Excitation energies of diatomic molecules#
In this section, we show how to compute the first electronic excitation energies and oscillator strengths of three diatomic molecules, namely NO, N2 and O2. These simple examples cover three different spin multiplicities of the ground state: doublet, singlet and triplet, respectively.
Nitrogen molecule#
[Input: recipes/linresp/diatomic/N2]
First, we examine the N2 case. This is a close-shell diatomic molecule. We are interested in the low-lying electronic transitions to excited states with both singlet and triplet multiplicities. The solution of the Casida equation is performed after a single-point (static) DFTB ground-state calculation for the optimised geometry. All this is done in a single DFTB run (one single input file). The dftb_in.hsd input file should look like this:
Geometry = GenFormat {
<<< "geo_in.gen"
}
Driver = {}
Hamiltonian = DFTB {
SCC = Yes
SCCTolerance = 1.0E-10
MaxAngularMomentum = {
N = "p"
}
SlaterKosterFiles = Type2FileNames {
Prefix = "../../../slakos/mio-ext/"
Separator = "-"
Suffix = ".skf"
}
SpinConstants = {
N = {-0.026} # HOMO Wpp
}
}
ExcitedState {
Casida {
NrOfExcitations = 10
Symmetry = both
Diagonaliser = Arpack{}
}
}
The input shows a standard spin-unpolarised static DFTB calculation,
except for a new block denoted as ExcitedState with an embedded
subblock called Casida. Once the SCC loop is converged, this block
instructs DFTB+ to build the response matrix according to Casida,
using the just-obtained DFTB orbitals and energies, and diagonalise
it.
Apart from this new block, there is another unusual thing in our input. The Hamiltonian block contains the spin constants, although our calculation is not spin-polarised. Those are actually not used for the ground state calculation, but necessary for the computation of the triplet excitations. That is, if we would be only interested in singlet transitions, we could just omit the spin constants in the input file.
Let us now take a closer look at the Casida block. The keyword
NrOfExcitations specifies how many transitions per spin symmetry, or
multiplicity, we want to compute. In Symmetry we specify the
multiplicity of the transition (either singlet, triplet or both). The
multiplicity of the transition is the difference between the
multiplicities of the excited state and the ground state. In our
example, we are instructing DFTB+ to calculate the first 10
excitations with singlet symmetry and the first 10 excitations with
triplet symmetry. Since the N2 ground state has multiplicity 1, this
means the first 10 singlet-to-singlet and 10 singlet-to-triplet
transitions.
Once the calculation is finished (it takes a fraction of a second), the output file EXC.DAT contains the excitation energies and oscillator strengths, as well as other valuable information. It should look like this:
w [eV] Osc.Str. Transition Weight KS [eV] Sym.
================================================================================
7.321 0.00000000 4 -> 7 0.648 9.014 T
8.118 0.00000000 5 -> 7 1.000 8.118 T
8.118 0.00000000 5 -> 6 1.000 8.118 T
9.014 0.00000000 4 -> 6 0.826 9.014 T
9.014 0.00000000 3 -> 6 0.715 9.014 T
9.014 0.00000000 3 -> 7 0.930 9.014 T
12.062 0.00000000 2 -> 6 0.997 12.062 T
12.062 0.00000000 2 -> 7 0.997 12.062 T
22.118 0.00000000 1 -> 6 0.795 22.118 T
22.118 0.00000000 1 -> 7 0.795 22.118 T
8.118 0.00000000 5 -> 7 0.804 8.118 S
8.118 0.00000000 5 -> 6 0.804 8.118 S
9.014 0.00000000 3 -> 7 0.911 9.014 S
9.014 0.00000000 4 -> 6 0.969 9.014 S
9.014 0.00000000 4 -> 7 0.772 9.014 S
12.062 0.00000000 2 -> 7 0.978 12.062 S
12.062 0.00000000 2 -> 6 0.978 12.062 S
12.750 0.80120458 3 -> 6 0.647 9.014 S
22.118 0.00000000 1 -> 6 0.884 22.118 S
22.118 0.00000000 1 -> 7 0.884 22.118 S
The triplet transitions are listed first, followed by the singlet
ones. They can be identified by the letter T or S in the last
column.
