Self-interaction in Green's-function theory of the hydrogen atom

0.16 MB·English

by  W. Nelson

submitted to Phys.Rev. A
Self-interaction in Green’s-function theory of the hydrogen atom
W. Nelson,1,∗ P. Bokes,2,3 Patrick Rinke,3,4 and R. W. Godby1,3,†
1Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom
2Department of Physics, Faculty of Electrical Engineering and Information Technology,
Slovak University of Technology, Ilkoviˇcova 3, 841 04 Bratislava, Slovak Republic
7 3European Theoretical Spectroscopy Facility (ETSF)
0 4Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4–6, 14195 Berlin, Germany
0 (Dated: February 6, 2008)
2
Atomichydrogenprovidesauniquetestcaseforcomputationalelectronicstructuremethods,since

n itselectronic excitation energiesareknownanalytically. Withonlyoneelectron,hydrogencontains
a no electronic correlation and is therefore particularly susceptible to spurious self-interaction errors
J
introducedby certain computational methods. In thispaper we focus on many-bodyperturbation-

4 theory (MBPT) in Hedin’s GW approximation. While the Hartree-Fock and the exact MBPT
2 self-energyarefreeofself-interaction, thecorrelation partoftheGW self-energydoesnothavethis
property. Here we use atomic hydrogen as a benchmark system for GW and show that the self-
] interaction part of the GW self-energy, while non-zero, is small. The effect of calculating the GW
i
c self-energy from exact wavefunctions and eigenvalues, as distinct from those from the local-density
s approximation, is also illuminating.
-
l
r
PACSnumbers: 31.25.Jf,31.15.Lc,31.15.Ar
t
m
.
t I. INTRODUCTION analyzed the transition between the high-pressure solid
a
m phase and the low density, atomic-like limit. For indi-
vidual atoms, GW electron removal and addition ener-
- Abinitiomany-bodyquantummechanicalcalculations
d gies(weusetheterm“quasiparticle”energiesbyanalogy
are crucially important to our understanding of the be-
n with the solid-state situation) have been investigated by
o havior of atomic, molecular and condensed matter sys- ShirleyandMartin[3],Dahlenetal. [4,5],Stanetal. [6]
c tems. It is well known that prediction of the behavior and Delaney et al. [7], although hydrogen was not con-
[ of these systems requires the description of electronic
sidered. These studies have shown that GW, in general,
correlation. Whilst density-functional theory (DFT) in
1 gives quasiparticle properties which are much improved
v the local-density approximation (LDA) does this with over DFT and Hartree-Fock methods, even for atoms.
2 startlingsuccessinmany cases,it doessoatthe expense
Inthispaperweusethehydrogenatomasabenchmark
9 of a non-physical electron self-interaction. For delocal-
system to quantify the self-interaction error in the GW
5 ized electron systems this self-interactionbecomes negli-
approach. Since the self-energy diagrams beyond GW,
1
gible, but in atomic or strongly localized electronic sys-
0 known as the vertex correction, must by definition cor-
tems it plays an important role. If one is interested in
7 rect this self-interaction error, our findings are relevant
thecalculationofquasiparticleexcitationspectra,many-
0 for research into vertex functions for the many-electron
/ body perturbation-theory (MBPT) is formally a correct problem.
t
a way to proceed. For solids, MBPT in Hedin’s GW ap-
Attentionhasrecentlyfocusedontheprospectsforim-
m proximation[1]hasbecomethemethodofchoice,butitis
provingtheusualnon-self-consistentGW calculationsby
alsoincreasinglybeing appliedto molecular systems and
- choosing an initial Green’s function, G , that is physi-
d clusters. The GW self-energy can be decomposed into 0
cally more reasonable than the LDA (e.g. [2, 8, 9]). We
n correlation and exchange parts, where the latter is the
o explore this here by determining the sensitivity of the
same as the Fock operator encountered in Hartree-Fock
c self-interaction error to the use of the exact hydrogenic
theory and thus self-interaction free. While the exact
: orbitals and energies in place of those from the local-
v self-energy must also be free of self-interaction, the cor-
i density approximation (LDA). We also assess the error
X relation part of the GW self-energy does not have this
introducedintoGW calculationsbyemployingfirst-order
property. To investigate the influence of self-interaction
r perturbationtheoryinsolvingthequasiparticleequation
a intheGW approachthehydrogenatomprovidesanideal
(as opposed to the full numerical solution), and we ana-
case, because the exact solution is known analytically.
