Second quantization - Chemisorption

Literature: Schwabl, Advanced Quantum Mechanics, Chapter 1

We are going to be dealing with many-body electron states: we have a Hamiltonian of the form

H = H(1, 2, ...)

(1)

where $1, 2,...$ denote the coordinates of particles 1, 2, etc. The wavefunction is

\Psi = \Psi(1, 2,...).

(2)

For indistinguishable fermonic particles such as electrons, the wavefunction is antisymmetric under particle exchange:

\psi(...,\alpha,...,\beta...) = -\psi(...,\beta...,\alpha,...)

(3)

For $N$ states and $N$ electrons, each electron with coordinates $\alpha$ can be in a state $i$ ; this is denoted as $|i\rangle_\alpha$ . The many-electron wave function can be written as a determinant to satisfy antisymmetry.

\Psi(1,2,...,N)= {1 \over \sqrt{N!}} \begin{vmatrix} |1\rangle_1 & |1\rangle_2 & \cdots & |1\rangle_N \\ |2\rangle_1 & |2\rangle_2 & \cdots & |2\rangle_N \\ \vdots & \vdots & \ddots & \vdots \\ |N\rangle_1 & |N\rangle_2 & \cdots & |N\rangle_N \end{vmatrix}

(4)

={1 \over \sqrt{N!}} \left( |1\rangle_1 |2\rangle_2\cdots|N\rangle_N + \text{permut.} \right)

(5)

where permut. denotes that we sum over all $N!$ permutations.

This notation is very cumbersome, so the idea of second quantization is to denote the many-electron wave function in terms of the occupation of energy levels:

\Psi(1, 2, ...) = |n_1n_2\cdots\rangle

(6)

where $n_i=0$ or 1 is the occupation of state $\phi_i$ .

These wave functions can be thought of as vectors $[1 1 1 1 ... 0 0 0]$ . The space of these vectors is called Fock space.

Creation and annihilation operators¶

Creation operators create an electron; annihilation operators remove one:

c_k^\dag |...,0_k,...\rangle = \theta_k |..., 1_k, ...\rangle

(7)

where $0_k$ and $1_k$ denote occupation 0 or 1 for state $k$ .

Similarly, annihilation operators remove an electron from a state:

c_k |...,1_k,...\rangle = \theta_k |..., 0_k, ...\rangle

(8)

Applying a creation operator to a level that is already 1 or an annihilation operator to a level that is already 0, yields zero. Using that property, the number operator can be defined in terms of creation/annihilation operators,

n_k = c_k^\dag c_k

(9)

which yields the occupation (0 or 1) of state $k$ .

Below we sometimes need to ‘translate’ between molecular orbital states and atomic orbital states. Let’s say the creation operator $c_k^\dag$ creates a particle in atomic orbital $k$ :

c_k^\dag |0\rangle = |k \rangle

(10)

The molecular orbitals $m$ are related to atomic orbitals $k$ as

|m\rangle = \sum_k C_{mk} |k\rangle

(11)

where $C_{mk} = \langle m|k\rangle$ are the basis-change or LCAO coefficients. Thus, to create a particle in atomic orbital $m$ , we need a creation operator

c_m^\dag|0\rangle = |m\rangle

(12)

c_m^\dag = \sum_k C_{mk} c_k^\dag.

(13)

Hence, the basis change carries over directly to second quantization.

Operators in second quantization¶

The kinetic energy can be written as a sum of 1-particle operators $t_\alpha=p_\alpha^2/2m$ :

T = \sum_\alpha t_\alpha

(14)

We can insert completeness relations / identity operators

I=\sum_i |i\rangle_\alpha\langle i |_\alpha

(15)

Then

T=\sum_\alpha t_\alpha = \sum_\alpha \sum_{i,j} | i \rangle_\alpha \langle i |_\alpha t_\alpha | j \rangle_\alpha \langle j |_\alpha

(16)

Then $t_{ij}=\langle i |_\alpha t_\alpha | j \rangle_\alpha$ (it’s the same for all identical particles $\alpha$ ), and

T = \sum_{i,j} t_{ij} \sum_\alpha | i \rangle_\alpha \langle j |_\alpha.

(17)

The operator $\sum_\alpha | i \rangle_\alpha \langle j |_\alpha$ basically moves the electron in state $j$ to state $i$ . To see this, we can apply it to a Slater determinant:

\sum_\alpha | i \rangle_\alpha \langle j |_\alpha\Psi(1,2,...) = \sum_\alpha | i \rangle_\alpha \langle j |_\alpha \left(\cdots |k\rangle_\alpha \cdots + \text{permut.}\right)

(18)

Every time the operator encounters particle $\alpha$ occupying state $j$ , i.e., $|j\rangle_\alpha$ , it ‘annihilates’ it ( $\langle j | j\rangle=1)$ and replaces it with a particle $\alpha$ occupying state $i$ , i.e., $|i\rangle_\alpha$ . Otherwise, it returns zero. It does this for all particles $\alpha$ .

This operation is exactly equivalent to what the creation and annihilation operators do. In second quantization, we can therefore replace this logic with

\sum_\alpha | i \rangle_\alpha \langle j |_\alpha=c_i^\dag c_j

(19)

and we do not care about keeping track of the individual particles $\alpha$ anymore, because we operate on the occupation representation (Fock space).

Thus, in second quantization, one-electron operators acting on all electrons in the system can be written as

T = \sum_{i,j} t_{ij} c_i^\dag c_j

(20)

This directly holds for the kinetic energy; the potential energy can also be written in this form following exactly the same argument:

V = \sum_{i,j} V_{ij} c_i^\dag c_j

(21)

A similar logic can be applied to two-particle operators, such as:

W = \frac{1}{2} \sum_{ijkm} w_{ijkl} c_i^\dag c_j^\dag c_m c_k

(22)

which removes two particles from the states $k$ and $m$ , and adds them into states $i$ and $j$ . Here, $w_{ijkl}=\langle i, j | w | k, m \rangle$ is the matrix element of the two-electron operator. The factor 1/2 is necessary because if you count a two-electron interaction for each of the electrons involved, you’re overcounting the total energy by a factor 2.

For the next part, we choose $|i\rangle$ to be eigenfunctions of $t=p^2/{2m}$ . These are plane waves, a common basis choice for a periodic crystal:

|i\rangle = e^{i \mathbf{k}_i \cdot \mathbf{r}}

(23)

In this eigenbasis, the $t$ -operator is diagonal:

t_{ij}=\varepsilon_i\delta_{ij}

(24)

with eigenvalues

\varepsilon_i = \frac{\hbar^2 k_i^2}{2m}.

(25)