## Hartree method for Helium ground state

After long preparation, I am ready to do this problem.

The two electron in the helium ground state occupy same spacial orbital but difference spin. Thus, the total wavefunction is

$\displaystyle \Psi(x,y) = \frac{1}{\sqrt{2}}(\uparrow \downarrow - \downarrow \uparrow) \psi(x) \psi(y)$

Since the Coulomb potential is spin-independent, the Hartree-Fock method reduce to Hartree method. The Hartree operator is

$F(x) = H(x) + \langle \psi(y)|G(x,y) |\psi(y) \rangle$

where the single-particle Hamiltonian and mutual interaction are

$\displaystyle H(x) = -\frac{\hbar^2}{2m} \nabla^2 - \frac{Ze^2}{4\pi\epsilon_0 x} = -\frac{1}{2}\nabla^2 - \frac{Z}{x}$

$\displaystyle G(x,y) = \frac{e^2}{4\pi\epsilon_0|x-y|} = \frac{1}{|x-y|}$

In the last step, we use atomic unit, such that $\hbar = 1, m=1, e^2 = 4\pi\epsilon_0$. And the energy is in unit of Hartree, $1 \textrm{H} = 27.2114 \textrm{eV}$.

We are going to use Hydrogen-like orbital as a basis set.

$\displaystyle b_i(r) = R_{nl}(r)Y_{lm}(\Omega) \\= \sqrt{\frac{(n-l-1)!Z}{n^2(n+l)!}}e^{-\frac{Z}{n}r} \left( \frac{2Z}{n}r \right)^{l+1} L_{n-l-1}^{2l+1}\left( \frac{2Z}{n} r \right) \frac{1}{r} Y_{lm}(\Omega)$

I like the left the $1/r$, because in the integration $\int b^2 r^2 dr$, the $r^2$ can be cancelled. Also, the $i = nlm$ is a compact index of the orbital.

Using basis set expansion, we need to calculate the matrix elements of

$\displaystyle H_{ij}=\langle b_i(x) |H(x)|b_j(x)\rangle = -\delta \frac{Z^2}{2n^2}$

$\displaystyle G_{ij}^{hk} = \langle b_i(x) b_h(y) |G(x,y) |b_j(x) b_k(y) \rangle$

Now, we will concentrate on evaluate the mutual interaction integral.

Using the well-known expansion,

$\displaystyle G(x,y) = \frac{1}{|x-y|}=\frac{1}{r_{12}} = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \frac{4\pi}{2l+1} \frac{r_<^l}{r_>^{l+1}} Y_{lm}^{*}(\Omega_1)Y_{lm}(\Omega_2)$

The angular integral

$\displaystyle \langle Y_i(x) Y_h(y)| Y_{lm}^{*}(x) Y_{lm}(y) | Y_j(x) Y_k(y) \rangle \\ = \big( \int Y_i^{*}(x) Y_{lm}^{*}(x) Y_j(x) dx \big) \big( \int Y_h^{*}(y) Y_{lm}(y) Y_k(y) dy \big)$

where the integral $\int dx = \int_{0}^{\pi} \int_{0}^{2\pi} \sin(\theta_x) d\theta_x d\phi_x$.

From this post, the triplet integral of spherical harmonic is easy to compute.

$\displaystyle \int Y_h^{*}(y) Y_{lm}(y) Y_k(y) dy = \sqrt{\frac{(2l+1)(2l_k+1)}{4\pi (2l_h+1)}} C_{l0l_k0}^{l_h0} C_{lm l_km_k}^{l_hm_h}$

The Clebsch-Gordon coefficient imposed a restriction on $l,m$.

$\displaystyle \langle R_i(x) R_h(y)| \frac{r_<^l}{r_>^{l+1}} | R_j(x) R_k(y) \rangle \\ = \int_0^{\infty} \int_{0}^{\infty} R_i(x) R_h(y) \frac{r_<^l}{r_>^{l+1}} R_j(x) R_k(y) y^2 x^2 dy dx \\ = \int_0^{\infty} R_i(x) R_j(x) \\ \left( \int_{0}^{x} R_h(y) R_k(y) \frac{y^l}{x^{l+1}} y^2dy + \int_{x}^{\infty} R_h(x)R_k(x) \frac{x^l}{y^{l+1}} y^2 dy \right) x^2 dx$

The algebraic calculation of the integral is complicated, but after the restriction of $l$ from the Clebsch-Gordon coefficient, only few terms need to be calculated.

