Characterizing metastable states beyond energies and lifetimes: Dyson orbitals andtransition dipole momentsThomas-C. Jagau and Anna I. Krylov Citation: The Journal of Chemical Physics 144, 054113 (2016); doi: 10.1063/1.4940797 View online: http://dx.doi.org/10.1063/1.4940797 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/144/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ab initio study of the RbSr electronic structure: Potential energy curves, transition dipole moments, andpermanent electric dipole moments J. Chem. Phys. 141, 234309 (2014); 10.1063/1.4903791 Accurate potential energy, dipole moment curves, and lifetimes of vibrational states of heteronuclear alkalidimers J. Chem. Phys. 140, 184315 (2014); 10.1063/1.4875038 Electronic transition dipole moments and dipole oscillator strengths within Fock-space multi-referencecoupled cluster framework: An efficient and novel approach J. Chem. Phys. 138, 094108 (2013); 10.1063/1.4793277 Wannier-ridge states of He and H − with symmetries P 3 e and D 1 o and the validity of classificationschemata J. Chem. Phys. 132, 154111 (2010); 10.1063/1.3385316 The HOOH UV spectrum: Importance of the transition dipole moment and torsional motion from semiclassicalcalculations on an ab initio potential energy surface J. Chem. Phys. 132, 084304 (2010); 10.1063/1.3317438

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THE JOURNAL OF CHEMICAL PHYSICS 144, 054113 (2016)

Characterizing metastable states beyond energies and lifetimes:Dyson orbitals and transition dipole moments

Thomas-C. Jagau and Anna I. KrylovDepartment of Chemistry, University of Southern California, Los Angeles, California 90089, USA

(Received 30 October 2015; accepted 6 January 2016; published online 5 February 2016)

The theoretical description of electronic resonances is extended beyond calculations of energies andlifetimes. We present the formalism for calculating Dyson orbitals and transition dipole momentswithin the equation-of-motion coupled-cluster singles and doubles method for electron-attachedstates augmented by a complex absorbing potential (CAP-EOM-EA-CCSD). The capabilities of thenew methodology are illustrated by calculations of Dyson orbitals of various transient anions. Wealso present calculations of transition dipole moments between transient and stable anionic states aswell as between different transient states. Dyson orbitals characterize the differences between theinitial neutral and final electron-attached states without invoking the mean-field approximation. Byextending the molecular-orbital description to correlated many-electron wave functions, they deliverqualitative insights into the character of resonance states. Dyson orbitals and transition moments arealso needed for calculating experimental observables such as spectra and cross sections. Physicallymeaningful results for those quantities are obtained only in the framework of non-Hermitian quantummechanics, e.g., in the presence of a complex absorbing potential (CAP), when studying resonances.We investigate the dependence of Dyson orbitals and transition moments on the CAP strength andillustrate how Dyson orbitals help understand the properties of metastable species and how they areaffected by replacing the usual scalar product by the so-called c-product. C 2016 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4940797]

I. INTRODUCTION

Depending on the sign of the electron affinity,1,2 elec-tron attachment to neutral molecules yields stable ortransient anions. The latter are metastable electron-moleculecompound states with finite lifetime that can decay throughautodetachment. Although their survival time often is as shortas a few femtoseconds, transient anions play an importantrole as doorway states opening new reaction channels that areinaccessible in neutral species.

The attached electron can act as a catalyst by modifyingthe bonding pattern. The catalytic process can proceedvia either bound3,4 or transient anions.5,6 For example, indissociative electron attachment,7–9 an initially populatedtransient anionic state undergoes internal conversion toanother anionic state in which bond dissociation becomespossible. In this way, barriers that are insurmountable on thepotential energy surface of the neutral species can be overcomein the presence of low-energy electrons, leading to unexpectedproducts. Metastable anionic states are also often encounteredin photodetachment experiments on stable anions. Populationof these higher-lying states followed by autodetachment cancompete with direct detachment into the continuum, leavingdistinct spectroscopic fingerprints.10–13

The analysis of the underlying anion wave functions interms of molecular orbital (MO) theory is important for aqualitative understanding of such processes. For example, theshape of the MO hosting the attached electron determines thechanges in bonding pattern and trends in photoelectron angulardistributions. Dyson orbitals14–19 for electron-attached states

are defined as overlaps between neutral and anionic wavefunctions thus extending MO concepts to correlated many-electron wave functions. Dyson orbitals and transition dipolemoments between different anionic states are also needed forthe quantitative modeling of experimental observables suchas total and differential photodetachment cross sections.18,20,21

These quantities as well as accurate detachment energiescan be routinely computed for bound anionic states usingthe equation-of-motion coupled-cluster singles and doublesmethod for electron-attached states (EOM-EA-CCSD).22

To be able to model processes involving transient anionsincluding the rates of competing pathways and spectroscopicobservables, robust electronic structure methods developedfor bound states need to be extended to resonances. The targetquantities are energies, lifetimes, transition dipole moments,and Dyson orbitals.

