First-Principle Approach to Electronic States and Metal-Insulator Transition in Selected Correlated Model Systems
Andrzej P. Kądzielawa
Ph.D. Thesis
Promotor: Prof. dr hab. Józef Spałek
Promotor pomocniczy: dr Andrzej Biborski
Uniwersytet Jagielloński
Instytut Fizyki im. Mariana Smoluchowskiego
Zakład Teorii Materii Skondensowanej i Nanofzyki
Kraków 2015
Wydział Fizyki, Astronomii i Informatyki Stosowanej Uniwersytet Jagielloński
Oświadczenie
Ja niżej podpisany Andrzej P. Kądzielawa (nr indeksu: 1014332) doktorant Wydziału Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiellońskiego oświadczam, że przedłożona przeze mnie rozprawa doktorska pt. "FirstPrinciple Approach to Electronic States and Metal�Insulator Transition in Selected Correlated Model Systems" jest oryginalna i przedstawia wyniki badań wykonanych przeze mnie osobiście, pod kierunkiem prof. dr hab. Józefa Spałka. napisałem samodzielnie.
Oświadczam, że moja rozprawa doktorska została opracowana zgodnie z Ustawą o prawie autorskim i prawach pokrewnych z dnia 4 lutego 1994 r. (Dziennik Ustaw 1994 nr 24 poz. 83 wraz z późniejszymi zmianami).
Jestem świadom, że niezgodność niniejszego oświadczenia z prawdą ujawniona w dowolnym czasie, niezależnie od skutków prawnych wynikających z ww. ustawy, może spowodować unieważnienie stopnia nabytego na podstawie tej rozprawy.
Kraków, dnia 7 VIII 2015
.............................. podpis doktoranta
Abstract
In this Thesis we present a fully microscopic, frstprinciple approach to describe the systems of molecular and atomic hydrogen. We apply the Exact Diagonalization Ab Initio approach (EDABI) both with the employment of the exact Hamiltonian diagonalization procedures and the socalled StatisticallyConsistent Gutzwiller Approximation (SGA), to model the system described by the socalled extended Hubbard model. We analyze the selected hydrogenic systems, and study their stability under high pressure, obserVing the insulatortometal transition of the MottHubbard type and moleculartoatomic transition, as well as assessing the zeropoint motion amplitude and energy and electron�lattice coupling constants. These all are important steps in description of superconductiVity in the hydrogenic systems with both the inclusion of interelectronic correlations and the electron lattice interaction. From the last point of View, the present work represents the frst step towards this goal.
The EDABI method proVides a realistic depiction of manyelectron states, starting from the renormalization of the singleparticle waVefunction basis. This allows us to obtain quality results while using rather small number of basis functions. Also, we are able to characterize the manybody waVefunction in the resultant correlated state.
We study metallization of solid atomic hydrogen by applying the EDABI method combined with SGA to the simplecubic lattice with halfflled 1s Slatertype orbitals. This allows us to describe the insulator�metal transition of the weakly frstorder MottHubbard type, at the external pressure „ 100GP a. We also examine the efect of applied magnetic feld and the critical scaling of the groundstate energy and the inVerse waVefunction size (one of the Variational parameters in EDABI). We haVe also made a frst assessment of the zeropoint motion energy in the correlated state.
Next, we take steps to include the Vast spectrum of the solid molecularhydrogen phases. We start from the complex description of the H2 molecule, taking into account all electronic interactions, electron�proton coupling, and estimating the zeropoint energy with the groundstate energy contribution up to the ninth order. The zeropoint motion energy is assessed to be „ 1% of the molecular binding energy, as is obserVed experimentally. We also create the frst model of molecular crystal a onedimensional molecular chain in the extended Hubbard model with longrange interactions and periodic boundary conditions. We fnd that, when nonzero force is applied, there exist two phases one of the molecular and one of the quasiatomic nature, leading, at some point, to the molecular�quasiatomic transition. The relation of this transition to the MottHubbard insulator metal transition is proposed. We also test the software and proposed computational solutions used to obtain the results presented in this Thesis.
This Thesis is supplemented with the computer animation on CD of the molecular to quasiatomichydrogen transition, carried out on the example of linear chain (cf. http:// th-www.if.uj.edu.pl/ztms/download/supplementary material/molecular to quasiatomic transition-hydrogen chain.avi).
Additionally,
both
the
cited
and
the
original
works
haVe
been linked to the original sources of the publications.
V
Keywords: solid hydrogen, metal�insulator transition, molecular�atomic transition, ab-initio quantum computations, correlated electron systems, extended Hubbard model, long-range interaction, molecular crystals, Exact Diagonalization Ab Initio approach {EDABI}, Statistically-Consistent Gutzwiller Approximation {SGA}
Streszczenie
W rozprawie zaprezentowano w pełni mikroskopowe podejście z pierwszych zasad do opisu układów molekularnego i atomowego wodoru. Zastosowano metodę EDABI (ang. Exact Diagonalization Ab Initio approach) do modelowania układów opisywanych rozszerzonym modelem Hubbarda, stosując zarówno dokładne procedury diagonalizacji hamiltonianu, jak i tak zwane statystycznie konsystentne przybliżenie Gutzwillera (ang. Statistically-Consistent Gutzwiller Approximation SGA). W rozważanych układach wodorowych, przy uwzględnieniu wysokich ciśnień, obserwowano przejście izolator�metal typu MottaHubbarda i przejście z układu molekularnego do atomowego. Oszacowano również energię drgań punktu zerowego, jak i stałe sprzężenia elektron�sieć. Są to niezbędne komponenty spójnego opisu nadprzewodnictwa takich układach, uwzględniając zarówno korelacje międzyelektronowe, jak i oddziaływanie elektronsieć. Ta rozprawa jest pierwszym krokiem do osiągnięcia tego celu.
Metoda EDABI zapewnia realistyczny opis stanów wieloelektronowych, wychodząc z renormalizacji bazy jednocząstkowych funkcji falowych, co pozwala na otrzymanie dobrych jakościowo wyników przy dosyć małej liczbie funkcji bazowych. Dzięki temu możemy scharakteryzować właściwości wielocząstkowej funkcji falowej w stanie skorelowanym.
Przebadano metalizację stałego, atomowego wodoru, stosując metodę EDABI jednocześnie z SGA do opisu sieci prostej kubicznej z w połowie wypełnionymi orbitalami Slatera 1s. Doprowadziło to do modelu przejścia pierwszego rodzaju z fazy izolatora to fazy metalicznej przy ciśnieniu zewnętrznym „ 100GP a. Przeanalizowano też wpływ pola magnetycznego na stan podstawowy układu i przedstawiono skalowanie krytyczne energii stanu podstawowego i odwrotności rozmiaru funkcji falowej (jednego z parametrów wariacyjnych w metodzie EDABI). Wyprowadzono też pierwsze przybliżenie na energię drgań punktu zerowego.
Następnym krokiem było uwzględnienie szerokiego spektrum molekularnych faz stałego wodoru. W tym celu opracowano kompleksowy opis molekuły H2 jak układu dwuwęzłowego z uwzględnieniem wszystkich oddziaływań międzyelektronowych i sprzężenia elektron�proton. Wyliczono energię drgań punktu zerowego z uwzględnieniem wkładu energii stanu podstawowego do wyrazów stopnia dziewiątego i stanowi ona „ 1% energii wiązania cząsteczki H2, co jest w pełnej zgodności z eksperymentem. Stworzono też pierwszy model molekularnego kryształu jednowymiarowy łańcuch molekuł H2 opisany rozszerzonym modelem Hubbarda z periodycznymi warunkami brzegowymi z uwzględnieniem oddziaływań długozasięgowych. Pokazano, że przy niezerowej sile zewnętrznej działającej na łańcuch układ ma dwie fazy molekularną i kwaziatomową, co prowadzi do przejścia od fazy molekularnej do kwaziatomowej dla dostatecznie wysokiej wartości zewnętrznej siły. Zaproponowano odniesienie pomiędzy tą przemianą, a przejściem izolator�metal typu MottaHubbarda. Przeanalizowano i przetestowano oprogramowanie i metody obliczeniowe użyte do otrzymania wyników prezentowanych w pracy.
Do ninejszej rozprawy dołączono dysk CD z animacją komputerową przejścia stałego wodoru molekularnego do fazy kwaziatomowej, zamodelowaną na przykładzie takiego przejścia dla łańcucha liniowego (zobacz http://th-www.if.uj.edu.pl/ztms/download/supplementary material/molecular to quasiatomic transition-hydrogen chain.avi).
Dodatkowo
dołączono
linki do cytowanych oraz oryginalnych prac wchodzących w skład pracy.
