Quantitative El-Sayed Rules for Many-Body Wave Functions

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Spectroscopy and Photochemistry; General Theory

Quantitative El-Sayed Rules for Many-Body Wavefunctions from Spinless Transition Density Matrices Pavel Pokhilko, and Anna I. Krylov J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b02120 • Publication Date (Web): 06 Aug 2019 Downloaded from pubs.acs.org on August 6, 2019

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The Journal of Physical Chemistry Letters

Quantitative El-Sayed Rules for Many-Body Wavefunctions from Spinless Transition Density Matrices Pavel Pokhilko and Anna I. Krylov∗ Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482 E-mail: [email protected]

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The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract One-particle transition density matrices and natural transition orbitals enable quantitative description of electronic transitions and interstate properties involving correlated many-body wave functions within molecular orbital framework. Here we extend the formalism to the analysis of tensor properties, such as spin–orbit couplings (SOCs), which involve states of different spin projection. By using spinless density matrices and Wigner–Eckart’s theorem, the approach allows one to treat the transitions between states with arbitrary spin projections in a uniform way. In addition to a pictorial representation of the transition, the analysis also yields quantitative contributions of hole-particle pairs into the overall many-body matrix elements. In particular, it helps to rationalize the magnitude of computed SOCs in terms of El-Sayed’s rules. The capabilities of the new tool are illustrated by the analysis of the equation-of-motion coupled-cluster calculations of two transition metal complexes.

Graphical TOC Entry Lz-allowed

L+/--allowed

2


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Molecular orbital theory is of central importance in chemistry. Chemists use molecular orbital concepts to explain molecular structure and properties. 1 Orbitals are used to rationalize systematic trends in series of compounds and chemical reactivity in the ground and electronically excited states. 2,3 Molecular orbitals are also commonly invoked to describe electronic transitions and interstate properties, such as transition dipole moments, non-adiabatic and spin-orbit couplings. 4–7 Molecular orbitals are often associated with (pseudo)-non-interacting electrons. In the Hartree–Fock mean-field theory, an N -electron wave function is given by an antisymmetrized product on N molecular orbitals and the differences between two electronic states (e.g., neutral and cation or ground and excited) can be described in terms of individual orbitals. Consequently, state and interstate properties can be analyzed in terms of matrix elements between the orbitals. Yet, a quantitative description of an N -electron system entails multi-determinantal ansätze. Even the simplest excited-state theory, configuration interaction singles (CIS), goes beyond the single-determinant representation. One can analyze such wave functions in terms of molecular orbitals by focusing on the dominant configurations, however, such approach is imprecise and quickly becomes impractical. Moreover, the choice of individual orbitals (and, consequently, the values of the individual amplitudes in the multi-configurational wave functions) is not unique. For example, energies and all physical observables of the Hartree–Fock and CIS wave functions are invariant with respect to any unitary orbital transformation within the occupied and virtual orbital spaces. This is also the case for coupled-cluster (CC) and equation-of-motion coupled-cluster (EOM-CC) theories. Despite these unsettling observations, molecular orbital theory and rigorous description of electron correlation are perfectly compatible with each other. Molecular orbital theory can be extended to correlated many-electron wave functions via generalized one-electron quantities such as Dyson 8–10 and natural transition orbitals (NTOs). 11–20 In this paper, we focus on one-electron transition properties. The key quantity, which

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allows one to characterize the differences between two states (e.g., ΨF and ΨI ) in terms of one-electron excitations and to compute interstate properties, is one-particle transition density matrix: FI γpq ≡ hΨF |a†p aq |ΨI i,

(1)

where a†p and aq are creation and annihilation operators associated with orbitals p and q. P The interstate matrix element for a one-electron operator Aˆ = pq Apq a†p aq is then: 21 ˆ Ii = hΨF |A|Ψ

X

Apq hΨF |a†p aq |ΨI i =

pq

X

FI Apq γpq = T r[γA],

(2)

pq

ˆ I i evaluated using the above where the sum runs over all orbitals. While the value of hΨF |A|Ψ ˆ q i and the expression is orbital-invariant, the values of the individual integrals Apq = hφp |A|φ respective elements of γ depend on the choice of orbitals. The most compact representation of γ and, consequently, of Eq. (2) is achieved by using diagonal representation of the transition density matrix via singular value decomposition (SVD): 11,14 γ F I = UΣV† ,

(3)

where Σ is a diagonal matrix with non-negative numbers (σk ) on the diagonal, k th columns of U and V are the pair of right and left singular vectors corresponding to k th singular value σk . In this way, one can represent the transition density between states F and I as a (usually small) number of particle-hole pairs: ξ F I (xp , xh ) =

X

σk φ˜pk (xp )φ˜hk (xh ),

(4)

k

where x denotes spatial and spin coordinates of an electron (x ≡ r, s) and φ˜pk and φ˜hk , which

4


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are called particle and hole NTOs, are defined by the singular vectors from Eq. (3): φ˜pk =

X

φ˜hk

X

Uqk φq ,

(5)

q

=

Vqk φq .

(6)

q

(7)

ξ F I (xp , xh ), which sometimes is called exciton wave function, 11,22 describes the difference between the two many-electron states, ΨF and ΨI , in terms of one-electron (i.e., hole-pair) excitations between the orbitals φ˜pk and φ˜hk . In this way, NTOs provide a rigorous extension of molecular orbital theory to general correlated wave functions. In the basis of NTOs, Eq. (2) becomes ˆ Ii = hΨF |A|Ψ

X

ˆ φ˜h i σk . hφ˜pk |A| k

(8)

k

Thus, an interstate property can be represented as a sum of the matrix elements between the hole and particle orbitals and the weight of each term is given by the respective singular value. 23 Often, only one or two singular values are significant in the expansion, while the contribution of the remaining singular values is small. NTOs are now routinely used to analyze various aspects of bound 16,19,20,24 and metastable 18 excited states, including non-linear optical phenomena. 17 In this contribution, we extend the formalism to the analysis of tensor properties, such as spin–orbit couplings (SOCs), which involve states of different spin projection. Qualitative molecular orbital analysis has been used to rationalize the rates of the spinforbidden transitions since early days of quantum chemistry. 5,25 As succinctly explained by El-Sayed, 5,25 the angular momentum operator of the SOC term induces an orbital torque, which means that large values of the SOC matrix element can be attained only when the transition involve the change of orbital orientation. 26 In the context of organic photochem-