The theoretical description of transient anions ischallenging as they are embedded in the detachmentcontinuum.23,24 In Hermitian quantum mechanics, they canonly be represented as wave packets and not as discreteeigenstates of the Hamiltonian. However, the problem can bereformulated such that resonances can be described as discreteeigenstates with complex energies. This can be achieved byimposing outgoing boundary conditions for the resonancewave function25 or by treating the resonance as a boundstate diabatically coupled to the continuum.26,27 Both of thesemethodologies lead to a non-Hermitian Hamiltonian.

Several approaches have been proposed for the compu-tation of resonances. The list comprises artificial stabi-lizing potentials,28–31 stabilization methods,32,33 complex

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scaling,34–38 exterior complex scaling,39,40 complex basisfunctions,41,42 and complex absorbing potentials (CAPs).43–48

Here, we employ the CAP approach, where an imaginarypotential is added to the Hamiltonian in order to describe theresonance as an L2-integrable discrete state. CAP methods areconceptually simple and easy to apply within the quantumchemistry framework. Yet, they have a solid theoreticalfoundation as they can be related to the rigorous theoryof exterior complex scaling.45,46,49–51

CAPs have been combined with several electronic-structure methods including density-functional theory,52

electron-propagator theory,53,54 multireference configurationinteraction,55,56 Fock-space coupled-cluster theory,57,58 andEOM-CCSD.59–63 Our recent implementation of CAPs withinthe EOM-CCSD family of methods61–63 enabled severalapplications that have demonstrated that this approachdescribes transient anions of small and medium-sizedmolecules with an accuracy allowing for quantitativecomparisons with experimental data.64,65 Importantly, inCAP-EOM-CCSD, bound and metastable anionic statesare obtained as biorthonormal eigenfunctions of the samemodel Hamiltonian, which ensures their balanced description.Consequently, the conversion of resonances into bound statesis described correctly, which is important for the computationof complex potential-energy surfaces.66 Another attractivefeature of the CAP-EOM-CCSD approach is that it doesnot involve solving the Hartree-Fock (HF) equations for thetransient anion and thus is not subject to the convergenceproblems often encountered in other methods.

In this article, we use CAP-EOM-CCSD wave functionsto compute Dyson orbitals and transition dipole momentsinvolving transient anionic states. This is a key step towardsa quantitative modeling of processes involving metastablestates. We note that electronic resonances rarely havebeen characterized beyond energy and lifetime. For somespecies, contour plots of HF orbitals42,64,65,67 and attachmentdensities68 have been reported. Our article presents the firstimplementation and calculation of transition moments andDyson orbitals for metastable states.

The article is structured as follows. Section II describesthe theory for the evaluation of Dyson orbitals and transitiondipole moments within the CAP-EOM-CCSD framework withan emphasis on the differences relative to standard EOM-CCSD theory for bound states. In Section III, we presentillustrative calculations for transient anions of different types.We consider several diatomics as well as benzoquinone,benzene, and N-methylformamide. Our concluding remarksare given in Section IV.

II. THEORY

In CAP methods, the position ER (i.e., energy) andwidth Γ (i.e., inverse lifetime) of a resonance are obtainedfrom the discrete complex eigenvalues E = ER − iΓ/2 of thenon-Hermitian Hamiltonian

H(η) = H0 − iηW , (1)

where H0 stands for the usual molecular Hamiltonian and Wfor the CAP, which is characterized by its strength η andonsets r0

x, r0y, r0

z,

W = Wx +Wy +Wz ,

Wα =

0 if |rα | < r0α

(rα − r0α)2 if |rα | > r0

α

, α = x, y, z.(2)

For a broad class of CAPs, the exact resonance position andwidth are obtained in the limit η → 0 for the exact solution ofthe Schrödinger equation in a complete basis.44 In a practicalcalculation, in which a finite basis set is employed and theSchrödinger equation is solved approximately, a finite optimalvalue for η has to be determined. η should be sufficiently largeto stabilize the resonance; however, using finite η perturbs theresonance energy. The strategy for finding ηopt is based on aperturbative analysis of the energy, which yields

η · dE/dη = iη⟨Ψ|W |Ψ⟩ (3)

for the perturbation of the energy in first order.44 Thus,the standard condition for finding ηopt as min [η · dE/dη] isequivalent to minimizing the perturbation due to the CAP,η⟨Ψ|W |Ψ⟩. We recently showed61 that better results, whichare also less dependent on the CAP onset, are obtained byconsidering the first-order corrected energy

U = E − η · dE/dη (4)

and determining the optimal η by minimizing the second-orderterm η · dU/dη. Moreover, since the energy is complex, theoptimal η values may be different for the real and imaginaryparts. Thus, we evaluate ER and Γ at different CAP strengths.An energy-only scheme achieving the same goal has beenproposed by Moiseyev and coworkers who employed Padéapproximants of the resonance energy to remove the CAPperturbation.69

