Analyzing cases of significant nondynamic correlation with DFT using the atomic populations of effectively localized electrons

Abstract

Quantum multireference effects are associated with degeneracies and near-degeneracies of the ground state and are critical to a variety of systems. Most approximate functionals of density functional theory (DFT) fail to properly describe such effects. A number of diagnostics have been proposed to estimate in advance the reliability of a given single-reference solution in this respect. Some of these diagnostics, however, lack size-consistency while remaining computationally expensive. In this work, we propose the DFT method of atomic populations of effectively localized electrons (APELE) as a novel diagnostic in this vein. It is compared with existing diagnostics of nondynamic correlation on select exemplary systems. The APELE method is on average in good agreement with the popular T₁ index, while being size-consistent and less costly. It becomes particularly informative in cases involving bond stretching or bond breaking. The APELE method is applied next to organic diradicals like the bis-acridine dimer and the p-quinodimethane molecule which possess unusually high nonlinear optical response, and to the reaction of ethylene addition to Ni dithiolene. Our results for this reaction are consistent with the T₁ diagnostics and in addition, shed some light on the degree of d-electron localization at the Ni center.

1 Introduction

In a typical theoretical study, one would start with some popular functional of density functional theory (DFT) or with a low level wavefunction-based correlation method and eventually would advance to the coupled cluster singles and doubles with perturbative triples (CCSD(T)) if the results are in doubt. These methods assume the dominance of a single-reference configuration (single-determinant- based methods). Single-determinant methods are unreliable for systems with strong multireference character in which more than one electronic configuration plays a significant role. The electron correlation due to multireference effects is called nondynamic correlation (NDC), and strong correlation when the NDC largely dominates. In multicenter systems, NDC tends to decouple a bond electron pair into two single unpaired electrons maximally separated from each other. This effect is also known as ‘left–right’ correlation [1,2,3]. The “simplest” example is the dissociating hydrogen molecule. At equilibrium, its two electrons with opposite spins are paired in a bond pair. Upon dissociation, the electrons become more and more “effectively unpaired” in the sense of virtual unpairing and localization on the H centers. This is due to the increased left–right correlation upon dissociation, which provides an energy gain by avoiding the two electrons to be close by. A proper description of this dissociation should take such a transformation into account without breaking the spin symmetry of the system.

Mainstream contemporary DFT functionals are known to poorly treat nondynamic/strong correlation, albeit some recent developments have shown great promise [4,5,6,7,8,9]. Instead, wavefunction-based methods like CASSCF (complete active space self-consistent field), MRCI (multireference configuration interaction), and their variations are usually employed. These methods are computationally expensive and can only be afforded to problems of relatively small size. It is desirable to be able to estimate a priori, the degree of multireference character of a given state, in order to estimate the reliability of a given single-determinant method. A few single-reference-based descriptors (diagnostics) have been proposed for this purpose. The most commonly used are based on CCSD(T), such as the T₁ [10], D₁ [11], and %TAE[12] indexes. These diagnostics estimate the strength of NDC in a system averaged way, averaging it per electron or per bond. However, NDC is often localized in specific subdomains of the system, like a transition-metal center in a complex or solid for example. In such cases, the NDC should remain approximately about the same upon increasing the system size by adding larger ligands for example, provided the latter does not change drastically the metal–ligand bonding pattern. Averaging the NDC over all electrons and/or bonds is not a size-consistent approach since some of these may not be directly involved in the multireference process. This leads to an artificial dilution of the NDC estimate.

Alternative methods of treating such effects in a local fashion have been developed, starting with the effectively unpaired (‘localized’, or ‘odd’) electron models of Takatsuka, Fueno and Yamaguchi [13], Lain et al. [14], Staroverov and Davidson [15], and Head-Gordon [16]. Upon stretching a bond, the bond electrons become partially decoupled due to the increased left–right NDC and tend to virtually localize as unpaired electrons on separate atoms. These electrons are referred to as ‘effectively unpaired,’ or ‘effectively localized’ electrons here. One example of effectively localized electrons are the electrons with radical character in diradicals. The above methods include, in principle, both nondynamic and dynamic correlation, while in fact localized radical density emerges mainly due to NDC. Recently, Ramos-Cordoba, Salvador and Matito [17] emphasized the importance of separating the electron correlation into dynamic and nondynamic parts and derived expressions of these, based on a cumulant expansion of the second-order reduced density matrix (RDM-2). These methods require correlated first-order reduced density matrix (RDM-1) which is not directly accessible from the single-determinant Kohn–Sham (KS) or Hartree–Fock (HF) solutions. More recently Liu, Duan and Kulik [18] proposed a local index of NDC based on a finite-temperature DFT formulation by Chai [19] that provides fractionally occupied orbitals and a non-idempotent RDM-1 beyond the standard KS method. Their approach is built on the earlier works of Matito and co’s [17, 20], and Grimme and Hansen’s [21], and is essentially similar to the ‘odd electrons’ approach, since both use the degree of non-idempotency of RDM-1 evaluated one way or another.

