Algorithms for discrete and
continuous quantum systems
Soumyodipto Mukherjee
Submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
under the Executive Committee
of the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2018
c© 2018
Soumyodipto Mukherjee
All rights reserved
ABSTRACT
Algorithms for discrete and continuous quantum systems
Soumyodipto Mukherjee
This thesis is divided into three chapters. In the first chapter we outline a simple
and numerically inexpensive approach to describe the spectral features of the single-
impurity Anderson model. The method combines aspects of the density matrix em-
bedding theory (DMET) approach with a spectral broadening approach inspired by
those used in numerical renormalization group (NRG) methods. At zero temperature
for a wide range of U , the spectral function produced by this approach is found to be
in good agreement with general expectations as well as more advanced and complex
numerical methods such as DMRG-based schemes. The theory developed here is
simply transferable to more complex impurity problems
The second chapter outlines the density matrix embedding methodology in the
context of electronic structure applications. We formulate analytical gradients for
energies obtained from DMET focusing on two scenarios: RHF-in-RHF embedding
and FCI-in-RHF embedding. The former involves solving the small embedded system
at Restricted Hartree-Fock (RHF) level of theory. This serves to check the validity
of the formulas by reproducing the RHF results on the full system for energies and
gradients. The latter scenario employs full configuration interaction (FCI) as a high
level solver for the small embedded system. Our results show that only Hellmann-
Feynman terms, which involve derivatives of one- and two-electron terms in the
atomic orbital basis, are required to calculate energy gradients in both cases. We
applied our methodology to the problem of H10 ring dissociation where the analytical
gradients matched those obtained numerically. The gradient formulation is applicable
to geometry optimization of strongly correlated molecules and solids. It can also be
used in ab initio molecular dynamics where forces on nuclei are obtained from DMET
energy gradients.
In the final chapter we focus on the study of finite-temperature equilibrium prop-
erties of quantum systems in continuous space. We formulate an expansion of the
partition function in continuous-time and use Monte Carlo to sample terms in the
resulting infinite series. Such a strategy has been highly successful in quantum lat-
tice models but has found scant application in off-lattice systems. The Monte Carlo
estimate of the average energy of quantum particles in continuous space subject to
simple model potentials is found to converge with low statistical error to the exact
solutions even when very high perturbation order is required. We outline two ways
in which the algorithm can be applied to more complex problems. First, by drawing
an analogy between the formulation in continuous-time with the discretized, Trotter
factorized version of standard path integral Monte Carlo (PIMC). This allows one to
use the suite of standard PIMC moves to carry out the position sampling required
to obtain the weight of each time configuration. Finally, we propose an alternate
route by fitting the many-particle potential with multidimensional Gaussians which
provides an analytical form for the position integrals.
Contents
List of Figures iii
1 Ground state and spectral properties via Density Matrix embed-
ding 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background and Methodology . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Model and Embedding approach . . . . . . . . . . . . . . . . . 5
1.2.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Peak spectrum and broadening . . . . . . . . . . . . . . . . . 11
1.2.4 Comparison to numerical renormalization-group methods . . . 13
1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Finite-sized SIAM . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.1 Choice of η1 to ensure convergence of delta peaks . . . . . . . 23
2 Analytic gradient method for Density Matrix Embedding Theory 26
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Ground state energy from Density Matrix Embedding . . . . . . . . . 28
2.2.1 The embedding Hamiltonian for molecular systems . . . . . . 28
2.2.2 Calculating the ground state energy . . . . . . . . . . . . . . . 32
i
2.2.3 Self-consistency in DMET . . . . . . . . . . . . . . . . . . . . 34
2.3 Analytical gradients for DMET energies . . . . . . . . . . . . . . . . 36
2.3.1 RHF-in-RHF embedding . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 FCI-in-RHF embedding . . . . . . . . . . . . . . . . . . . . . 38
2.4 Results: Hydrogen ring dissociation . . . . . . . . . . . . . . . . . . . 40
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Continuous-time Quantum Monte Carlo algorithm for thermal equi-
librium properties of continuous-space systems 44
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Path Integral Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 Partition function via imaginary time discretization . . . . . . 47
3.2.2 PIMC sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Continuous-time Quantum Monte
Carlo (CTQMC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 Partition function via continuous-time expansion . . . . . . . 53
3.3.2 CTQMC sampling . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.3 Thermal average energy . . . . . . . . . . . . . . . . . . . . . 58
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.1 Gaussian well . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.2 Harmonic Potential . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 Analogy with standard discrete-time path integral methods . . . . . . 67
3.6 Many-particle problems via Gaussian fitting of Potential . . . . . . . 69
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Bibliography 73
ii
List of Figures
1.1 Comparison of the local density of states between a four-site cluster
DMET calculation and a (six-impurity six-bath) CDMFT + ED calcu-
lation of the half-filled 1-d Hubbard model. Here, η1 = η2 = 0.05 has
been used for both CDMFT + ED and DMET calculations. Figure
taken from Ref. [40] . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Ground-state energy per-site Egs/L (L = 8) as a function of U for the
symmetric SIAM from single-site and cluster embedding. Vk = V =
0.1 and Emb(n) refers to the number of sites included in the fragment. 15
1.3 Double occupancy of the impurity 〈nd↑nd↓〉 for the symmetric SIAM
with the same parameters as used in Fig. 1.2. . . . . . . . . . . . . . 16
1.4 T = 0 single-particle impurity spectral function A(ω) obtained from
one and two-site embedding for the symmetric SIAM with L = 8,
Vk = V = 0.1 and η1 = η2 = 0.005 for different interaction strengths. . 17
1.5 Magnified view of Fig. 1.4 showing details of the high-energy spectral
features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
iii
1.6 T = 0 impurity spectral function obtained from two-site embedding
(Emb(2)) and Chebyshev-MPS for the symmetric SIAM with logarith-
mically discretized conduction band[30]. The total number of sites in
the chain, including the impurity, is 300. Λ = 1.05, Γ = 0.05 and
D = 1. c1 ≈ 0.3 − 0.4 and c2 ≈ 6.8 − 18.8. The different pan-
els correspond to different values of interaction. Bottom right panel
also includes the spectral function obtained from three-site embedding
(Emb(3)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 T = 0 impurity spectral function obtained from single-site embedding
(Emb(1)) for the symmetric SIAM with a linearly discretized conduc-
tion band. 299 conduction band levels are evenly spaced in ω ∈ [−1, 1].
Γ = 0.05 and the range of interactions considered is the same as in
Fig. 1.6. c1 ≈ 0.3− 0.35 and c2 ≈ 6.1− 12.0. . . . . . . . . . . . . . . 20
1.8 Magnified view of the region near ω = 0 for the spectral functions
considered in Fig. 1.6 (top left) and Fig. 1.7(top right) and the half-
width at half-maximum, dHWHM , for the different interactions showing
the exponential narrowing of the Kondo resonance with increasing U
(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.9 E ′n(ω, η1) for n = 4, 5, 6 and different values of η1 at U/Γ = 10. Top
row refers to Emb(1) and bottom row to Emb(2). . . . . . . . . . . . 24
1.10 Peak spectrum for different values of η1 at U/Γ = 10. Top row refers
to Emb(1) and bottom row to Emb(2). . . . . . . . . . . . . . . . . . 24
2.1 The total system is tiled with various fragments. When the circles de-
note orbitals, each fragment is a Fock space of single-particle fragment
orbitals. Figure taken from Ref. [41] . . . . . . . . . . . . . . . . . . 29
2.2 The DMET algorithm flowchart. Figure taken from Ref. [41] . . . . . 35
iv
2.3 Ground-state energy per atom for H10 ring vs the H−H bond dis-
tance r. DMET(1) refers to a fragment with only one hydrogen atom.
DMET(1) and 1-shot DMET(1) refers to DMET variants with and
without self-consistency, respectively. . . . . . . . . . . . . . . . . . . 40
2.4 Analytical energy gradients (Anal.) compared to numerical gradients
(Num.) for RHF-in-RHF (DMET(1)-RHF) and FCI-in-RHF (1-shot
DMET(1)-FCI) embedding without self-consistency. . . . . . . . . . . 41
3.1 CTQMC values of average energy 〈E〉 and standard error for the Gaus-
sian well as a function of T for λ ranging from low to high values. MC
average over 1000 independent trajectories. The exact results are ob-
tained from the numerical solution of the Schrodinger equation in one
dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Variation of CTQMC order distribution with T and λ for the Gaussian
well for a given trajectory with 106 MC steps. . . . . . . . . . . . . . 61
3.3 At T = 0.001, variation of CTQMC 〈E〉 (top panel) and the standard
error σ (bottom panel) as a function of λ for the quantum harmonic
oscillator obtained from 1000 independent trajectories. λ = 0 extrap-
olated answer is 〈E〉MC = 0.5051 ± 0.00847 and the exact analytical
value is 〈E〉exact = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Variation of CTQMC order distribution with λ at T = 0.001 for the
quantum harmonic oscillator for a given trajectory with 106 MC steps. 63
3.5 CTQMC average energy 〈E〉 and standard error as a function of T
for the harmonic oscillator benchmarked against exact analytical so-
lutions. MC average calculated from 1000 independent trajectories. . 65
v
3.6 Schematic of the weight of a time configuration for a single parti-
cle with n = 3 represented as a ring polymer and all the associated
CTQMC moves. Each spring corresponds to a free particle propagator
with an imaginary time equal to the difference in τ ’s with subscripts
corresponding to the bead index. . . . . . . . . . . . . . . . . . . . . 66
vi
Acknowledgements
I have been very fortunate to have interacted and collaborated with exceptional scien-
tists throughout my academic journey, and especially during my time at Columbia.
My graduate advisor, David Reichman, has been a mentor in every sense. I have
imbibed the art of identifying interesting problems and the responsibility of commu-
nicating my results effectively. I have always admired his way of telling a scientific
story that inevitably leaves an impression on the listener. He has been very support-
ive in my future endeavors and has always looked out for my overall well-being. He
ensured his students are happy and I cannot thank him enough for his continuing
support. I thank my committee members for taking out time to examine my thesis
and providing valuable feedback.
It has been a pleasure to have collaborated with my former colleague, Guy Cohen
and I enjoyed the various useful discussions we have had. He has always been willing
to spare time to clariy doubts via discussions in person, and email. I thank Garnet
Chan for discussions around density matrix embedding and quantum many-body
algorithms in general. He has been kind enough to share his computer code which
has often been the starting point of my research work. I also thank Gerald Knizia
for proactively responding to my email inquiries in impeccable detail. This exchange
of emails have certainly cleared some doubts and helped me move forward in my
research.
I must thank my current and former colleagues in the Reichman Group, particu-
larly Andres Montoya Castillo, Hsing-Ta Chen, Liesbeth Janssen, Roel Templaar, and
Ian Dunn for friendship both inside and outside the office. My friends at Columbia
have been key to my experience here in New York. Matthew Mayers and I have
not only been colleagues but also roommates for five years. I am going to cherish
our memories and I cannot imagine my graduate life without him. Tyler Evans has
been a close friend throughout the five years. I always look forward to discussing
experimental research with him and I have been a fan of his sense of humor which
vii
has been a great respite during difficult times in graduate school. I am grateful to
Erick Andrade for keeping up the spirits at all times. Being an international student,
I have found it incredibly easy to feel at home and it could not have been possible
without the support of my peer group and friends in Columbia Chemistry. I grateful
to my fiance, Ishani Roychowdhury, for being there through thick and thin. It has
been far from easy to have lived across continents for the most part of my PhD but
for her determination and belief in our relationship, and her neverending love and
support. The doctoral journey could not have been possible without the uncondi-
tional support of my parents who believed in my abilities at all times. I admire their
patience and am forever grateful for their love and affection.
viii
For my friends and family
ix
Chapter 1
Ground state and spectral
properties via Density Matrix
embedding
1.1 Introduction
Many-body methods in Physics and Chemistry primarily focus on a quantum me-
chanical treatment of many interacting particles[1]. When interactions are not present,
a single-particle picture yields exact information about the ground state and dynam-
ical properties of the system. However, in all practical cases, interactions need to
be considered and a framework to describe the physics of these interacting parti-
cles, both accurately and at a low numerical cost, is of paramount importance. The
discovery of heavy fermion[2] and high-temperature superconductors[3] has revived
the interest in strongly-correlated systems. These are materials where the strength
of the electron-electron interactions is comparable to or greater than the kinetic en-
ergy of the electrons. Single-particle methods like Hartree-Fock Theory[4] fail to
make correct predictions about the properties of these strongly-correlated materials,
particularly, the ones that are driven by the electron-electron interaction like the
1
metal-to-insulator transition (MIT)[5]. In the context of these strongly correlated
solids, quantum impurity models have taken on renewed importance as it provides a
window into the properties of wide range of strongly correlated materials as discussed
below.
Quantum impurity models play a major role in modern condensed matter physics.
As proxies of physical reality, impurity models encapsulate complex physical behav-
ior within the simple framework of a small subsystem hybridized with an otherwise
noninteracting bath. Such models have laid the foundation for our understanding of
qualitative phenomena such as quantum dissipation, namely the generic features of
how a bath induces dephasing and relaxation in a small subsystem, as well as detailed
specific and quantitative phenomena such as the description of magnetic impurities
in metals and the relaxation of tunneling centers in low-temperature dielectric me-
dia[6]. Due to central role played by impurity problems in condensed matter physics,
the development of methods for the the accurate description of their properties, in
particular their real-time or real-frequency behavior, remains at the forefront of the
field.
One of the most important and well-studied impurity models is the Anderson
model[7]. The Anderson Hamiltonian describes an electron-correlated quantum dot
hybridized with a non-interacting bath of fermions. The physics of the Anderson
model subsumes that of the Kondo model, and thus describes the physics of dilute
magnetic impurities in metals, including the appearance of a resistivity minimum and
the eventual formation of a T = 0 singlet state induced by the complete screening of
the impurity spin by the conduction electrons[8].
