The exact molecular wavefunction as a product of an electronic and a nuclearwavefunctionLorenz S. Cederbaum

Citation: The Journal of Chemical Physics 138, 224110 (2013); doi: 10.1063/1.4807115 View online: http://dx.doi.org/10.1063/1.4807115 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/138/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Response to “Comment on ‘Correlated electron-nuclear dynamics: Exact factorization of the molecularwavefunction”' [J. Chem. Phys.139, 087101 (2013)] J. Chem. Phys. 139, 087102 (2013); 10.1063/1.4818523 Molecular structure calculations: A unified quantum mechanical description of electrons and nuclei usingexplicitly correlated Gaussian functions and the global vector representation J. Chem. Phys. 137, 024104 (2012); 10.1063/1.4731696 Electronic excited-state energies from a linear response theory based on the ground-state two-electron reduceddensity matrix J. Chem. Phys. 128, 114109 (2008); 10.1063/1.2890961 Cumulant reconstruction of the three-electron reduced density matrix in the anti-Hermitian contractedSchrödinger equation J. Chem. Phys. 127, 104104 (2007); 10.1063/1.2768354 A computationally efficient exact pseudopotential method. II. Application to the molecular pseudopotential of anexcess electron interacting with tetrahydrofuran (THF) J. Chem. Phys. 125, 074103 (2006); 10.1063/1.2218835

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THE JOURNAL OF CHEMICAL PHYSICS 138, 224110 (2013)

The exact molecular wavefunction as a product of an electronicand a nuclear wavefunction

Lorenz S. Cederbauma)

Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229,D-69120 Heidelberg, Germany

(Received 10 January 2013; accepted 6 May 2013; published online 13 June 2013)

The Born-Oppenheimer approximation is a basic approximation in molecular science. In this ap-proximation, the total molecular wavefunction is written as a product of an electronic and a nuclearwavefunction. Hunter [Int. J. Quantum Chem. 9, 237 (1975)] has argued that the exact total wave-function can also be factorized as such a product. In the present work, a variational principle isintroduced which shows explicitly that the total wavefunction can be exactly written as such a prod-uct. To this end, a different electronic Hamiltonian has to be defined. The Schrödinger equation forthe electronic wavefunction follows from the variational ansatz and is presented. As in the Born-Oppenheimer approximation, the nuclear motion is shown to proceed in a potential which is theelectronic energy. In contrast to the Born-Oppenheimer approximation, the separation of the centerof mass can be carried out exactly. The electronic Hamiltonian and the equation of motion of thenuclei resulting after the exact separation of the center of mass motion are explicitly given. A simpleexactly solvable model is used to illustrate some aspects of the theory. © 2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4807115]

I. INTRODUCTION

The Born-Oppenheimer approximation1, 2 is a milestonein the theory of molecules and of electronic matter in gen-eral. The much larger masses of the nuclei compared to thatof electrons allow for an approximate separation of the elec-tronic and nuclear motions and this separation simplifies thequantum as well as classical treatment of molecules substan-tially. One should be aware that even the notion of molecularelectronic states is connected to this approximation.

As usual, one writes the molecular Hamiltonian

H = TN + Hel (1)

as a sum of the kinetic energy operator TN of the nucleiand the electronic Hamiltonian Hel which governs the elec-tronic motion at fixed nuclear configurations. Molecules arerather complicated quantum objects and the solution � of theSchrödinger equation for the total molecular Hamiltonian Hin (1) is, in general, rather involved. In the Born-Oppenheimerapproximation, the ansatz

�BO = ϕ(r; R)χ (R) (2)

for the wavefunction is a product of an electronic wave-function ϕ and a nuclear wavefunction χ . Here, r and Rstand symbolically for all electronic and nuclear coordinates,respectively. The electronic Schrödinger equation

Helϕ = Eel(R)ϕ (3a)

defines an electronic energy Eel(R) which, of course, dependson the nuclear coordinates. The electronic wavefunction

a)Electronic mail: lorenz.cederbaum@pci.uni-heidelberg.de

ϕ(r; R) is normalized at all values of the nuclear coordinates∫|ϕ|2dr = 〈ϕ|ϕ〉el = 1. (3b)

The index “el” indicates that the integration is over the elec-tronic coordinates r only.

