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there are still many interesting problems involving the electrostatics of cubic geometries .
these have to do with cubic ionic crystals , with the force and potential from cubic charge and mass distributions , and with the electric capacitance of the cube @xcite . here
we will discuss the evaluation of the electrostatic interaction energy of two coinciding homogeneous cubic charge distributions . for unit charge distributions in unit cubes this energy is given by , @xmath1 where , @xmath2 , and @xmath3 this integral arises naturally in the free electron gas theory of conduction electrons in metals , see raimes @xcite .
its actual value is usually not needed for most applications of that theory . in an extension of the theory by essn @xcite ,
however , it is needed .
another application of this type of integral will be given below .
the value of @xmath4 can be calculated exactly .
put , @xmath5 for the electrostatic potential energy from a homogeneous cubic charge distribution .
this potential has been discussed by waldvogel @xcite , by hummer @xcite , and by seidov and skvirsky @xcite . using it we can write ( [ eq.c.def ] ) in the form ,
@xmath6and this makes it possible to find the analytical expression , @xmath7   \right\},\ ] ] ( seidov and skvirsky @xcite ) .
this evaluates to , @xmath8 using twenty digits .
another expression for @xmath4 in terms of a one dimensional integral has been derived by essn and nordmark .    if we displace one of the cubes in ( [ eq.cubes.defs ] ) one unit along the @xmath9-axis the integral ( [ eq.c.def ] ) changes into an integral for the interaction energy of cubes with one face touching ( see fig .
[ fig1 ] ) .
let us call this integral @xmath10 .
if we displace one cube one unit along both the @xmath9 and the @xmath11-axis we get the integral for cubes with an edge in common , call it @xmath12 . if we finally displace one of the cubes one unit along all three directions of space we get the integral for cubes touching at one corner , call it @xmath13 .
one of the results found below then says that , @xmath14 this might be a new result .
coulomb interaction energy integrals find one of their main applications in hartree and hartree - fock self - consistent field studies of many electron systems , see for example raimes @xcite . as an application of the results of this paper we use them for crude estimates of the energy of electrons moving in a cubic background of smeared out positive charge .
in particular we compare the energies of delocalized electron states with those of localized states . when the density is small the localized states are found to have lower energy .
this is the phenomenon of wigner crystallization @xcite .
consider two electrons of charge @xmath15 in a cubic box with edges of length @xmath16 .
assume that both electrons have constant charge density , @xmath17 in this box .
the coulomb , electrostatic , interaction energy of these charge distributions is then : @xmath18 here @xmath19 denote the cubic boxes over which the integration variables , @xmath2 , take their values .
we now introduce units so that @xmath20 .
the integral can then be expressed in the form , @xmath21 which shows explicitly that this is a six - dimensional integral .
nowadays we are spoilt by systems for doing mathematics by computer .
it is therefore tempting to try these systems whenever some cumbersome integral arises , and frequently they do deliver sensible answers . for the integral ( [ eq.for.c.six.dim.explicit ] ) , however , those that i have tried fail .
brute force ca nt handle the coulomb singularity .
let us see what happens if we start by dividing each cube into @xmath0 sub - cubes : @xmath22 where the indices , @xmath23 and @xmath24 , run from 1 to @xmath25 .
our integral can then be written as the sum , @xmath26 over @xmath27 terms , integrals over pairs of sub - cubes , @xmath28 for sufficiently large @xmath25 most integrals are over pairs of spatially separated sub - cubes and can be easily approximated .
this leads to a brute force approach .
fairly large contributions should , however , come from pairs of cubes that coincide or touch since they are strongly affected by the singularity .
such an approach is clearly clumsy .
the awkward singularity occurs only in the interior of those @xmath0 terms of this sum for which the integration sub - cubes are equal .
if we thus write , @xmath29 where the terms with all three indices the same ( @xmath30 ) are excluded in the double sum , we see that the interior singularities occur in the first sum over coinciding sub - cubes . but these integrals are all identical and equal to , @xmath31 from formula ( [ eq.two.sums ] ) one thus gets , @xmath32 apart from being over a smaller cube , the integral @xmath33 is essentially like the original integral . in fact one easily finds the scaling property , @xmath34 using this equation ( [ eq.simp.sums ] ) becomes , @xmath35 solving for @xmath4 we thus finally have , @xmath36 here the original integral with its singularity has been written as a sum of @xmath37 integrals without ( interior ) singularities .
the non - singular integrals can be approximated by the product of the two cubic volumes divided by the distance between their midpoints .
a simple calculation gives , @xmath38 if we introduce the notation , compare equation ( [ eq.scaling ] ) , @xmath39 we now have , @xmath40 since the number of terms in the sum grows as @xmath27 it is of interest to take advantage of symmetries to reduce it as much as possible . doing this
we find that , @xmath41 is an alternative way of writing equation ( [ eq.no.sing ] ) .
putting , @xmath42 we have that , @xmath43 assuming that @xmath44 .
if we put this into ( [ eq.no.sing ] ) , or ( [ eq.i.sum.rewritten ] ) , we get , @xmath45 the smaller the box , the smaller the error , so there is hope that this expression will converge to the correct value of @xmath4 when @xmath25 goes to infinity , i.e.  that @xmath46 the approximation ( [ eq.basic.approximation ] ) then immediately gives the following estimate for @xmath4 , when @xmath47 , @xmath48 a value which turns out to be correct to two significant digits .
this is encouraging but the convergence for increasing @xmath25 is slow , see table [ table.1st.approx ] . when @xmath49 the error is still @xmath50 .
the exact value is from eqs.([eq.c.analytical.expression ] ) and ( [ eq.c.20.dig ] ) .