The need to develop an accurate quantitative treatment for strongly correlated
solids has resulted in the development of dynamical mean-field theory (DMFT)[9–
11], a quantum embedding approach where Hubbard-like models[12–14] are self-
consistently mapped to Anderson-like impurity problems. In this latter regard, the
development of methods to obtain numerical solutions of the Anderson model has re-
2
ceived a lot of interest. DMFT is a Green’s function-based approach which, for many
applications, requires the solution of the spectral function of the Anderson model on
the real-frequency axis. Despite the reduction of complexity afforded by the mapping
from the Hubbard model to the Anderson model, the determination of real-time or
real-frequency spectral behavior is a difficult task. Simple quasi-analytical approaches
like the non-crossing approximation[15–17] are often not sufficiently accurate in the
most interesting physical regimes[18]. Analytic continuation of exact imaginary-time
quantum Monte Carlo data is an ill-conditioned numerical problem which can often
produce artifacts[19–22], while exact diagonalization is restricted to a small num-
ber of bath states, rendering a detailed accurate description of features such as the
high frequency behavior of the spectral function difficult[22, 23]. Quantum chem-
istry methods such as truncated configuration interaction (CI) and complete active
space configuration interaction (CAS-CI) have been employed as impurity solvers
in DMFT[24]. Their accuracy is comparable to exact diagonalization solvers at a
much lower cost. However, they share similar drawbacks due to the finite number of
orbitals used in the CI expansion.
Recently great progress has been made in the formulation and use of time depen-
dent density matrix renormalization group (DMRG) and related approaches[25, 26],
as well as sophisticated renormalization group approaches to study the behavior of the
Anderson model on the real time and frequency axis[27–31]. These matrix product-
based methods have been used as impurity solvers in one and multi-orbital DMFT
calculations to obtain spectral functions of the one and two-band Hubbard models
[32–34]. Such techniques, while powerful, are also theoretically and numerically in-
volved. Since, calculating stationary states is much easier than calculating Green’s
functions, an embedding framework based on a frequency-independent variable would
be potentially easier to work with. On these lines, Density Matrix Embedding The-
ory (DMET) was introduced, where, the infinite bulk problem is self-consistently
mapped to an impurity model which consists of an interacting impurity coupled to
3
a bath which is of the same size as the impurity[35–41]. A significant advantage of
this construction is that it renders the impurity model finite and high-level meth-
ods could be employed to solve this finite impurity problem. The quantum variable
in DMET is the time-independent density matrix. Although, DMET is built on a
time-independent platform, it has been extended to obtain dynamical quantities[40]
as well.
In this chapter we propose a different and extremely simple method for the calcu-
lation of the real-frequency spectral function of the Anderson model. The approach
combines the use a density-matrix embedding theory (DMET) -like decomposition
without self consistency[41] with a simple and physical protocol motivated by some
implementations of the numerical renormalization group (NRG) for assigning line
widths[27, 42]. The results are comparable to those produced by far more sophis-
ticated theoretically and numerically involved approaches. In principle the method
outlined here is extendable to more complex situations such as those that arise in
cluster DMFT where the embedded impurity-like system contains multiple sites or
orbitals.
The chapter is organized as follows: In Sec.1.2 we introduce the single-impurity
Anderson model (SIAM) and briefly discuss the ground state embedding approach
and the calculation of T = 0 spectral functions. We propose a way to obtain the peak
spectrum and a frequency-dependent broadening scheme is employed that generates
smooth spectral functions. Sec.1.3.1 compares the ground state and spectral proper-
ties of a finite-sized SIAM to numerically exact results. In Sec.1.3.2 we analyze the
spectral properties for the bulk case. Sec.2.5 is devoted to our concluding remarks.
4
1.2 Background and Methodology
1.2.1 Model and Embedding approach
The single-impurity Anderson model (SIAM)[7] describes a single interacting im-
purity coupled to a host conduction band of noninteracting fermionic levels. The
Hamiltonian reads
H = Hcb + Himp + Hhyb
=∑kσ
εknkσ +∑σ
(εd +
U
2nd−σ
)ndσ
+∑kσ
Vk(c†dσ ckσ + c†kσ cdσ) , (1.1)
where εk denotes the energy levels of the conduction band, εd the impurity level,
U the interaction on the impurity, ndσ = c†dσ cdσ is the occupancy of the impurity,
nkσ = c†kσ ckσ is the band occupancy and Vk the hybridization between the impurity
and the conduction band. At this stage we defer the discussion of the choice of the
hybridization parameter until later to keep the discussion general.
The approach we employ is based on a simplified version of the density matrix
embedding theory (DMET) and its spectral extension[35–40]. At the heart of the
embedding procedure lies the Schmidt decomposition of the wave function. Formally,
the Schmidt decomposition of a state is given by
|Ψ〉 =mα∑i,j=1
λij |αi〉 |βj〉 , (1.2)
where the states |αi〉 represent the states that span the part of the system of interest,
the fragment and the states |βj〉 represent the states that span the rest of the system,
the bath. mα is the dimension of the fragment space assumed smaller than the bath.
The Schmidt decomposition renders the wavefunction expansion in a compact form.
Even for a complex state, the number of many-body states, |βj〉, required to exactly
5
define |Ψ〉, depends only on the size of the fragment which is generally much smaller
than the bath. The projector P =∑
ij |αiβj〉 〈αiβj| projects the full Hamiltonian
onto the basis obtained from the Schmidt decomposition, Hemb = PHP . If |Ψ〉 were
the exact ground state of the system, Hemb, a much smaller Hamiltonian, would
yield the exact ground-state energy. However, this procedure is purely formal as we
would need to know the exact ground-state wavefunction from the outset, which is
impossible to obtain for large correlated systems.
The mean-field solution for the ground state is the single Slater determinant |Φ(0)〉.
It’s Schmidt decomposition can be obtained from single-particle linear algebra[43, 44]
at a cost no greater than the diagonalization of the single-particle Hamiltonian. It
has the same form as Eq. 1.2, where the many-body fragment state |αi〉 is constructed
from the single-particle states corresponding to the fragment sites (fragment orbitals).
For the case of a Slater determinant, the many-body bath states |βj〉 take a particu-
larly simple form[36]. In particular, they are constructed from a single-particle bath
state (bath orbital) multiplied by a determinant of core electrons. These core or-
bitals have no overlap with the fragment space. Although, there are several ways to
construct the single-particle fragment and bath orbitals[35–39], we have followed the
procedure outlined in Ref. [39].
The single-particle states from which the many-both states |αi〉 and |βj〉 are
constructed define the embedding basis. This new basis generates an active space
spanned by the entangled fragment and bath single-particle orbitals and an inactive
space given by the unentangled environment orbitals which are the doubly occupied
core orbitals and empty virtual orbitals. The single-particle fragment orbitals are
chosen to be the same as the single-particle basis for the sites that include the frag-
ment. In this work, the single-particle bath orbitals, the unentangled core and virtual
orbitals are constructed from the mean-field density matrix which is obtained from
|Φ(0)〉, the RHF ground state solution to the full Hamiltonian in Eq. 1.1. |Φ(0)〉 is
6
given by
|Φ(0)〉 =Nocc∏µ
a†µ |0〉 , (1.3)
where Nocc is the total number of electrons. a† creates a hole (occupied) state |φµ〉
and |0〉 is a bare vacuum. The product runs from 1 to the number of electrons. The
hole creation operators are defined by the underlying Hartree-Fock transformation C
from physical fermions c†
a†µ =∑k∈AB
Ckµc†k , (1.4)
where A and B defined the fragment and rest of the system respectively. The mean-
field density matrix is given by,
Dkl = 〈Φ(0)|c†kcl|Φ(0)〉 =
Nocc∑µ
CkµC†µl . (1.5)
The eigenvalues of this idempotent density matrix are all either 0 or 1. Assuming L
and LA to be the size of the full system and the fragment respectively, we consider the
(L−LA)×(L−LA) subblock where k and l belong to the environmentB only. Dkl(kl ∈
B) is a projector of the environment orbitals onto the occupied orbitals. We have
hence removed the LA rows and columns corresponding to the local fragment. Due to
MacDonald’s theorem[45], at most, LA eigenvalues of the (L−LA)×(L−LA) subblock
will lie between 0 and 1. The corresponding eigenvectors are the orthonormal bath
orbitals. The Nocc−LA eigenvectors with eigenvalue 1 are the unentangled occupied
core orbitals. The empty virtual orbitals are constructed from the null space of the
fragment, bath and core orbitals. Thus the many-body bath states |βj〉 span the same
space as F(|b〉)⊗ det(e1e2 . . . eN−|A|), where det(e1e2 . . . eN−|A|) is the determinant of
pure environment orbitals or the core states. In other words, when split across a
fragment and environment, the determinant |Φ(0)〉 appears as a CAS-CI (complete
active space configuration interaction) expansion in a half-filled active embedding
basis of fragment plus bath orbitals, |f〉 ⊕ |b〉, with a core determinant of pure
7
environment orbitals |e〉; the total number of electrons in the active space is |A|.
The ground-state embedding approach for SIAM is summarized as follows:
1. First, a restricted Hartree-Fock (RHF) calculation which yields a single Slater
determinant, |Φ(0)〉, is used as an approximate ground-state of the SIAM Hamil-
tonian.
2. A set of local fragment site(s), which includes the interacting impurity, is cho-
sen. The Schmidt decomposition of |Φ(0)〉 is performed. Importantly, this
decomposition yields a single-particle basis which is used to construct the inter-
acting embedding Hamiltonian, Hemb. This can be understood as a simple basis
transformation of the full SIAM Hamiltonian from the single-particle fermionic
site basis to the single-particle embedding basis given by the entangled frag-
ment and bath orbitals and the unentangled occupied core and empty virtual
orbitals.
3. Hemb is easily diagonalized to obtain an approximate correlated ground state
wavefunction, |Ψemb〉. All expectation values are calculated using 〈Ψemb|O|Ψemb〉.
It is to be noted that |Ψemb〉 has the same form as the Schmidt decomposed |Φ(0)〉,
the only difference being that |Ψemb〉 is obtained after including the correlations on
the fragment whereas the latter included them in a mean-field way.
For the SIAM, there is no self-consistency as the interactions are localized on the
impurity and the bath states made out of the conduction band levels is, by default,
noninteracting. At T = 0, µ = 0 in the SIAM Hamiltonian in Eq. 1.1 maintains
half-filling on the impurity. Since the two-electron Coulomb term appears only on
the impurity in Hemb, µ = 0 still ensures the required half-filling. In this respect the
procedure outlined is a ‘single-shot’ embedding analogous to the u = 0 case discussed
in [39].
A translationally-invariant system like the Hubbard model can be divided into
identical fragments each of which is embedded in its own bath. Once the self-
consistency has been achieved, the fragment’s contribution to the ground state energy
8
is evaluated. The total ground state energy is a sum of such identical contributions
from each of the fragments. For SIAM, a non-translationally-invariant system, we
have only one fragment which includes the impurity site and its corresponding bath.
The contribution of the unentangled core orbitals to the total ground state energy
cannot be neglected in this case. Performing DMET with this unique partitioning
also results in an approximate correlated wavefunction for the full system which in a
‘single-shot’ gives variational estimates of the ground state energy.
1.2.2 Dynamics
The quantum embedding approach has been generalized to dynamic properties, par-
ticularly for the evaluation of bulk spectral functions[40, 46]. Here, we apply the
method to evaluate the single-particle, local density of states (LDOS) for the SIAM
and provide extensions to the current formulation. The technique builds on the
ground-state formalism where a bath space of frequency-dependent many-body states
is constructed by the Schmidt decomposition of the linear response vector given by
|Φ(1)(ω, η1)〉 =1
ω − (h− ε0) + iη1
V |Φ(0)〉 , (1.6)
where h is the single-particle Hamiltonian with ε0 it’s ground-state energy. V = c(†)d
is used for the evaluation of the impurity LDOS. |Φ(1)(ω, η1)〉 can be expressed in a
Schmidt decomposed form [40]
|Φ(1)(ω, η1)〉 =∑ijm
λijA(m)(ω, η1) |αi〉 B(m)(ω, η1) |βj〉 , (1.7)
where A(m)(ω, η1) and B(m)(ω, η1) are operators that act on the fragment and bath
states, respectively. This defines a frequency-dependent basis given by
|Kijm(ω, η1)〉 = |αi〉 ⊗ B(m)(ω, η1) |βj〉 (1.8)
9
and the corresponding projector
P =∑ijm
|Kijm(ω, η1)〉 〈Kijm(ω, η1)| . (1.9)
Defining H ′(ω, η1) = P(H − Egs1
)P , where Egs is the energy corresponding
to the frequency-independent ground state |Ψemb〉 discussed in the previous section.
The approximate embedding Green’s function is given by
G(ω, η1, η2) = 〈Ψemb| X ′1
ω − H ′(ω, η1) + iη2
V ′ |Ψemb〉 , (1.10)
where X = V †, X ′ = PXP and V ′ = PV P gives the single-particle impurity Green’s
function with it’s imaginary part as the single-particle density of states, namely
A(ω, η1, η2) = − 1
π=[G(ω, η1, η2)] . (1.11)
It should be noted that the formulation can be extended to more complex two-particle
Green’s functions with an appropriate choice for V . For example, V =∑
σ = c†iσ ciσ
is used to evaluate the local density-density response function[40].
This methodology generates a Lorentzian broadened spectral function dependent
on the parameters η1 and η2. A common choice is to set η1 = η2[27, 40]. This choice
might not always be desirable, especially for a discretized SIAM, where appropriate
broadening of the peak spectrum leads to better resolution of the spectral function at
both high and low energies[42, 47, 48]. For example, in a previous work, the spectral
function obtained for the one-dimensional Hubbard model from Ref. [40] using the
embedding approach just outlined is shown in Fig. 1.1. These spectral functions
have been generated using η1 = η2 = 0.05. This choice, although reproduces the
gaps in the spectral functions at various interaction strengths U accurately, fails
to reproduce the high energy spectral features when compared to a more accurate
(although not exact) cluster DMFT + ED calculation[49]. This emphasizes the need
10
Figure 1.1: Comparison of the local density of states between a four-site clusterDMET calculation and a (six-impurity six-bath) CDMFT + ED calculation of thehalf-filled 1-d Hubbard model. Here, η1 = η2 = 0.05 has been used for both CDMFT+ ED and DMET calculations. Figure taken from Ref. [40]
for a more detailed construction of spectral broadening to be able to benchmark
against numerically exact results at both high and low energies.