The CAP-EOM-EA-CCSD ansatz for the electronic wavefunction of a transient anion is

|Ψanion) = REA|Ψref) = REAeT |ΦHF) , (5)

where |Ψref) and |ΦHF) stand for the CAP-CCSD and CAP-HF wave functions of the parent stable neutral state. Thecorresponding bra states are

(Ψanion| = (ΦHF|e−TLEA† , (6)

(Ψref| = (ΦHF|e−T(1 + Λ†) . (7)

The operators T and Λ are particle-conserving excitationoperators, whereas REA and LEA are electron-attachingexcitation operators, i.e., of 1p and 2p1h (1-particle and2-particle-1-hole) type in EOM-EA-CCSD. The amplitudesof the operators T , Λ, REA, and LEA are complex; theyare determined by solving the usual CC, Λ, and EOM-CC equations.22,70 The form of the equations need not bemodified if the CAP is added at the HF stage, as is the casein our implementation. With a different choice of R and L,Eqs. (5)–(7) can describe metastable excited and ionizedstates. 1h1p and 2p2h give rise to CAP-EOM-EE-CCSD,whereas 1h and 2h1p define the CAP-EOM-IP-CCSD model.

In all equations, it is necessary to use the c-product insteadof the proper scalar product because H(η) is not Hermitianbut complex-symmetric.24,71 Consequently, here and in thefollowing A† denotes the transpose of operator A and not itsadjoint. We indicate this change of metric by using parenthesesinstead of chevrons. The c-product of two wave functions is

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defined as

(Ψi |Ψj) =

d1d2 . . . dn

Re(Ψi)Re(Ψj) − Im(Ψi)Im(Ψj)

+ i Re(Ψi)Im(Ψj) + i Im(Ψi)Re(Ψj)

(8)

and is, in general, complex. In contrast to the usual norm, thec-norm is not invariant to multiplication of the wave functionby an arbitrary complex number of the form eiα. Moreover,Eq. (8) implies that the real and the imaginary parts of ac-normalized wave function are orthogonal to each other,

1 = (Ψ|Ψ) =

d1d2 . . . dnRe(Ψ)Re(Ψ) − Im(Ψ)Im(Ψ)

=1

+ 2i

d1d2 . . . dn Re(Ψ)Im(Ψ) =0

, (9)

which also differs from the Hermitian case, where the relationbetween the real and imaginary parts is not fixed. We note thatWhitenack and Wasserman emphasized a similar behavior ofthe real and imaginary parts of the electron density in density-functional theory calculations of resonances.72,73 On the otherhand, the expectation value of an operator X , (Ψ|X |Ψ)/(Ψ|Ψ),is still invariant with respect to multiplying the resonancewave function by a phase factor. It was also pointed out that,once the resonance wave function is available, the resonanceposition and width can be obtained as expectation values ofH0 and W with respect to the Hermitian scalar product.44,74,75

The orthogonality relationship for the many-body wavefunction, Eq. (9), carries over to one-electron quantities suchas the CAP-HF orbitals and, in particular, the Dyson orbitalsφd. The latter are defined for an anionic state—be it stable ormetastable—with respect to a neutral reference state as

φd(n + 1) = √n + 1

d1d2 . . . dn Ψanion(1, . . . ,n + 1)×Ψref(1, . . . ,n) (10)

and are analogues to transition densities for states withdifferent number of electrons. A Dyson orbital can beexpanded in terms of the HF orbitals of the reference state as

φd =p

γp φp . (11)

For truncated CAP-EOM-CC schemes, two sets of expansioncoefficients γR and γL are obtained as17

γLp = (ΦHF|(1 + Λ†)e−T ap eTREA|ΦHF), (12)

γRp = (ΦHF|LEA†e−T a†peT |ΦHF), (13)

with a†p and ap as the usual creation and annihilation operatorsbecause the similarity-transformed Hamiltonian e−THeT hasdifferent left and right eigenvectors. The differences betweenγL and γR are small as long as the involved states are wellrepresented within the CCSD/EOM-CCSD approximations.Dyson orbitals defined by Eq. (11) are not normalized, andtheir norm is a measure for the one-electron character ofthe attachment process. When starting from a HF descriptionof the neutral state and applying Koopmans’ theorem to theattached state, the Dyson orbitals reduce to the canonical HF

orbitals, and their norm is one. Due to the biorthogonal natureof EOM-CC, the norms of the left and right Dyson orbitalsare arbitrary, and it is only the geometric mean of the two thatcan be related to observables,17,18 similar to other interstateproperties.76 For visualization purposes, we use normalizedDyson orbitals.

