The Born–Oppenheimer approximation: A toy versionGautam Gangopadhyay and Binayak Dutta-Roy

Citation: American Journal of Physics 72, 389 (2004); doi: 10.1119/1.1625927 View online: http://dx.doi.org/10.1119/1.1625927 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/72/3?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in On the mathematical treatment of the Born-Oppenheimer approximation J. Math. Phys. 55, 053504 (2014); 10.1063/1.4870855 Finite-temperature electronic simulations without the Born-Oppenheimer constraint J. Chem. Phys. 137, 134112 (2012); 10.1063/1.4755992 Student Understanding of Some Quantum Physical Concepts: Wave Function, Schrödinger’s Wave Equationand WaveParticle Duality AIP Conf. Proc. 899, 479 (2007); 10.1063/1.2733245 Born-Oppenheimer expansion at constant energy J. Chem. Phys. 125, 204109 (2006); 10.1063/1.2370992 Born–Oppenheimer invariants along nuclear configuration paths J. Chem. Phys. 117, 7405 (2002); 10.1063/1.1515768

This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.250.144.144 On: Sun, 14 Sep 2014 11:12:13

This ar

The Born–Oppenheimer approximation: A toy versionGautam Gangopadhyaya) and Binayak Dutta-RoyS. N. Bose National Centre for Basic Sciences, JD Block, Sector-III, Salt Lake, Kolkata-700098, India

~Received 27 May 2003; accepted 23 September 2003!

The Born–Oppenheimer approximation, which is central to the physics and chemistry of moleculesand solids, is illustrated by a one-dimensional toy model that is easily solved. ©2004 American

Association of Physics Teachers.

@DOI: 10.1119/1.1625927#

os

acib

denlo

diof

ok

tosa

i

b

ol

isth

m

func-d

-s.-full

ra-

wes of

heon

ur-

a-–the

e

nt

I. BACKGROUND AND MOTIVATION

Molecules and solids are describable in terms of the mtion of the electrons and nuclei of the constituent atomAlthough this description is immensely complicated, the fthat nuclei are much heavier than electrons makes it possto adopt an approach in which the nuclei, in the lowest orof approximation, are taken to be at rest, and their motioconsidered subsequently in higher orders. This methodowas first put forward by Born and Oppenheimer1 in 1927 andapplied to the hydrogen molecule by Heitler and London2 inthe same year. An appreciation of the depth of understaninvolved in this theory is to be found in this journal by onethe stalwarts of atomic physics, E. U. Condon.3 Excellentdiscussions of the approximation are available in textboby Davidov,4 Bethe and Jackiw,5 and Morrison, Estle, andLane.6 In the latter text the authors have presented aversion of the Born–Oppenheimer method; our model ifurther simplification and makes the solution more transpent and analytically tractable.

For molecules the underlying Hamiltonian of the system

H5(i

pi2

2m2(

m,i

Zme2

uRm2r i u1(

i . j

e2

ur i2r j u1(

m

Pm2

2Mm

1 (m.n

ZmZne2

uRm2Rnu, ~1!

where the upper-case letters,R and P, denote the positionand momentum operators of the nuclei which are indexedGreek letters. Lower-case letters,r and p, and Latin sub-scripts are reserved for the electrons;m is the mass of anelectron,Mm is the mass of themth nucleus,2e is the elec-tron charge, and1Zme is the charge on themth nucleus. TheSchrodinger equation for the stationary states of the mecule is

F2\2

2m(i

“ i22(

m,i

Zme2

uRm2r i u1(

i . j

e2

ur i2r j u2(

m

\2

2Mm“m

2

1 (m.n

ZmZne2

uRm2RnuGc~$r%,$R%!5Ec~$r%,$R%!. ~2!