The main idea behind the Born-Oppenheimer approxima-tion is that due to the smallness of the mass of the electronscompared to that of the nuclei, the electronic wavefunctionvaries very smoothly with R and is not affected by the oper-ation of TN. If so, one can insert the ansatz (2) into the fullSchrödinger equation, integrate out the electronic contribu-tion and obtain the illuminating result

[TN + Eel(R)]χ = EBO χ, (4)

where EBO is the total energy of the molecule in the Born-Oppenheimer approximation. The essence of this centralequation is that the nuclei move in the potential provided bythe electronic energy Eel(R). This quantity is hence called thepotential energy surface. In their recent paper, Sutcliffe andWoolley3 discuss that this surface does not arise naturallyfrom the solution of the Schrödinger equation for the molecu-lar Coulomb Hamiltonian and present arguments substantiat-ing their view.

The Born-Oppenheimer approximation as provided byEqs. (2)–(4) has been extremely useful in numerous applica-tions and is generally widely applied. It has become a standardreference even in cases where it fails. This approximationdoes fail, often severely, in particular for polyatomicmolecules whenever the potential energy surfaces belong-ing to different electronic states come close to each other.The most dramatic failure is encountered when these sur-faces exhibit conical intersections.4 The nuclear motion then

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224110-2 Lorenz S. Cederbaum J. Chem. Phys. 138, 224110 (2013)

proceeds over the potential surfaces of the involved elec-tronic states which are now coupled by this motion. To de-scribe the nuclear motion, one usually introduces a groupBorn-Oppenheimer approximation for the manifold of cou-pled electronic states.4–8 The total wavefunction is then a sumof products of electronic and nuclear wavefunctions.

Can the Born-Oppenheimer ansatz (2) be improved with-out resorting to a sum of products of electronic and nu-clear wavefunctions? The idea to search for an improved totalwavefunction which is a single product as in (2) is rather old.Pack and Hirschfelder9 searched in 1970 for the best Born-Oppenheimer-like approximation. Hunter10 argued that theexact wavefunction � can be factorized as in (2)

� = φ(r; R)f (R) (5a)

defining a nuclear wavefunction f (R) via the so-calledmarginal probability

|f (R)|2 = 〈�|�〉el . (5b)

An electronic wavefunction φ can be constructed using

φ = �/f (5c)

provided that f (R) is nodeless and the normalization condition

〈φ|φ〉el = 1 (5d)

is chosen. Then, the electronic wavefunction is well definedand a potential energy surface for f (R) could be defined aswell as

〈φ|H |φ〉el . (5e)

The nodeless function f (R) has been further analyzed byHunter11 and several numerical studies on the potential inEq. (5e) have been performed for the H2 molecule and itsion,12, 13 see also Ref. 14 on a related potential. The situationis nicely described in a recent general paper by Sutcliffe.15

The total wavefunction � must be known already before-hand in order to construct by the above procedure the elec-tronic and nuclear wavefunctions φ and f as well as the po-tential in Eq. (5e). That is, these quantities are constructeda posteriori after the full solution of the Schrödinger equationhas been determined.

In the present work, we would like to determine the elec-tronic and nuclear wavefunctions entering the product ansatzof the exact total wavefunction from first principles. To thatend, we introduce a variational principle which leads to theexact factorized result and provides the Schrödinger equationsfor the electronic and nuclear wavefunctions. The electronicwavefunction will, of course, not be the eigenfunction of theelectronic Hamiltonian Hel we are used to. For that purpose,we will have to introduce a different, more involved electronicHamiltonian. We shall show that the nuclear motion in thatcase still proceeds over a potential surface provided by theelectronic energy. The basic ideas are collected in Sec. II. InSec. III, we apply them to a simple exactly solvable model16

which has been used in the literature to study the ingredientsof the Born-Oppenheimer approximation. An application to amore realistic problem has been performed but is beyond thescope of the present paper.

