The Quantum Chemistry of Open-Shell Species

Home , Doublet state, Spin contamination

4 THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES

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

INTRODUCTION AND OVERVIEW k k The majority of stable chemical compounds have closed-shell electronic configurations. In this type of electronic structure, all electrons are paired and manifolds of orbitals that have the same energy are either completely filled or empty, as in dinitrogen shown in Figure 1. Depending on the type of MO hosting a pair of electrons, 𝜋 𝜋∗ each pair gives rise to bonding (such as -orbitals in N2), antibonding (e.g., CC in vinyl), or nonbonding (lone pairs) interactions. Due to the Pauli exclusion principle, electrons can occupy the same MO only when they have opposite spins. Thus, species that have different number of 𝛼 and 𝛽 electrons form an open-shell pattern. The examples include doublet radicals and triplet electronic states (see Figure 1). These types are often described as high-spin open-shell species, where “high spin” refers to the nonzero Ms values of the underlying wave functions. (The terms high spin and low spin are also used to distinguish different components of a multiplet. In some cases, all the components may have nonzero Ms. For example, for a system with 3-electrons-on-3-orbitals, the high-spin state is a quartet and the low-spin state is a doublet.) Open-shell character can be found even when N𝛼 = N𝛽 . This happens when a degenerate orbital manifold is only partially filled. Consider, for example, a molecule with 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

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* π CC

lp(C)

πCC

σCC

4 [core] Vinyl Methylene N2 (a) (b) (c) FIGURE 1 Examples of closed-shell and open-shell species. Dinitrogen in its ground electronic state is a closed-shell molecule. All electrons are paired and degenerate molecular orbitals (MOs) such as the two 𝜋-orbitals are completely filled. Vinyl is an example of a doublet radical with one unpaired electron (N𝛼 − N𝛽 = 1). Triplet methylene, which has two unpaired

electrons, is a diradical (the high-spin Ms = 1 component in which N𝛼 − N𝛽 = 2 of the triplet state is shown). (See color plate section for the color representation of this figure.)

R HH σ∗ k k σ σ∗ σ

ΨΨ= −λ = −

FIGURE 2 Closed-shell molecules acquire open-shell character at stretched geometries, where degeneracy between frontier MOs develops. At equilibrium geometry, the coefficient 𝜆 is small and the wave function is dominated by (𝜎)2. At the dissociation limit, both configurations . . become equally important resulting in an open-shell singlet radical pair, HA + HB.(See color plate section for the color representation of this figure.)

two orbitals form a degenerate manifold, which is only partially filled. The resulting wave 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 wave . . function corresponds to the two radicals, HA and HB, as one can easily verify by expanding the two determinants in terms of the atomic orbitals (AOs), sA and sB: 1 1 Ψ=√ [|𝜎𝛼𝜎𝛽⟩ + |𝜎∗𝛼𝜎∗𝛽⟩]= [𝜎𝜎 − 𝜎∗𝜎∗]×(𝛼𝛽 − 𝛽𝛼) 2 2 1 = [(s + s )(s + s )−(s − s )(s − s )] × (𝛼𝛽 − 𝛽𝛼) 2 A B A B A B A B 𝛼𝛽 𝛽𝛼 =[sAsB + sBsA]×( − ) [1]

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 153

1 The stretched H2 molecule is an example of a singlet diradical. Diradical species can be formed when two radical centers reside on a rigid molecular scaffold, which restricts their ability to form a chemical bond, such as in para-benzyne shown in Figure 3. Breaking chemical bonds often leads to formation of pairs of radicals or diradical intermediates, as depicted in Figure 4. Similarly, molecules at transition states can be described as diradicals.2 Species containing transition metals often feature open-shell character. Molecules with multiple radical centers, such as molecular magnets,3,4 are polyradicals. Two examples of molecular magnets with multiple transition metal centers hosting unpaired electrons are shown in Figure 5. Open-shell species can also be formed in various photoinduced processes. In photoexcitation, electrons promoted by light to higher orbitals become unpaired (consider for example the 𝜋𝜋∗ excited state in ethylene). Removing electrons from closed-shell species yields radicals. For example, photoionization of closed-shell neutral molecules or photodetachment from closed-shell anions leads to charged or

−λ k C k

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

. . >C–C<

>CC< >CC<

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

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154 REVIEWS IN COMPUTATIONAL CHEMISTRY

(a) (b) FIGURE 5 Examples of molecular magnets. (a) Molecular structure of a Cr(III) horseshoe complex. Each of the six chromium atoms has three unpaired electrons. (b) Structure of a nickel

complex with the nickel-cubane core (Ni4O8). Each of the four nickel atoms has two unpaired electrons. Reproduced with permission of American Chemical Society from Ref. 5.

neutral doublet radicals. Electron attachment to closed-shell species also produces radicals. Although open-shell character is associated with highly reactive and k k unstable 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 and spectroscopy. To computationally describe reaction profiles, intermediates, and catalytic centers, we need to be able to describe open-shell states on the same footing (and with the same accuracy) as closed-shell states. This, it turns out, is hard to achieve. In this chapter, we describe the challenges posed by open-shell species to 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 standard quantum chemistry methods, outlining their scope of applicability, and explaining why open-shell species require special treatment. We will then discuss several topics relevant to open-shell systems: spin and spin contamination, Jahn–Teller (JT), and pseudo-Jahn–Teller (PJT) effects. The following section will discuss the electronic structure of high-spin states and contrast cases with and without electronic degeneracies. Methods designed to tackle various types of open-shell electronic structure (simple high-spin states, charge-transfer systems, diradicals and triradicals, excited states of open-shell molecules) will be introduced, and their applications will be illustrated by examples. We will then discuss calculations of molecular properties relevant to spectroscopy and excited-state processes. We hope that this chapter will provide a practical guide for computational chemists interested in open-shell systems.

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 155

QUANTUM CHEMISTRY METHODS FOR OPEN- AND CLOSED-SHELL SPECIES

Quantum chemistry offers practical methods for solving the electronic Schrödinger equation.6,7 The exact wave function is represented by a linear combination of all possible Slater determinants that can be generated for a given orbital basis. Systematic approximations to this factorially complex object give rise to practical computational methods balancing accuracy and cost.6–8 The simplest member of this hierarchy (shown in Figure 6) is the Hartree–Fock (HF) method, in which the wave function is a single Slater determinant. It is often referred to as the pseudo noninteracting electrons model because it approximates the electron–electron interaction by a mean-field Coulomb potential. The missing electron correlation canbe gradually turned on by including selected sets of excited determinants into the wave function. In the simplest correlated method, Møller–Plesset theory of the second order (MP2), the contributions of all doubly excited determinants are evaluated by means of perturbation theory. Including singly and doubly excited determinants explicitly by using the exponential ansatz gives rise to the CCSD (coupled-cluster with single and double substitutions) method, and so on. Climbing up this ladder of many-body theories, one can improve the accuracy of the description, systematically approaching the exact solution. Importantly, this series relies on the zero-order wave function, Φ0, being a good approximation to the exact solution. This is usually k k

Virtual MOs: Ground-state methods a,b,c,d,... Ψ Φ ϕ ϕ > Hartree–Fock: = 0 = | 1... n T MP2: HF + 2 by PT Ψ T T Φ CCSD: = exp( 1+ 2) 0 T Occupied MOs: CCSD(T): CCSD + 3 by PT i j k l Ψ T T T Φ , , , ,... CCSDT: = exp( 1+ 2+ 3) 0

C C C Φ T T T Φ Φ FCI: (1+ 1+ 2+.... N) 0 = exp( 1+ 2+...+ n) 0 0 (a) (b) FIGURE 6 (a) Hierarchy of ab initio methods for ground electronic states. The simplest

approximation of an N-electron wave function is a single Slater determinant, Φ0, built of orbitals that are variationally optimized. The description can be systematically improved by including excited determinants up to the exact solution of the Schrödinger equation by full configuration interaction (FCI). Operators∑ Tk and Ck generate k-tuply excited determinants a a a ab from the reference, for example, T1Φ0 = iati Φi . The amplitudes (ti , tij , … ) are determined either by solving coupled-cluster equations or by perturbation theory. In the case of closed-shell species, 𝛼 and 𝛽 spin-orbitals have the same spatial parts and the resulting wave functions

are naturally spin-pure. (b) Reference determinant Φ0 (also called the Fock-space vacuum state) 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 used for 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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156 REVIEWS IN COMPUTATIONAL CHEMISTRY

the case when the energy gap between the ground and electronically excited states is large, as is the case in stable closed-shell molecules. The success of quantum chemistry can be attributed to the fact that the Hartree–Fock approximation works remarkably 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 configuration interaction (FCI) method, the excited states are just higher eigenroots of the secular equation. These higher eigenroots are orthogonal to the lowest one (Ψ0) and can be described as excitations from Ψ0. Thus, for an approximate wave function, one can represent excited states as excitations from the given ground-state wave function:

Ψex = RΨ0 [2]

where R is a linear excitation operator, that is, ∑ a + R1 = ri a i [3] ia ∑ 1 ab + + R2 = rij a b ji [4] 4 ijab ··· k k Here second-quantization operators p+∕p create/annihilate electrons on the respec- 6,9 tive MOs. Operators R1 and R2 thus can be described as excitation operators of hole-particle (hp) and 2-hole-2-particle (2h2p) types. A many-body operator is called an excitation operator only when it includes creation operators that create electrons in virtual orbitals and annihilation operators that remove electrons from occupied orbitals. This is a very important requirement, which makes single-reference CC and EOM-CC methods simple and elegant. Such an ansatz can be justified by, for example, considering the response ofan approximate wave function to a time-dependent perturbation.6 The level of excitation in R is determined by the correlation treatment in the ground state, as illustrated in a ab Figure 7. The amplitudes of excitation operators R, ri , and rij are found by solving an eigenproblem with a model Hamiltonian, which can be derived by applying a variational (or bivariational) principle to the original Schrödinger equation. For the CIS method (configuration interaction singles), this is simply the diagonalization of the bare Hamiltonian in the basis of all singly excited determinants. In the case of EOM-CC, EOM amplitudes R are found by diagonalizing the so-called similarity transformed Hamiltonian:10,11

HR = RΩ [5] H ≡ e−T HeT [6]

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 157

Excited-state methods Ψ R Φ Hartree–Fock -> CIS: = 1 0 Ψ R R R T T Φ CCSD -> EOM-CCSD: = ( 0 + 1 + 2) exp( 1 + 2) 0 Ψ R R R R T T T Φ CCSDT -> EOM-CCSDT: = ( 0 + 1 + 2 + 3) exp( 1 + 2 + 3) 0

FIGURE 7 Hierarchy of ab initio methods for electronically excited states. The excited-state model corresponding to the Hartree–Fock ground-state wave function is configuration interaction singles (CIS) in which excited∑ states are described as a linear combination of all a a singly excited Slater determinants, ri Φi . For CCSD, the excited-state ansatz is a linear combination of all single and double excitations from the reference CCSD wave function, and so on.

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

⟨ | | ⟩ ECC = Φ0 H Φ0 [7] ⟨ | | ⟩ Φ휇 H − ECC Φ0 = 0[8]

and 휇 denotes the level of excitation in the CC ansatz (e.g., 휇 = 1, 2inCCSD). H has the same spectrum as the original Hamiltonian, H. Thus, when diagonalized over the full many-electron basis, Eq. [4] produces energies identical to FCI. k However, when diagonalized in an incomplete many-electron basis (e.g., within k a, ab singly and doubly excited manifold, {Φi Φij }, in the case of EOM-CCSD), it delivers higher accuracy than the equivalent CI method (CISD) due to rigorous size-extensivity and correlation effects that are folded into H through the similarity transformation employing the CC amplitudes. What happens with the Hartree–Fock approximation and the whole hierarchy of many-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 first problem is spin-contamination, which stems from the different mean-field potentials for 훼 and 훽 electrons when N훼 ≠ N훽 . The second type of problem, which is much harder to deal with, occurs when wave functions become multiconfigurational due to electronic degeneracies, such as HOMO–LUMO (highest occupied and lowest unoccupied 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) shown in Figure 8. Multiconfigurational wave functions also appear due to spin-adaptation of open-shell electronic configurations (e.g., two Slater determinants with equal weights are needed to describe the Ms = 0 component of a triplet state). Because the hierarchy of methods in Figure 6 is built assuming that the wave function is dominated by a single determinant, it fails to provide a balanced description of the multiple leading configurations (of course, the imbalance is reduced asmore correlation is included through higher excitations and it disappears at the FCI level). An alternative and altogether different route toward high accuracy is pursued by DFT, which, in principle, could deliver the exact energy from electron density

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Ψ 1 =

Ψ 2 = (H-H....H-H)+ H–H+....H-H (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 states + of the ionized dimer, (H2)2 (the electronic configurations of the two cationic states are shown in the box). When the two H2 molecules are identical (a), the charge is equally delocalized in the ground (Ψ1) and lowest excited (Ψ2) states of the cation. When the bond is stretched in one H2, the MOs become localized, that is, in the ground state of the cation, the positive charge is localized on the stretched H2 moiety. At low symmetry, Ψ1 and Ψ2 can mix, resulting in multiconfigurational wave functions. (See color plate section for the color representation of this figure.)

