# Minimal Active Space for Diradicals Using Multistate Density Functional Theory

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{S}= 0, are of multiconfigurational character that cannot be adequately treated by Kohn–Sham density functional theory (KS-DFT) [6,7,8]. However, the energy-degenerate M

_{S}= ±1 spin multiplets of the triplet state can be represented by a single determinant using KS-DFT [6,7]. Furthermore, electron correlation plays an important role between the two closed-shell configurations in a two-electron and two-orbital active space, which ultimately determines the relative energies of the singlet and triplet ground states [9,10]. Consequently, the need to use a multiconfigurational method to treat open-shell diradicals poses a significant challenge to KS-DFT, and often a broken-symmetry approach is used to estimate the singlet–triplet-energy gap [11,12,13]. On the other hand, the difficulties of KS-DFT can be easily overcome using multiconfiguration self-consistent-field (MCSCF) methods in wave function theory (WFT), such as the complete-active-space self-consistent-field (CASSCF) approach, but, in this case, it is necessary to use a large active space, followed by including corrections for dynamic correlation to obtain quantitative results [14,15,16,17]. In this study, we present a multistate density functional theory (MSDFT) [18], in which a minimal active space (MAS) is sufficient to describe biradical species and to determine the singlet–triplet-energy gaps.

**S**

^{2}[27]. NOSI may be considered as one variant of the dynamic-then-static ansatz described by Liu and coworkers [28]. Moreover, diabatic states can be constructed within MSDFT framework [23,29,30].

_{4}by a p-type (cyclobutadiene) and an s-type (p-benzyne) diradical.

## 2. Theoretical Background

^{N}$\in $$\mathscr{H}$, is given by [18,20,23]

**r**.

**h**and $G$ are standard matrices of one-electron integrals (kinetic and nuclear attraction integrals) and two-electron Coulomb-exchange integrals. For the last term of Equation (1), Kohn–Sham exchange-correlation functional can be used to approximate the diagonal element of the multistate matrix functional, ${E}_{AA}^{xc}[{D}_{AA}(r)]={E}_{xc}^{KS}[{D}_{AA}(r)]$, but the off-diagonal elements are the transition density correlation functional (TDF), which does not exist in KS-DFT [20]. In special situations, such as the present singlet–triplet-state energy difference involving spin-pairing interactions between two unpaired electrons, the TDF energy can be obtained by enforcing the spin-multiplet degeneracy condition of the triplet states [24,25,26,29].

_{S}= 0 is degenerate with the M

_{S}= +1 component [25,26]. In turn, the pure singlet state energy is also determined. We note that the M

_{S}= +1 component of a triplet state can be adequately treated by KS-DFT with a single Slater determinant. Since the KS exchange-correlation functional leads to the exact ground-state energy, the TDF energy for diabatic electronic coupling in Equation (5) is exact, even though the specific functional form of TDF is unknown.

## 3. Computational Details

## 4. Results and Discussion

#### 4.1. Potential Energy Curves of Hydrogen Molecule

_{S}= 0 manifold (three singlet and one triplet) of a hydrogen molecule (H

_{2}) [1,43,44]. The minimal representation of these four states includes two doubly occupied closed-shell (CS) configurations, and a pair of singly occupied open-shell (OS) configurations by placing two electrons in the two lowest-energy orbitals. Thus, the state densities of the subspace $\mathcal{R}$

^{4}for the lowest four states of H

_{2}are represented by the following configuration state functions:

_{2}, and $\left\{{a}_{x}^{y}\right\}$ are the coefficients of CSFs. Note that all adiabatic states are of a multiconfigurational character, including the closed-shell states, although ${\Theta}_{\mathrm{CS}}^{20}$ makes the predominant contribution to the ground state near bonding distances.