Finally, we would like to mention that the Born-Oppenheimer approximation is also widely used in the con-text of nuclear dynamics. In this case, the time-dependentSchrödinger equation is solved for the nuclear motion in thepotential surface Eel(R). In recent years, theoretical interestin electronic dynamics has emerged as well,17–29 first drivenby curiosity and later on motivated by the advent of sub-femtosecond and attosecond experimental technologies.30–35

To incorporate both the nuclear and the electronic dynam-ics, very recently the full time-dependent Born-Oppenheimerapproximation has been introduced and investigated.36 Here,both the electronic and nuclear wavefunctions depend ontime. To go beyond this approximation, a time-dependentgroup Born-Oppenheimer approximation36 and an exact fac-torization of the wavepacket into an electronic and a nuclearwavepackets37, 38 have been put forward.

II. BASIC IDEAS

A. The factorization

We start with the total Hamiltonian (1) and its eigenvalueequation

H� = E�. (6)

The basic idea is to exactly factorize � into an electronicwavefunction ϕ(r; R) and a nuclear wavefunction χ(R)

� = ϕ(r; R)χ(R), (7a)

where the new electronic function is normalized

〈ϕ|ϕ〉el = 1 (7b)

similar to the eigenfunction ϕ of Hel in Eqs. (3a) and (3b).Choosing the total wavefunction � to be normalized, i.e.,〈�|�〉 = 1, the nuclear function χ is normalized in nuclearspace. To proceed, we have to specify the kinetic energy op-erator TN. For transparency, we use conveniently scaled rect-angular nuclear coordinates, for which TN can be expressedas

TN = − ¯2

2M∇ · ∇ = − ¯

2

2M�, (8)

where ∇ is the gradient in nuclear space, the dot is the com-mon scalar product, and M is an average nuclear mass.

Inserting the ansatz (7) into the Schrödinger equation (6)and using the above expression for the kinetic energy oper-ator, we obtain a lengthy equation coupling ϕ and χ . Thisequation simplifies considerably if we introduce an electronicoperator Hel which has ϕ as its eigenstate

Helϕ = Eel(R)ϕ. (9)

This is readily accomplished by choosing

Hel = Hel − ¯2

2M(∇ ln χ) · ∇ + TN. (10)

The electronic energy Eel(R) can be easily found by takingthe expectation value 〈ϕ|Hel|ϕ〉el of the effective electronicHamiltonian Hel with the electronic state ϕ. We do not con-fine ourselves to real solutions of Eq. (10), i.e., ϕ can be com-plex. But, for real functions ϕ, the term 〈ϕ|∇|ϕ〉el vanishes

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224110-3 Lorenz S. Cederbaum J. Chem. Phys. 138, 224110 (2013)

because of the normalization condition (7b), and hence

Eel(R) = 〈ϕ|H |ϕ〉el, (11)

i.e., the electronic energy is just the electronic expectationvalue of the full Hamiltonian H taken with the new electronicfunction ϕ.

Making use of Eq. (9), the Schrödinger equation (6) with� = ϕχ boils down exactly to

[TN + Eel(R)]χ = E χ. (12)

The operator TN + Eel(R) governs the nuclear motion in thestate � and provides the exact energy E of the system. Asin the Born-Oppenheimer approximation, Eq. (4), the nuclearmotion proceeds in a potential which is provided by the elec-tronic energy.

At this point, a number of issues have to be discussed.Some authors call Eq. (4) the Born-Oppenheimer adiabaticapproximation7 and reserve the name Born-Oppenheimer ap-proximation to an equation as (4), but with a diagonal cor-rection term � = 〈ϕ|TN|ϕ〉el added to the electronic energyEel = 〈ϕ|Hel|ϕ〉el. One notices that Eel + � is nothing butthe expectation value 〈ϕ|H|ϕ〉el of the full Hamiltonian takenwith the usual electronic wavefunction ϕ. The new electronicenergy Eel in (11) is nothing but the analogous quantity com-puted, however, with the exact electronic wavefunction ϕ. Seealso Eq. (5e).

