UNIVERSITY OF CALIFORNIA,
IRVINE
Exact and peculiar aspects of time-dependent density-functional theory
DISSERTATION
submitted in partial satisfaction of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
in Chemistry
by
Zenghui Yang
Dissertation Committee:Professor Kieron Burke, Chair
Professor Craig MartensProfessor Vladimir Mandelshtam
2011
Portion of Chapter 2 and Appendix B c© 2009 American Institute of PhysicsAll other materials c© 2011 Zenghui Yang
DEDICATION
To
my grandmother and my parents
ii
TABLE OF CONTENTS
LIST OF FIGURES v
LIST OF TABLES vii
ACKNOWLEDGEMENTS viii
CURRICULUM VITAE ix
ABSTRACT OF THE DISSERTATION x
1 Introduction 1
1.1 A brief review of the density-functional theory . . . . . . . . . . . . . 3
2 Non-exactness of the Kohn-Sham oscillator strength at the ioniza-
tion threshold 8
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 High-frequency limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Fitting the spectra of hydrogen and helium using sum rules . . . . . . 192.5 Results and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Prelude to the next chapter: the half-power decay of atomic high-
frequency spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Time-non-analyticities in time-dependent systems originating from
density cusps 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.1 1-d disappearing nucleus case . . . . . . . . . . . . . . . . . . 323.1.2 Non-applicability of the t-TE . . . . . . . . . . . . . . . . . . 343.1.3 Runge-Gross theorem of TDDFT . . . . . . . . . . . . . . . . 36
3.2 Analyzing the time-non-analyticities . . . . . . . . . . . . . . . . . . 373.2.1 Reason of the non-applicability of the t-TE . . . . . . . . . . . 373.2.2 The cause of the non-analytic short-time behavior . . . . . . . 383.2.3 The s-expansion method . . . . . . . . . . . . . . . . . . . . . 44
3.3 One dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.1 1-d disappearing nucleus case . . . . . . . . . . . . . . . . . . 453.3.2 1-d hydrogen in turned-on static electric field . . . . . . . . . 47
3.4 Three dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . 523.4.1 3-d hydrogen in turned-on static electric field . . . . . . . . . 53
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
iii
A The non-analytic short-t behavior in µ(t) from the non-analytic high-
ω behavior in ℑ[α(ω)] 59
B Fit parameters for real, exact KS, and ALDA helium 61
C Relation between the initial wavefunction and time-non-analyticities 62
D 1st order perturbative treatment of the 1-d hydrogen in static elec-
tric field case 66
E Method of dominant balance 68
F Borel summation 71
G Stationary phase approximation of the non-analytic short-time be-
havior 73
H 3-d disappearing nucleus case 75
Bibliography 77
iv
LIST OF FIGURES
2.1 Exact hydrogen oscillator strength spectrum. The ionization thresholdis at 0.5 Hartree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Exact hydrogen spectrum. The ionization threshold is at 0.5 Hartree.Note that the discrete bound-to-bound spectrum has been renormal-ized so that it joins smoothly with the continuum-region spectrum. . 14
2.3 KS and experimental single-electron oscillator strength of He nearthreshold[1, 2]. The ionization threshold is at 0.9036 Hartree. Thediscrete spectrum is renormalized with the factor n3
f , in which nf isthe principal quantum number of the final state. . . . . . . . . . . . . 16
2.4 g(ξ) of hydrogen(Eq. (2.18)). The ionization threshold is at ξ = 0. . . 212.5 Exact/ALDA oscillator strength and fitted curve of He. These oscil-
lator strength data are obtained from an ALDA calculation with theexact KS ground state. We use a box code[3] to calculate these data.There is a kink in our data near the ionization threshold because thecontinuum near the ionization threshold mixes with higher Rydbergstates, which are not well-described by the box code. . . . . . . . . . 24
2.6 KS and experimental single-electron oscillator strength and fitted curveof He near threshold[1, 2]. The ionization threshold is at 0.9036 Hartree(24.59eV). The curves are converted from g(x) fit(Eq. (2.21)). The uppercurve represents the exact KS helium oscillator strength data and fittedcurve, and the lower curve represents the experimental helium oscilla-tor strength data and fitted curve. . . . . . . . . . . . . . . . . . . . . 24
2.7 KS and experimental single-electron oscillator strength and fitted curveof He[1, 2]. This figure show the overall shape of the oscillator strengthcurves. The solid dots and curve represents the exact KS helium os-cillator strength data and fitted curve, and the cross dots and dashedcurve represents the experimental helium oscillator strength data andfitted curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 g-fit of the experimental helium oscillator strengths near the auto-ionizing resonances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.9 Illustration of the oscillator strengths of a system without continuum,in which the η in Eq. (1.11) is assumed to be finite. The black line isthe discrete oscillator strength spectrum. The blue line is the envelopeof the oscillator strength spectrum. The denser the discrete spectrumis, the better description of the high-frequency behavior the envelope is. 29
3.1 Electronic density of the 1-d disappearing nucleus case. Left panel:exact case. Right panel: t-TE result. . . . . . . . . . . . . . . . . . . 34
3.2 The time-dependent wavefunction of the 1-d disappearing nucleus casein (x, t) variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
v
3.3 The time-dependent wavefunction of the 1-d disappearing nucleus casein (s, x) variables. The cusp at the origin is removed under this change-of-variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 ψ(x, t = 0.1) of the 1-d disappearing nucleus case, plotting the trun-cated inner-region expansion and the outer-region expansion. . . . . . 41
H.1 Time-dependent exact density and t-TE density of the ground-statewavefunction of hydrogen atom under free-propagation. . . . . . . . . 75
vi
LIST OF TABLES
2.1 Sum rules from σ(ω) and g(ξ) fit . . . . . . . . . . . . . . . . . . . . 25
B.1 Fit parameters of the oscillator strength fit . . . . . . . . . . . . . . . 61
vii
ACKNOWLEDGEMENTS
I thank my research advisor Prof. Kieron Burke, who introduced me to the wonderfulworld of time-dependent density-functional theory, and guided me through all kindsof impossibly-looking obstacles. I never imagined about the stuff discussed in thisdissertation before I met Kieron, and his experience helped a lot during the time whenwe tried a lot of things and all didn’t work. Again, I thank Kieron for everything he didas an advisor and as a friend. I thank Dr. Meta van Faassen for co-authoring the paper“Must Kohn-Sham oscillator strengths be accurate at threshold?”, published in 2009in the Journal of Chemical Physics(Chapter 2 in this dissertation). Meta producedmost of the helium oscillator strength data in the paper, and without her work, Iwouldn’t have the material to work on in the first place. I also thank Prof. AdamWasserman for generously providing me his data on oscillator strengths. I thank Prof.Neepa Maitra, my collaborator on the time-non-analyticity project(Chapter 3 in thisdissertation), for her active involvement in the project and really helpful discussionsand rigorous mathematical considerations. I thank Prof. Filipp Furche, Dr. HenkEshuis and Prof. Barbara Finlayson-Pitts for introducing me into the AirUCI NO2
project which gave me valuable experience on real world stuff and real computations,and for all the help I received from them. I thank all my wonderful group-mates andex-group-mates and those in Filipp’s group for the great time we spent together.
I thank the American Institute of Physics for their general policy on author rights,which allows me to include Chapter 2 and Appendix B in my dissertation. Financialsupport was provided by the University of California, Irvine, and U.S. Departmentof Energy grant DE-FG02-08ER46496.
viii
CURRICULUM VITAE
Zenghui Yang
2006 B.S. in Chemistry, Fudan University, China2011 Ph.D. in Chemistry, University of California, Irvine
PUBLICATIONS
“Must Kohn-Sham oscillator strengths be accurate at threshold?” Zenghui Yang,Meta van Faassen, and Kieron Burke, J. Chem. Phys., 131, 114308(2009)
ix
ABSTRACT OF THE DISSERTATION
Exact and peculiar aspects of time-dependent density-functional theory
by
Zenghui Yang
Doctor of Philosophy in Chemistry
University of California at Irvine, 2011
Professor Kieron Burke, Chair
Ground-state density-functional theory (DFT) and time-dependent density-functional
theory(TDDFT) are popular electronic structure methods among chemists, thanks to
their balance between accuracy and computational cost. DFT methods achieve this
balance through a mapping from the real world system with interacting electrons
to a fictitious non-interacting Kohn-Sham(KS) system with the same electronic den-
sity, allowing one to represent the properties of the real system as functionals of the
density. The connections between the real world and the non-interacting world are
unclear in many aspects, and exact physical constraints for these connections must
be derived to make DFT and TDDFT work better.
This dissertation contains two closely related parts studying the connections be-
tween the real systems and the KS systems. In the first part, I study the oscillator
strength at the ionization threshold of helium, and I reach the conclusion that the KS
oscillator strength at the ionization threshold is not exact, despite the exactness of
the position of the ionization threshold. I observe that the high-frequency behavior
of the oscillator strength of all atoms decays as ω−7/2, implying non-time-Taylor-
x
expandability in the time-domain. This observation leads to the second part of the
dissertation, in which I demonstrate that the time-dependent wavefunctions are not
time-Taylor-expandable whenever the initial wavefunction has derivative discontinu-
ities at any order, due to specific time-non-analyticities occurring at the initial time.
The analysis of these time-non-analyticities is given in this dissertation, and I propose
a method of obtaining the correct short-time behavior of such systems.
xi
Chapter 1
Introduction
Within the Born-Oppenheimer approximation, a vast amount of useful informa-
tion about a molecule or solid can be extracted from the electronic ground state,
such as geometry, thermodynamics, and vibrations. In addition to these properties,
electronic excitations are extremely important for many areas, from photochemistry
to photoemission spectra[4]. Ground-state density functional theory(DFT)[5, 6, 7]
has been very successful for the former problem, and its time-dependent analog
(TDDFT)[8] has recently become popular for the latter[9]. Within linear response,
TDDFT yields useful predictions for the excited states of many molecules[4], and
extensions to solids are a keen area of research[10, 11].
Modern DFT calculations employ a Kohn-Sham(KS) scheme[6], in which a fic-
titious set of non-interacting electrons reproduces the one-electron density of the
real system. The non-interacting KS system reproduces the electronic density of the
real system by definition, and it does so through the use of an effective one-body
potential(the KS potential), derived from the density functional of ground-state en-
1
ergy. Knowing the exact KS potential is equivalent to exactly solving the Schrodinger
equation, so approximations to the functional are required in practical calculations.
One usually develops approximated functionals by enforcing a chosen set of exact
conditions[9], which are physical constraints derived exactly from quantum mechan-
ics, in order to have systematic performance over a wide range of systems. Many
such exact conditions for the ground-state DFT are known, but those for TDDFT
are much less developed and more understanding about what is exact and what is not
in TDDFT is important. In Chapter 2, I investigate one such aspect, the exactness of
the KS oscillator strength at the ionization threshold. I show that the KS oscillator
strength is not exact at the ionization threshold, though the ionization threshold is
exactly given by the KS system.
All density functional theories rely on establishing a one-to-one correspondence
between the one-body potential, such as the Coulomb potential −Z/r for an atom,
and the density n(r), under some restrictions. For the ground-state, the restrictions
are that the particles be fermions and the interaction be fixed (Coulomb repulsion).
For the time-dependent case, one also specifies the initial wavefunction. The origi-
nal proof in the ground-state case of Hohenberg and Kohn[5] has been refined over
the decades[12, 13, 14], but its essence remains unchanged. In the time-dependent
problem, after several pioneering works by others[15, 16, 17, 18, 19, 20, 21], Runge
and Gross[8] gave a proof assuming the one-body potential is time-analytic, i.e., that
it equals its Taylor expansion in time at the initial time for a finite time-interval.
Despite several recent attempts[22, 23, 24, 25, 26], no generally applicable proof has
been found that avoids this expansion.
2
Besides the one-to-one mapping problem, another fundamental problem in ground-
state DFT and TDDFT is the so-called v-representability problem. It questions the
existence of the KS potential, under whose influence the density of a given interacting
system is reproduced by the fictitious non-interacting KS system. This problem
has not yet been generally solved for the ground-state case. By assuming that the
time-dependent density is also time-analytic, van Leeuwen[27] gave a constructive
procedure for finding the time-dependent KS potential for a given system.
All this might be fine, if the static potentials of nuclei were sufficiently smooth.
But, in the usual treatment, matter has cusps in its ground-state electronic wave-
functions at the nuclei due to the singular Coulomb potential[28], and the resulting
spatial non-analyticities are coupled to the time-dependence in the time-dependent
Schrodinger equation(TDSE)[24]. Even in the most mundane example, a hydrogen
atom in a static electric field, this coupling leads to non-analytic short time behavior,
and its time-dependent density is not time-analytic, as shown definitively in Chapter
3.
Atomic units are used throughout this dissertation unless otherwise noted, so that
energies are in Hartrees and distances in Bohr radii.
1.1 A brief review of the density-functional theory
I provide a short review of the DFT and TDDFT for the purpose of the following
chapters. DFT is based on the observation that the 3N dimensional N -electron wave-
function Ψ(r1, . . . , rN) contains the same amount of information as the 3 dimensional
3
electronic density n(r), defined as
n(r) = N
∫
d3r1 . . . d3rN |Ψ(r1, . . . , rN)|2 . (1.1)
The above observation is the Hohenberg-Kohn theorem[5, 12, 13, 14] which proves
that the density of a system has a unique mapping to its external potential. The ex-
ternal potential in turn completely determines the N -electron wavefunction, and thus
shows that the N -electron wavefunction and the density are equivalent in describing
the electronic system.
