Top page (correct Bohr model including helium).
Back to hydrogen molecule ion page.
If you copy and paste the program source code below into a text editor, you can easily compile and run this.
(This class file name is hyion, so save this text editor as "hyion.java", and compile it.)
Here we use the new unit of 1 MM = 1 × 10-14 meter.
In this program, first we input the internuclear distance (in MM ) of H2+.
And then input y-coordinate "r" (in MM) , and x-coordinate "a" ( in MM ) of electron.
From the inputted values, this program outputs the binding energy (eV) of H2+, the ratio of force F1 to centrifugal force, the ratio of forces ( from electron and n1 ) acting on n2.
F1 is the force acting on electron toward the nucleus 1 (n1).
From the force acting on electron and its velocity, we calculate the number of de Broglie wave in one orbit around n1.
( Here Virial theorem E = -T = 1/2 V is used. )
Furthermore, considering precession around x axis by the force of nucleus 2, this de Broglie wave is computed.
Total wave means the sum of them.
We also calculate the de Broglie waves influenced by the force in the x direction by nucleus 2 (= new total waves ).
import java.util.Scanner;
class hyion {
public static void main(String[] args) {
Scanner stdIn=new Scanner(System.in);
System.out.println("nuc (MM) ? (= internuclear distance of H2+ ion");
double nnuc=stdIn.nextDouble();
System.out.println("r (MM)？ = electron y coordinate ");
double rr=stdIn.nextDouble();
System.out.println("a (MM) ? = electron x coordinate ");
double aa=stdIn.nextDouble();
double me=9.1093826e-31;
double pai=3.141592653589793; double epsi=8.85418781787346e-12;
double h=6.62606896e-34; double ele=1.60217653e-19;
double r=rr*1.0e-14;double nuc=nnuc*1.0e-14; // change MM to meter
double ke=(ele*ele)/(4*pai*epsi);
for (int i=1;i < 10;i++){ // repeat calculation from a to a + 10*10 (MM)
double a=aa*1.0e-14;
double rra=Math.sqrt(a*a+r*r); // rra =distance between electron and n1
double rrb=Math.sqrt(r*r+(nuc-a)*(nuc-a)); // rrb= distance between electron and n2
double poten=ke*(-1.0/rra-1.0/rrb+1.0/nuc); // poten=potential energy (J)
double ppot=poten*6.241509*1.0e18; // ppot = potential energy (eV)
double kinetic=-0.5*poten; // kinetic energy (J)
double velo=Math.sqrt((2.0*kinetic)/me); // velo = electron's velocity (m/s)
double binding=-ppot*0.5-13.606; // binding energy (eV)
double fyna=(ke*r)/(rra*rra*rra); // n1's force acting on electron in -y direction
double fxna=(ke*a)/(rra*rra*rra); // n1's force acting on electron in -x direction
double fynb=(ke*r)/(rrb*rrb*rrb); // n2's force acting on electron in -y direction
double fxnb=(ke*(nuc-a))/(rrb*rrb*rrb); // n2's force acting on electron in x direction
double internuc=ke/(nuc*nuc); // force between two nuclei
double ntwoforce = fxnb/internuc; // ratio of forces acting on n2
double debrogli = h/(me*velo); // de Broglie wavelength of electron
double wave=(2.0*pai*rra)/debrogli; // de Broglie's waves around nucleus 1
double tim=(2*pai*rra)/velo; // tim = time period of electron around n1
double vp=Math.sqrt((fynb*r)/me); // vp = electron precession veleocity around x axis
double debropre = h/(me * vp); // de Broglie wavelength of precession
double wavepre=(vp*tim)/debropre; // de Broglie's wave number of precession
double cf=(me*velo*velo)/rra; // cf= centrifugal force around n1
double fgen=ke/(rra*rra); // fgen = n1's force acting on the electron
double cosb=(rra*rra+nuc*nuc-rrb*rrb)/(2*rra*nuc); // cosi = cos b
double fnb=ke/(rrb*rrb);
fnb=(fnb*(rra-nuc*cosb))/rrb; // fnb= n2's force * cos (theta)
double fone=fnb+fgen; // total force acting on electron toward n1 (=F1)
double goux=fone*cosb; // x component of F1
double gouy=fone*Math.sqrt(1-cosb*cosb); // y component of F1
double gouxx=goux-fxnb; // subtract n2 force from goux
double fonea=Math.sqrt(gouxx*gouxx+gouy*gouy); // changed F1
double diele=Math.sqrt((nuc+a)*(nuc+a)+r*r); // between n2 and electron at position 2
double fxnbb=(ke*(nuc+a))/(diele*diele*diele); // x component of force from n2
double gouxxx=goux+fxnbb; // add n2 force to goux
double foneb=Math.sqrt(gouxxx*gouxxx+gouy*gouy); // changed F1 at position 2
fonea=(fonea+foneb)/2.0; // changed average F1
double rah=fone/cf; // rah = ratio of F1 to centrifugal force around n1
double newgoux = Math.sqrt(fonea*fonea-gouy*gouy);
double newx=(r*newgoux)/gouy;
double newradius=Math.sqrt(r*r+newx*newx); // radius influened by n2 force in x direction
double newvelo= Math.sqrt((fonea*newradius)/me);
double newtim=(2.0*pai*newradius)/newvelo; // time period
double newprewav=(vp*newtim*me*vp)/h; // new de Broglie's wave of precession
double newwav=(2.0*pai*newradius*me*newvelo)/h; // new de Broglie's waves number
System.out.printf("a:%.1f", aa);
System.out.printf(" binding energy (eV): %.4f ", binding);
System.out.printf("ratio of F1 to centrifugal: %.4f ", rah);
System.out.printf("ratio of forces acting on n2: %.3f \n", ntwoforce);
System.out.printf("waves around n1: %.3f ", wave);
System.out.printf("+ waves around x axis: %.3f ", wavepre);
int la=(int)(1000.0*wave); int lb=(int)(1000.0*wavepre);la=la+lb;
double twaves=la/1000.0;
System.out.printf(" = total waves: %.3f \n ", twaves);
System.out.printf("new waves around n1: %.3f ", newwav);
System.out.printf("+ new precession wave: %.3f ", newprewav);
la=(int)(1000.0*newwav); lb=(int)(1000.0*newprewav);la=la+lb;
twaves=la/1000.0;
System.out.printf(" = new total waves: %.3f \n ", twaves);
System.out.printf(" \n");
aa=aa+10;}}}