Although ϕ is an electronic wavefunction which dependson R, it is not true in the conventional sense that this is a pureparametrical dependence, since the nuclear momentum oper-ators appear in the electronic Hamiltonian (10). This impliesthat the electronic energy in Eq. (10) which serves as the po-tential surface for the nuclear motion in Eq. (11) implicitlyalready contains some information on the nuclear motion. Forgeneral molecules, it will be quite challenging to solve theproblem, since the product ansatz basically undoes the BornOppenheimer separation.

An important issue is the appearance of ln χ in the elec-tronic Hamiltonian Hel . For the ground state of the nuclearmotion, the nuclear wavefunction can be expected to be node-less. Then, ln χ is well defined everywhere. For other states, χmay have nodes, and ln χ will have singularities. Apart fromthese singularities, Eq. (10) for Hel is well defined. This equa-tion can be used to determine ϕ everywhere except at thesesingularities. This should be sufficient to solve the problem.In Subsection II B, we shall derive the working equations alsofrom another perspective, namely, by employing variationaltheory. It will be shown that the singularities are suppressedby a multiplicative term χ2.

Another aspect of the appearance of ln χ in Hel is thefact that the electronic wavefunction in the exact factorization� = ϕχ depends on χ . Consequently, Eq. (9) for the elec-tronic wavefunction and Eq. (12) for the nuclear wavefunc-tion are intimately coupled. This lowers the practicability ofthe ansatz in general. Nevertheless, the ansatz provides addi-tional insight into the complex problem of understanding themolecular Schrödinger equation. It could also be that the ap-proach is amenable to further simplifications, exact or inexactbut accurate. Here, we would just like to mention that as ϕ

depends on χ , the electronic energy is a functional of χ , i.e.,

Eel = Eel[χ]. In turn, the appealing equation (12) for the nu-clear motion can also be seen as

{TN + Eel[χ]}χ = E χ. (13)

If one finds a scheme to determine the functional form ofEel[χ], this equation could become of practical relevance.

B. Variational approach

In this subsection, we start from the Schrödingerequation (6) for the exact wavefunction and attempt to solve itvariationally with the product ansatz (7). Since ϕ depends onr and “parametrically” on R, which is expressed by the nor-malization condition (7b) valid at any value of R, a variationalansatz is not trivial.

We introduce the following functional:

τ [ϕ, χ ] = 〈ϕχ |H |ϕχ〉 + λ(1 − 〈ϕχ |ϕχ〉)+μ(1 − 〈χ |χ〉R), (14)

where λ and μ are Lagrange parameters ensuring that 〈�|�〉= 1 and

∫ |χ |2dR = 1. If we want to vary 〈�|H|�〉 with re-spect to ϕ(r; R), we cannot use the constraint 〈ϕ|ϕ〉el = 1 be-cause this constraint is only over r and we have to vary overr and R. With 〈ϕ|ϕ〉el = 1 the constraints in (14) are easy tofulfill. On the other hand, 〈ϕ|ϕ〉el = 1 does not follow imme-diately from those in (14) because ϕ depends not only on r,but also on R.

Varying with respect to ϕ∗, i.e.,

δτ

δϕ∗ = 0, (15a)

immediately gives

χ∗(H − λ)ϕχ = 0. (15b)

Remembering that H = Hel + TN and applying TN on ϕχ

leads to

|χ |2[Hel − ¯

2

M(∇ ln χ ) · ∇ + TN − λ

]ϕ = −(χ∗TNχ)ϕ,

(15c)where (χ∗TNχ)ϕ implies that TN is applied to χ only anddoes not act on ϕ. In parenthesis, we re-discover the effectiveelectronic Hamiltonian Hel (10).