The most important quantity in ground-state theories is the ground-state energy.
It is a density functional according to the Hohenberg-Kohn theorem. In the modern
form, it is written as[12, 7]
Egs = minn
F [n] +
∫
d3r vext(r)n(r)
, (1.2)
in which vext(r) is the external potential of the real system, and F [n] is
F [n] = minΨ→n
⟨
Ψ∣
∣
∣T + Vee
∣
∣
∣Ψ⟩
, (1.3)
with T and Vee are the kinetic energy operator and the electron-electron interaction
potential operator, respectively. F [n] is called the universal functional as it does not
change over different systems. F [n] is decomposed as the following:
F [n] = TS[n] + U [n] + EXC[n], (1.4)
in which TS[n] is the non-interacting kinetic energy, U [n] is the Hartree energy, EXC[n]
is the exchange-correlation energy. In practical computations, the KS scheme is em-
ployed, in which one solves the KS equations[6]:
−1
2∇2 + vs(r)
ψi(r) = ǫiψi(r). (1.5)
4
KS orbitals and KS eigenenergies are denoted as ψi(r) and ǫi, and vs(r) is the KS
potential. The KS density is defined as
n(r) =
N∑
i=1
|ψi(r)|2 , (1.6)
and is by definition equal to the exact density. vs(r) is related to the energy functionals
in Eq. (1.4) by[6, 7].
vs(r) = vext(r) +δU [n]
δn(r)+δEXC[n]
δn(r)
= vext(r) + vH(r) + vXC(r).
(1.7)
The main purpose of DFT research is to find better approximations to EXC[n]. Though
one can certainly approximate EXC[n] by fitting experimental data, a more physically
meaningful way is by making the approximated EXC[n] satisfy a certain set of ex-
act conditions. The exact conditions are important since they serve as connections
between the fictitious KS system and the real system.
Though the ground-state energy determines many properties of the system[29],
such as geometry, IR and NMR spectra, thermodynamics, dipole moment, static
polarizability and so on, important properties like excited-state energies and the
optical spectrum cannot be derived from the ground-state energy alone. These
properties in principle are ground-state density-functionals as well, and there have
been several recent studies trying to extend the ground-state DFT to excited state
properties[30, 31, 32, 33, 34, 35], but the most applicable method for excited-state
calculations is TDDFT, which goes beyond the ground-state formalism.
TDDFT is based on the Runge-Gross theorem[8] stating that there is a unique
mapping between the time-dependent density and the time-dependent external po-
5
tential. Its derivation is briefly discussed in Sect. 3.1.3. The Runge-Gross theorem
implies that the time-evolution of the density of the real interacting system can be ex-
actly reproduced by the time-evolution of the density of an fictitious non-interacting
system. For a practical calculation of the time-dependent density, one is required to
find an approximation to the effective external potential of the non-interacting system.
For excited-state calculations, the TDDFT formulation is Fourier transformed from
time domain to frequency domain, which leads to the linear response TDDFT[36, 37].
The linear response TDDFT studies the frequency-domain linear response function
χ(r, r′, ω) defined as the Fourier transform of the time-domain linear response function
χ(r, r′, t, t′), which is
χ(r, r′, t, t′) =δn[vext](rt)
δvext(r′t′)
∣
∣
∣
∣
vext[n0]
. (1.8)
The KS linear response function χKS(r, r′, ω) is defined similarly. The KS linear
response function is related to the real(or ‘exact TDDFT’) linear response function
by a Dyson-like equation[38, 39]:
χ(r, r′;ω) = χKS(r, r′;ω) +
∫
d3r1
∫
d3r2 χKS(r, r1;ω)fHxc(r1, r2;ω)χ(r2, r′;ω),
(1.9)
in which fHxc(r, r′;ω) is the so-called Hartree-exchange-correlation(HXC) kernel in
the frequency domain, defined as the Fourier transform of the HXC kernel in the
time domain:
fHxc(r, r′; t− t′) =
1
|r − r′| +δvxc(r, t)
δn(r′, t′). (1.10)
The linear response function is also represented in Lehmann representation[23]:
χ(r, r′;ω) = limη→0+
∑
α
gα(r)g∗α(r
′)
ω − Ωα + iη− g∗α(r)gα(r
′)
ω + Ωα + iη
, (1.11)
6
in which gα(r) = 〈Ψgs|n(r)|Ψα〉 with Ψα being an excited state, and Ωα = Eα − Egs.
7
Chapter 2
Non-exactness of the Kohn-Sham
oscillator strength at the ionization
threshold
This chapter is a partial rewrite of my previously published paper Ref. [40], with
co-authors Meta van Faassen and Kieron Burke. ‘We’ in this chapter refers to my
co-authors and me.
2.1 Preliminaries
By virtue of the Hohenberg-Kohn theorem, all atomic and molecular properties
are functionals of the ground-state density in principle, including the properties of
excited states[41]. In practice, however, only the ground-state energy functional has
been well-approximated. The excited-state properties of the non-interacting KS ref-
8
erence system are often employed for understanding and even approximating those
of the true interacting system, but in most cases this has no theoretical justification.
Thus the results of excited-state calculations with ground-state DFT must be care-
fully examined, since the KS orbitals and energies are(within ground-state DFT) by
definition artificial constructs designed only to reproduce the ground-state density.
The more we understand about the differences between the KS system and the real
system, the better we can determine whether an excited property of the KS system
can be justified as an approximation to the real property. To this end, we study the
exactness of the KS oscillator strength at the first ionization threshold in this chapter.
On the other hand, TDDFT gives the exact properties of time-dependent systems
in principle[8]. Its frequency-domain Fourier transform, the linear response TDDFT,
couples ground-state KS transitions to give correct properties of excited states[37, 36].
If the exact time-dependent functional is available, the TDDFT method would exactly
generate the properties of the real system from the results of the ground-state KS
calculation of systems with non-interacting electrons. Thus our study of the exactness
of the KS oscillator strength is also related to the difference between ground-state
DFT and TDDFT.
This may seem to be a simple problem, since ground-state DFT is not designed
to yield the correct oscillator strength at the ionization threshold. The oscillator
strengths can be extracted from strength of the poles of the linear response func-
tion, and the KS linear response function does not involve the Hartree-exchange-
correlation(HXC) kernel(Refer to Eq. (1.9) and (1.10)). There is no a priori reason
to expect the exact KS system to give the correct oscillator strength at the ionization
9
threshold. By ‘exact KS’ we mean the KS potential as extracted from an extremely
accurate ground-state density[42], thereby avoiding the difficulty of distinguishing
the effect of approximate ground-state XC functionals from that of KS-DFT itself.
However, the ionization threshold of the exact KS system, which is an excited-state
property, is equal to the ionization threshold of the real system, since Koopmans’ the-
orem is exact for the exact KS system, unlike for the Hartree-Fock method in which
the Koopmans theorem is an approximation[7]. Thus this specific excited-state prop-
erty is given exactly by the KS system, despite the lack of input from the Hartree and
XC kernels. This is the only known direct link between real excited-state properties
and their KS counterparts, and such links have proven invaluable in studying and
understanding both ground-state and TDDFT[4].
Given the usefulness of such links, it is important to study the exactness of the KS
oscillator strength at the ionization threshold to see if some unknown exact condition
might be lurking beneath the surface. To do this, one specific case is studied in
this chapter, which is the helium atom. If, for helium, we find definitively that the
threshold oscillator strength is not given by the KS system, the KS oscillator strength
at the ionization threshold cannot be exact in general. If we did find it to match, the
question would remain open, and we would look for other cases and/or a proof of the
equality. As shown in this chapter, the answer is indeed no.
This has important consequences for the unknown exact XC kernel of TDDFT.
To shift an ionization threshold, the kernel would need to be complex, with a branch
cut at the position of the KS threshold. This is not the case for the first ionization
threshold, but is for all higher ionizations. On the other hand, the oscillator strength
10
of the KS system typically is corrected by TDDFT, meaning the HXC kernel must
have some non-zero off-diagonal matrix elements at the threshold. To understand
this, Ref. [43] showed that, in the absence of off-diagonal matrix elements, the KS
oscillator strengths are unchanged by the action of the kernel.
2.2 Background
In this section we provide a brief definition of notation and concepts used in this
chapter. The exact dependence of the XC contribution on the density is unknown,
and many approximation schemes are available, but in this chapter the exact value is
used[42]. This is calculated by first obtaining the accurate density from a quantum
Monte-Carlo calculation, then inserting the density into the KS equations and finding
the potential that gives this density[42].
The absorption spectrum in terms of photoabsorption cross-section σ is defined
as below[44]:
σ(ω) =2π2
c
∑
q
fqδ(ω − ωq) + σcont(ω), (2.1)
in which δ(ω − ωq) is the Dirac delta function, q denotes discrete bound-to-bound
transitions from occupied state i to unoccupied state f , fq is the oscillator strength
of transition q, and σcont(ω) = (2π2/c)(df/dω) is the photoionization cross-section of
the continuum region which begins at ω = I. For bound-to-bound transitions, the
oscillator strengths are defined as:
fq = 2ωq |〈Ψf |r|Ψi〉|2 . (2.2)
11
For spherically symmetric systems, the oscillator strengths are related to the dy-
namic polarizability α(ω) by the following equations:
α(ω) =
∫
d3r
∫
d3r′ z z′χ(r, r′;ω), (2.3)
σ(ω) =4πω
cℑ[α(ω)], (2.4)
The ionization threshold is a special position in the spectrum. For frequencies
lower than the threshold, the spectrum is discrete, and the oscillator strength de-
cays to 0 as the frequency approaches the threshold. For frequencies higher than the
threshold, the spectrum is continuous, and the oscillator strength starts at a finite
value at the threshold and eventually decays to 0 as the frequency increases. Consid-
ering the KS spectrum and the real spectrum are usually not the same for both these
parts, and the oscillator strength at the ionization threshold is a limiting property,
one cannot directly compare the oscillator strength at the ionization threshold.
To illustrate these features in an exactly solvable case, we use the hydrogen spec-
trum as an example. The exact form of the hydrogenic oscillator strength is available
using Eq. (2.2):
f1s→np = 256n5
(
n+ 1
n− 1
)−2n
/[
3(n2 − 1)4]
, (2.5)
df
dω
∣
∣
∣
∣
1s→kp
= 128 exp[h(k)] csch(π
k
)
/[
3(1 + k2)4]
, (2.6)
in which h(k) = π + 2 tan−1 [2k/(k2 − 1)] − 2πθ(k − 1) /2, n is the principal quan-
tum number, k =√
2E =√
2(ω + 1/2) is the wavevector of the continuum wave-
function, and θ is the Heaviside step function. Fig. 2.1 plots f(ω) for the hydrogen
atom, in which the bound-to-bound transitions are represented by vertical lines whose
height is f(ω).
12
0
0.5
1
1.5
12 16 20 24os
cilla
tor
stre
ngth
ω(eV)
Exact hydrogenIonization threshold
Figure 2.1: Exact hydrogen oscillator strength spectrum. The ionization threshold isat 0.5 Hartree.
To avoid the discontinuity of f(ω) at the ionization threshold, we define the renor-
malized photoabsorption cross-section σ(ω) as the analytical continuation of f(ω) for
ω < I. This can be found easily by considering the oscillator strength of bound-to-
bound transitions as a continuous function of ωq, yielding [44]
σ(ω) = f(ω)/
(
dE
dn
)∣
∣
∣
∣
E=ω−I, (2.7)
in which n is the principal quantum number. In reverse, the usual oscillator strength
for transition q = 1s→ np is given by
fq =
(
dE
dn
)
σ(I + E), (2.8)
in which E is the energy of the np state. The phrases “renormalized photoabsorption
cross-section” and “oscillator strength” are used interchangeably from now on, and
both refer to σ(ω).
For the hydrogen atom, Eq. (2.7) means the bound-to-bound oscillator strengths
have to be multiplied by n3, so that the discrete and the continuum parts of the
oscillator strength spectrum agree with each other at the ionization threshold. Then
13
the renormalized oscillator strength spectrum for the hydrogen atom is as shown in
Fig. 2.2.
0
0.5
1
1.5
2
2.5
3
3.5
12 16 20 24
reno
rmal
ized
osc
illat
or s
tren
gth
ω(eV)
Exact Hydrogen
Figure 2.2: Exact hydrogen spectrum. The ionization threshold is at 0.5 Hartree.Note that the discrete bound-to-bound spectrum has been renormalized so that itjoins smoothly with the continuum-region spectrum.
To prove that the KS oscillator strength does not have to be exact at the ion-
ization threshold, only a counter-example is needed. The helium oscillator strength
spectrum is studied in this chapter. The helium atom is the simplest multi-electron
system, and thus a theorist’s favorite. There is no trivial formula for calculating the
energies of bound-states in multi-electron atoms. Helium atom has 2 electrons, so the
renormalization of the oscillator strengths Eq. (2.7) cannot be performed directly.
In order to characterize these energies for the renormalization, we use quantum de-
fect theory[44]. In quantum defect theory, the energy of the orbital with principal
quantum number n in a multi-electron atom is expressed thus:
En = − 1
2(n− µn)2, (2.9)
in which µn is the quantum defect of the state n. The quantum defect is a smooth
function of energy, and can be very accurately approximated[45] by its Taylor expan-
14
sion around µ = 0:
µ(p)(E) =
p∑
i=0
µiEi, E = ω − I. (2.10)
For helium, this curve is essentially linear, so µ ≃ µ0 + µ1E, in which µ0 = 0.0164
and µ1 = 0.0289 for KS helium, and µ0 = −0.0122 and µ1 = −0.0227 for real helium.
