# Implicit and Explicit Coverage of Multi-reference Effects by Density Functional Theory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1 Introduction

^{1}A

_{1}), C

_{2}(${}^{1}\mathsf{\Sigma}_{g}^{+}$),dissociating A-A molecules, etc.), and finally systems with multi-determinantal, multi-configurational wave functions (type III systems such as the Myer’s biradical). [9] The current article will concentrate on the correct description of type II and type III systems because this implies the inclusion of long-range Coulomb correlation effects (non-dynamicor static electron correlation) into DFT. It will not consider type I systems although their correct description is already problematic at the DFT level and requires methods such as ROSS [33] or ROKS [34].However, type I systemsdonotpossess(bydefinition)anylong-rangecorrelationandthereforearenotconsideredtorepresent multi-referenceproblems.

## 2 Implicit Account of Multi-reference Effects by Standard DFT

_{XC}is defined by Eq. (1) (see, e.g., Ref.2):

**r**

_{1}denotes the position of the first electron (reference electron), and

**r**

_{2}the position of the second electron. The exchange-correlation hole can be split up into exchange hole h

_{X}and correlation hole h

_{C}, which are related to exchange and Coulomb correlation, respectively.The exchange hole is negative or equal to zero everywhere and fulfills the sum rule [2]:

_{X}(

**r**

_{1},

**r**

_{2}) ≤ 0,

^{3}r

_{1}d

^{3}r

_{2}h

_{X}(

**r**

_{1},

**r**

_{2}) = − 1.

_{J}[ϱ] + E

_{X}[ϱ

_{α}, 0] = 0

_{C}[ϱ

_{α}, 0] = 0

_{α}(

**r**) d

**r**= 1, and ϱ

_{β}(

**r**) = 0, the self-exchange energy E

_{X}[ϱ

_{α}, 0] exactly cancels the self-repulsion E

_{J}[ϱ] of the electron and that there is no correlation of an electron with itself. All commonly used approximate exchange functionalsviolate condition (3) and lead to a relatively large exchange SIE. The SIE of approximate correlation functionals is smaller and even zero in the case of those correlation functionals, which were derived from WFT methods as for example theLYP functional.Therefore,wewillconsiderinthefollowingjusttheexchangeSIE.

- a)
- The SIE converts the delocalized HF exchange hole into a localized DFT exchange hole.Although it is an error by nature, it introduces in this way some useful electron correlation effects (to be discussed in the following), which significantly influence the performance of DFT. Self-interaction corrected DFT (SIC-DFT) annihilates the SIE and in this way also destroys the electron correlation effects mimicked by the SIE. Accordingly, SIC-DFT is closer to HF as directly reflected by the exchange holes generated by the two methods.
- b)
- The LDA and GGA exchange approximations are constructed to genuinely describe both exchange and left- right correlation. The self-interaction correction brings the DFT model closer to HF and destroys the left-right correlation LDA (or GGA) has built in (see, e.g. Ref. 12a).

_{2}in its ${}^{1}\mathsf{\Sigma}_{g}^{+}$ ground state and its ${}^{1}\mathsf{\Sigma}_{u}^{-}$ excited state as calculated at different levels of theory with Dunning’s cc-pVTZ basis set [35]: RHF (restricted Hartree-Fock), UHF (unrestricted Hartree-Fock), and BS(broken-symmetry)-UHF, RLDA, UDLA, and BS-ULDA, RSIC/LDA, USIC/LDA, and BS- USIC/LDA where for the exchange-correlation functional SVWN5 was used (S: Slater exchange [36]; VWN5: Vosko- Wilk-Nusair correlation [37]). The self-interaction corrected DFT (SIC-DFT) was obtained using the method of Perdew and Zunger [17,18] in a self-consistent-field version. [19,20,21,22,38]

_{e}of 83.7 kcal/mol (calculated relative to twice the UHF energy of H(

^{2}S)).

