Download Response Theory for Linear and Non-Linear X

Response Theory for Linear and Nonlinear X-ray
Spectroscopies
Exact State Response Theory
Patrick Norman
Division of Theoretical Chemistry and Biology
KTH Royal Institute of Technology, Stockholm
A perspective on nonresonant and resonant electronic response theory for
time-dependent molecular properties
P. Norman, Phys. Chem. Chem. Phys. 13, 20519 (2011)
Dalton2015
X-ray absorption spectrum
Cu-phthalocyanine
Cu
B
6
X-ray absorption
**DALTON INPUT
.RUN RESPONSE
**WAVE FUNCTION
.DFT
CAMB3LYP alpha=0.190 beta=0.810 mu=0.330
**RESPONSE
*ABSLRS
.ALPHA
.DAMPING
0.0045566
# Carbon K-edge
.FREQ INTERVAL
10.01 10.04 0.0025
*END OF
4
expt
D
A
2
0
284
C
285
286
E
F
286
287
288
Photon energy [eV]
288
290
289
292
General Frame of Reference
I
In a simplistic and generalized view of spectroscopy, the observable
is the number of detected particles (e.g. photons or electrons) per
unit time in a narrow energy interval and into a small solid angle.
I
The observable is recorded under certain conditions regarding
parameters such as temperature, pressure, concentration, and there
exist a model (or connection) from which one can deduce molecular
properties from the set of measured data.
I
Under typical circumstances in molecular spectroscopies, these
connections can be viewed upon as measures of changes in an
observable due to the presence of electromagnetic fields with origins
attributed to external sources.
I
Compared to atomic fields, the electric fields of conventional lasers
are relatively weak. A laser delivering pulses of 10 ns duration and 1
mJ in energy and with a spot size of 100 µm produces an intensity
of about 0.3 GW/cm2 . This intensity corresponds to an electric
field amplitude of some F ! = 5 ⇥ 10 5 a.u., which is several orders
of magnitude smaller than the internal electric fields that bind
electrons in atomic and molecular systems.
Reasons to employ Perturbation Theory
Why do we not simply determine time propagate the Schrödinger
equation in a direct and nonperturbational manner?
I
The response functions defined in perturbation theory provide
the natural meeting point between experiment and theory
with a distinct separation of one-, two-, three-photon, etc.,
optical processes.
I
It is a numerically elaborate process to separate out
nonlinearities from dominant lower-order components in the
polarization.
I
Only a small number of states (n ⇡ 101 102 ) can be
included in nonperturbational approaches.
I
Calculation of vibrational contributions can hardly be made
practical in a nonperturbational approach.
Response Theory and Perturbation Theory
Response theory may be thought of as a reformulation of standard
time-dependent perturbation theory into a form suitable for
approximate state theory.
Virtually all spectroscopic properties are encompassed by the
theory.
Possible perturbations are:
I
time-independent or time-dependent
I
electric or magnetic
I
internal or external
I
geometric distorsions
Selection of molecular properties
Electric
I
Polarizability
I
Hyperpolarizabilities
Magnetic
I
Magnetizability
Electric–Magnetic
I
Optical rotation
I
Circular dichroism
I
Faraday rotation
I
Magnetic circular
dichroism
Electric–Geometric
I
IR intensities
I
Raman intensities
I
Vibrational polarizabilities
I
ZPVA polarizabilites
Internal fields
I
g -tensor
I
Fine-structure
I
Spin-spin coupling
I
Shieldings
Historical Background
Dalgaard & Monkhorst
Coupled Cluster
Hättig et al.
Coriani et al.
Sekino & Bartlett
Christiansen et al.
Koch & Jørgensen
ADC
Hättig
Schirmer
Starcke et al.
Trofimov & Schirmer
Hättig & Hess
MP2
Aiga et al.
Rice & Handy
Runge & Gross
Kohn-Sham
Deb & Ghosh
Zangwill & Soven
Multi-Configuration
Olsen & Jørgensen
Yeager & Jørgensen
SOPPA
Hartree-Fock
Jonsson et al.
Hettema et al.
Vahtras et al.
Oddershede et al.
Dunning & McKoy
Gao et al.
Salek et al.
Maitra & Burke
van Gisbergen et al.
Bast et al.
van Leeuwen
Norman et al.
Packer et al.
Norman & Jensen
Dalgaard
Sekino & Bartlett
Saue & Jensen
Linderberg & Öhrn
General Theory
Langhoff et al.
