from orbitals to observables and backiopenshell.usc.edu/pubs/pdf/jcp_perspective.pdfquantum...

From orbitals to observables and back

Anna I. Krylov

Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482

[email protected]

Molecular orbital framework is of central importance in chemistry. Often used by

chemists and physicists to gain insight into molecular properties, Hartree–Fock or

Kohn–Sham orbitals are obtained from rather crude treatments and, strictly speak-

ing, are not observables. Yet, quantum mechanics offers a route for connecting

general many-electron wavefunctions with reduced quantities—density matrices and

orbitals—that give rise to observable properties. Such mapping makes possible,

in principle, reconstruction of these objects from sufficiently detailed experimental

data. This perspective discusses Dyson orbitals and various types of natural transi-

tion orbitals and illustrates their role in modeling and interpreting different types of

spectroscopic measurements.

I. INTRODUCTION

Quantum mechanics explains chemical and physical phenomena by using abstract math-

ematical constructs, such as point charges and masses, spins, Hamiltonians, and wavefunc-

tions. Today it is rightfully considered to be the foundation of chemistry, and most general

chemistry textbooks and courses begin with a brief foray into quantum mechanics. However,

back in the nineteenth century, this trait of quantum mechanics, its reliance on mathemat-

ical language and abstractions to describe physics, was not unanimously embraced by the

scientific community. Some of the best minds of the day denied quantum mechanics the

legitimacy of being a scientific discipline because it operates with objects that cannot be

observed.

Such was the opinion of one influential man from the 19th century, Ernst Mach1, a founder

of the Vienna Circle. Mach was not only an accomplished physicist (he even has a principle

named after him) but also a philosopher of science. The school of thought that he helped to

shape was called Logical Positivism, and one of its key principles was that science should be

based entirely on directly observable phenomena2. This sounds like something most of us

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subscribe to today, but there was a caveat in what Mach meant by observable phenomena.

Mach defended a type of phenomenalism that recognized only sensations as real. That is,

things you can see, hear, feel, or taste. Consequently, Mach considered atoms and molecules

to be artificial constructs of the mind. He famously declared after an 1897 lecture by Ludwig

Boltzmann at the Imperial Academy of Science in Vienna: “I don’t believe that atoms exist!”

These debates were eventually settled, and we now have a much broader view of what

observables are. We do not need to rely on our unaugmented faculties to interact with

the physical world. We accept that what we can see through microscopes, telescopes, or

spectrometers is real. But the main idea so ardently promoted by Mach is still valid—

that science should only deal with phenomena that can be observed. That is, from the

point of view of physical reality, a mathematical construct is only valid if one can design

an experiment that can, in principle, probe it. With this in mind, let us talk about a very

basic concept—orbitals. Are they real or just a mathematical construct?

When a common chemist hears “Quantum Mechanics”, she thinks “Orbitals”. Orbitals

are used to explain the structure of the periodic table of elements, the existence of chemical

bonds, trends in reactivity, and the colors of compounds. The role of molecular orbitals

in providing rational explanation of chemical transformations was recognized by the 1981

Nobel Prize given to Kenichi Fukui and Roald Hoffmann3.

Molecular orbital theory is now a truly native language of chemistry. Just as a small

child, without understanding its grammar and rules, learns to speak his native language

with great effectiveness, chemists learn how to speak orbitals before they learn quantum

mechanics. Indeed, some chemists never learn quantum mechanics, but use molecular orbital

theory with great effectiveness.

In this perspective, I wish to discuss molecular orbital theory with the experts, chemists

who not only speak the language, but also understand and appreciate its grammar. So let

us revisit the fundamentals and discuss the place of orbitals in quantitative and rigorous

physical chemistry.

In one-electron systems orbitals are wavefunctions. In a hydrogen-like atom, orbitals

correspond to its electronic states. In molecules, atomic orbitals give rise to delocalized

molecular orbitals, as shown in Fig. 1 for H+2 . The delocalization lowers the kinetic energy

of the electrons, gluing atoms into molecules. This is the quantum mechanical nature of

what we call a chemical bond4,5. Known as MO–LCAO theory (molecular orbital-linear

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FIG. 1: Orbital diagram explaining bonding in H+2 . Two atomic orbitals give rise to the in-phase

(σ) and out-of-phase (σ∗) molecular states, called molecular orbitals. The energy splitting between

σ and σ∗, which arises due to the different nodal structure (orbital shapes), explains chemical bond

formation.

combination of atomic orbitals), a qualitative generalization of this exact result is used to

explain trends in molecular properties at all levels, from freshman textbooks to state-of-the-

art research papers.

Being wavefunctions, orbitals of one-electron systems are connected to observables. Their

squares give the probability of finding an electron at a certain position in space, known as

electron density. Electron density can be probed directly by, for example, STM (scanning

tunneling microscopy), classic and ultrafast electron diffraction or X-ray scattering (see,

for example, Refs. 6,7). Different shapes of electron densities give rise to concrete prop-

erties, such as multipole moments. Differences in the shapes of orbitals corresponding to

different states encode optical properties—for example, one-photon transition between σ

and σ∗ orbitals in H+2 is induced only by light polarized along the molecular axis. Thus,

experiments that probe these properties probe the wavefunctions, and, consequently, orbital

shapes. Given sufficiently detailed information, orbital shapes can be reconstructed from

the experimental data. Hence, these orbitals are observables.

But what about many-electron systems? Their wavefunctions depend on the coordinates

of all electrons. They are explicitly non-separable and entangled. Orbitals are no longer

formally required to describe electronic states and their properties, rather, they are invoked

when discussing approximations to the true wavefunctions. For example, molecular orbitals

appear in the mean-field approximation, the Hartree–Fock model in which an N -electron

4

many-body wavefunction is approximated by the antisymmetrized product of N orbitals

(called the Slater determinant)8:

Φ0 =1√N !P−1,2,3,...,n[φ1(1)φ2(2) . . . φN(n)] (1)

In this picture, orbitals describe states of pseudo-independent electrons, a very crude repre-

sentation of real correlated electrons. Even more troubling is that the key properties—energy

and electron density—do not depend on a specific choice of orbitals—any linear combination

of the occupied Hartree–Fock orbitals is equally good. So are orbitals just a mathematical

instrument devoid of concrete physical meaning?

The meaning of orbitals becomes even murkier when one considers strongly correlated

wavefunctions, which contain many Slater determinants, so that one can no longer rely

on the pseudo-independent electron model to assign a physical meaning to the individual

orbitals. So, one may wonder, do orbitals make any sense at all in many-electron systems?

In fact, orbitals do make sense and we can connect them to experimental observables.

And we can do this without invoking approximations—molecular orbital concepts are valid

within rigorous many-body formalisms. Molecular orbital theory is fully compatible with the

exact quantum mechanical description of many-electron systems, and orbitals are related to

physical observables, so that they are themselves observables. This contribution reviews the

formalism behind this rigorous connection and presents examples highlighting the utility of

quantitative molecular orbital theory in modern quantum chemistry.

Our strategy is to follow a specific observable and to define orbitals from the perspective

of how they relate to the chosen observable. This means that different observables give

rise to different orbitals. For example, Dyson orbitals are related to photodetachment and

photoionization cross sections probed by photoelectron spectroscopy (or to electron-impact

ionization cross sections), natural transition orbitals are related to linear spectroscopies (UV-

VIS, XAS, etc), natural (density) difference orbitals (giving rise to attachment-detachment

densities) are related to spectroscopies probing differences in states’ properties (e.g., Stark-

effect spectroscopies), and so on. Thus, we abandon the idea of a single definition of the

best molecular orbitals—in the author’s opinion, they do not exist. I also do not consider

orbital definitions that cannot be related to potentially measurable quantities—in this re-

spect, I share the sentiment that science should concern itself with observables. Different

experiments probe different types of orbitals—hence, what you see depends on the tool you

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use to interrogate your system. This lack of a single definition of the true orbitals is not a

flaw of the theory, rather, it is fully in accord with the quantum mechanical world, in which

the probe and the system that is probed are interconnected.

II. TRUE ORBITALS FOR EVERY OCCASION

A. Dyson orbitals

As the first example of experimentally defined orbitals, let us consider a photoioniza-

tion/photodetachment experiment. Within dipole approximation, the probability of ejecting

an electron is given by the photoelectron matrix element9:

Dk = u〈ΨNI |µ|ΨN−1

F Ψelk 〉, (2)

where µ is the dipole-moment operator, u is a unit vector along the polarization of ionizing

radiation, ΨNI and ΨN−1

F denote the initial and the ionized target state, and Ψelk is the

wavefunction of the ejected electron. Both ΨNI and ΨN−1

F are many-body wavefunctions,

but their explicit knowledge is not required for computing matrix elements of one-electron

operators, such as, for example, the dipole moment

µ = −N∑

i=1

ri. (3)

Eq. (2) is equivalent to10:

Dk = −u〈φd(r)|r|Ψelk (r)〉. (4)

Here φd(r) is a one-electron function, which contains all the necessary information about

the molecular system before and after it is ionized,

φd(1) =√N

∫ΨNI (1, . . . , n)ΨN−1

F (2, . . . , n)d2 . . . dn =∑

p

〈ΨNI |p†|ΨN−1

F 〉φp(1), (5)

where p† is the creation operator corresponding to orbital φp from the set of orthonormal

spin-orbitals used to construct the many-electron basis functions (Slater determinants used

to expand Ψ). In the derivation of this result11–14 we relied on the indistinguishability of

the electrons, which allowed us to carry out the integration over N electrons in two steps,

thus reducing the many-body integral in Eq. (2) to integration over the coordinates of just

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one electron. The only approximation here is the assumption that Ψelk is orthogonal to

the molecular core, which is usually justified by the large size of the free-electron state (this

assumption can be lifted, giving rise to a slightly bulkier expression, with overlap corrections

to the main term).

This function, φd(r), defined by Eq. (5), is called a Dyson orbital (or a generalized overlap

amplitude)11,14–20. Dyson orbitals can be defined for any type of wavefunction, including the

exact one—thus, they do not rely on the pseudo-non-interacting electron model21. They

simply tell you what the difference between the N and N -1 electron system is. Thus, they

can be interpreted as the initial state of the ejected electron.

Dyson orbitals also appear in the electron-impact ionization (e, 2e) cross sections22,23.

