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1 Electronc Supplementary Materal (ESI) for Physcal Chemstry Chemcal Physcs. Ths journal s the Owner Socetes 01 Supplemental document Behnam Nkoobakht School of Chemstry, The Unversty of Sydney, Sydney, Australa S1 Dervaton of the electrostatc Hamltonan Ths secton s devoted to the evaluaton of the electrostatc effectve Hamltonan H es (.e. Eq. () of the text) for the trply degenerate electronc state T g n an octahedral system, whch has JT actve modes e g and t g wth = 1,. To the best of our knowledge, the dervaton of the electrostatc Hamltonan up to quadratc terms contanng blnear and couplng terms among JT actve modes were not reported n the lterature. To set up the electrostatc Hamltonan, we select the followng electronc bass set 1] ψ ξ = ηζf(r) ψ η = ξζf(r) ψ ζ = ξηf(r), (S1) where f(r) s an exponental or Gaussan radal functon. The electrostatc Hamltonan H es was expanded at the reference structure of the T g normal coordnates Q ǫ, Q θ, Q ξ, Q η and Q ζ for each JT actve mode up to second order ncludng all possble couplng between e g S1

2 and t g H es = =1 H (1) Q ǫ Q θ +H () Q +H () Q Q ǫ Q θ ] +H ǫ (1) 1 ǫ Q ǫ1 Q ǫ θ 1 θ Q θ1 Q θ ǫ θ 1 Q ǫ Q θ1 θ ǫ 1 Q θ Q ǫ1 + H (1) Q ξ +H η (1) Q η ζ Q ζ +H () Q +H η () Q η +H () ζ Q ζ =1 + H (1) ζ Q ξ Q ζ η Q ξ Q η η ζ Q η Q ζ ] Q ξ1 Q ξ Q η1 Q η Q ζ1 Q ζ ξ 1 ζ Q ξ1 Q ζ ξ ζ 1 Q ξ Q ζ1 ξ 1 η Q ξ1 Q η ξ η 1 Q ξ Q η1 η 1 ζ Q η1 Q ζ η ζ 1 Q ζ1 Q η, ( H es Q τ )0 ( ( where H τ (1) H = es Q τ, H τ )0 () = 1 and H (1) H τ τ j = es that the superscrpt τ,j {,j,,j,,j,η,j,ζ,j }, where and j are 1 and. (S) Q τ Q τj )0. Note NextstepstocalculatethematrxelementsofoperatorsoftypesH τ (1) Q τ, H τ () Q τ and H τ (1) τ j Q τ Q τj usng electronc bass set of T g of Eq. (S1). These matrx elements transform as do the components of the rreducble representaton T g of the symmetry pont group O h, namely, ξ, η and ζ. Snce Q τ, Q τ and Q τ Q τj do not operate on the electronc bass sets, t s requred to calculate matrx elements of H τ (1) τ j, H τ () and H τ (1) τ j. Q τ, Q τ Q τj and Q τ are consdered as multplyng factors. For the evaluaton of matrx elements, we have used the method descrbed n Ref. ]. Snce operators H τ (1) τ j, H τ (1) τ j have the same transformaton propertes as Q τ, Q τ and Q τ Q τj, we and H () τ should fnd rreps and ther components accordng to whch the operators Q τ and Q τ Q τj transform. Ths can be understood easly by usng the formula of the rreducble products of operators Q τ and Q τj ] M c γ := (Q a Q b ) c γ = λ(c) 1/ αβ V a b c Q a α β γ αq b β, (S) where := means equal by defnton. Note that operators Q a α and Q b β transformasdocomponentsαandβ ofrreducblerepresentatonsaandb, respectvely, V coeffcents correspondng to the the octahedral group a b c α β γ O h can be found n Ref. ]. λ(c) s the dmenson of rreducble representatons c and the sum s over all possble components of a and b. For example, S