The general consideration is done. now, we use the first 2 even states as a basis set.

$\displaystyle b_{1s}(r) = R_{10}(r)Y_{00}(\Omega) = 2Z^{3/2}e^{-Zr}Y_{00}(\Omega)$

$\displaystyle b_{2s}(r) = R_{20}(r)Y_{00}(\Omega) = \frac{1}{\sqrt{8}}Z^{3/2}(2-Zr)e^{-Zr/2}Y_{00}(\Omega)$

These are both s-state orbital. Thus, the Clebsch-Gordon coefficient

$\displaystyle C_{lm l_k m_k}^{l_h m_h} = C_{lm00}^{00}$

The radial sum only has 1 term. And the mutual interaction becomes

$\displaystyle G(x,y) = \frac{1}{|x-y|}=\frac{1}{r_{12}} = 4\pi \frac{1}{r_>} Y_{00}^{*}(\Omega_1)Y_{00}(\Omega_2)$

The angular part

$\displaystyle \langle Y_i(x) Y_h(y)| Y_{lm}^{*}(x) Y_{lm}(y) | Y_j(x) Y_k(y) \rangle = \frac{1}{4\pi}$

Thus, the mutual interaction energy is

$G_{ij}^{hk} = \displaystyle \langle b_i(x) b_h(y) |G(x,y) |b_j(x) b_k(y) \rangle = \langle R_i(x) R_h(y)| \frac{1}{r_>} | R_j(x) R_k(y) \rangle$

$G_{ij}^{hk} = \displaystyle \langle R_i(x) R_h(y)| \frac{1}{r_>} | R_j(x) R_k(y) \rangle \\ \begin{pmatrix} G_{11}^{hk} & G_{12}^{hk} \\ G_{21}^{hk} & G_{22}^{hk} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} G_{11}^{11} & G_{11}^{12} \\ G_{11}^{21} & G_{11}^{22} \end{pmatrix} & \begin{pmatrix} G_{12}^{11} & G_{12}^{12} \\ G_{12}^{21} & G_{12}^{22} \end{pmatrix} \\ \begin{pmatrix} G_{21}^{11} & G_{21}^{12} \\ G_{21}^{21} & G_{21}^{22} \end{pmatrix} & \begin{pmatrix} G_{22}^{11} & G_{22}^{12} \\ G_{22}^{21} & G_{22}^{22} \end{pmatrix} \end{pmatrix} \\= \begin{pmatrix} \begin{pmatrix} 1.25 & 0.17871 \\ 0.17871 & 0.419753 \end{pmatrix} & \begin{pmatrix} 0.17871 & 0.0438957 \\ 0.0439857 & 0.0171633 \end{pmatrix} \\ \begin{pmatrix} 0.17871 & 0.0438957 \\ 0.0438957 & 0.0171633 \end{pmatrix} & \begin{pmatrix} 0.419753 & 0.0171633 \\ 0.0171633 & 0.300781 \end{pmatrix} \end{pmatrix}$

We can easy to see that $G_{ij}^{hk} = G_{ji}^{hk} = G_{ij}^{kh} = G_{hk}^{ij} = G_{ji}^{kh}$. Thus, if we flatten the matrix of matrix, it is Hermitian, or symmetric.

Now, we can start doing the Hartree method.

