Supporting materials
SI Theoretical backgrounds
S.I.1 Symmetry breaking and its recovery via quantum resonance and valence-bond CI model via LMO in diradicals
The orbital symmetry breaking in the independent particle models such as the Hartree–Fock (HF) model arises from the strong electron correlation effect. Moreover, the symmetry breaking in the model entails the concept of quantum resonance that recovers the broken symmetry in finite systems. The resonance concept is familiar in quantum chemistry in relation to the valence-bond (VB) theory; to this end the delocalized molecular orbital (MO) picture can be transformed into localized MO (LMO) picture for the VB explanation of diradical species. The BS MOs in eqs. 2 in the text are indeed re-expressed with LMO(LNO)s in eq. 5 in the text as follows
(s1a)
(s1b)
where the mixing parameter is given by . Therefore the BS MO configuration can be expanded with using LMOs as
(s2a)
(s2b)
where the pure singlet diradical (SD) and triplet diradical (TD) covalent terms are given by
(s3a)
(s3b)
On the other hand, zwitterionic (ZW) configurations result from the charge transfer from to (vice versa) as follows:
(s4)
The low-spin (LS) BSI MO configuration involves the pure triplet covalent term, showing the spin-symmetry breaking property. Similarly, the LS BSII MO configuration is expressed by
(s5)
The LS BSII MO solution also involves the pure triplet term. Thus the spin symmetry breaking is inevitable in the case of the single-determinant (reference) BS solution for diradical species. However, both orbital and spin symmetries are conserved in finite quantum systems [24]; for example, the error arising from the triplet term in eq. s2(s5) is easily eliminated by the AP procedure that eliminate the pure triplet term in eq. s2(s5).
As shown in eqs. (s2) and (s5), the BSI and BSII solutions are degenerate in energy. Then the quantum resonance of them is permitted as follows [24]:
(s6a)
(s6b)
(s7a)
(s8b)
where the normalizing factor is neglected for simplicity. Thus the in(+)- and out-of-phase(-) resonating BS (RBS) solutions are nothing but the pure singlet and triplet states wave functions, respectively; the broken-symmetries are recovered via the quantum resonance. The chemical bonding between a and b sites is expressed with the mixing of the SD and ZW terms under the LMO approximation. The VB type explanation of electronic structures becomes feasible under the LMO approximation [24]. For example, the effective bond order becomes zero for the pure covalent term, but it increases with the increase of mixing with the ZW term until the ZW/SD ratio becomes 1.0, namely closed-shell limit. Therefore the diradical character defined with the weight of the covalent term in the LMO description in the VB approach is in turn quite different from our definition of the diradical character based on the delocalized NO (DNO) model [24]. The VB-type descriptions and explanations of unstable molecules are obtained using the LMO in our approach. Here LMO(LNO) is utilized as reference orbitals for MkMRCC in relation to elimination of the size-consistent error [104, 105].
The mutual transformation from the localized MO picture to the conventional delocalized MO picture is easy because of the mathematical relations in eqs. 2, 5 and 6 in the text. In order to obtain the effective bond order for the pure singlet RBS(+) solution, it is transformed into the symmetry-adapted NO expression as
(s9a)
where Ti is the orbital overlap between the up- and down-spin orbitals in eq. s1, and the first and second terms denote the ground and doubly excited configurations, respectively. In the text, the notation Si is utilized for Ti to avoid the confusion of notations. The effective bond order (B) for is given by
(s10a)
(s10b)
(s10c)
The B value, namely the elimination of the triplet state, is larger than b. The diradical character (y) is defined by twice of the weight of the doubly excited configuration (WD) under the delocalized NO approximation as
(s11)
(s12)
Thus the diradical character y (=Y by the MR approach such as MkCC) is directly related to the decrease of the effective bond order B. The chemical indices, b, B, and y(=Y), are mutually related in the present BS MO approach; in fact, these are key conceptual bridges for MR CI, MkCC, BS UHF(UB) and BS DFT approaches in the present paper. These indices are also useful for elucidation of scope and applicability of the computational schemes of the effective exchange integrals (J) in eq. 8 in the text.