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istry, this implies that SOCs between the states that have the same orbital character (e.g., ππ∗ triplet and singlet) are small and SOCs between the states that have different orbital character (e.g., nπ∗ triplet and ππ ∗ singlet) are large. Known as El-Sayed’s rule, this guideline is often invoked to explain many aspects of spin-forbidden processes, such as diradical reactivity, 4,27–29 reaction of triplet oxygen with unsaturated compounds, 28,29 and phosphorescent properties of photovoltaic materials. 30–35 Molecular orbital analysis is also invoked to explain the breakdown of El-Sayed’s rules due to vibronic effects 30,36,37 — since the molecular vibrations affect the shape of molecular orbitals, they can result in mixing of nπ ∗ and ππ ∗ states, strongly modulating the magnitude of SOCC and the rates of intersystem crossing. Despite a wide-spread use of NTOs and natural orbitals to describe characters of the states involved in the spin-forbidden transitions (as in, for example, Refs. 30,31,38 ), there have been only a couple of studies reporting the NTOs between spin-coupled states 24 or between particular components of a multiplet. 29 In this contribution, we introduce a new type of NTOs suitable for the analysis of SOCs. The definition is based on the spinless density matrices and Wigner–Eckart’s theorem, which are used for the calculation of SOCs for the entire multiplet from just one transition. 39,40 This approach allows one to treat the transitions between states with arbitrary spin projections in a uniform way and to quantitatively describe the contributions of specific orbital pairs into the overall SOC. We use this tool to rationalize the magnitude of the SOCs computed using EOM-CC wave functions in terms of El-Sayed’s rules. 5,25 To introduce the key concept, NTOs of the spinless transition density matrix, we first briefly outline the formalism for SOC calculations 40 using Wigner–Eckart’s theorem with the Breit-Pauli Hamiltonian and EOM-CC wave functions. It is convenient to describe spinˆ S,M . These operators are defined as a dependent operators through spin-tensor operators O

6


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ˆ S,M , satisfying the following relations: 41,42 set of O h

i p ˆ S,M = S(S + 1) − M (M ± 1)O ˆ S,M ±1 , Sˆ± , O h i S,M ˆ ˆ ˆ S,M , Sz , O = MO

(9) (10)

where S and M label the operator’s spin and spin projection. Spin-independent operators are always singlet operators, because they commute with Sˆz and Sˆ± . We can split all oneelectron excitation operators into the singlet and triplet subsets: 43 1 0,0 Tˆpq = √ a†pα aqα + a†pβ aqβ , 2

1,0 Tˆpq

1,1 Tˆpq = −a†pα aqβ , 1 = √ a†pα aqα − a†pβ aqβ , 2 1,−1 Tˆpq = a†pβ aqα .

(11) (12) (13) (14)

Tˆ0,0 , Tˆ1,−1 , Tˆ1,0 , and Tˆ1,1 form a basis in the space of one-particle excitation operators. Because of this partitioning, the transition density matrix γ F I can be broken down into the singlet and triplet components. The singlet density matrices have identical αα and ββ parts, singlet singlet = γββ ; M = 0 triplet density matrices have the opposite signs of αα and ββ parts γαα triplet, M = 0 triplet, M = 0 γαα = −γββ . Because of this permutational property, the singlet and triplet

density matrices are orthogonal to each other. 44 To evaluate the transition matrix element, the operators of the singlet type require only singlet part of the transition density matrix from Eq. (2), while the triplet operators require only the triplet part. Since different parts of the density matrix give rise to different physical observables, one should analyze the part that is relevant for the investigated property. That is, for rationalizing transition dipole moments or non-adiabatic couplings between spin-coupled states, 45 one should consider the singlet part of γ, whereas for computing SOC between the non-relativistic zero-order states one should consider the triplet part of γ. The latter is the key quantity in our analysis. 7



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One-electron and mean-field Breit–Pauli spin–orbit operators have the following form: 40,46 ˆ SO H

1 = 2

X

† hSO L+ ,pq apβ aqα +

pq

1 2

X

† † hSO z,pq apα aqα − apβ aqβ +

pq

X pq

! X

† hSO L− ,pq apα aqβ

=

pq

√ X SO 1,0 X SO 1,−1 ˆ1,−1 + 2 hSO T hz,pq Tˆ − hL− ,pq Tˆpq L+ ,pq pq pq

! ,

(15)

pq

where L+ and L− parts are spherical components: , + ih1eSO = h1eSO h1eSO y x L+

(16)

h1eSO = h1eSO − ih1eSO . L− x y

(17)

The spherical components for the spin–orbit mean-field (SOMF) operator are defined in a similar way. 40 In practical calculations 39,40 of SOCs, it is convenient to apply Wigner–Eckart’s theorem ˆ S,M : for matrix elements with spin-tensor operators O ˆ S,M |S 00 M 00 i = hS 00 M 00 ; SM |S 0 M 0 i hS 0 ||O ˆ S,· ||S 00 i , hS 0 M 0 |O

(18)

ˆ S,· ||S 00 i is reduced matrix where hS 00 M 00 ; SM |S 0 M 0 i is a Clebsh–Gordan coefficient, hS 0 ||O element, which does not depend on spin projections. Applying Wigner–Eckart’s theorem to the triplet density matrices, we obtain a spinless triplet transition density matrix u1,· : 0 ˆ1,· ˆ1,m |IS 0 M 0 i / hS 0 M 0 ; 1m|SM i u1,· pq = hJS||Tpq ||IS i = hJSM |T

8


(19)

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Doing the same for the spin–orbit operator, we obtain the following relations: 1 X SO 1,· h u , 2 p,q L− ;pq pq √ X 0 ˆ L0 ||IS i = 2 hSO u1,· , hJS||H 2 pq Lz ;pq pq X 1,· ˆ L+ ||IS 0 i = + 1 hJS||H hSO L+ ;pq upq . 2 pq

ˆ L− ||IS 0 i = − hJS||H

(20) (21) (22)

As one can see, all relevant matrix elements are computed from u1,· . Thus, to analyze the entire spin–orbit matrix between the two multiplets, we perform SVD of u1,· and obtain an analogue of Eq. (2). The reduced spin–orbit matrix elements can be then written as ˆ L− ||IS 0 i = − hJS||H

1 X p SO h hζk |hL− |ζk i ωk , 2 k

√ X p h ˆ L0 ||IS i = 2 hζk |hSO hJS||H Lz |ζk i ωk , 2 k X p h ˆ L+ ||IS 0 i = + 1 hJS||H hζk |hSO L+ |ζk i ωk , 2 k 0

