# To Multireference or not to Multireference: That is the Question?

## Abstract

**:**

## 1 INTRODUCTION

## 2 NATURE OF THE PROBLEM

**χ**›, the full CI is the right answer. So, our first objective is to achieve that answer. (A second objective is the basis set itself, which is a topic for a different discussion.) With any reference function, and any set of orbitals, |

**ϕ**› = |

**χ**›

**T**we know that we can get the ’exact’ answer in the basis by doing the full CI. Hence, the question we have to address is whether we get to that limit quicker and more efficiently when we introduce a ’multi-reference’ description.

_{1},φ

_{2},...φ

_{m}› referred to as the P space, where P is a projec-tor, P=|φ›‹φ|φ›

^{−1}‹φ|, improves the description as m is increased. Hence, the total (full CI) wavefunction for any state, k, Ψ

_{k}= (P + Q)Ψ

_{k}, leads to a model state reference,

**h**›‹

**h**|

**h**›

^{−1}‹

**h**|. Ω

_{(k)}is the wave operator that takes the model function into the exact result. It might be universal, where it will not depend explicitly upon (k), or it might be k dependent in some approaches. The better Φ

_{k}, the better the approximation to Ψ

_{k}(II) The subset of energies and model functions with coefficients, {c

_{µ}}, should typically be the eigenvalues and eigenvectors to an effective Hamiltonian matrix, $\overline{\mathbf{H}}$ = ‹φ|HΩ|φ›

**c**=

**cE**

_{k}, whose precise definition depends upon the particular ansatz for the wave operator of the eigenstate. The determination of the coefficients in Φ

_{k}should be accomplished in an unbiased way, i.e., typically the solution of an eigenvalue problem as here, where each c

_{µ}can immediately assume any weight it wants rather than have to ’grow’ into that weight very slowly via some poor initial, typically perturbative, approximation. The latter is what single reference CC typically does. The equation of Ω

_{(k)}, the Bloch equation, HΩ − ΩHΩ = Ω$\overline{\mathbf{H}}$ necessarily introduces coupling with $\overline{\mathbf{H}}$, which adds substantial complications to the theory. (III) In less than a full CI approximation, the orthogonal complement to |φ› should consist of some category of excitations like all single and double excitations out of |φ› that expands |

**h**› from what would be included if only a single function were in |φ› .The above three desiderata are obviously characteristic of MRCI, but are not all satisfied by what are called MRCC methods. IV, if we are to speak of MRCC, then we should insist upon size-extensivity, or that the approximations obtained from the method only consist of linked diagrams, since the latter is the principal rationale for CC theory, itself; and fundamentally distinguishes it from CI. Other issues of correct separation (size-consistency), size-intensivity, etc. are also highly desirable, but typically require further considerations than simply linked-ness. A fifth point (V) that is worth bearing in mind, is the distinction between achieving spin eigenfunction character and the necessity of a MR description that introduces ’non-dynamic’ correlation. I usually consider the spin (or symmetry) eigenfunction issue to be ’static’ correlation [5,6,7,8]. All correlation is in the full CI, so static, non-dynamic, and dynamic are nebulous terms that only attain relevance in distinguishing among approximations.

_{0}›, which makes its c0 chosen to be 1, irrelevant; and write the wavefunction as

_{1}+ T

_{2}+ T

_{3}...

**h**›, however their weighting in the final wavefunction is limited by the truncation. If we limit ourselves to T

_{2}, eg, we have doubles, quadruples, sextuples, or all even excitations in our wavefunction until ${\mathrm{T}}_{2}^{n}$ exceeds the number of electrons. This, of course, is the enormous power of CC theory compared to CI. Not only does the theory recognize that disconnected double excitations in electronic structure theory are more important than connected ones, like T

_{4}and T

_{6}, but we also ensure the size-extensive property that is absolutely essential in accurate applications.

