Comparison of Methods for Active Orbital Selection in Multiconfigurational Calculations

Abstract

Several methods of constructing the active orbital space for multiconfigurational wave functions are compared on typical moderately strongly or strongly correlated ground-state molecules. The relative merits of these methods and problems inherent in multiconfigurational calculations are discussed. Strong correlation in the ground electronic state is found typically in larger conjugated and antiaromatic systems, transition states which involve bond breaking or formation, and transition metal complexes. Our examples include polyenes, polyacenes, the reactant, product and transition state of the Bergman cyclization, and two transition metal complexes: Hieber’s anion [(CO)₃FeNO]⁻ and ferrocene. For the systems investigated, the simplest and oldest selection method, based on the fractional occupancy of unrestricted Hartree–Fock natural orbitals (the UNO criterion), yields the same active space as much more expensive approximate full CI methods. A disadvantage of this method used to be the difficulty of finding broken spin symmetry UHF solutions. However, our analytical method, accurate to fourth order in the orbital rotation angles (Tóth and Pulay J. Chem. Phys. 2016, 145, 164102.), has solved this problem. Two further advantages of the UNO criterion are that, unlike most other methods, it measures not only the energetic proximity to the Fermi level but also the magnitude of the exchange interaction with strongly occupied orbitals and therefore allows the estimation of the correlation strength for orbital selection in Restricted Active Space methods.

Graphical Abstract

I. INTRODUCTION

There has been much interest recently in defining active spaces for multiconfigurational wave functions.^1–10 The latter are necessary because the basic Hartree–Fock approximation becomes inaccurate for systems which have strong multiconfigurational character or static correlation. Although the transition between them is continuous, static correlation differs qualitatively from dynamical correlation. Unlike the latter, which arises from countless small contributions, static correlation involves fewer substituted configurations with significant weights. Dynamical correlation tends to cancel out for the nuclear gradient¹¹ (but not for the energy) and thus does not greatly affect potential surfaces locally. Static correlation has generally significant nonlocal contributions to the potential surface, sometimes changing it qualitatively. In most cases, static correlation is restricted to relatively few orbitals. Molecules with static correlation typically include the ground states of larger conjugated systems, most transition states, and essentially all transition metal compounds, particularly those of the first transition period (Sc – Cu). Most excited states are affected by static correlation. For such systems, the usual methods of including electron correlation by perturbational or coupled cluster methods (which typically include explicitly double or at most triple substitutions) are not sufficient, and a multireference correlation treatment is called for. Density functional theory (DFT), although much simpler computationally, does a fair job of describing static electron correlation. Analyzing some simple examples, say the bond breaking in H₂, makes it clear that the local treatment of the exchange term is responsible for the approximately correct description of static correlation. Unfortunately, essentially all DFT variants break down for very strong correlation because of the self-interaction error. Moreover, DFT is not a convergent theory, and it is difficult to identify, among hundreds of exchange-correlation functionals, the variant which will work in a given case. This, and increasing availability of computer resources, focused attention on wave function methods.

The standard method for static correlation is to replace the Hartree–Fock reference by a multireference wave function, generally multiconfigurational SCF (MC-SCF) which describes strong correlation within a limited orbital space, and add dynamical correlation by perturbational, coupled cluster, or variational (CI) techniques. This procedure has several difficulties. The first is the problem of identifying the orbital subspace, called active space, which contains the strongly correlating orbitals. Ideally, a relatively inexpensive procedure should yield all orbitals involved in strong correlation, with minimum human attention. Because of the steep scaling of most multiconfigurational calculations with the number of active orbitals, the active space should be as small as possible. Including weakly correlated orbitals in an MC-SCF calculation is deleterious for other reasons as well. They make the wave function rather arbitrary, and since they are not well determined, they can flip suddenly along the potential surface, causing discontinuities. MC-SCF by itself is not quantitative and often overestimates static correlation because dynamical correlation competes with static one. The inclusion of dynamical correlation by perturbation theory, coupled cluster theory, or configuration interaction is necessary to obtain quantitative results. Experience shows that the accuracy achieved by adding dynamical correlation to an MC-SCF wave function (“diagonalize then perturb” in the physicists’ parlor) is inferior to the analogous single-reference case. In our opinion, the reason is that the MC-SCF wave function rearranges in the presence of dynamical correlation, sometimes quite strongly, an effect which cannot arise in the single-reference case.

Early MC-SCF methods had serious convergence difficulties. The situation was much improved by the introduction of the Complete Active Space concept¹² which carries out a full CI treatment in a limited orbital space. This eliminates the main source of convergence problems, the strong coupling between the orbital coefficients within the active space, and the CI coefficients. In a CAS wave function, only the subspace spanned by the active orbitals matters, not the orbitals themselves. The drawback of CAS is that the number of configurations, and the computational effort, scales factorially with the dimension of the active space. Modern programs, for instance MOLPRO¹³ and MOLCAS,¹⁴ have largely overcome the MC-SCF convergence problem, particularly for CAS-SCF. There is intensive effort underway to develop approximate full CI methods which scale better than traditional techniques which are tractable only up to ~18 active orbitals and electrons.

We will consider primarily the determination of the active space for a single electronic state (generally the ground state) of a molecule, at a given geometry. A related problem is to generate a set of active orbitals for the simultaneous description of several excited states. We will comment on this problem later.

II. COMPARISON OF METHODS

Early suggestions, summarized in ref 15, and also many later ones relied heavily on chemical intuition,¹⁶ making them unsuitable for use by nonspecialists.¹⁷ For routine use, a black-box method, or as close to a black box as possible, is needed. The main problem with black-box methods for calculating potential surfaces is that static correlation is strongly geometry dependent, and the minimum active space varies over the potential surface. The reference potential surface becomes discontinuous when static correlation becomes dynamic or vice versa. Subsequent addition of dynamic correlation should, in principle, restore continuity, but in practice, small discontinuities remain. This problem is not specific to the UHF NO criterion but is common to all methods which use a threshold to determine the active space. One commonly used strategy is to determine the active space at the most strongly correlated geometry and continue the MC-SCF wave function, hoping that the correlating orbitals retain their identity. We have included an example.

Another early suggestion was to use the Unrestricted Natural Orbital (UNO) criterion which postulates that the fractionally occupied UHF charge natural orbitals span the active space.¹⁸ Fractional occupancy generally means electron population between 0.02 and 1.98 or 0.01 and 1.99. The UHF natural orbitals approximate the optimized CAS-SCF orbitals very well, in general: the error in the energy is typically below 1 mE_h/active orbital, and the nonparallelity error is less.¹⁹ As pointed out by one of the reviewers, fractionally occupied orbitals may not be active in heavy elements with deep-lying partially occupied f orbitals which have constant occupancy and interact little with the valence shell.