Vii
Słowa kluczowe: stały wodór, przejście metal�izolator, przejście kryształ molekularny atomowy, obliczenia z pierwszych zasad, skorelowane układy elektronowe, rozszerzony model Hubbarda, oddziaływania długozasięgowe, kryształy molekularne, metoda EDABI, statystycznie konsystentne przybliżenie Gutzwillera {SGA}
Contents
Abstract
v
Abstract
in
Polish
vii
Acknowledgements
xi
List
of
abbreviations
and
symbols
xiii
1
Introduction
and
Motivation
1
1.1
MottHubbard
Systems
and
their
Phase
Transitions
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1.1.1
MottHubbard
Criterion
of
Metal�Insulator
Transition
. . . . . . . . 2
1.1.2
Original
Mott
Localization�Delocalization
Criterion
. . . . . . . . . 3
1.2
Hydrogen
systems:
Principal
QualitatiVe
Features
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
1.3
AimandtheScopeoftheThesis
........................ 7
2
Methods
of
approach
9
2.1
FirstPrinciple
QuantumMechanical
Description
ofMatter
..................................... 9
2.2
ExactDiagonalizationabInitioApproach
. . . . . . . . . . . . . . . . . . . 10
2.2.1
Principal
points
of
Exact
Diagonalization
Ab
Initio
Approach
(EDABI).................................... 10
2.2.2
Manyparticle
waVefunction
and
the
particledensity
profle
. . . . . 12
2.2.3
Testingcase:Descriptionoflightatoms
................ 12
2.3
Gutzwiller
(GA)
and
Statistically
consistent
Gutzwiller
Approximations
(SGA)
17
2.3.1
GutzwillerwaVefunctionmethod
.................... 17
2.3.2
Gutzwiller
approximation:
Groundstate
energy
.
.
.
.
.
.
.
.
.
.
.
.
18
2.3.3
EfectiVeHamiltonianinGA
...................... 19
2.3.4
GA:
Grand
Canonical
Ensemble
Approach
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
19
2.3.5
Statistical
consistency
for
the
Gutzwiller
approximation
(SGA)
. . . 20
3
Published
Papers
with
their
summaries
22
3.1 Paper
A1
Extended
Hubbard
model
with
renormalized
Wannier
waVe
functions
in
the
correlated
state
III:
Statistically
consistent
Gutzwiller
approximation
and
the
metallization
of
atomic
solid
hydrogen
. . . . . . . . . . . . 22
3.2 Paper
A2
Metallization
of
Atomic
Solid
Hydrogen
within
the
Extended
Hubbard
Model
with
Renormalized
Wannier
WaVefunctions
. . . . . . . . . 33
3.3 Paper
A3
H2
and
pH2q2
molecules
with
an
ab
initio
optimization
of
waVefunctions
in
correlated
state:
electron
�proton
couplings
and
intermolecular
microscopicparameters
............................. 39
3.4 Paper
A4
Combined
shared
and
distributed
memory
ab-initio
computations
of
molecularhydrogen
systems
in
the
correlated
state:
process
pool
solutionandtwoleVelparallelism
....................... 67
3.5 Paper
A5
Discontinuous
transition
of
molecularhydrogen
chain
to
the
quasiatomic
state:
Exact
diagonalization
ab
initio
approach
. . . . . . . 82
4
Summary
and
Conclusions
88
5
Bibliography
90
Acknowledgements
I would like to express my gratitude to my superVisor, Prof. Józef Spałek, for suggesting the subject of the Thesis, all the discussions, inspiration to do research, help and good word, as well as for the critical reading of this Thesis and the multitude of his corrections.
I would also like to thank Dr. Andrzej Biborski, for all the time we spent programming, his support and openness towards me.
I am thankful to Marcin Abram, Agata Bielas, Dr. Danuta GocJagło, Dr. Jan Kaczmarczyk, Ewa KądzielawaMajor, Dr. Jan Kurzyk, Prof. Maciej Maśka, Grzegorz Rut, Dr. hab. Adam Rycerz, Dr. Leszek Spałek, Prof. Włodzimierz Wójcik, Marcin Wysokiński, and Dr. Michał Zegrodnik, for all the discussions we had.
I would like to thank Dr. Marcello Acquarone for his hospitality and help during my Visit at the UniVersity of Parma.
I also would like to show my appreciation for my family for their unconditional support, especially Kasia and Kostek without whom this Thesis would be fnished earlier.
The work was supported by the Project TEAM awarded by the Foundation for Polish Science (FNP) for the years 20112015 and by the project MAESTRO from the National Science Centre, Grant No. DEC2012/04/A/ST3/00342.
The principal part of the computations in this Thesis has been carried out on TERA ACMIN supercomputer located at Academic Centre for Materials and Nanotechnology, AGH UniVersity of Science and Technology.
Frequently used abbreviations
DFT Density Functional Theorem
DMFT Dynamical Meanfeld Theory
EDABI Exact Diagonalization Ab Initio Approach
GA Gutzwiller Approximation
LDA LocalDensity Approximation
MIT Metal�Insulator Transition
QMT Quantum Metallization Tools
SGA Statisticallyconsistent Gutzwiller Approximation
STO Slatertype Orbital
ZPM Zeropoint Motion
Frequently used symbols
EG groundstate energy
:
cˆiσpcˆiσq creation (annihilation) operator of a fermion with spin σ on site i H Hamiltonian wiprq singleparticle basis waVe function tij onebody microscopic parameter of Hamiltonian Vijkl interaction microscopic parameter of Hamiltonian U onsite Hubbard interaction (Viiii) Kij Coulomb interaction between sites i and j (Vijij) Jij exchange interaction between sites i and j (Vijji) W bandwidth |ψGy Gutzwiller waVe function |ψ0y uncorrelated waVe function PpPiq (local) Gutzwiller correlator qσ bandnarrowing factor d2 aVerage double occupancy nσ aVerage number of particles with spin σ per site n band flling (aVerage number of particles per site) m magnetic moment (spin polarization) λn Lagrange multiplier (molecular feld) coupled with n λm Lagrange multiplier (molecular feld) coupled with m
FGA
Landau functional in Gutzwiller Approximation
Chapter 1
lntroduction and Motivation
In this Chapter we introduce a general notion of fermionic localization as induced by the repulsiVe interactions among particles, here shown on the example of the Coulomb interactions among electrons. In particular, we discuss how the competition between the kinetic (band) and the Coulomb energies leads to the metal�insulator transition which at temperature T “ 0 represents a quantum phase transition.
1.1 Mott-Hubbard Systems and their Phase Transitions
Defning an insulator as a material with a Vanishing electrical conductiVity at T “ 0 in the electrical feld is not precise. If inVestigated carefully, the microscopic behaVior of electrons in the ionic lattice, brings seVeral mechanisms of the eVanescence of the metallicity. Namely, the Coulomb interaction between system elements will contribute to its macroscopic behaVior in a diferent way. We
can
separate
insulators
into
two
groups
[1],
in
which
the
leading
role will be played by
1.
electron�ion interactions,
2.
electron�electron interactions.
Within the frst group we can distinguish: the band insulator [2�4�
(also
called
the
BlochWilson insulator), where the conductance is suppressed by interaction between electrons and the lattice periodic potential, the Peierls insulator [5,
6],
with
the
interaction
between
electron and static lattice deformation, and the Anderson insulator [7], were the main contribution comes from electron�impurity interaction. A new kind of physics emerge from collectiVe
electron
behaVior
in
2
the Mott insulator [8�12]. This phenomenon can be obserVed
in
a
Variety
of
system,
including
peroVskites
[13�17],
NiS2´xSex [18�21], as well as VO2 and V2O3 [22�25].
1.1.1 Mott-Hubbard Criterion of Metal-Insulator Transition
One can consider a Mott insulator as a system, where an intriguing competition takes place: the kinetic (band) energy of the electrons in the lattice (much alike in the freeelectron case), experience mutual repulsiVe Coulomb interactions. EVen a rather crude approximation of locality of the electronic interaction allows us to model a system with metallic or insulating features. Namely, to asses the interaction, we take the socalled Hubbard U
U ” Ep´q ` Ep´(1.1)
Öq´pEp´Òq ` Ep´Óqq ,
where Ep´q, Ep´Òq, Ep´Óq, and Ep´
Öq are energies of onesite with zero, one, and two electron
respectiVely. From the other side, we take the bandwidth W related to the dispersion
relation Epkq
W ” maxpEpkqq ´ minpEpkqq. (1.2)
As
both
these
quantities
depend
on
the
system
topology,
we
list
some
examples
in
Tab.
1.1.