_{1}and T

_{2}, we add single, disconnected triple, etc excitations. We also add a very important property of CC theory that is not shared by CI, namely that

_{1})|φ

_{0}› = |ψ

_{0}›

_{0}›represents a determinant composed of rotated orbitals. This feature, coupled with the introduction of higher excitations via the disconnected cluster operators, makes CCSD and beyond, highly insensitive to orbital choice. Both CCSD and CISD are invariant to transformations among just the occupied or excited orbitals, but only the full CI is invariant to any orbital transformation. However, the above equation permits occupied-excited mixing, causing additional insensitivity. Much of the traditional education in electronic structure theory emphasizes orbital choice as being essential, as in MCSCF and MCSCF+CI. However, the insensitivity to the choice of orbitals in CC theory enables many convenient approximations to be introduced that would never be contemplated by CI aficionados, and this has a pivotal role in providing easily applied, but effectively multi-reference descriptions to be achieved in various CC approaches.

_{2}molecule, so the prime example cited as requiring a multi-reference description has to include the necessity of its being done with a wavefunction that is an exact eigenfunction of S

^{2}and potentially other operators like that due to inversion symmetry. See Fig 3 for RHF based results. Of course, all bond breaking cannot be adequately described by a UHF (O

_{2}is a case in point), but many can. Besides condemning

**Figure 1.**Differences in correlation energies for various CI, MBPT, and CC methods with respect to the full CI (average of BH, HF, H

_{2}O at Re, 1.5Re, 2.0Re, DZP basis).

_{2}requires essentially the full CI solution [17], and especially through hextuple excitations as one would expect. Hence, it is true that we need a way to accelerate the convergence

_{2}[18]. We reported some GVB based MRCC results a few years ago [19, 20], and before that, there was pertinent work from Dykstra’s group [21], who observed that simply using GVB orbitals in CCD gave improved potential curves. Recognizing that the perfect-paired GVB wavefunction arises from a restricted exp(T

_{2})|φ

_{0}› ansatz [22], using such a CCD procedure coupled to orbital optimization to get the GVB solution, Head-Gordon and co-workers have been able to obtain a qualitatively correct N

_{2}potential curve [18]. So this is another type of operationally single reference CC solution for this problem.

^{(µ)})|Φ

_{µ}›‹Φ

_{µ}|. Except in the limit of a complete space, unlike MRCI, there is a manifold

**Figure 3.**RHF-based CISD, CCSD, CCSDT-1, MCSCF, MR-CISD, and MR-LCCM potential curves for N

_{2}(absolute scale). The various curves are identified as follows: (•) CISD, (◦) CCSD, (△) CCSDT-1, (☐) MCSCF, (+) MR-CISD, and (*) MR-LCCM.

^{(µ)}, which is difficult to manage. Some realizations of this approach for special cases have been achieved. One was the open-shell singlet work of Balkova and me [24], which is easy, since the effective Hamiltonian matrix when implemented in spin-orbitals, can be block diagonalized by symmetry into two essentially single reference problems. The next level of difficulty is to use a GVB reference. Unlike the former, we do not know the values of c

_{µ}just by symmetry, so we have to develop a more general structure[19, 20, 24]. Even more complete methods were implemented by Kaldor and co-workers who report N

_{2}results [25].

^{(m,n)}=${\sum}_{k,l}^{m,n}$T

^{(k.l)}, with (k,l) starting from (0,0) being the sectors of Fock space up to (m,n). Ω = exp(T

^{(0,0)}){exp($\stackrel{\sim}{S}$

^{(m,n)}}, after separating T

^{(0,0)}, which is the usual SRCC T. EOM and Fock space have many aspects in common. Basic to both is that some reference state defines T=T

^{(0.0)}, a SRCC solution, defines a new reference function from which all other states can be extracted. This gives us the opportunity to introduce some multi-reference character into the target state, even if the reference state comes from SRCC. More interconnections and distinctions will become apparent below.