3 n the case of trgonal coordnates Q ξ, Q η and Q ζ, the sum s over the components trply degenerate rrep T g of symmetry group O h. Note that f c a b, Eq. (S) spans the rrep c, otherwse s zero. By employng the method descrbed above, we proceed to derve the T g (t g +t g ) part of JT problem. For other parts, the same method s applcable. To handle ths problem, we frst consder the lnear terms of the t g components of Eq. (S). In ths case, H τ (1) s transform as T g. Snce operators H τ (1) s have the same transformaton propertes as Q τ and the coordnate Q τ s transforms accordng to the components of ξ, η and ζ, thus non-zero matrx elements of lnear JT Hamltonan read ψ µ Q τ ψ ν = κ T ǫ αµν Q τ, (S4) where κ T s constant and ǫ τµν s the Lev-Cvta symbol, and µ, ν and τ {ξ, η, ζ}. Therefore, the non-zero matrx elements are ψ ξ Q ζ ψ η = κ T Q ζ (S5) In the next stage, we should consder blnear terms such as H (1) α β Q α Q β. We need to know the transformaton propertes of Q τ Q τ. Ths can be understood by usng Eq. (S). Therefore, we have M T g = Tg T V g T g Q T g η η ζ Q T g ζ = ] Q T g η Q T g ζ M T g η = Tg T V g T g Q T g ζ η Q T g ζ = ] Q T g ζ Q T g M T g ζ = Tg T V g T g Q T g η ζ Q T g η = ] Q T g Q T g η (S) Tg T Coeffcents lke V g T g n Eq. (S) can be found n Ref. ]. ζ η For operators such as H (1) α β j Q α Q βj, we have smlar stuatons. Eq. (S) tells that the correspondng matrx elements of operators Q T g η Q T g ζ, Q T g Q T g ζ and Q T g Q T g η n the dabatc electronc bass ξ, η and ζ are proportonal to M T g, M T g η and M T g ζ, respectvely. Usng ths knowledge and the electronc bass S

4 set of Eq. (S1) help to evaluate of the matrx elements as follows, Tg T ψ µ H α β Q α Q β ψ ν = T g H α β T g V g T g Q γ α β α Q β = T g H α β T g ( 1 ) Q α Q β }{{} B = B Q α Q β (S7) We have used the followng relaton n the evaluaton of Eq. (S7) ]: Tg T V g T g = 1 ǫ γ α β γαβ (S8) If we employ Eq. (S7), the matrx elements n the dabatc electronc bass ψ ξ, ψ η and ψ ζ read, ψ ξ H (1) η Q ξ Q η ψ η = B Q ξ Q η ψ ξ H (1) ζ Q ξ Q ζ ψ ζ = B Q ξ Q ζ ψ η H (1) η ζ Q η Q ζ ψ ζ = B Q η Q ζ (S9) Usng Eq. (S9) leads to the followng results: ψ ξ H ξ1 η Q ξ1 Q η +H ξ η 1 Q ξ Q η1 ψ η = b T (Q ξ1 Q η +Q ξ Q η1 ) ψ ξ H ξ1 ζ Q ξ1 Q ζ +H ξ ζ 1 Q ξ Q ζ1 ψ ζ = b T (Q ξ1 Q ζ +Q ξ Q ζ1 ) ψ η H η1 ζ Q η1 Q ζ +H ζ1 η Q ζ1 Q η ψ ζ = b T (Q η1 Q ζ +Q ζ1 Q η ), (S10) where coeffcent b T s proportonal to T g H α β T g. Fnally, we should evaluate the correspondng matrx elements of the quadratc terms n Eq.(S). Strctlyspeakng,wearenterestedntermssuchasH α () Q α andh α (1) α j Q α Q αj. For the quadratc terms, we should fnd rreducble representatons of O h pont group of the operators H α () and H α (1) α j. Let consder the rreps E g and A 1g and ther components and use Eq. (S). Thus, we have = 1 Qζ Q ξ Q η ] = 1 Qξ Q η ] M A 1g = 1 Qξ +Q η +Q ζ ] (S11) S4