The general solution of the wave function is

$\psi(x) = a_1 b_{1s}(x) + a_2 b_{2s}(x)$

The Hartree matrix is

$F_{ij} = H_{ij} + \sum_{h,k} a_h a_k G_{ij}^{hk}$

The first trial wave function are the Hydrogen-like orbital,

$\psi^{(0)}(x) = b_{1s}(r)$

$F_{ij}^{(0)} = \begin{pmatrix} -2 & 0 \\ 0 & -0.5 \end{pmatrix} + \begin{pmatrix} 1.25 & 0.17871 \\ 0.17817 & 0.419753 \end{pmatrix}$

Solve for eigen system, we have the energy after 1st trial,

$\epsilon^{(1)} = -0.794702 , (a_1^{(1)}, a_2^{(1)}) = (-0.970112, 0.242659)$

After 13th trial,

$\epsilon^{(13)} = -0.880049 , (a_1^{(13)}, a_2^{(13)}) = (-0.981015, 0.193931)$

$F_{ij}^{(13)} = \begin{pmatrix} -2 & 0 \\ 0 & -0.5 \end{pmatrix} + \begin{pmatrix} 1.15078 & 0.155932 \\ 0.155932 & 0.408748 \end{pmatrix}$

Thus, the mixing of the 2s state is only 3.7%.

Since the eigen energy contains the 1-body energy and 2-body energy. So, the total energy for 2 electrons is

$E_2 = 2 * \epsilon^{(13)} - G = -2.82364 \textrm{H} = -76.835 \textrm{eV}$

In which ,

$G = \langle \psi(x) \psi(y) |G(x,y) |\psi(x) \psi(y) \rangle = 1.06354 \textrm{H} = 28.9403 \textrm{eV}$

So the energies for

From He to He++.  $E_2 = -2.82364 \textrm{H} = -76.835 \textrm{eV}$
From He+ to He++, $E_1^+ = -Z^2/2 = 2 \textrm{H} = -54.422 \textrm{eV}$.
From He to He+, is $E_1 = E_2 - E_1^+ = -0.823635 \textrm{H} = -22.4123 \textrm{eV}$

The experimental 1 electron ionization energy for Helium atom is

$E_1(exp) = -0.90357 \textrm{H} = -24.587 \textrm{eV}$
$E_1^+(exp) = -1.99982 \textrm{H} = -54.418 \textrm{eV}$
$E_2(exp) = -2.90339 \textrm{H} = -79.005 \textrm{eV}$

The difference with experimental value is 2.175 eV. The following plot shows the Coulomb potential, the screening due to the existence of the other electron, the resultant mean field, the energy, and $r \psi(x)$

Usually, the Hartree method will under estimate the energy, because it neglected the correlation, for example, pairing and spin dependence. In our calculation, the $E_2$ energy is under estimated.

From the $(a_1^{(13)}, a_2^{(13)}) = (-0.981015, 0.193931)$, we can see, the mutual interaction between 1s and 2s state is attractive. While the interaction between 1s-1s and 2s-2s states are repulsive. The repulsive can be easily understood. But I am not sure how to explain the attractive between 1s-2s state.

Since the mass correction and the fine structure correction is in order of $10^{-3} \textrm{eV}$, so the missing 0.2 eV should be due to something else, for example, the incomplete basis set.

If the basis set only contain the 1s orbit, the mutual interaction is 1.25 Hartree = 34.014 eV. Thus, the mixing reduce the interaction by 5.07 eV, just for 3.7% mixing

I included the 3s state,

$\epsilon^{(13)} = -0.888475 , (a_1^{(13)}, a_2^{(13)}, a_3^{(13)}) = (0.981096, -0.181995, -0.06579)$

The mutual energy is further reduced to 1.05415 Hartree = 28.6848 eV. The $E_2 = -77.038 \textrm{eV}$. If 4s orbital included, the $E_2 = -77.1058 \textrm{eV}$. We can expect, if more orbital in included, the $E_2$ will approach to $E_2(exp)$.

## Hartree-Fock method for 1D infinite potential well

Following the previous post, I tested my understanding on the Hartree method. Now, I move to the Hartree-Fock method. The “mean field” energy of the Hartree-Fock is

$\displaystyle G_{\alpha \beta}= \sum_{j=i+1}^{N} \langle \psi_{\alpha}(i) \psi_{\nu} (j) | G(i,j) \left(| \phi_{\beta}(i) \phi_{\nu}(j) \rangle - |\phi_{\nu}(i) \phi_{\beta}(j) \rangle \right) \\ = \langle \alpha \nu | \beta \nu \rangle - \langle \alpha \nu | \nu \beta \rangle$

I also use the same method, in which, the trial wave function is replaced every iteration and the integration is calculated when needed.