alone,12–14 subject to the unknown energy functional. Current implementations of DFT use the Kohn–Sham formulation in which the density is described by a single determinant and the orbitals are found by solving Hartree–Fock type equations. Although lacking the ability to systematically converge to the exact solution, practi- k cal DFT implementations with various representations of the exchange-correlation k functional are very useful. The success of the Kohn–Sham formulation of DFT stems from the same fundamental reason that the mean-field description captures ∼90 percent of the physics in normal molecules. Consequently, although formally exact for any type of wave functions, in the case of open-shell species, Kohn–Sham DFT experiences problems similar to those of the wave function-based methods. Below we consider different types of open-shell species and explain the key methodological challenges and solutions. This chapter focuses on the CC/EOM-CC methods whose elegant formalism, attractive mathematical properties, and robust performance are now well recognized.10,11,15–20 We note that the so-called linear-response CCSD (LR-CCSD) methods21–23 yield identical equations for the energies and employ slightly different expressions for calculating molecular properties.20,24 The SAC-CI (symmetry-adapted cluster CI) methods25,26 can be described as EOM-CC in which threshold-based selection of configurations is employed. There are also alternative approaches that have similar properties. In particular, ADC (algebraic-diagrammatic construction) and propagator-based families of methods, albeit derived in a very different spirit, share many attractive features, such as the ability to describe multiple states and to tackle multiconfigurational wave functions.27,28 Finally, we would like to highlight the difference between the terms “multiconfigurational” and “multireference.” The former refers to a concrete physical phenomenon, wave functions of complex character that cannot be captured by 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 using multireference formalism. Methods described in this chapter—such as EOM-CC and ADC—treat multiconfigurational wave functions in a strictly single-reference framework. This leads to elegant formalism and robust numeric performance. Most important, these methods do not involve system-dependent parameterization such as a problem-specific choice of an active space and, therefore, fulfill Pople’s criteria of the theoretical model chemistry.29

SOME ASPECTS OF ELECTRONIC STRUCTURE OF OPEN-SHELL SPECIES

Spin Contamination of Approximate Open-Shell Wave Functions The exact solutions of the electronic Schrödinger equation have certain spin and spatial symmetry, which stems from the basic quantum mechanical result that commuting operators share a common set of eigenstates. Relevant operators are spatial trans- formations of symmetry-equivalent nuclei that do not change the electron-nuclear 2 operator and the spin operators, Sz (projection of the total spin) and S (the magnitude of the spin): k ∑N k 휎 Sz = z(i) [9] i=1 ∑N 2 3N 휎 휎 휎 휎 휎 휎 S = + ( x(i) x(j)+ y(i) y(j)+ z(i) z(j)) [10] 4 i, j=1 휎 where x,y,z are the Pauli matrices and N is the total number of the electrons. It is desirable 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 훼 or 훽 spin leads to Slater determinants that are eigenstates of Sz. In the case of the closed-shell species, one can trivially enforce the requirement that the wave function be an eigenstate of S2 by making the spatial parts of the 훼 and 훽 spin orbitals the same. Similarly, spatial symmetry can be enforced by not allowing the MOs to break point-group symmetry. A principal difference between closed- and open-shell electronic structure is that in the former enforcing spin and spatial symmetry is trivial, whereas in the latter, this task is more difficult. In fact, a special effort is often required tofind symmetry-broken solutions of the Hartree–Fock equations for systems with even number of electrons, for example, by deliberate mixing of MOs when forming the guess density for the self-consistent procedure. For closed-shell systems, starting from spin- and symmetry-pure Φ0 automatically results in pure correlated wave 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 called spin contamination; it often signals poor quality of the results. In open-shell wave functions, spin contamination can arise for two distinct reasons: (i) different mean-field experienced by 훼 and 훽 electrons in the N훼 ≠ N훽 configurations and (ii) spin contamination due to an incomplete set of configurations in model wave 2| 휙 2 휙 1⟩ 2| 휙 1 휙 1 휙 1⟩ functions. Consider two doublet states: ( 1) ( 2) and ( 1) ( 2) ( 3) .Inthe former, spin contamination is usually minor even at the Hartree–Fock level because a spin-pure wave function of this character can be described by a single Slater determinant (in which only one electron is unpaired). However, in the latter case (an open-shell doublet state with three unpaired electrons), constructing a spin-pure wave function requires taking a linear combination of several Slater determinants, with coefficients determined by spin-adaptation rules. Consequently, spin contamination of just one determinant of this character is large, and attempts to describe these types of 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 degenerate electronic states are not stationary points on potential energy surfaces (PESs) in k nonlinear molecules, which means that degenerate electronic states follow first-order k distortions to a lower symmetry at which the degeneracy is lifted. The JT effect often occurs in radicals derived by removing or adding an electron from/to a high-symmetry closed-shell reference system, such that the unpaired electron resides on one of the doubly degenerate MOs. The total charge is of no significance—the JT pattern can be found in radical anions, radical cations, and neutral radicals. Specifically, non-Abelian point-group symmetries containing irreducible representations (irreps) of order higher than one give rise to symmetry-required degeneracies between the MOs belonging to these irreps. Two examples of JT systems, the benzene radical cation and the cyclic N3 radical, are shown in Figure 9. The JT effect also occurs in systems with N훼 = N훽 , 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 displacement along which the linear derivative coupling matrix element between the unperturbed states and the difference between the diagonal matrix elements of the perturbation are not required to be zero. Because of the linear term, the JT degenerate PESs form a conical intersection. This leads to singular points on adiabatic PESs clearly seen in 38,39 Figure 10, which illustrates the JT effect for the cyclic N3 radical. An important feature of such an intersection is that the symmetry of the lowest electronic state changes as one passes through the CI, as one can see in Figure 10b. Consequently, the real electronic wave function calculated within the Born–Oppenheimer approximation gains a phase (sign change) along any path on the PES, which encircles a conical intersection and thus contains the singularity

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e′ a e′ b ( 1) ( 2) σ* a ′ b 2 ( 2)

e″ a e″ b π* ( 2) ( 1) e π 1g, g

y x

b π a b π o e′ a e′ b ( 2g, g ) ( 3g, g ) ( 1) ( 2)

lp e σ 2g, a ″ b π 2 ( 1) a ′ a 1 ( 1)

a σa b σo ( g, ) ( 1g, ) a π 2u, u e′ a e′ b ( 1) ( 2) σ a ′ a 1 ( 1) b π ( 1u, u) k (a) (b) k FIGURE 9 Two examples of Jahn–Teller systems. (a) Highest occupied MOs of

benzene molecule (both D6h and D2h labels are given). (Reproduced with permission from Ref. 36. Copyright 2008 American Institute of Physics). The two lowest states of the benzene radical-cation form a Jahn–Teller pair. (b) Molecular orbitals and the ground-state electronic + configuration of cyclic3 N (D3h structure, C2v labels are given in parentheses). Both HOMO and 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 Physics from 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 change of sign of the electronic wave function needs to be properly accounted for in the calculations of the nuclear wave function. This is illustrated in Figure 11, which 39 compares the vibrational wave functions of N3 computed with and without geometric phase effect (these are denoted by BO (Born–Oppenheimer) and GBO (generalized BO), respectively). The geometric phase has a profound effect on the nodal structure of the vibrational wave functions, the order of vibrational states, and the selection rules for the vibrational transitions. The correct description of JT states requires approaches that treat the pairs of degenerate (or nearly degenerate) JT states on an equal footing. The geometric phase issue can be avoided by employing a diabatic representation of the electronic states using vibronic Hamiltonians.43,44

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0.6 CI (i) 0.5 0.4 0.3 2A 2B (eV) 2 1

E 0.2 0.1 0.0 TS MIN –0.2 –0.1 0.0 0.1 0.2 x (y = 0)

(ii) 0.2 2A 2B 2 0.5 1 ϕ 0.4 0.3 0.1 0.2 0.1 2A 2B

0.2 Y 0.0 2 1

0.1 –0.1 0.2 –0.0 x 2B 0.1 1 2A –0.2 2 –0.0 –0.1 y –0.1 –0.2 –0.2 –0.1 0.0 0.1 0.2 (a) –0.2 (b) x

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

Vibronic Interactions and Pseudo-Jahn–Teller Effect The PJT effect arises from configuration interaction between electronic configurations that are close in energy;31 it is very common in radicals and has a major effect on the shape of PESs, for example, it may lead to structures of lower symmetry. Flawed description of these interactions by an approximate electronic structure method can lead to poor, and occasionally ridiculous, results and is responsible for the so-called artificial symmetry breaking. To illustrate the effect, consider the para-benzyne anion. Figure 12 shows its frontier MOs and the two nearly degenerate electronic configurations that give rise to the two lowest electronic states. At D2h symmetry, the two configurations are 2 of different symmetry and therefore give rise to two distinct electronic states, Ag

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BO GBO BO A (0,0,0) 1 2A + 2B 2 – 1 + – ++ – + – – + – + +

+ + – 2A + 2B 2 – 1 A + (1,0,0) 2 – + + – + + + + –

+ – + – – 2B + + 1 2A 2

(a) (b) FIGURE 11 Geometric phase effect. (a) Illustration of the sign change in the lowest state adiabatic (Born–Oppenheimer) electronic wave function. The drawing represents a contour

plot of the cyclic N3 PES as in Figure 10, but with only two contour lines: one contour line is given at 0.16 eV, which is also the highest energy contour line in Figure 10. It forms two concentric loops, which encompass the white area, at the inner and outer parts of the well. The second contour line is given exactly at the energy of the transition state 0.0386 eV; it defines the colored area. This contour line exhibits a Möbius-like shape with the nodes at the transition k state points and serves here as a reflection of the electronic wave function symmetries. The k 2 2 wave function is symmetric across the B1 lines and is antisymmetric across the A2 lines. This is indicated as “+/+”or“+/−” across each C2v symmetry line. (b) Demonstration of the changes in the nodal structure of vibrational wave functions due to the geometric phase , , effect. 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 phase changes the nodal structure of the vibrational wave function. The correct nodal structure of the generalized 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 the correct permutation symmetry. Reproduced with permission of American Institute of Physics from Ref. 39. (See color plate section for the color representation of this figure.)

2 and B1u. However, at lower symmetries (e.g., C2v), the two configurations are of the same symmetry (A1) and can, therefore, mix, resulting in two-configurational wave functions for both states. As follows from the 2 × 2 eigenproblem, this configuration interaction lowers the energy of the lower electronic state and increases the energy of the upper. Consequently, the PES of the lower state softens along such symmetry-lowering displacements. If the interaction is sufficiently strong, a minimum at lower symmetry structure may develop, as illustrated in Figure 13. Since here nuclear displacements modulate electronic couplings, this phenomenon is called vibronic interaction. While vibronic interaction is a real physical phenomenon, it may or may not be accurately described by an approximate

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D C 2h 2v

C1C1 C1C1 a a g 1 6ag a 111 1 A2B A2A C4C4 C4C4 1u 1 b a 5 1u 101 1

C1C1 C1C1 a a 6 g 111 1 b a 2A X2A 1u 1 X g 1 b a 5 1u 110 1 CC44 CC4 4

(a) (b) FIGURE 12 Frontier MOs (a) and leading electronic configurations (b) of the two lowest

electronic states of the para-benzyne anion at D2h and C2v geometries. Reproduced with permission of Springer from Ref. 45.

k k

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

method. If vibronic interactions are overestimated, calculations may yield incorrect lower symmetry structure, and vice versa. For example, in the para-benzyne anion,45 many DFT methods lead to artificial symmetry lowering, yielding incorrect C2v structures. Many other examples can be found in the works of Stanton, Gauss, and + Crawford. Classic examples include NO3,BNB,C3 , and the cyclic C3H radical. In-depth analysis of the PJT effects and its description by approximate electronic structure methods can be found in Refs 46–48. Formally, this effect of the interactions between electronic states on the shape of the PES can be described46 by the Herzberg–Teller expansion of the potential energy

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 165

of electronic state i: ⟨ ⟩ ⟨ ⟩ ∑ | | ∑ | 2 | | 휕H | 1 | 휕 H | 2 V (Q)=V + Ψ | | Ψ Q훼 + Ψ | | Ψ Q훼 i 0 i 휕 i i | 2 | i 훼 | Q훼 | 2 훼 |휕 Q훼 | |⟨ | 휕H | ⟩|2 ∑ ∑ Ψi 휕 Ψj Q훼 2 − Q훼 [11] 훼 j≠i Ej − Ei

It is the last term that describes the vibronic couplings between two different electronic 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 displacements might be PJT active. In order for this matrix element to be nonzero, the product of the irreps for Ψi, Ψj, and Q훼 should contain a fully symmetric irrep. Thus, in Abelian groups, the nuclear displacement should be of the same symmetry as the product of the irreps of the two electronic states. For example, in the para-benzyne anion, any mode of the ag × b1u = b1u symmetry might be PJT active.