The first step in the Born–Oppenheimer approach consin asserting that the kinetic energy term associated withheavy particles may be neglected in the lowest approxition and their coordinates treated as parameters~and not asdynamical variables!. We write the full Hamiltonian asH

5H01TR , whereH0 is the electronic Hamiltonian in which

389 Am. J. Phys.72 ~3!, March 2004 http://aapt.org/ajticle is copyrighted as indicated in the article. Reuse of AAPT content is su

128.250.144.144 On: Sun

-.tler

isgy

ng

s

yar-

s

y

-

tse

a-

the nuclear coordinates appear as parameters. The eigentions and eigenvalues ofH0 describe the electrons for fixevalues of the coordinates of the nuclei:

H0fn~$r%;$R%!5en~$R%!fn~$r%;$R%!. ~3!

We write $R% after a semicolon to indicate that$R% appearonly as parameters. The eigenvaluesen($R%) depend on theparticular values of$R% of the nuclei and yield energy surfaces that are labeled byn, the electronic quantum numberBecause the statesfn($r%;$R%) are a complete set of orthonormal vectors, we can use them as a basis to write thewave function as

cn~$r%;$R%!5(n

xn~$R%!fn~$r%;$R%!, ~4!

with the understanding that the summation includes integtion over continuum states. If we substitute the expansion~4!into Eq.~2! and integrate over the electronic coordinates,obtain a set of coupled equations describing the dynamicthe heavy nuclei, namely,

F2(m

\2

2Mm“m

2 1en~$R%!2EGxn~$R%!

5(m,m

F E fn* ~$r%;$R%!“mfm~$r%;$R%!P id3r i G

3“mxm~$R%!1(m,m

F E fn* ~$r%;$R%!

3S 2\2

2Mm“m

2 Dfm~$r%;$R%!P id3r i Gxm~$R%!, ~5!

wheremÞn. The terms on the left-hand side describe tmotion driven by the kinetic energy operator of the nucleia given electronic energy surfaceen($R%), whereas those onthe right-hand side represent ‘‘jumps’’ from one energy sface to another (n→m) resulting in the coupling of the dif-ferent electronic states.

Equation~5! represents a formidable set of coupled equtions which involve no approximations. The BornOppenheimer approximation asserts that the terms onright-hand side of Eq.~5! may be neglected. Hence, thequation for the heavy nuclei, which are confined to thenthelectronic energy surface, becomes

F2(m

\2

2Mm“m

2 1en~$R%!2EGxn~$R%!50. ~6!

Equation~6! is tantamount to the statement that the differeelectronic energy surfacesen($R%) do not intersect or come

389p © 2004 American Association of Physics Teachersbject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

, 14 Sep 2014 11:12:13

anoleredsmcbet

fo

s

b

a

us

e

rs

.

u

n,

-cle

par-

aing

gen-

uste ofdi-

tq.

and

ic

g

n-ak

outhisnot

by

This ar

close to each other in the relevant region of parameter spand are sufficiently distant in energy so that they areappreciably connected by the kinetic motion of the nucThat is, the time scale of nuclear motions is slow compato the movements of the light electrons or, in other worthe amplitudes of the nuclear motion are negligible in coparison with the equilibrium internuclear distances. Onthis simplification is made, the motion of the nuclei canfurther investigated via Eq.~6! to obtain the vibrations abouthe minima ofen($R%) and the rotations of the molecules.

Real molecular systems are very complicated exceptsimple examples such as the hydrogen molecular ion H2

1 . Toillustrate the basic ideas in a simple setting, we discusone-dimensional model of the actual physical system.

II. THE TOY MODEL

Imagine a system of two particles: one light~of massm)with coordinatex and the other heavy~massM ) located atX. The particles are confined between two impenetrawalls a distanceL apart, that is,2L/2<x,X<L/2. We as-sume that these two particles interact with each other viaattractive Dirac delta function interaction of strengthl. TheHamiltonian of the system within the allowed range is th

H5p2

2m1

P2

2M2ld~x2X!, ~7!

where p and P are the momenta of the two particles. Thstationary states are given by

F2\2

2m

]2

]x2 2\2

2M

]2

]X2 2ld~x2X!Gc~x,X!5Ec~x,X!.

