DENSITY FUNCTIONALS ON THE BASIS OF THE RELATIVISTIC FIELD THEORY

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DENSITY FUNCTIONALS ON THE BASIS OF THE RELATIVISTIC FIELD THEORY

E. Engel, R. Dreizler, P. Malzacher

To cite this version:

E. Engel, R. Dreizler, P. Malzacher. DENSITY FUNCTIONALS ON THE BASIS OF THE REL- ATIVISTIC FIELD THEORY. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-321-C2-328.

�10.1051/jphyscol:1987249�. �jpa-00226518�

JOURNAL DE PHYSIQUE

Colloque C2, suppl6ment au n o 6, Tome 48, juin 1987

DENSITY FUNCTIONALS ON THE BASIS OF THE RELATIVISTIC FIELD THEORY

E. ENGEL, R.M. DREIZLER and P. MALZACHER

Institut ffir Theoretische Physik, Universitat Frankfurt.

0-6000 Frankfurt-am-Main, F.R.G.

RCsum6.- Nous exposons un sch6ma syst6matique pour obtenir la fonctionnelle de la densit6 dt6nergie relativiste sur la base de la limite Hartree-Fock de 1'Blectrodynamique quantique. En particulier, un analogue relativiste du modsle T.F.D.W. est pr6sent6.

Abstract.- We outline a systematic scheme for the derivation of relati- vistic energy density functionals on the basis of the Hartree-Fock limit of QED. In particular, a relativistic analogue of the nonrelativistic TFDW-model is presented.

1.

Introduction

Density functional methods have become quite a useful tool for the discussion of groundstate properties of nonrelativistic many body systems1. In view of the complexity of the field theoretical many particle problem a generalization of these methods to the relativistic domain would be desirable. As a natural testing ground one would consider (heavy) atomic systems for which QED, the best developped relativistic field theory, applies, and a large number of Dirac-Fock results are available for comparison. In this note we attempt to set up a relativistic TFDW model. The corresponding nonrelativistic limit has been used and discussed extensively. We will, however, keep the arguments general enough in order to approach possible extensions.

2. The groundstate of a relativistic N-electron system in the Hartree-Fock approximation

For the discussion of atomic systems we consider the QED Lagrangian

x = - ^{? F Y'v} + L A B v - 1 (avAv)2-!~

4

2 v 2 F""

4 ext,pv ext

+

4

T, ^{(i7- m} - ^efext)~I

⁵ (-6 - ^m - eJext)

3 1

We work in the Gupta-Bleuler formalism and use the Feynman dagger notation

while the vector bars on the partial derivatives indicate in which direction the derivatives have to be taken. The commutator representation of the Lagrangian ensures that the corresponding Hamiltonian density

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987249

C2-322 JOURNAL DEPHYSIQUE

is hermitian gnd invariant under charge conjugation2. The external (electron- nucleus) interaction Vext ", which is assumed to be time independent, is treated classically. The photon mass

is introduced to avoid infrared divergencies at intermediate steps and will be removed by taking the limit

at the final stage.

The binding energy of an atom is given by the energy difference of the groundstate

Ig> , which is the state of lowest energy in the sector of Fock-space with

(with N being the number of bound electrons), and the vacuum state Iv> , the state of lowest energy with

Even if no normalordered representation of H is used this construction leads to finite energies as all contributions of the 'Fermi-sea', the negative continuum, cancel.

As the nonrelativistic TFDW model can be considered as an approximation of the Hartree-Fock limit we will argue on the basis of the HF limit of QED~. The energy of each state (Eg as well as Ev) can then be expressed in terms of the HF approxi- mated electron propagator

In graphic form we have the well known diagrams

The cross (E3 represents the kinetic energy operator, the second term being the potential energy due to the external potential. The third contribution is readily

identified as the Coulomb energy while the last graph characterizes the exchange energy.

The four current density is related to the Greens function via

Looking at the lowest order diagramms contributing to jV ,

or at the exchange energy graph above, one immediately recognizes that one cannot discuss QED problems without addressing renormalization.

3.

Renormalization

We thus should have started with the renormalized Lagrangian 4

containing the standard renormalization constants Z ,, ^Z2,

and 6m of QED.

3

Renormalization directly modifies the vacuum polarization, the self energy and the irreducible vertex function. Using, for example, the counterterm technique and dimensional regularization we obtain for the four current density in lowest order of

The second term compensates the UV-divergence of the first graph of

( o ) ' reg

Taking into account the renormalization of the lowest orderself energy graph

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where the cross represents a regularized UV-divergent insertion that keeps the self energy bubble finite, we find two identical counterterms for the exchange energy density

Charge renormalization has altered our ~amiltonian(dr0~ index R for brevity).

z ₊

= { [

v,( ^-iy_ - ¹

^-

efeXt)$l+1~(ir -

^{m -}

^{eTfext),$ 1 1}

Here the energy of the external field contains a renormalization constant which will compensate a divergence in the kinetic energy.