Within the expectation-value formulation of properties,77

the transition dipole moment between two electronic states Aand B is defined as

µAB = (ΨA|µ|ΨB),µBA = (ΨB|µ|ΨA), (14)

µ ≡ √µAB · µBA,

with µ as the dipole operator. Within the c-product metric,µ is in general complex. When evaluating transition dipolemoments between two anionic states described by truncatedCAP-EOM-EA-CC wave functions, two different expressionsfor µAB and µBA are obtained as76

µAB = (ΨHF|LEA†A

e−T µ eTREAB |ΨHF) , (15)

µBA = (ΨHF|LEA†B e−T µ eTREA

A |ΨHF) . (16)

These expressions are identical to the CAP-free case apartfrom the different metric. Again, the quantity that isunambiguously defined is µAB · µBA.

We have implemented the computation of CAP-EOM-CCSD Dyson orbitals and transition dipole moments in the Q-Chem program package.78 The implementation builds on thegeneral implementation of CAP-EOM-CCSD in Q-Chem62,63

and uses the libtensor library for high-performance tensoroperations.79 In addition to the evaluation of Eqs. (12), (13),(15), and (16), our implementation includes the computationof analogous expressions involving CAP-EOM-IP-CCSD andCAP-EOM-EE-CCSD states.

III. APPLICATIONS

To illustrate the utility of Dyson orbitals for the analysisof resonance wave functions, we consider shape resonancesin polar (CO−, CuF−) and non-polar (N−2 , H−2) diatomics, aswell as larger molecules (anions of benzoquinone, benzene,and N-methylformamide). We then proceed to the analysisof transition dipole moments. Computational details such asbasis sets, structures, and CAP parameters are provided in thesupplementary material.80

A. Dyson orbitals

As a first application, we discuss a shape resonance in anon-polar molecule, N−2 . We begin with CAP-free EOM-EA-CCSD calculations and consider the six lowest-lying solutionsof 2Πg symmetry in N−2 at the equilibrium distance of neutralN2 (1.1 Å). The attachment energies of these solutions are0.18 eV, 0.53 eV, 1.20 eV, 2.23 eV, 2.97 eV, and 5.22 eV. Thepositive values indicate that the attached states lie above theparent neutral state and are, therefore, unbound.

Figure 1 shows Dyson orbitals of these electron-attachedstates. Here, and in all following figures, we show thec-normalized “left” Dyson orbital computed according to

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FIG. 1. Dyson orbitals for the six lowest-lying solutions of 2Πg symmetry in N2+e− computed by EOM-EA-CCSD/aug-cc-pVTZ+6s6p6d(C) at R(NN) = 1.1

Å and ordered by increasing energy from left to right. On the far right, the Dyson orbital of the bound 2Πg state of N−2 at R(NN) = 1.6 Å is shown. The orbitalsare plotted at an isovalue of 0.004.

Eq. (12). In addition, Figure 1 shows the Dyson orbital of thebound 2Πg state of N−2 at an NN distance of 1.6 Å (attachmentenergy = −0.92 eV). This orbital looks like a regular π∗

orbital in contrast to the Dyson orbitals computed at R(NN)= 1.1 Å, which have very different shapes. This is becausethere are no bound anionic states at R(NN) = 1.1 Å: Thespectrum of the Hamiltonian is continuous when representedin a complete basis. The solutions of the EOM-EA-CCSDequations computed here are pseudocontinuum states, wherethe excess electron is not bound to the N2 molecule. Asexpected, the number of nodal planes grows with increasingenergy and the excess electron density is located farther awayfrom the molecule for solutions with lower energy. We pointout that the energies and Dyson orbital shapes are determinedby the discretization brought about by the finite basis set. Inparticular, the energy and Dyson orbital of the lowest solutionare governed by the most diffuse function in the basis set.In the complete basis set limit, the attachment energy willapproach zero and the Dyson orbital will become a planewave. In sum, Figure 1 demonstrates that a resonance cannotbe associated with a single eigenstate of the Hamiltonian inHermitian quantum mechanics. However, it has been shownthat resonance positions and widths can be extracted fromsuch calculations by means of stabilization methods32,33 orStieltjes moment theory.81,82

The changes brought about by turning on the CAP areillustrated in Figure 2 where we show the non-Hermitian

CAP-EOM-EA-CCSD Dyson orbital of the 2Πg shaperesonance of N−2 at different CAP strengths η. In the plotsof the imaginary part, the onset of the CAP is indicatedas a dashed line. Figure 2 demonstrates how one of thepseudocontinuum states (the one corresponding to the fifthDyson orbital from the left in Figure 1 in this particular case)turns into a discrete resonance state once the CAP strengthhas reached a critical value. The Dyson orbitals of the otherpseudocontinuum states in Fig. 1 also undergo changes withincreasing CAP strength that are in some cases similar tothose shown in Figure 2. An unambiguous identification ofthe resonance state is, however, always possible by analyzingthe behavior of the energy as a function of η.61–63