.[table.1st.approx ] this table illustrates the slow convergence of @xmath51 .
note that computation time goes as @xmath27 .
[ cols="^,<,<",options="header " , ]     in table [ table.2nd.approx ] some results of this approach are shown . after finding four different sets of quantities
a standard linear equation solver delivers four solutions to the linear set of equations . for @xmath4
this is clearly seen to give excellent values .
the three other integrals converge much more slowly but seem to approach @xmath52 , and @xmath53 , respectively .
here we will use crude estimates of the hartree energy @xcite of electrons that move in a cube of homogeneous positive charge density . using this crude theory
we will investigate whether the electrons tend to delocalize in the cube or if a state with localized electrons has lower energy .
we assume that the electrons either are delocalized in the cube and have constant charge density in the cube or that they localize in one octant of the cube and have constant charge density there .
this means that we can treat either 8 electrons or 8 electron pairs . with these assumptions
the electrostatic interaction energy can be found from the results above .
the kinetic energy is estimated essentially by means of the uncertainty principle and the pauli exclusion principle .
we start with 8 electrons in delocalized states .
they are assumed to move in a cube of side @xmath16 and positive charge @xmath54 .
the energy is then the sum of the kinetic energy , @xmath55 , \ ] ] and the electrostatic energy , @xmath56 in the kinetic energy the two first electrons are assumed delocalized over the cube without nodes in the wave function .
the remaining six must then go into the three degenerate states with one node .
the first term in the electrostatic energy is the self energy of the positive background . then follows the attraction between the background and the eight delocalized electrons .
the final term is the sum of the @xmath57 electron - electron pair repulsion terms .
simplifying this gives the total energy @xmath58 the energy of this closed shell delocalized state should now be compared to the energy of the ferromagnetic localized state with the electrons in one corner each .
we find , @xmath59 for the kinetic energy since all 8 electrons now sit in cubes ( octants ) of side @xmath60 .
they are however alone in their corners ( octants ) so the pauli principle is automatically obeyed .
the electrostatic energy becomes @xmath61 simplification of this using eq .
( [ eq.exact.expression ] ) gives the total energy @xmath62 if we introduce atomic units ( @xmath63 ) so that length is measured in units of the bohr radius we can plot the two energy curves , @xmath64 and get the results of fig .
[ fig8 ] .
finally we give the corresponding results for 16 electrons sharing orbitals pairwise . in the electrostatic energy one
can then essentially change the particle charge @xmath15 to @xmath65 and add the contributions from the repulsion within the pairs .
this gives the two curves , @xmath66 for the delocalized and localized energies respectively .
these curves are plotted in fig .
[ fig16 ] .
one notes that the localized states always have lower electrostatic energy simply because in these states the electrons are better at avoiding each other . for small @xmath16-values
the delocalized states always have lower energy because of the uncertainty principle .
the curves in these plots resemble those of wigner @xcite who predicted that localization gives lower energy in metals at low densities
. this phenomenon is called wigner crystallization .
i am not aware of any comparable study of the electrostatic interaction energies of homogeneous cubic charge distributions .
the algebraic and combinatoric tricks used to eliminate the coulomb singularities in the integrals seem partly new , as well as the result of eq .
( [ eq.result.for.the.four ] ) .
it is possible that these ideas can be generalized to more general integration problems involving the coulomb singularity .
it is a further bonus that these insights into the electrostatics of cubes and their sub - cubes can be used to make simple estimates for the hartree energy of electrons distributed in cubes in different ways .
such simple model systems are of value for the qualitative understanding of more complex systems .                z.  f. seidov and p.  i. skvirsky . gravitational potential and energy of homogeneous rectangular parallelepiped .
e - print : astro - ph/0002496 , at : cornell university library , arxiv.org e - print archive , feb 2000 .
the starting point is the problem of finding the interaction energy of two coinciding homogeneous cubic charge distributions . 
 the brute force method of subdividing the cube into @xmath0 sub - cubes and doing the sums results in slow convergence because of the coulomb singularity . using symmetry and algebra the coulomb singularities can be eliminated . 
 this leads to an accurate numerical algorithm as well as an interesting exact result relating the desired interaction energy to three other interaction energies , namely those of cubes touching each other at a face , at an edge , and at a corner , respectively . as an application 
 a simple model illustrating wigner crystallization is presented .