To proceed, we now vary τ [ϕ, χ ] with respect to χ∗ andset

δτ

δχ∗ = 0. (16a)

This readily leads to

〈ϕ|H |ϕχ〉el = [μ + λ〈ϕ|ϕ〉el]χ. (16b)

By applying again TN on ϕχ , we can rewrite this equation togive

〈ϕ|ϕ〉el[TN + 〈ϕ|Hel|ϕ〉el

〈ϕ|ϕ〉el

]χ = [μ + λ〈ϕ|ϕ〉el]χ. (16c)

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224110-4 Lorenz S. Cederbaum J. Chem. Phys. 138, 224110 (2013)

Multiplying from the left by χ∗ allows us to obtain

(χ∗TNχ ) = |χ |2〈ϕ|ϕ〉el

[μ + λ〈ϕ|ϕ〉el − 〈ϕ|Hel|ϕ〉el

〈ϕ|ϕ〉el

],

(16d)

which can be inserted into the right hand side of Eq. (15c) andnow this equation takes on the appearance

|χ |2Helϕ = |χ |2〈ϕ|ϕ〉el

[〈ϕ|Hel|ϕ〉el

〈ϕ|ϕ〉el − μ

]ϕ. (17a)

This equation can be used to obtain μ by multiplying it fromthe left by ϕ∗ and integrating over the electronic coordinates

|χ |2μ = |χ |2[

〈ϕ|Hel|ϕ〉el〈ϕ|ϕ〉el − 〈ϕ|Hel|ϕ〉el

]. (17b)

Since μ should be a number and all other quantities appearingin (17b) are functions of R, the only solution is

〈ϕ|ϕ〉el = 1

and hence μ = 0.We are now in the position to write down the final results

of the variational computation. Equation (17a) now reads

|χ |2[Hel − Eel(R)]ϕ = 0, (18a)

which is identical to Eq. (9) derived in Subsection II A exceptfor the overall factor |χ |2 in front of the equation. This factorsuppresses the possible singularities in Hel . Equation (16c)now takes on the appearance

[TN + Eel(R)]χ = λχ. (18b)

This central equation which governs the nuclear motion inthe electronic state ϕ is identical to Eq. (12) obtained inSubsection II A. From the comparison, it is evident thatλ = E, the exact total energy of the system.

Note added: After this work was submitted, it was foundthat there is an unpublished work that provides and discussesthe equation of motion determining the exact electronic wave-function before the separation of the center of mass.47

C. Center of mass motion and separationof coordinates

It is well known that in the absence of external forces,the center of mass motion of the whole system separatesfrom the internal motion. In the Born-Oppenheimer approx-imation, this separation is violated. Here, only the center ofmass of the nuclei separates. In an external magnetic field,the center of mass does not separate39, 40 and the ensuing cou-pling between this motion and the internal motion gives rise todramatic effects in strong fields.41–43 In magnetic fields,the Born-Oppenheimer approximation severely fails and hasto be corrected.44–46

Since the present approach of factorizing the total wave-function into a product of an electronic and a nuclear wave-function is formally exact, the separation of the center of massis exact too. The explicit example presented in Sec. III illus-trates this nicely. To exploit this separation, we express TN

as the sum of the translational kinetic energy and the kineticenergy Tn of the internal nuclear motion

TN = Tn + 1

2MNT

(∑k

Pk

)2

, (19a)

where MNT is the total mass of the nuclei and Pk is the mo-mentum operator of the kth nucleus. Let the total momentumof the system be PCM. The translational kinetic energy thenreads

1

2MNT

(PCM −

∑i

pi

)2

, (19b)

where pi is the momentum of the ith electron. Because of theseparability of the center of mass motion in field free space,PCM� = ¯K� where K is the good quantum number of thetranslational motion. This helps to remove exactly all the cen-ter of mass operators from the calculations.