1970
1980
1990
2000
2010
Motivation for derivation of exact state response functions
I
I
I
I
I
I
First and foremost, it reveals the dependence of molecular
response properties on intrinsic properties of the system, such
as excitation energies and transition moments, and it thereby
connects di↵erent spectroscopies.
It o↵ers the possibility of constructing few-states-models upon
which one can base rational molecular design.
It provides explicit formulas for response functions in the
configuration interaction (CI) approximation.
It reveals general properties and symmetries of response
functions that are also valid in approximate state theories.
It suggests the identification of excited state properties from a
study of poles and residues of ground state response
functions, e.g., excitation energies are identified as poles of
linear response functions.
The formulations of quantum mechanics and perturbation
theory that are suitable for approximate state theories are best
illustrated in exact state theory.
A few words about the Schrödinger equation
The Schrödinger equation reads as:
i~
@
(t) = Ĥ(t) (t).
@t
The time-independent Schrödinger equation reads as:
Ĥ0
n (x)
= En
(t) =
n (x),
iEn t/~
n (x)e
.
The wave function (t) is always time-dependent, but for a
stationary state hp̂i = 0 and the density ⇢(t) is time-indepedent.
(t0 )
-
i~ ˙ = Ĥ
(t)
-
t > t0
Time-evolution for conservative systems; Ĥ(t) = Ĥ0
The evolution of the wave function from some initial time t0 to
time t is given by
(t) = Û(t, t0 ) (t0 ),
where
Û(t, t0 ) = e
i Ĥ0 (t t0 )/~
.
The operator Û(t, t0 ) is referred to as the time evolution operator,
or time propagator.
Alternatively, the time-dependent wave function can be written in
terms of the eigenfunctions of the Hamiltonian:
X
(t) =
cn (t0 ) n (x)e iEn (t t0 )/~
n
Note:
For cases when the Hamiltonian is time-dependent, a time
propagator cannot be constructed in such a simple manner.
Molecular properties from the “exact” wave function
We will now study a simple system for which we, for an arbitrary
external electric field turned on at t = 0, will have access to the
exact time-dependent wave function that solves the Schrödinger
equation.
By exact, we mean ”for all practical purposes exact” and limited
only by the numerical representation on the computer, but this
aspect is a side-issue to the present discussion.
(t0 )
-
i~ ˙ = Ĥ
(t)
-
t > t0
This toolbox will let us explore the di↵erent formulations of the
theory of molecular properties without relying on perturbation
theory.
Two-level atom in static electric field (conservative system)
In the electric dipole approximation, the
Hamiltonian of the system will equal
µ̂F 0
Ĥ = Ĥ0
where Ĥ0 is the Hamiltonian of the
isolated atom, µ̂ is the electric dipole
operator, and F 0 is the amplitude of the
external static electric field.
E 6
E = Eb
Ea
Eb
b
Ea
a
Ĥ0 =

Ea 0
0 Eb
µ̂ =

0 µab
µba 0
The energies of the ground and excited states are given as the
eigenvalues of the Hamiltonian:
⇣
⌘
det Ĥ ~! Iˆ = 0
The two eigenvalues become
Ea + Eb
~! =
±
2
r
Ea ) 2
(Eb
4
+ (µab F 0 )2
from which, for small fields, the electric-field dependent energies
are found to be
(µab F 0 )2 (µab F 0 )4
+
+ ···
E
( E )3
(µab F 0 )2 (µab F 0 )4
Eb0 (F 0 ) = Eb +
+ ···
E
( E )3
Ea0 (F 0 ) = Ea
From these energy expansions we readily determine the electric
polarization properties by taking the field derivatives of the energy
in the limit of zero field strength. For the ground state, we get
µa =
↵a =
a
=
a
=
@Ea0
@F 0
=0
F 0 =0
@ 2 Ea0
@(F 0 )2 F 0 =0
@ 3 Ea0
@(F 0 )3 F 0 =0
@ 4 Ea0
@(F 0 )4 F 0 =0
=2
(µab )2
E
=0
=
24
(µab )4
( E )3
where we have introduced the electric dipole moment (µ),
polarizability (↵), first-order hyperpolarizability ( ), and
second-order hyperpolarizability ( ).
Induced polarization
The permanent dipole moment of the atom is zero, but, in the
presence of the external field, there will appear an induced
polarization.
Polarizability
With E = 0.5 Eh and µab = 1.0 a.u., the static ground state
polarizability ↵a is equal to 4.0 a.u., and, with F 0 = 5 ⇥ 10 5 a.u.,
the induced dipole moment amounts to µ0a = 2.0 ⇥ 10 4 a.u.