Electron-impact ionization is an example of electron-momentum spectroscopy, formulated

in the momentum space. The (e, 2e) cross section is proportional to the energy-momentum

spectral function:

S(Ei,p) =∑

av

|φdi (p)|2, (6)

where∑

av denotes averaging over the initial and final state degeneracies and φdi (p) is the

Dyson orbital corresponding to ionized state i (with ionization energy Ei) in the momentum

space:

φdi (p) =1

(2π)3/2

∫e−iprφdi (r)dr. (7)

The Dyson orbital in the coordinate space can be obtained by the reverse Fourier transform

of φdi (p).

Because Dyson orbitals are directly related to the

photoionization/photodetachment/electron-impact ionization cross sections, they can

be reconstructed from the experimental data and are, therefore, observable quantities.

Although the process of the reconstruction is not at all trivial (more on this below), here

we merely point out that experimental imaging of Dyson orbitals is, in principle, possible.

Alternatively, we can compute experimental observables from a given Dyson orbital and

judge the quality of the computed wavefunctions based on how well the theoretical and

experimental results agree. An example in Fig. 2 illustrates this point. Absolute pho-

toionization cross sections are very sensitive to the quality of Dyson orbitals. In a detailed

benchmark study14, we compared experimental absolute cross sections of the photoionization

of small molecules with the theoretical values based on the Dyson orbitals computed from

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0.0

0.2

0.4

0.6

0.8

10.5 11.0 11.5 12.0 12.5

Theory

Tota

l Cro

ss-S

ectio

n (a

.u.)

Energy of Ionizing Radiation (eV)

FIG. 2: Absolute photoionization yield of formaldehyde. In this energy range, photoelectron is

removed from the in-plane lone pair orbital (shown on the graph). The experimental values of the

cross section (shown in atomic units) are well reproduced by calculation using correlated Dyson

orbitals14.

coupled-cluster (CC) and equation-of-motion coupled-cluster (EOM-CC) wavefunctions24–26.

Figure 2 shows such a comparison for formaldehyde. As one can see, theory and experiment

are in perfect agreement. Importantly, using lower-quality orbitals (e.g., Kohn–Sham or

Hartree–Fock) leads to noticeable discrepancies. Thus, the ability to reproduce the absolute

photoionization cross sections provides a measure of the closeness of the shape of the Dyson

orbital to the exact one (of course, the calculation also requires the photoelectron wavefunc-

tion, as per Eq. (4); hence, the quality of the computed cross section depends also on the

quality of Ψelk ).

However, although total cross sections are highly sensitive to the shape of the Dyson

orbital, we cannot really reconstruct the orbital from the total cross section, which is a

too-deeply integrated quantity. Indirect information about orbital shapes can be deduced

from vibrational progressions and Franck–Condon factors, but to really see the orbitals,

one needs more detailed experimental observables. Differential cross sections, which provide

angular-resolved photoelectron spectra, contain such information. For randomly oriented

molecules, the photoelectron angular distribution (PAD) is completely characterized by the

dipole anisotropy parameter β: β ∼0 corresponds to isotropic distribution of photoelectrons,

β ∼ 2 corresponds to the photoelectrons aligned parallel to the polarization of the ionizing

radiation, and so on. Patterns in the PADs can be rationalized from dipole selection rules.

8

As illustrated in Fig. 3, ionization from an s-orbital gives rise to a p-wave (with β=2),

ionization of a p-orbital gives rise to interfering s- and d-waves, and so on.Photoelectron Spectroscopy and Imaging

referring the reader to ref. 16 for a description of the rest ofour instrumentation.

In the idealized case of a monoenergetic electron and in theabsence of an external field, the position of the photoelectronat a given time t (measured from the photodetachment event)will lie on the surface of a sphere of radius vt. The center of thesphere coincides with the photoelectron origin. From thequantum mechanical perspective, the probability distributionof the position of the electron on the surface of the sphereconforms to |c|2, where c is the photoelectron wavefunction.Only when the position is measured, can we talk about alocalized position of the electron. Suppose for a moment thatwe employ a spherical position-sensitive detector surroundingthe photoelectron origin. Measurement with this detector willlocate the electron at a particular point on the sphere. That is,the act of measurement collapses the delocalized probabilitydistribution on the single point of observed electron impact.Repetitions of the experiment under exactly the sameconditions generally yield different impact positions. A largenumber of such repetitions will eventually map out the |c|2

probability density on the surface of the detector.In reality a spherical detector is impractical for photo-

electron spectroscopic measurements. Instead, the inherentcharge of the electron allows the use of electric fields to collectall of the photoelectrons by accelerating them towards a planardetector located some distance away from the photodetachmentorigin (see Fig. 1). The detector usually consists of a pair ofmicrochannel plates acting as a position-sensitive electronmultiplier, equipped with a phosphor screen. Impacts appearingon the screen are accumulated using a digital camera.

For a point electron source and uniform electric fielddirected along the y axis, which is perpendicular to thedetector plane (xz), this accumulation would eventually yielda 2-D projection of the 3-D probability density onto thedetector. Furthermore, since the applied field is perpendicularto the detector face the image would also represent the velocitydistribution within the xz plane. For multiple electron sources(ions spread throughout the interaction volume), any imagerecorded using a uniform field directed along y will consist of aseries of overlapping velocity distributions, resulting in spatialblurring. The ‘‘velocity mapping’’ electrode arrangement firstintroduced by Eppink and Parker2 minimizes this problem.The three electrodes shown in Fig. 1 create an inhomogeneouselectric field. In classical terms we might think of this asfunneling electrons onto specific points on the detector,which depend only on the velocity vector, regardless of thestarting point. In quantum mechanical terms the part of theprobability density corresponding to a particular electronvelocity is mapped onto a specific point on the detector(the reader is referred to ref. 2 for a full description). Withinthis ‘‘velocity mapping’’ arrangement the final imagerepresents the momentum–space distribution of the electronsin the plane of the detector.2,17

The photoelectron distribution produced with a polarized(or even an unpolarized, but directional) laser beam is notnecessarily isotropic. It is these anisotropic characteristicsof photodetachment that enable one to gain insights intothe symmetry properties of the parent-ion electron orbitalsand the dynamics of the detachment process. For linear

polarization, the electric field vector (ep) defines a unique axisin space (z). The resulting 3-D velocity distribution of thephotoelectrons varies with respect to y, the angle between theelectron velocity vector and the z axis, but is cylindricallysymmetric about z. The cylindrical symmetry allows amathematical reconstruction of the nascent 3-D velocitydistribution from its 2-D projection onto the detector planeusing the inverse Abel transformation.

3. The photoelectron image

The following discussion is based on the photoelectron imagesshown in Fig. 2 and 3, corresponding to the photodetachmentof H! at 800 nm and I! at 267 nm, respectively. For H!, asimilar 796 nm result was reported by Reichle et al.18

The process of image acquisition makes a compellingillustration of the Born interpretation of the wavefunctionand also demonstrates the nature of measurement in quantummechanics. As discussed in the previous section, individualelectrons are observed as localized impacts on the detectorsurface (Fig. 2a). With limited statistics, the impact spotsappear randomly distributed, but after many experimentalcycles, a pattern emerges (Fig. 2b) reflecting the probabilitydensity distribution. The radial intensity distributions in these

Fig. 2 Photoelectron imaging of H! at 800 nm. (a) An image

accumulated for 7 laser pulses, showing B10 randomly distributed

electron impacts (seen as bright spots). (b) 140 experimental cycles.

(c) 1.1 " 105 experimental cycles. The image is a 2-D projection of the

3-D distribution, which can be recovered from the inverse Abel

transformation, shown in (d). The photoelectron spectrum (e) and the

angular distribution (f) are extracted from the inverse Abel transformed

image. The laser polarization is vertical in the figure plane.

This journal is #c The Royal Society of Chemistry 2009 Chem. Soc. Rev., 2009, 38, 2169–2177 | 2171

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Mabbs, R.; Grumbling, E. R.; Pichugin, K.; Sanov, A. Chem. Soc. Rev., 2009, 38, 2169-2177




















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H-

I-

vθ

images show sharp maxima, while the angular distributionsare distinctly anisotropic with respect to the laser polarizationdirection. We will use these results to illustrate the basics ofimage analysis and interpretation.

3.1 The photoelectron spectrum

In the Abel-inverted image, the electron speed is proportionalto the distance r from the center. Integration over y at eachparticular r yields the radial distribution, or the photo-electron momentum spectrum, which can be converted to theconventional photoelectron energy spectrum via Jacobiantransformation. The spectra of H! and I! in Fig. 2e and 3care shown as functions of electron kinetic energy (eKE).

The peaks in a photoelectron spectrum correspond toelectron ejection accompanied by the population of specificenergy levels of the residual neutral. The single peak in thephotoelectron spectrum of H! is due to the removal of anelectron from the fully populated 1s orbital yielding the 2S1/2ground state of the neutral atom.

For I!, two spin–orbit states, significantly different inenergy, are accessed upon removal of a p-electron using267 nm radiation. The higher-eKE peak corresponds tophotoelectrons produced in coincidence with the ground(2P3/2) electronic state of the iodine atom, while the lowerpeak corresponds to the excited state (2P1/2). The 0.94 eVspacing between the peaks reflects the magnitude of thespin–orbit interaction in the iodine atom. The relativeintegrated peak intensities are proportional to the transitionprobabilities, including the degeneracies of the 2P3/2 and2P1/2 states of iodine.19 The finite peak widths in the energydomain reflect experimental broadening factors, such as laserlinewidth, parent-ion velocity spread, etc.16

3.2 The photoelectron angular distribution

The photoelectron angular distribution (PAD) is obtained byintegrating the intensity of the Abel-inverted image over achosen spectral (i.e. radial) range. By inspection, the H! PAD(Fig. 2f) has a ‘‘parallel’’ character, peaking at y = 0 and p,while the I! PADs (Fig. 3d) for both detachment channelspeak at y = p/2, corresponding to ‘‘perpendicular’’ photo-detachment transitions.The PADs in one-photon photodetachment with linearly

polarized light are generally described by the function:20

I(y) = (s/4p) [1 + bP2(cosy)], (2)

where the proportionality constant s corresponds to the totalphotodetachment cross-section and P2(cosy) " 1/2(3cos2y ! 1)is the second-order Legendre polynomial. The angulardependence is completely defined by b, the anisotropy para-meter, which can be determined by fitting eqn (2) to theexperimental PAD. The b values range from !1 to +2, withnegative and positive values corresponding to perpendicularand parallel polarization, respectively. The b value observed inH! photodetachment is very close to the +2 limit of a purelyparallel transition. For I! photodetachment, however, therings in Fig. 3(a) correspond to b E !0.6.

