0 Ang 0.402 Ang]
5       Li      [0.0 Ang 0.0 Ang 2.768 Ang]
6       Li      [0.0 Ang 0.0 Ang 5.668 Ang]
7       Li      [0.0 Ang 0.0 Ang 8.568 Ang]
==================================================

This confirms that the atoms chosen as projection_atoms indeed were our hydrogen atoms.

By comparing the molecular projected energy spectrum for the hydrogen atoms in the two-probe system with that of molecular hydrogen, we can see that the energy gap between the HOMO and the LUMO is smaller for the hydrogen atoms in the two-probe system. This reduction of the HOMO–LUMO gap is directly responsible for the large value of the transmission coefficient of the hydrogen molecule observed experimentally.

Figure 41 shows the energy spectrum of the hydrogen atoms in the two-probe system along with the transmission spectrum for the entire Li-H₂-Li two-probe system. Here, we see that only the LUMO of the hydrogen atoms take part in conducting electrons from one electrode to the other, as that is the only orbital which is located at energies where transmission is noticeable.

The calculateProjectedHamiltonianEigenstates() function calculates wave-functions (i.e. eigenstates) of the projected molecular Hamiltonian discussed just above. This way, one can study how the molecular orbitals for a molecule are changed when the molecule is placed between between the two electrodes.

The script below shows how to calculate the eigenstates of a molecule in a two-probe scattering region using the calculateProjectedHamiltonianEigenstates() method. We want to analyze the HOMO and LUMO eigenstates, corresponding to the first and second eigenstate for a hydrogen molecule. The complete script for doing this is shown below.

from ATK.TwoProbe import *

# Restore calculation
scf = restoreSelfConsistentCalculation('lih2li-scf.nc')

# Reopen old two-probe configuration
vnl_file=VNLFile("lih2li.vnl")
atomic_configuration = vnl_file.readAtomicConfigurations()["lih2li"]

# Calculate HOMO and LUMO eigenstates for the H atoms in the two-probe system
eigenstates = calculateProjectedHamiltonianEigenstates(
    scf, projection_atoms = [3,4],
    quantum_numbers = [0,1]
    )

# Store results in VNL files
vnl_file = VNLFile("lih2li-mpsh.vnl")
vnl_file.addToSample(atomic_configuration,'lih2li_mpsh')
for eigenstate in eigenstates:
    vnl_file.addToSample(eigenstate, 'lih2li_mpsh')

lih2li-mpsh.py

Remember, that in Python the first element in a list has the index zero. So when we specify the quantum states as

quantum_numbers = [0,1]

we mean the first and the second eigenstate, which in a hydrogen molecule correspond to the HOMO and LUMO states.

The produced file can now be loaded into VNL where the orbitals can be visualized.