HIGH-SPIN OPEN-SHELL STATES k Let us begin by considering a relatively easy case of high-spin open-shell states, k such as doublet radicals, triplet diradicals, and quartet triradicals. Often, the exact wave function for such species is dominated by a single determinant; thus, one may expect performance of quantum chemistry methods to be similar to the case of closed-shell molecules. One complication, however, is that the direct application of the variational principle to the single-determinant ansatz leads to different MOs 훼 훽 2 for and electrons, giving rise to Φ0, which is not an eigenstate of the S operator (Eq. [10]). This spin-contamination phenomenon affects the quality of post-Hartree–Fock methods that use the reference determinant produced by the unrestricted Hartree–Fock calculations (“unrestricted” refers to the full variational optimization of the single-determinant wave function without restricting it to be spin- or symmetry- pure). This compromises MP2 theory often rendering the results unreliable (except for several special cases such as solvated electron49 or when MOs are optimized for the MP2 wave function50). When electron correlation is included 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 in CCSD(T), deteriorates considerably (the EOM methods are also very sensitive to spin-contamination of the reference). One can introduce a spin-purity constraint into the Hartree–Fock equations, which gives rise to the restricted open-shell HF (ROHF) method.53 Unfortunately, this does not resolve all problems, and even brings new ones, such as relatively poor convergence of the self-consistent field procedure 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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166 REVIEWS IN COMPUTATIONAL CHEMISTRY

coupled-cluster ansatz to an ROHF determinant does not produce a spin-pure wave function.51,52 Spin-adaptation can be achieved by imposing additional constraints on the amplitudes.54–56 The spin-contamination of ROHF- and UHF-based CCSD wave functions is similar and is relatively small. Consequently, the quality of the results delivered by these methods is practically the same, which can be explained by 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. Other choices of the reference determinants (or orbitals) mitigating spin-contamination include QRHF (quasi-restricted HF),57 Kohn–Sham orbitals, and orbitals optimized for correlated wave functions, such as in Brueckner or optimized orbitals CCD methods.58–61 When spin-contamination is relatively small at the UHF level, it does not cause significant decline in the quality of the coupled-cluster calculations. A related problem is symmetry breaking, which stems from the solutions of the HF equations that do not have correct spatial symmetry. For example, a UHF solution for + He2 may have the hole localized on one atom. We will discuss this issue later, because it originates from a different phenomenon, that is, multiconfigurational character of the 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 operator, as clearly seen from Eq. [10], cannot be computed from the density alone. The k commonly reported values of ⟨S2⟩ are computed for the Kohn–Sham determinant; k they do not reflect the spin-contamination of the unknown many-electron wave function 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 훼 and 훽 orbitals,62 so that areas with an excess 훽 density in the M = 1 states can be described. s 2 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–Sham or TDDFT “wave functions” is indicative of a problematic behavior justifying some degree of spin-adaptation. Interestingly, DFT orbitals often show good performance in wave function calculations, 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 functional (specifically, the amount of the HF exchange). The reasons for the usefulness of 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, they are 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 overdelocalization of the unpaired spin.63–66 Thus, in cases when UHF solutions break symmetry and overlocalize the unpaired electron (or hole), the restoration of symmetry in Kohn–Sham calculations often occurs for a wrong reason. Overdelocalization due to SIE 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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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 167

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

OPEN-SHELL STATES WITH MULTICONFIGURATIONAL CHARACTER

Here we consider high-spin open-shell states whose description is complicated by electronic 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-transfer system (a similar example can be constructed by considering electron-attached states instead of ionized ones). When the H2 fragments are identical, the hole is equally delocalized 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) leads to preferential charge localization. Koopmans’ theorem predicts that in the lowest k electronic state of the cation, the charge should reside on the stretched H2 moiety, k whereas in the excited state, on the other one. The energy gap between the two states depends on the distance between the fragments. By varying the bond lengths and the separation between the two H2 molecules, the character of these two states can be controlled. One can consider a charge-transfer coordinate along which the bond length of the stretched H2 is decreased while the bond length on the other one increases. Along this coordinate, the charge in the ground electronic state flows from the initially stretched H2 (donor) to the second fragment (acceptor). Importantly, the two Koopmans configurations, Ψ1 and Ψ2, can mix, giving rise to multiconfigurational wave functions. Thus, except for the high-symmetry structures where the two Koopmans configurations have different symmetry, a single-determinantal description is flawed. Hartree–Fock typically overlocalizes the hole, which is then carried over 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 remedy the problem.70 In order to treat this type of electronic structure correctly, one needs to incorporate the multiconfigurational character in the ansatz. This is naturally achieved 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 standard single-reference and Kohn–Sham methods, can be elegantly circumvented by the EOM-CC formalism. Instead of following the strategy outlined in Figure 6, in the EOM-CC framework, one does not attempt to construct an open-shell wave

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168 REVIEWS IN COMPUTATIONAL CHEMISTRY

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

T1+T2 , Ψ(N)=exp Φ0(N) [12]

where N denotes the number of electrons in the reference state and Φ0 is the reference determinant (typically, the Hartree–Fock one). Then the target open-shell states are described by acting on this reference state by general excitation operators, which are not particle conserving:

IP IP T1+T2 Ψ(N−1)=(R1 + R2 )exp Φ0(N) [13]

EA EA T1+T2 Ψ(N+1)=(R1 + R2 )exp Φ0(N) [14]

where ∑ IP R1 Φ0(N)= riΦi [15] i ∑a IP 1 a a R2 Φ0(N)= rijΦij [16] 2 ij ∑ k EA a a k R1 Φ0(N)= r Φ [17] a ∑ab EA 1 ab ab R2 Φ0(N)= ri Φi [18] 2 i

Both CC operators T and EOM operators R are net excitation operators with respect to the reference determinant Φ0 (see Figure 6b), which defines the separation of orbital space into the occupied and virtual orbitals. Just as in the case of excited states, the diagonalization 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 than diagonalization of the bare Hamiltonian, because the correlation effects are folded into H. A detailed introduction to EOM-CC theory and discussion of its various aspects can be found in several reviews.10,11,15–19,48 So what do we gain by following this strategy? Figure 14 shows the reference and target configurations in the EOM-IP/EA methods employing closed-shell reference 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 determinants are naturally spin complete, resulting in spin-pure wave functions. Furthermore, possible multiconfigurational character of the target wave functions can be described—for example, the two configurations from Figure 8 appear in the EOM-IP expansion at the same excitation level (R1). Their relative weights are determined by the 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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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 169

ψ N R ψ N EOM-IP: ( ) = (–1) 0( + 1)

Φ Φ Φa 0 i ij

ψ N R ψ N EOM-EA: ( ) = (+1) 0( – 1)

Φ a ab 0 Φ Φi

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

EOM states can be computed at once, such that one obtains both ground and excited states of an open-shell system. Because these target states are produced in the same calculation, using the same set of orbitals (determined by the choice of Φ0), and the same T-amplitudes, the calculations of interstate properties, such as transition dipole moments, Dyson orbitals, nonadiabatic, and spin-orbit couplings (NACs and SOCs, respectively) is straightforward. Because of the good behavior of the closed-shell k reference and flexible target-state ansatz, EOM-IP/EA provide qualitatively and k quantitatively correct description of charge-transfer, JT, and PJT systems including cases of extensive degeneracies, as illustrated by the examples given here. EOM-IP has also been successfully used for constructing vibronic Hamiltonians for systems with strong nonadiabatic interactions.43 The choice of a specific EOM-CC method depends on the type of target states that are sought, together with the MO pattern. The best description is attained when the reference is of a closed-shell type in which manifolds of degenerate (and nearly degenerate) orbitals are completely filled and the dominant character of the target states can be described by a single-excitation EOM operator (1h in EOM-IP or 1p in EOM-EA). For example, core and valence ionized states of molecules (or electron-detached states of closed-shell anions) and molecular clusters are well described by EOM-IP. However, satellite states that have doubly excited character (2h1p) are described less accurately. Ground and selected excited states of many doublet radicals are well described by EOM-IP starting from the anionic closed-shell 휋 휎 → reference. For example, the vinyl radical (Figure 1) and its CC∕ CC lp(C) excited − states can be described by EOM-IP using the C2H3 reference. However, its Rydberg 휋 → 휋∗ states or lp(C)∕ CC 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 unpaired electron can be either localized on different parts of the system or delocalized, such

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170 REVIEWS IN COMPUTATIONAL CHEMISTRY

as a model charge-transfer system in Figure 8. A benchmark study71 of charge localization and electronic properties in small charge-transfer systems, such as + + + + (He)2 , (H2)2 , (Be-BH) , and (Li-H2) , has demonstrated that EOM-IP-CCSD is indeed capable of reproducing the trends correctly, whereas methods with similar level correlation treatment (such as open-shell CCSD) often show larger + errors. Figure 15 shows relevant properties for (H2)2 (see Figure 8) along the charge-transfer coordinate (q), which is defined as a linear interpolation between the two structures: one in which the left fragment has the geometry of the cation and the 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 localization depends on the distance between the fragments, that is, at q = 0, the charge should be completely localized on the left fragment at very large separations. At 3.0 Å separation, the FCI charge on the left fragment at q = 0 equals 0.957, and at 5.0 Å 0.995. 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 delocalized due to symmetry. As one can see from Figure 15, the differences between the EOM-IP-CCSD and FCI charges are tiny, whereas the open-shell CCSD results show much larger deviations. Other properties, such as ground and excited-state energies (shown in Figure 15) and the respective transition dipole moments,71 are affected as well. The difference between EOM-IP-CCSD and CCSD is striking given k the small size of this system with only three electrons. Thus, one would expect that k CCSD 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 along the charge-transfer coordinate in the ethylene dimer cation at two different interfragment 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 flow of the charge from ∼0.75 at q = 0to0.5atq = 0.5. B3LYP overdelocalizes the charge, whereas ROHF- and UHF-based CCSD show much larger charge localization (0.8 at q = 0). Consequently, the shape of the PES is different. Instead of the minimum at the symmetric configuration, CCSD yields a noticeable barrier, while B3LYP overestimates the energy gain at the symmetric configuration. At 6 Å separation, the charge is fully localized at q = 0 and the symmetric structure corresponds to the maximum on the potential energy curve. As one can see, B3LYP fails to reproduce both charge localization and the barrier. Long-range corrected functionals improve the behavior considerably, but do not completely remedy the problem.70

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

k k

800 ) 0.03 ) –1 120

–1 600 EOM-EE-CCSD 90 0.02 CCSD 400 60 CCSD 200 0.01 30 0 EOM-IP-CCSD Ground state charge (e) Excitation energy (cm 0.00

Ground state energy (cm EOM-IP-CCSD 0 EOM-IP-CCSD –200 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 (a) Reaction coordinate (b) Reaction coordinate (c) Reaction coordinate

82 800

) CCSD ) –1 80 750 0.0009 –1 EOM-EE-CCSD CCSD 78 700

76 650 0.0006 600

0 0 0.0003 –50 EOM-IP-CCSD –2 EOM-IP-CCSD Excitation energy (cm Ground state charge (e) 0.0000

Ground state energy (cm –100 –4 EOM-IP-CCSD –150 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 (d) Reaction coordinate (e) Reaction coordinate (f) Reaction coordinate

+ FIGURE 15 Comparison of EOM-IP-CCSD and CCSD/EOM-EE-CCSD description of the electronic states of (H2)2 along charge-transfer coordinate 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.

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172 REVIEWS IN COMPUTATIONAL CHEMISTRY

ROHF-CCSD 0 0.8 ROHF-CCSD )

–1 UHF-CCSD –200 UHF-CCSD 0.6 B3LYP –400 MR-CISD+Q 0.4

–600 EOM-IP-CCSD Ground state charge (e) EOM-IP-CC(2,3) Ground state PES (cm 0.2 EOM-IP-CC(2,3) EOM-IP-CCSD –800 B3LYP MR-CISD

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 (a) Reaction coordinate Reaction coordinate

400 1.00 EOM-IP-CCSD ) EOM-IP-CCSD –1 0.75 0 B3LYP 0.50

–400 0.25 Ground state charge (e) Ground state PES (cm B3LYP –800 0.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 k (b) Reaction coordinate Reaction coordinate k FIGURE 16 Comparison of EOM-IP-CCSD, CCSD, and B3LYP description of the ground electronic state of the ethylene dimer cation along the charge-transfer coordinate at the 4.0 Å (a) and 6.0 Å (b) separations using aug-cc-pVTZ. (a) Highly reliable MR-CISD and EOM-IP-CC(2,3) results. On the left, potential energy along the charge-transfer coordinate is shown. On the right, the charge on the left moiety is shown. Reproduced with permission of American Institute of Physics from Ref. 71.

Structure-dependent electronic properties give rise to concrete spectroscopic signatures. Multistate methods capable of describing multiple electronic states in a balanced way are required for reliable modeling of such properties. Figure 17 shows the MO diagram explaining the nature of the lowest ionized states of the symmetric 휋-stacked dimers of nucleobases and their electronic spectroscopy. The two highest occupied MOs of the dimer can be described as bonding and antibonding pairs of the fragments’ MO, similarly to a diatomic molecule.36 Thus, their energies are split and, since the HOMO of the dimer is destabilized relative to the monomer, 휋-stacking reduces the IE. For the uracil dimer shown in Figure 17, the EOM-IP-CCSD/6-311(2+)G(d, p) VIEs of the monomer and the dimer are 9.48 and 9.14 eV, respectively.73 Similar red shifts in VIE have been observed in other nucleic acid base dimers.77,78 Removing the electron from the dimer’s HOMO, which has an antibonding character with respect to the two units, leads to relatively strong bonding between the two fragments and initiates structural relaxation—the two moieties are moving closer to each other.74,75 Importantly, new electronic transitions appear

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0.28 0.26 0.24 0.22 0.20 φA–φB 0.18 0.16 0.14 0.12 –0.372–0.361 –0.372 0.10 Oscillator strength Oscillator 0.08

–0.384 0.06 φ π π φA 1p π π 0.04 A=1p(N) + CC + CO = (N) + CC + CO 0.02 0.00 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 φA+φB Energy (eV) (a) (b) FIGURE 17 Ionized states of 휋-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 follow the same trend. The ionization from the antibonding orbital changes the bonding from noncovalent to covalent and enables a new type of electronic transitions, which are unique to the ionized dimers. (b) Vertical electronic spectra of the stacked uracil dimer cation at two different geometries: the geometry of the neutral (bold lines) and the cation’s geometry (dashed lines). MOs hosting the unpaired electron in each electronic states are shown for each tran- k sition. Reproduced with permission of Royal Society of Chemistry from Ref. 73. (See color k plate section for the color representation of this figure.)

in the ionized dimer, such as a bright HOMO-1 → HOMO band; their energies and strengths depend sensitively on the interfragment distances (Figure 17). Figure 18, which shows ionized states of the two 휋-stacked thymine dimers, illustrates the effect of structure on the charge localization. In a symmetric dimer, orbitals are delocalized (and the shift in IEs relative to the monomer is larger), whereas in a lower symmetry structure, the ionized states are localized on one of the moieties. Note that in both structures, the EOM-IP states are dominated by Koopmans-like configurations (as illustrated by the magnitude of the leading R1 amplitude); thus, canonical Hartree–Fock MOs provide a good representation of the correlated ionized states. This example illustrates EOM-IP’s ability to describe multiple ionized states, to treat hole localization/delocalization correctly, and to compute electronic spectra.