4. Interpreting photoelectron images

4.1 Relating photoelectron angular distributions to parentorbitals: the H! example

In a pulsed laser experiment the electron is described by asuperposition of waves spanning a finite energy range. In thelaboratory frame, this wavepacket is described as:

C(r,y,f,t) =Ra(k)Ck (r,y,f,t)dk (3)

where a(k) is a shape function related to the laser lineshape.Assuming a central, i.e. spherically symmetric, potentialCk(r,y,f,t) can be separated into radial and angular components,

Ck(r,y,f,t) = Rk(r,t)Y(y,f) (4)

The time dependence is carried in the radial function and thecorresponding electron probability density can be likened to apattern on a spherical balloon. As the ‘‘balloon’’ is inflated,the pattern expands, but its angular dependence remainsunchanged.The connection between a photoelectron image and the

parent orbital can be demonstrated using three different(yet closely inter-related) approaches based on: (1) angularmomentum conservation; (2) transition dipole momentoperations; (3) general symmetry considerations. In thefollowing, we adopt the one-electron picture of photo-detachment and discuss the uniquely simple case of H! toillustrate the key ideas in photoelectron image interpretation.

4.1.1 Conservation of angular momentum. The selectionrule for one-photon, one-electron atomic transitions isDc = #1, where c is the orbital angular momentum quantumnumber. In detachment from the hydrogen 1s orbital (ci = 0),the free-electron wave is characterized by a single c value of 1,corresponding to a p wave.

Fig. 3 Photoelectron imaging of I! at 267 nm. (a) The photoelectron

image. (b) Inverse Abel transformation. The two distinct rings corres-

pond to the different peaks in the photoelectron spectrum (c). The

photoelectron angular distribution (d) corresponding to the 2P3/2

transition peaks at p/2, in contrast to the H! detachment in Fig. 2

and contrary to classical expectations.

2172 | Chem. Soc. Rev., 2009, 38, 2169–2177 This journal is $c The Royal Society of Chemistry 2009

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I(y) = (s/4p) [1 + bP2(cosy)], (2)


















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View Article OnlinePhotoelectron Spectroscopy and Imaging




















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εP

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Mabbs, R.; Grumbling, E. R.; Pichugin, K.; Sanov, A. Chem. Soc. Rev., 2009, 38, 2169-2177




















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I(y) = (s/4p) [1 + bP2(cosy)], (2)


















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I(y) = (s/4p) [1 + bP2(cosy)], (2)


















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µ<latexit sha1_base64="YUIm7Tsc5b5P2pMGLKWRa4Nw0cU=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvgqexWQY9FLx4r2g9ol5JNs21okl2SrFCW/gQvHhTx6i/y5r8xbfegrQ8GHu/NMDMvTAQ31vO+UWFtfWNzq7hd2tnd2z8oHx61TJxqypo0FrHuhMQwwRVrWm4F6ySaERkK1g7HtzO//cS04bF6tJOEBZIMFY84JdZJDz2Z9ssVr+rNgVeJn5MK5Gj0y1+9QUxTyZSlghjT9b3EBhnRllPBpqVealhC6JgMWddRRSQzQTY/dYrPnDLAUaxdKYvn6u+JjEhjJjJ0nZLYkVn2ZuJ/Xje10XWQcZWklim6WBSlAtsYz/7GA64ZtWLiCKGau1sxHRFNqHXplFwI/vLLq6RVq/oX1dr9ZaV+k8dRhBM4hXPw4QrqcAcNaAKFITzDK7whgV7QO/pYtBZQPnMMf4A+fwBerI3a</latexit>

el<latexit sha1_base64="PnLKfJSMMef3ecrAM3g5TVHWYXU=">AAAB8HicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGC/ZA2ls120i7dbMLuRiihv8KLB0W8+nO8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6mfqtJ1Sax/LejBP0IzqQPOSMGis9dOuaP2YoJr1S2a24M5Bl4uWkDDnqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZwRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjlZ1wmqUHJ5ovCVBATk+n3pM8VMiPGllCmuL2VsCFVlBmbUdGG4C2+vEya1Yp3XqneXZRr13kcBTiGEzgDDy6hBrdQhwYwiOAZXuHNUc6L8+58zFtXnHzmCP7A+fwB/IOQhw==</latexit>x =

FIG. 3: Dipole selection rules in photoionization/photodetachment. Ionization/detachment from

an s-orbital yields a pure p-wave, giving rise to β=2, as in photodetachment from H−. Ion-

ization/detachment from a p-orbital yields interfering s- and d-waves, giving rise to a distinctly

different PAD. Images on the right show PADs for photodetachment from H− and I− (reproduced

from Ref. 27 with permission from the Royal Society of Chemistry).

Thus, PADs (or β’s) encode the information about the shapes of the Dyson orbitals from

which the photoelectrons originate27–29. PADs can be used to quantify the extent of orbital

deformation in the molecular environment due to hybridization and polarization. For ex-

ample, PADs of gas-phase water molecules reveal that the molecular Dyson orbitals can be

described as slightly deformed 2p(O) orbitals, giving rise to β ≈1.5 at high photoelectron

energies (above 100 eV). PADs are also used to interrogate the extent to which a solute’s

wavefunction is perturbed by the solvent29. A recent study30 used angular-resolved pho-

toelectron spectroscopy and the microjets technique to understand the electronic structure

of bulk water, i.e., to assess whether water molecules in bulk retain their electronic identi-

ties or give rise to qualitatively different delocalized band-like states. As shown in Fig. 4,

delocalized states are expected to give rise to an isotropic distribution of photoelectrons,

in contrast to β ≈1.5 of localized states of individual water molecules. So which picture

applies to bulk water? It turns out, the answer depends on the energy of the ionizing ra-

9

diation: in the low-energy regime, one sees delocalized states, whereas in the high-energy

regime one observes the states of the individual waters. The two regimes and the transition

between them can be understood in terms of interference between photoelectrons coming

off individual molecules. When the de Broglie wavelength of photoelectrons is shorter than

the distance between the individual molecules (20 eV, corresponding to 3.5 A, first peak

in g(r) of the liquid water), then there is no interference, and one observes an incoherent

superposition of photoelectrons from the individual centers. Such interplay between the de

Broglie wavelength of photoelectrons and the distance between the emitting centers has been

illustrated by Sanov’s molecular interferometer experiment, showing the transition from the

delocalized and entangled molecular state of I−2 to the incoherent atomic states in the course

of dissociation31. Surprisingly, the microjet measurements of neat water30 show persistent

anisotropy of the photoelectrons, with only moderate reduction of β relative to the gas

phase, due to transport scattering. In stark contrast to the energetic consequences of hy-

drogen bonding and electrostatic interactions, manifested by large (1-2 eV) solvent-induced

shifts and inhomogeneous broadening of the valence bands, these interactions do not result

in reduction of anisotropy, meaning that the valence states of water in bulk can be thought

of as energy-shifted valence states of the isolated water molecules.

The utility of Dyson orbitals goes far beyond the interpretation of photoelectron or elec-

tron momentum (e, 2e) spectra. Because they show the difference between the N and N -1

electron states, the shape of Dyson orbitals can be related to the differences in electron den-

sity and, consequently, bonding patters. In the formaldehyde example (Fig. 2), the Dyson

orbital corresponding to the lowest ionization energy can be described as a lone pair on oxy-

gen (with smaller contributions from σOH). Thus, the ionization of this mostly non-bonding

orbital leads to relatively small changes in the geometry and short vibrational progression

(a small feature at the onset is due to the respective Franck–Condon factors).

In open-shell species (e.g., radicals), Dyson orbitals can be used to describe states of

the unpaired electrons. Fig. 5 shows Dyson orbitals of the four lowest electronic states

in SrOH. The orbitals have clear atomic-like characters, revealing that these species can

be described as a cationic core with the unpaired electron residing on the Sr. Because of

this feature, the electronic transitions in SrOH are atomic-like and the change in the state

of the unpaired electron does not perturb the core. Consequently, the geometry does not

change upon excitation, leading to diagonal Franck-Condon factors. This type of electronic

10

-1.0

0.0

1.0

2.0

0 100 200 300 400 500 600 700

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1p2

1b1,1 + 1b1,2

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1p2

1b1,1 1b1,2

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+ + + +

β

β 1

2

1

2

Kinetic energy [eV]A

niso

tropy

par

amet

er

and/or

A

B

D

C

FIG. 4: How do molecular orbitals of water clusters and bulk water look like? The answer is

encoded in photoelectron dipole anisotropy parameter β. For gas-phase water, β ≈1.5 at high

energies, as expected for ionization from a p-type orbital. Panel A shows model calculations of

water dimer: delocalized states treated with single-center expansion of the photoelectron state

give rise to isotropic distribution (β ≈ 0), whereas the treatment in terms of localized states

with no coherences results in β ≈1.5, as in isolated molecules. The domains of validity of the

two frameworks depend on de Broglie’s wave-length of photoelectrons in relation to the distance

between the centers. In high-energy regime, there is no interference between photoelectrons coming

off the individual waters, such that the resulting PAD reports on the valence orbitals of individual

waters in bulk (shown in panel B). Panels C and D illustrate that delocalized Dyson orbitals of bulk

water yield strikingly different results when treated within the single-center model (which ignores

the coherences in energy domain) and the multi-center model (which ignores the coherences in the

spatial domain). In the low-energy regime, delocalized picture is correct (panel C), whereas in the

high-energy regime, localized picture is correct (panel D). Reprinted with permission from Ref. 30.

Copyright (2020) American Chemical Society.

structure enables multiple optical transitions, which was instrumental in extending laser-

cooling techniques to polyatomic molecules33. The design of laser-coolable molecules has

been guided by the analysis of orbitals involved in optical transitions32,34–38, highlighting

the utility of molecular orbital theory in the emerging field of quantum information science.

11

FIG. 5: Dyson orbitals showing the shape of the unpaired electron in SrOH. Reproduced from Ref.

32 with permission from the Royal Society of Chemistry.