Inserting this expression into the En formula and solving self-consistently yields highly
accurate excitation energies[3]. The discrete part of the oscillator strength spectrum
is then renormalized with (n− µn)3, instead of n3 for hydrogen.
Fig. 2.3 shows the renormalized photoabsorption spectrum of helium near the
ionization threshold(24.6 eV) for real helium and exact ground-state KS helium. Fig.
2.3 suggests the KS oscillator strength is not exact at the ionization threshold, but
there could conceivably be near-degeneracies near the ionization threshold. We wish
to demonstrate that the oscillator strength curve can be expected to be smooth near
the ionization threshold explicitly. Hence a fitting method is developed to explicitly
show that the oscillator strength curves of the real helium and the KS helium are
smooth across the ionization threshold, showing there are no near-degeneracies at the
threshold. This allows the comparison of the value of the oscillator strengths at the
ionization threshold of these two systems, showing that the difference between the
spectra near the threshold of KS and real systems is inherent. Since our purpose is
to understand the difference between KS system and the real system, the fit is not
done to the data points but to the general properties, such as the oscillator strength
sum rules. The fit is tested on the hydrogen oscillator strength spectrum, and then
applied to KS helium, real helium, and the result of using approximated TDDFT on
15
the exact ground-state KS spectrum.
0
0.3
0.6
0.9
1.2
1.5
21 24 27 30 33
reno
rmal
ized
osc
illat
or s
tren
gth
ω(eV)
ExperimentalExact KS
1st ionization threshold
Figure 2.3: KS and experimental single-electron oscillator strength of He nearthreshold[1, 2]. The ionization threshold is at 0.9036 Hartree. The discrete spectrumis renormalized with the factor n3
f , in which nf is the principal quantum number of
the final state.
The fitting method is based on the oscillator strength sum rules. These sum
rules are moments of the oscillator-strength spectrum, and they are related to various
theoretical or experimental physical properties of the ground-state atom. They are
expressed with the following formula[46, 47]:
Sj =∑
q
ωjqfq +
∫ ∞
I
dω ωjf(ω), (2.11)
in which q denotes the discreet 1s → np transitions and j is an integer. We only
use −2 ≤ j ≤ 2 in this chapter. These sum rules have simple relations to physical
properties, such as the ground-state density, polarizability, and kinetic energy, and
thus they are easily calculated or determined from experiment. The specific relations
used are:
S−2 = α(ω = 0), S−1 =2
3
⟨
∣
∣
∣
∑
j rj
∣
∣
∣
2⟩
0
, S0 = N,
S1 =2
3
⟨
∣
∣
∣
∑
j pj
∣
∣
∣
2⟩
0
, S2 =4π
3
∑
atom A
ZA n(r = 0),
(2.12)
16
where α(ω = 0) is the static polarizability, and ZA is the nuclear charge of atom A.
Sn>2 does not exist, due to the ω−7/2 asymptotic decay of f(ω → ∞)[48], and Sn<−2
are related to various other properties of the system[47]. Eq. (2.11) and Eqs. (2.12)
not only provide connections between the spectrum and several physical properties,
but also imply that S0 and S2 are identical in the KS and real spectra, since the
ground-state density in exact DFT is by definition equal to that of the real system.
These equations also suggest the possibility of a fit which takes general physical
properties as input and is able to generate the entire spectrum for the H atom.
2.3 High-frequency limit
Fig. 2.3 suggests the KS oscillator strength and exact oscillator strength share the
same asymptotic form. Real oscillator strength spectra of atoms decay as ω−7/2[48,
49, 47]. Here we derive the decay of the KS oscillator strength.
The oscillator strength is related to the transition dipole matrix element 〈Ψf|z|Ψi〉
by Eq. (2.2). In the KS system, the matrix element is greatly simplified, and is written
with one-electron KS orbitals as 〈ψf|z|ψi〉. For the absorption spectrum of the KS
helium atom, the final orbital is a p orbital with wavevector k, and the initial orbital is
the 1s orbital. In the high frequency limit of the absorption spectrum(ω → ∞), k →
∞ as well, and φkp(r) is highly oscillatory(φ denotes radial wavefunctions[44]). This
means the matrix element is determined by the integrand near the nucleus[50]. The
matrix element is evaluated with following approximation to the initial KS orbital:
17
φi(r) = exp(−αr)[φi(r) exp(αr)]
∼ exp(−αr)
φ′i(0)r +
1
2[2αφ′
i(0) + φ′′i (0)]r2 + · · ·
,
(2.13)
in which φi is the spherical wavefunction of the initial KS orbital, and α is a positive
real number characterizing the decay of the wavefunction. The cusp condition[28]
holds in KS helium, so φ′′i (0) = −2Zφ′
i(0), in which Z = 2 is the nucleus charge.
Then φi is rewritten as
φi(r) ≈ exp(−αr)r + (α− Z)r2φ′i(0). (2.14)
For k → ∞ limit, only the −Z/r Coulomb well in the KS potential is important
to φkp. If φkp is approximated with hydrogenic wavefunctions, the approximation be-
comes exact for k → ∞. The transition dipole matrix element can then be evaluated
for k → ∞ limit.
〈ψf|z|ψi〉 ∼(
4
√
2
3πZφ′
i(0)
)
k−9/2 , k → ∞. (2.15)
The k−9/2 term is the leading term in 〈ψf|z|ψi〉. Being unrelated to the arbitrary
shape parameter α supports the observation that the high-frequency part of this
matrix element is determined by the region near the nucleus. Moreover, replacing
the exponential envelope in Eq. (2.13) with an arbitrary Gaussian envelope does not
change the result Eq. (2.15) as well, further justifying this result.
The oscillator strength spectrum then decays as
σ(ω) ∼ 2√
2
3π[φ′
i(0)Z]2ω− 7
2 , ω → ∞. (2.16)
18
Eq. (2.16) implies the asymptotic decay of the oscillator strength only depend on
the properties at the nucleus. For hydrogen and helium, Eq. (2.16) is related to the
electronic density by
σ(ω) ∼ 8√
2
3Z2n(0)ω− 7
2 , ω → ∞. (2.17)
For hydrogen, the coefficient of the ω−7/2 term is 8√
2/(3π). Eq. (2.17) yields
the exact result. With these equations, the asymptotic behavior of the oscillator
strength spectrum is determined. The discussion of the high-frequency part of the
KS helium oscillator strength spectrum can be extended to other KS atoms easily,
as the KS system is a one-electron picture. Following a similar procedure, it can be
easily verified that the high-frequency part of the KS oscillator strength of other KS
atoms can be expressed in terms of the density at the nucleus. As only s orbitals
have non-zero contribution to the density at the nucleus, Eq. (2.17) also holds for
other atoms, using the corresponding Z and n(0).
The half-power decay of Eq. (2.17) differs noticeably from the integer-power decay
mentioned in Ref. [23]. This is discussed further in Sect. 2.6 and Chapter 3.
2.4 Fitting the spectra of hydrogen and helium us-
ing sum rules
We fit the oscillator strength spectra to show the non-exactness of KS oscillator
strength at the ionization threshold. Since we want to study the near-threshold
behavior of the oscillator strength spectrum, the position of the ionization threshold
19
is treated explicitly in our fit. We define ξ and g(ξ) as
ξ = 2(ω − I),
g(ξ) =3ω4
8σ(ω) =
3(ξ + 2I)4
128σ(ξ + 2I).
(2.18)
The shape of the g(ξ) function is shown in Fig. 2.4.
The fit has to satisfy a few criteria to generate the correct shape for the oscillator
strength spectrum. Since the fit is employed to study the exactness of the KS oscillator
strength at the ionization threshold, it needs to have the correct series expansion. We
take the expansion of the hydrogen oscillator strength at the ionization threshold:
σ(ω → I) = c0 + c1(ω − I) + c2(ω − I)2 + · · · . (2.19)
The fit formula should have the same expansion near the ionization threshold.
This is justified by the following consideration. Near the ionization threshold, the
oscillator strength spectrum of real helium is determined by the Rydberg states,
which resemble the hydrogenic states. KS helium is a system with non-interacting
electrons, so the oscillator strength spectrum resembles that of one-electron systems.
We use the fit to show that the oscillator strength spectrum around the ionization
threshold is smooth, that no near- degeneracies exist around the ionization threshold.
Thus the fit also needs to accurately generate the entire oscillator strength spectrum,
including both discrete and continuum regions, so that the conclusions from the fit
are reliable. To generate the correct continuum spectrum, the fit needs to have the
correct series expansion for ω → ∞. As shown in Sect. 2.3, the asymptotic series
expansion of helium has the same form as hydrogen:
20
σ(ω → ∞) = d1ω− 7
2 + d2ω−4 + d3ω
− 9
2 + · · · . (2.20)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 100 200 300
g(x)
x(eV)
Exact HydrogenIonization threshold
Figure 2.4: g(ξ) of hydrogen(Eq. (2.18)). The ionization threshold is at ξ = 0.
Our g-fit formula is
g(ξ) = a+ b[1 − exp(−cξ)] + d√
e+ ξ, (2.21)
in which a, b, c, d, e are fit parameters. Note that aside from giving the correct
series expansion at the ionization threshold and asymptotically, the form does not
have other explicit physical motivation. It is solely designed to recover the shape of
the oscillator strength curves. We determine the parameters by the process below.
Since we use the fit to study the oscillator strength around the ionization threshold,
we fix the value and the first derivative of the oscillator strength at the ionization
threshold. The asymptotic coefficient in Sect. 2.3 is not fixed, and instead is used
as the initial point of search. The remaining three parameters are determined by
applying oscillator strength sum rules(Eq. (2.11)) to the fitted curve. We evaluate
the sum rule of a fitted curve by adding the contributions from the discrete transitions
21
and that from the continuum. For the discrete region, we calculate the frequency of a
transition with quantum defect theory(refer to Eq. (2.9) and (2.10)). The oscillator
strength of the transition is then evaluated with the fit formula(with a certain initial
choice of parameters). We add the contribution of different discrete transitions up to
n = 1000. For the continuum region, we carry out a numerical integration over the
entire continuum.
The exact values of the oscillator strength sum rules are available for both KS
and real helium, since these sums are related to various physical properties (refer
to Eq. (2.12)). To fit the oscillator strength spectrum, we choose an initial set of
the parameters. Only three fit parameters are independent, so we choose three sum
rules to fit. We minimize the difference between the sums evaluated on the fitted
curves and the exact sums obtained from physical properties by varying the three
parameters numerically. The search ends when the accuracy of the fitted sums reach
a predetermined goal. In our application, the difference between the sums of the fit
and the exact sums is smaller than 10−8. The accuracy of the fit is also checked by
evaluating the unused sum rules (Table. 2.1).
The fit uses two to four sum rules depending on how many points are fixed in the
beginning. Using more sum rules increases the overall accuracy of the fit, but the
process of numerically fitting sum rules becomes more difficult. All results in this
paper are obtained with three sum rules. With Eq. (2.21), the sum rules of the fitted
22
curve are written out in terms of the parameters.
Sj = Sdisj +
8
3
23−jd (2I − e)j−5/2B1−e/2I
(
5
2− j,
3
2
)
+Ij−3[a + b+ b (j − 3) exp (2cI)E4−j (2cI)]/(3 − j)
,
Sdisj =
8
3
∞∑
n=2
β3
a+ b[
1 − exp(
cβ2)]
+ d√
e− β2
γ.
(2.22)
in which β = (−µ0 − µ1/n2 + n)
−1, γ = (I − β2/2)
−4+j, µ0 and µ1 are the parameters
in the quantum defect formula(Eq. (2.10)), B is the incomplete beta function, and
E is the exponential integral function[51].
With Eq. (2.21), we obtain the oscillator strength curves of KS helium and real
helium. We also apply our method to the ALDA helium(with exact KS ground state)
as the first step of studying the threshold behavior in TDDFT (Fig. 2.5). The
comparison of results and figures of oscillator strength curves are shown in Sect. 2.5
and in Fig. 2.6.
Note that the fit is not designed to be used as an interpretation tool, but to recover
the shape of the oscillator strength spectrum. Thus comparing the fit parameters of
different curves(exact KS, ALDA, and experimental) is largely meaningless as there
is no visible trend. An exception is the fit parameter d, which describes the shape of
the asymptotic part of the oscillator strength curve, as it is related to the coefficient
of the leading term(ω−7/2) of the asymptotic expansion of the oscillator strength. The
fit parameters of related systems are provided in Appendix B.
23
0
0.2
0.4
0.6
0.8
1
20 30 40 50
reno
rmal
ized
osc
illat
or s
tren
gth
ω(eV)
ALDA fitALDA data
Experiment data1st ionization threshold
Figure 2.5: Exact/ALDA oscillator strength and fitted curve of He. These oscillatorstrength data are obtained from an ALDA calculation with the exact KS groundstate. We use a box code[3] to calculate these data. There is a kink in our data nearthe ionization threshold because the continuum near the ionization threshold mixeswith higher Rydberg states, which are not well-described by the box code.
0.7
0.8
0.9
1
1.1
1.2
22 24 26 28 30
reno
rmal
ized
osc
illat
or s
tren
gth
ω(eV)
KS fitExperiment fit
1st ionization thresholdExperiment data
KS data
Figure 2.6: KS and experimental single-electron oscillator strength and fitted curve ofHe near threshold[1, 2]. The ionization threshold is at 0.9036 Hartree(24.59 eV). Thecurves are converted from g(x) fit(Eq. (2.21)). The upper curve represents the exactKS helium oscillator strength data and fitted curve, and the lower curve representsthe experimental helium oscillator strength data and fitted curve.
