LINEAR RESPONSE THEORY
t = t0 : Interacting system in ground state of potential v0(r) with density 0(r)
t > t0 : Switch on perturbation v1(r t) (with v1(r t0)=0).
Density: (r t) = 0(r) + (r t)
Consider functional [v](r t) defined by solution of the interacting TDSE
Functional Taylor expansion of [v] around vo:
ρ  v  rt   ρ  v0  v1   rt 
 ρ  v0   rt 

δρ  v   rt 
δv  r't' 
ρo  r 
v1  r ' t ' d 3r'dt'
ρ1  rt 
v0

δ 2ρ  v   rt 
1
 
v1  r ', t ' v1  r", t" d 3r'd 3r"dt'dt"
2 δv  r't' δv  r"t" v
0
ρ2  rt 
1(r,t) = linear density response of interacting system
  rt, r ' t '  :
δρ  v   rt 
δv  r't' 
= density-density response function of
interacting system
v0
Analogous function s[vs](r t) for non-interacting system
ρS  vS   rt   ρS  vS,0  vS,1  rt   ρS  vS,0   rt   
S  rt, r ' t '  :
δρS  vS   rt 
δvS  r't' 
δρS  vS   rt 
δvS  r't'
vS,1  r't' d 3r'dt' 
v S,0
= density-density response function of
non-interacting system
vS,0
GOAL: Find a way to calculate 1(r t) without explicitly evaluating
(r t,r't') of the interacting system
starting point: Definition of xc potential
vxc ρ  rt  : vS ρ  rt   vext ρ  rt   vH ρ  rt 
Notes:
• vxc is well-defined through non-interacting/ interacting 1-1
mapping.
• vS[] depends on initial determinant 0.
• vext[] depends on initial many-body state 0.
 In general, vxc  vxc  ρ,Φ0 ,Ψ0 
only if system is initially in ground-state then, via HK, 0
and 0 are determined by 0 and vxc depends on  alone.
δv xc ρ   rt 
δρ  r't'

ρ0
δvS ρ   rt 
δρ  r't'

ρ0
δvext ρ   rt 
δρ  r't'

ρ0
δ  t  t' 
r  r'
δv xc ρ   rt 
δρ  r't'
f xc  rt, r't'

ρ0
δvS ρ   rt 
δρ  r't'
S1  rt, r't'

ρ0
δvext ρ   rt 
δρ  r't'
1  rt, r't'

ρ0
δ  t  t' 
r  r'
WC  rt, r't'
δv xc ρ   rt 
δρ  r't'
f xc  rt, r't'

ρ0
δvS ρ   rt 
δρ  r't'

ρ0
S1  rt, r't'
δvext ρ   rt 
δρ  r't'
1  rt, r't'
1
S
f xc  WC    
1

ρ0
δ  t  t' 
r  r'
WC  rt, r't'
δv xc ρ   rt 
δρ  r't'
f xc  rt, r't'

ρ0
δvS ρ   rt 
δρ  r't'

ρ0
S1  rt, r't'
δvext ρ   rt 
δρ  r't'
1  rt, r't'
1
S
S • f xc  WC    
1
•

ρ0
δ  t  t' 
r  r'
WC  rt, r't'
δv xc ρ   rt 

δρ  r't'
ρ0
f xc  rt, r't'
δvS ρ   rt 
δρ  r't'

ρ0
S1  rt, r't'
δvext ρ   rt 
δρ  r't'
1  rt, r't'
1
S
S • f xc  WC    
1
χ S  f xc  WC  χ  χ  χ S
•

ρ0
δ  t  t' 
r  r'
WC  rt, r't'
χ  χ S  χ S  WC  f xc  χ
Act with this operator equation on arbitrary v1(r t) and use  v1 = 1 :


3
ρ1  rt  =  d r'dt'χ S  rt, r't'   v1  rt  +  d r"dt"WC  r't', r"t" + f xc  r't', r"t" ρ1  r"t" 


