H2 molecule
[Shortcut to examples
01
02
03
04
05
06
and exercises
]
by Andrea Marini & Daniele Varsano
The material
- The molecule is along the x direction in the center of a 25 × 25 × 25 (a.u.) cubic cell.
- 2 atom in the cell (2 electrons)
- Plane waves cutoff 14 Hartree (40000 RL vectors)
Introduction
The H2 monomer constitutes an example of a perfectly zero dimensional system. The key feature of this molecule is that the polarizability is well described within the ALDA approximation, in contrast with what we have seen the in the previous systems.
[01] Initialization: 01_init(yambo -i)
During the initialization run we need to calculate the closed shells of the Reciprocal space vectors. However if we enter the 0D_h2/yambo directory and type
localhost:>yambo -D [RD./SAVE/ns.db1]------------------------------------------- Bands : 250 K-points : 1 G-vectors [RL space]: 39127 Components [wavefunctions]: 39127 Symmetries [spatial+T-rev]: 16 Spinor components : 1 Spin polarizations : 1 Temperature [ev]: 0.000000 Electrons : 2.000000 - S/N 009258 ---------------------------- v.03.00.00 r.000 -
we notice that the G-vectors needed to describe the charge of the H2 molecule are almost 40000! This is a common feature of localized systems. To reduce the compuatatio time in the Inputs/01_init we have defined
MaxGvecs= 10000 RL # [INI] Max number of G-vectors planned to useTo run the initialization type
localhost:>yambo -F Inputs/01_init -J 01_init
[02-03] Random-Phase approximation: 02_RPA_no_LF(yambo -o c) 03_RPA_LF(yambo -o c) 03_RPA_LF_r_space(yambo -o b -y d) 03_RPA_LF_xl(yambo -o b -y d) 03_RPA_LF_QP(yambo -o b -y d -V qp)
To calculate the polarizability spectrum in the simple independent particle approximation we follow the same scheme described previously. In Inputs/02_RPA_no_LF a simple RPA calculation is obtained when
NGsBlkXd=1 RL # (Xd) Response block sizeTo run this example type
localhost:>yambo -F Inputs/02_RPA_no_LF -J 02_RPA_no_LF
Now to include Local Fields effects (and later on xc-effects
in the ALDA approximation) we should increase
the NGsBlkXd variable. Unfortunately with
almost 40000 RL vectors in the charge we would reach
the convergence using NGsBlkXd=2000.
When the TDDFT equation is solved in reciprocal space this
value for NGsBlkXd corresponds to the inversion
of a 2000 × 2000 matrix, times as many
frequencies at which we want to calculate the polarizability.
This cumbersome calculation can be avoided using
the Bloch representation of the TDDFT equation,
corresponding, in yambo, to a BS equation with a modified
exchange part that may include the ALDA kernel.
To use this option we calculate again the RPA polarizabilty
using the input file Inputs/03_RPA_LF_r_space
where, because of the RPA, we have
BSresKmod= "x" # [BSK] Resonant Kernel mode. (`x`;`c`;`d`)
that means: no correlation terms. The next figure confirms that the two procedures lead to very similar results but the real space calculation needs much less cpu-time.
localhost:>yambo -F Inputs/02_RPA_LF -J 02_RPA_LF localhost:>yambo -F Inputs/03_RPA_LF_r_space -J 03_RPA_LF_r_space
Still using the Bloch representation of the TDDFT equation we include Local Field effects up to 9000(!) RL components with and without a QP gap correction of 7.6 eV.
localhost:>yambo -F Inputs/03_RPA_LF -J 03_RPA_LF localhost:>yambo -F Inputs/03_RPA_LF_QP -J 03_RPA_LF_QP
obtaining
From this plot we see how strong the Local Field effects are in H2. This is due to the strong inhomogeneity of the cell charge that has a peak in the cell center while it is very small at the cell boundaries.
[04] ALDA: 04_ALDA_r_space(yambo -o b -t a)
As mentioned before the TDDFT polarizability in the simple ALDA approximation can be obtained in the Bloch representation using the input file Inputs/04_alda_r_space
localhost:>yambo -F Inputs/04_alda_r_space -J 04_alda_r_space
We see the the effect of including xc effects is to counteract the Local Field effects increasing the axial polarizability :
| RPA (no LF) | RPA | ALDA | RPA-QP | |
| α(ω=0) | 20.78 | 9.70 | 12.51 | 6.60 |
While the TDDFT calculations (RPA and ALDA) give a very similar axial polarizability the QP corrections strongly reduce it not in agreement with the Hartree-Fock calculations.
[05] The Statically screened Electron-electron interaction: 05_W(yambo -b)
As a first step to introduce more elaborate approximations for fxc we calculate the statically screened electron-electron interaction of H2: using (remember to not use the -J so that the database ndb.em1s is stored where all subsequent runs can read it):
localhost:>yambo -F Inputs/05_W
[06] The Bethe-Salpeter equation, Excitons: 06_BSE(yambo -o b -y d)
The input file Inputs/06_BSE can now calculate the BS polarizability. We notice the use of the BS kernel including exchange scatterings in the coupling part
BScplKmod= "x" # [BSK] Coupling Kernel mode. (`x`;`c`;`d`)At difference with 2D and 3D systems, the presence of strong charge inhomogeneity enhances the strength of exchange scatterings between electron-hole pairs with opposite energy. While generally negligible these terms cannot be neglected in the H2 molecule.
localhost:>yambo -F Inputs/06_BSE -J 06_BSE
The resulting BS polarizability is very similar to the ALDA spectrum, showing that the QP gap correction is almost entirely compensated by the "excitonic" effects.
Additional Exercises
- Calculate the absorption spectra like in section 02-03 both in RL space and the Bloch representation but using a resonant only kernel. Check the two methods yield the same results, and see the effect of the coupling term (compare it with the linear chain).