_{e}= 113.3 kcal/mol) where this difference is caused by a) the VWN5 correlation functional (major effect) and b) the self-interaction error of the Slater exchange functional (minor effect). [14,15,16] The two effects can be separated carrying out a SIC-LDA calculation, again using the HF density. The SIC-LDA calculation leads to the same exchange energy as the HF calculation and since the same correlation functional is used, [40] the difference between LDA and SIC- LDA (Figure 1) reflects the consequences of the self-interaction error of Slater exchange. Although differences are relatively small, the self-interaction error causes a) the HH bond length to increase from 0.727 Å (SIC-LDA) to 0.766 Å (LDA) and b) the D

_{e}value to decrease from 114.7 to 113.3 kcal/mol (Figure 1).

_{g}orbital. - The consequences of the self- interaction error of the exchange functional can be assessed by inspection of Figure 2.Exact RHF exchange implies a delocalized exchange hole with deep pockets at the positions of the nuclei and a shallow (close to zero) depression in the internuclear region (Figure 2a).Since, for ${\text{H}}_{2}\left({}^{1}\mathsf{\Sigma}_{g}^{+}\right)$ or any other spin-coupled 2-electron system, exchange is equal to the self-exchange (intra-electronic exchange), the exchange hole is static (independent of the position of the reference electron) and equal to -1/2 the electron density distribution ρ(

**r**). When SIC-LDA is calculated with the same density, it leads to exactly the same delocalized, static exchange hole. In this way, the form of both the HF and the SIC-LDA exchange hole provides electronic structure information.

**r**), but otherwise it does not contain any electronic structure information. Since the sum rule implies that both LDA and HF exchange hole integrate to -1, a deep LDA exchange hole is narrow while a shallow LDA exchange hole is much more diffuse.

- 1)
- With increasing distance R(H-H), the exchange hole becomes more and more delocalized. Accordingly the self-interaction error of the LDA exchange functional increases in magnitude and mimics more strongly left-right correlation.
- 2)
- As a consequence of 1), the RLDA solution becomes more stable than the RHF solution for the dissociating ${\text{H}}_{2}\left({}^{1}\mathsf{\Sigma}_{g}^{+}\right)$ molecule. The bifurcation point, at which RLDA becomes unstable relative to ULDA is shifted to a larger R(H-H) value (from 1.212 to 1.745 Å, Figure 1).The more long-range correlation effects are covered by a given method, the more stable it becomes.
- 3)
- The energy difference between RLDA and RSIC/LDA is increased due to the increase of the self-interaction error.

- 4)
- The explicit two-configurational description of long-range electron correlation as introduced by the BS-UDFT solution reduces the self-interaction error for increasing distance R(H-H) to a small value, i.e. the self-interaction error for the isolated H atoms is close to zero. This is reflected by the difference BS-ULDA - BS-USIC/LDA (Figure 1).
- 5)
- With increasing R(H-H), the self-interaction error mimics more short-range rather than long-range electron correlation. In the region of spin-recoupling from a closed-shell to an open-shell singlet, both long-range and short-range correlation effects are simulated by the exchange functional.

_{2}+ D

_{2}cycloaddition reaction via a D

_{4h}-symmetrical transition state leading to 2 HD molecules (for reasons of simplicity we discuss the degenerate reaction H2 + H2). This is a well-known example for the failure of standard Kohn-Sham DFT as was demonstrated by Baerends and co-workers. [5b] The reaction barrier can only be described by mixing the CSF of the ground state A

_{1g}[(a

_{1g})

^{2}(b

_{2u})

^{2}] with the CSF of the doubly excited state A

_{1g}[(a

_{1g})

^{2}(b

_{3u})

^{2}].

_{2u}-symmetrical HOMO and the b

_{3u}-symmetrical LUMO obtained in the BS-UHF calculations: 0.931 and 0.068, respectively.

