Cluster Functions and the Exponential Ansatz

The complicated notation of Eq. [2.12] can be drastically reduced by using a simple analytic form for the cluster functions. Note again that each determinant involving a cluster function is actually a linear combination of determinants each of which differs from the reference, $\Phi_0$, by a specific number of orbitals. For example, the 27th term in Eq. [2.12] expands to become

$$\vert f_{ij} \phi_k f_l \rangle = \sum_{a > b} \sum_c t_{ij}^{ab} t_l^c \vert \phi_a \phi_b \phi_k \phi_c \rangle,$$ (2.13)

where we have inserted the definition of the two-electron cluster function in Eq. [2.4] and its one-electron counterpart to indicate the pairwise correlation of electrons in orbitals $\phi_i$and $\phi_j$ as well as the ``correlation'' of electrons in orbital $\phi_l$. Note that each determinant in the above summation differs from the reference by exactly three orbitals: orbitals $\phi_i$, $\phi_j$, and $\phi_l$ are replaced by orbitals $\phi_a$, $\phi_b$, and $\phi_c$, respectively. Hence, each term can be written as the result of some substitution operator (or products of such operators) acting on $\Phi_0$. This task is perhaps most easily accomplished using the mathematical technique known as second quantization.[84,82,80]

We will define a creation operator by its action on a Slater determinant:

$$a_{p}^{\dagger} \vert \phi_q \ldots \phi_s \rangle = \vert \phi_p \phi_q \ldots \phi_s \rangle,$$ (2.14)

where we have added one more column (orbital) and one more row (electron) to form the new determinant on the right-hand side. We may define an annihilation operator in a similar manner to obtain

$$a_{p}^{\ }\vert \phi_p \phi_q \ldots \phi_s \rangle = \vert \phi_q \ldots \phi_s \rangle,$$ (2.15)

where we have removed the first column (orbital) and the first row (electron) from the original function.^2.2 A given Slater determinant may be written as a chain of creation operators acting on the true vacuum (a state containing no electrons or orbitals), i.e.,

$$a_{p}^{\dagger} a_{q}^{\dagger} \ldots a_{s}^{\dagger} \vert \ \rangle = \vert \phi_p \phi_q \ldots \phi_s \rangle.$$ (2.16)

Note also that an annihilation operator acting on the vacuum state gives a zero result,

$$a_{p}^{\ } \vert \ \rangle = 0.$$ (2.17)

Pairwise permutations of the operators introduce changes in the sign of the resulting determinant, e.g.,

$$a_{q}^{\dagger} a_{p}^{\dagger} \vert \ \rangle = \vert \phi_... ...q \rangle = - a_{p}^{\dagger} a_{q}^{\dagger} \vert \ \rangle.$$ (2.18)

Therefore, the anticommutation relation for a pair of creation operators is simply

$$a_{p}^{\dagger} a_{q}^{\dagger} + a_{q}^{\dagger} a_{p}^{\dagger} = 0.$$ (2.19)

The analogous relation for a pair of annihilation operators is

$$a_{p}^{\ } a_{q}^{\ } + a_{q}^{\ } a_{p}^{\ } = 0.$$ (2.20)

Therefore, if we change the ordering of a pair of annihilation or creation operators, we must also change the sign of the resulting expression. Finally, it may be shown that the anticommutation relation for the ``mixed'' product is

$$a_{p}^{\dagger} a_{q}^{\ } + a_{q}^{\ } a_{p}^{\dagger} = \delta_{pq},$$ (2.21)

where $\delta_{pq}$ is the conventional Kronecker delta which equals 1 if p = q and 0 if $p \neq q$.

Using these so-called second-quantized operators, we may define the single-orbital cluster operator

$$\hat{t}_i \equiv \sum_a t_i^a a_{a}^{\dagger} a_{i}^{\ },$$ (2.22)

where the operator $a_{i}^{\ }$ deletes the orbital $\phi_i$ from the determinant on which the operator acts, whereas $a_{a}^{\dagger}$ introduces the orbital $\phi_a$ in its place. (The $\hat{\ }$ is used to indicate a second-quantized operator.) Similarly, a two-orbital cluster operator which substitutes orbital $\phi_a$ for $\phi_i$ and $\phi_b$ for $\phi_j$ is given by

$$\hat{t}_{ij} \equiv \sum_{a>b} t_{ij}^{ab} a_{a}^{\dagger} a_{b}^{\dagger} a_{j}^{\ } a_{i}^{\ },$$ (2.23)

(Again note that the order of replacement is important for the sign of the resulting determinant.) Hence, the 27th term of Eq. [2.12] shown explicitly in Eq. [2.13] may be written simply as

$$\vert f_{ij} \phi_k f_l \rangle = \hat{t}_{ij} \hat{t}_l \vert \Phi_0 \rangle.$$ (2.24)