(23) (24) (25)

where ζkh,p (r) and ωk are the new spinless NTOs and the respective weights (singular values of u1,· ). Because all reduced spin–orbit matrix elements are expressed through the spin–orbit integrals over the spinless NTO pairs and the respective weights ωk are the same for all three reduced spin–orbit matrix elements, one can analyze different components of the spin–orbit matrix in a uniform way. We note that, in contrast to the parent γ, the norm of u1,· is not bounded by one because of the division by the Clebsh-Gordan coefficient, as per Eq. (19). Consequently, the magnitude of the respective singular values (ωk ) and their sum can exceed one. However, the extent of the collectivity of the transition (i.e., the number of significant NTO pairs) can be characterized by the participation ratio computed using the same expression as for regular

9



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NTOs: 20 P RN T O

P ( k ωk2 )2 = P 4 k ωk

(26)

For example, for a transition dominated by a single NTO pair, P RN T O =1, for a transition described by two NTO pairs with equal weights, P RN T O =2, and so on. Below we illustrate the utility of the spinless NTOs by considering several examples of open-shell species described using EOM-CC wave functions. EOM-CC target electronic states are parameterized using linear excitation (ionization, electron attachment) operators acting on the exponential coupled-cluster ansatz describing the reference state: ˆ Tˆ |Φ0 i , |ΨEOM i = Re

(27)

ˆ −Tˆ , hΨEOM | = hΦ0 | Le

(28)

ˆ and L ˆ are the EOM right excitation and where Tˆ is a coupled-cluster excitation operator, R left de-excitation operators. In the singles and doubles variant 47 (EOM-CCSD), Tˆ is truncated after single and double excitations. In the electron8 excitation (EOM-EE-CCSD) and 8 8 ÊOM-EA-CCSD states (EOM-SF-CCSD) of methylethylgalium states of methylethylgalium states were described of methylethylgalium by were EOM-EA-CCSD described werebydescribed from the byclosed-shell EOM-EA-CCSD from the cationic closed-shell from the closed-shell cationic cationic spin-flip 48,49 variants, R comprises 1h1p (one-hole-one-particle) and 2h2p reference; thesereference; calculations these reference; were calculations carried theseout calculations were at the carried doublet were out carried state at thegeometry doublet out at the state optimized doublet geometry with stateoptimized geometrywith optimized with !B97X-D (given !B97X-D in SI). (given !B97X-D in SI). (given in SI). In the ionization 50,51 (EOM-IP-CCSD) and electron(two-holes-two-particles) operators.

ˆ comprises 1h, 2h1p and attached 52 (EOM-EA-CCSD) variants, R 1p, 1h2p operators, re8 8 8 8 (3) (3) (3) spectively.

(1)of methylethylgalium (1) (1) (2)of methylethylgalium states states (2) of methylethylgalium were described states (2) of bymethylethylgalium EOM-EA-CCSD were described states by from were EOM-EA-CCSD the described closed-shell bywere EOM-EA-CCSD from cationic described the closed-shell by EOM-EA-CCSD fromcationic the closed-shell fromcationic the closed-shell cationic Reference State 1 out State 2 reference; these calculations reference; were thesecarried calculations reference; out at these were the calculations doublet reference; carried state these were atgeometry the calculations carried doublet optimized outstate were at the geometry carried with doublet out optimized state at the geometry doublet with optimized state geometry with optimized with

!B97X-D (given in!B97X-D SI). (given !B97X-D in SI). (given !B97X-D in SI). (given in SI).

(3)

(3)

(1)

(1) (2)

(3)

(3)

(1)(2)

(1)(2)

(2)

FIG. 1: Top, left: FIG. schematic 1: Top, FIG. picture left:1:schematic Top, of the left: hextet picture schematic reference of the picture hextet of (tpa)Fe. of reference the hextet Electron of reference (tpa)Fe. attachment of Electron (tpa)Fe. is attachment Electron attachment is is Reference State 1 State 2 State 3 shown by dashedshown lines. by The dashed attachments shownlines. by dashed The to the attachments lines. orbitals The(1) attachments toand the (2) orbitals generate to (1) the and orbitals the (2) leading generate (1) and configura(2) thegenerate leading configurathe leading configurations of the doubly tions degenerate of the doubly tions quintet ofdegenerate the ground doublystate quintet degenerate of the ground anion, quintet state (tpa)Fe ground of the.anion, state the anion, (tpa)Fe Theseof(tpa)Fe electronic . These states electronic . These states electronic states are denoted as State are denoted 1 and State as areState denoted 2 later. 1 and The as State State attachment 21 later. and State The to the 2attachment later. orbitalThe (3)to attachment gives the the orbital leading to(3) thegives configorbital the(3) leading givesconfigthe leading config153. schematic 4 electron 15picture uration of the non-degenerate uration of the uration State non-degenerate 3. of Top, the non-degenerate right: State schematic 3. Top,State right: picture Top, of right: schematic attachment of electron picture to the attachment of electron to attachment the to the + closed-shell + reference, reference, generating closed-shell the leading generating reference, configurations the generating leadingofconfigurations the theleading two lowest configurations ofdoublets the two lowest of thedoublets two lowest doublets EtMeGa+ closed-shell EtMeGa EtMeGa of the neutral EtMeGa of the neutral molecule. of EtMeGa theBottom: neutral molecule. EtMeGa considered Bottom: molecule. geometries considered Bottom: of thegeometries considered consideredofgeometries systems. the considered of thesystems. considered systems.

