Second quantization#

Fock space#

Consider a system of electrons in \(M\) spin orbitals. An occupation number vector lists the spin-orbital occupation

\[ | \mathbf{k} \rangle = | k_1 \ldots k_p\ldots k_M \rangle \]

where the occupation numbers, \(k_p\), are restricted to 0 or 1.

The Fock space, \(F(M)\), is defined as the vector space of all possible occupation vectors.

There is a unique mapping between occupation number vectors and Slater determinants, but the Fock space includes states with different number of electrons.

The Fock space is a linear vector space with the following properties:

  • occupation number vectors are orthonormal

    \[ \langle \mathbf{k} | \mathbf{m} \rangle = \prod_{p=1}^{M} \delta_{k_p m_p} \]

  • the identity operator can be written as

    \[ \hat{I} = \sum_\mathbf{k} | \mathbf{k} \rangle \langle \mathbf{k} | \]

  • it can be partitioned into sub-spaces according to

    \[ F(M) = F(0,M) \oplus F(1,M) \oplus \cdots \oplus F(M,M) \]

    where \(F(N, M)\) is the set of occupation number vectors with \(N\) occupied spin-orbitals.

  • subspace \(F(0, M)\) corresponds to a completely unoccupied Slater determinant known as the vacuum state, \(| \mathrm{vac} \rangle\). For the vacuum state, we have

    \[\begin{equation*} | \mathrm{vac} \rangle = | 0_1 0_2 \ldots 0_M \rangle ; \qquad \langle \mathrm{vac} | \mathrm{vac} \rangle = 1 \end{equation*}\]

Creation and annihilation operators#

Second quantization introduces a convenient way to manipulate states by means of creation and annihilation operators. As their name imply, these operators create and destroy electrons in specific spin-orbitals as follows

  • creation operator \(\hat{a}_p^\dagger\):

\[ \hat{a}_p^\dagger | \mathbf{k} \rangle = \delta_{k_p 0} \prod_{i = 1}^{p} (-1)^{k_p} | k_1 k_2...1_p ...k_M \rangle \]

  • annihilation operator \(\hat{a}_p\):

\[ \hat{a}_p | \mathbf{k} \rangle = \delta_{k_p 1} \prod_{i = 1}^{p} (-1)^{k_p} | k_1 k_2...0_p ...k_M \rangle \]

where the factor \(\delta_{k_p 0}\) prevents the addition of an electron to a state that is already populated and \(\delta_{k_p 1}\) prevents the destruction of a non-existing electron.

The phase factor \(\prod_{i = 1}^{p} (-1)^{k_p}\) ensures the anti-symmetry of the associated wave function and it is a direct consequence of the anti-commutator relationships:

\[\begin{eqnarray*} \hat{a}_p^\dagger \hat{a}_q + \hat{a}_q \hat{a}_p^\dagger &=& \delta_{pq} \\ \hat{a}_p^\dagger \hat{a}_q^\dagger + \hat{a}_q^\dagger \hat{a}_p^\dagger &=& 0 \\ \hat{a}_p \hat{a}_q + \hat{a}_q \hat{a}_p &=& 0 \end{eqnarray*}\]

Let us consider the closed-shell ground state for a system with four electrons in five molecular orbitals, or ten spin orbitals,

\[ |0\rangle = |\psi_1, \psi_\bar{1}, \psi_2, \psi_\bar{2}\rangle \]
import numpy as np

M = 10  # number of spin orbitals in our Fock space

We implement a class of occupation number vectors.

class OccupationNumberVector:
    def __init__(self, occ_orbs=[]):

        self.phase = 1
        self.occ_num_vec = np.zeros(M, dtype=int)

        for i in occ_orbs:

            self.occ_num_vec[i] = 1

    def __mul__(self, other):

        if np.array_equal(self.occ_num_vec, other.occ_num_vec):
            inner_prod = self.phase * other.phase
        else:
            inner_prod = 0

        return inner_prod

    def creation(self, p):

        if len(self.occ_num_vec) == 0:
            return

        if self.occ_num_vec[p] == 1:
            self.occ_num_vec = []
        else:
            self.phase *= np.prod((-1) ** self.occ_num_vec[:p])
            self.occ_num_vec[p] = 1

    def annihilation(self, p):

        if len(self.occ_num_vec) == 0:
            return

        if self.occ_num_vec[p] == 0:
            self.occ_num_vec = []
        else:
            self.phase *= np.prod((-1) ** self.occ_num_vec[:p])
            self.occ_num_vec[p] = 0

The ground state can formed by applying a series of creation operators on the vacuum state

\[ | 0 \rangle = \prod_{i}^\mathrm{occ} \hat{a}^\dagger_i |\mathrm{vac}\rangle \]
ket = OccupationNumberVector()

for i in [3, 2, 1, 0]:
    ket.creation(i)

print("ground state:", ket.occ_num_vec)

ground state: [1 1 1 1 0 0 0 0 0 0]