We shall start from our basic equations (9) and (12) andseek for wavefunctions ϕK and χK which satisfy these equa-tions and lead to the expected motion of the center of massof the system. Since ϕK is an electronic and χK a nuclearwavefunction, we set

ϕK = ei K · m

MT

∑i (r i−RN )

ϕ, (20a)

χK = ei K ·RN χ, (20b)

where m is the electron mass and MT is the total mass of thesystem, i.e., the nuclei plus the electrons. The coordinate ofthe ith electron is r i and RN is the center of mass of the nucleialone. Obviously, the total wavefunction �K = ϕKχK fulfillsthe known relationship

�K = ei K ·RCM�. (20c)

Here, RCM is the center of mass of the whole system includingthe electrons.

Inserting the ansatz (20a) into Eq. (9) for the electronicwavefunction, one can eliminate the exponential function. Forthat purpose, one has also to let the electronic kinetic energyoperator in Hel operate on this exponential function and toevaluate the term (∇ ln χK ) · ∇ appearing in the electronicHamiltonian Hel in (10) using (20b). Finally, one has alsoto compute the impact of the nuclear kinetic energy TN on theexponential function. After a somewhat lengthy calculation,we obtain

HelϕK = ei K · m

MT

∑i (r i−RN )

[Hel − ¯

2

M(∇n ln χ) · ∇n

+ ¯

MTK · PCM − ¯

2K2

2MT

Nm

MNT

]ϕ

and since ϕ does not depend on the center of mass of thewhole system, PCMϕ = 0. The final equation for ϕ now readsagain

Helϕ = EK

el ϕ, (21a)

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224110-5 Lorenz S. Cederbaum J. Chem. Phys. 138, 224110 (2013)

where the electronic Hamiltonian Hel now takes on the ap-pearance

Hel = Hel − ¯2

M(∇n ln χ ) · ∇n + TN + CK, (21b)

which, using Eqs. (19a) and (19b), can be further reduced togive

Hel =Hel − ¯2

M(∇n ln χ ) · ∇n+Tn+ 1

2MNT

(∑i

pi

)2

+CK.

(21c)

The only dependence on the total translational momentum ¯Kis via the constant

CK = −¯2K2

2MT

Nm

MNT, (21d)

where N is the number of electrons. The quantity Nm/MNT isthus the ratio of the total electron mass to that of the nuclei.The electronic energy becomes

EK

el (R) = Eel(R) + CK. (21e)

The electronic energy Eel fulfills as before (11) forreal ϕ.

We now turn our attention to the nuclear wavefunctionχK . Inserting the ansatz (20b) into the basic equation (12) andrecognizing that the translational kinetic energy of the nucleican be expressed as

− ¯2

2MNT

∂2

∂ R2N

,

one readily arrives at[Tn + E

K

el (R) + ¯2K2

2MNT

]χ = E χ, (22a)

which leads via (21e) and (21d) to the final result[Tn + Eel(R) + ¯

2K2

2MT

]χ = E χ. (22b)

Equations (21)–(22b) complete our separation of the cen-ter of mass motion. In these equations, only the internal nu-clear degrees of freedom appear. Equations (22a) and (22b)are as one would expect: The exact energies are the sum ofthe internal energy and the total center of mass energy of thesystem. The product ansatz � = ϕχ holds also in the pres-ence of external fields. The pseudo-separation of the center ofmass motion is expected, however, to lead to intricate work-ing equations reflecting the residual interaction between theinternal and center of mass motions.

III. A SIMPLE EXACTLY SOLVABLE MODEL

In order to examine the quality of the Born-Oppenheimerapproximation, Moshinsky and Kittel (MK)16 devised a sim-ple model of a light particle (the electron) and two heavy parti-cles (the nuclei) which are coupled to each other by harmonicforces. The MK-model can be solved exactly and also explic-

itly in the Born-Oppenheimer approximation. The Hamilto-nian of the system reads

H = p2

2m+ P 2

1

2M+ P 2

2

2M+ k

2(r − R1)2

+ k

2(r − R2)2 + �

2(R1 − R2)2, (23)

where p is the momentum of the light particle of mass m, P1

and P2 the momenta of the heavy particles of mass M, and kand � are spring constants. The meaning of the coordinates r,R1, and R2 is obvious.

