Real-space Greens function approach for intrinsic losses in x-ray spectra J. J. Kas and J. J. Rehr Dept. of Physics Univ. of Washington Seattle WA 98195

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Real-space Green’s function approach for intrinsic losses in x-ray spectra∗

J. J. Kas and J. J. Rehr

Dept. of Physics, Univ. of Washington Seattle, WA 98195

(Dated: October 27, 2022)

Intrinsic inelastic losses in x-ray spectra originate from excitations in an interacting electron

system due to a suddenly created core-hole. These losses characterize the features observed in x-ray

photoemission spectra (XPS), as well as many-body eﬀects such as satellites and edge-singularities

in x-ray absorption spectra (XAS). However, they are usually neglected in practical calculations.

As shown by Langreth these losses can be treated within linear response in terms of a cumulant

Green’s function in momentum space. Here we present a complementary ab initio real-space Green’s

function (RSGF) generalization of the Langreth cumulant in terms of the dynamically screened core-

hole interaction Wc(ω) and the independent particle response function. We ﬁnd that the cumulant

kernel β(ω) is analogous to XAS, but with the transition operator replaced by the core-hole potential

with monopole selection rules. The behavior reﬂects the analytic structure of the loss function, with

peaks near the zeros of the dielectric function, consistent with delocalized quasi-boson excitations.

The approach simpliﬁes when Wc(ω) is localized and spherically symmetric. Illustrative results and

comparisons are presented for the electron gas, sodium, and some early transition metal compounds.

I. INTRODUCTION

Intrinsic losses in x-ray spectra are fundamental to the

photoabsorption process.1They originate from the dy-

namic response of the system to a suddenly created core-

hole, leading to dynamic screening by local ﬁelds and

inelastic losses. This transient response is responsible for

observable eﬀects in x-ray photoemission spectra (XPS)

and x-ray absorption specta (XAS). These include satel-

lites due to quasi-bosonic excitations such as plasmons,

charge-transfer, and shake-processes, as well as particle-

hole excitations responsible for edge-singularity eﬀects.

These features are signatures of electronic correlation be-

yond the independent particle approximation.2Various

theoretical techniques have been developed for treating

these losses, including plasmon models, quasi-boson ap-

proximations, ﬂuctuation potentials, determinantal ap-

proaches, dynamical-mean-ﬁeld theories, conﬁguration-

interaction methods, and coupled-cluster approaches.3–15

Recently cumulant Green’s function methods have been

developed16 based on a real-space real-time (RSRT) gen-

eralization of the Langreth cumulant.3While the ap-

proach gives good results for the satellites observed

in XPS, even for moderately correlated systems such

as transition metal oxides,16–18 it depends on compu-

tationally demanding real-time time-dependent density

functional theory (TDDFT) calculations of the density-

density response function. Thus despite these advances,

quantitative calculations remain challenging, and intrin-

sic losses are usually neglected in current calculations of

x-ray spectra.