One-electron operators#

A one-electron operator takes the form

\[ \hat{\Omega} = \sum_{p,q} \omega_{pq} \, \hat{a}_p^\dagger \hat{a}_q \]

where

\[ \omega_{pq} = \langle \psi_p | \hat{\omega} | \psi_q \rangle \]

One-particle density#

The one-particle density matrix becomes

\[ D_{pq} = \langle 0| \hat{a}_p^\dagger \hat{a}_q| 0\rangle \]
bra = OccupationNumberVector([0, 1, 2, 3])

D = np.zeros((M, M), dtype=int)

for p in range(M):
    for q in range(M):

        ket = OccupationNumberVector([0, 1, 2, 3])

        ket.annihilation(q)
        ket.creation(p)

        D[p, q] = bra * ket

print("One-particle denisty matrix:\n", D)

One-particle denisty matrix:
 [[1 0 0 0 0 0 0 0 0 0]
 [0 1 0 0 0 0 0 0 0 0]
 [0 0 1 0 0 0 0 0 0 0]
 [0 0 0 1 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0 0]]

Two-electron operators#

A two-electron operator takes the form

\[ \hat{\Omega} = \sum_{p,q,r,s} \omega_{pqrs} \, \hat{a}_p^\dagger \hat{a}_q^\dagger \hat{a}_s \hat{a}_r \]

where, in the physicist’s notation,

\[ \omega_{pqrs} = \langle \psi_p \psi_q | \hat{\omega} | \psi_r \psi_s \rangle \]

Hamiltonian#

The Hamiltonian becomes

\[ \hat H = \sum_{p,q} h_{pq} \hat{a}_p^\dagger \hat{a}_q + \frac{1}{2} \sum_{p,q,r,s} g_{pqrs} \hat{a}_p^\dagger \hat{a}_r^\dagger \hat{a}_s \hat{a}_q \]

where, in the chemist’s notation,

\[ g_{pqrs} = (\phi_p \phi_q | \hat{\omega} | \phi_r \phi_s ) \delta_{\sigma_p\sigma_q} \delta_{\sigma_r\sigma_r} \]

Here \(\phi_p\) is the spatial part (MO) of spin orbital \(\psi_p\) and the Kronecker deltas are the result of the spin integration.

Illustration#

The energy of a single-determinant state equals

\[ \langle 0| \hat H | 0\rangle = \sum_{pq} h_{pq} \langle 0| \hat{a}_p^\dagger \hat{a}_q| 0\rangle + \frac{1}{2} \sum_{pqrs} g_{pqrs} \langle 0| \hat{a}_p^\dagger \hat{a}_r^\dagger \hat{a}_s \hat{a}_q | 0\rangle \]

We thus need to evaluate the one- and two-particle density matrices.

For the one-electron term, we notice that orbital \(q\) has to be occupied to obtain a nonzero result as the annihilation operator acts to the right on \(| 0\rangle\). Using the anti-commutator relation, we get

\[ \langle 0| \hat{a}_p^\dagger \hat{a}_i | 0 \rangle = \delta_{pi} \langle 0| 0 \rangle - \langle 0| \hat{a}_i \hat{a}_p^\dagger | 0 \rangle = \delta_{pi} \]

This results in

\[ \sum_{p,q} h_{pq} \langle 0| \hat{a}_p^\dagger \hat{a}_q| 0\rangle = \sum_i^\mathrm{occ} h_{ii} \]

For the two-electron term, we notice that orbitals \(q\) and \(s\) both have to be occupied to obtain a nonzero result. We focus on \(s\) and will use the same trick to move \(\hat{a}_i\) to the left

\[\begin{align*} \hat{a}_p^\dagger \hat{a}_r^\dagger \hat{a}_i \hat{a}_q & = \delta_{ri} \hat{a}_p^\dagger \hat{a}_q - \hat{a}_p^\dagger \hat{a}_i \hat{a}_r^\dagger \hat{a}_q \\ &= \delta_{ri} \hat{a}_p^\dagger \hat{a}_q - \delta_{pi} \hat{a}_r^\dagger \hat{a}_q + \hat{a}_i \hat{a}_p^\dagger \hat{a}_r^\dagger \hat{a}_q \end{align*}\]

The last term will vanish in the expectation value and we are left with only terms involving only pairs of creation or annihilation operators. We get

\[\begin{align*} \sum_{p,q,r,s} g_{pqrs} \langle 0| \hat{a}_p^\dagger \hat{a}_r^\dagger \hat{a}_s \hat{a}_q | 0\rangle &= \sum_{i,p,q} g_{pqii} \langle 0|\hat{a}_p^\dagger \hat{a}_q | 0\rangle - \sum_{i,q,r} g_{iqri} \langle 0| \hat{a}_r^\dagger \hat{a}_q | 0\rangle \\ & = \sum_{i,j} \big( g_{jjii} - g_{ijji} \big) \end{align*}\]

and the total energy thus becomes equal to

\[ \langle 0| \hat H | 0\rangle = \sum_i h_{ii} + \frac{1}{2} \sum_{i,j} \big( g_{jjii} - g_{ijji} \big) \]