lyze the quasiparticle wavefunctions that emerge from a
Hydrogen in its solid phase has previously been stud- full solution.
ied within the GW approximation by Li et al. [2], who
II. HARTREE-FOCK VS. DFT-LDA
∗Presentaddress: Department ofPhysics, King’sCollegeLondon,
Strand,LondonWC2R2LS,UnitedKingdom In many-body perturbation theory the quasiparticle
†Electronicaddress: [email protected] excitationenergies ǫiσ and wavefunctionsψiσ are the so-
2
lutions of the quasiparticle equation potential v , which is formally given as the functional
xc
derivative of the exchange-correlationenergy E :
xc
H0(r)ψiσ(r)+ dr′Mσσ′(r,r′;ǫqiσp)ψiσ′(r′)=ǫqiσpψiσ(r) δE [n]
Xσ′ Z (1) vxc(r)= δnx(cr) . (8)
where, in Hartree atomic units, H (r)= 1 2+v (r)
and v (r) is the external poten0tial. I−t 2is∇custoemxtary In analogy to the quasiparticle equation (1) the Kohn-
ext Sham eigenvalues ǫKS are often interpreted as excitation
to divide the mass operator M into the local Hartree iσ
energies, although this is not formally justified.
potential (v ) and the non-local self-energy (Σ)
H One of the most common approximations for E is
xc
Mσσ′(r,r′;ǫ)=vH(r)δ(r r′)δσσ′ +Σσσ′(r,r′;ǫ). (2) thelocal-densityapproximation(LDA)[10],inwhichthe
− many-bodyexchangeandcorrelationcontributionstothe
Omitting correlation contributions from Σ yields the total energy (Exc) are included by comparison with the
exact-exchange or Hartree-Fock case, where the self- homogeneous electron gas (HEG):
energy takes the form
ELDA[n]= drn(r)ǫHEG[n(r)] . (9)
occ ψ (r)ψ∗ (r′) xc xc
Σxσσ′(r,r′)=− iσr riσ′ δσσ′ (3) Z
Here [12] we follow the parameterization of Perdew and
i | − |
X
Zunger [13] for the exchange-correlation energy density
and the Hartree potential is given by ǫHEG[n(r)]ofthehomogeneouselectrongasbasedonthe
xc
occ ψ (r′)ψ∗ (r′) data of Ceperley and Alder [14].
v (r)= dr′ iσ iσ . (4) TheLDAhasbeenremarkablysuccessfulataccounting
H r r′
forcorrelationeveninsystemsthatarehighlyinhomoge-
i,σ Z | − |
X
neous. However,itiswellknownthattheintroductionof
Since the sum runs over all occupied states the Hartree correlationintheLDAcomesattheexpenseoftheexact
potential contains an artificial interaction of electrons treatment of the self-interaction. Because the exchange
with themselves. For the hydrogen atom this so-called functional is taken from the homogeneous electron gas
self-interaction of the electron is the only content of the it no-longer cancels the spurious self-interaction present
Hartree potential and may be calculatedanalytically us- in the Hartree term. In most systems this is a minor
ing the exact ground-state wavefunction effect and is more than compensated by the improved
treatment of the electron correlation. The LDA can be
1
ψ↑ (r)= e−r (5) improvedby explicitly removingthe self-interaction[13],
1s √π
however this becomes increasingly difficult as the sys-
as tem’s complexity is increased.