3
• Exact integral equation for 1(r t), to be solved iteratively
• Need approximation for f xc  r't', r"t" 
(either for fxc directly or for vxc)
δv xc ρ   r't'
δρ  r"t"
ρ0
Adiabatic approximation
adiab
xc
v
ρ  rt  := v
static DFT
xc
 ρ  t   rt 
In the adiabatic approximation, the xc potential vxc(t) at time t
only depends on the density ρ(t) at the very same point in time.
e.g. adiabatic LDA:
 f xcALDA  rt, r't'  
v
ALDA
xc
 rt  : v
δv ALDA
 rt 
xc
δρ r't'
ρ0
LDA
xc
 ρ  rt   α ρ  rt 
13
 v ALDA
 δ  r  r' δ  t  t'  xc
 ρ r 
 2ehom
xc
 δ  r  r'δ  t  t' 
n 2

ρ0  r 
ρ0  r 
Total photoabsorption cross section of the Xe atom versus photon
energy in the vicinity of the 4d threshold.
Solid line: self-consistent time-dependent KS calculation [A. Zangwill and P.
Soven, PRA 21, 1561 (1980)]; crosses: experimental data [R. Haensel, G. Keitel, P.
Schreiber, and C. Kunz, Phys. Rev. 188, 1375 (1969)].
continuum
states
I1
I2
unoccupied
bound states
occupied
bound states
photo-absorption cross section
Photo-absorption in weak lasers
(ω)
Laser frequency 
No absorption if  < lowest excitation energy
Standard linear response formalism
H(t0) = full static Hamiltonian at t0
H  t 0  m  Em m
 exact many-body eigenfunctions
and energies of system
full response function
 0 ˆ  r  m m ˆ  r  0
0 ˆ  r '  m m ˆ  r '  0 
  r, r ';   lim 




   E m  E 0   i
   E m  E 0   i
0 m 

 The exact linear density response
1    ˆ   v1  
has poles at the exact excitation energies  = Em - E0
Discrete excitation energies from TDDFT
exact representation of linear density response:

ˆ     fˆ     
1    ˆ S   v1    W
C 1
xc
1

“^” denotes integral operators, i.e. f̂ xc1  f xc  r, r '  1  r '  d 3r '
where ˆ S  r, r ';   
j,k


M jk  r, r ' 
    j   k   i

with M jk  r, r '   f k  f j  j  r   j  r '  k  r '  k  r 
1 if m
fm  
0 if m
*
is occupied in KS ground state
is unoccupied in KS ground state
 j   k KS excitation energy
*
1ˆ  ˆ  Wˆ  fˆ
S
C
xc
   1    ˆ S   v1  
1()   for    (exact excitation energy) but right-hand side
remains finite for   
hence


ˆ  fˆ             
1ˆ  ˆ S    W
 C xc

()  0 for   
This condition rigorously determines the exact excitation energies, i.e.,


ˆ  fˆ          0
1ˆ  ˆ S     W
 C xc

This leads to the (non-linear) eigenvalue equation
(See T. Grabo, M. Petersilka, E. K. U. G., J. Mol. Struc. (Theochem) 501, 353 (2000))
  M       
qq'
q qq'
q'
 q
q'
where
M qq'
 1

 α q'  d r  d r'Φ q  r  
 f xc  r, r', Ω   Φ q'  r' 
 r  r'



3
3
q   j,a  double index
q  f a  f j
q  r     r   j  r 
q  a   j
*
a
Atom
Exp erimental Excitation
Energies 1S1P
(in Ry)
KS energy
differences
KS (Ry)
KS + K
Be
0.388
0.259
0.391
Mg
0.319
0.234
0.327
Ca
0.216
0.157
0.234
Zn
0.426
0.315
0.423
Sr
0.198
0.141
0.210
Cd
0.398
0.269
0.391
from: M. Petersilka, U. J. Gossmann, E.K.U.G., PRL 76, 1212 (1996)
E = KS + K
j - k
 1

