Posted: March 9th, 2010 | Author: obm | Filed under: Physics/Theories, TDDFT | Tags: decomposition, response charge, TDDFPT | No Comments »
The electric susceptibility in the in time domain and in terms of the interaction picture bra-ket notation of (ref [2]) is given as
=\left\langle\phi_v(t)\right|T.r_{i}\rho^{\prime}_{j}(t)\left|\phi_v(t)\right\rangle)
where 
is the dipole operator towards direction “i” (e=1), )
is the response charge density along “j” direction, T is the cross-term equivalent of 
that contains all the probabilities of transition from an initial energy state 1 of all the electrons in the system to state 2 and
\right\rangle=e^{i\epsilon_{j}t/\hbar}\left(\left|\phi^0\right\rangle+\left|\phi^{\prime}(t)\right\rangle\right))
(eq 26 in ref [2]) explicitly expanding the operator one obtains
=\sum_{v}\left(p_{a}r_{i}\left|\phi^{0}_v\right\rangle\left\langle\phi^{\prime}_{j}(t)\left|+p_{e}r_{i}\left|\phi^{\prime}_{j}(t)\right\rangle\left\langle\phi^{0}_v\right|\right))
(eq.34 in ref [2]) (subscript v denotes an index that runs up to but no more than filled (valance) states.) In microcanonical case
where 
and 
are the total energy of all the electrons in initial and final state correspondingly, 
is the number of accessible states if the transition is permitted (otherwise p=0) and 
is the energy added to the system due to perturbation.
This term is equivalent to the projection.
Noting that:
\left|\left|\phi^0\right\rangle=0)
(eq.6 in ref[1])
since this represents a valance to valance transition which has no accessible state, and
\right|r\left|\phi^{\prime}(t)\right\rangle\approx0)
being a second order in perturbation, the susceptibility becomes
=\sum_{v}\left(\left\langle\phi^{\prime}_j(t)\right|r_i\left|\phi^{0}_v\right\rangle+\left\langle\phi^{0}_v\right|r_i\left|\phi^{\prime}_j(t)\right\rangle\right)p)
returning back to frequency domain, at gamma point where the ground state wavefunctions are real, the above expression becomes
=\sum_{v}2\left\langle\phi^{0}_v\right|r_i\left|x_j(\omega)\right\rangle)
where
\right\rangle=\frac{1}{2}p\left(\left|\phi^{\prime}_j(\omega)\right\rangle + \left|\phi^{\prime\star}_j(-\omega)\right\rangle\right))
in the not-so-pedantic but more shorthand notation of ref[2]. The presence of p in the description here is due to presence of projectors in the definition of batches.
Now, in order to investigate this a little bit more one can use the completeness of KS orbitals
=\sum_{v}2\left\langle\phi^{0}_v\right|r_i\sum\left|\phi^{0}\right\rangle\left\langle\phi^{0}\right|\left|x_j(\omega)\right\rangle)
due to selection rule embedded in x batch
\right\rangle=0)
hence
=\sum_{v,c}2\left\langle\phi^{0}_v\right|r_i\left|\phi^{0}_c\right\rangle\left\langle\phi^{0}_c\right|\left|x_j(\omega)\right\rangle)
or
=\sum_{v,c}R_{vc}.F_{vc}.)
where
\right\rangle)
where I have explicitly stated that there is one response orbital per occupied Kohn-Sham state.
The calculation of R coefficients is straightforward in the code. The F coefficients are calculated using the two-pass approach I use to calculate response charge density. In the first pass, the coefficients )
are calculated, in the second pass the coefficients are calculated from
_v\phi_c\right>.W_m(\omega))
Since q of x appears every other lanczos iteration, the operation is performed in even steps, in the begining of the iterator.
References
- Dario Rocca, Ralph Gebauer, Yousef Saad, and Stefano Baroni “Turbo charging time-dependent density-functional theory with Lanczos chains“, J. Chem. Phys. 128, 154105 (2008)
- Ralph Gebauer, Brent Walker “Ultrasoft pseudopotentials in time-dependent density-functional theory” J. Chem. Phys., 127, 164106 (2007)
Posted: February 16th, 2010 | Author: obm | Filed under: Project 21: Extreme, TDDFT | Tags: flavonoid | No Comments »
The following spectra is obtained for the dry system without counter-ion
The cell-size has no significant impact on the spectra of the dry system. The M-T correction does not work, since the cell size is not at least twice the molecule size.