Table
1.1:
Dispersion
relation
in
the
tightbinding
approximation
[26]
and
the
corresponding
bandwidth for seVeral exemplary systems with intersite distance a and nearestneighbor hopping t.
lattice type dispersion relation Epkq bandwidth W
chain 2t cospkxaq 4|t|
simple quadratic 2trcospkxaq ` cospkyaqs 8|t|
simple cubic 2trcospkxaq ` cospkyaq ` cospkzaqs 16|t|
hypercubic (d dimensions) 2t řd i“1 cospkiaq ? 4d|t|
triangular 2trcospkxaq ` 2 cosp1 2kxaq cosp3 2kyaqsb 9|t|
honeycomb ˘t 3 ` 2 cosp ? 3kxaq ` 4 cosp3 2kyaq cosp3 2kxaq ? 3|t|
facecentered cubic 4trcosp1 2kxaq cosp1 2kyaq ` cosp1 2kyaq cosp1 2kzaq` cosp1 2kzaq cosp1 2kxaqs 16|t|
bodycentered cubic 1 21 21 2
8t cospkxaq cospkyaq cospkzaq
16|t|
Let us construct a criterion for metallicity. We take the halfflled (the number of electrons Ne equal to the number of sites Λ) lattice and defne gap as
´
Δ “ µ ` ´ µ, (1.3) where µ˘ is the energy required to add/remoVe one electron to the system `
µ “ EGpNe ` 1q´ EGpNeq, (1.4a)
´
µ “ EGpNeq´ EGpNe ´ 1q, (1.4b)
where EG is the groundstate energy. We
can
rewrite
(1.4)
in
terms
of
U and bandwidth
Wu
`
µ “ U ´ , (1.5a)
2
Wd
´
µ “ , (1.5b)
2
where Wu is socalled upper Hubbard band and Wd is socalled lower Hubbard band (cf. Fig.
1.1).
Wu corresponds to the charge excitation in the system, whereas Wd with its charge
remoVal.
Combining
(1.5)
and
(1.3)
we
get
Wu ` Wd
´
Δ “ µ ` ´ µ “ U ´« U ´ W. (1.6)
2
It is rather straightforward that for W ! U the system is insulating as the Hubbard interaction U ą 0. Also, for W " U the system is metallic as the system can be treated with the freeelectron model (and the interaction consecutiVely included as perturbation). From the modeling point of View, the interesting physics happens when W « U and the system approaches the metal�insulator transition (MIT) in the halfflled band case.
1.1.2 Original Mott Localization-Delocalization Criterion
One
can
look
at
the
localization�delocalization
phenomenon
in
a
more
general
manner
[27],
as the competition between the kinetic energy of the electrons, dominating for the delocalized (metallic) state, and the electronic Coulomb repulsion, connected with the localized (insulating) state. Let us consider a 3dimensional, simplecubic lattice. AVerage kinetic energy in such a system will be
3
¯
Ek “ EF , (1.7)
5
where EF is Fermi energy defned Via dispersion relation
n2k2
Ek ” , (1.8)
2m
and Fermi momentum
˜
`˘¸
5
2
ρ
1 3
3
23π
Γ
2
.
(1.9)
kF ”
2S ` 1
S “ 1{2 is the electron spin, Γpxq is the Euler Gamma function, and ρ is the particle density. Inserting
(1.8)
and
(1.9)
into
(1.7)
we
haVe
3 n2 `˘
¯
Ek “ 3π2ρ
2
3
.
(1.10)
52m
On the other hand, the Coulomb repulsion will take form
2
e
Ee´e « , (1.11)
2εr¯
where e is the electron charge, ε is the absolute permittiVity, and r¯” ρ ´1{3 is the aVerage interelectronic distance.
We
can
now
compare
(1.10)
with
(1.11)
`
˘
22
¯3 n2 3π2Ek 52m
“
n2ε
`˘
3
ρ
3
2
31
3π2
.
(1.12)
“
3
ρ
312
2
Ee´e
5
e
me
ρ
3
2ε
Both
energies
are
comparable
when
(1.12)
is
equal
to
unity.
Taking Bohr radius aB ” n2ε{me2 we obtain the socalled Mott (or MottWigner) criterion for critical particle density ρC
5 `˘´
3π2
12
“ 0.174 « 0.2.
(1.13)
“
33
aBρ
C
3
One sees clearly that for ρ ă ρC the interaction energy is dominant, whereas for ρ ą ρC the kinetic energy is. In other words, for the lowdensity (ρ ă ρC ) the potential energy freezes the electrons in the hydrogeniclike orbits with the Bohr radius aB, whereas for ρ " ρC the particles are almost free, i.e. the metallic state is stable. Thus the Mott criterion
(1.13)
defnes
in
rough
terms
the
limiting
concentration
on
freeelectron
concept
applicability to the description of metallic state.
At this point, one should mention that there exist other related criteria of metallicity (cf. Mott
[10]);
here
we
should
mention
only
the
original
Mott
criterion.
This criterion addressed the discontinuous transition to the metallic phase in ed magnetic oxides. Namely, Mott argued that the transition from the (magnetic) insulating system to the metallic state as a function of e.g. pressure, should be discontinuous. This is because a small number of conduction electrons in the conduction band would increase largely the Coulomb repulsiVe energy (lack of mutual screening). Hence their number must be substantial so the emerging band energy oVercomes the repulsiVe interaction. A reasoning of that kind leads to the criterion
(1.13)
which
expresses
the
instability
of
the
atomic
bound
state
with
respect
to
the freeparticle (electrongas) state.
1.2 Hydrogen systems: Principal Qualitative Features
The atomic hydrogen solid would present itself as an ideal system to model bandelectron physics and the MottHubbard type of metal�insulator transitions. This is because such system has only one Valence electron, originally on 1stype orbital, as the distance of the frst excited (2s, 2p) state is located 3{4Ry « 10eV higher. So its admixture to the 1stype Wannier state should be rather small, particularly neat the metalinsulator transition, where according
to
(1.13)
we
haVe
that
the
intersite
distance
a „p4˜5qaB. It is quite amazing that studies of such hydrogeniclike solids has started only recently by incorporating the MottHubbard
physics
into
the
frstprinciple
electronic
structure
modeling
[28�32].
These
studies
brought eVen some estimates for the critical pressure p „ 102GP a for the atomic hydrogen metallization, as well as pointed out to the possibility of quantum critical behaVior of the atomic orbit size at the metallization threshold. The zero point motion of the lightest ions (protons) in this case has been estimated and shown to be sizable, but without destroying the ionic lattice, what would amount to the transition from an atomic Mott insulating solid to the electronproton plasma.
HaVing said that, one has to note one principal feature complicating any such atomichydrogen modeling. Namely, hydrogen as such is stable in the molecular state H2 at ambient pressure and likewise, it forms a Variety of molecular crystal structures at ambient and
applied
pressures
[33,
34].
There
are
at
least
three
known
solid
phases
[33,
35�37],
often
referred
to
as
phases
I,
II
and
III
(cf.
Fig.
1.2).
Appearing
for
relatiVely
small
pressures
closedpacked (hexagonal) molecular crystal (P 63{m) constitutes what is known as phase I (cf.
Fig
1.3.a).
Phases
II
and
III
are
not
well
recognized
experimentally;
neVertheless,
there
are
DFT
structural
calculations
[38,
39]
recognizing
phase
II
as
monoclinic
C2{c molecular
crystal
(cf.
Fig
1.3.b),
and
phase
III
as
monoclinic
Cmca ´ 12 molecular crystal (cf. Fig
1.3.c).
Whether
or
not
the
phase
III
is
metallic,
was
carefully
examined
experimentally
by Zha et al. [37]
in
the
broad
range
of
temperature
and
pressure
(up
to
300GP a), but no metallic behaVior had been obserVed. In 2011 a new phase IV was reported by Eremets and Troyan
[40],
claimed
to
be
an
atomic
fuid.
Both
its
atomic
character
and
metallic
properties
were
subsequently
questioned
[41�43].
The
ab initio Molecular Dynamic approach by GoncharoV et al. [41]
suggests
that
phase
IV
could
be
a
mixture
of
molecular
liquid
and
the
atomic, graphenelike twodimensional layers. Additionally, Howie et al. [36]
reported
in
an
experiment (Raman spectroscopy under high pressure) a new phase transition at 255GP a and 480K, which can be understood as a melting transition at surprisingly low temperature (giVen the magnitude of the pressure).
We can say that the natural starting situation is a dimerized hydrogen quantum solid which may metallize under applied pressure. The modeling of molecular phases at ambient and external pressures is one of the main purposes of the Thesis. To this aim, a specifc combination of exact diagonalization and ab initio approach has been deVeloped by us in recent
years
[44].
The
method
of
approach
is
discussed
in
detail
in
the
next
Chapter.
First,
we characterized the aim and scope of the Thesis in general terms.
It is essential to say few words about the importance of hydrogen metallization. First, it is a model system for the correlated systems, albeit with the complications introduced by the molecular binding. Second, and most important, metallized by pressure hydrogen is regarded
as
a
prospectiVe
roomtemperature
superconductor
[45].
For
that
purpose,
both
the correlation efects and the local electron�lattice coupling constants must be considered concomitantly. The
second
task
has
been
achieVed
already
[46].
As
for
the
frst,
there
is
still
a
road
ahead,
as
we
point
it
out
explicitly
in
the
Summary
and
Outlook
section.
1.3 Aim and the Scope ofthe Thesis
As already said, the principal aim of this Thesis is to deVelop and implement a reliable method of calculating hydrogen or hydrogeniclike systems up to the metallization and characterization of resultant metallic state. In this Thesis we start from a precise approach to molecular systems regarded as the principal and necessary frst step in that direction. We supplement the molecular considerations (cf. paper
A3)
with
rigorous
treatment
of
molecular
hydrogen
chains
(cf.
papers
A4
and
A5).