(nonlinear terms are ignored).
Charge dislocation
The static electric field (with the strength of a conventional laser
field) is e↵ectively capable of causing a displacement of an
elementary charge in this system by a distance of 2.0 ⇥ 10 4 a.u.
Two-level atom in optical field (nonconservative system)
In the electric dipole approximation, the
Hamiltonian of the system will equal
Ĥ(t) = Ĥ0
µ̂F (t)
where Ĥ0 is the Hamiltonian of the
isolated atom, µ̂ is the electric dipole
operator, and F (t) is the external
electric field.
E 6
E = Eb
Ea
Eb
b
Ea
a
Ĥ0 =

Ea 0
0 Eb
µ̂ =

0 µab
µba 0
Non-stationary states in laser fields
Pulse energy
Pulse duration
Spot size
Intensity
Electric field
b
1
10
100
0.3
4.6 ⇥ 10 5
ω
⇤E = 0.5 a.u.
F(t) = F sin(ωt)
µ ab= 1.0 a.u.
a
mJ
ns
µm
GW/cm2
a.u.
ψ(0) = ψa
i ψ(t) = H ψ(t)
H = H0 - µ F(t)
Nonconservative system: Study of Induced Dipole Moment
By (t) we will denote the time-dependent wave function that is a
solution to the Schrödinger equation
i~
@
(t) = Ĥ (t).
@t
We will assume that a nonresonant (! = 0.1 a.u.) external
perturbation is slowly swithed on (a = 1/100 a.u.) in accordance
with
F (t) = F ! sin !t ⇥ erf(at).
10 fs
F(t)
5 ×10
−5
0
-5
0
100
200
tim e
300
400
500
Initial Condition
The initial condition for our system is that it resides in the ground
state prior to exposure of the perturbation and with a phase that is
zero at t = 0, i.e.
(t) =
ae
iEa t/~
for t  0
Infinitesimal Time Propagation
With a time step t that is small, we can consider the external
field to be constant between t0 and t0 + t and thereby get
(t0 +
t) = Û(t0 , t0 +
t) (t0 )
where the time-evolution propagator equals
Û(t0 , t0 +
t) = e
i Ĥ(t0 ) t/~
Time Propagation
Repeated application of Û enables us to determine the wave
function in the region t > 0, and, given the time-dependent wave
function, the dipole moment is obtained as the expectation value
of the electric dipole operator according to
µ(t) = h (t)|µ̂| (t)i
Initialization
Time propagation
psi[:,0]=[1, 0]
for k in range(1,n):
psi[:,k]=dot(expm(-1j*(H-mu*F[k-1])*delta),psi[:,k-1])
P[k]=dot(conj(psi[:,k]),dot(mu,psi[:,k]))
H=array([[Ea, 0],
[0, Eb]])
mu=array([[0, muab],
[muba, 0]])
F=Fw*sin(w*t)*erf
Populations of States
As a measure of the e↵ect of the perturbation on the system, we
may study the population in the ground and excited states which
we denote by ⇢a (t) and ⇢b (t), respectively. The populations are
given by the projections of the wave function on the eigenfunctions
of Ĥ0 according to
⇢a (t) = |h
a|
(t)i|2
Program code
popa=conj(psi[0,:])*psi[0,:]
popb=conj(psi[1,:])*psi[1,:]
⇢b (t) = |h
b|
(t)i|2
Induced Dipole Moment
Molecular properties are defined by an expansion of the induced
dipole moment in orders the time-dependent electric field
µ(t) = ↵F (t) +
1 3
F (t) + · · ·
6
Polarizability
The linear polarizability can be read o↵ directly as the ratio of the
amplitudes of the polarization and the electric field
↵=
max[µ(t)]
⇡ 4.17 a.u.
F!
Dispersion
The dependency of the molecular property (in this case the
polarizability) to the frequency of the perturbation is referred to as
the dispersion of the property.
Where are the nonlinearities?
µ(t) = ↵F (t) +
1 3
F (t) + · · ·
6
There are three ways to enhance nonlinear responses:
I
Design system with large -value
I
Increase the electric field strength
I
Tune the frequency to multi-photon resonances
Nonlinear responses
µ(t) = ↵( !; !)F ! sin(!t)+
1
( 3!; !, !, !)F ! F ! F ! sin(3!t)+. . .
6
Fourier transform
Let us choose: F ! = 0.02 a.u. and ! = 0.166 a.u.:
Rayleigh
scattering
Third-Harmonic
Generation (THG)
Dipole moment (a.u.)