The UNO criterion has been criticized as having problems when a strongly occupied orbital has several important correlation partners and not well suited for excited states.¹⁰ The first point is inaccurate. If there are several strongly correlated partners of a formally occupied orbital, an infrequent occasion, then there are multiple independent UHF solutions. The correct natural orbitals are the eigenfunctions of the average charge density matrices of these solutions.¹⁹ A simple example, Li₂, is shown in the Examples section. For excited states, a similar procedure should be used. A method based solely on the ground state wave function cannot be expected to yield reliably active orbitals for excited states calculation. A long time ago, one of us (P.P.) worked out a method in which a sequence of UHF calculations, Schmidt-orthogonalized to all lower-lying states, is carried out, and the average natural orbitals of the ground state and the first N excited state UHF charge density matrices are averaged. A brief description of this method was given previously.¹ The fractionally occupied natural orbitals of the average density span the active space for the ground and the first N excited states. Although initial tests were promising, we did not pursue it further because the then-emerging time-dependent density functional theory was simpler and equally accurate. Bao and co-workers proposed recently a method for selecting active orbitals for excitation energies, using UHF calculations.⁸ Their goal is to offer a balanced description of the ground and excited states and is somewhat different from ours. This is also true for the method of Shu and co-workers³ which uses state-averaged natural orbitals of singly excited Configuration Interaction (CIS) calculation. Methods based on excited state UHF calculations may run into difficulties if there is a dense manifold of such solutions, which may happen in molecules with several transition metal centers.

The most obvious case of change in the active space with molecular geometry using the UNO criterion is when the system suddenly becomes triplet stable. E.g., the F₂ molecule is strongly correlated at the equilibrium and stretched geometries but becomes triplet stable at short F–F bond lengths, leading to a discontinuity on the potential energy surface.¹ As mentioned above, MC-SCF wave functions may be extended in these regions. An alternative strategy is to try to use analytic continuation. The energy expectation value is not an analytic function because it uses complex conjugation, but there is promising development of the holomorphic Hartree–Fock theory.²⁰

Recent contributions to active space selection belong to two classes. Sayfutyarova and co-workers⁷ and Sayfutyarova andHammes-Schiffer⁹ revert to chemical intuition based methods. In Knizia’s AVAS method,⁷ a small set of n initial active (often atomic) orbitals is selected manually. The occupied and virtual orbitals of an SCF-type wave function are projected separately to this initial active space. By diagonalizing the overlap matrix of the projected orbitals, in the spirit of the Density Matrix Embedding Theory,²¹ a set of n occupied and n virtual active orbitals, overlapping with the initial active space, is obtained (assuming that the number of both occupied and virtual orbitals exceeds n). This 2n dimensional active space can be truncated further, depending on the degree of overlap with the initial active space. The identification of the initial active space in AVAS⁷ is fairly straightforward in the case of bond breaking or forming and in transition metal compounds with partially occupied d or f orbitals. It is less clear if strong correlation arises from a small HOMO/LUMO gap in large conjugated systems. The basis of the AVAS criterion is that strong correlation generally occurs in the valence shell. However, it uses no physical input characterizing the strength of the correlation, that is the exchange matrix element K_ia = (ia|ia) where |i > is a formally occupied and |a > a formally unoccupied orbital and the Mulliken convention is used for the two-electron integrals. It has been shown for a (2e,2o) (2-electron, 2-orbital) model case that the emergence of strong correlation in an electron pair is determined by the relative magnitude of the SCF double excitation energy ΔE(ii → aa) and the exchange matrix element K_ia. The former is generally dominated by the orbital energy difference 2(ε_a — ε_i), but the latter is more difficult to estimate. AVAS is advantageous in large molecules when only the electronic structure in a localized fragment is of interest. However, the same result can be obtained by determining the active space for the full system and localizing the active orbitals. The application of this method to planar π systems⁹ gives the space of the minimum basis π orbitals as the active space. As will be shown below, in some larger systems, for instance, free base porphine, this space includes weakly correlated orbitals. Application to nonplanar π systems requires the (necessarily arbitrary) definition of the local out-of-plane direction.

Another class of methods aims at determining the active space without chemical intuition. The FOD method of Grimme and Hansen² uses the fractional occupation numbers from high (~10000 K) temperature DFT calculations to identify the density of active orbitals. This technique picks up the orbitals close to the HOMO–LUMO frontier but, just like AVAS, uses no information to identify strong correlation, i.e., a large exchange matrix element between a strongly and a weakly occupied orbital pair. Both of these methods (AVAS and FOD) can be combined with preliminary correlated calculations which do include information about orbital entanglement to refine the active space. It yields a useful real-space indicator of the density of active orbitals and thus the spatial localization of strong correlation in a large molecule. Similar real-space information is obtained from the fictitious (in a single system) spin density, as shown in the Examples section for Thiele’s hydrocarbon. The FOD technique can be used to define active spaces based on the fractional occupation numbers of the natural orbitals of the thermally averaged density.²² These orbitals and occupation numbers have not yet been compared to other methods like CASSCF or UNO-CAS. A good figure of merit for FOD (and for AVAS) would be the CAS-CI energy obtained using the FOD orbitals. During the revision of this manuscript, we have implemented this method and plan to carry out such comparisons. In a few cases which we considered, for instance, strongly twisted ethylene and cyclobutadiene near the square planar geometry, the fractionally occupied finite-temperature orbitals are visually indistinguishable from the corresponding canonical frontier orbitals, showing that the finite temperature method essentially selects the frontier orbitals.