One has to note that this is not a simple task as for example singleparticle DFT calculations
[39,
47�49]
proVide
conficting
results
[38].
PreViously, promising results in this
direction
haVe
been
obtained
within
the
Monte
Carlo
[33,
50�52],
and
other
[53�56]
techniques. In this respect, our approach may be regarded as systematic. This is because of few reasons. First of them, unlike the other approaches including the electronic correlations (LDA+U
[57,
58],
LDA+DMFT
[59�63])
our
approach
does
not
haVe
their
drawback
of
counting twice the repulsiVe Coulomb interaction among the particles. Second, the method allows for determination of singleparticle Wannier waVe functions in the correlated state. Third, the microscopic parameters such as the intersite hopping integrals (tij) or magnitude of the Hubbard interaction (U) are eValuated also in the correlated state. In that manner, one can relate explicitly the ab initio electronic structure to the MottHubbard criteria of localization.
As to the scope of the Thesis apart from two methodological papers setting the method correctly
both
from
the
molecular
[46]
and
the
computational
[64]
sides,
we
haVe
also
demonstrated
its
efectiVeness
already
on
the
nanoscopic
(multimolecular)
leVel
[65].
For
infnite
system, so far only the socalled statistically consistent Gutzwiller approximation (SGA) has
been
formulated
[32].
The
zero
point
motion
problem
has
been
tackled
also
successfully
[46,
66].
In
the
near
future,
an
extension
of
our
results
to
the
socalled
diagrammatic
expansion for the Gutzwiller waVe function (DEGWF), formulated in our group and successfully applied
to
a
number
of
problems
[67
72],
is
planned.
The
formal
approach
deVeloped
for
the
present problem of hydrogen metallization, allows utilizing both exact and approximated methods of diagonalization of the secondquantized Hamiltonians. Within this perspectiVe, our results represent a frst step towards the problem solution.
In general, the method is applicable to atoms, molecules, and correlated solids. Some of the examples of application to each type of the aboVe systems are proVided in the next Chapter.
Chapter 2
Methods of approach
In this Chapter we characterize in detail the methods used throughout the Thesis. Namely, the Exact Diagonalization Ab Initio Approach (EDABI) and the StatisticallyConsistent Gutzwiller Approximation (SGA). Both of these methods haVe been deVeloped and used in our group in the context of electron localization in the correlated systems (EDABI) and in the context of unconVentional superconductiVity (SGA).
2.1 First-Principle Quantum-Mechanical Description of Matter
The starting manyelectron Hamiltonian for calculating the stationary states has the following form within the frst quantization scheme
electron-electron repulsion
hkkkkkkkikkkkkkkjÿÿÿÿ
2Zj 12 12ZiZj
H “´ 2 ´` ` , (2.1)
i
|ri ´ Rj|2 |ri ´ rj| 2 |Ri ´ Rj|
iij i‰ji‰j
looooooomooooooon loooooooomoooooooon
electron-ion ttr ction ion-ion repulsion
where electronic coordinates are denoted by ri, whereas ions are statically at positions Ri. The
consecutiVe
terms
are
represented
schematically
in
Fig.
2.1.
We
utilize
the
BornOppenheimer
approximation
[73]
and
regard
the
ionic
coordinates
tRiu as fxed and determine the groundstate energy EG Via the corresponding Nparticle Schrodinger equation for the electronic part
HΨpr1, r2, r3,... rN q“ EGΨpr1, r2, r3,... rN q. (2.2)
We
call
a
method
allowing
to
solVe
(2.2)
the
frstprinciple
(of
Quantum
Mechanics)
or
ab initio method. It is today almost synonymic with use of modernday Density Functional Theorem
[74,
75]
and
its
many
Variations
(e.g.,
LDA+U
[57,
58],
LDA+DMFT
[59�63]),
Figure 2.1: A schematic representation of system in a metallic state close to the frstorder metal�insulator transition, as composed of frozen ions (in BornOppenheimer approximation), electron�ion (B), electron�electron (C), and Coulomb ion�ion (D) interactions.
which are not exact by any precise means. There is a Vast number of tools in the frstquantization
language
dealing
with
Hamiltonian
(2.1),
worth
mentioning
is
the
HartreeFock
method
[76�78]
and
its
many
deriVatiVes
[79,
80],
as
well
as
the
M0ller�Plesset
method
[81],
and
the
ConfgurationInteraction
methods
[82].
In
the
next
Section
we
present
an
original
method, called Exact Diagonalization Ab Initio approach (EDABI), which allows for frstprinciple description of the system, while employing both the secondquantization language and the renormalization of singleparticle basis of waVe functions. Instead of working with (2.1),
we
describe
next
our
system
in
terms
of
secondquantized
Hamiltonian.
2.2 Exact Diagonalization ab Initio Approach
As a method to tackle hydrogen systems we haVe selected the socalled Exact Diagonalization Ab Initio
approach
(EDABI)
[83�85],
proVed
to
be
efcient
to
describe
hydrogen
systems
both asymptotically (ionization energy of a free atom difers ă 0.1% from experimental Value, binding energy of a H2 molecule is calculated with the accuracy of 2% [46,
83])
and
as
an
atomic
solid
[30�32].
The method successfully combines frst and secondquantization pictures, and allows for employment Vast number of algorithms of diagonalizing the parametrized Hamiltonian in its secondquantization form..
2.2.1 Principal points of Exact Diagonalization -Ab Initio Approach (EDABI)
To understand the method we start from the feld operators in the form
ÿ
ν
ˆ
Ψσprq“ wi prqχσˆ (2.3)
ciνσ, iν ν
where wi prq is the set of singleparticle νth band orthogonal and normalized waVefunctions centered on the lattice site i, ˆis the corresponding annihilation operator, and χσ is the
ciνσ
spin waVefunction (σ “˘1) with the single (global) spin quantization axis (zaxis). Band index ν can be dropped for clarity of formulas, and from now on we treat i as an arbitrary index (not necessarily connected only with the lattice site).
We
can
rewrite
our
manyparticle
Hamiltonian
(2.1)
in
the
secondquantized
form
˜
ż
¸
Ψˆσprq (2.4)
σ
`
ÿ
2Zi
i
:
1Ψˆ: σprqΨˆσ1 pr 1
ÿ
d3 Ψˆ: σprq 2 ´
r ´
ˆ
H “
|r ´ Ri|
ij
ÿ
σσ1 i‰j
All terms are expressed in the atomic units (n “ e2{2 “ 2me “ 1, where e is the charge
ř
of electron and me is its mass). The last term 1{22ZiZj{|Ri ´ Rj| is the classical
i‰j
Coulomb repulsion between ions located respectiVely at the positions Ri{j, and with the atomic numbers Zi{j. Note that we assumed the classical behaVior of the ions, i.e., the electrons interact with frozen ionic centers. If
we
proceed
with
including
(2.3)
into
(2.4),
we
obtain
the
explicit
secondquantized
form
of
the
Hamiltonian
[86,
87]
i.e.,
ÿ
2ZiZj
1
21
1
σ1 pr Ψσprq`
d3rd3
ˆ
Ψ
q ˆ
q
r
.
|r ´ r1| |Ri ´ Rj|
2
2
ÿÿ
ÿ
ijσ ijklσ,σ1
ÿ
where tij and Vijkl are microscopic parameters represented by integrals
:
::
cˆ(2.5)
iσ jσ1 cˆlσ1 cˆkσ,
H “
cˆjσ `
tijcˆ
Vijklcˆiσ
ˇˇˇˇˇ
ˇˇˇˇˇ
C
˜
G
¸
2Zi
ÿ
i
” d3 rw i ˚prq´ 2 ´
2 ´
(2.6a)
”
´
tij wiprq
wjprq|r ´ Ri|
˜
ż
¸
wjprq,
F
2Zi
|r ´ Ri|
ˇˇˇˇ
ˇˇˇˇÿ
i
B
2
1
1
q (2.6b)
Vijkl ”
ij
wiprqwjpr
1˚ ˚1
i prqwj pr
q
wkprqwlpr|r ´ r1|
2
1
” d3rd3
q wkprqwlpr
|r ´ r1|
q.
rw
Up to this point the discussion is general as long as the singleparticle basis twiprqu is complete, i.e.,
ÿ
i
As
the
basis
is
infnite,
it
is
not
possible
to
diagonalize
(2.5)
exactly.
To do so, one must select rich enough fnite basis to describe giVen system, which introduces an unknown error
11
˚1
wi prqwipr 1q“ δpr ´ r q. (2.7)
to the calculations. The EDABI method ofers a workaround to this problem: One assumes fnite basis with a set of adjustable parameters tαiu and minimizes the system energy to fnd the optimal Values of these parameters. As we usually start from the singleparticle basis being a solution of the oneelectron situation, this approach allows us to obtain realistic results
with
relatiVely
small
bases.
Please
refer
to
Fig.
2.2
for
the
stepbystep
essentials
about the EDABI.
