document: the trihydrogen dication , @xmath1 , which consists of three protons and one electron , is among the simplest coulomb systems . 
   its stability has been studied intensely in the sixties and early seventies . in a series of articles , conroy  @xcite investigated the potential energy surfaces of the electronic ground state and the lowest excited states at linear and isosceles triangular configurations . 
   he employed a variational approach in which the electronic trial wavefunction is expanded around the center of the nuclear charges . analyzing the contour plots conroy concluded that @xmath1 is not stable . 
   schwartz and schaad @xcite , and somorjai and yue  @xcite , who reported single - point calculations of the system @xmath2 at the supposed equilibrium equilateral triangular configuration of @xmath3 , did not address the stability problem . to assess conroy s results , berkowitz and stocker  @xcite searched for this ion through charge stripping experiments on @xmath4 . 
   they could not find evidence of stable @xmath1 . 
   later , the issue was reconsidered also from the theoretical side , by shoucri and darling  @xcite , who examined equilateral configurations with the variational linear combination of atomic orbitals ( lcao ) method , and by hernndes and carb  @xcite , who studied two particular configurations with a more compact variational approach and obtained total energy values below those published before . 
   no bound state has been determined in these calculations . 
   johnson and poshusta  @xcite reported another single - point calculation in the context of gaussian basis set optimization for some one - electron systems . 
   about twenty years later ackermann _ et al . _ 
   @xcite revisited the question about the existence of @xmath1 using the finite element method which provided much higher accuracy than previously achieved . 
   the problem of the stability of @xmath1 was treated keeping the nuclear charge as a continuous parameter . 
   critical values of the charges for the existence of stable or metastable equilateral triangular configurations were obtained as @xmath5 and @xmath6 , respectively . 
   the authors excluded the possibility of stable @xmath1 in the electronic ground state . 
   however , the explicit electronic energy data are reported only for one particular equilateral triangular configuration at the triangle size @xmath7 . in conclusion , accurate _ ab initio _ 
   results on the basis of which the non - existence of @xmath1 can be demonstrated are scarce and not that convincing . 
   this question is thus addressed once again in the present study . 
   one of the motivations of our study is related to a fact that @xmath1 in equilateral triangular configuration may exist as metastable state in a magnetic field @xmath8 g @xcite . 
   we study a coulomb system of one electron and three protons @xmath2 which form an equilateral triangle of size @xmath9 . 
   the protons are assumed to be infinitely massive according to the born - oppenheimer approximation at zero order . 
   the schrdinger equation for the system is written as @xmath10\psi({\mathbf r } ) = e\psi({\mathbf r } )   \ , \ ] ] where @xmath11 is the electron momentum , @xmath12 and @xmath13 are the distances from each proton to the electron and @xmath9 is the interproton distance , see figure  [ trian ] . 
   atomic units are used throughout ( @xmath14=@xmath15=@xmath16=1 ) , although energies are expressed in rydbergs ( ry ) . 
   our goal is to study the stability of the molecular ion @xmath1 . 
   if such an ion exists , it implies the existence of the ground state of the system @xmath2 . 
   based on symmetry arguments it seems evident that the optimal geometry of @xmath2 in the case of existence of a bound state is the equilateral triangle . 
   two methods are used to explore the system : ( i ) variational with physically relevant trial functions ( see e.g.  @xcite ) which we will call _ specialized _ and ( ii ) _ standard _ variational based on using standard gaussian trial functions as implemented in _ ab initio _ quantum chemistry packages such as molpro  @xcite . 
   both methods lead to highly accurate quantitative results for total energy versus the size of the triangle .    in the first variational approach , a trial function is taken in a form of linear superposition of six basis functions @xmath17 where @xmath18 are linear parameters . 
   each function @xmath19 is chosen in such a way as to describe different physical characteristics of the system . 
   in general , @xmath19 has the form of a symmetrized product of three coulomb orbitals @xmath20 let us give a brief description of each of them :    @xmath21 : : :    all @xmath22 s are chosen to be equal to    @xmath23 , @xmath24 it is a heitler - london type    function . 
   this corresponds to _ coherent _ interaction between the    electron and all protons . 
   supposedly , it describes the system at small    interproton distances and , probably , the equilibrium configuration . 
   it    might be verified a posteriori . 
   @xmath25 : : :    two @xmath22 s are equal to zero and the remaining one    is set to be equal to @xmath26 ,    @xmath27    it is a hund - mulliken type function . 
   this function possibly describes    the system at large distances , where essentially the electron    interacts with only one proton at a time thus realizing _ incoherent _    interaction . 
   @xmath28 : : :    one @xmath22 is equal to zero , two others are different    from zero and equal to each other and to @xmath29 ,    @xmath30 it is assumed that this    function describes the system @xmath31 plus proton when    a triangle is of a sufficiently small size . 
   in fact , it is the    heitler - london function of @xmath31 symmetrized over    protons . @xmath32 : : :    one @xmath22 is equal to zero and two others are    different from each other being equal to @xmath33 ,    respectively , @xmath34    it is assumed that this function describes the system    @xmath31 plus one proton . 
   in fact , it is the    guillemin - zener function of @xmath31 symmetrized over    protons . 
   if @xmath35 , the function    @xmath32 is reduced to @xmath28 . if    @xmath36 , the function @xmath32 is    reduced to @xmath25 . 
   hence @xmath32 is    a non - linear interpolation between @xmath25 and    @xmath28 . 
   it has to describe intermediate interproton    distances . 
