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4THE QUANTUM CHEMISTRY OFOPEN-SHELL SPECIES

Anna I. KrylovDepartment of Chemistry, University of Southern California, Los Angeles, CA, United States

INTRODUCTION AND OVERVIEW

The majority of stable chemical compounds have closed-shell electronic configura-tions. In this type of electronic structure, all electrons are paired and manifolds oforbitals that have the same energy are either completely filled or empty, as in dini-trogen shown in Figure 1. Depending on the type of MO hosting a pair of electrons,each pair gives rise to bonding (such as 𝜋-orbitals in N2), antibonding (e.g., 𝜋∗CC invinyl), or nonbonding (lone pairs) interactions.

Due to the Pauli exclusion principle, electrons can occupy the same MO only whenthey have opposite spins. Thus, species that have different number of 𝛼 and 𝛽 electronsform an open-shell pattern. The examples include doublet radicals and triplet elec-tronic states (see Figure 1). These types are often described as high-spin open-shellspecies, where “high spin” refers to the nonzero Ms values of the underlying wavefunctions. (The terms high spin and low spin are also used to distinguish differentcomponents of a multiplet. In some cases, all the components may have nonzeroMs. For example, for a system with 3-electrons-on-3-orbitals, the high-spin state is aquartet and the low-spin state is a doublet.)

Open-shell character can be found even when N𝛼= N

𝛽. This happens when a

degenerate orbital manifold is only partially filled. Consider, for example, a moleculewith a stretched bond, such as H2 shown in Figure 2. At large internuclear separations,the energy gap between the bonding and antibonding 𝜎-orbitals disappears and the

Reviews in Computational Chemistry, Volume 30, First Edition.Edited by Abby L. Parrill and Kenny B. Lipkowitz.© 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

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152 REVIEWS IN COMPUTATIONAL CHEMISTRY

[core]4

N2

π*CC

lp(C)

σCC

πCC

Vinyl

(a) (b) (c)

Methylene

FIGURE 1 Examples of closed-shell and open-shell species. Dinitrogen in its groundelectronic state is a closed-shell molecule. All electrons are paired and degenerate molecularorbitals (MOs) such as the two 𝜋-orbitals are completely filled. Vinyl is an example of a doubletradical with one unpaired electron (N

𝛼− N

𝛽= 1). Triplet methylene, which has two unpaired

electrons, is a diradical (the high-spin Ms = 1 component in which N𝛼− N

𝛽= 2 of the triplet

state is shown). (See color plate section for the color representation of this figure.)

−λ

σ∗

σ∗

σσ

RHH

−Ψ Ψ= =

FIGURE 2 Closed-shell molecules acquire open-shell character at stretched geometries,where degeneracy between frontier MOs develops. At equilibrium geometry, the coefficient 𝜆 issmall and the wave function is dominated by (𝜎)2. At the dissociation limit, both configurationsbecome equally important resulting in an open-shell singlet radical pair, H.

A + H.

B. (See colorplate section for the color representation of this figure.)

two orbitals form a degenerate manifold, which is only partially filled. The resultingwave function is a linear combination of the two electronic configurations, (𝜎)2 and(𝜎∗)2. Although each of these configurations is of a closed-shell type, the total wavefunction corresponds to the two radicals, H.

A and H.

B, as one can easily verify byexpanding the two determinants in terms of the atomic orbitals (AOs), sA and sB:

Ψ = 1√2[|𝜎𝛼𝜎𝛽⟩ + |𝜎∗𝛼𝜎∗𝛽⟩] = 1

2[𝜎𝜎 − 𝜎∗𝜎∗] × (𝛼𝛽 − 𝛽𝛼)

= 12[(sA + sB)(sA + sB) − (sA − sB)(sA − sB)] × (𝛼𝛽 − 𝛽𝛼)

= [sAsB + sBsA] × (𝛼𝛽 − 𝛽𝛼) [1]

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 153

The stretched H2 molecule is an example of a singlet diradical.1 Diradical speciescan be formed when two radical centers reside on a rigid molecular scaffold, whichrestricts their ability to form a chemical bond, such as in para-benzyne shown inFigure 3. Breaking chemical bonds often leads to formation of pairs of radicals ordiradical intermediates, as depicted in Figure 4. Similarly, molecules at transitionstates can be described as diradicals.2 Species containing transition metals often fea-ture open-shell character. Molecules with multiple radical centers, such as molecularmagnets,3,4 are polyradicals. Two examples of molecular magnets with multiple tran-sition metal centers hosting unpaired electrons are shown in Figure 5.

Open-shell species can also be formed in various photoinduced processes. Inphotoexcitation, electrons promoted by light to higher orbitals become unpaired(consider for example the 𝜋𝜋∗ excited state in ethylene). Removing electrons fromclosed-shell species yields radicals. For example, photoionization of closed-shellneutral molecules or photodetachment from closed-shell anions leads to charged or

−λ

Para-benzyne

C

C

FIGURE 3 para-Benzyne is an example of a singlet diradical. Its open-shell characterderives from the fact that the two nearly degenerate frontier MOs are only partially occupiedin each of the two leading electronic configurations. (See color plate section for the colorrepresentation of this figure.)

>CC< >CC<

. .>C–C<

FIGURE 4 Molecules with partially broken bonds can be described as diradicals. Thiscartoon depicts a diradical transition state often encountered in cis–trans isomerization arounddouble bonds and a diradical intermediate in a cycloaddition reaction.

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(a) (b)

FIGURE 5 Examples of molecular magnets. (a) Molecular structure of a Cr(III) horseshoecomplex. Each of the six chromium atoms has three unpaired electrons. (b) Structure of a nickelcomplex with the nickel-cubane core (Ni4O8). Each of the four nickel atoms has two unpairedelectrons. Reproduced with permission of American Chemical Society from Ref. 5.

neutral doublet radicals. Electron attachment to closed-shell species also producesradicals. Although open-shell character is associated with highly reactive andunstable species, there exist a number of relatively stable radicals, such as dioxygen,ClO2, and NO.

In sum, open-shell electronic structure is ubiquitous in chemistry andspectroscopy. To computationally describe reaction profiles, intermediates,and catalytic centers, we need to be able to describe open-shell states on the samefooting (and with the same accuracy) as closed-shell states. This, it turns out, is hardto achieve. In this chapter, we describe the challenges posed by open-shell speciesto quantum chemistry methodology and provide a guide for practical calculations.We will discuss both wave function–based methods and density functional theory(DFT). Since open-shell species are vastly diverse, the problems and, consequently,the appropriate theoretical approaches vary. We will begin by summarizing standardquantum chemistry methods, outlining their scope of applicability, and explainingwhy open-shell species require special treatment. We will then discuss severaltopics relevant to open-shell systems: spin and spin contamination, Jahn–Teller(JT), and pseudo-Jahn–Teller (PJT) effects. The following section will discuss theelectronic structure of high-spin states and contrast cases with and without electronicdegeneracies. Methods designed to tackle various types of open-shell electronicstructure (simple high-spin states, charge-transfer systems, diradicals and triradicals,excited states of open-shell molecules) will be introduced, and their applications willbe illustrated by examples. We will then discuss calculations of molecular propertiesrelevant to spectroscopy and excited-state processes. We hope that this chapterwill provide a practical guide for computational chemists interested in open-shellsystems.

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QUANTUM CHEMISTRY METHODS FOR OPEN- AND CLOSED-SHELLSPECIES

Quantum chemistry offers practical methods for solving the electronic Schrödingerequation.6,7 The exact wave function is represented by a linear combination ofall possible Slater determinants that can be generated for a given orbital basis.Systematic approximations to this factorially complex object give rise to practicalcomputational methods balancing accuracy and cost.6–8 The simplest member ofthis hierarchy (shown in Figure 6) is the Hartree–Fock (HF) method, in which thewave function is a single Slater determinant. It is often referred to as the pseudononinteracting electrons model because it approximates the electron–electron inter-action by a mean-field Coulomb potential. The missing electron correlation can begradually turned on by including selected sets of excited determinants into the wavefunction. In the simplest correlated method, Møller–Plesset theory of the secondorder (MP2), the contributions of all doubly excited determinants are evaluated bymeans of perturbation theory. Including singly and doubly excited determinantsexplicitly by using the exponential ansatz gives rise to the CCSD (coupled-clusterwith single and double substitutions) method, and so on. Climbing up this ladder ofmany-body theories, one can improve the accuracy of the description, systematicallyapproaching the exact solution. Importantly, this series relies on the zero-order wavefunction, Φ0, being a good approximation to the exact solution. This is usually

Ground-state methods

Hartree–Fock: Ψ = Φ0 = |ϕ1...ϕn>MP2: HF + T2 by PT

CCSD: Ψ = exp(T1+T2)Φ0

CCSDT: Ψ = exp(T1+T2+T3)Φ0

(a) (b)

CCSD(T): CCSD + T3 by PT

FCI: (1+C1+C2+....CN)Φ0 = exp(T1+T2+...+Tn)Φ0

Virtual MOs:a,b,c,d,...

Occupied MOs:i,j,k,l,...

Φ0

FIGURE 6 (a) Hierarchy of ab initio methods for ground electronic states. The simplestapproximation of an N-electron wave function is a single Slater determinant, Φ0, built oforbitals that are variationally optimized. The description can be systematically improved byincluding excited determinants up to the exact solution of the Schrödinger equation by fullconfiguration interaction (FCI). Operators Tk and Ck generate k-tuply excited determinantsfrom the reference, for example, T1Φ0 =

∑iata

i Φai . The amplitudes (ta

i , tabij , … ) are determined

either by solving coupled-cluster equations or by perturbation theory. In the case of closed-shellspecies, 𝛼 and 𝛽 spin-orbitals have the same spatial parts and the resulting wave functionsare naturally spin-pure. (b) Reference determinant Φ0 (also called the Fock-space vacuumstate) determines the separation of the orbital space into the occupied and virtual subspaces.Traditionally, indices i, j, k, … are used to denote occupied orbitals and a, b, c, … are usedfor virtuals. General-type orbitals are denoted by p, q, r, … . In single-reference methods,excitation operators (such as Tk and Ck) are determined with respect to Φ0.

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the case when the energy gap between the ground and electronically excited statesis large, as is the case in stable closed-shell molecules. The success of quantumchemistry can be attributed to the fact that the Hartree–Fock approximation worksremarkably well for closed-shell molecules with a large gap between the occupied(in the reference determinant, Φ0, shown in Figure 6(b)) and virtual MOs.

How does one describe excited states? In the case of the exact full configurationinteraction (FCI) method, the excited states are just higher eigenroots of the secularequation. These higher eigenroots are orthogonal to the lowest one (Ψ0) and can bedescribed as excitations from Ψ0. Thus, for an approximate wave function, one canrepresent excited states as excitations from the given ground-state wave function:

Ψex = RΨ0 [2]

where R is a linear excitation operator, that is,

R1 =∑

ia

rai a+i [3]

R2 = 14

∑ijab

rabij a+b+ji [4]

· · ·

Here second-quantization operators p+∕p create/annihilate electrons on the respec-tive MOs.6,9 Operators R1 and R2 thus can be described as excitation operators ofhole-particle (hp) and 2-hole-2-particle (2h2p) types. A many-body operator is calledan excitation operator only when it includes creation operators that create electronsin virtual orbitals and annihilation operators that remove electrons from occupiedorbitals. This is a very important requirement, which makes single-reference CC andEOM-CC methods simple and elegant.

Such an ansatz can be justified by, for example, considering the response of anapproximate wave function to a time-dependent perturbation.6 The level of excitationin R is determined by the correlation treatment in the ground state, as illustrated inFigure 7. The amplitudes of excitation operators R, ra

i , and rabij are found by solving

an eigenproblem with a model Hamiltonian, which can be derived by applying avariational (or bivariational) principle to the original Schrödinger equation. For theCIS method (configuration interaction singles), this is simply the diagonalization ofthe bare Hamiltonian in the basis of all singly excited determinants. In the case ofEOM-CC, EOM amplitudes R are found by diagonalizing the so-called similaritytransformed Hamiltonian:10,11

HR = RΩ [5]

H ≡ e−THeT [6]

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Excited-state methods

Hartree–Fock -> CIS: Ψ = R1Φ0

Ψ = (R0 + R1 + R2) exp(T1 + T2)Φ0

Ψ = (R0 + R1 + R2 + R3) exp(T1 + T2 + T3)Φ0

CCSD -> EOM-CCSD:

CCSDT -> EOM-CCSDT:

FIGURE 7 Hierarchy of ab initio methods for electronically excited states. The excited-statemodel corresponding to the Hartree–Fock ground-state wave function is configurationinteraction singles (CIS) in which excited states are described as a linear combination of allsingly excited Slater determinants,

∑ra

i Φai . For CCSD, the excited-state ansatz is a linear

combination of all single and double excitations from the reference CCSD wave function, andso on.

where T are the CC amplitudes computed for the reference state:

ECC = ⟨Φ0|H|Φ0⟩ [7]

⟨Φ𝜇|H − ECC|Φ0⟩ = 0 [8]

and 𝜇 denotes the level of excitation in the CC ansatz (e.g., 𝜇 = 1, 2 in CCSD).H has the same spectrum as the original Hamiltonian, H. Thus, when diagonal-

ized over the full many-electron basis, Eq. [4] produces energies identical to FCI.However, when diagonalized in an incomplete many-electron basis (e.g., withinsingly and doubly excited manifold, {Φa

i ,Φabij }, in the case of EOM-CCSD), it

delivers higher accuracy than the equivalent CI method (CISD) due to rigoroussize-extensivity and correlation effects that are folded into H through the similaritytransformation employing the CC amplitudes.

What happens with the Hartree–Fock approximation and the whole hierarchy ofmany-body methods from Figures 6 and 7 in the case of open-shell systems? Depend-ing on the type of open-shell wave function, different problems emerge. The firstproblem is spin-contamination, which stems from the different mean-field potentialsfor 𝛼 and 𝛽 electrons when N

𝛼≠ N

𝛽. The second type of problem, which is much

harder to deal with, occurs when wave functions become multiconfigurational due toelectronic degeneracies, such as HOMO–LUMO (highest occupied and lowest unoc-cupied MOs, respectively) degeneracies in a molecule with a partially broken bond(Figures 2 and 3) or in charge-transfer type states, for example, (H2 … H2)+ shownin Figure 8. Multiconfigurational wave functions also appear due to spin-adaptationof open-shell electronic configurations (e.g., two Slater determinants with equalweights are needed to describe the Ms = 0 component of a triplet state). Becausethe hierarchy of methods in Figure 6 is built assuming that the wave function isdominated by a single determinant, it fails to provide a balanced description ofthe multiple leading configurations (of course, the imbalance is reduced as morecorrelation is included through higher excitations and it disappears at the FCI level).

An alternative and altogether different route toward high accuracy is pursuedby DFT, which, in principle, could deliver the exact energy from electron density

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(H-H....H-H)+ H–H+....H-H

Ψ1 =

Ψ2 =

(a) (b)

FIGURE 8 Molecular orbitals in symmetric and asymmetric (H2)2. By virtue of Koopmans’theorem, the shapes of the MOs represent the charge localization in the two electronic statesof the ionized dimer, (H2)+2 (the electronic configurations of the two cationic states are shownin the box). When the two H2 molecules are identical (a), the charge is equally delocalized inthe ground (Ψ1) and lowest excited (Ψ2) states of the cation. When the bond is stretched in oneH2, the MOs become localized, that is, in the ground state of the cation, the positive chargeis localized on the stretched H2 moiety. At low symmetry, Ψ1 and Ψ2 can mix, resulting inmulticonfigurational wave functions. (See color plate section for the color representation ofthis figure.)

alone,12–14 subject to the unknown energy functional. Current implementations ofDFT use the Kohn–Sham formulation in which the density is described by a singledeterminant and the orbitals are found by solving Hartree–Fock type equations.Although lacking the ability to systematically converge to the exact solution, practi-cal DFT implementations with various representations of the exchange-correlationfunctional are very useful. The success of the Kohn–Sham formulation of DFTstems from the same fundamental reason that the mean-field description captures∼90 percent of the physics in normal molecules. Consequently, although formallyexact for any type of wave functions, in the case of open-shell species, Kohn–ShamDFT experiences problems similar to those of the wave function-based methods.

Below we consider different types of open-shell species and explain the keymethodological challenges and solutions. This chapter focuses on the CC/EOM-CCmethods whose elegant formalism, attractive mathematical properties, and robustperformance are now well recognized.10,11,15–20 We note that the so-calledlinear-response CCSD (LR-CCSD) methods21–23 yield identical equations forthe energies and employ slightly different expressions for calculating molecularproperties.20,24 The SAC-CI (symmetry-adapted cluster CI) methods25,26 can bedescribed as EOM-CC in which threshold-based selection of configurations isemployed. There are also alternative approaches that have similar properties. In par-ticular, ADC (algebraic-diagrammatic construction) and propagator-based familiesof methods, albeit derived in a very different spirit, share many attractive features,such as the ability to describe multiple states and to tackle multiconfigurational wavefunctions.27,28

Finally, we would like to highlight the difference between the terms “multi-configurational” and “multireference.” The former refers to a concrete physicalphenomenon, wave functions of complex character that cannot be capturedby a single Slater determinant. The latter refers to the methodology choice,that is, abandoning the single-determinantal definition of the vacuum state and

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employing a reference that is a linear combination of several determinants. Impor-tantly, describing multiconfigurational wave functions does not require usingmultireference formalism. Methods described in this chapter—such as EOM-CCand ADC—treat multiconfigurational wave functions in a strictly single-referenceframework. This leads to elegant formalism and robust numeric performance. Mostimportant, these methods do not involve system-dependent parameterization such asa problem-specific choice of an active space and, therefore, fulfill Pople’s criteria ofthe theoretical model chemistry.29

SOME ASPECTS OF ELECTRONIC STRUCTURE OF OPEN-SHELLSPECIES

Spin Contamination of Approximate Open-Shell Wave Functions

The exact solutions of the electronic Schrödinger equation have certain spin and spa-tial symmetry, which stems from the basic quantum mechanical result that commutingoperators share a common set of eigenstates. Relevant operators are spatial trans-formations of symmetry-equivalent nuclei that do not change the electron-nuclearoperator and the spin operators, Sz (projection of the total spin) and S2 (the magnitudeof the spin):

Sz =N∑

i=1

𝜎z(i) [9]

S2 = 3N4

+N∑

i, j=1

(𝜎x(i)𝜎x(j) + 𝜎y(i)𝜎y(j) + 𝜎z(i)𝜎z(j)) [10]

where 𝜎x,y,z are the Pauli matrices and N is the total number of the electrons. It isdesirable that an approximate wave function also satisfies these symmetries. How-ever, owing to the nonlinear nature of the approximate ansatz, it is not guaranteed.Possible spin and symmetry breaking is most pronounced at the Hartree–Fock level.Using a spin-orbital basis in which each orbital has pure 𝛼 or 𝛽 spin leads to Slaterdeterminants that are eigenstates of Sz. In the case of the closed-shell species, onecan trivially enforce the requirement that the wave function be an eigenstate of S2

by making the spatial parts of the 𝛼 and 𝛽 spin orbitals the same. Similarly, spatialsymmetry can be enforced by not allowing the MOs to break point-group symmetry.

A principal difference between closed- and open-shell electronic structure isthat in the former enforcing spin and spatial symmetry is trivial, whereas in thelatter, this task is more difficult. In fact, a special effort is often required to findsymmetry-broken solutions of the Hartree–Fock equations for systems with evennumber of electrons, for example, by deliberate mixing of MOs when formingthe guess density for the self-consistent procedure. For closed-shell systems,starting from spin- and symmetry-pure Φ0 automatically results in pure correlatedwave functions, even if no constraints on the amplitudes are imposed. In contrast,

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maintaining the correct spin and spatial symmetry is cumbersome (to say the least)in the case of open-shell species.

The deviation of approximate wave functions from spin-pure forms is calledspin contamination; it often signals poor quality of the results. In open-shellwave functions, spin contamination can arise for two distinct reasons: (i) differentmean-field experienced by 𝛼 and 𝛽 electrons in the N

𝛼≠ N

𝛽configurations and

(ii) spin contamination due to an incomplete set of configurations in model wavefunctions. Consider two doublet states: 2|(𝜙1)2(𝜙2)1⟩ and 2|(𝜙1)1(𝜙2)1(𝜙3)1⟩. In theformer, spin contamination is usually minor even at the Hartree–Fock level becausea spin-pure wave function of this character can be described by a single Slaterdeterminant (in which only one electron is unpaired). However, in the latter case (anopen-shell doublet state with three unpaired electrons), constructing a spin-pure wavefunction requires taking a linear combination of several Slater determinants, withcoefficients determined by spin-adaptation rules. Consequently, spin contaminationof just one determinant of this character is large, and attempts to describe these typesof states by single-reference methods from Figure 6 are bound to be problematic.

Jahn–Teller Effect

According to the JT theorem,30–33 high-symmetry structures with degenerateelectronic states are not stationary points on potential energy surfaces (PESs) innonlinear molecules, which means that degenerate electronic states follow first-orderdistortions to a lower symmetry at which the degeneracy is lifted. The JT effectoften occurs in radicals derived by removing or adding an electron from/to ahigh-symmetry closed-shell reference system, such that the unpaired electron resideson one of the doubly degenerate MOs. The total charge is of no significance—theJT pattern can be found in radical anions, radical cations, and neutral radicals.Specifically, non-Abelian point-group symmetries containing irreducible represen-tations (irreps) of order higher than one give rise to symmetry-required degeneraciesbetween the MOs belonging to these irreps. Two examples of JT systems, thebenzene radical cation and the cyclic N3 radical, are shown in Figure 9. The JTeffect also occurs in systems with N

𝛼= N

𝛽, such as high-symmetry diradicals and

triradicals or electronically excited closed-shell molecules.34,35

More precisely, the JT theorem states that for any geometry with symmetry-required degeneracy between the electronic states, there exists a nuclear displacementalong which the linear derivative coupling matrix element between the unperturbedstates and the difference between the diagonal matrix elements of the perturbationare not required to be zero. Because of the linear term, the JT degenerate PESs forma conical intersection. This leads to singular points on adiabatic PESs clearly seen inFigure 10, which illustrates the JT effect for the cyclic N3 radical.38,39

An important feature of such an intersection is that the symmetry of thelowest electronic state changes as one passes through the CI, as one can see inFigure 10b. Consequently, the real electronic wave function calculated within theBorn–Oppenheimer approximation gains a phase (sign change) along any path onthe PES, which encircles a conical intersection and thus contains the singularity

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e1g, πg

(b2g, πga) (b3g, πg

o)

e2g, σ

(ag, σa) (b1g, σo)

a2u, πu

(b1u, πu)

y x

e′ (a1)

e″ (a2) e″ (b1)

e′ (a1) e′ (b2)

a1′ (a1)

e′ (a1) e′ (b2)

a1′ (a1)

lp

(a) (b)

σ

σ*

π*

πa2″ (b1)

e′ (b2)

a2′ (b2)

FIGURE 9 Two examples of Jahn–Teller systems. (a) Highest occupied MOs ofbenzene molecule (both D6h and D2h labels are given). (Reproduced with permission fromRef. 36. Copyright 2008 American Institute of Physics). The two lowest states of the benzeneradical-cation form a Jahn–Teller pair. (b) Molecular orbitals and the ground-state electronicconfiguration of cyclic N+

3 (D3h structure, C2v labels are given in parentheses). Both HOMOand LUMO are doubly degenerate. The two lowest electronic states of the doublet N3

radical form a Jahn–Teller pair. Reproduced with permission of American Institute of Physicsfrom Ref. 37.

point.40–42 This is shown in Figure 11a. Thus, in order for the total wave function,which is a product of electronic and nuclear parts, to be single valued, this changeof sign of the electronic wave function needs to be properly accounted for in thecalculations of the nuclear wave function. This is illustrated in Figure 11, whichcompares the vibrational wave functions of N3

39 computed with and withoutgeometric phase effect (these are denoted by BO (Born–Oppenheimer) and GBO(generalized BO), respectively). The geometric phase has a profound effect on thenodal structure of the vibrational wave functions, the order of vibrational states,and the selection rules for the vibrational transitions. The correct description ofJT states requires approaches that treat the pairs of degenerate (or nearly degen-erate) JT states on an equal footing. The geometric phase issue can be avoidedby employing a diabatic representation of the electronic states using vibronicHamiltonians.43,44

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0.2

0.1

–0.0

–0.1

–0.2 (b)(a)–0.2

–0.1

–0.0

0.1

0.2

0.60.50.40.30.2E

(e

V)

Y

0.10.0

0.2

0.1

0.0

–0.1

–0.2

–0.2

2A2 2B1

2A2

2A2

2B1

2B1

2B1

ϕ2A2

TS MIN

CI

–0.1 0.0x (y = 0)

0.1 0.2

(ii)

(i)

–0.2 –0.1 0.0x

0.1 0.2

0.10.20.30.40.5

y

x

FIGURE 10 Potential energy surfaces illustrating JT effect in cyclic N3. Exact definition ofcoordinates x and y can be found in Ref. 38. Point x = y = 0 corresponds to equilateral triangle;this is the point of conical intersection. At y = 0, x < 0 displacements lead to an obtuse isosce-les triangle, whereas x > 0 displacements result in an acute isosceles triangle. The motionconnecting the three symmetry-equivalent minima is called pseudo-rotation. (a) A surface plotof the PES (in eV). (b) Electronic state symmetries. (i) A cross section of the PES shown on theleft along the y = 0 line. TS, MIN, and CI mark the points of the transition state, minimum, andconical intersection, respectively. The electronic state symmetry of the adiabatic ground-statePES changes at the point of CI. (ii) A contour map of the PES shown in (a). Three solid andthree dashed lines cross at the point of CI and indicate 2A2 and 2B1 electronic state symmetries,respectively. Open and filled circles mark minima and transition states. Reproduced with per-mission of American Institute of Physics from Ref. 38. (See color plate section for the colorrepresentation of this figure.)