2.2.2 Many-particle wavefunction and the particle-density profle
In
EDABI
method
we
diagonalize
the
Hamiltonian
(2.5)
in
the
Fock
space,
obtaining
the
ground state defned as
i1,...,iN
We can reVerse the procedure, and retrieVe the frstquantization picture of resultant, physical
state
by
employing
feld
operators
(2.3).
The
exact
Nbody waVefunction can be written in the following form
ÿ
::
... cˆ|0y . (2.8)
i1 iN
|Φ0y“
Ci1,...,iN
cˆ
ÿ
ÿ
i1,...,iN
Out of all features of manybody picture, the possibility of obtaining the particle density profles is of particular signifcance. Namely, we can introduce the particle density operator
σ
ˇˇ
ˇˇ
@
D
1
?
(2.9)
Ψpr1,..., rN q”
0
ˆ... ˆ
ciN ci1 Φ0 wi1 pr1q ...wiN prNq.
N!
Ψˆ: σprqΨˆσprq, (2.10)
nˆprq”
where ˆ
Gˇˇˇˇˇ
ÿ
ˇˇˇˇˇ
Ψσprq is
the
feld
operator
(2.3).
(2.10)
reduces
to
the
particle
density
function
C
rN´1 |Ψpr1,..., rN´1, rq|2
σ
ż
Ψˆ:
d3 r1 ...d3
(2.11)
nprq”
“ N
Φ0
σprqΨˆσprq
Φ0
.
E
A
:
By knowing the optimized basis twiprqu and correlation functions cˆiÒcˆjÒ , one can draw the particle density profles for the system at hand.
2.2.3 Testing case: Description of light atoms
The natural language to describe electrons in atoms is proVided by the Slatertype orbitals (STO)
[88],
hence
the
choice
of
them
as
our
basis.
This
causes
an
issue,
as
Slater
functions
d
ψnlmprq” p2αnlmq2n`1 r n´1 e ´αnlmrY mprq, (2.12)
l
p2nq!
Figure 2.2: Flowchart of the EDABI method. The essence of the method is as follows: We start with setting the rules of the basis composition. This includes geometry of considered system and the singleparticle waVefunction renormalization parameters tαiu. We set the parameters, construct the basis with giVen rules (this usually consists of basis orthogonalization), then we calculate Hamiltonian parameters tij and Vijkl. This is the most timeconsuming
part
and
usually
requires
some
computational
treatment
(cf.
paper
A4).
Next,
we must diagonalize the parametrized Hamiltonian (standard approach will include exact diagonalization, Lanczos algorithm, or statisticallyconsistent Gutzwiller approximation). Finally, the essential step in EDABI is inVoked basis is renormalized Via (nongradient) optimization scheme to obtain the physical groundstate energy EG.
where n “ 1, 2, 3,... , l “ 0, 1,...,n ´ 1, m “´l, . . . , ´1, 0, 1,...,l are quantum numbers and Y mprq are spherical harmonics (for simplicity, we select the real spherical harmonics),
l
are non always orthogonal (in fact when we consider more than one atom in the system they are almost always nonorthogonal). This can be fxed by employing the socalled linearcombination
atomic
orbitals
(LCAO)
[89]
and
orthogonalized
Via
selected
method:
either
the
Lowdin
symmetrical
orthogonalization
[90]
or
the
bilinear
forms
method
see
paper
A4.
The other necessity of the EDABI is to select a number of parameters tαiu, renormalizing basis' singleparticle waVefunctions. It can obViously be done in seVeral ways, but we chose the natural coefcients the inVerse waVefunction size αnlm embedded in Slatertype orbitals. For clarity we will use only n explicitly, l we will traditionally refer to as s (0), p (1), d (2), etc. and instead of m we will write main symmetry axes (i.e., α210 becomes α2pz ).
See
Fig.
2.3
for
examples
of
Slatetype
orbitals
for
diferent
α coefcients.
Gaussian contraction of Slater-type orbitals
As
the
calculation
of
integrals
(2.6)
for
the
Slatertype
orbitals
based
singleparticle
basis
is problematic in general case (especially when it comes to three and four�site terms) it is conVenient to use approximated and integrable representation of STO in terms of Gaussian orbitals (the so called STOpG basis, where p is the number of Gaussian function approximating a single Slatertype orbital)
ψiprq« ψG
i
prq“ α
3 2
˜¸
p
ÿ2Γ2
q
Bq
π
q“1
3
4
e
´α2Γ
2
q
|r´Ri|2
,
(2.13)
where Ri denotes ionic coordinates, α is the STO inVerse waVefunction size, and Bq are the contraction coefcients and Γq are the inVerse Gaussian sizes, both obtain by a proper minimization procedure of the error function
ż
d32
E ” rpψiprq´ ψGprqq. (2.14)
i
Results
Employing the EDABI approach with concomitant optimization of all the Slatertype orbitals (represented as p “ 5 Gaussians (1s,2s) and p “ 10 Gaussians (2p) contractions each) we are able to reproduce manyelectron states of atoms of the frst two periods of the MendeleeV periodic table with all ionization energies for the full Hamiltonian in the form
ÿÿÿÿ
: ::
H “ tijcˆiσcˆjσ ` Vijklcˆiσcˆjσ1 cˆlσ1 cˆkσ. (2.15) ij σ ijkl σ,σ1
Figure 2.3: Slater functions ψ1sprq and ψ2px prq on the xyplane for diferent Values of inVerse waVefunction size α1s and α2px respectiVely.
Table 2.1: Groundstate energy EG and ionization energies Eion for selected light atoms in
i
xz
Rydbergs (Ry). Optimiziation of 1s, 2s, 2p, 2py, and 2preal Slatertype orbitals. Note that systems with morethanhalfflling (number of electrons ą 5, marked with red color) haVe
their
Values
difer
from
the
results
presented
in
literature
[91]
(cf.
Table
2.2).
This
is
due to the fact, that the singleparticle basis is no longer rich enough for the purpose of ab initio description.
EG E1
H He Li Be B C N O F Ne
Table 2.2: Groundstate EG and ionization Ei energies
from
Lide
[91]
for
comparison
with
results
presented
in
Table
2.1.
Please
note
that
for
system
not
difering
much
from
halfflling the results are similar.
EG E1
H He Li Be B C N O F Ne
xy
Table 2.3: Hydrogeniclike atoms in the EDABI approach (optimiziation of 1s, 2s, 2p, 2p,
z
and 2preal Slatertype orbitals). Note that the exact solution of Schrodinger equation proVides α1s “ Zpa0q, EHL “´Z2pRyq.
Z α1s pa ´1 0 q EHL pRyq
H 1 1.00802 0.999878
He1` 2 1.99596 3.9995
Li2` 3 3.02355 8.9989
Be3` 4 4.0038 15.9981
B4` 5 5.04009 24.997
C5` 6 6.0057 35.9957
N6` 7 7.00665 48.9941
O7` 8 8.0076 63.9923
F8` 9 9.00855 80.9903
Ne9` 10 10.0095 99.988
The
results
are
listed
in
Table
2.1.
As
a
test
of
the
method,
one
may
also
compare
results
for
the Hydrogeniclike atoms obtained by the solVing the singleelectron Schrodinger equation with
the
results
obtained
from
EDABI
method
(listed
in
Table
2.3).
2.3 Gutzwiller (GA) and Statistically consistent Gutzwiller Approximations (SGA)
When we consider a system with the strong electronic correlations, the perturbation theory is no longer a efectiVe approach, opening the doors to the Variational treatment with the trial
waVe
functions.
Such
an
approach
was
deVised
by
Gutzwiller
[92�94]
and
generalized
oVer
the
years
to
fulfll
requirements
of
diferent
models,
e.g.,
multiband
[95,
96],
timedependent
case
[97],
as
well
as
tJ,
tJU
[98�101],
and
periodic
Anderson
model
[102�105].
2.3.1 Gutzwiller wave function method
Let us take the Hubbard Hamiltonian
ÿÿ
:
H “ tijcˆiσcˆjσ ` UnˆiÒnˆiÓ, (2.16) ijσ i
: :
where cˆand ˆare the fermionic creation and annihilation operators, ˆ” cˆˆis the
iσ ciσ niσ iσciσ fermionic particle number operator, tij is the hopping amplitude, and U is the Hubbard onsite repulsion term. Its expectation Value is calculated with respect to the Gutzwiller trial waVe function
ź
|ψGy” P |ψ0y” Pi |ψ0y , (2.17)
i
where |ψ0y is the uncorrelated singleparticle product state (Slater determinant) to be determined explicitly later, and Pi is the local Gutzwiller correlator defned for the singleband case in the form
˘
`
Pi ” 1 ´p1 ´ gqnˆiÒnˆiÓ , (2.18)
where g is the Variational parameter. The case g “ 0 refers to the situation with no double occupancy, as all these terms are projected out from the waVe function |ψ0y. Following Binemann et al. [96],
we
can
redefne
the
local
Gutzwiller
correlator
according
to
Γ
where λiΓ are the Variational parameters describing the occupation probabilities of possible local states |Γy i (|Γy i P t|´y i, |´Òy i, |´Óy i, |´
Öy iu for oneband case), where |´y i is the empty
ÿÖy i
site i confguration, |´σy i is that with the single occupation with spin σ “Ò or Ó, and |´is the site occupied by an electron pair.