In an eﬀort to facilitate these calculations, we present

here an ab initio real-space Green’s function (RSGF) gen-

eralization of the Langreth cumulant complementary to

the RSRT approach, in which the calculations are car-

ried out using a discrete site-radial coordinate basis. The

formalism of the cumulant kernel β(ω) is analogous to x-

ray absorption spectra µ(ω), except that the transition

operator is replaced with the core-hole potential Vc(r),

and the transitions are between valence and conduction

states with monopole selection rules. The generalized

RSGF approach thereby permits calculations of many-

body eﬀects in x-ray spectra in parallel with RSGF cal-

culations of XAS.1,19 Several representations of β(ω) are

derived, which are useful in the analysis and comparison

with other approximations. For example, we show that

β(ω) can be expressed either in terms of the bare core-

hole potential Vc(r) and the full density response function

χ(r,r0, ω), or the dynamically screened core hole poten-

tial Wc(r, ω) and the independent particle response func-

tion χ0(r,r0, ω). In adddition, we derive the link between

the Langreth cumulant, and the commonly used approxi-

mation based on the GW self-energy.5,20,21 The potential

Wc(r, ω) is a key quantity of interest in this work. How-

ever, its real-space behavior does not appear to have been

extensively studied heretofore. This quantity is closely

related to the dynamically screened Coulomb interaction

W(r,r0, ω) used e.g., in Hedin’s GW approximation for

the one-electron self-energy.5The RSGF approach sim-

pliﬁes when Wc(r, ω) is well localized and spherically

symmetric. This leads to a local model for the cumu-

lant kernel on a 1-dradial grid. The local approach is

tested with calculations for the homogeneous electron gas

(HEG), and illustrative results are presented for nearly-

free-electron systems and early 3d transition metal com-

pounds. We ﬁnd that the local model provides a good

approximation for β(ω). The model also accounts for the

Anderson edge-singularity exponent in metals. The be-

havior of the cumulant kernel reﬂects the analytic struc-

ture of the loss function, with pronounced peaks near the

zeros of the dielectric function. This structure is consis-

tent with interpretations of intrinsic excitations in terms

of plasmons or charge-transfer excitations.

The remainder of this paper is organized as follows.

Sec. II. summarizes the Langreth cumulant and the

RSGF and RSRT generalizations. Sec. III. and IV. re-

spectively describe the calculation details and results for

the HEG and charge-transfer systems. Finally Sec. V.

arXiv:2210.14423v1 [cond-mat.other] 26 Oct 2022

contains a summary and conclusions.

II. THEORY

A. Cumulant Green’s function and x-ray spectra

Intrinsic inelastic losses in x-ray spectra including

particle-hole, plasmons, shake-up, etc., are characterized

by features in the core-hole spectral function A(ω) =

Σn|Sn|2δ(ω−εn), where Snis the amplitude for an exci-

tation of energy εndue to the creation of the core-hole.

Equivalently Ac(ω) is given by the Fourier transform

of the core-hole Green’s function, i.e., the one-particle

Green’s function with a deep core-hole ccreated at t= 0

gc(t) = h0|c†

ceiHtcc|0iθ(t),

Ac(ω) = −1

πIm Zdt e−iωtgc(t).(1)

Here His the Hamiltonian of the system while c†

cand

ccare creation and annihilation operators, respectively.

The core-level XPS photocurrent Jk(ω)∼Ac(ω) is di-

rectly related to the spectral function, which describes

both the asymmetry of the quasi-particle peak and satel-

lites in the spectra. Intrinsic losses in x-ray absorption

spectra (XAS) µ(ω) and related spectra (e.g., EELS)

from deep core-levels6can be expressed in terms of a con-

volution of Ac(ω) and the single (or quasiparticle) XAS

µ1(ω)6

µ(ω) = Zdω0Ac(ω0)µ1(ω−ω0).(2)

This accounts for eﬀects such as satellites and the re-

duction factor S2

0in the XAS ﬁne structure.1Here and

below, unless otherwise noted, we use atomic units e=

¯h=m= 1 with distances in Bohr = 0.529 ˚

A and ener-

gies in Hartrees = 27.2 eV. A cumulant Green’s function,

which is a pure exponential in the time-domain, is par-

ticularly appropriate for the treatment of intrinsic losses,

gc(t) = g0

c(t)eCc(t),(3)

where g0

c(t) = eictis the independent particle Green’s

function for a given core-level c, and Cc(t) is the cu-

mulant. This representation naturally separates single

(or quasi-particle) and many-particle aspects of the ﬁnal-

state of the system with a deep core hole following pho-

toabsorption, where many-body eﬀects are embedded in

the cumulant. It is convenient in the interpretation to

represent the cumulant in Landau form,22

Cc(t) = Zdω β(ω)eiωt −iωt −1

ω2.(4)

where the cumulant kernel β(ω) characterizes the

strength of the excitations at a given excitation en-

ergy ω. This representation yields a normalized spectral

function, with a quasi-particle renormalization constant

Z= exp(−a), where a=Rdω β(ω)/ω2is a dimensionless

measure of correlation strength, and ∆ = Rdωβ(ω)/ω is

the relaxation energy shift of the core-level.55

B. Langreth cumulant

For a deep core-hole coupled to the interacting elec-

tron gas Langreth showed that within linear response,

the intrinsic inelastic losses can be treated in terms of

a cumulant Green’s function, with a cumulant kernel in

frequency and momentum space given by

β(ω) = −X

q|Vq|2S(q, ω).(5)