_{2}+ H

_{2}cycloaddition reaction.The reference electron (denoted by X) is at the center of the reaction complex. The exchange hole is delocalized possessing deep pockets at the positions of the four nuclei. For a vertical movement (toward the center of the H1,H3-distance or toward the center of the H2,H4 distance) of the reference electron, the RHF exchange hole is static while for a horizontal movement (toward the center of the H1-H2 bond or toward the center of the H3-H4 bond) of the reference electron the RHF exchange hole of the dissociating H1-H2 or the dissociating H3-H4 molecule is approached (see Figure 2a).

_{2}molecule. Any movement of the reference electron from the central position adopted in Figure 4a to the left or right (Figure 4d) changes the situation to that of the corresponding H

_{2}molecule approached, i.e. the part of the exchange hole due to the self-interaction error becomes identical to that shown in Figure 2a. Hence, RLDA with the SVWN5 exchange- correlation functional can provide an improved, although not correct description of the H

_{2}+ H

_{2}cycloaddition reaction because multi-reference effects are simulated by the Slater exchange functional.

## 3 Explicit Coverage of Multi-reference Effects: Ensemble DFT

_{s}in eq. (10), hasbeendemonstratedutilizingthe adiabatic connection between the real system of interacting electrons and a fictitious system with the interelectronic interaction switched off [41,42]. Such an adiabatic route does exist only if the corresponding density at the intermediate or zero electron-electron coupling strength is the ground state density [27].

_{i}: weighting factor for state i with density ρ

_{i}) rather than the density of a single Slater determinant. Indeed, theoretical arguments were given that the ensemble representation of the density is the only one that guarantees the existence of the corresponding potential V

_{s}in eq. (10). In terms of the one-electron orbitals, the ensemble representation translates to the fractional occupation numbers of the orbitals φ

_{k}(

**r**), i. e.

_{2}molecule and for the

^{1}A

_{g}ground state of the H

_{2}+H

_{2}system it is impossible to fit the KS density to the exact one unless fractional occupation numbers are invoked for certain KS orbitals. Taken together with the theoretical arguments, these results showed unambiguously that the ensemble representation has practical relevance andis theonly rigorous representation for thedensity ofa systemwith strong multi-reference character.

_{r}and another with doubly occupied orbital φ

_{s}, where φ

_{r}and φ

_{s}can be the HOMO and the LUMO from the conventional single determinant Kohn-Sham calculation. The inactive core orbitals are occupied with 2 electrons each, such that the ground state density is given by eq. (13).

^{KS}(...${\phi}_{r}^{2}$...) and E

^{KS}(...${\phi}_{s}^{2}$...) of the two configurations described above and a coupling term D(φ

_{r}, φ

_{s}) which is expressed as a linear combination of KS energies of the singly excited configurations generated within the same (2,2) active space.

_{r}≈ 2 and n

_{s}≈ 0 and open-shell singlet system with n

_{r}= n

_{s}= 1, the results of the conventional spin-restricted Kohn-Sham calculations are reproduced by REKS. [29] The one-electron orbitals and the FON values are obtained in REKS variationally from minimization of the REKS total energy (14). [29]

## 4 Implicit and Explicit Coverage of multi-reference Effects: The Problem of Double Counting of Electron Correlation

_{ST}were calculated for a number of biradicals with distinct multi-reference character both with restricted Kohn-Sham DFT, BS-UDFT, and the ensemble DFT REKS(2,2) method. The biradicals investigated (see Scheme 1) include p-didehydrobenzene (p-benzyne,

**1**),m-didehydrobenzene (m-benzyne,

**2**), propane-1,3-diyl (trimethylene,

**3**), 2,2-difluoropropane-1,3-diyl (

**4**), 4,5-dimethylene-cyclopentane-1,3-diyl (

**5**), and 3,4-dimethylene-oxolane-2,5-diyl (

**6**). The open-shell singlet states of these molecules represent type II systems, which can be described by mixing the ground state with the doubly-excited state formed by exciting two electrons from the HOMO to the LUMO, i.e.the biradical character obtained by this two-configurational description can be assessed from the FON values of HOMO and LUMO. The triplet states are high-spin systems without anymulti- reference character (type 0 systems). They constitute appropriate reference systems, which are needed to assess the multi-reference effects of the singlet biradicals. Note that forsomeofthesemolecules,REKS/BLYPcalculations based on the 6-31G(d) basis set were previously published. [29]

