Quantum entanglement between electronic and vibrational degrees offreedom in moleculesLaura K. McKemmish, Ross H. McKenzie, Noel S. Hush, and Jeffrey R. Reimers Citation: J. Chem. Phys. 135, 244110 (2011); doi: 10.1063/1.3671386 View online: http://dx.doi.org/10.1063/1.3671386 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i24 Published by the American Institute of Physics. Related ArticlesRotational dynamics of solvated carbon dioxide studied by infrared, Raman, and time-resolved infraredspectroscopies and a molecular dynamics simulation J. Chem. Phys. 136, 014508 (2012) Vibrationally averaged post Born-Oppenheimer isotopic dipole moment calculations approaching spectroscopicaccuracy J. Chem. Phys. 135, 244313 (2011) Quantum entanglement between electronic and vibrational degrees of freedom in molecules JCP: BioChem. Phys. 5, 12B606 (2011) An activated scheme for resonance energy transfer in conjugated materials J. Chem. Phys. 135, 244512 (2011) Efficient electron dynamics with the planewave-based real-time time-dependent density functional theory:Absorption spectra, vibronic electronic spectra, and coupled electron-nucleus dynamics J. Chem. Phys. 135, 244112 (2011) Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 135, 244110 (2011)

Quantum entanglement between electronic and vibrational degreesof freedom in molecules

Laura K. McKemmish,1 Ross H. McKenzie,2 Noel S. Hush,3 and Jeffrey R. Reimers1,a)

1School of Chemistry, The University of Sydney, NSW 2006, Australia2School of Mathematics and Physics, The University of Queensland, QLD 4072, Australia3School of Molecular Biosciences and School of Chemistry, The University of Sydney, NSW 2006, Australia

(Received 6 July 2011; accepted 1 December 2011; published online 30 December 2011)

We consider the quantum entanglement of the electronic and vibrational degrees of freedom inmolecules with tendencies towards double welled potentials. In these bipartite systems, the vonNeumann entropy of the reduced density matrix is used to quantify the electron-vibration entan-glement for the lowest two vibronic wavefunctions obtained from a model Hamiltonian based oncoupled harmonic diabatic potential-energy surfaces. Significant entanglement is found only in theregion in which the ground vibronic state contains a density profile that is bimodal (i.e., containstwo separate local maxima). However, in this region two distinct types of density and entanglementprofiles are found: one type arises purely from the degeneracy of energy levels in the two potentialwells and is destroyed by slight asymmetry, while the other arises through strong interactions be-tween the diabatic levels of each well and is relatively insensitive to asymmetry. These two distincttypes are termed fragile degeneracy-induced entanglement and persistent entanglement, respectively.Six classic molecular systems describable by two diabatic states are considered: ammonia, benzene,BNB, pyridine excited triplet states, the Creutz-Taube ion, and the radical cation of the “specialpair” of chlorophylls involved in photosynthesis. These chemically diverse systems are all treatedusing the same general formalism and the nature of the entanglement that they embody is elucidated.© 2011 American Institute of Physics. [doi:10.1063/1.3671386]

I. INTRODUCTION

Entanglement is one of the quintessential “quantum”phenomena. Here, we develop an understanding of the quan-tum entanglement between electrons and nuclei in moleculesby an analysis of a simple model involving two coupled in-tersecting potential-energy surfaces. Such a system was firstintroduced by Horiuti and Polanyi in 1935 to describe pro-ton and hydrogen transfer reactions1 and was subsequentlyextended by Hush to oxidation-reduction processes2 in 1953,now forming the basis of modern electron-transfer theory3–6

describing for example exciton and charge transport throughmolecules, organic conductors and organic photovoltaics aswell as electron-transfer reactions in biochemistry.7–18 It alsodescribes general racemization processes19, 20 and has beenwidely used in spectroscopic analyses,21–23 forming the coreof Herzberg-Teller theory.24–26 Indeed, Bersuker has sug-gested that all chemical processes can be described in this wayas a pseudo Jahn-Teller effect.26 These chemical processescause the vibrational and electronic motions of the moleculeto become entangled.

In quantum information theory, the simplest type of sys-tem is a bipartite pure state27 given by |ψ〉 = ∑

ij cij |ai〉|bj 〉where |a1 〉, |a2 〉, . . . , |an 〉 and |b1 〉, |b2 〉, . . . , |bn 〉 form anyorthonormal set of basis vectors for subsystem A and B, re-

a)Author to whom correspondence should be addressed. Electronicmail: reimers@chem.usyd.edu.au. Telephone: +61(2)93514417. Fax:+61(2)93513329.

spectively. We consider these subsystems as the electronic andvibrational degrees of freedom of a molecule, and the entan-glement is, qualitatively, a measure of the connection betweenthem. While many, related, definitions of entanglement havebeen proposed (see, e.g., Refs. 28–30), the von Neumannentropy of the reduced density matrix is the most commonmethod of quantifying this entanglement, and we apply it togain understanding of entanglement in chemical systems.

The chemical model used involves the interactionof a doubly degenerate electronic state with a non-degenerate vibration (E ⊗ B), a Herzberg-Teller systemthat can be considered26 as a special case of Jahn-TellerHamiltonian24–26, 31 that normally involves the interaction ofa doubly degenerate electronic state with a doubly degen-erate vibration (E ⊗ E). Such an approach can be used todescribe molecules with symmetric double-welled potential-energy surfaces such as that observed for ammonia as a func-tion of its inversion vibrational coordinate, as well as thoseobserved for say inversion of molecular stereochemistry. Insymmetric systems the diabatic potential-energy surfaces de-picting the two localized wells have the same force constantsand energy. We also consider pseudo E ⊗ B systems25, 26 inwhich an asymmetry E0 is introduced in the relative ener-gies of the two wells, and the general scenario consideredis sketched in Fig. 1. In this figure, the red and blue dashedcurves represent diabatic potential-energy surfaces describ-ing two non-interacting wells that are coupled by a resonanceenergy J. These give rise to the classic Born-Oppenheimerground-stated adiabatic potential-energy surface and its

0021-9606/2011/135(24)/244110/11/$30.00 © 2011 American Institute of Physics135, 244110-1

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244110-2 McKemmish et al. J. Chem. Phys. 135, 244110 (2011)

FIG. 1. The chemical model used to describe electron-vibration entangle-ment. Harmonic diabatic surfaces (blue and red dashed lines), located atminima separated by 2δ in some generalized dimensionless vibrational co-ordinate with energy asymmetry E0, are coupled by a resonance interactionJ. The ensuing Born-Oppenheimer ground-state and excited-state adiabaticpotential-energy surfaces are denoted by purple and green solid lines, respec-tively. The reorganization energy λ is also indicated.

associated excited state shown in purple and green in this fig-ure, respectively.