Since the total wave function must be anti-symmetry under permutation, therefore one state can be occupied by only one particle. Thus, if we use the ground state in the mean field, the “meaningful” wave functions are the other states.

It is interesting that the mean field energy is zero when $\mu = \nu$, the consequence is no mean field for the same state. Suppose the mean field energy is constructed using the ground state, and we only use 3 states, the direct term is

$G_D = \begin{pmatrix} \langle 11|11 \rangle & \langle 11|21 \rangle & \langle 11|31 \rangle \\ \langle 21|11 \rangle & \langle 21|21 \rangle & \langle 21|31 \rangle \\ \langle 31|11 \rangle & \langle 31|21 \rangle & \langle 31|31 \rangle \end{pmatrix}$

The exchange term is

$G_E = \begin{pmatrix} \langle 11|11 \rangle & \langle 11|12 \rangle & \langle 11|13 \rangle \\ \langle 21|11 \rangle & \langle 21|12 \rangle & \langle 21|13 \rangle \\ \langle 31|11 \rangle & \langle 31|12 \rangle & \langle 31|13 \rangle \end{pmatrix}$

Due to the symmetry of the mutual interaction. We can see that some off-diagonal terms are cancelled. For example,

$\displaystyle \langle 1 1 | 3 1 \rangle = \int \psi_1^*(x) \psi_1^*(y) \cos(x-y) \psi_3(x) \psi_1(y) dy dx$

$\displaystyle \langle 1 1 | 1 3 \rangle = \int \psi_1^*(x) \psi_1^*(y) \cos(x-y) \psi_1(x) \psi_3(y) dy dx$

These two integrals are the same. In fact,

$latex \langle \alpha \nu | \beta \nu \rangle – \langle \alpha \nu | \nu \beta \rangle =$

whenever $\alpha = \nu$

$\displaystyle \langle \nu \nu | \beta \nu \rangle = \int \psi_\nu^*(x) \psi_\nu^*(y) \cos(x-y) \psi_\beta(x) \psi_\nu(y) dy dx$

$\displaystyle \langle \nu \nu | \nu \beta \rangle = \int \psi_\nu^*(x) \psi_\nu^*(y) \cos(x-y) \psi_\nu(x) \psi_\beta(y) dy dx$

We can see, when interchange $x \leftrightarrow y$, the direct term and the exchange term are identical, and then the mean field energy is zero. Also, when $\beta = \nu$ the mean field energy is also zero.

Due to the zero mean field, the off-diagonal terms of the Hamiltonian $H_{\alpha \beta}$ with $\alpha = \nu$ or $\beta = \nu$ are zero. Then, the eigen energy is the same as the diagonal term and the eigen vector is un-change.

Back to the case, the direct matrix at the 1st trial is,

$G_D = \begin{pmatrix} 0.720506 & 0 & -0.144101 \\ 0 & 0.576405 & 0 \\ -0.144101 & 0 & 0.555819 \end{pmatrix}$

The exchange matrix is

$G_E = \begin{pmatrix} 0.720506 & 0 & -0.144101 \\ 0 & 0.25 & 0 \\ -0.144101 & 0 & 0.0288202 \end{pmatrix}$

Thus, the Fock matrix is

$F = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4.3264 & 0 \\ 0 & 0 & 9.527 \end{pmatrix}$

Therefore, the eigen states are the basis, unchanged

$\displaystyle \psi_1(x) = \sqrt{\frac{2}{\pi}} \sin(x)$

$\displaystyle \psi_2(x) = \sqrt{\frac{2}{\pi}} \sin(2x)$

$\displaystyle \psi_3(x) = \sqrt{\frac{2}{\pi}} \sin(3x)$

Only the eigen energies are changed, as $\epsilon_1 = 1$$\epsilon_2 = 4.3264$$\epsilon_3 = 9.527$

The total wave function for 2 particles at state 1 and state-\mu is

$\Psi(x,y) = \frac{1}{\sqrt{2}} ( \psi_1(x) \psi_\mu(y) - \psi_\mu(x) \psi_1(y) )$

I found that the “mean field” function is not as trivial as in the Hartree case, because of the exchange term. In principle, the mean field for particle-i at state-$\mu$ is,

$G(i) = \int \phi_\nu^*(j) G(i,j) \phi_\nu(j) dj - \int \phi_\nu^*(j) G(i,j) \phi_\mu(j) dj$

However, the direct term is multiply with $\psi_\mu(i)$, but the exchange term is multiply with $\psi_\nu(i)$, which are two different functions. i.e., the “mean field” is affecting two functions, or the “mean field” is shared by two states.