The first column w [eV] is the excited state energy we are looking
for, the second one Osc.Str. lists the corresponding oscillator
strength. The column Transition reports the indices of the dominant
molecular orbitals involved in the electronic transition. In our
example, the singlet state at 12.75 eV features a transition from the
occupied Kohn-Sham orbital 3 (HOMO-2) to the virtual orbital 6 (the
LUMO). The next column Weight indicates the weight of the
corresponding singly excited determinant in the CIS expansion of the
excited state. Values close to one indicate that the excited state is
well described by a single electronic excitation, while small values
speak for a collective excitation. Column KS [eV] provides the
Kohn-Sham transition energy difference \(\omega_{ia\sigma} =
\epsilon_{a\sigma} - \epsilon_{i\sigma}\) (see above).
Oxygen molecule#
[Input: recipes/linresp/diatomic/O2]
For the O2 molecule, we will consider its triplet ground state. This
is specified in the input file through the
Hamiltonian/SpinPolarisation block:
SpinPolarisation = Colinear {
UnpairedElectrons = 2
}
Our excited state block will in this case looks like this:
ExcitedState {
Casida {
NrOfExcitations = 10
Diagonaliser = Arpack{}
}
}
We are instructing DFTB+ to compute the first 10 excitations. Note that since our system is not closed-shell, we can no longer separate our eigenvalue problem into two independent singlet and triplet equations, so we have to build and diagonalise the entire response matrix in this case. But, how do we know the spin multiplicities of the computed transitions? We get this information from the last column of the EXC.DAT file:
w [eV] Osc.Str. Transition Weight KS [eV] D<S*S>
================================================================================
6.353 0.00000000 5 -> 6 0.829 6.353 0.000
6.353 0.00000000 4 -> 6 0.787 6.353 0.000
6.353 0.00000000 5 -> 7 0.722 6.353 0.000
6.793 0.00000000 3 -> 6 0.993 6.793 0.000
6.793 0.00000000 3 -> 7 0.993 6.793 0.000
8.204 0.23976646 4 -> 7 0.617 6.353 0.007
14.567 0.00000000 2 -> 7 0.989 14.567 0.000
14.567 0.00000000 2 -> 6 0.989 14.567 0.000
22.424 0.00000000 6 -> 8 0.800 22.424 0.000
22.424 0.00000000 7 -> 8 0.800 22.424 0.000
In the last column are the expectation values of the square of the total spin operator for the transitions. A value of zero means we have a singlet transition (triplet to triplet). Note that we may have transitions with some spin contamination (transitions leading to unphysical states). In our next example, we will explore this in more detail.
Nitric oxide molecule#
[Input: recipes/linresp/diatomic/NO]
Finally, we have the NO molecule, with one unpaired electron (doublet ground state). The symmetry of NO leads to degenerate orbitals, which causes problems with the SCC convergence. We therefore additionally provide a small electronic temperature to ease the ground state computation:
Filling = Fermi {
Temperature [K] = 40
}
NOTE: If your structure features a band gap, it is typically neither necessary nor advisable to set the electronic temperature different from 0 K. The code also works with fractional occupations, but the response matrix will be much larger then necessary and this will cause long calculations. In addition, the memory need increases significantly.
In this case, the first 10 excitations are:
w [eV] Osc.Str. Transition Weight KS [eV] D<S*S>
================================================================================
7.478 0.00209868 4 -> 7 0.556 8.534 1.418
7.772 0.00000000 5 -> 6 1.000 7.772 -0.000
7.772 0.00000000 5 -> 7 1.000 7.772 -0.000
7.793 0.00000000 5 -> 6 0.779 7.793 0.999
7.793 0.00000000 5 -> 7 0.779 7.793 0.999
8.534 0.00000000 4 -> 6 0.707 8.534 0.999
8.534 0.00000000 3 -> 6 0.728 8.534 0.999
8.636 0.00000000 4 -> 6 0.984 8.636 0.000
8.636 0.00000000 3 -> 7 0.657 8.636 -0.000
11.652 0.49971991 3 -> 6 0.600 8.636 -0.597
Let us pay attention to the last column of the EXC.DAT
file. Contrary to the previous case, here we obtain large non-zero
\(\Delta S^2\) values. When \(\Delta S^2 = 0\), we are in the
presence of a doublet-to-doublet transition. Likewise, if
\(\Delta S^2 = 2\), we would have an excitation to a quadruplet
state. Otherwise, we have some extent of spin contamination in our
obtained transitions. The last column should help us determine which
excitations are to be trusted. We can set an arbitrary spin
contamination threshold to establish which transitions we will
consider leading to a physical excited state. In our NO-TiO2
recipe, we will compute the absorption spectrum of a system where
transitions with a spin contamination beyond an imposed threshold are
excluded.