5 Solvng Eq. (S11) n terms of, Mǫ Eg and M A 1g yelds Q ξ = 1 M A 1g Q η = 1 M A 1g 1 1 Q ζ = + 1 M A 1g. (S1) We can repeat ths calculaton for term such as H α (1) α j and summarze the results as follows, Q ξ1 Q ξ = MA 1g 1 MEg + 1 MEg Q η1 Q η = Q ζ1 Q ζ = MA 1g MA 1g + 1 MEg MEg 1 MEg (S1) where can be chosen 1 or. Eqs. (S1) and (S1) ndcate that the correspondng matrx elements of the operators H α () Q α and H α (1) α j Q α Q αj are proportonal to,mǫ Eg and M A1g. Thus, non-zero matrx elements of the quadratc terms of Eq. (S) reads ψ ξ H () Q ξ +H () η Q η +H () ζ Q ζ ψ ξ = A (Q ξ Q η Q ζ )+ ω T (Q ξ +Q η +Q ζ ) ψ η H () Q ξ +H () η Q η +H () ζ Q ζ ψ η = A (Q η Q ξ Q ζ )+ ω T (Q ξ +Q η +Q ζ ) ψ ζ H () Q ξ +H η () Q η +H () ζ Q ζ ψ ζ = A (Q ζ Q ξ Q η )+ ω T (Q ξ +Q η +Q ζ ) (S14a) ψ ξ H (1) Q ξ1 Q ξ Q η1 Q η Q ζ1 Q ζ ψ ξ = a T 1 (Q ξ1 Q ξ +Q η1 Q η +Q ζ1 Q ζ )+ +a T (Q ξ1 Q ξ Q η1 Q η Q ζ1 Q ζ ) ψ η H (1) Q ξ1 Q ξ Q η1 Q η Q ζ1 Q ζ ψ η = a T 1 (Q ξ1 Q ξ +Q η1 Q η +Q ζ1 Q ζ )+ +a T (Q η1 Q η Q ξ1 Q ξ Q ζ1 Q ζ ) ψ ζ H (1) Q ξ1 Q ξ Q η1 Q η Q ζ1 Q ζ ψ ζ = a T 1 (Q ξ1 Q ξ +Q η1 Q η +Q ζ1 Q ζ )+ S5 +a T (Q ζ1 Q ζ Q ξ1 Q ξ Q η1 Q η ) (S14b)

6 Here,coeffcentsA areproportonalto T g H () Ω T g. Notethatcoeffcents a T T g H (1) Ω Ω j T g where Ω {,η,ζ } and Ω j {ξ j,η j,ζ j } wth,j = 1,. We used followng relatons ] Eg T V g T g Eg T = V g T g = 1 θ ξ ξ θ η η V Eg T g T g = 1 θ ζ ζ A1g b b V = λ(b) 1/ δ β γ βγ (S15) where λ(b) s the dmenson of rreducble representaton b and δ refers to the Kronecker delta. So far, we dscussed how to calculate the matrx elements for the the T g (t g + t g ) part of the JT Hamltonan; Eq. (S4) refers to the matrx elements for lnear JT Hamltonan, Eqs. (S9), (S10) and (S14b) refer to the matrx elements of the blnear terms. Fnally, Eq. (S14a) are the matrx elements for the quadratc terms of the JT Hamltonan. If one follows the same computatonal method for the T g e g part of JT Hamltonan, the correspondng matrx Hamltonans for ths part JT Hamltonan wll be obtaned. For ths part of JT Hamltonan, we dd not present the detals of calculatons and restrct ourselves to the fnal results for matrx elements. In ths way, the electrostatc Hamltonan H es can be obtaned by the aforementoned matrx elements. The fnal form of H es was wrtten down n Appendx A. S Potental energy surfaces References 1] F. A. Cotton, Chemcal Applcatons of Group Theory, ( Wley- Interscence, New York 1971). ] J. S. Grffth, The Irreducble Tensor Method for Molecular Symmetry Groups (PRENTICE-HALL, INC, New Jersy, 19). S

7 Fgure S1: Adabatc PESs of the T g electronc state of W(CO) +. the both components of e g modes and components of t g modes. The computed DFT data and the correspondng ftted lnes are represented by crcles and sold lnes, respectvely. S7