We consider the properties of six molecular examplespertaining to this Hamiltonian: ammonia, BNB, benzene, theCreutz-Taube ion (CT), the bacterial photosynthetic reactioncenter radical cation (PRC), and pyridine excited triplet states(3PYR), see Fig. 2 and Table I. These systems display a widerange of chemical and physical properties. For ammonia andBNB,32 2|J| is less than the reorganization energy λ requiredto distort the molecule in one well to the nuclear coordi-nates of the minimum of other well (see Fig. 1) and so theBorn-Oppenheimer surface is double welled.33 In contrast, forbenzene the opposite is true, producing aromaticity with thecarbon-carbon bonds taking on equal lengths rather than thealternating single-bond and double-bond pattern expected forKekulé structures.

The precise nature of the Creutz-Taube ion34 has been de-bated for over 40 years,35 this molecule being the first mixed-valence compound investigated for which it was apparent thatthe molecule cannot be simply described as comprising anion in each of two standard valence states (i.e., an Ru(II)and an Ru(III)), and this molecule became the paradigmthrough which biological electron transfer processes includ-ing those involved in solar-energy conversion during photo-

synthesis was subsequently interpreted.36 Molecules showingthese types of effects are often classified under the Robin-Day system37 as either Class II (localized double well), ClassIII (delocalized single well), or Class II–III (some mixture).38

Solar to electrical energy conversion occurs in the PRC whenthe “special pair” of bacteriochlorophylls shown in Fig. 2ejects an electron to become a dimer radical cation. This ioncan be thought of as a “mixed-valence complex” in whicheach bacteriochlorophyll could take the charge 0 or +1, and,like CT, the charge could alternatively be delocalized overboth functionalities. Energy asymmetry in the PRC is inducednaturally through asymmetric coordination with the surround-ing protein as well as from the asymmetric protein electricfield. The excited states of pyridine also display asymme-try but in this case its cause is chemical in origin as nitro-gen substitution for CH in benzene makes the associated lo-calized structures inequivalent. While this modifies the forceconstants of the diabatic states as well as their energies, theeffect of the energy variation is the most profound and it is re-alistic to neglect force-constant variations in a simple modeldescription. This scenario is appropriate to a very wide rangeof excited-state molecular spectroscopy and photochemistry.

We consider only the coupling of electronic statesthrough a single vibrational mode, though in general manymodes could contribute to the coupling. In practical situations,generalization to multiple modes is typically straightforwardand is essential in quantitative analyses. Nevertheless, the es-sential physics of electron-vibration interaction in moleculescan very often be described by the basic one-mode modelusing appropriately chosen effective vibrational parameters.Actually, the one-mode model is a good approximation formost properties of three of the molecules considered herein:ammonia, BNB, and benzene. For the excited triplet states ofpyridine,39 at least 6 modes and 3 electronic states (includesthe crossing (n, π*) state) are required in a quantitative anal-ysis, whilst for CT a continuum of solvent modes is critical35

and 4 electronic states with 70 modes have been used tomodel PRC.40 In all cases, some important molecular proper-ties such as the shape, central frequency, and intensity of thecharacteristic intervalence electronic transition are known tobe independent of the number of modes used in the analysis.41

The simple model thus provides a useful general startingposition for considering electron-vibration entanglement. TheHamiltonian also describes a superconducting qubit coupledto the resonant microwave mode in a cavity, sometimes called

N

N N RuRu

AA

AA A AAA

AAA= NH3 5+

Ammonia

NH H

HN

HH

H

N

NN

NO

O

O

O

OPHY

Mg

ON

N N

N O

O

O

O

OPHY

Mg

O

Cretutz-Taube ion (CT)

Bacteriochlorophyll-a dimer cation (PRC)

Benzene

Pyridine triplet states (3PYR)

+

+B N- B B N- B+

BNB

FIG. 2. Some sample molecular systems with electronic states that can be described using two coupled diabatic potential-energy surfaces.

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244110-3 Electron-vibration entanglement J. Chem. Phys. 135, 244110 (2011)

TABLE I. Estimates of parameters values for the coupled harmonic potential-energy surfaces of some different molecular systems, along with the deducedvibronic entanglements S0 and S1.

System E0(eV) ¯ω(cm−1) J (eV) λ(eV)2 |J |

λ

¯ω

�E

E0

¯ωS0 S1

dS0

dE0¯ω

dS1

dE0¯ω

Ammoniaa 0 1580 − 11.0 27.5 0.80 0.006 0 0.45b 0.45b −420 −420Benzenea 0 1564 − 9.5 5.7 3.3 0.010 0 0.01c 0.03c −5 × 10−5 −12 × 10−5

3PYR (Ref. 39) 0.26 1620 − 0.33 2.0 0.3 0.095 1.3 0.03 0.29 −0.017 −0.85CT (Ref. 35) 0 800d 0.35e 0.87e 0.80 0.089 0 0.37 0.59 −0.050 −0.038PRC (Ref. 40) 0.069 ∼980f − 0.13 0.14 1.8 0.41 0.6 0.17 0.57 −0.10 −0.19BNB (Ref. 32) 0 1800 − 0.36 0.98 0.74 0.18 0 0.44 0.73 −0.007 −0.006

aDerived from experimental data,51, 97, 102, 103 see supplementary material.56

bUsing full 3-state 2-electron model S0 = 0.67 and S1 = 0.67, see supplementary material.56

cUsing full 5-state 4-electron model S0 = 0.03 and S1 = 0.09, see supplementary material.56

dThis value corresponds to the vibrational frequency of the libration mode of water, which calculations39 indicate to be the primary carrier of the distortion.eThese values are debated.fThis value of the vibrational frequency is a one-mode approximation to the 70 modes used in quantitative simulations.104