   @xmath37 : : :    two @xmath22 s are equal but the third one is different ,    @xmath38 it describes a `` mixed '' state of three hydrogen    atoms . 
   if @xmath39 , the function    @xmath37 is reduced to @xmath21 . if    @xmath40 , the function @xmath37 is    reduced to @xmath25 . 
   if @xmath41 ,    the function @xmath37 is reduced to    @xmath28 . hence @xmath37 is a    non - linear interpolation between @xmath21 ,    @xmath25 and @xmath28 . 
   as    function  ( [ psi4 ] ) this is a type of guillemin - zener function and    should describe intermediate interproton distances . 
   @xmath42 : : :    all @xmath22 s are different ,    @xmath43    this is a general non - linear interpolation of all functions    @xmath44 . 
   the total number of the parameters of the function ( [ trial ] ) is equal to 15 , where five of them are linear ones . 
   note that @xmath45 can be fixed at @xmath46 . 
   + in standard _ 
   ab initio _ calculations , @xmath47 is most commonly expanded in terms of gaussian basis functions  @xcite @xmath48 centered at atoms @xmath49 , @xmath50 whose coefficients @xmath51 are then determined variationally . 
   the basis functions @xmath52 themselves are built up by primitive gaussians  @xcite @xmath53 with contraction coefficients @xmath54 held fixed . 
   our calculations were performed using the hartree - fock code implemented in the molpro suite of programs  @xcite with the correlation consistent cc - pv6z and modified mcc - pv7z basis sets  @xcite . 
   the cc - pv6z basis set contains 91 contracted gaussians per atom , with @xmath55 quantum numbers up to @xmath56 , i.e. @xmath57 $ ] , yielding a total of 273 basis functions . 
   the mcc - pv7z basis includes @xmath58 functions , leading to 140 contracted gaussians per atom , @xmath59 $ ] , or 420 basis functions in total . 
   calculations were carried out for a range of equilateral triangular configurations using @xmath60 symmetry . in this point group , 
   the lowest electronic state is @xmath61 . 
   the total number of contracted gaussians of this symmetry is 168 for the cc - pv6z basis set and 255 for the mcc - pv7z basis set , respectively . 
   the cc - pv6z results , which are not reported here explicitly , have been generated to assess the accuracy of this type of calculations . 
   judging from such a comparison , we estimate the accuracy of the mcc - pv7z data to about @xmath62 over a large range of distances , deteriorating somewhat at short distances where the basis functions tend to become linearly dependent . 
   in framework of the specialized variational method ( i ) some numerical computations were made . 
   the minimization routine minuit  @xcite from the cern - lib library was used as well as d01fcf routine from the nag - lib  @xcite for three - dimensional numerical integration . 
   numerical values of the total energy @xmath63 of the system @xmath2 for different values of the interproton distance @xmath9 were obtained , see table  [ tmedel ] . 
   the results from the molpro calculation with a huge standard - type basis set ( mcc - pv7z ) are given for comparison . 
   a problem of the standard apprach is its slow convergence with respect to the angular momentum quantum number @xmath55 , requiring the use of large basis sets . 
   the method based on the specially tailored trial function , eq . 
   ( [ trial ] ) , leads to systematically lower variational energy values with considerably less terms . 
   it should be noted that this method relies on a careful optimization of non - linear parameters . 
   .variational results obtained with the specialized method ( [ trial ] ) and with a standard quantum chemistry method molpro for the total energy @xmath63 as a function of the internuclear distance @xmath9 for the system @xmath64 in the equilateral triangular geometry . for @xmath65 , in  @xcite @xmath66 . 
   [ tmedel ] [ cols="^ , > , > " , ]      as a conclusion we have to state that the total energy @xmath63 as a function of the internuclear distance @xmath9 does not indicate either to a minimum or even slight non - adiabatic irregularities for finite @xmath9 . 
   thus , the molecular ion @xmath1 does not exist in equilateral triangular configuration in the born - oppenheimer approximation . 
   hmc expresses his deep gratitude to j.c . 
   lpez vieyra for the valuable comments and for their interest in the present work , avt thanks universite libre de bruxelles for the hospitality extended to him where this work was completed . 
   this work was supported in part by fenomec and papiit grant * in121106 * ( mexico ) . 
   turbiner and j.c . 
   lopez vieyra , _ phys.rev . * 
   a66 * _ , 023409 ( 2002 ) h .- j . 
   werner , p.  j. knowles , r.  lindh , f.  r. manby , m.  schtz , p.  celani , t.  korona , g.  rauhut , r.  d. amos , a.  bernhardsson , a.  berning , d.  l. cooper , m.  j.  o. deegan , a.  j. dobbyn , f.  eckert , c.  hampel , g.  hetzer , a.  w. lloyd , s.  j. mcnicholas , w.  meyer , m.  e. mura , a.  nicklass , p.  palmieri , r.  pitzer , u.  schumann , h.  stoll , a.  j. stone , r.  tarroni , and t.  thorsteinsson , molpro , version 2006.1 , a package of ab initio programs ( 2006 ) , see http://www.molpro.net
summary: it is shown that the molecular ion @xmath0 does not exist in a form of the equilateral triangle . to this end , 
   a compact variational method is presented which is based on a linear superposition of six specially tailored trial functions containing non - linear parameters . 
   careful optimization of a total of fifteen parameters gives consistently lower variational results for the electronic energy than can be obtained with standard methods of quantum chemistry even with huge basis sets as large as mcc - pv7z . 
   _ dedicated to professor rudolf zahradnik on the occasion of his 80th birthday _