1
v (r)= 1 (1+r)e−2r . (6)
H
r − III. THE GW APPROXIMATION
(cid:8) (cid:9)
Theself-interactionisapositive,decreasingfunctionand
hence tends to delocalize the wavefunction (i.e. incor- In Hedin’s GW approximation [1] the self-energy in
rectly pushes its weight away from the nucleus). Eq. 2 is given by
In Hartree-Fock the self-interaction terms introduced
intheHartreepotentialareexactlycanceledbytheFock Σ (r,r′;ǫ)= i ∞dǫ′eiǫ′δG (r,r′;ǫ+ǫ′)W(r,r′;ǫ)
σσ σσ
operator Σx. This makes Hartree-Fock exact for one- 2π Z−∞
electron systems such as the hydrogen atom. However, (10)
the lack of correlation renders Hartree-Fock unsuitable where δ is an infinitesimal positive time. At the level of
for many polyatomic systems of interest. GW, spin flips are not accounted for [2], and the input
Of more practical use is Kohn-Sham [10] density- Green’s function is diagonal in its spin representation
functional theory, which by virtue of the Hohenberg- Gσσ =Gσσ′δσσ′
Kohn theorem [11] establishes an exact and universal IncommonwithusualGW calculations,aKohn-Sham
functional relationshipbetween the ground-state density Green’s function G0 is used for G, given by [15]
n(r) and the total energy E of a system. Mapping the
φ (r)φ∗ (r′)
system of interacting electrons onto a fictitious system G0 (r,r′,ǫ)= iσ iσ , (11)
of non-interacting electrons, that reproduces the exact σσ i ǫ−ǫKiσS∓iδ
X
density, yields the Kohn-Sham equations:
and makes the non-interacting polarizability
[H (r)+v (r)+v (r)]φ (r)=ǫKSφ (r). (7)
0 H xc iσ iσ iσ i ∞
χ0 (r,r′,ǫ)= dǫ′G0 (r,r′;ǫ′ ǫ)G0 (r′,r;ǫ′)
All electron-electron interactions beyond the Hartree σσ −π σσ − σσ
Z−∞
mean field are encompassed by the exchange-correlation (12)
3
and the dielectric function spin-dependent (though spin-
diagonal). Theinversedielectricfunctionintherandom- 0.02
phase approximation 0.5
0.00
y
−1lit 0.4 -0.02
ε−1(r,r′,ǫ)="δ(r−r′)−Z dr′′v(r,r′′) σ χ0σσ(r′′,r′,ǫ)# babi -0.04 difference
X (13) o 0.3 -0.06 to exact
r
p
and thus the screened Coulomb interaction
l 0 1 2 3 4
a 0.2
di exact+GW radius [bohr]
W (r,r′,ǫ)= dr′′ε−1(r,r′′,ǫ)v(r′′,r′). (14) a
0 r exact
Z 0.1 LDA+GW
then emerge as spin-independent quantities, giving rise LDA
to the simple spin dependence in the GW self-energy 0.0
(Eq. 10). 0 1 2 3 4 5
Fornumericalconvenienceandphysicalinsightwesep- radius [bohr]
arate the GW self-energy (10) according to
FIG. 1: Radial probability distributions of the hydrogen 1s
Σ = i[G v+G (W v)]=Σx +Σc . (15)
σσ − σσ σσ − σσ σσ state: the quasiparticle wavefunction deviates only slightly
from the exact wavefunction, when the latter is used as a
The first term (Σxσσ) corresponds to the Fock operator starting point (exact+GW). The LDA wavefunction, on the
inEq.3andwillexactlycanceltheself-interactionintro-
otherhand,ismoredelocalizedasaresultoftheinherentself-
duced by the Hartree potential. It is therefore immedi- interaction. Adding quasiparticle corrections (LDA+GW)
ately clear that any deviation from the exact result for bringstheresultingquasiparticlewavefunction slightly closer
hydrogencan only come from the correlationpart of the to theexact one again. The inset shows thedifference tothe
self-energy (Σc ). exact wavefunction.