0
0.1
0.2
0.3
0.4
Frequency (a.u.)
0.5
0.6
0.1
0
-0.1
0
200
400
600
Time (a.u.)
800
1000
Nonconservative system: expectation value summary
I
The energy of the system is not well defined as energy transfer
occurs between field and system
I
The dipole moment and other observables (expectation
values) are well defined and time-dependent
I
Molecular properties (response functions) are defined as
expansion coefficients of the observable in terms of field
strengths
I
The overall phase of the wave function is of no concern
I
Linear response functions are directly identified as ratios of
induced dipole moments and field strengths (in the small field
limit)
I
Nonlinear response functions can be obtained in the same
manner but the procedure is hampered by numerical issues
I
This time-resolved procedure is applicable to real systems but
limited by the need to maintain low matrix ranks
Molecular properties from perturbation theory
expansions of the wave function
Expansions are made in the basis of the exact eigenstates of the
zeroth-order Hamiltonian
The response theory cookbook recipe
1. Find an efficient parameterization of wave function
I
I
I
redundancy
state vector normalization
phase isolation and secular divergencies
2. Choose an appropriate equations-of-motion
I
I
I
based on Schrödinger equation
equivalent in exact state theory
important in approximate state theory
3. Apply perturbation theory
I
not much to say, mostly boring work...
4. Form a well-defined quantity of interest, e.g. µ(t), and
identify response functions
Parametrization by projections
(t) =
X
cn (t)
n
requires that
X
n
n;
cn (t) = hn| (t)i
|cn (t)|2 = 1.
Parametrization by rotations
| ¯(t)i = e
e
i P̂(t)
i P̂(t)
|0i;
|0i = |0i cos ↵
P̂(t) =
X
Pn (t)|nih0| + Pn⇤ (t)|0ihn|
n>0
i
X
n>0
sin ↵
Pn |ni
;
↵
↵=
sX
n>0
|Pn |2 .
Energy
2
1
0
0
0
-
0
-
-
0
The Hamiltonian and the Generator-of-Rotations for
system equal
0
1
0
E0 0 0
0 P1⇤
Ĥ = @ 0 E1 0 A
P̂ = @ P1 0
0 0 E2
P2 0
a three-states
1
P2⇤
0 A
0
and the energy as a function of transition state amplitudes is
E (P1 , P2 ) = h0|e i P̂ Ĥe
i P̂
|0i.
Electronic Hessian
E (P1 , P2 ) = h0|Ĥ|0i + ih0|[P̂, Ĥ]|0i
1
h0|[P̂, [P̂, Ĥ]]|0i + . . . ,
2
which gives an expression for the second-order derivatives of the
energy (or Hessian) with respect to the transition state amplitides
that equals
@2E
@Pn @Pm
=
P=0
h
ii
1⇣ h
h0| |nih0|, |mih0|, Ĥ |0i
2
h
h
ii ⌘
+ h0| |mih0|, |nih0|, Ĥ |0i
= 0,
@2E
⇤
@Pn @Pm
=
P=0
h
ii
1⇣ h
h0| |nih0|, |0ihm|, Ĥ |0i
2
h
h
ii ⌘
+ h0| |0ihm|, |nih0|, Ĥ |0i
= (En
E0 )
nm .
Choice of Equation of Motion
I
I
Rayleigh–Schrödinger
@
i~ @t
(t) = Ĥ (t);
Ehrenfest theorem
@ ¯
ˆ ¯(t)i =
h (t)|⌦|
@t
I
I
Quasi-energy
⇣
Q(t) = h ¯| Ĥ
(t) =
P
n cn (t) n
1 ¯
ˆ
¯
i~ h (t)|[⌦, Ĥ]| (t)i
⌘
@
i~ @t
| ¯i
Liouville equation
@
1
ˆ]mn
@t ⇢mn = i~ [Ĥ, ⇢
mn (⇢mn
⇢eq
mn )
Perturbation Theory
The not quite so fun part...
Hamiltonian and initial conditions
The Hamiltonian can be divided:
Ĥ = Ĥ0 + V̂ (t);
V̂ (t) =
X
V̂ ! F ! e
i!t ✏t
e .
!
The solutions to the eigenvalue problem of Ĥ0 are known:
Ĥ0 |ni = En |ni,
where |ni are the exact eigenstates and En the respective energies.
Before being exposed to the perturbation, we assume the molecule
to be in a reference state |0i—in most cases the molecular ground
state—which corresponds to Pn ( 1) = 0 for all n > 0.