Vibronic Interactions and Pseudo-Jahn–Teller Effect

The PJT effect arises from configuration interaction between electronicconfigurations that are close in energy;31 it is very common in radicals andhas a major effect on the shape of PESs, for example, it may lead to structuresof lower symmetry. Flawed description of these interactions by an approximateelectronic structure method can lead to poor, and occasionally ridiculous, results andis responsible for the so-called artificial symmetry breaking.

To illustrate the effect, consider the para-benzyne anion. Figure 12 shows itsfrontier MOs and the two nearly degenerate electronic configurations that give riseto the two lowest electronic states. At D2h symmetry, the two configurations areof different symmetry and therefore give rise to two distinct electronic states, 2Ag

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BO

(a) (b)

2B1

2A2

(0,0,0) A1

+ +

–

––

––

–

–

–

+

+ +

+–

+

++–

–

+

(1,0,0) A2

– –

++

++

+

+

+

+

+ +

++

2B1

2A2

2B1

2A2

BO GBO

–

FIGURE 11 Geometric phase effect. (a) Illustration of the sign change in the lowest stateadiabatic (Born–Oppenheimer) electronic wave function. The drawing represents a contourplot of the cyclic N3 PES as in Figure 10, but with only two contour lines: one contour lineis given at 0.16 eV, which is also the highest energy contour line in Figure 10. It forms twoconcentric loops, which encompass the white area, at the inner and outer parts of the well. Thesecond contour line is given exactly at the energy of the transition state 0.0386 eV; it defines thecolored area. This contour line exhibits a Möbius-like shape with the nodes at the transitionstate points and serves here as a reflection of the electronic wave function symmetries. Thewave function is symmetric across the 2B1 lines and is antisymmetric across the 2A2 lines.This is indicated as “+/+” or “+/−” across each C2v symmetry line. (b) Demonstration ofthe changes in the nodal structure of vibrational wave functions due to the geometric phaseeffect. The wave functions for the (0,0,0) state of the A1 symmetry and (0, 0, 1) state of the A2

symmetry obtained from calculations that neglect or account for geometric phase are shown on(a) (marked as BO) and (b) (marked as GBO), respectively. Accounting for the geometric phasechanges the nodal structure of the vibrational wave function. The correct nodal structure of thegeneralized Born–Oppenheimer wave functions leads to the total molecular wave functions(a product of electronic and vibrational wave functions) that are single valued and have thecorrect permutation symmetry. Reproduced with permission of American Institute of Physicsfrom Ref. 39. (See color plate section for the color representation of this figure.)

and 2B1u. However, at lower symmetries (e.g., C2v), the two configurations are ofthe same symmetry (A1) and can, therefore, mix, resulting in two-configurationalwave functions for both states. As follows from the 2 × 2 eigenproblem, thisconfiguration interaction lowers the energy of the lower electronic state andincreases the energy of the upper. Consequently, the PES of the lower state softensalong such symmetry-lowering displacements. If the interaction is sufficientlystrong, a minimum at lower symmetry structure may develop, as illustrated inFigure 13. Since here nuclear displacements modulate electronic couplings, thisphenomenon is called vibronic interaction. While vibronic interaction is a realphysical phenomenon, it may or may not be accurately described by an approximate

�

� �

�


C2vD2h

a1

a1

ag

b1u

C1C1

C1C1 C1C1

C1C1

C4C4 C4C4

C1

C4C4C4 C4C4C4

C1 C1

C1

C4 C4

X2Ag

A2B1u

X2A1

A2A1

6

5

6ag

5b1u

6ag

55b1u

1

1

11a1

10a1

1

1

11a1

10a1

(a) (b)

FIGURE 12 Frontier MOs (a) and leading electronic configurations (b) of the two lowestelectronic states of the para-benzyne anion at D2h and C2v geometries. Reproduced with per-mission of Springer from Ref. 45.

Q Q

(a) (b) (c)

Q

FIGURE 13 Schematic depictions of (a) weak, (b) intermediate, and (c) strong second-orderJahn–Teller interactions along a symmetry-breaking coordinate Q. Reproduced with permis-sion of American Institute of Physics from Ref. 47.

method. If vibronic interactions are overestimated, calculations may yield incorrectlower symmetry structure, and vice versa. For example, in the para-benzyne anion,45

many DFT methods lead to artificial symmetry lowering, yielding incorrect C2vstructures. Many other examples can be found in the works of Stanton, Gauss, andCrawford. Classic examples include NO3, BNB, C+

3 , and the cyclic C3H radical.In-depth analysis of the PJT effects and its description by approximate electronicstructure methods can be found in Refs 46–48.

Formally, this effect of the interactions between electronic states on the shape ofthe PES can be described46 by the Herzberg–Teller expansion of the potential energy

�

� �

�


of electronic state i:

Vi(Q) = V0 +∑𝛼

⟨Ψi

|||| 𝜕H𝜕Q

𝛼

||||Ψi

⟩Q𝛼+ 1

2

∑𝛼

⟨Ψi

|||||𝜕

2H𝜕

2Q𝛼

|||||Ψi

⟩Q2𝛼

−∑𝛼

∑j≠i

|⟨Ψi| 𝜕H𝜕Q

𝛼

|Ψj⟩|2Ej − Ei

Q2𝛼

[11]

It is the last term that describes the vibronic couplings between two differentelectronic states, Ψi and Ψj. Contrary to the JT effect, PJT is a second-order effect,as seen from Eq. [11]. The form of the last term also clarifies which displacementsmight be PJT active. In order for this matrix element to be nonzero, the product ofthe irreps for Ψi, Ψj, and Q

𝛼should contain a fully symmetric irrep. Thus, in Abelian

groups, the nuclear displacement should be of the same symmetry as the product ofthe irreps of the two electronic states. For example, in the para-benzyne anion, anymode of the ag × b1u = b1u symmetry might be PJT active.

HIGH-SPIN OPEN-SHELL STATES

Let us begin by considering a relatively easy case of high-spin open-shell states,such as doublet radicals, triplet diradicals, and quartet triradicals. Often, the exactwave function for such species is dominated by a single determinant; thus, onemay expect performance of quantum chemistry methods to be similar to the caseof closed-shell molecules. One complication, however, is that the direct applicationof the variational principle to the single-determinant ansatz leads to different MOsfor 𝛼 and 𝛽 electrons, giving rise to Φ0, which is not an eigenstate of the S2

operator (Eq. [10]). This spin-contamination phenomenon affects the quality ofpost-Hartree–Fock methods that use the reference determinant produced by theunrestricted Hartree–Fock calculations (“unrestricted” refers to the full variationaloptimization of the single-determinant wave function without restricting it to bespin- or symmetry- pure). This compromises MP2 theory often rendering the resultsunreliable (except for several special cases such as solvated electron49 or whenMOs are optimized for the MP2 wave function50). When electron correlation isincluded through the coupled-cluster ansatz, the spin-contamination is reduced,and the CCSD method performs relatively well for high-spin states.51,52 However,the performance of perturbative methods, such as perturbative triples correction inCCSD(T), deteriorates considerably (the EOM methods are also very sensitive tospin-contamination of the reference). One can introduce a spin-purity constraintinto the Hartree–Fock equations, which gives rise to the restricted open-shell HF(ROHF) method.53 Unfortunately, this does not resolve all problems, and evenbrings new ones, such as relatively poor convergence of the self-consistent fieldprocedure and an increased propensity, relative to UHF, for symmetry breaking.ROHF-based MP2 is often a disaster. Interestingly, straightforward application of the

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�


coupled-cluster ansatz to an ROHF determinant does not produce a spin-pure wavefunction.51,52 Spin-adaptation can be achieved by imposing additional constraintson the amplitudes.54–56 The spin-contamination of ROHF- and UHF-based CCSDwave functions is similar and is relatively small. Consequently, the quality of theresults delivered by these methods is practically the same, which can be explainedby the exp (T1) operator, whose effect is similar to orbital optimization. However,using ROHF references improves the quality of CCSD(T) (and, significantly,EOM-CCSD), and is, therefore, recommended for high-spin open-shell states. Otherchoices of the reference determinants (or orbitals) mitigating spin-contaminationinclude QRHF (quasi-restricted HF),57 Kohn–Sham orbitals, and orbitals optimizedfor correlated wave functions, such as in Brueckner or optimized orbitals CCDmethods.58–61 When spin-contamination is relatively small at the UHF level, it doesnot cause significant decline in the quality of the coupled-cluster calculations.

A related problem is symmetry breaking, which stems from the solutions of the HFequations that do not have correct spatial symmetry. For example, a UHF solution forHe+2 may have the hole localized on one atom. We will discuss this issue later, becauseit originates from a different phenomenon, that is, multiconfigurational character ofthe wave function.

DFT generally provides a reasonable description of the high-spin wave functions.We note that the expectation value of the S2 operator, which is a two-electron oper-ator, as clearly seen from Eq. [10], cannot be computed from the density alone. Thecommonly reported values of ⟨S2⟩ are computed for the Kohn–Sham determinant;they do not reflect the spin-contamination of the unknown many-electron wave func-tion that has the same density. As has been discussed by Pople, Gill, and Handy,for open-shell species, the Kohn–Sham determinant must have different 𝛼 and 𝛽

orbitals,62 so that areas with an excess 𝛽 density in the Ms =12

states can be described.However, the bulk of computational experience suggests that large spin contamination(e.g., ⟨S2⟩ values deviating by ∼0.5 or more from the exact values) of the Kohn–Shamor TDDFT “wave functions” is indicative of a problematic behavior justifying somedegree of spin-adaptation.

Interestingly, DFT orbitals often show good performance in wave function calcula-tions, that is, they can be used in lieu of the ROHF or Brueckner orbitals. In particular,they reduce spin contamination and symmetry breaking relative to UHF. Of course,the difference between the two depends strongly on the exchange-correlation func-tional (specifically, the amount of the HF exchange). The reasons for the usefulnessof Kohn–Sham orbitals in wave function calculations are two-fold. On one hand,one can expect Kohn–Sham orbitals to include some correlation effects; thus, theyare closer to orbitals optimized for a correlated wave function. On the other hand,they may be affected by self-interaction error (SIE), which results in overdelocal-ization of the unpaired spin.63–66 Thus, in cases when UHF solutions break symme-try and overlocalize the unpaired electron (or hole), the restoration of symmetry inKohn–Sham calculations often occurs for a wrong reason. Overdelocalization due toSIE is a well-known phenomenon63–66 complicating the description of, for example,charge-transfer-like systems, which we will discuss below.

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In sum, high-spin open-shell states such as doublets, triplets, and quartetsare often well behaved and can be described by the standard hierarchy ofsingle-reference methods (Figure 6) or by DFT, because their wave functions aredominated by a single electronic configuration. It is desirable to use a spin-pure(or not-too-spin-contaminated) reference determinant, which can be achieved byusing ROHF orbitals, orbitals optimized for a correlated wave function, or, asa quick but efficient trick—Kohn–Sham orbitals. Spin-contamination needs tobe carefully monitored—large deviations from spin-pure values signal potentialmulticonfigurational character, which requires a different approach.

OPEN-SHELL STATES WITH MULTICONFIGURATIONAL CHARACTER

Here we consider high-spin open-shell states whose description is complicated byelectronic degeneracies. Figure 8 shows the MOs in the H2 dimer and electronic con-figurations of the cation, (H2)+2 . This is a prototypical example of a charge-transfersystem (a similar example can be constructed by considering electron-attached statesinstead of ionized ones). When the H2 fragments are identical, the hole is equallydelocalized between them, both in the ground and the excited state of the cation.However, making the two molecules different (e.g., by stretching one H2 bond) leadsto preferential charge localization. Koopmans’ theorem predicts that in the lowestelectronic state of the cation, the charge should reside on the stretched H2 moiety,whereas in the excited state, on the other one. The energy gap between the twostates depends on the distance between the fragments. By varying the bond lengthsand the separation between the two H2 molecules, the character of these two statescan be controlled. One can consider a charge-transfer coordinate along which thebond length of the stretched H2 is decreased while the bond length on the other oneincreases. Along this coordinate, the charge in the ground electronic state flows fromthe initially stretched H2 (donor) to the second fragment (acceptor). Importantly, thetwo Koopmans configurations, Ψ1 and Ψ2, can mix, giving rise to multiconfigura-tional wave functions. Thus, except for the high-symmetry structures where the twoKoopmans configurations have different symmetry, a single-determinantal descrip-tion is flawed. Hartree–Fock typically overlocalizes the hole, which is then carriedover to correlated methods; Kohn–Sham DFT tends to overdelocalize the hole.63–66

Long-range corrected functionals67–69 show some improvement, but do not fully rem-edy the problem.70 In order to treat this type of electronic structure correctly, oneneeds to incorporate the multiconfigurational character in the ansatz. This is naturallyachieved by EOM-IP/EA-CC (and ADC-IP/EA) methods, described next.

EOM-IP and EOM-EA Methods for Open-Shell Systems

The problems arising due to open-shell character, which plague standardsingle-reference and Kohn–Sham methods, can be elegantly circumvented bythe EOM-CC formalism. Instead of following the strategy outlined in Figure 6, inthe EOM-CC framework, one does not attempt to construct an open-shell wave

�

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�


function at the Hartree–Fock level. Instead, we begin by choosing a closed-shellreference state and computing the correlated (e.g., CCSD) wave function:

Ψ(N) = expT1+T2Φ0(N), [12]

where N denotes the number of electrons in the reference state and Φ0 is the referencedeterminant (typically, the Hartree–Fock one). Then the target open-shell states aredescribed by acting on this reference state by general excitation operators, which arenot particle conserving:

Ψ(N−1) = (RIP1 + RIP

2 )expT1+T2Φ0(N) [13]

Ψ(N+1) = (REA1 + REA

2 )expT1+T2Φ0(N) [14]

where

RIP1 Φ0(N) =

∑i

riΦi [15]

RIP2 Φ0(N) = 1

2

a∑ij

raijΦ

aij [16]

REA1 Φ0(N) =

∑a

raΦa [17]

REA2 Φ0(N) = 1

2

ab∑i

rabi Φab

i [18]

Both CC operators T and EOM operators R are net excitation operators with respect tothe reference determinant Φ0 (see Figure 6b), which defines the separation of orbitalspace into the occupied and virtual orbitals. Just as in the case of excited states, thediagonalization of the similarity-transformed Hamiltonian over the truncated basis(e.g., 1h and 2h1p states in the case of EOM-IP-CCSD) delivers better accuracy thandiagonalization of the bare Hamiltonian, because the correlation effects are foldedinto H. A detailed introduction to EOM-CC theory and discussion of its variousaspects can be found in several reviews.10,11,15–19,48

So what do we gain by following this strategy? Figure 14 shows the referenceand target configurations in the EOM-IP/EA methods employing closed-shell ref-erence states. First, the orbitals, which are obtained from closed-shell calculations,are spin-pure. Second, as one can see from Figure 14, the target sets of determinantsare naturally spin complete, resulting in spin-pure wave functions. Furthermore,possible multiconfigurational character of the target wave functions can bedescribed—for example, the two configurations from Figure 8 appear in the EOM-IPexpansion at the same excitation level (R1). Their relative weights are determined bythe EOM-IP eigenproblem and can vary from pure Koopmans-like character (i.e.,one configuration is dominant) to an equal mixture of the two. Moreover, several

�

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�


EOM-IP: ψ(N) = R(–1)ψ0(N + 1)

EOM-EA: ψ(N) = R(+1)ψ0(N – 1)

Φi ΦaijΦ0

Φa bΦai

Φ0

FIGURE 14 Reference (Φ0) and target configurations for EOM-IP-CCSD andEOM-EA-CCSD wave functions.

EOM states can be computed at once, such that one obtains both ground and excitedstates of an open-shell system. Because these target states are produced in the samecalculation, using the same set of orbitals (determined by the choice of Φ0), and thesame T-amplitudes, the calculations of interstate properties, such as transition dipolemoments, Dyson orbitals, nonadiabatic, and spin-orbit couplings (NACs and SOCs,respectively) is straightforward. Because of the good behavior of the closed-shellreference and flexible target-state ansatz, EOM-IP/EA provide qualitatively andquantitatively correct description of charge-transfer, JT, and PJT systems includingcases of extensive degeneracies, as illustrated by the examples given here. EOM-IPhas also been successfully used for constructing vibronic Hamiltonians for systemswith strong nonadiabatic interactions.43

The choice of a specific EOM-CC method depends on the type of target statesthat are sought, together with the MO pattern. The best description is attained whenthe reference is of a closed-shell type in which manifolds of degenerate (and nearlydegenerate) orbitals are completely filled and the dominant character of the targetstates can be described by a single-excitation EOM operator (1h in EOM-IP or1p in EOM-EA). For example, core and valence ionized states of molecules (orelectron-detached states of closed-shell anions) and molecular clusters are welldescribed by EOM-IP. However, satellite states that have doubly excited character(2h1p) are described less accurately. Ground and selected excited states of manydoublet radicals are well described by EOM-IP starting from the anionic closed-shellreference. For example, the vinyl radical (Figure 1) and its 𝜋CC∕𝜎CC → lp(C) excitedstates can be described by EOM-IP using the C2H−

3 reference. However, its Rydbergstates or lp(C)∕𝜋CC → 𝜋

∗CC states would require a different approach. This will be

discussed in the following section on the excited states of open-shell species.

Examples

Charge-Transfer States Let us begin with open-shell species in which the unpairedelectron can be either localized on different parts of the system or delocalized, such

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as a model charge-transfer system in Figure 8. A benchmark study71 of chargelocalization and electronic properties in small charge-transfer systems, such as(He)+2 , (H2)+2 , (Be-BH)+, and (Li-H2)+, has demonstrated that EOM-IP-CCSDis indeed capable of reproducing the trends correctly, whereas methods withsimilar level correlation treatment (such as open-shell CCSD) often show largererrors. Figure 15 shows relevant properties for (H2)+2 (see Figure 8) along thecharge-transfer coordinate (q), which is defined as a linear interpolation betweenthe two structures: one in which the left fragment has the geometry of the cation andthe right fragment has the geometry of the neutral and the other its mirror image.Thus, q = 0 corresponds to the charge preferentially localized on the left moiety;q = 1, the charge localized on the right one. The extent of charge localizationdepends on the distance between the fragments, that is, at q = 0, the charge shouldbe completely localized on the left fragment at very large separations. At 3.0 Åseparation, the FCI charge on the left fragment at q = 0 equals 0.957, and at 5.0 Å0.995. At q = 0.45, the FCI charge is 0.648 and 0.991 at 3 and 5 Å, respectively.At q = 0.5, both fragments are identical and the charge should be delocalizeddue to symmetry. As one can see from Figure 15, the differences between theEOM-IP-CCSD and FCI charges are tiny, whereas the open-shell CCSD resultsshow much larger deviations. Other properties, such as ground and excited-stateenergies (shown in Figure 15) and the respective transition dipole moments,71 areaffected as well. The difference between EOM-IP-CCSD and CCSD is striking giventhe small size of this system with only three electrons. Thus, one would expect thatCCSD is very close to FCI. This is indeed the case for EOM-IP-CCSD; however,open-shell CCSD shows much larger deviations because of its less-balanced ansatz.

Figure 16 shows the potential energy scan and charge delocalization alongthe charge-transfer coordinate in the ethylene dimer cation at two differentinterfragment separations.71 At small separations, the charge is never fully localized.EOM-IP-CCSD as well as EOM-IP-CC(2,3) and MR-CISD show smooth flowof the charge from ∼0.75 at q = 0 to 0.5 at q = 0.5. B3LYP overdelocalizesthe charge, whereas ROHF- and UHF-based CCSD show much larger chargelocalization (0.8 at q = 0). Consequently, the shape of the PES is different. Insteadof the minimum at the symmetric configuration, CCSD yields a noticeable barrier,while B3LYP overestimates the energy gain at the symmetric configuration. At6 Å separation, the charge is fully localized at q = 0 and the symmetric structurecorresponds to the maximum on the potential energy curve. As one can see, B3LYPfails to reproduce both charge localization and the barrier. Long-range correctedfunctionals improve the behavior considerably, but do not completely remedy theproblem.70

Systems with Multiple Ionized States and Their Electronic Properties Afterconsidering toy charge-transfer systems, let us now switch gears and move to morerealistic and challenging examples. Owing to its attractive features, EOM-IP-CCSDhas been extensively used in studies of ionized states of nucleobases72 and theirdimers and clusters with water.73–81 These systems feature multiple ionized stateswhose nature (i.e., hole delocalization) depends sensitively on the structure.

�

�

0.0 0.1

CCSD

EOM-IP-CCSD

EOM-IP-CCSD

EOM-IP-CCSD

EOM-EE-CCSD

EOM-IP-CCSD

CCSD

EOM-EE-CCSD

EOM-IP-CCSD

EOM-IP-CCSD

CCSD

CCSD

0.2 0.3

Reaction coordinate(a) (b) (c)

(d) (e) (f)

0.4 0.5 0.0 0.1 0.2 0.3

Reaction coordinate

0.4 0.5 0.0 0.1 0.2 0.3

Reaction coordinate

0.4 0.5

0.0 0.1 0.2 0.3

Reaction coordinate

0.4 0.5 0.0 0.1 0.2 0.3

Reaction coordinate

0.4 0.5 0.0 0.1 0.2 0.3

Reaction coordinate

0.4 0.5

0

30

60

90

120

Gro

und s

tate

energ

y (

cm

–1)

Gro

und s

tate

charg

e (

e)

Gro

und s

tate

charg

e (

e)

Excitation e

nerg

y (

cm

–1)

Excitation e

nerg

y (

cm

–1)

4–

–2

0

76

78

80

82

Gro

und s

tate

energ

y (

cm

–1)

200–

0

200

400

600

800

0.00

0.01

0.02

0.03

150–

–100

–50

0

600

650

700

750

800

0.0000

0.0003

0.0006

0.0009

2FIGURE 15 Comparison of EOM-IP-CCSD and CCSD/EOM-EE-CCSD description of the electronic states of (H2)+ along charge-transfercoordinate at the 3.0 Å (a–c) and 5.0 Å (d–f) separations using aug-cc-pVTZ. Errors against FCI in ground-state energy (a and d), excitation energy(b and e), and ground-state charge (c and f) are shown. Reproduced with permission of American Institute of Physics from Ref. 71.

�

� �

�


0.00

(a)

(b)

0.25 0.50 0.75 1.00

–800

Reaction coordinate Reaction coordinate


Gro

und s

tate

PE

S (

cm

–1)

Gro

und s

tate

PE

S (

cm

–1)

Gro

und s

tate

charg

e (

e)

Gro

und s

tate

charg

e (

e)

–600

–400

–200

ROHF-CCSD

UHF-CCSD

MR-CISD+Q

EOM-IP-CCSD

EOM-IP-CC(2,3)B3LYP

EOM-IP-CC(2,3)

EOM-IP-CCSD

MR-CISD

ROHF-CCSD

EOM-IP-CCSDEOM-IP-CCSD

B3LYP

B3LYP

UHF-CCSD

B3LYP

0

0.00 0.25 0.50 0.75 1.00

0.2

0.4

0.6

0.8

0.00 0.25 0.50 0.75 1.00

–800

–400

0

400

0.00 0.25 0.50 0.75 1.00

0.00

0.25

0.50

0.75

1.00

FIGURE 16 Comparison of EOM-IP-CCSD, CCSD, and B3LYP description of the groundelectronic state of the ethylene dimer cation along the charge-transfer coordinate at the4.0 Å (a) and 6.0 Å (b) separations using aug-cc-pVTZ. (a) Highly reliable MR-CISD andEOM-IP-CC(2,3) results. On the left, potential energy along the charge-transfer coordinate isshown. On the right, the charge on the left moiety is shown. Reproduced with permission ofAmerican Institute of Physics from Ref. 71.

