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Lessons Lesson 9

Lesson 9 - Intro To Quantum Mechanics

Written by Jeremy Schroeder.

Introduction

All levels of theory (DFT, HF etc.) are simplifications and special cases of Quantum Mechanics. So in this lesson, we will do a deep dive in quantum mechanics to understand where the equations that define DFT come from.

Time Independent Schrödinger Equation

This is the equation that defines all particle interactions for DFT. There is a time dependent Schrödinger Equation but that is more complicated and we don't use it in DFT but it is used in AIMD (Ab-Initio Molecular Dynamics) where time is considered.

Hamiltonionian Case:

\(\hat{H} \psi = E \psi\)

1-D Case:

\(-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x)\)

where \(\psi(x) = (\frac{2}{l})^{1/2} \sin(\frac{n\pi x}{l})\)

Particle in a One-Dimensional Infinite Well

This is the first analytical solution to the Schrödinger equation. Lets say we have a 1-D Box where the potential well is infinty from x < 0 and x > l.

In Region I and Region III,

  • \(\psi_I(x) = \psi_{III}(x) = 0\).
  • \(V_I(x) = V_{III}(x) = \infty\)

In Region II,

  • \(\psi_{II}(x) = (\frac{2}{l})^{1/2} \sin(\frac{n\pi x}{l})\)
  • \(V_{II}(x) = 0\)
  • \(n = 1,2,3,4,5...\)

Figure 2.1

If you notice there is n in the wavefunction equation. n is the principle quantum number and is any non zero positive integer. This process of assigning n is quantization. This is where quantum comes from in quantum mechanics. The below image is showing the wave function for n = 1,2,3.

Figure 2.3 and 2.4

\(\psi\) is the location of where a particle and \(|{\psi}^2|\) is the probability density of where that particle will be found. For the 1-D Box, \(\int_{0}^{l} |{\psi(x)}^2| dx = 1\).

Particle in a One-Dimensional Rectangular Well

If the Potential function is not infinite, then we have:

  • \(V_I(x) = V_{III}(x) = V_0\)
  • \(V_{II}(x) = 0\)
  • \(\psi_I(x) = C \exp[(\frac{2m}{\hbar^2})^{1/2}(V_0 - E)^{1/2}x]\)
  • \(\psi_{II}(x) = A \cos((\frac{2m}{\hbar^2})^{1/2}E^{1/2}x) + B \sin((\frac{2m}{\hbar^2})^{1/2}E^{1/2}x)\)
  • \(\psi_{III}(x) = G \exp[-(\frac{2m}{\hbar^2})^{1/2}(V_0 - E)^{1/2}x]\)

If \(\psi \rightarrow 0\) as \(x \rightarrow -\infty\) and \(x \rightarrow \infty\), this is a bound state. For a bound state, you are guranteed to find the particle in a finite region. For an unbound state which is where \(\psi\) does not go to zero as \(x \rightarrow -\infty\) and \(x \rightarrow \infty\). For this case, if \(E > V_0\) it is unbound and if \(E < V_0\) is a bound state.

Figure 2.5

If you notice in the picture above, in image (b), the probability density of the wave function exists in Region I and Region II which is different than in the infinite well case. This is the theory behind quantum tunnelling of particles. This picture is a bound state so \(E < V_0\), in classical theory, a particle would not be able to exist being the potential well but here in quantum mechanics, there is a chance the particle will escape the potential well.

A Note on Operators

Angular Momentum of a Particle

Figure 5.5

Below are Angular Momentum Equations for a particle in 3D coordinates. Notice they are operators and that \(\hat{L_z}\) is a much simplier equation than \(\hat{L_x}\) and \(\hat{L_y}\).