Here Vqis the Fourier transform of the core-hole poten-

tial and S(q, ω) is the dynamic structure factor, which is

directly related to time-Fourier transform of the density-

density correlation function χ(q, ω) and the loss function

L(q, ω) = −Im −1(q, ω), i.e.,

S(q, ω)≡ −1

πIm χ(q, ω)θ(ω)

=−1

πvq

Im −1(q, ω)θ(ω),(6)

χ(q, ω) = Zdt eiωthρq(t)ρq(0)i.(7)

The response function χ(q, ω) can be expressed in terms

of the non-interacting response χ0(q, ω) using the rela-

tion χ=χ0+χ0Kχ. Here the particle-hole interac-

tion kernel Kwithin TDDFT is given by K=v+fxc,

where v= 4π/q2is the bare Coulomb interaction, and

fxc =δvxc[n]/δn is the TDDFT kernel; this is ob-

tained from the exchange-correlation potential used in

the deﬁnition of the independent particle response func-

tion χ0(ω).

Although Langreth’s expression for β(ω) in Eq. (5) is

elegant for its simplicity, calculations of the full density

response function χ(q, ω) are challenging, being compa-

rable to that for a particle-hole Green’s function or the

Bethe-Salpeter equation (BSE). Moreover, the core-hole

potential Vc(r) has a long range Coulomb tail to deal

with. On the other hand, the Thomas-Fermi screening

length r0= 0.64 r1/2

sis short-ranged, so one may wonder

to what extent a local approximation for the dynamically

screened interaction might be applicable?

To this end, we note that the cumulant kernel can

be expressed equivalently in terms of the dynamically

screened core-hole interactions Wq(ω) = Vq/(q, ω) and

the independent particle response function χ0(q, ω),

β(ω) = 1

πX

q|Vq|2Im χ(q, ω)θ(ω),(8)

≡1

πX

q|Wq(ω)|2Im χ0(q, ω)+

+|χ0(q, ω)|2Imfxcθ(ω).(9)

This equivalence is implicit in Langreth’s analysis of the

low energy behavior of β(ω) for the homogeneous electron

gas (HEG) in the random phase approximation (RPA),

where an adiabatic approximation is also valid (q, ω)≈

(q,0). If the exchange correlation kernel fxc is taken

to be real, as in typical implementations of TDDFT or

ignored as in the RPA (fxc = 0), Eq. (10) reduces to

β(ω) = 1

πX

q|Wq(ω)|2Im χ0(q, ω)θ(ω).(10)

This result in terms of the screened-core-hole potential

Wq(ω) can be advantageous for practical calculations in

inhomogeneous systems. For example, in the adiabatic

approximation Wq(ω)≈Wq(0), only a single matrix in-

version is needed to obtain −1(ω= 0), rather than an

inversion at each frequency.

As a practical alternative to momentum-space, calcu-

lations of the Langreth cumulant for inhomogeneous sys-

tems have recently been carried out by transforming to

real-space and real-time (RSRT).16 The approach ﬁrst

calculates the time-evolved density response δρ(r, t) with

the Yabana-Bertsch reformulation of TDDFT that builds

in a DFT exchange-correlation kernel23

δρ(r, t)≡Zd3r0dt0χ(r, t;r0, t0)Vc(r0, t0).(11)

A Fourier transform then yields the cumulant kernel β(ω)

β(ω) = Re ω

πZ∞

dt e−iωt Zd3r Vc(r)δρ(r, t)θ(ω).(12)

This approach has been shown to give good results for

a number of systems.16 However, the method requires

a computationally demanding long-time evolution of the

density response using a large supercell, with Kohn-Sham

DFT calculations at each time-step. In addition, a post-

processing convolution is needed for XAS calculations.