Although the mean field for single state can be defined using exchange operator symbolically, but I don’t know how to really implement into a calculation. Thus, I don’t know how to cross-check the result.

## Method on solving differential equation

The foundation of using Hartree-Fock method to get the self-consistence wave function and potential is solving the Hartree-Fock equation, which is kind of differential equation.

The idea of the Hartree-Fock method is

1. Assume the trial wave function $\psi^{(0)}_\mu(i)$
2. put the trial wave function into the Hartree-Fock equation $\displaystyle F(\psi_{\nu}^{(0)}(j))$
3. Solve the equation and obtain a new wave function $\psi^{(1)}_\mu(i)$
4. Go to step 2. until converge.

In this post, we will discuss at least 2 method on the step 3.

One method is solve the Hartree-Fock equation using numerical method, for example, Rungu-Kutta method, or Numerov’s method. Those method can also obtain the energy when impose some conditions, for example, the behaviour of the wave function at long range.

The 2nd method is basis expansion. We assume the wave function is a superposition of some basis,

$\displaystyle \psi_\mu(i) = \sum_{\alpha}^{N_\alpha} a_{\mu\alpha} \phi_\alpha(i)$

Substitute into the Hartree equation (Hartree-Fock equation is similar),

$\displaystyle F \left(\sum_{j=i+1}^{N}\psi_\nu(j) \right) \psi_\mu(i) = H(i) \psi_\mu(i) + \sum_{j=i+1}^{N}\langle \psi_\nu(j) | G(i,j) | \psi_\nu(j) \rangle \psi_\mu(i)$

$\displaystyle \sum_{\alpha} a_{\mu\alpha} F |\phi_{\alpha}(i) \rangle \\ = \sum_{\alpha} a_{\mu\alpha} \left( H(i) |\phi_{\alpha}(i) \rangle + \sum_{k,l} \sum_{j=i+1}^N a_{\nu k}^{*} a_{\nu l}\langle \phi_{\nu k}(j) | G(i,j) | \phi_{\nu l} (j) \rangle |\phi_{\mu \alpha}(i) \rangle \right) \\ = \sum_{\alpha} a_{\mu\alpha} \left( H(i) + G(i) \right) |\phi_{\alpha}(i) \rangle \\ = \sum_{\alpha} a_{\mu \alpha} E |\phi_{\alpha} \rangle$

Acting $\langle \phi_{\mu \beta}(i)|$

$\displaystyle \sum_{\alpha} a_{\mu \alpha} \langle \phi_{\mu \beta} (i) | F | \phi_{\mu \alpha} (i) \rangle = \sum_{\alpha} F_{\beta \alpha} a_{\mu \alpha} = E a_{\mu \beta}$

Thus, we can solve the coefficient $a_{\mu \alpha}$, which is the eigen vector. Then, new trial functions with the new coefficients are used.

In the 2nd method, we can see the integral

$H_{\beta \alpha} = \langle \phi_\beta(i) | H(i)| \phi_\alpha (i) \rangle$

$G_{\beta \alpha}^{k l} = \langle \phi_\beta(i) \phi_k(j) | G(i,j) | \phi_l(j) \phi_\alpha(i) \rangle$

can be pre-calculated. The calculation becomes iterating the coefficient $a_{\mu \alpha}$.

In method 2, the basis does not change, but only the coefficients. The advantage, also the disadvantage of this method is that, all possible integrals are pre calculated, even though some of them will never be used. Once the integrals are calculated, it can be used for solving many similar problems. But the calculation time for the integrals may be long and consume memory.

In previous post, I did not pre-calculate the integral, I simply put the new trial function and integrate again.

This method is very similar to the method 2. However, the basis is the wave function itself. After one calculation, a new basis, which is the wave function, will be used. And the old basis will be replaced. In method 2, only the coefficient is replaced.