## 3 EQUATION-OF-MOTION CC METHODS

_{k}› = R

_{k}|Ψ

_{g}›

_{k}= R

_{k0}+ R

_{k1}+ R

_{k2}+ ...

_{k}› = E

_{k}|Ψ

_{k}›

_{g}› = E

_{g}|Ψ

_{g}›

^{−T}He

^{T}R

_{k}]

_{c}|φ

_{0}› = ω

_{k}R

_{k}|φ

_{0}›

_{g}› = exp(T)|φ

_{0}›. A similar equation provides the left-hand eigenfunction, ‹φ

_{0}|L

_{k}to $\overline{H}$ = e

^{−T}He

^{T}, a non-Hermitian Hamiltonian, but has the same eigenvalue. It is not connected to $\overline{H}$ like the right-hand eigenfunction. We can allow the electron put into orbital |a› by the operator a

^{†}to be a continuum function, giving us a net ionization, and reducing the operators in eqn. 6 accordingly. That is R(i)

_{k}= r

_{0}+ ∑

_{i}r

_{i}{i} + ∑

_{i<j,b}${r}_{\mathit{ij}}^{b}${ib

^{†}j} + ... We can obviously do the same kind of thing to describe electron attachment [40], or to describe double ionizations, etc. So EOM is enormously flexible in this regard.

_{k}operator. The eigenfunctions are obtained from the generalized CI-like matrix, so any determinant can assume any weight it needs in the ’target’ eigenstate. The reference CC state, however, is still considered to be described by a SRCC solution. Failing that for some reference state, EOM-CC can be subject to error. For closed shell reference states, all the EOM ’excited’ eigenstates are spin eigenfunctions. If we use a symmetry broken CC reference state, like a UHF based one, this does not follow, and special attention needs to be paid to the resultant EOM-CC eigenvalues and eigenfunctions, where even their spin eigenvalues may be difficult to discern. Although excited states for many cases involving simple open shells are about as accessible [36, 41] as ground states typically are with UHF, ROHF or quasirestricted (QRHF) reference determinants. I might add that all of these reference determinants result in breaking spin symmetry in the reference CC solution even if the single determinant itself, like ROHF or QRHF, is a spin eigenfunction.

_{k}exp(T)|φ

_{0}› = exp(T)Rk|φ

_{0}›, since [T,R

_{k}] = 0. The first viewpoint suggests taking excitations out of a correlated reference state which is clearly MR, while the second suggests first make the ’CI’ like excitations and then operate with exp(T); but, of course, T does not depend upon this choice, as it might with some generalized MRCC ansatz. So we have conditions II and III satisfied, but subject to the underlying limitations in the SRCC reference solution. Condition I, though formally satisfied, is less relevant, since for this part EOM-CC is a single reference theory, where the P function is the reference SRCC state, which is what gives its ease of application. However, as the ansatz shows, it is a correlated reference state, so it is multi-reference in that regard, and will be improved as more functions are added in describing the P space for the underlying SRCC. (The degree of dependence of EOM-CC results on the P space of the reference CC has been considered by independently varying it and the complementary configuration (R

_{k}) space [42].) (If we partition the EOM $\overline{\mathbf{H}}$ matrix, we can formally introduce another P’ and a Q’ space, that is pertinent to Fock space, discussed below.) However, for condition IV, strictly speaking, EOM-CC is not a fully linked method [43, 44]. This should not be surprising, as we are diagonalizing a CI-like matrix for the excited states, and they are not generally described by an exponential operator. They are, however, in the [1,0] and [0,1] sectors, since in those cases we can show the equivalence with an exponential ansatz [7, 45] (see below). This raises the issue of ’size-intensivity’ between two eigenstates, which would follow automatically if both states were described in a fully linked way. EOM-CC does have the property that the excited states for AB go smoothly into those for A + B, but not those for A

^{+}and B

^{−}[7, 43]. (Fock space CC does.) In an infinite system, the difference due to one electron is meaningless, so for that situation there is no problem.