Wouters and co-workers²³ and Stein and Reiher^4,5 use the Density Matrix Renormalization Group (DMRG)²⁴ technique to converge toward full CI. This method picks up in the limit all correlation, strong or weak, and thus, with a proper threshold, yields an unambiguous definition of the active space, by the occupation numbers²³ or the von Neumann entropy.⁵ It is, however, very expensive computationally, and, unless fully converged, shows strong dependence on the initial set of orbitals and their ordering. Indeed, if DMRG is fully converged, there is no need for an active space. It is seldom noted, but DMRG becomes essentially equivalent to local correlation for linear or tree-like systems (which is its original application area). Khedkar and Roemelt, in a very useful paper,¹⁰ use a less expensive method, NEVPT2,²⁵ on the top of an initial DMRG in their ASSIST method to generate approximate multireference wave functions which yield the active space. Although this method also requires an initial active space guess, it appears to converge to a definite orbital space if applied iteratively, as discussed for [Fe(CO)₃NO]⁻. ASSIST is a generalization of an older method by Jensen et al.²⁶ that uses MP2 natural orbital occupation numbers to identify active orbitals. MP2 itself is inaccurate in the presence of strong correlation, but its natural orbitals may be still useful. Unfortunately, the method of Jensen et al. was used for typical dynamical correlation problems (H₂O at equilibrium geometry, HCl) where MC-SCF is neither necessary nor desirable; the only example showing moderately strong correlation is benzene. A common problem of all methods which use the natural orbital occupation numbers of an approximate Configuration Interaction (CI) wave function (MP2, DMRG, ASSIST) is that the natural orbital occupation numbers do not discriminate between two cases: (a) a strongly occupied orbital has a number of weakly coupled correlation partners, giving it fractional occupancy, and (b) a strongly occupied orbital has one (or at most a few) strongly coupled correlation partners. Only case (b) represents static correlation. This problem is evident from the benzene example of Jensen et al.²⁶ which fails to pick up the chemically obvious π space as emerging static correlation. The UNO criterion clearly defines 6 π orbitals as the active space, without any chemical intuition. It also defines a clear (6e, 6o) active space if the molecule is distorted from planarity.

The purpose of the present paper is to show that the original Unrestricted Natural Orbital (UNO) criterion, if applied properly, reproduces the correct active space for a selected state at a fraction of the computational cost, compared to the DMRG and DMRG-NEVPT2 methods. As some examples below show, it is also capable to select a correct active space which may be smaller than methods which only identify the frontier orbitals. It does not rely on chemical intuition, not even in the initial Ansatz, and requires little or no human intervention. The UNO criterion has another advantage compared to AVAS:^7,9 it gives an approximate measure of the correlation strength. If CAS is intractable, Restricted Active Space (RAS)²⁷ and its generalizations²⁸ can be used to reduce the dimension of the configuration space. RAS partitions the orbital space to the strongly occupied RAS1, the variably occupied RAS2, and the weakly occupied RAS3 subspaces. Only a restricted excitation level (say, doubles) out of RAS1 and a restricted number of excitations into RAS3 are allowed. One can define these subspaces by UHF NO occupation numbers, say RAS1 between 1.99 and 1.90, RAS2 between 1.90 and 0.1, and RAS3 between 0.1 and 0.01. The FOD method²² could also be used this way, but it measures only the distance of the orbital to the Fermi level, not the correlation strength. The AutoCAS method⁵ is a good way to define these subspaces but at considerable computational cost. Another way of reducing the dimension of the configuration space is to use a wave function based on the Generalized Valence Bond (GVB) concept.²⁹ A UHF wave function which gives a (2e,2o) active space becomes a GVB wave function after spin projection.³⁰ The orbital pairs are generally evident from the UHF NO occupation numbers (they add up to exactly 2), but in the case of degeneracy, one should switch to the corresponding orbital representation (these are simply normalized sums and differences of the charge natural orbitals³¹). The corresponding orbitals give the nonorthogonal form of the GVB wave function;²⁹ this can be easily transformed to the orthogonal form. Corresponding pairs of UHF NOs should be useful initial orbitals for methods which use paired excitations to avoid the factorial scaling of CAS, for instance, the pair coupled cluster method.^32,33

The usefulness of UHF occupation numbers for identifying strong correlation has a long history, starting with Dohnert and Koutecky.³⁴ Yamaguchi pioneered the use of UHF charge natural orbitals in Configuration Interaction (CI) calculations^35–37 anticipating the concept of active space. The use of triplet (UHF) instability as a criterion for strong correlation was put on a firm base when Bofill and Pulay¹⁹ proved that triplet instability in a 2-orbital, 2-electron system implies that the minor component of the wave function has a weight exceeding about 5% (0.05). This corresponds to a natural orbital occupation number for a formally unoccupied orbital of 0.1. One can argue about what “strong” correlation means, but 5% weight or 0.1 electron occupancy appear to be reasonable thresholds. The original UNO criterion suggested a threshold of 0.02 (occupancies between 0.02 and 1.98) for inclusion in the active space. For calculations on large systems with many strongly correlated orbitals, this threshold may yield so many active orbitals that a traditional CAS becomes impracticable. For these cases, is useful to introduce a second, higher threshold, say 0.075 (occupancies between 0.075 and 1.925). Improvements in the treatment of dynamical correlation, for instance, NEVPT2²⁵ and the emerging multireference coupled cluster theories, also allow the treatment of milder static correlation by traditional dynamical correlation techniques.

Other types of instabilities: singlet, complex, and noncollinear spin instabilities^38,39 are also signs of strong correlation. This is clear from the fact that the eigenfunctions of a one-electron operator must conform to the symmetry of the physical system. Spontaneous symmetry breaking arises when electron–electron repulsion overcomes the one-electron terms. Hartree–Fock instability was a very active area of research in the 1970s and early 1980s. More recently Jiménez-Hoyos and co-workers studied more general symmetry breakings.⁴⁰ It appears, however, that these additional instabilities seldom generate new active orbitals compared to triplet instability. General Hartree–Fock wave functions are able to generate the active orbitals of multiple triplet-unstable UHF wave function from a single symmetry-breaking wave function which requires the average density of several UHF solutions. Complex instability may be more important in periodic systems where it can lead to charge-density waves.

We have compared the active spaces predicted by the UNO criterion with active spaces from DMRG calculations, using natural orbital occupation numbers, in a previous paper.¹ The systems considered were difluorine (F₂) at several geometries, nitric oxide (NO₂), ozone, hexatriene, octatetraene, decapentaene, naphthalene, azulene, anthracene, nitrobenzene, the phenoxy and benzyl radicals, o-, m-, and p-benzyne (C₆H₄), nickel-acetylene, and dichromium (Cr₂). The present paper extends this study to larger and more difficult cases and compares the active spaces with those obtained by other techniques.