Ehrenfest theorem
The time-evolution of an observable (expectation value) is
determined from the Ehrenfest theorem. For a general
ˆ we have
time-independent operator ⌦,
@
ˆ (t)i
h (t)|⌦|
@t
ˆ @ (t)i + h @ (t)|⌦|
ˆ (t)i
= h (t)|⌦|
@t
@t
1
ˆ Ĥ]| (t)i.
=
h (t)|[⌦,
i~
Example: For a conservative system, the application of the above
ˆ = Ĥ0 gives the result:
equation for ⌦
@
h (t)|Ĥ0 | (t)i = 0,
@t
which shows that the energy is conserved.
Response Functions from the Polarization
The time-evolution of Pn (t) is determined from the Ehrenfest
ˆ = |0ihn|:
theorem applied to ⌦
@ ¯
ˆ ¯(t)i = 1 h ¯(t)|[⌦,
ˆ Ĥ0 + V̂ (t)]| ¯(t)i.
h (t)|⌦|
@t
i~
Expansion is made with
ˆ
e i P̂(t) ⌦e
i P̂(t)
ˆ + i[P̂, ⌦]
ˆ
=⌦
1
ˆ
[P̂, [P̂, ⌦]]
2
i
ˆ + ...,
[P̂, [P̂, [P̂, ⌦]]]
6
and
ˆ
h0|[P̂, ⌦]|0i
ˆ
h0|[P̂, [P̂, ⌦]]|0i
ˆ
h0|[P̂, [P̂, [P̂, ⌦]]]|0i
as well as
h
=
=
=
Pn ,
0,
4Pn
i
ˆ Ĥ0 = ~!n0 ⌦.
ˆ
⌦,
X
m>0
|Pm |2 .
Perturbation expansion
With
(1)
(2)
(3)
Pn (t) = Pn + Pn + Pn + · · · ,
the first-order equation reads
@
1
ˆ
ˆ Ĥ0 ]]|0i
h0|[P̂ (1) , ⌦]|0i
= h0|[P̂ (1) , [⌦,
@t
i~
1
ˆ V̂ (t)]|0i.
h0|[⌦,
~
or, equivalently,
@ (1)
Pn =
@t
1
(1)
i!n0 Pn + hn|V̂ (t)|0i,
~
which, by direct time-integration, yields
Z t
1
0
(1)
i!n0 t
Pn
= e
hn|V̂ (t 0 )|0ie i!n0 t dt 0
~
1 X hn|V̂ ! |0iF ! e i!t
=
.
i~ !
!n0 !
Linear response function
ˆ are defined by:
The response functions of the observable ⌦
X
ˆ ¯(t)i = h0|⌦|0i
ˆ +
ˆ V̂ !1 iiF !1 e i!1 t
h ¯(t)|⌦|
hh⌦;
!1
1 X ˆ !1 !2
+
hh⌦; V̂ , V̂ iiF !1 F !2 e
2!!
i(!1 +!2 )t
1 2
+ ···
We identify:
ˆ
ih0|[P̂ (1) , ⌦]|0i
=
X
!
ˆ V̂ ! iiF ! e
hh⌦;
i!t
or, equivalently,
ˆ V̂ ! ii =
hh⌦;
"
#
ˆ
ˆ
1 X h0|⌦|nihn|
V̂ ! |0i h0|V̂ ! |nihn|⌦|0i
+
.
~
!n0 !
!n0 + !
n>0
Two-level atom: response function value
Observable:
Perturbation:
E 6
ˆ = µ̂
⌦
V̂ ! = µ̂
E = Eb
Ea
E = 0.5 a.u.
Eb
b
Ea
a
↵(!) =
µab = 1.0 a.u.
~! = 0.1 a.u.
|µab |2
+
E ~!
|µab |2
= 4.16666...
E + ~!
Secular divergencies
The Rayleigh–Schrödinger expression for the linear response
function is:
"
#
! |0i
! |nihn|⌦|0i
X h0|⌦|nihn|
ˆ
ˆ
1
V̂
h0|
V̂
!
ˆ V̂ ii =
hh⌦;
+
~ n
!n0 !
!n0 + !
This response function is not convergent as ! ! 0, and the
sum-over-states expression in this form cannot be used in the static
limit.
Response functions free of secular divergencies are obtained with
derivations based on phase-isolated wave functions.
Overall permutation symmetry
The linear response can be compactly written as
ˆ V̂ !1 ii =
hh⌦;
X
P
,1
X h0|⌦|nihn|
ˆ
V̂ !1 |0i
n>0
!n0
!
where ! = !1 is the sum of optical frequencies and P
permutes pairs of operators and frequencies:
ˆ and (!1 ,V̂ !1 ).