At CAP strengths larger than ∼0.004 a.u., the shape ofthe real part of the resonance Dyson orbital is similar to theDyson orbital of the bound N−2 shown on the right of Figure 1.Importantly, past this value of η, the orbital does not undergosignificant changes anymore, indicating that the resonancehas been stabilized. The imaginary part of the resonanceDyson orbital grows quickly when the CAP is turned onand then shrinks slowly when η is increased further. Thisshrinkage mainly affects the part of the Dyson orbital in theregion where the CAP is operational. Interestingly, even atCAP strengths above the stabilization point, both the real andimaginary parts of the Dyson orbital have finite amplitudes inregions exposed to the CAP. As one can see, the imaginarypart has one more nodal plane than the real part, which is

FIG. 2. 2Πg resonance of N−2 com-puted by CAP-EOM-EA-CCSD/aug-cc-pVTZ+6s6p6d(C). Upper panel:Real (upper row) and imaginary (lowerrow) parts of the Dyson orbital atdifferent CAP strengths η plotted atan isovalue of 0.003. The dashed linesindicate the onset of the CAP. Lowerpanels: Real (left) and imaginary (right)parts of the resonance energy and theexpectation value of R2 as a function ofCAP strength.

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054113-5 T.-C. Jagau and A. I. Krylov J. Chem. Phys. 144, 054113 (2016)

a consequence of the orthogonality relation from Eq. (9).We add that the c-norm of the Dyson orbital depends onlyweakly on η. In the present case, the real part of the averagednorm of the left and right Dyson orbitals is between 0.90 and0.95 for η = 0–0.008 and the imaginary part does not exceed0.02.

The lower panels of Figure 2 illustrate that the behaviorof the Dyson orbital is consistent with the changes in theresonance energy and the expectation value of R2. We showzeroth-order energies E, i.e., the uncorrected eigenvalues ofthe CAP-EOM-EA-CCSD equations and first-order energiescomputed according to Eq. (4). The difference between E andU provides a measure of the perturbation of the resonancewave function by the CAP. The expectation value of R2 is ameasure of the spatial extent of the resonance wave function61

and is approximately constant at larger CAP strengths, similarto the behavior of the Dyson orbital. It is also noteworthythat the rapid growth of the imaginary part of the Dysonorbital at small η is paralleled by pronounced changes inIm(E) and Im(⟨R2⟩). The optimal CAP strengths determinedfrom the uncorrected and first-order corrected energies are0.0022 and 0.0036 a.u., respectively. The resonance positionsand widths evaluated at these CAP strengths are ER = 2.511eV, Γ = 0.407 eV and ER = 2.458 eV, Γ = 0.419 eV in zerothorder and first order, which is in agreement with the seminalvalue from Ref. 83 (ER = 2.316 eV, Γ = 0.414 eV). We notethat both ⟨R2⟩ and the Dyson orbital are not yet stationaryat the zeroth-order optimal CAP strength, but rather reachstability around the optimal η determined from the first-order

corrected energy. This provides additional justification forusing this criterion.61

To further illustrate their properties, we computed CAP-EOM-EA-CCSD Dyson orbitals for several shape resonancesof small to medium-sized molecules. These are compiled inFigure 3 along with the corresponding resonance positionsand widths. The c-norms of all Dyson orbitals computedhere have a real part well above 0.90 confirming distinctone-electron character of the electron attachment, which isexpected for shape resonances. The corresponding imaginaryparts are loosely connected to the widths of the resonances;for the present examples, they are around or below 0.01.

The upper panel of Figure 3 shows Dyson orbitals forattachment to the diatomics N2, CO, CuF, and H2. The realparts of these orbitals can be described as a symmetric π∗

orbital (N−2), a slightly distorted π∗ orbital (CO−), a highlydistorted π∗ orbital (CuF−), and a symmetric σ∗ orbital (H−2).The shape of the imaginary parts can be rationalized interms of Eq. (9) as discussed above. We emphasize that forall anions, calculations on the basis of Hermitian quantummechanics would yield results similar to those shown for N−2in Figure 1.

The symmetry of a Dyson orbital can be related tothe corresponding resonance lifetime, which is particularlyinteresting for the isoelectronic species N−2 (Γ = 0.42 eV) andCO− (Γ = 0.64 eV). Since the stabilized resonance state can beinterpreted as the initial state of the autodetaching electron in atime-dependent framework, the symmetry of its Dyson orbitalis related to the wave function of the outgoing electron.64

FIG. 3. CAP-EOM-EA-CCSD Dysonorbitals of various transient anions.For N -methylformamide (lower rightpanel), CAP-HF orbitals are shown aswell. The upper and lower plots in eachpanel show the real and imaginary parts,respectively. Isovalues are 0.003 for alldiatomics, real parts of orbitals for ben-zoquinone and benzene, and the imagi-nary part of the 2B2g benzene resonanceorbital; 0.001 for the imaginary partsof the 2E2u benzene resonance orbitaland the benzoquinone resonance or-bital; 0.005 for all N -methylformamideorbitals.