Figure 1: Left: Structure of (tpa)Fe (C N H Fe). Iron is shown in red, nitrogens in blue, carbons in gray, and hydrogens in white. Right: Frontier MOs and electronic configuration FIG. 1: Top, left: schematic FIG. 1: Top, picture left:of FIG. schematic the1:hextet Top, picture reference left:FIG. schematic of the of 1: (tpa)Fe. Top, hextet picture left: reference Electron schematic of theofhextet attachment (tpa)Fe. picture reference of Electron istheofhextet (tpa)Fe. attachment reference Electron is of (tpa)Fe. attachment Electron is attachment is of the d5 hextet reference and relevant states. The target are obtained by the shown by dashed lines. shown Theby attachments dashed target lines. shown to the The byorbitals dashed attachments (1) lines. shown and to The by (2) thedashed attachments generate orbitals lines. (1) theto and leading The the (2) attachments orbitals generate configura(1)states the to andthe leading (2)orbitals generate configura(1)the andleading (2) generate configurathe leading configuraWe consider theWe three consider lowest We the quintet consider three states lowest the three of quintet (tpa)Fe lowest states . quintet The of generation (tpa)Fe states . of The of (tpa)Fe these generation states . The is of generation these states of these is states is tions of the doubly degenerate tions of thequintet doublyground tions degenerate of state the doubly quintet of thetions degenerate anion, ground of the (tpa)Fe state doubly quintet of. the degenerate ground anion, state (tpa)Fe quintet ofstates the anion,electronic state (tpa)Fe of the anion,electronic (tpa)Fe .states These electronic . ground These .states These These electronic states attachment of a β-electron to one of the three MOs marked by2 later. the dashed box: attachment are denoted as in State are 1 and denoted as 2 later. State are 1denoted and attachment State asdegenerate 2State later. are tois 1the denoted The and orbital State attachment asdegenerate (3) 2State later. gives and the the State leading attachment orbital (3) configtoThe the the attachment orbital leading gives to the the orbital leading (3)configgives the leading configdepicted schematically depictedinschematically Figure depicted 1. The schematically ground Figure state 1.State inThe Figure quintet ground 1.The is The doubly state ground quintet state doubly quintet because is doubly of1toThe because degenerate ofgives because of(3)configuration of the non-degenerate uration ofState the non-degenerate 3. uration Top, right: of the schematic State non-degenerate 3. uration Top, picture right: of the of State schematic non-degenerate electron 3. Top, attachment picture right: State schematic of electron to3.the Top,picture attachment right: schematic of electron to thepicture attachment of electron to the attachment to the point-group symmetry point-group (C point-group symmetry (C symmetry (C ); these states are ); these denoted states ); as these State are denoted states 1 and are 2. as The denoted State non-degenerate 1 and as State 2. The 1 and non-degenerate 2. The non-degenerate to the two lowest MOsEtMeGa givesclosed-shell riseEtMeGa to closed-shell degenerate states 1theconfigurations and 2doublets and attachment the next reference, generating reference, the leading closed-shell generating configurations reference, theclosed-shell leading of generating two reference, lowest the leading generating of the configurations twothe lowest leading doublets of the configurations two lowest doublets of to the two lowest doublets EtMeGa EtMeGa of the neutral EtMeGa of the molecule. neutralBottom: EtMeGa of theconsidered molecule. neutral EtMeGa Bottom: geometries of themolecule. neutral considered of theEtMeGa considered Bottom: geometries molecule. considered systems. of the Bottom: considered geometries considered systems. of the considered geometries systems. of the considered systems. denoted quintet as State is denoted 3. quintet We have asis State denoted shown 3. We [37] as State have recently 3. shown We that have [37] taking recently shown into [37] that account recently taking only that into these account taking into onlyaccount these only these MO givesquintet riseis to state 3. 3

+

3

+ 3

+

+

three states leads three to the states spin-reversal three leads states to the barrier, leads spin-reversal toWe close the spin-reversal tobarrier, the experimental close barrier, toof the value close experimental [43]. toThe the Figure experimental value 2 [43]. Figure value [43]. 2 is Figure 2(tpa)Fe We consider the three lowest consider quintet the three We states consider lowest (tpa)Fe quintet the three We . states consider lowest generation of (tpa)Fe quintet the three ofstates . these lowest Theof generation states (tpa)Fe quintet states .ofThe these of generation states is .ofThe these generation states is of these states is 1,· shows the leading shows spinless the depicted leading shows NTO the pair spinless leading of the NTO reduced spinless pair of NTO triplet pair reduced transition the triplet density reduced transition matrix triplet density utransition u1,·The matrix u1,· schematically depicted in Figure schematically 1.the The depicted ground inof Figure schematically state 1.quintet depicted The ground inis Figure schematically doubly state 1. degenerate The quintet ground inmatrix Figure isdensity because doubly state 1. quintet degenerate of ground is doubly state because quintet degenerate of is doubly because degenerate of because of

for (tpa)Fe . We for show (tpa)Fe the for . transitions We (tpa)Fe show .the between We transitions show manifolds transitions between of the between degenerate the of manifolds degenerate nonof and nonand nonpoint-group symmetry point-group (Cthe symmetry point-group (C symmetry point-group (C symmetry (C these states are denoted );manifolds these as states State are 1and );denoted and these 2. states The asthe State non-degenerate are );denoted 1these and states 2. asThe State are non-degenerate denoted 1 and 2. asThe State non-degenerate 1 and 2. The non-degenerate 3 );the 3the 3the 3degenerate

To illustrate the we consider tris(pyrrolylmethyl)amine Fe(II) complex (which quintet isThe denoted as quintet State 3. is denoted We have quintet as shown State is[37] 3. denoted We recently have quintet asare State shown that is taking 3. denoted [37] Weorders recently into have asby account State shown that 3.orders [37] taking only Werecently have these into shown account that [37] taking only recently into these account that taking onlyinto these account only these degenerate states. degenerate Theanalysis, singular states. degenerate values states. of singular the remaining The values singular ofNTOs the values remaining are of by the two NTOs remaining orders ofby magnitude NTOs two are of two magnitude of magnitude three states leads three theFigure spin-reversal states leads three tobarrier, the states spin-reversal close leads tothree to thethe barrier, experimental states spin-reversal close leads to tovalue the the barrier, experimental spin-reversal [43].close Figure to 2the barrier, value experimental [43]. close Figure to the value 2 experimental [43]. Figurevalue 2 [43]. Figure 2 smaller than of smaller those shown than smaller of in those Figure than shown 2.of to those in shown2. in Figure 2. 1,· 1,· shows shows spinless the NTO leading pair shows spinless ofthe the the reduced NTO leading pair shows triplet spinless of the the transition NTO leading reduced pair spinless density triplet of thematrix NTO transition reduced pair u1,·triplet density of the transition reduced matrix utriplet density transition matrix udensity matrix u1,· Table I illustrates Table howI well illustrates Table thethe leading Ileading how illustrates well NTO the pair how leading represents well the NTO leading pair reduced represents NTO pair spin–orbit the represents reduced matrix the spin–orbit reduced matrix spin–orbit matrix

10

for (tpa)Fe . We show for (tpa)Fe the transitions . Weforshow (tpa)Fe between the transitions .the Wefor manifolds show (tpa)Fe between the oftransitions . the We thedegenerate show manifolds between the transitions and of the the nonmanifolds degenerate between ofand the the nonmanifolds degenerateofand the nondegenerate and non-

elements (full EOM-EA-MP2 elements (full elements EOM-EA-MP2 values). (full AsEOM-EA-MP2 onevalues). can see,As thevalues). one truncation can As see,one of thethe can truncation sums see, the from of truncation Eqs. the sumsoffrom the Eqs. sums from Eqs.


