How to know if the bisection orbitals localized or not?

[ngoclinh84phys]

Hi all,

I am using the Qbox to compute the bisection orbitals for Si nano-particles (about 5\AA).

By starting with the trial values for the bisection augments lx=2 ly=2 lz=2 and threshold=0.02, I obtain an output of the bisection part as below.
Would you mind to help me to explain how can I know if the bisection orbitals are localized or not?


Thanks for your time.

Best,

Linh


[qbox] [qbox] <cmd>bisection 2 2 2 0.02 </cmd>
BisectionCmd: lx=2 ly=2 lz=2 threshold=0.02
localization[0]: 1386 000000000000000000010101101010 size: 0.01562 overlaps: 65
localization[1]: 1386 000000000000000000010101101010 size: 0.01562 overlaps: 65
localization[2]: 1387 000000000000000000010101101011 size: 0.03125 overlaps: 73
localization[3]: 1386 000000000000000000010101101010 size: 0.01562 overlaps: 65
localization[4]: 1370 000000000000000000010101011010 size: 0.01562 overlaps: 29
localization[5]: 1382 000000000000000000010101100110 size: 0.01562 overlaps: 27
localization[6]: 1406 000000000000000000010101111110 size: 0.0625 overlaps: 80
localization[7]: 3434 000000000000000000110101101010 size: 0.03125 overlaps: 65
localization[8]: 1390 000000000000000000010101101110 size: 0.03125 overlaps: 73
localization[9]: 1386 000000000000000000010101101010 size: 0.01562 overlaps: 65
localization[10]: 1402 000000000000000000010101111010 size: 0.03125 overlaps: 72
localization[11]: 1898 000000000000000000011101101010 size: 0.03125 overlaps: 65
localization[12]: 3434 000000000000000000110101101010 size: 0.03125 overlaps: 65
localization[13]: 1407 000000000000000000010101111111 size: 0.125 overlaps: 88
....


Dr. Ngoc Linh Nguyen

Postdoctoral Associates at the Galli group, University of Chicago, IL, USA

[fgygi]

Hi Linh,

The "size" value printed for each state indicates the fraction of the total volume of the cell occupied by the bisected orbital. If this fraction is smaller than 1.0, it means that the orbital could be bisected into a smaller domain. For example, in the above output, orbital 0 has a size of 0.01562 (=1/64). This means that this orbital is contained within a volume of 1/64 of the cell. On the other hand, orbital 13 has a size of 0.125, which means that it is enclosed in a volume of 1/8 of the cell.

Orbitals can be visualized using the plot command and the VMD visualization software. For example, to plot orbital 13, use:

Code: Select all

plot -wf 13 file.cube

This writes the plot data in "cube" format on the file "file.cube".
You can then invoke VMD using:

Code: Select all

vmd file.cube

The localization string of zeros and ones printed with each state gives details about the effect of each of the bisecting projectors used (bisecting projectors correspond to the Walsh functions represented on Fig. 1 of the paper Phys. Rev. Lett 102, 166406 (2009):

Walsh projectors used in the bisection algorithm.

fig1.png (37.64 KiB) Viewed 18244 times

The effect of each bisecting projector is represented in the output by a pair of digits. The first digit indicates whether the orbital has a finite amplitude in the domain where the projector is 1 (or "inside" the projector), and the second digit indicates whether the orbital has a finite amplitude where the projector is 0 (or "outside" the projector).
There are 3 possible situations:

01: The orbital is only on the inside (bisection by that plane was successful)
10: The orbital is only on the outside (bisection by that plane was successful)
11: The orbital is on both sides (bisection by that plane is unsuccessful).

In the "2 2 2 0.02" example above, there are two bisecting projectors (Walsh functions W1(x) and W3(x)) in each of the x,y,z directions, i.e. a total of 6 projectors. The rightmost 12 digits of the localization string are meaningful. For example for state 0, we have 010101101010 (all bisections successful), whereas for state 13, we have 010101111111 (only 3 bisections successful).

The smallest possible volume described by these projectors is (1/4)*(1/4)*(1/4) = 1/64 when using blHF= 2 2 2.

The "overlap" value on the right gives the number of orbitals that have an overlapping domain with the current orbital.

Francois