K   d r  d r '  j  r    r '  k  r '    r  
 f r, r '  
 r  r ' xc 



3
3

j

k
Excitation energies of CO molecule
State
expt
KS-transition
52
A
1

0.3127
a
3

0.2323
I
D
a'
e
d
1
1

-
0.3631
KS
0.2523
KS + K
0.3267
0.2238
12
0.3626
0.3626

0.3759
0.3812

0.3127
0.3181
0.3631
0.3626
0.3440
0.3404
3 +
3

3

-
T. Grabo, M. Petersilka and E.K.U. Gross, J. Mol. Struct. (Theochem) 501, 353 (2000)
LDA
approximations made: vxc
and fxcALDA
Quantum defects in Helium
En  
1
2  n  n 
2
a.u.
Bare KS
exact
x-ALDA
xc-ALDA (VWN)
x-TDOEP
M. Petersilka, U.J. Gossmann and E.K.U.G., in: Electronic Density
Functional Theory: Recent Progress and New Directions, J.F. Dobson, G. Vignale,
M.P. Das, ed(s), (Plenum, New York, 1998), p 177 - 197.
(M. Petersilka, E.K.U.G., K. Burke, Int. J. Quantum Chem. 80, 534 (2000))
The above equations are from the Refs.
M. Petersilka, U. J. Gossmann, E.K.U.G., PRL 76, 1212 (1996);
T. Grabo, M. Petersilka, E. K. U. G., J. Mol. Struc. (Theochem)
501, 353 (2000)
They yield the same excitation spectrum as the
Casida equations:
2
W

v


vq
 qq   q
q
Wqq      qq  4 q q M qq   
2
q
M. E. Casida, in Recent Developments and Applications in
Density Functional Theory, edited by J. M. Seminario
(Elsevier, Amsterdam, 1996), p. 155-192.
Failures of ALDA in the linear response regime
• H2 dissociation is incorrect:



E 1  u  E 1 g

(in ALDA)


0
R 
(see: Gritsenko, van Gisbergen, Görling, Baerends, JCP 113, 8478 (2000))
• response of long chains strongly overestimated
(see: Champagne et al., JCP 109, 10489 (1998) and 110, 11664 (1999))
• in periodic solids,
f xcALDA  q, ,   c  whereas,
exact
2
f