~8!

We follow the Born–Oppenheimer approach and fisolve for the energy levels of the light particle,En(X), treat-ing X as a parameter, that is,

F2\2

2m

d2

dx2 2ld~x2X!Gf~x;X!5ef~x;X! ~9!

or

F d2

dx2 1k2Gf~x,X!52ld~x2X!f~x;X!, ~10!

wherek2[2me/\2 and l[2ml/\2. The wave functionfsatisfies the boundary conditions,f(x56L/2,X)50. Thusfor x,X the solution isA sink(x1L/2) while for x.X wemust haveB sink(L/22x), where A and B are constantsThe continuity of the wave function atx5X implies that

A sinkS X1L

2D5B sinkS L

22XD . ~11!

The first derivative of the wave function must be discontinous so as to yield the delta function in Eq.~10!, whose inte-gral betweenx5X2 and x5X1 yields the conditionf8(x

5X1;X)2f8(x5X2;X)52lf(x5X;X), or

2Bk coskS L

22XD2Ak coskS X1

L

2D52lA sinkS X1

L

2D . ~12!

390 Am. J. Phys., Vol. 72, No. 3, March 2004ticle is copyrighted as indicated in the article. Reuse of AAPT content is su

128.250.144.144 On: Sun

cet

i.d,-e

r

a

le

n

t

-

If we eliminateA andB from Eqs.~11! and~12!, we find theeigenvalue condition fork ~and hencee5\2k2/2m):

k sinkL5l sinkS X1L

2D sinkS L

22XD . ~13!

For a givenl and L, Eq. ~13! is to be solved fork as afunction of X to determine the energy surfaces~here‘‘curves’’ as there is only one parameterX). We observe thatfor X56Ł/2 the right-hand side of the eigenvalue conditioEq. ~13!, vanishes and sinkL50 has the solutionsk5np/Landen5\2n2p2/(2mL2). In other words, the infinite repulsion at the walls makes the wave function of the light parti~and hence the probability of finding it! zero atX56L/2,and hence it must disregard the presence of the heavyticle if the latter is located at either of these two places.

As we consider values ofX away from X56L/2, thevalue ofk must be determined by solving Eq.~13! and wouldbe less than or equal tonp/L because of the attractive deltfunction potential. Furthermore, the energy curves belongto different n cannot cross each other asX varies, becausesuch one-dimensional systems cannot possess any deeracy. Thus we may use the integern to label the statescorresponding to the motion of the light particles analogoto the set of quantum numbers used to designate the staan electron in a molecule. Moreover, the eigenvalue contion in Eq. ~13! is symmetric under the transformationX→2X, and hence it suffices to examine the range ofX between0 to L/2.

In the above discussion we have tacitly assumed thae.0 by taking trigonometric functions as the solutions of E~10!, which generally holds except for the lowest state~andthat too only if l is large enough!. The condition for cross-over from positive to negativee is easily derived by notingthat if e becomes negative, it must pass through zerohence we can consider the eigenvalue condition, Eq.~13!, forsmall k to obtain the critical potential strength

l5lc5L

L2

42X2

. ~14!

If l.lc for a range ofX values, we must use the hyperbolsolutions of the Schro¨dinger equation with ueu52e5\2k2/2m, and the eigenvalue condition becomes

k sinhkL5l sinhkS X1L

2D sinhkS L

22XD . ~15!

Equation~15! admits only one solution for the correspondink. The reason is physically transparent. Note that forL

→`, lc→0, which means that as the walls recede to infiity, the attractive delta function potential no matter how wewill have a solution with a negative energy eigenvalue~ascan be seen by solving the delta function potential withwalls!. Hence, in the presence of the repulsive walls, tenergy eigenvalue can be negative provided the walls aretoo close and the potential not too weak.