SI.2 MRCC methods as direct extension of BS methods for quasi-degenerated electronic systems
The theoretical description of diradicals is closely related to the strong electron correlation problem that has been investigated during past decades. In late 1960s and early 1970s the extended Hückel MO and conventional restricted HF (RHF) methods had been applied to the elucidation of concerted reactions predicted on the basis of the orbital-symmetry conservation rules proposed by Woodward and Hoffmann; the electronic structures of key transition structures in these reactions had been assumed to be nonradical in nature, namely in weak correlation regime. On the other hand, in 1970s, we had been concerned with theoretical studies on strongly correlated electron systems that were interesting targets for many body theories developed in late 1950 and 1960s as shown in many references (s1-s27). The instability of closed-shell RHF solutions had been investigated in relation to more stable BS HF solutions that are chemically related to the instability of chemical bonds （namely broken-chemical bonds）in diradical species and their clusters. Particularly, as shown in the text, we were interested in BS unrestricted HF (UHF) and unrestricted HF-Slater (UHFS) models by the use of different-orbitals-for-different spins (DODS) and more generalized HF (GHF) and generalized HFS (GHFS) based on the general spin orbitals (GSO); two component spinors, that arise from the static electron and spin correlations. In late 1970s the extension and refinement of these independent models (namely post RHF, UHF and GHF models) have been our interesting theoretical problem. To this end, we have performed the natural orbital (NO) analysis of these BS solutions to obtain the natural orbitals and their occupation numbers as shown in the supporting Fig. S1. The NO analysis has indeed elucidated active orbitals that are closely related to nondynamical correlation corrections; partitioning active orbitals have entailed the necessity of the genuine MR approach in strongly correlated electron systems.
Therefore we considered that the MRCI and MRCC schemes are the most natural extensions of the BS methods, namely independent particle model, because the reference MR space can be constructed so as to describe quasi-degenerated electronic systems related to the BS solutions. We presented our MRCC scheme at Sanibel 1980 (ref. 23) on the basis of the MRCC by Offermann in nuclear physics, Mukherjee and Sinanoglu in quantum chemistry; a lot of reference papers on the CC methods are cited in the text for understanding of historical developments. Roos also proposed CASSCF at the same Sanibel conference. For the natural extension of the BS computations, the active space is limited so as to include nondynamical (static) correlations that are origins of instabilities in the RHF solution. Under this approximation, only the minimum reaction NO (MinRNO) = principal active space (PAS), namely CAS in our definition in Fig. 1 in the text, was considered instead of the maximum RNO (MaxRNO) = PAS+SAS (secondary active space). Therefore the reference function in MinRNO was taken to be UNO(=UHF-NO) and GNO(=GHF-NO) CASCI in our MR CI and MR CC schemes. On the other hand, Max RNO is often necessary for CASPT2 and related PT theories to include the higher-order excitations.
The CC excitation operator was considered for the reference state to obtain the UNO(GNO) CASCC as
CAS (MR CC) = exp (T) | YNO CASCI > (Y=U, G or D).(s13)
where T = Ti (i=1-4). Before Jeziorski and Monkhorst [29] proposed their CC scheme in 1981, the uniform excitation operator formalism had been employed in the MRCC approach. If we consider only the one-electron excitation operator responsible for semi-internal correlation for full NO space, UNO (GNO) CASCCS is refined to UNO(GNO) CASSCF after the convergence of the CC equation because of the Thouless theorem (ref. s13).
(CASSCF) = exp (T1) | YNO CASCI > (Y=U, G or D).(s14)
However, the inclusion of the double excitation operator (D) in UNO (GNO) CASCCS is crucial for dynamical correlation correction, namely UNO(GNO) CASCCSD, as
CAS(MR CCSD) = exp (T1 + T2) | YNO CASCI > (Y=U, G or D).(s15)
The UNO(GNO) CASCCSD approach starting from UNO(GNO) CASCI was our chemical picture at that time. However, in the next year (1981), Jeziorski and Monkhorst in ref. 29 in the text proposed a more general state universal (SU) MR CC scheme; excitation operator is used for each configuration this scheme as shown below. The SU MRCC scheme is furthermore specified into the state specific (SS) MRCC version: which is now employed by several groups; in fact, it is developed by Mukherjee, Kalláy, Paldus, Evangelists, developers of PSIMRCC program (Crawford et al) and many research groups cited in references. However, the extension of the SS MRCC scheme to quasi-degenerated systems with large CAS space is still difficult.
