Excitons in the Hydrogen chain: The Bethe-Salpeter Equation
Introduction
The H2 Hydrogen chain
constitutes an example of a perfectly one-dimensional
system. This property causes some tricky numerical problems
that are, however, related to a precise physical process: the
extreme confinement of 3D electrons in a small region of space.
So we start using yambo to perform a BS calculation, and you will immediately
see where the problem is !
We start by generating the input file. At the command line we have to tell yambo to construct the BSE (-o b) , to calculate the static screened interaction (-b), and to diagonalize the BS matrix (yambo -y d). The input line will be:
localhost> yambo -o b -b -y d -V qp
The generated input file describes our first attempt to calculate the excitonic polarization spectrum of the H chain:
optics # [R OPT] Optics bse # [R BSK] Bethe Salpeter Equation. em1s # [R Xs] Static Inverse Dielectric Matrix bss # [R BSS] Bethe Salpeter Equation solver KfnQPdb= "none" # [EXTQP BSK BSS] Database KfnQP_N= 1 # [EXTQP BSK BSS] Interpolation neighbours % KfnQP_E 0.000000 | 1.000000 | 1.000000 | # [EXTQP BSK BSS] E parameters (c/v) % % KfnQP_W 0.000 | 0.000 | 0.000 | 0.000 | # [EXTQP BSK BSS] W parameters (c/v) % KfnQP_Z= ( 1.000000 , 0.000000 ) # [EXTQP BSK BSS] Z factor (c/v) BSresKmod= "xc" # [BSK] Resonant Kernel mode. (`x`;`c`;`d`) % BSEBands 1 | 2 | # [BSK] Bands range % BSENGBlk= 1 RL # [BSK] Screened interaction block size BSENGexx= 7659 RL # [BSK] Exchange components XfnQPdb= "none" # [EXTQP Xd] Database XfnQP_N= 1 # [EXTQP Xd] Interpolation neighbours % XfnQP_E 0.000000 | 1.000000 | 1.000000 | # [EXTQP Xd] E parameters (c/v) % % XfnQP_W 0.000 | 0.000 | 0.000 | 0.000 | # [EXTQP Xd] W parameters (c/v) % XfnQP_Z= ( 1.000000 , 0.000000 ) # [EXTQP Xd] Z factor (c/v) % QpntsRXs 1 | 41 | # [Xs] Transferred momenta % % BndsRnXs 1 | 20 | # [BSK] Bands range % NGsBlkXs= 1 RL # [Xs] Response block size % LongDrXs 1.000000 | 0.000000 | 0.000000 | # [Xs] [cc] Electric Field % BSSmod= "d" # [BSS] Solvers `h/d/i/t` % BEnRange 0.00000 | 10.00000 | eV # [BSS] Energy range % % BDmRange 0.10000 | 0.10000 | eV # [BSS] Damping range % BEnSteps= 100 # [BSS] Energy steps % BLongDir 1.000000 | 0.000000 | 0.000000 | # [BSS] [cc] Electric Field %
Remember to change the highlighted values. The first change is to include a QP gap correction of 3.5 eV
% KfnQP_E 3.50000 | 1.000000 | 1.000000 | # [EXTQP BSK BSS] E parameters (c/v) %Then we decide some reasonable size of the inverse dielectric function that defines the statically screened interaction.
NGsBlkXs= 111 BSENGexx= 111
Finally you should change
% BSEBands 1 | 2 | # [BSK] Bands range %2 bands are enough to get reasonable results.
Remember also to uncomment the variable WRbsWF if you want to plot the excitonic wave function afterwords
so we run
localhost> yambo
To our great surprise the polarization spectrum we obtain is completely wrong, with just a poor signal around 3eV which is nonsense.