B. Orbitals for excited states

It is difficult to imagine a conversation about excited states and photoinduced processes

without invoking molecular orbitals39,40. Yet, here again we face conceptual challenges: in

many-electron systems, electrons interact with each other, and changing the state of one

electron affects the states of the remaining ones. Hence, a naive Koopmans’ picture of

electronic excitations as removing an electron from an occupied orbital and attaching it to a

virtual orbital is wrong quantitatively and often even qualitatively. For example, in a typical

molecule estimates of the lowest excited-state energy as orbital energy gaps are off by many

electron-volts, due to a large electron–hole interaction energy neglected within the pseudo-

independent electron picture and a poor description of virtual orbitals. Qualitatively, the

character of excited states also often differs from orbital-energy based predictions, especially

when realistic basis sets with diffuse orbitals are used.

Within the lowest level of theory capturing the physics of excited states, configuration

interaction singles (CIS), the ground state is given by a Hartree–Fock determinant (Φ0)

and the target wavefunction is described as a linear combination of singly excited Slater

determinants, Φai :

ΨCIS =∑

ia

caiΦai . (8)

(Here we follow the standard convention: indices i, j, k, . . . denote orbitals occupied in the

reference determinant Φ0 and a, b, c, . . . denote orbitals from the virtual subspace.) The

analysis of the leading amplitudes cai in this expansion enables the interpretation of elec-

12

tronic transitions in terms of hole and particle orbitals. For example, a CIS/6-31G* calcu-

lation of ethylene shows that the lowest excited state can indeed be described as a π → π∗

transition (the weight of the respective Φai ≈0.94), as expected from Koopmans’ theorem

(HOMO→LUMO transition); this is illustrated in Fig. 6. Unfortunately, this type of anal-

ysis (based on the wavefunction amplitudes) is prone to several problems. First, it is not

unique. The CIS energy and excited-state properties are invariant with respect to any

unitary transformation of orbitals within the occupied or virtual orbital spaces, but the

wavefunction amplitudes are not. Thus, choosing different orbitals changes the apparent

character of the wavefunction. One consequence of this is that the weights of the leading

amplitudes are strongly basis-set dependent. Often, in large bases the wavefunction features

several amplitudes with similar weights, which may or may not reflect the true character

of the transition. Second, this analysis quickly becomes intractable when both states ac-

quire multi-configurational character. For example, already at the CIS level, the analysis

of the transitions between two excited states is much more challenging than between the

ground and excited states. In correlated methods, both the ground-state and excited-state

wavefunctions become multiconfigurational and include multiply excited determinants.

To find a way out of this predicament, we begin by considering a relevant experimental

observable. The wavefunction amplitudes are not observables, but the transition density

is41. The transition density is an object that connects the two states:

ρ(xe, xh) =∑

pq

γFIpq φp(xe)φq(xh), (9)

where xe and xh denote electron (particle) and hole coordinates and γFIpq is a one-particle

transition density matrix between the initial and final states:

γFIpq = 〈ΨF |p†q|ΨI〉. (10)

Here p† and q are creation and annihilation operators corresponding to the φp and φq molec-

ular orbitals. The transition density ρ(xe, xh) is related to observable properties because it

defines the probability of transition. For example, the transition dipole moment between

two many-body states is:

〈ΨF |µ|ΨI〉 =

∫ρ(xe, xh)|xe=xh=rrdr = Tr[γFIµ]. (11)

In deriving this expression, we again used electron indistinguishability, following the same

steps as above. Because ρ(xe, xh) is mapped into the observable, it can, in principle be

13

FIG. 6: Lowest excited triplet state in ethylene. In a small basis set (6-31G*), the CIS wavefunction

is dominated by the HOMO→LUMO excitation (the weight of the leading amplitude is 0.94), and

can be easily identified as the π → π∗ transition (the canonical Hartree–Fock MOs are shown on

the left). When the basis is extended to 6-31(2+)G*, several Rydberg orbitals appear below the π∗

valence MO and the CIS wavefunction of the lowest triplet state has two comparable amplitudes

involving both Rydberg and valence like orbitals. What is the true character of this state? The

NTO analysis reveals that the character of the lowest triplet state in the 6-31(2+)G* basis is

the same as in the 6-31G* basis: 3ππ∗. Rydberg triplet state, corresponding to the π → Ry(s)

transition, appears several electron-volt above 3ππ∗. In the singlet manifold, the ordering of the

π → Ry(s) and π → π∗ states is reversed.

reconstructed from the experimental data. When plotted in spatial coordinates, ρ(xe, xh)

provides a visual map of the changes in the electron density. By virtue of Eq. (11), one

can interpret ρ(xe, xh) as an exciton’s wavefunction and use it to rationalize the computed

properties42. For example, the shape of ρ(xe, xh) for different excited states in a bacte-

riochlorophyll dimer shown in Fig. 7 explains the relative intensities and polarization of

the main spectral features and clearly distinguishes local, entangled, and charge-transfer

transitions43.

γFI , the one-particle transition density matrix (1PTDM), contains the coefficients of the

expansion of ρ over molecular orbitals. It is a much simpler object than two many-body

wavefunctions. One can think of it as of a map connecting the two states by moving one

electron at a time, thus extending the CIS-like picture to correlated states:

ΨF =∑

pq

γFIpq p†qΨI + higher excitations. (12)

14

charge-transfer excitations. These states lie just above the Qxtransitions and form a band that overlaps with the Qx Frenkel-exciton band in higher aggregates. Even though the CT modesare essentially forbidden in linear absorption, they show up innonlinear optical studies of these aggregates (e.g., electroab-sorption).We have performed a similar analysis of the higher frequency

Tx1-Tx3 and By transitions (see the second column of Table2), but in the present paper we limit our discussion to the low-frequency Q, Bx, and CT states that participate in the light-harvesting function of LH2.Finally we compare several different Bchl-a dimers with

strong electronic couplings. The third column of Table 2 showsthe 2R1! B850 aggregate (intersubunit). Since the intersubunitand intrasubunit aggregates are similar, the excitation energies

and splittings of these dimers (second and third column) arevery close. In contrast, a much weaker interaction is found indimers 2RB850-1B800 (upper-lower ring) and 2B800-1B800(lower-lower ring) since the center-to-center separation of ∼25Å is larger compared to that of the neighboring molecules ofthe upper ring. Consequently the CT modes of these aggregateslie∼1 eV higher in energy (compare columns 2, 3 with columns4, 5 of Table 2). Additional Bchl dimers and their electroniccouplings will be discussed in the next section.

V. Frenkel-Exciton Hamiltonian for LH2

The CEO analysis of the dimer’s electronic modes presentedin the previous section shows that the interaction between

Figure 3. (A) Structure of the R! B850 dimer. (B) Calculated linear absorption spectrum of the R! B850 dimer. Contour plots the electronicmodes that dominate the optical absorption of the B850 dimer. The axis labels represent the individual atoms. Atoms of each monomer are labeledaccording to panel A of Figure 2. The panels indicate the electronic mode according to panel B. The color code is given in Figure 2. Modefrequencies are given in Table 2.

4524 J. Phys. Chem. B, Vol. 104, No. 18, 2000 Tretiak et al.

FIG. 7: Analysis of excited states using transition density matrices in a bacteriochlorophyll dimer

from the LH2 antenna complex of purple bacteria. (A) Structure of the dimer. (B) Calculated

linear (i.e., one-photon) absorption spectrum of the dimer. Contour plots of the transition densities

corresponding to the main spectral transitions (the first panel shows the ground-state density).

The axis labels represent the individual atoms. The plots clearly show the extent of exciton

delocalization. Charge-transfer transitions (CT1 and CT2) are revealed by the off-diagonal pattern.

Reprinted with permission from Ref. 43. Copyright (2000) American Chemical Society.

Although it is not sufficient for complete reconstruction of ΨF from ΨI , γ contains all

the information needed to compute the transition dipole moment and, hence, relates to an

15

experimentally observable property. For the CIS transitions from the ground state, i.e., when

ΨF = ΨCIS and ΨI = Φ0, only the occupied-virtual block of γ is non-zero and γia = cai .

The difference between two electronic states can be also described by density difference44:

∆(r) =∑

pq

∆FIpq φp(r)φq(r), (13)

∆FIpq = 〈ΨF |p†q|ΨF 〉 − 〈ΨI |p†q|ΨI〉. (14)

This quantity provides an alternative measure of the change in electronic structure upon

the transition and is related to the change in the one-electron state properties (such as

permanent dipole moment); hence, it is also related to observables. At the CIS level (for

the transitions from the ground state), γ and ∆ are closely related and contain identical

information45; however, for correlated wavefunctions (or transitions between the CIS states)

this is no longer the case. For example, ∆ can be used to characterize the differences between

the states that are multiply excited with respect to each other, whereas γ only captures the

differences in terms of one-electron excitations.

While ρ(xe, xh) and ∆(r) are invariant with respect to different orbital choices—any set

of orthonormal orbitals can be used in Eqs. (11)-(14)—the values of the individual elements

of γpq and ∆pq depend on the specific orbital choice in the same way the wavefunction

amplitudes do. Rather than obscuring the intrinsic nature of electronic transitions, this

arbitrariness is the key to deriving a rigorous and clear orbital picture. One can find the

best set of orbitals, which represents ∆pq and γFI in the most compact way.

An important difference between the two objects is that the state density matrix is Her-

mitian, whereas the transition density matrix is not46. Hence, the most compact representa-

tion of ∆pq is achieved by simple diagonalization, yielding natural density-difference orbitals

(NDOs) and their occupations, which add up to zero. By interpreting negative occupations

as electron detachment from state ΨI and positive occupations as electron attachment to

state ΨF , one can compute promotion numbers, which quantify how many electrons need

to be simultaneously excited to describe the transition. NDOs and their occupation num-

bers yield attachment and detachment densities, which compactly represent the difference

between the two states44,47.

In the case of the non-Hermitian 1PTDM, γFI , the most compact representation is ac-

complished by the singular value decomposition

γ = VΣUT, (15)

16

where Σ is the diagonal matrix of singular values, σK , and matrices V and U contain the

hole and particle (electron) NTOs according to

ψhK (r) =∑

q

UqKφq (r) (16)

and

ψeK (r) =∑

q

VqKφq (r) . (17)

(Note that hole and particle orbitals are not orthogonal to each other). In the NTO repre-

sentation, ρ has the most compact form:

ρ(xe, xh) =∑

K

σKψeK(xe)ψ

hK(xh). (18)

Usually, only a few σK are non-vanishing. The squares of the σKs can be interpreted as the

weights of the respective NTO pair when divided by the square of the Frobenius norm (Ω)

of γ:

Ω ≡ ||γ||2 =∑

pq

γ2pq =∑

k

σ2K . (19)

Ω is also useful for identifying transitions that have predominantly one-electron character

(Ω=1 for a pure single-electron transition, such as between the ground state and a CIS

excited state)48.