24
2.5 Results and summary
The g-fit curves of KS helium and real helium are shown in Fig. 2.6 and 2.7. The
fit is very accurate in the entire range of ω. The accuracy is also checked with the
unused sum rules, listed in Table. 2.1. The results of the hydrogen atom are listed as
a reference, and it shows that the inherent error of the method is small. With these
curves, we explicitly show that the oscillator strength spectrum is a smooth curve
around the ionization threshold, and thus the oscillator strength of the exact KS
helium is not that of real helium. The auto-ionizing resonances in real helium are not
included in our fit, but the fit is still accurate even near the resonances (Fig. 2.8). The
errors in the sum rules are small, so the fitted curve can be used as a background for
studying these auto-ionizing resonances, and the pure resonance peaks are obtained
by subtracting the fitted curve from the experimental spectrum.
Table 2.1: Sum rules from σ(ω) and g(ξ) fit
S−2 S−1 S1 S2
g-fit 4.4999 2c 0.6667 1.3371Hexact 4.5 2 2/3 4/3g-fit 0.7563 0.7952 1.9114c 15.167 c
He KSa
exact 0.7579[52] 0.7957d 1.9114[53] 15.167d
g-fit 0.691 0.7504 2.09c 15.167 cHe Exp.ab
exact 0.698 0.754 2.09 15.167d
g-fit 0.6912c 0.7519 2.0414 15.167cHe ALDAa
exact 0.6912[52] 0.7957 1.9114[53] 15.167
One reason for the good performance near the resonances is that the auto-ionization
aAll the sums are converted to corresponding single-electron sums.bThe expected value of experimental data are listed in Ref. [47].cThis sum rule is a constraint. S0 is always a constraint.(S0 = 1 for all systems after converted
to single-electron model.)dThe expected value of S
−1 and S2 are calculated from the exact helium density[42].
25
0
0.3
0.6
0.9
1.2
20 40 60 80 100re
norm
aliz
ed o
scill
ator
str
engt
h
ω(eV)
KS fitExperiment fit
1st ionization threshold2nd ionizaiton threshold
Experiment dataKS data
Figure 2.7: KS and experimental single-electron oscillator strength and fitted curveof He[1, 2]. This figure show the overall shape of the oscillator strength curves. Thesolid dots and curve represents the exact KS helium oscillator strength data andfitted curve, and the cross dots and dashed curve represents the experimental heliumoscillator strength data and fitted curve.
resonances occur at relatively high frequencies, so their contributions to the smaller
sum rules are negligible. The other reason is the shape of the auto-ionization res-
onances in He is asymmetric, which have both a dip and a peak in the resonance
region[54, 55]. The contribution of these two parts to the sum rules cancels, so
the values of the sum rules are not influenced by the auto-ionization resonances too
much(even for S2), and thus the fit accurately generates the oscillator strength curves
for He.
0
0.1
0.2
0.3
0.4
40 50 60 70 80
reno
rmal
ized
osc
illat
or s
tren
gth
ω(eV)
Experiment fit2nd ionization threshold
Experiment data
Figure 2.8: g-fit of the experimental helium oscillator strengths near the auto-ionizingresonances.
26
If the KS oscillator strength at the ionization threshold was exact, then it would
yield a strong exact condition on the XC kernel in TDDFT, which many approxima-
tions would fail. Thus we studied this problem. We have shown that Kohn-Sham
oscillator strength of He is not exact at the ionization threshold (even though the po-
sition of the threshold is exact). This implies that the Hartree-XC kernel in TDDFT
has non-zero off-diagonal matrix elements at the threshold, and simple approxima-
tions such as the single-pole approximation are insufficient in this region.
We also developed a numerical fit to generate the spectrum near the ionization
threshold from a few physical conditions such as sum rules. The fit is accurate for all
frequencies due to the smoothness of the oscillator strength near the threshold, but
also works well for the spectrum far from the ionization threshold due to its correct
asymptotic behavior. The fit is not physically motivated, but is a simple and accurate
representation of the curves.
These results are not general since we only studied atoms with one or two elec-
trons, and multi-electron resonances are ignored as in Fig. 2.6. However, obvious
generalizations can be performed for atoms with more electrons since we only use
general properties(the asymptotic behavior, the value and first derivative of the spec-
trum at ionization threshold, and sum rules) in our method. Thus multi-electron
resonances are handled by subtracting their contribution from sum rules, and our
method can be extended to other atoms by following the methods of Sect. 2.3 and
2.4.
27
2.6 Prelude to the next chapter: the half-power
decay of atomic high-frequency spectra
The ω−7/2 decay of the atomic oscillator strength spectrum in Eq. (2.17) is un-
usual, considering that σ(ω) is related to χ(r, r′, ω) by Eq. (2.3) and (2.4), and that
χ(r, r′, ω) contains no such ω-half-powers in Eq. (1.11).
This unusual behavior is understood by considering putting a hard-wall box
around any system with a continuum. Such a boxed system has no continuum, as only
those continuum states of the original system that satisfy the hard-wall box boundary
condition are left, and these states becomes bound states[3]. The separation between
these states become smaller as the box is chosen to be bigger, and in the limit of
taking the size of the hard-wall box to infinity, the original system with continuum is
restored. For such a boxed system, the high-frequency behavior of the linear response
function is completely determined by that of the single transitions in Eq. (1.11), that
Eq. (1.11) can be expanded for ω → ∞ before taking the sum-over-states. Only
integer powers of ω are in the asymptotic series of the oscillator strength for ω → ∞,
with the leading order term being O(ω−2)[23].
However, as the box becomes bigger and bigger, the above-mentioned asymptotic
series(obtained from expanding before doing the sum-over-states in Eq. (1.11)) be-
comes more and more unable to describe the high-frequency behavior of the system,
and the envelope of the discrete poles in χ(r, r′, ω) takes over and becomes the correct
asymptotic behavior. Fig. 2.9 provides an illustration of this situation.
In the continuum limit, uniform convergence is required to interchange the order
28
ω
osc
illa
tor
stre
ngth
Oscillator strength of a system without continuumEnvelope of the oscillator strength
ω-7/2
ω-2
Figure 2.9: Illustration of the oscillator strengths of a system without continuum,in which the η in Eq. (1.11) is assumed to be finite. The black line is the discreteoscillator strength spectrum. The blue line is the envelope of the oscillator strengthspectrum. The denser the discrete spectrum is, the better description of the high-frequency behavior the envelope is.
of summing-over-states in Eq. (1.11) and taking asymptotic series of χ(r, r′, ω) for
ω → ∞. Due to the close relation between the frequency domain and the time domain,
time-dependent systems have similar order-of-limit problems. Consider the time-
dependent dipole moment µ(t) for a system exposed to a suddenly turned-on linear
static electric field with field strength E . It is related to the dynamic polarizability
α(ω) by
µ(t) =2Eπ
∫ t
0
dt′∫ ∞
0
dω ℑ[α(ω)] sin[ω(t− t′)]. (2.23)
The high-frequency behavior ℑ[α(ω → ∞)] has been derived in Sect. 2.3. For
hydrogen-like atoms, Eq. (2.17) yields
ℑ[α(ω → ∞)] ∼ 4√
2Z5
3ω9/2. (2.24)
29
This result agrees with Ref. [46]. Since the objects in the frequency domain and in
the time domain are connected by Fourier transform, the short-time behavior of µ(t)
is determined by the high-frequency behavior of α(ω). Knowing the leading ω-half-
power term in ℑ[α(ω → ∞)], I obtain the leading t-half-power in µ(t → 0+) for a
hydrogen-like atom in a suddenly turned-on linear static electric field as
µ(t→ 0+) ∼ t-integer-power terms +256EZ5
2835√πt9/2. (2.25)
See Appendix A for details. It is often assumed that the time-evolution operator
of a time-independent Hamiltonian is Taylor-expandable at the initial time, and the
TDDFT is formulated with such time-Taylor-expansions. The existence of such time-
half-powers suggests that the time-Taylor-expansion is not applicable to all systems.
This problem is addressed in the next chapter.
30
Chapter 3
Time-non-analyticities in
time-dependent systems
originating from density cusps
3.1 Introduction
In this chapter, I show that the structure of the TDSE implies non-analytic short-
time behaviors, and I develop a method for extracting the exact non-analytic short-
time behavior of the TDSE in the presence of density cusps. (In this chapter, cusps
refer to discontinuities in the space-derivatives of any order of the density). I show that
there are distinct spatial regions with different asymptotic behavior for short times.
Two 1-dimensional example systems are used to demonstrate the method. I calculate
the exact short-time behavior for a real system as well, that of a hydrogen atom in an
electric field. I do this without using linear response theory, but I also demonstrate
31
agreement with linear response theory in the limit of weak fields. While the examples
show that the constructive procedure for the time-dependent KS potential does not
apply to these systems(i.e., that the v-representability question is not solved), I show
the original proof of the Runge-Gross theorem remains valid, nonetheless.
The theorems of TDDFT are proven for general systems, but only a specific case
is needed to demonstrate the limitations of their applicability. I consider only single
particle systems in very specific time-dependent fields in this chapter. The essence of
possible difficulties can be explained even when considering the free-propagation of a
single particle system in the following section.
3.1.1 1-d disappearing nucleus case
Consider a one-dimensional one-electron system, with the initial wavefunction
ψ(x, t = 0) =√Z exp(−Z |x|), Z > 0. (3.1)
Notice that this wavefunction has a cusp at the origin x = 0. Let this wavefunction
propagate in time(begins at t = 0) under the influence of the free-particle Hamiltonian
H = T = −1
2
∂2
∂x2. (3.2)
Since this initial wavefunction Eq. (3.1) is not an eigenfunction of Hamiltonian Eq.
(3.2), it is evident that the wavefunction spreads out in space as the time increases;
moreover, the cusp in the initial wavefunction vanishes immediately as the time be-
comes non-zero, as such a feature in the wavefunction requires a singular potential to
balance it in the TDSE. We call this specific system the ‘1-d disappearing nucleus case’
32
since it can be viewed as taking a ground state 1-d hydrogen and removing the nucleus
immediately at t = 0[24]. (The 1-d hydrogen has the Hamiltonian H = T − Zδ(x),
since its ground-state wavefunction Eq. (3.1) resembles that of the 3-d hydrogen
atom.) The benefit of using such a model system is that its exact time-dependent
wavefunction is solvable. Applying the exact free-particle time-evolution operator
U(t) = exp(−iHt) to the initial wavefunction, I get
ψ(x, t) = exp(−iHt) |ψ(t = 0)〉
=
√Z
2exp(iZ2t/2)
[
exp(Zx) erfc
(
x+ iZt√2it
)
+ exp(−Zx) erfc
(−x+ iZt√2it
)]
,
(3.3)
in which erfc(z) = 1 − erf(z) is the complementary error function[51].
The left panel of Fig. 3.1 plots the time-dependent density of this system at
different time slices, which shows that the time-dependent wavefunction spreads out
over time. On the other hand, since the Hamiltonian Eq. (3.2) does not contain
explicit time-dependence, it is usual practice to write the time-evolution operator as
a time-Taylor expansion(t-TE) and evaluate the time-dependent wavefunction with
the Taylor-expanded operator:
ψTE(x, t) = TE[exp(−iHt)] |ψ(t = 0)〉
=
∞∑
j=0
(−iH)j
j!tj |ψ(t = 0)〉
=√Z exp(−Z |x| + iZ2t/2), (x 6= 0).
(3.4)
The t-TE wavefunction is a formal solution to the TDSE for all systems, yet it is clear
that the t-TE wavefunction does not equal the exact wavefunction Eq. (3.3). The
time-dependent density of ψTE is shown in the right panel of Fig. 3.1. The astounding
33
fact is, though the initial wavefunction is certainly not an eigenfunction of the free-
particle Hamiltonian, the t-TE density for x 6= 0 appears to not be changing over
time(for x 6= 0)! This discrepancy between the t-TE wavefunction and the exact
wavefunction is the problem addressed in this chapter.
-3 -2 -1 0 1 2 3
x(a.u.)
0
0.2
0.4
0.6
0.8
1
time-dependent density n(x,t)
t=0t=0.1t=0.2t=0.5t=1
-3 -2 -1 0 1 2 3
x(a.u.)
0
0.2
0.4
0.6
0.8
1
time-Taylor-expansion density n(x,t) t=0
t=0.1t=0.2t=0.5t=1
Figure 3.1: Electronic density of the 1-d disappearing nucleus case. Left panel: exactcase. Right panel: t-TE result.
3.1.2 Non-applicability of the t-TE
Considering the 1-d disappearing nucleus case is rather unrealistic, one may be
tempted to discard the failure of t-TE as a pathological behavior of this specific
system. However, this non-applicability of t-TE is more general than it appears to be.
To my knowledge, the failure of t-TE was first studied in a 1972 paper by Holstein and
Swift[56]. In this paper a 1-d 1-electron initial wavefunction with compact support is
constructed, and its analytic expression is
ψ(x, t = 0) =
exp[−a2/(a2 − x2)], |x| < a
0, |x| >= a.
(3.5)
34
The time-dependent wavefunction is obtained by pplying the free-particle time-evolution
operator to Eq. (3.5). It spreads out as time increases, but the t-TE wavefunction
behaves strangely, as it never spreads out of the interval x ∈ [−a, a]. This is seen
easily by considering that the t-TE of the time-evolution operator only contains dif-
ferential operators, and the derivatives to all orders of the initial wavefunction vanish
at x = ±a.
