B.G. Carlsson and J. Dobaczewski
July 28, 2010
The description of finite many-body systems is a continuous exciting challenge in physics. In view of the exploding dimensionality of many-particle Hilbert spaces, direct approaches are however often untractable. Instead, density functional theory (DFT)  provides an alternative by showing that a much simpler description in terms of the density alone is in principle possible. The main challenge is finding or deriving the best density functionals that are able to consistently describe large classes of physical systems. In this work we address a related question, namely to what extent the physics of the exact nonlocal exchange potential can be captured by an approximate density functional? This question is not only of fundamental importance but also of great practical interest, and attempts to solve it date back to the early 1950s, involving the Slater as well as the optimized effective potential approximations .
For nuclear-physics applications, Negele and Vautherin developed an alternative method known as the density-matrix expansion (DME) , which takes advantage of the short-range of the nuclear two- and three-body forces. It is particularly convenient as it was originally constructed to generate quasi-local density functionals  of a form similar to the successful phenomenological Skyrme functionals used extensively today . The method have recently gone through a revival [5,6,7,8,9] and appears to be a promising starting point for the construction of ab initio nuclear DFT models [7,8]. The idea is to start from effective forces, derived from first principles, and then transform the expression for the interaction energies into quasi-local density functionals using the DME. Because the starting point is an effective force, this method is able to take correlations into account in a way that goes much beyond the Hartree-Fock treatment of the bare interaction.
In conjunction with these ab initio methods, there are empirical efforts to improve nuclear density functionals, such as the systematic method we recently proposed to construct nuclear density functionals using higher-order derivatives of densities . In such an approach, the DME is helpful as it can serve as a method of classifying different terms according to their importance, similarly to what is available in the effective-field-theory power-counting scheme .
In this Letter, we explore the ideas of the DME to transform the direct and exchange energies resulting from the finite-range Gogny interaction into quasi-local density functionals. We then extend these methods beyond the second order expansions used so far to make the first tests of convergence and accuracy in function of the expansion order.
We begin our analysis by recalling that the nuclear one-body density matrix
which depends on the space-spin-isospin nucleon
can be separated into four nonlocal spin-isospin densities
For a local interaction, the energy corresponding to the direct (Hartree) term depends on the
in Eq. (1) as
Introducing the standard
coordinates, we have
the following expansion of the local densities,
Expanded in this way, and after a partial integration and recoupling needed
to make all derivative operators act on the same density, the direct
interaction energy of Eq. (2) can
be written as a
To test the quality and convergence properties of this expansion, we generated a set of realistic test densities for doubly magic nuclei by solving the self-consistent equations for the nuclear SLy4 functional . For these test densities, the sums in Eq. (4) were evaluated to different orders by calculating the higher-order derivatives of densities needed using the program HOSPHE (v1.00) . The coupling constants in Eq. (4) were calculated from moments of the finite-range part of the Gogny D1S interaction . Then for the same set of test densities, the exact direct and exchange energies of the Gogny interaction were determined using the capability of treating Gaussian interactions with the code HFODD (v2.45h) .
As an example, in Table 1 we show the exact energies calculated in Pb for the four spin-isospin channels. In the ground states of even-even nuclei, the vector terms of the direct energies vanish due to the conserved time-reversal symmetry, and thus our calculations here only test the scalar terms of the expansion in Eq. (4). In Fig. 1 we show percentage deviations between the gradient approximations and the exact results. It is extremely gratifying to see that in light nuclei the sixth-order expansion allows for reaching the precision of 0.1-0.01% or 0.1MeV. Even more important, in each higher order the precision increases by a large factor, which is characteristic to a rapid power-law convergence. One also sees that in heavier, isospin-unsaturated nuclei the precision slightly deteriorates, probably owing to a slower convergence for the isovector densities, which fluctuate more than the isoscalar ones.
One can, of course, quite easily calculate the direct terms exactly, but still it is interesting to see that a few of the lowest-order coupling constants appearing in the expansion are able to capture the essential physics contained in the nuclear finite-range force. For the exchange (Fock) terms whereto we turn now, the situation is significantly different, because here the expansion into quasi-local densities dramatically speeds up the calculations.
In order to deal with the exchange term, we propose
a new approximation, which we call a damped
Taylor (DT) expansion, and we define it in the following way:
The model density can be chosen in different ways by the requirement that the truncated expansions must become exact in certain limits. The model used in the following leads to exact values in SNM, and is obtained by assuming that the density in each point has the SNM nonlocal dependence, For large relative distances, when the expansion around can no longer be trusted, the bracketed term in Eq. (5) is damped to zero and the asymptotic behavior is then given entirely by this model part. In this way the parameter interpolates between placing trust in the model part or in the expansion part, and the optimal balance is determined by the range in which the expansion converges, as estimated above.
By inserting the DT expansion of Eq. (5) into the expression for the
exchange energy, we obtain the quasi-local expansion
In Fig. 3 we show results obtained with the DT expansion compared to the exact values, similar to those listed in Table 1. The comparison is made by considering the RMS deviations between the exact and approximate exchange energies, and is also shown for the NV  and phase-space averaging (PSA)  expansions. As seen in this figure, the NV Bessel-function expansion used in the scalar channels works very well, whereas the alternative method used for the vector channels [see Eq. (4.23) in Ref. ] does not improve when evaluated beyond the second order.
We employ the recently proposed PSA method in the INM-DME version  and generalize it to higher orders by incorporating the model density in the same way as in Eq. (5). Other modifications are probably also possible and could be studied as well. At second order, our prescription reproduces that proposed in Ref.  with the PSA performed over SNM. Fig. 3 shows that this is the best expansion at second order, where the total RMS deviation for the exchange energy is 14.09 MeV. When used for the vector part, the PSA expansion is similar to the NV expansion and also has problems when evaluated beyond second order.
As seen in Fig. 3, the DT expansion exhibits excellent convergence properties both for the vector and scalar parts, with every next order improving the results by large factors. Note that in the vector channels, the exact exchange energies of Table 1 are quite small, so at second order their relative description is poor. In contrast, at higher orders their description becomes rather accurate. The results obtained for the DT expansion are not so sensitive to the value of the damping width , and values in the range of - all lead to improvements over the NV expansion.
The total RMS deviations obtained in the DT case for the exchange energies at fourth and sixth orders are 1.54 and 0.36MeV, respectively . Calculations using a fixed value of fm give results that are not significantly different . They lead to substantial simplifications, because then the density dependencies of coupling constants can be neglected.
In summary, we presented the first analysis of density-matrix expansions extended beyond second order, with derivatives of densities included up to sixth order. We also proposed a new type of expansion that has excellent convergence properties. The results demonstrate that its possible to construct expansions into quasi-local density functionals that converge for the description of low-energy nuclear observables. This also facilitates the difficult adjustments of phenomenological higher-order nuclear functionals by providing inital guesses for free parameters as well as a method to derive them.
We thank Johan Grönqvist for useful discussions. This work is supported in part by the Academy of Finland and the University of Jyväskylä within the FIDIPRO programme, and by the Polish Ministry of Science under Contract No. N N202328234.