As an alternative, Nakano et al. [22] have proposed an index of diradical character (y) based on the HOMO–LUMO gap in natural-orbital representation of the spin-projected unrestricted self-consistent field (SCF) solution.

We have developed previously a DFT method of describing the formation of radical-like density [23]. It leads to a compact expression of the atomic population of effectively localized electrons (APELE) within a single-determinant KS DFT approach. The method retains the computational efficiency of a single-determinant KS solution while incorporating explicitly NDC in the evaluation of APELE. It does not involve fractional orbital occupancies, unlike the methods of refs. [18, 21], and is simpler to implement within the standard KS framework. Also, the calculations do not require spin-symmetry breaking, thus avoiding the side effects of spin contamination. By design, the APELE values reflect the strength of NDC locally which leads to size-consistent comparisons among systems of different size. A typical example is a singlet diradical, where two electrons with opposite spin, that formally would be paired in a bond pair, are in fact localized, each on a separate center. Larger APELE of the radical atoms reflect stronger NDC acting locally among the radical electrons. It requires two-reference configurations or more in a correlated wavefunction treatment of such a situation without spin-symmetry breaking.

Indeed, strong electron correlation manifests most notably in singlet diradical molecules. Much attention has been devoted recently to a particular class of organic diradicaloids that have unusually high nonlinear optical (NLO) response [24]. Molecular design of efficient NLO systems is a subject of intense research due to their importance for photonics and optoelectronics such as optical switches, 3D microfabrics, photodynamic therapy and others. Recent studies have shown that the degree of diradicaloid character in systems like diphenalenyls, π-electron systems involving imidazole rings and in other compounds that involve quinoid and benzenoid resonance structures like p-quinodimethane (PQM), is a key factor for the large second-order hyperpolarizability (γ) of these systems [22, 25,26,27,28]. Similar features have been found also in some graphene nanoflakes [29, 30].

2 The APELE method within the Kohn–Sham DFT

The formation of open-shell singlet systems is driven by strong left–right NDC that tends to partly decouple a covalent bond electron pair and form radical-like effectively localized electrons. This formation has been related to the non-idempotency of the correlated RDM-1 [13, 31]. At single-determinant SCF level (HF or KS), RDM-1 remains idempotent (integer orbital occupation numbers of 1 or 0) and APELE formation is not feasible at that level. We have proposed a way to remedy this situation without using fractional orbital occupancies [23]. The (non) idempotency condition of the RDM-1 $\gamma ({\mathbf{r}};{\mathbf{r}}^\prime)$ reads:

$$ D_{\sigma } ({\mathbf{r}}) = \,\rho_{\sigma } ({\mathbf{r}})\, - \,\int {\gamma_{\sigma } ({\mathbf{r}};{\mathbf{r}}^\prime)} \,\gamma_{\sigma } ({\mathbf{r}}^\prime;{\mathbf{r}}){\text{d}}{\mathbf{r}}^\prime\,\, \ge \,0 $$

(1)

where $\rho_{\sigma }$ is the electron density of spin σ, D_σ is the APELE density per spin σ. The total density of effectively localized electrons D_u and the total number of these electrons,$\overline{N}_{u}$, are then given by:

$$ D_{u} ({\mathbf{r}})\, = \,2\,\sum\limits_{\sigma } {D_{\sigma } ({\mathbf{r}})};\,\,\,\,\,\,\,\,\,\,\,\,\overline{N}_{u} \, = \,\int {D_{u} ({\mathbf{r}})\,{\text{d}}{\mathbf{r}}} $$

(2)

At the single-determinant level (KS or HF), RDM-1 is idempotent meaning $D_{\sigma } \equiv 0$ everywhere as a limiting case of Eq. (1). We generalize the integrand of Eq. (1) by introducing a model effective exchange-hole that is otherwise compatible with the single-determinant KS scheme but having a specific form and relaxed normalization [23]:

$$ \gamma_{\sigma } ({\mathbf{r}};{\mathbf{r}}^\prime)\gamma_{\sigma } ({\mathbf{r}}^\prime;{\mathbf{r}}) \approx \,\rho_{\sigma } ({\mathbf{r}})\,\overline{h}_{X\sigma }^{eff} ({\mathbf{r}},{\mathbf{r}}^\prime)\,\, $$

(3)

such that:

$$ \int {\overline{h}_{X\sigma }^{eff} ({\mathbf{r}},{\mathbf{r}}^\prime){\text{d}}{\mathbf{r}}^\prime\,\, \le \,1\,\,} $$

(4)

The existence of such a form was justified using an approximate cumulant expansion of the two-body density matrix [23]. In the APELE method, the effective exchange hole in Eq. (3) is represented by the relaxed exchange hole of Becke–Roussel [32] introduced in the B05 functional formalism [3, 33], $h_{X\sigma }^{B05}$, which estimates the NDC locally by its relaxed normalization $\overline{N}_{X\sigma }^{{e{\text{ff}}}}$ (similar to Eq. (4)):

$$ \int {h_{X\sigma }^{B05} ({\mathbf{r}},{\mathbf{r}}^\prime)\,{\text{d}}{\mathbf{r}}^\prime\, = \,\overline{N}_{X\sigma }^{{e{\text{ff}}}} } ({\mathbf{r}})\, \le \,1 $$