Jahn–Teller Systems As discussed in Section Jahn–Teller Effect, ionization of molecules that have degenerate HOMOs, such as benzene, benzene dimer,36 and triazine, 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 9 shows relevant MOs for benzene and N3. Attempts to describe such JT pairs of states by ground-state single-reference methods (such as open-shell CCSD) yield flawed results. The lowest adiabatic PES could be computed by open-shell CCSD,

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8.99 8.78 (–0.14; 0.917) (–0.35; 0.957) 9.13 (0.00; 0.919)

9.30 (0.17; 0.955) VIE (eV) VIE (eV)

10.02 10.12 10.13 10.17 (–0.11; 0.931) (–0.01; 0.934) (0.00; 0.942) (0.04; 0.947) 10.39 10.31 10.50 (–0.14; 0.931) (–0.21; 0.944) (–0.02; 0.949) 10.54 (0.02; 0.931) (a) (b) FIGURE 18 The MOs and respective VIEs (eV,EOM-IP-CCSD/6-311+G(d,p) extrapolated to cc-pVTZ) for the two stacked thymine dimers, symmetric (a) and nonsymmetric (b). The energy difference (eV) between the dimer IE and corresponding IE of the monomer and leading EOM amplitudes for the shown orbitals are given in parenthesis. Reproduced with permission of Royal Society of Chemistry from Ref. 77. (See color plate section for the color representation of this figure.)

by following the lowest energy HF solution, but the shape of the surface is likely to be pathologically wrong, especially in regions where the reference determinant k changes symmetry. Computing the second JT state by this strategy is only possible k at high-symmetry structures, where the two states are of different symmetry, so that one can relatively easily perform separate HF and CCSD calculations for each state. 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 this exact degeneracy). Similar to the charge-transfer systems discussed above, this type of electronic structure is best described by EOM-IP or EOM-EA methods, which have sufficient flexibility to represent the two JT states at arbitrary geometries. For example, the ionized states of benzene (and even a more challenging system, the benzene dimer) can be accurately described by EOM-IP using a closed-shell neutral Refs 36, 82. In the case of N3, the best description is given by EOM-EA using a closed-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 reaction intermediates. An example of an organic diradical, para-benzyne, is shown in Figure 3. Polyradical character, which is of interest in the context of molecular magnets, can be realized in organic molecules with multiple NO groups or transition metal centers. Owing to orbital degeneracy, the wave functions of polyradical species contain several (at least two in diradicals) leading electronic configurations. Obviously, the hierarchy of methods built on the single-determinantal

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 175

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 energy curves along bond-breaking coordinates can collapse to negative infinity, as one should expect from the MP2 correlation energy expression, which contains orbital energy differences in the denominator. When the extent of the diradical character is mild (such as in singlet methylene), correlated CC methods provide a reasonable description, however eventually they also fail. On the other hand, methods that use a multiconfigurational zero-order wave function (such as CASSCF, complete active space self-consistent field) often have difficulties due to separate treatment of nondynamical and dynamical correlation, intruder states, sensitivity to the active-space selection, etc. To illustrate the type of problems that one may encounter in diradicals, consider meta-benzyne,83 an isomer of para-benzyne. In this diradical, shown in Figure 19, the interaction between the unpaired electrons and, consequently, the distance between the two radical centers depend sensitively on the correlation treatment. The CASSCF and valence bond methods exaggerate the diradical character and fail to correctly account for a partial bond formation between the two centers. CCSD, on the other hand, overestimates the bonding interactions and yields an incorrect bicyclic structure. 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, it would be unwise to rely on such calculations without validating the results against a k more appropriate approach. Rather than resorting to a brute-force approach, the cor- k rect description of such systems is achieved when both configurations are treated on an equal footing and dynamical correlation is included. This can be accomplished by employing spin-flip (SF) methods.84,85

H7 H7

C2 C2 C1 C3 C1 C3

C6 C4 C6 C4 H10 H10 H8 C5 H8 C5 H9 H9 12

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

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In the SF approach, the target states are described as spin-flipping excitations from a high-spin reference state. Figure 20 illustrates how it works for di- and triradicals. As one can see, single spin-flipping excitations generate a spin-complete set of the determinants needed to describe ground and low-lying excited states corresponding to 2-electrons-in-2-orbitals and 3-electrons-in-3-orbitals types of electronic 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 (restricted active space CI with SF) and CAS-SF formulations with general number of spin flips.93–95 Contrary to low-spin states, high-spin reference states used in SF methods (such as 훼훼 or 훼훼훼) are not affected by orbital degeneracies because all frontier orbitals are occupied in the respective Hartree–Fock wave functions. In EOM-SF-CCSD, when using the 훼훼 reference state, the target wave functions are SF SF T1+T2 Ψ(Ms = 0)=(R1 + R2 )e Φ0(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 the 1h1p and 2h2p type), but EOM-SF ones change the spin of an electron. That is, they conserve the total number of electrons but change the number of 훼 and 훽 electrons. Electronic configurations shown in Figure 20 give rise to the primary manifolds k of diradical and triradical states. These configuration expansions are identical to the k sets generated in CAS for the 2-electrons-in-2-orbitals and 3-electrons-in-3-orbitals active spaces. However, target SF wave functions also contain configurations that

ψ M M ψ M ( s =0) = R( s = –1) 0( s =1)

Φ Φa 0 i (a)

ψ M M ψ M ( s = 1/2) = R( s = –1) 0( s = 3/2)

Φ Φa 0 i (b)

FIGURE 20 High-spin reference (Φ0) and low-spin target configurations for the EOM-SF-CCSD wave functions. (a) EOM-SF using the Ms = 1 triplet reference describes the Ms = 0 states of diradicals. Note that both open-shell and closed-shell states can be described. (b) The M = 1 electronic states of triradicals can be described by using the high-spin quartet s 2 reference (M = 3 ). s 2

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 177

are excitations outside singly occupied space; these configurations are not present in the CASSCF expansion. Such configurations describe dynamical correlation and interactions between the doubly and singly occupied orbital spaces. In the context of aromatic diradicals (as para-benzyne from Figure 3), these configurations describe 휎–휋 interactions, which are omitted in minimal active-space CASSCF calculations. When implemented in a straightforward spin-orbital formulation, the target SF states 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 for constructing spin-pure target states from the primary diradical (or triradical) manifolds. However, the set of single excitations from doubly occupied orbitals (or into unoccupied ones) is not spin-complete because the counterparts of some of these configurations appear at a higher excitation level.96 Consequently, SF target states are spin-contaminated even when using a ROHF reference. The spin-contamination of the primary SF states is usually minor even at the SF-CIS or SF-TDDFT levels; it is further reduced when double excitations are included, as in EOM-SF-CCSD. How- ever, states dominated by excitations outside of the singly occupied orbital manifold may be strongly spin-contaminated. The SF expansions can easily be spin adapted by including a small subset of higher excitations.93–96 Importantly, since the number of such configurations is proportional to the number of single excitations, their inclusion does not lead to increased scaling of the method. The difference between the states from the primary SF manifolds (Figure 20) k and other electronic states persists even in spin-adapted versions. Thus, users should k remember that while all electronic states that are dominated by configurations from Figure 20 are well described by SF methods, the states that are dominated by excitations outside of the singly occupied orbital manifold are not treated on the same footing as the primary states and that the accuracy of SF description deteriorates in this case. Thus, when computing excited states of diradicals and triradicals, one should always distinguish between these two groups of states. The utility of the SF approach has been illustrated by numerous studies of organic diradicals and triradicals.34,35,45,84–87,97–102 To quantify the numeric performance of SF 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 separation between them is notoriously difficult to compute reliably.1,2 The main reason is that the gap depends sensitively on the correlation energy, which is very different in states of different multiplicity. That is, high-spin states tend to have much less correlation energy relative to low-spin states of similar orbital character because the Pauli exclusion principle restricts the possibilities of distributing same-spin electrons within the given orbital space and also reduces the probability of the two electrons approaching each other. The former results in smaller nondynamical correlation and the latter effect (Pauli hole) reduces dynamical correlation.84–86,88 Table 1 presents experimental103 and calculated values of singlet–triplet gaps in benzyne isomers. The distance between the radical centers affects their interaction and, consequently, the singlet–triplet gap. The smallest overlap between the orbitals hosting the unpaired electrons occurs in para-benzyne (see Figure 3), which has the 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-, and para-Benzyne Isomers

Method ortho-C6H4 meta-C6H4 para-C6H4 EOM-SF-CCSDb 1.58 0.78 0.16 EOM-SF-CCSD(dT)b 1.62 0.89 0.17 NC-SF-TDDFT/PBE50b 1.59 0.78 0.12 COL-SF-TDDFT/B505LYPb 1.88 0.98 0.16 Experimentc 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. In ortho-benzyne, a partial bond is formed between the two adjacent centers, leading to the singlet–triplet gap of 1.66 eV. The energies of these partial bonds between the radical 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 in para-benzyne is indeed very weak, whereas in ortho-benzyne the interaction energy is equal to roughly one-third of the energy of a covalent bond. The benzynes are a k good test for a computational method because of their variable diradical character. k 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 offers an inexpensive alternative to EOM-SF-CCSD and can be applied to much larger systems, also delivers good accuracy.91,92 We stress that SF works for various degrees of orbital degeneracies, including cases with exactly degenerate frontier MOs such as trimethylmethane.34 Importantly, in SF calculations, one is not restricted to the lowest singlet state and can also compute higher singlet states in the same framework. Thus, all primary diradical states can be described with uniform accuracy. As an example, consider methylene. The MO diagram and electronic configurations of the low-lying states are shown in Figure 21; the relevant energies are collected in Table 2. For small basis set calculations (TZ2P), we can compare SF against FCI. We see that when the triples correction is included, the errors do not exceed 0.02 eV for any of the three states. To compare against experiment, larger basis sets need to be employed. The EOM-SF-CCSD(dT)/cc-pVQZ values for the two lowest states are within 0.03 eV from the experimental excitation energies. Electronic degeneracies and a high density of states are even more pronounced in triradicals.35,87,97–101 Figure 22 shows the MOs and vertical state ordering for three prototypical triradicals—trimethylenebenzene (TMB), an all-휋 triradical, and the two isomers of dehydro-m-xylylene (DMX), a 휎-휋 triradical. As one can see, the density of states is rather large and most of the states have heavily multiconfigurational wave

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 179

Orbitals Reference Target states

b 1 + −λ − λ +

a 3B 11A 1B 21A 1 1 1 1 1

FIGURE 21 Frontier MOs and the leading configurations of the low-lying electronic states 3 1 1 1 of methylene, X B1,1A1 (closed-shell singlet), B1 (open-shell singlet), and 2 A1 (also 1 closed-shell singlet, which is doubly excited with respect to 1 A1). The wave functions of the Ms = 0 components of all four states can be described as single SF excitations from the high-spin (Ms = 1) triplet reference. (See color plate section for the color representation of this figure.)

TABLE 2 Adiabatic Excitation Energies (eV) Relative 3 to the X B1 State of Methylene 1 1 1 Method 1 A1 1 B1 2 A1 EOM-SF-CCSD/TZP 0.52 1.56 2.72 EOM-SF-CCSD/TZP 0.50 1.55 2.68 FCI/TZPa 0.48 1.54 2.67 EOM-SF-CCSD/cc-pVQZ 0.45 1.43 2.58 EOM-SF-CCSD(dT)/cc-pVQZ 0.42 1.41 2.53 Expb 0.39 1.42 k k 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 excitations outside the singly occupied orbital manifold. The EOM-SF approach describes all these states in a single calculation, giving rise to a balanced description of the states and accurate energy differences. All-휎 triradicals35,97 as well as substituted and heteroatom analogues99,101 are also well described. Can we extend this approach beyond diradicals and triradicals to tackle polyradical species, which are of interest in the context of molecular magnets? There are two strategies toward this goal. One is to employ an ansatz with more spin flips and higher spin references. For example, double spin-flip (Ms =−2) from a quin- tet (Ms = 2) reference can describe tetraradicals (and simultaneous breaking of two bonds).93 An EOM-CC implementation with double spin flip and RASCI and CAS implementations including an arbitrary number of spin flips exist.93–95 The problem with this strategy is that using a higher number of spin-flips increases the cost of the calculations. Consider, for example, double spin-flip. Since the spins of two electrons 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, R3 need to be included in the EOM ansatz, so the correlation of the target states can be properly described (i.e., the CCSD double SF method can be viewed as a variant of EOM-CCSDT). In TDDFT, which only admits single excitations within the adiabatic connection, inclusion of double excitations would require theoretical justification

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H H H

H H

H H a b a 1 1 2 H H

H H H H a b a H 1 1 2 HH

H H H H

H H H a″/b e″/a e″/b 2 1 2 1 (a)

2B 4 2 1 2A 1 2 22B 2A − 1 2 1 22A −λ 1 2B 3 1 3 k λ + k 2B 1 1 2B 1 1 + 2 22B 22B 22B 2 − 1 2 2A − 2A 2 2 1 2 −

−λ + 1 2B 2A 2A 1 1 1 1 1 1 2A Excitation energy (eV) energy Excitation λ + 1 2 2B 1 2 + 4B 1 2 2B 14B 2 −− 14A 0 1 2 2 2

2 −−

–1 5-DMX 2-DMX 1,3,5-TMB

(b) 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 permission of American Chemical Society from Ref. 87.