Therefore, several MRCC schemes have indeed been presented as shown in many references in refs. s1-s76. For example, more detailed formulation of the CASCC-type scheme can be obtained by eliminating redundant excitations with the Feynman- diagram techniques developed by Adamowicz at al. in refs. s69 and s73. For the purpose, Adamowicz et al. have divided the CC excitation operators into the internal (Tint) and external (Text) types like the Silberstone-Oktuz-Sinanoglu classification in refs. S21 and s22
CAS(CASCC) = exp (Text) exp (Tint) | 0 > (s16a)
= exp (Text) ( 1 + Ci (i=1-n) )| 0 > (s16b)
where |0> is taken as the most doubly occupied determinant. If active m-orbitals n-electrons are used for CAS, CASCCSD is constructed by using both Tint and Text excitation operators as
CAS(CASCC) = exp (Ti(i=1-(n+2)) ( 1 + Ci (i=1-n) )| 0 > (s17)
This CC scheme is therefore employed as a direct extension of MRSDCI. The total excitation operators are described with the pure external and mixed (Tint x Text) excitations in their scheme. The derivations of the MRCC equations and calculated results are given in several papers by Adamowizc (ref.s69). Judging from the numerical results for diradical species such as monocentric diradical (I), antiaromatic molecules (II) and 1,3-diradicals (III), SR CCSDTQ and MRCCSD provide similar results. This implies that the truncation of the excitation operators is possible at the SD level if the MR part in eq. s17 has been appropriately selected. However, SR CCSDTQ is indeed necessary for complex diradical as shown in the text. Our selection of the MR part is one of such reasonable procedures starting from the BS calculations for quasi-degenerated systems.
As mentioned above, Jeziorski and Monkhorst [s27] have developed more general state universal (SU) MR CC scheme on the basis of the Bloch wave-operator technique. Their MR CC scheme is given by a simple formula
(SU MR CC) = Cim (i=1-n)exp (Tim) im . (s18)
As is apparent from this equation, the CC excitation operator is applied to each configuration involved in the MR zero-order function as mentioned above. In fact, the CI coefficient Ci and amplitude in the excitation operators Ti are determined in an iterative manner. This in turn means that the CASSCF part may be skipped if the reference orbitals are appropriately determined. For example, UNO CAS CI can be used for the purpose. Therefore diradical character can be also defined even in this SU MRCC scheme. However, expansion of CAS (RNO) space for polyradicals (IV) is not so easy in this scheme because of too many amplitude equations, though UHF-CC is easily applicable to them.
Past decades several groups cited in references in the text have performed further derivations of the MRCC schemes as shown in refs. s40-s76. Now, the MRCC approach is classified into (a) state-universal (SU) or Hilbert space type, (b) valence-universal (VU) or Fock space type and (c) state-specific (SS) type. The delocalized UNO (UHF-NO) (GNO=GHF-NO) can be used for reference orbitals of these CC schemes. The size-inconsistent errors however are not negligible in the case of Mk-type UNO-MRCC (SS) approach, leading to use of the localized UNO (ULO) (see eq. 5 in the text) for elimination of such errors; ULO has been introduced to obtain the VB CI like pictures of the ground and excited states of diradical species; Yamaguchi K, Fueno T(1977) Chem Phys 23:375. In fact, ULO-MRCC provided reasonable potential curves of F2, CH2 and others as shown in this supporting material.
Fig. S1 Computational schemes proposed in the paper:
K. Yamaguchi, Int. J. Quant. Chem. S14, 269 (1980).
The Brueckner double (BD) method often applied to polyenic diradicals with moderate spin polarization effects. The BD results are approximately reproduced with those of hybrid DFT. The hybrid DFT natural orbitals can be used as an alternative to the UBD natural orbitals, giving the delocalized DFT-NO (DNO) and localized DFT-LNO (DLO) MRCC (SS) as shown. The excitation energies of these systems can be calculated by the linear response (LR), equation of motion (EOM) or time-dependent energy derivative (TDEG) (which has been called as the quasi energy derivative (QED)) method for MRCC if correlation effects involved in the ground state are not drastically changed upon electronic excitations. As shown in the text, the MRCC results for parent systems have been used to confirm AP-UHF-CC, AP-UBD and AP hybrid UDFT approximations that have been applied to much more larger systems with chemical interests, for example molecule-based magnetic materials. Early papers concerning with symmetry breaking and CC methods are given as follows.
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SI.3 . Dynamical correlations via genuine multi-reference coupled-cluster method: the MkCCSD method
Independent particle models incorporate the internal (static) electron correlation effects via symmetry breaking [s1-s6]; however, beyond the model is crucial for inclusion of dynamical correlation effect. As shown previously [24], active space constructed of UNOand/or DNOcan be used for reference functions for MRCI and MRCC computations of quasi-degenerated systems [24]; here genuine MRCC is employed as a successive reliable and refined procedure for the BS computations. This in turn provides a theoretical background for approximate spin projection scheme for the BS UHF-CCSD and UBD solutions. Therefore the Mukherjee’s type MRCC (MkCC) theories are briefly described for the present purpose: namely spin correction to the BS methods.
In the Hilbert space MRCC, a model space
containing determinants is defined to construct a projection operator to ,