**1**-

**6**. Also given are the FON values of HOMO and LUMO (Table 2), which reflect the biradical character and, by this, the amount of long-range correlation effects introduced by REKS(2,2). Calculated REKS geometries are given in Figure 5.

**1**into two allyl units hold together by two long C2-C3/C5-C6 bonds that disturb π-delocalization and destabilize the molecule. Consequently, the singlet state is higher in energy than the triplet state, which benefits from the exchange interactions of the two unpaired electrons with aligned spin (thus reducing Coulomb repulsion).This is reflected by the calculated singlet-triplet splittings of -3.7 (SVWN5), 1.3 (BLYP), 14.9 (B3LYP), and 36.7 kcal/mol (BH&HLYP; all calculations with the 6-31G(d) basis; for cc-pVTZ results, see Table 1), respectively where the triplet state was calculated at the UDFT level of theory.

_{ST}. The energy difference between singlet ground state and the doubly excited state increasesand,bythis,thepossibilityofstatemixingisreduced.StatemixinginensembleDFTleads to an occupation of the a

_{g}-symmetrical LUMO (Scheme 2) which is C2-C3 and C5-C6 bonding thus reducing the influence of through-bond coupling. Thus, the parameter ∆ reflects the amount of CSF-mixing and the corresponding reduction of through-bond coupling.

_{ST}value of -3.5 kcal/mol [60,53] are obtained with B3LYP using a larger basis set (BS/UDFT/cc-pVTZ: -2.6), REKS(2,2)/cc- pVTZ: -2.5kcal/mol, Table 1) while with the BLYP functional values of singlet-triplet splittings are themore exaggerated the larger the basis set used (-4.5 and -5.7 kcal/mol, Table 1). Since a larger basis set should lead to the better results, we conclude that B3LYP provides the more reliable description. The biradical character predicted by REKS(2,2)/B3LYP (ca60%, Table 1) is comparable to that suggested on the basis of CCSD(T) calculations (65% [52b]) while the corresponding REKS(2,2)/BLYP value (32%, Table 1) is much smaller.

**2**, a through-space 1,3-bond can be establishedto avoid a biradical structure. [52,57,58,59] Hence, restricted theory with HF or hybrid functional falsely predicts bicyclo [3.1.0] hexatriene to be the only isomer of

**2**. [58,59] This is also true for those functionals that exaggerate the strength of abond. Hence, LDA functionals, despite of the multi-reference effects mimicked by the Slater exchange functional, predict a bicyclic rather than biradical form thus indicating that the nature of the functional dominates.

**2**than for

**1**by at least a factor of five (see FON values for B3LYP in Table 2), which reflects the fact that the biradical (multi-reference) character of

**2**is considerably reduced compared to that of

**1**. Thus, the (absolute) corrections in the singlet-triplet splitting of

**2**caused by REKS are smaller: a) BH&HLYP: -4 kcal/mol (from -7.4 to -11.4 kcal/mol, Table 2); b) B3LYP: -2.3 kcal/mol (from -14.3 to -16.6 kcal/mol); c) BLYP: -2.2 kcal/mol (from -19.7 to -21.9 kcal/mol).

