Inconsistent kinetic energy functionals of electron gases in the presence of inhomogeneous magnetic fields

The two formally equivalent kinetic energy functionals of an electron gas in the presence of inhomogeneous magnetic fields give inconsistent results when used in a Gordon-Kim (interacting closed-shell atom) calculation. This inconsistency is a direct measure of the accuracy of the Thomas-Fermi (slowly varying potential) assumption for the system studied.


RESEARCH NOTE
Inconsistent kinetic energy functionals of electron gases in the presence of inhomogeneous magnetic fields (Received 17 August 1990; accepted 22 August 1990) The two formally equivalent kinetic energy functionals of an electron gas in the presence of inhomogeneous magnetic fields give inconsistent results when used in a Gordon-Kim (interacting closed-shell atom) calculation.This inconsistency is a direct measure of the accuracy of the Thomas-Fermi (slowly varying potential) assumption for the system studied.
Recently we have constructed a ground state energy functional for an inhomogeneous electron gas in the presence of a weak inhomogeneous magnetic field [1].We used the functional to calculate the nuclear magnetic shielding tensor of the 3Z~+ state of H2 [2].The method of calculation was similar to an earlier calculation of the 3 + magnetic susceptibility tensor of g u H2, which used the ideas of Gordon and Kim [3][4][5].Both calculations lead to gauge-invariant physical quantities.In this note we point out that an alternative method for calculating the gauge-invariant kinetic energy is not consistent with our earlier method when the magnetic field is inhomogeneous and we use a Gordon-Kim type theory.
The relevant portion of the kinetic energy functional has been calculated using the following expression (for a somewhat modified version of this expression and its use in Thomas-Fermi theory see [6]): ( T O is a functional of the density p(r) and the magnetic field B(r).The density also depends upon the magnetic field.
To is written in the form (1) because V(r) (and p(r)) are assumed to vary slowly in space.Now the interacting closed shell aspect of the calculation means that for, say, two atoms [4,5] AT0 m To(PA + PB) --T0(PA) --T0(P6). ( Suppose that we calculate To using the alternative textbook method.Under the 0026-8976190 $3.00 ~ 1990 Taylor & Francis Lid The kinetic energy contribution to the parallel and perpendicular components of the interaction nuclear magnetic shielding tensor of the 3 + 2:u state of He, shown versus internuclear separation, as calculated using the electron gas theory of [2].The curves labelled 'asymmetric' result from the use of To, while those labelled 'symmetric' result from the use of To'.assumptions of the last paragraph, we have h 2 /" t' e -i vt,),/h T~ = + ~ 2 /d3r / dt--lira V,,.V,(rtlr'O>A. (4) J Jc 2~it ,,., . 4 priori we can see, by integrating by parts, that T~ and To differ by V V(r), as expected.However, when .4 is zero or arises from a constant or locally constant magnetic field, the two forms To and T~ are identical.This result may be shown by direct calculation.When B is inhomogeneous, such as is the case in our nuclear shielding calculations [2], the results obtained do actually differ, as may be seen in the figure.The inconsistency described above may be looked upon as a test of the assumption of a slowly varying V(r) made in the electron gas theory of interacting closed shell systems.3 + Y~u H2 is an 'electron desert', as we have said previously [2].When the interacting systems are more elecron-rich, like 129Xe, then we expect the disagreement between To and T~ to diminish.
Given the inconsistency, which form is superior?We may only give a weak answer: To arises out of the natural method of expectation values and also gives more realistic results.
One final point is of interest.Although the kinetic energies are consistent when B is locally homogeneous, the exchange energy diverges [3].This divergence cannot be corrected by a simple gradient expansion.On the other hand, treating the inhomogeneous B field 'exactly' renders the exchange energy finite.Thus we may correct for the inconsistencies in the kinetic energy by gradient expansions that do not qualitatively modify the exchange energy.
By C. J. GRAYCE and R. A. HARRIS Department of Chemistry, University of California, Berkeley, California 94720, U.S.A.
where V(r) is the effective angle particle potential, itself a function of the density of electron and magnetic field.(rt[r'O)~ is the single-particle propagator of a free electron in the presence of a vector potential A; [ it (p e A~][r,).