III. COMPUTATIONAL DETAILS

Unless noted otherwise, all calculations were carried out at geometries optimized at the RB3LYP/def2-TZVP⁴¹ level with Grimmes D3 dispersion correction,⁴² using the PQS⁴³ (for optimization, UHF, and CCSD(T)) and the TEXAS⁴⁴ (for UNO-CAS and CAS-SCF) suites of programs; the latter were checked against MOLPRO¹³ in most cases. The molecular geometries are given in the Supporting Information. Although UHF is considered an inexpensive method (according to Roemelt¹⁰ “it comes at almost no cost at all”), finding a stable UHF solution (i.e., a local minimum with respect to orbital rotations) may be difficult, particularly in even-electron systems. UHF convergence is often poor near the triplet instability (Coulson-Fischer⁴⁵) point. A Restricted Hartree–Fock (RHF) wave function is always a solution of the UHF equations for an even number of electrons, although not a minimum but a saddle point if the RHF wave function is triplet unstable. UHF also can have saddle point solutions in which orbital pairs, which are significantly split in the lowest energy solution, are essentially doubly occupied. The widely used DIIS^46,47 convergence acceleration method, which minimizes the norm of the electronic gradient, tends to converge to the RHF solution, unless it is started close to the UHF minimum. Various laborious methods have been proposed to find UHF minima based on the eigenvectors of the electronic Hessian, but the real solution to this problem (for the time restricted to singlets) was given by Tóth and Pulay.⁴⁸ Based on the serendipitous discovery that a Taylor expansion of the UHF energy as a function of orbital mixing angles x, y, … in the eigenvector basis of the electronic Hessian does not contain terms with odd powers, such as xy³ (even though the total power is even), they were able to determine the minima of a quartic model of the UHF energy noniteratively, using only linear algebra. The resulting solution is generally close enough to the minimum for DIIS to converge smoothly to the minimum. As quintic terms vanish automatically, ref 48 determines the lowest UHF minimum at the fifth-order level. In most systems, especially in organic ones, this is all that is needed. However, it may happen that there are multiple independent UHF solutions of equal or nearly equal energy, particularly for transition metal compounds. As pointed out earlier, in such cases, the eigenfunctions of the average density matrix of the UHF solutions span the correct active space. A simple example, Li₂, is given below. In the case of degeneracy, i.e., when there are two or more UHF solutions with equal energy, this averaging is absolutely necessary.

A further point to watch for is that UHF orbitals often break spatial symmetry, and thus symmetry, which is automatically detected and enforced in most quantum chemistry programs, must be disabled to obtain UHF solutions which are true minima. Perhaps these simple difficulties gave rise to the notion that the UHF NO criterion is not robust.¹⁰

For the singlet systems, we also give the Yamaguchi diradical character.⁴⁹ This can be defined in terms of the Unrestricted Hartree–Fock frontier natural orbital (NO) occupation numbers⁵⁰ as y = (2 – Δ)²/(4 + Δ²) where Δ = n(HOMO) – n(LUMO); HOMO is the NO with the lowest occupation number equal or greater than 1, and LUMO is the NO with the highest occupation equal or less than 1.

IV. EXAMPLES

Multiple Correlation Partners: Li₂.

As pointed out in the Introduction, strong correlation generally occurs between orbital pairs. A survey of CAS calculations (for singlet states) shows that the number of electrons and number of orbitals are equal in most of them. CAS calculations which violate this are often found to contain orbitals which are either almost doubly occupied or almost empty and therefore do not belong to the active space. It is not easy to find a simple example of an (me,no) CAS wave function with m ≠ n for a singlet state, except for transition metal complexes. Dilithium is such a system. Unlike the isoelectronic H₂, Li₂ is triplet unstable at its equilibrium geometry and therefore (moderately) strongly correlated. Indeed, high-quality calculations show that the Hartree–Fock reference has only 90% weight in this effectively two-electron wave function,⁵¹ which is a clear sign of nondynamical correlation. Our standard scheme yields an equilibrium distance of 2.722 Å, slightly longer than in high-level calculations (e.g., 2.68 Å,⁵¹ 2.67 Å experiment⁵²), but the potential curve is very shallow. The electronic Hessian of the closed-shell Restricted Hartree–Fock (RHF) solution has three negative eigenvalues, the smaller two (in absolute value) of which are degenerate. The minima were determined to fourth order using the analytical procedure of ref 48, followed by UHF iteration. This gives three independent UHF solutions, the valence orbitals of which are shown in Figure 1. The average of the three UHF charge densities¹⁹ gives a (2e,4o) active space. This example demonstrates that the UNO criterion can handle more than one strongly correlating partner for a formally doubly occupied reference orbital.

Figure 1.

Fractionally occupied average natural orbitals and occupation numbers in Li₂ from three UHF calculations, yielding a (2e,4o) CAS. The Li—Li distance is 2.721926 Å.

Pi-Electron Systems.

The active space for planar π electron systems at the usual occupation number threshold (~0.02) is similar to the valence p_z orbitals of the atoms and should be correctly identified by methods which do not test for correlation strengths, like the method of Sayfutyarova and Hammes-Schiffer⁹ or Grimme and Hansen. However, in larger systems, these methods may generate active spaces that are too large. A particularly interesting case is free base porphine which will be discussed below.

Conjugated trans-Polyenes.

Conjugated all-E polyenes exhibit moderately strong correlation which gets stronger with increasing chain length. Hexatriene, octatetraene, and decapentaene were considered in a previous paper.¹ The present paper extends this up to eicosadecaene (C₂₀H₂₂). At the default UNO threshold (0.02), the active space consists of a set of valence π orbitals on the conjugated carbon atoms. At higher threshold (0.1), the dimension of the active space is only about half of this: 2n for polyenes C_4n+2H_4n+4 and C_4nH_4n+2 (4 for C₁₀H₁₂ and C₁₂H₁₄, 6 for C₁₄H₁₆ and C₁₆H₁₈, etc.). Figure 2 shows the occupation numbers of the Unrestricted Natural Orbitals (UNOs) in the series C₁₀H₁₂ to C₂₀H₂₂. The quality of the active orbitals can be assessed by comparing the multiconfigurational SCF energy of the initially generated orbitals (the UNO-CAS energy for the UNO criterion) with the converged CAS-SCF energy. In general, the UNO-CAS energy is only about 1 mE_h above the corresponding CAS-SCF energy per active orbital.¹⁹ Reference 1 has many examples of this kind, and it would be instructive to evaluate the CAS-CI energy using the unoptimized orbitals by the methods of refs 6, 9, and 2. For instance, the PiOS active orbitals obtained by Sayfutyarova and Hammes-Schiffer⁹ are qualitatively similar to those obtained by the UNO criterion, but the CAS energies obtained with the initial active orbitals most likely do not approximate the CASSCF energies as well as the UNO orbitals. Figure 3 shows the UNO natural orbitals for octatetraene; they can be compared with Figure 2 of ref 9. Table 1 shows the UNO-CAS and CASSCF energies of E,E-octatetraene for two different UNO occupation number thresholds: 0.02 and 0.075, yielding (8e,8o) and (4e,4o) actives spaces, respectively. This table and analogous results for decapentaene (Table S-B1 of the Supporting Information) show that the UNO-CAS energies and occupation numbers are close to the converged CAS results. However, reducing the active space by raising the threshold has a significant effect and reduces the strength of the correlation in this case. This shows that orbitals which have marginal open shell character (occupation number between 0.01 and 0.075 and between 1.925 and 1.99) should be included in the active space, either fully or approximately by a RAS construct or DMRG.