- a)
- The error in the REKS/B3LYP singlet-triplet splitting of -17 kcal/mol (cc-pVTZ basis, Table 2) corresponds to the B3LYP destabilization energy of 3.5 kcal/mol caused by stretching the C1-C3 bond. Since the B3LYP functional [46] was optimized to reproduce the thermochemistry of suitable reference molecules and, by this, the strength of the bonds in these compounds, it makes little sense to draw the conclusion that the B3LYP functional, because of its local contributions, exaggerates the strength of the C1C3 interactions and therefore has to be reoptimized.
- b)
- A (2,2) active space provides a poor description of the multi-reference effects of
**2**which can only be correctly described if the π-space is included thus leading to a (8,8) active space (see Ref. 58). This will lead to a stabilization of the singlet state and, by this, an increase of the singlet-triplet splitting. For the B3LYP functional, the (absolute) experimental value of 21.1 kcal/mol would be approached in this way from above. In contrast, for the BLYP functional the singlet-triplet splitting would be significantly more exaggerated than with the (2,2) description (Table 2).

**3**-

**6**.

**Biradicals 3 - 6**. The biradical character of the singlet states of

**1**-

**6**can be assessed by the FON values of the LUMO. It increases with the value of the singlet-triplet splitting. If the latter is significantly negative (strong stabilization of the singlet state relative to the triplet state by spin-coupling), the biradical character is small while a singlet state of comparable or even higher energy than that of the corresponding triplet state possesses high biradical character. This is reflected by Figure 6 which suggests that the biradical character increases exponentially with the singlet triplet splitting in the range −25 ≤ ∆E

_{ST}≤ 5 kcal/mol.The REKS and BS-UDFT ∆E

_{ST}values reproduce the trend in the experimental (or CASPT2) reference value of the splittings [60,61,62,63,64,65] where in the case of

**5**and

**6**no exact values are known.

**3**(Scheme 1), the single electrons can only interact by hyperconjugation via the CH

_{2}group, which leads to little stabilization and, therefore a singlet energy slightly above the triplet energy (0.7 kcal/mol, Table 2) results. Substitution of the H atoms by F atoms (

**4**) increases hyperconjugation due to a lowering of the pseudo-π

^{*}(CF

_{2}) (or σ

^{*}(CF)) orbitals thus coupling the single electrons more effectively and stabilizing the singlet state; the biradical character is reduced from 79 to 55 % (B3LYP; BLYP: from 64 to 34 %, Table 2).

**5**(Scheme 1), the situation is similar to that in

**3**: Now two allyl units are connected by a CH

_{2}group so that again spin-coupling can only be mediated by hyperconjugation.Bond C4-C5 is relatively long (1.50 to 1.51 Å, Figure 5) indicating that π-conjugation via this bond is moderate.Steric repulsion between the exocyclic CH

_{2}-groups plays an important role in this connection (Figure 5). The singlet-triplet splitting for

**5**is larger than zero. [64] Calculations indicate that it has a similar value (0.7 kcal/mol, REKS/B3LYP, Table 2) as biradical

**3**because of the same coupling mechanism between the unpaired electrons.

_{2}group is replaced by an oxygen atom as in

**6**, the two allyl units can conjugate via the π orbital of the heteroatom thus leading to an effective coupling of the single electrons. The singlet state is stabilized below the triplet state and the singlet-triplet splitting becomes -2.5 kcal/mol (REKS/B3LYP/cc-pVTZ, Table 2). This is in agreement with the experimental observation that the singlet-triplet splitting is negative for

**6**. [66]

**1**-

**6**(Table 2, Figure 6).

_{ST}< 0) beyond or reducing them (∆E

_{ST}> 0) below the experimental value (Table 2).

_{r}, φ

_{s}) of Eq. (14). However, a more precise knowledge is necessary of the dependence of the weighting factor in front of the coupling term D(φ

_{r}, φ

_{s}) of Eq. (14) on the fractional occupation numbers of orbitals to avoid precisely the double counting of the dynamic correlation in REKS.

