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I'm trying to solve this problem with Mathematica.

Im not a Mathematica expert, but my program works perfect when vhartree={0,0,..,0}.

This is the program I wrote:

The problem arises when vhartree is not zero. In books: 2energy-integrate2= -2.861, I get: -3.89561. I would appreciate any suggestion, thanks in avance for your help. In this page, you can find the code in Python for the whole problem (including the exchange potential):

<https://www.leetspeak.org/teaching/cqp_fs14/09-dft.html>

Im not a Mathematica expert, but my program works perfect when vhartree={0,0,..,0}.

This is the program I wrote:

Code:

```
Clear[u, poisson, vhartree]
h = 10^(-2);(*step integration*)
rmax = 20;
rmin = 10^(-30);
Z = 2; (*atomic number*)
points = rmax/h;
vhartree = Table[0, {i, 1, points + 1}];
(*numerov integration*)
g[energy_, j_] :=
Module[{a},
If [j < rmax/h,
a = N[2 (energy + Z/(rmax - j h)) - vhartree[[points - j]]],
a = N[2 (energy + 1/rmin)] + vhartree[[1]]];
Return[a];
];
numerov[energy_] := Module[{j},
u = {N[rmax*Exp[-rmax]]};
PrependTo[u, N[(rmax - h)*Exp[-(rmax - h)]]];
alpha = N[h^2/12];
x = u[[1]];
y = u[[2]];
z = (2 (1 - 5 alpha g[energy, 1]) y + (1 +
alpha g[energy, 0]) x)/(1 + alpha g[energy, 2]);
PrependTo[u, z];
For[ j = 2, j < points, j++,
x = y;
y = z;
z = (2 (1 - 5 alpha g[energy, j]) y - (1 +
alpha g[energy, j - 1]) x)/(1 + alpha g[energy, j + 1]);
PrependTo[u, z];
];
Return
];
searchenergy := Module[{k, n, a, b, M, j},
(*searching range where numerov[[1]] change its sign*)
k = 1;
a = -10 ;(*min. energy*)
While[numerov[a][[1]]*numerov[a + 0.1][[1]] > 0,
a = a + 0.1;
k++;
];
b = a + 0.1; (*max energy*)
(*bisection*)
M = 50;(*max number of iterations*)
wave = {};
n = 0;
epsilon = 10^(-8);
While[(n < M) && (Abs[a - b] > epsilon),
energy = N[(a + b)/2];
If[(numerov[a][[1]])*(numerov[energy][[1]]) > 0, a = energy,
b = energy];
n++;
];
wave = numerov[energy];
(*normalization, Simpson's rule*)
hsimpson = rmax/(Length[wave]);
integrate = (hsimpson/3) ((wave[[1]])^2 +
2 Sum[wave[[2 j]]^2, {j, 1, Length[wave]/2 - 1}] +
4 Sum[wave[[2 j - 1]]^2, {j, 1, Length[wave]/2}]);
wave = wave/Sqrt[integrate];
Return[wave];
]
Poisson := Module[{i, beta, t, v, w},
(*Verlet algorithm*)
t = 0;
v = h;
poisson = {t, v};
For[i = 1, i <= puntos - 1, i++,
w = 2 v - t - (h) wave[[I]]^2/(i);
AppendTo[poisson, w];
t = v;
v = w;
];
(*adding homogeneus solution*)
beta = (1 - poisson[[Length[poisson]]])/rmax;
For[i = 1, i <= Length[poisson], i++,
poisson[[I]] = poisson[[I]] + beta (i - 1) h;
];
Return[poisson];
]
Clear[k]
(*adding vhartre \neq 0*)
lista = {energy};
For[k = 0, k < 5, k++,
vhartree[[1]] = poisson[[1]]/rmin;
For[m = 2, m <= Length[poisson], m++,
vhartree[[m]] = 2 poisson[[m]]/((m + 1) h);
];
(*print the number of cycles before self consistency*)
Print[k];
searchenergy;
AppendTo[lista, energy];
Poisson;
]
integrate2 = (hsimpson/3) *(vhartree[[1]] (wave[[1]])^2 +
vhartree[[Length[vhartree]]] (wave[[Length[vhartree]]])^2 +
2 Sum[vhartree[[2 j]] (wave[[2 j]])^2, {j, 1,
Length[vhartree]/2 - 1}] +
4 Sum[vhartree[[2 j - 1]] (wave[[2 j - 1]])^2, {j, 1,
Length[vhartree]/2}]);
2energy-integrate2[/I][/I][/I]
```

The problem arises when vhartree is not zero. In books: 2energy-integrate2= -2.861, I get: -3.89561. I would appreciate any suggestion, thanks in avance for your help. In this page, you can find the code in Python for the whole problem (including the exchange potential):

<https://www.leetspeak.org/teaching/cqp_fs14/09-dft.html>

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