As usual, one defines the electronic Hamiltonian

Hel = p2

2m+ Vel(r),

(24)

Vel(r) = k

2(r − R1)2 + k

2(r − R2)2 + �

2(R1 − R2)2,

which depends parametrically on the nuclear positions R1 andR2. The electronic potential is easily rewritten to give

Vel(r) = k

2(r − RN )2 + 2� + k

4(R1 − R2)2, (25)

where RN = (R1 + R2)/2 is the center of mass of the nu-clei. Since the Ri , i = 1, 2, are parameters, the electronicSchrödinger equation

Helϕ = Eel(|R1 − R2|)ϕ (26a)

is readily solved for all electronic states. The solutions ϕq(r −RN ) are harmonic oscillator eigenstates with energies

Eel,q(|R1 − R1|) = ¯ωel(q + 3/2) + 2� + k

4(R1 − R2)2,

(26b)ωel = [2k/m]1/2,

where, as usual, q = 0, 1, 2, . . . .In the next step of the Born-Oppenheimer approach, one

solves Eq. (4) for the nuclear function, which in the MKmodel reads[

P 21

2M+ P 2

2

2M+ Eel,q(|R1 − R2|)

]χ = EBO χ. (27a)

The nuclear kinetic energy operator can be written as

TN = P 21

2M+ P 2

2

2M= π2

n

M+ π2

N

4M, (27b)

where πn = (P1 − P2)/2 is the momentum conjugated tothe relative coordinate R1 − R2 describing the internal nu-clear motion, and πN = P1 + P2 is conjugated to the centerof mass coordinate RN = (R1 + R2)/2. By introducing thesemomenta, these two motions are separated and the solutionsof Eq. (27a)

χq,ν,K (|R1 − R2|, RN ) = ei K ·RN χq,ν(|R1 − R2|) (27c)

are plane waves which describe the translational motion ofthe center of mass of the nuclei and the three-dimensionalharmonic oscillator functions describing the vibrational androtational motion of the nuclei. The total energies in the

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224110-6 Lorenz S. Cederbaum J. Chem. Phys. 138, 224110 (2013)

Born-Oppenheimer approximation simply take on the appear-ance

EBO,q,ν,K = ¯ωel(q + 3/2) + ¯�(ν + 3/2) + ¯2K2

4M,

(27d)� = [(2� + K)/M]1/2.

It is seen that for each electronic state specified by the quan-tum number q, there is a manifold of rotational-vibrationalstates specified by ν = 0, 1, 2, . . . which arise from the mo-tion of the nuclei in the potential provided by the electronicenergy (26b). It is also seen that only the center of mass ofthe nuclei and not that of the whole system separates fromthe electronic and internal nuclear motion. This fact is alsoreflected in the total energy (27d) which exhibits ¯2K2/4M asthe translational energy.

We now turn to the exact factorization of the total wave-function of the system into an electronic and a nuclear wave-function discussed in Sec. II. The new electronic HamiltonianHel defining the electronic state ϕ is given in (21b) and readsin the present example

Hel = Hel + 2

M(πn ln χ) · πn + π2

n

M+ π2

N

4M+ CK. (28a)

Here, Hel is that in (24). As seen above, in the Born-Oppenheimer approximation the electronic states ϕq dependin the MK-model only on r − RN . This is also the case for theϕq . Applying Hel to ϕq , one immediately sees that because ofπnϕ = 0, one arrives at

Helϕ =(

Hel + π2N

4M+ CK

)ϕ, (28b)

which can be further simplified, either by employing the gen-eral result (21c) or explicitly. Indeed,

π2N

4Mϕ = − ¯

2

4M

∂2

∂ R2N

ϕ = − ¯2

4M

∂2

∂ r2ϕ = p2

4Mϕ. (28c)

One immediately sees from (28b) and (24) that in the MK-model

Hel = 1

2μp2 + Vel(r) + CK, (28d)

where

μ = 2Mm

2M + m

is the reduced mass of the electron. Consequently, ϕ and Eel

differ from ϕ and Eel only by the reduced mass and the latteralso by CK.