“circuit QED,”42, 43 a technology of significant interest forpractical quantum information processing.44–47

Also in the main text we consider only descriptions ofchemical processes involving two electronic states. For prob-lems involving electron or hole transport that serves to changethe location of a free radical, the simplest description of thebasic physical interactions does indeed involve just two elec-tronic states. However, for more general chemical reactionsinvolving closed-shell systems, multiple excitations involv-ing the same molecular orbitals are possible and so multi-ple states can be thought of as being intrinsically involvedin the reaction. For ammonia, the electronic-vibrational cou-pling mixes the nitrogen non-bonding orbital with an NH an-tibonding orbital48–50 and so the vibronic coupling betweenthe ground state and the singly excited state involving theseorbitals should parallel that between this singly excited stateand its associated doubly excited state. Hence three states, theground state, the singly excited state, and the doubly excitedstate, all share the same strong vibronic coupling. Alterna-tively, the Kekulé distortion of benzene mixes the doubly de-generate HOMO and LUMO orbitals51–55 and so 4 electronsexperience the same vibronic coupling scenario, generating astrongly coupled network of 7 electronic states. However, forboth of these systems it is possible to reduce the multi-stateinteractions to give an effective two-state model involvingrenormalized parameters that accurately reproduces the prop-erties of the electronic ground state of the system, as demon-strated in supplementary material.56 In the main text we uti-lize only the resulting effective 2-state model. Significantly,the transformations from the full Hamiltonians, in which elec-tronic and nuclear degrees of freedom are fully separated, tothe effective two-state models mix the electronic and nuclearmotions, thus generating entanglement. For these systems, weprovide results for the full minimalist multi-state models insupplementary material56 to demonstrate that the two-statedescriptions of the entanglement to actually depict all signifi-cant qualitative properties.

To manifest the electron-vibration entanglement we ex-pand the exact, entangled, wavefunctions of the system interms of unentangled basis states expressed diabatically as aproduct of an electronic wavefunction, φ, dependent only onelectronic coordinates, r, and a nuclear wavefunction, χ , de-

pendent only on nuclear coordinates, R, i.e.,

|ψCA (r, R)〉 = |φ (r)〉|χ (R)〉. (1)

This representation of the vibronic wavefunctions is knownas the crude-adiabatic approximation57 and differs from theBorn-Oppenheimer approximation58 in that no nuclear depen-dence of the electronic wavefunctions is allowed (i.e., thesame electronic wavefunctions are used for all geometries).Exact numerical wavefunctions for the full Hamiltonian arethen obtained, using these crude adiabatic wavefunctions as abasis set, in the form

|ψexact (r, R)〉 =∑ij

cij |φi (r)〉|χj (R)〉 (2)

from which the von Neumann entropy of the reduced densitymatrix S can be readily obtained as the electronic and vibra-tional basis sets are orthonormal.

Such entanglement S has been previously quantified31

in detail for the vibronic ground state of this model forE0 = 0, and some results in the adiabatic limit59 are alsoavailable for E0 �= 0. Here we examine in detail how the en-tanglement changes when E0 �= 0, as well as considering forthe first time entanglement within the lowest-energy vibronicexcited state. These results bring into prominence the exis-tence of two distinct types of entanglement in the parameterspace: entanglement that persists despite the introduction ofasymmetry (persistent entanglement) and entanglement thatdisappears (fragile degeneracy-induced entanglement).

At the moment no simple relationship is known linkingthe amount of entanglement manifested in a quantum stateand its usefulness as a basis for qubits in some device; despitethis, entanglement is almost universally referred to as a use-ful quantum information resource. In principle, any amountof entanglement is sufficient to allow quantum states to beused in some device for quantum information processing.60

The actual amount of entanglement in any one state is not nec-essarily a good indicator of performance as a typical experi-ment could, for example, mix an initially unentangled ground-state wavefunction with an initially unentangled excited-statewavefunction to produce a maximally entangled state.

However, entanglement must be very insensitive toenvironmental disturbances, such as dynamic motion of the

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244110-4 McKemmish et al. J. Chem. Phys. 135, 244110 (2011)

surrounding solvent, for all relevant quantum states in orderto minimize device errors.61 As variations in the environmentcan be modeled as providing fluctuations in E0, sensitivity ofthe entanglement to the environment can be depicted in a gen-erally useful way by considering dS/dE0. Understanding ofthe consequences of asymmetry is critical to any applicationof electron-vibration entanglement to quantum informationprocessing or quantum information transport and will particu-larly influence the design of experimental apparati and condi-tions used for measuring electron-vibration entanglement. Anadvanced description of such an apparatus has only been pro-posed to measure entanglement between the electronic statesof pairs of ammonia molecules,62 but means for measuringrelated properties for large molecules and nanoparticles withstrong environment interactions have been conceived.63–67

We determine the basic molecular properties required to make

the entanglement insensitive to environment, thus indicatingwhich systems are likely to be useful in practice.

Our results also allow a broad picture to be developed ofthe significance of entanglement to the understanding of basicchemical bonding and reactivity. The magnitude of vibroniccoupling is related to the entanglement between the elec-trons and nuclei. While the magnitude of vibronic couplingcannot easily be compared between molecules, entanglementcan be.

