A molecule contains both nuclei and electrons, but the nuclear and electronic motions may be considered separately because the nuclei are much heavier than the electrons and their motion is therefore more restricted. The separability of nuclear and electronic motion is known as the Born-Oppenheimer approximation.
Jules Robert Oppenheimer (1904–1967) was a young American who went to Europe to study physics after graduating from Harvard. He worked with Born in Go¨ttingen, where he was awarded a Ph.D. degree in 1927. The separability of nuclear and electronic motion was formulated in 1927 by Born and Oppenheimer
THE BORN-OPPENHEIMER APPROXIMATION 161
by means of a precise but rather complex mathematical analysis. We present a less comprehensive but simpler version of their work.
We first describe a quantum mechanics theorem that deals with the separability of the Hamiltonian. We consider a HamiltonianHðX;Yịthat depends on two sets of coordinates,XandY, and that may be written as a sum of two parts:
HðX;Yị ẳH1ðXị ỵH2ðYị ð12-1ị Here the first part,H1 depends only on the coordinatesX and the second part,H2, depends only on the coordinatesY.
We define the eigenvalues ofH1 andH2 as H1ðXịnðXị ẳlnnðXị
H2ðYịmðYị ẳmmmðYị ð12-2ị It is then easily verified that
HðX;YịnðXịmðYị ẳ ðlnỵmmịnðXịmðYị ð12-3ị We see that the eigenfunctions of the operatorHðX;Yịof Eq. (12-1) are products of the eigenfunctions ofH1andH2, while its eigenvalues are sums of the eigenvalues ofH1 andH2.
A diatomic molecule contains two nuclei, a and b, with massesMaandMb, elec- tric chargesZaeandZbe, and coordinatesRaandRb. In addition, there areNelec- trons with coordinatesri. The molecular Hamiltonian may now be written as a sum of a nuclear and an electronic part,
HmolẳHnuclỵHel ð12-4ị We write the nuclear part as
Hnuclẳ h2
2Maa h2
2MbbþZaZbe2
Rab ð12-5ị
Hereaandbare Laplace operators
aẳ q2 qX2aþ q2
qY2aþ q2
qZ2a etc: ð12-6ị andRabis the internuclear distance. The electronic part is
Helẳ X
j
h2
2mjþe2 rajþe2
rbj
þX
j>i
e2
rij ð12-7ị
In order to analyze the eigenfunctions of the molecular Hamiltonian, we replace the nuclear coordinatesRaandRbby the coordinatesRcof their center of gravity andRof their distance:
RcẳMaRaỵMbRb
MaþMb
RẳRbRa ð12-8ị It is easily shown that the motion of the center of gravity may then be separated; it may therefore be disregarded. The nuclear Hamiltonian is then reduced to
Hnuclẳ h2 2m
q2 qX2þ q2
qY2þ q2 qZ2
þZaZbe2
R ð12-9ị
wheremis the reduced mass of the nuclei a and b:
1 mẳ 1
Ma
þ 1 Mb
ð12-10ị We now define the nuclear center of gravity as the origin of the electron coordinates and the vectorRas their Z axis (see Figure 12-1). We note that the electronic Hamiltonian Hel of Eq. (12-7) contains the electron coordinates that we denote byr, but it also depends implicitly on the internuclear distance R. We write it therefore as Hel (r;R). The total molecular Hamiltonian of Eq. (12-4) may now be represented as
Hmolðr;Rị ẳHnuclðRị ỵHelðr;Rị ð12-11ị The separation of nuclear and electronic motion is not immediately obvious because Eq. (12-11) differs from Eq. (12-1) insofar asHeldepends on the nuclear coordinateRin addition to the electronic coordinatesr. However, the two types of motion may still be separated if we make the two assumptions that are equivalent with the Born-Oppenheimer approximation.
R B
C x
y
A
Figure 12-1 Definition of nuclear and electronic coordinate systems in a diatomic molecule.
THE BORN-OPPENHEIMER APPROXIMATION 163
We first define the eigenvaluesEn and corresponding eigenfunctionsFn of the operatorHel(r;R) by means of
Helðr;RịFnðr;Rị ẳenðRịFnðr;Rị ð12-12ị We note that both the eigenvalues and eigenfunctions depend onR.
Our first assumption is that the eigenfunctions of the molecular Hamiltonian Hmol may be represented as products of a functionfnðX;Y;Zịof the nuclear coor- dinates only and of one of the eigenfunctionsFnðr;Rị
Hmolðr;RịfnðX;Y;ZịFnðr;Rị ẳEnfnðX;Y;ZịFnðr;Rị ð12-13ị Our second assumption is
q2
qX2ẵfnðX;Y;ZịFnðr;Rị ẳFnðr;Rịq2fnðX;Y;Zị
qX2 ; etc: ð12-14ị In other words, the derivatives of the electronic eigenfunctionFnwith respect to the nuclear coordinates (X;Y;Z) are much smaller than the corresponding derivatives of the nuclear functionsfn. It may be helpful to present a simple physical interpre- tation of the Born-Oppenheimer approximation. First, we may derive the electronic part of the electronic wave function on the assumption that the nuclei are stationary at a fixed internuclear distanceR. Second, even though the electronic wave function is dependent on the internuclear distanceR, its changes as a function ofRare neg- ligible compared to the changes in the nuclear wave function. In summary, even though the nuclear and electronic motions may not be separated in the strictest sense, for all practical purposes it is permitted to separate them.