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 181

and significant coding effort.107,108 In the CI formulations, general excitations are possible, but the cost increases as well. Despite these shortcomings, RASCI-SF methods have been successfully employed to describe polyradicals.95,109,110 The problem can be circumvented by an elegant approach based on the Heisenberg Hamiltonian. Mayhall and Head-Gordon have demonstrated that one can construct and parameterize the Heisenberg Hamiltonian for an arbitrary number of unpaired electrons by performing only a single spin-flip calculation from the highest multiplicity state.5,111 In the case of two-center systems,111 the exchange coupling constant 112 (JAB) is simply taken from the Landé interval rule:

− 2SJAB = E(S)−E(S−1) [20] where S = N . For multiple-center systems, in which the couplings between the cen- 2 ters may differ, a single SF calculation is followed by additional steps summarized in Figure 23. Then the Heisenberg Hamiltonian can be diagonalized, providing energies of all the Ms components. Importantly, since only single SF calculations are required, this approach scales linearly with the number of unpaired spins, in contrast to the factorial scaling of the full SF calculation, which requires flipping the spins of N electrons. 2 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 this approach, one should compare the results with the full SF calculations that directly k computes all the multiplet components. For the nickel-cubane complex from Figure 5, k 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 by the 4SF-CAS(S) method with the results from the Heisenberg Hamiltonian param- eterized by 1SF-CAS(S) calculation, illustrates the excellent performance of this approach in the present example. Mayhall and Head-Gordon also applied this approach to a molecular magnet with 18 unpaired electrons (Cr(III) horseshoe complex from Figure 5) for which full SF calculations 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 agreement 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 the excited-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 working equations that are similar to the CIS ones.113 In all these methods, the target wave functions are described by linear parameterization. Consequently, in the case of closed-shell references, excited states are naturally spin-adapted. For example, spin-conserving HOMO–LUMO excitations give rise to two Slater determinants (in one, an 훼 electron is excited, and in the other, 훽), which are combined to yield the Ms = 0 triplet and singlet states of this character. The reliability of the

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182 REVIEWS IN COMPUTATIONAL CHEMISTRY

1 z 2 3/2 3 1/2 x/y 4

1 E J J 6 1 12 13 7 2 E J 2 23 8 3 E 9 3 (a) ROHF (b) Neutral determinant (c) Local ground- (d) Heff basis state basis

Spin-flip + projection Block diagonalization Effective Hamiltonian

FIGURE 23 Procedure for extracting exchange coupling constants from single SF calculations. (a) Start with the Boys-localized ROHF orbitals for the highest multiplicity state (M = N ). (b) A single SF excitation operator generates all configurations necessary to s 2 describe the M = N − 1 manifold. From this set, only the neutral (noncharge-transfer) subset s 2 is retained. (c) After site-by-site block diagonalization, the M = N − 1 eigenstates are now s 2 linear combinations of three local quartet states, | 1 , 2 , 2 ⟩, | 2 , 1 , 2 ⟩,and| 2 , 2 , 1 ⟩. The tilted 2 3 3 3 2 3 3 3 2 k gray arrows indicate the M = 1 component of the local quartet spin states. (d) In the local k s 2 eigenstate basis, the effective Hamiltonian is now full rank and contains the exchange coupling

constants, JAB, as off-diagonal elements divided by −2SASB =−3, where SA denotes local spin on center A. Reproduced with permission of American Chemical Society from Ref. 5.

60 1SF-CAS(S) + Heisenberg

50 4SF-CAS(S) )

–1 40

(cm 30 E 20 Delta Delta 10 0 20 12 12 12 666662262022020 FIGURE 24 Comparison of the low-energy spectrum taken from direct ab initio 4SF-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.Thex-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 Slater determinant, it is well described by the ground-state methods from Figure 6, and, consequently, the excited-state methods from Figure 7 also perform well. The correlated EOM-EE-CCSD method performs very well for excited states, which are dominated by a single excitation (i.e., the magnitude of R1 should be much larger than R2). Owing to the linear parameterization, EOM-EE-CC can describe states of mixed character, for example, 휋휋∗ mixed with n휋∗ or states of Rydberg-valence character, and even states of charge-transfer character. TDDFT (when used with caution114) performs very well for valence electronic states (typical errors ∼0.2 eV); however, the description of Rydberg, charge-resonance, and charge-transfer states may be affected by SIE.115,116 How does the excited-state methodology apply to the excited states of open-shell species? The answer depends on the type of open-shell character. Let us revisit diradicals, for example, methylene from Figure 21. In the case of a modest diradical character, the lowest singlet-state wave function is dominated by a single Slater determinant and correlated methods such as CCSD may yield reasonable description. Consequently, EOM-EE-CCSD can describe low-lying states of singly excited 3 character, such as B1 (this state will appear with negative excitation energy when 1 1 doing EOM-EE-CCSD calculations using the closed-shell A1 reference) and B1. The EOM-EE-CCSD/TZ2P adiabatic gaps86 for these states are 0.54 and 1.57 eV 105 k (to be compared with the FCI values of 0.48 and 1.54 eV, see Table 2). However, k 1 the EOM-EE-CCSD excitation energy for the second excited state, 2 A1, which 1 is doubly excited with respect to the reference (1 A1), is of very poor quality: 3.84 eV with the TZP basis, to be compared with the FCI value105 of 2.67 eV. If the degeneracy between the frontier orbitals increases, then the multiconfigurational character of the lowest closed-shell singlet will increase (up to the case when both configurations have equal weights, as at the bond dissociation limit shown in Figure 2 or in methylene at linear geometries). Thus, EOM-EE-CCSD will eventually break down. A solution to this problem is given by EOM-SF described above. SF can describe all four low-lying states of diradicals from the high-spin triplet reference, which is not affected by orbital degeneracies since both orbitals are occupied. Thus, in the case of diradicals and triradicals, the manifold of low-lying excited states (primary diradical/triradical states) can be accurately described by SF methods, provided that possible spin-contamination is mitigated by using ROHF references, as discussed above. However, the states that lie outside of the 2-electrons-in-2-orbitals manifold in diradicals (or 3-electrons-in-3-orbitals manifold in triradicals) are described less accurately. One issue is that spin-complements of some of the configurations from the SF set96 (i.e., the excitations outside the 2 × 2 space) are missing their counterparts in the straightforward, nonspin-adapted formulation. This can be tackled by including selected higher excitations,96 which improves the description of these states. This is implemented in RASCI-SF and CAS-SF methods.93–95,117 Thus, one should exercise caution when performing SF calculations of excited states in diradicals and triradicals. 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 the leading configurations of the computed states. The ⟨S2⟩ values of the target states also provide a useful diagnostic. The situation is different in the case of doublet radicals. As explained, EOM-IP and 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 target ≳ state is dominated by R1 (e.g., weight 90%), then it is usually well described by the chosen EOM-CC method. Examples include various valence ionized states of neutral molecules that have predominantly Koopmans-like character (or are linear combination of Koopmans-like configurations). The excited charge-transfer type states, such as Ψ2 in Figure 8, are also well described by EOM-IP/EA. However, other types of excited states of doublet radicals are not amenable to that treatment and require other approaches. Consider, for example, the vinyl radical from Figure 1. Its ground state and the 휋 → 휎 → CC lp(C) and CC lp(C) states are well described by EOM-IP-CCSD. How- → 휋 → 휋∗ ever, its Rydberg states, lp(C) Ry, or valence states of, for example, CC CC character, cannot be described by a single EOM-IP operator; rather, these configurations are double excitations (2h1p). Can we utilize the EOM-EE-CCSD approach to compute these states? For example, can we compute the CCSD wave function 휎2 휋2 1 for the doublet reference state, CC CClp , and then perform EOM-EE-CCSD using this reference? Yes, with some caveats. The first one is spin-contamination k of the reference wave function. Just as the CCSD wave function of an open-shell k reference is spin-contaminated, so will be the target EOM-EE-CCSD states (the problem is exacerbated at the EOM level). In many cases, the problem can be mitigated by using ROHF-like references. The second issue, which is more difficult to address, is conceptually similar to the problem with the description of higher excited states by the SF approach. As always, EOM-EE is reliable only if all leading electronic configurations of the target state appear as single excitations. In the case of doublet radicals, this means that all electronic states that can be 휋∗ described by excitation of the unpaired electron to either valence (e.g., CC)or Rydberg orbitals will be reasonably well described by EOM-EE-CCSD. Similarly, excitations from low-lying doubly occupied orbitals to the singly occupied orbital can be tackled well (these give rise to the same manifold of excited states that is well described by EOM-IP-CCSD). However, states derived by excitations from the 휋 → 휋∗ doubly occupied orbitals into the unoccupied ones, such as CC CC, will cause difficulties.118–120 To understand the problem, consider the set of configurations 휋 1 1 휋∗ 1 that are needed to correctly describe the state of the ( CC) (lp) ( CC) character. Figure 25 shows all single and double EOM-EE excitations generated from the 휎 2 휋 2 1 doublet ( CC) ( CC) (lp) reference. To construct a spin-adapted wave function for the doublet or quartet (M = 1 ) state of the (휋 )1(lp)1(휋∗ )1 character, one needs s 2 CC CC to combine configurations (4), (5), and (6) from Figure 25. While (4) and (5)are singly excited determinants with respect to the doublet reference (1), determinant (6) is doubly excited. This leads to an unbalanced description of this state at the EOM-EE-CCSD level, which manifests itself in strong spin contamination. Methods based on single-excitation expansions, such as CIS, CIS(D), and TDDFT, produce

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 185

π∗ CC R 1 lp(C)

π CC R 2 (1) (2) (3) (4) (5)

(6) (7) (8) FIGURE 25 Doublet reference, (1), and target configurations for the EOM-EE-CCSD wave functions for vinyl radical. Configurations (2)–(5) are single excitations from reference (1), whereas determinants (6)–(8) are formally double excitations. Note that M = 1 open-shell s 2 휋 1 1 휋∗ 1 configurations, (4)–(6), needed to describe doublet and quartet ( CC) (lp) ( CC) states of vinyl radical, appear at different excitation levels in the EOM-EE-CCSD ansatz.

TABLE 3 Different Types of the M = 1 Electronic States in the Vinyl s 2 k Radical and Appropriate CC/EOM-CC Models k State Ansatz Reference

2|휎2 휋2 1⟩ 휎2 휋2 1 CC CClp CCSD CC CClp 휎2 휋2 0 EOM-EA CC CClp 휎2 휋2 2 EOM-IP CC CClp 2|휎2 휋1 2⟩ 2|휎1 휋2 2⟩ 휎2 휋2 2 CC CClp , CC CClp EOM-IP CC CClp 휎2 휋2 1 EOM-EE CC CClp 2|휎2 휋2 휋∗ 1⟩ 2|휎2 휋2 1⟩ 휎2 휋2 1 CC CC CC , CC CCRy EOM-EE CC CClp 휎2 휋2 0 EOM-EA CC CClp 2|휋1 1휋∗ 1⟩ 4|휋1 1휋∗ 1⟩ 휋1 1휋∗ 1 훼훼훼 CClp CC , CClp CC EOM-SF CClp CC , component

disastrous results because they lack this class of configurations.118–120 As in the case of diradicals, careful monitoring of the state characters and the ⟨S2⟩ values for the target states helps to detect the problematic behavior. We note that these states can be well described by EOM-SF using a quartet reference (see Figure 20). This strategy has been exploited, for example, in the calculations of substituted vinyl radicals121 and, more recently, in calculations of excited states of open-shell organometallic compounds.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 provide access to specific states. As one can see, sometimes different EOM-CC models need to 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-shell species. That is, all molecules discussed above are stable with respect to ejecting an electron (although some of these states can be metastable with respect to nuclear rearrangements, such as isomerization or dissociation). An important class of radicals involves electronically metastable species, often called resonances.123–126 Examples − − include transient anions that do not support a bound anionic state, such as N2 ,CO , − and H2 . Yet, such electron-attached states can be formed and have finite lifetime, sufficient to be detected experimentally. Another class of electronic resonances is exemplified by core-ionized (or core-excited) states. These states decay onafem- tosecond scale via Auger processes. How can we describe these autoionizing states? Their metastable character poses significant challenges to standard quantum chemistry methods. The difficulty stems from the fact that these states belong to the continuum and their wave functions are not L2-integrable. Attempts to compute these states using standard electronic structure methods employing Gaussian-type bases lead to problematic behavior—that is, in compact bases, such states are artificially stabilized (and look like normal bound states), but upon basis set increase, they dissolve in the continuum. Distinguishing real resonances from artificially stabilized ones is hard and determining their lifetimes and converged energies is even harder. k An elegant solution to this problem is offered by complex-variable approaches, in k which the original Hamiltonian is modified such that its bound spectrum is unchanged and 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 using complex absorbing potentials.130,131 The latter appear to be particularly promising for practical applications. In this method, the electronic Hamiltonian is modified by adding an artificial pure imaginary potential, i휂W (the parameter 휂 controls the strength of the potential). We have implemented this approach within the EOM-CCSD family of methods, which enables calculation of various types of metastable open-shell states using standard EOM-CC approaches.132–134 Figure 26 shows a calculation135 of bound and unbound anionic states using EOM-EA-CCSD. In this study,135 electron-attached states of silver and copper fluorides were investigated. Because AgF and CuF are strongly ionic, they have large dipole moments and are able to support two bound anionic states: a regular valence-attached state and a dipole-bound type.136 In AgF−, three low-lying resonances have been identified by calculations and from features observed in photoelectron angular distributions.135 Particularly interesting is an upper state in Figure 26, a dipole-stabilized 휋-shape resonance. In contrast to regular 휋-type − shape resonances (such as in N2 ), the attached electron resides mostly outside of the diatomic core, which leads to very different PESs. Whereas the resonance in − N2 is stabilized at stretched NN geometries, the dipole-stabilized resonance in AgF− does not show noticeable variation in lifetime with nuclear motions. The electron-attachment energies listed in Figure 26 illustrate robust and consistent performance of EOM-EA-CCSD for both bound and unbound states. The largest

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Energy Pi resonance 12Π+ (dipole stabilized) E = 3.88 eV (3.8 eV)

32∑+ E = 3.63 eV (3.5 eV) Sigma resonance (dipole stabilized) 22∑+ E = 2.71 eV (2.7 eV)

AgF (E = 0.0 eV) 12∑+ Dipole-bound E = –0.014 eV (–0.012 eV) state

Anion GS X2∑+ E = –1.36 eV (–1.46 eV) FIGURE 26 Electron-attached states of AgF (휇 = 5.3 debye) (see Ref. 135). Theoretical values are computed by EOM-EA-CCSD augmented by CAP. Experimental values are shown in parentheses. The ground state of the anion, AgF−, is strongly bound. Because of the strongly polar character, AgF also supports a dipole-bound state (bound by 0.014 eV). Three resonance states have been identified by the CAP-EOM-EA-CCSD calculations and experimentally.135 k The character of the target orbital hosting the attached electron reveals dipole-stabilized nature k of the two upper resonance states. (See color plate section for the color representation of this figure.)

error (0.1 eV) was observed for the lowest state of the anion and was attributed to the effect of higher excitations (triples) that are missing in EOM-CCSD.135 Other types of open-shell resonances include core-ionized states and highly excited states; they can be described by a complex-variable variant of EOM-IP or EOM-EE, respectively.