1.2.3 Peak spectrum and broadening
The size of the frequency-dependent Schmidt basis generally makes it manageable
to perform full exact diagonalization (ED) on H ′(ω, η1) giving access to its entire
eigenspectrum. This allows one to express Eq. 1.10 in an explicit Lehmann represen-
tation. Since the functional form of the eigenvectors, |Ψ′n(ω, η1)〉 and the eigenvalues
E ′n(ω, η1) of H ′(ω, η1) are not known, η1 is chosen to be very small to obtain results
independent of it. The value of η1 determines the number of delta peaks observed
when calculating the peak spectrum under the following guideline. Hence, for all the
results in Sec.1.3.2, we have used η1 = 10−8 which ensured the convergence of the
number of delta peaks. For a more detailed discussion on the behavior of E ′n(ω, η1) for
different values of n and η1, see Appendix 1.5.1. The single-particle Green’s function
11
is given by
G(ω, η2) =∑n
| 〈Ψ′n(ω)|c†d|Ψemb〉 |2
ω − E ′n(ω) + iη2
, (1.12)
where we have used
H ′(ω) =∑n
E ′n(ω) |Ψ′n(ω)〉 〈Ψ′n(ω)| . (1.13)
If ω − E ′n(ω) = 0 has m solutions given by ω(n)m , the denominator of the imaginary
part of G(ω, η2) in Eq.1.12 can be approximated using a Taylor series around ω(n)m
up to second order in ω. Integrating over all frequencies gives the spectral function
in terms of delta functions at the peak positions and it’s corresponding weights,
A(ω) =∑mn
Amnδ(ω − ω(n)m ) , (1.14)
and the spectral weight is given by
Amn =|Cn(ω
(n)m )|2∣∣∣∣1− dE′n
(ω(n)m
)dω
∣∣∣∣ , (1.15)
where |Cn(ω)|2 = | 〈Ψ′n(ω)|c†d|Ψemb〉 |2. The peak spectrum is convoluted with a
Gaussian kernel[27, 47, 48]
K(ω, ω′) =1
b√π
exp[−(ω − ω′)2/b2
]. (1.16)
The width, b is chosen to be frequency dependent [27, 42] and of the form
b = c1ωmin + c2|ω′| , (1.17)
where ωmin is the position of the lowest lying peak above the Fermi energy, and c1 and
c2 are positive constants. We delay our discussion on how to appropriately choose
12
these constants until Sec.1.3.2. The smooth spectral function is then given by
A(ω) =
∞∫−∞
K(ω, ω′)A(ω′) (1.18)
=∑mn
Amnb√π
exp[−(ω − ω(n)
m )2/b2(ω(n)m )], (1.19)
with Amn obtained from Eq.1.15.
1.2.4 Comparison to numerical renormalization-group meth-
ods
Since the first applications to the SIAM[50], non-perturbative approaches like the nu-
merical renormalization-group method(NRG)[51] have been successful in describing
both static thermodynamic properties and dynamical response and spectral func-
tions[8, 47]. These approaches have seen the introduction of a variety of techniques
aimed at improving accuracy and resolution in both the high and low frequency re-
gions [42, 47]. NRG relies on a logarithmic discretization of the conduction band
which maps the Hamiltonian of Eq. 1.1 onto a chain geometry. The mapping is ana-
lytically performed with the energy levels of the conduction band arranged according
to εn = ±DΛ−n, where D is the conduction band edge and Λ > 1 is the discretization
parameter. The mapping is exact in the limit Λ→ 1. For a conduction band with con-
stant hybridization Vk = V and a flat density of states ρ(ω) =∑
k δ(ω − εk) = 1/2D
for ω ∈ [D,D], the mapping leads to the chain Hamiltonian[51]
H =∑σ
(εd +
U
2nd−σ
)ndσ + V
∑σ
(c†dσ c1σ + H.c
)+
∞∑σ,n=1
tn(c†nσ cn+1σ + H.c
), , (1.20)
13
with Γ = πV 2ρ(0) and
tn =DΛ−n/2 (1 + Λ−1) (1− Λ−n−1)(2√
(1− Λ−2n−1) (1− Λ−2n−3)) . (1.21)
In NRG, the Hamiltonian in Eq.1.20 is iteratively diagonalized increasing the number
of sites from the impurity at each iteration and truncating the high-energy states if
the number of states in the Fock space exceeds a certain chosen limit.
More recent methods such as DMRG and related approaches[26, 52] use a vari-
ational optimization to obtain the ground state, allowing for feedback from lower
to higher energies, which are absent in the NRG approach. DMRG-like methods
allow for an arbitrary discretization of the conduction band. This has the advan-
tage of improving the spectral resolution at high energies by incorporating a linear
discretization instead of a logarithmic one which has fewer states at high energies.
In NRG calculations, the discretization parameter ranges from Λ = 1.5 to 2.5[47].
Lower values of Λ imply that more states are retained in each NRG iteration, making
it computationally challenging with increasing number of iterations. This is not the
case for DMRG or MPS-based methods where values as low as Λ = 1.05 have been
used[30]. These advantages of DMRG, however, come at the cost of the loss of direct
access to the full spectrum of excited states.
The embedding scheme proposed here is similar to DMRG or MPS-like methods
in terms of the advantages it bears, but is far simpler. If a logarithmic discretization
is employed, although not necessary, the infinite chain in Eq.1.20 may cut-off at a
finite N . The process of obtaining the spectral functions starts with the evaluation
of the embedded ground-state. In this chapter, we have used N ∼ 300. Such large
values can be handled easily because the embedded system that ultimately needs
to be fully diagonalized is independent of N , and only depends on the size of the
fragment embedded.
14
0 2 4 6 8 10
U/V
−0.59
−0.57
−0.55
−0.53
Eg
s/L
EDUHFRHFEmb(1)Emb(2)Emb(4)
Figure 1.2: Ground-state energy per-site Egs/L (L = 8) as a function of U for thesymmetric SIAM from single-site and cluster embedding. Vk = V = 0.1 and Emb(n)refers to the number of sites included in the fragment.
1.3 Results
1.3.1 Finite-sized SIAM
In order to assess the performance of the prescribed embedding method for the SIAM,
we first investigate a small-sized system and the results are compared against mean-
field approaches like RHF and UHF and exact results obtained from Exact Diago-
nalization (ED). For this case, the SIAM is represented by an interacting impurity
coupled to seven noninteracting conduction band levels (L=8) at half-filling. We
consider the symmetric case with εd = −U/2. The seven conduction band levels are
evenly spaced on [−1, 1]. The hybridization energy from the impurity to the conduc-
tion band is taken to be same for all the conduction band levels with Vk = V = 0.1.
Fig.1.2 shows the ground-state energy per-site Egs/L as a function of U . The various
levels of embedding depend on the number of sites included in the fragment. Emb(n)
refers to a fragment with n sites which is coupled to a bath of the same size. The
cost amounts to solving a 2n sized system which is achieved via ED for small n. For
15
0 2 4 6 8 10
U/V
0.06
0.10
0.14
0.18
0.22
0.26
〈nd↑
n d↓〉
EDEmb(1)Emb(2)Emb(4)
Figure 1.3: Double occupancy of the impurity 〈nd↑nd↓〉 for the symmetric SIAMwith the same parameters as used in Fig. 1.2.
n = 1, only the impurity is part of the fragment. For higher values of n, the fragment
consists of the impurity and the n − 1 conduction band levels closest to the Fermi
energy.
The embedding approach performs better than mean-field methods like RHF and
UHF for the entire range of U . This is expected as the embedding improves over
RHF by taking into account the interactions on the impurity explicitly. Even Emb(1)
which is equivalent to solving a two-site system is significantly better than standard
mean-field methods. Higher levels of embedding systematically improve the energies
becoming numerically exact when n = L/2. This can be seen in Emb(4) which
reproduces the ED result at the same cost as doing the ED calculation.
Fig.1.3 shows the double occupancy of the impurity 〈nd↑nd↓〉 as a function of U .
Emb(1) does not capture the correct curvature due to the lack of short-range pairing
in the small embedded system. However, cluster embedding generates systematic
improvements over the single-site case. Emb(2) is visibly indistinguishable from the
exact answer while Emb(4) is numerically exact, as expected.
Fig.1.4 compares the spectral functions obtained from different levels of embed-
16
0
10
20
A(ω
)
EDEmb(1)Emb(2)
-1 -0.5 0 0.5 1
ω
0
10
20
A(ω
)
-1 -0.5 0 0.5 1
ω
U/V = 0 U/V = 2.5
U/V = 5.0 U/V = 7.5
Figure 1.4: T = 0 single-particle impurity spectral function A(ω) obtained fromone and two-site embedding for the symmetric SIAM with L = 8, Vk = V = 0.1 andη1 = η2 = 0.005 for different interaction strengths.
0
2
4
6
8
A(ω
)
EDEmb(1)Emb(2)
0.2 0.4 0.6 0.8
ω
0
2
4
6
8
A(ω
)
0.2 0.4 0.6 0.8
ω
U/V = 0 U/V = 2.5
U/V = 5.0 U/V = 7.5
Figure 1.5: Magnified view of Fig. 1.4 showing details of the high-energy spectralfeatures.
17
ding approximations with exact results from ED for varying strength of interaction,
U . The Lorentzian broadened spectral functions are obtained from Eq. 1.11 with
η1 = η2 = 0.005. The embedding results are exact in the noninteracting case, as
expected, for both single-site and cluster embedding as seen in the top left panel of
Fig.1.4. Both Emb(1) and Emb(2) correctly predict the positions of the low energy
excitations. As U increases, the weight of excitations around the Fermi energy de-
creases and peaks at higher energies are observed. Emb(2) displays this general trend
whereas Emb(1) shows some oscillating behavior for the weight of these low energy
excitations. Emb(2), as expected, is able to resolve features in the range 0 ≤ ω ≤ 0.4
better than Emb(1) as seen in Fig. 1.5. The accuracy of Emb(2) is high at low values
of U . At high values of U , particularly at U/V = 7.5, Emb(2) is able to resolve the
positions of some of the peaks in this range but does not quantitatively capture the
weights of these excitations.
The systematic improvement with cluster embedding and its promising perfor-
mance on small systems motivates the application of the technique to systems in the
thermodynamic limit.
1.3.2 Thermodynamic limit
For a single impurity coupled to a continuous conduction band, we take a conduction
band with D = 1 and use a logarithmic discretization with 299 conduction band
states distributed in the interval [−D,D]. The discretization parameter is taken as
Λ = 1.05 and the hybridization as Γ = 0.05. Fig. 1.6 shows the T = 0 impurity
spectral function using two-site embedding in the wide-band limit D Γ, where in
each panel we have considered different values of the interaction strength from the
weak to strong-coupling regime. Our results are compared to calculations using the
Chebyshev-MPS method in the wide-band limit where the computed Chebyshev mo-
ments are post-processed with linear prediction[30]. These results achieve similar pre-
cision as the dynamical density matrix renormalization group method(DDMRG)[53]
18
0
0.2
0.4
0.6
0.8
1
πΓ
A(ω
)
MPSEmb(2)
-1 -0.5 0 0.5 1
ω
0
0.2
0.4
0.6
0.8
1
πΓ
A(ω
)
-1 -0.5 0 0.5 1
ω
U/Γ = 2 U/Γ = 6
U/Γ = 10 U/Γ = 14 MPSEmb(2)Emb(3)
Figure 1.6: T = 0 impurity spectral function obtained from two-site embedding(Emb(2)) and Chebyshev-MPS for the symmetric SIAM with logarithmically dis-cretized conduction band[30]. The total number of sites in the chain, including theimpurity, is 300. Λ = 1.05, Γ = 0.05 and D = 1. c1 ≈ 0.3− 0.4 and c2 ≈ 6.8− 18.8.The different panels correspond to different values of interaction. Bottom right panelalso includes the spectral function obtained from three-site embedding (Emb(3)).
19
0
0.2
0.4
0.6
0.8
1
πΓ
A(ω
)
MPSEmb(1)
-1 -0.5 0 0.5 1
ω
0
0.2
0.4
0.6
0.8
1
πΓ
A(ω
)
-1 -0.5 0 0.5 1
ω
U/Γ = 2 U/Γ = 6
U/Γ = 10 U/Γ = 14
Figure 1.7: T = 0 impurity spectral function obtained from single-site embedding(Emb(1)) for the symmetric SIAM with a linearly discretized conduction band. 299conduction band levels are evenly spaced in ω ∈ [−1, 1]. Γ = 0.05 and the range ofinteractions considered is the same as in Fig. 1.6. c1 ≈ 0.3−0.35 and c2 ≈ 6.1−12.0.
at a lower cost, and serves as a good benchmark for low-cost embedding methods.
The delta peaks and their corresponding weights are obtained from Eq. 1.14 and
Eq. 1.15 respectively. As discussed in Sec. 1.2.3, the frequency-dependent Gaussian
broadening kernel has two parameters, c1 and c2 (see Eq. 1.17). At high values
of U , the resolution of the Hubbard satellites is almost entirely governed by c2 as
c1 primarily affects the low energy resolution. As a result, c2 is chosen such that
for strong interactions the high-energy Hubbard satellites have a width of order 2Γ
as predicted from strong-coupling results[8]. c2 ≈ 0.3 to 0.4 is sufficient to ensure
this. With c2 fixed for all interactions, c1 is chosen such that the spectral functions
essentially recover the Friedel sum rule, namely A(0) ' 1/πΓ. In a simple Anderson
model the Friedel sum rule and the parameter Γ which governs the width of the
Hubbard bands for strong interactions are both readily available. For more complex
Anderson-like models information via a Hartree-Fock calculation in the limit of large
20
-0.1 -0.05 0 0.05 0.1
ω
0
0.2
0.4
0.6
0.8
1
πΓ
A(ω
)
Emb(2)U/Γ = 2U/Γ = 6U/Γ = 10U/Γ = 14
-0.1 -0.05 0 0.05 0.1
ω
0
0.2
0.4
0.6
0.8
1Emb(1)
2 6 10 14
U/Γ
-5
-4.5
-4
-3.5
-3
-2.5
ln(d
HW
HM
)
Emb(1)Emb(2)linear fit
Figure 1.8: Magnified view of the region near ω = 0 for the spectral functionsconsidered in Fig. 1.6 (top left) and Fig. 1.7(top right) and the half-width at half-maximum, dHWHM , for the different interactions showing the exponential narrowingof the Kondo resonance with increasing U (bottom).
U for the Hubbard bandwidth and generalizations of the Friedel sum rule should
still be obtainable. This implies a general applicability of the proposed broadening
scheme.