Example input file:
&CONTROL
calculation = "scf",
restart_mode="from_scratch",
prefix="flav-14-nowat-nocl-x2",
pseudo_dir = "/u/cm/mbaris/work/arrigo/pseudo/"
outdir="./out"
/
&SYSTEM
ibrav=8,
celldm(1)=37.794538
celldm(2)=1.0
celldm(3)=0.75
nat=53,
ntyp=3,
ecutwfc=25.0,
ecutrho=200.0,
tot_charge=1.0
occupations="smearing",
smearing="gaussian",
degauss=0.006
/
&ELECTRONS
diagonalization = "cg"
electron_maxstep = 50,
tqr=.true.
/
ATOMIC_SPECIES
C 12.011 C.pbe-van_bm.UPF
O 16.00 O.pbe-van_bm.UPF
H 1.001 H.pbe-van_bm.UPF
ATOMIC_POSITIONS (angstrom)
H 16.022422 8.404178 4.026815
H 16.927685 5.434648 2.237667
H 14.746682 3.281691 1.359805
H 10.229997 3.426490 1.603412
H 11.849385 5.171335 0.566569
H 12.384277 3.542142 1.022973
O 10.665086 4.279735 1.982355
O 12.659925 7.182996 4.571237
O 16.462485 6.291544 2.537711
H 15.173886 5.567365 4.038213
H 14.870297 7.913472 2.112808
O 15.071861 8.237275 4.170719
H 12.596159 8.423016 2.813131
H 12.796823 4.959934 3.511343
C 11.928708 4.470731 1.405801
O 14.875801 3.751071 2.205858
H 14.410707 5.360034 0.955741
C 15.187645 6.001185 2.989469
C 14.553003 7.461131 3.121187
C 13.008388 7.475260 3.162092
O 12.571817 6.611538 2.187258
C 12.876253 5.208181 2.444845
C 14.354138 5.030383 2.071934
H 8.221876 3.334249 5.640921
H 9.458457 13.276892 5.709185
H 4.658234 6.122102 6.725152
H 14.055153 13.956144 4.956817
H 10.915334 5.178283 4.960759
H 6.806275 8.872129 5.448724
H 6.445641 4.601972 6.539093
H 9.439671 10.941687 5.060393
H 14.344719 11.546695 4.865238
H 13.538306 9.221439 5.004162
O 9.383525 8.744544 5.025901
O 8.894354 4.064138 5.738343
O 10.332657 13.506396 5.370300
O 5.096403 7.003869 6.303451
O 12.960550 13.824802 5.203096
C 10.738854 8.618018 4.931856
C 8.577166 7.648407 5.406602
C 11.321160 7.294758 4.831180
C 8.318398 5.251129 5.780359
C 11.073399 12.356283 5.173758
C 9.179369 6.372297 5.447613
C 6.398120 6.752775 6.016743
C 12.493869 12.540966 5.143188
C 11.373125 9.907093 4.943921
C 10.496861 6.146920 5.064049
C 7.222102 7.841186 5.575991
C 6.927668 5.447613 6.087758
C 10.569993 11.097001 5.025261
C 13.265356 11.384291 5.019503
C 12.818361 10.091406 4.991848
K_POINTS {gamma}
Posted: January 19th, 2010 | Author: obm | Filed under: Seminar Notes | Tags: Casimir | No Comments »
Today, Robert J. Jaffe gave a very inspiring talk on Casimir effect. He started from the classical grounded planes and ended up describing a method for calculating more general shapes. Some highlights from his talks are:
- Divergences removed using renormalization theory lead to unphysical results, as there are no terms in the expansion that can cancel the divergence.
- Divergences often result from unphysical conditions imposed on the system, like unphysical boundary conditions. For example assuming a metal to be rigid at every frequency is unphysical.
- The transition from VdW to Casimir forces is continuous, and can be approximated by adding retardation effects to Quantum-Relativistic calculation of VdW.
- Using T matrices and dynamical polarizability of the system, just like how engineers use them to identify material based on their radar signature, Casimir forces can be calculated.
- Dark energy/Casimir correlation is overstated, S matrix theories show casimir effects even without vacuum.
- Many different methods of calculating casimir forces each lead to interesting physics
- Experimental manifestations?
T. Emigetal et Al. PRL 99 170403 (2007) might be interesting to read.
