Cluster Expansion of the Wavefunction

Consider a model system of four electrons moving in an arbitrary electrostatic field generated by the nuclei in a molecule. For our purposes, it is not necessary to specify the number of these nuclei, their types, or positions; only the general form of the electronic wavefunction is of interest. It is convenient to describe the motions of each electron separately by assigning them to one-electron functions, $\phi_i({\mathbf{x}}_1)$, where x₁ is a vector of the coordinates (including spin) of electron 1. In addition, electrons are fermions, so the electronic wavefunction must be antisymmetric with respect to interchange of the coordinates of any pair of electrons. A traditional and very useful starting point for such a four-electron wavefunction is the so-called Slater determinant

$$\Phi_0 = \frac{1}{\sqrt{4!}}\left\vert \begin{array}{cccc} \p... ... {x}}_4) & \phi_l({\mathbf {x}}_4) \\ \end{array}\right\vert,$$ (2.1)

where the $1/\sqrt{4!}$ is a normalization constant. Expansion of this determinant reveals a linear combination of products of the four functions, $\phi_i$, $\phi_j$, $\phi_k$, and $\phi_l$, with the electronic coordinates x_n distributed among them in all possible ways. Since permutation of any two rows in the determinant -- which is equivalent to interchanging the coordinates of any two electrons -- changes the sign of $\Phi_0$, the antisymmetry principle is maintained.

The component functions $\phi_i$ may be chosen in a variety of ways. For example, if the nuclear field were only a single beryllium nucleus, the one-electron spatial functions could be constructed to mimic the atomic 1s and 2s orbitals. For a molecular system, the functions can be constructed as a linear combination of atomic orbitals (AOs) in which each one-electron function represents a molecular orbital (MO) whose AO coefficients are optimized via the Hartree-Fock self-consistent-field (SCF) procedure.[82] A convenient shorthand notation for this wavefunction consists of a Dirac-notation ket containing only the diagonal elements of the above matrix,

$$\Phi_0 = \vert \phi_i({\mathbf {x}}_1) \phi_j({\mathbf {x}}_2) \phi_k({\mathbf {x}}_3) \phi_l({\mathbf {x}}_4) \rangle,$$ (2.2)

where the normalization factor is included implicitly. As discussed in detail elsewhere in Reviews in Computational Chemistry[77], the single-determinant wavefunction fails to account for the instantaneous Coulombic interactions which keep the electrons of opposite spin apart.[82]

How can we improve this so-called independent-particle approximation such that the motions of the electrons are correlated? Often the set of occupied orbitals (i.e., those functions which compose the Slater determinant above) is chosen from a larger set of one-electron functions. These ``extra'' functions are frequently referred to as virtual orbitals and may, for example, arise as a byproduct of the SCF procedure.^2.1 Within the space described by the full set of orbitals, any function of N variables may be written in terms of N-tuple products of the $\phi_p$. For example, a function of two variables may be constructed by using all possible binary products of the set of one-electron functions, e.g.,

$$f({\mathbf x}_1, {\mathbf x}_2) = \sum_{p>q} c_{pq} \phi_p({\mathbf x}_1) \phi_q({\mathbf x}_2),$$ (2.3)

where the double-summation runs over the entire set of one-electron functions and the notation p > q indicates that only unique pairs of functions are included. Instead of correlating the motions of a specific pair of electrons, however, we may use a modified form of this expansion to correlate the motions of any two electrons within a selected pair of occupied orbitals -- say functions i and j -- using a two-particle cluster function,

$$f_{ij} ({\mathbf x}_m, {\mathbf x}_n) = \sum_{a > b} t_{ij}^{ab} \phi_a({\mathbf x}_m) \phi_b({\mathbf x}_n),$$ (2.4)

where the t_ij^ab are the cluster coefficients whose specific values are determined via the electronic Schrödinger equation (see the next section on formal coupled cluster theory). Inserting this into $\Phi_0$ leads to the somewhat-improved electronic wavefunction,

$$\Psi = \vert \left[ \phi_i({\mathbf x}_1) \phi_j({\mathbf x}_... ...) \right] \phi_k({\mathbf x}_3) \phi_l({\mathbf x}_4) \rangle,$$ (2.5)

where the Dirac shorthand implies a correctly antisymmetrized wavefunction including normalization factors as in Eq. [2.2]. Inclusion of the cluster function, f_ij, in the wavefunction produces a linear combination of Slater determinants involving replacement of occupied orbitals $\phi_i$ and $\phi_j$ by virtual orbitals $\phi_a$ and $\phi_b$, such that

$$\Psi = \Phi_0 + \sum_{a > b} t_{ij}^{ab} \vert \phi_a({\mathb... ...hbf x}_2) \phi_k({\mathbf x}_3) \phi_l({\mathbf x}_4) \rangle.$$ (2.6)

In addition, the determinantal form of the individual terms in this expansion implies antisymmetrization of the cluster coefficients, such that t_ij^ab = -t_ji^ab = -t_ij^ba = t_ji^ba.

It should be carefully noted here that the cluster function, f_ij(x₁, x₂), is intended to correlate the motions of any pair of electrons placed in orbitals i and j, and not just the motions of electrons 1 and 2. Since the Slater determinant produces a linear combination of orbital products, including terms such as

$$\left[\phi_i({\mathbf x}_1) \phi_j({\mathbf x}_2) + f_{ij} ({... ...thbf x}_2) \right] \phi_k({\mathbf x}_3) \phi_l({\mathbf x}_4)$$ (2.7)

and

$$\left[\phi_i({\mathbf x}_3) \phi_j({\mathbf x}_4) + f_{ij} ({... ...hbf x}_4) \right] \phi_k({\mathbf x}_1) \phi_l({\mathbf x}_2),$$ (2.8)

which differ only in their distribution of electronic coordinates, the cluster function correlates the motion of every pair of electrons found in orbitals $\phi_i$ and $\phi_j$.

