Truncated CI is not Size Extensive

As we have previously pointed out, full CI --being the matrix formulation of the Schrödinger equation--is an exact theory for nonrelativistic electronic structure problems. If we truncate the CI (either in the one-electron or N-electron space), we no longer have an exact theory. Of course either of these truncations will introduce an error in the wavefunction, which will cause errors in the energy and all other properties. One particularly unwelcome result of truncating the N-electron basis is that the CI energies obtained are no longer size extensive or size consistent.

These two terms, size extensive and size consistent, are used somewhat loosely in the literature. Of the two, size extensivity is the most well-defined. A method is said to be size extensive if the energy calculated thereby scales linearly with the number of particles N. The word ``extensive'' is used in the same sense as in thermodynamics, when we refer to an extensive, rather than an intensive property. A method is called size consistent if it gives an energy E_A + E_B for two well separated subsystems A and B. While the definition of size extensivity applies at any geometry, the concept of size consistency applies only in the limiting case of infinite separation. In addition, size consistency usually also implies correct dissociation into fragments; this is the source of much of the confusion arising from this term. Thus restricted Hartree-Fock (RHF) is size extensive, but it is not necessarily size consistent, since it cannot properly describe dissociation into open-shell fragments. It can be shown that many-body perturbation theory (MBPT)  and coupled-cluster  (CC) methods are size extensive, but they will be size consistent only if they are based on reference wavefunction which dissociates properly.

As previously stated, truncated CI's are neither size extensive nor size consistent. A simple (and often used!) example is sufficient to make the point. Consider two noninteracting hydrogen molecules. If the CISD  method is used, then the energy of the two molecules at large separation will not be the same as the sum of their energies when calculated separately. In order for this to be the case, we would have to include quadruple excitations in the supermolecule calculation, since local double excitations could happen simultaneously on A and B.

We would tend to think that size extensivity and size consistency are important, physical properties that all quantum mechanical models should have (indeed, full CI , an exact theory, has these properties), but perhaps they are not as essential as all that. Duch and Diercksen have claimed that ``making size extensivity the most important requirement of quantum chemical methods, although it does not guarantee correct physical description, seems to be based not that much on physical as on esthetical criteria'' [24]. Indeed, they show that quantum mechanics is a ``holistic'' theory, not well-suited toward the description of separated subsystems:

Hilbert space of antisymmetric, many particle functions, describing the total system, can not be decomposed into separate subspaces. Consider two systems, S_A and S_B, with N_A and N_B electrons, respectively. Each system is described by its own function, $\Psi_A$ antisymmetric in N_A particles and $\Psi_B$ in N_B. Assuming that both functions are normalized to unity it is easy to show that the product function $\Psi_{AB} = \Psi_A \Psi_B$ is always ``far'' from the antisymmetric function $\Psi = {\cal A} \Psi_{AB}$, as measured by the overlap $\langle \Psi_{AB} \vert \Psi \rangle$ or the norm of the difference $2 - \sqrt{2} \leq \vert\vert \Psi_{AB} - \Psi \vert\vert ^2 \leq 2$.

 

Such arguments notwithstanding, it is clear that the fraction of the correlation energy  recovered by a truncated CI will diminish as the size of the system increases, making it a progressively less accurate method. There have been many attempts to correct the CI energy to make it size extensive. The most widely-used (and simplest) of these methods is referred to as the Davidson correction  [25], which is

$$\Delta E_{DC} = E_{SD}(1 - c_0^2)$$ (tex2html_deferredtex2html_deferred4.tex2html_deferred26 E_DC = E_SD(1 - c_0^2) )

where E_SD is the basis set correlation energy  recovered by a CISD  procedure. This correction approximately accounts for the effects of ``unlinked quadruple'' excitations (i.e. simultaneous pairs of double excitations). The multireference version [24] of this correction is

$$\Delta E_{DC} = \left( 1 - \sum_{i \in {\rm Ref}} \vert c_i\vert^2 \right) (E_{MRCI} - E_{MR})$$ (tex2html_deferredtex2html_deferred4.tex2html_deferred27 E_DC = ( 1 - _i Ref |c_i|^2 ) (E_MRCI - E_MR) )

where E_MRCI is the multireference CI energy and E_MR is the energy obtained from the set of references (MCSCF energy if the references are obtained as all references in an MCSCF procedure). We have simply replaced the CISD  correlation energy in equation (4.26) with the analogous multireference correlation energy, and we have replaced c₀² with the analogous sum of squares of all the reference coefficients.

There are a number of other size extensivity corrections, and most of them do not take any significant amount of computation. Reference [24] provides a nice comparison of several of the more common CI size extensivity correction methods. We should also mention that Malrieu and co-workers have presented a self-consistent dressing of the Hamiltonian which gives size extensive results for selected CI procedures [26].