It is nearly impossible to encounter a wavefunction with compact support in
real situations, but as shown in Chapter 2, the oscillator strength spectrum f(ω)
of all atoms decays as ω−7/2 in the high-frequency part, which suggests non-Taylor-
expandability in time-domain. The detailed analysis is given later in this chapter,
but what I find is that the time-dependent behavior of any system with density cusps
cannot be obtained from t-TE. This category includes all real world systems, such as
atoms, molecules and solids, since all these systems has singular Coulomb potentials,
thereby guaranteeing density cusps due to Kato’s cusp condition[28].
Though the t-TE may not give the correct short-time expansion of the wavefunc-
tion, it is not practical to solve for the exact time-dependent wavefunction for most
of the problems in order to obtain the correct time-dependent behavior. Moreover,
if one is only interested in the short-time behavior, the exact time-dependent wave-
function contains too much unrelated information. We seek a method that extracts
the exact short-time behavior(at least to the leading-order time-non-analyticity) of
the time-dependent wavefunction in situations to which the t-TE is not applicable,
without solving for the entire wavefunction.
35
3.1.3 Runge-Gross theorem of TDDFT
As discussed in Chapter 1, TDDFT is based on the Runge-Gross theorem which
proves the one-to-one correspondence between the density and the external potential.
For the sake of the following discussions, the original Runge-Gross proof[8] is briefly
reviewed here. With given statistics and given interaction between particles, consider
two systems with the same initial wavefunction Ψ(r1, . . . , rN , t = 0) but different
external potentials, v(r, t) and v′(r, t), which are Taylor-expandable at the initial
time t = 0. The time-dependent current density j(r, t) is
j(r, t) = N
∫
d3r2 . . .
∫
d3rN ℑΨ(r, r2, . . . , rN , t)∇Ψ∗(r, r2, . . . , rN , t) . (3.6)
Using the assumption that the external potentials are Taylor-expandable, one can
compare the two systems and write
∂k+1
∂tk+1j(r, t) − j′(r, t)|t=0 = −n(r, t = 0)∇ ∂k
∂tkv(r, t) − v′(r, t)|t=0 . (3.7)
By inserting the continuity equation ∂tn(r, t) = −∇ · j(r, t), one reaches
∂k+2
∂tk+2n(r, t) − n′(r, t)t=0 = ∇ ·
[
n0(r)∇∂k
∂tkv(r, t) − v′(r, t)t=0
]
. (3.8)
The Runge-Gross proof concludes with showing the t-TE of the two densities are dif-
ferent to each other, thus proving the one-to-one correspondence between the external
potential and the density. The problem in the proof is, as shown in all the examples
in this chapter, the t-TE may not give the correct time-dependent behavior. After the
original Runge-Gross paper[8], there have been attempts to reformulate the theorem
without using t-TE[22, 23, 24, 25, 26], but none of them is as general as the original
36
proof. It is well known that TDDFT is accurate in actual computations, so another
purpose of this study is to find out how the Runge-Gross theorem is affected by this
discrepancy and why TDDFT works despite this problem in its foundation.
3.2 Analyzing the time-non-analyticities
3.2.1 Reason of the non-applicability of the t-TE
It is not difficult to see why the time t-TE does not apply to the previous examples.
A function of an operator is correctly defined by using the operator’s eigenfunctions.
The time-evolution operator U(t) is an exponential function of the Hamiltonian op-
erator H , and the correct definition is
U(t) = exp(−iHt) =∑
j
exp(−iǫjt) |φj〉 〈φj| , (3.9)
in which ǫj and φj are the eigenvalue and the eigenstate of H. The exact time-
dependent wavefunction is then
ψ(r, t) =∑
j
cj exp(−iǫjt)φj(r)
=∑
j
cj
( ∞∑
p=0
(−iǫj)pp!
tp
)
φj(r),
(3.10)
in which cj = 〈φj|ψ0〉. On the other hand, the t-TE of the wavefunction is
ψTE(r, t) =∞∑
p=0
(
∑
j
cj(−iǫj)pp!
φj(r)
)
tp, (3.11)
which is obtained by interchanging the order of the two summations. If the system
only has a finite number of eigenstates, such an interchange is valid; otherwise one
37
requires uniform convergence for two summations of infinite number of terms to be
interchangeable. Noticing the expansion exp(−iǫjt) =∑∞
p=0(−iǫj)ptp/p! does not
converge uniformly, the t-TE wavefunction should not be expected to work in general.
In practice, I observe that the t-TE wavefunction works when the initial wavefunction
is space-analytic, from which the wavefunctions with cusps are excluded. An analysis
of this point is provided in Appendix C.
The t-TE wavefunction is always a formal solution of the TDSE. This does not
contradict with its non-applicability in certain systems. For all the examples in this
chapter to which the t-TE is not applicable, the t-TE wavefunctions have infinite
norm. As an example, the 1-d disappearing nucleus case ψTE including x = 0 point
has the following form:
ψTE(x, t) = ψ(x, t = 0) − it
[
−Z2
2+ Zδ(x)
]
ψ(x, t = 0) − t2
2
− Zδ′(x)ψ′(x, t = 0)
[
+Z4
4− Z
2δ′′(x) − Z3δ(x) + Z2δ2(x)
]
ψ(x, t = 0)
+O(t3). (3.12)
The t-TE wavefunctions of such systems do not satisfy the boundary conditions of
the TDSE, making them unqualified as wavefunctions, thus resolving the issue that
the TDSE appears to have more than one solution.
3.2.2 The cause of the non-analytic short-time behavior
1-d disappearing nucleus case
For the sake of simplicity, I continue to use the 1-d disappearing nucleus case to
analyze the problem. Fig. 3.2 shows the time-dependent wavefunction of this system.
38
-4-2
0 2
4 0 0.02
0.04 0.06
0.08 0.1
-0.2
0
0.2
0.4
0.6
0.8
1
ψ(x,t)
Real partImaginary part
|x|=t1/2
x(a.u.)
t(a.u.)
ψ(x,t)
Figure 3.2: The time-dependent wavefunction of the 1-d disappearing nucleus case in(x, t) variables.
The density cusp in the initial wavefunction vanishes and the wavefunction be-
comes smooth immediately after the time becomes non-zero. Considering the only
difference between this case and the 1-d hydrogen is the lack of the δ-well potential
at the origin, all the changes in the time-dependent wavefunction of this case from
that of the 1-d hydrogen are due to the vanishing cusp. As the cusp vanishes, the
curvature of the wavefunction at the origin becomes finite and increases over time.
Fig. 3.2 shows the region away from the origin(|x| ≫√t) becomes oscillatory, show-
ing the plane-wave nature of the eigenstates of the free-particle Hamiltonian, yet the
region near the origin(|x| ≪√t) is non-oscillatory, resembling the spread-out cusp. I
denote the |x| ≫√t region as the outer region, and the |x| ≪
√t region as the inner
region. The correct short-time behavior is composed of the short-time behaviors of
these two regions.
39
The short-time series expansions for these two regions can be obtained by changing
the variables from (x, t) to the following reduced variables:
s = Z√t, x =
x√2t. (3.13)
Fig. 3.3 shows such a change-of-variables effectively zooms in to the inner region,
and the cusp in the initial wavefunction is removed in the reduced variables.
-4-2
0 2
4 0
0.1
0.2
0.3
-0.2
0
0.2
0.4
0.6
0.8
1
ψ(x/(2t)1/2,Zt1/2)
Real partImaginary part
|x|=t1/2
x/(2t)1/2 (a.u.)
Zt1/2 (a.u., Z=1)
ψ(x/(2t)1/2,Zt1/2)
Figure 3.3: The time-dependent wavefunction of the 1-d disappearing nucleus case in(s, x) variables. The cusp at the origin is removed under this change-of-variables.
The inner-region expansion can be obtained by Taylor-expanding ψ(s, x) at s →
0+ while holding x fixed:
ψinner(s, x)s→0+∼
√Z +
[
−(1 + i) exp(ix2)
√
Z
π−
√2Zx erf
(
1 − i√2x
)
]
s+O(s2)
=√Z +
Z5/2x2
2− Z3/2x erf
(
1 − i
2√tx
)
− 1 + i√πZ3/2 exp
(
ix2
2t
)√t+ · · · .
(3.14)
The outer-region expansion can be obtained by expanding ψ(s, x) at x→ ±∞ while
40
holding s fixed:
ψouter(s, x)x→±∞∼
√Z exp(−
√2s |x| + is2/2) +
(1 − i)√Z exp(ix2)s
2√π
x−2 +O(x−4)
= exp(−Z |x|)(√
Z +iZ5/2
2t
)
+(1 − i)Z3/2
√πx2
exp
(
ix2
2t
)
t3/2 + · · ·
(3.15)
-0.2
0
0.2
0.4
0.6
0.8
-4 -2 0 2 4
ψ(x
,t=0.
1)
x(a.u.)
Re[ψ]Im[ψ]
Re[ψinner]Im[ψinner]Re[ψouter]Im[ψouter]
Figure 3.4: ψ(x, t = 0.1) of the 1-d disappearing nucleus case, plotting the truncatedinner-region expansion and the outer-region expansion.
The truncated inner-region and outer-region expansions are plotted in Fig. 3.4.
These two together define the correct short-time behavior of the wavefunction. Since
these two expansions both contain time-non-analyticities, such as half-powers in t
and exp[ix2/(2t)], it is clear that the t-TE cannot describe the correct short-time
behavior.
41
An analysis of the TDSE
The analysis of the 1-d disappearing nucleus case is only for that specific case. Here
I provide a heuristic demonstration that the time-non-analyticities originate from the
form of the TDSE, and show whether the time-analyticity of the time-dependent
wavefunction is determined by the space-analyticity of the initial wavefunction.
Consider a perturbed 1-d 1-electron model system described by the following
TDSE:
−1
2
∂2
∂x2ψ(x, t) + V0(x)ψ(x, t) + Exnθ(t)ψ(x, t) = i
∂
∂tψ(x, t). (3.16)
The structure of the problem is exposed by taking a space-Fourier transform and a
time-Laplace transform of the TDSE.
Lt
Fx
[
−1
2
∂2
∂x2ψ(x, t) + V0(x)ψ(x, t) + Exnθ(t)ψ(x, t) − i
∂
∂tψ(x, t)
]
(k)
(ν) = 0
⇓
k2
2ψν(k) + V0(k) ∗ ψν(k) + Einψ(n)
ν (k) − iνψν(k) + iψ(k, t = 0) = 0,
(3.17)
in which ∗ denotes convolution. This is an integro-differential equation. It can be
simplified by only considering analytic V0(x), and then the V0(k) ∗ ψν(k) term is
again derivatives of ψν(k). The goal is to find out the short-time behavior of the
time-dependent wavefunction. Notice the TDSE can be rearranged as
k2
2νψν(k) +
1
νV0(k) ∗ ψν(k) +
Einνψ(n)ν (k) − iψν(k) +
i
νψ(k, t = 0) = 0. (3.18)
t and ν are conjugate variables of the Laplace transform, and small t behavior cor-
responds to large ν behavior. Thus Eq. (3.18) is a differential equation with a small
42
parameter 1/ν before the highest order derivative, and the solution of such an equa-
tion has so-called boundary layer behavior[57, 58]. This means the solution changes
its behavior rapidly in a narrow region whose thickness is determined by the small
parameter. Using boundary layer theory, I obtain a crude estimate of the outer-region
expansion of the time-dependent wavefunction by dropping all derivative terms[57]:
k2
2ψν(k) = iνψν(k) − iψ(k, t = 0), (3.19)
and obtaining
ψν(k) ∼2iψ(k, t = 0)
k2 − 2iν. (3.20)
This specific pole structure is due to the specific form of the TDSE, that of a 2nd
order differential equation in space, but a 1st order differential equation in time. This
pole structure generates the time-non-analyticities shown in the previous examples.
One recognizes this by doing the inverse Laplace/Fourier transform of the pole:
F−1k
L−1ν
[
1
k2 − 2iν
]
(t)
(x) =1 + i
4√πt
exp
(
ix2
2t
)
. (3.21)
Though the form of the TDSE implies time-non-analyticities, it is not neces-
sary that such non-analyticities show up in every system. Whether they show up
is determined by the initial wavefunction. I show in Appendix C that if the initial
wavefunction is analytic in space, then the time-dependent wavefunction of the sys-
tem described by Eq. (3.16) is analytic in time; if the initial wavefunction has cusps,
the time-dependent wavefunction is not time-analytic, and the time-non-analyticities
have the form tn/2 and exp[ix2/(2t)]. Such an analysis can be extended to 3-d systems
easily, which yields similar results as in the 1-d case.
43
3.2.3 The s-expansion method
I present here a method for extracting the correct short-time behavior. According
to the form of the time-non-analyticities shown in the previous section, I begin by a
change-of-variables:
s = Z√t, r =
r√2t. (3.22)
With these reduced variables, I can describe the time-non-analyticities which are not
covered by the form of the t-TE. In these variables, the time-dependent Schrodinger
equation becomes
∇2rψ − 4s2
Z2vψ + 2i
s∂ψ
∂s− r · ∇
rψ
= 0. (3.23)
In the disappearing nucleus case, ψ is equal to its Taylor expansion in powers of s for
fixed r, and thus I assume the following s-expansion ansatz in the more general case:
ψ(r, s) =∞∑
p=0
ψn(r) sn. (3.24)
This yields a set of differential equations:
∇2rψn − 2ir · ∇
rψn + 2niψn −
4
Z2
n−2∑
p=−2
vpψn−p−2 = 0, (3.25)
in which I assume the potential has a simple form of v(r, t) =∑∞
p=−2 vp(r) sp. Thus
each half-power of s produces a second-order differential equation for a function of r.