II. MODEL HAMILTONIAN AND ITS PARAMETERS

Expressed in terms of a localized diabatic electronic basisset |φ1 〉, |φ2 〉, the coupled harmonic diabatic surfaces can bewritten as a function of a single generalized dimensionlessnuclear coordinate Q as

H (Q) =

⎡⎢⎢⎢⎣¯ω

2(Q + δ)2 − ¯ω

2

d2

dQ2J

J¯ω

2(Q − δ)2 + E0 − ¯ω

2

d2

dQ2

⎤⎥⎥⎥⎦ , (3)

where ω is the fundamental vibrational frequency on the twodiabatic surfaces (assumed herein to have equivalent forceconstants), ¯ is Planck’s constant, and δ is the dimensionlessdisplacement of each diabatic surface away from the symmet-ric configuration. The dimensionless variables Q and δ can berelated to mass-weighted Cartesian-type coordinates using therelationships68 Q = q/qzpt and δ = q0/qzpt, where q is the vibra-tional coordinate in units of

√mass × length, q0 indicates the

vibrational coordinate where the minima of the diabatic sur-faces lie again in units of

√mass × length, and qzpt is a scaling

factor known as the zero-point vibrational length given by68

qzpt = √¯/ω in units of

√mass × length.

A variety of useful quantities may be described in termsof the basic model parameters. Firstly, the reorganizationenergy is defined as

λ = 2¯ωδ2 (4)

and specifies the energy necessary to distort a molecule indiabatic electronic state 1 to the equilibrium structure ofdiabatic state 2 and vice versa (see Fig. 1). This energy isoften expressed alternatively via the Huang-Rhys factor,λ¯ω

= 2δ2, that indicates the effective number of vibrationalquanta excited during electronic transitions between purelylocalized electronic states. The electronic energy spacing inthe presence of the resonance interaction increases to

�E =√

λ2 + 4J 2 (5)

at the geometry of a diabatic minimum for a symmetric (E0

= 0) system. The Born-Oppenheimer adiabatic approxima-tion provides a good description of the system properties

whenever

¯ω

�E� 1, (6)

i.e., the vibrational energy-level spacing is much smallerthan the electronic energy-level spacing. This approximationyields the ground-state (GS) and excited-state (ES) electronicpotential-energy surfaces shown in Fig. 1 as it allows theeffect of the nuclear momentum operator on the electronicwavefunction to be ignored so that H can simply be diagonal-ized parameterically as a function of the nuclear coordinatesQ. For E0 = 0, the curvatures of these surfaces at Q = 0 aregiven by

∂2

∂Q2EGS = ¯ω

(1 − λ

2 |J |)

and

∂2

∂Q2EES = ¯ω

(1 + λ

2 |J |) (7)

so that the ground-state surface becomes double welledwhenever33 2|J |

λ< 1.

The diabatic Hamiltonian Eq. (3) is not unique as allphysical properties, including electron-vibration entangle-ment, are invariant69 to the rotation of the electronic basis set|φ1 〉, |φ2 〉 that produces

H′ = RHRT , (8)

where

R =[

cos θ − sin θ

sin θ cos θ

]. (9)

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244110-5 Electron-vibration entanglement J. Chem. Phys. 135, 244110 (2011)

FIG. 3. The optimized diabatic and adiabatic molecular potential energy surfaces (Energy vs. Q) and associated lowest energy levels in each state for differentvalues of 2|J|/λ and ¯ω/�E; the optimized rotation angles θ are indicated for the diabatic surfaces.

In particular, a rotation of the localized diabatic Hamil-tonian (Eq. (3)) typically used to describe double-welled sys-tems such as ammonia by θ = 45◦ produces the delocalizeddiabatic Hamiltonian typically used by spectroscopists to de-scribe the ground and excited states of aromatic molecules.69

The essential feature of such a diabatic transformation ma-trix is that its matrix elements are independent of the nu-clear coordinate Q, unlike say the transformation to the Born-Oppenheimer adiabatic description of the problem.

While four parameters E0, ω, J, and δ are specified inEq. (3), entanglement is independent of the absolute energyscale and hence we simplify the problem by considering onlythe three independent parameters 2J/λ, ¯ω/�E, and E0/¯ω.Descriptive parameter values for the iconic systems ammo-nia, BNB, benzene, CT, PRC, and 3PYR are given in Table I.

Figure 3 indicates the physical significance of the ra-tios ¯ω/�E and 2J/λ by plotting the diabatic and adiabaticpotential-energy surfaces generated with ¯ω/�E = 0.1, 1, or10 and 2|J|/λ = 0.1, 1, or 10, all at E0/¯ω = 0. As the shapesof the diabatic surfaces are not invariant to electronic-state ro-tation (Eq. (8)), we seek here a “best-possible” diabatic rep-resentation of the low-energy parts of the potential-energysurfaces. To do this, we evaluate the lowest 10 vibronic en-ergy levels from the diabatic surfaces as a function of θ ateach point in the plots shown in the figure, optimizing theangle to minimize the difference between these energy lev-els and those produced following full diagonalization of thecoupled electron-vibration Hamiltonian (which are physicalobservables and hence independent of θ ). In Fig. 3, the vi-brational levels of the lower-energy diabatic surface are indi-cated by blue solid lines, while the vibrational levels of thehigher-energy diabatic surface indicated by red dashed lines;the optimized angles θ are also indicated.

An optimized angle near θ = 0◦ indicates that fully lo-calized states provide the best-possible diabatic description ofthe intersecting potential-energy surfaces. This result is pro-duced whenever ¯ω/�E < 1 and 2|J|/λ< 1, roughly the re-gion in which the ground-state adiabatic potential-energy sur-face is double-welled and supports below-barrier zero-pointvibration. Optimized angles near θ = 45◦ result whenever¯ω/�E < 1 and 2|J|/λ > 1, the region in which the adiabaticpotential-energy surfaces are well-separated from each otherand are single welled, typical, say, of the delocalized aromaticstates of benzene. In the intermediate region with ¯ω/�E < 1and 2|J|/λ ∼ 1 the ground-state adiabatic potential-energy sur-face becomes flat and very anharmonic, leading to highly un-usual molecular properties, e.g., for the Creutz-Taube ion.35, 70

