On the mutual exclusion of variationality and size consistency

Abstract

Why do variational electron-correlation methods such as truncated configuration-interaction methods tend to be non-size-consistent (non-size-extensive)? Why are size-consistent (size-extensive) methods such as Møller–Plesset perturbation and coupled-cluster methods non-variational? We conjecture that the variational and size-consistent properties are mutually exclusive in an ab initio electron-correlation method (which thus excludes the Hartree–Fock and density-functional methods). We analyze some key examples that support the truth of this conjecture.

Appendix: Questions and answers

In the following, we list questions, many of which are from the reviewers and colleagues who have read a version of this paper and challenged the conjecture. We offer our answers to these questions. The answers are increasingly speculative as we go down the list, and some may not have the rigor normally expected in a scientific paper. It is our hope that this serves as a basis of a fuller discussion and the ultimate proof or disproof of the conjecture.

Q: What is the difference between size consistency and size extensivity?

A: There is none, as regard to a method that computes the total energy in the ground state; they are synonymous and are based on one and the same diagrammatic criterion [7, 8]. One of the authors has extensively discussed the inadequacy of the supermolecule criterion often associated with the term “size consistency” in the past.

Q: Is DFT size consistent and variational?

A: No. DFT is not diagrammatically size consistent, but its energy is extensive, which can be shown as follows. An exchange-correlation energy per unit cell in a crystal is written as $\int f[\rho(\user2{r})] \hbox{d}\user2{r}$, where the domain of integration is one unit cell and f is the so-called exchange-correlation functional of electron density ρ. When one increases the volume of the crystal under the periodic boundary condition without changing ρ, the exchange-correlation energy per unit cell remains the same, which means that it is thermodynamically intensive. This, in turn, proves the extensivity of the total energy, assuming that the rest of the energy expressions are formally the same as in HF and are extensive.

DFT is not variational, but merely stationary, according to our adopted definitions of these terms [4]. This is because its energy can go below the full CI energy in the same basis set. An exception is the optimized effective potential (OEP) method [32, 33], which is diagrammatically size consistent and variational; OEP is sometimes viewed as the exact-exchange DFT [34–36]. It does not describe any electron correlation and is, therefore, not a counterexample of the conjecture.

Q: Is the Hylleraas functional size consistent and variational?

A: No. The Hylleraas functional [37, 38] (see also the Sinanoǧlu equation [39]) is given by

$$ E = \langle \Upphi_0 | \hat{T}_2^\dagger \hat{H}_0 \hat{T}_2 |\Upphi_0 \rangle + 2 \langle \Upphi_0 | \hat{T}_2^\dagger \hat{H}_1|\Upphi_0 \rangle, $$

(30)

where $\hat{H}_0$ and $\hat{H}_1$ are the zeroth-order Hamiltonian and perturbation operator in the Møller–Plesset partitioning, $\hat{T}_2$ is a two-electron excitation operator, and $\Upphi_0$ is the HF reference wave function. The coefficients in $\hat{T}_2$ are determined variationally so as to minimize E, which can be easily shown to be the energy of second-order MP (MP2) [37, 38]. Being MP2, the Hylleraas functional is size consistent, but not variational. Extending the variational space spanned by $\hat{T}_2$ is equivalent to enlarging the basis set. In this sense and considering the fact that an MP2 energy can be lower than the full CI energy, we contend that the Hylleraas functional is only stationary [4].