Eq. (3.25) requires proper boundary conditions for the solution to be well-defined.
Eq. (3.25) is equivalent to the TDSE whenever the wavefunction ansatz Eq. (3.24)
is applicable. This requires the boundary conditions to be derived from the initial
condition of the TDSE, which is the initial wavefunction ψt=0(r) = ψt=0(√
2rs/Z).
44
For finite argument r, as s→ 0, r → ∞, so the expansion of the initial wavefunction
at s → 0 determines the large r behavior of the ψm(r), i.e., provides the boundary
conditions of Eq. (3.25). The short-time behavior of the time-dependent wavefunction
is then obtained order-by-order from the solutions of Eq. (3.25) without the hassle
of solving the entire TDSE. I explain the details of the method using the example
systems in Sect. 3.3 and 3.4.
This method is not meant to be a general method, as one can easily find systems
to which this method is not applicable, such as systems whose external potential
explicitly contains other types of time-non-analyticities like t1/3. Rather, the method
described here is presented as a first attempt at tackling the delicate features of
TDSE. To my knowledge, these non-analytic short-time behaviors have never been
systematically analyzed before.
3.3 One dimensional systems
3.3.1 1-d disappearing nucleus case
By applying the s-expansion method to 1-d disappearing nucleus case, Eq. (3.25)
becomes the following set of differential equations:
ψ′′n − 2ixψ′
n + 2inψn = 0. (3.26)
45
These equations are simple enough to solve directly, and the solutions has the follow-
ing general formulas are
ψ2n(x) = a2nH2n(√ix) + b2nH2n(
√ix)
∫ x
0
dx′exp[i(x′)2]
H2n(√ix′)2
ψ2n+1(x) = a2n+1H2n+1(√ix) + b2n+1 1F1
(
−n− 1
2,1
2, ix2
)
,
(3.27)
in which H is the Hermite polynomial, and 1F1 is Kummer’s confluent hypergeometric
function[51].
The coefficients an and bn in Eq. (3.27) need appropriate boundary conditions to
be determined. The boundary conditions ψn(|x| → ∞) are determined by expanding
the initial wavefunction at s→ 0+:
ψ(x, t = 0) =√Z exp(−Z |x|) =
√Z exp(−
√2s |x|) =
∞∑
n=0
(−√
2)n√Z |x|n
n!sn.
(3.28)
Thus the boundary conditions are
ψn(x) →√Z(−
√2)n|x|n/n!, |x| → ∞. (3.29)
Eq. (3.29) yields
a2n+1 = b2n = 0, a2n =(−i)n
√Z
(2n)!2n, b2n+1 = −
√2in+1/2
√Z
(2n+ 1)!!√π. (3.30)
With Eq. (3.27) and (3.30) inserted into Eq. (3.24), the inner-region expansion is
obtained. It equals the previously shown Eq. (3.14), which is obtained from exactly
solving the entire TDSE.
The short-time behavior of the time-dependent wavefunction is fully described
with both the inner-region and the outer-region expansion. The inner-region expan-
sion corresponds to expanding the exact time-dependent wavefunction at s→ 0 while
46
holding r = r/√
2t constant, but there is no requirement on the magnitude of the
constant. Therefore no extra calculations are required for the outer-region expansion,
and it is obtained by expanding the inner-region expansion for r → ∞. For the 1-d
disappearing nucleus case, expanding Eq. (3.14) for |x| → ∞ yields
ψouter(s, x)|x|→∞∼
√Z(1 −
√2s |x| + s2x2 + · · · ) +
i√Zs2
2(1 −
√2s |x| + · · · )
+(1 − i)s
√Z exp(ix2)
2√πx2
+ · · ·
=√Z(1 − Z |x| + Z2x2
2+ · · · ) +
iZ5/2t
2(1 − Z |x| + · · · )
+(1 − i)Z3/2
√πx2
exp
(
ix2
2t
)
t3/2 + · · · .
(3.31)
This result agrees with Eq. (3.15), except that the exp(−Z |x|) envelope of the regular
terms in Eq. (3.15) is expanded at x → 0, as the price paid for obtaining the outer-
region expansion from the inner-region expansion.
3.3.2 1-d hydrogen in turned-on static electric field
Here I provide another 1-d 1-electron example. Consider a system with the fol-
lowing Hamiltonian:
H = H0 + H1(E) = −1
2
∂2
∂x2− Zδ(x) + Exθ(t). (3.32)
The system described by H0 is denoted as 1-d hydrogen as before. The system stays
in ψgs for t < 0, and a linear static electric-field with field strength E is turned on at
t = 0. For this system, the 1st order perturbative wavefunction is exactly solvable.
Its explicit formula is listed in Appendix D. Its outer expansion to the leading time-
47
non-analytic order is
ψ(x, t→ 0+) ∼
−i√Zxt +
Z3/2
2[Zx− sgn(x)]t2 +
iZ7/2
8[Zx− sgn(x)]t3
−Z11/2
48[Zx− sgn(x)]t4
exp(−Z |x|) − (4 + 4i)EZ3/2
√πx5
exp
(
ix2
2t
)
t9/2 +O(t5),
(3.33)
Below I show that the s-expansion method reproduces the t9/2 term in Eq. (3.33).
The potential in the reduced variables is
V (x, t > 0) = −Zδ(x) + Ex = − Z2
s√
2δ(x) + E s
√2
Zx. (3.34)
Although V have no explicit time-dependence for t > 0, the potential in the reduced
variables has explicit s-dependence. With the s-dependent potential Eq. (3.34), the
differential equations Eq. (3.25) become inhomogeneous differential equations:
ψ′′n − 2ixψ′
n + 2inψn + 2√
2δ(x)ψn−1 −4√
2E xZ3
ψn−3 = 0. (3.35)
The boundary conditions for Eq. (3.35) is still Eq. (3.29), as the initial condition
does not change from that of the 1-d disappearing nucleus case. Solving Eq. (3.35) is
not as easy as Eq. (3.26) of the disappearing nucleus case, and a general formula for
ψn(x) like Eq. (3.27) is not available. Converting the t9/2 term in Eq. (3.33) to (s, x)
variables, I observe this non-analyticity occurs at the s4 order of the wavefunction,
and solving for ψ4(x) yields this leading-order time-non-analyticity. Since the form
of Eq. (3.35) suggests that ψ4(x) depends on all the previous ψn(x)’s, I need ψ0(x)
to ψ3(x) to solve for ψ4.
In this case, ψ0(x) and ψ3(x) can be obtained easily from their differential equa-
tions; but for a more complicated system, there may be more such extra work to do
48
before reaching the leading-order time-non-analyticity, and it is cumbersome having
to solve for the first few ψn’s which are analytic in time. I observe that though the
t-TE wavefunction does not have the correct short-time behavior, it can be used to
facilitate the process of obtaining the leading-order time-non-analyticity in ψ(x, t).
ψTE(x, t) of this system is
ψTE(x, t) = ψgs(x)
[
1 − it
(
−Z2
2+ Ex
)
−t2
2
(
Z4
4+ EZ sgn(x) − EZ2x+ E2x2
)]
+O(t3). (3.36)
Converting Eq. (3.36) to (s, x) variables and collecting the sn terms gives a set of
ψTEn (x):
ψTE0 (x) =
√Z, ψTE
1 (x) = −√
2Z |x| , ψTE2 (x) =
i√Z
2+ x2
√Z,
ψTE3 (x) =
−3i√
2Z3 |x| − 2√
2Z3 |x|3 − 6i√
2E x6Z5/2
,
ψTE4 (x) =
−3Z3 + 12iZ3x2 + 4Z3x4 − E sgn(x)(12 − 48ix2)
24Z5/2, and so on.
(3.37)
ψTE0 (x) to ψTE
3 (x) satisfy both the differential equations Eq. (3.35) and the boundary
conditions Eq. (3.29), which is expected since the outer-expansion Eq. (3.33) suggests
the leading-order time-non-analyticity occurs in ψ4. Inserting ψTE4 (x) into Eq. (3.35)
yields −Eδ′(x)/Z5/2, showing ψTE4 (x) does not satisfy the differential equation. Then
I only needs to solve the differential equations starting from ψ4(x).
With the help of t-TE, the leading time-non-analytic term is obtained by solving
Eq. (3.35) for ψ4(x). Here I present a method of solving the complicated differential
equation occurred here and in the following examples. I define
∆(x) = ψ4(x) − ψTE4 (x), (3.38)
49
the differential equation for ∆(x) is
∆′′ − 2ix∆′ + 8i∆ − Eδ′(x)Z5/2
= 0. (3.39)
Eq. (3.39) is a inhomogeneous differential equation, so ∆(x) = ∆g(x) + ∆p(x), with
∆g(x) and ∆p(x) being the general and the particular solution respectively. I obtain
the complete asymptotic expansion of ∆g(x) for x → ∞ by the so-called method of
dominant balance[57](see Appendix E for details):
∆g(x) ∼ c1
(
x4 + 3ix2 − 3
4
)
+ c2exp(ix2)
x5
[
1 +1
3x2
∞∑
m=0
(2m+ 6)!(−i)m+1
(m+ 1)!22m+5x−2m
]
,
(3.40)
in which c1 and c2 are coefficients to be determined later. This asymptotic series
diverges for all x as most asymptotic series do. However, noticing the divergence
growing less rapid than m!, I apply the so-called Borel summation[59, 57](see Ap-
pendix F for details) to the Eq. (3.40), which yields the exact formula for the general
solution of ∆(x):
∆g(x) = c1(x4 + 3ix2 − 3/4) + c2
[
−1
3exp(ix2)x(5i+ 2x2)
+1 − i
6
√
π
2(−3 + 12ix2 + 4x4) erfc
(
1 − i√2x
)]
. (3.41)
With ∆g(x) known, ∆p(x) is obtained by the usual method of undetermined coeffi-
cients. The coefficients c1 and c2 are determined using the boundary conditions Eq.
50
(3.29). In the end, the expression of ψ4(x) is
ψ4(x) =
√Z(4x4 + 12ix2 − 3)
24
+E
−4√πx4 sgn(x) + 2
√i exp(ix2)x(5i+ 2x2) +
√π(4x4 + 12ix2 − 3) erf(1−i√
2x)
6√πZ5/2
.
(3.42)
I obtain the leading time-non-analytic term in the outer-region expansion similarly
as in Sect. 3.3.1, and it agrees with Eq. (3.33).
Eq. (3.33) is obtained from 1st order perturbation theory, and the higher-order
terms in t(which are not written out) are at most proportional to E1. These higher-
order terms in t would fail if the strength of the electric field E was taken too large,
due to not containing higher-order(in E) contributions. In contrast, the s-expansion
method contains the effects of all the higher-order(in E) responses. Eq. (3.35) implies
that E2 first appears in ψ6(x), and E3 first appears in ψ9(x) and so on. Thus one
obtains the correct short-time behavior beyond linear response regime by solving for
the higher-order(in s) ψn(x)’s.
The analytic form of the time-dependent dipole moment of the 1-d hydrogen in
static electric field case is known(refer to Appendix D). The time-dependent dipole
moment is a linear response property, and its analytic form is obtained from the linear
response function:
µ(t) =
∫ t
−∞dt′∫ ∞
−∞dx
∫ ∞
−∞dx′ x δV (x′, t′)χ(x, x′, t− t′), (3.43)
in which δV (x′, t′) = Ex′θ(t′), and χ(x, x′, t− t′) for 1-electron systems[60] is
χ(x, x′, t− t′) = ψgs(x)ψ∗gs(x
′) exp[−iEgs(t− t′)]G∗(x, x′, t− t′) + h.c. (3.44)
51
The Green’s functionG(x, x′, t−t′) for the 1-d hydrogen has a closed-form expression[61],
with which I obtain the short-time behavior of µ(t) as
µ(t→ 0+)/E ∼ −t2
2+
32Z3
105√πt7/2 − Z4t4
12+O(t9/2). (3.45)
ψ4(x) in the s-expansion method contributes to the leading half-power term(the
t7/2 term) in µ(t→ 0+). According to the relation
µ(t) = 2ℜ⟨
ψ(t = 0) |x|ψ(1)(t)⟩
, (3.46)
the contribution of ψ4(x) to µ(t→ 0+) is obtained by
µψ4(t) = 2ℜ
⟨
ψ(t = 0) |x| s4ψ4(x)⟩
. (3.47)
This yields the 32EZ3t7/2/(105√π) term in Eq. (3.45).
3.4 Three dimensional systems
Aside from the dimensionality change, the main change from 1-d cases to 3-d cases
is that the Coulomb potential replaces the δ-well potential as the singular potential.
Unlike the δ-well potential, the Coulomb potential is long-ranged, and this makes 3-d
wavefunction much more complicated than 1-d cases.
One point need to be changed for the s-expansion method in 3-d. Consider a sys-
tem whose initial wavefunction equals the ground-state wavefunction of the hydrogen
atom:
ψ(r, t = 0) =Z3/2
√π
exp(−Zr). (3.48)
52
Free-propagation of this wavefunction(Appendix H) yields a similar situation as in
the 1-d disappearing nucleus case, as the system is effectively 1-d due to the spherical
symmetry. By expanding the initial wavefunction Eq. (3.48) as
ψ(s→ 0+, r → ∞) ∼ Z3/2
√π
− s
√
2
πZ3/2r + s2Z
3/2r2
√π
+ · · · , (3.49)
I only obtain one boundary condition(ψn(r → ∞)) for Eq. (3.25) instead of two
boundary conditions as in 1-d cases(ψn(x → ±∞)). Eq. (3.25) requires another
boundary condition to be well-defined, and it is related to how t-TE behaves in 3-d
cases. For the 3-d disappearing nucleus case, the t-TE wavefunction is
ψTE(r, t) =Z3/2
√π
exp(−Zr + iZ2t/2)
(
1 − iZt
r
)
= ψTE0 (r) + ψTE
1 (r)s+ ψTE2 (r)s2 +O(s3)
=Z3/2
√π
− Z3/2(i+ 2r2)√2πr
s+Z3/2(3i+ 2r2)
2√π
s2 +O(s3).
