I'm trying to determine a numerical method for calculating the position
of k electrons on a (n-1) hypersphere in R^n (n<k<2n) in such a way
that the potential energy of the system is minimized. This is a known
hard problem even when n=3, but I'm not looking for exact solutions. I
just want to determine a good way of coding up a programming language
like C or Fortran or Matlab to iterate until the points settle in their
equilibrium position.

Can anyone shed any insight into how to numerically solve this problem?
I'm interested in n<=80.

Thanks,

Troy Henderson

In article <1165426816.931546.106200@n67g2000cwd.googlegroups.com>,
"tlhiv" <thenders@gmail.com> writes:
I'm trying to determine a numerical method for calculating the position
of k electrons on a (n-1) hypersphere in R^n (n<k<2n) in such a way
that the potential energy of the system is minimized.

you can write this as a nonlinear optimization problem . the need of
the nonlinear equality constraint can be avoided by using n-dim spherical
coordinates at the cost of using box constraints and getting an even higher
nonlinearity.

In article <1165426816.931546.106200@n67g2000cwd.googlegroups.com>,
"tlhiv" <thenders@gmail.com> writes:
I'm trying to determine a numerical method for calculating the position
of k electrons on a (n-1) hypersphere in R^n (n<k<2n) in such a way
that the potential energy of the system is minimized. This is a known
hard problem even when n=3, but I'm not looking for exact solutions. I
just want to determine a good way of coding up a programming language
like C or Fortran or Matlab to iterate until the points settle in their
equilibrium position.

Can anyone shed any insight into how to numerically solve this problem?
I'm interested in n<=80.

Thanks,

Troy Henderson


you can write this as a nonlinear optimization problem . the need of
the nonlinear equality constraint can be avoided by using n-dim spherical
coordinates at the cost of using box constraints and getting an even higher
nonlinearity. since there will be local minimizer, you need a global minimizer
which will be hard if the dimension is as large as you indicate. but maybe that
simulated annealing does the job.
out there are ready to use codes for this.
check out
http://plato.la.asu.edu/guide.html
under "global optimization"
hth
peter

[ stereographic projection ] The coordinates
are singular at the south pole, x= [0,0 ... 0,1] maps to u=
infinity, but are regular everywhere else.

In article <1165426816.931546.106200@n67g2000cwd.googlegroups.com>,
"tlhiv" <thenders@gmail.com> writes:
I'm trying to determine a numerical method for calculating the position
of k electrons on a (n-1) hypersphere in R^n (n<k<2n) in such a way
that the potential energy of the system is minimized. This is a known
hard problem even when n=3, but I'm not looking for exact solutions. I
just want to determine a good way of coding up a programming language
like C or Fortran or Matlab to iterate until the points settle in their
equilibrium position.

Can anyone shed any insight into how to numerically solve this problem?
I'm interested in n<=80.

Thanks,

Troy Henderson


you can write this as a nonlinear optimization problem . the need of
the nonlinear equality constraint can be avoided by using n-dim spherical
coordinates at the cost of using box constraints and getting an even
higher
nonlinearity. since there will be local minimizer, you need a global
minimizer
which will be hard if the dimension is as large as you indicate. but maybe
that
simulated annealing does the job.

Thanks for the replies. However, I was looking for an actual "plan of
attack" because I've not done any (non-)linear optimzation. I'm very
fluent in programming, and perfectly fluent in solving linear systems
of equations.

If anyone has an idea for pseudocode, I'd be much appreciative.

Troy

In article <1165426816.931546.106200@n67g2000cwd.googlegroups.com>,
"tlhiv" <thenders@gmail.com> writes:
I'm trying to determine a numerical method for calculating the position
of k electrons on a (n-1) hypersphere in R^n (n<k<2n) in such a way
that the potential energy of the system is minimized. This is a known
hard problem even when n=3, but I'm not looking for exact solutions. I
just want to determine a good way of coding up a programming language
like C or Fortran or Matlab to iterate until the points settle in their
equilibrium position.

Can anyone shed any insight into how to numerically solve this problem?
I'm interested in n<=80.

Thanks,

Troy Henderson


you can write this as a nonlinear optimization problem . the need of
the nonlinear equality constraint can be avoided by using n-dim spherical
coordinates at the cost of using box constraints and getting an even higher
nonlinearity. since there will be local minimizer, you need a global minimizer
which will be hard if the dimension is as large as you indicate. but maybe that
simulated annealing does the job.
out there are ready to use codes for this.
check out
http://plato.la.asu.edu/guide.html
under "global optimization"
hth
peter

How doese your solution look for one "electron" (or two)?

D. Baruth


John Herman <John_W_Herman@yahoo.com> wrote in message
news:bJAeh.52353$Fg.6292@tornado.socal.rr.com...
: This is a naive question but I was wondering if the solution for 80 electrons
: close to the solution for 79? In other words, is starting with a lower
: dimensional problem and working up to the dimension needed reasonable?

This is a naive question but I was wondering if the solution for 80 electrons
close to the solution for 79? In other words, is starting with a lower
dimensional problem and working up to the dimension needed reasonable?

this should work if you move all points a little towards the one fixed

In article <el74hb$hdj$1@fb04373.mathematik.tu-darmstadt.de>,
spellucci@fb04373.mathematik.tu-darmstadt.de (Peter Spellucci) wrote:

In article <1165426816.931546.106200@n67g2000cwd.googlegroups.com>,
"tlhiv" <thenders@gmail.com> writes:
I'm trying to determine a numerical method for calculating the position
of k electrons on a (n-1) hypersphere in R^n (n<k<2n) in such a way
that the potential energy of the system is minimized. This is a known
hard problem even when n=3, but I'm not looking for exact solutions. I
just want to determine a good way of coding up a programming language
like C or Fortran or Matlab to iterate until the points settle in their
equilibrium position.

Can anyone shed any insight into how to numerically solve this problem?
I'm interested in n<=80.

Thanks,

Troy Henderson


you can write this as a nonlinear optimization problem . the need of
the nonlinear equality constraint can be avoided by using n-dim spherical
coordinates at the cost of using box constraints and getting an even higher
nonlinearity. since there will be local minimizer, you need a global minimizer
which will be hard if the dimension is as large as you indicate. but maybe that
simulated annealing does the job.
out there are ready to use codes for this.
check out
http://plato.la.asu.edu/guide.html
under "global optimization"
hth
peter

I was just wondering if a solution was found for 79, would that be a help in
solving the problem for 80 electrion. Or would it be necessary to start over
again for 9- electrons. So I was thinking that start with 3, then 4, ... and
so on until the desired degree is obtained.