_{2}along the bond-distance coordinate can be roughly divided into three regions, corresponding to closed-shell (I), diradicaloid (II) and diradical (III) [1]. In region I, at the bonding distance, the energy of the ${\Theta}_{\mathrm{CS}}^{20}$ state is below the T

_{1}state (${\Psi}_{1u}^{T}$), contributing dominantly to the singlet ground state ${\Psi}_{1u}^{{S}_{0}}$ (Figure 2a). In region II, the ${\Psi}_{1u}^{{S}_{0}}$ singlet ground state is characterized as a diradicaloid [1], which has an increasing amount of multireference character with a large and rapidly increasing diradical index in Figure 2a [9]. Finally, in region III, the ${\Theta}_{\mathrm{CS}}^{20}={\varphi}_{\mathrm{CS}}^{20}$ configuration (Equation (8)) has a higher energy than that of the triplet state (Figure 1), but ${\Psi}_{1u}^{{S}_{0}}$ is still below the triplet state as a diradical species. In this region, the singlet and triplet ground states are nearly degenerate (degenerate at infinite separation). Here, the singlet state can be equivalently viewed as a multiconfiguration combination of two CS configurations (Equation (12)) and two localized open-shell diradicals as in valence bond (VB) representation. Here, the triplet state results from the combination of two spin-mixed OS determinants (Equation (15)). Nevertheless, the geometry at which the energies of the closed-shell CSF ${\Theta}_{\mathrm{CS}}^{20}$ and the triplet-state ${\Psi}_{1u}^{T}$ switch order may be used as the transition point from the closed-shell diradicaloid bonding character to a singlet diradical species.

_{2}from MSDFT-NOSI calculations along with the FCI results (dotted curves). The agreement between MSDFT and FCI results is good, although the relative energy between the singlet and triplet states from MSDFT-NOSI is somewhat greater than that of FCI, suggesting that electronic coupling between the two CS configurations may be overestimated. Significantly, the correct trend of the potential energy curve for H-H dissociation is obtained by MSDFT-NOSI through state interaction between the incorrect dissociation behavior of the spin-mixed determinants (${\Theta}_{\mathrm{CS}}^{20}$ and ${\Theta}_{\mathrm{CS}}^{02}$). The area in blue between 1.45 Å and 2.2 Å in Figure 2b corresponds to region II in Figure 2a, which extends to longer bond distances than the latter (CASSCF) calculations without including the dynamic correlation. Figure 2 highlights the gradual transition in bonding character from a closed-shell configuration to a diradical state. The crossing point between the ${\Theta}_{\mathrm{CS}}^{20}$ diabatic state and the triplet state ${\Psi}_{1u}^{T}$ is located at 1.98 Å from MSDFT, consistent with the earlier studies [1,46].

#### 4.2. Singlet–Triplet-Energy Gap

#### 4.2.1. Benzyne Isomers

^{2}values (Table S5) and the large difference between $\Delta {E}_{ST}$ from UDFT (underestimate) and RDFT (overestimate). Clearly, static correlation plays an important role to adequately describe the singlet diradicals.

#### 4.2.2. Cyclobutadiene

_{4h}symmetry on the singlet-state potential energy surface was a minimum of the triplet state [53]. At this geometry, the lowest singlet state corresponds to a combination of two doubly occupied (closed-shell) configurations [54,55], which can undergo Jahn–Teller distortion, leading to two rectangular geometries with D

_{2h}symmetries [56]. The potential energy profiles determined with the CASSCF, multi-reference configuration interaction (MRCI) and MSDFT methods are shown in Figure 3 for the lowest singlet and triplet states. In Figure 3, the reaction coordinates for the interconversion between the two rectangular structures is interpolated to match the difference between the two rectangular sides (bond lengths). At the square geometry (D

_{4h}), the interpolation reaction coordinate is 0, and the two rectangular structures correspond to a unitless value of ±5. We used a minimum active space of two electrons in two orbitals (Equations (8)–(11)) both for MSDFT and CASSCF, whereas an active space of 4 electrons in 4 orbitals was also employed in the multireference MRCI calculations. However, we emphasized that a common set of orbitals were used both in CASSCF and in MRCI calculations, but nonorthogonal orbitals were adopted in the present MSDFT-NOSI.