C. Real-space Green’s function Cumulant

Our primary goal in this work is to develop an al-

ternative, real-space Green’s function formulation of the

Langreth cumulant and it’s key ingredients. In particu-

lar we aim to investigate the behavior of the cumulant

kernel β(ω) and the dynamically screened core-hole po-

tential Wc(r.ω). The method is complementary to the

RSRT formulation but is based on a similar approach for

the response function, and can be carried out in parallel

with RSGF calculations of XAS.1,24 Within the RPA, the

RSGF formulation of β(ω) can be derived from the per-

haps better known GW approximation to the cumulant,

based on the core-level self-energy Σc(ω),20,21,25

β(ω) = 1

πIm Σc(c−ω),(13)

where Σc(ω) = hc|Σ(ω)|ci. This matrix element can be

evaluated in real-space using the GW approximation

Im Σ(r,r0, ω) = −

occ

ψi(r)ψi(r0)×

Im[W(r,r0, i−ω)]θ(i−ω),(14)

where W(r,r0, ω) is the dynamically screened Coulomb

interaction. Within the decoupling approximation,21 the

core-level wave-function is assumed to have no overlap

with any other electrons, and Eq. (13) becomes

β(ω) = 1

πIm

Zd3rd3r0ρc(r)W(r,r0, ω)ρc(r0)θ(ω)(15)

πIm

Zd3r ρc(r)Wc(r, ω)θ(ω),(16)

where Wc(r, ω) = Rd3r0W(r,r0, ω)ρc(r0) and ρc(r) =

|ψc(r)|2. Then noting that Im Wc= Im [KχVc] (indices

suppressed for simplicity) and within the RPA (K=v),

we obtain a real-space generalization of the Langreth cu-

mulant in Eq. (8)

β(ω) = 1

πZd3rd3r0Vc(r)Vc(r0) Im χ(r,r0, ω)θ(ω).(17)

This result is also equivalent to the RSRT expression in

Eq. (12). Alternatively, in analogy with Eq. (10), and

again within the RPA, β(ω) can be expressed in terms of

Wc(r, ω) and the independent particle response function

β(ω) = 1

πZd3rd3r0W∗

c(r, ω)Wc(r0, ω)×

×Im χ0(r,r0, ω)θ(ω).(18)

While Eqs. (16-18) are formally equivalent within the

RPA, they diﬀer if the interaction kernel Kis complex

(which is the case for most non-adiabatic kernels), in

which case it is not obvious which approximation is best.

Here we focus on the RPA expressions only, although a

generalization to adiabatic fxc would be relatively sim-

ple.

The static limit of Eq. (18) is interesting in itself.

At frequencies well below ωp, the core-hole potential is

strongly screened beyond the screening length r0and

nearly static. On expanding χ(r,r0, ω) about ω= 0 and

keeping only the leading terms, one obtains the adiabatic

approximation

β(ω)≈1

πZd3rd3r0W∗

c(r,0)Wc(r0,0) Im χ0(r,r0, ω).

(19)

This limiting behavior is similar to the adiabatic TDDFT

approximation for XAS,26 but with the replacement of

the dipole operator d(r) by the statically screened core-

hole potential Wc(r,0). This potential is also used in

calculations of XAS to approximate the particle-hole in-

teraction, and is similar to the ﬁnal state rule approxi-

mation for the static core-hole potential.

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Real-spaceGreen'sfunctionapproachforintrinsiclossesinx-rayspectraJ.J.KasandJ.J.RehrDept.ofPhysics,Univ.ofWashingtonSeattle,WA98195(Dated:October27,2022)Intrinsicinelasticlossesinx-rayspectraoriginatefromexcitationsinaninteractingelectronsystemduetoasuddenlycreatedcore-hole.Theselossescharacterizeth...

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Real-space Greens function approach for intrinsic losses in x-ray spectra J. J. Kas and J. J. Rehr Dept. of Physics Univ. of Washington Seattle WA 98195

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