(3.50)
Unlike in the 1-d examples, all ψTEn (r) satisfy Eq. (3.25), but they diverge at r = 0
for any non-zero time, which is not acceptable as a wavefunction. Thus the other
boundary condition for Eq. (3.25) is ψn(r)’s being regular at r = 0.
3.4.1 3-d hydrogen in turned-on static electric field
In this section I apply the s-expansion method to a real-world system, a ground-
state hydrogen-like atom perturbed by a suddenly turned-on linear static electric field.
The system is described by the following Hamiltonian:
H = −1
2∇2 − Z
r+ Ezθ(t), (3.51)
53
in which Z is the nuclear charge. I use the (r, z, φ, t) coordinate system in this section,
with
r =√
x2 + y2 + z2, φ = tan(y/x). (3.52)
One can easily check that the t-TE wavefunction of this system does not have a
convergent norm. I obtain the leading order time-non-analytic term in the outer-
region expansion of this system from the 1st order perturbation theory, which is
ψouter(r, t)r/
√2t→∞∼ · · · − (8 − 8i)EZ5/2z
πr8exp
(
ir2
2t
)
t11/2 + · · · , (3.53)
see Appendix G for details. I show below that the s-expansion method reproduces
this result.
I begin with performing a change-of-variables:
s = Z√t, r =
r√2t, z =
z√2t. (3.54)
The external potential in these reduced variables is
V = −Zr
+ Ez = − Z2
√2sr
+ E√
2sz
Z. (3.55)
Inserting the wavefunction ansatz Eq. (3.24) into Eq. (3.25) gives
∂2ψn∂r2
+∂2ψn∂z2
− 2ir∂ψn∂r
− 2iz∂ψn∂z
+2
r
∂ψn∂r
+2z
r
∂2ψn∂z∂r
+ 2inψn
+2√
2
rψn−1 −
4√
2E zZ3
ψn−3 = 0. (3.56)
The ψTEn (r)’s are correct before the leading-order time-non-analyticity occurs, and
I only need to solve for ψn(r) if ψTEn (r) fails to satisfy the differential equation Eq.
54
(3.56) and the boundary conditions. The ψTEn ’s of this system are
ψTE0 (r) =
Z3/2
√π, ψTE
1 (r) = −√
2
πZ3/2r, ψTE
2 (r) =Z3/2(i+ 2r2)
2√π
,
ψTE3 (r) = −3iZ3r + 2Z3r3 + 6iE z
3√
2πZ3/2,
ψTE4 (r) =
Z3/2
24√π
(−3 + 12ir2 + 4r4) + E iz
12√πZ3/2r3
(1 + 6ir2 + 24r4), and so on.
(3.57)
ψTE1 (r) to ψTE
4 (r) all satisfies Eq. (3.56), but ψTE4 (r) does not satisfy the boundary
conditions since it diverges at r → 0. The leading-order time-non-analyticity is in
ψ4(r).
I use the method of dominant balance and Borel summation to solve for ψ4(r).
Since ψTE4 satisfies Eq. (3.56), the equation can be rewritten as
∂2∆
∂r2+∂2∆
∂z2− 2ir
∂∆
∂r− 2iz
∂∆
∂z+
2
r
∂∆
∂r+
2z
r
∂2∆
∂z∂r+ 8i∆ = 0, (3.58)
in which ∆(r) = ψ4(r)−ψTE4 (r). As r → 0, the divergence in ψTE
4 is proportional to z,
and ∆(r) has to cancel this divergence to satisfy the boundary conditions. Therefore
∆(r) must have the following form:
∆(r, z) = exp[S(r)]z. (3.59)
The method of dominant balance yields the entire asymptotic expansion of ∆(r):
∆(r)/z = c1
(
r3 +9ir
2− 9
4r+
3i
8r3
)
+ c2exp(ir2)
r8
[
1 +1
9r2
∞∑
m=0
(−i)m+1(m+ 4)(2m+ 6)!
(m+ 1)!22m+5r2m
]
. (3.60)
55
Replacing the sum over m with Borel summation, the result is
∆(r)/z = c1
(
r3 +9ir
2− 9
4r+
3i
8r3
)
+c21 + i
72r3
[
(2 + 2i) exp(ir2)r(−3 + 16ir2 + 4r4)
−√
2π(3i− 18r2 + 36ir4 + 8r6) erfc
(
1 − i√2r
)]
. (3.61)
The coefficients c1 and c2 are determined by the boundary conditions. One boundary
condition is Eq. (3.49), and the other one is ψ4(r) being non-divergent at r = 0,
which requires the divergence in Eq. (3.61) cancels the divergence in ψTE4 (r) as in
Eq. (3.57). The resulting coefficients are
c1 = 0, c2 = − 1 − i√2πZ3/2
E . (3.62)
Eq. (3.53) is reproduced by expanding s4ψ4(r) for r → ∞.
3.5 Summary
The examples are all single-electron systems. However, the nuclear potential dom-
inates over the electron-electron repulsion in the region of the nucleus, so that the
cusp condition on the time-dependent density near a nucleus remains valid regardless
of the electron-electron repulsion. Thus I expect the qualitative features to remain
(i.e., the half-powers of t and exp[ir2/(2t)] in the short-time behavior) true even in
the presence of interaction. In any event, the theorems of TDDFT apply (or not)
even for N = 1.
I conclude with a discussion of the relevance of the results in this chapter for
time-dependent density functional theory (TDDFT). The proof of a one-to-one corre-
spondence given by Runge and Gross[8] shows that, for two time-analytic potentials
56
with differing Taylor expansions in t, one can calculate the corresponding difference
in the Taylor expansion for the densities and show it to be non-zero. The densities
themselves need not be time-analytic: different densities differ in at least one of their
Taylor-coefficients, even if they are not equal to their Taylor expansions. The converse
is not true: if two Taylor expansions were identical, their originating functions are
not necessarily identical. In the present context, studying differences in densities via
their Taylor coefficients corresponds to Taylor expansion of the evolution operator.
In the examples given here, the true density does not equal its t-TE, but this does
not disqualify the Runge-Gross proof.
As an example, consider the 3-d illustration with the disappearing nucleus(Appendix
H), and compare the t-TE for the density if the nucleus does not change, and if it
disappears. In the case where the nucleus does not change, the t-TE density equals
the initial density trivially; and in the case where the nucleus disappears, the t-TE
density changes with time regarding Eq. (H.3). While the true density does not
match its t-TE density when the nucleus vanishes, that the t-TE densities differ is
sufficient to show the true densities differ as well, and the Runge-Gross proof holds.
On the other hand, the method for constructing the KS potential used in Ref.
[27] relies on the density equalling its Taylor expansion. Thus it constructs the wrong
density for the examples given here, and likely fails whenever the initial wavefunction
has a cusp and undergoes a non-trivial time evolution. The explicit construction
algorithm of the time-dependent KS potential implies a positive solution to the v-
representability problem of TDDFT, i.e., for a given interacting n(r, t), does the time-
dependent KS potential exist? It does, but not necessarily for cases where the initial
57
wavefunction has a cusp. In the cases considered here, a much more sophisticated
procedure would be needed. This is important as the constructive procedure has been
invoked or applied to several other situations[62, 63, 64, 65, 66]. Recent progress in
this direction has been made[24, 25, 26].
Finally, one might argue that real atoms have finite nuclei, or that real molecules
and solids have nuclear wavefunctions that smear the cusps due to nuclei. But such
arguments miss the basic point. Time-dependent DFT is an exact mapping of the
quantum mechanics of electrons, for which there are no difficulties with the BO ap-
proximation or need for finite nuclei. There is a logical error if the functional requires
avoiding cusps for its definition.
58
Appendix A
The non-analytic short-t behavior
in µ(t) from the non-analytic
high-ω behavior in ℑ[α(ω)]
In quantum mechanics, the frequency domain and time domain are connected
by Fourier transform. As a consequence, the high-frequency behavior of ℑ[α(ω)]
determines the short-time behavior of µ(t). Knowing the ℑ[α(ω → ∞)] behavior in
Eq. (2.24), µ(t → 0+) of a 3-d hydrogen in turned-on static electric field can be
written as
µ(t→ 0+) ∼ limΩ→∞
2Eπ
∫ t
0
dt′∫ ∞
Ω
dω ℑ[α(ω)] sin[ω(t− t′)]. (A.1)
59
Inserting Eq. (2.24) into Eq. (A.1) yields
µ(t→ 0+) ∼ limΩ→∞
2Eπ
∫ t
0
dt′∫ ∞
Ω
dω sin[ω(t− t′)]4√
2Z5
3ω9/2
∼ limΩ→∞
E∫ t
0
dt′16√
2Z5(t− t′)
15πΩ5/2− 8
√2Z5(t− t′)3
9πΩ1/2+
128Z5(t− t′)7/2
315√π
+ · · · ,
(A.2)
in which Ω is an arbitrary cut-off frequency. Noticing that the only term that does
not depend on Ω is the (t − t′)7/2 term, I obtain the leading time-non-analytic term
in µ(t→ 0+):
µ(t→ 0+) ∼ · · ·+ 256EZ5
2835√πt9/2 + · · · . (A.3)
The leading regular-power terms in µ(t → 0+) cannot be obtained this way, as seen
in Eq. (A.2), in which all other terms depend on the arbitrary chosen Ω. The inverse
procedure is also valid, which reproduces the Eq. (2.24) from Eq. (A.4).
For the 1-d hydrogen in turned-on static electric field case, the same procedure
yields
µ(t→ 0+) ∼ · · · + 32EZ3
105√πt7/2 + · · · , (A.4)
which is verified by the exactly calculated µ(t→ 0+) in Eq. (3.45).
60
Appendix B
Fit parameters for real, exact KS,
and ALDA helium
Table B.1: Fit parameters of the oscillator strength fit
a b c d e
H -0.9161 -0.0020 2.2094 0.3183 8.9594He KSa -64.343 -0.7140 0.3482 8.7386 54.646He Exp. -10.004 -3.5809 0.3313 5.5406 3.4113He ALDAb -35.160 -0.3795 0.9617 7.0277 26.790
aThe exact KS potential is obtained from Ref. [42].bALDA is applied to the exact KS ground state instead of LDA ground state.
61
Appendix C
Relation between the initial
wavefunction and
time-non-analyticities
Inverse Laplace transform of Eq. (3.20) is
L−1ν
[
2iψ(k, t = 0)
k2 − 2iν
]
(t) = − exp(−ik2t/t)ψ(k, t = 0). (C.1)
The outer-region asymptotic behavior of ψ(x, t → 0+) is obtained from the inverse
Fourier transform of Eq. (C.1):
F−1k
[
− exp(−ik2t/t)ψ(k, t = 0)]
(x) = −exp[ix2/(2t)]√it
∗ ψ(x, t = 0), (C.2)
in which ∗ denotes convolution. If ψ(x, t) is space-analytic, it equals its Taylor ex-
pansion:
ψ(x, t = 0) =∞∑
j=0
ψ(j)(0)
j!xj . (C.3)
62
Then the convolution in Eq. (C.2) can be evaluated term by term:
−exp[ix2/(2t)]√it
∗ xj = 2j/2(−1)(j+3)/4x
t(j−1)/2jΓ(j/2) 1F1
(
1−j2
; 32; ix
2
2t
)
j is odd,
(1 + i)tj/2Γ[(j + 1)/2] 1F1
(
− j2; 1
2; ix
2
2t
)
j is even.
(C.4)
The confluent hypergeometric functions appeared in Eq. (C.4) can be expressed with
more familiar functions as the following:
1F1(−n; 3/2; z) =(−1)nn!
(2n+ 1)!2√zH2n+1(
√z),
1F1(−n; 1/2; z) =(−1)nn!
(2n!)H2n(
√z).
(C.5)
Inserting Eq. (C.5) into Eq. (C.4) while noticing H2n+1(ζ) ∼ O(ζ1) + O(ζ3) + · · ·
and H2n(ζ) ∼ O(ζ0) +O(ζ2) + · · · , I arrive at the conclusion that there are no time-
non-analyticities starting from a space-analytic initial wavefunction.
For initial wavefunctions with cusps, I use the following ansatz:
ψ′(x, t = 0) =
∞∑
j=0,j 6=m
ψ(j)(0)
j!xj + c+x
mθ(x) + c−xmθ(−x) (c+ 6= c−), (C.6)
in which a derivative discontinuity is manually added in. The convolution in Eq.
(C.2) for the discontinuity yields
− exp[ix2/(2t)]√it
∗ xmθ(x) = −2(m−1)/2eimπ/4tm/2Γ[(m+ 1)/2] 1F1
(
−m2
;1
2;ix2
2t
)
+ (−1 + i)2(m−1)/2 exp(imπ/4)t(m−1)/2xΓ[(m+ 2)/2] 1F1
(
1 −m
2;3
2;ix2
2t
)
. (C.7)
The convolution with the θ(−x) part yields a similar result. With Eq. (C.5), I
conclude that the initial wavefunction with cusps has time-non-analyticities in its
short-time behavior.