_{4h}geometry by CASSCF (2,2). The correct $\Delta {E}_{ST}$ difference is obtained from MRCI and MSDFT, both using a minimal active space of two electrons in two orbitals. This is confirmed further by the comparison with MRCI calculations using a larger active space of four electrons in four orbitals [57]. Interestingly, the correct $\Delta {E}_{ST}$ difference at the square geometry can also be achieved by expanding the active space to four electrons in four orbitals, even without perturbation correction (Figure 3). Clearly, the amount of dynamic correlation introduced by expanding the size of the active space is sufficient to recover the correct energy difference. The effect introduced by using a larger active space has been called spin-dynamic correlation in the literature [58,59,60]. Of all the methods examined, MSDFT-NOSI, employing M06-HF/cc-pVTZ, yields a somewhat large $\Delta {E}_{\mathrm{ST}}$ value at 23.2 kcal/mol. SF-EOM-CCSD results are found at 16 kcal/mol [57]. In closing this discussion, we note that Jahn–Teller effects are found in a variety of systems, for example, recent computational studies of gold nanoclusters that include 25 gold atoms in different oxidation states [61], a classical example of the compressed and elongated conformers resulting from ionization of benzene [20,25,62], and coronene radical cation [63].

#### 4.2.3. Polyacenes

#### 4.3. Hydrogen-Atom Transfer Reactions

_{4}by the singlet diradicals of cyclobutadiene and para-benzyne may be formally considered as a concerted electron–proton transfer (CEPT) process [1], and we previously introduced a diabatic state representation of the different natures in electronic structure between HAT and CEPT using MSDFT [30]. Since these two reactions are known to be hydrogen atom abstraction, we restricted our discussion to the HAT diabatic states.

_{4}) in the reactant state and those of the HAT product (DRH) and the SiH

_{3}free radicals (Scheme 1) are grouped in parentheses, the superscript core denotes a product of doubly occupied orbitals that do not change occupation during the reaction, the superscripts α and β specify the electron spin, the orbitals ${\chi}_{H}$ and ${\chi}_{L}$ are the highest-occupied and lowest-unoccupied block-localized KS (BLKS) orbitals of the diradical fragment, and ${\gamma}_{hyd}$ represents the 1 s orbital of the hydrogen atom (it is a BLKS orbital of Si-H bond in the reactant state) [30].

- The diagonal elements of $H[D(r)]$ are directly determined as the BLKS-DFT energies of the corresponding determinants. The two reactant diabatic states can be separately obtained as the lower root of a 2 × 2 NOSI diagonalization of the two states in Equations (16) and (17) for illustration in Figure 6, although it is not needed to determine the potential energy surface of the adiabatic ground state. The four determinants in Equations (16) and (17) involve block-local excitations, for which the optimization has been detailed in reference [41]. For the product state, only ground-state BLKS optimization is sufficient;
- The spin-coupling matrix element between the spin-pair determinants in Equation (18) is evaluated using Equations (1) and (5), which requires a separate BLKS calculation of the triplet state with M
_{S}= +1; - For all other off-diagonal matrix elements of $H[D(r)]$, we used the DFT energy-scaled nonorthogonal determinant value to approximate ${H}_{AB}[{D}_{AB}(r)]$ (Equation (6));
- To examine the variations of state interactions as the HAT occurs, we also computed the effective diabatic coupling values between the reactant (Equations (16) and (17)) and product (Equation (18)) states, denoted as ${V}_{13}$ and ${V}_{23}$ according to ${V}_{RP}={H}_{RP}-{\epsilon}_{g}{S}_{RP}$, where R = 1 and 2, P = 3, and ${\epsilon}_{g}$ is the adiabatic ground-state energy.