_{k},we do not have an exp(R

_{1}) so we cannot expect the same insensitivity to orbital choice that we can for the ground state. For condition V, we know that as long as the reference SRCC state is a closed shell, all $\overline{\mathbf{H}}$ eigenstates are spin-eigenfunctions. Of course, the overwhelming advantage of the EOM-CC approach is its ease of application. Essentially, no more decisions are required in doing EOM-CC than in doing SRCC, so EOM-CC methods are suited to black-box application and calibration. Hence, EOM-CC has most of the desiderata we want.

^{†}j. For the [0,1] and [1,0] sectors, the principal ionization eigenvalues (electron attached energies) of the Fock space problem are precisely those for the IP-EOM-CC (EA-EOM-CC) methods (the proof by partitioning is presented elsewhere [40]), so increasing the target space (adding more 1h configurations does not change their values, but does permit more to be obtained. Hence, item I is not really satisfied. The Fock space wave operator is fully exponential, however, unlike that in EOM-CC. Hence, the equations consist solely of linked diagrams and have all of the consequent properties, being size-extensive and size-intensive, which is not shared by EOM-CC for the higher sectors. For [0,1] and [1,0] the exponential operator reduces to a linear form which corresponds to the EOM-CC ansatz. FS-MRCC has properties II, III, and V, as does EOM-CC, and as we go to other sectors like [1,1], there is also benefit from increasing the P’ space. Importantly, FS-CC adds IV, and can do so for even incomplete P’ spaces [32]. In fact, one of the best ways to introduce triple excitations into EOM-CCSD, is to add selective elements missing that would arise from triples in a Fock space method, and that also ensures that the resulting equations are fully linked. Such an approach has been offered by Meissner [46, 47].

_{k}operator [49]. Then by building the double similarity transformed Hamiltonian,

_{3}, we will benefit from the double transformations above.

_{k}|φ

_{0}›

_{k}creates some linear combination of determinants and at least static correlations, {exp(S)} introduces non-dynamic correlation, and finally exp(T) introduces dynamic correlation. Such a MRCC approach has been pursued by Nooijen [51]. Such a form will satisfy all five of the conditions above, but does not appear to be able to offer the same ease of implementation as the SRCC, EOM-CC, or FS-CC since an active space selection will probably be a necessity. If that selection can be made unambiguously, as discussed in the last section, then there would be hope.

## 4 THE CI COMPROMISE

_{0}>. This introduces unlinked diagrams, because it is not built upon a fully connected expression like,

_{0}|[exp(T

^{†})H exp(T)]C|φ

_{0}› = ‹φ

_{0}|(1 + Λ) exp(−T)H exp(T)|φ

_{0}›

_{f}) and their Λ based alter-natives [69]. However, for some truncation of the exp(T)|φ

_{0}>, the expectation value in Eqn 17, could be used instead, which leads to “renormalization”. We recognize that once a perturbative approximation is made before insisting upon a fully connected structure as in Eqn. 18, a consistent cancellation of terms will not completely remove unlinked diagrams [70]. So a more flexible ap-proach might be thought to be obtained from their retention. Certainly when it comes to potential energy curves where there can be pathological consequences of the non-variational character of SRCC, having a few unlinked terms greatly aids in getting bounded curves, as they are ultimately responsible for the variational bound of truncated CI theory. In fact renormalized CC is a hybrid of CC and CI. So as the CC result turns over due to perturbation theory failures, the variation-aly bounded CI can take over! Hence, there is a CI compromise; but it might be recommended because we get better curves. On the other hand, it has not yet been shown that these kind of renormalized approximations actually converge to the right answer in a satisfactory way [70], as you can get quite a variety of results, short of putting in so many higher excitations that the dis-tinction between CI and CC is moot. Without systematic convergence we threaten the paradigm of converging CC methods CCD<CCSD<CCSD(T)<CCSDT<CCSDT(Q

_{f}) < CCSDTQ < Full CI that is responsible for their predictive quality, a cornerstone of the theory.