Figure 2.

Unrestricted Natural Orbital (UNO) occupation numbers of the all-E polyenes C₁₀H₁₂ to C₂₀H₂₂ at the def2-TZVP level. To improve readability, the quantity plotted on the y axis is not the occupation number n but f(n) = c[(a-n)⁻¹-(b+n)⁻¹+c⁻¹] with a = 2.25, b = 0.25, and c = 0.28125. f(n) maps the interval (0,2) onto itself but expands the scale near small (~0) and large (~2) occupation numbers. Ten threshold lines are shown, at 0.02, 0.05, 0.1, 0.2, and 0.4 electrons, and their complementary values (1.98, 1.95,1.9, 1.8,1.6). The formal HOMO is orbital 0, and the formal LUMO is orbital 1. The x axis is the orbital count, 0, −1, −2 correspond to HOMO, HOMO–1, HOMO–2, etc., and 1, 2, 3 correspond to LUMO, LUMO+1, LUMO+2. See Table S-B2 of the Supporting Information.

Figure 3.

Fractionally occupied UHF natural orbitals (orbitals 26–33) of E-E-octatetraene (def2-TZVP basis).

Table 1.

Natural Orbital Occupation Numbers and Total Energies (E_h) for E,E-Octatetraene with the 6–31G* and the def2-TZVP Basis Sets

wavef basis	UHF 6–31G*	UHF def2	UNO-CAS (8e,8o) def2	CASSCF (8e,8o) def2	UNO-CAS (4e,4o) def2	CAS-SCF (4e,4o) def2
26	1.9536	1.9577	1.9464	1.9454
27	1.9342	1.9405	1.9323	1.9322
28	1.8645	1.8762	1.9000	1.9054	1.9578	1.9479
29	1.6002	1.6209	1.8373	1.8518	1.9067	1.9178
30	0.3998	0.3791	0.1699	0.1569	0.0943	0.0857
31	0.1355	0.1238	0.0998	0.0949	0.0412	0.0494
32	0.0658	0.0595	0.0646	0.0639
33	0.0464	0.0423	0.0497	0.0495
E+308		−0.8288185	−0.9146307	−0.9179873	−0.8376242	−0.8417055

Polyacenes.

Linear polyacenes exhibit increasingly strong correlation as the chain length grows. We compared previously¹ the UNO criterion with DMRG-generated active spaces for naphthalene and anthracene. Both agree that, at a sharp threshold, the correct active space is a minimum π space, (10e,10o) and (14e,14o), respectively. In this paper, we extend this to tetracene, pentacene, and hexacene; benzene was discussed in ref 48. The UNO criterion at the default threshold (0.02) gives active spaces of (10e,10o), (14e,14o), (18e,18o), (22e,22o), and (26e,26o) for the series naphthalene to hexacene, see Figure 4. The UHF natural orbital occupation numbers and the Yamaguchi free radical indices are shown in Table S-B3 of the Supporting Information.

Figure 4.

Unrestricted Natural Orbital (UNO) occupation numbers of the linear polyacenes naphthalene to hexacene (def2-TZVP). See the caption of Figure 1.

Porphine.

Free base porphine, C₂₀H₁₄N₄, is the parent compound of porphyrins. It has 26 electrons and 24 orbitals in a minimum π basis, and the PiOS algorithm of ref 9 yields accordingly a (26e,24o) active space. However, it is generally accepted that the ground state of porphine is an 18-electron aromatic system. Both the UNO criterion and, more conclusively, the DMRG+SC-NEVPT2 calculations of Khedkar and Roemelt⁵³ confirm this. According to the UNO criterion, the active space at a sharp (0.02) threshold has only 18 electrons in 18 orbitals. At a high threshold (0.1), the active space has only 4 electrons in 4 orbitals. The latter were identified by Gouterman⁵⁴ a long time ago. It is not clear how refs 2 and 7 could identify the 4 Gouterman orbitals in this case. The occupation numbers of the UHF natural orbitals are shown in Figure 5.

Figure 5.

UHF charge natural orbital occupation numbers of porphine with small (6–31G*) and large (def2-TZVP) basis sets. See the caption of Figure 1.

The small 6–31G* and the large def2-TZVP basis sets show little difference in occupation numbers, showing that static correlation is mainly a valence shell phenomenon. The four strongly correlated Gouterman⁵⁴ orbitals are clearly separated from the rest, although the UNO occupancies, as usual, exaggerate the open-shell character of the state. The fractionally occupied UHF natural orbitals are shown in Figure 6. We have generated the UHF wave function from the RHF using our new deterministic technique which finds UHF solutions reliably.⁴⁸ The instability matrix of the RHF wave function has two negative eigenvalues (−0.0827 and −0.0366 E_h at the 6–31* level). However, discarding equivalent solutions, there is only one UHF minimum. This is analogous to the case of ozone⁴⁸ and can be understood by considering the UHF energy as a function of the orbital mixing angles corresponding to the two negative eigenvalues, shown in Figure 7.

Figure 6.

Fractionally occupied UHF natural orbitals of porphine, with occupation numbers at the 6–31G* level. The orbitals at the def2-TZVP level are visually identical at this resolution. Top rows: formally virtual orbitals. Bottom rows: formally occupied orbitals. The four Gouterman frontier orbitals have occupation numbers 1.549, 1.509, 0.491, and 0.451.

Figure 7.

UHF energy of porphine (6–31G* basis) as a function of orbital rotation angles (rad) along the eigenvectors corresponding to the two negative instability eigenvalues: the larger one (−0.48) is plotted along the horizontal axis, and the smaller one (−0.16) is plotted along the vertical axis. The 2D potential surface is symmetrical to sign changes of the angles, and only part of it is shown to save space. Energies are in Eh and contain the additive constant 983.296 Eh. The RHF wave function corresponds to (0,0). Although there are two distinct instabilities, there is only one symmetry-unique minimum at (−0.793,0). The minimum energy is higher than the optimized UHF energy, because the other orbitals are fixed at their RHF form.

We have calculated the active space using the FON (finite temperature DFT) method, following the recommendations of Bauer et al.,²² using the BHLYP density functional at 15000 K temperature, with the 6–31G(d), def2-SVPP, and def2-TZVP basis sets. At the recommended natural orbital occupation number thresholds between 0.02 and 1.98, this procedure yields a 20 electron, 13 orbital active space, rather different from the (18e,18o) space given by the DMRG+SC-NEVPT2 results.⁵³

Table 2 compares UHF, UNO-CAS, and CASSCF correlation energies of porphine for various CAS-type wave functions; the total energies and natural orbital occupation numbers are listed in Tables S-B4 and S-B5. The DMRG energy⁵³ of (8e,8o) CAS agrees closely with the full CASSCF energy. As for polyenes, inclusion of the less strongly correlated active orbitals has a significant effect on the wave function. The most important result for porphine is that the DMRG+SC-NEVPT2 calculations agree fully with the (much less expensive) UNO criterion, and both predict an active space that is significantly smaller than the minimum basis π space.