(5)

The smaller the relaxed normalization $\overline{N}_{X\sigma }^{{e{\text{ff}}}}$, the stronger the NDC locally at a given reference point r. Using this relaxed exchange hole normalization in Eqs. (3),(1), the density of effectively localized electrons takes the form (see Appendix of ref. [23] for details):

$$ D_{u} ({\mathbf{r}})\, = \,2\,\sum\limits_{\sigma } {\rho_{\sigma } ({\mathbf{r}})} \,\left( {1 - \overline{N}_{X\sigma }^{{e{\text{ff}}}} ({\mathbf{r}})} \right) $$

(6)

The smaller the relaxed normalization $\overline{N}_{X\sigma }^{{e{\text{ff}}}}$, the larger the density of effectively localized electrons.

To obtain the APELE of an individual atom A (denoted here as $F_{r} (A)$), we partition the molecular space into atomic regions (Ω_A) using the grid-based atomic partitioning of Becke’s grid integration scheme [34].

$$ F_{r} (A) = \int\limits_{{\Omega_{A} }} {D_{u} ({\mathbf{r}})} \,{\text{d}}{\mathbf{r}},\,\,\,\sum\limits_{A} {F_{r} (A)\, = \,\overline{N}_{u} } $$

(7)

The values of the individual APELE, F_r(A), and the sum of all APELE (total APELE $\overline{N}_{u}$), may be used as NDC diagnostics measuring the strength of NDC locally (F_r(A)) and globally ($\overline{N}_{u}$) per Eq. (7). The APELE density, Eq. (6), can be used to generate 2D and 3D plots illustrating the NDC distribution locally in the molecular space, similar to the plots presented in refs.[20, 21]. The APELE analysis was applied previously to elucidate more fully the bonding nature of Cr₂, C₂ and (NO)₂ dimer [23]. It was found that the bond in C₂ is unlikely to have diradical character in its ground state, while the singlet ground state of the (NO)₂ dimer does have a diradical character. The APELE method correctly estimated the magnetic moment localized on the Cr atoms in Cr₂ and predicted that the bond in Cr₂ may have a quad-radical nature [23]. It is worth reminding that the total APELE $\overline{N}_{u}$ is a quantum-statistical averaged number [14], in contrast to the total number of electrons in the system which is considered fixed here.

3 Computational details

The APELE-based NDC indexes were calculated with the KP16/B13 functional [5] which take into account NDC via real-space corrections to the exchange hole. This allows open-shell singlets to be described with restricted single-determinant KS (RKS) solution even at large bond stretches. This way we avoid the painful symmetry breaking procedure that is not always feasible.

Psi4, an open-source quantum chemistry software package [35], was used to optimize the structures at B3LYP/cc-pVQZ level using very tight convergence criteria. Psi4 was also used to evaluate the CCSD(T) diagnostics T₁, D₁, and %TAE[(T)], as well as, the A_λ diagnostic for M06, and M06-HF [35]. Instances in which the Psi4 package failed to deliver accurate results were circumvented using the Gaussian’09 software package [36]. The APELE results were obtained with the KP16/B13 functional using cc-pVTZ and G3LARGE basis sets [37] using the xTron code, an in-house computational program that allows the efficient computation of the HF exchange energy density [38,39,40]. It allows users to run various datasets with single commands in a cluster environment.

4 Diagnostics of nondynamic correlation

Wavefunction-based methods that include both dynamic and nondynamic correlation are expensive and quickly out-scale the current computational resources. This motivated the need for diagnostic tools to predetermine whether NDC must be added to a given single-reference solution. There have been a number of techniques created for this purpose. They have been quite useful and have received wide popularity but often involve computationally complex methods or require the help of other techniques. Here we give a gist of the more commonly used diagnostics that will next be compared with the proposed APELE-based diagnostics.

4.1 Summary of current diagnostics

4.1.1 Methods based on unrestricted SCF

Nakano et al. [22] have proposed an index of diradical character (y) based on the HOMO–LUMO gap in natural-orbital representation of the spin-projected unrestricted SCF (HF or Kohn–Sham) solution.

$$ y = 1 - \frac{2T}{{1 + T^{2} }}\,\,\,, $$

(8)

where T is the orbital overlap between the HOMO and LUMO natural orbitals. This analysis relies heavily on spin symmetry breaking, which brings large spin contamination. Moreover, a small HOMO–LUMO gap does not necessarily correspond to a large NDC, e.g., in a metallic system. Another problem is that systems with significant NDC do not always possess a spin-symmetry broken stable SCF solution with energy lowering.

4.1.2 CI based

The configuration interaction (CI) expansion of the wave function contains the expansion coefficients for each determinantal configuration involved. A system with a large amount of NDC would have a relatively small contribution to the total CI wave function from the main ground-state determinant. This diagnostic is represented by the leading coefficient (C₀) in the CI wave function expansion. It is quite informative, but it has an inherent bias tending to give higher C₀ values when the method uses single-determinant HF orbitals [41]. Complete Active Space SCF (CASSCF) calculations are used to remedy these shortcomings. It is generally accepted that a system with a C₀ < 0.95 or C₀² < 0.90 has multireference character and a multireference method should be used.