Structure-dependent electronic properties give rise to concrete spectroscopicsignatures. Multistate methods capable of describing multiple electronic states in abalanced way are required for reliable modeling of such properties.

Figure 17 shows the MO diagram explaining the nature of the lowest ionized statesof the symmetric 𝜋-stacked dimers of nucleobases and their electronic spectroscopy.The two highest occupied MOs of the dimer can be described as bonding and anti-bonding pairs of the fragments’ MO, similarly to a diatomic molecule.36 Thus, theirenergies are split and, since the HOMO of the dimer is destabilized relative to themonomer, 𝜋-stacking reduces the IE. For the uracil dimer shown in Figure 17, theEOM-IP-CCSD/6-311(2+)G(d, p) VIEs of the monomer and the dimer are 9.48 and9.14 eV, respectively.73 Similar red shifts in VIE have been observed in other nucleicacid base dimers.77,78 Removing the electron from the dimer’s HOMO, which has anantibonding character with respect to the two units, leads to relatively strong bond-ing between the two fragments and initiates structural relaxation—the two moietiesare moving closer to each other.74,75 Importantly, new electronic transitions appear

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φA–φB

φA+φB

φA=1p(N) + πCC + πCOφA=1p(N) + πCC + πCO

–0.372 –0.372

–0.384

–0.361

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

Oscill

ato

r str

ength

Energy (eV)

(b)(a)

FIGURE 17 Ionized states of 𝜋-stacked uracil dimer. (a) Molecular orbitals diagram.In-phase and out-of-phase overlap between the fragments’ MOs results in bonding (lower)and antibonding (upper) dimer’s MOs. Changes in the MO energies, and, consequently, IEs,are demonstrated by the Hartree–Fock orbital energies (hartrees). The correlated IEs followthe same trend. The ionization from the antibonding orbital changes the bonding from non-covalent to covalent and enables a new type of electronic transitions, which are unique to theionized dimers. (b) Vertical electronic spectra of the stacked uracil dimer cation at two dif-ferent geometries: the geometry of the neutral (bold lines) and the cation’s geometry (dashedlines). MOs hosting the unpaired electron in each electronic states are shown for each tran-sition. Reproduced with permission of Royal Society of Chemistry from Ref. 73. (See colorplate section for the color representation of this figure.)

in the ionized dimer, such as a bright HOMO-1 → HOMO band; their energies andstrengths depend sensitively on the interfragment distances (Figure 17).

Figure 18, which shows ionized states of the two 𝜋-stacked thymine dimers, illus-trates the effect of structure on the charge localization. In a symmetric dimer, orbitalsare delocalized (and the shift in IEs relative to the monomer is larger), whereas in alower symmetry structure, the ionized states are localized on one of the moieties. Notethat in both structures, the EOM-IP states are dominated by Koopmans-like configu-rations (as illustrated by the magnitude of the leading R1 amplitude); thus, canonicalHartree–Fock MOs provide a good representation of the correlated ionized states.

This example illustrates EOM-IP’s ability to describe multiple ionized states, totreat hole localization/delocalization correctly, and to compute electronic spectra.

Jahn–Teller Systems As discussed in Section Jahn–Teller Effect, ionization ofmolecules that have degenerate HOMOs, such as benzene, benzene dimer,36 andtriazine, leads to degenerate states of the cation exhibiting JT distortions. Similarly,attaching an electron to a species with degenerate LUMOs results in JT states.Some neutral radicals also feature JT structure, for example, cyclic N3. Figure 9shows relevant MOs for benzene and N3. Attempts to describe such JT pairs ofstates by ground-state single-reference methods (such as open-shell CCSD) yieldflawed results. The lowest adiabatic PES could be computed by open-shell CCSD,

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174 REVIEWS IN COMPUTATIONAL CHEMISTRYV

IE (

eV

)

8.78 (–0.35; 0.957)

9.30 (0.17; 0.955)

10.12 (–0.01; 0.934)10.17

(0.04; 0.947)

(b)(a)

10.39 (–0.14; 0.931)10.50

(–0.02; 0.949)

8.99(–0.14; 0.917)

9.13(0.00; 0.919)

10.02(–0.11; 0.931) 10.13

(0.00; 0.942)

10.31(–0.21; 0.944)

10.54(0.02; 0.931)

VIE

(eV

)

FIGURE 18 The MOs and respective VIEs (eV, EOM-IP-CCSD/6-311+G(d,p) extrapolatedto cc-pVTZ) for the two stacked thymine dimers, symmetric (a) and nonsymmetric (b). Theenergy difference (eV) between the dimer IE and corresponding IE of the monomer and leadingEOM amplitudes for the shown orbitals are given in parenthesis. Reproduced with permissionof Royal Society of Chemistry from Ref. 77. (See color plate section for the color representa-tion of this figure.)

by following the lowest energy HF solution, but the shape of the surface is likelyto be pathologically wrong, especially in regions where the reference determinantchanges symmetry. Computing the second JT state by this strategy is only possibleat high-symmetry structures, where the two states are of different symmetry, sothat one can relatively easily perform separate HF and CCSD calculations for eachstate. However, the exact degeneracy of the two JT states will not be reproduced(we note that state-specific CASSCF-based calculations also fail to reproduce thisexact degeneracy). Similar to the charge-transfer systems discussed above, this typeof electronic structure is best described by EOM-IP or EOM-EA methods, whichhave sufficient flexibility to represent the two JT states at arbitrary geometries. Forexample, the ionized states of benzene (and even a more challenging system, thebenzene dimer) can be accurately described by EOM-IP using a closed-shell neutralRefs 36, 82. In the case of N3, the best description is given by EOM-EA using aclosed-shell cation reference.

DIRADICALS, TRIRADICALS, AND BEYOND

Molecules with stretched bonds (Figure 2) or partially broken bonds (see, e.g.,Figure 4) develop diradical character. Diradicals are often encountered as reactionintermediates. An example of an organic diradical, para-benzyne, is shown inFigure 3. Polyradical character, which is of interest in the context of molecularmagnets, can be realized in organic molecules with multiple NO groups or transitionmetal centers. Owing to orbital degeneracy, the wave functions of polyradicalspecies contain several (at least two in diradicals) leading electronic configu-rations. Obviously, the hierarchy of methods built on the single-determinantal

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zero-order wave function summarized in Figures 6 and 7 is deficient by design.MP2 theory in such cases fails dramatically; for example, MP2 potential energycurves along bond-breaking coordinates can collapse to negative infinity, as oneshould expect from the MP2 correlation energy expression, which contains orbitalenergy differences in the denominator. When the extent of the diradical characteris mild (such as in singlet methylene), correlated CC methods provide a reasonabledescription, however eventually they also fail. On the other hand, methods thatuse a multiconfigurational zero-order wave function (such as CASSCF, completeactive space self-consistent field) often have difficulties due to separate treatmentof nondynamical and dynamical correlation, intruder states, sensitivity to theactive-space selection, etc.

To illustrate the type of problems that one may encounter in diradicals, considermeta-benzyne,83 an isomer of para-benzyne. In this diradical, shown in Figure 19, theinteraction between the unpaired electrons and, consequently, the distance betweenthe two radical centers depend sensitively on the correlation treatment. The CASSCFand valence bond methods exaggerate the diradical character and fail to correctlyaccount for a partial bond formation between the two centers. CCSD, on the otherhand, overestimates the bonding interactions and yields an incorrect bicyclic struc-ture. Inclusion of higher excitations, such as triples in CCSDT, would, of course,solve the problem. In meta-benzyne, CCSD(T) happens to be sufficient; however, itwould be unwise to rely on such calculations without validating the results against amore appropriate approach. Rather than resorting to a brute-force approach, the cor-rect description of such systems is achieved when both configurations are treated onan equal footing and dynamical correlation is included. This can be accomplished byemploying spin-flip (SF) methods.84,85

C1

C6

H10 C5

H9

C4

H8

1 2

H10C6

C5

H9

C1

C4H8

C3

C2

H7

C3

C2

H7

FIGURE 19 Possible structural forms of meta-benzyne: a monocyclic singlet diradical 1versus a bicyclic closed-shell singlet 2. Structure 1 can be described as the 𝜎-allylic motif.Structure 2 has 𝜎 delocalized character. The distance between the two radical centers pro-vides a measure of the competition between the two structures. Reproduced with permissionof American Institute of Physics from Ref. 83.

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In the SF approach, the target states are described as spin-flipping excitationsfrom a high-spin reference state. Figure 20 illustrates how it works for di- andtriradicals. As one can see, single spin-flipping excitations generate a spin-completeset of the determinants needed to describe ground and low-lying excited statescorresponding to 2-electrons-in-2-orbitals and 3-electrons-in-3-orbitals types ofelectronic structure.86,87

The SF idea can be used within the EOM-CC formalism88 as well as within CI,88,89

PT,90 and TDDFT.91,92 The CI variants include spin-adapted RASCI-SF (restrictedactive space CI with SF) and CAS-SF formulations with general number of spinflips.93–95 Contrary to low-spin states, high-spin reference states used in SF meth-ods (such as 𝛼𝛼 or 𝛼𝛼𝛼) are not affected by orbital degeneracies because all frontierorbitals are occupied in the respective Hartree–Fock wave functions.

In EOM-SF-CCSD, when using the 𝛼𝛼 reference state, the target wave functionsare

Ψ(Ms= 0) = (RSF1 + RSF

2 )eT1+T2Φ0(Ms=1) [19]

where the operators T1 and T2 are the same CC operators as in other EOM methods.Operators RSF have the same form as the EOM-EE operators (i.e., both are of the1h1p and 2h2p type), but EOM-SF ones change the spin of an electron. That is, theyconserve the total number of electrons but change the number of 𝛼 and 𝛽 electrons.

Electronic configurations shown in Figure 20 give rise to the primary manifoldsof diradical and triradical states. These configuration expansions are identical to thesets generated in CAS for the 2-electrons-in-2-orbitals and 3-electrons-in-3-orbitalsactive spaces. However, target SF wave functions also contain configurations that

ψ(Ms = 0) = R(Ms = –1)ψ0(Ms = 1)

(a)

(b)

ψ(Ms = 1/2) = R(Ms = –1)ψ0(Ms = 3/2)

Φ0 Φai

Φ0 Φai

FIGURE 20 High-spin reference (Φ0) and low-spin target configurations for theEOM-SF-CCSD wave functions. (a) EOM-SF using the Ms = 1 triplet reference describes theMs = 0 states of diradicals. Note that both open-shell and closed-shell states can be described.(b) The Ms =

1

2electronic states of triradicals can be described by using the high-spin quartet

reference (Ms =3

2).

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are excitations outside singly occupied space; these configurations are not presentin the CASSCF expansion. Such configurations describe dynamical correlation andinteractions between the doubly and singly occupied orbital spaces. In the context ofaromatic diradicals (as para-benzyne from Figure 3), these configurations describe𝜎–𝜋 interactions, which are omitted in minimal active-space CASSCF calculations.

When implemented in a straightforward spin-orbital formulation, the target SFstates are not spin-pure. The sets of determinants in Figure 20 are spin-complete;thus, at the single-excitation level, the SF expansion contains all the terms needed forconstructing spin-pure target states from the primary diradical (or triradical) mani-folds. However, the set of single excitations from doubly occupied orbitals (or intounoccupied ones) is not spin-complete because the counterparts of some of theseconfigurations appear at a higher excitation level.96 Consequently, SF target statesare spin-contaminated even when using a ROHF reference. The spin-contaminationof the primary SF states is usually minor even at the SF-CIS or SF-TDDFT levels; itis further reduced when double excitations are included, as in EOM-SF-CCSD. How-ever, states dominated by excitations outside of the singly occupied orbital manifoldmay be strongly spin-contaminated. The SF expansions can easily be spin adapted byincluding a small subset of higher excitations.93–96 Importantly, since the number ofsuch configurations is proportional to the number of single excitations, their inclusiondoes not lead to increased scaling of the method.

The difference between the states from the primary SF manifolds (Figure 20)and other electronic states persists even in spin-adapted versions. Thus, users shouldremember that while all electronic states that are dominated by configurations fromFigure 20 are well described by SF methods, the states that are dominated by exci-tations outside of the singly occupied orbital manifold are not treated on the samefooting as the primary states and that the accuracy of SF description deterioratesin this case. Thus, when computing excited states of diradicals and triradicals, oneshould always distinguish between these two groups of states.

The utility of the SF approach has been illustrated by numerous studies of organicdiradicals and triradicals.34,35,45,84–87,97–102 To quantify the numeric performance ofSF methods, consider singlet–triplet gaps in diradicals. Because of the orbital (near-)degeneracy, the singlet and triplet states are close in energy and the small energy sep-aration between them is notoriously difficult to compute reliably.1,2 The main reasonis that the gap depends sensitively on the correlation energy, which is very differ-ent in states of different multiplicity. That is, high-spin states tend to have much lesscorrelation energy relative to low-spin states of similar orbital character because thePauli exclusion principle restricts the possibilities of distributing same-spin electronswithin the given orbital space and also reduces the probability of the two electronsapproaching each other. The former results in smaller nondynamical correlation andthe latter effect (Pauli hole) reduces dynamical correlation.84–86,88

Table 1 presents experimental103 and calculated values of singlet–triplet gaps inbenzyne isomers. The distance between the radical centers affects their interactionand, consequently, the singlet–triplet gap. The smallest overlap between the orbitalshosting the unpaired electrons occurs in para-benzyne (see Figure 3), which hasthe smallest singlet–triplet gap, 0.14 eV. The interaction between the unpaired

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TABLE 1 Adiabatic Singlet–Triplet Gapsa (eV) in ortho-, meta-, andpara-Benzyne Isomers

Method ortho-C6H4 meta-C6H4 para-C6H4

EOM-SF-CCSDb 1.58 0.78 0.16EOM-SF-CCSD(dT)b 1.62 0.89 0.17NC-SF-TDDFT/PBE50b 1.59 0.78 0.12COL-SF-TDDFT/B505LYPb 1.88 0.98 0.16Experimentc 1.66 0.87 0.14

All calculations employ the cc-pVTZ basis.aZPE subtracted.bFrom Ref. 92.cFrom Ref. 103.

electrons is larger in meta-benzyne, in which the singlet–triplet gap is 0.87 eV. Inortho-benzyne, a partial bond is formed between the two adjacent centers, leadingto the singlet–triplet gap of 1.66 eV. The energies of these partial bonds between theradical centers are: 31.8, 17.6, and 2.1 kcal/mol for ortho-, meta-, and para-benzyne,respectively.97 Thus, the bonding interaction between the unpaired electrons inpara-benzyne is indeed very weak, whereas in ortho-benzyne the interaction energyis equal to roughly one-third of the energy of a covalent bond. The benzynes are agood test for a computational method because of their variable diradical character.

As one can see by comparing the computed values against experiment,EOM-SF-CCSD with a perturbative account of triples,104 EOM-SF-CCSD(dT),is capable of reproducing the gaps with chemical accuracy (< 1 kcal∕mol).EOM-SF-CCSD performance is also consistently good.86 SF-TDDFT, which offersan inexpensive alternative to EOM-SF-CCSD and can be applied to much largersystems, also delivers good accuracy.91,92 We stress that SF works for variousdegrees of orbital degeneracies, including cases with exactly degenerate frontierMOs such as trimethylmethane.34

Importantly, in SF calculations, one is not restricted to the lowest singlet stateand can also compute higher singlet states in the same framework. Thus, all primarydiradical states can be described with uniform accuracy. As an example, considermethylene. The MO diagram and electronic configurations of the low-lying statesare shown in Figure 21; the relevant energies are collected in Table 2. For smallbasis set calculations (TZ2P), we can compare SF against FCI. We see that whenthe triples correction is included, the errors do not exceed 0.02 eV for any of the threestates. To compare against experiment, larger basis sets need to be employed. TheEOM-SF-CCSD(dT)/cc-pVQZ values for the two lowest states are within 0.03 eVfrom the experimental excitation energies.

Electronic degeneracies and a high density of states are even more pronounced intriradicals.35,87,97–101 Figure 22 shows the MOs and vertical state ordering for threeprototypical triradicals—trimethylenebenzene (TMB), an all-𝜋 triradical, and the twoisomers of dehydro-m-xylylene (DMX), a 𝜎-𝜋 triradical. As one can see, the densityof states is rather large and most of the states have heavily multiconfigurational wave

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Orbitals Reference

b1

3B1 11A1 21A11B1

++ λ−λ −

a1

Target states

FIGURE 21 Frontier MOs and the leading configurations of the low-lying electronic statesof methylene, X3B1, 11A1 (closed-shell singlet), 1B1 (open-shell singlet), and 21A1 (alsoclosed-shell singlet, which is doubly excited with respect to 11A1). The wave functions ofthe Ms = 0 components of all four states can be described as single SF excitations from thehigh-spin (Ms = 1) triplet reference. (See color plate section for the color representation ofthis figure.)

TABLE 2 Adiabatic Excitation Energies (eV) Relativeto the X3B1 State of Methylene

Method 11A1 11B1 21A1

EOM-SF-CCSD/TZP 0.52 1.56 2.72EOM-SF-CCSD/TZP 0.50 1.55 2.68FCI/TZPa 0.48 1.54 2.67EOM-SF-CCSD/cc-pVQZ 0.45 1.43 2.58EOM-SF-CCSD(dT)/cc-pVQZ 0.42 1.41 2.53Expb 0.39 1.42

aFrom Ref. 105.bFrom Ref. 106.

functions. Note that only the leading electronic configurations are shown in the figure;the actual SF expansion comprises a larger set of configurations including excita-tions outside the singly occupied orbital manifold. The EOM-SF approach describesall these states in a single calculation, giving rise to a balanced description of thestates and accurate energy differences. All-𝜎 triradicals35,97 as well as substitutedand heteroatom analogues99,101 are also well described.

Can we extend this approach beyond diradicals and triradicals to tackle polyrad-ical species, which are of interest in the context of molecular magnets? There aretwo strategies toward this goal. One is to employ an ansatz with more spin flipsand higher spin references. For example, double spin-flip (Ms = −2) from a quin-tet (Ms = 2) reference can describe tetraradicals (and simultaneous breaking of twobonds).93 An EOM-CC implementation with double spin flip and RASCI and CASimplementations including an arbitrary number of spin flips exist.93–95 The problemwith this strategy is that using a higher number of spin-flips increases the cost ofthe calculations. Consider, for example, double spin-flip. Since the spins of two elec-trons need to be changed, the EOM (or CI) ansatz cannot include single excitations(no R1) and the primary target states require R2. Thus, in the EOM-CCSD variant, R3need to be included in the EOM ansatz, so the correlation of the target states can beproperly described (i.e., the CCSD double SF method can be viewed as a variant ofEOM-CCSDT). In TDDFT, which only admits single excitations within the adiabaticconnection, inclusion of double excitations would require theoretical justification

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(a)

H H

Ha1 b1 a2

a1 b1 a2

a″2/b1 e″/a2 e″/b1

H

H

H

H

H

H H

HH

H H

HHH

H

H

HHH

H

(b)

4

3

2

1

0

–1

5-DMX

12A2

22B122A1

λ

λ

−λ

−

−λ−

−

− −

− −

−

12B1

22B2

12A1

14B2

12B2

2

2

+

+

+

+

22B1

32B1

22B1

22A2

12B1

12A2

14A2

22A1

+12B1

22B2

12A2

12A1

12B2

14B2

2-DMX 1,3,5-TMB

Exc

itation e

nerg

y (

eV

)

FIGURE 22 Molecular orbitals (a) and low-lying electronic states (b) of the 1,3,5-TMB,5-DMX, and 2-DMX triradicals. 1,3,5-TMB and 2-DMX feature a quartet ground state,whereas 5-DMX has an unusual open-shell doublet ground state. Reproduced with permissionof American Chemical Society from Ref. 87.

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and significant coding effort.107,108 In the CI formulations, general excitations arepossible, but the cost increases as well. Despite these shortcomings, RASCI-SF meth-ods have been successfully employed to describe polyradicals.95,109,110

The problem can be circumvented by an elegant approach based on the HeisenbergHamiltonian. Mayhall and Head-Gordon have demonstrated that one can constructand parameterize the Heisenberg Hamiltonian for an arbitrary number of unpairedelectrons by performing only a single spin-flip calculation from the highest multi-plicity state.5,111 In the case of two-center systems,111 the exchange coupling constant(JAB) is simply taken from the Landé interval rule:112

− 2SJAB = E(S) − E(S−1) [20]

where S = N2

. For multiple-center systems, in which the couplings between the cen-ters may differ, a single SF calculation is followed by additional steps summarizedin Figure 23. Then the Heisenberg Hamiltonian can be diagonalized, providing ener-gies of all the Ms components. Importantly, since only single SF calculations arerequired, this approach scales linearly with the number of unpaired spins, in contrastto the factorial scaling of the full SF calculation, which requires flipping the spins ofN2

electrons.Obviously, this scheme only works when the idealized Heisenberg physics is valid;

it will break down if the unpaired electrons are strongly coupled. To validate thisapproach, one should compare the results with the full SF calculations that directlycomputes all the multiplet components. For the nickel-cubane complex from Figure 5,which has eight unpaired electrons, this requires flipping the spins of four electrons(4SF). Figure 24, which compares the energy spectrum of this complex computed bythe 4SF-CAS(S) method with the results from the Heisenberg Hamiltonian param-eterized by 1SF-CAS(S) calculation, illustrates the excellent performance of thisapproach in the present example.

Mayhall and Head-Gordon also applied this approach to a molecular magnet with18 unpaired electrons (Cr(III) horseshoe complex from Figure 5) for which full SFcalculations are not possible.5 When using NC-SF-TDDFT with the PBE0 functional,they obtained5 two exchange constants, −4.38 and −4.52 cm−1, in reasonable agree-ment with the experimental values of −5.65 and −5.89 cm−1.

EXCITED STATES OF OPEN-SHELL SPECIES

Excited states of closed-shell systems can be reliably described by the hierarchy of theexcited-state approaches shown in Figure 7, that is, CIS, CIS(D), EOM-EE-CCSD,etc. Within DFT, excited states can be described by TDDFT, which leads to workingequations that are similar to the CIS ones.113 In all these methods, the target wavefunctions are described by linear parameterization. Consequently, in the case ofclosed-shell references, excited states are naturally spin-adapted. For example,spin-conserving HOMO–LUMO excitations give rise to two Slater determinants(in one, an 𝛼 electron is excited, and in the other, 𝛽), which are combined toyield the Ms = 0 triplet and singlet states of this character. The reliability of the

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(a) ROHF

1

2

3

4

5

3/2

1/2x/y

z

6

7

8

9

1

2

3

(b) Neutral determinant

basis

Spin-flip + projection Block diagonalization Effective Hamiltonian

(c) Local ground-

state basis

(d) Heff

E1 J12 J13

E2 J23

E3

FIGURE 23 Procedure for extracting exchange coupling constants from single SFcalculations. (a) Start with the Boys-localized ROHF orbitals for the highest multiplicitystate (Ms =

N

2). (b) A single SF excitation operator generates all configurations necessary to

describe the Ms =N

2− 1 manifold. From this set, only the neutral (noncharge-transfer) subset

is retained. (c) After site-by-site block diagonalization, the Ms =N

2− 1 eigenstates are now

linear combinations of three local quartet states, | 1

2,

2

3,

2

3⟩, | 2

3,

1

2,

2

3⟩, and | 2

3,

2

3,

1

2⟩. The tilted

gray arrows indicate the Ms =1

2component of the local quartet spin states. (d) In the local

eigenstate basis, the effective Hamiltonian is now full rank and contains the exchange couplingconstants, JAB, as off-diagonal elements divided by −2SASB = −3, where SA denotes local spinon center A. Reproduced with permission of American Chemical Society from Ref. 5.

60

Delta E

(cm

–1)

20 12 12 12

1SF-CAS(S) + Heisenberg

4SF-CAS(S)

6 6 6 6 6 2

<S2>

2 6 2 0 2 2 0 2 0

50

40

30

20

10

0

FIGURE 24 Comparison of the low-energy spectrum taken from direct ab initio4SF-CAS(S) calculations (gray) and the 1SF-CAS(S) + Heisenberg diagonalization (black).The y-axis is the Δ energy from the ground state in cm−1. The x-axis is the ⟨S2⟩ for each state.Reproduced with permission of American Chemical Society from Ref. 5.