The result reads

Eel,q,K (|R1 − R2|)

= ¯ωel(q + 3/2) + 2� + k

4(R1 − R2)2 + CK, (28e)

ωel = [2k/μ]1/2.

Using this input data, we are now able to complete the solu-tion of the problem. The general equation (22a) takes on the

following appearance in the MK-model:(π2

n

M+ Eel,q,K (|R1 − R2|) + CK

)χq,ν(|R1 − R2|)

= Eq,ν,Kχq,ν(|R1 − R2|), (29a)

which is again an harmonic oscillator problem, readily givingthe following final result:

Eq,ν,K = ¯ωel(q + 3/2) + ¯�(ν + 3/2) + ¯2K2

2(2M + m).

(29b)

Needless to say, � = ϕχ with the above solutions are the ex-act eigenfunctions of the MK-model and the energies Eq, ν, K

in (29b) are the corresponding exact energies. We would liketo add that exact results can be similarly obtained for systemsof many light and heavy particles interacting via harmonicforces. The calculations and the presentation of the results are,however, rather lengthy.

IV. SUMMARY AND CONCLUSION

In the widely used Born-Oppenheimer approximation, anelectronic Hamiltonian is a priori defined which gives riseto its electronic eigenfunction. The molecular wavefunctionis then approximated as a product of this electronic wave-function and a nuclear wavefunction. The latter and the ap-proximate total energy are found as solutions of a nuclearSchrödinger equation with the electronic energy serving asthe potential driving the nuclear motion. In this work, we in-vestigate the finding that the molecular wavefunction can ex-actly be expressed as a product of an electronic and a nuclearwavefunction originally discussed by Hunter.10 In the work ofHunter, these quantities are constructed a posteriori after thefull solution of the Schrödinger equation has been determined.In the present work, we determine the electronic and nuclearwavefunctions entering the product ansatz of the exact totalwavefunction from first principles. This electronic function isan eigenfunction of a new rather involved electronic Hamilto-nian, which depends on nuclear momentum operators. As inthe Born-Oppenheimer approximation, the nuclear wavefunc-tion is determined by the electronic energy as the underlyingpotential. The energy corresponding to this nuclear wavefunc-tion is, however, the exact total energy of the system.

The electronic Hamiltonian and the working equationshave been derived by two complementary methods. Directlyfrom the Schrödinger equation for the total wavefunction andvia a variational ansatz. Since the electronic wavefunction de-pends on the electronic and parametrically also on the nu-clear coordinates, and is normalized in the electronic space ateach nuclear configuration, the variational ansatz is not trivial.Interestingly, one does not have to impose a priori this nor-malization condition for the electronic function. This normal-ization condition rather follows as a result of the variationalcomputation.

A simple exactly solvable model is analyzed in the Born-Oppenheimer approximation and in the exact factorization ap-proach. The study of the model is presented to didactically

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224110-7 Lorenz S. Cederbaum J. Chem. Phys. 138, 224110 (2013)

illuminate the physics behind the approach in a rather trans-parent case.

Until now it is unclear whether the exact factorization ap-proach will have practical value in solving complex problems.The main reason lies in the new electronic Hamiltonian whichis rather involved and of unusual structure. It has already beenargued before that the dependence of the potential energy sur-face for the nuclear motion on the state in question makes thefactorization approach less practical.13, 15 We hope that the in-teresting underlying basic idea of factorization will becomefruitful once the electronic Hamiltonian and its physics arefurther studied and better understood.

ACKNOWLEDGMENTS

The author thanks Alexander Kuleff and Ying-Chih Chi-ang for discussions, and the Deutsche Forschungsgemein-schaft (DFG) for financial support. The author also thanksone of the referees for pointing out the unpublished work inRef. (47).

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