## 5 Explicit Coverage of Multi-reference Effects: Merging of DFT and WFT

_{J}is replaced by the full electron interaction energy V

_{ee}of the CAS trial wave function, which covers exact exchange and the non-dynamic electron correlation effects.Hence, the CAS-DFT functional is given by Eq.(15):

_{α}and ρ

_{β}to determine F[ρ] because they lead to errors in the description of state multiplets.Therefore, these densities are replaced by the total CAS density ρ(

**r**) and the CAS on-top pair density P (

**r**,

**r**) (the second electron is on-top of the reference electron thus leading to a simplified pair density distribution) as input quantities for the correlation functional [32] as was originally suggested by Moscardo and SanFabian. [72] In this way, differences between states with different multiplicity can be distinguished. By em- ploying the Colle-Salvetti functional [49], which uses the total density and pair density directly as input quantities, there is no need for a conversion of the functional as would be the case for any other gradient-corrected correlation functional based on spin densities ρ

_{α}and ρ

_{β}.

_{ref}(

**r**) is determined by considering all core orbitals and active space orbitals to be doubly occupied so that the magnitude of ρ

_{ref}(

**r**) reflects the size of the active space.If compared with the true density of the system in question, ρ(

**r**), the ratio ρ

_{ref}(

**r**)/ρ(

**r**) ≥ 1 will increase with the size of the active space as will the amount of dynamic electron correlation already covered by the CASSCF wave function. Hence, one has to scale the DFT correlation energy locally by a scaling factor f (0 ≤ f ≤ 1) to avoid this double counting.

_{ref}. For the determination of the scaling factor f, the resulting correlation energy ${\epsilon}_{C}^{\text{HEG}}(\rho ,{\rho}_{\text{ref}})$ is related to the value ${\epsilon}_{C}^{\text{HEG}}(\rho ,\mathsf{\infty})$ for all virtual orbitals being included.[32] The correlation energy is defined by Eq. (17).

_{C}depends on the total and the on-top pair density as well as on their gradients (apart from higher terms in the most general case).

_{ST}> 0) or overestimating (∆E

_{ST}< 0) the singlet-triplet splitting because of an exaggerated down-scaling of dynamic electron correlation in the triplet. [73] However, once these problems are solved, CAS-DFT can provide a reliable and computationally feasible account of both non-dynamic and dynamic electron correlation effects.

## 6 Conclusions

## Acknowledgment

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**Figure 1.**Potential curves of the ${}^{1}\mathsf{\Sigma}_{g}^{+}$ ground state and the ${}^{3}\mathsf{\Sigma}_{g}^{-}$ excited state of H

_{2}calculated with a cc-pVTZ basis set at various levels of theory.

**Figure 2.**Graphical representation of the exchange hole calculated for the dissociating ${\text{H}}_{2}\left({}^{1}\mathsf{\Sigma}_{g}^{+}\right)$ molecule along the bond axis for the situation that the reference electron is at the left H nucleus (position P1). The SIE part of the LDA exchange hole is given by the line in bold print. a) R(H-H) = 1.5 Å: RHF, RSIC/LDA (both the normal line), and RLDA (dashed line) calculations. b) R(H-H) = 1.5 Å: BS-UHF, BS-USIC/LDA (both the normal line), and BS-ULDA calculations. c) R(H-H) = 3.0 Å: RHF, RSIC/LDA (both the normal line), and RLDA (dashed line) calculations. d) R(H-H) = 3.0 Å: BS-UHF, BS-USIC/LDA (both the normal line), and BS-ULDA calculations. All calculations with a cc-pVTZ basis set using in all cases the HF density to avoid differences because of differences in the density.

**Figure 3.**Contour line diagram of the difference electron density distribution ∆ρ(

**r**) = ρ(method I) - ρ(method II) of the reaction complex H

_{2}+ H

_{2}calculated withthecc-pVTZbasisatageometryclosetothetransition state (R(H1-H2) = 1.212; R(H2-H4) = 1.244 Å). Solid (dashed) contour lines are in regions of positive (negative) difference densities. Reference plane is the plane containing the four nuclei. The positions of the atoms are indicated by numbers 1 to 4. The contour line levels have to be multiplied by the scaling factor 0.01 and are given in e bohr

^{−3}.

