Second Quantization

   Much of the literature in CI theory makes use of the notation of second-quantization. Szabo and Ostlund [1] give a good introduction to second-quantized operators. Here we will only summarize the anticommutation relations between creation and annihilation operators, and then proceed to express the Hamiltonian in second quantized form for spatial orbitals, rather than for spin orbitals. Then we will use these results to derive the Hamiltonian in terms of the unitary group generators.

The anticommutation relations for two annihilation operators is

$$\{ a_{j}, a_{i} \} = a_{j} a_{i} + a_{i} a_{j} = 0$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred1 { a_j, a_i } = a_j a_i + a_i a_j = 0 )

and the anticommutation relation for two creation operators is similarly

$$\{ a_{j}^{\dagger}, a_{i}^{\dagger} \} = a_{j}^{\dagger} a_{i}^{\dagger} + a_{i}^{\dagger} a_{j}^{\dagger} = 0$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred2 { a_j^, a_i^ } = a_j^ a_i^ + a_i^ a_j^ = 0 )

The anticommutation relation between a creation and an annihilation operator is

$$\{ a_{i}, a_{j}^{\dagger} \} = a_{i} a_{j}^{\dagger} + a_{j}^{\dagger} a_{i} = \delta_{ij}$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred3 { a_i, a_j^ } = a_i a_j^ + a_j^ a_i = _ij )

Now we will find an expression for the Hamiltonian in terms of creation and annihilation operators over spatial orbitals. We begin with the second-quantized form of the one- and two-electron operators (see Szabo and Ostlund [1] p. 95)

$${tex2html_deferred}{{tex2html_deferred}5.{tex2html_deferred}4}\ha... ...l O}_1 = \sum_{ij}^{2n} \langle i \vert h \vert j \rangle a_{i}^{\dagger} a_{j}$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred4Ô_1 = _ij^2n i | h | j a_i^ a_j )

$${tex2html_deferred}{{tex2html_deferred}5.{tex2html_deferred}5} \h... ...l}^{2n} \langle ij \vert kl \rangle a_{i}^{\dagger} a_{j}^{\dagger} a_{l} a_{k}$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred5 Ô_2 = 12 _ijkl^2n ij | kl a_i^ a_j^ a_l a_k )

where the sums run over all spin orbitals $\{ \chi_{i} \}$. Thus the Hamiltonian is

$$\hat{H} = \sum_{pq}^{2n} a_{p}^{\dagger} a_{q} [p\vert h\vert... ...^{2n} a_{p}^{\dagger} a_{r}^{\dagger} a_{s} a_{q} [pq\vert rs]$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred6H = _pq^2n a_p^ a_q [p|h|q] + 12 _pqrs^2n a_p^ a_r^ a_s a_q [pq|rs] )

From the previous equation we can see that the second-quantized form of the Hamiltonian is independent of the number of electrons in the system. Now integrate over spin, assuming that spatial orbitals are constrained to be identical for $\alpha$ and $\beta$ spins. A sum over all 2n spin orbitals can be split up into two sums, one over n orbitals with $\alpha$spin, and one over n orbitals with $\beta$ spin. Symbolically, this is

$$\sum_{a}^{2n} = \sum_{a}^{n} + \sum_{\bar{a}}^{n}$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred7_a^2n = _a^n + _a^n )

The one-electron part of the Hamiltonian becomes

$$\hat{H}_{\rm one} = \sum_{pq}^{n} [p \vert h\vert q] a_{p \a... ...\bar{p}\vert h\vert\bar{q}] a_{p \beta }^{\dagger} a_{q \beta}$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred8H_one = _pq^n [p |h| q] a_p ^ a_q + [p |h| q ] a_p ^ a_q + [p |h| q ] a_p ^ a_q + [p|h|q] a_p ^ a_q )

After integrating over spin, this becomes

$$\hat{H}_{\rm one} = \sum_{pq}^{n} (p\vert h\vert q) \lbrace... ...r} a_{q \alpha} + a_{p \beta }^{\dagger} a_{q \beta} \rbrace}$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred9H_one = _pq^n (p|h|q) { a_p ^ a_q + a_p ^ a_q } )

