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In this tutorial, you will learn how to do a Reduced Density Matrix Functional Theory (RDMFT) calculation with Octopus.

In contrast to density functional theory or Hartree-Fock theory, here we do not try to find one optimal slater-determinant (single-reference) to describe the ground-state of a given many-body system, but instead approximate the one-body reduced density matrix (1RDM) of the system. Thus, the outcome of an RDMFT minimization is a set of eigen-vectors, so called natural orbitals (NOs,) and eigenvalues, so-called natural occupation numbers (NONs), of the 1RDM. One importance aspect of this is that we need to use more' orbitals than the number of electrons. This additional freedom allows to include static correlation in the description of the systems and hence, we can describe settings or processes that are notoriously difficult with any single-reference method. One such process is the dissociation of a molecule, which thus becomes our example for this tutorial: You will learn how to do a properly converged dissociation study of H_2. A good recap of RDMFT can be found in ...

Basis Steps of an RDMFT Calculation

The first thing to notice for RDMFT calculations is that, we will explicitly make use of a basis set, which is in contrast to the minimization over the full real-space grid that is performed normally with Octopus. So we always need to do a preliminary calculation to generate our basis set. So let us start with the hydrogen molecule in equilibrium position. Create the following inp file:

CalculationMode = gs
TheoryLevel = independent_particles

Dimensions = 3
Radius = 8
Spacing = 0.15

# distance between H atoms
d = 1.4172 # (equilibrium bond length H2)

 'H' | 0 | 0 | -d/2
 'H' | 0 | 0 | d/2

ExtraStates = 14

You should be already familiar with all these variables, so there is no need to explain them again, but we want to stress the two important points here:

  1. We chose TheoryLevel = independent_particles, which you should always use as a basis in an RDMFT calculation (see actual octopus paper?)
  2. We set the ExtraStates variable, which controls the size of the basis set that will be used in the RDMFT calculation. Thus in RDMFT, we have with the basis set a new numerical parameter besides the Radius and the Spacing that needs to be converged for a successful calculation. The size of the basis set M is just the number of the orbitals for the ground state (number of electrons divided by 2) plus the number of extra states.
  3. All the numerical parameters depend on each other: If we want to include many ExtraStates to have a sufficiently large basis, we will need also a larger Radius and especially a smaller Spacing at a certain point. The reason is that the additional states will have function values bigger than zero in a larger region than the bound states. Additionally all states are orthogonal to each other, and thus the grid needs to resolve more and more nodes.

With this basis, we can now do our first rdmft calculation. For that, you just need to change the TheoryLevel to 'rdmft' and add the part ExperimentalFeatures = yes. In the standard out there will be some new information:

  • Calculating Coulomb and exchange matrix elements in basis --this may take a while--
  • Occupation numbers: they are not zero or one/two here but instead fractional!
  • anything else important?

Basis Set Convergence

To find good value for the Radius and Spacing, one should perform convergence sets with a larger number of ExtraStates, in our case say 14 (thus M=1+19=20) and make sure that they are converged. Having done this, we can go on and converge the basis set. So let us perform a series.

For that, we first create large enough basis by repeating the above calculation with say ExtraStates = 29. We copy the restart folder in a new folder that is called 'basis' and execute the following script (if you rename anything you need to change the respective part in the script):


echo "#ES    Energy    1. NON  2.NON" > $outfile
list="4 9 14 19 24 29"

export OCT_PARSE_ENV=1 
for param in $list 
  mkdir $folder
  cd $folder
   # here we specify the basis folder
   cp -r ../basis/restart .
   # create inp file
cat <<-EOF
calculationMode = gs
TheoryLevel = rdmft
ExperimentalFeatures = yes

ExtraStates = $param

Dimensions = 3

Radius = 8
Spacing = 0.15

# distance between H atoms
d = 1.4172 # (equilibrium) 

  'H' | 0 | 0 | -d/2
  'H' | 0 | 0 | d/2

Output = density
OutputFormat = axis_x
} > inp
  # put the octopus dir here
  [octopus-dir]/bin/octopus > $out_tmp

   energy=`grep -a "Total energy" $out_tmp  | tail -1 | awk '{print $3}'`
   seigen=`grep -a " 1  " static/info |  awk '{print $2}'`
   peigen=`grep -a " 2  " static/info |  awk '{print $2}'`
   echo $param $energy $seigen $peigen >> ../$outfile
 cd ..

If everything works out fine, you should find the following values in the 'ES-series.log' file:

#ES    Energy    1. NON  2.NON
4 -1.1476150018E+00 1.935750757008 0.032396191164
9 -1.1498006205E+00 1.932619193929 0.032218954381
14 -1.1609241676E+00 1.935215440985 0.032526426664
19 -1.1610006378E+00 1.934832116929 0.032587169713
24 -1.1622104536E+00 1.932699204653 0.032997371081
29 -1.1630839347E+00 1.932088486112 0.032929702131

So we see that even with M=30, the calculation is not entirely converged! Thus for a correctly converged result, one would need to further increase the basis and probably resort to a cluster calculation. For the purposes of this tutorial, we will use not entirely converged parameters.

H2 Dissociation

Now, we can go on with our original goal: the H2 dissociation curve.