_{3}and T

_{4}are included in the determination of T

_{1},T

_{2}, eg, then we have an exact expression for the energy from the latter. Hence, a God-given T

_{3}and T

_{4}offers a great deal. Paldus et al. have shown the often spectacular results that emerge from such a method [55], so it is telling us a lot about higher order correlation. However, the CI calculation should be more time-consuming that the corresponding CC one, as it is affected by unlinked diagrams. Furthermore, I cannot see how such a method would be very widely applicable because of an inability to get analytical gradients easily.

_{2}, e.g., as shown in Fig. 3. Also, even in 1986 we were already using a quite extensive reference space (P space of 176 determinants) compared to those that have actually been considered computationally in other variants of state-specific, or Hilbert space CC. This, of course, was made possible by the extensive work and experience devoted to MRCI methods. MR-AQCC, as implemented into the COLUMBUS program system [78], has obtained a following, with several investigators who obtain their best results from the method [60]. It has been further extended by Szalay and co-workers to fine-tune the EPV-like approximation with further improvements [58].

_{α}= ‹φ

_{0}|(1 + Λ) exp(−T)H

^{α}exp(T)|φ

_{0}›

^{α}could be replaced by solving for only one Λ operator, P$\overline{H}$ΛQ − PEΛQ = 0 [81]. This is a necessity for realistic gradient evaluation for non-variational MRCC methods, too.

## 5 NUMERICAL RESULT

_{3}. The prob-lem is well known. The two symmetric, a

_{1}, vibrations of O

_{3}are qualitatively correct, while the asymmetric stretch vibration of a

_{2}symmetry is not. In Figure 4 and Figure 5, I show the results of many single and multi-reference methods in a DZP basis set. In Figure 6, I show some results in the large cc-pVTZ basis Unfortunately, the full CI result is not known, so we cannot provide an unambiguous comparison. We have to appeal to experimental harmonic frequencies. The essence of the problem, is that for the asymmetric stretch, certain configurations are allowed to mix, which cannot contribute to the symmetric displacements [82]. So the normal single reference function is a lesser part of the whole wavefunction for asymmetric displacements, causing a misbalance that affects the vibrational frequency. Consequently, it is usually thought that a proper multi-reference P space will resolve all these issues. Naively, there are two quasi-degenerate orbitals in O

_{3}, so we need at least a two-determinant reference space. For asymmetric displacements, we would have a third determinant in the complete active space that could not mix for symmetric displacements, but excitations can be created out of it that should be in the Q space manifold.

_{f}) and CCSDT(Q

_{f}) close behind [84]. For the difference, the latter is the best result, and the most theoretically complete method.

^{n−1}potential.

_{3}. But notice what this does for us. Since we have at least two important orbital levels in O

_{3}, of which one is doubly occupied in its SCF reference with the other nearby in energy, by filling that orbital too, and then kicking two electrons out with our EOM operator, R

_{k}= ∑ r

_{ij}{ij} + ∑ ${r}_{\mathit{ij}}^{a}${ija

^{†}}+ ... we create a CC wavefunction that treats the two quasi-degenerate orbitals completely equivalently; the essence of MR theory, but done is a operationally SR way. The CASSCF is presented for comparative purposes [87].

_{f}) is very good, but is slightly poorer than the simpler CCSD(TQ

_{f}). Of course, this is still far from the basis set limit, though a fairly extensive basis set study [88] pointed to the cc-pVTZ basis as having the best combination of quality and applicability. The fact that I only have one MRCC method, DIP-STEOM-CC, on this figure attests to the difficulty in obtaining these for most MRCC methods. Obviously, we can apply

_{3}has proven to be an outstanding example to illustrate failures of theories, the ground and excited states of (NO)

_{2}offers another demanding example. See Tobita, et al. [89], for a recent comparative study of DIP-STEOM, MR-AQCC, MR-BWCC, and other methods applied to this highly difficult system.