4.1.3 Coupled cluster based

The coupled cluster method employs a priori measures based on the $\vec{t}_{1}$ amplitudes and/or uses %TAE[(T)] (atomization energy due to triple excitations) as indicators of near-degeneracy. The Frobenius norm of the $\vec{t}_{1}$ amplitudes taken as a base and divided by the square root of the number of correlated electrons define the T₁ diagnostic [10]:

$$ T_{1} = \frac{{\left\| {\vec{t}_{1} } \right\|}}{\sqrt n } $$

(9)

The ‘number of correlated electrons’ n here implies only the valence electrons but not the core electrons. A few thresholds for T₁ have been suggested, which are largely based on the types of systems studied [10, 41] It is generally accepted that T₁ > 0.02 for organic molecules, T₁ > 0.05 for 3d transition metals, and T₁ > 0.045 for 4d transition metals indicates large NDC [41, 42].

Another diagnostic rooted in the coupled cluster approach is %TAE[(T)]. It uses the atomization energy from CCSD(T) compared with that from CCSD. This gives the percentage of atomization energy coming from the triple excitations alone:

$$ \% {\text{TAE}}[({\text{T}})] = \frac{{{\text{TAE}}({\text{CCSD(T)}}) - {\text{TAE}}({\text{CCSD}})}}{{{\text{TAE}}({\text{CCSD}}({\text{T}}))}}\,100 $$

(10)

The %TAE[(T)] index is an energy-based diagnostic. It has been shown to correlate well with the more comprehensive estimates based on higher levels of the theory, such as %TAE[(T₄ + T₅)] [41, 43]. %TAE[(T)] indicates mild, moderate, or strong NDC at less than 5%, greater than 5% but less than 10%, and greater than 10%, respectively [41]. This classification remains more or less consistent across the different types of systems.

4.1.4 DFT-based diagnostics

There are only a handful of methods developed for gauging NDC within DFT. The B₁ method developed by Schultz, Zhao and Truhlar has been shown to correlate well with the T₁ diagnostic for systems such as BeO and MgO [44]. Another diagnostic rooted in DFT that displays similar qualities is the A_λ diagnostic [43]. It uses the ratio of the atomization energy (AE) from a hybrid functional vs the atomization energy from 100% of HF exchange alone:

$$ A_{\lambda } = \frac{{\left( {1 - \frac{{{\text{AE}}[(X_{\lambda } C)]}}{{{\text{AE}}[(X_{100} )]}}} \right)}}{\lambda }, $$

(11)

where X and C denote here exchange and correlation components, respectively. The fraction of model DFT exchange (100 − λ) in the hybrid functional acts to some extent as a partial substitute for the left–right correlation. The fraction of HF exchange λ in the denominator of Eq. (11) normalizes this diagnostic.

For A_λ < 0.10, the system is primarily dominated by dynamic correlation, A_λ ≈ 0.15 indicates that it is mildly ND correlated, A_λ ≈ 0.30 means moderately ND correlated and for A_λ > 0.50 the system should be strongly ND correlated. It has been shown that A_λ agrees relatively well with %TAE[(T₄ + T₅)] when certain specific functionals are used. A problem with the A_λ diagnostics is that the results sensibly depend on the approximate functionals used, and also that it is a system averaged index prone to size inconsistency.

Recently a DFT-based local indexes of NDC [21] has been proposed based on a finite-temperature DFT [19]. The main ingredients of this method are the electronic temperature $T_{{{\text{el}}}}$ and the fractional occupations $f_{i}$ of the temperature-dependent one-electron states governed by the Fermi–Dirac statistical distribution:

$$ f_{i} = \frac{1}{{e^{{(\varepsilon_{i} - E_{F} )/kT_{{{\text{el}}}} }} \, + 1}}\,\,,\,\,\,\,\,0 \le \,f_{i} \, \le \,1 $$

(12)

Using this formalism, they proposed the so-called fractional orbital density (FOD) as a local index of NDC:

$$ \rho^{{{\text{FOD}}}} ({\mathbf{r}})\, = \,\sum\limits_{i}^{N} {(\delta_{1} } - \delta_{2} \,f_{i} )\left| {\varphi_{i} ({\mathbf{r}})} \right|^{2} , $$

(13)

where $\delta_{1} ,\,\delta_{2}$ are constants chosen so that only the fractionally occupied orbitals are included in $\rho^{{{\text{FOD}}}} ({\mathbf{r}})\,.$ Although physically appealing, there seems no quite clear connection between this FOD density of ‘hot’ electrons and the strength of the NDC as defined within the zero-temperature quantum mechanics.