Fig. 1.6 shows the transfer of spectral weight from low to high-energies at in-
termediate to strong interactions which marks the canonical distinct features of the
SIAM at strong coupling and T = 0, namely the upper and lower Hubbard satellites,
and a zero-frequency peak, the Abrikosov-Suhl or Kondo resonance. A close exam-
ination of Fig. 1.6 shows that at weak interactions, the two-site embedding method
(Emb(2)) overestimates the transfer of spectral weight from the Kondo resonance to
the Hubbard bands. However, this is mostly a broadening artifact as the parameter
c2, which was optimized to resolve the Hubbard bands accurately, now affects the
low energy resonance. A larger value of c2 would remedy this defect. However, this
would make the proposed broadening scheme inconsistent. For strong interactions
21
(U/Γ = 14), the peaks of the Hubbard bands are located at ω ≈ ±U/2 which would
appear to produce an improvement over the Chebyshev-MPS results. The bottom
right panel of Fig. 1.6 shows the spectral function obtained with a three-site frag-
ment, Emb(3), using the impurity, the site adjacent to the impurity and the terminal
site of the Wilson chain as part of the fragment. The results show that the position of
the Hubbard band peaks converge to ω = ±U/2 for high values of U with increasing
fragment size, n.
Fig. 1.7 shows single-site embedding results for the symmetric SIAM with a lin-
early discretized conduction band where 299 conduction band levels are uniformly
distributed in ω ∈ [−D,D]. We have considered the wide-band limit analogous to
previous two-site embedding calculations shown in Fig.1.6 with Γ = 0.05 and D = 1.
With increasing interactions, the narrowing of the Kondo resonance and the appear-
ance of the Hubbard bands is observed. We have used the same broadening scheme to
convolute the delta peaks in Fig. 1.6 and Fig. 1.7. For strong interactions (U/Γ = 14),
the Hubbard band peak is located at slightly higher energies than ±U/2. For the
intermediate coupling regime (U/Γ = 6), the Hubbard bands seem to be slightly
overdeveloped compared to Emb(1) in Fig. 1.6 and MPS results. For weak interac-
tions (U/Γ = 2), however, the Emb(1) spectra appears to be more accurate than
Emb(2).
The width of the Kondo resonance decreases exponentially with increasing inter-
action. This trend is captured by both Emb(1) and Emb(2) as seen in Fig. 1.8 where
the region near ω = 0 is combined for all four panels of Fig. 1.6 and Fig. 1.7 and
the half-width at half-maximum, dHWHM is plotted on a log scale for the values of
interaction considered in Fig. 1.6 and Fig. 1.7. For Emb(2), the width of the Kondo
peak is somewhat overestimated in the intermediate-coupling regime. For Emb(1),
the narrowing of the Kondo resonance is not as sharp as the two-site case but still
follows the correct trend. For all the spectral functions, the Friedel sum rule is nearly
trivially obeyed with deviations below 1% with an appropriate choice of c1. This con-
22
dition, being imposed, does not automatically reflect the accuracy of the low-energy
features obtained from the embedding method.
1.4 Conclusion
In this chapter we outline and investigate a very simple embedding approach for cal-
culating the spectral properties of the single impurity Anderson model. The method
essentially combines features from density matrix embedding theory (DMET) with
discretization and broadening schemes adopted from numerical renormalization group
(NRG) approaches. We advocate a ‘single-shot’ embedding scheme whereby no self-
consistency is required, even for cases where multiple bath sites are included in the
embedding. The approach provides results compatible with much more advanced
DMRG-based algorithms at a fraction of the numerical complexity and expense.
The approach discussed in this chapter should find useful application as an im-
purity solver for DMFT, as a prior for analytical continuation of imaginary-time
quantum Monte Carlo, and for investigation of the spectral features of more com-
plex impurity problems in their own right. With respect to the latter, it should be
noted here that much more complex impurity problems, such as those with multiple
orbitals and coupled sites are amenable to the method discussed here with limited
additional expense. Future work will be devoted to study of both more complex im-
purity problems such as multi-orbital cases that emerge in DMFT as well as testing
our approach on the more subtle and intricate two-impurity Anderson model.
1.5 Appendix
1.5.1 Choice of η1 to ensure convergence of delta peaks
In this short appendix we provide more details concerning the determination of broad-
ening parameters and the sensitivity of the spectra to these parameters. The value
23
0.0
0.2
0.4
0.6
0.8
1.0
1.2
En(ω,η
1)
n=4n=5n=6
−1.0−0.5 0.0 0.5 1.0
ω
0.0
0.1
0.2
0.3
0.4
0.5
0.6
En(ω,η
1)
−1.0−0.5 0.0 0.5 1.0
ω
−1.0−0.5 0.0 0.5 1.0
ω
−1.0−0.5 0.0 0.5 1.0
ω
η1 = 0.05
η1 = 10−6 η1 = 10−8 η1 = 10−10
η1 = 0.05 η1 = 10−6 η1 = 10−8 η1 = 10−10
Figure 1.9: E ′n(ω, η1) for n = 4, 5, 6 and different values of η1 at U/Γ = 10. Top rowrefers to Emb(1) and bottom row to Emb(2).
−1.0 −0.5 0.0 0.5 1.0−1.0 −0.5 0.0 0.5 1.0−1.0 −0.5 0.0 0.5 1.0−1.0 −0.5 0.0 0.5 1.0
−1.0 −0.5 0.0 0.5 1.0
ω
−1.0 −0.5 0.0 0.5 1.0
ω
−1.0 −0.5 0.0 0.5 1.0
ω
−1.0 −0.5 0.0 0.5 1.0
ω
η1 = 0.05η1 = 10−6 η1 = 10−8 η1 = 10−10
η1 = 0.05 η1 = 10−6 η1 = 10−8η1 = 10−10
Figure 1.10: Peak spectrum for different values of η1 at U/Γ = 10. Top row refersto Emb(1) and bottom row to Emb(2).
24
of η1 determines the number of delta peaks in the peak spectrum obtained from
the Lehmann representation of the single-particle Green’s function, see Eq. 1.10 and
Eq. 1.12. Fig. 1.9 shows the behavior of En=4,5,6(ω, η1) for different values of η1. For
Emb(1), En(ω, η1) is not sensitive below η1 = 10−8 while, for Emb(2), η1 = 10−6
ensures convergence for En(ω, η1). The choice of η1 is primarily determined by ex-
amining the peak spectrum in Fig. 1.10. The structure and the number of peaks in
Fig. 1.10 converges for η1 = 10−8 which is the value used throughout this chapter.
25
Chapter 2
Analytic gradient method for
Density Matrix Embedding Theory
2.1 Introduction
The variation of molecular properties based on its nuclear geometries is one of the
most widely studied problems in quantum chemistry. The total energy of the molec-
ular system determines a range of macroscopic properties including structure and
reactivity. Ab initio electronic structure theory is primarily devoted to finding the
best estimate of the electronic energy of these molecular systems from first principles
quantum mechanics[4]. This coupled with classical or quantum molecular dynam-
ics provides a rigorous numerical framework to understand macroscopic behavior in
chemical systems at the quantum mechanical level[54]. In spite of such promise, ac-
curate quantum mechanical treatment of molecular systems come with the burden
of high computational cost. The electron correlations play a central role in creating
this computational burden. For most chemical or biological applications a single-
particle or mean-field picture, that doesn’t explicitly handle electron correlations, is
sufficient. However, in strongly correlated electron systems, the single-particle ap-
proximation breaks down and electron correlations need to be included beyond the
26
mean-field level[11]. Strongly correlated electronic systems are typically molecules
or solids where the molecular geometry is very sensitive to changes in the electronic
energy[55]. This leads to interesting phase changes in these materials, such as typi-
cally seen in high-temperature superconducting materials[3, 55]. Much of Chapter1
was devoted to developing a computationally efficient and robust method based on
density matrix embedding to evaluate electronic properties of prototypical model
systems displaying strong electron correlations. In this chapter we study molecular
systems focusing on it’s ground state electronic energy which directly determines its
geometry.
Geometry optimization in molecules is typically done with the use of analytical
gradients[56]. Analytical gradients refer to formulas for evaluating the gradient of
the electronic energy with respect to nuclear coordinates. It is the most practical
way of evaluating the force exerted on the nuclei due to the electrons. Energy gra-
dients can be evaluated numerically by calculating energy differences. However, the
direct calculation of the energy derivatives offers two main advantages: increased
computational efficiency and increased numerical precision. For a system with N
atoms, a difference method would roughly need 2N energy calculations to calculate
the gradient in all independent directions. Analytical formulas for energy derivatives
can achieve that in a single energy calculation. This is particularly useful in ab initio
molecular dynamics when the electronic energy and its gradient is required at every
time-step to propagate the nuclei.
In this chapter we formulate the gradient of electronic energies obtained from
density matrix embedding theory (DMET). DMET, as introduced in it’s simpler
version in Chapter 1, is a systematically improvable electronic structure method that
captures strong electron correlations in a computationally efficient manner [35–41].
As a result, it’s derivatives would find practical use in the study of strongly correlated
systems. Sec. 2.2 outlines the methodology to evaluate the ground state energy of
molecular systems via density matrix embedding. The self-consistent optimization of
27
a correlation potential is outlined in Sec. 2.2.3 which leads to more accurate energy
estimates. Analytical gradients are introduced for RHF-in-RHF embedding in Sec.
2.3.1. It is the case when the small embedded system is solved at the Restricted
Hartree-Fock level of theory. This reproduces the energies obtained from a RHF
calculation on the full system. RHF-in-RHF gradients serve to compare our results
to those obtained numerically or from RHF analytical gradients and to check the
validity of the formulas. Sec. 2.3.2 outlines the more practical version, that is, FCI-
in-RHF embedding which refers to the case when the embedded system is solved
at full configuration interaction (FCI) level of theory. In both these cases, only the
Hellmann-Feynman terms, that is, derivatives of the one- and two-electron terms in
the atomic orbital basis are required to calculate the energy gradients. Sec. 2.4 is
devoted to the application of different variants of DMET and its gradient formulas
to a molecular system with strong correlations, the H10 ring. Sec. 2.5 is devoted to
our concluding remarks.
2.2 Ground state energy from Density Ma-
trix Embedding
2.2.1 The embedding Hamiltonian for molecular systems
The Hamiltonian for a molecular system is given by
H = Enuc +∑kl
tkla†kal +
1
2
∑klmn
(kl|mn)a†ka†manal , (2.1)
where tkl and (kl|mn) are the one- and two-electron integrals in the atomic orbital
basis. tkl and (kl|mn) are diagonal in the spin indices k and l and, (kl|mn) is diagonal
in the spin indices m and n as well.
The DMET process starts with dividing the total system into various fragments
28
Figure 2.1: The total system is tiled with various fragments. When the circlesdenote orbitals, each fragment is a Fock space of single-particle fragment orbitals.Figure taken from Ref. [41]
as seen in Fig. 2.1. For each fragment Ax, a Hermitian single-particle operator ux is
introduced. ux solely acts within the fragment Ax.
ux =
LAx∑kl
uxklak†al , (2.2)
where LAx is the number of orbitals in the fragment Ax. The sum of all these
operators form the DMET correlation potential u.
u =∑x
ux . (2.3)
The Hamiltonian H + u is solved using a low-level or a mean-field method, such
as RHF, which yields a single Slater determinant, |Φ(0)(~u)〉 as the ground state
|Φ(0)(~u)〉 =Nocc∏µ
a†µ |0〉 , (2.4)
where Nocc is the total number of electrons, ~u denotes all the variables in u and a†µ
29
creates a RHF spin-orbital |φµ〉 and |0〉 is a bare vacuum. The RHF orbital creation
operators are defined by the underlying Hartree-Fock transformation C. The RHF
orbitals are a linear combination of atomic orbitals.
a†µ =L∑k
Ckµa†k , (2.5)
where L is the total number of atomic orbitals or the total number of atoms in
a minimal basis context. The subscripts k, l,m, n and µ, ν, ρ, σ denote indices in
the atomic orbital and RHF orbital basis, respectively. The mean-field one-particle
density matrix follows from the RHF solution. It is given by
Dkl = 〈Φ(0)(~u)|a†kal|Φ(0)(~u)〉 =
Nocc∑µ
CkµC†µl . (2.6)
The Schmidt decomposition of |Φ(0)(~u)〉 allows the construction of the embedding
space which forms a Complete Active Space (CAS) ansatz with:
1. LA fragment orbitals and and equal number of bath orbitals that make up the
active space.
2. Nocc − LA frozen core orbitals without any overlap on the fragment.
3. L−Nocc−LA unoccupied virtual orbitals without any overlap on the fragment.
The construction of the single-particle orbitals that define the embedding basis is
outlined in Sec. 1.2.1. It follows from single-particle linear algebra and scales poly-
nomially with respect to the number of basis functions. The orbitals of the embed-
ding basis denoted by subscripts p, q, r, s can be expressed as a linear combination of
atomic orbitals similar to Eq. 2.5 with
a†p =L∑k
Ckpa†k . (2.7)
All variables in ~u influence the bath orbitals for each fragment Ax. The electrons
in the the half-filled active space interact with the external frozen core electrons
30
through Coulomb and exchange terms and this interaction is incorporated while
constructing the embedding Hamiltonian. Since the core orbitals are frozen and
doubly occupied, their interaction with the active space electrons can be expressed
via a Fock contribution as seen in RHF theory. The embedding Hamiltonian, Hxemb,
is a much smaller Hamiltonian defined in the active space for each fragment Ax. It
is given by
Hxemb =
LAx+LBx∑pq
(tpq + f core,x
pq
)a†paq +
1
2
LAx+LBx∑pqrs
(pq|rs)a†pa†rasaq
+ µglob
LAx∑r
a†rar , (2.8)
where LAx , LBx are the number of fragment and bath orbitals, respectively, with
LAx = LBx and, together defining the active space. tpq and (pq|rs) are the one- and
two-electron terms in the embedding basis given by
tpq =L∑kl
CkptklClq (2.9)
and,
(pq|rs) =L∑
klmn
CkpCql(kl|mn)CmrCnq , (2.10)
with tkl and (kl|mn) are one- and two-electron terms of the original Hamiltonian, H
in the atomic orbital basis. The core Fock matrix, f core,xpq , is given by
f core,xpq = f full,x
pq − f act,xpq , (2.11)
where f full,xpq is the RHF Fock matrix elements of the full system transformed into the
active embedding basis according to
f full,xpq =
L∑kl
Ckpf fullkl Clq , (2.12)
31
with
f fullkl = tkl +
L∑mn
[(kl|mn)− 1
2(km|ln)
]Dmn (2.13)
and, f act,xpq is the Fock matrix elements in the active space given by
f act,xpq = tpq +
LAx+LBx∑rs
[(pq|rs)− 1
2(pr|qs)
]Dxpq , (2.14)
with
Dxpq =
L∑kl
CkpDklClq , (2.15)
where Dkl given by Eq. 2.6.