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Experiments have shown that the latter is of pure d-characterfor N−2 but contains p-wave contributions for CO−.84 Thislower angular momentum barrier results in a shorter lifetimeof CO− compared to N−2 . We add that the resonance positionsand widths of all species considered here roughly agreewith those from previous theoretical studies55,62–64,66,85,86 andexperiment.64,83,84,87–89

The lower left panel of Figure 3 shows Dyson orbitalsfor attachment to benzoquinone (2Au resonance) and benzene(2E2u and 2B2g resonances). The real parts of all orbitalsand the imaginary part of the 2B2g benzene resonance orbitalwere plotted at the same isovalue as the previous examples,whereas the imaginary parts of the remaining two orbitalsare of so low magnitude that their visualization required alower isovalue. This illustrates again that the magnitude of theDyson orbital’s imaginary part is connected to the width of theresonance. For all three examples, the shape of the real parts isconsistent with qualitative predictions by Hückel theory, i.e.,attachment to the lowest-lying π∗ orbital of benzoquinoneand the lower- and higher-lying π∗ orbitals of benzene.The imaginary parts can again be rationalized in terms ofEq. (9).

Our results for ER (2.88 eV) and Γ (0.012 eV) of the2Au resonance of the benzoquinone anion are in very goodagreement with recent CAP-EOM-EA-CCSD results65 andalso with the experimental value for ER

90 (2.5 eV). Ourvalues for the position and width of the 2E2u resonanceof the benzene anion (ER = 1.64 eV, Γ = 0.04 eV) alsoshow reasonable agreement with experimental results91

(ER = 1.12 eV, Γ ≈ 0.12 eV) and EOM-EA-CCSD valuesobtained with the stabilization method.85,92 However, forthe higher-lying 2B2g resonance of the benzene anion, weobserve a substantial discrepancy between our results(ER = 6.75 eV, Γ = 0.35 eV), EOM-EA-CCSD stabiliza-tion calculations92 (ER = 5.93 eV), and experiment91 (ER

= 4.82 eV). A possible reason could be that the 2B2gresonance is not fully stabilized in our calculations andis affected by the artificial stabilization of a lower-lyingpseudocontinuum state. This points at deficiencies in the basisset and a suboptimal choice of the CAP onset. However,we did not modify these parameters as they are adequatefor the 2E2u resonance, and the evaluation of transitionproperties between the two resonances necessitates usingthe same parameters for both states. This issue is discussedagain in Section III B in conjunction with transition dipolemoments.

The lower right panel of Figure 3 displays CAP-HF andCAP-EOM-EA-CCSD Dyson orbitals for attachment to N-methylformamide. In terms of MO theory, this 2A′′ resonancecan be interpreted as attachment to the π∗ orbital of theC-O bond with appreciable additional contributions locatedat the nitrogen atom. Two aspects of this example should behighlighted. First, the imaginary parts of the Dyson and the HForbital have one nodal plane less than the real parts, whereasthe opposite applies to all previous examples. Second, theplots illustrate visible differences between the CAP-EOM-EA-CCSD Dyson orbital and the CAP-HF orbital. This deviationfrom Koopmans’ theorem is also apparent from the EOM-EAvectors, which feature several sizable elements corresponding

to attachment to different HF orbitals. It should be added thatthe composition of the EOM-EA vector always changes as afunction of η because different MOs are affected to a differentextent by the CAP. In particular, analysis at too low η wouldsuggest deviations from Koopmans’ theorem for most of theresonances discussed before. However, upon increasing theCAP strength, Koopmans’ character is recovered for all casesexcept N-methylformamide.

Little experimental information on the 2A′′ resonance ofN-methylformamide is available from the literature. However,a recent study93 on dissociative electron attachment to thatmolecule reports low-lying resonances at 1.4-1.7 eV and at2.9 eV depending on the dissociation channel. Although ourcomputational value from Figure 3 (2.91 eV) seems to agreewith the second value, we point out that structural changes ofthe anion and the basis-set error ought to be quantified to allowfor a rigorous comparison to these experimental data. Hence,it cannot be ruled out that the experimentally observed peakat 1.4-1.7 eV stems from the 2A′′ resonance characterizedhere.

B. Transition dipole moments

Transition dipole moments involving metastable statescan be interpreted in the same way as transition momentsbetween bound states, that is, as a measure of the probabilityof an optical transition. For anions with a stable groundstate, the transition dipole moment between that state anda higher-lying resonance is a key quantity for modelingphotodetachment experiments. If no stable ground state exists,the transition moment between two metastable anion statescould be accessed experimentally in a two-step processin which electron attachment is followed by an opticaltransition. Another area where transition moments involvingresonances are important is non-Hermitian Floquet theory,94,95

which has been applied, for example, to laser-inducedionization in quantum dots.96 Moreover, transition momentsare important not only for optical transitions but also play arole in radiationless energy transfer processes,97–99 which mayinvolve metastable states as well.