**a)**Method I: BS-UHF; method II: RHF.

**b)**Method I: RLDA;methodII:RHF.

**c)**MethodI:BS-ULDA; method II: RLDA.

**d)**Method I: BS-ULDA; method II: BS-UHF. The LDA calculations were carried out with the Slater exchange-only functional.

**Figure 4.**Contour line diagrams of the exchange hole of the reaction complex H

_{2}+ H

_{2}calculated with the cc-pVTZ basis at a geometry close to the transition state (R(H1-H2) = 1.212; R(H2-H4) = 1.244 Å). The contour line levels have to be multiplied by the scaling factor 0.01 and are given in e bohr

^{−3}.

**a)**RHF exchange hole. The reference electron (X) is at the center of the reaction complex.

**b**

**- d)**The part of the Slater exchange hole due to the self-interaction error. The reference electron X is at nucleus H

_{2}(

**b**), between H2 and H4 (

**c**) or close to the bond H1-H2 (

**d**).

**e)**One-dimensional representation of the exchange hole calculated along the internuclear axis H2,H4. The reference electron X has the same position as in

**c**.

**Figure 5.**REKS(2,2)/6-31G(d) geometries of the lowest singlet and triplet state of biradicals

**1**-

**6**. B3LYP: values in bold print, BLYP: normal print, BH&HLYP: italics. Numbers in parentheses were obtained with the cc-pVTZ basis set. Distances in Å, angles in degree.

**Scheme 2.**Frontier orbitals of p-benzyne (

**1**).

**a**) Interactions of the orbitals occupied by the single electrons with the σ(C2C3;C5C6) and σ*(C2C3;C5C6) orbitals as needed for through-bond coupling between the single electrons.

**b**) Frontier orbitals MO20 and MO21 of

**1**resulting from the interactions shown in

**a**) taken from a HF calculation.

**Figure 6.**The biradical character of molecules

**1 -5**in their lowest singlet state calculated as 100 × FON(LUMO). REKS(2,2)/6-31G(d) calculations.

∆E_{ST} | RDFT ∆ | ∆E_{ST} | BS-UDFT ∆ | %Birad | ∆E_{ST} | REKS(2,2) ∆ | %Birad | |
---|---|---|---|---|---|---|---|---|

SVWN5 | -3.7 | 0.087 | -5.7 | 0.047 | 39 | -6.4 | 0.064 | 19 |

(-5.8) | (0.091) | (-6.6) | (0.059) | (34) | (-7.2) | 0.070 | (16) | |

BLYP | 1.3 | 0.097 | -4.1 | 0.037 | 51 | -5.0 | 0.056 | 32 |

(-0.1) | (0.105) | (-4.5) | (0.041) | (50) | (-5.7) | (0.063) | (28) | |

B3LYP | 14.9 | 0.106 | -2.4 | 0.019 | 74 | -2.1 | 0.031 | 59 |

(13.1) | (0.109) | (-2.6) | (0.022) | (72) | (-2.5) | (0.033) | (57) | |

BH&HLYP | 36.7 | 0.117 | -1.8 | 0.010 | 84 | -0.7 | 0.015 | 77 |

(34.3) | (0.120) | (-1.7) | (0.010) | (84) | (-0.8) | (0.018) | (76) |

^{a}Calculations with the 6-31G(d) basis; numbers in parentheses with Dunning’s cc-pVTZ basis. Energy differences ∆E

_{ST}between singlet and triplet state in kcal/mol (a negative value indicates that the singlet state is more stable), ∆ is the difference [R(C2−C

_{3}) − R(C2−C1)]

^{S}− [R(C2−C

_{3}) − R(C2−C1)]

^{T}given in Å, %Birad gives the calculated biradical character in percent as 100 x FON(LUMO).