The nature of the energy surfaces is easily revealedanalyzingk as a function ofX as given by Eq.~13!. Thelocation of the extrema is obtainable from

390G. Gangopadhyay and B. Dutta-Roybject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

, 14 Sep 2014 11:12:13

aAr-

tio

a

rsss

sheat

ecclr-

cleehea

r-

thby

uer

viathe

he-

en

ger

stic

f

els

d a

gons,

This ar

dk

dX5

2lk sin 2kX

sinkL1kL coskL1lX sin 2kX2lL

2sinkL

50,

~16!

which yields 2kX56Np (N50,1,2,...). For the lowestpositive energy state,n51, there will be only one extremumat X50 ~corresponding toN50), and this energy must beminimum as can be seen from the following argument.X56L/2 we havek5p/L, and as we move the heavy paticle away from the wall, the energy and hencek must de-crease because of the attractive nature of the interacThus for then51 energy surface,X50 is a minimum. Forthe first excited electronic state,n52, we have three extremat N50 and61. The pointX50 ~corresponding toN50) isa maximum, because the vanishing of the wave functionx50 with X50 ~the wave function is an odd function! im-plies that the derivative is continuous and hence the enecorresponds tok52p/L ~as if the light particle disregardthe presence of the heavy one!. As the heavy particle moveaway fromX50, the energy must decrease due to the onof the nonzero attraction. Thus then52 energy surface giverise to a double well. Similarly, it is easy to see that for tsecond excited electronic state, there are three minimaX50 andX56p/k and two maxima atX56p/2k.

The wave functions for the ground and first excited eltronic states are shown in Fig. 1 with the heavy partilocated atX5L/4 and the strength of the delta function co

responding tol52/L. We have takenL51. Attention isdrawn to the discontinuity in the derivative of the wave funtion of the light particle at the location of the heavy partic~indicated by the arrow!. The shape and symmetry of thwave functions are clearly affected by the location of theavy particle and the strength of the delta function intertion.

Now that we have solved for the motion of the light paticle with the heavy particle atX, we may, in the spirit of theBorn–Oppenheimer approach, turn our attention toslower vibrational motion of the heavy particle governedthe potential surfaces,en(X), obtained frome5\2k2/2m,wherek are solutions~labeled by the quantum numbern) ofEq. ~13!. The vibrational energies may be determined nmerically given the values of the relevant paramet

Fig. 1. Wave function for the light particle when the heavy one is locate

X5L/4 taking l52/L.

391 Am. J. Phys., Vol. 72, No. 3, March 2004ticle is copyrighted as indicated in the article. Reuse of AAPT content is su

128.250.144.144 On: Sun

t

n.

at

gy

et

-e

-

c-

e

-s

(m,M ,l,L). However, these energies can be estimatedthe harmonic approximation by using the curvature ofpotential at the minima

d2e~X!

dX2 5\2k2

2mF2

4lk cos 2kX

C

12l2 sin 2kX~sin 2kX12kX cos 2kX!

C2 G ,

~17!

where the dimensionless quantityC5kL coskL1lXsin 2kX

1(12 lL/2)sinkL.

For convenience we choose the valuel52/L. For theground state electronic configuration, the minimum of tpotentiale(X)5e1(X) is at X50 and the corresponding solution of Eq. ~13! yields kL52.33 ~and coskL520.69). Atthe minimum of the potential, the curvature is

d2e~X!

dX2 524\2k2

mL2 coskL, ~18!

leading to the classical angular frequency

v5Ad2e~X!/dX2

M. ~19!

The ensuing spectra is a ladder with spacing\v, and accord-ingly, the vibrational energy levels are approximately givby

\vS v11

2D5\2

2mL2A16m

M

k2L2

ucoskLuS v11

2D ,

wherev is the vibrational quantum number and takes intevalues 0,1,2,... .

To make contact between our toy model and the realisituation of molecules, we takeL to be of the order of chemi-cal bond lengths (L;1 Å) andm andM to be the masses othe electron and the proton, respectively,m510230 kg andM;2000 m. The separations between the electronic levare thus of the order ofDe;(\2/mL2);6 eV, and the vi-brational level spacings are \v;\2/(mL2)Am/M;0.1 eV.