Table 2.

UNO-CAS and CASSCF Correlation Energies of Porphine^a

active space	UNO-CAS 6–31G*	CASSCF 6–31G*	UNO-CAS def2-PVTZ	CASSCF def2-PVTZ	DMRG-SC-NEVPT2 def2-PVTZ
(4e,4o)	−24.238	−34.088	−23.820	−33.787	−33.79b
(6e,6o)					−44.47b
(8e,8o)	−62.244	−72.763	−61.254	−72.164	−72.175b
(14e,14o)	−128.514	−134.7c	−125.173	−131.3c
(18e,18o)		−204.80b			−202.49b

^aIn millihartrees (10⁻³ E_h). All energies contain the additive constant 983.0 E_h. Closed-shell SCF energies are −983.247320 E_h (6–31G*) and −983.589185 E_h (def2-TZVP).

^bDMRG-CAS calculations by Khedkar and Roemelt, ref 53.

^cConverged only to 0.1 mE_h.

Magnesium Porphyrin.

Deprotonating the porphyrin ring, as in Mg porphyrin, gives a structure with D_4h symmetry which is not compatible with a single 18-electron aromatic system. The instability matrix, unlike in porphine, which has a single dominant negative eigenvalue, has a pair of degenerate negative eigenvalues at −0.0584 E_h (6–31G basis). Rotating the orbitals along the corresponding eigenvectors leads to two distinct degenerate (l8e, 18o) UHF solutions. The average UHF charge density matrix of these two solutions gives, at a sharp threshold of 0.01, a (26e,24o) active space: the full π space. However, very strong correlation is restricted to the four Gouterman orbitals. The occupation numbers are shown in Table S-B6 and Figure S-1.

Nonplanar π Systems: Corannulene, Tetraphenylqui-nodimethane (Thiele’s Hydrocarbon), and Buckminsterfullerene.

Corannulene.

One advantage of the Unrestricted Natural Orbital criterion is that there is no need to define the local out-of-plane direction, as in the method of Sayfutyarova and Hammes-Schiffer.⁹ Corannulene, C₂₀H₁₀, is a mildly nonplanar aromatic hydrocarbon. Its triplet instability matrix has a pair of degenerate negative eigenvalues (−0.0424 at the 6–31G* level). The two UHF solutions corresponding to these instabilities both contain 18 electrons in 18 orbitals, but they are not strictly degenerate. The average UHF density matrix yields a (20e,20o) active space, i.e., the full local π space at the default threshold 0.02. Like other aromatic ring systems with odd-membered rings, corannulene is a spin-frustrated system and is probably best treated starting with a General Hartree–Fock wave function (with noncollinear spins). This is discussed below for buckminsterfullerene. Figures S-2 and S-3 show the occupation numbers and the active orbitals.

Thiele’s Hydrocarbon (Tetraphenyl Quinodimethane), C₃₂H₂₄.

This strongly nonplanar hydrocarbon has two geometry minima at the B3LYP/def2-TZVP+D3 level, one with D₂ and one with C_2v symmetry, the latter of which (see Figure S-4) is slightly lower in energy by 1.15 mEh (0.72 kcal/mol). The central ring in both forms is quinoidal. By the UNO criterion, the molecule is strongly correlated, as the UHF NO occupation numbers in Figure S-5 show: the most fractionally occupied frontier orbitals (roughly HOMO and LUMO) have occupancies of 1.403 and 0.597 using the def2-TZVP basis. The strong correlation is associated with the quinodimethane moiety, as the pictures of the frontier orbitals in Figure 8 show. In the parent quinodimethane, the occupancies of the analogous frontier orbitals at the UHF/def2-TZVP level are 1.575 and 0.425, more closed-shell like but still strongly fractional.

Figure 8.

Two most fractionally occupied (frontier) UHF natural orbitals in Thiele’s hydrocarbon, with occupation numbers (6–31G* basis; see text for the def2-TZVP basis).

Buckminsterfullerene.

In this strongly spin-frustrated system, the best starting wave function is probably a General Hartree–Fock (GHF, i.e., noncollinear spins) wave function.⁵⁵ However, although this method has been known for almost 40 years⁵⁶ and used occasionally,^57–59 it is still not generally available in most programs. There is also a perception that it is difficult to converge. We note that the theorem of ref 48 holds probably also for GHF wave functions and would afford a straightforward way of finding solutions.

Without using GHF, we have multiple triplet instabilities and corresponding solutions. The triplet instability matrix of C₆₀ has 7 negative eigenvalues: two degenerate sets of 4 and 3 components. With the 6–31G basis, the eigenvalues are −0.0565 E_h and −0.0370 E_h. An analysis of the principal components of the corresponding eigenvectors⁴⁸ shows that they are dominated by excitation from the 5-fold degenerate HOMO orbitals to the 3-fold degenerate LUMOs. We have generated 12 different but degenerate UHF solutions by mixing the 5 HOMOs and the 3 LUMOs which belong to different representations under the D_2h subgroup of the I_h symmetry group (the other three mixings revert to the RHF wave function). The UNO criterion of each of these 12 solutions yields a (52e,52o) active space. The natural orbitals of the average of the 12 charge densities give a (60e,60o) active space at the default 0.02 threshold. However, strong correlation is restricted to a much smaller subset of orbitals, as the occupation numbers in Table S-B7 show. The individual UHF solutions show rather strong correlation (occupation numbers of 1.499 and 0.501; Yamaguchi diradical character y = 0.201); but the lowest ocupancy of a formally occupied natural orbital is 1.701, and the highest occupancy of a formally vacant natural orbital is 0.369 for the averaged density, with a Yamaguchi index of only y = 0.077. This shows that, while many orbitals are involved in moderately strong correlation (33 orbitals have UHF NO occupancies between 0.075 and 1.925), C₆₀ does not have very strong polyradical character.

Reaction Paths and Transition States.

We will discuss in detail only the Bergman cyclization, as it exemplifies the difficulties which arise when the active space changes along the reaction coordinates.

Bergman Cyclization.

