Perspective Relativity Correlation QED Experiment Wenjian Liu

2025-05-02 0 0 522.87KB 44 页 10玖币

侵权投诉

Perspective: “Relativity + Correlation + QED

= Experiment”

Wenjian Liu∗

Qingdao Institute for Theoretical and Computational Sciences, Institute of Frontier and

Interdisciplinary Science, Shandong University, Qingdao, Shandong 266237, China

E-mail: liuwj@sdu.edu.cn

Abstract

The ultimate goal of electronic structure calculations is to make the left and right

hand sides of the titled “equation” as close as possible. This requires high-precision

treatment of relativistic, correlation, and quantum electrodynamics (QED) effects si-

multaneously. While both relativistic and QED effects can readily be built into the

many-electron Hamiltonian, electron correlation is more difﬁcult to describe due to

the exponential growth of the number of parameters in the wave function. Compared

with the spin-free case, spin-orbit interaction results in the loss of spin symmetry and

concomitant complex algebra, thereby rendering the treatment of electron correlation

even more difﬁcult. Possible solutions to these issues are highlighted here.

Keywords: Relativity; correlation; quantum electrodynamics; static-dynamic-static Ansatz

1 Introduction

Any electronic structure calculation ought to ﬁrst choose an appropriate Hamiltonian

(equation) and then an appropriate wave function Ansatz (method), according to the tar-

get problem and accuracy. After a brief summary of relativistic Hamiltonians in Sec. 2,

arXiv:2210.03470v1 [physics.chem-ph] 7 Oct 2022

we discuss in Sec. 3 the construction of many-electron wave functions, focusing mainly

on the static-dynamic-static (SDS) framework1for strongly correlated systems of elec-

trons. Possible ways for a balanced and adaptive treatment of electron correlation and

spin-orbital coupling (SOC) are further highlighted therein. The paper ends up with per-

spectives in Sec. 4.

2 Relativistic Hamiltonians

As the highest level of theory for electromagnetic interactions between charged particles

(electrons, positrons, and nuclei), quantum electrodynamics (QED) has achieved great

success in ultrahigh-precision electronic structure calculations of few-body systems (up

to ﬁve electrons2). However, it can hardly be applied to many-body systems due to its

tremendous computational cost and complexity on the one hand, and its underlying phi-

losophy on the other hand: QED assumes from the outset that relativistic and QED ef-

fects are dominant over electron correlation, thereby following the “ﬁrst relativity and

QED then correlation” paradigm. It is obvious that this is only true for heavy ions of

few electrons. Moreover, the underlying time-dependent perturbation formalism is not

suited to a high-order treatment of electron correlation: The more electrons and higher

orders, the more diagrams need to be included. It is clear that, for many-electron sys-

tems, it is relativity and correlation that should be accounted for prior to QED effects, in

a time-independent manner, thereby leading to the “ﬁrst relativity and correlation and

then QED” paradigm.3This requires a many-body effective QED (eQED) Hamiltonian,

HeQED, that is linear in the electron-electron interaction, such that HeQEDΨ=EΨis just a

standard eigenvalue equation. HeQED is an effective Hamiltonian in the sense that it acts

on the fermion Fock space as compared with the much larger fermion-photon Fock space

of QED. A natural route to obtain such a no-photon Hamiltonian is to extract appropriate

operators from the lowest-order QED energies.4Interestingly, a completely bottom-up

procedure (i.e., without recourse to QED at all) can also be invoked, either algebraically5

or diagramatically.6Brieﬂy, the procedure starts with the famous ﬁlled Dirac picture7

but incorporates charge conjugation (see, e.g., Ref. 8) in a proper way. To see this, let us

start with the following second-quantized Hamiltonian (under the Einstein summation

convention),

H=Dq

pap

q+1

2grs

pq apq

rs ,p,q,r,s∈PES, NES, (1)

p=hψp|D|ψqi,grs

pq =hψpψq|g12|ψrψsi, (2)

q=a†

paq,apq

rs =a†

pa†

qasar, (3)

which is normal ordered with respect to the genuine vacuum |0iof no particles nor

holes (PES: positive-energy states; NES: negative-energy states). Here, Dconsists of the

Dirac operator and nuclear attraction, g12 represents the electron-electron interaction, and

{ep,ψp}are eigenpairs of the mean-ﬁeld operator D+Veff(r). The ﬁlled Dirac picture can

then be realized in a ﬁnite basis representation by setting the Fermi level below the ener-

getically lowest of the ˜

Noccupied NES {˜

i}. The physical energy of an N-electron state

can be calculated9as the difference between those of states Ψ(N;˜

N)and Ψ(0; ˜

N),

E=hΨ(N;˜

N)|H|Ψ(N;˜

N)i − hΨ(0; ˜

N)|H|Ψ(0; ˜

N)i, (4)

provided that the charge-conjugation symmetry is incorporated properly. To do so, we