This method may somehow faster then method 2, because only nesscary integrals are calculated. In Mathematica, method 2 is generally slower, especially for larger basis.

## Some comments on Hartree-Fock method

Most confusing thing for the method is that, weather the index is represent particle or state.

Look back the method. The total wave function is product of the wave function of each particle.

$\Psi = \phi_1(1) \phi_2(2) .. \phi_N(N)$

Here, we fixed our notation that the subscript is state, the parentheses is particle ID.

The single particle energy is,

$\displaystyle \langle \Psi |\sum_{i}^{N} H(i)|\Psi \rangle = \sum_{i}^{N} \langle \phi_\mu(i) | H(i)|\phi_\mu(i) \rangle$

Here, the $i$-particle in state $\mu$

The mutual interaction energy is

$\displaystyle \langle \Psi |\sum_{i}^{N-1}\sum_{j=i+1}^{N} G(i,j)|\Psi \rangle = \sum_{i}^{N-1}\sum_{j=i+1}^{N} \langle \phi_\mu(i) \phi_\nu(j) | G(i,j)|\phi_\mu(i) \phi_\nu(j) \rangle$

In both energy terms, we can factor out the sum of particle, we get the Hartree equation

$\displaystyle \left( H(i) + \sum_{j=i+1}^{N} \langle \phi_\nu(j) | G(i,j)| \phi_\nu(j) \rangle \right) |\phi_\mu(i) \rangle = \epsilon_\mu | \phi_\mu(i) \rangle$

Now, we can see, the sum is sum over the number of particle, not state.

When solving the Hartree equation using a given basis $b_k(x)$, we set the first trial of the solution to be

$| \phi_\mu(i) \rangle = | b_\mu(i) \rangle$

Then, in the first trial, the Hartree matrix is

$\displaystyle \hat{F}_{\alpha \beta} = \langle b_\alpha(i) | \left( H(i) + \sum_{j=i+1}^{N} \langle b_\nu(j) |G(i,j) | b_\nu(j) \rangle \right) | b_\beta(i) \rangle$

In here, we keep the particle ID. Inside the mean field integral, the trial state $b_\nu(j)$ is used for the particle-j.

After a trial, we will get the eigen vector for each state $v_\mu$, then we construct a new trial function and iterate until converge.

We can see, for multi-particle wave function, the sum is sum over the particle, even thought there are particles share the same state. In fact, since each particle should be at a state, the state should be a function of particle ID. So, the general Hartree matrix is

$\displaystyle \hat{F}_{\alpha \beta} = \langle \psi_\alpha(i)|\left( H(i) + \sum_{j=i+1}^{N} \langle \psi_{\nu(j)}(j) | G(i,j)| \psi_{\nu(j)}(j) \rangle \right) |\psi_\beta(i) \rangle$

where $\alpha, \beta = 1,2,3,..., k$, $k$ is number state to use.

In Hartree-Fock method, the sum is also over all other particle, not state.

## Hartree method for 1D infinite potential well

In order to test my understanding on the Hartree method. I played a simple case. The potential is

$V(x) = \left\{ \begin{matrix} 0 , 0 \leq x \leq \pi \\ \infty , else \end{matrix} \right\}$

The mutual interaction between 2 particles is, for simple calculation,

$G(x_1, x_2) = g_0 \cos(x_1 - x_2)$

This mutual interaction is repulsive, we can expect the eigen wave function will be more spread out.

We use the eigen state of the $V(x)$ potential to be the basis,

$\displaystyle \phi_n(x) = \sqrt{\frac{2}{\pi}} \sin(nx)$

The Hartree method is basically solving the Hartree equation

$\displaystyle \left( -\frac{\hbar^2}{2m} \frac{d^2}{dx_1^2} + V(x_1) + \int G(x_1, x_2) (\psi_k(x_2))^2 dx_2 \right) \psi_n(x_1) = \epsilon_n \psi_n(x_1)$

We assume the two particles are in the same ground state, thus, $k = n = 1$. We assume,

$\psi_1(x) = \sum a_i \phi_i(x)$

The algorithm is that, for the 1 particles matrix

$\displaystyle F_{ij} = \int \psi_i(x) \left( -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right) \psi_j(x) dx$