NTOs are an old and extremely useful concept44,45,47,49–57, which provide a way to describe

electronic excitations in terms of molecular orbitals for any type of wavefunction. NTOs are

also directly related to the observables:

〈ΨF |µ|ΨI〉 = Tr[γµ] =∑

K

σK〈ψeK |µ|ψhK〉, (20)

that is, the matrix element between the two many-body states can be expressed by a (short)

sum of matrix elements between the hole and particle NTOs. This allows one to use an

orbital picture to explain trends. For example, the orbitals shown in Fig. 6 make it immedi-

ately clear that in the singlet manifold both types of transitions carry noticeable oscillator

strength, but are polarized differently: π → π∗ has the transition dipole along the molecular

axis, whereas π → Ry(3s) has the transition dipole perpendicular to the molecular plane.

Fig. 8 illustrates the ability of NTOs to unambiguously determine the character of the

excited states. The CIS/aug-cc-pVDZ wavefunction of the lowest singlet state features 5

17

leading amplitudes (which collectively account for only 75% of the wavefunction), suggesting

a naive interpretation of this transition as having multi-configurational Rydberg-valence

character. However, the NTOs computed for this wavefunction reveal a pure (99.1 %)

valence lp(O) → π∗ transition. Thus, the mixed multi-configurational appearance in the

basis of canonical orbitals is just an artifact of the representation.

0.991

0.069

0.305 0.149 0.120 0.108

FIG. 8: Lowest excited singlet state in uracil, CIS/aug-cc-pVDZ. Top: weights of the leading

CIS amplitudes in the basis of canonical Hartree–Fock orbitals show multiple configurations with

similar weights (these configurations account for 75% of the wavefunction). Bottom: NTO analysis

of this wavefunction reveals simple lp(O) → π∗ valence transition (99.1 % of the wavefunction is

accounted for by this NTO pair).

The ability of NTOs to provide an essential picture of the transition becomes even more

important when using correlated wavefunctions. For example, in EOM-CCSD, both the

ground-state and excited-state wavefunctions include single and double excitations, making

amplitude-based analysis even more ambiguous. Yet, NTOs distill these complex wavefunc-

tions into a simple orbital picture. Fig. 9 shows NTOs for the bright XAg →1Bu transition

and dark XAg →4Ag transition in trans-stilbene computed using EOM-CCSD/daug-cc-

pVDZ wavefunctions. The analysis shows that 1Bu can be described as a simple π → π∗

excitation, whereas 4Ag has more complex character and requires 2 pairs of NTOs. Being

18

4Ag

Trans stilbene, EOM-CCSD/daug-cc-pVDZ basis:

0.70

1Bu

0.54

0.21

FIG. 9: Trans-stilbene, EOM-CCSD/daug-cc-pVDZ. NTO analysis of the bright XAg →1Bu tran-

sition (top) and dark XAg →4Ag transition (bottom). NTO weights (σ2K) are shown for each NTO

pair.

independent of the correlation treatment and the basis, NTOs enable comparisons of wave-

functions computed by different methods55, including comparisons between wavefunction

treatments and TD-DFT.

The number of NTOs with significant weights reveals the inherent collectivity of the tran-

sition and can be used to quantify quantum entanglement of the exciton. Several collectivity

indices are used. For example, the participation ratio (PRNTO) is defined as:

PRNTO =(∑

K σ2K)2∑

K σ4K

=Ω2

∑K σ

4K

(21)

where Ω = ||γ||2. For a transition between a reference and a CIS target state, PRNTO equals

1 for a CIS target state with exactly one configuration, 2 for a CIS state represented by two

configurations with equal weights, and so on. Thus, pure excitations (such as ππ∗ or π →Ry(s) transitions in ethylene) are revealed by PRNTO ≈1, whereas states of mixed character

(e.g., Rydberg-valence) would have larger values. Plasmonic excitations are identified by

very large PRNTO.

PRNTO is closely related to the number of entangled states, an alternative metric58 also

19

defined on the basis of the 1PTDMs:

ZHE = 2SHE = 1/∏

i

λλii , (22)

where λi = σ2i /Ω are renormalized squares of the singular values of 1PTDM and SHE is the

hole-electron entanglement entropy:

SHE = −∑

i

λi log2 λi. (23)

This definition gives the same answer in the idealized example above, but has an advantage

of being grounded in thermodynamics, which gives rise to the correct scaling with the system

size. For example, utilizing the additivity of entropy allows one to properly account for spin

entanglement in spin-adapted calculations (i.e., the total SHE is the sum of the αα and

ββ parts of the transition). Thus, for singlet–singlet transitions ZHE accounting for spin-

entanglement is the square of ZHE computed from the spinless transition density. A recent

study36 of novel molecular frameworks for quantum information science used these metrics

to investigate the entanglement between two optical cycling centers attached to the same

molecular scaffold. The calculations on model systems (Me-O-CC-OMe′, Me=Mg, Ca, Ba,

Sr) showed that the extent of entanglement and electronic communication can be controlled

by the diradical character.

NTOs obtained from 1PTDM can always be used to describe the difference between the

two states, however, their connection to spectroscopic observables requires further discussion.

As per Eq. (20), NTOs can be used to understand the trends in one-particle transition

properties. For example, from Fig. 9, one can easily see why the XAg →1Bu transition is

bright and the XAg →4Ag is dark in linear (UV-VIS) spectroscopy. However, it is not at

all obvious why the XAg →4Ag has a very large two-photon absorption (2PA) cross section,

simply because the NTOs shown in Fig. 9 do not directly relate to the 2PA cross section.

The probability of a two-photon transition between two states depends not only on the

initial and final state, but also on all other states of the system. For example, the 2PA

transition moments are given by the following sum-over-states (SOS) expressions59–61:

M f←ixy (ω1x, ω2y) = −

∑

n

(〈Ψf |µy|Ψn〉〈Ψn|µx|Ψi〉Ωni − ω1x

+〈Ψf |µx|Ψn〉〈Ψn|µy|Ψi〉

Ωni − ω2y

), (24)

M i←fxy (−ω1x,−ω2y) = . . .

20

Here, Ωni = En−Ei and ω1 and ω2 are the frequencies of the two absorbed photons (polarized

along x- and y-directions) such that ω1 + ω2 = Ωfi.

Eq. (24) reflects important physics of the 2PA and other nonlinear optical processes—the

properties of the transition can no longer be understood by considering only the initial and

final states. Rather, one needs to consider all states of the system. Except for resonant

regimes (such as a 2PA transition in which one of the photons is resonant with a bright

excited state Ψn), the SOS expressions cannot be truncated to a few terms without significant

loss of accuracy, presenting an obstacle to deriving molecular orbital pictures of non-linear

phenomena, such as 2PA or RIXS (resonant inelastic X-ray scattering).

This obstacle can be overcome by exploiting response theory, the same approach used

to circumvent the calculation of all excited states entering the SOS expressions in practical

calculations of the 2PA and RIXS cross sections. By using response theory, Eqs. (24) can

be recast as:

M f←ixy (ω1x) = −

(〈Ψf |µy|Xω1x

i 〉+ 〈Xω1xf |µy|Ψi〉

), (25)

M i←fxy (−ω2y) = . . .

where the response states, X and X, are computed by solving auxiliary response

equations60–63. Within the formalism outlined in Refs. 61 and 63, the EOM-CC response

states are:

Xω1xk (H − Ek + ω1x) = 〈Ψk|µx|ΦI〉, (26)

(H − Ek − ω1x)Xω1xk = 〈ΦI |µx|Ψk〉, (27)

where ΦIs denote Slater determinants from the target-state manifold. The physical mean-

ing of response states becomes clear once we recognize that they describe the first-order

response64 of Ψk to the perturbation with frequency ωx:

|Xω1xk 〉 ≡ |Ψ

(1)k 〉 =

∑

n

|Ψn〉〈Ψn|µx|Ψk〉Ωnk − ω1x

, (28)

〈Xω1xk | ≡ 〈Ψ

(1)k | =

∑

n

〈Ψn|µx|Ψk〉Ωnk + ω1x

〈Ψn|. (29)

Hence, one can express (see Eq. (25)) 2PA cross sections as matrix elements of the dipole

operator between the initial state and the first-order perturbed final state plus the matrix

element between the perturbed initial state and the final state. The resulting approach

21

is formally and numerically equivalent to the SOS expressions. The perturbed states X,

which assimilate the contributions from all real excited states of the system, provide con-

crete meaning to the so-called “virtual states”, which are commonly invoked in non-linear

spectroscopies.

Following Eq. (25), one can define and compute the respective 1PTDMs (called perturbed

1PTDMs) and extract the NTOs. Such analysis delivers simple molecular orbital pictures

of nonlinear processes. Fig. 10 shows NTOs for the bright 2PA transition in ethylene

(XAg →2Ag). As one can see from the regular (linear or one-photon) NTOs, the 2Ag state

has πRy(pz) character. The NTO analysis of the Mzz and Mxx 2PA moments shows that

the former moment is dominated by the pathway via the Ry(s) virtual state, whereas the

latter moment involves the π∗ virtual state.

Recently, following the same steps, we extended the NTO analysis of perturbed density

matrices to the RIXS process65. Fig. 11 shows the orbital picture of the RIXS transitions

in benzene derived from the NTOs computed from perturbed density matrices. As in the

case of 2PA, these NTOs are directly mapped onto the cross sections. Without making

simplifying assumptions, they reveal the character of the virtual state responsible for the

transitions. In the benzene example, the character of the virtual state changes from valence

π∗-like to Rydberg s-like depending on the frequency of the excitation pulse.

22

One-photon NTOsTwo-photon NTOs (Mzz)

Two-photon NTOs (Mxx)

One-photon NTOsTwo-photon NTOs (Mzz)

Two-photon NTOs (Mxx)

FIG. 10: One-photon (top) and two-photon (bottom) NTOs for the XAg →2Ag transition in

ethylene; EOM-CCSD/daug-cc-pVDZ. The final excited state can be described as Ry(pz). 2PA

NTOs reveal virtual states responsible for 2PA: Mzz and Mxx moments can be described as π →

Ry(s)→ Ry(pz) and π → π∗ → Ry(pz) transitions, respectively.

