Set $R_{10}(r)=Ce^{-r^2}$ , then normalized the guess to get C $$\int_{0}^{\infty}C^2e^{-2r^2}r^2dr = 1$$, $$C = \sqrt \frac{1}{\int_{0}^{\infty}e^{-2r^2}r^2dr}$$
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import sympy as sy
import numpy as np
from sympy import *
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r = Symbol('r')
I = integrate(exp(-2*r**2)*r**2,(r,0,+oo))
C = sqrt(1/I)
print(latex(simplify(C)))
So the normalized 1s wavefunction is $\tilde{R}_{10}(r) = \frac{2}{\sqrt[4]{\pi}} 2^{\frac{3}{4}} e^{-r^2} = (\frac{128}{\pi}) ^ {\frac{1}{4}} e^{-r^2} $.
$\because$ 1s orbital, n=1,l=0 $$\therefore \hat{H} = -\frac{1}{2}\frac{d^2}{dr^2} - \frac{1}{r}\frac{d}{dr}-\frac{1}{r}$$
$$\frac{d(\tilde{R}_{10})}{dr}= -2rCe^{-r^2} $$$$\frac{d^2(\tilde{R}_{10})}{dr^2}=-2Ce^{-r^2}+4r^2Ce^{-r^2}$$$$\hat{H}\tilde{R}_{10}(r) = -\frac{1}{2}\frac{d^2(\tilde{R}_{10})}{dr^2} - \frac{1}{r}\frac{d(\tilde{R}_{10})}{dr}-\frac{1}{r}(\tilde{R}_{10}) = Ce^{-r^2}-2r^2Ce^{-r^2} +2Ce^{-r^2} -\frac{Ce^{-r^2}}{r} = \frac{Ce^{-r^2}(-2r^3+3r-1)}{r}$$The expectation value of the energy: $$\langle E\rangle = \int_{0}^{\infty}\tilde{R}_{10}(r)\hat{H}\tilde{R}_{10}(r)r^2dr = \int_{0}^{\infty} C^2(-2r^4+3r^2-r)e^{-2r^2} dr$$
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E = C**2*integrate((-2*r**4+3*r**2-r)*exp(-2*r**2),(r,0,oo))
print('Expected value is %0.4f Ha.'%E)
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# Hydrogen atom energy equation is given in class notes
n=1
E_true = -1/(2*n**2) # unit Ha
print('The ture value is %0.4f Ha. So the expected value is greater than the true value.' %E_true)
The variational principle says true wavefunction energy is always lower bound on energy of any trial wavefunction. $$\langle ψ_\text{trial}^λ | \hat{H} | ψ_\text{trial}^λ\rangle =E_\text{trial}^λ \geq E_0$$
We can get the "best" $\gamma$ by optimizing the wavefunction with respect to variational parameter: $$\frac{\partial\langle E\rangle}{\partial\gamma} = 0$$
Normalize $R_{10}$: $\int_{0}^{\infty}r^2C^2e^{-2\gamma r^2}dr = 1$ $$C = \sqrt \frac{1}{\int_{0}^{\infty}e^{-2\gamma r^2}r^2dr}$$
$\langle E\rangle = \int_{0}^{\infty}\tilde{R}_{10}(r)\hat{H}\tilde{R}_{10}(r)r^2dr = \int_{0}^{\infty} C^2(-2\gamma^2 r^4+3\gamma r^2-r)e^{-2\gamma r^2} dr$
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gamma = symbols('gamma',positive=True) # We know the gamma has to be positive, or the R10 would be larger when r increase.