_{23}) with the product state (dotted green curves) is weak with little variations along the reaction coordinate (Figure 6). On the other hand, interactions between the CS reactant state and product state (V

_{13}) is strong, reaching its maxima near the transition states for the two HAT reactions. The substantial difference in the effective diabatic couplings between ${\mathsf{\Theta}}_{\mathrm{CS}}^{\mathrm{R}}$ and ${\mathsf{\Theta}}_{\mathrm{OS}}^{\mathrm{R}}$ diabatic states indicates that the reactivities of both the π and σ types of biradicals in the singlet states are dominantly determined by the closed-shell states. This result is in good accord with previous analyses by Hoffman and coworkers who found that both singlet diradicals are best characterized by two closed-shell configurations [1].

_{4}by cyclobutadiene is 12 kcal/mol MSDFT-NOSI employing the M06-HF functional and the cc-pVTZ basis set (Figure 6), which is in reasonable accord with a value of 15 kcal/mol using CCSD(T)/cc-pVTZ//B3LYP/cc-pVTZ [1]. Employing the same structures reported by Hoffmann and coworkers [1], we found that the difference in the reaction energy between the two methods is greater (2 vs. −2 kcal/mol) [1]. There is no reported computational study of the HAT between SiH

_{4}and p-benzyne. We optimized the reaction pathway using M06-HF/cc-pVTZ and obtained an energy barrier of 24 kcal/mol, significantly greater than the reaction by cyclobutadiene, and an energy of reaction of −3 kcal/mol. For comparison, they are, respectively, 28 kcal/mol and −1 kcal/mol from MSDFT-NOSI calculations. Interestingly, the reaction involving the CS diabatic state (Equation (16)) is formally an electron transfer from the σ

_{Si-H}bond to the lowest unoccupied BLKS orbital of the diradical reactants. Although both reactions occur dominantly via the charge transfer pathway, it is remarkable to note that through-bond interactions between the two σ-frontier orbitals of p-benzyne lead to significant energy separations between the in-phase and out-phase combinations, resulting in a CS diradical character. In the case of cyclobutadiene, on the other hand, the breaking of orbital degeneracy is due to the Jahn–Teller distortion of the molecular geometry. For comparison, the nature of orbital interactions in the OS diabatic state (Equation (17)) follows a spin-exchange mechanism between the SiH

_{4}and the diradical configurations (Figure 5).

## 5. Conclusions

**D**(

**r**) can be performed variationally. In the present study, we adopted a nonorthogonal state interaction (NOSI) approach to approximate the self-consistent-field optimization in which both orbitals and state coefficients were simultaneously changed. We note here that since the Slater determinants in the MAS were used to represent the matrix density

**D**(

**r**) of interacting states of the subspace, the procedure was a state interaction rather than a configuration interaction. In this article, we first illustrated the computational procedure and its performance on the well-understood case of hydrogen molecule dissociation, and then we extended the same MAS (in terms of constrained Kohn–Sham determinant states) to other prototypical diradical cases, including singlet–triplet-energy splitting of benzyne isomers, Jahn–Teller structural tautomerization of cyclobutadiene and polyacenes up to 15 fused rings. The method was also used to investigate the competition between the charge transfer and spin exchange mechanism in the hydrogen abstraction reaction of SiH

_{4}by the π-type diradical cyclobutadiene and σ-type diradical p-benzyne. In comparison with accurate results, we found that MAS-MSDFT can be used to adequately model the energies and reactivities of the diradicals examined in this work, and we anticipate that this trend can be extended to other diradical species. The overall computational cost was slightly greater than that needed for N-sperate KS-DFT calculations, with N being the number of determinants in the MAS, plus the time needed to evaluate the nonorthogonal matrix element.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

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**Figure 1.**Schematic illustration of the relative energies of four diabatic (basis states) and adiabatic states of H

_{2}, varying from a predominantly closed-shell (

**I**) configuration, to diradicaloid (

**II**), ending at a diradical state (

**III**) at infinite separation.

**Figure 2.**(

**a**) Potential energy curves of the singlet ground and first triplet states, S0 and T1, and diradical index N

_{D}/2 for H

_{2}determined by RHF, UHF and CASSCF methods. (

**b**) Computed potential energy curves for the configuration state functions (colored symbols) and the lowest singlet and triplet ground states using MSDFT-NOSI (solid curves) along with the FCI results (dotted curves) for comparison. All calculations were performed with the cc-pVQZ basis set and the M06-HF functional was used in MSDFT calculations.