## 6 EXPECTATIONS FOR MRCC

- Size-extensivity
- Unambiguous active orbital space
- Single state(SS) MRCC as a special case of multi state (MS) MRCC
- SRCC as a special case of (SS)-MRCC and (MS)-MRCC

- (SS) and (MS) MRCC functional analogous to that for SRCC, to facilitate analytical gradi-ents and properties
- Reference for EOM approach to obtain all other states than obtained in the MRCC itself, with appropriate correspondence when a (SS) MRCC is used plus EOM to give a state that could also be obtained from (MS) MRCC.
- Automatic avoidance of intruder states possibly via an intermediate Hamiltonian framework
- Spin and spatial symmetry adapted
- Routine applicability for non expert use and calibration
- Satisfy conditions, I, II, and III from text.

_{0}› by higher excitations chosen from the consideration that if a second double excitation function, | ν›, that is actually treated in the Q space in CCSD, e.g., had been in the P space in a MR approach, then it would logically add new triples and quadruple excitations at its single and double excitation level that would not be included in the usual Q space for the SR-CCSD problem. In this way, some augmentation of the Q space, consistent with condition III, effects somewhat more balance indicative of a MR treatment. However, all operations are still those of SRCC. Of course, there is still substantial misbalance between treating the |ν› in Q space while the |φ

_{0}› is the real reference function. Some of this misbalance can be further ameliorated if we also fix the coefficient for |ν› to be what it should become based upon independent (or symmetry) considerations. I refer to this as a ’coefficient restricted’ approach, and for static correlation cases like open-shell singlets, or triplets, such a restriction to make both φ

_{0}and ν be constrained to have the same (±) coefficient could be easily incorporated to accelerate convergence of the CC equations. Generalizations to low-spin doublets, etc. also can be made. If we already have CCSDT or CCSDTQ, then there would be little to gain from such an augmentation, except convergence, which is where the ’coefficient restricted’ approach still has merit.

_{4}, which would normally require an ∼ n

^{10}algorithm, and similar higher connected terms at the cost of introducing ${\mathrm{T}}_{2}^{\u2020}$ terms. The factorization is beyond that of ordinary CC theory, like C

_{4}≈ (T

_{2})

^{2}/2, eg. The T

_{4}operator shows an exact factorization in its fifth-order energy expression, but does not in the corresponding amplitude expression. However, we have observed that we can force the latter at no significant cost in accuracy. This means that we can evaluate most of the effect of T

_{4}with only an ∼n

^{6}procedure, and if we insist upon including the initial T

_{3}, a ∼ n

^{7}method, or ∼ n

^{8}for the full T

_{3}. Hence we have a series of approximations including CCSDTQ

_{f}, CCSDT(Q

_{f}), CCSD(TQ

_{f}) etc. We have suggested a possible ansatz Ψ = exp(T

_{f}) exp(T

_{u}) where we separate the amplitude diagrams that are factorizable, T

_{f}, from those that are not, T

_{u}. If the effect of the latter could be shown to be bound in its consequences for calculations, then we could introduce the fully factorized terms without changing the asymptotic dependence of the computation. Clearly, such tricks can be combined with the active orbital methods above and should also be possible in the MRCC framework.

## 7 ACKNOWLEDGMENTS

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Bartlett, R.J.
To Multireference or not to Multireference: That is the Question? *Int. J. Mol. Sci.* **2002**, *3*, 579-603.
https://doi.org/10.3390/i3060579

**AMA Style**

Bartlett RJ.
To Multireference or not to Multireference: That is the Question? *International Journal of Molecular Sciences*. 2002; 3(6):579-603.
https://doi.org/10.3390/i3060579

**Chicago/Turabian Style**

Bartlett, Rodney J.
2002. "To Multireference or not to Multireference: That is the Question?" *International Journal of Molecular Sciences* 3, no. 6: 579-603.
https://doi.org/10.3390/i3060579