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excited-state methods depends on the quality of the underlying wave functions.That is, when the ground-state wave function is dominated by a single Slaterdeterminant, it is well described by the ground-state methods from Figure 6, and,consequently, the excited-state methods from Figure 7 also perform well. Thecorrelated EOM-EE-CCSD method performs very well for excited states, which aredominated by a single excitation (i.e., the magnitude of R1 should be much largerthan R2). Owing to the linear parameterization, EOM-EE-CC can describe statesof mixed character, for example, 𝜋𝜋∗ mixed with n𝜋∗ or states of Rydberg-valencecharacter, and even states of charge-transfer character. TDDFT (when used withcaution114) performs very well for valence electronic states (typical errors ∼0.2 eV);however, the description of Rydberg, charge-resonance, and charge-transfer statesmay be affected by SIE.115,116

How does the excited-state methodology apply to the excited states of open-shellspecies? The answer depends on the type of open-shell character. Let us revisitdiradicals, for example, methylene from Figure 21. In the case of a modest diradicalcharacter, the lowest singlet-state wave function is dominated by a single Slaterdeterminant and correlated methods such as CCSD may yield reasonable descrip-tion. Consequently, EOM-EE-CCSD can describe low-lying states of singly excitedcharacter, such as 3B1 (this state will appear with negative excitation energy whendoing EOM-EE-CCSD calculations using the closed-shell 1A1 reference) and 1B1.The EOM-EE-CCSD/TZ2P adiabatic gaps86 for these states are 0.54 and 1.57 eV(to be compared with the FCI values105 of 0.48 and 1.54 eV, see Table 2). However,the EOM-EE-CCSD excitation energy for the second excited state, 21A1, whichis doubly excited with respect to the reference (11A1), is of very poor quality:3.84 eV with the TZP basis, to be compared with the FCI value105 of 2.67 eV. If thedegeneracy between the frontier orbitals increases, then the multiconfigurationalcharacter of the lowest closed-shell singlet will increase (up to the case whenboth configurations have equal weights, as at the bond dissociation limit shownin Figure 2 or in methylene at linear geometries). Thus, EOM-EE-CCSD willeventually break down.

A solution to this problem is given by EOM-SF described above. SF can describeall four low-lying states of diradicals from the high-spin triplet reference, which isnot affected by orbital degeneracies since both orbitals are occupied. Thus, in thecase of diradicals and triradicals, the manifold of low-lying excited states (primarydiradical/triradical states) can be accurately described by SF methods, provided thatpossible spin-contamination is mitigated by using ROHF references, as discussedabove. However, the states that lie outside of the 2-electrons-in-2-orbitals manifoldin diradicals (or 3-electrons-in-3-orbitals manifold in triradicals) are described lessaccurately. One issue is that spin-complements of some of the configurations from theSF set96 (i.e., the excitations outside the 2 × 2 space) are missing their counterparts inthe straightforward, nonspin-adapted formulation. This can be tackled by includingselected higher excitations,96 which improves the description of these states. This isimplemented in RASCI-SF and CAS-SF methods.93–95,117 Thus, one should exercisecaution when performing SF calculations of excited states in diradicals and triradi-cals. While the dominant SF states are described well, the energies of higher lying

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states are less reliable. To detect potential problems, one should always monitor theleading configurations of the computed states. The ⟨S2⟩ values of the target states alsoprovide a useful diagnostic.

The situation is different in the case of doublet radicals. As explained, EOM-IPand EOM-EA can describe ground and selected excited states of doublet radicals.Here again the rule of thumb is to monitor the weight of R1 amplitudes. If the targetstate is dominated by R1 (e.g., weight ≳ 90%), then it is usually well described by thechosen EOM-CC method. Examples include various valence ionized states of neutralmolecules that have predominantly Koopmans-like character (or are linear combina-tion of Koopmans-like configurations). The excited charge-transfer type states, suchas Ψ2 in Figure 8, are also well described by EOM-IP/EA. However, other types ofexcited states of doublet radicals are not amenable to that treatment and require otherapproaches.

Consider, for example, the vinyl radical from Figure 1. Its ground state and the𝜋CC → lp(C) and 𝜎CC → lp(C) states are well described by EOM-IP-CCSD. How-ever, its Rydberg states, lp(C) → Ry, or valence states of, for example, 𝜋CC → 𝜋

∗CC

character, cannot be described by a single EOM-IP operator; rather, these configu-rations are double excitations (2h1p). Can we utilize the EOM-EE-CCSD approachto compute these states? For example, can we compute the CCSD wave functionfor the doublet reference state, 𝜎2

CC𝜋2CClp1, and then perform EOM-EE-CCSD

using this reference? Yes, with some caveats. The first one is spin-contaminationof the reference wave function. Just as the CCSD wave function of an open-shellreference is spin-contaminated, so will be the target EOM-EE-CCSD states (theproblem is exacerbated at the EOM level). In many cases, the problem can bemitigated by using ROHF-like references. The second issue, which is more difficultto address, is conceptually similar to the problem with the description of higherexcited states by the SF approach. As always, EOM-EE is reliable only if allleading electronic configurations of the target state appear as single excitations.In the case of doublet radicals, this means that all electronic states that can bedescribed by excitation of the unpaired electron to either valence (e.g., 𝜋∗CC) orRydberg orbitals will be reasonably well described by EOM-EE-CCSD. Similarly,excitations from low-lying doubly occupied orbitals to the singly occupied orbitalcan be tackled well (these give rise to the same manifold of excited states that iswell described by EOM-IP-CCSD). However, states derived by excitations from thedoubly occupied orbitals into the unoccupied ones, such as 𝜋CC → 𝜋

∗CC, will cause

difficulties.118–120 To understand the problem, consider the set of configurationsthat are needed to correctly describe the state of the (𝜋CC)1(lp)1(𝜋∗CC)

1 character.Figure 25 shows all single and double EOM-EE excitations generated from thedoublet (𝜎CC)2(𝜋CC)2(lp)1 reference. To construct a spin-adapted wave function forthe doublet or quartet (Ms =

12) state of the (𝜋CC)1(lp)1(𝜋∗CC)

1 character, one needsto combine configurations (4), (5), and (6) from Figure 25. While (4) and (5) aresingly excited determinants with respect to the doublet reference (1), determinant(6) is doubly excited. This leads to an unbalanced description of this state at theEOM-EE-CCSD level, which manifests itself in strong spin contamination. Methodsbased on single-excitation expansions, such as CIS, CIS(D), and TDDFT, produce

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πCC∗

lp(C)

πCC

R1

(1) (2) (3) (4)

(6)

(5)

(7) (8)

R2

FIGURE 25 Doublet reference, (1), and target configurations for the EOM-EE-CCSD wavefunctions for vinyl radical. Configurations (2)–(5) are single excitations from reference (1),whereas determinants (6)–(8) are formally double excitations. Note that Ms =

1

2open-shell

configurations, (4)–(6), needed to describe doublet and quartet (𝜋CC)1(lp)1(𝜋∗CC)

1 states of vinylradical, appear at different excitation levels in the EOM-EE-CCSD ansatz.

TABLE 3 Different Types of the Ms =1

2Electronic States in the Vinyl

Radical and Appropriate CC/EOM-CC Models

State Ansatz Reference

2|𝜎2CC𝜋

2CClp1⟩ CCSD 𝜎

2CC𝜋

2CClp1

EOM-EA 𝜎2CC𝜋

2CClp0

EOM-IP 𝜎2CC𝜋

2CClp2

2|𝜎2CC𝜋

1CClp2⟩, 2|𝜎1

CC𝜋2CClp2⟩ EOM-IP 𝜎

2CC𝜋

2CClp2

EOM-EE 𝜎2CC𝜋

2CClp1

2|𝜎2CC𝜋

2CC𝜋

∗CC

1⟩, 2|𝜎2CC𝜋

2CCRy1⟩ EOM-EE 𝜎

2CC𝜋

2CClp1

EOM-EA 𝜎2CC𝜋

2CClp0

2|𝜋1CClp1

𝜋∗CC

1⟩, 4|𝜋1CClp1

𝜋∗CC

1⟩ EOM-SF 𝜋1CClp1

𝜋∗CC

1, 𝛼𝛼𝛼 component

disastrous results because they lack this class of configurations.118–120 As in the caseof diradicals, careful monitoring of the state characters and the ⟨S2⟩ values for thetarget states helps to detect the problematic behavior. We note that these states can bewell described by EOM-SF using a quartet reference (see Figure 20). This strategyhas been exploited, for example, in the calculations of substituted vinyl radicals121

and, more recently, in calculations of excited states of open-shell organometalliccompounds.122

This analysis, which uses vinyl’s states as an example, is summarized in Table 3.The table lists the combinations of EOM-CC models and reference states that provideaccess to specific states. As one can see, sometimes different EOM-CC models needto be combined in order to characterize all electronic states of interest.

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METASTABLE RADICALS

Up to this point, our discussion has been focused on electronically bound open-shellspecies. That is, all molecules discussed above are stable with respect to ejectingan electron (although some of these states can be metastable with respect to nuclearrearrangements, such as isomerization or dissociation). An important class of radicalsinvolves electronically metastable species, often called resonances.123–126 Examplesinclude transient anions that do not support a bound anionic state, such as N−

2 , CO−,and H−

2 . Yet, such electron-attached states can be formed and have finite lifetime,sufficient to be detected experimentally. Another class of electronic resonances isexemplified by core-ionized (or core-excited) states. These states decay on a fem-tosecond scale via Auger processes.

How can we describe these autoionizing states? Their metastable character posessignificant challenges to standard quantum chemistry methods. The difficulty stemsfrom the fact that these states belong to the continuum and their wave functions arenot L2-integrable. Attempts to compute these states using standard electronic struc-ture methods employing Gaussian-type bases lead to problematic behavior—that is,in compact bases, such states are artificially stabilized (and look like normal boundstates), but upon basis set increase, they dissolve in the continuum. Distinguishing realresonances from artificially stabilized ones is hard and determining their lifetimes andconverged energies is even harder.

An elegant solution to this problem is offered by complex-variable approaches, inwhich the original Hamiltonian is modified such that its bound spectrum is unchangedand the resonances appear as L2-integrable solutions with complex energies.123,125–127

This can be achieved through the complex-scaling formalism128,129 or by usingcomplex absorbing potentials.130,131 The latter appear to be particularly promisingfor practical applications. In this method, the electronic Hamiltonian is modifiedby adding an artificial pure imaginary potential, i𝜂W (the parameter 𝜂 controlsthe strength of the potential). We have implemented this approach within theEOM-CCSD family of methods, which enables calculation of various types ofmetastable open-shell states using standard EOM-CC approaches.132–134

Figure 26 shows a calculation135 of bound and unbound anionic states usingEOM-EA-CCSD. In this study,135 electron-attached states of silver and copperfluorides were investigated. Because AgF and CuF are strongly ionic, they havelarge dipole moments and are able to support two bound anionic states: a regularvalence-attached state and a dipole-bound type.136 In AgF−, three low-lyingresonances have been identified by calculations and from features observed inphotoelectron angular distributions.135 Particularly interesting is an upper statein Figure 26, a dipole-stabilized 𝜋-shape resonance. In contrast to regular 𝜋-typeshape resonances (such as in N−

2 ), the attached electron resides mostly outside ofthe diatomic core, which leads to very different PESs. Whereas the resonance inN−

2 is stabilized at stretched NN geometries, the dipole-stabilized resonance inAgF− does not show noticeable variation in lifetime with nuclear motions. Theelectron-attachment energies listed in Figure 26 illustrate robust and consistentperformance of EOM-EA-CCSD for both bound and unbound states. The largest

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Energy

AgF (E = 0.0 eV)

E = 3.88 eV (3.8 eV)

E = 3.63 eV (3.5 eV)

E = 2.71 eV (2.7 eV)

E = –0.014 eV (–0.012 eV)

Pi resonance

(dipole stabilized)

Sigma resonance

(dipole stabilized)

Dipole-bound

state

Anion GS

E = –1.36 eV (–1.46 eV)

12Π+

32∑+

22∑+

12∑+

X2∑+

FIGURE 26 Electron-attached states of AgF (𝜇 = 5.3 debye) (see Ref. 135). Theoreticalvalues are computed by EOM-EA-CCSD augmented by CAP. Experimental values are shownin parentheses. The ground state of the anion, AgF−, is strongly bound. Because of the stronglypolar character, AgF also supports a dipole-bound state (bound by 0.014 eV). Three resonancestates have been identified by the CAP-EOM-EA-CCSD calculations and experimentally.135

The character of the target orbital hosting the attached electron reveals dipole-stabilized natureof the two upper resonance states. (See color plate section for the color representation of thisfigure.)

error (0.1 eV) was observed for the lowest state of the anion and was attributed tothe effect of higher excitations (triples) that are missing in EOM-CCSD.135

Other types of open-shell resonances include core-ionized states and highlyexcited states; they can be described by a complex-variable variant of EOM-IP orEOM-EE, respectively.

BONDING IN OPEN-SHELL SPECIES

From the very outset of this chapter, we employed an MO framework, which isa cornerstone of the theory of chemical bonding, to explain salient features ofopen-shell electronic structure. However, the connection between MOs and the wavefunctions of interacting electrons deserves some discussion. Strictly speaking, MOsare wave functions only for one-electron systems (e.g., H+

2 ) or in a hypothetical caseof noninteracting electrons. In such simple cases, one can unambiguously explainelectronic properties by using the MO-LCAO (MOs linear combination of atomicorbitals) framework.

In the many-electron case, the meaning of MOs changes. In the frameworkof pseudo noninteracting electrons (mean-field Hartree–Fock model), the MOsrepresent an approximate many-electron wave function; thus, their shapes can be

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used to rationalize bonding in the same manner as in the case of noninteractingelectrons. We note that the specific choice of orbitals is arbitrary, as the Hartree–Fockenergy and state properties (such as dipole moments) are invariant with respect toany unitary transformation of orbitals within the occupied space. That is, any linearcombination of occupied Hartree–Fock MOs leads to the same observable propertiesand to the same electron density. Yet, there is one specific choice—canonical HForbitals—that is of special significance. The canonical orbitals are those that diago-nalize the Fock operator. By virtue of Koopmans’ theorem, these orbitals provide adirect view of the states of the ionized electrons and the respective orbital energiesapproximate the IEs. Thus, canonical HF orbitals do have physical significance,and, therefore, their use for analysis of the wave function and molecular propertiesis well justified. For example, just such an analysis is at the root of the well-knownWoodward–Hoffman rules that govern electrocyclic reactions in organic chemistry.

Is there a way to rigorously extend the MO framework beyond the mean-field HFmodel? In cases when the electronic wave function is dominated by the Hartree–Fockdeterminant, using HF orbitals for the analysis is reasonable. Canonical Hartree–Fockorbitals can also be used to analyze the characters of EOM states, when the tar-get wave function is dominated by one or two leading configurations. More rigor-ous description can be achieved by using correlated Dyson orbitals, as explainedbelow. Dyson orbitals can also be used for heavily multiconfigurational wave func-tions including strongly correlated systems.

Dyson Orbitals

Dyson orbitals are defined as an overlap between N-electron and N−1-electronstates:137–141

𝜙dIF(1) =

√N ∫ ΨN

I (1, … , n)ΨN−1F (2, … , n)d2 … dn. [21]

Thus, they represent the difference between the two wave functions. In the context ofphotodissociation/photodetachment experiments, Dyson orbitals connect the initial(I) and final (F) states of the system and contain all the information about the respec-tive wave functions that is needed to compute photoionization/photodetachment crosssections. In this respect, they are analogues to transition density matrices for stateswith different number of electrons. Since both are related to concrete experimentalobservables, they can be considered observables themselves.

Dyson orbitals can, in principle, be computed for any many-electron wave functionusing Eq. [21]. If one takes a Hartree–Fock determinant built of the canonical orbitalsfor ΨN

I and constructs ΨN−1F using the same orbitals (in the framework of Koopmans’

theorem), then 𝜙dIF(1) is just the canonical Hartree–Fock MO that was removed from

Φ0. Thus, Dyson orbitals provide a generalization of Koopmans’ theorem to cor-related many-electron wave functions. As such, they provide a convenient way tocharacterize the spatial distributions of unpaired electrons. Dyson orbitals can be

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3.05 2.23 2.22 2.17

FIGURE 27 Dyson orbitals and vertical IEs (eV) for the four lowest states of the Na(NH3)3cluster computed using EOM-EA-CCSD/aug-cc-pVDZ. The two lowest IEs of the bare sodiumatom are 4.96 and 2.98 eV. The orbitals reveal that the unpaired electron resides on the sur-face of the cluster and that its states resemble atomic s and p states perturbed by the solventmolecules. (See color plate section for the color representation of this figure.)

computed by using various EOM-CCSD wave functions140,141 including those ofmetastable electronic states.142 Unlike Hartree–Fock MOs, Dyson orbitals includecorrelation effects.

Figure 27 shows Dyson orbitals for the four lowest electronic states of Na(NH3)3computed using EOM-EA-CCSD. The orbitals resemble atomic s and p orbitals, butthey are distorted by the ammonia molecules and shifted away from the parent Na+

core. These electronic states can be described as surface states of solvated electronsstabilized by the micro-solvated sodium cation.

How large is the difference between Dyson orbitals and canonical Hartree–FockMOs? As always, the answer depends on the system. For ionization from the groundelectronic states, Hartree–Fock MOs usually provide a very good approximation tothe correlated Dyson orbitals. The EOM-IP amplitudes quantify how well HF MOsrepresent the correlated states—the weight of a given MO in the correlated Dysonorbital is roughly equal to the weight of the respective EOM-IP amplitude. As onecan see from the values of the leading EOM-IP amplitude shown in Figure 18, theionized states for thymine dimers are of Koopmans-like character and, therefore, theHF MOs are sufficient for qualitative analysis. For quantitative purposes (such ascalculations of absolute cross sections), the difference between the HF and Dysonorbitals becomes important. Similar to EOM-IP, the EOM-EA amplitudes can informthe user as to whether HF MOs provide a good approximation to Dyson orbitals. InEOM-EA calculations, the difference between the HF and correlated orbitals is usu-ally more significant. In the case of ionization from electronically excited (or stronglycorrelated) states, Dyson orbitals become indispensable even for qualitative analysis.

Density-Based Wave Function Analysis

There is no need to invoke noninteracting electrons to understand bonding patterns.Because the electrons are indistinguishable, the state properties are determined bydensity matrices, which provide a compact representation of many-electron wavefunctions. One-electron properties can be computed from one-electron density

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matrix, 𝜌(1, 1′), alone:

𝜌(1, 1′) ≡ N ∫ Ψ∗(1, 2, … ,N)Ψ(1′, 2, … ,N)d2 · · · dN [22]

𝜌(1, 1′) =∑pq

𝛾pq𝜙p(1)𝜙q(1′) [23]

𝛾pq ≡ ⟨Ψ|p+q|Ψ⟩ [24]

where 𝜙p denotes molecular spin-orbitals and 𝛾 is a one-particle density matrix. Forsome operators, only the electron density is needed:

𝜌(1) = 𝜌(1, 1′)|1′=1 [25]

Thus, the bonding pattern can be determined by using the density, 𝜌(1). This isexploited in natural bond orbital (NBO) analysis,143,144 which can be performedfor an arbitrary many-electron wave function. NBO allows one to partition the totalelectron density into the contributions corresponding to bonds, core electrons, andlone pairs.

For excited states, the analyses based on transition density matrix provide a way tointerpret the excited-states characters in terms of local excitations, charge-resonance,and charge-transfer configurations.145–149 The transition density matrix connectingtwo states, ΨI and ΨF, is defined as follows:

𝛾IFpq ≡< ΨI|p+q|ΨF > [26]

It contains information needed to compute one-electron transition properties suchas transition dipole moments and NACs. It also presents a compact descrip-tion of the differences in electronic structure between ΨI and ΨF. Importantly,density-matrix-based analysis145–149 of the excited states does not depend onthe choice of MOs and can be applied to general wave functions. The norm of𝛾

IF is a useful quantity; it measures the degree of one-electron character in theΨI → ΨF transition. For example, ||𝛾|| = 1 for those transitions which are of purelyone-electron character.150

In the case of open-shell species, commonly asked questions are (i) what is thecharacter of the unpaired electron(s) and (ii) to what extent are the unpaired electronsunpaired. In the case of high-spin open-shell species (doublets, triplets, and quar-tets), computing the difference between the 𝛼 and 𝛽 densities provides a simple wayto analyze the unpaired electrons. In these states, the number of unpaired electronsis defined unambiguously, as the difference between the number of 𝛼 and 𝛽 elec-trons, and the electron density difference provides a means to visualize their states.Figure 28 shows the spin-density difference plot for a bicopper molecular magnet. Asone can see, the excess 𝛼 density, which is located mostly on the Cu atoms, extendstoward the organic ligands.

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(a) (b)

FIGURE 28 Bicopper molecular magnet. (a) Molecular structure (copper atoms are shownin gold). (b) Difference spin-density plot for the Ms = 1 triplet state computed usingB5050LYP/LANL2DZ. The excess 𝛼 electrons (purple) can be clearly associated with cop-per atoms, with some degree of delocalization along the Cu-ligand bonds. (See color platesection for the color representation of this figure.)

The difference in spin-densities, however, is not directly related to experimentalobservables other than the Ms value. We note that Dyson orbitals, described previ-ously, do provide such a link.

In the case of low-spin states, such as singlet states of diradicals, the characteriza-tion of unpaired electrons becomes more difficult. Inspired by the analysis of modelwave functions, various ways of computing the effective number of unpaired electronshave been proposed.151 Most of the definitions are based on the one-particle densitymatrix, Eq. [24]. The trace of density matrix equals the number of electrons, N

𝛼+

N𝛽. Diagonalization of 𝛾 yields natural orbitals and natural occupations, 𝜆k ∈ [0, 1].

Because of the trace properties,∑

k𝜆k = N𝛼+ N

𝛽. For single-determinant wave func-

tions, the 𝜆k equal 1 for occupied orbitals and 0 for unoccupied. In a perfect diradical,such as the stretched H2 molecule from Figure 2, described by a two-determinantalwave function, the natural occupation of each of the four spin-orbitals is 0.5. For sys-tems with N

𝛼= N

𝛽= N, a spinless density matrix can be defined by summing the

𝛼𝛼 and 𝛽𝛽 blocks of one-particle density matrix, which are identical in the case ofspin-pure wave functions:

D ≡ 𝛾𝛼𝛼 + 𝛾𝛽𝛽 [27]

Natural occupations for D vary from 0 to 2; for a perfect 2 × 2 diradical they equal 1.Thus, the quantity

𝜆oddk = 2𝜆k − 𝜆2

k [28]

is zero for a single determinant. This observation led to the so-called Yamaguchiindex,152 which can be interpreted as the effective number of unpaired electrons:

Nodd = 2N − Tr[D2] = 2N −∑

k

𝜆2k [29]

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Unfortunately, this index shows incorrect behavior in some cases. A better-behavingquantity is the Head-Gordon index:153

Nodd =∑

k

Min[𝜆k, 2 − 𝜆k] [30]

Other means for analyzing unpaired electrons are reviewed in Ref. 151.

Insight into Bonding from Physical Observables

The wave function analyses described above are useful for deriving insight from elec-tronic structure calculations, but one should remember that quantities such as atomiccharges, bond orders, and particle-hole separations are not experimental observablesand, therefore, are not uniquely defined. Strictly speaking, the only rigorous wayto describe bonding is to characterize relevant observables, such as bond lengths,force constants, and bond dissociation energies. These quantities can be comparedto reference values for systems with prototypical electronic structure and from thesecomparisons quantitative aspects of bonding can be derived.