An example where optical transitions between boundstates and resonances are possible is the benzoquinone radicalanion with its stable 2B2g ground state and the 2Au shaperesonance. Figure 4 shows the transition dipole momentbetween these two states calculated as geometric mean of the“right” and “left” CAP-EOM-EA-CCSD transition moments(cf. Eqs. (15) and (16)) as a function of CAP strength. Alsoshown are the expectation value of R2 for the resonance stateand its energy.

Figure 4 illustrates that the real part of the transitionmoment increases from 0.985 to 1.030 a.u. as the CAP is turnedon and changes very little when η is increased further. Theinitial increase is due to the different description of resonancesin Hermitian and non-Hermitian quantum mechanics. In acalculation at η = 0, the resonance character is distributedover several pseudocontinuum states all of which featuresizable contributions to the transition moment. At finite η,a single resonance state emerges and comprises the overalltransition moment. As the 2Au resonance of the benzoquinone

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054113-7 T.-C. Jagau and A. I. Krylov J. Chem. Phys. 144, 054113 (2016)

FIG. 4. 2Au resonance of the benzoqui-none anion computed by CAP-EOM-EA-CCSD /aug-cc-pVTZ+24s. Upperpanels: Real (left) and imaginary (right)parts of the resonance energy as a func-tion of CAP strength. Lower panels:Real (left) and imaginary (right) parts ofthe transition dipole moment betweenthe 2Au resonance and the bound 2B2gground state and expectation value ofR2 for the 2Au resonance as a functionof CAP strength.

anion is quite narrow (Γ = 0.012 eV), the initial increase ofthe transition moment is small. One may expect that broaderresonances to which more pseudocontinuum states contributeat η = 0 feature a more substantial initial increase of thetransition moment. This is confirmed by Figure 5 that showsthe transition moment between the stable 1Σ+ ground stateand the 2Π resonance of CuF− (Γ = 0.60 eV, cf. Figure 3)as a function of η. In this example, the value of the real partof the transition moment more than doubles from 1.03 to2.25 a.u. when going from η = 0 to finite η.

The imaginary part of the transition moment is moredifficult to interpret. The imaginary part of the expectationvalue of an operator can be rationalized in terms ofthe Hellmann-Feynman theorem as the response of theresonance width to that operator;24,71 however, extending suchinterpretation to transition properties is not straightforward.As Figure 4 shows for benzoquinone, Im(µ[2Au ↔ 2B2g])decreases quickly at low η, then passes a minimum andasymptotically assumes a value of −0.007i a.u. Thus, ourbest value for the transition moment is (1.030-0.007i) a.u.,which yields an oscillator strength of 0.072 a.u. We notethere is no fundamental reason for the asymptotic value of theimaginary part of the transition dipole moment to be negative,which complicates a qualitative interpretation as an intrinsicuncertainty in the transition moment due to metastability. This

is also illustrated by calculations for the 2Π resonance of CuF−

reported in Figure 5. Our best value for the transition momentµ[2Π ↔ 2Σ+] is (2.251 + 0.111i) a.u. yielding an oscillatorstrength of 0.246 a.u.

Figure 4 also shows that the changes in the raw andfirst-order corrected energies E and U and the spatial extent⟨R2⟩ are consistent with the behavior of µ[2Au ↔ 2B2g]. Atlarger η, the real part of the raw energy E depends linearlyon η, whereas the real parts of U, ⟨R2⟩, and µ[2Au ↔ 2B2g]become stationary. This can be interpreted as stabilization ofthe resonance wave function as explained in Ref. 61. Thecorresponding imaginary parts exhibit a similar pattern withthe exception that Im(U) is not as stationary as Re(U), abehavior known from previous studies.61 We add that the realand imaginary parts of all quantities become stationary atdifferent CAP strengths, which was also observed previously.

An interpretation of the dependence of the transitionmoment on η in terms of one resonance state is only possibleif the other state involved in the transition is not affected bythe CAP. This is the case for the stable 2B2g ground state of thebenzoquinone anion; in the present calculation, its artificialwidth is less than 0.0003 eV. The situation is different iftransitions between two metastable states are investigated. Anexample for such a system is the benzene anion that featuresa 2E2u and a 2B2g resonance (cf. Figure 3). Figure 6 shows

FIG. 5. 2Π resonance of CuF− com-puted by CAP-EOM-EA-CCSD/aug-cc-pVTZ+6s6p6d(A). Upper panels:Real (left) and imaginary (right) partsof the resonance energy as a functionof CAP strength. Lower panels: Real(left) and imaginary (right) parts of thetransition dipole moment between the2Π resonance and the bound 2Σ+ groundstate as a function of CAP strength.