4. The gradient expansion

In order to set up the TFDW model we imagine (in analogy to the nonrelativistic case) that the HF propagator can be replaced by a propagator of an effective field theory

where the simple line stands for the free propagator and the insertion for an effective four potential, representing both the electron nucleus and the electron- electron interaction together.

The effective propagator can be expressed in terms of the solutions of the

effective Dirac equation

where s is the threshold which distinguishes occupied and unoccupied states. This sum can be reexpressed in terms of Dirac plane waves,

with the standard spinors u 1

and v'"(~) (in the notation of ref.4).

We now restrict ourselves to a pure electrostatic effective potential Veff such that Heff becomes

The index x indicates,

that Heff,x acts on the coordinate 5 . As the commutator

- ^ig-x + Bm, Ve,,(5) 1

does not vanish,Geff can be evaulated via a gradient e ~ ~ a n s i o n ~ ' ~ ( t o second order in the gradient terms) yielding a functional

Gef f

t veff9ai veff>ai a ^j v eff I .

With Geff we can directly compute the charge density

and the energy density

veff,ai ajVeff I

of our interacting problem. Inverting

to second order in the gradient terms we obtain the desired energy density

In detail, the current indicated above has the form

with the notation

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E ( x ) = S - V (x) eff -

d E 2 (5)- m

where E(x)

m p(tf)

0

elsewhere

The divergent contribution of the lowest order vacuum polarization graph

which occurs in the gradient expanded current exactly as in the original QED current has been removed by renormalization.

The total energy density is given by

2

2 4 2 a

+ A

2m2aZ + a2b2 - 6m a6 Arsh(;) + 3m Areh

1

32a

with

2 1/3 a

(3n

The renormalization procedure has lead to a finite second order contribution of the kinetic ener y and a finite exchange energy (compare the functional to former results in ref. s%-~).

In the nonrelativistic limit

p<<m in our units) this functional goes over into the well known TFDW energy density.

5. The variational equation and preliminary results

It is straightforward to derive the relativistic TFDW variational equation via

Implimenting charge conservation

we obtain

(

! a ) ' - 2

Aa

aa

12@(@+~+j~-l) -

1 + a ^{+ 4} L A r s h a

[l+ZE Arsh a] 2a f32 B~

+ 24 e (a@ 2 - ^{3 Arsh a) 1}

where U represents the total electrostatic potential,

is the Lagrange multiplier of the charge conservation condition and

has been set equal to one for brevity.

This nonlinear differential equation has to be solved in conjunction with the Poisson equation

for appropriate boundary conditions.

A few preliminary results for the case of neutral atoms with high nuclear charge

where one expects to see the influence of relativistic effects, are given in the following table. For comparison we also list the nonrelativistic TFDW ground- state energies9 , which agree with the nonrelativistic HE-ener ies within an error of less than 1%. The relativistic DF data are taken from ref.18 . All energies are given in atomic units.

Z

TFDW RTFDW DF-results deviation

(nonrelativistic) (preliminary) Z

6. Final remarks

We have demonstrated that the model presented above is the correct extension of the nonrelativistic TFDW model to the relativistic domain. Further numerical work with the variational equation is required. The main task in addition is the generali- zation of the RTFDW model to arbitrary external fields by using a full effective four potential rather than a pure electrostatic one when computing the effective Greensfunction. The result will be an energy momentum tensor dependent on a full four current. Such a functional should have all the properties one expects as eg.

gauge invariance.

References 1. See eg.

a) R.M. Dreizler and J. da Providencia, eds.

" Density Functional Methods in Physics ", NATO AS1

Series B 123, Plenum Press, New York (1985)

C2-328 JOURNAL DE PHYSIQUE

b) S. Lundqvist and N.H. March, eds.

"

Theory of the Inhomogeneous Electron Gas ",

Plenum Press, New York (1983)

2. G. Kallen,

Quantum Electrodynamics ", Springer (1972)

3. P.G. Reinhard, W. Greiner, H. ArenhBvel, Nucl. Phys. A166,173(1971) 4. See eg. C.Itzykson, J.B. Zuber,

"

Quantum Field Theory ", McGraw-Hill (1980)

5. D.A. Kirznits,

Field Theoretical Methods in Many Body Systems ",

Pergamon Press (1967)

6. E.K.U. Gross, R.M. Dreizler in ref .I p. 81 - ¹⁴⁰

7. E.E. Salpeter, Astrophys. J 134,669(1961) 8. B. Jancovici, Nuovo Cim. 25,428(1962)

9. W. Stich, E.K.U. Gross, P. Malzacher, R.M. Dreizler, Z. Phys. A309,5(1982)

10.

J.P. Desclaux, At. Data Nucl. Data Tab. 12,-311(1973)