The embedding Hamiltonian, Hxemb in Eq. 2.8 can be viewed as a molecular
Hamiltonian in the small active space obtained for each fragment Ax. The elements
of the DMET correlation potential ~u appears only indirectly through its effect on
the form of the bath and external core orbitals. A global chemical potential µglob,
independent of the choice of fragment Ax, is introduced in Hxemb. It ensures that the
total number of electrons in all fragments add up to Nocc.
2.2.2 Calculating the ground state energy
The embedding Hamiltonian, Hxemb, is a much smaller Hamiltonian constructed in
the active space spanned by the fragment and bath orbitals. Hxemb is usually solved
with a high-level method, such as, full configuration interaction (FCI), density matrix
renormalization group (DMRG) or coupled-cluster theory (CC). The ground state of
Hxemb is denoted by |Ψx
emb〉. The total energy is evaluated by summing of the energies
obtained from each fragment’s contribution obtained from its high-level solution. We
have
Etot = Enuc +∑x
Ex , (2.16)
32
where
Ex =
LAx∑p
[LAx+LBx∑
q
(tpq +
1
2f core,xpq
)Demb,xpq +
1
2
LAx+LBx∑qrs
(pq|rs)P emb,xpq|rs
]. (2.17)
The factor of 12
appears before the core Fock contribution because it is an interaction
term that is factored into the one-particle sector of the energy and is scaled like other
interaction terms. Demb,x and P emb,x are the one- and two-particle density matrices
obtained from the high level solution. We have
Demb,xpq = 〈Ψx
emb|a†paq|Ψxemb〉 (2.18)
and,
P emb,xpq|rs = 〈Ψx
emb|a†pa†rasaq|Ψxemb〉 . (2.19)
Unlike the case of the single-impurity Anderson model (SIAM) in Chapter 1, the
energy cannot be evaluated as an expectation value of Hxemb with |Ψx
emb〉, that is,
Ex 6= 〈Ψxemb|Hx
emb|Ψxemb〉 . (2.20)
The expectation value on the right-side of Eq. 2.20 includes both one- and two-
particle terms that are exclusive to the bath. These terms need to be neglected while
evaluating the energy contribution from fragment Ax exclusively. This is the reason
the sum over at least one of the indices in Eq. 2.17 is restricted to run over the
fragment orbitals only.
However, if one wishes to evaluate the individual fragment’s energy contribution,
Ex, as an expectation value, a new Hamiltonian, Hxproj, is constructed with the one-
and two-particle terms as in Eq. 2.17 being appropriately scaled by constant factors.
This serves the purpose of projecting the embedding Hamiltonian onto the fragment
subspace. For example, all one- and two-particle terms with all indices on the bath
are set to 0 and terms with all indices on the fragment are left as is. The fragment-
33
bath terms in the one-particle segment are scaled by a factor of 12. The two-particle
terms with one, two and three indices on the fragment are multiplied by factors of
14, 1
2and 3
4, respectively.
With this new projected Hamiltonian, the energy can be evaluated as an expec-
tation value given by
Ex = 〈Ψxemb|Hx
proj|Ψxemb〉 . (2.21)
This construction is going to be useful for evaluation of energy gradients as shown
in Sec. 2.3.
2.2.3 Self-consistency in DMET
The DMET correlation potential ux is determined by matching the one-particle den-
sity matrices obtained from the low-level and high-level calculations. This amounts
to a least squares minimization of ∆pq(~u) and ∆N(µglob), where
∆pq(~u) = Dlow,xpq (~u)−Demb,x
pq (2.22)
and,
∆N(µglob) = Ntot(µglob)−Nocc , (2.23)
where Ntot(µglob) is the sum over the number of electrons on all the fragments. It
is to be noted that the high-level density matrix, Demb,x, implicitly depends on the
correlation potential ~u. In the minimization however it is kept stationary. Analytical
gradients of Dlow,xpq (~u) with respect to ~u can be found in the Appendix of Ref. [39].
This speeds up the minimization and ensures better convergence. There are various
choices of achieving this minimization:
1. Minimize the difference between all the elements of the active space projected
low- and high-level density matrices.
2. Minimize the difference between the fragment block of the active space pro-
34
Figure 2.2: The DMET algorithm flowchart. Figure taken from Ref. [41]
35
jected low- and high-level density matrices.
3. Minimize the difference between the diagonal elements of the fragment block
of the active space projected low- and high-level density matrices.
4. Minimize the difference between the total number of electrons in all the frag-
ments from the high-level calculation and Nocc.
For choices 1 and 2, the full DMET correlation potential in optimized while only the
diagonal elements are optimized and for 3. For 4, only µglob is optimized.
In practice, the mean-field problem is solved for the full system and the calculated
density matrix is projected to the different active spaces. The desired parts are
matched simultaneously for all fragments. The algorithm is summarized in Fig. 2.2
2.3 Analytical gradients for DMET ener-
gies
2.3.1 RHF-in-RHF embedding
RHF-in-RHF embedding refers to the case when both the high-level and low-level
solutions are derived from RHF level of theory. In this case the total energy of the full
system obtained from the initial RHF calculation matches the total energy obtained
from all the individual fragment’s contribution. In this case the one- and two-particle
density matrices for the embedded system are obtained from RHF theory. The energy
contribution from fragment Ax is given by
Ex = 〈Φxemb|Hx
proj|Φxemb〉 , (2.24)
36
where |Φxemb〉 is the RHF ground state solution for Hx
emb and Hxproj is constructed as
shown in Sec. 2.2.2. Defining Hxproj as
Hxproj =
LAx+LBx∑pq
hproj,xpq a†paq +
1
2
LAx+LBx∑pqrs
(pq|rs)proja†pa†rasaq , (2.25)
the energy contribution from fragment Ax is given by
Ex =1
2
LAx+LBx∑pq
(hproj,xpq + fproj,x
pq
)Demb,xpq , (2.26)
where fproj,x is the Fock matrix given by
fproj,xpq = hproj,x
pq +
LAx+LBx∑rs
[(pq|rs)proj −
1
2(pr|qs)proj
]Demb,xrs , (2.27)
where Demb,xpq = 〈Φx
emb|a†paq|Φxemb〉 refers to the RHF one-particle density matrix. For
the RHF-in-RHF embedding case, Demb,xpq is, by default, equal to the one-particle
density matrix obtained from the RHF calculation of the full system projected onto
the active space constructed for fragment Ax. This implies there is no self-consistency,
that is, ~u, µ = 0. Hence, a “one-shot” embedding calculation yields the ground state
energy for the RHF-in-RHF embedding case.
The gradient of the energy is obtained from differentiating Eq. 2.26 with respect
to some parameter r.
∂Ex
∂r=
1
2
LAx+LBx∑pq
(∂hproj,x
pq
∂r+∂fproj,x
pq
∂r
)Demb,xpq . (2.28)
Since the idempotent RHF one-particle density matrix, Demb,x, is stationary, its
derivative is zero. The only non-zero contributions come from the derivatives of
hproj,xpq and fproj,x
pq . These matrix elements are constructed from the one- and two-
electron terms in Eq. 2.17 scaled by appropriate constant factors. These one- and
two-electron terms in the active embedding space are in turn built from the one- and
37
two-electron terms in the atomic orbital basis projected onto the active embedding
space via the unitary transformation matrix C, see Eq. 2.9−Eq. 2.14.
The gradient calculation for DMET energies is further simplified because the uni-
tary transformation matrix C is constructed from the idempotent RHF one-particle
density matrix as outlined in Sec. 1.2.1. This density-matrix is stationary which
makes C stationary as well. Hence the only derivate terms with non-zero contri-
bution comes from the derivatives of the one and two-electron terms in the atomic
orbital basis. As a result, we have
∂tpq∂r
=L∑kl
Ckp∂tkl∂rClq (2.29)
and,
∂(pq|rs)∂r
=L∑
klmn
CkpCql∂(kl|mn)
∂rCmrCnq . (2.30)
The derivatives on the right side of Eq. 2.29 and Eq. 2.30 are just the Hellmann-
Feyman terms or the “skeleton” terms referred to in the literature of analytic gradient
theory. These derivatives have an analytical form which depends on the basis set and
can be easily evaluated. The standard Pulay terms with arise from the derivative of
C are vanishing as the fragment and bath single-particle orbitals that make up the
columns of C are stationary which do not contribute to the energy gradient.
In summary, the only derivative terms required to evaluate the energy gradient
in Eq. 2.28 are the Hellmann-Feynman terms or the derivatives in Eq. 2.29 and Eq.
2.30 for which analytical formulas are available depending on the atomic orbital basis
used in building these second-quantized one- and two-electron matrix elements.
2.3.2 FCI-in-RHF embedding
FCI-in-RHF embedding is the practical implementation of DMET. FCI-in-RHF em-
bedding refers to the case when full configuration interaction (FCI) is used as the
high-level method to solve Hxemb in the small active space while the low-level method
38
from which the single-particle fragment and bath orbitals, spanning the active space,
are derived from RHF level of theory. The total energy of the full system obtained
from FCI-in-RHF embedding includes the electron correlations in the active space
and provides a better estimate compared the the mean-field RHF energy.
The individual fragment’s contribution to the total energy, Ex is given by Eq.
2.21. Taking its derivative with respect to some parameter r, we get
∂Ex
∂r= 〈Ψx
emb|∂Hx
proj
∂r|Ψx
emb〉+ 2 〈∂Ψxemb
∂r|Hx
proj|Ψxemb〉 . (2.31)
In standard analytical gradient theory for FCI the second term on the right side
of Eq. 2.31 is zero because the FCI wavefunction is an eigenstate of the Hamiltonian
and is normalized[57]. However, in the case of DMET, |Ψxemb〉 is not an eigenstate of
Hxproj. It is an eigenstate of Hx
emb. As a result, the extra term with the wavefunction
derivative needs to be accounted for.
The first term on the right side of Eq. 2.31 involves the derivative of Hxproj which
can be calculated from the Hellmann-Feynman terms only as argued in Sec. 2.3.1.
This requires the derivatives of the one- and two-electron matrix elements in the
atomic orbital basis as shown in Eq. 2.29 and Eq. 2.30.
The wavefunction derivative can be obtained using analytical expressions for
eigenvector derivatives given the derivative of the corresponding matrix. We have,
Hxemb |Ψx
emb〉 = E0 |Ψxemb〉 (2.32)
Differentiating both sides of Eq. 2.32 with respect to r and solving for the wavefuction
derivative, we get
|∂Ψxemb
∂r〉 = −
(Hx
emb − E0
)−1 ∂Hxemb
∂r. |Ψx
emb〉 (2.33)
(Hx
emb − E0
)−1
, when expressed in matrix form, is singular. Hence, a Moore-Penrose
39
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
H-H distance r (a.u.)
−0.54
−0.52
−0.50
−0.48
−0.46
Ene
rgy
per
atom
(a.u
.)
RHF1-shot DMET(1)SC DMET(1)FCI
Figure 2.3: Ground-state energy per atom for H10 ring vs the H−H bond distance r.DMET(1) refers to a fragment with only one hydrogen atom. DMET(1) and 1-shotDMET(1) refers to DMET variants with and without self-consistency, respectively.
pseudoinverse is required. We note that derivative of Hxemb requires only Hellman-
Feynman terms. Therefore, the FCI-in-RHF energy gradient is given by Eq. 2.31,
Eq. 2.33 and Eq. 2.32.
2.4 Results: Hydrogen ring dissociation
The DMET gradient formulation is tested on a small molecular system, the H10 ring,
with all hydrogen atoms equally spaced. This system has been studied in Ref.[36,
39] in the context of DMET ground state energies. Fig. 2.3 shows the ground state
energies from RHF and several variants of DMET compared to the exact FCI energies.
The case of uniform stretching of the hydrogen atoms is considered. The calculations
employ a minimal hydrogen basis consisting of orthogonalized 1s-like atomic orbitals
obtained from an underlying cc-pVTZZ basis.
The RHF results, as expected, is not able to capture the bond dissociation and
diverges at large bondlengths. It gives a single-particle answer which doesn’t capture
40
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
r (a.u.)
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
-dE
/dr
RHF (Num.)DMET(1)-RHF (Anal.)1-shot DMET(1) (Num.)1-shot DMET(1)-FCI(Anal.)SC DMET(1) (Num.)
Figure 2.4: Analytical energy gradients (Anal.) compared to numerical gradients(Num.) for RHF-in-RHF (DMET(1)-RHF) and FCI-in-RHF (1-shot DMET(1)-FCI)embedding without self-consistency.
the electron correlations. DMET(1) refers to the embedding situation when only one
hydrogen atom is included in the fragment. Since the bath is of the same size as the
fragment embedded, DMET(1) is carried out effectively at the cost of solving a system
with two hydrogen atoms. For such a small active space, the results are surprisingly
accurate. The self-consistent version of DMET is referred to as DMET(1)-SC in
Fig. 2.3. The self-consistency is achieved by matching the the one-particle density
matrices from the RHF and FCI calculation in the fragment block only. Finally, a
simpler variant of DMET, referred to as the “one-shot” version or 1-shot DMET(1)
Fig. 2.3 doesn’t involve the self-consistency cycle. It is a one step process with
~u, µ = 0. This simpler version of DMET is very accurate at lower bond distances, all
the way up to the equilibrium distance and, tends to overestimate the correlations
at larger bond distances. However, it is a significant improvement over the RHF
answer.
Fig. 2.4 shows the energy derivatives calculated at various levels of theory. The
smooth curves are obtained from numerical differentiation and provides a benchmark
41
to compare our analytical answers. The negative of the gradient gives the forces
on the hydrogen atom. The point at which the force is zero is the equilibrium
bondlength. Although the ground state energies from RHF are significantly different
from the DMET(1) and FCI values, RHF gradient gets close to predicting the correct
equilibrium bondlength. The analytic energy gradients for RHF-in-RHF embedding
in DMET is referred to as DMET(1)-RHF in Fig. 2.4 where the energy gradients
are calculated from Eq. 2.28. Since the RHF-in-RHF energies are identical to the
energies obtained from the original RHF calculation for the full system, the RHF-
in-RHF gradients also match the original RHF answer. This confirms the fact that
Eq. 2.28 is the correct expression for the RHF-in-RHF analytic gradients. For the
FCI-in-RHF embedding we calculated analytical gradients for the simpler 1-shot
version of DMET. It is labeled as 1-shot DMET(1)-FCI (Anal.) in Fig. 2.4. The
energy gradients are calculated from Eq. 2.31, Eq. 2.33 and Eq. 2.32. Indeed, for
both types of embedding considered, only Hellmann-Feynman terms are required to
calculate energy gradients. It amounts to getting derivatives of one- and two-electron
terms in the atomic orbital basis for which analytical formulas are easily constructed
given a particular basis set. The analytical gradients for FCI-in-RHF embedding
agree with exact numerical gradients, see 1-shot DMET(1)-FCI (Anal.) in Fig. 2.4
which confirms the validity of our formulation.