In our studies of bonding patterns in diradicals and triradicals, we focusedon physically motivated quantities. In particular, we characterized the degree ofinteraction between the unpaired electrons in terms of its structural and energeticconsequences. For example, quantities such as distance between radical centers, fre-quencies of the relevant vibrational modes, and the so-called diradical and triradicalseparation energies (DSE and TSE, respectively) reflect the strength of the interac-tions between the unpaired electrons. Figure 29 shows the results for benzynes andtridehydrobenzenes.97 In high-spin states, the distances between the radical centersare within 4% of the parent benzene structure, however, in low-spin states, contrac-tion up to 17% is observed. DSE and TSE quantify these interactions in terms ofenergies, such that they can be compared with typical bond energies. Using benzynesas an example, the DSE is defined as the energy change in the following reaction:

C6H4 + C6H6 → 2C6H5 + DSE [31]

Although reaction [31] is hypothetical, the energetics is real, as it can be computedfrom experimental thermochemical data such as heats of formation. DSE representsenergy gain due to having the two radical centers within the same moiety. If the twocenters are noninteracting, the DSE is zero. To illustrate the utility of this quantity,consider the DSE in the ortho-, meta-, and para-benzyne series. In triplet states,the DSEs are small (∼1 kcal∕mol), indicating insignificant interactions betweenthe unpaired electrons. For the singlet states, the theoretical values (which are inexcellent agreement with the experimental ones103) are 31, 17, and 1.3 kcal/mol.97

The trend is consistent with the distance between the radical centers (Figure 29).The DSEs suggest that para-benzyne can be described as an almost perfect diradicalwith nearly noninteracting unpaired electrons, whereas in ortho-benzyne the twoelectrons form ∼ 1

3of a covalent bond. The conclusions based on structures and

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0

–2

–4

–6

–8

–10

–12

High-spin states

Low-spin states

–14

Δ r/

r benz (

%)

–16

–18

1,2-C6H4

(3B2, 1A1)

1,3-C6H4

(3B2, 1A1)

Short Long O m Om

p1,4-C6H4

(3B1u, 1A1g)

1,3,5-C6H3

(4B2, 2A1)

1,2,3-C6H3

(4B2, 2B2)

1,2,4-C6H3

(4A′, 2A′)

FIGURE 29 Relative change of the distance between radical centers in benzynes (C6H4)and tridehydrobenzenes (C6H3) with respect to benzene. For each species, Δr∕rbenz (in %) forthe ground (low-spin) state and for the lowest high-spin state is shown. For the C6H3 isomers,all possible distances between two radical centers are considered; for example, in 1,2,4-C6H3,the distances between centers in ortho (o), meta (m), and para (p) positions are used. Cristianet al.97. Reproduced with permission of American Chemical Society.

DSEs are consistent with the wave function composition (i.e., relative weights of thetwo leading configurations).

To give another example of unpaired electrons giving rise to interesting bondingpatterns, consider the CH2Cl radical and its homologues, CH3, CH2F, and CH2OH. InCH2Cl and CH2OH, the carbon’s pz orbital, which hosts the unpaired electron, formsa bonding and antibonding pair with the adjacent lone pair on Cl (O), which leads toincreased bond order of the C–Cl bond. Consequently, the bond length shrinks andthe respective frequency increases, relative to saturated compounds.154–158 In CH2F,such bonding interactions effectively do not occur because of the energy mismatch(the lone pairs of F are too low to interact with pz(C)). These bonding interactionsalso manifest themselves spectroscopically, giving rise to new intense transitions,159

which are not present in CH3, CH2F, or CH3Cl.

PROPERTIES AND SPECTROSCOPY

Electronic structure calculations rarely stop at computing energies and structures.To make connections with experimental observables, the ability to compute variousproperties is desired. In the case of open-shell species, which are often highly reac-tive and, therefore, difficult to prepare and study experimentally, the ability to modelexperimental measurements is crucially important.

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Spectroscopy is often used for detection and characterization of closed- andopen-shell species. Below we discuss computational aspects of modeling commonspectroscopic measurements, highlighting caveats relevant to open-shell species.

Vibrational Spectroscopy

Vibrational spectra—infrared or Raman—provide detailed information aboutmolecular structure. The positions of the lines depend on the shape of the PES(its curvature around the minimum), and the intensities contain information aboutelectronic properties, that is, changes in charge distributions upon nuclear displace-ments. Calculations of vibrational spectra in quantum chemistry are most oftenperformed within the harmonic approximation; they require the construction of thesecond-derivative (Hessian) matrix. The most robust way to compute the Hessian isby using analytic derivative techniques. Unfortunately, analytic second derivativesare not commonly available for advanced methods and, instead, finite differences (ofenergies or gradients) are used, which entails ∼M single-point gradient calculationsor ∼M2 energy evaluations (where M is the number of nuclear degrees of freedom).The drawbacks of the finite-differences approach are that (i) point-group symmetryis lowered upon some displacements, which increases the cost of single-pointcalculations; (ii) staying on the same electronic state might be problematic in casesof nearly degenerate electronic states. The second problem is particularly importantin open-shell species, where there are multiple electronic states.

While harmonic spectra provide a useful starting point for analysis, they areusually not sufficient for rigorous and full assignment, due to anharmonicity effects.Anharmonicity can also strongly affect the intensities. The anharmonic effects canbe accounted for by more advanced techniques.160–163 For quick semiquantitativecomparison, empirical scaling factors are often used. When high accuracy is notneeded, CCSD or standard DFT often provides a reasonable description of thevibrational spectra in radicals.

Electronic and Photoelectron Spectroscopy

The probability of a transition between two states is proportional to the square of thetransition dipole matrix element between the initial (I) and final (F) states:

PIF ∝ |∫ ΨI(r,R)𝜇ΨF(r,R)dr dR|2, [32]

where 𝜇 is the electronic dipole moment operator, r and R are the electronic andnuclear coordinates, respectively, and ΨI(r,R) and ΨF(r,R) are the initial and finalwave functions. In the case of photoionization or photodetachment, the final wavefunction is a product of the N−1 electron state of the molecular core and the wavefunction of the free electron, whereas in electronic spectroscopy, it is an electronicallyexcited state of the molecule. Within the Born–Oppenheimer approximation, ΨI(r,R)

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and ΨF(r,R) are separated into electronic and nuclear parts such that

Ψ(r,R) ≈ Φ(r;R)𝜉(R), [33]

where 𝜉(R) is the vibrational wave function and the dependence of Φ(r;R) on thenuclear coordinates is parametric. Substituting this back into Eq. [32] gives

PIF ∝ |∫ ΦI(r;R)𝜉I(R)𝜇ΦF(r;R)𝜉F(R)dr dR|2. [34]

Introducing the electronic transition moment,

𝜇IF(R) = ∫ ΦI(r;R)𝜇ΦF(r,R)dr, [35]

the expression for the transition probability becomes

PIF ∝ |∫ 𝜉I(R)𝜇IF(R)𝜉F(R)dR|2. [36]

This expression takes into account both the spatial overlap of the initial and finalvibrational wave functions as well as the dependence of the transition moment onthe nuclear geometry. In the Condon approximation, the dependence of 𝜇IF on thenuclear coordinates is neglected (i.e., 𝜇IF(R) ≈ 𝜇

IF(Req)), giving rise to the familiarFranck–Condon factors (FCFs), defined as overlaps of the vibrational wave functionsof the initial and final states:

∫ 𝜉I(R)𝜉F(R)dR = ⟨𝜉In|𝜉Fn′⟩ [37]

Figure 30 shows a diagram illustrating energy levels and intensity profiles character-istic of electronic transitions. It also defines relevant energy differences, adiabatic andvertical ΔE. Adiabatic energy corresponds to the 00 transition; thus, it determines theonset of a band (provided that the respective FCFs are nonzero). The vertical energydifference is well defined theoretically; however, its relationship to the spectral fea-tures is less clear. In the case of a single FC-active mode, it corresponds roughlyto the maximum of the vibrational progression. However, in polyatomic molecules,the maximum of the spectra often corresponds to the 00 transition, as illustrated bythe photoelectron spectrum of phenolate in the right panel of Figure 30. In a largermolecule, a model chromophore of the green fluorescent protein, the computed peakmaximum (which corresponds to the 00 transition) is shifted relative to the verticalenergy by 0.11 eV. The reason for such large deviations from the simplified picture inFigure 30 is that in polyatomic molecules the ionization-induced structural changesare relatively small (because the MOs are delocalized), giving rise to the large FCFsfor the 00 transitions. However, small relaxation along many modes adds up to asignificant difference between vertical and adiabatic ΔE.

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0-0

0-1

0-2

0-3 0

-4 0-5

0-6

E

Initial state

Final state

Vertical ΔE

Adiabatic ΔE

(a) (b)

2.2 2.3 2.4 2.5 2.6

Energy (eV)

0

0.1

0.2

0.3

Inte

nsity (

au

)

FIGURE 30 Illustration of Franck–Condon progressions in electronic and photoelectronspectroscopy. (a) Initial and final electronic states and the corresponding photoelectron spec-trum for a diatomic molecule (or, more generally, a single Franck-Condon active mode). Thedefinitions of adiabatic and vertical energy differences are shown. In this case, the most intensevibronic peak corresponds to vertical ΔE. (b) Photodetachment spectrum of the phenolateanion: calculated Franck–Condon factors (gray) and experimental spectrum (black).164 Themost intense band corresponds to the 00 transition, which is shifted by 0.05–0.08 eV fromthe vertical ΔE. The theoretical spectrum is computed using T = 300 K. Reproduced withpermission of American Chemical Society from Ref. 165.

Photoelectron Spectroscopy Photoionization and photodetachment experiments arebroadly used in chemical physics.166–168 Photoionization is often used as a tool toidentify transient reaction intermediates, whereas photodetachment can be employedto create molecules that are reactive intermediates or otherwise unstable in order tostudy their properties and spectroscopic signatures.169–171 Such experiments providedetailed information about the energy levels of the target system and, often equallyimportant, about the underlying wave functions.

There are several types of spectra produced in photoionization/photodetachmentexperiments.166–168 Most detailed information is provided by spectra that are resolvedwith respect to both the kinetic energy and the angular distribution of ejected elec-trons. If all photoejected electrons with a given energy are collected without resolvingtheir direction with respect to the ionizating photon, one obtains a spectrum of the typeshown in Figure 30; these are called photoelectron spectra. In some experiments,the total yield of ions as a function of ionizing energy is measured, producing thephotoionization efficiency (PIE) curves. They can be described as integrated photo-electron spectra (conversely, differentiation of the PIE curves produces a spectrumwhich can be used as a proxy for the photoelectron spectrum).

For interpretation of photoionization/photodetachment measurements, the follow-ing quantities are needed:

• Energies of electronic states of the target system. These determine which statesare accessible at a given photon wavelength and determine the onsets of pho-toelectron kinetic energies for each band. The definition of relevant energies isgiven in Figure 30.

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• FCFs, the overlaps between the vibrational wave functions defined in Eq. [37].FCFs determine lineshape and vibrational progressions.

• Photoelectron dipole matrix element, Dk = u⟨𝜙d|r|Ψelk ⟩, a quantity entering

the expression of the total and differential cross sections. Here r is the dipolemoment operator, u is a unit vector in the direction of the polarization of light,Ψel

k is the wave function of the ejected electron, and 𝜙d is a Dyson orbital

defined by Eq. [21] connecting the initial N-electron and the final N−1-electronstates.

As follows from the selection rules for the harmonic oscillator, vibrationalprogressions arise due to differences in equilibrium geometries of the initial and finalstates (i.e., for the two identical harmonic potentials, only the diagonal transitions,n = n′, are allowed). The FCFs can be easily computed within the double-harmonicapproximation by, for example, the ezSpectrum software.172

As input, such calculations require equilibrium geometries, normal modes, andfrequencies for the initial and final states. The double-harmonic approximation withor without Duschinsky rotations provides a good description of photoelectron spectrafor rigid molecules (treating free-rotor displacements requires special care). Figure 31shows the photoelectron spectrum of the para-benzyne anion. The spectrum containstwo overlapping bands corresponding to the triplet and singlet target states. Fromthe 00 peaks of each band, the adiabatic singlet–triplet gap can be determined. Thevibrational progressions contain the information about the respective equilibriumstructures. The SF calculations combined with the double-harmonic parallel modeapproximation yield excellent agreement with the experimental spectrum and allowone to extract structural information from the latter.

1.2 1.4 1.6 1.8 2.0 2.2 2.4

Inte

nsity

Electron binding energy (eV)

1.2

(a) (b)

1.4 1.6 1.8 2.0 2.2 2.4

Inte

nsity


FIGURE 31 Photodetachment spectrum for para-benzyne anion. (a) Singlet (black)and triplet (gray) bands of the photoelectron spectra computed using SF methods anddouble-harmonic approximation. (b) The experimental (black) and the calculated (gray) spec-tra, which include singlet and triplet states of the diradical, using the CCSD-optimized geom-etry and normal modes for the anion. Reproduced with permission of Springer from Ref. 45.

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When electron ejection leads to large displacements in modes of the target statethat are largely curvilinear in character (especially torsional angles), as may happenin clusters, a different treatment is required.76 An example of such a case, thethymine–water cluster, is shown in Figure 32. The calculations revealed that theionization induces large changes in the relative position of the water and thyminemoieties. Consequently, double-harmonic calculations (even with inclusion ofDuschinsky rotations) of the spectrum yield very poor agreement with experiment.To account for strongly coupled water–thymine displacement, the water–thyminemodes need to be treated separately, by performing model low-dimensionality quan-tum calculations (three degrees of freedom were included in quantum calculations76).The total vibrational state can then be described as a product of the harmonic statesof the two fragments and a three-dimensional wave function corresponding to

1.2

H3C

N

N

CH3

O O

O

O

N

NO

O

T1 T2 and T3

1.198

(–0.023)

1.486

(0.030)

1.391

(–0.009)

1.012

(0.001)

1.201

(–0.031)1.044

(0.023)

1.670

(–0.226)1.357

(–0.024)

1.190

(–0.025)1.446

(–0.063)

1.312

(–0.059)

1.401

(–0.050)

1.015

(–0.009)

1.383

(–0.005)

0.958

(–0.013)0.958

(0.001)1.490

(0.048)

1.366

(–0.009)1.192

(–0.035)

1.433

(–0.062)

2.764

(0.893)

0.961

(–0.011)

1.625

(–0.245)

1.051

(–0.023)

1.308

(–0.064)

1.404

(–0.052)

0.958

(0.000)

T-H2Oexp. DPIET

1T

2T

31.0

0.8

0.3Water–thymine motion

Thymine

Overall

0.2

0.1

0.0

1.0

0.5

Inte

nsity (

au

)

0.0

1.0

0.8

0.6

0.4

0.2

0.0

8.4 8.5 8.6 8.7 8.8 8.9

Energy (eV)

9.0 9.1 9.2 9.3 9.4

0.6

Inte

nsity (

au

)

0.4

0.2

0.08.0 8.5 9.0 9.5 10.0 10.5

Photon energy (eV)

(b) (c)

(a)

11.0 11.5 12.0 12.5

FIGURE 32 Photoionization of thymine–water clusters. (a) Equilibrium structures of ion-ized thymine monohydrate (lowest energy structures, T1,T2, and T3). Bond lengths andchanges in bond lengths due to ionization are shown (in Å). (b) Computed and experimen-tal photoelectron spectra. The latter is derived from the photoionization efficiency curves.(c) Computed photoelectron spectrum of water–thymine T1 cluster. Calculated FCFs for thefirst ionized state of T1 (lower panel). Upper panel: FCFs due to water–thymine motion com-puted using curvilinear coordinates. Middle panel: FCFs due to thymine moiety computedusing double-harmonic parallel mode approximation. Bottom: Overall photoelectron spectrumcomputed as a product of the thymine and water–thymine FCFs. Reproduced with permissionof Royal Society of Chemistry from Ref. 76. (See color plate section for the color representa-tion of this figure.)

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water–thymine modes. This description results in good agreement between thetheoretical and experimental spectra, as one can see from the bottom-left panel inFigure 32.

Modeling of photoionization spectra requires more advancement treatments insystems with strong vibronic interactions, which result in the breakdown of theFranck–Condon approximation. Due to the large density of states, such cases arerather common in radicals. To model such situations, the FCFs must be computedin a multistate representation and no longer are nonzero only for modes that do notbreak the symmetry of the molecular framework. This can be achieved, for example,by employing vibronic Hamiltonians. An example44 of a photoelectron spectrumthat is noticeably distorted by vibronic interactions is shown in Figure 33.

Total and Differential Cross Sections To model quantities such as PIE curvesand angular distributions of photoelectrons, electronic cross sections are needed.173

The key quantity is the photoelectron dipole matrix element defined above. Inorder to compute it, one needs Dyson orbitals, which contain all the necessary

0.5

(a)

(b)

1200

1000

800

600

400

200

1000

800

600

400

200

03.1

hg

f e d

cb

a

3.0 2.9 2.8


2.7 2.6 2.5

0

0.6 0.7

Electron kinetic energy (eV)

Photo

ele

ctr

on c

ounts

Photo

ele

ctr

on c

ounts

0.8 0.9 1.0

FIGURE 33 Simulated photoelectron spectra of the 1-imidazolyl radical, superimposedon the experimental spectrum (dots). (A) Adiabatic simulations roughly similar todouble-harmonic calculations within the Franck–Condon model. The spectrum shown in(b) includes vibronic interactions between the radical states that lead to the breakdown of theCondon approximation. Inclusion of vibronic effects has noticeable effect on relative peakintensities (peaks g, h, and f ) and new (“vibronic”) peaks appear in the spectrum, resulting inbetter agreement with the experiment. Reproduced with permission of American Institute ofPhysics from Ref. 44.

�

� �

�


2.0

1.5

1.0

0.5

Tota

l cro

ss-s

ectio

n (M

b)

Tota

l cro

ss-s

ectio

n (M

b)

Tota

l cro

ss-s

ectio

n (M

b)T

ota

l cro

ss-s

ection (

au)

Tota

l cro

ss-s

ection (

au)

Tota

l cro

ss-s

ection (

au)

0.00.5 1.0

H– Ne Formaldehyde

1.5 2.0 2.5 3.0

Energy of ionizing radiation (eV) Energy of ionizing radiation (eV) Energy of ionizing radiation (eV)

560.6 0.8

0.6

0.4

0.2

0.010.5 11.0 11.5 12.0 12.5

22.4

16.8

11.2

5.6

0.0

0.4

0.2

0.010 30 50 70 90 110 130

16.8

11.2

5.6

0.0

42

28

14

0

FIGURE 34 Absolute total cross sections for H−, Ne, and formaldehyde. The theoreticalvalues are computed using EOM-IP-CCSD Dyson orbitals (shown in the insets) and ezDyson.Light and dark gray lines denote cross sections computed using plane wave (Z = 0) andCoulomb wave (Z = 1) wave functions of the free electron. Plane waves give a good descrip-tion of photodetachment cross sections, while Coulomb waves give a better description forphotoionization from atoms (and some small molecules). For formaldehyde, the orange linefollowing the experimental points denotes cross section computed using a Coulomb wave withpartial charge (Z = 0.25). Reproduced with permissions of the American Chemical Societyfrom Ref. 174. (See color plate section for the color representation of this figure.)

information about the initial and final states, and the wave function of the freeelectron. Commonly used approximate treatments of the latter are either plane wavesor Coulomb waves.174 The calculations of the cross sections and photoelectronangular distributions can be performed with the stand-alone ezDyson program,175

which requires information about molecular structure, Dyson orbitals, and specificexperimental details. Figure 34 shows absolute cross sections for several anionicand neutral systems. Using EOM-CC Dyson orbitals along with a plane or Coulombwave description of the free electron leads to good agreement with the experimentalcross sections.

Electronic Spectroscopy Similar to photoelectron/photodetachment spectroscopy,modeling lineshapes of electronic transitions also requires FCFs, which can be com-puted in exactly the same way as for photoionization, by using the structural informa-tion of the initial and final electronic states. In addition, one needs transition dipolemoments between the states involved. In cases with strong vibronic interactions, tran-sition dipole moments computed along Franck–Condon-active modes are needed.176

Electronic Transitions

Studying chemical processes often requires going beyond computing PES. Owing toelectronic degeneracy, which is common in open-shell species, multiple electronicstates are often involved in the dynamics. In this context, the ability of an elec-tronic structure method to compute multiple states and to describe their interactionsis important. In this respect, multistate methods such as EOM-CC are very attractive.In addition to computing multiple states, the electronic couplings between them canbe easily computed, which is important for dynamics, since these couplings governelectronic transitions between the states.

Nonradiative transitions between different electronic states fall into two cat-egories. The first category results from the breakdown of Born–Oppenheimerseparation of nuclear and electronic motions; they are governed by NACs (derivative

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couplings). The second category occurs due to relativistic effects, which are mostoften neglected in standard quantum chemistry; these transitions are governedby SOCs.

The rates of transitions between different electronic states can be estimated usingFermi’s Golden Rule. In this framework, derived using perturbation theory, therate of a transition between the initial I and final F electronic states of a system isgiven by

kI→F = 2𝜋⟨∑F′

|⟨ΦI𝜉I′ |A|ΦF𝜉F′⟩|2𝛿(EII′ − EFF′ )⟩T [38]

where ΦI𝜉I′ and ΦF𝜉F′ denote the initial and final states, the sum runs over the finalvibrational states, and the thermal averaging is performed over the initial vibrationalstates. EII′ and EFF′ denote total (electronic and nuclear) energies of the initial andfinal states. A is the coupling operator, for example, NAC for radiationless transi-tions (internal conversion) and SOC for spin-forbidden ones (intersystem crossing).Depending on the physics of the problem, different approximations to this expressionare possible.177–179

There are two alternative strategies for computing properties and transition prob-abilities for approximate wave functions.24 In the response-theory formulation, oneapplies time-dependent PT to an approximate state (e.g., the CCSD wave function); inthis approach, transition moments are defined as the residues of respective responsefunctions (and excitation energies as poles). Identical expressions for properties canbe derived for static perturbations using an analytic derivative formalism.18,180–185

In the expectation-value approach, one begins with expressions derived for exactstates and then uses approximate wave functions to evaluate the respective matrix ele-ments. Of course, the two approaches give the same answer for exact states; however,the expressions for approximate states are different.24 For example, one can com-pute transition dipole moments for EOM-EE using the expectation-value approach;in this case, a so-called unrelaxed one-particle transition-density matrix, Eq. [26],will be used.186 Alternatively, a full response derivation gives rise to expressions thatinclude amplitude (and, optionally, orbital) relaxation terms; this is how the proper-ties are computed within the linear-response formulation of CC theory. Numerically,the two approaches are rather similar,24,183,187 although the expectation-value formu-lation of properties within EOM-CC is not size intensive188 (the deviations, however,are relatively small189). The response formulation can become singular in the casesof degenerate states,190 whereas the expectation-value approach does not have such aproblem. For the purpose of this chapter, the differences between the two approachesare insignificant. Following the author’s personal preference, the following discussionis based on the expectation-value approach.

The non-Hermitian nature of EOM-CC theory requires additional steps when com-puting interstate properties because the left and right eigenstates of H are not thesame, but rather form a biorthogonal set. Thus, in EOM-CC, the matrix element ofan operator A between states ΨM and ΨN is defined as follows:

|AMN| = √⟨ΨM|A|ΨN⟩⟨ΨN|A|ΨM⟩ [39]

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AMN is independent of the choice of the norms of the left and right EOM states(because the left and right EOM states are biorthogonal, the norms of the left or rightvectors are not uniquely defined).

Spin-Orbit Couplings Within the expectation-value approach, SOCs can becomputed as the matrix elements of the respective part of the Breit-PauliHamiltonian173,191 (spin-same-orbit and spin-other-orbit terms) using zero-ordernonrelativistic wave functions. The SO part of the Breit-Pauli Hamiltonian containsone- and two-electron operators:

Hso = ℏe2

2m2ec2

[∑i

hso(i) ⋅ s(i) −∑i≠j

hsoo(i, j) ⋅ (s(i) + 2s(j))

][40]

hso(i) =∑

K

ZK(ri − RK) × pi|ri − RK|3 [41]

hsoo(i, j) =(ri − rj) × pi|ri − rj|3 [42]

where ri and pi are the coordinate and momentum operators of the ith electron andRK and ZK denote the coordinates and the charge of the Kth nucleus.

Thus, evaluation of the respective matrix elements requires one- and two-particlereduced density matrices. The two-electron part is significant—it can account forabout 50% of the SOC in molecules composed of light atoms (for heavy atoms, therelative magnitude of the two-electron part is reduced). Fortunately, the dominantcontributions from the two-electron part can be computed using the separable part ofthe two-particle density matrix (which consists of the product of the one-particle den-sity matrix for the reference state and the one-particle transition density matrix con-necting the two states of interest) giving rise to the mean-field approximation.192–196

Since such calculations do not involve the contraction of the full two-particle densitymatrix with the two-particle spin-orbit integrals, they have significantly reduced com-putational and storage requirements. The differences between the full SOC and themean-field approximation are consistently less than 1 cm−1 for all examples studiedto date.