( ! ,⌦)
,1
Poles and residues
A residue analysis provides a means to obtain excited state
properties from the ground state response function. The poles
equal excitation energies and the residues are given by
lim
!1 ! !f 0
(!f 0
ˆ V̂ !1 ii = h0|⌦|f
ˆ ihf |V̂ !1 |0i
!1 ) hh⌦;
Summary: perturbation theory expansions
I
Response functions identified from perturbation theory
expansions equal sum-over-states expressions
I
An understanding is provided as to how response properties
depend on excitation energies and transition moments, which
allows for rational material design
I
With use of phase isolated wave functions, response functions
are free of secular divergences
I
From residue analyses of ground state response functions we
are able to identify excited state properties
I
A typical calculation of an excitation energy is made by
finding a pole of the linear response function (polarizability)
rather than an eigenvalue of the molecular Hamiltonian
I
This approach can be somewhat complicated for system
and/or spectral regions showing a high density of states
Resonant fields
b
ω
F(t) = F sin(ωt)
⇤E = 0.5 a.u.
µ ab= 1.0 a.u.
a
ψ(0) = ψa
i ψ(t) = H ψ(t)
H = H0 - µ F(t)
What happens if ~! =
I
E?
Response functions of the form

1 X h0|µ̂|nihn|µ̂|0i h0|µ̂|nihn|µ̂|0i
↵(!) =
+
~
!n0 !
!n0 + !
n>0
I
are not well defined and do not relate to the polarization of
the system
Perturbation theory is only applicable for a short period of
time after the field is turned on
Absorption
Let us ask the Schrödinger equation:
Perturbation theory in the short-time region gives the result:
1
⇢B (t) = 2 |hf |V̂ ! |0iF ! |2 t 2
~
Rabi Oscillations
Relaxation
Sn
IC
ESA
S1
ISC
ESA
OPA
TPA
IC
T1
Fluoresc.
Phosphoresc.
S0
Tn
Density matrix formalism
The density operator is defined as
X
⇢ˆ =
p(s) | s (t)ih
s (t)|,
s
where the sum denoted a classical ensemble average. If all systems
in the ensemble are identical then the summation contains a single
term with unit probability.
With wave function expansions in the form of projections onto the
eigenstates
X
| s (t)i =
cns (t)|ni,
n
the matrix elements of the density operator becomes equal to
X
s s⇤
⇢mn =
p(s) cm
cn .
s
Coherence
An ensemble of systems with equal state populations, but for
which the phases of the wave function components varies in an
incoherent manner, will have vanishing elements ⇢mn (m 6= n):
✓
◆
1
1 0
⇢= ⇥
.
0 1
2
An ensemble that can be described by a single wave function, on
the other hand, is fully coherent and is said to be in pure state:
1 ⇣
| i = p ei a |
2
ai
+ ei b |
bi
⌘
with a density matrix:
✓
1
1
⇢ = ⇥ i( b
e
2
e i(
a)
a
1
b)
◆
.
Time evolution and the Liouville equation
Let us consider a pure state situation. Then
@
@
@
⇢ˆ = |
ih | + | ih
|
@t
@t
@t
1
=
[H, ⇢ˆ],
i~
which is known as the Liouville equation. For all intents and
purposes, this equation provides an identical description of the QM
system as if one chooses to work with the Schrödinger equation
and a wave function approach.
The di↵erence lies in the possibility to modify the Liouville
equation as to treat e↵ects of spontaneous and collision-induced
relaxation and heat bath interactions.
Liouville equation with relaxation
@
1
⇢mn = [Ĥ, ⇢ˆ]mn
⇢eq
mn (⇢mn
mn )
@t
i~
The diagonal elements of the damping parameter matrix will
govern the spontaneous population decays:
⌧n = 1/
nn ;
n
=
nn ,
n 6= 0;
0
=0
we can, for a pure state, draw a conclusion that the o↵-diagonal
elements of the density matrix will depend on time according to
|⇢mn (t)| = |cm (t)cn⇤ (t)| = |cm (0)cn⇤ (0)|e
and we must therefore have
mn
=(
m
+
n )/2.