63
I provide the free-propagation of a Gaussian initial wavefunction as an example
in which there is no non-analytic short-time behavior starting from a smooth initial
wavefunction. The initial wavefunction is
ψ(x, t = 0) =exp[−x2/(2σ)2]
π1/4√σ
, (C.8)
in which σ characterizes the width of the Gaussian. Combining Eq. (C.1) and Eq.
(C.2) gives
L−1ν
F−1k
[
2iψ(k, t = 0)
k2 − 2iν
]
(x)
(t) = −i√
2σπ1/4
√t− iσ2
exp
[
ix2
2(t− iσ2)
]
. (C.9)
Eq. (C.9) has no time-non-anlayticities at the initial time. The radius of convergence
of the t-TE at the initial time is clearly σ2 due to the pole at t = iσ2. In the limit
of σ → 0, the Gaussian becomes a δ-function and no longer smooth. The pole in Eq.
(C.9) coincides with t = 0, and as a consequence the radius of convergence of the
t-TE becomes exactly zero.
The 1-d and 3-d nucleus disappearing cases resemble this situation. The wave-
functions of the nucleus disappearing cases for finite t are smooth in space. If one
take their wavefunctions at a certain time t0 and denote them as the new initial
wavefunction, the time propagations of these new initial wavefunctions contain no
non-analytic short-time behavior, and the t-TE propagations of them have finite ra-
dius of convergences depending on t0. In the limit of taking t0 → 0, one returns to
the nucleus disappearing cases, and the radius of convergence of the t-TE becomes
exactly 0.
In principle, the time-evolution of the nucleus disappearing cases can be obtained
by taking a short time-step away from the initial time(so that the wavefunction be-
64
comes smooth). The time-dependent wavefunctions can be obtained with t-TE if t is
within the radius of convergence, and it can be obtained by analytic continuation if t
is outside the radius of convergence. In contrast, the advantange of the s-expansion
method is that it does not suffer the radius of convergence problem. The inherent
time-non-analyticities of the TDSE are removed using the reduced variables in Eq.
(3.22), and thus the radius of convergence of the s-expansion is infinity for all the
examples in Chapter 3.
65
Appendix D
1st order perturbative treatment of
the 1-d hydrogen in static electric
field case
For the 1-d hydrogen in a turned-on static electric field case described in Sect.
3.3.2, the exact 1st order perturbative wavefunction ψ(1)(x, t) can be written out in
closed-form. ψ(1)(x, t) in time-dependent perturbation theory is
ψ(1)(x, t) =
∫ t
−∞dt′∫ ∞
−∞dx′ G(x, x′, t− t′)δV (x′, t′)ψ(0)(x′, t′), (D.1)
in which δV (x′, t′) = Ex′θ(t′), ψ(0)(x′, t′) =√Z exp(−Z |x′| + iZ2t′/2), and the 1-d
hydrogen Green’s function G(x, x′, t− t′)[67] is
G(x, x′, t− t′) = −i√
1
2πi(t− t′)exp
[
i(x− x′)2
2(t− t′)
]
− iZ
2exp
[
−Z(|x| + |x′|) +iZ2(t− t′)
2
]
erfc
[
|x| + |x′|√
2i(t− t′)− Z
√
i(t− t′)
2
]
. (D.2)
66
The explicit formula for ψ(1)(x, t) is
ψ(1)(x, t) = −exp(−Zx+ iZ2t/2)
4Z3/2(x+Zx2−2iZ2xt−Z3t2) erf
[
(1 − i)(−x+ iZt)
2√t
]
− exp(Zx+ iZ2t/2)
4Z3/2(x− Zx2 − 2iZ2xt + Z3t2) erf
[
(1 − i)(x+ iZt)
2√t
]
+exp(iZ2t/2)
2Z3/2
[
(|x| + Z3t2) sinh(Zx) − Zx(|x| + 2iZt) cosh(Zx)]
+(1 + i) exp[ix2/(2t)]x
√t
2√πZ
. (D.3)
The exact time-dependent dipole moment of this system is obtained by Eq. (3.43):
µ(t) = − t2
12(Z4t2 + 6) +
√t cos(Z2t/2)
12√πZ3
(Z6t3 − 3Z4t2 + 7Z2t+ 15)
−√t sin(Z2t/2)
12√πZ3
(Z6t3 + 3Z4t2 + 7Z2t− 15)
1
24Z4(Z8t4 + 4iZ6t3 + 6Z4t2 − 12iZ2t− 15) erf
(
1 − i
2Z√t
)
+1
24Z4(Z8t4 − 4iZ6t3 + 6Z4t2 + 12iZ2t− 15) erf
(
1 + i
2Z√t
)
. (D.4)
67
Appendix E
Method of dominant balance
The method of dominant balance[57] extracts the asymptotic behavior of the solu-
tion of a differential equation. This method assumes certain terms in the differential
equation to be dominant(termed ‘balance’), and only keeping these terms gives a
reduced differential equation determining the asymptotic behavior of the solution,
which is usually easier to solve than the original. The solution of the reduced differ-
ential equation is then substituted back into the original differential equation to check
the validity of the assumed balance. If there are more than one consistent balances,
the asymptotic expansions from these different balances need to be added up.
I use the 1-d disappearing nucleus case to demonstrate the method of dominant
balance. For this case the leading-order time-non-analyticity is in ψ1(x), and I use
the following ansatz:
ψ1(x) = exp[S(x)] (E.1)
68
Inserting the ansatz into Eq. (3.26) yields
S ′′(x) + [S ′(x)]2 − 2ixS ′(x) + 2i = 0 (E.2)
One consistent balance is assuming S ′′(x) ≪ [S ′(x)]2. Removing S ′′(x) from Eq.
(E.2), I obtain
S(x) ∼ ix2. (E.3)
The next order correction is found by the ansatz
S(x) ∼ ix2 + C(x) (E.4)
Insert Eq. (E.4) into Eq. (E.2) yields
C(x) ∼ −2 ln(x) (E.5)
Thus ψ1 has the following asymptotic behavior from the balance S ′′(x) ≪ [S ′(x)]2:
ψ1(x) = c1 exp[S(x)] ∼ cexp(ix2)
x2(E.6)
More terms are obtained by inserting the ansatz ψ1 ∼ c exp(ix2)x−2[1+ ǫ(x)] into Eq.
(3.26), and using the method of dominance several times yields
ψ1(x) ∼ c1exp(ix2)
x2
(
1 − 3i
2x2− 15
4x4
)
. (E.7)
At this point the structure of the asymptotic series becomes clear, and the following
ansatz is used:
ψ1(x) ∼ c1exp(ix2)
x2
∞∑
n=0
anx−2n. (E.8)
With this ansatz, the entire asymptotic series coming from the balance S ′′(x) ≪
[S ′(x)]2 is obtained as
ψ1(x) ∼ c1exp(ix2)
x2
∞∑
n=0
(2n+ 1)!!(−i)n2n
x−2n. (E.9)
69
Another consistent balance is assuming −2ixS ′(x) ≫ S ′′(x), [S ′(x)]2, I obtain
S(x) ∼ ln(x). (E.10)
Similar as what has been done for the previous balance, the asymptotic series coming
from the balance −2ixS ′(x) ≫ S ′′(x), [S ′(x)]2 is
ψ1(x) ∼ c2x, (E.11)
which only has 1 term in contrast to Eq. (E.9). The complete asymptotic behavior
is then
ψ1(x) ∼ c1exp(ix2)
x2
(
1 − 3i
2x2− 15
4x4+ · · ·
)
+ c2x. (E.12)
70
Appendix F
Borel summation
The usual behavior of an asymptotic series is that it appears to be converging for
the first few terms, and then becomes divergent when one add more terms. It is a
common practice to approximate a function using the first few terms of its asymptotic
series with moderate accuracy, but if one requires higher accuracy, one has to find
a method to extract the information from the divergent series. Borel summation is
a method of extracting such information and obtain the closed-form formula of a
function from its asymptotic series under certain restrictions[57, 59].
Considering the divergent series
S(p) =
∞∑
n=0
βnpn, (F.1)
the Borel sum of the original series f(x) is defined as
SBorel(p) ≡∫ ∞
0
dξ exp(−ξ)φ(pξ), (F.2)
in which
φ(p) =
∞∑
n=0
βnpn
n!. (F.3)
71
It is evident that the Borel summation requires the original divergent series diverges
at most as fast as n!. For a uniformly convergent series, the Borel summation is
equivalent to summing up the series directly. The Borel method can be generalized
to divergent series diverges as (n!)m, m > 1 as well[57].
As an example, I do the Borel sum of the series in Eq. (E.12). The original
divergent series is
S(x) =exp(ix2)
x2
(
1 − 3i
2x2− 15
4x4+ · · ·
)
=exp(ix2)
x2
∞∑
n=0
(2n+ 1)!!(−i)n2n
x−2n.
(F.4)
Borel sum of f(x) is
SBorel(x) =exp(ix2)
x2
∫ ∞
0
dξ exp(−ξ)∞∑
n=0
(2n+ 1)!!(−i)n2nn!
(ξ/x2)n
=exp(ix2)
x2
∫ ∞
0
dξ exp(−ξ) 1
(1 + iξ/x2)3/2
= −2i exp(ix2) + (1 + i)√
2πx erfc
(
1 − i√2x
)
.
(F.5)
Then Eq. (E.12) becomes the exact form of ψ1:
ψ1(x) = c2x+ c1
[
−2i exp(ix2) + (1 + i)√
2πx erfc
(
1 − i√2x
)]
, (F.6)
and c1, c2 are obtained by matching with Eq. (3.29).
c1 =1 − i
2
√
Z
π, c2 = −
√2Z. (F.7)
This result agrees with Eq. (3.27).
72
Appendix G
Stationary phase approximation of
the non-analytic short-time
behavior
Stationary phase approximation[57] yields the leading term in asymptotic expan-
sion of an integral. It is applicable to integrals of the following form:
I(x) =
∫ b
a
dt f(t) exp[ixψ(t)], (G.1)
and it yields the leading behavior of I(x) for x→ ∞. It has the following formula[57]:
∫ b
a
dt f(t) exp[ixψ(t)] ∼ 2f(c) exp[ixψ(c) ± iπ
2p]
(
p!
x |ψ(p)(c)|
)1/pΓ(1/p)
p. (G.2)
c is the stationary point that ∂tψ(t = c) = 0, and c ∈ (a, b). p is the power of the
first non-vanishing term in the Taylor expansion of ψ(t) after the linear term at the
stationary point. The leading non-analytic term in outer-region expansion can be
obtained through the stationary phase method.
73
Using the 1-d disappearing nucleus case as an example, write out the time-
dependent wavefunction using Green’s function:
ψ(x, t) =i
2π
∫
kdk
∫
dx′ exp(−ik2t/2)G(x, x′,k2
2)ψ0(x
′)
=
√Z
π
∫ ∞
0
dk exp(−ik2t/2)
[
ik exp(−Z |x|)k2 + Z2
+Z exp(ik |x|)k2 + Z2
]
.
(G.3)
The first term in the integral gives the t-TE wavefunction. The second term is in Eq.
(G.1)’s form, and I apply the stationary phase approximation to it. The stationary
point is k = |x| /t. Noticing the stationary point corresponds to the imaginary part
of the exponential in Eq. (G.1) goes to infinity, this approximation in Eq. (G.3) is
equivalent to taking the y = x/√
2t→ ∞ limit, which is the outer-region expansion.
Applying the stationary phase approximation to Eq. (G.3), it correctly yields the
leading time-non-analyticity:
ψ(x, t)x/
√2t→∞∼ · · · + (1 − i)Z3/2
√πx2
exp
(
ix2
2t
)
t3/2 + · · · . (G.4)
For the 3-d hydrogen in turned-on static electric field case, the leading-order time-
non-analyticity in the outer-region expansion can be obtained by applying stationary
phase approximation to the 1st order perturbation theory. The 1st order change in
the wavefunction is
ψ(1)(r, t) =
∫ t
−∞dt′∫
d3r1 G(0)(r, r1, t− t1)δV (r1, t1)ψ
(0)(r1, t1), (G.5)
in whichG(0)(r, r1, t−t1) is the Green’s function of the 3-d hydrogen atom, δV (r1, t1) =
Ez1θ(t1) is the perturbing potential. Using the stationary phase approximation, the
leading non-analytic short-time behavior is obtained as
δψ(1)(r, t)r/
√2t→∞∼ · · ·+ (8 − 8i)EZ5/2z
πr8exp
(
ir2
2t
)
t11/2 + · · · . (G.6)
74
Appendix H
3-d disappearing nucleus case
I start from the ground-state wavefunction of the hydrogen atom:
ψ(0) =Z3/2
√π
exp(−Zr). (H.1)
0 1 2 3
r(a.u.)
0
0.1
0.2
0.3
n(r,t)
n(t=0)n(t=0.1)n(t=0.5)n(t=1)TE n(t=0.1)TE n(t=0.5)TE n(t=1)
Figure H.1: Time-dependent exact density and t-TE density of the ground-statewavefunction of hydrogen atom under free-propagation.
The exact time-dependent wavefunction is found by applying the free-particle
time-dependent Green’s function to Ψ(0). It is
ψ(r, t) =Z3/2 exp(iZ2t/2)
2√πr
[f(r, t) − f(−r, t)], (H.2)
75
in which f(r, t) = (r+ iZt) exp(Zr) erfc[(r+ iZt)/√
2it]. I plot in Fig. H.1 the exact
solution Eq. (H.2) and the power-series solution Eq. (H.3), which is summed to all
orders:
ψTE(r, t) =Z3/2
√π
exp(−Zr + iZ2t/2)(1 − iZt/r), r > 0. (H.3)
76
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