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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