ﬁrst shift the Fermi level just above the top of the NES. This amounts to normal ordering

the Hamiltonian H(1) with respect to |0; ˜

Ni, the non-interacting, zeroth-order term of

|Ψ(0; ˜

N)i. Here, the charge-conjugated contraction (CCC) of fermion operators,5e.g.,

apaq=h0; ˜

N|1

2[ap,aq]|0; ˜

Ni,p,q∈PES, NES (5)

2h0; ˜

N|a˜

pa˜

q|0; ˜

Ni|e˜

p<0,e˜

q<0−1

2h0; ˜

N|aqap|0; ˜

Ni|ep>0,eq>0(6)

=−1

2δp

qsgn(eq),∀p,q, (7)

must be invoked, so as to obtain

H=HeQED

n+Cn, (8)

HeQED

n=Dq

p{ap

q}n+1

2grs

pq{apq

rs }n+Qq

p{ap

q}n, (9)

p=˜

p+¯

p=−1

2¯

gqs

pssgn(es), (10)

p=−1

2gqs

pssgn(es), (11)

p=1

2gsq

pssgn(es). (12)

The constant, zero-body term Cn=h0; ˜

N|H|0; ˜

Nidoes not contribute to the physical en-

ergy and will be ‘renomalized away’. Noticeably, the two effective one-body terms, ˜

and ¯

Q, involve summations over all positive- and negative-energy states, a direct conse-

quence of the CCC (5) (which treats the ﬁlled negative-energy electron and positron seas

on an equal footing, as it should be). Had the standard contraction of fermion opera-

tors, h0; ˜

N|apaq|0; ˜

Ni=δ˜

qn˜

q, been taken, an inﬁnitely repulsive potential, ¯

gq˜

p˜

in˜

i, would

be obtained, such that no atom would be stable! As can be seen from Fig. 1(c) and 1(d),

the direct term ˜

Q(11) and exchange term ¯

Q(12) are precisely the vacuum polarization

(VP) and electron self-energy (ESE), respectively. It also deserves to be mentioned that,

although the diagrams Fig. 1(c) and 1(d) are asymmetric, their weight factors are still 1/2

instead of 1, again due to the averaging of the negative-energy electron and positron seas.

The expression (9) can be rewritten in a more familiar form known from nonrela-

tivistic quantum mechanics, by further normal ordering with respect to |N;˜

Ni, the non-

interacting, zeroth-order term of |Ψ(N;˜

N)i. Since only PES are involved here, the stan-

dard contraction of fermion operators, e.g.,

apaq=hN;˜

N|{apaq}n|N;˜

Ni=hN; 0|apaq|N; 0i=δp

qnq,eq>0, (13)

should be invoked, thereby leading to

HQED

n=EQED

ref +fQED

pq {ap

q}F+1

2grs

pq{apq

rs }F, (14)

fQED

pq =f4C

pq +Qq

p, (15)

f4C

pq =Dq

p+ (VHF)q

p,(VHF)q

p=¯

gqj

pj, (16)

EQED

ref =hN;˜

N|HeQED

n|N;˜

Ni=E4C

ref +Qi

i, (17)

E4C

ref = (D+1

2VHF)i

i. (18)

For more comprehensive elucidations of the above bottom-up construction5,6 of HeQED

(9)/(14), we refer the reader to Refs. 3,6,10–13. Some brief remarks are sufﬁcient here.

HeQED

ndiffers from the eQED Hamiltonian obtained in a top-down fashion4primarily

in that the latter adopts the no-pair approximation (NPA) from the outset (via projec-

tion operators that have to be chosen carefully), but which is not invoked in HeQED

n. As

such, HeQED

n(9)/(14) represents the most accurate many-electron relativistic Hamiltonian,

and hence serves as the basis of “molecular QED” and new physics beyond the standard

model of physics (see recent investigations14,15 in this context, although QED effects were

treated only approximately therein). Given its short-range nature, the Qpotential (10) can

readily be ﬁtted into a model operator for each atom,16,17 so as to treat the VP-ESE (Lamb

shift) variationally at the mean-ﬁeld level. Overall, the present time-independent formu-

lation is not only elegant but also simpliﬁes greatly the derivation of QED energies: The

full, Møller-Plesset (MP)-like second-order QED energy involves only three Goldstone

diagrams in the present case (cf. Fig. 2), but involves in total 28 Feynman diagrams in the

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摘要：

Perspective:Relativity+Correlation+QED=ExperimentWenjianLiuQingdaoInstituteforTheoreticalandComputationalSciences,InstituteofFrontierandInterdisciplinaryScience,ShandongUniversity,Qingdao,Shandong266237,ChinaE-mail:liuwj@sdu.edu.cnAbstractTheultimategoalofelectronicstructurecalculationsistomaketh...

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