23

15

virtual molecular orbitals (not shown) are doubly degenerate LUMO and di↵use s and p

Rydberg orbitals.

The two dominant features in the XAS spectrum of benzene42–49 are peak A and peak B at

285.97 eV and 287.80 eV, respectively (theoretical values21 computed with fc-CVS-EOM-EE-

CCSD/uC-6-311(2+,+)G**). When the incoming photon’s energy is tuned to the peak-A

resonance, the dominant inelastic feature is the energy-loss peak at 10.67 eV, characterized

by equal contributions from the degenerate XAg ! 13B2g and XAg ! 12B3g transitions. In

contrast, when the incoming photon’s energy is tuned to the peak-B resonance, the dominant

inelastic feature is the energy-loss peak at 6.45 eV, characterized by equal contributions

from the degenerate XAg ! 1B2g and XAg ! 1B3g transitions. Below we show the NTO

analysis of only the XAg ! 13B2g and XAg ! 1B2g RIXS transitions with incoming photon

energies tuned at peak-A and peak-B resonances, respectively. The NTO analyses for the

two transitions are similar except for the di↵erences in symmetry labels of the orbitals.

The NTO analysis for the dark one-photon XAg ! 13B2g transition (given in the SI)

suggests that this valence transition is made up of two orbital transitions: b2u ! au and

b3u ! b1u. Similarly, the NTO analysis of XAS peak A (XAg ! c2B1u) transition in

Fig. 6 shows two dominant orbital transitions: cb1g ! au and cag ! b1u. Similarly, the

FIG. 6: NTO analysis of the XAg ! 13B2g RIXS transition in benzene. Comparison of important

NTO pairs computed separately for the x-ray absorption (XAg ! c2B1u) and x-ray emission

(c2B1u ! 13B2g) transitions with the NTO pairs computed from the RIXS 1PTDMs shows that

the three-states model is adequate for this RIXS transition. Both orbital channels contribute

significantly into this transition.

16

NTO analysis of the c2B1u ! 13B2g x-ray emission shows two dominant orbital transitions:

b2u ! cb1g and b3u ! cag. Based on these analyses, the three-states model for the XAg !13B2g RIXS transition identifies two important orbital channels: b2u ! cb1g ! au and

b3u ! cag ! b1u.

The g ! f and f ! g RIXS moment tensors are dominated by the imaginary zx

components. This is also reflected in the norms of the imaginary 1PTDMs given in the SI,

which are a few orders of magnitude larger than those of real 1PTDMs (Im = 1.00). The

NTO analyses of the z,Im and x,Im RIXS 1PTDMs of the XAg ! 13B2g shown in Fig.

6 and in the SI identifies two dominant near-resonance orbital channels: b2u ! cb1g ! au

and b3u ! cag ! b1u. In other words, cb1g and cag are the intermediate core orbitals that

facilitate the two-photon inelastic scattering, driving the electronic density from the b2u

orbital to the 2au orbital and from the b3u orbital to the 4b1u orbital, respectively. This

orbital analysis of the RIXS 1PTDMs is consistent with the approximate analysis from the

three-states model.

The one-photon XAg ! 1B2g transition is dark; its NTO analysis given in the SI reveals

its dominant HOMO-LUMO character (b2g ! ag) and a miniscule contribution from the

b1u ! b3u transition. The NTO analysis of XAS peak B transition (XAg ! c2B3u) shown

FIG. 7: NTO analysis of the 1Ag ! 1B2g RIXS transition in benzene. Comparison of impor-

tant NTO pairs computed separately for the x-ray absorption (1Ag ! c2B3u) and x-ray emission

(c2B3u ! 1B2g) transitions with the NTO pairs computed from the RIXS 1PTDMs shows that

the three-states model is adequate for this RIXS transition. Orbital channel 1 provides dominant

contributions.

FIG. 11: NTO analysis of perturbed densities reveals different channels giving rise to RIXS emission

in benzene. Top: Orbital picture of RIXS transition due to resonant excitation at 286 eV. Bottom:

Orbital picture of RIXS transition due to resonant excitation at 288 eV. Reprinted from Ref. 65,

with the permission of AIP Publishing.

24

C. Orbital picture of spin-forbidden transitions and magnetic properties

Molecular orbital theory has been used to explain relativistic phenomena, ranging from

orbital contraction in heavy elements due to scalar relativistic effects to inter-system crossing

(ISC), intensity borrowing, and magnetic anisotropies facilitated by spin-orbit interactions.

The last effect plays an important role even in molecules composed of light atoms, as it cou-

ples otherwise non-interacting states (e.g., singlets and triplets), opening up new relaxation

pathways.

In photochemistry, the rates of ISC are often explained by El-Sayed’s rules66,67, which

state that a large SOC (spin-orbit coupling) can be expected for transitions involving a

flip of an orbital, such as in [(π)2(n)1(π∗)1] → [(π)2(n)2(π∗)0], whereas transitions that

do not involve orbital flip, such as in [(π)1(n)2(π∗)1] → [(π)2(n)2(π∗)0], would have much

smaller SOC and, consequently, slower ISC. El-Sayed’s rules are based on the analysis of

the one-electron part of the Breit–Pauli Hamiltonian, which is responsible for the SOC. As

illustrated in Fig. 12, the SOC term has the form of a local angular momentum operator,

whose matrix representation reveals the need for orbital torque—for example, in the basis

of three p-orbitals, the only non-zero matrix elements of Lz are between px and py.

Can we use the concept of NTOs to develop a quantitative version of El-Sayed’s rules

applicable to correlated many-body wavefunctions? The conceptual difficulty is that the

SOC is a tensorial property that depends on the matrix elements between all multiplet

components. For example, the SOC constant (SOCC), which enters the expressions for ISC

rates and spectral line splittings, involves the sum over all spin-projections within the two

coupled multiplets (e.g., the three Ms components of a triplet state, in the case of singlet-

triplet couplings). Thus, calculation of NTOs between one multiplet component (such as

the Ms=0 component, most often available from electronic structure calculations) is not

sufficient for quantitative analysis of the SOCC. This obstacle can be overcome by using

Wigner-Eckart’s theorem and introducing a new type of NTOs appropriate for the analysis

of spin-forbidden phenomena68. The definition of spin-orbit NTOs68 is based on the spinless

density matrices that are used for the calculation of the SOCs for the entire multiplet from

just one transition69,70. This approach allows one to treat the transitions between states with

arbitrary spin projections in a uniform way and to quantitatively describe the contributions

of specific orbital pairs to the overall SOC; hence, these NTOs are linked to experimentally

25

observable quantities (e.g., ISC and oscillator strengths). Fig. 12 shows NTOs computed for

the SOC within the quintet manifold in an iron-contained single-molecule magnet described

by the EOM-EA-CC ansatz68,70. A large spin-reversal barrier71 of this single-molecule mag-

net can be explained by the shapes of these NTOs. In the spirit of El-Sayed’s rule, these

two orbitals, which can be described as dzy and dz2 , give rise to non-zero matrix elements of

the angular momentum operator.

Analysis: Extended NTOs

Example: tris(pyrrolylmethyl)amine Fe(II) complex,quintet-quintet transition, EOM-EA-MP2

ω=0.87

cm1 NTO integrals ! Reduced matrix elements (exact)

hS||HSOL ||Si 31.79 + 224.48i 32.07 + 228.23i

hS||HSOL0

||Si 0.04i 0.07i

hS||HSOL+

||Si 31.79 224.44i 32.07 228.23i

12/19

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(n)<latexit sha1_base64="bLd8oljV8VqhTI6XlWU5qwwy0NA=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRaheii7VdBj0YvHCvZD2rVk02wbmmSXJCuUpb/CiwdFvPpzvPlvTNs9aOuDgcd7M8zMC2LOtHHdbye3srq2vpHfLGxt7+zuFfcPmjpKFKENEvFItQOsKWeSNgwznLZjRbEIOG0Fo5up33qiSrNI3ptxTH2BB5KFjGBjpYey7Mbs8ewU9Yolt+LOgJaJl5ESZKj3il/dfkQSQaUhHGvd8dzY+ClWhhFOJ4VuommMyQgPaMdSiQXVfjo7eIJOrNJHYaRsSYNm6u+JFAutxyKwnQKboV70puJ/Xicx4ZWfMhknhkoyXxQmHJkITb9HfaYoMXxsCSaK2VsRGWKFibEZFWwI3uLLy6RZrXjnlerdRal2ncWRhyM4hjJ4cAk1uIU6NICAgGd4hTdHOS/Ou/Mxb8052cwh/IHz+QNZj490</latexit>

Analysis: Angular momentum

h1e(i) =X

K

ZK (ri RK) pi

|ri RK |3

Angular momentum around K

Rapid decay with distance

Real harmonics, the same center:

Lz =

0B@

py pz px

py 0 0 i

pz 0 0 0

px i 0 0

1CA L+ =

0B@

py pz px

py 0 i 0

pz i 0 1

px 0 1 0

1CA

El-Sayed rules: large SOC is achieved when the orientation ofp-orbitals changes, e.g. px ! pz.

9/19


h1e(i) =X

K

ZK (ri RK) pi

|ri RK |3




Lz =

0B@

py pz px

py 0 0 i

pz 0 0 0

px i 0 0

1CA L+ =

0B@

py pz px

py 0 i 0

pz i 0 1

px 0 1 0

1CA


9/19

FIG. 12: El-Sayed’s rules for spin-forbidden transitions. Top: One-electron part of the Breit–Pauli

Hamiltonian has a form of local angular momentum on center K (marked by purple), owing to the

rapid decay with distance (denominator marked by pink). Matrix representation of Lz illustrates

that non-zero matrix elements involve the change of angular momentum (flip of the orbital). Top

right: By using this analysis, El-Sayed’s rules explain small magnitude of SOC for ππ∗ transitions

(no orbital flip) and large magnitude of SOC for nπ∗ transitions in organic molecules. Bottom:

NTOs of the spinless density matrix of Fe(II) single-molecule magnet explain large magnitude of

SOC and illustrate El-Sayed’s rules for d-orbitals. Reproduced with permission from Ref. 68.