BONDING IN OPEN-SHELL SPECIES

From the very outset of this chapter, we employed an MO framework, which is a cornerstone of the theory of chemical bonding, to explain salient features of open-shell electronic structure. However, the connection between MOs and the wave functions of interacting electrons deserves some discussion. Strictly speaking, MOs + are wave functions only for one-electron systems (e.g., H2 ) or in a hypothetical case of noninteracting electrons. In such simple cases, one can unambiguously explain electronic properties by using the MO-LCAO (MOs linear combination of atomic orbitals) framework. In the many-electron case, the meaning of MOs changes. In the framework of pseudo noninteracting electrons (mean-field Hartree–Fock model), the MOs represent an approximate many-electron wave function; thus, their shapes can be

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188 REVIEWS IN COMPUTATIONAL CHEMISTRY

used to rationalize bonding in the same manner as in the case of noninteracting electrons. We note that the specific choice of orbitals is arbitrary, as the Hartree–Fock energy and state properties (such as dipole moments) are invariant with respect to any unitary transformation of orbitals within the occupied space. That is, any linear combination of occupied Hartree–Fock MOs leads to the same observable properties and to the same electron density. Yet, there is one specific choice—canonical HF orbitals—that is of special significance. The canonical orbitals are those that diago- nalize the Fock operator. By virtue of Koopmans’ theorem, these orbitals provide a direct view of the states of the ionized electrons and the respective orbital energies approximate the IEs. Thus, canonical HF orbitals do have physical significance, and, therefore, their use for analysis of the wave function and molecular properties is well justified. For example, just such an analysis is at the root of the well-known Woodward–Hoffman rules that govern electrocyclic reactions in organic chemistry. Is there a way to rigorously extend the MO framework beyond the mean-field HF model? In cases when the electronic wave function is dominated by the Hartree–Fock determinant, using HF orbitals for the analysis is reasonable. Canonical Hartree–Fock orbitals can also be used to analyze the characters of EOM states, when the target wave function is dominated by one or two leading configurations. More rigorous description can be achieved by using correlated Dyson orbitals, as explained below. Dyson orbitals can also be used for heavily multiconfigurational wave functions including strongly correlated systems. k k Dyson Orbitals Dyson orbitals are defined as an overlap between N-electron and N−1-electron states:137–141 √ 휙d N , , N−1 , , . IF(1)= N ∫ ΨI (1 … n)ΨF (2 … n)d2 … dn [21]

Thus, they represent the difference between the two wave functions. In the context of photodissociation/photodetachment experiments, Dyson orbitals connect the initial (I) and final (F) states of the system and contain all the information about the respective wave functions that is needed to compute photoionization/photodetachment cross sections. In this respect, they are analogues to transition density matrices for states with different number of electrons. Since both are related to concrete experimental observables, they can be considered observables themselves. Dyson orbitals can, in principle, be computed for any many-electron wave function using Eq. [21]. If one takes a Hartree–Fock determinant built of the canonical orbitals N N−1 for ΨI and constructs ΨF using the same orbitals (in the framework of Koopmans’ 휙d theorem), then IF(1) is just the canonical Hartree–Fock MO that was removed from Φ0. Thus, Dyson orbitals provide a generalization of Koopmans’ theorem to correlated many-electron wave functions. As such, they provide a convenient way to characterize the spatial distributions of unpaired electrons. Dyson orbitals can be

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 189

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)3 cluster computed using EOM-EA-CCSD/aug-cc-pVDZ. The two lowest IEs of the bare sodium atom are 4.96 and 2.98 eV. The orbitals reveal that the unpaired electron resides on the surface of the cluster and that its states resemble atomic s and p states perturbed by the solvent molecules. (See color plate section for the color representation of this figure.)

computed by using various EOM-CCSD wave functions140,141 including those of metastable electronic states.142 Unlike Hartree–Fock MOs, Dyson orbitals include correlation effects. Figure 27 shows Dyson orbitals for the four lowest electronic states of Na(NH3)3 computed using EOM-EA-CCSD. The orbitals resemble atomic s and p orbitals, but they 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 electrons stabilized by the micro-solvated sodium cation. k k How large is the difference between Dyson orbitals and canonical Hartree–Fock MOs? As always, the answer depends on the system. For ionization from the ground electronic states, Hartree–Fock MOs usually provide a very good approximation to the correlated Dyson orbitals. The EOM-IP amplitudes quantify how well HF MOs represent the correlated states—the weight of a given MO in the correlated Dyson orbital is roughly equal to the weight of the respective EOM-IP amplitude. As one can see from the values of the leading EOM-IP amplitude shown in Figure 18, the ionized states for thymine dimers are of Koopmans-like character and, therefore, the HF MOs are sufficient for qualitative analysis. For quantitative purposes (suchas calculations of absolute cross sections), the difference between the HF and Dyson orbitals becomes important. Similar to EOM-IP, the EOM-EA amplitudes can inform the user as to whether HF MOs provide a good approximation to Dyson orbitals. In EOM-EA calculations, the difference between the HF and correlated orbitals is usually more significant. In the case of ionization from electronically excited (or strongly correlated) 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 by density matrices, which provide a compact representation of many-electron wave functions. One-electron properties can be computed from one-electron density

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190 REVIEWS IN COMPUTATIONAL CHEMISTRY

matrix, 휌(1, 1′), alone:

휌 , ′ ≡ ∗ , , , ′, , , (1 1 ) N ∫ Ψ (1 2 … N)Ψ(1 2 … N)d2 ···dN [22] ∑ 휌 , ′ 훾 휙 휙 ′ (1 1 )= pq p(1) q(1 ) [23] pq 훾 ≡ ⟨ | + | ⟩ pq Ψ p q Ψ [24]

휙 훾 where p denotes molecular spin-orbitals and is a one-particle density matrix. For some operators, only the electron density is needed:

휌 휌 , ′ | (1)= (1 1 ) 1′=1 [25]

Thus, the bonding pattern can be determined by using the density, 휌(1).Thisis exploited in natural bond orbital (NBO) analysis,143,144 which can be performed for an arbitrary many-electron wave function. NBO allows one to partition the total electron density into the contributions corresponding to bonds, core electrons, and lone pairs. For excited states, the analyses based on transition density matrix provide a way to interpret the excited-states characters in terms of local excitations, charge-resonance, k and charge-transfer configurations.145–149 The transition density matrix connecting k two states, ΨI and ΨF, is defined as follows:

훾IF ≡< | + | > pq ΨI p q ΨF [26]

It contains information needed to compute one-electron transition properties such as transition dipole moments and NACs. It also presents a compact description of the differences in electronic structure between ΨI and ΨF. Importantly, density-matrix-based analysis145–149 of the excited states does not depend on the choice of MOs and can be applied to general wave functions. The norm of 훾IF is a useful quantity; it measures the degree of one-electron character in the → ||훾|| ΨI ΨF transition. For example, = 1 for those transitions which are of purely one-electron character.150 In the case of open-shell species, commonly asked questions are (i) what is the character of the unpaired electron(s) and (ii) to what extent are the unpaired electrons unpaired. In the case of high-spin open-shell species (doublets, triplets, and quartets), computing the difference between the 훼 and 훽 densities provides a simple way to analyze the unpaired electrons. In these states, the number of unpaired electrons is defined unambiguously, as the difference between the number of 훼 and 훽 electrons, 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. As one can see, the excess 훼 density, which is located mostly on the Cu atoms, extends toward the organic ligands.

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 191

(a) (b) FIGURE 28 Bicopper molecular magnet. (a) Molecular structure (copper atoms are shown

in gold). (b) Difference spin-density plot for the Ms = 1 triplet state computed using B5050LYP/LANL2DZ. The excess 훼 electrons (purple) can be clearly associated with copper atoms, with some degree of delocalization along the Cu-ligand bonds. (See color plate section for the color representation of this figure.)

The difference in spin-densities, however, is not directly related to experimental observables other than the Ms value. We note that Dyson orbitals, described previ- ously, do provide such a link. k In the case of low-spin states, such as singlet states of diradicals, the characteriza- k tion of unpaired electrons becomes more difficult. Inspired by the analysis of model wave functions, various ways of computing the effective number of unpaired electrons have been proposed.151 Most of the definitions are based on the one-particle density matrix, Eq. [24]. The trace of density matrix equals the number of electrons, N훼 + 훾 휆 , N훽 . Diagonalization of yields∑ natural orbitals and natural occupations, k ∈[0 1]. 휆 Because of the trace properties, k k = N훼 + N훽 . For single-determinant wave func- 휆 tions, the k 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-determinantal wave function, the natural occupation of each of the four spin-orbitals is 0.5. For systems with N훼 = N훽 = N, a spinless density matrix can be defined by summing the 훼훼 and 훽훽 blocks of one-particle density matrix, which are identical in the case of spin-pure wave functions: D ≡ 훾훼훼 + 훾훽훽 [27]

Natural occupations for D vary from 0 to 2; for a perfect 2 × 2 diradical they equal 1. Thus, the quantity 휆odd 휆 휆2 k = 2 k − k [28]

is zero for a single determinant. This observation led to the so-called Yamaguchi index,152 which can be interpreted as the effective number of unpaired electrons: ∑ 2 휆2 Nodd = 2N − Tr[D ]=2N − k [29] k

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192 REVIEWS IN COMPUTATIONAL CHEMISTRY

Unfortunately, this index shows incorrect behavior in some cases. A better-behaving quantity is the Head-Gordon index:153 ∑ 휆 , 휆 Nodd = Min[ k 2 − k] [30] k

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 electronic structure calculations, but one should remember that quantities such as atomic charges, bond orders, and particle-hole separations are not experimental observables and, therefore, are not uniquely defined. Strictly speaking, the only rigorous way to describe bonding is to characterize relevant observables, such as bond lengths, force constants, and bond dissociation energies. These quantities can be compared to reference values for systems with prototypical electronic structure and from these comparisons quantitative aspects of bonding can be derived. In our studies of bonding patterns in diradicals and triradicals, we focused on physically motivated quantities. In particular, we characterized the degree of interaction between the unpaired electrons in terms of its structural and energetic k consequences. For example, quantities such as distance between radical centers, fre- k quencies of the relevant vibrational modes, and the so-called diradical and triradical separation energies (DSE and TSE, respectively) reflect the strength of the interactions between the unpaired electrons. Figure 29 shows the results for benzynes and tridehydrobenzenes.97 In high-spin states, the distances between the radical centers are within 4% of the parent benzene structure, however, in low-spin states, contraction up to 17% is observed. DSE and TSE quantify these interactions in terms of energies, such that they can be compared with typical bond energies. Using benzynes as 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 computed from experimental thermochemical data such as heats of formation. DSE represents energy gain due to having the two radical centers within the same moiety. If the two centers 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 between the unpaired electrons. For the singlet states, the theoretical values (which are in excellent 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 diradical with nearly noninteracting unpaired electrons, whereas in ortho-benzyne the two electrons form ∼ 1 of a covalent bond. The conclusions based on structures and 3

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 193

High-spin states Low-spin states

–2

–4

–6

(%) –8 benz r /

r –10

Δ –12

–14

–16

–18 m O m Short Long O p 1,2-C H 1,3-C H 1,4-C H 1,2,3-C H 1,2,4-C H 6 4 6 4 6 4 1,3,5-C6H3 6 3 6 3 (3B , 1A ) (3B , 1A )(3B , 1A ) 4 2 (4B , 2B ) 4 ′ 2 ′ 2 1 2 1 1u 1g ( B2, A1) 2 2 ( A , A )

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 %) for the 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, k the distances between centers in ortho (o), meta (m), and para (p) positions are used. Cristian k et al.97. Reproduced with permission of American Chemical Society.

DSEs are consistent with the wave function composition (i.e., relative weights of the two leading configurations). To give another example of unpaired electrons giving rise to interesting bonding patterns, consider the CH2Cl radical and its homologues, CH3,CH2F, and CH2OH. In CH2Cl and CH2OH, the carbon’s pz orbital, which hosts the unpaired electron, forms a bonding and antibonding pair with the adjacent lone pair on Cl (O), which leads to increased bond order of the C–Cl bond. Consequently, the bond length shrinks and 154–158 the respective frequency increases, relative to saturated compounds. 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 interactions also 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 various properties is desired. In the case of open-shell species, which are often highly reactive and, therefore, difficult to prepare and study experimentally, the ability to model experimental measurements is crucially important.

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194 REVIEWS IN COMPUTATIONAL CHEMISTRY

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

Vibrational Spectroscopy Vibrational spectra—infrared or Raman—provide detailed information about molecular structure. The positions of the lines depend on the shape of the PES (its curvature around the minimum), and the intensities contain information about electronic properties, that is, changes in charge distributions upon nuclear displacements. Calculations of vibrational spectra in quantum chemistry are most often performed within the harmonic approximation; they require the construction of the second-derivative (Hessian) matrix. The most robust way to compute the Hessian is by using analytic derivative techniques. Unfortunately, analytic second derivatives are not commonly available for advanced methods and, instead, finite differences (of energies or gradients) are used, which entails ∼M single-point gradient calculations or ∼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 symmetry is lowered upon some displacements, which increases the cost of single-point calculations; (ii) staying on the same electronic state might be problematic in cases of nearly degenerate electronic states. The second problem is particularly important k in open-shell species, where there are multiple electronic states. k While harmonic spectra provide a useful starting point for analysis, they are usually not sufficient for rigorous and full assignment, due to anharmonicity effects. Anharmonicity can also strongly affect the intensities. The anharmonic effects can be accounted for by more advanced techniques.160–163 For quick semiquantitative comparison, empirical scaling factors are often used. When high accuracy is not needed, CCSD or standard DFT often provides a reasonable description of the vibrational spectra in radicals.