**Table 2.**Singlet-triplet energy splitting ∆EST and fractional occupation numbers (FON) of orbitals as calculated with the REKS(2,2) method using different functionals and basis sets.

^{a}

Molecule | ∆ E_{ST} | BLYP FON MO1 | FON MO2 | ∆ E_{ST} | B3LYP FON MO1 | FON MO2 | ∆ E_{ST} | BH&H-LYP FON MO1 | FON MO2 | Ref. Value ∆ E_{ST} | Ref. |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | -5.0 | 1.68 | 0.32 | -2.1 | 1.41 | 0.59 | -0.7 | 1.23 | 0.77 | 3.5 | 60 |

p−C_{6}H_{4} | (-5.7) | (1.72) | (0.28) | (-2.5) | (1.43) | (0.57) | (-0.8) | (1.24) | (0.76) | ||

2 | -21.9 | 1.96 | 0.04 | -16.6 | 1.88 | 0.12 | -11.4 | 1.62 | 0.38 | 21.1 | 60 |

m−C_{6}H_{4} | (-22.0) | (1.97) | (0.03) | (-17.0) | (1.90) | (0.10) | (-11.7) | (1.65) | (0.35) | ||

3 | 0.7 | 1.36 | 0.64 | 1.0 | 1.21 | 0.79 | 1.0 | 1.12 | 0.88 | 0.7 ^{b} | 62 |

C_{3}H_{6} | (0.1) | (1.47) | (0.53) | (0.8) | (1.27) | (0.73) | (1.0) | (1.15) | (0.85) | ||

4 | -5.5 | 1.66 | 0.34 | -3.7 | 1.45 | 0.55 | -2.7 | 1.31 | 0.69 | -4.8 ^{b} | 62 |

C_{3}H_{4}F_{2} | (-7.0) | (1.75) | (0.25) | (-4.6) | (1.51) | (0.49) | (-3.2) | (1.35) | (0.65) | ||

5 | 0.5 | 1.26 | 0.74 | 0.7 | 1.15 | 0.85 | 0.6 | 1.10 | 0.90 | >0 | 64 |

C_{5}H_{4}(CH_{2})_{2} | (0.6) | (1.28) | (0.72) | (0.7) | (1.17) | (0.83) | (0.6) | (1.12) | (0.88) | ||

6 | -4.7 | 1.86 | 0.14 | -2.8 | 1.61 | 0.39 | -1.6 | 1.35 | 0.65 | <0 | 66 |

OC_{4}H_{2}(CH_{2})_{2} | (-4.7) | (1.89) | (0.11) | (-2.5) | (1.63) | (0.37) | (-1.2) | (1.35) | (0.65) |

^{a}Calculations with the 6-31G(d) basis set. In parentheses, results obtained with the cc-pVTZ basis set. Energy differences ∆E

_{ST}between singlet and triplet state in kcal/mol (a negative value indicates that the singlet state is more stable). MO1 corresponds to the HOMO, MO2 to the LUMO. -

^{b}CASPT2 results.

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Cremer, D.; Filatov, M.; Polo, V.; Kraka, E.; Shaik, S.
Implicit and Explicit Coverage of Multi-reference Effects by Density Functional Theory. *Int. J. Mol. Sci.* **2002**, *3*, 604-638.
https://doi.org/10.3390/i3060604

**AMA Style**

Cremer D, Filatov M, Polo V, Kraka E, Shaik S.
Implicit and Explicit Coverage of Multi-reference Effects by Density Functional Theory. *International Journal of Molecular Sciences*. 2002; 3(6):604-638.
https://doi.org/10.3390/i3060604

**Chicago/Turabian Style**

Cremer, Dieter, Michael Filatov, Victor Polo, Elfi Kraka, and Sason Shaik.
2002. "Implicit and Explicit Coverage of Multi-reference Effects by Density Functional Theory" *International Journal of Molecular Sciences* 3, no. 6: 604-638.
https://doi.org/10.3390/i3060604