The reason of this serious failure of the BS calculation is due to the peculiar geometry of the H chain, and of corresponding BZ sampling that is strictly one dimensional, This is readily detected in any report file, like r_setup
[...] [02.01] K-grid lattice ====================== Compatible Grid might be 1D B1 [rlu]= -0.01250 0.00000 0.00000 Grid dimensions : 80 K lattice UC volume [au]: 0.0011
As a consequence the region of space assigned to each k-point is strongly compressed in one of the dimensions, like the thin slices of this picture
The drastic consequence of this compression is that each region receives a piece of the electron-electron interaction that is multiplied by a form factor of the region, that, in general is assumed spatially constant. With such a severe sampling of the BZ this assumption is no longer valid and the screened interaction is anomalously enhanced.
To avoid this anomalous electron-electron interaction we use the Random Integration Method (-c option) described here:
localhost> yambo -c -o b -b -y dand use for the RandQpts and for the RandGvec:
RandQpts= 1000000 # [RIM] Number of random q-points in the BZ RandGvec= 1 RL # [RIM] Coulomb interaction RS components
RandQpts specifies the number of random points to use and RandGvec the number of RL components to evaluate.
After typing
localhost:>yambo
we notice a new section in the report file r_optics_bse_em1s_bss_rim_cut:
[04] Coulomb potential Random Integration (RIM) =============================================== [RD./SAVE/ndb.RIM]------------------------------------------ Brillouin Zone Q/K grids (IBZ/BZ): 41 80 41 80 Coulombian RL components : 1 Coulombian diagonal components :yes RIM random points : 1000000 RIM RL volume [a.u.]: 0.08864 Real RL volume [a.u.]: 0.08820 Eps^-1 reference component :0 Eps^-1 components : 0.00000 0.00000 0.00000 RIM anisotropy factor : 0.00000 - S/N 003292 ---------------------------------- v.02.09.09 - Summary of Coulomb integrals for non-metallic bands |Q|[au] RIM/Bare: Q [1]:0.1000E-4 0.7653 * Q [2]: 0.01745 0.06494 Q [3]: 0.03491 0.17408 * Q [4]: 0.05236 0.28906 Q [5]: 0.06981 0.39545 * Q [6]: 0.08727 0.48820 Q [7]: 0.104720 0.566608 * Q [8]: 0.122173 0.631844 Q [9]: 0.139626 0.685750 * Q [10]: 0.157080 0.730229 Q [11]: 0.174533 0.767000 * Q [12]: 0.191986 0.797519 Q [13]: 0.209440 0.822983 * Q [14]: 0.226893 0.844352 Q [15]: 0.244346 0.862396 * Q [16]: 0.261799 0.877727 Q [17]: 0.279253 0.890832 * Q [18]: 0.296706 0.902101 Q [19]: 0.314159 0.911847 * Q [20]: 0.331613 0.920320 Q [21]: 0.349066 0.927727 * Q [22]: 0.366519 0.934233 Q [23]: 0.383972 0.939973 * Q [24]: 0.401426 0.945061 Q [25]: 0.418879 0.949588 * Q [26]: 0.436332 0.953633 Q [27]: 0.453786 0.957260 * Q [28]: 0.471239 0.960523 Q [29]: 0.488692 0.963470 * Q [30]: 0.506145 0.966138 Q [31]: 0.523599 0.968561 * Q [32]: 0.541052 0.970768 Q [33]: 0.558505 0.972783 * Q [34]: 0.575959 0.974629 Q [35]: 0.593412 0.976322 * Q [36]: 0.610865 0.977879 Q [37]: 0.628319 0.979314 * Q [38]: 0.645772 0.980640 Q [39]: 0.663225 0.981867 * Q [40]: 0.680678 0.983004 Q [41]: 0.698132 0.984061
The interesting result is that 2nd and the 4th column of numbers reflects the effect of the BZ compression: 1 means little effect, small numbers mean large effect. We see that the effect is huge for points in the BZ with small modulus (1st and 3rd columns).
The final BS polarizability is, now, physically correct and substantially different from ALDA.
Additional Exercises
- The H chain is a one dimensional system, and,
consequently it should not absorb when the light
is polarized perpendicularly to the chain axis
(the x direction).
Repeat the RPA, ALDA, BSE calculations for light polarized perpendicular to the chain axis to verify that the polarization is much smaller than in the parallel case. - Use ypp to plot the excitonic wave function.