2.5 Conclusion
In this chapter we outline the ground state energy calculation for molecular systems
using the density matrix embedding formalism and introduce formulas for the cal-
culation of analytical energy gradients. The gradients are obtained from Hellmann-
Feynman terms which are derivatives of the one- and two-electron terms in the atomic
orbital basis. The analytical energy gradients agree with their numerical counterparts
across the entire range of bond distances explored.
42
The gradient formulation should find useful application in geometry optimization
problems of strongly correlated molecular systems and solids and in the implementa-
tion of ab intio molecular dynamics where Density Matrix Embedding Theory can be
used as a high-level electronic structure method. It outlines a way to get the forces
on-the-fly from the gradients of the ground state energy. Future work will be devoted
to studying systems where accurate description of quantum mechanical forces is re-
quired to understand the correct macroscopic properties obtained from the nuclear
motion. It could be applied to study temperature and pressure dependent proper-
ties of high-temperature superconducting solids. DMET gradients are expected to
achieve better accuracy than standard DFT energy gradients for such materials.
43
Chapter 3
Continuous-time Quantum Monte
Carlo algorithm for thermal
equilibrium properties of
continuous-space systems
3.1 Introduction
Many processes in chemistry, physics, biology and materials science involve light
nuclei for which a classical treatment is inadequate. Since these processes occur at
finite temperatures, a quantum statistical mechanical treatment is required to deal
with this challenge. Using Feynman’s path integral approach in imaginary time,[58]
a quantum particle may be mapped to a ring-polymer of pseudoparticles or “beads”
connected by harmonic springs. This mapping, called the “classical isomorphism”,
formally reproduces the exact quantum Boltzmann distribution of the particle from
the classical statistics of ring-polymers in the limit of an infinite number of beads.[59]
For practical purposes, simulations are performed with a finite number of beads,
M . This imaginary time discretization introduces a formal error of O (β2/M2) in the
44
primitive approximation of the Trotter-Suzuki decomposition of the thermal density
operator,[60, 61] where β = 1/kBT is the inverse temperature. The description of
spectroscopy experiments carried out in superfluid helium environments[62–64] at
temperatures on the order of 1 K implies a large value of M to control the error
and obtain converged estimates of thermal observables. The study of electron solva-
tion in polar fluids provides another example of path integral simulations requiring
M ∼ O(103) beads even at room temperature, to correctly predict the spread of the
electron charge density in the solvent medium.[65–68] As a final example, we note
that, the accurate description of the thermal properties of CH+5 , a highly fluxional
molecule with an unusually shallow potential energy surface that enables large am-
plitude nuclear motion, requires M = 16000 beads at 1.67 K when the standard
discretized path integral approach is used.[69]
A large number of beads introduces ergodicity issues in path integral molecular
dynamics (PIMD) simulations.[70] It also results in inefficient sampling of config-
urations in path integral Monte Carlo (PIMC) simulations by increasing the force
constant between the polymer beads resulting in the acceptance of only small par-
ticle moves. In such cases, convergence can be accelerated through normal-mode
sampling[71] or the use of staging algorithms.[72] In addition, the number of beads
can be reduced by using the higher-order Trotter-Suzuki decomposition,[73, 74] via
the pair product action instead of the high temperature propagator[75, 76] and, in
some cases via the use of, path integral methods based on the colored noise gen-
eralized Langevin equation.[77, 78] Although these approaches are promising, their
applicability remains system specific.
An alternate method for the exact calculation of thermal properties is to avoid
the imaginary-time discretization entirely by expanding the partition function in
continuous-time and stochastically sampling terms in the resulting perturbation ex-
pansion. A general scheme to sample diagrams with continuous variables using Monte
Carlo sampling has been formulated for lattice and impurity models[79–83] where
45
systematic errors from the Trotter-Suzuki decomposition are eliminated and a signif-
icant gain in computational efficiency achieved. This line of investigation has resulted
in a class of diagrammatic Monte Carlo techniques called continuous-time quantum
Monte Carlo (CTQMC) and the reader is directed to the review[83] and the references
therein for further details. CTQMC algorithms have mainly been applied to discrete
quantum systems or lattice models providing exact solutions for unfrustrated bosonic
lattice models.[82] Application to fermionic systems has been largely restricted to im-
purity models,[83] providing exact solutions within different expansion techniques for
a wide range of parameters.[84–86]
This chapter explores discrete-time and the continuous-time Quantum Monte
Carlo algorithm for continuous quantum systems. Sec. 3.2 outlines the mapping
of the partition function of a quantum particle to that of a classical ring-polymer
using the imaginary time discretization. It motivates the need for discretization-free
representations of the partition function especially for the case of ultra-low temper-
atures. In that context, borrowing ideas from Ref. [87], a general formulation for
sampling the partition function in continuous-time and the basic algorithm behind
the continuous-time sampling is laid out in Sec. 3.3. In Sec. 3.4 we consider simple
model potentials in the form of the Gaussian well and the harmonic oscillator in one
dimension to benchmark CTQMC results against those obtained from exact solu-
tions. In Sec. 3.4.2 a general extrapolation scheme is introduced which renders the
sampling free of a potential sign issue which can reduce sampling very high orders
at low temperatures. To formulate CTQMC for many-particle systems applicable to
real molecules, Sec. 3.5 draws analogy between the continuous-time formulation with
the discretized version in standard PIMC as in Sec. 3.2. Expressions are derived for
the partition function and energy observables making it amenable to the entire suite
of PIMC sampling moves for the evaluation of the weight of different time configu-
rations. Sec. 3.6 introduces a Gaussian fitting of the many-particle potential. This
opens another route to sample different diagrams in the perturbation expansion of
46
the partition function. Sec. 3.7 is devoted to our concluding remarks.
3.2 Path Integral Monte Carlo
Path Integral Monte Carlo (PIMC) is a finite-temperature Quantum Monte Carlo
(QMC) technique most commonly used to simulate boltzmannons and bosons at low
temperatures at which quantum effects become significant. Instead of sampling the
ground state wavefunction, PIMC, a finite-temperature technique, samples the finite-
temperature partition function. In PIMC, the partition function is re-expressed as
a path-integral via imaginary time discretization. This reduces the task of sampling
the partition function into that of sampling configurations of polymers that rep-
resent different quantum particles[59]. For boltzmannons, these polymers close on
themselves, a manifestation of their distinguishability; for bosons and fermions, they
become inter- connected, a manifestation of their indistinguishability. The ability to
represent quantum particles as classical polymers makes PIMC amenable to many
of the sampling techniques commonly used to study classical polymers and likewise
significantly more straightforward than other QMC algorithms.
3.2.1 Partition function via imaginary time discretization
We consider a system of N distinguishable particles in continuous space, with a
Hamiltonian given by
H = H0 + V =N∑i
p2i
2mi
+ V (R) , (3.1)
where pi, mi are the momentum and mass of the i-th particle, respectively, and
V (R) is a general potential expressed as a function of the particle coordinates R =
(r1, . . . , rN). The partition function of a quantum system in thermal equilibrium at
47
temperature T is given by
Z =
∫dRρ(R,R; β) =
∫dR 〈R| e−βH |R〉 , (3.2)
where ρ(R,R′; β) = 〈R| e−βH |R′〉 is the thermal density matrix with β as the inverse
temperature and |R〉 = |r1 . . . rN〉 represents a many-body position state.
In order to evaluate the partition function in this basis, one must likewise be able
to evaluate the thermal density matrices. This cannot be done for “long-time” density
matrices with large β. The word “time” comes from expressing β as a Wick rotation
from real time to imaginary time which maps the time evolution operator in real
time representation to the thermal density operator in imaginary time. Nevertheless,
accurate approximations can be made for short-time density matrices with small β.
We exploit the following identity to represent the “long-time” density matrix as a
product a product of “short-time” density matrices.
e−(β1+β2)H = e−β1He−β2H . (3.3)
Dividing β into M evenly-spaced time slices, τ , where τ = β/M , the partition
function in Eq. 3.2 can be expressed as
Z =
∫. . .
∫ρ(R(0),R(1); τ)ρ(R(1),R(2); τ) . . . ρ(R(M−1),R(0); τ)dR(0) . . . dR(M−1) ,
(3.4)
where R(p) = (r(p)1 , . . . r
(p)N ). Each of the coordinate vectors in this convolution may be
thought of as coordinates of a path at different imaginary times, τ . The superscript
in r(p)i refers to the imaginary time index and the subscript denotes the particle index.
If M is a finite number, the path is a discrete-time path. If M is infinite, the path
becomes a continuous-time path as outlined in Sec. 3.3.1. Using the Baker-Campbell-
Hausdorff (BCH) formula, the short-time thermal density matrix can be expressed
48
as
e−τ(H0+V ) = e−τH0e−τV +O(τ 2) , (3.5)
where we assume[H0, V
]6= 0. The primitive approximation of the thermal density
matrix is given by
e−τ(H0+V ) ≈ e−τH0e−τV . (3.6)
For ultra-low temperature or large β, small systematic errors can only be achieved
using he primitive approximation with large M . This results in time-consuming
calculations. While higher than second-order approximation of the thermal density
matrix can be employed to reduce the value of M [73, 74], we assume for simplicity
the primitive approximation is sufficient. The free-particle thermal density matrix
or the free-particle propagator is given by
ρ0(R,R′; τ) = 〈R| e−τH0 |R′〉 =N∏i
[ mi
2π~τ
] 32
exp
(−
N∑i
mi(ri − r′i)2
2τ~
). (3.7)
Typically, V is diagonal in the position representation, where
〈R| e−τV |R′〉 = e−τV (R)δ(R−R′) . (3.8)
Using Eq. 3.7 and Eq. 3.8 in Eq. 3.4, we obtain the partition function as a discrete
path-integral given by
Z =N∏i
[ mi
2π~τ
] 3M2
∫. . .
∫exp
[−βHeff(R(0) . . .R(M−1))
]dR(0) . . . dR(M−1) , (3.9)
where
Heff(R(0) . . .R(M−1)) =1
2β
N∑i=1
M∑p=1
[mi(r
(p−1)i − r
(p)i )2
~τ
]+τ
β
M∑p=1
V (R(p)) , (3.10)
where R(0) = R(M) meaning that the path closes on itself. The exponent in Eq. 3.9,
49
the system’s action, resembles the Hamiltonian of a classical ring polymer. The ki-
netic portion of the Hamiltonian resembles the interaction between stretched springs
where all the springs have the same spring constant; the potential term resembles the
potential terms in classical simulations. The final partition function in Eq. 3.9 de-
scribes a system of particles that consist of chains of M beads connected by springs.
The only difference between this system of polymers and a typical classical polymer
is that the beads at each imaginary time only interact with other beads at the same
imaginary time. We evaluate observables in PIMC by sampling Eq. 3.9 in the same
way as we would for a classical simulation. The computational cost of simulating N
quantum particles amounts to the cost of simulating NM classical particles.
3.2.2 PIMC sampling
The properties of distinguishable particles obeying Boltzmann statistics are obtained
by sampling configurations according to Eq. 3.9 and evaluating observables expressed
as a function of those configurations.
〈O〉 =1
Z
∫dRdR′ρ(R,R′; β)O(R′,R) , (3.11)
where O(R′,R) = 〈R′|O|R〉. Since the ratio of the “long-time” density matrix to
the partition function, which appears in Eq. 3.11, can be viewed as a probability,
Monte Carlo techniques are can be used to sample the partition function in Eq.
3.9. If we consider a Monte Carlo update from configuration ~P = R(0) . . .R(M−1)
to ~P ′ = R′(0) . . .R′(M−1), the acceptance probability is given by the Metropolis
criterion.
W acc~P→~P ′
= min
[1,ρ(~P ′)T (~P ′ → ~P )
ρ(~P )T (~P → ~P ′)
], (3.12)
where T (~P → ~P ′) is the transition probability and,
ρ(~P ) = ρ(R(0) . . .R(M−1)) = exp[βHeff(R(0) . . .R(M−1))
]. (3.13)
50
In PIMC, we are allowed to choose the transition probability however we like as long
as Eq. 3.12 is satisfied. If we choose the transition probability proportion to the
kinetic portions of density matrices
T (~P → ~P ′) ∝ exp
[−
N∑i=1
M∑p=1
(mi(r
′(p−1)i − r′
(p)i )2
~τ
)], (3.14)
the acceptance probability can be expressed in a highly simplified form
W acc~P→~P ′
= min
[1, exp
(−τ
M∑p=1
(V (R′
(p))− V (R(p))
))]. (3.15)
The PIMC transition probability in Eq. may be sampled in a variety of ways.
The simplest choice is to displace one bead at a time. With this single-slice update, it
may be shown that the ring-polymer center-of-mass displacement scales as M−3 and
hence for large M the configuration space is sampled really slowly because the springs
become really stiff. To circumvent this problem, multislice moves are used in PIMC
simulations in which chunks of polymers across multiple time slices are sampled all at
once. There are various multislice sampling techniques, including Levy construction-
based techniques[88], normal-mode sampling methods[71], the staging algorithm[72]
and the bisection algorithm[76]. The bisection algorithm is built in stages. In the
first stage the time slices at the two ends of the chunk of the polymer to be replaced
are sampled and accepted with probability 1. The second stage samples the bead at
the mid-point of the two ends and if accepted the third stage samples the bead at the
mid-point between either end and the new mid-point accepted in the second stage.
The algorithm continues to sample midpoints of midpoints until a configuration is
rejected. If a move is rejected at any point, the entire polymer is returned to its
initial configuration. If all of the moves are accepted at all time slices, the polymer
assumes its newly sampled form.
The advantage of the bisection algorithm is that it enables one to build a new path
between two fixed points with the ability to reject the path at any stage before the
51
entire path is constructed. Generally, the configurations least likely to be accepted
are those at the first midpoint between the two fixed ends since this is the time slice at
which the new configurations most differ from the old configurations. Configurations
at each successive stage become increasingly more likely to be accepted. As such, the
bisection algorithm is highly efficient, as it can reject the least likely configurations
first before continuing to sample the whole path.