  • \(\hat{L_x} = i\hbar(\sin\phi\frac{\delta}{\delta\theta} + \cot\theta\cos\phi\frac{\delta}{\delta\phi})\)
  • \(\hat{L_y} = -i\hbar(\cos\phi\frac{\delta}{\delta\theta} - \cot\theta\sin\phi\frac{\delta}{\delta\phi})\)
  • \(\hat{L_z} = -i\hbar\frac{\delta}{\delta\phi}\)
  • \(\hat{L}^{2} = -i\hbar^{2}(\frac{\delta^2}{\delta\theta^2} + \cot\theta\frac{\delta}{\delta\theta} + \frac{1}{\sin^2\theta}\frac{\delta^2}{\delta\phi^2})\)

Because \([\hat{L_z}, \hat{L}^2] = 0\), operators \(\hat{L_z}\) and \(\hat{L}^2\) commute. Therefore we can write the equations below:

  • \(\hat{L_z}Y(\theta,\phi) = m\hbar Y(\theta,\phi)\)
  • \(\hat{L}^2Y(\theta,\phi) = l(l+1)\hbar^2Y(\theta,\phi)\)

Where \(Y(\theta,\phi)\) is an arbitrary eigenfunction and \(m\hbar\) and \(l(l+1)\hbar^2\) are eigenvalues of \(\hat{L_z}\) and \(\hat{L}^2\) respectively.

Then we find the eigenfunction for the above set of equations:

  • \(Y_{l}^{m}(\theta, \phi) = \frac{1}{\sqrt{2\pi}} S_{l,m}(\theta)e^{im\phi}\)

Where \(m\), and \(l\) are other quantum numbers.

  • \(l = 0, 1, 2,...\)
  • \(m = -l, -l+1,...-1,0,1,...+l-1,+l\)

\(S_{l,m}(\theta)\) is given by the equations below. This is called a Legrende Equation by the person who discovered it.

  • \(S_{l,m}(\theta) = \sin^{|m|}\theta\sum_{\substack{j=1,3,...\\or\:j=0,2...}}^{l-|m|}a_j\cos^j\theta\)
  • \(a_{j+2} = \frac{(j+|m|)(j+|m|+1)-l(l+1)}{(j+1)(j+2)}\)

This table shows the above series expansion for different l and m quantum numbers.

Table 5.1

Describing the Hydrogen Atom

Now we have a set of eigenfunctions to describe the one central atom problem.

  • \(\hat{H}\psi = E\psi\)
  • \(\hat{L}^2\psi = l(l+1)\hbar^2\psi\) where \(l = 0, 1, 2,...\)
  • \(\hat{L_z}\psi = m\hbar \psi\) where \(m = -l, -l+1,...-1,0,1,...+l-1,+l\)

Where \(\psi = R(r)Y_{l}^{m}(\theta, \phi)\)

There is a lot more math and derivation to get to orbital functions. Below is what orbital the quantum number l denotes.

Table 6.110

The solution of \(R(r)\) for the bond state of the Hydrogenlike atom is below:

  • \(R_{nl}(r) = r^le^{-\frac{Zr}{na}}\sum_{j=0}^{n-l-1}b_jr^j\)

Where \(a = \frac{4\pi\epsilon_0\hbar^2}{\mu e^2}\) and Z is atomic number (Z = 1 for Hydrogen).

So the complete wavefunction for a Hydrogenlike atom is:

  • \(\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_l^m(\theta, \phi)\)
  • \(\psi_{nlm}(r, \theta, \phi) = R_{nl}(r)S_{l,m}(\theta)\frac{1}{\sqrt{2\pi}}e^{im\phi}\)
  • \(\psi_{nlm}(r, \theta, \phi) = r^le^{-\frac{Zr}{na}}\sum_{j=0}^{n-l-1}[b_jr^j] * \sin^{|m|}\theta\sum_{\substack{j=1,3,...\\or\:j=0,2...}}^{l-|m|}[a_j\cos^j\theta] * \frac{1}{\sqrt{2\pi}}e^{im\phi}\)

Table 6.2

The constant a stated earlier is actually the Bohr radius, which is the radius of the circle in which the electron moved in the grpund state of the hydrogen atom in Bohr Theory.

  • \(a = \frac{4\pi\epsilon_0\hbar^2}{\mu e^2}\)
  • \(\mu = \frac{m_em_p}{m_e + m_p}\)

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.13

Figure 6.14

Work Still Needed

  • Add Quantum Chemical Theorems and Assumptions
  • Table 15.1
  • Chapter 15.17 Solvent Effects

Table 15.1

Citations

  • All Figures and Images are from I. Levine, Quantum Chemistry, 7th Edition, Pearson