Define

$\displaystyle G_0(x) = \int G(x, x_2) (\psi_1(x_2))^2 dx_2$

The 2-particle matrix

$\displaystyle G_{ij} = \int \psi_i(x) G_0(x) \psi_j(x) dx$

For the 1st trial, we set

$\psi_i^{(0)}(x) = \phi_{(2i+1)}(x)$

We only use “even” function with respect to $x = \pi/2$, because mixing parity will result non-reasonable result. ( why ? )

Then we get the matrix of the Hartree Hamiltonian $H_{ij} = F_{ij} + G_{ij}$, and solve for the eigen value and eigen vector,

$\vec{v}^{(0)} = (v_1^{(0)}, v_2^{(0)}, .... v_n^{(0)})$

The next trial is

$\displaystyle \psi_i^{(1)}(x) = \sum_i v_i^{(0)} \psi_i^{(0)}$

The process repeated until converge. The convergent can be exam using the Hartree Hamiltonian that the non-diagonal terms are vanished, i.e.

$F_{ij} + G_{ij} = 0, i\neq j$

I tried with 2 basis and 6 basis, I set $\hbar^2/2m = 1$ for simplicity, the Hartree ground state is 1.7125 for both basis. The eigen wavefunction for the 2-basis is

$\psi(x) = 0.797752 \sin(x) + 0.0145564 \sin(3x)$

with the mean field

$V_m(x) = 0.842569 \sin(x)$

For 6 basis, the wave function is

$\psi(x) = 0.797751 \sin(x) + 0.0145729 \sin(3 x) + 0.000789143 \sin(5 x) + 0.000125871 \sin(7 x) + 0.0000336832 \sin(9 x) + 0.0000119719 \sin(11 x)$

we can see, the contribution of the higher energy states are getting smaller and smaller. The higher energy states only contributes 0.03%. Following is the plot compare $\phi_1(x)$ (orange) and $\psi(x)$ (blue). We can see that the wave function becomes little spread out.

If we increase the strength of the mutual interaction by 10 times, the wave function becomes more spread out.

To cross check the result,  we can use the mean field and solve the Schrodinger equation using variational method. The method gives the ground state energy be 1.7125 and converge.

This is interesting that the variational method needs 4 or 5 basis to converge, while Hartree method only need 2.

At last, we substitute the wave functon back to the Hartree equation, it turns out the result is not satisfied 100%. The total wave function is

$\Psi(x_1, x_2) = \psi(x_1) \psi(x_2)$

The Schrodinger equation is

$\displaystyle H \Psi = -\frac{\hbar^2}{2m}\left(\frac{d^2}{dx_1^2} + \frac{d^2}{dx_2^2}\right) \Psi(x_1,x_2) + G(x_1,x_2)\Psi(x_1,x_2) = E \Psi(x_1,x_2)$

However, the left side and the right side are not equal, but the integration

$\langle \Psi |H|\Psi\rangle = \langle |\Psi|\Psi \rangle$

is equal.

I plot the $\Psi H\Psi$ (orange) and $E \Psi^2$ (blue), and there difference is the following

We can imagine, the integration of the bottom plot is zero. I don’t know the reason, but I guess it is not because of the number of basis, because the higher energy states only contributes for 0.03%. I guess, it is because of the energy is the quantity that is optimized rather then the wave function. And I am not sure that the reverse of the following is true.

$H|\Psi\rangle = E |\Psi\rangle \rightarrow \langle \Psi|H|\Psi \rangle = E \langle \Psi|\Psi \rangle$

For the case of many particles, the $G_0$ has to sum over all other particle. When all particles are in the same states, the sum is simple. For particles distribute among different states, in general, the sum is

$\displaystyle G_0 =\sum_{k\neq i} \int G(x_1,x_2) \psi_k(x_2)^2 dx$

Here, the sum is excluded the particle $x_1$, which is in state $i$. For example, if the system consists of 2 particles on state-1, and 2 particles on state-2. The mean field for the particle in state-1 is constructed using 1 particles on state-1 and 2 particles on state-2. Thus the energy for the state-2 may be not correct.