degenerate states. The degenerate singularstates. values degenerate The of the singular remaining states. values degenerate The NTOs of singular the areremaining states. by values two The orders NTOs of singular theofremaining are magnitude by values two NTOs orders of the remaining are of magnitude by two NTOs orders are of magnitude by two orders of magnitude

(23), (24), and (23), (25), (24), by retaining and (23),(25), (24), only byand the retaining (25), first by leading only retaining the term first only recovers leading the first 99% term leading ofrecovers the respective term 99% recovers of the 99% respective of the respective smaller than of those smaller shownthan in Figure of those smaller 2. shown thaninofFigure those smaller 2. shown thaninofFigure those 2. shown in Figure 2.

full value CHECK. full value CHECK. full valueisSometimes CHECK. Sometimes is 97%. Sometimes For is is small 97%. values isFor is 97%. it small is worse. For values small In it other is values worse. it is In worse. other In other

Table I illustrates how Table well I illustrates the leading Table how NTO well I illustrates pair therepresents leading Table howNTO well Ithe illustrates reduced the pairleading represents how spin–orbit NTO wellthe the pair matrix reduced leading represents spin–orbit NTO the pair reduced matrix represents spin–orbit the reduced matrixspin–orbit matrix

terms, only oneterms, spinless only NTO terms, onepair spinless only is needed one NTO spinless to pair describe isNTO needed the pair to physics isdescribe needed of these to thedescribe physics systems the ofcan and these physics to systems oftruncation these and systems toofthe and to elements (full EOM-EA-MP2 elements (full values). EOM-EA-MP2 elements As one(full canvalues). EOM-EA-MP2 see,elements theAs truncation one (full values). EOM-EA-MP2 see, of the the Assums one from can values). see, Eqs. the As truncation one sumscan from see, of Eqs. the thetruncation sums from of Eqs. the sums from Eqs.

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8

8

8

states of methylethylgalium states of methylethylgalium states were described of methylethylgalium by were EOM-EA-CCSD described werebydescribed EOM-EA-CCSD from the byclosed-shell EOM-EA-CCSD from the cationic closed-shell from the closed-shell cationic cationic reference; thesereference; calculations these reference; were calculations carried theseout calculations were at the carried doublet were out carried state at thegeometry doublet out at the state optimized doublet geometry with stateoptimized geometrywith optimized with !B97X-D (given !B97X-D in SI). (given !B97X-D in SI). (given in SI).

(3)

(3)

(3)

8

8

8

(1)of methylethylgalium (1) (1) (2)of methylethylgalium states states (2) of methylethylgalium were described states (2) of bymethylethylgalium EOM-EA-CCSD were described states by from were EOM-EA-CCSD the described closed-shell bywere EOM-EA-CCSD from cationic described the closed-shell by EOM-EA-CCSD fromcationic the closed-shell fromcationic the closed-shell cation

Reference State 1 out State 2 reference; these calculations reference; were thesecarried calculations reference; out at these were the calculations doublet reference; carried state these were atgeometry the calculations carried doublet optimized outstate were at the geometry carried with doublet out optimized state at the geometry doublet with optimized state geometry with optimized wit

!B97X-D (given in!B97X-D SI). (given !B97X-D in SI). (given !B97X-D in SI). (given in SI).

Figure 2: Left: Structure of the methylethylgalium (EtMeGa) radical (GaC3 H8 ). Gallium is shown in pink, carbons in gray, and hydrogens in white. (3) (3) Right: (3)Electronic (3) configuration of the closed-shell cationic reference and relevant target states. (1) The target states are (2) obtained (1) (1) (2) (2) (1)(2) by electron attachment toFIG. the two lowest unoccupied orbitals ofofElectron the by FIG. 1: Top, left: FIG. schematic 1: Top, picture left:1:schematic Top, of the left: hextet picture schematic reference of the picture hextet of (tpa)Fe. of reference the hextet Electron of reference (tpa)Fe. attachment (tpa)Fe. is cation attachment Electronmarked attachment is is dashed Reference State 1 State 2 State 3 shown by dashedshown lines. by The dashed attachments shownlines. by dashed The to the attachments lines. orbitals The(1) attachments toand the (2) orbitals generate to (1) the and orbitals the (2) leading generate (1) and configura(2) thegenerate leading configurathe leading configurabox. tions of the doubly tions degenerate of the doubly tions quintet ofdegenerate the ground doublystate quintet degenerate of the ground anion, quintet state (tpa)Fe ground of the.anion, state the anion, (tpa)Fe Theseof(tpa)Fe electronic . These states electronic . These states electronic states are denoted as State are denoted 1 and State as areState denoted 2 later. 1 and The as State State attachment 21 later. and State The to the 2attachment later. orbitalThe (3)to attachment gives the the orbital leading to(3) thegives configorbital the(3) leading givesconfigthe leading configuration of the non-degenerate uration of the uration State non-degenerate 3. of Top, the non-degenerate right: State schematic 3. Top,State right: picture 3. schematic Top, of electron right:picture schematic attachment of electron picture to the attachment of electron to attachment the to the + closed-shell + −reference, 53lowest doublets reference, generating closed-shell the leading generating reference, configurations the generating leadingofconfigurations the theleading two lowest configurations ofdoublets the two lowest of thedoublets two EtMeGa+ closed-shell EtMeGa EtMeGa of the neutral EtMeGa of the neutral molecule. of EtMeGa theBottom: neutral molecule. EtMeGa considered Bottom: molecule. geometries considered Bottom: of thegeometries considered consideredofgeometries systems. the considered of thesystems. considered systems.

we hereafter denote as (tpa)Fe ), exhibiting a large spin-reversal barrier,

and methylethyl-

FIG. 1: Top, left: schematic FIG. 1: Top, picture left:of FIG. schematic the1:hextet Top, picture reference left:FIG. schematic of the of 1: (tpa)Fe. Top, hextet picture left: reference Electron schematic of theofhextet attachment (tpa)Fe. picture reference of Electron istheofhextet (tpa)Fe. attachment reference Electron is of (tpa)Fe. attachment Electron is attachment

galium (EtMeGa) radical; their structures inandtoThe Figures 1toand and 2, (1)respectively. Togenerate shown by dashed lines. shown Theby attachments dashedare lines. shown to shown the The byorbitals dashed attachments (1) lines. shown by (2) thedashed attachments generate orbitals lines. (1) the leading The the (2) attachments orbitals generate configurathe to andthe leading (2)orbitals generate configura(1)the andleading (2) configurathe leading configur