The basic Bergman cyclization is the transformation of cis-1,3,5-hexaenediyne to p-benzyne. The reaction path was calculated at the (U)B3LYP/def2-TZVP level with D3 dispersion correction, varying the C⋯C distance between the terminal C atoms of the endiyne between 1.3 and 4.5 Å. The transition state, shown in Figure S-7, has a C₁⋯C₆ distance of about 1.93 Å. A reaction path calculated at the UHF level is qualitatively similar, although it has a longer C⋯C bond. The occupation numbers of the fractionally occupied UHF natural orbitals along the reaction path are shown in Figure 9 and illustrate the problem: p-benzyne has 8 active orbitals, two of which are very strongly correlated, while the endiyne has 10 weakly or moderately correlated active orbitals.

Figure 9.

UHF natural orbital occupation numbers along the reaction path in the Bergman rearrangement (def2-TZVP basis). The left side corresponds to p-benzyne, and the right side corresponds to Z-hexaendiyne; the x axis is the C₁⋯C₆ distance in Å.

There is one well-defined UHF solution with both the 6–31G* and the def2-TZVP basis sets at the B3LYP/def2-TZVP optimized transition state. The UHF natural orbital occupation numbers (Table S-B8 and Figure S-8) are, as usual, virtually the same with both basis sets and give a (10e, 10o) active space, consisting of 4 σ (two a1 and two b2) and 6 π (three a2 and three b1) orbitals under C_2v symmetry, see Figure S-9. This reaction exemplifies the main difficulty of multireference methods: the active space changes along the reaction coordinate. We would like to emphasize that this problem is not specific to the UNO or any other criterion. Any procedure which uses thresholds to select orbitals for special treatment has this problem. The reactant UHF natural orbitals correspond, both in number an symmetry under C_2v, to the transition state, see Figure S-10, although some of them have relatively low (~0.03) or high (~1.97) occupations. However, the product, p-benzyne has only an (8e,8o) active space,¹ (a₁ + 3a₂ + 3b₁ + b₂ under C_2v). The occupation numbers and symmetries (under the C_2v group of the transition state) of the active orbitals are shown in Tables S-B8, S-B9, and S-B10 for the transition state, the reactant, and the product, respectively. By continuing the orbitals along the reaction path in MC-SCF calculations, one can create a (10e,10o) active space for p-benzyne with orbitals which correspond to the transition state symmetry-wise. However, the UNO criterion shows that two of the orbitals are closed-shell like. Including them in the active space means that a particular type of dynamical correlation is treated by a multireference method which is, as discussed in the Introduction, arbitrary and may lead to instability or root flipping (see Figure S-11). Table S-B10 and Figure S-11 include, unlike the other tables of occupation numbers, a few orbitals with very low (<0.005) and very high (>1.995) occupation, to show that the natural active space of p-benzyne is (8e,8o) in the ground state. The geometry of p-benzyne was optimized, unlike all other systems, with unrestricted B3LYP because the molecule is an almost perfect biradical, with a Yamaguchi index of 0.83. The system is triplet stable for the B3LYP exchange-correlation functional along the reaction path from the reactant to the transition state, but the α and β orbitals split right after the transition state. Although the RB3LYP and UB3LYP geometries are fairly different, the UHF occupation numbers at the RB3LYP geometry give qualitatively the same active space as at the UB3LYP geometry.

The energy profile along the reverse cyclization reaction path is displayed in Figure 10 for the UHF, the (10e,10o) CASSCF and CASSCF augmented with the Average Coupled Pair Functional⁶⁰ (ACPF) dynamical correlation energy, calculated using MOLPRO.¹³ In this case, the highest and lowest occupancy orbitals retained their identity in p-benzyne but dynamical correlation significantly (about 10 kcal/mol) to the reaction enthalpy. We will not discuss here the reaction energetics, as MC-SCF wave functions are not quantitatively accurate; only note the significant difference in dynamical correlation energy along the reaction path (Figure 10).

Figure 10.

Energy profile of the Bergman cyclization at the UHF, (10e,10o) CASSCF and CASSCF+Average Coupled Pair Functional levels. Energies are in E_h, offset by +229 E_h (UHF), +229.1284534 E_h (CASSCF), and +229.9506090 E_h (ACPF). All calculations used the cc-pVTZ basis. See Figure 9.

Transition Metal Complexes.

The Hieber Anion, Fe(CO)₃NO⁻.

Tricarbonyl nitrosyl ferrate, commonly called the Hieber anion,^61,62 has been known for a long time,⁶³ but its significance as catalyst based on Earth-abundant elements was recognized relatively recently.⁶⁴ It is a strongly correlated system with a unique Fe–N bonding pattern:⁶⁵ a double π bond but no σ bond. Previous theoretical studies used a variety of active spaces: (4e,4o),⁶⁵ (l0e,8o) and (l6e,14o),⁷ and (l4e, 9o).⁶⁶ However, these active spaces were based mainly on chemical intuition, and show that intuition can lead to widely different results. Khedkar and Roemelt¹⁰ have determined an active space based on an objective criterion, orbital occupancies using their ASSIST technique.¹⁰ This is, however, a highly demanding method, both in terms of human and computer effort, and still requires the use of chemical insight initially. They start with an active space of the iron atom: 8 d electrons in 5 d orbitals. Perturbation theory indicates the need to include 8 more electrons and 8 orbitals, leading to a (16e, 13o) active space. A CAS calculation with this active space reveals that 3 orbitals are essentially doubly occupied; removing them leads to a final (10e, 10o) active space. By contract, the UNO criterion gives the (10e, 10o) space directly. The triplet instability matrix has 6 negative eigenvalues at the m6–31G* level: −0.0945, −0.0833, −0.0561, −0.0351, −0.0179, −0.0116. The m6–31G* basis⁶⁷ was used because the standard 6–31G basis and its derivatives are deficient in the d shell of first-row transition metals, particularly the late ones. Solving for the UHF solution algebraically in the space of the 4 largest negative eigenvalues⁴⁸ gives a (10e, 10o) active space. The UHF (m6–31G* and def2-TZVP), UNO-CAS, and CASSCF occupation numbers are listed in Table 3, and the unrestricted natural orbitals are shown in Figure 11. Some caution is still required because there are two very close-lying RHF solutions with all basis sets we have tried. The two RHF energies differ by only 4–5 mE_h, see the footnote to Table 3. However, the lowest UHF solution derived from both RHF wave function is the same. The geometry was optimized at the B3LYP/def2-tzvp level (without dispersion correction) and shows a somewhat better agreement with the X-ray data than the geometry of ref 65.

Table 3.