3.3 Continuous-time Quantum Monte
Carlo (CTQMC)
Sec. 3.2.1 discussed the mapping of the partition function of a quantum particle to
that of a classical ring-polymer with M beads using the classical isomorphism. This
mapping introduced formal errors O(τ 2) where τ = β/M if the primitive approxi-
mation of the thermal density matrix is used. The mapping is formally exact when
τ → 0 or M → ∞. The numerical approach is to however approximate the path
integral by (i) retaining a non-zero τ and (ii) using a Monte Carlo method to estimate
the sums over all intermediate states. The exact partition function is recovered after
the twin steps of converging the Monte Carlo method and extrapolating the results
to τ = 0. The time-step extrapolation is problematic and discretization errors are
large for cases where quantities of interest change rapidly as τ → 0 and has discon-
tinuous derivatives at τ = 0 as seen in solving the impurity problem in Dynamical
Mean-Field Theory (DMFT) applications of lattice problems[83]. This motivates the
need for methods which do not involve an explicit imaginary time discretization.
The basic idea behind the continuous-time expansion of the partition function
is to avoid the time discretization entirely by expanding the partition function in
a diagrammatic perturbation expansion and stochastically sampling the different
diagrams using a Monte Carlo scheme. The continuous-time methods in use now
stem from the work of Ref. [79] and Ref. [80]. It was shown that simulations
52
of bosonic lattice models can be implemented simply and efficiently in continuous
time by a stochastic sampling of a diagrammatic perturbation theory for the parti-
tion function. The general scheme for treating diagrams with continuous variables
of arbitrary nature (diagrammatic Monte Carlo) was formulated in Ref. [81] and
Ref. [82]. In these methods the systematic errors associated with time discretiza-
tion and the Suzuki-Trotter decomposition were eliminated. With significant boost
in computational efficiency, compared to the discretized version, exact results were
obtained for bosonic lattice problems. Application of the continuous-time expansion
to general fermionic problems involve the notorious fermionic sign problem. It has
been applied to a range of sign problem-free models, with attractive interaction, to
investigate the Bose-Einstein condensation to Bardeen-Cooper-Schreiffer crossover in
ultracold atomic gases[89]. Continuous-time Quantum Monte Carlo (CTQMC) has
been widely used as impurity solvers in the context of DMFT applications. Exact so-
lutions have been obtained for impurity models where the sign problem is mild and in
some cases absent[83]. To the best of my knowledge, continuous-time expansion has
been used mainly for discrete quantum problems. Its promising performance in that
realm and the significant advantages it bears over the Trotter discretized version mo-
tivates its application to quantum systems in continuous-space. In the following we
formulate the partition function expansion in continuous-time for quantum particles
in continuous space.
3.3.1 Partition function via continuous-time expansion
We consider a system of N distinguishable particles in continuous space, with a
Hamiltonian given by Eq. 3.1. Unlike the imaginary-time discretized version of
the thermal density operator in Eq. 3.6, the thermal density operator is formally
expressed as the evolution operator in imaginary time β~ as
e−βH = e−βH0T[e−
1~∫ β~0 V (τ)dτ
], (3.16)
53
where T is the time-ordering operator and V (τ) = eτH0V e−τH0 follows from the
interaction representation. Using the expansion of the time-ordered exponential in
Eq. 3.16, the partition function in Eq. 3.2 is given by
Z =∞∑n=0
(−1
~
)n ∫ β~
0
dτ1
∫ τ1
0
dτ2 . . .
∫ τn−1
0
dτn×∫dR(0) 〈R(0)|e−βH0V (τ1) . . . V (τn)|R(0)〉 , (3.17)
where n is the term order and τ1 ≥ τ2 ≥ . . . ≥ τn. Using the interaction representation
and inserting a complete set of position states before each of the V in Eq. 3.17, the
integrand in Eq. 3.17 may be expressed as
∫dR(1) . . .
∫dR(n)ρ0(R(0),R(1); β − τ1)V (R(1)) . . .
ρ0(R(n−1),R(n); τn−1 − τn)V (R(n))ρ0(R(n),R(0); τn) , (3.18)
where R(p) = (r(p)1 , . . . r
(p)N ). The superscript denotes the time order index and the
subscript the particle index. ρ0(R,R′; τ) = 〈R| e−τH0 |R′〉 is the free-particle thermal
density matrix, also known as the free-particle propagator, given by Eq. 3.7. Using
Eq. 3.18, the partition function in Eq. 3.17 reduces to
Z =∞∑n=0
(−1
~
)n ∫ β~
0
dτ1 . . .
∫ τn−1
0
dτn
∫dR(1) . . .
. . .
∫dR(n)ρ0(R(n),R(1); β~− τ1 + τn)V (R(1)) . . .
. . . ρ0(R(n−1),R(n); τn−1 − τn)V (R(n)) , (3.19)
54
where the convolution property of free-particle propagators has been used to integrate
over R(0). Substituting Eq. 3.7 into Eq. 3.19, we find
Z =∞∑n=0
(−1
~
)n ∫ β~
0
dτ1 . . .
∫ τn−1
0
dτnA(τ1, . . . , τn)
×∫dR exp
[−
N∑i
1
2R
T
i Mi(τ1, . . . , τn)Ri
]
× V (R(1)) . . . V (R(n)) , (3.20)
where R = (R1, . . . , RN), Ri = (r(1)i , . . . , r
(n)i ) and R(n) is as defined in Eq. 3.18.
A(τ1, . . . , τn) is given by
A(τ1, . . . , τn) =N∏i
n∏p=1
(mi
2π~(τp−1 − τp)
) d2
, (3.21)
where τ0 = β~ + τn and Mi is expressed as
Mi =mi
~
Ω1 + Ω2 −Ω2 . . . −Ω1
−Ω2 Ω2 + Ω3. . .
...
.... . . . . . −Ωn
−Ω1 . . . −Ωn Ωn + Ω1
, (3.22)
where
Ωp =
1
β~−τp+τnp = 1
1τp−1−τp p = 2, . . . , n .
(3.23)
Eq. 3.20 gives the continuous-time expansion of the partition function for particles
in continuous space. In the following we use Monte-Carlo sampling[83, 90] to sample
the infinite series in Eq. 3.20 and calculate thermal equilibrium observables.
It should be emphasized that by expanding the partition function in continuous
time we trade the problem of finite-time discretization errors, as discussed in the Sec.
3.1, for sampling high-order perturbation terms. We draw motivation from strongly
55
coupled lattice problems where the continuous-time expansion often is superior to the
discretized approach.[82, 83] In the following we briefly review the CTQMC procedure
used to sample the infinite series in Eq. 3.20.
3.3.2 CTQMC sampling
The partition function Z is of the form
Z =∞∑n=0
(−1
~
)n ∫ β~
0
dτ1 . . .
∫ τn−1
0
dτnP (τ1, . . . , τn) . (3.24)
Eq. 3.24 is written assuming the position-integrals appearing as the integrand of the
time-integrals in Eq. 3.20 can be evaluated analytically. In the remainder of the paper
we consider model systems where this assumption holds true. However, for applying
the algorithm to more general cases, numerical evaluation of the multidimensional
position integral is required as outlined in Sec. 3.5 and 3.6.
The basic CTQMC algorithm is now outlined, with further details found in Ref.
[83]. The Monte Carlo sampling is used to evaluate the series in Eq. 3.24 by stochas-
tically moving between different time configurations τ1, . . . , τn based on the weight
it contributes to the partition function. The basic steps are as follows:
1. A Markov chain starts via choosing a random time-order n, generating a random
time-ordered configuration τ1, . . . , τn and calculating its weight P (τ1, . . . , τn).
2. A move is selected from three kinds of Monte Carlo moves: insertion, removal
and “stay” moves. Insertion moves add a new time value chosen randomly to
the existing time configuration to increase the order of the considerred term.
The process of elimination of a random time from the existing configuration
which decreases the order of the considered term defines a removal move. The
stay move is defined as changing one or more time values from the existing
configuration maintaining the same time order.
3. After a move is selected, the weight of the new time configuration is calculated
56
and is accepted or rejected based on the Metropolis acceptance criterion. The
Metropolis acceptance ratio for an insertion move is given by
W accn→n+1 = min(1, Rn→n+1) , (3.25)
where
Rn→n+1 =P (τ ′1, . . . , τ
′n+1)
P (τ1, . . . , τn)
β
(n+ 1)(3.26)
and τ ′1, . . . , τ ′n+1 is the new time configuration with an extra time index. For
the removal move we have
W accn+1→n = min(1, Rn+1→n) , (3.27)
where
Rn+1→n =1
Rn→n+1
. (3.28)
For the stay move we have
W accn→n = min(1, Rn→n) , (3.29)
where
Rn→n =P (τ ′1, . . . , τ
′n)
P (τ1, . . . , τn). (3.30)
If the move is accepted based on Eq. 3.25, 3.27 or 3.29, the Markov chain is
updated with the new time configuration else the old configuration is retained.
4. After an initial equilibration period, observables are recorded at finite intervals.
The insertion and removal updates are necessary to satisfy the ergodicity require-
ment. The stay moves typically not required for ergodicity are used to improve the
sampling efficiency.
57
3.3.3 Thermal average energy
The average energy, calculated from the partition function, is given by
〈E〉 =Tr(He−βH
)Tr(e−βH
) = − 1
Z
∂Z
∂β. (3.31)
The β-derivative of Z in Eq. 3.24 generates two terms - one from the derivative of
the τ1 integral and the other from the derivative of P . Using this, the average energy
can be compactly expressed as the following ratio
〈E〉 =
∑n
(−1~
)n β~∫0
dτ1 . . .τn−1∫0
dτn Q(τ1, . . . , τn)
∑n
(−1~
)n β~∫0
dτ1 . . .τn−1∫0
dτn P (τ1, . . . , τn)
. (3.32)
Assuming we are sampling configurations based on its weight P , the Monte Carlo
sum for 〈E〉 in Eq. 3.32 is given by
〈E〉MC =
NMC∑i
(−1)niQ(τ1,...,τni )
P (τ1,...,τni )
NMC∑i
(−1)ni, (3.33)
where NMC is the total number of observable recordings, and ni is the order of the
i-th time configuration. Eq. 3.33 has an alternating sign in the denominator. This
“sign problem” can become problematic for cases where the contribution from odd
and even orders are comparable to each other. We outline a generic strategy for
removal of this problem in Sec. 3.4.2.
3.4 Results
While a general many-body formulation has been laid out, this work only deals with
the model case of a single particle in one dimension. Generalization to more complex
58
-1
-0.5
0
0.5
〈E〉
ExactCTQMC
0 0.1 0.2 0.3 0.4 0.5 0.6
T
-1
-0.5
0
0.5
〈E〉
0 0.1 0.2 0.3 0.4 0.5 0.6
T
λ = 0.1 λ = 0.25
λ = 0.5 λ = 1.0
Figure 3.1: CTQMC values of average energy 〈E〉 and standard error for the Gaus-sian well as a function of T for λ ranging from low to high values. MC average over1000 independent trajectories. The exact results are obtained from the numericalsolution of the Schrodinger equation in one dimension.
cases is trivial and merely comes at the expense of potentially more Monte Carlo
integration. The partition function in Eq. 3.20 can be simplified as
Z =∞∑n=0
(−1
~
)n ∫ β~
0
dτ1 . . .
∫ τn−1
0
dτnA(τ1, . . . , τn)
×∫dX exp
[−1
2XTM(τ1, . . . , τn)X
]× V (x(1)) . . . V (x(n)) , (3.34)
where X = (x(1), . . . , x(n)). A and M is given by Eq. 3.21 and Eq. 3.22, respectively,
for a single particle in one dimension with N = 1 and d = 1.
In the following we benchmark our CTQMC approach with two model potentials,
the Gaussian well and the harmonic oscillator for which exact solutions are easily
obtained. The Gaussian well is a simple case where the position integral in Eq. 3.34
59
can be evaluated analytically and the sampling is free of alternating signs as discussed
in Sec 3.4.1. For the harmonic potential the analytic solution for the position integral
is given by a sum of terms, the number of which scales factorially with time-order
n making it difficult to compute high orders. In order to overcome the high order
sampling and remove the “sign problem”, we propose a general strategy by mapping
the harmonic potential to a Gaussian well as discussed in 3.4.2.
3.4.1 Gaussian well
The Gaussian well is defined by the potential
V (x) = −e−λx2 . (3.35)
For this potential, the position integral in Eq. 3.34 are evaluated analytically
∫dX exp
[−1
2XTMX
]V (x(1)) . . . V (x(n)) = (−1)n
√(2π)n
det(M + 2λI). (3.36)
The functional dependence of M has been omitted for brevity of notation. The (−1)n
in Eq. 3.36 cancels the identical factor in Eq. 3.34. This eliminates alternation
of signed terms and the weights can be interpreted as probability densities on the
configuration space.
Fig. 3.1 shows the average energy 〈E〉 as a function of temperature T . In our
calculations we have used atomic units where ~ = 1, m = 1. The four panels
correspond to four different values of λ. The CTQMC results are benchmarked
against exact answers obtained by numerically solving the Schrodinger equation on a
one dimensional position grid.[91] The Monte Carlo sampling converges to the exact
answer within error bars for both low and high-T for all values of λ. MC averages
are calculated from 1000 independent trajectories. Each trajectory corresponds to a
total of 106 steps with an initial 20% used for equilibration, following which, energy
60
λ = 0.1
T = 0.05T = 0.25T = 0.5
λ = 0.25
λ = 0.5
0 5 10 15 20 25 30
λ = 1.0
0 5 10 15 20 25 30
Cou
nt(a
rb.
units
)C
ount
(arb
.un
its)
n n
Figure 3.2: Variation of CTQMC order distribution with T and λ for the Gaussianwell for a given trajectory with 106 MC steps.
observables are recorded every 5 steps. The error bars denote the standard error of
the mean. For λ = 0.1, the relative error or the deviation of the mean values from the
exact answer is less than 0.2% for T ≤ 0.2. It increases only slightly with temperature,
being less than 3% for T = 0.5. For λ = 1.0, the relative error is less than 1.2% for
T ≤ 0.2 and, is maximum at 3.5% for T = 0.25. Fig. 3.2 shows the distribution of
time-orders sampled for low, intermediate and high T denoted by T = 0.05, 0.25, 0.5
respectively and the same values of λ considered in Fig. 3.1. Higher values of λ,
implying a weaker perturbation, causes an overall shift towards lower orders sampled.