When ¯ω/�E > 1, the scenario is that of well-separated vi-brational levels split by small electronic effects, making theBorn-Oppenheimer approximation a poor descriptor of sys-tem properties. In this regime, the optimum diabatic angle θ

is no longer a physically significant indicator, and the effectsof vibrations not included in the Hamiltonian may dominateany real chemical scenario, bringing into operation say theJahn-Teller effect.71, 72

III. NUMERICAL DETERMINATIONOF THE HAMILTONIAN EIGENFUNCTIONS

To find converged numerical solutions to the eigenvec-tors of H, this operator is represented using a product basisof the form φi ⊗ χ j, with {φ1, φ2} forming the (localized)diabatic electronic basis and {χ1, χ2, . . . , χn} forming atruncated harmonic-oscillator vibrational basis centeredaround Q = 0. The Hamiltonian matrix elements are thengiven by

H1i,1j = H1j,1i = 〈χiφ1|H|φ1χj 〉 = −δ¯ω√

i+12 δj,i+1 + (i + 1

2 )¯ωδi,j

H2i,2j = H2j,2i = 〈χiφ2|H|φ2χi〉 = δ¯ω√

i+12 δj,i+1 + [E0 + (i + 1

2 )¯ω]δi,j

H1i,2j = 〈χiφ1|H|φ2χj 〉 = Jδi,j

(10)

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244110-6 McKemmish et al. J. Chem. Phys. 135, 244110 (2011)

Diagonalizing the Hamiltonian matrix then allows 2n inde-pendent wavefunctions to be written in the form

|ψj 〉 =n∑

i=1

c1i,j |φ1χi〉 + c2i,j |φ2χi〉. (11)

In particular, we are interested in the properties of the ground-state vibronic wavefunction |ψ0〉 and the wavefunction ofnext highest energy, |ψ1〉.

IV. ENTANGLEMENT AS THE VONNEUMANN ENTROPY

The entanglement between the electronic and nucleardegrees of freedom can be expressed as the associated vonNeumann entropy of the reduced density matrix,27, 31 obtainedby re-expressing the eigenfunctions from Eq. (11) in the form

|ψj 〉 =2∑

k=1

φk(q)

[n∑

i=1

cki,jχi(Q)

]=

2∑k=1

φk(q)χ ′k(Q)

or

|ψj 〉 =n∑

i=1

[2∑

k=1

cki,jφk(q)

]χi(Q) =

n∑i=1

φ′i(q)χi(Q),

(11a)where q and Q are the electronic and vibrational co-ordinates,respectively. Based on these expansions, reduced electronicand vibrational density matrices for eigenstate j can be definedas

ρjE

kl =∫ ∞

−∞χ ′

k(Q)χ ′l (Q) dQ =

n∑i=1

cki,j cli,j

and

ρjV

kl =∫ ∞

−∞φ′

k(q)φ′l(q) dq = c1k,j c1l,j + c2k,j c2l,j . (12)

While ρ jE is a 2 × 2 matrix and ρ jV is an n × n matrix, bothshare the same set of at-most-two non-zero eigenvalues ρ

j−

and ρj+ = 1 − ρ

j− (with 0 ≤ ρ

j− ≤ 1/2). The commonality of

these two eigenvalues can be seen by writing the quantumstate as a Schmidt decomposition, which can have at most twoterms.30 The von Neumann entropy of eigenstate j can thus beexpressed as

Sj = − ρj− log2 ρ

j− − (1 − ρ

j−) log2(1 − ρ

j−). (13)

If ρj− = 0, the wavefunction can be expressed as a single prod-

uct of an electronic wavefunction and a vibrational wavefunc-tion and as a result there is no entanglement, Sj = 0. Alter-natively, if ρ

j− = 1/2, the wavefunction is maximally entan-

gled and Sj = 1. It is hence convenient to express the twoeigenvalues as

ρj− = 1

2(1 − �ρj )

and

ρj+ = (1 − ρ

j−) = 1

2(1 + �ρj ). (14)

The entanglement within eigenstate j is then maximal when�ρ j = 0 and minimal when �ρ j = 1.

V. FRAGILE VS. PERSISTENT ENTANGLEMENT

The entanglements S0 for the ground vibronic-state wave-function and S1 for the first-excited vibronic-state wavefunc-tion are shown in Fig. 4, calculated over the whole parameterspace of the model Hamiltonian. For symmetric systems (E0

= 0), the entanglement is large whenever 2|J| < λ and �E> ¯ω. From Fig. 3 it is clear that this region correspondsto double-well potentials that support localized vibrationalmotions.

However, Fig. 4 also shows that the introduction of asmall amount of asymmetry, manifest at say E0/¯ω = 0.01,results in a dramatic reduction of the ground-state entangle-ment, with significant entanglement becoming restricted pre-dominantly to a region within 0.02 < 2|J|/λ < 0.5 and 0.1< ¯ω/�E < 1. The entanglement that remains at E0/¯ω= 0.01 is said to be persistent whilst that which is lost issaid to be degeneracy-induced entanglement. After the ini-tial dramatic reduction in the ground-state entanglement forsmall values of E0/¯ω, the entanglement continues to decreaseslowly with increasing asymmetry. The greatest entanglementbecomes concentrated in the regions near 2|J|/λ just less than1 and ¯ω/�E values of 0.3 for E0/¯ω = 0.1, 0.2 for E0/¯ω= 0.6 and 0.1 for E0/¯ω = 1.3, with always a tail extendingto low E0/¯ω for 2|J|/λ = 1. Similar results have also beenobserved using adiabatic calculations on a qubit coupled to anoscillator59 and in applications of the spin-boson model.73, 74

The feature that we stress here is that while asymmetric sys-tems inherently manifest less entanglement in the ground-state wavefunction than do symmetric double-welled systems,the sensitivity of this entanglement to external perturbations ismuch less than that which symmetric systems can sometimesdevelop.