D. Orbital concepts in non-Hermitian quantum chemistry

Standard quantum chemistry deals with electronically bound states (i.e., states stable

with respect to electron ejection). These states are discrete and their wavefunctions are

finite (L2-integrable). Ground states and low-lying excited states of many molecules are of

this type. Yet, many important physical phenomena take us outside the bound part of the

spectrum into the electronic continuum72–75. Examples include highly excited states (any

26

excited state above the ionization threshold), core-level states, excited and sometimes even

ground states of anions (i.e., transient anions), and molecules on metal surfaces. In the

continuum, there are no discrete states and all wavefunctions are decaying asymptotically.

Yet, the continuum has a structure, revealed spectroscopically. It contains states, called

resonances, that behave as bound states, but with a finite lifetime. In Hermitian quantum

mechanics, these states correspond to an increased density of states in certain parts of the

continuum. Hence, standard quantum chemistry is not applicable—neither for a quantitative

treatment of these states nor for their conceptual analysis. The increased density of states

does not have a potential energy surface governing nuclear motion. The increased density of

states cannot be easily interpreted using molecular orbital theory. Yet, spectral signatures

of resonances, such as vibrational progressions and angular distributions of photoelectrons,

suggest that the concepts of Born–Oppenheimer separation of nuclear and electronic motion

and a molecular orbital picture of the transitions, do apply to these states hidden in the

continuum.

Non-Hermitian quantum mechanics72,73,75 offers an elegant framework for tackling reso-

nances. By recasting the Schrodinger equation into a non-Hermitian form, one can project

out the continuum and describe the resonances as isolated states with L2-integrable wave-

functions and complex energies. The real part of the resonance energy corresponds to the

resonance position and the imaginary part corresponds to the resonance width (inverse life-

time). By using complex potential energy surfaces76–79, one can describe nuclear dynamics

associated with resonances. Similar to bound states, observable properties of resonances are

encoded in the respective state and transition density matrices. Within the non-Hermitian

formalism, these densities become complex-valued80,81. For example, Dyson orbitals and

NTOs acquire real and imaginary components.

What can we learn from these complex-valued orbitals? In order to understand the

physical meaning of real and imaginary densities, one needs to relate them to experimental

observables. The lifetime of a resonance is the key observable. It is related to the imaginary

density, opening a route to interpretation. Further insight into the meaning of imaginary

densities and orbitals can be obtained by following the Feshbach-Fano formalism82. Within

this framework81, wavefunctions of metastable excited states (such as, for example, excited

states in many anions83) can be analyzed in terms of real and imaginary excitons via direct

extension of Eq. (12). Figure 13 shows such real and imaginary NTOs for two excited

27

states of CN−. The real NTO pairs reveal that the two states can be described as π → π∗

and σ → π∗ excitations. The imaginary NTOs characterize the decay channels due to the

coupling to the continuum—the hole orbitals are very similar to the real hole orbitals, but

the particle orbitals have different nodal structure and are more diffuse. Figure 14 shows real

and imaginary NTOs for C3N− and C7N

−. The NTOs in the 1∆ state of C3N− are similar

to CN−, revealing π → π∗ character of the bound part of the exciton and a p-like decay

channel. In the 1Σ+ state of C7N−, the NTOs have more complex character: the bound

part of the exciton is a mixture of π → π∗ and σ → σ(Ry) excitation and the imaginary

part shows two different decay channels. Note that the relative weights of the configurations

involving the π and σ holes are flipped in the real and imaginary parts—in the real exciton,

the contribution of the σ hole is minor, whereas in the imaginary part it becomes the leading

component. Recall that, while the singular values for the real part have similar meaning as

in the bound-state calculations (they describe the character of the transition), the singular

values of the imaginary part are related to the partial widths of the resonance81. Hence,

this flip of leading σK reveals that this state of ππ∗ character decays predominantly via the

σσ(Ry) channel.

24



cm1 NTO integrals Reduced matrix elements (exact)

hS||HSOL ||Si 31.79 + 224.48i 32.07 + 228.23i

hS||HSOL0

||Si 0.04i 0.07i

hS||HSOL+

||Si 31.79 224.44i 32.07 228.23i

12/19




h1e(i) =X

K

ZK (ri RK) pi

|ri RK |3




Lz =

py pz px

py 0 0 i

pz 0 0 0

px i 0 0

L+ =

py pz px

py 0 i 0

pz i 0 1

px 0 1 0


9/19


h1e(i) =X

K

ZK (ri RK) pi

|ri RK |3




Lz =

py pz px

py 0 0 i

pz 0 0 0

px i 0 0

L+ =

py pz px

py 0 i 0

pz i 0 1

px 0 1 0


9/19

FIG. 12: El-Sayed rules for spin-forbidden transitions. Top: One-electron part of the Breight-Pauli


rapid decay with distance (denominator marked by pink). Matrix representation of Lz isssutrates

that non-zero matrix elements involve the change of anumar moemnum (flip of the orbital). Top

right: By using this analysis, El-Sayed rules expain small magntitude if SOC for transitions

(no orbital flip) and large magnitude of SOC for n transitions in organic molecles. Bottom:

NTOs of the spinless density matrix of Fe(II) SMM explain large magnitude of SOC and illustrate

El-Sayed rules for d-orbitals.

Orbital characterization of the excited states

Resonances correspond to valence transition of the type (+, , ) or ()

1+ resonance in C7N is an exception – it has a strongly mixed and characterW. Skomorowski Real and imaginary excitons 9 / 14

Redo diagrams

FIG. 13: Real and imaginary NTOs in the two excited states of CN.

24



cm1 NTO integrals Reduced matrix elements (exact)

hS||HSOL ||Si 31.79 + 224.48i 32.07 + 228.23i

hS||HSOL0

||Si 0.04i 0.07i

hS||HSOL+

||Si 31.79 224.44i 32.07 228.23i

12/19




h1e(i) =X

K

ZK (ri RK) pi

|ri RK |3




Lz =

py pz px

py 0 0 i

pz 0 0 0

px i 0 0

L+ =

py pz px

py 0 i 0

pz i 0 1

px 0 1 0


9/19


h1e(i) =X

K

ZK (ri RK) pi

|ri RK |3




Lz =

py pz px

py 0 0 i

pz 0 0 0

px i 0 0

L+ =

py pz px

py 0 i 0

pz i 0 1

px 0 1 0


9/19

FIG. 12: El-Sayed rules for spin-forbidden transitions. Top: One-electron part of the Breight-Pauli


rapid decay with distance (denominator marked by pink). Matrix representation of Lz isssutrates

that non-zero matrix elements involve the change of anumar moemnum (flip of the orbital). Top

right: By using this analysis, El-Sayed rules expain small magntitude if SOC for transitions

(no orbital flip) and large magnitude of SOC for n transitions in organic molecles. Bottom:

NTOs of the spinless density matrix of Fe(II) SMM explain large magnitude of SOC and illustrate

El-Sayed rules for d-orbitals.

Orbital characterization of the excited states

Resonances correspond to valence transition of the type (+, , ) or ()

1+ resonance in C7N is an exception – it has a strongly mixed and characterW. Skomorowski Real and imaginary excitons 9 / 14

Redo diagrams

FIG. 13: Real and imaginary NTOs in the two excited states of CN.

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<latexit sha1_base64="UyKAkdDl8Y1enA6pUASuPNFQvcY=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSKIh5JUQY9FLx4rmLbQxrLZbtqlm03YnQil9Dd48aCIV3+QN/+N2zYHbX0w8Hhvhpl5YSqFQdf9dlZW19Y3Ngtbxe2d3b390sFhwySZZtxniUx0K6SGS6G4jwIlb6Wa0ziUvBkOb6d+84lrIxL1gKOUBzHtKxEJRtFKficVj+fdUtmtuDOQZeLlpAw56t3SV6eXsCzmCpmkxrQ9N8VgTDUKJvmk2MkMTykb0j5vW6pozE0wnh07IadW6ZEo0bYUkpn6e2JMY2NGcWg7Y4oDs+hNxf+8dobRdTAWKs2QKzZfFGWSYEKmn5Oe0JyhHFlCmRb2VsIGVFOGNp+iDcFbfHmZNKoV76JSvb8s127yOApwDCdwBh5cQQ3uoA4+MBDwDK/w5ijnxXl3PuatK04+cwR/4Hz+AGvUjm0=</latexit>


FIG. 13: Real and imaginary NTOs for the Π (top) and Σ+/∆/Σ resonances in CN−. CAP-EOM-

EE-CCSD.

28

C3N-

C7N-

1

Partial widths: Γ(Σ) = 0.10 eVΓ(Π) = 0.03 eVΓtot = 0.13 eV

14

Real

Real

Real

Imaginary

Imaginary

Imaginary

! = #. %&

! = #. %&

! = #. '(

! = #. )(

! = #. #%

! = #. #%

FIG. 3: Real and imaginary NTOs and their corresponding singular values K for the 1+ resonance

state in C7N.

Figure 4 shows average hole sizes for the valence triplet and singlet states in C2n+1N,

n = 0, . . . , 3. As one can see, in the states the hole size remains nearly the same (or

even shrinks slightly), because orbital is localized, whereas in and + states it increases

with n, indicating the delocalized nature of the orbital. The shapes and sizes of real

and imaginary hole NTOs are similar. In contrast, the sizes of the real and imaginary

particle orbitals are di↵erent (see Fig. S2 in SI), the latter being more di↵use because of

their continuum origin. These di↵erent trends in sizes of holes and particles are also clearly

reflected in the average hole-particle separation, as shown in Fig. 5. Imaginary excitons

are characterized by significantly larger hole-particle distance than the real excitons, which

illustrates our point of Im enhancing the decay channels via the virtual states.

We also analyze electron-hole correlation, Reh, (shown in Fig. S3 in SI). As expected for

molecular excitonic states, the real excitons show positive correlation, indicative of electron-

hole attraction resulting in a correlated motion of a hole and a particle. The magnitude of

Reh(Re) is similar for all states and is close to the values in Mg porphyrin reported recently in

Ref. 33. Interestingly, the imaginary excitons show much smaller correlation and sometimes

even negative values, meaning that an electron and a hole move independently (or away

from each other) and that the imaginary exciton is unbound. This is again consistent with

the interpretation of the imaginary part of the exciton as a virtual decay channel.

Finally, Fig. 6 shows the correlation between the magnitude of the imaginary part of

the exciton with the resonance width. Indeed, as suggested by Eqs. (30) and (31), there

13

in Fig. 3, we estimate = 0.13 0.142

0.142+2·0.052 = 0.10 eV and = 0.13 2·0.0.052

0.142+2·0.052 = 0.03 eV.