The shape of the distribution shows distinct patterns for the different temperature
regimes. For T = 0.05, the distribution has little contribution from low orders and
perturbation order up to n ≈ 30 are noticeable. The T = 0.25 distribution shows a
strong zeroth order contribution followed by Gaussian-like behavior. For T = 0.5,
the distribution shows a similar strong zeroth order contribution followed by a decay.
61
0.4
0.45
0.5
0.55
〈E〉
0 0.1 0.2 0.3 0.4 0.5
λ
0
0.005
0.01
0.015
σ
T = 0.001
T = 0.001
Figure 3.3: At T = 0.001, variation of CTQMC 〈E〉 (top panel) and the stan-dard error σ (bottom panel) as a function of λ for the quantum harmonic oscil-lator obtained from 1000 independent trajectories. λ = 0 extrapolated answer is〈E〉MC = 0.5051± 0.00847 and the exact analytical value is 〈E〉exact = 0.5
62
Cou
nts
(arb
.un
its)
0 500 1000 1500 2000
n
Cou
nts
(arb
.un
its)
0 500 1000 1500 2000
n
λ = 0.25 λ = 0.3
λ = 0.35 λ = 0.4
Figure 3.4: Variation of CTQMC order distribution with λ at T = 0.001 for thequantum harmonic oscillator for a given trajectory with 106 MC steps.
3.4.2 Harmonic Potential
For the quantum harmonic oscillator, the potential is defined as
V (x) =1
2mω2x2 . (3.37)
For this potential, the position integral in Eq. 3.34 is given by
∫dX exp
[−1
2XTMX
](x(1))2 . . . (x(n))2 =√
(2π)n
detM
1
2nn!
∑σ∈S2n
(M−1)σ(1),σ(2) . . . (M−1)σ(2n−1),σ(2n) , (3.38)
where σ is a permutation of 1, . . . , 2n and the extra factor on the right-hand side
is the sum over all combinatorial pairings of 1, . . . , 2n of n copies of M−1. At large
n, the number of terms inside the sum in Eq. 3.38 grows factorially with n making
its exact calculation numerically difficult. In addition to this, the alternating nature
63
of the series leads to a type of sign problem as discussed in Sec. 3.4.1. In order to
overcome this, we propose a general solution that is applicable to any potential by
exploiting the following identity
V = limλ→0
1
λ(1− e−λV ) . (3.39)
For the harmonic potential, we have
V (x) =1
2mω2x2 = lim
λ→0
mω2
2λ
(1− e−λx2
). (3.40)
The first term on the right-hand side of Eq. 3.40 is a constant resulting in a shift
in the value of the observables calculated. We use the second term as the potential
for the continuous-time sampling. Computing Monte Carlo averages for small values
of λ, the final answer is obtained by extrapolating to λ = 0. Using the potential
as expressed in Eq. 3.40, the negative sign cancels out in Eq. 3.34 eliminating sign
alternation and the position integral can be evaluated analogous to Eq. 3.36.
We have used ~ = m = ω = 1. Fig. 3.3 shows the low-temperature variation of
〈E〉 and the standard error σ for small λ at T = 0.001. MC averages are calculated
from 1000 independent trajectories where each trajectory is created in a manner
identical to those described in Sec. 3.4.1. Using the linear fit, the λ = 0 extrapolated
CTQMC result converges to the exact analytical answer within error bars. The
relative error of the extrapolated answer is 1% . Exploiting this linearity not only
gives the exact answer but also avoids sampling high orders for very small λ as
seen in Fig. 3.4, thereby reducing the error. Fig. 3.4 shows the CTQMC order
distribution for T = 0.001 for the same values of λ considered in Fig. 3.3. For
such temperatures, the Monte Carlo trajectories converged even though sampling
high orders (nmax ≈ 2000) is required. With increasing λ, the distribution becomes
narrower and shifts towards smaller values of n. Fig. 3.5 shows the average energy
〈E〉 for the quantum harmonic oscillator for a range of temperatures. For each Monte
64
0.0 0.1 0.2 0.3 0.4 0.5 0.6
T
0.4
0.5
0.6
0.7
0.8
〈E〉
ExactCTQMC
Figure 3.5: CTQMC average energy 〈E〉 and standard error as a function of T forthe harmonic oscillator benchmarked against exact analytical solutions. MC averagecalculated from 1000 independent trajectories.
65
Figure 3.6: Schematic of the weight of a time configuration for a single particle withn = 3 represented as a ring polymer and all the associated CTQMC moves. Eachspring corresponds to a free particle propagator with an imaginary time equal to thedifference in τ ’s with subscripts corresponding to the bead index.
Carlo data point, the extrapolation method, as previously discussed, has been used
to obtain energies that agree with the exact analytical answer within error bars. The
relative error is less than 0.15% for all values of T in Fig. 3.5.
It should be noted that the avoidance of sign alternation as proposed in this
section comes at the expense of performing additional simulations and performing an
extrapolation. This approach is only advantageous if the relative magnitude of terms
n and n + 1 in the expansion of the partition function are of similar magnitude. If
this is not the case, then it is simply preferable to use the direct approach of Sec.
3.4.1.
66
3.5 Analogy with standard discrete-time
path integral methods
In the following we rewrite the partition function in a manner analogous to the ring-
polymer representation in the standard discretized path integral version.[59] Starting
from Eq. 3.17, following the steps outlined in Sec. 3.3.1 and using the potential in
Eq. 3.39, the partition function is given by
Z =∞∑n=0
(1
λ~
)n ∫ β~
0
dτ1 . . .
∫ τn−1
0
dτnA′(τ1, . . . , τn)
×∫dR(0) . . .
∫dR(n) exp [−βHeff] , (3.41)
where
Heff = − 1
2β
n+1∑p=1
N∑i=1
mi(r(p−1)i − r
(p)i )2
~(τp−1 − τp)− λ
β
n∑p=1
V (R(p)) , (3.42)
and R(n+1) = R(0), τ0 = β~, τn+1 = 0 and,
A′(τ1, . . . , τn) =N∏i=1
n+1∏p=1
(mi
2π~(τp−1 − τp)
) d2
. (3.43)
Here we have not integrated over R(0). This allows one to estimate observables as
an average over the canonical distribution function in Eq. 3.41 and the case for the
average energy is shown below.
Eq. 3.41 has two useful features. Firstly, using the modified potential, the sign
problem has been eliminated. Secondly, the configuration weight resembles that
of a ring polymer obtained from the Trotter decomposition of the thermal density
operator expressed in path integral form. For every order n sampled, the weight
is given by a ring polymer with n + 1 beads where the zeroth bead does not have
the potential acting on it. Unlike the ring polymer case where the imaginary time
is discretized into a fixed number of evenly spaced time slices, the continuous-time
67
expansion samples over different configurations with varying number of time slices
which can be unequally spaced. This type of sampling eliminates the Trotter error
as discussed in Sec. 3.1.
The average energy from Eq. 3.31 is given by,
〈E〉 − 1
λ=
∑n
(1~λ
)n ∫d~τ∫dRU(~τ ,R)P (~τ ,R)∑
n
(1~λ
)n ∫d~τ∫dR P (~τ ,R)
, (3.44)
where ~τ = (τ1, . . . , τn), R = (R(0), . . . ,R(n)) and R(p) = (r(p)1 , . . . , r
(p)N ). We have
P (~τ ,R) = A′(~τ) exp [−βHeff] , (3.45)
with A′ and Heff given by Eq. 3.43 and Eq. 3.42, respectively, and
U(~τ ,R) = U(τ1,R(0),R(1)) =
~2(β~− τ1)
− e−λV (R(0))
λ−
N∑i=1
mi(r(0)i − r
(1)i )2
2(β~− τ1)2. (3.46)
With this “ring-polymerized” expression for the partition function, all CTQMC
moves can be easily incorporated as seen in Fig. 3.6 which illustrates possible up-
dates from a time configuration with n = 3. For a particular time configuration
which defines the spring constants of the ring polymer, all special PIMC moves[71]
like center-of-mass displacement, staging and bisection moves can be deployed to
carry out the positional sampling. Therefore continuous-time becomes analogous to
standard PIMC with insertion, removal and stay moves that allow updates between
different time configurations by adding, removing or changing a particular node in the
ring-polymer, respectively. Future work will be devoted to investigating the efficacy
of such a representation for non-trivial many-body problems.
68
3.6 Many-particle problems via Gaussian
fitting of Potential
Sampling the partition function in Eq. 3.20 via CTQMC requires the evaluation of the
multidimensional position integrals to obtain the weight of each time configuration.
In Sec. 3.5, we outlined one possible strategy to achieve this by drawing on an analogy
with the discretized path integral representation. Here we outline another approach
which involves fitting of the many-body potential with multidimensional Gaussians
allowing for an analytical evaluation of the position integrals. The partition function
in Eq. 3.20 can be more compactly expressed as
Z =∞∑n=0
(−1
~
)n ∫ β~
0
dτ1 . . .
∫ τn−1
0
dτnA(τ1, . . . , τn)
×∫dR exp
[−1
2R
TM(τ1, . . . , τn)R
]V (R(1)) . . . V (R(n)) , (3.47)
where R is a 3nN dimensional vector given by, R = (R(1), . . . , R
(n)) and, R
(p)=
(R(p)
1 , . . . , R(p)
N ). M is a 3nN × 3nN matrix obtained from restructuring the ele-
ments of a block-diagonal matrix with Mi, given by Eq. 3.22, as the blocks. This
restructuring, however, removes the block-diagonal property of M. The many-body
potential can be approximately fitted as a sum of multidimensional Gaussians with
mean R0α and covariance matrix W−1α given by
V (R) ≈αmax∑α
Cα exp
[−1
2(R−R0α)TWα(R−R0α)
], (3.48)
where R,R0α are 3N dimensional vectors and Wα is a 3N×3N matrix. The numer-
ical procedure to obtain the fitting parameters Cα,R0α and Wα may be obtained in
69
a variety of ways.[92, 93] Thus, we find
V (R(1)) . . . V (R(n)) ≈αmax∑α1
. . .
αmax∑αn
Cα1 . . . Cαn exp
[−1
2(R− R0)TW(R− R0)
],
(3.49)
where R0 is a 3nN dimensional vector given by R0 = (R0α1 , . . . ,R0αn), and W is
3nN × 3nN block-diagonal matrix with the blocks given by Wα1 , . . . ,Wαn . The
explicit dependence on indices α1, . . . , αn for R0 and W has been omitted for brevity
of notation. Substituting Eq. 3.49 in Eq. 3.47 gives
Z ≈∞∑n=0
(−1
~
)n ∫ β~
0
dτ1 . . .
∫ τn−1
0
dτnA(τ1, . . . , τn)
×αmax∑α1...αn
Cα1 . . . Cαn exp
[−1
2R
T
0 WR0
]×∫dR exp
[−1
2R
TM′R + zT R
], (3.50)
where M′
= M + W, and zT = RT
0 W. The multidimensional position integral has
an analytic form. As a result, Eq. 3.50 simplifies to
Z ≈∞∑n=0
(−1
~
)n ∫ β~
0
dτ1 . . .
∫ τn−1
0
dτnA(τ1, . . . , τn)
×αmax∑α1...αn
P (τ1, . . . , τn; α1, . . . , αn) , (3.51)
where
P (τ; α) = Cα1 . . . Cαn
√det(
2πM′−1)
exp
[−1
2R
T
0 UR0
], (3.52)
and U = W + W[(M + W)−1]TWT
. The α-dependence of P arises from the
Gaussian fitting parameters.
This approach provides expressions for the multidimensional position integrals.
However, in order to find the weight of the time configurations for the CTQMC sam-
70
pling, the sum in Eq. 3.51 needs to be evaluated. Since the number of terms in the
sum grows factorially with time order n, a Monte Carlo estimate of the sum can be
used for high orders. For this MC sampling, only the set of indices α1, . . . , αn need
to be sampled as the weight for a particular set is completely deterministic. The MC
sum can be evaluated via importance sampling by drawing samples from a multi-
nomial distribution. Since, α1, . . . , αn each range from 1 to αmax, the multinomial
probability density function is given by
f(w1, . . . , wαmax ;n, p1, . . . , pαmax) =n!
w1! . . . wαmax !pw1
1 . . . pwαmaxαmax
, (3.53)
where wi is an integer that counts the number of times the value of index αi appears in
the set of α1, . . . , αn with∑αmax
i wi = n, and pi is the probability of i-th outcome
with∑αmax
i pi = 1. A choice of pi = 1/αmax allows us to draw samples from the
multinomial distribution with the probability density function given by Eq. 3.53
Therefore, the MC estimate of the sum is obtained by taking the average of Eq. 3.52
from all the samples generated from the distribution. This approach could be applied
to investigate the thermodynamics of small molecules where the number of fitting
parameters is likely small implying both an accurate Gaussian fit as well as facile
sampling of high-order terms.
3.7 Conclusion
In this work we have formulated an exact, continuous-time Monte Carlo evaluation
of the thermodynamics of particles in continuous space. This formalism is free of the
Trotter error associated with standard discretized PIMC techniques. The CTQMC
algorithm samples various time-configurations based on the weight it contributes to
the partition function. A general way to express the potential along with an extrap-
olation scheme is proposed which renders the sampling free of alternating signs. For
simple model potentials considered, the CTQMC results easily converge to the exact
71
solutions even for low temperatures where Monte Carlo runs require the sampling
of very high orders. For more complex systems, we advocate writing the partition
function in a way that resembles the ring-polymer representation of the discretized
path integral expressions. Standard PIMC technology can be used to evaluate the
weight of the time configurations. Finally, we show that fitting the many-body po-
tential with a sum of multidimensional Gaussians leads to an analytical solution for
the position integrals which may be of use in more realistic problems.
The approach discussed in this work should find useful application in various
condensed phase problems where standard PIMC may not be the optimal choice.[65,
94, 95] The promising performance of CTQMC sampling on simple model systems
along with it’s compatibility with the entire PIMC suite of sampling moves motivates
us to apply this methodology to problems where the Trotter error is problematic,
such as the low-temperature behavior of highly fluxional molecules like CH+5 . Future
work is also devoted to studying the low temperature behavior in liquids where the
quantum nature of the interacting nuclei leads to interesting and non-trivial behavior.
72
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