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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 threle, so save this text editor as "threle.java", and compile it.)
In this program, we first input atomic number Z = 3, 4, 5, 6, 7, 8, 9, or 10.
Next we input the initial x-coordinate r1 (in MM) of electron 1, and the absolute value of the total energy E (in eV) of three-electron atoms (ions).
From the inputted values, this program outputs the y component of electron 1 velocity after a quarter of its orbit, and WN (the number of de Broglie's waves included in one quarter of the orbital).
Here 1 SS = 1 × 10-23 second and 1 MM = 1 × 10-14 meter.
The initial x-coodinate is automatically increased per calculation until +100.
import java.util.Scanner;
class threle {
public static void main(String[] args) {
Scanner stdIn=new Scanner(System.in); // input r1 and |E|
System.out.println("Atomic number Z ? (Li=3, Be+=4, B2+=5, C3+=6, N4+=7...)");
double Z=stdIn.nextDouble();
System.out.println("r1 between nucleus and electron 1 (MM)？ ");
double r=stdIn.nextDouble();
System.out.println("total energy |E| in three-electron-atom (eV) ? ");
double E=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 ab=(ele*ele)/(4.0*pai*epsi); double ac=ab/me;
double Eb=-((Z-2)*(Z-2)*ele*ele*ele*ele*me)/(32*epsi*epsi*h*h); // Eb=2S energy
double Rb=(4.0*epsi*h*h)/((Z-2.0)*pai*ele*ele*me); // Rb=2S radius
double Rbb=Rb*1.0e14; // change meter to MM
double Rbc=Rbb/(Math.sqrt(2.0));
System.out.printf("Rbb:%.2f\n", Rbb);
for (int i=1;i < 100;i++) { // repeat until r1=initial r1+100
double rr=r*1.0e-14; // change r(MM) to rr(meter)
// disb=distance (m) between initial e1S and e2S
double disb=Math.sqrt((rr*rr)+Rb*Rb);
double poten=-(2.0*Z*ab)/rr+ab/(2.0*rr)+(2.0*ab)/disb-(Z*ab)/(Rb);
//vya= total E - potential energy - 2S kinetic energy(-Eb)
double vya=-(E*1.60217646e-19)-poten+Eb;
if (vya > 0) {
// vyb= initial velocity (m/s) of 1S electron
double vyb=Math.sqrt(vya/me);
double VY=vyb*1.0e-9; // change m/sec to MM/SS
double prexx=r; double VX=0.0; double WN=0.0; double preyy=0.0;
double xx,yy,vk,preVY,preWN,midWN, leng, wav; xx=0.0;
do {
xx=prexx+VX; yy=preyy+VY; //electron 1 position after 1SS
preVY=VY;preWN=WN ;
vk=VX*VX+VY*VY;
leng=Math.sqrt(vk)*1.0e-14; // moving length (m) for 1 SS
wav=h/(me*Math.sqrt(vk)*1.0e9); // de Broglie wavelength (m)
WN=WN+leng/wav; // add de Broglie wavelength
double ra=Math.sqrt(prexx*prexx+preyy*preyy); // between nucleus and 1S ele
double rb=Math.sqrt(4.0*prexx*prexx+2.0*preyy*preyy); // between 1S electrons
// rc=distance between 1S ele and 2S ele
double rc=Math.sqrt(prexx*prexx+(Rbb*Rbb)/2.0+(preyy+Rbc)*(preyy+Rbc));
// change MM to meter
ra=ra*1.0e-14; rb=rb*1.0e-14; rc=rc*1.0e-14;
prexx=prexx*1.0e-14; preyy=preyy*1.0e-14;
// acceleration ( MM/SS^2 )
VX=VX+1.0e-32*ac*prexx*(-Z/(ra*ra*ra)+2.0/(rb*rb*rb)+1.0/(rc*rc*rc));
VY=VY+1.0e-32*ac*((-Z*preyy)/(ra*ra*ra)+preyy/(rb*rb*rb)+(preyy+Rbc*1.0e-14)/(rc*rc*rc));
prexx=xx;preyy=yy;
} while (xx >= 0); //electron has moved a quater of an orbit?
if (VY > -0.0001 && VY < 0.0001) { // last VY condition
System.out.print("r1: "+r+" ");
System.out.printf("VX:%.6f", VX);
System.out.printf("preVY:%.6f", preVY);
System.out.printf("VY:%.6f", VY);
midWN=(preWN+WN)/2.0; System.out.printf("midWN:%.6f\n", midWN);
}} r=r+1;
}}}