σσ
The incorrect self-interaction affects the electron re-
moval energies (here, the ionization potential). For elec-
tron addition energies such as the electron affinity, the function G . The expression for the self-energy on the
0
entireHartreepotentialhasaphysicallyreasonableinter- real frequency axis is obtained by analytic continuation
pretation, since it acts on the wavefunction of the origi- [16]. The current implementation has been successfully
nally unoccupied state which has not contributed to the applied to jellium clusters [17] and light atoms [7].
electron density. Our code allows us to solve the quasiparticle equation
(1) for the GW self-energy with no further approxima-
tion. However,inordertoseparatethecontributionthat
IV. COMPUTATIONAL APPROACH arises from the correlation part of the self-energy from
thatoftheexchangepartandthe Hartreeandexchange-
Wesolvethequasiparticleequation(1)withtheG W correlationpotentialwealsosolvethequasiparticleequa-
0 0
self-energy (10) for the quasiparticle energies and wave- tion with the frequently made approximation that the
functions by fully diagonalizing the quasiparticle Hamil- quasiparticlewavefunctionsaregivenbytheKohn-Sham
tonian in the basis of the single particle orbitals of the wavefunctions. The resulting equation for the quasipar-
non-interacting system. Since the ground state of the ticle energies is
hydrogen atom (Eq. 5) is spherically symmetric, it is
ǫqp =ǫKS+ Σx + Σc (ǫqp) vxc , (17)
sufficient to describe all non-local operators in the GW iσ iσ h σσi h σσ iσ i−h σ i
formalismbytworadialandonespincoordinates,r,r′,σ
where the brackets denote matrix elements with re-
and one angular coordinate, θ, that denotes the angle hi
spect to the Kohn-Sham wavefunction φ .
between the vectors r and r′. The self-energy (Eq. 10) iσ
In orderto explore the role ofthe starting points for a
then assumes the much simpler form
GW calculation, two possible Kohn-Sham input Green’s
functions are chosen. First, the familiar LDA, and, sec-
∞
Σ (r,r′,θ;ω)= [Σ (r,r′;ω)]P(cosθ)δ (16) ond, the exact Kohn-Sham solution for the hydrogen
σσ lσ l σ,σ
atom which has the exact wavefunction of the hydrogen
l=0
X 1s state (5) and v (r) = v (r). (This exact Kohn-
xc H
−
where P (cosθ) is a Legendre polynomial of order l. Sham Green’s function, incidentally, differs from the ex-
l
The Legendre expansion coefficients of the self-energy act Green’s function of the hydrogen atom, because the
are calculateddirectly, thereby surpassingthe need for a exact Kohn-Sham unoccupied eigenvalues do not signify
numerical treatment of the angular dependence. We use electron affinities. The exact Green’s function cannot
a real-space and imaginary time representation [16] to be constructed from any orthonormal set of one-particle
calculatetheself-energyfromthenon-interactingGreen’s wavefunctions.)
4
Exact HF LDA LDA+GW Exact+GW significantly more delocalized. The GW self-energy cor-
−13.61 −13.61 −6.36 −12.66 −13.40 rects this to a small extent (as reflected in the quasipar-
ticle wavefunction), but the remaining discrepancy reit-
erates the unsuitability of the LDA as a starting point
TABLE I: Quasiparticle energies (eV) for the 1s state of hy- for GW in this self-interaction-dominated atom.
drogen (the ionization potential) obtained by diagonalizing
For an analysis of the contributions to the self-energy
thequasiparticle Hamiltonian (1). Two GW calculations are
we turn to the perturbative solution of the quasiparti-
shown,startingfromtheLDAandfromexactKohn-Sham,re-
cle equation using Eq. 17, shown in Table II. When
spectively. For comparison, the Hartree-Fock (HF) and LDA
the exact Kohn-Sham wavefunction and eigenvalues are
eigenvalues are also shown.