In SOC calculations, one computes the matrix elements of Hso given by Eq. [40]between the nonrelativistic wave functions, Ψ(s,ms) and Ψ′(s′,m′

s), of the statesinvolved:

⟨Hso⟩ = ⟨Hsox ⟩ + ⟨Hso

y ⟩ + ⟨Hsoz ⟩ [43]

⟨Hso𝛼⟩ ≡ ⟨Ψ(s,ms)|Hso

𝛼|Ψ′(s′,m′

s)⟩ 𝛼 = x, y, z [44]

Following these equations, ⟨Hso⟩ is a complex number given by the sum of the matrixelements of the three Cartesian components of the spin-orbit operator from Eqs [41]and [42]. The couplings between different multiplet components, as well as the indi-vidual Cartesian contributions, are not invariant with respect to molecular orientation

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(assuming that the spin quantization axis is fixed along the z-axis). However, thequantity called the SOC constant194,197 (SOCC) is invariant:

|SOCC(s, s′)|2 ≡s∑

ms=−s

s′∑m′

s=−s′

[|⟨Ψ(s,ms)|Hsox |Ψ′(s′,m′

s)⟩|2+ |⟨Ψ(s,ms)|Hso

y |Ψ′(s′,m′s)⟩|2 + |⟨Ψ(s,ms)|Hso

z |Ψ′(s′,m′s)⟩|2]

[45]

It is the SOCC rather than individual components that is needed in the context ofFermi’s Golden Rule calculations of rates for spin-forbidden processes.

SOC calculations are available for various EOM-CCSD wave functions.190,195,196

A recent study196 analyzed the performance of EOM-CCSD for SOCs calculations. Itwas found that when used with an appropriate basis set, the errors against experimentare below 5% for light elements. The errors for heavier elements (elements up toBr and Se were considered) are larger (up to 10–15%). This study196 also analyzedthe impact of different correlation treatments in zero-order wave functions and foundthat EOM-IP-CCSD, EOM-EA-CCSD, EOM-EE-CCSD, and EOM-SF-CCSD wavefunctions yield SOCs, which agree well with each other (and with the experimentalvalues when available). Not surprising, using an EOM approach that provides a morebalanced description of the target states10 yields more accurate results.

Nonadiabatic (Derivative) Couplings The NAC calculation for the exact wave func-tions (or, more precisely, when Hellman–Feynman conditions are satisfied) requiresonly an unrelaxed one-particle transition density matrix:

1EI − EF

⟨ΨI|Hx|ΨF⟩ = 1ΔIF

∑pq

Hxpq𝛾

IFpq [46]

where 𝛾 IFpq is the transition density matrix defined by Eq. [26], ΔIJ is the difference

between the adiabatic energies, and Hx denotes the full derivative of the electronicHamiltonian with respect to nuclear coordinate x.

When using approximate wave functions such as CIS, TDDFT, or EOM-CC,one may invoke full response formalism for evaluating the derivative of the wavefunction, leading to the Pulay terms that needs to be added to the expectation-valueexpression.24 Such theoretically complete formalism and its computer implemen-tation for the MR-CI wave functions were first developed by Yarkony.198–202 Morerecently, this formalism has been applied to the CIS and TDDFT methods;203–205

analytic derivative matrix elements have also been computed for EOM-CC wavefunctions.43,206,207

Using the expectation-value approach, in which orbital-response terms areneglected, has an important benefit in the context of NAC calculations. When usingincomplete atomic orbital bases, NACs computed using full derivative formalismare not translationally invariant, which may cause serious problems in the context of

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calculating rates of nonadiabatic transitions in ab initio molecular dynamics.204,205

It has recently been shown for CIS and TDDFT that the correction restoring thetranslational invariance is exactly equal to the Pulay term. Thus, an “approximate”Hellman–Feynman expression for the NAC is, in fact, identical to the full NACexpression corrected to achieve translational invariance. This strongly argues in favorof adopting the expectation-value approach for computing NACs for the EOM-CCwave functions.

Finally, it is worth mentioning that ||𝛾||, the norm of the transition one-particle den-sity matrix, can be used as a crude proxy for NAC,150 owing to the Cauchy–Schwartzinequality:

1EI − EF

⟨ΨI|Hx|ΨF⟩ = 1ΔIF

∑pq

Hxpq𝛾

IFpq ≤ 1

ΔIF||Hx|| ⋅ ||𝛾 IF|| [47]

More generally, ||𝛾|| quantifies the degree of one-electron character in the I → Ftransition and can be exploited to monitor changes in the states upon structuralchanges. This tool has been used to investigate the effect of molecular packing onthe singlet-fission rates.208–210 It also provides a simple metric for analyzing the typeof electronic transitions.150

Charge-Transfer Systems and Marcus Rates Marcus theory provides a recipe forcalculating the rates of charge transfer between donor and acceptor states:211,212

k = 2𝜋ℏ

|HIF|2 1√4𝜋𝜆kBT

exp

{−(ΔG + 𝜆)2

4𝜆kBT

}[48]

where ΔG, 𝜆, and HIF are the free energy change, reorganization energy, and thecoupling between the electronic states involved in the transfer (I and F denote theinitial and final states, respectively). The key quantity here is the electronic couplingelement between the diabatic donor and acceptor states, HIF.

Several alternative approaches based on a simplified two-state model existfor computing HIF.213 Marcus theory is formulated for interacting diabaticcharge-localized states. In this representation, the electronic Hamiltonian is non-diagonal. Diagonalizing it produces adiabatic states (these are the states that areproduced by regular ab initio calculations):

(EI HIF

HIF EF

)→

(E1 00 E2

)[49]

with the eigenvalues

E1,2 =EI + EF

2±

√(EI − EF

2

)2

+ H2IF [50]

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At the transition state (such as symmetric configuration of the (H2)+2 ), EI = EFleading to a simple connection between the energy gap between the adiabatic statesand the coupling:

|HIF| = |E1 − E2|2

[51]

Equation [51] can be used for computing the couplings only for symmetricdonor–acceptor pairs and only at transition states.

One can compute the coupling element directly, if the localized-charge states areavailable (these can be computed by fragment-based calculations):

HIF = ⟨ΨI|H|ΨF⟩ [52]

SIF = ⟨ΨI|ΨF⟩ [53]

Here SIF denotes the overlap matrix between the fragment-localized states, ΨI andΨF. When these states are nonorthogonal, the eigenproblem [49] needs to be appro-priately modified, leading to the following expression for the electronic coupling:213

VIF =HIF − (EI + EF)SIF∕2

1 − S2IF

[54]

One can also obtain the transformation from adiabatic to diabatic charge-transferstates by diagonalizing a property matrix. For example, it is reasonable to expectthat for the localized charge-transfer states, the matrix of the dipole moment opera-tor is diagonal. Thus, the desired diabatic states can be found by diagonalizing thematrix of the dipole moment operator computed in the basis of adiabatic states. Thisis the essence of the Mulliken–Hush scheme.214–217 The resulting expression for thecoupling is

HIF =𝜇12ΔE12√

(𝜇1 − 𝜇2)2 + 4𝜇212

[55]

where indices 1 and 2 denote the respective adiabatic states. Thus, the Mulliken–Hushapproach requires the ability to compute state and transition dipole moments for givenadiabatic wave functions. Provided these quantities are available, one can computeHIF for any type of adiabatic wave functions.

Figure 35 compares the values of HIF computed for a model system using theenergy-gap scheme, Eq. [51], with various wave functions. Couplings for hole andelectron transfer were computed for the ethylene dimer cation and anion, respec-tively. The crudest model, HF-KT, used Hartree–Fock orbital energies in lieu of thetarget-state energies of the ionized or electron-attached states. Correlation effectswere included by performing CCSD calculations. A more balanced description wasachieved by using EOM-CC energies. Figure 36 shows various EOM-CC schemes forcomputing energies of the target states. Finally, direct coupling (DC) calculation wasperformed by computing the overlap between the localized fragment Hartree–Fock

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0.03

0.025

0.02

0.015

Couplin

g (

eV

)

0.01

0.005

0

H

H

H

H

H

H

C C

H

CH3

CH3

9.4

1 Å

3.9

4 Å

H

C C

HT ET

HF-KTSF-CCSDIP(EA)-EOMΔCCSDDC

FIGURE 35 The values of coupling elements for hole transfer (HT) and electron transfer(ET) for model ethylene dimers. Reproduced with permission of American Chemical Societyfrom Ref. 218. (See color plate section for the color representation of this figure.)

Spin-Flip

π∗+

π∗–

π+

π–

π∗+

π∗–

π+

π–

Reference

(a)

(A)

(B)

Target 1

(b)

Target 2

(c)

Reference

(d)

Reference

(a)

Target 1

(b)

Target 2

(c)

Reference

(d)

IP-EOM

Spin-Flip EA-EOM

Mo

lecu

lar

orb

ita

l e

ne

rgy

Mo

lecu

lar

orb

ita

l e

ne

rgy

FIGURE 36 Schemes for computing the couplings using EOM-SF, EOM-IP, and EOM-EAwave functions, with 𝜋 orbitals depicted for two ethylenes as an example. Shown in panel (A) isthe scheme for hole transfer and in panel (B) that for electron transfer. Configurations in panels(a) are the charged quartet configuration (for SF) and those in panels (d) are neutral singlet (IPand EA) reference states. In panels (b) and (c) are the target charged, doublet configurations.Reproduced with permission of American Chemical Society from Ref. 218.

�

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�


0.0 0.1 0.2

Reaction coordinate(a) (b) (c)

Couplin

g (

cm

–1)

Couplin

g (

cm

–1)

Couplin

g (

cm

–1)


0.3 0.4 0.5–60

–30

EOM-IP-CCSD

EOM-IP-CCSD

CCSD/EOM-EE-CCSD

CCSD/EOM-EE-CCSD

EOM-IP-CCSD

CCSD/EOM-EE-CCSD

0

30

0.0 0.1 0.2 0.3 0.4 0.5–60

–58

–560.00.51.01.52.0

0.00 0.25 0.50 0.75 1.0031.75

32.00

38.25

38.50

38.75

FIGURE 37 Errors against FCI for diabatic Mulliken–Hush coupling in model charge-transfer systems. (a and b) (H2)+2 at 3.0 and 5 Å separation. EOM- IP-CCSD/aug-cc-pVTZ(solid line) and EOM-EE-CCSD/aug-cc-pVTZ (dotted line) results are shown. (c) (Be-BH)+.EOM-IP-CCSD/aug-cc-pVDZ (solid line) and EOM-EE-CCSD/aug-cc-pVDZ (dotted line)results are shown. Pieniazek et al.71. Reproduced with permission of American Institute ofPhysics.

wave functions. As one can see, correlation is clearly important. When using theenergy-gap scheme, all CCSD methods give similar results. The reasonable perfor-mance of open-shell CCSD can be explained by the fact that these calculations arebased on symmetric structures at which the problems due to unbalanced descriptionof the important configurations do not occur, that is, the leading configurations of thetwo charge-transfer states (i.e., Ψ1 and Ψ2 from Figure 8) are of different symmetryand cannot mix.

The advantage of the Mulliken–Hush scheme is that it can be applied to allstructures including the nonsymmetric ones. For example, one can computeMulliken–Hush couplings along a charge-transfer reaction coordinate. Examplesof such calculations for (H2)+2 and (Be-H2)+ are shown in Figure 37. In thesecalculations, the couplings are computed using two different approaches: oneemploying EOM-IP-CCSD wave functions and open-shell CCSD/EOM-EE-CCSDones. Absolute errors against FCI of the CCSD and EOM-EE-CCSD calculationsare comparable with those of EOM-IP-CCSD; however, the latter behaves in a muchmore uniform way.

OUTLOOK

The electronic structure of open-shell species is rich and complex; it leads to uniquechemical and spectroscopic properties. However, it also brings methodological chal-lenges. Often, open-shell systems feature close-lying multiple electronic states. Theseelectronic near-degeneracies result in multiconfigurational wave functions that causethe breakdown of the standard hierarchy of single-reference methods and Kohn–ShamDFT. A skilled computational chemist should be able to identify these situations andemploy more appropriate treatments, such as a particular EOM-CC approach thatsuits the task at hand. Some types of open-shell structure can be tackled by standardapproaches; however, complications due to spin-contamination may occur. We hopethat this review provides a useful guide for navigation through the myriad problems

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posed by open-shell species. We also highlighted computational aspects relevant toapplications, such as modeling spectroscopy and nonadiabatic processes in open-shellspecies.

Of course, a single chapter does not provide complete coverage of the topic. Thus,we would like to conclude by suggesting additional reading. For a general introduc-tion to practical quantum chemistry, we suggest the textbook Essentials of Com-putational Chemistry: Theories and Models.7 For readers seeking more-advancedunderstanding of methodology, we recommend the book by Szabo and Ostlund.9 Avery detailed and rigorous description of many-body methods can be found in Molec-ular Electronic Structure Theory.6

There are also more focused reviews and tutorials. An excellent introduction tocorrelated methods, with an emphasis on CC approaches, is given in Refs. 16 and219. To learn more about EOM-CC approaches, we recommend Refs. 10 and 11. Analternative strategy based on ADC is reviewed in Ref. 27.

More-specific reviews include an overview of the SF approach,85 excited-statemethods targeting large molecules,114 Rydberg-valence interactions in radicals,220

and the quantum chemistry of loosely bound electrons.221 Different aspects relevantto electronic structure of radicals are discussed in Ref. 48.

This review would not be complete without a brief discussion of software. Manyexcellent quantum chemistry packages exist offering both standard and highlyspecialized features. The author is familiar with the functionality of (and has utilized,on various occasions) CFOUR,222 ACES3,223 Turbomole,224 Dalton,225 Psi4,226

GAMESS,227 Molpro,228 Molcas,229 Spartan,230 and Q-Chem.231 The choice ofpackage is a matter of personal preferences. The author has been a long-term userof and contributor to Q-Chem,231,232 a versatile electronic structure suite of tools,offering efficient implementations of methods targeting open-shell species describedin this chapter. For a detailed list of features, readers are referred to Q-Chem UserManual and to the series of webinars describing practical aspects of calculations.233

Useful auxiliary packages include the graphic visualization tool, IQmol,234 andspectroscopy modeling tools such as ezSpectrum172 and ezDyson.175

ACKNOWLEDGMENTS

The work on open-shell species, which is a main focus of my research, has beenfunded by the National Science Foundation, Department of Energy, and ArmyResearch Office. The methodological developments have greatly benefited fromthe support by Q-Chem, Inc. The author is grateful to Prof. John F. Stanton for hisinsightful remarks and helpful suggestions. I would also like to thank Profs. A. V.Nemukhin, V. I. Pupyshev, T. D. Crawford, Drs. S. Gozem, T.-C. Jagau, S. Faraji, K.Nanda, and Mr. P. Pokhilko for their useful feedback regarding the manuscript. Myspecial thanks to Mr. Jay Tanzman for his meticulous proofreading and for making459 corrections, which greatly improved the readability of the manuscript.

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APPENDIX: LIST OF ACRONYMS

ADC: algebraic-diagrammatic constructionAO: atomic orbital

CAS: complete active spaceCASSCF: complete active-space self-consistent fieldCOL: collinear

CC: coupled-clusterCCD: coupled-cluster with doublesCCSD: coupled-cluster with singles and doublesCIS: Configuration interaction singles

CISD: Configuration interaction with singles and doublesDFT: density functional theoryDSE: diradical separation energyEOM: equation-of-motion

EOM-EA-CCSD: equation-of-motion coupled-cluster with single and double sub-stitutions for electron attachment

EOM-EE-CCSD: equation-of-motion coupled-cluster with single and double sub-stitutions for excitation energies

EOM-IP-CCSD: equation-of-motion coupled-cluster with single and double substi-tutions for ionization potentials

EOM-SF-CCSD: equation-of-motion coupled-cluster with single and double sub-stitutions with spin-flip

FCI: full configuration interactionFCF: Franck–Condon factor

HF: Hartree–FockIE: ionization energyJT: Jahn–TellerMO: molecular orbital

NAC: nonadiabatic couplingNC: noncollinearHOMO: highest occupied molecular orbitalLR-CCSD: linear-response coupled-cluster with single and double substitutions

LUMO: lowest unoccupied molecular orbitalMO-LCAO: molecular orbital-linear combination of atomic orbitalsMP2: Møller–Plesset theory of the second orderMR-CI: multireference configuration interaction

NBO: natural bond orbitalsPES: potential energy surface

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PIE: photoionization efficiency

PJT: pseudo-Jahn–Teller

QRHF: quasi-restricted Hartree–Fock

RASCI: restricted active space configuration interaction

ROHF: restricted open-shell Hartree–Fock

SAC-CI: symmetry-adapted cluster configuration interaction

SF: spin-flip

SIE: self-interaction error

SOC: spin-orbit coupling

SOCC: spin-orbit coupling constant

TDDFT: time-dependent density functional theory

TSE: triradical separation energy

UHF: unrestricted Hartree–Fock

VIE: vertical ionization energy

ZPE: zero-point energy

REFERENCES

1. L. Salem and C. Rowland, Angew. Chem. Int. Ed. Engl., 11(2), 92 (1972). The ElectronicProperties of Diradicals.

2. V. Bonacic-Koutecký, J. Koutecký, and J. Michl, Angew. Chem. Int. Ed. Engl., 26, 170(1987). Neutral and Charged Biradicals, Zwitterions, Funnels in S1, and Proton Translo-cation: Their Role in Photochemistry, Photophysics, and Vision.

3. P. M. Lahti, Ed., Magnetic Properties of Organic Materials, Marcel Dekker, Inc., 1999.

4. M. R. Pederson and T. Baruah, in Handbook of Magnetism and Magnetic Materials, S.Parkin, Ed., John Wiley & Sons, Ltd., London, UK, 2007, pp. 1–22. Molecular Magnets:Phenomenology and Theory.

5. N. J. Mayhall and M. Head-Gordon, J. Phys. Chem. Lett., 6, 1982 (2015). ComputationalQuantum Chemistry for Multiple-Site Heisenberg Spin Couplings Made Simple: StillOnly One Spin-Flip Required.

6. T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic Structure Theory,John Wiley & Sons, Ltd., 2000.

7. C. J. Cramer, Essentials of Computational Chemistry: Theories and Models, John Wiley& Sons, Inc., New York, 2002.

8. M. Head-Gordon, J. Phys. Chem., 100(31), 13213 (1996). Quantum Chemistry andMolecular Processes.

9. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to AdvancedElectronic Structure Theory, McGraw-Hill, New York, 1989.

10. A. I. Krylov, Annu. Rev. Phys. Chem., 59, 433 (2008). Equation-of-MotionCoupled-Cluster Methods for Open-Shell and Electronically Excited Species: The Hitch-hiker’s Guide to Fock Space.

�

� �

�


11. R. J. Bartlett, WIREs Comput. Mol. Sci., 2(1), 126–138 (2012). Coupled-Cluster Theoryand its Equation-of-Motion Extensions.

12. W. Kohn, A. D. Becke, and R. G. Parr, J. Phys. Chem., 100, 12974 (1996). Density Func-tional Theory of Electronic Structure.

13. N. C. Handy, in Lecture Notes in Quantum Chemistry II, B. Roos, Ed., Springer-Verlag,Berlin, 1994. Density Functional Theory.

14. W. Koch and M. C. Holthausen, A Chemist’s Guide to Density Functional Theory,Wiley-VCH, 2001.

15. R. J. Bartlett, Annu. Rev. Phys. Chem., 32, 359 (1981). Many-Body Perturbation Theoryand Coupled Cluster Theory for Electron Correlation in Molecules.

16. R. J. Bartlett and J. F. Stanton, in Reviews in Computational Chemistry, K. B. Lipkowitzand D. B. Boyd, Eds., Wiley-VCH, New York, 1994, Vol. 5, pp. 65–169. Applications ofPost-Hartree-Fock Methods: A Tutorial.

17. R. J. Bartlett, Int. J. Mol. Sci., 3, 579 (2002). To Multireference or Not to Multireference:That is the Question?

18. R. J. Bartlett, Mol. Phys., 108, 2905 (2010). The Coupled-Cluster Revolution.

19. K. Sneskov and O. Christiansen, WIREs Comput. Mol. Sci., 2, 566 (2012). Excited StateCoupled Cluster Methods.

20. J. F. Stanton and R. J. Bartlett, J. Chem. Phys., 98, 7029–7039 (1993). The equation ofmotion coupled-cluster method. A systematic biorthogonal approach to molecular exci-tation energies, transition probabilities, and excited state properties.

21. H. Sekino and R. J. Bartlett, Int. J. Quantum Chem., 26, 255 (1984). A Linear Response,Coupled-Cluster Theory for Excitation Energy.

22. H. Koch, H. J. Aa. Jensen, P. Jørgensen, and T. Helgaker, J. Chem. Phys., 93(5), 3345(1990). Excitation Energies from the Coupled Clusters Singles and Doubles LinearResponse Functions (CCSDLR). Applications to Be, CH+, CO, and H2O.

23. M. Head-Gordon and T. J. Lee, in Modern Ideas in Coupled Cluster Theory, R. Bartlett,Ed., World Scientific, Singapore, 1997. Single Reference Coupled Cluster and Perturba-tion Theories of Electronic Excitation Energies.

24. T. Helgaker, S. Coriani, P. Jørgensen, K. Kristensen, J. Olsen, and K. Ruud,Chem. Rev., 112, 543 (2012). Recent Advances in Wave Function-Based Methods ofMolecular-Property Calculations.

25. H. Nakatsuji and K. Hirao, J. Chem. Phys., 68, 2053 (1978). Cluster Expansion ofthe Wavefunction. Symmetry-Adapted-Cluster Expansion, its Variational Determination,and Extension of Open-Shell Orbital Theory.

26. H. Nakatsuji, Chem. Phys. Lett., 177(3), 331 (1991). Description of Two- and Many-Electron Processes by the SAC-CI Method.

27. A. Dreuw and M. Wormit, WIREs Comput. Mol. Sci., 5, 82 (2015). The AlgebraicDiagrammatic Construction Scheme for the Polarization Propagator for the Calculationof Excited States.

28. J. V. Ortiz, WIREs Comput. Mol. Sci., 3, 123 (2013). Electron Propagator Theory: AnApproach to Prediction and Interpretation in Quantum Chemistry.

29. J. A. Pople, in Energy, Structure and Reactivity: Proceedings of the 1972 Boulder Sum-mer Research Conference on Theoretical Chemistry, D. Smith and W. McRae, Eds.,John Wiley & Sons, Inc., New York, 1973, pp. 51–61. Theoretical Models for Chemistry.

�

� �

�


30. H. Jahn and E. Teller, Proc. R. Soc. London, Ser. A, 161, 220 (1937). Stability ofPolyatomic Molecules in Degenerate Electronic States. I. Orbital Degeneracy.

31. I. B. Bersuker, Chem. Rev., 101, 1067 (2001). Modern Aspects of the Jahn-Teller Effect.Theory and Applications to Molecular Problems.

32. V. I. Pupyshev, Int. J. Quantum Chem., 104, 157 (2005). Jahn-Teller Theorem and NodalPoints of Wave Functions.

33. V. I. Pupyshev, Int. J. Quantum Chem., 107, 1446 (2007). Simplest Proof of theJahn-Teller Theorem for Molecular Systems.

34. L. V. Slipchenko and A. I. Krylov, J. Chem. Phys., 118, 6874 (2003). Electronic Structureof the Trimethylenemethane Diradical in its Ground and Electronically Excited States:Bonding, Equilibrium Structures and Vibrational Frequencies.

35. L. V. Slipchenko and A. I. Krylov, J. Chem. Phys., 118, 9614 (2003). Electronic Structureof the 1,3,5,-Tridehydrobenzene Triradical in its Ground and Excited States.

36. P. A. Pieniazek, S. E. Bradforth, and A. I. Krylov, J. Chem. Phys., 129, 074104 (2008).Charge Localization and Jahn-Teller Distortions in the Benzene Dimer Cation.

37. V. A. Mozhayskiy, D. Babikov, and A. I. Krylov, J. Chem. Phys., 124, 224309 (2006).Conical and Glancing Jahn-Teller Intersections in the Cyclic Trinitrogen Cation.

38. D. Babikov, P. Zhang, and K. Morokuma, J. Chem. Phys., 121(14), 6743 (2004).Cyclic-N3. I. An Accurate Potential Energy Surface for the Ground Doublet ElectronicState up to the Energy of the 2A2∕B2(1) Conical Intersection.

39. D. Babikov, B. K. Kendrick, P. Zhang, and K. Morokuma, J. Chem. Phys., 122(4), 044315(2005). Cyclic-N3. II. Significant Geometric Phase Effects in the Vibrational Spectra.

40. H. C. Longuet-Higgins, Proc. R. Soc. London, Ser. A, 344(1637), 147 (1975). TheIntersection of Potential Energy Surfaces in Polyatomic Molecules.