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

| , 휇 , |2, PIF ∝ ∫ ΨI(r R) ΨF(r R)dr dR [32]

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

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 195

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

Ψ(r, R)≈Φ(r; R)휉(R), [33]

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

| 휉 휇 휉 |2. PIF ∝ ∫ ΦI(r; R) I(R) ΦF(r; R) F(R)dr dR [34]

Introducing the electronic transition moment,

휇IF 휇 , , (R)=∫ ΦI(r; R) ΦF(r R)dr [35]

the expression for the transition probability becomes

| 휉 휇IF 휉 |2. PIF ∝ ∫ I(R) (R) F(R)dR [36]

This expression takes into account both the spatial overlap of the initial and final vibrational wave functions as well as the dependence of the transition moment on k the nuclear geometry. In the Condon approximation, the dependence of 휇IF on the k 휇IF 휇IF nuclear coordinates is neglected (i.e., (R)≈ (Req)), giving rise to the familiar Franck–Condon factors (FCFs), defined as overlaps of the vibrational wave functions of the initial and final states:

휉 휉 ⟨휉 |휉 ⟩ ∫ I(R) F(R)dR = In Fn′ [37]

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

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196 REVIEWS IN COMPUTATIONAL CHEMISTRY

Final state E

0-6 0.3 0-5

0-4

0-3

0-2 0-1 0.2 Vertical ΔE 0-0 Intensity (au) 0.1 Adiabatic ΔE

0 2.2 2.3 2.4 2.5 2.6 Initial state Energy (eV) (a) (b) FIGURE 30 Illustration of Franck–Condon progressions in electronic and photoelectron spectroscopy. (a) Initial and final electronic states and the corresponding photoelectron spectrum for a diatomic molecule (or, more generally, a single Franck-Condon active mode). The definitions of adiabatic and vertical energy differences are shown. In this case, the most intense vibronic peak corresponds to vertical ΔE. (b) Photodetachment spectrum of the phenolate anion: calculated Franck–Condon factors (gray) and experimental spectrum (black).164 The most intense band corresponds to the 00 transition, which is shifted by 0.05–0.08 eV from the vertical ΔE. The theoretical spectrum is computed using T = 300 K. Reproduced with permission of American Chemical Society from Ref. 165. k Photoelectron Spectroscopy Photoionization and photodetachment experiments are k broadly used in chemical physics.166–168 Photoionization is often used as a tool to identify transient reaction intermediates, whereas photodetachment can be employed to create molecules that are reactive intermediates or otherwise unstable in order to study their properties and spectroscopic signatures.169–171 Such experiments provide detailed information about the energy levels of the target system and, often equally important, about the underlying wave functions. There are several types of spectra produced in photoionization/photodetachment experiments.166–168 Most detailed information is provided by spectra that are resolved with respect to both the kinetic energy and the angular distribution of ejected electrons. If all photoejected electrons with a given energy are collected without resolving their direction with respect to the ionizating photon, one obtains a spectrum of the type shown 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 the photoionization efficiency (PIE) curves. They can be described as integrated photoelectron spectra (conversely, differentiation of the PIE curves produces a spectrum which can be used as a proxy for the photoelectron spectrum). For interpretation of photoionization/photodetachment measurements, the following quantities are needed:

• Energies of electronic states of the target system. These determine which states are accessible at a given photon wavelength and determine the onsets of photoelectron kinetic energies for each band. The definition of relevant energies is given 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. ⟨휙d| | el⟩ • Photoelectron dipole matrix element, Dk = u r Ψk , a quantity entering the expression of the total and differential cross sections. Here r is the dipole moment operator, u is a unit vector in the direction of the polarization of light, el 휙d Ψk is the wave function of the ejected electron, and is a Dyson orbital defined by Eq. [21] connecting the initial N-electron and the final N−1-electron states.

As follows from the selection rules for the harmonic oscillator, vibrational progressions arise due to differences in equilibrium geometries of the initial and final states (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-harmonic approximation by, for example, the ezSpectrum software.172 As input, such calculations require equilibrium geometries, normal modes, and frequencies for the initial and final states. The double-harmonic approximation with or without Duschinsky rotations provides a good description of photoelectron spectra for rigid molecules (treating free-rotor displacements requires special care). Figure 31 shows the photoelectron spectrum of the para-benzyne anion. The spectrum contains two overlapping bands corresponding to the triplet and singlet target states. From the 00 peaks of each band, the adiabatic singlet–triplet gap can be determined. The k vibrational progressions contain the information about the respective equilibrium k structures. The SF calculations combined with the double-harmonic parallel mode approximation yield excellent agreement with the experimental spectrum and allow one to extract structural information from the latter. Intensity Intensity

1.2 1.4 1.6 1.8 2.0 2.2 2.4 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Electron binding energy (eV) Electron binding energy (eV) (a) (b) FIGURE 31 Photodetachment spectrum for para-benzyne anion. (a) Singlet (black) and triplet (gray) bands of the photoelectron spectra computed using SF methods and double-harmonic approximation. (b) The experimental (black) and the calculated (gray) spectra, which include singlet and triplet states of the diradical, using the CCSD-optimized geometry 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 state that are largely curvilinear in character (especially torsional angles), as may happen in clusters, a different treatment is required.76 An example of such a case, the thymine–water cluster, is shown in Figure 32. The calculations revealed that the ionization induces large changes in the relative position of the water and thymine moieties. Consequently, double-harmonic calculations (even with inclusion of Duschinsky rotations) of the spectrum yield very poor agreement with experiment. To account for strongly coupled water–thymine displacement, the water–thymine modes need to be treated separately, by performing model low-dimensionality quantum calculations (three degrees of freedom were included in quantum calculations76). The total vibrational state can then be described as a product of the harmonic states of the two fragments and a three-dimensional wave function corresponding to

Water–thymine motion 0.3

0.958 1.198 1.201 O O 1.044 (–0.013) (–0.023) 1.012 (–0.031) 0.958 0.2 1.383(0.023) 1.486 1.391 (0.001) 1.490 O (0.001) H C (0.030) (–0.009) CH (0.048) (–0.005) 3 3 1.670 N 1.366 N (–0.226) 1.404 1.401 1.357 0.1 (–0.009) (–0.052) (–0.050) (–0.024) 1.192 1.190 (–0.035) 1.308 1.312 (–0.025) N 1.433 O N 1.446 O (–0.064) (–0.059) (–0.062) (–0.063) 0.0 1.051 1.015 (–0.023) 2.764 (–0.009) (0.893) 1.625 Thymine (–0.245) O 1.0 0.961 0.958(–0.011) (0.000) k T T T k 1 2 and 3 (a) 0.5 Intensity (au)

exp. DPIE T T 0.0 1.2 -H2O T1 2 T 1.0 3 1.0 Overall

0.8 0.8

0.6 0.6

0.4 0.4 Intensity (au) 0.2 0.2 0.0 0.0 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3 9.4 Photon energy (eV) Energy (eV) (b) (c) FIGURE 32 Photoionization of thymine–water clusters. (a) Equilibrium structures of ion- , , ized thymine monohydrate (lowest energy structures, T1 T2 and T3). Bond lengths and changes in bond lengths due to ionization are shown (in Å). (b) Computed and experimental photoelectron spectra. The latter is derived from the photoionization efficiency curves.

(c) Computed photoelectron spectrum of water–thymine T1 cluster. Calculated FCFs for the first ionized state of T1 (lower panel). Upper panel: FCFs due to water–thymine motion computed using curvilinear coordinates. Middle panel: FCFs due to thymine moiety computed using double-harmonic parallel mode approximation. Bottom: Overall photoelectron spectrum computed as a product of the thymine and water–thymine FCFs. Reproduced with permission of Royal Society of Chemistry from Ref. 76. (See color plate section for the color representation of this figure.)

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 199

water–thymine modes. This description results in good agreement between the theoretical and experimental spectra, as one can see from the bottom-left panel in Figure 32. Modeling of photoionization spectra requires more advancement treatments in systems with strong vibronic interactions, which result in the breakdown of the Franck–Condon approximation. Due to the large density of states, such cases are rather common in radicals. To model such situations, the FCFs must be computed in a multistate representation and no longer are nonzero only for modes that do not break the symmetry of the molecular framework. This can be achieved, for example, by employing vibronic Hamiltonians. An example44 of a photoelectron spectrum that is noticeably distorted by vibronic interactions is shown in Figure 33.

Total and Differential Cross Sections To model quantities such as PIE curves and angular distributions of photoelectrons, electronic cross sections are needed.173 The key quantity is the photoelectron dipole matrix element defined above. In order to compute it, one needs Dyson orbitals, which contain all the necessary

Electron kinetic energy (eV) 0.5 0.6 0.7 0.8 0.9 1.0 1200 1000 k 800 k 600 400 200 Photoelectron counts Photoelectron (a) 0 a 1000

800

600 b c 400 f g e d 200 h Photoelectron counts Photoelectron 0 3.1 3.0 2.9 2.8 2.7 2.6 2.5 (b) Electron binding energy (eV) FIGURE 33 Simulated photoelectron spectra of the 1-imidazolyl radical, superimposed on the experimental spectrum (dots). (A) Adiabatic simulations roughly similar to double-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 the Condon approximation. Inclusion of vibronic effects has noticeable effect on relative peak intensities (peaks g, h,andf ) and new (“vibronic”) peaks appear in the spectrum, resulting in better agreement with the experiment. Reproduced with permission of American Institute of Physics from Ref. 44.

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0.6 16.8 0.8 22.4 2.0 – 56 Ne H Total cross-section (Mb) Total cross-section (Mb) Formaldehyde Total cross-section (Mb)

1. 5 42 0.6 16.8 0.4 11.2

1. 0 28 0.4 11.2 0.2 5.6 0.5 14 0.2 5.6 Total cross-section (au) cross-section Total (au) cross-section Total (au) cross-section Total

0.0 0 0.0 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 30 50 70 90 110 130 10.5 11.0 11.5 12.0 12.5 Energy of ionizing radiation (eV) Energy of ionizing radiation (eV) Energy of ionizing radiation (eV) FIGURE 34 Absolute total cross sections for H−, Ne, and formaldehyde. The theoretical values 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) and Coulomb wave (Z = 1) wave functions of the free electron. Plane waves give a good description of photodetachment cross sections, while Coulomb waves give a better description for photoionization from atoms (and some small molecules). For formaldehyde, the orange line following the experimental points denotes cross section computed using a Coulomb wave with partial charge (Z = 0.25). Reproduced with permissions of the American Chemical Society from 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 free electron. Commonly used approximate treatments of the latter are either plane waves or Coulomb waves.174 The calculations of the cross sections and photoelectron angular distributions can be performed with the stand-alone ezDyson program,175 which requires information about molecular structure, Dyson orbitals, and specific experimental details. Figure 34 shows absolute cross sections for several anionic k and neutral systems. Using EOM-CC Dyson orbitals along with a plane or Coulomb k wave description of the free electron leads to good agreement with the experimental cross sections.

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

Electronic Transitions Studying chemical processes often requires going beyond computing PES. Owing to electronic degeneracy, which is common in open-shell species, multiple electronic states are often involved in the dynamics. In this context, the ability of an electronic structure method to compute multiple states and to describe their interactions is important. In this respect, multistate methods such as EOM-CC are very attractive. In addition to computing multiple states, the electronic couplings between them can be easily computed, which is important for dynamics, since these couplings govern electronic transitions between the states. Nonradiative transitions between different electronic states fall into two cat- egories. The first category results from the breakdown of Born–Oppenheimer separation of nuclear and electronic motions; they are governed by NACs (derivative

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THE QUANTUM CHEMISTRY OF OPEN-SHELL SPECIES 201

couplings). The second category occurs due to relativistic effects, which are most often neglected in standard quantum chemistry; these transitions are governed by SOCs. The rates of transitions between different electronic states can be estimated using Fermi’s Golden Rule. In this framework, derived using perturbation theory, the rate of a transition between the initial I and final F electronic states of a system is given by ∑ 휋⟨ |⟨ 휉 | ̂ | 휉 ⟩|2훿 ⟩ kI→F = 2 ΦI I′ A ΦF F′ (EII′ − EFF′ ) T [38] F′ 휉 휉 where ΦI I′ and ΦF F′ denote the initial and final states, the sum runs over the final vibrational states, and the thermal averaging is performed over the initial vibrational states. EII′ and EFF′ denote total (electronic and nuclear) energies of the initial and final states. Â is the coupling operator, for example, NAC for radiationless transitions (internal conversion) and SOC for spin-forbidden ones (intersystem crossing). Depending on the physics of the problem, different approximations to this expression are possible.177–179 There are two alternative strategies for computing properties and transition probabilities for approximate wave functions.24 In the response-theory formulation, one applies time-dependent PT to an approximate state (e.g., the CCSD wave function); in this approach, transition moments are defined as the residues of respective response functions (and excitation energies as poles). Identical expressions for properties can k k be derived for static perturbations using an analytic derivative formalism.18,180–185 In the expectation-value approach, one begins with expressions derived for exact states and then uses approximate wave functions to evaluate the respective matrix elements. 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 compute 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 that include amplitude (and, optionally, orbital) relaxation terms; this is how the properties are computed within the linear-response formulation of CC theory. Numerically, the two approaches are rather similar,24,183,187 although the expectation-value formulation of properties within EOM-CC is not size intensive188 (the deviations, however, are relatively small189). The response formulation can become singular in the cases of degenerate states,190 whereas the expectation-value approach does not have such a problem. For the purpose of this chapter, the differences between the two approaches are insignificant. Following the author’s personal preference, the following discussion is based on the expectation-value approach. The non-Hermitian nature of EOM-CC theory requires additional steps when computing interstate properties because the left and right eigenstates of H are not the same, but rather form a biorthogonal set. Thus, in EOM-CC, the matrix element of an 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 right vectors are not uniquely defined).