The creation operators in Eqs. [2.22] and [2.23] are restricted to act only on the virtual orbitals, and the annihilation operators may act only on the occupied orbitals. Therefore, by Eq. [2.21], the creation-annihilation operator pairs exactly anticommute:

$$a_{a}^{\dagger} a_{i}^{\ } + a_{i}^{\ } a_{a}^{\dagger} = \delta_{ia} = 0,$$ (2.25)

since the occupied orbital $\phi_i$ and the virtual orbital $\phi_a$cannot be the same. Therefore, by the above equation as well as the anticommutation relations given in Eqs. [2.19] and [2.20], all of the creation and annihilation operators in $\hat{t}_{i}$ and $\hat{t}_{ij}$ anticommute. Given the additional fact that the cluster operators always contain even numbers of second-quantized operators, the $\hat{t}_{i}$ and $\hat{t}_{ij}$operators themselves will exactly commute.^2.3

Equations [2.22] and [2.23] may be used to rewrite the long one- and two-orbital cluster wavefunction in Eq. [2.12] above as

$$\Psi \left( 1 + \sum_{i} \hat{t}_i + \frac{1}{2} \sum_{ij} \hat{t}_i\h... ...{ijk} \hat{t}_i\hat{t}_j\hat{t}_k + \frac{1}{2}\sum_{ij} \hat{t}_{ij} + \right.$$ =

$$\left. \frac{1}{8} \sum_{ijkl} \hat{t}_{ij} \hat{t}_{kl} + \frac{... ...{t}_k + \frac{1}{4} \sum_{ijkl} \hat{t}_{ij} \hat{t}_k\hat{t}_l \right) \Phi_0.$$ (2.26)

We may simplify this expression even further by defining the total one- and two-orbital cluster operators

$$\hat{T}_1 \equiv \sum_i \hat{t}_i = \sum_{ia} t^{a}_{i} a_{a}^{\dagger}a_{i}^{\ },$$ (2.27)

and

$$\hat{T}_2 \equiv \frac{1}{2} \sum_{ij} \hat{t}_{ij} = \frac{1... ...^{ab}_{ij} a_{a}^{\dagger}a_{b}^{\dagger}a_{j}^{\ }a_{i}^{\ },$$ (2.28)

respectively.^2.4 More generally, an n-orbital cluster operator may be defined as

$$\hat{T}_n = \left(\frac{1}{n!}\right)^2\sum_{ij\ldots ab\ldot... ... a_{a}^{\dagger} a_{b}^{\dagger} \ldots a_{j}^{\ } a_{i}^{\ }.$$ (2.29)

This reduces the wavefunction expression to

$$\Psi = \left( 1 + \hat{T}_1 + \frac{1}{2!}\hat{T}^2_1 + \frac... ...2\hat{T}_1 + \frac{1}{2!} \hat{T}_2\hat{T}^2_1 \right) \Phi_0.$$ (2.30)

Higher-order terms (e.g., $\hat{T}_2^3$) do not appear, of course, because our example system contains only four electrons. If we remember that $\hat{T}_1$ and $\hat{T}_2$ commute, then all of the terms from the above equation match those from the power series expansion of an exponential function! Thus, the general expression for Eq. [2.30] is

$$\Psi = e^{\hat{T}_1 + \hat{T}_2} \Phi_0 \equiv e^{\hat{T}} \Phi_0,$$ (2.31)

which is a rather convenient reduction from the original Eq. [2.12].

The ``exponential ansatz'' given in Eq. [2.31] is one of the central equations of coupled cluster theory. The exponentiated cluster operator, $\hat{T}$, when applied to the reference determinant, produces a new wavefunction containing cluster functions, each of which correlates the motion of electrons within specific orbitals. If $\hat{T}$ includes contributions from all possible orbital groupings for the N-electron system (that is, $\hat{T}_1, \hat{T}_2, \ldots, \hat{T}_N$), then the exact wavefunction within the given one-electron basis may be obtained from the reference function. The cluster operators, $\hat{T}_n$, are frequently referred to as excitation operators, since the determinants they produce from $\Phi_0$ resemble excited states in Hartree-Fock theory. Truncation of the cluster operator at specific substitution/excitation levels leads to a hierarchy of coupled cluster techniques (e.g., $\hat{T} \equiv \hat{T}_1 + \hat{T}_2 \rightarrow$ CCSD; $\hat{T} \equiv \hat{T}_1 + \hat{T}_2 + \hat{T}_3 \rightarrow$ CCSDT, etc., where ``S'', ``D'', and ``T'', indicate that single-, double-, and triple-excitations, respectively, are included in the wavefunction expansion).