The fragility of the degeneracy-induced entanglementcan be understood by expanding the non-zero eigenvalues ofthe density matrices for the ground state and first vibronic ex-cited state using perturbation theory in the localized limit of2|J|/λ � 1, giving

�ρ0 = �ρ1 =(

1 − 1 − F00

1 + E0/(2 |J | F00)

)1/2

, (15)

where

F00 = exp−λ

2¯ω(16)

is the Franck-Condon overlap of the two localized-wellharmonic-oscillator diabatic ground-state functions, see sup-plementary material.56 Note that these equations actually re-main useful to even up to at least 2|J|/λ = 0.5. As ground-stateentanglement is large whenever �ρ0 � 1, significant entan-glement can only occur whenever

E0 < 2 |J | exp−λ

2¯ωor

E0

¯ω� 2 |J |

λ

�E

¯ωexp

−�E

2¯ω. (17)

This equation indicates that E0/¯ω is maximized when¯ω/�E = 1/2 and when 2|J|/λ is maximized, qualitativelyexplaining the results presented in Fig. 4 for 2|J|/λ < 1,

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244110-7 Electron-vibration entanglement J. Chem. Phys. 135, 244110 (2011)

0 0.2 0.4 0.6 0.8 1

S .

E0.01

0.1

110

100

E0/ = 0

E0.01

0.1

110

E0/ = 0.01

E0.01

0.1

110

E0/ = 0.1

E0.01

0.1

110

E0/ = 0.6

E0.01

0.1

110

E0/ = 1

E0.01

0.1

110

E0/ = 1.3

2J/0.1 1 10

0.01

0.1

110

E0/ = 10

E0/ = 0

E0/ = 0.01

E0/ = 0.1

E0/ = 0.6

E0/ = 1

E0/ = 1.3

2J/0.01 0.1 1 10 100

E0/ = 10

S0 = 0.99

S0 = 0.93

S0 = 0.67

S0 = 0.29

S0 = 0.20

S0 = 0.16

S0 = 0.03

FIG. 4. Persistent versus degeneracy-induced entanglement. Center andright: Electron-nuclear entanglement (von Neumann entropy) is shown as acontour plot for the vibronic ground state (S0, center) and the first vibronicexcited state (S1, right) wavefunctions vs. 2|J|/λ and ¯ω/�E at various val-ues of E0/¯ω. The black lines denote regions in which the ground-state vi-brational probability density is bimodal or unimodal, while the crosses in-dicate parameter values relevant to (see Table I): red—ammonia; green—benzene, blue—CT, purple—BNB, 3PYR, PRC. Left: the Born-Oppenheimerpotential-energy surfaces (purple and green) and the ground-state vibrationalprobability density (black) at the indicated parameters.

for example the maximum entanglement at E0/¯ω = 0.01 inFig. 4 occurs near 2|J|/λ = 0.1, ¯ω/�E = 0.3 at which pointthe right-hand side of Eq. (17) evaluates to 0.005–0.01. Inany practical scenario, the entanglement of the eigenfunc-tions in a symmetric system will be robust to environmentalfluctuations only if these are very much less that the oscil-lator frequency, demanding high-frequencies and/or very lowtemperatures, and the symmetric double well must support aminimum number of locally bound vibrational levels.

A very rapid change occurs in the nature of the eigen-vectors of H occurs as 2|J|/λ increases towards unity and asa result for 2|J|/λ ≥ 1 the extreme sensitivity of the entan-glement to E0 is lost and the entanglement remains persistent.In this region of the parameter space, the eigenvalues of theground-vibronic-level density matrix can be expanded usingperturbation theory based on the delocalized diabatic Hamil-tonian and through Eq. (14) become specified by

�ρ0 = 1 − ¯ωλ

2(¯ω − 2 |J |)2(1 + E20/16J 2)

. (18)

This expression is exact in the limit 2|J|/λ � 1. As the en-tanglement becomes large whenever �ρ0 � 1, this equationindicates that ¯ω → 2|J| ∼ �E will maximize entanglementwhenever 2|J|/λ > 1, as indicated in Fig. 4. Asymmetry E0

damps this entanglement for large values of ¯ω/�E but when-ever (E0/¯ω)2 � 4(¯ω/�E)−2 the entanglement becomes in-sensitive to asymmetry, e.g., for ¯ω/�E � 1,

S0S̃1 ∼ −ρ0− log2 ρ0

−,

where (19)

ρ0−

1̃

4

λ

2 |J |¯ω

�E

(1 + 1

2

λ

2 |J |)

.

This explains the very rapid attenuation of the entanglementshown in Fig. 4 as 2|J|/λ increases above unity for small¯ω/�E � 1 whilst also indicating how the very fragile en-tanglement shown for 2|J|/λ just less than unity quickly turnsinto persistent entanglement.

Also shown in Fig. 4 are the Born-Oppenheimerpotential-energy surfaces at a salient point in the parame-ter space for each illustrated value of �E/¯ω, along withthe vibrational densities calculated from the exact wavefunc-tions at these points: for E0 = 0, the point at 2|J|/λ = 0.1,¯ω/�E = 0.1 in the strongly entangled region is illustrated,whilst for E0 �= 0 the points manifesting the largest ground-state entanglement are selected. The selected points for bothE0/¯ω = 0 and 0.01 display double-welled potential-energysurfaces that are sufficiently deep to support zero-point vi-bration and thus generate bimodal density profiles (i.e., havea local minimum in the density at or near Q = 0 separatingtwo local maxima75, 76). The entanglement S0 at each of thesepoints is also large, exceeding 0.93. However, the parametersthat maximize the entanglement for E0/¯ω ≥ 0.1 all resultin unimodal ground-state density profiles indicating that thewavefunctions have become localized into the lower-energydiabatic well. At E0/¯ω = 0.1, the entanglement remains sig-nificant, S0 = 0.67, but this reduces quickly as E0/¯ω in-creases, becoming just S0 = 0.16 at E0/¯ω = 1.3.