Charge Natural Orbital Occupation Numbers and Correlation Energies^a in the Hieber Anion, Fe(CO)₃NO⁻

natural orbital	UHF m6–31G*	UHF def2-TZVP	UNO-CAS m6–31G*	CASSCF m6–31G*
38	1.7499	1.7250	1.8956	1.9132
39	1.7047	1.6823	1.8855	1.9008
40	1.7047	1.6823	1.8855	1.9008
41	1.4060	1.3996	1.7767	1.8542
42	1.4060	1.3996	1.7767	1.8542
43	0.5940	0.6004	0.2233	0.1553
44	0.5940	0.6004	0.2233	0.1553
45	0.2953	0.3177	0.1143	0.0895
46	0.2953	0.3177	0.1143	0.0895
47	0.2501	0.2750	0.1048	0.0872
correlation energy/Eh	−0.148460	−0.156598	−0.348306	−0.380054

^aCorrelation energies are referenced to the lowest RHF energy with the same basis set, −1729.473904 E_h for the m6–31G* basis and −1729.847982 E_h for def2-TZVP. There is a qualitatively different RHF solution at a slightly higher energy and larger dipole moment (−1729.4690806 E_h and 1.8534 au at the m6–31G* level; −1729.843726 and 1.9223 au at the def2-TZVP level). The calculated dipole moments of the lower energy solutions are 1.4033 and 1.4198 au, respectively.

Figure 11.

Fractionally occupied UHF natural orbitals of the Hieber anion, [Fe(CO)₃NO]⁻ with occupation numbers (6–31G* basis; see Table 3 for the def2-TZVP basis). Note the two frontier orbitals with occupation number 1.406 form a double Fe=N π bond.

Ferrocene, (C₅H₅)₂Fe.

The ground state of ferrocene is frequently characterized as dominated by a single configuration.⁶⁸ However, the substantial fractional UHF NO and CAS NO (Table 4) occupation numbers, the fact that a Hartree–Fock calculation overestimates the ring-Fe distance by over 20 pm while closed-shell MP2 underestimates it by nearly the same amount, leave little doubt that the singlet ground state is quite strongly correlated. Like the Hieber anion, ferrocene also has two energetically close, qualitatively different RHF solutions.⁶⁹ The triplet instability matrix has two degenerate pairs of negative eigenvalues. With the def2-TZVP basis, these are −0.0576 and −0.0259 E_h. Natural orbital occupation numbers and UHF, UNO-CAS, and CASSCF correlation energies are listed in Table 4 for three different thresholds: 0.1, 0.02, and 0.01. These give 4-, 8-, or 12-orbital active spaces, respectively. Table 4 shows that only the 4 frontier orbitals in Figure 12 are strongly correlated. The correlation is essentially (d_xx-yy, d_xy,) → (d_xz,d_yz). The rest of the orbitals describe the correlation intermediate between dynamical and static. However, the occupation numbers of the frontier orbitals are significantly affected by expanding the active space, most likely because the diagonal elements of the Hamiltonian are shifted, as discussed in the Introduction.

Table 4.

Natural Charge Density Occupation Numbers and Correlation Energies in Ferrocene (D_5h)^a

Orb	UHF m6–31G*	UHF	UNO-CAS CASSCF (4e,4o)	UNO-CAS CASSCF (8e,8o)	UNO-CAS CASSCF (12e,12o)
43	1.9906	1.9892	2	2	1.9652
			2	2	1.9659
44	1.9878	1.9863	2	2	1.9640
			2	2	1.9628
45	1.9767	1.9726	2	1.8972	1.8753
			2	1.9386	1.9208
46	1.9732	1.9694	2	1.9140	1.8868
			2	1.9384	1.9162
47	1.6999	1.6743	1.9310	1.8848	1.8700
			1.9341	1.9167	1.9111
48	1.6999	1.6742	1.9310	1.8785	1.8623
			1.9341	1.9157	1.8981
49	0.3001	0.3258	0.0690	0.1260	0.1430
			0.0659	0.0792	0.0955
50	0.3001	0.3257	0.0690	0.1100	0.1269
			0.0659	0.0797	0.0910
51	0.0268	0.0306	0	0.0984	0.1266
			0	0.0664	0.0904
52	0.0233	0.0274	0	0.0910	0.1117
			0	0.0652	0.0868
53	0.0122	0.0137	0	0	0.0420
			0	0	0.0339
54	0.0094	0.0108	0	0	0.0262
			0	0	0.0275
correlation energy/Eh	−0.016818	0.020261	−0.016723	−0.120631	−0.168690
			−0.018692	−0.155574	−0.193235

^aAll results except those in column 2 use the def2-TZVP basis. Correlation energies are relative to the RHF energy (−1646.5976007 E_h for the m6–31G* basis;⁶⁷ −1646.9046652 Eh for the def2-TZVP basis). In columns 4–6, the upper number is the UNO-CAS result (full CI in the fractionally occupied UHF charge natural orbital space without orbital optimization), and the lower number is the CASSCF result (both orbitals and CI coefficients are optimized).

Figure 12.

Four frontier natural orbitals of ferrocene, with UHF occupation numbers (m6–31G* basis).

V. CONCLUSIONS

We have compared the UHF natural orbital criterion for selecting active spaces in strongly correlated systems with alternative criteria. Although our primary aim was ground-state molecules, we have discussed briefly the determination of active spaces for excited states. Methods which are aimed at selecting the frontier orbitals,² or orbitals which interact with a set of manually selected valence atomic orbitals,^7,9 are frequently successful. However, they do not consider the exchange matrix elements which are responsible for strong correlation and therefore may include orbitals in the active space which are essentially doubly occupied, as some of our examples show. Methods utilizing a conceptual atomic valence basis^7,9 do not give an approximate measure of the correlation strength and are therefore less helpful in calculations which do not use the complete active space, e.g., in Restricted Active Space. Methods which approximate full CI, such as Density Matrix Renormalization Group methods^4–6 or a combination of DMRG and perturbation theory,¹⁰ are reliable but expensive, both in terms of human effort and computer resources. In the cases we were able to check, the UHF natural orbital criterion has always determined the correct active space, like in a previous study,¹ in comparison with high-level calculations. In most strongly correlated cases, there is a single low-lying UHF solution, and each formally occupied orbital has only a single dominant correlation partner, giving an active space with n electrons in n orbitals. If there are degenerate or multiple low-lying UHF solutions, the natural orbitals of the average density should be used.^18,19 We would be very interested in finding counterexamples where the UHF NO criterion does not work. However, the relation between triplet instability and multiconfigurational character for a (2e,2o) model¹⁹ makes it unlikely that significant counterexamples exist. The main problem of multireference calculations is not finding the active space at a given molecular geometry but defining an active space for a reactive potential surface where the occupation numbers vary strongly with the geometry.