We consider theWe three consider lowest the quintet consider three states lowest the three ofquintet lowest states . ground quintet The ofofgeneration (tpa)Fe states .oftions The of (tpa)Fe these states . The is of generation these states of these is states isof the tionsWe of the doubly degenerate tions of(tpa)Fe thequintet doubly tions degenerate state the doubly quintet of the degenerate anion, ground ofgeneration the (tpa)Fe state doubly quintet of. the degenerate ground anion, state (tpa)Fe quintet ofstates the anion, state (tpa)Fe anion,electronic (tpa)Fe .states These electronic . ground These electronic .states These These electronic stat

are denoted as in State 1 and denoted as 2 later. State are 1denoted and attachment State asdegenerate 2State later. are tois 1the denoted The and orbital State attachment asdegenerate (3) 2State later. gives and the the State leading attachment orbital 2 later. (3) configtoThe the the attachment orbital gives to the the orbital leading (3)configgives the leading confi depicted schematically depictedinschematically Figure depicted 1. The schematically ground Figure state 1.State inThe Figure quintet ground 1.The is The doubly state ground quintet state doubly quintet because is doubly of1toThe because degenerate ofgives because of(3)config−are 5leading

describe quintet states ofuration (tpa)Fe , weofState used EOM-EA-MP2/cc-pVDZ from dattachment ref-of electron of the non-degenerate uration the non-degenerate 3. uration Top, right: of the schematic State non-degenerate 3. uration Top, picture right: of the of State schematic non-degenerate electron 3. Top, attachment picture right: State schematic of electron to3.the Top,picture right:hextet schematic of electron to thepicture attachment to the attachment to th

+ closed-shell + leading + closed-shell point-group symmetry point-group (C3EtMeGa point-group symmetry (C symmetry (C ); these+ states );EtMeGa these denoted states as these State areEtMeGa denoted states 1reference, and are 2. as generating The denoted State non-degenerate 1reference, and as State 2. The 1 configurations and non-degenerate 2. non-degenerate 3are 3 ); closed-shell reference, generating the closed-shell configurations the leading of generating the two reference, lowest theThe leading doublets generating of the configurations twothe lowest leading doublets of the configurations two lowest doublets of the two lowest double EtMeGa

of the neutral EtMeGa of the molecule. neutralBottom: EtMeGa of theconsidered molecule. neutral EtMeGa Bottom: geometries of themolecule. neutral considered of theEtMeGa considered Bottom: geometries molecule. considered systems. of the Bottom: considered geometries considered systems. of the considered geometries systems. of the considered systems. quintet is1, denoted quintet as State is denoted 3. quintet We have asis State denoted shown 3. We [37] ascalculation, State have recently 3. shown We that have [37] taking recently shown that account recently taking only that into these account taking into onlyaccount these onlyand these followed erence (Fig. right panel). In this weinto[37] used the same geometry

three states leads three to the states spin-reversal three leads states to the barrier, leads spin-reversal toWe close the spin-reversal tobarrier, the experimental close barrier, toof the value close experimental [43]. toThe the Figure experimental value 2 [43]. Figure value [43]. 2 is Figure 2(tpa)Fe We consider the three lowest consider quintet the three We states consider lowest (tpa)Fe quintet the three We . states consider lowest generation of (tpa)Fe quintet the three ofstates . these lowest Theof generation states (tpa)Fe quintet states .ofThe these of generation states is .ofThe these generation states is of these states

40 1,· 1,· the sameshows protocol asspinless in The relevant states of methylethylgalium described by the leading shows the Ref. leading shows NTO the pair spinless leading of the NTO reduced spinless pair of NTO triplet pair reduced transition the triplet density reduced transition matrix triplet density utransition uwere matrix u1,· depicted schematically depicted in Figure schematically 1.the The depicted ground inof Figure schematically state 1. quintet depicted The ground inis Figure schematically doubly state 1. degenerate The quintet ground inmatrix Figure isdensity because doubly state 1. The quintet degenerate of ground is doubly state because quintet degenerate of is doubly because degenerate of because

for (tpa)Fe . We for show (tpa)Fe the for . transitions We (tpa)Fe show .the between We transitions show manifolds transitions between of the between degenerate the of manifolds degenerate nonof and nonand nonpoint-group symmetry point-group (Cthe symmetry point-group (C symmetry point-group (C symmetry (C these states are denoted );manifolds these as states State are 1and );denoted and these 2. states The asthe State non-degenerate are );denoted 1these and states 2. asThe State are non-degenerate denoted 1 and 2. asThe State non-degenerate 1 and 2. The non-degenerat 3 );the 3the 3the 3degenerate

EOM-EA-CCSD from the closed-shell cationic reference (Fig. 2,orders right panel); these calculaquintet isThe denoted as quintet State 3. is denoted We have quintet as shown State is[37] 3. denoted We recently have quintet asare State shown that is taking 3. denoted [37] We recently into have asby account State shown that 3.orders [37] taking only Werecently have these into shown account that [37] taking only recently into these account that taking onlyinto these account only thes degenerate states. degenerate The singular states. degenerate values states. of singular the remaining The values singular ofNTOs the values remaining are of by the two NTOs remaining orders ofby magnitude NTOs two are of two magnitude of magnitude three states leads three theFigure spin-reversal states leads three tobarrier, the states spin-reversal close leads tothree to thethe barrier, experimental states spin-reversal close leads to tovalue the the barrier, experimental spin-reversal [43].close Figure to 2the barrier, value experimental [43]. close Figure to the value 2 experimental [43]. Figurevalue 2 [43]. Figure smaller than of smaller those shown than smaller of in those Figure than shown 2.of to those in shown2. in Figure 2.

tions wereTable carried out atillustrates the doublet state geometry optimized with (given in the shows shows spinless the NTO leading pair shows spinless ofthe the the reduced NTO leading pair shows triplet spinless of the the transition NTO leading reduced pair spinless density triplet of ωB97X-D thematrix NTO transition reduced pair u triplet density of the transition reduced matrix utriplet density transition matrix udensity matrix u I illustrates Table howI well Table thethe leading Ileading how illustrates well NTO the pair how leading represents well the NTO leading pair reduced represents NTO pair spin–orbit the represents reduced matrix the spin–orbit reduced matrix spin–orbit matrix 1,·

SI).

