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@xmath7 , where @xmath8 are the spin coordinates and @xmath9 are the @xmath10-dimensional spatial coordinates of the @xmath11 electrons , the nodal hypersurface ( or simply `` nodes '' ) is a @xmath12-dimensional manifold defined by the set of configuration points @xmath13 for which @xmath14 .
the nodes divide the configuration space into nodal cells or domains which are either positive or negative depending on the sign of the electronic wave function in each of these domains . in recent years
, strong evidence has been gathered showing that , for the lowest state of any given symmetry , there is a single nodal hypersurface that divides configuration space into only two nodal domains ( one positive and one negative ) .
@xcite in other words , to have any chance to have _ exact _ nodes , a wave function must have only two nodal cells .
for simplicity , in the remainder of this paper , we will say that a wave function has exact nodes if it has the same nodes as the exact wave function .
the number of systems for which the exact nodes are known analytically is very limited . for atoms , it includes two triplet  @xmath15 and @xmath16
 and two singlet  @xmath17 and @xmath18  states of the helium atom and the three - electron atomic state @xmath19 .
@xcite    the quality of fermion nodes is of prime importance in quantum monte carlo ( qmc ) calculations due to the fermion sign problem , which continues to preclude the application of in principle exact qmc methods to large systems .
the dependence of the diffusion monte carlo ( dmc ) energy on the quality of the trial wave function is often significant in practice , and is due to the fixed - node approximation which segregates the walkers in regions defined by the trial or guiding wave function .
recently , mitas and coworkers have discovered a interesting relationship between electronic density , degree of node nonlinearity and fixed - node error . @xcite    here , we study the topology and properties of the nodes in a class of systems composed of electrons located on the surface of a sphere . due to their high symmetry and their mathematical simplicity , these systems are the ideal `` laboratory '' to study nodal hypersurface topologies in electronic states .
moreover , mitas showed that the non - interacting wave function of spin - polarized electrons on a sphere has only two nodal cells which is probably a necessary condition for exactness @xcite ( see above ) .    in this paper ,
we report the analytic expression of the exact nodes for two - electron triplet states ( @xmath0 , @xmath1 , @xmath2 and @xmath3 ) .
we also show that , as in the atomic case , the nodes of the non - interacting wave function for the three - electron state @xmath20 are identical to the nodes of the exact wave function .
in addition to these systems where the non - interacting wave function has exact nodes , we study the quality of the non - interacting nodes for the @xmath5 and @xmath6 configurations . for the @xmath5 configuration , we show that , although not exact , the non - interacting nodes are very accurate and , based on numerical evidences , we conjecture that the non - interacting and exact nodes of the @xmath6 configuration are identical .
we use atomic units throughout .    [
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in this paper , we have studied the fermionic nodes for various electronic states of the `` electrons on a sphere '' paradigm .
we have rigorously demonstrated that , for the @xmath0 , @xmath1 , @xmath2 , @xmath3 and @xmath4 configurations , the non - interacting wave function has the same nodes as the exact wave function .
we have shown that this peculiar feature can be attributed to exact interdimensional degeneracies .
interdimensional degeneracies also explain why the exact nodes of various atomic and molecular systems are known analytically .
therefore , we could potentially used new interdimensional degeneracies to discover the exact nodes for new atomic and molecular systems .
even when the non - interacting nodes are not exact , we have shown that most of the features of the exact nodal surface are captured by the non - interacting nodes .
thus , we expect the fixed - node error to be quite small for these systems . this could be a new , alternative way to obtain accurate near - exact energies for finite and infinite uniform electron gases .
although we have not been able to prove it rigorously , we have conjectured that the nodes of the non - interacting wave function for the @xmath6 configuration are exact .
this claim is supported by numerical evidence .
p.f.l .  and d.b .
would like to thank mike towler for the hospitality at the tti institute during the `` quantum monte carlo in the apuan alps ix '' conference where this work has begun .
thanks the nci national facility for generous grants of supercomputer time and the australian research council for a discovery early career researcher award ( grant no .
de130101441 ) and a discovery project grant ( dp140104071 ) .
thanks peter gill for valuable discussions .
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the aim of this paper is to shed light on the topology and properties of the nodes ( i.e. the zeros of the wave function ) in electronic systems . using the `` electrons on a sphere '' model , we study the nodes of two- , three- and four - electron systems in various ferromagnetic configurations ( @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 , @xmath5 and @xmath6 ) . in some particular cases ( @xmath0 , @xmath1 , @xmath2 , @xmath3 and @xmath4 ) 
 , we rigorously prove that the non - interacting wave function has the same nodes as the exact ( yet unknown ) wave function . 
 the number of atomic and molecular systems for which the exact nodes are known analytically is very limited and we show here that this peculiar feature can be attributed to interdimensional degeneracies . 
 although we have not been able to prove it rigorously , we conjecture that the nodes of the non - interacting wave function for the @xmath6 configuration are exact .