**Figure 3.**Computed potential energy curves for the ground-state singlet S

_{0}and triplet T

_{1}states of cyclobutadiene along the interpolated reaction coordinate between the two rectangular (D

_{2h}) geometries at a unitless value of ±5 through the transition state geometry (D

_{4h}). The cc-pVTZ basis set was used in all calculations. The energy of triplet state at the square geometry was set as the reference, since it has the least multiconfigurational character. The M06-HF functional was used in MSDFT calculations.

**Figure 4.**The adiabatic singlet-triplet-energy gaps $\Delta {E}_{\mathrm{ST}}$ in kcal/mol for the polyacene series. Restricted (RDFT) and unrestricted (UDFT) Kohn–Sham density functional theory and multistate density functional theory (MSDFT), employing the B3LYP functional and the 6-31G(d) basis set are shown, along with computational results from the pp-RPA approach [67] calculations.

**Figure 5.**Depiction of the reactant diabatic states, representing the closed-shell (CS) and open-shell (OS) diradicals, and the product diabatic state for the hydrogen-atom transfer (HAT) reactions from SiH

_{4}to cyclobutadiene (CBD) and p-benzyne (PBZ).

**Figure 6.**Adiabatic ground-state (${\mathrm{S}}_{0}^{\mathrm{ad}}$) and diabatic state potential energy profiles for the hydrogen atom abstraction of SiH

_{4}by cyclobutadiene (CBD) in (

**a**), and by p-benzyne (PBZ) in (

**b**). The adiabatic potential energies are presented as black curves and those for the reactant diabatic states (${\mathsf{\Theta}}_{\mathrm{CS}}^{\mathrm{R}}$ and ${\mathsf{\Theta}}_{\mathrm{OS}}^{\mathrm{R}}$ ) and product diabatic state (${\mathsf{\Theta}}_{\mathrm{HAT}}^{\mathrm{P}}$ ) are shown as colored solid curves. The effective coupling values between the product state and the closed-shell (CS) reactant state (V

_{13}) and the open-shell (OS) reactant state (V

_{23}) are shown as dotted curves. The geometries are obtained by migrating the transferring hydrogen atom in the framework of the transition-state structure. The reaction coordinate is defined as $\mathsf{\Delta}\mathrm{r}={\mathrm{r}}_{\mathrm{Si}-\mathrm{H}}-{\mathrm{r}}_{\mathrm{C}-\mathrm{H}}$. The M06-HF functional and the cc-pVTZ basis set were used in MSDFT calculations.

**Table 1.**Computed adiabatic singlet–triplet-energy gaps ($\Delta {E}_{ST}={E}_{S}-{E}_{T}$ ) of meta-, ortho- and para-benzyne, along with experimental values. Energies are presented in kcal/mol. The cc-pVTZ basis set was used in all calculations, and the M06-HF correlation functional is used in Kohn–Sham and multistate density functional theory (MSDFT) calculations.

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**MDPI and ACS Style**

Han, J.; Zhao, R.; Guo, Y.; Qu, Z.; Gao, J.
Minimal Active Space for Diradicals Using Multistate Density Functional Theory. *Molecules* **2022**, *27*, 3466.
https://doi.org/10.3390/molecules27113466

**AMA Style**

Han J, Zhao R, Guo Y, Qu Z, Gao J.
Minimal Active Space for Diradicals Using Multistate Density Functional Theory. *Molecules*. 2022; 27(11):3466.
https://doi.org/10.3390/molecules27113466

**Chicago/Turabian Style**

Han, Jingting, Ruoqi Zhao, Yujie Guo, Zexing Qu, and Jiali Gao.
2022. "Minimal Active Space for Diradicals Using Multistate Density Functional Theory" *Molecules* 27, no. 11: 3466.
https://doi.org/10.3390/molecules27113466