(
m + n )t/2
,
Time propagation of density matrix
@
1
⇢mn = [Ĥ, ⇢ˆ]mn
@t
i~
mn (⇢mn
⇢eq
mn )
Initialization
Time Propagation
H=array([[Ea, 0],
[0, Eb]])
mu=array([[0, muab],
[muba, 0]])
F=Fw*sin(w*t)*erf
for k in range(1,n):
rho[:,:,k] = rho[:,:,k-1] delta*1.0j*(
dot(H - mu*F[k-1],rho[:,:,k-1]) dot(rho[:,:,k-1],H - mu*F[k-1])
) delta*gamma*(rho[:,:,k-1]-rho_eq)
rho_eq=array([[1,0],
[0,0]])
rho[:,:,0]=[[1,0],
[0,0]]
P[k] = dot(rho[:,:,k],mu).trace()
popa=rho[0,0,:]
popb=rho[1,1,:]
⇢(0) =
✓
◆
0 0
;
0 1
⇢eq =
✓
◆
1 0
;
0 0
=
⇥
✓
◆
1 1/2
1/2 1
Induced polarization in two-level atom (resonant field, relaxation)
= 0.025 a.u.
↵=i
max[µ(t)]
⇡ 80i a.u.
F!
Density matrix from perturbation theory
(0)
(1)
(2)
⇢mn (t) = ⇢mn + ⇢mn + ⇢mn + . . .
with
(0)
⇢mn =
m0 n0
leads to
(N)
⇢mn
From
=e
(i!mn +
mn )t
Z
t
1
1
[V̂ , ⇢ˆ(N
i~
1)
]mn e (i!mn +
mn )t
0
dt 0 .
ˆ (1) = Tr(ˆ
ˆ
h⌦i
⇢(1) ⌦)
we identify the linear response function as
"
#
! |0i
! |nihn|⌦|0i
X h0|⌦|nihn|
ˆ
ˆ
1
V̂
h0|
V̂
ˆ V̂ ! ii =
hh⌦;
+
.
~ n
!n0 ! i n0
!n0 + ! + i n0
Two-level atom: response function value
Observable:
Perturbation:
E 6
↵(!) =
ˆ = µ̂
⌦
V̂ ! = µ̂
E = Eb
Ea
E = 0.5 a.u.
Eb
b
Ea
a
|µab |2
+
E ~! i~
µab = 1.0 a.u.
~! = 0.5 a.u.
~ = 0.0125 a.u.
|µab |2
|µab |2
⇡
= 80i
E + ~! + i~
i~
Complex Polarizability
Pole structure
Im(z)
Re(z)
** * *
hhÂ; V̂ iiz
=
* * **
"
#
1 X h0|Â|nihn|V̂ |0i h0|V̂ |nihn|Â|0i
+
~
!n z i n
!n + z + i n
n>0
I
Linear response function is convergent for all real frequencies
(but will be complex).
I
No poles in the upper half plane as required for causal
propagators.
Relaxation in wave function theory
0
E0
0
B 0 E1 + i
B
Ĥ0 = B .
..
@ ..
.
0
0
···
/2
···
1
..
.
0
0
..
.
···
En + i
1
n /2
C
C
C
A
I
Requires a resolution of all states
I
Non-Hermitian Hamiltonian is not norm conserving
Ehrenfest theorem
@
ˆ (t)i = 1 h (t)|[⌦,
ˆ Ĥ]| (t)i
h (t)|⌦|
@t
i~
ˆ = |nihm|:
Apply the Ehrenfest theorem to ⌦
@
1 h
hm| (t)ih (t)|ni =
hm|Ĥ| (t)ih (t)|ni
@t
i~
hm| (t)ih (t)|Ĥ|ni
The above equation is thus a mere repetition of the Liouville
equation for density matrix element ⇢mn , and a suitable
equation-of-motion with relaxation in wave function theory is:
@
ˆ nm | i = 1 h |[⌦
ˆ nm , Ĥ]| i
h |⌦
@t
i~
mn
h
ˆ nm | i
h |⌦
h
eq
ˆ nm |
|⌦
eq
i
i
i
Linear response functions
Molecular spectroscopies are interpreted as responses to
electromagnetic fields. The linear responses of a molecular
property to a perturbation are given by
hhÂ; V̂ ii! =
"
#
1 X h0|Â|nihn|V̂ |0i h0|V̂ |nihn|Â|0i
+
~
!n ! i n0
!n + ! + i n0
n>0
Selection of spectroscopies properties and
associated linear response functions
Refraction
Absorption
Magnetizability
Optical rotation
Circular dichroism
Re
Im
Re
Im
Re
hhµ̂; µ̂ii!
hhµ̂; µ̂ii!
hhm̂; m̂ii0
hhµ̂; m̂ii!
hhµ̂; m̂ii!