The two-electron term can be expanded similarly to give

$$\hat{H}_{\rm two} = \frac{1}{2} \sum_{pqrs}^{n} (pq\vert rs)... ...ger} a_{r \beta }^{\dagger} a_{s \beta } a_{q \beta } \rbrace}$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred10H_two = 12 _pqrs^n (pq|rs) { a_p ^ a_r ^ a_s a_q + a_p ^ a_r ^ a_s a_q + a_p ^ a_r ^ a_s a_q + a_p ^ a_r ^ a_s a_q } )

Now we make use of the anticommutation relation (5.1) and we swap the order of a_s and a_q, introducing a minus sign. This yields

$$\hat{H}_{\rm two} = - \frac{1}{2} \sum_{pqrs}^{n} (pq\vert rs) \l... ..._{p \beta }^{\dagger} a_{r \beta }^{\dagger} a_{q \beta } a_{s \beta } \rbrace}$$

Now we use the anticommutation relation between a creation and an annihilation operator, which is given by (5.3). This relation allows us to swap the a_q and $a_{r}^{\dagger}$ in each term, to give

$$\hat{H}_{\rm two} \frac{1}{2} \sum_{pqrs}^{n} (pq\vert rs) \left[ a_{p \alpha}^{\da... ...eta } - \delta_{q \alpha, r \beta } a_{p \alpha}^{\dagger} a_{s \beta } \right.$$ =

$$\left. a_{p \beta }^{\dagger} a_{q \beta } a_{r \alpha}^{\dagger}... ...beta } - \delta_{q \beta, r \beta } a_{p \beta }^{\dagger} a_{s \beta } \right]$$ tex2html_deferredtex2html_deferred5.tex2html_deferred11 + (tex2html_deferredtex2html_deferred5.tex2html_deferred11 & + & . a_p ^ a_q a_r ^ a_s - _q , r a_p ^ a_s + a_p ^ a_q a_r ^ a_s - _q , r a_p ^ a_s ] )

Now we observe that $\delta_{q \alpha, r \alpha}$ and $\delta_{q \beta, r \beta}$ can both be written $\delta_{qr}$, and also that $\delta_{q \alpha, r \beta }$ and $\delta_{q \beta, r \alpha}$ are both 0. This simplifies our equation to

$$\hat{H}_{\rm two} \frac{1}{2} \sum_{pqrs}^{n} (pq\vert rs) \left[ a_{p \alpha}^{\da... ...a_{p \beta }^{\dagger} a_{q \beta } a_{r \beta }^{\dagger} a_{s \beta } \right.$$ =

$$\left. \delta_{qr} a_{p \alpha}^{\dagger} a_{s \alpha} - \delta_{qr} a_{p \beta }^{\dagger} a_{s \beta } \right]$$ tex2html_deferredtex2html_deferred5.tex2html_deferred12 - (tex2html_deferredtex2html_deferred5.tex2html_deferred12 & - & . _qr a_p ^ a_s - _qr a_p ^ a_s ] )

Now we introduce the replacement (or shift) operator

$$\hat{E}_{ij} = a_{i \alpha}^{\dagger} a_{j \alpha} + a_{i \beta}^{\dagger} a_{j \beta}$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred13Ê_ij = a_i ^ a_j + a_i ^ a_j )

which Paldus has shown [5] to be isomorphic to the generators of the unitary group. This replacement operator is commonly referred to as a unitary group generator, but as Duch has pointed out [27], such usage is somewhat dubious in papers where no unitary group theory is employed.

$$\hat{H} = \sum_{pq}^{n} (p\vert h\vert q) \hat{E}_{pq} + \... ... \hat{E}_{pq} \hat{E}_{rs} - \delta_{qr} \hat{E}_{ps} \right)$$ (tex2html_deferredtex2html_deferred5.tex2html_deferred14 H = _pq^n (p|h|q) Ê_pq + 12 _pqrs^n (pq|rs) ( Ê_pq Ê_rs - _qr Ê_ps ) )

This is the Hamiltonian in terms of replacement operators. Contemporary papers on CI theory often express the Hamiltonian in the form of equation (5.15).⁴