used,asintheHartree-Fockcasetheexchangepartofthe
self-energyis seento cancelthe self-interactioncontribu-
Kohn-ShamG0 ǫG1sW hΣxi hvxci hΣci tion from the Hartree potential exactly. The correlation
LDA −12.93 −15.38 −8.25 0.56
part, on the other hand, is not zero, but amounts to a
Exact −13.35 −17.00 −17.00 0.25
self-polarization of 0.25 eV. When the LDA is used as
starting point the influence of the LDA wavefunction on
TABLEII:Quasiparticleenergies(eV)forthe1sstateofhy- the exchange operator becomes apparent and it reduces
drogenobtainedbysolvingEq. (17). Thecontributionsfrom from 17.00eVintheexactcaseto 15.38eV.Thiscor-
− −
theexchangehΣxiandcorrelationhΣcipartoftheself-energy rectsthehighlyoverestimatedLDAeigenvalueforthe 1s
are compared to that of the exchange-correlation potential stateof 6.36eV(seeTableI)to 13.49eV.However,in
hvxcifortheLDAandtheexactcase(vxc =−vH)asastart- thiscase−thecontributionfromthe−correlationpartofthe
ing point. Exact value for ǫ1s is −13.61 eV. GW self-energy is even larger than when starting from
the exact case and increases the quasiparticle energy to
12.93 eV.
V. RESULTS AND DISCUSSION −
The calculated ionization potentials (from a full solu-
VI. CONCLUSION
tion of the quasiparticle equation) are shown in Table
I. The self-interaction errors in the two GW quasipar-
We have performed spin-resolved benchmark calcula-
ticle energies are seen to be fairly small: 0.95 eV when
tionsfortheGW formalismusingtheanalyticallyknown
theapproximateLDAKohn-Shamstartingpointisused,
solutionsofthehydrogenatomasareference,makingthe
and the much smaller 0.21 eV when the exact Kohn-
self-interaction error introduced by the correlation part
Sham starting point is used. Clearly the LDA is such a
of the GW self-energy directly assessable. When the ex-
physically poor representation of the correct physics in
act Kohn-Sham Green’s function is used as the input
this extreme system (owing to the large self-interaction
to GW, the self-interaction error is small (0.21 eV, one-
present in the LDA calculation itself, as reflected in the
thirtieththe sizeofthatinthe LDA),but notnegligible.
large error in the LDA Kohn-Sham eigenvalue) that it
IftheLDAKohn-ShamGreen’sfunctionisused,asdone
forms a very unsuitable starting point for GW. How-
in many GW calculations for more complex systems, a
ever, a physically reasonable starting point reduces the
larger self-interaction error remains, inherited from the
GW self-interaction error to a small size.
LDA starting point.
Since Σ gives a non-vanishing contribution to the
c
h i
hydrogen1sstate,eveniftheanalyticsolutionisusedas
astartingpoint,thequasiparticlewavefunctionwilldiffer
from the exact one. Figure 1 shows that the GW corre- Acknowledgments
lationgivesrisetoaslightdelocalizationofthequasipar-
ticlewavefunctioninthiscase. Thisrelaxation,however, This work was supported by the NATO Security
now makes the quasiparticle wavefunction an eigenfunc- Through Science Programme (EAP.RIG.981521), the
tion of the quasiparticle Hamiltonian and reduces the Slovak grant agency VEGA (project No. 1/2020/05)
deviationfromtheexactenergyofthe1sstateto0.2eV, and the EU’s 6th Framework Programme through the
asshowninTableI. IntheLDAtheself-interactionerror NANOQUANTA Network of Excellence (NMP4-CT-
ismuchmorepronouncedandthewavefunctionbecomes 2004-500198).
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