All other resonances identified in cyanopolyyne anions8 are of shape type decaying almost

exclusively through a single decay channel, which is reflected by the respective NTOs and

their singular values.

In addition to these valence excited states, some cyanopolyynes also support dipole-

stabilized states. Figure S1 in SI shows NTOs for the 3+ dipole-stabilized resonance of

C5N. The shapes of NTOs clearly reveal the di↵erent nature of this state. The imaginary

part of the exciton has a d-wave character.

Real

Imaginary

Real

Real Real

Imaginary

CN- C3N-

C5N- C7N-

! = #. %&

! = #. '&

! = #. %(

! = #. #&

! = #. )% ! = #. )%

FIG. 1: Real and imaginary NTOs and their corresponding singular values K for the 3 state in

four cyanopolyyne anions: CN, C3N, C5N

, and C7N.

Real

Real

Real

Real

Imaginary

Imaginary

Imaginary

Imaginary

CN-

C3N-

! = #. %%

! = #. %%

! = #. &'

! = #. &'

! = #. &(

! = #. &(

! = #. #)

! = #. #)

FIG. 2: Real and imaginary NTOs and their corresponding singular values K for the 1 state in

CN and C3N.

1+

FIG. 14: Real and imaginary NTOs in C3N− (top) and C7N

− (bottom); CAP-EOM-EE-CCSD.

Reprinted with permission from Ref. 81. Copyright (2018) American Chemical Society..

29

III. CAN WE REALLY SEE THEM?

LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS3212

−0.05 V−0.26 V−0.72 V

−0.15 V−0.35 V−0.81 V

dI/d

V

Hig

hLo

w

−1.0 −0.5 0.0 0.50

1

2

3a

b

c

d

A4 A3 A2

A4v A3v A2v

−1.0 −0.5 0.00

1

2

3

dI/d

V (a

.u.)

E − E0N (eV)

LDO

S (a

.u.)

A2

A2v

A3A3v

A4A4v

Sample bias (V)

CoPC−1

ψ⎜ 2N ⟩

ψ⎜ 0N ⟩

ψ⎜ 2N ⟩ ψ⎜ 1

N ⟩ ψ⎜ 0N ⟩ ψ⎜ 0

N +1⟩

ψ⎜ 1N ⟩

Figure 3 | Wavefunction mapping on the negatively charged CoPC. a,b, Experimental high-resolution dI/dV spectra (a) and simulated LDOS (b) ondifferent points over the CoPC−1. Spectra are offset for clarity. The hump at ∼−0.65V in the blue spectrum in a was not reproducibly found on othermolecules (see Supplementary Information) and thus not assigned to a molecular resonance. c, Experimental dI/dV maps at different energies,corresponding to the different resonances (middle row) and their vibronic replica (upper row). The lower row shows simulated LDOS maps for the MBtransitions from CoPC−1 to CoPC0. d, STM image overlaid with the structural model of CoPC, indicating the positions of the experimental and simulatedspectra, respectively. Scale bar is 1 nm, tunnelling parameters are −1.51 V, 8 pA.

0.5 1.0 1.5 2.0

0.0 0.5 1.00

1

2

3 −1.01 VN3 1.93 VN1vv1.69 VN10.98 VN2

1.79 VN1v1.08 VN2v

dI/d

V

Hig

hLo

w

N2

N2v

N1

N1vN1vv

Sample bias (V)

E − E0N+1 (eV)

LDO

S (a

.u.)

0

1

2

3a c d

b

dI/d

V (a

.u.)

CoPC0

ψ⎜ 0N+1 ⟩

ψ⎜ 2N+1 ⟩

ψ⎜ 5N+1 ⟩

ψ⎜ 2N+1 ⟩ ψ⎜ 5

N+1 ⟩ψ⎜ 0N−1 ⟩ ψ⎜ 0

N ⟩ ψ⎜ 0N ⟩ ψ⎜ 0

N+1 ⟩

Figure 4 | Wavefunction mapping on the neutral CoPC. a,b, Experimental high-resolution dI/dV spectra (a) and simulated LDOS (b) on different pointsover the CoPC0. Spectra are offset for clarity. c, Experimental dI/dV maps at different energies, corresponding to the different resonances (middle row)and their vibronic replica (upper row). The lower row shows simulated LDOS maps for the MB transitions from CoPC0 to CoPC−1, as well as from CoPC0

into the CoPC+1 ground state at negative energy. d, STM image overlaid with the structural model of CoPC, indicating the positions of the experimental andsimulated spectra, respectively. Scale bar is 1 nm, tunnelling parameters are −2.00V, 2 pA.

of the states from which or into which the tunnelling occurs.The dI/dV spectra taken on different positions over the CoPCmolecule can be simulated by multiplying the total DOS shown inFig. 2 by the squares of the relevant wavefunctions at the positionof the STM tip. Figure 3a shows experimental high-resolutionspectra taken on three different positions over CoPC−1. Clearly, therelative intensities of the A2, A3 and A4 resonances change withposition. This is in good agreement with the simulated spectra for

the ground state |Ψ N0 ⟩ and the MB excited states |Ψ N

1 ⟩ and |Ψ N2 ⟩

(Fig. 3b), and directly reflects the different symmetries and nodesof the wavefunctions involved in the transitions (see SupplementaryInformation for simulated SP spectra). Their shape and symmetrycan be directly visualized by recording the dI/dV signal in theconstant-height mode, allowing comparison with simulated LDOSmaps (Fig. 3c). The wavefunctions of the excited states |Ψ N

1 ⟩ and|Ψ N

2 ⟩ both have four-fold symmetry but different nodal planes.

232 NATURE PHYSICS | VOL 11 | MARCH 2015 | www.nature.com/naturephysics

© 2015 Macmillan Publishers Limited. All rights reserved

FIG. 15: STM image of neutral cobalt phthalocyanine (CoPC) overlaid with the molecular struc-

ture. Experimental differential conductance maps at different energies, corresponding to the differ-

ent resonances (middle row) and their vibronic replica (upper row). The lower row shows simulated

local density of states maps for the transitions from CoPC0 to CoPC1 and from CoPC0 to CoPC+1,

representing squares of the Dyson orbitals of the electron-attached and ionized states, respectively.

Reproduced with permission from Ref. 84.

The theory provides a solid foundation for experimental orbital imaging, but design

and implementation of such experiments are far from trivial. The interpretation of orbital

imaging experiments is still controversial, highlighting the need for more robust theoretical

modeling and data processing frameworks. Despite these challenges, there are many exciting

examples illustrating our progress towards direct experimental orbital imaging.

STM enables atomic-scale imaging of molecules adsorbed on surfaces. By measuring

changes in the local molecular density of states under different applied voltages, STM can

provide real-space images of the electron density of the Dyson orbitals that connect the

neutral molecules with their respective negative and positive ions. Figure 15 shows images

for cobalt phthalocyanine (CoPC). The experimental differential conductance maps reveal

clear nodal structure of the Dyson orbitals, in good agreement with the computed maps.

However, the phase information is not available and, at present, STM imaging is limited to

the two-dimensional picture. Additional complications involve perturbation of the electronic

structure by the STM tip and the surface.

30

Advanced light sources offer new possibilities for orbital imaging, such as the use of high-

order harmonic generation85. An addition, strong fields can be used to align molecules in

space, which eliminates orientational averaging and, therefore, increases the amount of in-

formation that can be used for reconstruction. Angular-resolved photoelectron experiments,

especially when executed for oriented molecules, can deliver three-dimensional images of

orbitals86–89, however, the exact details of the reconstruction process remain a key chal-

lenge. Given that the interpretation of these experiments often involves certain simplifying

assumptions (such as a plane-wave treatment of photoelectrons) and multiple steps of con-

verting the raw data into the image, one may argue that we are not yet truly able to see

the orbitals, at least not in a way that would convince Mach and his fellow members of the

Vienna Circle. Challenges of the full tomographic orbital imaging aside, the mapping be-

tween orbitals and observables is indisputable, and this connection provides a sound guiding

principle for responsible use of orbital concepts in physics and chemistry.

IV. CONCLUSION

In this contribution, I have discussed molecular orbital concepts from the point of view

of many-body electronic structure and observable molecular properties. The examples il-

lustrate how simple molecular orbital pictures can be rigorously distilled from correlated

wavefunctions by using reduced quantities such as one-particle transition density matrices.

By using this formalism, molecular orbitals can be related to molecular properties and spec-

troscopic observables. Thus, orbitals are not a mere mathematical abstraction; they are

real and can be observed. However, the type of orbitals you see depends on which tool you

use to look at them. This is an intrinsic feature of quantum mechanics — the result of the

observation depends not only on the state of the system but also on the act of observation.

Orbital concepts are instrumental in deriving physical insight from spectroscopy. Here again

different spectroscopies relate to different types of orbitals. I conclude with a general guiding

principle for using molecular orbital concepts: Follow the observable. Or design a new one!

31

Acknowledgments

I wish to dedicate this work to Dr. Vladimir Ivanovich Pupyshev (08/06/1949-

06/16/2020), who passed away90 as this manuscript was being finalized. Dr. Pupyshev was

an inspiring teacher and a gentle mentor, who ignited my interest in quantum chemistry

and encouraged my initial clumsy attempts in theoretical work. Dr. Pupyshev was able to

convey the conceptual beauty of quantum mechanics while maintaining the highest level of

mathematical rigor, always spiced with a good joke, a literary quote, or a thought-provoking

sharp remark. His scientific style has provided a constant inspiration for my work.

This work was supported by the U.S. National Science Foundation (No. CHE-1856342).

Conflicts of interest

A.I.K. is the president and a part-owner of Q-Chem, Inc.

Data availability statement

Data sharing is not applicable to this article as no new data were created or analyzed in

this work.

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exams—I am completely blank on which part of Mach’s teaching Lenin deemed to be at odds

with Marxist’s philosophy. Its low intellectual content and poor scholarly quality notwithstand-

ing, ”Materialism and Empirio-criticism” impacted science in a profound way. The arrogance of

a poorly educated revolutionary zealot attacking competent scientists heralded the beginning

of a new era in which ideology trumps the science and politicians censor it. In the spirit of

”Materialism and Empirio-criticism”, Soviet commissars declared entire scientific disciplines as

32

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