The Hylleraas functional is often used as a basis of explicitly correlated MP2 (MP2-F12) [37, 38], in which the action of $\hat{T}_2$ yields,

$$ \hat{T}_2 \Upphi_0 = \frac{1}{(2!)^2} \sum_{i,j}^{{\rm occ.}} \sum_{a,b}^{{\rm virt.}} t^{ab}_{ij} \Upphi^{ab}_{ij} + \frac{\hat{Q}_{12}}{(2!)^3} \sum_{i,j}^{{\rm occ.}} \sum_{k,l}^{{\rm occ.}} \sum_{a,b}^{{\rm virt.}} F^{ab}_{kl} t^{kl}_{ij} \Upphi^{ab}_{ij}, $$

(31)

with $F^{ab}_{kl} = \langle ab | F_{12}| kl \rangle$, where F ₁₂ is an explicit function of inter-electronic distance r ₁₂ such as $\exp(-\gamma r_{12})$. See Ref. [37] for the definition of $\hat{Q}_{12}$, which is unimportant here. The functional is made stationary with respect to {t ^ab _ij }, {t ^kl _ij }, and γ to approach the MP2 energy in the complete-basis-set limit, which is not an upper bound of the full CI energy. It is, therefore, still not a counterexample of the conjecture. Note that two-electron basis functions such as r ₁₂ or $\exp(-\gamma r_{12})$ can only couple two electrons at a time and span the same space the two-electron excitation operator does. Hence, they cannot fundamentally alter the non-variationality or size consistency of MP2.

In the same token, a method using a single-determinant wave function multiplied by a Jastrow factor (an exponential of a function of electron--electron and electron--nucleus distances) should have the same characteristics as the projection or variational CC regarding variationality and size consistency. It cannot, therefore, simultaneously achieve size consistency and variationality, as we have seen in Sect. 3. It has been documented [26, 40] that nuclear calculations using a Jastrow factor do not have the upper-bound property and are, therefore, not variational according to our definition of the term. This is known as the “Emery difficulty” [26, 40].

Q: Can a size-consistency correction to a variational method restore size consistency?

A: There are size-consistency corrections to non-size-consistent methods such as CISD [41–46] and MRCI [45, 47]. These corrections, some related to Padé approximants [45], can “minimize” size-consistency errors and render the methods “approximately” size consistent, which means they are not size consistent. They may also be non-variational because the energies are not expectation values any more.

Q: Is local basis CI size consistent and variational?

A: A truncated CI method combined with an embedded-fragment scheme [48] can obviously yield an extensive total energy, giving a false impression that the method is simultaneously variational and size consistent. It is not variational because the energy thus obtained, which is not the energy expectation value in the global wave function, has no reason to be an upper bound of the exact energy of the whole molecule except in a special circumstance such as when there is no interaction between fragments. A counterexample for a special circumstance is, however, not a counterexample.

Another example of a variational local basis method is the generalized valence bond method. It is equivalent to MCSCF [1], which is not size consistent. Yet another example of a variational method that can employ a variety of bases including spatially local ones is the density matrix renormalization group method [49]. The underlying approximations in its wave functions are, however, often full CI [50] and CASSCF [51], neither of which is a counterexample, as explained in Sect. 2.

We have, however, been unable to generalize the definition of variationality in this conjecture to encompass all spaces including the determinant space and one-particle basis set. Classes of methods such as the Fock-space perturbation theory of Refs. [52, 53] are not easily subjected to this conjecture, though no claim seems to have been made that it is simultaneously size consistent and variational.

Q: If the conjecture is correct, how have variational methods been successfully applied to solids?

A: With a scalable algorithm running on a supercomputer, variational methods, which cannot be size consistent if the conjecture is correct, can still be applied to sufficiently large systems to address problems of condensed matter [11, 54, 55] including the issue of strong electron correlation in solids. The conjecture does not contradict this. Nonetheless, it is reasonable to expect a size-consistent method to be more accurate and efficient (these two properties are interchangeable) for solids than a similar non-size-consistent method; the development strategy emphasizing size consistency championed by Bartlett [56] has been successful in quantum chemistry despite the fact that it usually deals with relatively small molecules. In fact, the question of mutual exclusion and the conjecture emerged as a result of our desire to obtain size-consistent methods for strong correlation in solids. If a diagrammatically size-consistent method (that may or may not be variational) exists that can describe strong correlation in solids, we expect it to outperform the existing methods that are not size consistent. If a size-consistent method that works well for strong correlation is shown not to exist, it may even imply different size dependence of the strongly correlated portion of the energy.