41. M. V. Berry, Proc. R. Soc. London, Ser. A, 392, 45 (1984). Quantal Phase FactorsAccompanying Adiabatic Changes.

42. C. A. Mead, Rev. Mod. Phys., 64(1), 51 (1992). The Geometric Phase in MolecularSystems.

43. T. Ichino, J. Gauss, and J. F. Stanton, J. Chem. Phys., 130(17), 174105 (2009). Quasidi-abatic States Described by Coupled-Cluster Theory.

44. T. Ichino, A. J. Gianola, W. C. Linebrger, and J. F. Stanton, J. Chem. Phys., 125, 084312(2006). Nonadiabatic Effects in the Photoelectron Spectrum of the Pyrazolide-d(3)Anion: Three-State Interactions in the Pyrazolyl-d(3) Radical.

45. V. Vanovschi, A. I. Krylov, and P. G. Wenthold, Theor. Chim. Acta, 120, 45 (2008). Struc-ture, Vibrational Frequencies, Ionization Energies, and Photoelectron Spectrum of thepara-Benzyne Radical Anion.

46. J. F. Stanton, J. Chem. Phys., 115, 10382 (2001). Coupled-Cluster Theory,Pseudo-Jahn-Teller Effects and Conical Intersections.

47. N. J. Russ, T. D. Crawford, and G. S. Tschumper, J. Chem. Phys., 120, 7298 (2004). RealVersus Artifactual Symmetry Breaking Effects in Hartree-Fock, Density-Functional, andCoupled-Cluster Methods.

48. J. F. Stanton and J. Gauss, Adv. Chem. Phys., 125, 101 (2003). A Discussion onSome Problems Associated With the Quantum Mechanical Treatment of Open-ShellMolecules.

49. J. M. Herbert and M. Head-Gordon, Phys. Chem. Chem. Phys., 8, 68 (2006). Accuracyand Limitations of Second-Order Many-Body Perturbation Theory for Predicting VerticalDetachment Energies of Solvated-Electron Clusters.

�

� �

�


50. R. C. Lochan and M. Head-Gordon, J. Chem. Phys., 126, 164101 (2007).Orbital-Optimized Opposite-Spin Scaled Second-Order Correlation: An EconomicalMethod to Improve the Description of Open-Shell Molecules.

51. G. D. Purvis III, H. Sekino, and R. J. Bartlett, Collect. Czech. Chem. Commun., 53, 2203(1988). Multiplicity of Many-Body Wavefunctions Using Unrestricted Hartree-Fock Ref-erence Functions.

52. A. I. Krylov, J. Chem. Phys., 113(15), 6052 (2000). Spin-Contamination in CoupledCluster Wavefunctions.

53. C. C. Roothaan, Rev. Mod. Phys., 32, 179 (1960). Self-Consistent Field Theory for OpenShells of Electronic Systems.

54. C. L. Janssen and H. F. Schaefer, Theor. Chim. Acta, 79, 1 (1991). The AutomatedSolution of Second Quantization Equations with Applications to the Coupled ClusterApproach.

55. P. J. Knowles, C. Hampel, and H.-J. Werner, J. Chem. Phys., 99, 5219 (1993). CoupledCluster Theory for High Spin, Open Shell Reference Wavefunctions.

56. P. G. Szalay and J. Gauss, J. Chem. Phys., 107, 9028 (1997). Spin-Restricted Open-ShellCoupled-Cluster Theory.

57. M. Rittby and R. Bartlett, J. Phys. Chem., 92, 3033 (1988). An Open-ShellSpin-Restricted Coupled Cluster Method: Application to Ionization Potentials in N2.

58. G. D. Purvis and R. J. Bartlett, J. Chem. Phys., 76, 1910 (1982). A Full Coupled-ClusterSingles and Doubles Model: The Inclusion of Disconnected Triples.

59. G. E. Scuseria and H. F. Schaefer, Chem. Phys. Lett., 142, 354 (1987). The Optimizationof Molecular Orbitals for Coupled Cluster Wavefunctions.

60. N. C. Handy, J. A. Pople, M. Head-Gordon, K. Raghavachari, and G. W. Trucks, Chem.Phys. Lett., 164, 185 (1989). Size-Consistent Brueckner Theory Limited to DoubleSubstitutions.

61. C. D. Sherrill, A. I. Krylov, E. F. C. Byrd, and M. Head-Gordon, J. Chem. Phys., 109,4171 (1998). Energies and Analytic Gradients for a Coupled-Cluster Doubles ModelUsing Variational Brueckner Orbitals: Application to Symmetry Breaking in O+

4 .

62. J. A. Pople, P. M. W. Gill, and N. C. Handy, Int. J. Quantum Chem., 56, 303 (1995).Spin-Unrestricted Character of Kohn-Sham Orbitals for Open-Shell Systems.

63. V. Polo, E. Kraka, and D. Cremer, Mol. Phys., 100, 1771 (2002). Electron Correlationand the Self-Interaction Error of Density Functional Theory.

64. T. Bally and G. N. Sastry, J. Phys. Chem. A, 101, 7923 (1997). Incorrect DissociationBehavior of Radical Ions in Density Functional Calculations.

65. Y. Zhang and W. Yang, J. Chem. Phys., 109(7), 2604 (1998). A Challenge for DensityFunctionals: Self-Interaction Error Increases for Systems with a Noninteger Number ofElectrons.

66. M. Lundber and P. E. M. Siegbahn, J. Chem. Phys., 122, 224103 (2005). Quantifying theEffects of the Self-Interaction Error in DFT: When do the Delocalized States Appear?

67. H. Iikura, T. Tsuneda, T. Yanai, and K. Hirao, J. Chem. Phys., 115, 3540 (2001).A Long-Range Correction Scheme for Generalized-Gradient-Approximation ExchangeFunctionals.

68. J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 128, 084106 (2008). SystematicOptimization of Long-Range Corrected Hybrid Density Functionals.

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69. R. Baer, E. Livshits, and U. Salzner, Annu. Rev. Phys. Chem., 61, 85 (2010). TunedRange-Separated Hybrids in Density Functional Theory.

70. T. Körzdörfer and J.-L. Brédas, Acc. Chem. Res., 47, 3284 (2014). Organic ElectronicMaterials: Recent Advances in the DFT Description of the Ground and Excited StatesUsing Tuned Range-Separated Hybrid Functionals.

71. P. A. Pieniazek, S. A. Arnstein, S. E. Bradforth, A. I. Krylov, and C. D. Sherrill,J. Chem. Phys., 127, 164110 (2007). Benchmark Full Configuration Interaction andEOM-IP-CCSD Results for Prototypical Charge Transfer Systems: Noncovalent IonizedDimers.

72. K. B. Bravaya, O. Kostko, S. Dolgikh, A. Landau, M. Ahmed, and A. I. Krylov, J. Phys.Chem. A, 114, 12305 (2010). Electronic Structure and Spectroscopy of Nucleic AcidBases: Ionization Energies, Ionization-Induced Structural Changes, and PhotoelectronSpectra.

73. A. A. Golubeva and A. I. Krylov, Phys. Chem. Chem. Phys., 11, 1303 (2009). The Effectof 𝜋-Stacking and H-Bonding on Ionization Energies of a Nucleobase: Uracil DimerCation.

74. A. A. Zadorozhnaya and A. I. Krylov, J. Chem. Theory Comput., 6, 705 (2010).Ionization-Induced Structural Changes in Uracil Dimers and Their Spectroscopic Sig-natures.

75. A. A. Zadorozhnaya and A. I. Krylov, J. Phys. Chem. A, 114, 2001 (2010). Zooming into𝜋-Stacked Manifolds of Nucleobases: Ionized States of Dimethylated Uracil Dimers.

76. K. Khistyaev, K. B. Bravaya, E. Kamarchik, O. Kostko, M. Ahmed, and A. I. Krylov,Faraday Discuss., 150, 313 (2011). The Effect of Microhydration on Ionization Energiesof Thymine.

77. K. B. Bravaya, O. Kostko, M. Ahmed, and A. I. Krylov, Phys. Chem. Chem. Phys., 12,2292 (2010). The Effect of 𝜋-Stacking, h-Bonding, and Electrostatic Interactions on theIonization Energies of Nucleic Acid Bases: Adenine-Adenine, Thymine-Thymine andAdenine-Thymine Dimers.

78. O. Kostko, K. B. Bravaya, A. I. Krylov, and M. Ahmed, Phys. Chem. Chem. Phys., 12,2860 (2010). Ionization of Cytosine Monomer and Dimer Studied by VUV Photoioniza-tion and Electronic Structure Calculations.

79. A. Golan, K. B. Bravaya, R. Kudirka, S. R. Leone, A. I. Krylov, and M. Ahmed, Nat.Chem., 4, 323 (2012). Ionization of Dimethyluracil Dimers Leads to Facile Proton Trans-fer in the Absence of H-Bonds.

80. K. B. Bravaya, E. Epifanovsky, and A. I. Krylov, J. Phys. Chem. Lett., 3, 2726 (2012).Four Bases Score a Run: Ab Initio Calculations Quantify a Cooperative Effect ofh-Bonding and pi-Stacking on Ionization Energy of Adenine in the AATT Tetramer.

81. K. Khistyaev, A. Golan, K. B. Bravaya, N. Orms, A. I. Krylov, and M. Ahmed, J. Phys.Chem. A, 117, 6789 (2013). Proton Transfer in Nucleobases is Mediated by Water.

82. P. A. Pieniazek, A. I. Krylov, and S. E. Bradforth, J. Chem. Phys., 127, 044317 (2007).Electronic Structure of the Benzene Dimer Cation.

83. C. Smith, T. D. Crawford, and D. Cremer, J. Chem. Phys., 122, 174309 (2005). TheStructures of m-Benzyne and Tetrafluoro- m-Benzyne.

84. A. I. Krylov, L. V. Slipchenko, and S. V. Levchenko, in ACS Symposium Series, Vol.958, ACS Publications, pp. 89–102, 2007. Breaking the Curse of the Non-DynamicalCorrelation Problem: the Spin-Flip Method.

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85. A. I. Krylov, Acc. Chem. Res., 39, 83 (2006). The Spin-Flip Equation-of-MotionCoupled-Cluster Electronic Structure Method for a Description of Excited States,Bond-Breaking, Diradicals, and Triradicals.

86. L. V. Slipchenko and A. I. Krylov, J. Chem. Phys., 117, 4694 (2002). Singlet-Triplet Gapsin Diradicals by the Spin-Flip Approach: A Benchmark Study.

87. A. I. Krylov, J. Phys. Chem. A, 109, 10638 (2005). Triradicals.

88. A. I. Krylov, Chem. Phys. Lett., 338, 375 (2001). Size-Consistent Wave Functions forBond-Breaking: The Equation-of-Motion Spin-Flip Model.

89. A. I. Krylov, Chem. Phys. Lett., 350, 522 (2001). Spin-Flip Configuration Interaction:An Electronic Structure Model that is Both Variational and Size-Consistent.

90. A. I. Krylov and C. D. Sherrill, J. Chem. Phys., 116, 3194 (2002). Perturbative Correc-tions to the Equation-of-Motion Spin-Flip SCF Model: Application to Bond-Breakingand Equilibrium Properties of Diradicals.

91. Y. Shao, M. Head-Gordon, and A. I. Krylov, J. Chem. Phys., 118, 4807 (2003). TheSpin-Flip Approach within Time-Dependent Density Functional Theory: Theory andApplications to Diradicals.

92. Y. A. Bernard, Y. Shao, and A. I. Krylov, J. Chem. Phys., 136, 204103 (2012).General Formulation of Spin-Flip Time-Dependent Density Functional Theory UsingNon-Collinear Kernels: Theory, Implementation, and Benchmarks.

93. D. Casanova, L. V. Slipchenko, A. I. Krylov, and M. Head-Gordon, J. Chem. Phys., 130,044103 (2009). Double Spin-Flip Approach within Equation-of-Motion Coupled Clusterand Configuration Interaction Formalisms: Theory, Implementation and Examples.

94. F. Bell, P. M. Zimmerman, D. Casanova, M. Goldey, and M. Head-Gordon, Phys. Chem.Chem. Phys., 15, 358 (2013). Restricted Active Space Spin-Flip (RAS-SF) with ArbitraryNumber of Spin-Flips.

95. N. J. Mayhall and M. Head-Gordon, J. Chem. Phys., 141, 044112 (2014). IncreasingSpin-Flips and Decreasing Cost: Perturbative Corrections for External Singles to theComplete Active Space Spin Flip Model for Low-Lying Excited States and Strong Cor-relation.

96. J. S. Sears, C. D. Sherrill, and A. I. Krylov, J. Chem. Phys., 118, 9084 (2003). ASpin-Complete Version of the Spin-Flip Approach to Bond Breaking: What is the Impactof Obtaining Spin Eigenfunctions?

97. A. M. C. Cristian, Y. Shao, and A. I. Krylov, J. Phys. Chem. A, 108, 6581 (2004). Bond-ing Patterns in Benzene Triradicals from Structural, Spectroscopic, and ThermochemicalPerspectives.

98. L. Koziol, M. Winkler, K. N. Houk, S. Venkataramani, W. Sander, and A. I. Krylov, J.Phys. Chem. A, 111, 5071 (2007). The 1,2,3-Tridehydrobenzene Triradical: 2B or not 2B?The Answer is 2A!

99. P. U. Manohar, L. Koziol, and A. I. Krylov, J. Phys. Chem. A, 113, 2519 (2009).Effect of a Heteroatom on Bonding Patterns and Triradical Stabilization Energies of2,4,6-Tridehydropyridine Versus 1,3,5-Tridehydrobenzene.

100. L. V. Slipchenko, T. E. Munsch, P. G. Wenthold, and A. I. Krylov, Angew. Chem. Int. Ed.,43, 742 (2004). 5-Dehydro-1,3-Quinodimethane: A Hydrocarbon with an Open-ShellDoublet Ground State.

101. T. Wang and A. I. Krylov, Chem. Phys. Lett., 426, 196 (2006). Electronic Structure ofthe Two Dehydro-meta-Xylylene Triradicals and Their Derivatives.

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102. T. Wang and A. I. Krylov, J. Chem. Phys., 123, 104304 (2005). The Effect of Substituentson Singlet-Triplet Energy Separations in meta-Xylylene Diradicals: Qualitative Insightsfrom Quantitative Studies.

103. P. G. Wenthold, R. R. Squires, and W. C. Lineberger, J. Am. Chem. Soc., 120, 5279(1998). Ultraviolet Photoelectron Spectroscopy of the o-, m-, and p-Benzyne NegativeIons. Electron Affinities and Singlet-Triplet Splittings for o-, m-, and p-Benzyne.

104. P. U. Manohar and A. I. Krylov, J. Chem. Phys., 129, 194105 (2008). ANon-Iterative Perturbative Triples Correction for the Spin-Flipping and Spin-ConservingEquation-of-Motion Coupled-Cluster Methods with Single and Double Substitutions.

105. C. D. Sherrill, M. L. Leininger, T. J. Van Huis, and H. F. Schaefer III, J. Chem. Phys.,108(3), 1040 (1998). Structures and Vibrational Frequencies in the Full ConfigurationInteraction Limit: Predictions for Four Electronic States of Methylene Using Triple-ZetaPlus Double Polarization (TZ2P) Basis.

106. P. Jensen and P. R. Bunker, J. Chem. Phys., 89, 1327 (1988). The Potential Surface andStretching Frequencies of X3B1 Methylene (CH2) Determined from Experiment Usingthe Morse Oscillator-Rigid Bender Internal Dynamics Hamiltonian.

107. Z. Rinkevicius, O. Vahtras, and H. Årgen, J. Chem. Phys., 133, 114104 (2010). Spin-FlipTime Dependent Density Functional Theory Applied to Excited States with Single, Dou-ble, or Mixed Electron Excitation Character.

108. O. Vahtras and Z. Rinkevicius, J. Chem. Phys., 126, 114101 (2007). General Excitationsin Time-Dependent Density Functional Theory.

109. F. Bell, D. Casanova, and M. Head-Gordon, J. Am. Chem. Soc., 132, 11314 (2010). The-oretical Study of Substituted PBPB Dimers: Structural Analysis, Tetraradical Character,and Excited States.

110. P. M. Zimmerman, F. Bell, M. Goldey, A. T. Bell, and M. Head-Gordon, J. Chem. Phys.,137, 164110 (2012). Restricted Active Space Spin-Flip Configuration Interaction: Theoryand Examples for Multiple Spin Flips with Odd Numbers of Electrons.

111. N. J. Mayhall and M. Head-Gordon, J. Chem. Phys., 141(13), 134111 (2014). Compu-tational Quantum Chemistry for Single Heisenberg Spin Couplings Made Simple: JustOne Spin Flip Required.

112. A. Landé, Z. Phys., 15, 189 (1923). Termstruktur und Zeemaneffekt der Multipletts.

113. S. Hirata and M. Head-Gordon, Chem. Phys. Lett., 314, 291 (1999). Time-DependentDensity Functional Theory within the Tamm-Dancoff Approximation.

114. A. Dreuw and M. Head-Gordon, Chem. Rev., 105, 4009 (2005). Single-Reference AbInitio Methods for the Calculation of Excited States of Large Molecules.

115. A. Dreuw, J. L. Weisman, and M. Head-Gordon, J. Chem. Phys., 119, 2943 (2003).Long-Range Charge-Transfer Excited States in Time-Dependent Density FunctionalTheory Require Non-Local Exchange.

116. R. M. Richard and J. M. Herbert, J. Chem. Theory Comput., 7, 1296 (2011).Time-Dependent Density-Functional Description of the 1La State in Polycyclic AromaticHydrocarbons: Charge-Transfer Character in Disguise?

117. D. Casanova and M. Head-Gordon, Phys. Chem. Chem. Phys., 11, 9779 (2009).Restricted Active Space Spin-Flip Configuration Interaction Approach: Theory, Imple-mentation and Examples.

118. D. Maurice and M. Head-Gordon, Int. J. Quantum Chem. Symp., 56, 361 (1995). Config-uration Interaction with Single Substitutions for Excited States of Open-Shell Molecules.

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119. P. G. Szalay and J. Gauss, J. Chem. Phys., 112, 4027 (2000). Spin-Restricted Open-ShellCoupled-Cluster Theory for Excited States.

120. A. Ipatov, F. Cordova, L. J. Doriol, and M. E. Casida, J. Mol. Struct. THEOCHEM, 914,60 (2009). Excited-State Spin-Contamination in Time-Dependent Density-FunctionalTheory for Molecules with Open-Shell Ground States.

121. L. Koziol, S. V. Levchenko, and A. I. Krylov, J. Phys. Chem. A, 110, 2746 (2006). BeyondVinyl: Electronic Structure of Unsaturated Propen-1-yl, Propen-2-yl, 1-Buten-2-yl, andtrans-2-Buten-2-yl Hydrocarbon Radicals.

122. S. Xu, S. Gozem, A. I. Krylov, C. R. Christopher, and J. M. Weber, Phys. Chem. Chem.Phys. (2015). Ligand Influence on the Electronic Spectra of Bipyridine-Cu(I) Complexes.

123. W. P. Reinhardt, Annu. Rev. Phys. Chem., 33, 223 (1982). Complex Coordinates in theTheory of Atomic and Molecular Structure and Dynamics.

124. N. Moiseyev, Phys. Rep., 302, 212 (1998). Quantum Theory of Resonances: CalculatingEnergies, Widths and Cross-Sections by Complex Scaling.

125. N. Moiseyev, Non-Hermitian Quantum Mechanics, Cambridge University Press, 2011.

126. S. Klaiman and I. Gilary, Adv. Quantum Chem., 63, 1 (2012). On Resonance: A FirstGlance into the Behavior of Unstable States.

127. N. Moiseyev and J. O. Hirschfelder, J. Chem. Phys., 88, 1063 (1988). Representation ofSeveral Complex Coordinate Methods by Similarity Transformation Operators.

128. J. Aguilar and J. M. Combes, Commun. Math. Phys., 22, 269 (1971). A Class of AnalyticPerturbations for One-Body Schrödinger Hamiltonians.

129. E. Balslev and J. M. Combes, Commun. Math. Phys., 22, 280 (1971). Spectral Propertiesof Many-Body Schrödinger Operators with Dilatation-Analytic Interactions.

130. G. Jolicard and E. J. Austin, Chem. Phys. Lett., 121, 106 (1985). Optical Potential Sta-bilisation Method for Predicting Resonance Levels.

131. U. V. Riss and H.-D. Meyer, J. Phys. B, 26, 4503 (1993). Calculation of Resonance Ener-gies and Widths Using the Complex Absorbing Potential Method.

132. D. Zuev, T.-C. Jagau, K. B. Bravaya, E. Epifanovsky, Y. Shao, E. Sundstrom, M.Head-Gordon, and A. I. Krylov, J. Chem. Phys., 141(2), 024102 (2014). ComplexAbsorbing Potentials within EOM-CC Family of Methods: Theory, Implementation, andBenchmarks.

133. T.-C. Jagau, D. Zuev, K. B. Bravaya, E. Epifanovsky, and A. I. Krylov, J. Phys. Chem.Lett., 5, 310 (2014). A Fresh Look at Resonances and Complex Absorbing Potentials:Density Matrix Based Approach.

134. T.-C. Jagau and A. I. Krylov, J. Phys. Chem. Lett., 5, 3078 (2014). Complex Absorb-ing Potential Equation-of-Motion Coupled-Cluster Method Yields Smooth and InternallyConsistent Potential Energy Surfaces and Lifetimes for Molecular Resonances.

135. T.-C. Jagau, D. B. Dao, N. S. Holtgrewe, A. I. Krylov, and R. Mabbs, J. Phys. Chem.Lett., 6, 2786 (2015). Same But Different: Dipole-Stabilized Shape Resonances in CuF−

and AgF−.

136. J. Simons, J. Phys. Chem. A, 112, 6401 (2008). Molecular Anions.

137. J. Linderberg and Y. Öhrn, Propagators in Quantum Chemistry, Academic, London, UK,1973.

138. J. V. Ortiz, Adv. Quantum Chem., 35, 33 (1999). Toward an Exact One-Electron Pictureof Chemical Bonding.

�

� �

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139. H. R. Hudock, B. G. Levine, A. L. Thompson, H. Satzger, D. Townsend, N. Gador, S.Ulrich, A. Stolow, and T. J. Martínez, J. Phys. Chem. A, 111, 8500 (2007). Ab InitioMolecular Dynamics and Time-Resolved Photoelectron Spectroscopy of ElectronicallyExcited Uracil and Thymine.

140. C. M. Oana and A. I. Krylov, J. Chem. Phys., 127, 234106 (2007). Dyson Orbitals forIonization from the Ground and Electronically Excited States within Equation-of-MotionCoupled-Cluster Formalism: Theory, Implementation, and Examples.

141. C. M. Oana and A. I. Krylov, J. Chem. Phys., 131, 124114 (2009). Cross Sectionsand Photoelectron Angular Distributions in Photodetachment from Negative Ions UsingEquation-of-Motion Coupled-Cluster Dyson Orbitals.

142. T.-C. Jagau and A. I. Krylov, J. Chem. Phys., 144, 054113 (2016). CharacterizingMetastable States beyond Energies and Lifetimes: Dyson Orbitals and Transition DipoleMoments.

143. E. D. Glendening, J. K. Badenhoop, A. E. Reed, J. E. Carpenter, and F. Weinhold, NBO4.0 Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 1996.

144. F. Weinhold and C. R. Landis, Chem. Educ. Res. Pract. Eur., 2, 91 (2001). Natural BondOrbitals and Extensions of Localized Bonding Concepts.

145. A. V. Luzanov, Russ. Chem. Rev., 49, 1033 (1980). The Structure of the Electronic Exci-tation of Molecules in Quantum-Chemical Models.

146. M. Head-Gordon, A. M. Grana, D. Maurice, and C. A. White, J. Phys. Chem., 99, 14261(1995). Analysis of Electronic Transitions as the Difference of Electron Attachment andDetachment Densities.

147. A. V. Luzanov and O. A. Zhikol, in Practical Aspects of Computational Chemistry I: AnOverview of the Last Two Decades and Current Trends, J. Leszczynski and M. Shukla,Eds., Springer-Verlag, 2012, pp. 415–449. Excited State Structural Analysis: TDDFTand Related Models.

148. F. Plasser and H. Lischka, J. Chem. Theory Comput., 8, 2777 (2012). Analysis of Exci-tonic and Charge Transfer Interactions from Quantum Chemical Calculations.

149. F. Plasser, M. Wormit, and A. Dreuw, J. Chem. Phys., 141, 024106 (2014). New Toolsfor the Systematic Analysis and Visualization of Electronic Excitations. I. Formalism.