Spin-Orbit Couplings Within the expectation-value approach, SOCs can be computed as the matrix elements of the respective part of the Breit-Pauli Hamiltonian173,191 (spin-same-orbit and spin-other-orbit terms) using zero-order nonrelativistic wave functions. The SO part of the Breit-Pauli Hamiltonian contains one- and two-electron operators: [ ] ∑ ∑ ℏe2 Hso = hso(i) ⋅ s(i)− hsoo(i, j) ⋅ (s(i)+2s(j)) [40] 2 2 2mec i i≠j ∑ Z (r − R )×p hso(i)= K i K i [41] | |3 K ri − RK

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

where ri and pi are the coordinate and momentum operators of the ith electron and RK and ZK denote the coordinates and the charge of the Kth nucleus. Thus, evaluation of the respective matrix elements requires one- and two-particle k reduced density matrices. The two-electron part is significant—it can account for k about 50% of the SOC in molecules composed of light atoms (for heavy atoms, the relative magnitude of the two-electron part is reduced). Fortunately, the dominant contributions from the two-electron part can be computed using the separable part of the two-particle density matrix (which consists of the product of the one-particle density matrix for the reference state and the one-particle transition density matrix connecting 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 density matrix with the two-particle spin-orbit integrals, they have significantly reduced computational and storage requirements. The differences between the full SOC and the mean-field approximation are consistently less than− 1cm 1 for all examples studied to 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 ms), of the states involved:

⟨ so⟩ ⟨ so⟩ ⟨ so⟩ ⟨ so⟩ H = Hx + Hy + Hz [43] ⟨ so⟩ ≡ ⟨ , | so| ′ ′, ′ ⟩ 훼 , , H훼 Ψ(s ms) H훼 Ψ (s ms) = x y z [44]

Following these equations, ⟨Hso⟩ is a complex number given by the sum of the matrix elements 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 individual 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, the quantity called the SOC constant194,197 (SOCC) is invariant:

∑s ∑s′ | , ′ |2 ≡ |⟨ , | so| ′ ′, ′ ⟩|2 SOCC(s s ) [ Ψ(s ms) Hx Ψ (s ms) m =−s ′ ′ s ms=−s |⟨ , | so| ′ ′, ′ ⟩|2 |⟨ , | so| ′ ′, ′ ⟩|2 + Ψ(s ms) Hy Ψ (s ms) + Ψ(s ms) Hz Ψ (s ms) ] [45]

It is the SOCC rather than individual components that is needed in the context of Fermi’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. It was found that when used with an appropriate basis set, the errors against experiment are below 5% for light elements. The errors for heavier elements (elements up to Br and Se were considered) are larger (up to 10–15%). This study196 also analyzed the impact of different correlation treatments in zero-order wave functions and found that EOM-IP-CCSD, EOM-EA-CCSD, EOM-EE-CCSD, and EOM-SF-CCSD wave functions yield SOCs, which agree well with each other (and with the experimental values when available). Not surprising, using an EOM approach that provides a more balanced description of the target states10 yields more accurate results. k k Nonadiabatic (Derivative) Couplings The NAC calculation for the exact wave functions (or, more precisely, when Hellman–Feynman conditions are satisfied) requires only an unrelaxed one-particle transition density matrix: ∑ 1 ⟨ | x| ⟩ 1 x 훾IF ΨI H ΨF = Hpq pq [46] EI − EF ΔIF pq

훾IF where pq 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 electronic Hamiltonian 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 wave function, leading to the Pulay terms that needs to be added to the expectation-value expression.24 Such theoretically complete formalism and its computer implementation for the MR-CI wave functions were first developed by Yarkony.198–202 More recently, this formalism has been applied to the CIS and TDDFT methods;203–205 analytic derivative matrix elements have also been computed for EOM-CC wave functions.43,206,207 Using the expectation-value approach, in which orbital-response terms are neglected, has an important benefit in the context of NAC calculations. When using incomplete atomic orbital bases, NACs computed using full derivative formalism are 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 the translational invariance is exactly equal to the Pulay term. Thus, an “approximate” Hellman–Feynman expression for the NAC is, in fact, identical to the full NAC expression corrected to achieve translational invariance. This strongly argues in favor of adopting the expectation-value approach for computing NACs for the EOM-CC wave functions. Finally, it is worth mentioning that ||훾||, the norm of the transition one-particle density matrix, can be used as a crude proxy for NAC,150 owing to the Cauchy–Schwartz inequality: ∑ 1 ⟨ | x| ⟩ 1 x 훾IF ≤ 1 || x|| ⋅ ||훾IF|| ΨI H ΨF = Hpq pq H [47] EI − EF ΔIF pq ΔIF

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

Charge-Transfer Systems and Marcus Rates Marcus theory provides a recipe for k k calculating the rates of charge transfer between donor and acceptor states:211,212 { } 2휋 1 (ΔG + 휆)2 k = |H |2 √ exp − [48] ℏ IF 휋휆 4휆k T 4 kBT B

휆 where ΔG, , and HIF are the free energy change, reorganization energy, and the coupling between the electronic states involved in the transfer (I and F denote the initial and final states, respectively). The key quantity here is the electronic coupling element between the diabatic donor and acceptor states, HIF. Several alternative approaches based on a simplified two-state model exist 213 for computing HIF. Marcus theory is formulated for interacting diabatic charge-localized states. In this representation, the electronic Hamiltonian is non- diagonal. Diagonalizing it produces adiabatic states (these are the states that are produced by regular ab initio calculations): ( ) ( ) E H E 0 I IF → 1 [49] HIF EF 0 E2

with the eigenvalues √ ( ) 2 EI + EF EI − EF 2 E , = ± + H [50] 1 2 2 2 IF

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+ At the transition state (such as symmetric configuration of the (H2)2 ), EI = EF leading to a simple connection between the energy gap between the adiabatic states and the coupling: |E − E | |H | = 1 2 [51] IF 2 Equation [51] can be used for computing the couplings only for symmetric donor–acceptor pairs and only at transition states. One can compute the coupling element directly, if the localized-charge states are available (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 and ΨF. When these states are nonorthogonal, the eigenproblem [49] needs to be appro- priately modified, leading to the following expression for the electronic 213coupling:

H −(E + E )S ∕2 V = IF I F IF [54] IF 2 1 − SIF k One can also obtain the transformation from adiabatic to diabatic charge-transfer k states by diagonalizing a property matrix. For example, it is reasonable to expect that for the localized charge-transfer states, the matrix of the dipole moment operator is diagonal. Thus, the desired diabatic states can be found by diagonalizing the matrix of the dipole moment operator computed in the basis of adiabatic states. This is the essence of the Mulliken–Hush scheme.214–217 The resulting expression for the coupling is 휇 12ΔE12 HIF = √ [55] 휇 휇 2 휇2 ( 1 − 2) + 4 12

where indices 1 and 2 denote the respective adiabatic states. Thus, the Mulliken–Hush approach requires the ability to compute state and transition dipole moments for given adiabatic wave functions. Provided these quantities are available, one can compute HIF for any type of adiabatic wave functions. Figure 35 compares the values of HIF computed for a model system using the energy-gap scheme, Eq. [51], with various wave functions. Couplings for hole and electron transfer were computed for the ethylene dimer cation and anion, respectively. The crudest model, HF-KT, used Hartree–Fock orbital energies in lieu of the target-state energies of the ionized or electron-attached states. Correlation effects were included by performing CCSD calculations. A more balanced description was achieved by using EOM-CC energies. Figure 36 shows various EOM-CC schemes for computing energies of the target states. Finally, direct coupling (DC) calculation was performed by computing the overlap between the localized fragment Hartree–Fock

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0.03

H H 0.025 CC H H 0.02 HF-KT 3.94 Å SF-CCSD CH3 0.015 IP(EA)-EOM ΔCCSD

9.41 Å DC CH3 Coupling (eV) 0.01

H H 0.005 C C H H 0 HT ET 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 Society from Ref. 218. (See color plate section for the color representation of this figure.)

Spin-Flip IP-EOM

∗ π + ∗ π –

π+ π k – k Molecular orbital energy Molecular orbital

Reference Target 1 Target 2 Reference (a) (b) (c) (d) (A)

Spin-Flip EA-EOM

∗ π + ∗ π –

π+

π– Molecular orbital energy Molecular orbital

Reference Target 1 Target 2 Reference (a) (b) (c) (d) (B) FIGURE 36 Schemes for computing the couplings using EOM-SF, EOM-IP, and EOM-EA wave functions, with 휋 orbitals depicted for two ethylenes as an example. Shown in panel (A) is the 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 (IP and 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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30 2.0 EOM-IP-CCSD 38.75 ) EOM-IP-CCSD ) 1.5 ) –1 –1 1.0 –1 38.50 EOM-IP-CCSD 0 0.5 CCSD/EOM-EE-CCSD 0.0 38.25 –56 –30 CCSD/EOM-EE-CCSD –58 32.00 Coupling (cm Coupling (cm Coupling (cm CCSD/EOM-EE-CCSD –60 –60 31.75 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00 (a)Reaction coordinate (b)Reaction coordinate (c) Reaction coordinate 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 of Physics.

wave functions. As one can see, correlation is clearly important. When using the energy-gap scheme, all CCSD methods give similar results. The reasonable performance of open-shell CCSD can be explained by the fact that these calculations are based on symmetric structures at which the problems due to unbalanced description of the important configurations do not occur, that is, the leading configurations ofthe two charge-transfer states (i.e., Ψ1 and Ψ2 from Figure 8) are of different symmetry k and cannot mix. k The advantage of the Mulliken–Hush scheme is that it can be applied to all structures including the nonsymmetric ones. For example, one can compute Mulliken–Hush couplings along a charge-transfer reaction coordinate. Examples + + of such calculations for (H2)2 and (Be-H2) are shown in Figure 37. In these calculations, the couplings are computed using two different approaches: one employing EOM-IP-CCSD wave functions and open-shell CCSD/EOM-EE-CCSD ones. Absolute errors against FCI of the CCSD and EOM-EE-CCSD calculations are comparable with those of EOM-IP-CCSD; however, the latter behaves in a much more uniform way.

OUTLOOK

The electronic structure of open-shell species is rich and complex; it leads to unique chemical and spectroscopic properties. However, it also brings methodological challenges. Often, open-shell systems feature close-lying multiple electronic states. These electronic near-degeneracies result in multiconfigurational wave functions that cause the breakdown of the standard hierarchy of single-reference methods and Kohn–Sham DFT. A skilled computational chemist should be able to identify these situations and employ more appropriate treatments, such as a particular EOM-CC approach that suits the task at hand. Some types of open-shell structure can be tackled by standard approaches; however, complications due to spin-contamination may occur. We hope that 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 to applications, such as modeling spectroscopy and nonadiabatic processes in open-shell species. 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 introduction to practical quantum chemistry, we suggest the textbook Essentials of Com- putational Chemistry: Theories and Models.7 For readers seeking more-advanced understanding of methodology, we recommend the book by Szabo and Ostlund.9 A very 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 to correlated methods, with an emphasis on CC approaches, is given in Refs. 16 and 219. To learn more about EOM-CC approaches, we recommend Refs. 10 and 11. An alternative strategy based on ADC is reviewed in Ref. 27. More-specific reviews include an overview of the SF approach,85 excited-state methods targeting large molecules,114 Rydberg-valence interactions in radicals,220 and the quantum chemistry of loosely bound electrons.221 Different aspects relevant to electronic structure of radicals are discussed in Ref. 48. This review would not be complete without a brief discussion of software. Many excellent quantum chemistry packages exist offering both standard and highly specialized features. The author is familiar with the functionality of (and has utilized, k on various occasions) CFOUR,222 ACES3,223 Turbomole,224 Dalton,225 Psi4,226 k GAMESS,227 Molpro,228 Molcas,229 Spartan,230 and Q-Chem.231 The choice of package is a matter of personal preferences. The author has been a long-term user of and contributor to Q-Chem,231,232 a versatile electronic structure suite of tools, offering efficient implementations of methods targeting open-shell species described in this chapter. For a detailed list of features, readers are referred to Q-Chem User Manual and to the series of webinars describing practical aspects of calculations.233 Useful auxiliary packages include the graphic visualization tool, IQmol,234 and spectroscopy modeling tools such as ezSpectrum172 and ezDyson.175

ACKNOWLEDGMENTS

The work on open-shell species, which is a main focus of my research, has been funded by the National Science Foundation, Department of Energy, and Army Research Office. The methodological developments have greatly benefited from the support by Q-Chem, Inc. The author is grateful to Prof. John F. Stanton for his insightful 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. My special thanks to Mr. Jay Tanzman for his meticulous proofreading and for making 459 corrections, which greatly improved the readability of the manuscript.

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APPENDIX: LIST OF ACRONYMS

ADC: algebraic-diagrammatic construction AO: atomic orbital CAS: complete active space CASSCF: complete active-space self-consistent field COL: collinear CC: coupled-cluster CCD: coupled-cluster with doubles CCSD: coupled-cluster with singles and doubles CIS: Configuration interaction singles CISD: Configuration interaction with singles and doubles DFT: density functional theory DSE: diradical separation energy EOM: equation-of-motion EOM-EA-CCSD: equation-of-motion coupled-cluster with single and double substitutions for electron attachment EOM-EE-CCSD: equation-of-motion coupled-cluster with single and double substitutions for excitation energies k EOM-IP-CCSD: equation-of-motion coupled-cluster with single and double substi- k tutions for ionization potentials EOM-SF-CCSD: equation-of-motion coupled-cluster with single and double substitutions with spin-flip FCI: full configuration interaction FCF: Franck–Condon factor HF: Hartree–Fock IE: ionization energy JT: Jahn–Teller MO: molecular orbital NAC: nonadiabatic coupling NC: noncollinear HOMO: highest occupied molecular orbital LR-CCSD: linear-response coupled-cluster with single and double substitutions LUMO: lowest unoccupied molecular orbital MO-LCAO: molecular orbital-linear combination of atomic orbitals MP2: Møller–Plesset theory of the second order MR-CI: multireference configuration interaction NBO: natural bond orbitals PES: 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

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