Another local index was more recently introduced and explored in refs. [17, 20] based on natural orbitals cumulant expansion of RDM-2 and its decomposition of dynamic and nondynamic correlation parts. The correlation part was written in terms of (spin-dependent) fractional occupation numbers (FON) $n_{i}^{\sigma }$ of the natural spin orbitals $\phi_{i}^{\sigma }$:

$$ I_{{{\text{ND}}}} ({\mathbf{r}}) = \frac{1}{2}\sum\limits_{i,\sigma } {n_{i}^{\sigma } (1 - n_{i}^{\sigma } )\left| {\phi_{i}^{\sigma } } \right|}^{2} $$

(14)

Equation (14) is in fact equivalent to the natural orbital representation of the ‘effectively unpaired’ (‘odd’) electron density defined via the non-idempotency of RDM-1 [13, 16]. Both expressions are a different way of representing the APELE density given by the real-space Eq. (6) [23] (see paragraph 2 and Sect. 4.2).

Very recently Liu, Duan and Kulik [18] combined Eq. (14) with the finite-temperature DFT expression (12) to obtain more efficient and easier to use DFT variant of the FON method. None of these latter methods introduce any quantitative atomic population analysis based on the FOD or FON local functions.

4.2 Diagnostic based on the atomic population of effectively localized electrons (APELE)

A brief gist of the APELE method is given in Sect. 2. Summing the APELE of all the atoms of the system gives the total APELE index $\overline{N}_{u}$, Eq. (7). Given the specific setup of our model, $\overline{N}_{u}$ reflects the global strength of NDC in the system and can be used as global NDC diagnostic complementing the APELE density distribution $D_{u}$, Eq. (6), and the individual APELE $F_{r} (A)$, Eq. (7). Indexing the NDC based on the APELE analysis ($F_{r} (A)$ and $\overline{N}_{u}$) has some distinct advantages. Above all, $F_{r} (A)$ is a local characteristic and does not include averaging over the whole system like in the case of T₁ or %TAE. Another advantage is the computational efficiency since our method is based on single-determinant KS-DFT and does not require fractional orbital occupancies. Furthermore, the method directly accounts for the NDC by design, so that the APELE values directly reflect the strength of NDC locally ($F_{r} (A)$) or globally ($\overline{N}_{u}$), and do not include contributions from dynamic correlation.

4.2.1 Linear alkane chains with a stretched central C–C bond distance

A good example illustrating the size-consistency issue is a set of linear alkanes with a stretched carbon–carbon bond in the middle [21]. The APELE results for ethane (C2H6), butane, (C4H10), hexane (C6H14) and octane (C8H18) are presented in Fig. 1 and Table 1.

Fig. 1

APELE of the central C1 atom in the alkanes as a function of the C1-C2 distance

Table 1 Left–right correlation diagnostics. The APELE values are per C atom of the central C1-C2 bond

Table 1 also contains the values of several other diagnostics calculated as explained in the computational details. The molecular geometries are first optimized with B3LYP/ cc-pVQZ. Then, the middle C–C bond (‘C1–C2’) is stretched out, starting from equilibrium (which for these systems is at around 1.5Å), up to 3.5Å. The bond stretching induces strong left–right ND correlation mainly localized at the C1–C2 bond of each molecule. Figure 1 shows the change of APELE on one of the two central carbon atoms in each alkane with increasing the C1-C2 distance. The observed trend is consistent with one’s physical intuition that the NDC should increase when the bond is progressively stretched. This trend holds steady, and the values only slightly vary from ethane to octane, showing that the APELE index is size-consistent and reflects the increase in left–right correlation mainly at the C1–C2 bond. The coupled cluster-based diagnostics show the proper trend as the C1-C2 bond distance increases in each of the molecules. However, as we move from smaller to larger alkanes, the T₁, and %TAE[(T)] values tend to decline when comparing them at the same C1–C2 distance. As the size of the molecule increases, the values of these diagnostics become more diluted, which is a sign of size-inconsistency. A similar trend with the coupled cluster-based diagnostics on other occasions was reported in the literature as well [45].

Regarding the A_λ method, the M06, and M06-HF functionals were used for the evaluation of this index. These functionals include a fraction of 27% and 100% Hartree–Fock exchange, respectively. The obtained A_λ values all fall below the threshold suggested by the authors of ref. [43], and misleadingly would indicate that no significant NDC is present in these molecules at any bond stretch. However, the A_λ index does display qualitatively a proper trend with respect to the bond stretching. Similar to the coupled cluster-based diagnostics, the A_λ diagnostic is not size-consistent and reflects the situation is a system averaged way (see Eq. (11)).

To compare more fully the different diagnostics, a statistical correlation analysis was done on the results for the considered four alkanes. Table 2 contains the statistical correlation between the methods with respect to the trend of NDC in these alkanes. A relatively good agreement between the diagnostics is observed when the molecules are at their equilibrium geometry (small NDC, Table 2a), with an exception of A_λ[M06-HF]. The poor correlation of the latter with the rest of the diagnostics at equilibrium deserves more attention. Using 100% HF exchange in the M06-HF hybrid functional in the context of its multiparameter optimization leaves the A_λ[M06-HF] estimates without any simulation of NDC due to the lack of a model DFT exchange in the mixture. This is the main reason the A_λ[M06-HF] would correlate poorly with the rest of the diagnostics.