In Figs. 2 and 3 the potential energy surfaces~in units of

tFig. 2. Potential energy surface~in units of \2/2mL2) of the ground elec-

tronic state (n51) plotted againstX ~in units of L) for l52.0/L. We haveshown the vibrational bound states forM52000 m. Note the decreasinseparation between the adjacent levels, particularly for higher excitatiwhich reflects the anharmonicity of the potential surface.

391G. Gangopadhyay and B. Dutta-Roybject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

, 14 Sep 2014 11:12:13

ar

vi-

n

-e

erulucpniea

leen-to

it-alra--u-emsia-y

forsen-rd-y

n,’’

lar

J.

,

ubwe

e

This ar

\2/2mL2) of the ground and first excited electronic states

plotted againstX ~in units ofL) for l52.0/L. It can be seenthat for M52000 m, the energy separation of the lowerbrational levels is almost 0.26~in units of\2/2mL2). Thus aquadratic fit to the potential surface of the electronic groustate is very good at least for the lower vibrational levels~theagreement is within 5%!. Similarly, for the first excited electronic state (n52), we have a double well potential surfacand the curvature at the minima both correspond to an enseparation of 0.56. Indeed the results of a numerical calction reveal that the almost equidistant vibrational level strture is obtained for the low lying levels, but the energy gareduce with increasing vibrational energy as the anharmoity of the potential surface becomes important. Another f

Fig. 3. Potential energy surface of the first excited (n52) electronic statefor the same parameters as in Fig. 2. Each energy level is almost dodegenerate which undergoes splitting due to tunneling between thethrough the potential barrier. Splitting, though not discernible for the lowstates, is clearly evident for the highest vibrational pair of levels.

392 Am. J. Phys., Vol. 72, No. 3, March 2004ticle is copyrighted as indicated in the article. Reuse of AAPT content is su

128.250.144.144 On: Sun

e

d

gya--sc--

ture of the first excited state, which is typical of a doubwell potential, is the occurrence of a doublet due to a degerate pair of levels corresponding to each well split duetheir coupling from tunneling through the barrier. This splting is too small to be seen in Fig. 3 for the lower vibrationexcitation and barely discernible in the next to highest vibtional level ~shown by the thickening of this line in the diagram! and is clearly evident in the highest vibrational doblet. Hence, we see that many features of molecular systapart from the methodology of the Born–Oppenheimer adbatic approximation find a simple illustration in the tomodel that we have discussed.

ACKNOWLEDGMENTS

The authors are thankful to Professor S. Dattaguptadiscussions and suggestions for improvement in the pretation, and also to Professor Kamal Bhattacharyya of Buwan University for bringing to our attention the book bMorrison, Estle, and Lane.

a!Electronic mail: gautam@bose.res.in1M. Born and J. R. Oppenheimer, ‘‘Zur Quantentheorie der MolekeiAnn. Phys.~Leipzig! 84, 457–484~1927!.

2W. Heitler and F. London, ‘‘Interaction of neutral atoms and homopobinding according to the quantum mechanics,’’ Z. Phys.44, 455–472~1927!.

3E. U. Condon, ‘‘The Franck-Condon principle and related topics,’’ Am.Phys.15, 365–374~1947!.

4A. S. Davidov,Quantum Mechanics~Pergamon, Oxford, 1965!, pp. 472–478.

5H. A. Bethe and R. Jackiw,Intermediate Quantum Mechanics~Addison–Wesley, Reading, MA, 1997!, 3rd ed., pp. 179–189.

6M. A. Morrison, T. L. Estle, and N. S. Lane,Quantum States of AtomsMolecules and Solids~Prentice–Hall, Englewood Cliffs, NJ, 1976!, pp.259–300.

lyllsr

392G. Gangopadhyay and B. Dutta-Roybject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

, 14 Sep 2014 11:12:13