Anharmonic phonon behavior via irreducible derivatives self-consistent perturbation theory and molecular dynamics Enda Xiao1and Chris A. Marianetti2

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Anharmonic phonon behavior via irreducible derivatives: self-consistent perturbation

theory and molecular dynamics

Enda Xiao1and Chris A. Marianetti2

1Department of Chemistry, Columbia University, New York, New York 10027, USA and

2Department of Applied Physics and Applied Mathematics,

Columbia University, New York, New York 10027, USA

(Dated: November 21, 2023)

Cubic phonon interactions are now regularly computed from ﬁrst principles, and the quartic in-

teractions have begun to receive more attention. Given this realistic anharmonic vibrational Hamil-

tonian, the classical phonon Green’s function can be precisely measured using molecular dynamics,

which can then be used to rigorously assess the range of validity for self-consistent diagrammatic ap-

proaches in the classical limit. Here we use the bundled irreducible derivative approach to eﬃciently

and precisely compute the cubic and quartic phonon interactions of CaF2, systematically obtaining

the vibrational Hamiltonian purely in terms of irreducible derivatives. non frequency shifts and

linewidths, We demonstrate that the 4-phonon sunset diagram has an important contribution to

the optical phonon linewidths beyond T= 500 K. Reasonable results are obtained even at T= 900

K when performing self-consistency using the 4-phonon loop diagram and evaluating the 3-phonon

bubble and 4-phonon sunset diagrams post self-consistency. Further improvements are obtained by

performing quasiparticle perturbation theory, where both the 4-phonon loop and the real part of

the 3-phonon bubble are employed during self-consistency. Our irreducible derivative approach to

self-consistent perturbation theory is a robust tool for studying anharmonic phonons in both the

quantum and classical regimes.

I. INTRODUCTION

Lattice anharmonicity is essential to the understand-

ing of many physical properties of solids, such as ther-

mal expansion, thermal conductivity, etc. [1]. Therefore,

computing phonon interactions from ﬁrst principles is a

critical task. Preliminary calculations of cubic phonon

interactions from density functional theory (DFT) be-

gan decades ago using ﬁnite displacements [2], and den-

sity functional perturbation theory (DFPT) computa-

tions eventually followed [3, 4]. Cubic phonon interac-

tions have since been computed in a wide range of crystals

using both DFPT and ﬁnite displacements [5]. Quartic

phonon interactions were investigated early on as well us-

ing DFT with ﬁnite displacements [6, 7]. While quartic

phonon interactions have been formulated at the level of

DFPT [3], we are not aware of any explicit calculations.

Given that the number of derivatives increases drastically

with the order, precisely and eﬃciently executing ﬁnite

displacement calculations of quartic phonon interactions

is critical. Recently, an approach for computing phonons

and phonon interactions based on irreducible derivatives

was put forward, which maximally uses group theory to

reduce computational cost, and the resulting gain in ef-

ﬁciency can be converted into gains in accuracy [8]. En-

coding anharmonicity in terms of irreducible derivatives

has many advantages, including allowing a straightfor-

ward comparison between diﬀerent methods. Whenever

possible, it is critical to separately assess the quality of

the vibrational Hamiltonian versus the solution to the

vibrational Hamiltonian, as the latter will contain errors

from both aspects. Here we solely focus on computing

the interacting phonon Green’s function of a particular

realistic vibrational Hamiltonian containing up to quartic

phonon interactions.

The vibrational Hamiltonian poses a non-trivial many-

boson problem, and therefore it is challenging to assess

the quality of the method being used to solve the vibra-

tional Hamiltonian. Imaginary time Quantum Monte-

Carlo (QMC) techniques [9, 10] can be used to accu-

rately compute thermodynamic observables for relatively

large systems, and various applications exist in the lit-

erature using realistic tight-binding potentials [11, 12].