Table 2 Correlation (in %) between the diagnostics at the equilibrium and at stretched C1-C2 bond length across all four alkanes

While most of the methods perform well at equilibrium here, the statistical analysis at C1-C2 = 3.5^˚A, Table 2b, shows a lack of size-consistency by all the methods but APELE, with high negative correlations with respect to the correct trend.

We have also calculated the total APELE index $\overline{N}_{u}$ for a few selected molecules from the W4-11 database [12] (Table 3). It is interesting that the two global indexes $\overline{N}_{u}$ and %TAE[(T)] agree well qualitatively when comparing molecules with the same stoichiometric composition. Table 3 lists four pairs of such molecules. For each pair, the total APELE index $\overline{N}_{u}$ is larger whenever %TAE is larger.

Table 3 Calculated total APELE $\overline{N}_{u}$ vs the %TAE[(T)] index of selected small molecules

To keep in mind that in most singlet cases the $\overline{N}_{u}$ values have an upper limit of 2 here (diradical molecules), and that values of $\overline{N}_{u}$ below 0.15 are typical for singlet single-bonded molecules like H₂ at equilibrium.

4.2.2 Strength of NDC in acetylene vs vinylidene.

To extend the comparison among the different diagnostics, we consider next a pair of interesting molecules presented in Fig. 2 [46]. They are rather different in their properties but have formally the same stoichiometry. This allows a comparison of the relative strength of NDC in both structures using the individual APELE values as well as the total APELE index $\overline{N}_{u}$. For instance, the carbon atoms in acetylene have relatively stronger NDC than either of the carbons in vinylidene, while APELE of the hydrogen atoms vary only slightly. The total APELE index $\overline{N}_{u}$ for acetylene and vinylidene is 0.800 and 0.690, respectively, while the %TAE[(T)] values are 2.1 and 1.9. This confirms that $\overline{N}_{u}$ for relatively small systems with the same stoichiometry correlates well qualitatively with the wave-function-based diagnostics.

Fig. 2

Comparison of APELE in acetylene (left) vs vinylidene molecules

5 Other exemplary applications of the APELE method

5.1 Study of bis-acridine dimers

An interesting case study are the bis-acridine dimers considered by Nakano et al.[25]. The diradical character of these compounds has been verified and is analyzed here by applying the diagnostics to the Model 8 and Model 12 compounds considered in ref. [25]. All calculations are done at B3LYP/6-31G* optimized geometries. The only difference between the two compounds is that Model 8 has a positive charge of 2 while Model 12 is neutral [25]. Both molecules have a singlet ground state. Model 8 has been shown to have strong diradical character and high second-hyperpolarizability corresponding to high intensity of the two-proton absorption. Figure 3 shows our slightly simplified structure used for Model 8 (and similarly for Model 12) in our calculations. The obtained total APELE values $\overline{N}_{u}$ are 12.834 and 12.281 for Model 8 and Model 12, respectively, indicating that Model 8 may involve larger NDC than Model 12. However, in these systems size-consistency matters since the radical density formation is localized about certain sub-regions of these molecules. The spin-density distribution calculated in ref. [25] for Model 8 with spin-symmetry breaking is mainly localized on the two N atoms and their nearest neighbors, all carrying the bulk of the positive charge and radical charge altogether. Our APELE results are in good agreement with the findings of Nakano et al., even though we do not use spin-symmetry breaking. Table 4 contains the APELE of the atoms that are carrying most of the diradical formation: (HN1 + C2 + C3) on one side (the N atom with its nearest neighbors, see Fig. 3) and their symmetric counterparts (HN6 + C7 + C8) on the other side, each group carrying a total of 1.035 unpaired radical charge. We also find some noticeable accumulation of APELE around each of the two oxygen atoms (O4 + C5 = 0.76 = O9 + C10) (Table 4), each oxygen being situated on the symmetrically opposite side of the N atoms. This indicates a possibility for tetra-radical formation in this compound or at least that NDC has an increased significance in the vicinity of these oxygens too. Once more, the ability of probing the strength of NDC locally/regionally within a given system brings some more detailed insight.

Fig. 3

A slightly simplified structure of Model 8 used in the calculations

Table 4 Calculated APELE values of the atomic groups in Model 8 carrying most of the radical density

5.2 Study of the model system PQM

Much attention has been devoted to a particular class of organic diradicals with an unusually high nonlinear optical (NLO) response [22, 24, 27]. Recent studies have shown that the degree of diradical character in systems like p-quinodimethane (PQM) is related to the large second-order hyperpolarizability (γ) observed in these systems [22, 25,26,27,28].