1,·

1,·

for (tpa)Fe . We show for (tpa)Fe the transitions . Weforshow (tpa)Fe between the transitions .the Wefor manifolds show (tpa)Fe between the oftransitions . the We thedegenerate show manifolds between the transitions and of the the nonmanifolds degenerate between ofand the the nonmanifolds degenerateofand the nondegenerate and non

elements (full EOM-EA-MP2 elements (full elements EOM-EA-MP2 values). (full AsEOM-EA-MP2 onevalues). can see,As thevalues). one truncation can As see,one of thethe can truncation sums see, the from of truncation Eqs. the sumsoffrom the Eqs. sums from Eqs.

degenerate states. The degenerate singularstates. values degenerate The of the singular remaining states. values degenerate The NTOs of singular the areremaining states. by values two The orders NTOs of singular theofremaining are magnitude by values two NTOs orders of the remaining are of magnitude by two NTOs orders are of magnitude by two orders of magnitud

(23), (24), and (23), (25), (24), by retaining and (23),(25), (24), only byand the retaining (25), first by leading only retaining the term first only recovers leading the first 99% term leading ofrecovers the respective term 99% recovers of the 99% respective of the respective smaller than of those smaller shownthan in Figure of those smaller 2. shown thaninofFigure those smaller 2. shown thaninofFigure those 2. shown in Figure 2.

−small full value CHECK. full three value CHECK. full valueisSometimes CHECK. Sometimes isquintet 97%. Sometimes For isstates is small 97%. values is For is(tpa)Fe 97%. it small is worse. For values In it other is values worse. it is In worse. otherof In other states is We consider the lowest ofNTO The character these Table I illustrates how Table well I illustrates the leading Table how well I illustrates pair therepresents leading Table how.NTO well Ithe illustrates reduced the pairleading represents how spin–orbit NTO wellthe the pair matrix reduced leading represents spin–orbit NTO the pair reduced matrix represents spin–orbit the reduced matrixspin–orbit matr

terms, only oneterms, spinless only NTO terms, onepair spinless only is needed one NTO spinless to pair describe isNTO needed the pair to physics isdescribe needed of these to thedescribe physics systems the ofcan and these physics to systems oftruncation these and systems toofthe and to elements (full EOM-EA-MP2 elements (full values). EOM-EA-MP2 elements As one(full canvalues). EOM-EA-MP2 see,elements theAs truncation one (full values). EOM-EA-MP2 see, of the the Assums one from can values). see, Eqs. the As truncation one sumscan from see, of Eqs. the thetruncation sums from of Eqs. the sums from Eq

depicted schematically in(23),Figure 1.(23), The state state is a(25), doubly degenerate quintet (24), and (25), by retaining (24), ground and only (25), (23), the by(24), first retaining and leading (25), (23), only term by the (24), retaining recovers first andleading 99% only by term of theretaining the first recovers respective leading only 99% term the of first the recovers respective leading 99% term of the recovers respective 99% of the respectiv

full value CHECK. fullSometimes value CHECK. value CHECK. isfull is value Sometimes 97%.CHECK. For full small is is Sometimes 97%. values For it issmall is worse. Sometimes 97%. values InFor other it issmall is is worse. 97%. values For In itother small is worse. values In itother is worse. In othe

because of point-group symmetry (C3NTO ); we refer these states as State 1thesystems and 2.these The terms, only one spinless terms, onlypair one is spinless terms, neededonly NTO toto describe one pairspinless terms, is needed the only physics NTO toone pair describe ofspinless these is needed the systems NTO physics to pair describe and ofisto these needed physics to describe and of tothe systems physics and of these to systems and t next quintet state, State 3, is non-degenerate. As shown in a recent paper, 40 taking into account only these three lowest states is sufficient for accurate description of the spin-reversal barrier. 53 Figure 3 shows the leading spinless NTO pair of the reduced triplet transition density matrix u1,· for (tpa)Fe− . The singular values of the remaining NTOs are by two orders of magnitude smaller than of those shown in Figure 3 and the participation ratio for these transitions are 1.002 and 1.001. As one can see, the transitions between all three states can be described as transition between nearly perfect d-orbitals, dyz → dxz and dyz → dz2 . The respective value of SOC (given in Table 1) are large, which can be explained by generalized El-Sayed’s rules, as discussed below. The NTOs corresponding to the transitions between 11



the two doublet states in EtMeGa a pair of flipped p-like orbitals giving rise to a substantial value of SOC. The participation ratio for this transition is 1.008. As per Eq. (19), the normalization of u1,· is different from the normalization of regular one-particle transition densities (γ). Consequently, in contrast to the singular values of γ, the singular values of u1,· (ωk ) may exceed 1, as it happens in EtMeGa. u1,· connects the EE spin-flip (γ SF ) and spin-preserving triplet (γtriplet ) transition density matrices via Clebsh–

Gordan coefficients. The numeric consequences of this relationship are analyzed in the SI by using the electronic states of EtMeGa as an example. Table 1 also illustrates how well the leading NTO pair represents the reduced spin–orbit matrix elements (full EOM-EA values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25) by retaining only the first leading term recovers the full value with good accuracy. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to recover the full SOC matrix with high accuracy in both one-electron and mean-field cases, which is consistent with the respective participation ratios. State 2, particle (dxz)

State 1, hole (dyz)

ω=0.88 State 3, particle (dz2)

State 1, hole (dyz)

ω=0.87

Figure 3: Spinless triplet NTOs for the transitions between three lowest quintet states in (tpa)Fe− . The states 1 and 2 are degenerate. Red, green, and blue axes indicate X, Y , and Z coordinates axes, respectively. The isovalue of 0.050 was used in all the cases. We conclude our analysis by discussing the computed NTOs and SOCs in terms of ElSayed’s rules. 5,25 Originally introduced on the basis of the selection rules for p-orbitals, El-Sayed’s rule 5 explains why the change in orbital character (e.g., n → π) is needed for at-

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State 1, hole (pxy)

State 2, hole (pz)