150. S. Matsika, X. Feng, A. V. Luzanov, and A. I. Krylov, J. Phys. Chem. A, 118, 11943(2014). What We Can Learn from the Norms of One-Particle Density Matrices, and WhatWe Can’t: Some Results for Interstate Properties in Model Singlet Fission Systems.

151. A. V. Luzanov, in Practical Aspects of Computational Chemistry IV , J. Leszczynski andM. Shukla, Eds., Springer-Verlag, Chapter 6, pp. 151–206 (2016). Effectively UnpairedElectrons for Singlet States: From Diatomics to Graphene Nanoclusters.

152. K. Takatsuka, T. Fueno, and K. Yamaguchi, Theor. Chim. Acta, 48, 175 (1978). Distri-bution of Odd Electrons in Ground-State Molecules.

153. M. Head-Gordon, Chem. Phys. Lett., 372, 508 (2003). Characterizing Unpaired Electronsfrom the One-Particle Density Matrix.

154. D. W. Smith and L. Andrews, J. Chem. Phys., 55(11), 5295 (1971). Matrix Infrared Spec-trum and Bonding in the Monobromomethyl Radical.

155. T. Carver and L. Andrews, J. Chem. Phys., 50(10), 4235 (1969). Matrix Infrared Spec-trum and Bonding in Dichlormethyl Radical.

156. T. G. Carver and L. Andrews, J. Chem. Phys., 50(10), 4223 (1969). Matrix InfraredSpectrum and Bonding in Dibromomethyl Radical.

�

� �

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157. L. Andrews and D. W. Smith, J. Chem. Phys., 53(7), 2956 (1970). Matrix Infrared Spec-trum and Bonding in Monochloromethyl Radical.

158. S. V. Levchenko and A. I. Krylov, J. Chem. Phys., 115(16), 7485 (2001). ElectronicStructure of Halogen-Substituted Methyl Radicals: Excited States of CH2Cl and CH2F.

159. S. V. Levchenko, A. V. Demyanenko, V. Dribinski, A. B. Potter, and H. Reisler, J. Chem.Phys., 118, 9233 (2003). Rydberg-Valence Interactions in CH2Cl → CH2 + Cl Photodis-sociation: Dependence of Absorption Probability on Ground State Vibrational Excitation.

160. I. M. Mills, J. Mol. Spectrosc., 5, 334–340 (1961). Vibrational Perturbation Theory.

161. J. M. Bowman, S. Carter, and X. Huang, Int. Rev. Phys. Chem., 22(3), 533 (2003).Multimode: A Code to Calculate Rovibrational Energies of Polyatomic Molecules.

162. R. B. Gerber, G. M. Chaban, B. Brauer, and Y. Miller, in Theory and Applications of Com-putational Chemistry: The First Fourty Years, C. Dykstra, G. Frenking, K. Kim, and G.Scuseria, Eds., Elsevier, 2005, pp. 165–194. First-Principles Calculations of AnharmonicVibrational Spectroscopy of Large Molecules.

163. S. S. Xantheas, Int. Rev. Phys. Chem., 25, 719 (2006). Anharmonic Vibrational Spec-tra of Hydrogen Bonded Clusters: Comparison between Higher Energy Derivative andMean-Field Grid Based Methods.

164. R. F. Gunion, M. K. Gilles, M. L. Polak, and W. C. Lineberger, Int. J. Mass Spectrom.Ion Processes, 117, 601 (1992). Ultraviolet Photoelectron Spectroscopy of the Phenide,Benzyl and Phenoxide Anions, With Ab Initio Calculations.

165. K. B. Bravaya and A. I. Krylov, J. Phys. Chem. A, 117, 11815 (2013). On the Photode-tachment from the Green Fluorescent Protein Chromophore.

166. J. Simons, in Photoionization and Photodetachment, C. Ng, Ed., Part II of AdvancedSeries in Physical Chemistry, World Scientific Publishing Co., Singapore, 2000, Vol. 10.Detachment Processes for Molecular Anions.

167. A. Sanov and R. Mabbs, Int. Rev. Phys. Chem., 27, 53 (2008). Photoelectron Imaging ofNegative Ions.

168. K. L. Reid, Int. Rev. Phys. Chem., 27, 607 (2008). Picosecond Time-Resolved Photo-electron Spectroscopy as a Means of Gaining Insight into Mechanisms of IntramolecularVibrational Energy Redistribution in Excited States.

169. O. Welz, J. D. Savee, D. L. Osborn, S. S. Vasu, C. J. Percival, D. E. Shallcross, andC. A. Taatjes, Science, 335, 204–207 (2012). Direct Kinetic Measurements of CriegeeIntermediate (CH2OO) Formed by Reaction of CH2I with O2.

170. D. M. Neumark, Phys. Chem. Chem. Phys., 7, 433 (2005). Probing the Transition Statewith Negative Ion Photodetachment: Experiment and Theory.

171. D. M. Neumark, J. Chem. Phys., 125, 132303 (2006). Probing Chemical Dynamics withNegative Ions.

172. V. A. Mozhayskiy and A. I. Krylov, ezSpectrum, http://iopenshell.usc.edu/downloads.

173. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms,Plenum, New York, 1977.

174. S. Gozem, A. O. Gunina, T. Ichino, D. L. Osborn, J. F. Stanton, and A. I. Krylov, J.Phys. Chem. Lett., 6, 4532 (2015). Photoelectron Wave Function in Photoionization:Plane Wave or Coulomb Wave?

175. S. Gozem and A. I. Krylov, ezDyson user’s manual, 2015. ezDyson, http://iopenshell.usc.edu/downloads/.

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� �

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176. E. Kamarchik and A. I. Krylov, J. Phys. Chem. Lett., 2, 488 (2011). Non-Condon Effectsin One- and Two-Photon Absorption Spectra of the Green Fluorescent Protein.

177. O. V. Prezhdo and P. J. Rossky, J. Chem. Phys., 107, 5863 (1997). Evaluation of QuantumTransition Rates from Quantum-Classical Molecular Dynamics Simulations.

178. A. F. Izmaylov, D. Mendive-Tapia, M. J. Bearpark, M. A. Robb, J. C. Tully, and M.J. Frisch, J. Chem. Phys., 135, 234106 (2011). Nonequilibrium Fermi Golden Rule forElectronic Transitions through Conical Intersections.

179. M. H. Lee, B. D. Dunietz, and E. Geva, J. Phys. Chem. C, 117, 23391 (2013). Calculationfrom First Principles of Intramolecular Golden-Rule Rate Constants for Photo-InducedElectron Transfer in Molecular Donor-Acceptor Systems.

180. J. F. Stanton and J. Gauss, J. Chem. Phys., 101(10), 8938 (1994). Analytic Energy Deriva-tives for Ionized States Described by the Equation-of-Motion Coupled Cluster Method.

181. J. F. Stanton and J. Gauss, Theor. Chim. Acta, 91, 267 (1995). Analytic Energy Derivativesfor the Equation-of-Motion Coupled-Cluster Method—Algebraic Expressions, Imple-mentation and Application to the S1 State of HFCO.

182. J. F. Stanton and J. Gauss, J. Chem. Phys., 103(20), 88931 (1995). Many-Body Methodsfor Excited State Potential Energy Surfaces. II. Analytic Second Derivatives for ExcitedState Energies in the Equation-of-Motion Coupled-Cluster Method.

183. P. B. Rozyczko, S. A. Perera, M. Nooijen, and R. J. Bartlett, J. Chem. Phys., 107, 6736(1997). Correlated Calculations of Molecular Dynamic Polarizabilities.

184. M. Kállay and J. Gauss, J. Mol. Struct. THEOCHEM, 768, 71 (2006). Calculation ofFrequency-Dependent Polarizabilities Using General Coupled-Cluster Models.

185. D. P. O’Neill, M. Kállay, and J. Gauss, J. Chem. Phys., 127, 134109 (2007). Calculationof Frequency-Dependent Hyperpolarizabilities Using General Coupled-Cluster Models.

186. J. F. Stanton and R. J. Bartlett, J. Chem. Phys., 98, 7029 (1993). The Equation of MotionCoupled-Cluster Method. A Systematic Biorthogonal Approach to Molecular ExcitationEnergies, Transition Probabilities, and Excited State Properties.

187. K. Nanda and A. I. Krylov, J. Chem. Phys., 142, 064118 (2015). Two PhotonAbsorption Cross Sections within Equation-of-Motion Coupled-Cluster FormalismUsing Resolution-of-the-Identity and Cholesky Decomposition Representations: Theory,Implementation, and Benchmarks.

188. J. F. Stanton, J. Chem. Phys., 101(10), 8928 (1994). Separability Properties of Reducedand Effective Density Matrices in the Equation-of-Motion Coupled Cluster Method.

189. R. Kobayashi, H. Koch, and P. Jørgensen, Chem. Phys. Lett., 219, 30 (1994). Calculationof Frequency-Dependent Polarizabilities Using Coupled-Cluster Response Theory.

190. O. Christiansen, J. Gauss, and B. Schimmelpfennig, Phys. Chem. Chem. Phys., 2(5), 965(2000). Spin-Orbit Coupling Constants From Coupled-Cluster Response Theory.

191. P. Schwerdtfeger, Ed., Relativistic Electronic Structure Theory, Elsevier, Amsterdam,2002.

192. C. M. Marian, WIREs Comput. Mol. Sci., 2, 187 (2012). Spin-Orbit Coupling and Inter-system Crossing in Molecules.

193. B. A. Hess, C. M. Marian, U. Wahlgren, and O. Gropen, Chem. Phys. Lett., 251, 365(1996). A Mean-Field Spin-Orbit Method Applicable to Correlated Wavefunctions.

194. D. G. Fedorov and M. S. Gordon, J. Chem. Phys., 112(13), 5611 (2000). A Study of theRelative Importance of One and Two-Electron Contributions to Spin-Orbit Coupling.

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� �

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195. K. Klein and J. Gauss, J. Chem. Phys., 129, 194106 (2008). Perturbative Cal-culation of Spin-Orbit Splittings Using the Equation-of-Motion Ionization-PotentialCoupled-Cluster Ansatz.

196. E. Epifanovsky, K. Klein, S. Stopkowicz, J. Gauss, and A. I. Krylov, J. Chem. Phys., 143,064102 (2015). Spin-Orbit Couplings within the Equation-of-Motion Coupled-ClusterFramework: Theory, Implementation, and Benchmark Calculations.

197. D. G. Fedorov, S. Koseki, M. W. Schmidt, and M. S. Gordon, Int. Rev. Phys. Chem., 22,551 (2003). Spin-Orbit Coupling in Molecules: Chemistry beyond the Adiabatic Approx-imation.

198. P. Saxe and B. H. Lengsfield III, and D. R. Yarkony, J. Chem. Phys., 81, 4549 (1984). Onthe Evaluation of Nonadiabatic Coupling Matrix-Elements Using SA-MCSCF/CI WaveFunctions and Analytic Gradient Methods.1.

199. P. Saxe, B. H. Lengsfield III, and D. R. Yarkony, Chem. Phys. Lett., 113, 159 (1985). Onthe Evaluation of Non-Adiabatic Coupling Matrix-Elements for Large-Scale CI Wave-functions.

200. B. H. Lengsfield III and D. R. Yarkony, Adv. Chem. Phys., 82, 1 (1992). NonadiabaticInteractions between Potential Energy Surfaces: Theory and Applications.

201. H. Lischka, M. Dallos, P. G. Szalay, D. R. Yarkony, and R. Shepard, J. Chem. Phys., 120,7322 (2004). Analytic Evaluation of Nonadiabatic Coupling Terms at the MR-CI Level.I. Formalism.

202. M. Dallos, H. Lischka, R. Shepard, D. R. Yarkony, and P. G. Szalay, J. Chem. Phys., 120,7330 (2004). Analytic Evaluation of Nonadiabatic Coupling Terms at the MR-CI Level.II. Minima on the Crossing Seam: Formaldehyde and the Photodimerization of Ethylene.

203. E. Tapavicza, G. D. Bellchambers, J. C. Vincent, and F. Furche, Phys. Chem. Chem.Phys., 15, 18336 (2013). Ab Initio Non-Adiabatic Molecular Dynamics.

204. S. Fatehi, E. Alguire, Y. Shao, and J. E. Subotnik, J. Chem. Phys., 135, 234105 (2011).Analytic Derivative Couplings between Configuration-Interaction-Singles States withBuilt-in Electron-Translation Factors for Translational Invariance.

205. X. Zhang and J. M. Herbert, J. Chem. Phys., 141, 064104 (2014). AnalyticDerivative Couplings for Spin-Flip Configuration Interaction Singles and Spin-FlipTime-Dependent Density Functional Theory.

206. A. Köhn and A. Tajti, J. Chem. Phys., 127, 044105 (2007). Can Coupled-Cluster TheoryTreat Conical Intersections?

207. A. Tajti and P. G. Szalay, J. Chem. Phys., 131, 124104 (2009). Analytic Evaluation ofthe Nonadiabatic Coupling Vector between Excited States Using Equation-of-MotionCoupled-Cluster Theory.

208. X. Feng, A. V. Luzanov, and A. I. Krylov, J. Phys. Chem. Lett., 4, 3845 (2013). Fissionof Entangled Spins: An Electronic Structure Perspective.

209. A. B. Kolomeisky, X. Feng, and A. I. Krylov, J. Phys. Chem. C, 118, 5188 (2014). A Sim-ple Kinetic Model for Singlet Fission: A Role of Electronic and Entropic Contributionsto Macroscopic Rates.

210. X. Feng, A. B. Kolomeisky, and A. I. Krylov, J. Phys. Chem. C, 118, 19608 (2014). Dis-secting the Effect of Morphology on the Rates of Singlet Fission: Insights From Theory.

211. R. A. Marcus, J. Chem. Phys., 24, 966 (1956). On the Theory of Oxidation-ReductionReactions Involving Electron Transfer. I.

�

� �

�


212. R. A. Marcus, Annu. Rev. Phys. Chem., 15, 155 (1964). Chemical and ElectrochemicalElectron-Transfer Theory.

213. C.-P. Hsu, Z. You, and H. C. Chen, J. Phys. Chem. C, 112(4), 1204 (2008). Characteri-zation of the Short-Range Couplings in Excitation Energy Transfer.

214. R. S. Mulliken, J. Am. Chem. Soc., 74, 811 (1952). Molecular Compounds and TheirSpectra. II.

215. N. S. Hush, Prog. Inorg. Chem., 8, 391 (1967). Intervalence-Transfer Absorption. Part 2.Spectroscopic Data.

216. R. J. Cave and M. D. Newton, Chem. Phys. Lett., 249, 15 (1996). Generalization of theMulliken-Hush Treatment of the Calculation of Electron Transfer Matrix Elements.

217. R. J. Cave and M. D. Newton, J. Chem. Phys., 106(22), 9213 (1997). Calculation ofElectronic Coupling Matrix Elements for Ground and Excited State Electron TransferReactions: Comparison of the Generalized Mulliken-Hush and Block DiagonalizationMethod.

218. C.-P. Hsu, Acc. Chem. Res., 42(4), 509 (2009). The Electronic Couplings in ElectronTransfer and Excitation Energy Transfer.

219. R. J. Bartlett, in Theory and Applications of Computational Chemistry, C. Dykstra, G.Frenking, and G. Scuseria, Eds., Elsevier, 2005. How and Why Coupled-Cluster TheoryBecame the Preeminent Method in Ab Initio Quantum Chemistry.

220. H. Reisler and A. I. Krylov, Int. Rev. Phys. Chem., 28, 267 (2009). Interacting Rydbergand Valence States in Radicals and Molecules: Experimental and Theoretical Studies.

221. J. M. Herbert, in Reviews in Computational Chemistry, A. L. Parrill and K. B. Lipkowitz,Eds., John Wiley & Sons, Inc., Hoboken, NJ, 2015, vol. 28, 391–517. The QuantumChemistry of Loosely-Bound Electrons.

222. J. F. Stanton, J. Gauss, M. Harding, P. Szalay, CFOUR. With contributions from A. A.Auer, R. J. Bartlett, U. Benedikt, C. Berger, D. E. Bernholdt, Y. J. Bomble, L. Cheng,O. Christiansen, M. Heckert, O. Heun, C. Huber, T.-C. Jagau, D. Jonsson, J. Jusélius, K.Klein, W. J. Lauderdale, F. Lipparini, D. A. Matthews, T. Metzroth, L. A. Mück, D. P.O’Neill, D. R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiffmann, W. Schwalbach,C. Simmons, S. Stopkowicz, A. Tajti, J. Vázquez, F. Wang, J. D. Watts, and the integralpackages MOLECULE (J. Almlöf and P. R. Taylor), PROPS (P. R. Taylor), ABACUS (T.Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen), and ECP routines by A. V. Mitinand C. van Wüllen. For the current version, see http://www.cfour.de.

223. E. Deumens, V. F. Lotrich, A. Perera, M. J. Ponton, B. A. Sanders, and R. J. Bartlett,WIREs Comput. Mol. Sci., 1(6), 895–901 (2011). Software Design of ACES III with theSuper Instruction Architecture.

224. TURBOMOLE v6.5 2013, a development of University of Karlsruhe and Forschungszen-trum Karlsruhe GmbH, 1989-2007, TURBOMOLE GmbH, since 2007, http://www.turbomole.com.

225. K. Aidas, C. Angeli, K. L. Bak, V. Bakken, R. Bast, L. Boman, O. Christiansen, R. Cimi-raglia, S. Coriani, P. Dahle, E. K. Dalskov, U. Ekström, T. Enevoldsen, J. J. Eriksen, P.Ettenhuber, B. Fernández, L. Ferrighi, H. Fliegl, L. Frediani, K. Hald, A. Halkier, C. Hät-tig, H. Heiberg, T. Helgaker, A. C. Hennum, H. Hettema, E. Hjertenæs, S. Høst, I.-M.Høyvik, M. F. Iozzi, B. Jansik, H. J. Aa. Jensen, D. Jonsson, P. Jørgensen, J. Kauczor,S. Kirpekar, T. Kjærgaard, W. Klopper, S. Knecht, R. Kobayashi, H. Koch, J. Kongsted,A. Krapp, K. Kristensen, A. Ligabue, O. B. Lutnæs, J. I. Melo, K. V. Mikkelsen, R.H. Myhre, C. Neiss, C. B. Nielsen, P. Norman, J. Olsen, J. M. H. Olsen, A. Osted, M.

�

� �

�


J. Packer, F. Pawlowski, T. B. Pedersen, P. F. Provasi, S. Reine, Z. Rinkevicius, T. A.Ruden, K. Ruud, V. Rybkin, P. Salek, C. C. M. Samson, A. Sánchez de Merás, T. Saue,S. P. A. Sauer, B. Schimmelpfennig, K. Sneskov, A. H. Steindal, K. O. Sylvester-Hvid,P. R. Taylor, A. M. Teale, E. I. Tellgren, D. P. Tew, A. J. Thorvaldsen, L. Thøgersen, O.Vahtras, M. A. Watson, D. J. D. Wilson, M. Ziolkowski, and H. Ågren, WIREs Comput.Mol. Sci., 4, 269 (2014). The Dalton Quantum Chemistry Program System.

226. J. M. Turney, A. C. Simmonett, R. M. Parrish, E. G. Hohenstein, F. A. Evangelista, J.T. Fermann, B. J. Mintz, L. A. Burns, J. J. Wilke, M. L. Abrams, N. J. Russ, M. L.Leininger, C. L. Janssen, E. T. Seidl, W. D. Allen, H. F. Schaefer, R. A. King, E. F.Valeev, C. D. Sherrill, and T. D. Crawford, WIREs Comput. Mol. Sci., 2, 556 (2012).PSI4: An Open-Source Ab Initio Electronic Structure Package.

227. M. W. Schmidt, K. K. Baldridge, S. T. Elbert, J. A. Boatz, M. S. Gordon, S. Koseki,J. H. Jensen, N. Mastunaga, K. A. Nguyen, T. L. Windus, S. Su, M. Dupuis, and J. A.Montgomery, J. Comput. Chem., 14(11), 1347 (1993). General Atomic and MolecularElectronic Structure System.

228. H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz, P. Celani, W. Györffy,D. Kats, T. Korona, R. Lindh, A. Mitrushenkov, G. Rauhut, K. R. Shamasundar, T. B.Adler, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J.Dobbyn, F. Eckert, E. Goll, C. Hampel, A. Hesselmann, G. Hetzer, T. Hrenar, G. Jansen,C. Köppl, Y. Liu, A. W. Lloyd, R. A. Mata, A. J. May, S. J. McNicholas, W. Meyer, M. E.Mura, A. Nicklass, D. P. O’Neill, P. Palmieri, D. Peng, K. Pflüger, R. Pitzer, M. Reiher,T. Shiozaki, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson, and M. Wang, MOLPRO,www.molpro.net.

229. F. Aquilante, L. de Vico, N. Ferré, G. Ghigo, P.-Å. Malmqvist, P. Neogrády, T. B. Ped-ersen, M. Pitonak, M. Reiher, B. Roos, L. Serrano-Andrés, M. Urban, V. Veryazov, andR. Lindh, J. Comput. Chem., 31, 224 (2010). Molcas 7: The Next Generation.

230. Spartan, Wavefunction, Inc., 1991, www.wavefunction.com.

231. A. I. Krylov and P. M. W. Gill, WIREs Comput. Mol. Sci., 3, 317 (2013). Q-Chem: AnEngine for Innovation.

232. Y. Shao, Z. Gan, E. Epifanovsky, A. T. B. Gilbert, M. Wormit, J. Kussmann, A. W. Lange,A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P. R. Horn, L. D. Jacobson, I. Kaliman,R. Z. Khaliullin, T. Kus, A. Landau, J. Liu, E. I. Proynov, Y. M. Rhee, R. M. Richard,M. A. Rohrdanz, R. P. Steele, E. J. Sundstrom, H. L. Woodcock III, P. M. Zimmerman,D. Zuev, B. Albrecht, E. Alguires, B. Austin, G. J. O. Beran, Y. A. Bernard, E. Berquist,K. Brandhorst, K. B. Bravaya, S. T. Brown, D. Casanova, C.-M. Chang, Y. Chen, S. H.Chien, K. D. Closser, D. L. Crittenden, M. Diedenhofen, R. A. DiStasio Jr., H. Do, A. D.Dutoi, R. G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya,J. Gomes, M. W. D. Hanson-Heine, P. H. P. Harbach, A. W. Hauser, E. G. Hohenstein,Z. C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R. A. King,P. Klunzinger, D. Kosenkov, T. Kowalczyk, C. M. Krauter, K. U. Laog, A. Laurent, K.V. Lawler, S. V. Levchenko, C. Y. Lin, F. Liu, E. Livshits, R. C. Lochan, A. Luenser, P.Manohar, S. F. Manzer, S.-P. Mao, N. Mardirossian, A. V. Marenich, S. A. Maurer, N.J. Mayhall, C. M. Oana, R. Olivares-Amaya, D. P. O’Neill, J. A. Parkhill, T. M. Perrine,R. Peverati, P. A. Pieniazek, A. Prociuk, D. R. Rehn, E. Rosta, N. J. Russ, N. Sergueev,S. M. Sharada, S. Sharmaa, D. W. Small, A. Sodt, T. Stein, D. Stuck, Y.-C. Su, A. J.W. Thom, T. Tsuchimochi, L. Vogt, O. Vydrov, T. Wang, M. A. Watson, J. Wenzel, A.White, C. F. Williams, V. Vanovschi, S. Yeganeh, S. R. Yost, Z.-Q. You, I. Y. Zhang,X. Zhang, Y. Zhou, B. R. Brooks, G. K. L. Chan, D. M. Chipman, C. J. Cramer, W. A.

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Goddard III, M. S. Gordon, W. J. Hehre, A. Klamt, H. F. Schaefer III, M. W. Schmidt,C. D. Sherrill, D. G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A. T. Bell,N. A. Besley, J.-D. Chai, A. Dreuw, B. D. Dunietz, T. R. Furlani, S. R. Gwaltney, C.-P.Hsu, Y. Jung, J. Kong, D. S. Lambrecht, W. Z. Liang, C. Ochsenfeld, V. A. Rassolov, L.V. Slipchenko, J. E. Subotnik, T. Van Voorhis, J. M. Herbert, A. I. Krylov, P. M. W. Gill,and M. Head-Gordon, Mol. Phys., 113, 184 (2015). Advances in Molecular QuantumChemistry Contained in the Q-Chem 4 Program Package.

233. http://www.q-chem.com.

234. A. T. B. Gilbert, IQmol molecular viewer, http://iqmol.org.

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