The PQM (Fig. 4) system was studied first by Nakano and collaborators using their y-factor index (see Eq. (8)) in search for a connection between the degree of diradical character and the second hyperpolarizability [22]. Our APELE analysis allows to explore from somewhat different angle the strongly correlated nature of these NLO active diradicaloids. The p-quinodimethane (PQM) is an interesting example itself of a normal singlet state at equilibrium that transforms into an open-shell singlet upon stretching of certain key bonds. First we optimized the geometry of the PQM with B3LYP following the recommendation of ref.[22]. The lowest-energy geometry obtained with B3LYP has a typical alternating C–C bond pattern in the ring, similar to the one reported in ref.[22]. Bond length alternation in carbon ring systems is a known indication of π-electrons localization on the individual C–C bonds. This localization is sustained by strong ND left–right correlation. Similar effects have been observed in other polycarbonic systems [47], and in periodic systems [48]. In PQM, this effect is accompanied with loss of aromaticity, which is observed also in other carbon ring systems with alternating C–C bond lengths [47]. Our results presented in Table 5 show that the PQM diradical charge estimated by APELE is partly delocalized around the C manifold, with an emphasis on the two terminal C-CH₂ group of atoms. It gradually increases with the symmetric stretch of the C–CH₂ bond distance on both sides of the ring. More pronounced is the change of the group-APELE Fr(C–CH₂), which shows that the increased NDC upon bond stretching is mainly confined locally within these group of atoms where the bond stretching occurs. The global NDC index $\overline{N}_{u}$ is relatively less sensitive to the stretching here, which shows again the importance of examining the effect of NDC locally.

Fig. 4

The PQM structure optimized with B3LYP using G3LARGE basis set and (128,302) unpruned grid. The APELE values are also given

Table 5 APELE of individual atoms and groups in the PQM diradicaloid

Considering the diradical formation upon bond-stretching, a few DFT indexes have been proposed based on UKS calculations with broken spin-symmetry [22, 27, 28]. The y index was mentioned already (Eq. (8)). A difficulty with this approach is that not always the UKS or UHF calculations lead to a lower energy state with broken symmetry and localized radical spins. For example, we performed a stability analysis of the RKS vs UKS state of the PQM using RB3LYP and UB3LYP, respectively. At the equilibrium PQM geometry, there is no RKS-UKS instability of any kind, and the ground-state energy remains RKS. Our APELE analysis on the other hand shows that there is a good amount of NDC as evidenced by the $\overline{N}_{u}$ value of 2.78. In this case, one can hardly claim a diradical formation based on the UKS treatment alone, even though the y parameter value is also nonzero (Table 5) True RKS-UKS instability appears only when both C-CH₂ bond lengths are stretched to about 1.6 Å or so and longer.

Another group-APELE index in the case of PQM that shows a relative concentration of radical charge on the terminal C-CH₂ groups is Q_r = 2(F_r(CCH₂)/$\overline{N}_{u}$_, where F_r(CCH₂) is the group-APELE in the union of these four atomic regions as mentioned above, Fig. 4. These are the atoms of the C-CH₂ fragment where most of the APELE charge is located. At equilibrium geometry, our estimate of Q_r here is about 0.514. It gradually increases to about 0.633 at stretched C-CH₂ distance of 2.0 Å. It is worth noting that the total number of radicalized electrons $\overline{N}_{u}$ is larger than 2 even at equilibrium. This is because there is some APELE located also on other C atoms from the ring, besides the diradical formation (Fig. 4). This is mainly due to the above-mentioned localization of the ring π electrons. Those findings are not accessible from the values of the global y index alone presented in Table 5.

5.3 Study of the ethylene addition to Ni dithiolene reactions

Metal bis(dithiolene) complexes have been widely studied because of their unusual chemical, redox, and optical properties related to the specific nature of the dithiolene ligand [49, 50, 51, 52]. Their complex character makes it difficult to assign unambiguously the oxidation state of the metal center, as pointed out in the studies of Dang et al.[53] on the transition states and intermediates of the ethylene addition to Ni(edt)₂ (edt = S₂C₂H₂). A wide variation of calculated relative reaction and transition-state energies using different functionals has been reported, which was attributed mainly to the peculiar multireference character of the Ni-ligand bonding in these systems. The latter was confirmed both experimentally and by the reported T₁ diagnostic of the different reaction species and CASSCF relative energies [53].

We apply here the APELE analysis to estimate the degree of radical character of these reaction species in the notations of ref [53]. (see Figs. 5,6). Large APELE values also mean that the corresponding atoms are less involved in normal covalent bonding while they may remain highly reactive due to their unpaired radical-like electrons. The comparison between the $\overline{N}_{u}$ and T₁ diagnostics in this case shows a relatively good correlation between the two (Fig. 5).

Fig. 5

Correlation between $\overline{N}_{u}$ and T₁ for the different reaction species

Fig. 6

Reaction species in the ethylene addition to Ni dithiolene reactions per ref. [53]

The variations among different reaction species can be further analyzed based on the total APELE value $\overline{N}_{u}$ and the individual or group APELE values which are not available from the T₁ analysis alone. The results from the APELE analysis on some of the reaction species (the initial reactant 1H and the product 2H, Fig. 6) are summarized in Table 6. We observe a large decrease of APELE on the two upper S atoms (S1, S3, Fig. 6) when going from 1H to 2H reaction species. This is due to their bonding with the carbon atoms of ethylene in the 2H product. These new S–C bonds cause in turn some decrease of APELE on the Ni center, which means that a fraction of APELE on Ni in 1H is now engaged in enforcing the Ni-S bonds in 2H. Hence, we observe some decrease of the radical-like density on the Ni center when going from the initial 1H structure to the 2H product.