The mean field for the particle on state-2, the mean field should be constructed by 2 particles on state-1 and 1 particle on state-2.

For higher states that without particle, the mean field is built by 2 particles on state-1 and 2 particles on state-2.

In general, the mean field is constructed using other particles.

This fixed mean field method is simple but we have to define the mean field for each state and calculate again and again.

## Variational method on 1D potential well using plane wave basis

The potential is

$V(x) = \left \{ \begin{matrix} - V_0 & , -L/2 \leq x \leq L/2 \\ 0 & , else \end{matrix} \right \}$

The exact solution can be solved by matching boundary condition.

$\tan(\sqrt{\frac{2m}{\hbar^2}(E+V_0 ) \frac{L}{2}}) = \sqrt{\frac{|E|}{E+V_0}}$

The plane wave basis is

$b_n(x) =\frac{1}{\sqrt{a}} \exp\left( i \frac{2\pi}{a} n x \right)$

The matrix of the Hamiltonian is

$\displaystyle H_{ij} = \langle b_i | \frac{P^2}{2m} + V(x) | b_j \rangle$

$\displaystyle \langle b_i | \frac{P^2}{2m} | b_j \rangle = \delta_{ij} \frac{L}{a}\frac{\hbar^2}{2m} \left(\frac{2\pi}{a}\right)^2 i^2$

$\displaystyle \langle b_i | V(x) | b_j \rangle = -V_0 \frac{L}{a} \frac{\sin( \pi (i-j) L/a)}{\pi (i-j) L/a}$

Then, we solve for the eigen system.

I use mathematica for solving the eigen system.

For simplicity, I set $L = 2, V_0 = 1$. The exact solution is $E = -0.603898$.

The parameter $a$ controls the wavelength of the plane wave. By increasing the number of wave $n$, we effectively control the minimum wavelength $\lambda_0 = a/n$ Therefore, larger the $n$, the energy will approach to the actual value.

I generated $a$ from 10 to 200 with step 10, and $n$ from 5 to 60 with step 5. Here is the result. As we can see, the larger the $a$, we need more number of wave to go to the actual energy. When small $a$ and large $n$, the result quickly converge to the actual energy. For $n = 60, a = 40$, the energy is $E = -0.603792$, the difference is $\Delta E = 0.000106$ or 0.02% difference.

The following plot show $a = 40$

However, when the $a$ is too small, the plane wave cannot describe the actual wave function, then the converge fail. The following plot show $n = 60$

To investigate, I plot the case for $a = 3$ and $a = 10$.

We can see, since the well width is 2, if the maximum wavelength is only 3, then it cannot capture a longer wavelength component. And we can see the calculated wave functions are repeated.

## Variation method & Eigenvalue Problem

The relationship I haven’t stated explicitly, so, here.

In solving the Schrodinger equation (or a general Hamiltonian problem )

$H |\psi \rangle = E|\psi \rangle$

The variation method is a golden method to find the minimum energy. For the minimum energy solution $\psi_0$

$\displaystyle \frac{\langle \psi_0|H|\psi_0\rangle}{\langle \psi_0|\psi_0 \rangle} = E_0$

For other solution

$\displaystyle \frac{\langle \psi|H|\psi\rangle}{\langle \psi|\psi \rangle} \geq E_0$

This implies the diagonal elements of $H$ are always larger or equal the minimum energy.

If we expand the solution in to an orthogonal complete basis $\phi_i$

$\displaystyle |\psi\rangle = \sum_i a_i |\phi_i\rangle$

The Schrodinger equation

$\displaystyle \sum_i a_i H|\phi_i\rangle = \sum_i a_i E |\phi_i\rangle$

since the basis is orthogonal, multiple

$\displaystyle \sum_i a_i \langle \phi_j |H|\phi_i \rangle = \sum_i a_i E \langle \phi_j |\phi_i \rangle$

$\displaystyle \sum_i \langle \phi_j | H|\phi_i\rangle a_i = E a_j$

$\displaystyle \sum_i H_{ji} a_i = E a_j$

In matrix form

$H \cdot \vec{a} = E \vec{a}$

this is the eignevalue problem.

The eigenvalue problem does not solve the ground state, but also the excited states.