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244110-8 McKemmish et al. J. Chem. Phys. 135, 244110 (2011)

In Fig. 4, the regions in which the ground-vibronic-statedensity is bimodal are indicated and can be seen to correspondto regions in which the entanglement is in excess of ∼0.7.This is true at both E0/¯ω = 0, encompasing both persistentand fragile entanglement, and at E0/¯ω = 0.01 for which onlypersistent entanglement of this magnitude remains. Hencewe see that regions of fragile entanglement are also the re-gions in which bimodal density profiles become unimodal af-ter the application of very small asymmetries. This qualitativechange associated with the unimodal-bimodal changeover hassome similarities to a quantum phase transition in a systemwith an infinite number of degrees of freedom.77 Considera-tion of the density thus provides a possible means for under-standing entanglement within the ground vibronic wavefunc-tion. Similar results occur in other types of qubits, see, e.g.,Refs. 29, 42, 78–90. In particular, systems in which bimodaldensity profiles survive finite temperature65, 91 are clearly onesin which the entanglement is persistent.

For the first excited-state vibronic level, the overall ef-fect is similar to that for the ground vibronic state but the re-gion of persistent entanglement is much larger and survivesbeyond E0/¯ω = 1. For the symmetric double-well situationwith E0/¯ω = 0 and 2|J|/λ < 1, the first excited vibroniceigenfunction is very similar to the associated ground-statefunction except for the opposite phasing of the two localized-diabatic ground states, and so the entanglements S0 and S1

behave very similarly in this region: Eq. (16) again depictsthe origin of fragile symmetry-induced entanglement whenE0/¯ω < 0.1, while Eq. (19) again shows how the entangle-ment diminishes in magnitude but becomes persistent as soonas 2|J|/λ exceeds unity. However, the similarity between thebehavior of S0 and S1 is lost when E0/¯ω approaches 1, asin this scenario the localized diabatic ground-state from onewell becomes degenerate with the localized first excited levelof the other, as illustrated in Fig. 5. As a result, S1 remainslarge at all values of E0/¯ω, being large in the limit of 2|J|/λ� 1 whenever

(E0

¯ω

)2

+(

2J

¯ω

)2

≈ 1. (20)

This equation also indicates the effects of an additional reso-nance that occurs in the delocalized diabatic limit at 2|J|/¯ω= 1 in which the first excited vibrational level of the grounddelocalized state equals the zero-point level of the excited de-localized state; this gives rise to the large persistent entangle-ment near ¯ω/�E = 1 for 2|J|/λ > 1, see Fig. 5.

FIG. 5. Origin of the resonances that enhance entanglement S1 in the first ex-cited vibronic state: left—crude adiabatic surfaces at E0/¯ω = 1 and ¯ω/�E= 0.1 at θ = 0 and right—crude adiabatic surfaces at ¯ω/�E = 1 and 2|J|/λ= 10 at θ = 45◦.

VI. RELATION TO THE BOHM-AHARONOV TEST OFTHE EINSTEIN-PODOLSKY-ROSEN PARADOX

The Einstein-Podolsky-Rosen paradox92 involves a pairof entangled states of continuous variables (position and mo-mentum). Bohm and Aharonov93, 94 re-formulated a test of theEPR paradox in terms of discrete degrees of freedom, a pair ofsinglet-coupled spins. The chemical systems considered areintermediate between EPR and Bohm-Aharonov because onedegree of freedom is discrete (the two state electronic sys-tem) and the other involves a continuous degree of freedom(the vibrational coordinate). For coupled spins, interactionswith the environment destroy entanglement by “measuring”the spin state, whereas in a chemical system interactions withthe environment destroy symmetry to prevent entanglementfrom ever developing; to do this in a Bohm-Aharonov experi-ment, the environment would need to provide a magnetic fieldstrong enough to make a component of 3A2 the ground-stateof the molecule to eliminate the entanglement. Dissociating asinglet state provides just one example of the general chem-ical effect of static electron correlation, the type of electroncorrelation that arises when symmetry rather than interactiondetermines key features of a quantum state.95 Static electroncorrelation can thus be a source of fragile degeneracy-basedentanglement, entanglement that is destroyed by either weakintramolecular interactions or weak external fields that breakthe symmetry.

VII. APPLICATIONS TO SOME MODELMOLECULAR SYSTEMS

The properties of the 6 paradigm molecular systems con-sidered are listed in Table I, including the calculated val-ues of the ground vibronic level entanglement S0 and thefirst-excited vibronic level entanglement S1; the appropriatemolecular data points are also marked on Fig. 4. Ammonia,benzene, CT, and BNB all appear on the E0/¯ω = 0 dia-gram in Fig. 4 and are found reasonably close together, de-spite their qualitatively different chemical properties; this isindicative of the broad range of chemical systems that can bemodeled. CT and BNB fall in a region of persistent entangle-ment, there being no significant change in S0 and S1 at E0/¯ω= 0.01 compared to E0/¯ω = 0 for these molecules, but theentanglements for ammonia decrease dramatically with thissmall amount of asymmetry. This effect is quantified in Ta-ble I in which the derivatives dS/d(E0/¯ω) are listed for eachmolecule: for the ground vibronic level, these are −0.050 and−0.007 for CT and BNB, respectively, but −420 for am-monia. Indeed, the deduced sensitivity of the entanglementfor ammonia indicates that, in any conceivable quantum de-vice exploiting entanglement, robust operation will requirean extreme level of environmental isolation. For benzene theentanglement is highly insensitive to environment but its mag-nitude is very small for both the ground and first-excited vi-bronic levels, <0.03, and any practical quantum device wouldneed to mix say the ground vibronic with some very high-energy level in order to generate significant entanglement. In-trinsically asymmetric systems such as 3PYR and PRC cangenerate significant entanglement with intermediate levels of

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244110-9 Electron-vibration entanglement J. Chem. Phys. 135, 244110 (2011)

environmental sensitivity and so in this respect are not funda-mentally incompatible with the needs of a practical quantumdevice.