Depending on the chemical system of interest, however, it might be more prudent to correlate the motions of electrons in orbitals k and l rather than orbitals i and j. For example, $\phi_i$ and $\phi_j$ might correspond to molecular core orbitals, while $\phi_k$and $\phi_l$ might correspond to the atomic or molecular valence orbitals. Electron correlation can be particularly important in the latter set of functions because the valence orbitals are often directly involved in the formation of chemical bonds. In this case, the wavefunction would be written as

$$\Psi = \vert \phi_i({\mathbf x}_1) \phi_j({\mathbf x}_2) \lef... ... x}_4) + f_{kl}({\mathbf x}_3, {\mathbf x}_4) \right] \rangle.$$ (2.9)

On the other hand, a more intelligent approach might be to correlate all possible pairwise combinations of orbitals in this four-electron system, i.e.,

$$\Phi \vert \phi_i \phi_j \phi_k \phi_l \rangle + \vert f_{ij} \phi_k \... ...gle + \vert f_{il} \phi_j \phi_k \rangle + \vert \phi_i f_{jk} \phi_l \rangle -$$ =

$$\vert \phi_i f_{jl} \phi_k \rangle + \vert \phi_i \phi_j f_{kl} \... ...ij} f_{kl} \rangle - \vert f_{ik} f_{jl} \rangle + \vert f_{il} f_{jk} \rangle,$$ (2.10)

where the electronic coordinates are now implicit in the notation, and the signs on individual terms arise from the permutations in the orbital ordering needed to define the appropriate cluster functions. However, there is no need to limit this approach to only orbital pairs; following Harris et al.,[80] we could introduce three-orbital cluster functions and include these in our new wavefunction to give

$$\Phi \vert \phi_i \phi_j \phi_k \phi_l \rangle + \vert f_{ij} \phi_k \... ...gle + \vert f_{il} \phi_j \phi_k \rangle + \vert \phi_i f_{jk} \phi_l \rangle -$$ =

$$\vert \phi_i f_{jl} \phi_k \rangle + \vert \phi_i \phi_j f_{kl} \... ...j} f_{kl} \rangle - \vert f_{ik} f_{jl} \rangle + \vert f_{il} f_{jk} \rangle +$$

$$\vert f_{ijk} \phi_l \rangle - \vert f_{ijl} \phi_k \rangle + \vert f_{ikl} \phi_j \rangle + \vert \phi_i f_{jkl} \rangle.$$ (2.11)

If one continues this process to include all cluster functions for up to N orbitals (four in the case discussed here), as well as single-orbital ``cluster'' functions which account for adjustment of the one-electron basis as other cluster functions are added, we could obtain the exact wavefunction within the space spanned by the $\{\phi_p\}$. On the other hand, we might assume that clusters larger than pairs are less important to an adequate description of the system -- an assumption supported by the fact that the electronic Hamiltonian contains operators describing pairwise electronic interactions at most[75]. We could therefore write a four-electron wavefunction which includes all clusters of only one and two orbitals as[83,80]

 

$$\Psi \vert \phi_i \phi_j \phi_k \phi_l \rangle + \vert f_i \phi_j \phi... ...ert \phi_i \phi_j f_k \phi_l \rangle + \vert \phi_i \phi_j \phi_k f_l \rangle +$$ =

$$\vert f_i f_j \phi_k \phi_l \rangle + \vert f_i \phi_j f_k \phi_l... ...e + \vert \phi_i f_j f_k \phi_l \rangle + \vert \phi_i f_j \phi_k f_l \rangle +$$

$$\vert \phi_i \phi_j f_k f_l \rangle + \vert f_i f_j f_k \phi_l \r... ...\rangle + \vert f_i \phi_j f_k f_l \rangle + \vert \phi_i f_j f_k f_l \rangle +$$

$$\vert f_{ij} \phi_k \phi_l \rangle - \vert f_{ik} \phi_j \phi_l \... ...gle + \vert \phi_i f_{jk} \phi_l \rangle - \vert \phi_i f_{jl} \phi_k \rangle +$$

$$\vert \phi_i \phi_j f_{kl} \rangle + \vert f_{ij} f_{kl} \rangle ... ... f_{jl} \rangle + \vert f_{il} f_{jk} \rangle + \vert f_i f_j f_k f_l \rangle +$$ (2.12)

$$\vert f_{ij} f_k \phi_l \rangle + \vert f_{ij} \phi_k f_l \rangle... ...hi_l \rangle - \vert f_{ik} \phi_j f_l \rangle - \vert f_{ik} f_j f_l \rangle +$$

$$\vert f_{il} f_j \phi_l \rangle + \vert f_{il} \phi_j f_l \rangle... ...hi_l \rangle + \vert \phi_i f_{jk} f_l \rangle + \vert f_i f_{jk} f_l \rangle -$$

$$\vert f_i f_{jl} \phi_k \rangle - \vert \phi_i f_{jl} f_k \rangle... ..._{kl} \rangle + \vert \phi_i f_j f_{kl} \rangle + \vert f_i f_j f_{kl} \rangle.$$