ω=1.19

Figure 4: Spinless triplet NTOs for the transitions between the two lowest doublet states of EtMeGa system. Red, green, and blue axes indicate X, Y , and Z coordinates axes, respectively. The isovalue of 0.050 was used in all the cases. Table 1: Spin–orbit mean-field reduced matrix elements of the considered systems in the selected orientations. (tpa)Fe− , 1 → 2 ||Si hS||HLSO − SO hS||HL0 ||Si hS||HLSO ||Si + − (tpa)Fe , 1 → 3 hS||HLSO ||Si − SO hS||HL0 ||Si ||Si hS||HLSO + EtMeGa, 1 → 2 hS||HLSO ||Si − SO hS||HL0 ||Si hS||HLSO ||Si +

hζ p |hSO |ζ h i × ω full EOM-EA-MP2 −0.30 + 0.09i −0.29 + 0.10i −209.04 −211.34i −0.30 − 0.09i −0.29 − 0.10i p SO h hζ |h |ζ i × ω full EOM-EA-MP2 −31.79 + 224.48i −32.07 + 228.23i 0.04i 0.07i −31.79 − 224.44i −32.07 − 228.23i p SO h hζ |h |ζ i × ω full EOM-EA-CCSD 135.87 + 210.65i 132.69 + 205.38i −1.68i −1.80i 135.87 − 210.65i 132.69 − 205.38i

taining large SOCs. In the context of transition-metal photochemistry, following El-Sayed’s reasoning explains why mixing of metal-to-ligand charge-transfer configurations facilitates spin-forbidden transitions on the metal and that large values of SOC can be obtained when the respective singlet and triplet states involve different components of the d-orbital manifold. 32 Here we follow this strategy and generalize El-Sayed’s rules to the case of arbitrary orbitals involved in a localized transition. The generalization is based on the representations of the angular momentum operators in real spherical harmonics. The derivation and the key expressions are given in the SI. This generalization provides selection rules, which can be used to rationalize the magnitude of SOC for a particular transition and to also explain the effect of molecular vibrations on the SOCs. It also explains relative values of different 13



components of the SOC. Figure 4 shows the leading spinless NTO pair of the type of pxy → pz , where pxy is a linear combination of px and py orbitals in the shown coordinate system. According to ˆ ± |pz i the matrix representation of the angular momentum in the basis of p-orbitals, the hpx |L ˆ SO ||Si and hS||H ˆ SO ||Si components matrix elements are non-zero, which leads to large hS||H L− L+ ˆ SO ||Si is close to zero. of spin–orbit coupling, while hS||H Lz The generalized El-Sayed’s rules predict not only the transitions with large SOC, but also estimate the relative values of SOC for different transitions in the same molecule. For example, the transitions in Figure 3 correspond to the dyz → dxz and dyz → dz2 cases. The ˆ z operator with a relative magnitude of X. The case of dyz → dxz is allowed through L main contribution to the SOC from this transition in this orientation indeed comes from ˆ SO ||Si (Table 1). The dyz → dz2 transition is allowed through L ˆ + and L ˆ − with a relahS||H L0 √ tive magnitude of X 3. The computed spin–orbit matrix elements confirm this prediction as ˆ SO ||Si well: the main contribution to the couplings comes from the imaginary parts of hS||H L− ˆ SO ||Si. There is also an increase in the magnitude relative to the first case. The and hS||H L+ √ increase is not exactly 3, which is likely due to a small mismatch of the coordinate system ˆ SO ||Si) and deviation of the shape of the NTOs from ˆ SO ||Si and hS||H (real parts of hS||H L+ L− ideal d-orbitals, which is especially clear for dz2 -like orbital.

To conclude, we introduced a new type of NTOs, which are suitable for describing spinforbidden transitions. The new NTOs are obtained by the SVD procedure of the reduced spinless transition density matrix, the quantity determining the SOC values for the entire multiplet by virtue of Wigner–Eckart’s theorem. These spinless NTOs describe the transitions between states with arbitrary spin projections in a uniform way. In addition to pictorial representation of the transition, the analysis also yields quantitative contributions of hole-particle pairs into the overall many-body matrix elements and helps to rationalize the magnitude of computed SOCs in terms of El-Sayed’s rules. By providing a clear orbital

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picture of the transitions at different geometries, this analysis can also be used to explain vibronic effects on the rate of spin-forbidden transitions. We hope that these tools will be helpful in deriving insight from high-level electronic structure calculations of phenomena facilitated by SOCs.

Acknowledgement This work is supported by the Department of Energy through the DE-SC0018910 grant. A.I.K. is also a grateful recipient of the Simons Fellowship in Theoretical Physics, of a visiting professorship of the MAINZ graduate school of excellence, and of the Mildred Dresselhaus Award, which supported her sabbatical stay in Germany.

The authors declare the following competing financial interest(s): A.I.K. is a member of the Board of Directors and a part-owner of Q-Chem, Inc.

Supporting Information Available Derivation of generalized El-Sayed’s rules, comparison of one-electron and mean-field values of SOCs, analysis of normalization of the spinless density matrices, Cartesian coordinates of EtMeGa.

15



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(40) Pokhilko, P.; Epifanovsky, E.; Krylov, A. J. Chem. Phys. 2019, 151, 034106. (41) Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular electronic structure theory; Wiley & Sons, 2000. (42) Racah, G. Phys. Rev. 1942, 62, 438–462. (43) Here we assume a spin-restricted set of orbitals that have the same spatial parts for α and β spins. ˆ Bi ˆ = (44) In the sense of the inner product of two operators defined as hA,

P

pq

ˆpq = Aˆpq B

T r[A† B]. (45) Matsika, S.; Feng, X.; Luzanov, A. V.; Krylov, A. I. J. Phys. Chem. A 2014, 118, 11943–11955. (46) We assume that the artificial singlet part of SOMF has been already removed, as in Ref. 40 . (47) Stanton, J. F.; Bartlett, R. J. J. Chem. Phys. 1993, 98, 7029–7039. (48) Krylov, A. I. Chem. Phys. Lett. 2001, 338, 375–384. (49) Krylov, A. I. Acc. Chem. Res. 2006, 39, 83–91. (50) Stanton, J. F.; Gauss, J. J. Chem. Phys. 1994, 101, 8938–8944. (51) Pieniazek, P. A.; Bradforth, S. E.; Krylov, A. I. J. Chem. Phys. 2008, 129, 074104. (52) Nooijen, M.; Bartlett, R. J. J. Chem. Phys. 1995, 102, 3629–3647. (53) Freedman, D. E.; Harman, W. H.; Harris, T. D.; Long, G. J.; Chang, C. J.; Long, J. R. J. Am. Chem. Soc. 2010, 132, 1224–1225. (54) Head-Gordon, M.; Maslen, P.; White, C. J. Chem. Phys. 1998, 108, 616–625.

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Quantitative El-Sayed Rules for Many-Body Wave Functions

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