2.3.2 Gutzwiller approximation: Ground-state energy
For
giVen
(2.17)
trial
waVe
function
we
can
calculate
the
groundstate
energy
EG per site in the socalled Gutzwiller approximation (GA) and obtain
(2.19)
”
λiΓ |Γy ii xΓ| ,
Pi
EG
Λ
“
xψG| H |ψGyxψG| ψGy
”
ψ0
Dˇˇˇˇ@
xψ0| PHP |ψ0yP2
ψ0 « qÒEÒ ` qÓEÓ ` Ud2 , (2.20)
where Λ is the number of sites, d2 is the aVerage number of double occupancies,
ˇˇˇˇˇ
ˇˇˇˇˇ
C
G
ÿ
ÿ
ij k
is the aVerage bare band energy per site for particles of spin σ, and the band narrowing factor is
´a ? ¯ 2 pnσ ´ d2qp1 ´ nσ ´ nσ¯` d2q` dnσ¯´ d2 qσ ” qσpd, nσ,nσ¯q“ , (2.22)
nσp1 ´ nσq
1
1
:
(2.21)
”
“
Eσ ψ0
ˆ
ciσ
ψ0 Ektijcˆ
iσ
Λ
Λ
E
A
where nσ is the aVerage number of particles with spin σ, and d2 “ nˆiÒnˆiÓ . We see that the bare band energies are renormalized by the factor qσ. Additionally, an essential new feature appears, namely the twoparticle correlation function d2 . It is the Variational parameter for giVen ratio U{W (W is the bare bandwidth) and the band flling n.
2.3.3 Efective Hamiltonian in GA
It more conVenient to formulate the problem in a diferent manner. Namely, we assume that there exists an efectiVe, singleparticle Hamiltonian HGA, with its expectation Value equal
to
the
(2.20),
but
now
calculated
with
respect
to
the
uncorrelated
waVe
function
|ψ0y. Thus,
instead
of
the
diagonalizing
the
Hamiltonian
(2.16),
we
minimize
the
eigenValue
of
the efectiVe Hamiltonian
ÿÿ
::
HGA “ qσpd, n, mqtijcˆˆ´ σhcˆˆ` ΛUd2 (2.23)
iσcjσ iσciσ ijσ iσ
ÿ
:
“pqσpd, n, mqEk ´ σhq cˆcˆk ` ΛUd2 ,
kkσ
where Ek is the dispersion relation and depends on the lattice geometry (examples are
ř
:
listed
in
Table
1.1),
σhcˆˆis the Zeeman term with the reduced magnetic feld
iσ iσciσ h “ 1{2gµBHa (that we can take into account without complicating our case), whereas n ” nÒ ` nÓ and m “ nÒ ´ nÓ are the more conVenient Variational parameters, satisfying
$ AE &n ” 1 ř cˆ: ˆ,
Λ kσ kck
1 řA : E (2.24a)
%m ” σcˆcˆk .
Λ kσ k
The two aboVe quantities represent, respectiVely, the band flling (aVerage number of particles per site) and the magnetic moment (spin polarization) per site.
2.3.. GA: Grand Canonical Ensemble Approach
We describe the system at nonzero temperature by constructing the grand potential functional
1
FGA
“´ log Z, (2.25)
β
where β “ 1{kBT , kB is the Boltzmann constant, and the grand partition function Z
´¯
´βpHGA ´µnˆq
Z ” Tr e , (2.26)
where µ is
the
system
chemical
potential.
Using
(2.23)
we
can
write
that
We
can
insert
(2.27)
to
(2.25)
and
obtain
Z “ ź kσ Z1 “ ź kσ 1ÿni “0 e ´βniEGA kσ e ´βUd2 “ e ´βΛUd2 ź kσ ´ 1 ` e ´βEGA kσ ¯ , (2.27)
with the quasiparticle energy defned by
EGA kσ ” qσEk ´ σh ´ µ. (2.28)
ÿ´ ¯
FGA 1 ´βEGA
“´ log 1 ` e kσ ` λUd2 , (2.29)
β
kσ
related to the free energy functional Via a simple redefnition
F GA
“ FGA ` Λµn. (2.30)
In the next Section we describe in detail the minimization procedure of this efectiVe Landau functional for our fermionic system.
2.3.5 Statistical consistency for the Gutzwiller approximation (SGA)
To
fnd
the
groundstate
energy
of
Hamiltonian
(2.16)
we
minimize
the
potential
(2.29)
with respect to quantities d, n and m. This leads to the set of equations
$ř
’ Bqσ fpEGAqEk “´2ΛUd,
& kσ Bd kσ
ř
Bqσ fpEGA (2.31a)
%řkσ Bn kσ qEk “ 0,
’
Bqσ fpEGAqEk “ 0,
kσ Bm kσ
βE
where fpEq” 1{p1`eq is the FermiDirac distribution. On the other hand, we also haVe, directly
from
their
defnition
(2.24),
the
selfconsistent
equations
for
n and m
# ř
Λn “ kσ fpEGAq,
kσ
ř (2.31b)
Λm “ σfpEGAq,
kσ kσ
that produce constrains on the aVerage Value of the particle number and the magnetization operator per site, nˆand mˆ, respectiVely. At this point, it is essential to include in the minimization procedure the statistical consistency conditions of the solution. To do so, following Jędrak et al. [106],
we
employ
the
Lagrange
multiplier
method
and
defne
the
new
efectiVe Hamiltonian
HSGA “ HGA ´ λnpnˆ´ nq´ λmpmˆ´ mq, (2.32)
where Lagrange multipliers λn and λm are the molecular felds, coupled respectiVely to the total charge and the spin polarization. In efect, we defne the Landau grand potential functional in the form
´¯
1 ÿ´βESGA `˘
FSGA
“´ log 1 ` e kσ ` λ Ud2 ` λnn ` λmm, (2.33)
β
kσ
where the quasiparticle energy is now
ESGA ” qσEk ´ σph ` λmq´pµ ` λnq. (2.34)
kσ
To
minimize
(2.33)
we
need
now
to
solVe
the
modifed
set
of
selfconsistent
equations,
namely
$ř
’ Bqσ fpESGA qEk “´2ΛUd,
’ kσ Bd kσ
’
’ř
’ Bqσ fpESGA
’ qEk “´Λλn,
& kσ Bn kσ
ř
Bqσ fpESGA qEk “´Λλm, (2.35a)
kσ Bm kσ
’
’ř
’ fpESGA
’ q“ Λn,
’ kσ kσ
’ř
% σfpESGA q“ Λm,
kσ kσ
where the k summation goes oVer the frst Brillouin zone. Note that by enabling the Lagrange multipliers, we ensure the statistical consistency of the system. This is not the case
for
the
standard
set
of
equations
(2.31).
The consistency amounts to forcing the condition
that
the
aVerages
calculated
with
the
help
of
the
selfconsistent
equations
(2.35)
coincide
with
those
obtained
from
a
direct
minimization
of
the
Landau
functional
(2.33).
This feature is a fundamental correction to GA and has been elaborated earlier in our group
[106].
Here it will be applied to the interesting problem of Mott localization in hydrogeniclike systems. Parenthetically, only after including the statistical consistency conditions, our SGA results coincide with those obtained within the slaVeboson approach in the saddlepoint approximation.
Chapter 3
Published Papers with their summaries
In this chapter we present the published papers, constituting this Thesis, together with its brief summaries. The papers are in the chronological order.
3.1 Paper A-1 -Extended Hubbard model with renormalized Wannier wave functions in the correlated state III: Statistically consistent Gutzwiller approximation and the metallization of atomic solid hydrogen
In this paper we consider a 3dimensional, atomic model of hydrogen. This is a semirealistic approach to the problem of metallization, in the regime of atomic crystal, as we do not know the predictions of the atomic phase structure. Hence, we start our discussion with a model case of simple cubic (sc) lattice. For the frstprinciple system description, we employ the Exact
Diagonalization
Ab
Initio
Approach
(EDABI)
with
the
StatisticallyConsistent
Gutzwiller
Approximation
(SGA)
as
the
parametrized
Hamiltonian
diagonalization
scheme.
This is no longer an exact approach, neVertheless, it proVides useful information about an infnite system. We describe the model with the socalled extended Hubbard model with the singleparticle energy Ea, the nearest neighbor hopping term t, the onsite Hubbard interaction U and the intersite Coulomb repulsion Kij, where the microscopic parameters are obtained by integrating either singleparticle Hamiltonian (for onebody parameters Ea and t) or Coulomb potential (for twobody parameters U and Kij), both with properly prepared singleparticle basis waVe functions. This is a problematic case to be treated by means of SGA, therefore we make two more approximations. First, we only consider the basis
functions
as
linear
combinations
of
atomic
orbitals
(LCAO
[89])
up
to
the
certain
limit,
that is we require the hopping terms to be only between orthogonal functions. Secondly, we rearrange the Hamiltonian in a way, that intersite Coulomb interaction parameters Kij and ion�ion repulsion renormalize the singleparticle energy Ea, leaVing the remaining terms to be of negligible infuence, when the groundstate energy is calculated, taking adVantage of the oneelectronperlatticesite scenario. Hamiltonian prepared in that way is now tractable with the SGA, thus exchanging the problem of diagonalization the Hamiltonian to the minimization of corresponding Landau functional.