However, it is far more challenging to obtain highly ac-

curate solutions of dynamical quantities, such as the real

time phonon Green’s function. Imaginary time QMC re-

sults can be approximately analytically continued to the

real axis [13], but these approaches are uncontrolled and

are best restricted to determining peak locations in the

Green’s function [14]. There are several established tech-

niques which can be used to approximate real time quan-

tum correlation functions [15, 16], such as centroid molec-

ular dynamics and ring polymer molecular dynamics, but

these approaches have various limitations [17] and have

not yet been used to compute phonon linewidths in realis-

tic systems; though recent studies are beginning to make

average comparisons via thermal conductivity [18]. A

practical approach for obtaining the Green’s function on

the real axis is to use diagrammatic perturbation theory,

but the problem is assessing whether or not a suﬃcient

number of diagrams have been computed. A partial so-

lution to this problem is to revert to the classical limit

where the vibrational Hamiltonian can be solved accu-

rately using molecular dynamics, and then a perturba-

tive solution can be rigorously assessed. Success implies

that the diagrammatic approach is robust in the classi-

cal regime, and will likely be suﬃcient in the quantum

regime for suﬃciently weak anharmonicity.

arXiv:2210.15537v4 [cond-mat.mtrl-sci] 18 Nov 2023

Using molecular dynamics to measure classical dynam-

ical correlation functions is well established in the con-

text of empirical potentials [19]. There have been a small

number of studies which use an anharmonic vibrational

Hamiltonian based upon ﬁrst-principles calculations to

measure dynamical correlation functions within molec-

ular dynamics [20–25], as is in the present study. Our

molecular dynamics is based purely on irreducible deriva-

tives, which we refer to as irreducible derivative molecu-

lar dynamics (IDMD), and we have developed methods

to precisely compute the irreducible derivatives from ﬁrst

principles [8], ensuring that we are working with a real-

istic vibrational Hamiltonian. A key goal of this work is

to use the IDMD results to assess diagrammatic pertur-

bation theory in the classical limit, and we are not aware

of comprehensive comparisons in the existing literature.

One of the more popular approximations to the inter-

acting phonon problem is a variational theory which uses

a Gaussian ansatz for the density matrix, which was orig-

inally pioneered by Hooton [26]. This approach can be

viewed as the Hartree-Fock (HF) approximation for inter-

acting phonons [27], given that the theory variationally

determines the optimum non-interacting bosonic refer-

ence system which minimizes the free energy. Naturally,

Hartree-Fock for phonons also has a clear diagrammatic

interpretation, and therefore it is an integral approach

for solving our anharmonic Hamiltonian in this work.

There are several popular approaches which implement

this Hartree-Fock approximation for phonons, and it is

useful to contrast them with our own implementation.

The self-consistent phonon (SCP) approach of Ref. [28],

later referred to as the ﬁrst-order self-consistent phonon

(SC1) approach [29], uses compressive sensing [25, 30]

to ﬁt the anharmonic vibrational Hamiltonian, and then

the Hartree-Fock equations are solved self-consistently in

the usual manner. Compressive sensing is useful given

that it can eﬃciciently ﬁt the anharmonic terms to a

relatively small set of forces, but it is unclear how pre-

cisely it recovers the anharmonic terms as compared to

the numerically exact answer, such as what can be ob-

tained with our irreducible derivative approach [8]. The

SCP approach has been used to compute temperature de-

pendent phonon dispersions [28, 29, 31–33] and thermal

expansion [34, 35].

Another approach for executing Hartree-Fock is the

stochastic self-consistent harmonic approximation (SS-

CHA) [36, 37], which circumvents the need to Taylor

series expand the Born-Oppenheimer potential by per-

forming a stochastic sampling of the gradient of the trial

free energy. If executing the Taylor series is prohibitive,

potentially because very high orders are needed to cap-

ture the relevant physics, a stochastic approach might be

the only practical method to execute Hartree-Fock. An-

other important aspect of the SSCHA is that the full

variational freedom of the Hartree-Fock ansatz is ex-

plored, allowing for the expectation values of the nuclear

displacements to be included as variational parameters.