1
q
for insulators, xc
divergent.
q 0
• charge-transfer excitations not properly described
(see: Dreuw et al., JCP 119, 2943 (2003))
charge-transfer excitation
R
A
B
CT excitation energy
 A r   B r ' 
  d r  d r'
r  r'
2
 CT  IP
A 
 EA
B
3
3
Infinite R
Finite (but large) R
In exact DFT IP(A) = – (A)
HOMO
= – (A)
MOL
(B)
EA(B) = – (B)
–
∆
LUMO
xc
= – (B)
MOL
derivative discontinuity
 A r   B r '
 CT   MOL   MOL   d r  d r '
r  r'
2
B
A 
2
3
3
2
 A r   B r '
 CT   MOL   MOL   d r  d r '
r  r'
2
B
A 
3
3
~ 1/R
2
 A r   B r '
 CT   MOL   MOL   d r  d r '
r  r'
2
B
A 
3
2
3
~ 1/R
In TDDFT (single-pole approximation)
B
A 
CT  MOL  MOL   d3r  d3r' *A (r' )B (r' )fxc (r, r' , CT )A (r )*B (r )
 A r   B r '
 CT   MOL   MOL   d r  d r '
r  r'
2
B
A 
3
2
3
~ 1/R
In TDDFT (single-pole approximation)
B
A 
CT  MOL  MOL   d3r  d3r' *A (r' )B (r' )fxc (r, r' , CT )A (r )*B (r )
Exponentially
small
Exponentially
small
 A r   B r '
 CT   MOL   MOL   d r  d r '
r  r'
2
B
A 
3
2
3
~ 1/R
In TDDFT (single-pole approximation)
B
A 
CT  MOL  MOL   d3r  d3r' *A (r' )B (r' )fxc (r, r' , CT )A (r )*B (r )
Exponentially
small
Exponentially
small
CONCLUSIONS: To describe CT excitations correctly
• vxc must have proper derivative discontinuities
• fxc(r,r’) must increase exponentially as function of r and of r’
for ω→ΩCT
How good is ALDA for solids?
optical absorption (q=0)
ALDA
Solid Argon
L. Reining, V. Olevano, A. Rubio, G. Onida, PRL 88, 066404 (2002)
OBSERVATION:
In the long-wavelength-limit (q = 0), relevent for optical
absorption, ALDA is not reliable. In particular, excitonic
lines are completely missed. Results are very close to RPA.
EXPLANATION:
In the TDDFT response equation, the bare Coulomb interaction
and the xc kernel only appear as sum (WC + fxc). For q 0,
WC diverges like 1/q2, while fxc in ALDA goes to a constant.
Hence results are close to fxc = 0 (RPA) in the q
0 limit.
CONCLUSION:
Approximations for fxc are needed which, for q
0, correctly
diverge like 1/q2. Such approximations can be derived from
many-body perturbation theory (see, e.g., L. Reining, V.
Olevano, A. Rubio, G. Onida, PRL 88, 066404 (2002)).
WHAT ABOUT FINITE Q??
see: H.C. Weissker, J. Serrano, S. Huotari,
F. Bruneval, F. Sottile, G. Monaco, M. Krisch,
V. Olevano, L. Reining,
Phys. Rev. Lett. 97, 237602 (2006)
Silicon: Loss function Im χ(q,ω)
X-ray absorption spectroscopy of 3d metals
d band
EF
2
1
2p3/2
(4-fold degenerate
M3 ,  1 ,  1 , 3 )
2
2
2
2
2p1/2 (2-fold degenerate
M1 ,  1 )
2
2
Core levels localized: TDDFT works well
Pioneers: John Rehr
Hubert Ebert
Detailed analysis using two-pole approximation
M12 
 ω1  M11

 β  Ωβ
ω2  M22 
 M 21
From knowledge of the KS orbitals and the KS excitation
energies ω1 , ω2 and the experimental excitation
energies Ω1 , Ω2 and their branching ratio, one can deduce
experimental values for the matrix elements Mij, i.e. one can
“measure” fxc.
A. Scherz, E.K.U.G., H. Appel, C. Sorg, K. Baberschke, H. Wende, K.
Burke, PRL 95, 253006 (2005)
PRIZE QUESTION No 2
H(t0) = full static Hamiltonian (without laser field)
H  t 0  m  Em m
 exact many-body eigenfunctions
and energies of system
Whenever ρ1(ω) has a pole, the frequency of that pole
corresponds to an excitation energy (Em–Eo) of the many-body
Hamiltonian H(to)
Question: Does each excitation energy (Em–Eo) of the manybody Hamiltonian H(to) yield a pole of ρ1(ω)? In other words,
can we expect to get the complete spectrum of the many-body
Hamiltonian H(to) from linear-response TDDFT?
ANSWER to PRIZE QUESTION No 2
The answer to the question depends on the system under study,
but generally the answer is “NO”. The reason is that for some
excitation energies (Em-Eo) appearing in the denominator of the
response function the numerator may vanish (usually due to
symmetry considerations/selection rules, etc.), i.e. there is no pole.
It is an experimental fact, that certain states, so-called dark states,
are not accessible by a simple direct transition in a weak laser.
We do not observe these states in weak-laser experiments and,
correspondingly, we also do not observe them in linear-response
theory. One may observe these states experimentally, e.g., by
cascades from higher states. For the theoretical description, one
then needs to calculate non-linear processes. Another possibility
to “see” these states within the linear response regime -both
experimentally and theoretically- is to apply other probes, such
as magnetic fields (instead of lasers).