We calculated the groundstate and all its components for Varying lattice parameter (interatomic distance) R Pr3.25, 8s and magnetic feld Ha, and obserVed the frst order
AE
transition at RC “ 4.1a0 (Fig. 1), where the aVerage double occupancy d2 “ nˆiÒnˆiÓ drops discontinuously to zero with the increasing lattice parameter R (Fig. 2) indicating a metal -insulator transition. This claim can be backed up by the diVergence of magnetic susceptibility (Fig. 5) and the disappearance of Hubbard gap Egap “ U´W (Fig. 11), where U is the Hubbard repulsion and W “ 12|t| is the bandwidth. We obtain the microscopic parameters of the Hamiltonian (Fig. 3) in the correlated state (cf. the renormalized inVerse waVefunction size α (Fig. 9)), as well as the Landau functional minimization parameters, the most interesting being magnetization m (Fig. 6) and the molecular feld λm coupled with m (Fig. 4). For the sake of completeness we supply critical scaling of the groundstate energy (Fig. 7) and of the inVerse waVefunction size (Fig. 8).
As an extra result, we assessed the critical pressure to stabilize metallic phase and haVe found it to be „ 100GP a (Fig. 12). As we disregarded the molecular phases, this paper can be treated only as an frst estimate, as there can still be a stable, lowerenergy molecular structure. NeVertheless, eVen in the model case of simple cubic lattice, the metallization under pressure can be obtained.
The paper was published in European Physical Journal B (Eur. Phys. J. B 86, 252 (2013)), pp. 1 9.
3.2 Paper A-2 -Metallization of Atomic Solid Hydrogen within the Extended Hubbard Model with Renormalized Wannier Wavefunctions
This
is
a
followup
paper
to
paper
A1.
In
this
contribution
to
the
XVI
National
Conference
on SuperconductiVity in which we expand the discussion of the most important outcome of that preVious work, namely, the critical pressure of metallization of solid atomic hydrogen crystal. Using the extended Hubbard model, with the singleparticle energy Ea, the nearestneighbor hopping term t, the onsite Hubbard interaction U, and the intersite Coulomb repulsion Kij, we model the metal�insulator transition using better approximation for the singleparticle basis states with p “ 9 instead of p “ 3 Gaussians per waVe function. This is discussed briefy with Gaussian ft results presented in Table 1 and the basis waVefunctions profles displayed in Fig. 1. We show that although the results change slightly quantitatiVely, they do not change qualitatiVely the preVious results (cf. the groundstate energy EG in Fig. 2 and microscopic parameters t, U and K ” Ki,zpiq, where zpiq is the nearest neighbor of site i).
We also discuss a fne outcome of our approach the possibility of comparing both Mott
1{3
and MottHubbard criteria. The frst being nc aB « 0.2, where for critical concentration nc we haVe nc ” R ´3, and efectiVe Bohr radius aB “ α ´1, where RC is the critical Value of
CC lattice parameter R, and αC is the critical Value of the inVerse waVefunction size. This lead
1{3 α ´1
the Mott criterion to be nc aB “ R ´1 C « 0.22, close to the expected Value. Similarly,
C
we obtain the MottHubbard ratio U{W « 1.18, close to the critical Value of unity.
Finally, we assessed the zeropoint motion energy EZPM of the interacting ions in the simple cubic lattice. The zeropoint motion represents the primary efect of nonclassical
`
character of the H ions (protons). As a starting point, we assume a small ion displacement δR, so that only the nearest neighbor contribution to the total energy will change signifcantly. Then, we deriVe the formula for the total energy for such displacement, including estimation of the kinetic energy of an ion through the uncertainty principle δP2δR2 ě 3n2{4. This allowed us to fnd all modes, the one with smallest energy being diagonal. We display the Value of EZPM , as well as its ratio to the groundstate energy in Fig. 5.
The paper was published in Acta Physica Polonica A (Acta Phys. Polon. A 126, 4A (2014)), pp. A58A62.
3.3 Paper A-3 -H2 and pH2q2 molecules with an ab initio optimization of wavefunctions in correlated state: electronproton couplings and intermolecular microscopic parameters
In this paper we utilize the Exact
Diagonalization
Ab
Initio
Approach
(EDABI)
to
describe the H2 molecule
with
more
precise
approach
(in
comparison
to
that
in
paper
A2)
to
assess the zeropoint motion amplitude and corresponding energy, taking the contributions up to the ninth order. We also proVide a full microscopic picture of Van der Waalslike attraction between two H2 molecules, that eVentually led to the onedimensional molecular chain
discussed
in
the
papers
A4
and
A5.
We
also
present
the
methodology
of
including
higher (than 1s) orbitals into the model, thus opening the door for description of diferent atoms that is planned for the near future.
We
start
from
the
full
twosite
Hamiltonian
(2.5),
with
explicit
expressions
for
all
six
microscopic parameters: the singleparticle energy E, the hopping term t, the onsite Hubbard interaction U and the intersite Coulomb repulsion K, the exchange integral J and the socalled correlated hopping V . As we haVe realValued singleparticle basis waVe functions, the socalled pair hoppingterm amplitude is equal to the exchange integral. We calculate explicitly the form of these parameters as a function of inVerse waVefunction size α and the intersite distance R, for the singleparticlebasis waVe functions, composed of 1s Slatertype orbitals (all formulas can be found in Appendix A). Then, for giVen R, we minimize the system groundstate energy EG “ E´ ` 2{R with respect to the inVerse waVefunction size α, where E´ is the exact eigenValue for the Hamiltonian and 2{R is classical Coulomb repulsion of the ions. Within this simple model, we fnd the minimum of EG for RB “ 1.43042a0 and EB “´2.29587Ry, „ 2% from the almost exact result RK´W “ 1.3984a0, EK´W “´2.349Ry of
Kołos
and
Wolniewicz
[107].
The results are presented in Figs. 1�5. The next step is to estimate the zeropoint motion amplitude and energy. For this purpose we consider the oscillations of interionic distance R “ RB ` δR, and: {i} we expand the system energy around minimum, in terms of δR up to the ninth order; {ii} calculate the additional term δH of Hamiltonian, that coVers the change of energy caused by the change δΞ of the microscopic parameters, here labeled generically Via Ξ; these are calculated explicitly (the corresponding formulas can be found in Appendix B) and presented in Figs. 7�8; {iii} obtain explicitly the aVerages of the operator part of Hamiltonian in the groundstate (cf. Fig. 6); {iv} estimate the kinetic energy of the ions and the change of the interionic Coulomb repulsion. We combine steps {i} {iv} and minimize the expression for the total energy of the system with respect to the δR. The result is the zeropoint motion amplitude δR0 “ 0.189a0 and the energy EZPM “ 0.0241Ry.
We supplement the results with the solution of the pH2q2 system with all twosite parameters of interaction (three and foursite terms are of negligible magnitude) and present the results in Figs. 10�14. We obserVe the Van der Waalslike behaVior of the energy diference per molecule ΔEH2 paq“ EpH2q2 {2 ´ EB, with a shallow minimum for the intermolecular distance a “ 4.5a0. In Appendix D, we sketch how to incorporate higher orbitals to the model on the example of onebody microscopic parameters for 1s and 2s Slatertype orbitals.
The paper was published in New Journal of Physics (New J. Phys 16, 123022 (2014)), pp. 1�26. It has also been selected by the Institute of Physics Publishing as one of the papers of the year 2014 (the socalled IOPSelect).
51
3.4 Paper A-4 -Combined shared and distributed memory ab-initio computations of molecular-hydrogen systems in the correlated state: process pool solution and two-level parallelism
In this paper we present the computational scheme of the Exact
Diagonalization
Ab
Initio
Approach
(EDABI)
when
applied
to
realistic
hydrogen
systems.
In particular, we focus on the twoleVel parallelization of the problem of parameterizing the Hamiltonian, employing both shared (by means of Open MultiProcessing OMP) and distributed (by the Message Passing Interface MPI) memory models. We also reView why such elaborated scheme is required by comparing the results for H2 chain with periodic boundary conditions, both for diferent Slatertype orbital Gaussian representation (STONGG) resolution NG, with the socalled background feld size M (the neighborhood of the site, where electronic Coulomb repulsion, ion�ion interaction and electron�ion attraction are taken into account).