However, the SSCHA has its own computational limi-

tations for proper sampling, and the eﬃcacy of the SS-

CHA is inherently tied to the Hartree-Fock approxima-

tion. The SSCHA has been used in the computation of

soft-mode driven phase transitions [37, 38], charge den-

sity wave transitions [39], and superconducting proper-

ties [36, 40–47]. An earlier stochastic approach is the self-

consistent ab initio lattice-dynamical method (SCAILD)

[48, 49], which can be viewed as an approximation to

the classical limit of the SSCHA. The SCAILD approach

has been used in the computation of temperature de-

pendent phonon dispersion and structural phase transi-

tions [50–52], and was later extended to the quantum

case [53, 54]; moving the method closer to the SSCHA.

Another popular approach is the temperature-dependent

eﬀective potential approach (TDEP) [55], which uses the

forces from a classical ab initio molecular dynamics tra-

jectory as a source of data to parameterize an eﬀective

quadratic potential. TDEP is not based on the Hartree-

Fock approach: the intent is to faithfully recover the ex-

pectation value of the potential energy of an ab initio

molecular dynamics simulation, and use the resulting ef-

fective quadratic potential to evaluate the corresponding

harmonic quantum free energy [56]. A recent applica-

tion of the TDEP method changes course and adopts

the stochastic sampling method of the SSCHA [57]. It

should be emphasized that all of the aforementioned ap-

proaches could be directly applied to the anharmonic

Hamiltonian in the present study, but this would oﬀer

no beneﬁt as compared to our own Hartree-Fock solu-

tion. The application of SCP would only test whether the

compressive sensing approach faithfully reproduces our

irreducible derivatives, while the SSCHA would only test

the accuracy of the stochastic sampling. While Hartree-

Fock is a key method that is used to study interacting

phonon systems, it will also be important to go beyond

Hartree-Fock, motivating the evaluation of higher order

diagrams.

Diagrammatic perturbation theory is a conventional

method used to compute the phonon Green’s function on

the real axis [58, 59]. We consider all O(λ2) self-energy

diagrams and selected O(λ4) diagrams [60] (see Figure 1

for schematics and labels). The colloquial diagram names

of bubble, loop, tadpole, sunset, cactus, and ﬁgure-eight

are abbreviated as b,ℓ,t,s,c, and f, respectively. The

mathematical deﬁnitions for the b,ℓ,s,c, and fdiagrams

are given in equations 17, 12, 19, 21, and 24 of Reference

[60], respectively, and the tdiagram is deﬁned in equa-

tion 2 of Reference [61]. The classical limit of the O(λ2)

diagrams (i.e. b,ℓ, and t) yield a linear temperature de-

pendence for the self-energy, while the classical limit of

the O(λ4) diagrams yield a quadratic temperature de-

pendence. It should be noted that the tdiagram is zero

in the ﬂuorite crystal structure [59, 61]. The inclusion

of all O(λ2) diagrams is clearly essential. The imaginary

part of the bdiagram is widely used in the context of

perturbative thermal conductivity calculations [62, 63],

and classically will provide the only contribution to the

linewidth to ﬁrst order in temperature. The ℓdiagram is

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Anharmonicphononbehaviorviairreduciblederivatives:self-consistentperturbationtheoryandmoleculardynamicsEndaXiao1andChrisA.Marianetti21DepartmentofChemistry,ColumbiaUniversity,NewYork,NewYork10027,USAand2DepartmentofAppliedPhysicsandAppliedMathematics,ColumbiaUniversity,NewYork,NewYork10027,USA(Dated...

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Anharmonic phonon behavior via irreducible derivatives self-consistent perturbation theory and molecular dynamics Enda Xiao1and Chris A. Marianetti2

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