Work done by EM fields on particles
A point charge in an EM field experiences the Lorentz force:
F = qE + q (v ⇥ B)
The magnetic force is perpendicular to the magnetic field and the
velocity (or current j = qv) of the particle. The latter implies that
magnetic forces do not work. The work becomes
Z rt
Z t
Z t
Z t
0
0
W =
F · dr =
F · v dt =
qE · v dt =
E · j dt 0
We get an “instantaneous power”:
dW
=E·j
dt
Linear absorption cross section
In the electric dipole approximations, the average change of
absorbed energy per unit volume becomes
⌧
d absorbed energy
!
= hE(t) · j(t)iT = Im [ e (!)] I
dt
volume
c
T
where the angular brackets indicate the average over one period of
oscillation T . We have used
@P(t)
+ e i!t ); j(t) =
@t
1
I = "0 cE02
e (!) = N ↵(!);
2
E(t) = 12 E0 ✏(e
"0
i!t
We identify the linear absorption cross section as
(!) =
!
Im [↵(!)]
"0 c
Summary
I
Molecular properties are defined by expansions of energy,
quasi-energy, polarization, magnetization, etc. in orders of
field amplitudes
I
The diagonal elements of the electronic Hessian contains
excitation energies
I
Use of explicitly unitary parameterizations embeds the
requirement of conserving the norm of the wave function
I
Use of phase-isolated wave functions in expansions yields
response functions that are free of secular divergences
I
Poles and residues of response functions are connected to
excited state properties
I
With spontaneous relaxation, excited state populations may
remain small under resonance conditions. The corresponding
response functions become complex and describe absorptive
and dispersive spectroscopies
Some numerical examples with complex response functions
Restricted open-shell DFT for radicals
**DALTON INPUT
.RUN RESPONSE
**WAVE FUNCTION
.DFT
CAMB3LYP alpha=0.190 beta=0.810 mu=0.330
**RESPONSE
*ABSLRS
.ALPHA
.DAMPING
0.0045566
# Carbon K-edge
.FREQ INTERVAL
10.01 10.04 0.0025
*END OF
X-ray absorption spectrum
Cu-phthalocyanine
Cu
B
6
X-ray absorption
Dalton2015
4
expt
D
A
2
0
284
C
285
Linares et al., J. Phys. Chem. B 115, 5096 (2011)
286
E
F
286
287
288
Photon energy [eV]
288
290
289
292
Wave function correlation and self-interaction free spectra
Accuracy in absolute transition energies
CIS
STEX
Relaxation: 11.14 eV
1.2 eV
CCSD(R3)
IPCH2
Expt
285
290
IPCF2
295
Photon energy [eV]
300
305
Fransson, Coriani, Christiansen, Norman, J. Chem. Phys. 138, 124311 (2013)
Chemical shifts
σ(ω) [arb. units]
CCSD level of theory
triples corrected energies and scalar relativistic e↵ects included
additional common shift of 1.2 eV applied
Ethene
cis-1,2-difluoroethene
Vinylfluoride
Trifluoroethene
1,1-difluoroethene
Tetrafluoroethene
284
286
288
290
292
294 284
286
288
290
292
294
Fransson, Coriani,
Norman, J. Chem.Photon
Phys.energy
138, 124311
(2013)
PhotonChristiansen,
energy [eV]
[eV]
Silicon L-edge of silane derivatives
K- and L-edge X-ray absorption spectrum calculations of
closed-shell carbon, silicon, germanium, and sulfur
compounds using damped four-component density
functional response theory
spin-orbit
coupling
p3/2
p1/2
CAMB3LYP(100%)
Fransson, Burdakova, Norman, Phys. Chem. Chem. Phys. (2016)
X-ray one- and two-photon absorption
XTPA
1b
3p
3s
2a
2p
2s
Ne
1a
3d
Expt
3d
CPP/DFT
Ne
3s
2b
XTPA
CAMB3LYP(100%)
865.0
1s
XAS
4p
Absorption Cross Section
4p
4s
3p
Ne
K-edge selection rules
866.0
867.0
868.0
869.0
870.0
Transition Energy (eV)
Fahleson, Ågren, Norman, J. Phys. Chem. Lett. (2016)
X-ray one- and two-photon absorption
XAS
Absorption Cross Section
Probing the s and p character of the final state
Expt
2b2
4a1
CPP/DFT
XTPA
CAMB3LYP(100%)
wikipedia.org
533.0
534.0
535.0
536.0
537.0
538.0
539.0
Transition Energy (eV)
Fahleson, Ågren, Norman, J. Phys. Chem. Lett. (2016)