Firstly, we describe the Exact
Diagonalization
Ab
Initio
Approach
(EDABI)
for
a
general case, as well as present the procedure of orthogonalization of the singleparticle basis Via the corresponding bilinear form problem. This approach allows for error control for the case of infnite systems. Then we explain in detail the computational scheme, recognizing the calculation of microscopic parameters as the bottleneck of the problem solution and employing the twoleVel parallelism: {i} the process�pool or the root�worker solution, where the root process distributes the workload between the workers; this on the other hand allows, {ii} the second leVel parallelization of the integral calculations at each process (cf. Fig. 1 for the fowchart of the solution).
As a next step, we proVide an exemplary physical system and analyze Validity of the proposed numerical solution. We consider the pH2q3 chain with periodic boundary conditions within the extended Hubbard model, with the three nearest neighbor hoppings ti (Fig. 2) and the intersite electron�electron Coulomb interaction Kij included up to the interionic distance cutof rcut´off “ Ma, where M is a parameter (the socalled background feld size) and a is the intermolecular distance. We fnd the global minimum for a “ 4.12a0 and constant molecular bond length R “ Rfree “ 1.43042a0 (Figs. 3�5). The discussion of the system with optimized molecular size R is
the
subject
of
paper
A5.
Here,
we
study
the dependence of the global energy minimum on the background feld size M (Fig. 6) and Gaussian basis resolution NG (Fig. 7), as well as perform fnitesize scaling to determine the efectiVe Values. We also perform the analysis of the proposed twoleVel parallelism by calculating the socalled speedup (SU) as a function of the number of nodes P and compare it to the Amdahl law (Figs. 8�9). As we exchange only a small amount of data (the Values of microscopic parameters) through the Message Passing Interface, we haVe an almost perfect scaling of speedup with the increasing number of nodes and with 96.97% of the time spent in the parallelized part of the computations, allowing utilization of the Vast amount of computational power on the TERA�ACMIN supercomputer.
As of August 3, 2015 the paper was accepted for publication in Computer Physics Communication and is in press.
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
3.5 PaperA-5 -Discontinuoustransitionofmolecular-hydrogen chain to the quasi-atomic state: Exact diagonalization ab initio approach
This paper represents the frst implementation of the methods deVeloped for the purpose of this Thesis, to the solid molecular hydrogen metallization. Namely, we apply the Exact
Diagonalization
Ab
Initio
Approach
(EDABI)
to
the
case
of
onedimensional
H2 chain under external pressure (applied force). We optimize not only the singleparticle basis waVe functions, but also the molecular size (H2 bond length) R. We start with the extended Hubbard model with the singleparticle energy E, the hopping integrals ti (up to the thirdnearest neighbor), the Hubbard interaction U, and the intersite electron�electron Coulomb interaction Kij included up to the distance of rcut´off “ 250a, where a is the intermolecular distance
(selected
based
on
the
analysis
performed
in
paper
A4).
We fnd, that although there is only one minimum in the thermodynamic potential, connected to the molecular chain, when no force is applied, there are two minima in enthalpy when a sufciently large force („ 5nN) is applied to the molecules which stabilizes the system. Additional stable phase appearing under pressure has a quasiatomic structure (a comparison of electronic densities is aVailable in Fig. 3). EVolution of enthalpy of both minima is presented in Fig. 2, meaning in the regime of low applied force the molecular phase is stable, whereas for large forces quasiatomic chain has the lowest enthalpy. The transition comes with discontinuous change of lattice parameters, including intermolecular distance a (whole eVolution is showed in Fig. 4) and efectiVe (optimized) molecular size Reff (inset in Fig. 2). As of the nature of transition, we calculated the Hubbard ratio U{W (inset in Fig. 3) that discontinuously drops below 1 at the transition, as well as the aVerages C0 and C1 connected respectiVely with intramolecular and intermolecular hopping. These aVerages exhibit an interesting property: they change from the molecular phase case C0 « 1 and C1 « 0 to C1 « C0 in the quasiatomic phase. The principal conclusion of the paper is that the molecular -atomicsolid transition is concomitant with the MottHubbard transition. The eVolution from the molecular to the quasiatomic phase has also been illustrated on the computer animation enclosed to the Thesis (cf. http://th-www.if.uj.edu.pl/ztms/download/ supplementary material/molecular to quasiatomic transition-hydrogen chain.avi)
The paper has been submitted to the Physical ReView B.
83
84
85
86
87
Chapter 4
Summary and Conclusions
The principal aim of this Thesis is to deVelop and implement a reliable method of calculating the physical properties of solid hydrogen system, both in terms of its insulatortometal and moleculartoatomic transition. In
order
to
do
so
we
haVe
successfully
employed
the
Exact
Diagonalization
Ab
Initio
Approach
(EDABI),
both
in
its
original
exact-methodof-diagonalization based
Version
and
the
modifed,
using
the
socalled
the
StatisticallyConsistent
Gutzwiller
Approximation
(SGA).
We
haVe
also
created
the
open
access,
computational C++based library: Quantum Metallization Tools (QMT [44]),
that
proVides
a
generic description of quantummechanical system in terms of EDABI method.
We haVe used EDABI in combination with SGA to model the metallization of atomic hydrogen
(cf.
papers
A1
and
A2).
Rather
than
including
the
electron�electron
Coulomb
interaction explicitly, we haVe used an efectiVe approximation, working only close to the halfflling. As a fnal result, we obtained the metallization pressure „ 100GP a. This is still a rather simplistic model and it requires a more inVolVed approach to make realistic estimate of the critical pressure. This means explicitly to {i} describe the hydrogen as a molecular crystal (in all of phases I, II, III and IV discussed in the introductory Chapters), modify EDABI with a Hamiltonian diagonalization scheme that include {ii} longrange (Coulomb) interactions and {iii} a multiorbital picture, {iv} fnd the proper thermodynamic potential to describe the system under high pressure, {v} assess the zeropoint motion energy and calculate the electron�lattice coupling, and fnally {vi} use the outcome of steps {i}
{v} to study whether, and under which conditions, the system manifests superconducting properties. This path led us to the creation of QMT and
results
presented
in
papers
A3,
A4,
and
A5.
We started {i} with the simplest buildingblock of threedimensional molecular crystals a H2 molecule (cf. paper
A3).
This case was already well described in a static situation
[107],
which
gaVe
us
a
point
of
reference
for
further
deVelopment.
We used the full Hamiltonian with all twosite interaction terms, which in turn led to the results difering by „ 2%, eVen when omitting in the singleparticle basis waVe functions diferent from 1s. For the purpose of {v}, we deVised an approach to asses the zeropoint motion amplitude and energy, as well as electron�proton couplings, resulting in the Value close to that obserVed experimentally. We expanded {i} and
tested
our
computational
approach
in
paper
A4,
where we also made frst attempts to deal with the problem {ii}. This requires a separate study, as the standard Gutzwiller approximation increases the numerical complexity exponentially when the intersite interactions are included. For this purpose, we plan to employ modifed SGA or the socalled diagrammatic expansion for the Gutzwiller waVe function (DEGWF)
[68].
When
the
longrange
interactions
are
included,
the
most
timeconsuming
element of EDABI is the Hamiltonian parametrization, the part of the problem that can be efciently parallelized, thus allowing to use the modern supercomputer infrastructure. We proceeded with {i} by modeling the onedimensional H2 crystal (cf. papers
A4
and
A5).
It has been completed with semirealistic description of H2 chain including optimization of molecular size and renormalization of the singleparticle basis. Treating {iv}, we chose enthalpy, a thermodynamic potential allowing for description with pressure as a proper Variable to analyze the system behaVior. We haVe found two classes of enthalpy minima one connected to the H2 molecular crystal and one connected to 2H quasiatomic phase. MoreoVer, we determine a critical applied force causing the system transition from molecular to quasiatomic structure in a discontinuous (frstorder) manner. This is what we expect to happen when describing the hydrogenic systems. Up to this moment, we used Lanczos algorithm for the diagonalization of parameterized Hamiltonian in the Fock space. The next step will be to use modifed SGA in a way that it includes: {ii} longrange both Coulomb and exchange interaction. As of {iii}, we haVe prepared computational methods for incorporating higher s and p �type orbitals. This is not included in any of the papers here,
but
the
reader
may
refer
to
Sec.
2.2.3
with
the
results
concerning
the
groundstate
and the ionization energies of frst ten elements of the Periodic Table.
Only when steps {i}�{v} are completed, we can approach the superconducting state of these systems. This topic is the fnal goal of the whole project on the metal hydrogen. It is still a long way to go. Nonetheless, few decisiVe results towards this fnal goal haVe been accomplished. First of all, that the metallization can be understood in the terms of MottHubbard localization. In this respect, a successful model has been formulated. Second, the zeropoint motion energy has been estimated in realistic terms. Third, local electron�lattice interaction coefcients haVe been eValuated. The knowledge of these parameters is indispensable to estimate the Value of the superconducting transition temperature. HoweVer, a simple BCSlike estimate will not work, as we haVe shown that eVen in the metallic state the electrons are moderately to strongly correlated (U{W À 1), so there may appear an essential contribution to the pairing coming from the correlations. It would be Very interesting to see if the electron�phonon interaction alone can account for the superconductiVity with such a high transition temperature. The future will tell.
Chapter 5
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