Isomerization in closed-shell molecules such as ammo-nia and benzene intrinsically generate multiple strongly in-teracting excited-state potential-energy surfaces, and, while itis possible to accurately represent the ground electronic statefor these systems using effective two-state models, the trans-formation into this form is not diabatic and so generates ad-ditional entanglement between electronic and vibrational de-grees of freedom not embodied in the two-state model. Insupplementary material,56 we show the entanglements of theground vibronic level and first excited vibronic levels calcu-lated using multi-state models and compare them to those al-ready shown in Fig. 4. While adding the additional coupledstates introduces no significant qualitative changes, the regionof persistent entanglement does become restricted even fur-ther towards the region around 2 |J | /λ ∼ 1 and ¯ω/�E ∼ 1,with a tail extending to low vibration frequency for 2 |J | /λ∼ 1. Also, quantitatively changes to the entanglement near2 |J | /λ ∼ 1 are identified, and this effect could not be ig-nored during accurate modeling of device properties.

Often it is possible to use chemical or spectroscopicmeans to modify the basic molecular parameters, opening upthe possibility of dynamically switching entanglement on andoff. CT not only presents the largest persistent entanglementof all the molecules considered but also allows for this pos-sibility. X-ray photoelectron spectroscopy (XPS) can be usedto create a core hole that, because of the small overlap be-tween the core orbitals on the two Ru atoms, becomes 98%localized onto one of the two metal centers,33 introducing anasymmetry96 of E0 = 22 000 cm−1 making E0/¯ω = 28. Thecalculated entanglement for this switched CT is S1 = 0.0007,a reduction by a factor of 500.

VIII. CONCLUSIONS

Double-welled chemical systems by their nature embodyelectron-vibrational entanglement, and the entanglement ofthe lowest-energy eigenfunctions is found to be largest whenthe ground-state wavefunction has a bimodal density profile,with each maximum depicting localization of the wavefunc-tion on one side of the double well. Ammonia provides atypical example of this effect, displaying significant ground-state entanglement of 0.45 and a bimodal density profile, butits well is shallow and supports only 4 localized vibrationallevels.97 Similar entanglement is also manifest by BNB andCT which have double wells but unimodal density profilesthat display just one single maximum near or at the sym-metric configuration. However, the entanglement does de-crease rapidly when double-wells disappear altogether, fallingto <0.1 for benzene. Truly asymmetric systems such as thespecial-pair radical cation in natural photosynthesis and theexcited states of pyridine can also manifest significant entan-glement.

While the degree of entanglement in the ground-statewavefunction provides important information concerning itsnature, the sensitivity of the entanglement to environmen-tal fluctuations is a critical indicator of the suitability of a

molecule in some proposed quantum device. Symmetric sys-tems are shown to possess two types of entanglement, fragileentanglement that is lost in the presence of small environmen-tally induced asymmetries and persistent entanglement thatis robust to such effects. The sensitivity of the entanglementto environment is found to range over 7 orders of magnitudefor the chemical systems considered, being extremely smallfor benzene and extremely large for ammonia. This effectcan have significant consequences in devices that utilize thesewavefunctions. For example, mixing of the ground and first-excited vibronic levels in ammonia has been suggested62 asthe basis of a qubit in a quantum device, the instantaneousstate of such a qubit being of the form

= a |ψ0〉 + b |ψ1〉

=n∑

i=1

(ac1i,0 + bc1i,1)|φ1χi〉 + (ac2i,0 + bc2i,1)|φ2χi〉.

(21)

Sensitivity of the electron-vibration entanglement within eachstate to environment indicates that the contributions c1i, 0 etc.are highly sensitive to environmental fluctuations and there-fore so is the quantum state of the proposed qubit. This is anundesired feature that would demand extreme levels of iso-lation of the qubit from its environment (as happens say indilute gasses and utilized in the ammonia maser) but is un-likely in any solid-state device. However, BNB is shown toprovide a simple chemical system akin to ammonia for whichthe ground-state entanglement is large and persistent, suggest-ing that this may be a useful candidate molecule for use in aquantum device. Nevertheless, this molecule does not haveparameters in the optimal range of 0.02 < 2|J|/λ < 0.5 and0.1 < ¯ω/�E < 1 and so improved molecular candidatesshould be sought before any quantum device is implemented.Interestingly, large molecular systems such as CT and PRCshow both significant ground-state entanglement and low en-vironmental sensitivity, indeed utilizing environmental inter-actions to generate the vibronically active models that wouldactually be exploited in a quantum device. Entanglement isknown to be able to persist in such environments63, 65, 67, 91

and so they cannot be trivially eliminated as candidate sys-tems for quantum devices. Further, we describe a spectro-scopic method by which entanglement could be switched onand off for CT, a useful feature in any device.

Entanglement of the first vibrational level occurs overmuch the same region of the parameter space as does entan-glement of the ground state and shows similar environmentalsensitivity, developing from the oppositely phased linear com-bination of the localized-well zero-point vibrations, but alsooccurs in a much larger region of the parameter space ow-ing to specific new resonances involving locally excited vi-brations. While resonance-driven entanglement is persistent,its manifestation would be much more difficult for most sys-tems and also it would be sensitive to the multi-modal natureof real molecular motions.

Fragile entanglement in ammonia is indicative of thebasic chemical properties of this molecule which has of-ten been used as a model for complex chemical phenomena

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244110-10 McKemmish et al. J. Chem. Phys. 135, 244110 (2011)

such as charge and energy transfer through symmetric dou-ble wells.98, 99 The effect of minute asymmetry in blockingcoherent charge and energy transport has been described100

and can be considered as occurring as a consequence of frag-ile entanglement, as can effects associated with environmen-tally induced asymmetry in electronic states dominated bystatic electron correlation. Thus, there appears an intrinsicconnection between entanglement and fundamental chemi-cal processes. Indeed, a general theory connecting this ef-fect to the Rabi and Golden-rule limits of kinetics in ex-tended systems has been developed101 and clearly parallelsthe changeover between persistent and fragile entanglement,an internal “clock” setting the time scale over which entangle-ment must persist so that the transport obeys quantum ratherthan classical laws.

ACKNOWLEDGMENTS

We thank the National Computational Infrastructure(NCI) for providing computational resources for this project.This work was supported by the Australian Research Council.We thank S. Olsen and X. Huang for helpful discussions.

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