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054113-8 T.-C. Jagau and A. I. Krylov J. Chem. Phys. 144, 054113 (2016)

FIG. 6. 2E2u and 2B2g resonancesof the benzene anion computed byCAP-EOM-EA-CCSD / aug-cc-pVTZ+36s. Upper panels: Real (left) andimaginary (right) parts of the resonanceenergies as a function of CAP strength.Lower panels: Real (left) and imaginary(right) parts of the transition dipolemoment between the two resonancesand expectation values of R2 as afunction of CAP strength.

energies and ⟨R2⟩ values for these two states together with thetransition moment between them.

As observed for benzoquinone, the real part of thetransition moment µ[2E2u ↔ 2B2g] grows substantially from0.1 to 0.6 a.u. at low CAP strengths when discrete resonancestates are formed from pseudocontinuum states. However, incontrast to the previous case, no clear stabilization occursat higher CAP strengths. Also, the imaginary part of thetransition moment only exhibits a temporary stabilization ataround η = 0.015–0.030 and then diverges. It should be alsonoted that the imaginary part of the transition moment is oneorder of magnitude larger here than for the benzoquinoneanion. The behavior of µ[2E2u ↔ 2B2g] can be related to theplots of the resonance energies in Figure 6, which look similarto those from Figure 4 for the lower-lying and narrower2E2u state (albeit on a different η scale), but different for thehigher-lying and broader 2B2g state. This can be interpretedas insufficient stabilization of the latter resonance, as wasmentioned above. The same can be inferred from the differentscale on which the energies of the two resonances change.Interestingly though, the Re(⟨R2⟩) values of both resonancesare rather stable, and the asymptotic values are quite similar(482 and 489 a.u.). We note that a customization of basis setand CAP onset might stabilize the 2B2g resonance to a higherdegree, but would likely perturb the 2E2u resonance more. Forthe same reason, we refrained from going to CAP strengthslarger than η = 0.05. Our best value for µ[2E2u ↔ 2B2g] hasbeen evaluated at the optimal CAP strength for the 2B2g state,η = 0.03 a.u. and reads (0.724 + 0.254i) a.u., which results inan oscillator strength of 0.054 a.u.

IV. CONCLUDING REMARKS

In this article, we have presented the framework for evalu-ating Dyson orbitals and transition dipole moments usingCAP-EOM-EA-CCSD wave functions. Selected examplesillustrate that these quantities can be computed straight-forwardly within non-Hermitian quantum mechanics wherediscrete resonance states can be formulated. In contrast,calculations on the basis of Hermitian quantum mechanicsdo not provide direct access to Dyson orbitals or transi-tion moments involving metastable states.

Specifically, Dyson orbitals from CAP-free calculationsdescribe pseudocontinuum states rather than resonances,whereas CAP-EOM-EA-CCSD Dyson orbitals can beemployed to characterize electron attachment to transientanions. In contrast to their Hermitian counterparts, the relationbetween the real and imaginary parts of CAP-EOM-CCSDDyson orbitals is fixed. Their real parts meet expectationsbased on qualitative considerations from MO or Hückel theory,whereas the shape of the imaginary parts is governed by therequirement that the real and imaginary parts be orthogonal.

Similar differences between Hermitian and non-Hermitian quantum mechanics are exemplified by transitiondipole moments. In the former framework, the resonancecharacter is distributed over several pseudocontinuum statesall of which feature appreciable contributions to the overalltransition dipole moment. In contrast, in the presence ofa CAP, a discrete resonance state emerges for which thetransition dipole moment and oscillator strength can be readilyevaluated. For cases involving just one metastable state, nearlyCAP strength-independent results could be obtained, whereastransition moments between two resonances are subject toa more pronounced dependence on the CAP strength. Thelatter behavior arises from a principle feature of the CAPapproach in its present form, namely, that a set of optimalCAP parameters is specific to a particular resonance state.

In sum, we have presented, for the first time, Dysonorbitals and transition dipole moments for metastablestates. This enables a more comprehensive characterizationof resonances beyond position and lifetime. Our resultsdemonstrate that chemical concepts for stable states such asMO theory can be applied to transient anions and encouragethe implementation of other molecular properties for CAPmethods.

ACKNOWLEDGMENTS

This work has been supported by the Army ResearchOffice through Grant No. W911NF-12-1-0543 and theAlexander von Humboldt Foundation (Bessel Award toA.I.K. and Feodor Lynen fellowship to T.-C.J.). We thankProfessor Ksenia Bravaya and Anastasia Gunina for valuablediscussions.

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054113-9 T.-C. Jagau and A. I. Krylov J. Chem. Phys. 144, 054113 (2016)

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