Table 6 APELE values of the atoms in the 1H and 2H reaction species in the reaction of ethylene addition to Ni(edt)₂

More detailed studies of the electronic aspects of this reaction are in order, particular concerning the evolution of the oxidation state of the Ni center along the reaction path. A lot has already been done along these lines (see ref [53]. and references therein). Testing our APELE-based indexes of NDC in this case shows an encouraging agreement with the T₁ diagnostic and yields some additional information not directly assessable by the T₁ diagnostic alone (Table 6).

5.4 Study of multireference character of transition-metal compounds

A variety of materials contain transition metal (TM) elements as essential ingredients. Those compounds often have moderate-to-strong multireference character mainly due to the partly filled inner d-shells of the TM centers. The APELE method could be advantageous in studies of such compounds. We are currently exploring this possibility using the ccCA-TM-11 (correlation-consistent composite approximation for transition-metals 2011) database [54]. The multireference character of these complexes has been examined already by the same authors using the T₁, %TAE, and a few other diagnostics based on CASSCF [41]. It is still a work in progress from our side. Here we present just a small exemplary case of transition-metal-carbonyl complexes that have interesting bonding features. Table 7 lists the calculated APELE values of Ni and Cr in small series of their carbonyls. All these molecules are singlets (low-spin complexes). The metal–ligand bonding here is different from the bonding in organic diradicals and acquires a multicenter character upon increasing the number of ligands. The bonding of a given CO ligand to the transition metal center is done via the CO carbon atom. The lone electron pair of the ligand carbon atom induces a donation of electron density to the empty metal d-orbitals, forming a dative bond. The donated electron density enhances the energy of the metal d-electrons due to increased electron–electron repulsion. This in turn facilitates a π-acceptor bonding interaction between the metal d-electrons and the empty π* orbitals of the CO ligand. Such a two-process picture of the metal–ligand bonding shows a possibility of significant left–right correlation due to the rather localized nature of the metal d-electrons. The APLELE of the TM atoms reflects this localization and hence the left–right correlation between the metal and the ligand electrons.

Table 7 Calculated APELE values on Ni and Cr in a series of carbonyl complexes. The T₁ and %TAE values are from ref.[41]

The %TAE and T₁ values tend to decrease when more ligands are added to the Ni complex, which is in part due to the size inconsistency issue with these diagnostics. The APELE values on the other hand remain almost constant (a slight increase in the second decimals) for the Ni series. A decreasing pattern of the CC-based indexes is also observed in the Cr-based complexes when number of the ligands increases. The APELE values again remain almost unchanged in his case too, with a very slight decrease in the second decimals upon increasing the number of ligands.

Both APELE and T₁ show an increased multireference character going from Ni(CO)₃ to Cr(CO)₃ complex. This is mainly due to the difference in the metal electronic configuration in these two complexes, particularly in regard to the partly filled inner 3d shell (${\text{Ni}}(3d^{8} )$ vs. ${\text{Cr}}(3d^{5} )$) which results in different strength and bond order of the metal–ligand bonding. APELE maintains essentially the same difference between the larger Ni(CO)₄ and Cr(CO)₄ complexes while the difference in T₁ is smaller there, perhaps due to the increase of molecular size.

The distinction between these two types of complexes with the same number of ligands is less clear from the %TAE values.

6 Conclusions

Nondynamic correlation plays an important role in many cases, but it is expensive and hard to take into account. Most approximate DFT functionals fail to describe it properly. A number of diagnostics have been proposed that allow to estimate the quality of a given single-reference solution in this respect. Most of these diagnostics are, however, not size-consistent, while remaining computationally expensive. In this work, we use our DFT method of atomic population of effectively localized electrons (APELE) and the total number of effectively localized electrons [23] as novel diagnostics in this vein. It retains the simplicity of the single-determinant Kohn–Sham DFT in contrast to some more recent methods based on explicit fractional orbital occupancy [18, 21]. The APELE analysis is generally in good agreement with other currently used diagnostics like T₁ and %TAE, while having the advantage of being size-consistent, less costly and does not involve unrestricted spin-symmetry breaking. It becomes particularly informative in cases involving bond stretching or bond breaking as in the series of linear alkane chains. The APELE method is used also to explore some specifics of organic diradicaloids like the bis-acridine dimer and the p-quinodimethane having unusually high nonlinear optical response. The calculated APELE values point out on nonnegligible nondynamic correlation in these systems that is localized within certain specific subdomains of the molecules. We tested the method also on the complicated reaction of ethylene addition to Ni dithiolene that was already studied before with the T₁ diagnostic. The results with APELE are consistent with that of T₁, with additional atomic detail on how the degree of d-electron localization on the Ni center may change when going from the initial reactant to the final product.