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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 program is simple c language, so save this text editor as "filename.c", and compile it.)
In this program, we first input atomic number Z = 3 (= Li ), 4 (= Be+ ), 5 (= B2+ ), 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).
The 3rd electron in 2s outer orbit is approximately fixed in position of hydrogen-like orbit around +(Z-2) nucleus (= +Z nucleus and 2 inner electrons ).
After calculating 3rd-electron, like the two-electron atomic calculaton, from the inputted values, this program outputs the y component of electron 1 velocity (= VY ) after a quarter of its orbit, and WN (= the number of de Broglie's waves included in one quarter of the orbital).
Here units are time= 1 SS = 1 × 10-23 second, length = 1 MM = 1 × 10-14 meter, velocity = 1 MM/SS = 109 m/s, acceleration = 1 MM/SS2 = 1032 m/s2
The initial electron-1's x-coordinate r1 is automatically increased per calculation until +100.
#include <stdio.h>
#include <math.h>
int main(void)
{
/* All three-electron atoms, ions */
int i;
double Z,r,E,ab,ac,Eb,Rb,Rbb,Rbc ;
double vya,vyb,poten,VX,VY,prexx,preyy,WN,ra,rb,rc;
double xx,yy,vk,leng,wav,rr,disb;
/* me = electron mass (kg), pai = pi
epsi = electric constant, h=Planck constant,
ele = electron's charge
*/
double me=9.1093826e-31;
double pai=3.141592653589793;
double epsi=8.85418781787346e-12;
double h=6.62606896e-34;
double ele=1.60217653e-19;
/* inpu atomic number Z */
printf("Atomic number Z ? (Li=3, Be+=4, B2+=5, C3+=6, N4+=7...) ");
scanf("%lf",&Z);
/* input elecron-1's initial x-coordinate r1 */
printf("r1 between nucleus and electron 1 (MM)？ ");
scanf("%lf",&r);
/* input absolute value of total energy |E| of an three-electron atom or ion */
printf("absolute value of total energy |E| of three-electron-atom (eV) ? ");
scanf("%lf", &E);
ab=(ele*ele)/(4.0*pai*epsi); ac=ab/me;
/* Eb = approximate total energy of the 3rd outer electron of n = 2 feeling effective central charge = Z-2 = nucleus + two inner electrons */
Eb=-((Z-2)*(Z-2)*ele*ele*ele*ele*me)/(32*epsi*epsi*h*h);
/* Rb = approximate radius of the 3rd outer electron */
Rb=(4.0*epsi*h*h)/((Z-2.0)*pai*ele*ele*me);
/* change meter into MM (= 10^-14 meter )*/
Rbb=Rb*1.0e14;
Rbc=Rbb/(sqrt(2.0));
/* display the 3rd outer electron's radius */
printf("The outer electron's radius (MM) = %.2f\n", Rbb);
printf(" \n");
for (i=1; i < 100 ;i++) {
/* repeat until electron-1's initial x-coordinate r1 to r1 + 100 */
/* change r(MM) to rr(meter) */
rr=r*1.0e-14;
/* disb = distance (meter) between electrons-1 and 3 */
disb=sqrt(rr*rr+Rb*Rb);
/* poten = initial Coulomb potential energy (J) */
poten=-(2.0*Z*ab)/rr+ab/(2.0*rr)+(2.0*ab)/disb-(Z*ab)/(Rb);
/* vya = total energy - potential energy - the 3rd outer electron's kinetic energy (= -Eb ) */
vya=-(E*1.60217646e-19)-poten+Eb;
if (vya > 0) {
/* vyb=electron's initial velocity (m/sec) */
vyb=sqrt(vya/me);
/* VX,VY = electron-1's x,y-velocity (= MM/SS = 10^9 m/s )*/
VY=vyb*1.0e-9; VX=0.0;
/* prexx, preyy = electron-1's x,y-coordinate */
prexx=r; preyy=0.0;
/* WN = number of de Broglie wavelength */
WN=0.0;
do {
/* xx,yy = electron-1's x,y-coordinate after 1SS (= 1x10^-23 second )*/
xx=prexx+VX; yy=preyy+VY;
vk=VX*VX+VY*VY;
/* leng = electron's moving distance (= meter ) for 1SS */
leng=sqrt(vk)*1.0e-14;
/* wave = de Broglie wavelength (meter) = h/mv */
wav=h/(me*sqrt(vk)*1.0e9);
/* WN = sum of de Boglie wavelengths in all short segments */
WN=WN+leng/wav;
/* ra = distance between electron and nucleus */
ra=sqrt(prexx*prexx+preyy*preyy);
/* rb = distance between two electrons */
rb=sqrt(4.0*prexx*prexx+2.0*preyy*preyy);
/* rc = distance between electrons-1 and 3 */
rc=sqrt(prexx*prexx+(Rbb*Rbb)/2.0+(preyy+Rbc)*(preyy+Rbc));
/* change length unit from MM into 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;
/* calculating acceleration (= MM/SS^2 = 10^32 m/s^2 ) */
/* update electron-1's x,y-velocity VX,VY from Couloumb force */
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);
/* until electron-1 has moved its 1/4 orbit to reach y-axis */
/* display electron-1's initial x-coordinate r1 */
printf("r1= %.2f ", r );
/* display electron-1's last velocity VX, VY */
printf("VX= %.6f ", VX);
printf("VY= %.6f ", VY);
/* display total de Broglie wavelength WN of 1/4 orbit */
printf("WN= %.6f\n", WN);
} r=r+1;
} return 0;
}