r = symbols("r",positive=True)
C = sqrt(1/integrate(exp(-2*gamma*r**2)*r**2,(r,0,oo)))
E = C**2*integrate((-2*gamma**2*r**4+3*gamma*r**2-r)*exp(-2*gamma*r**2),(r,0,oo)) # expectation value of energy as a function of gamma
gamma_best=solve(diff(E,gamma),gamma)
print("Expectation of energy:");print(E)
print("Best value of gamma is: %s, which equals to %f."% (gammabest,8/(9*np.pi)))
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import math
gamma_best = 8/(9*np.pi)
E_best = 8*math.sqrt(2)*gamma_best**(3/2)*(-1/(4*gamma_best) + 3*math.sqrt(2)*math.sqrt(np.pi)/(32*math.sqrt(gamma_best)))/math.sqrt(np.pi)
print("Energy with the best gamma: %0.3f eV."%E_best)
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import matplotlib.pyplot as plt
gamma = np.linspace(0.001,1.5,10000)
E = []
for x in gamma:
E.append(8*math.sqrt(2)*x**(3/2)*(-1/(4*x) + 3*math.sqrt(2)*math.sqrt(np.pi)/(32*math.sqrt(x)))/math.sqrt(np.pi))
plt.plot(gamma,E)
plt.xlabel("Gamma")
plt.ylabel("Energy(eV)")
plt.axvline(x=8/(9*np.pi),color='k',linestyle='--')
plt.axhline(y=E_best,color='k',linestyle='--')
plt.annotate('Lowest energy spot', xy=(8/(9*np.pi), E_best), xytext=(0.5,-0.2), arrowprops=dict(facecolor='black'))
plt.show()
Schrödinger equation: $$\hat{H}\Psi(r_1,r_2)=E\Psi(r_1,r_2)$$ $$\hat{H}=\hat{h_1}+\hat{h_2}+\frac{e^2}{4\pi\epsilon_0}\frac{1}{|r_1-r_2|}$$ $$\hat{h_1}=-\frac{\hbar^2}{2m_e}\nabla^2_1-\frac{2e^2}{4\pi\epsilon_0}\frac{1}{r_1}$$ $$\hat{h_2}=-\frac{\hbar^2}{2m_e}\nabla^2_2-\frac{2e^2}{4\pi\epsilon_0}\frac{1}{r_2}$$
All the potential terms $\hat{\nu}$ depend on the $\psi$ solved. The self-consistently means we get a $\psi$ solution, then we use it to calculate new potential terms and solve for new $\psi$, and repeat until we get a $\psi$ same with the last one.
One solution for each orbital, so there needs to be only one solution to describe the electrons in the He atom.
Two possible set of quantum numbers for each electron: $$n=1, l=0, m_l=0, m_s=+\frac{1}{2}$$ $$n=1, l=0, m_l=0, m_s=-\frac{1}{2}$$
$-\frac{1}{2}\nabla^2$: Kinetic energy. Always positive.
$-\frac{2}{r}$: Due to the attraction between the electron and the nucleus. Negative.
$\hat{\nu}_{coul}$: Classical repulsion between distinguishable electron “clouds”. Positive.
$\hat{\nu}_{ex}$: Accounts for electron indistinguishability (Pauli principle for fermions). Decreases Coulomb repulsion because electrons of like spin intrinsically avoid one another. Negative.
$\hat{\nu}_{corr}$: Decrease in Coulomb repulsion due to dynamic ability of electrons to avoid one another; “fixes” orbital approximation. Negative.
B(doublet) | Energy (Hartree) | C(triplet) | Energy (Hartree) | N(quarlet) | Energy (Hartree) | O(triplet) | Energy (Hartree) | F(doublet) | Energy (Hartree) | Ne(singlet) | Energy (Hartree) |
---|---|---|---|---|---|---|---|---|---|---|---|
1s | -13.23430 | 1s | -20.01336 | 1s | -28.13652 | 1s | -37.71226 | 1s | -48.64339 | 1s | -30.45968 |
2s | -0.65884 | 2s | -0.93550 | 2s | -1.23866 | 2s | -1.63718 | 2s | -2.06968 | 2s | -1.26438 |
2p | -0.15147 | 2p | -0.43435 | 2p | -0.87602 | 2p | -1.30625 | 2p | -1.89967 | 2p | -1.34278 |
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# From http://www.genstrom.net/public/biology/common/en/em_spectrum.html
print("The 1s energies become increasingly negative with inceasing Z. Light must become increasingly energetic to kick out one of them.")
hc = 1239.8 #eV*nm
E = [13.23430*27.212, 20.01336*27.212, 28.13652*27.212, 37.71226*27.212,48.64339*27.212,30.45968*27.212] # eV
lamb = [] #nm
for e in E:
lamb.append(hc/e)
print(lamb,"nm.\nThey corresponds to X-rays.")
*There is a big energy decrease as it goes across the series because the electrostatic attraction of the nucleus for electrons increases as the the number of protons increases.
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hc = 1239.8 #eV*nm
E = [0.15147*27.212, 0.43435*27.212, 0.87602*27.212, 1.30625*27.212,1.89967*27.212,1.34278*27.212] # eV
lamb = [] #nm
for e in E:
lamb.append(hc/e)
print(lamb,"nm.\nThey corresponds to UVs")
There is a decrease as it goes across the series because the electrostatic attraction of the nucleus for electrons increases as the the number of protons increases.
And energy decrease is smaller because the outer electrons are partially shielded from the attractive force of the protons in the nucleus by inner electrons.