The Adiabatic and Born–Oppenheimer Approximations

3 Electronic Structure

3.1 The Adiabatic and Born–Oppenheimer Approximations

We will start by reviewing the Born–Oppenheimer approximation in more detail.²The total (non-relativistic) Hamiltonian operator can be written as kinetic and potential energies of the nuclei and electrons.

(3.3) The Hamiltonian operator is first transformed to the centre of mass system, where it may be written as (using atomic units, see Appendix D):

(3.4)

H T H H

H T V V V

H

tot n e mp

e n ne ee nn

mp

tot elec

= + +

= + + +

= −  ∇

 

∑

 1 2

2

M _i ⁱ

N

Htot=Tn+Te+Vne+Vee+Vnn

Ψ Ψ Ψ Ψ

Ψ Ψ Ψ Ψ

Ψ Ψ Ψ Ψ

≡ ≡

=

=

∫

∫

; *

*

* d d r

H r H

Here H_eis the electronic Hamiltonian operatorand H_mpis called the mass-polarization (M_tot is the total mass of all the nuclei). The mass-polarization term arises because it is not possible to rigorously separate the centre of mass motion from the internal motion for a system with more than two particles. We note that H_e only depends on the nuclear positions(via V_neand V_nn, see eq. (3.23)), but not on their momenta.

Assume for the moment that the full set of solutions to the electronic Schrödinger equation is available, where Rdenotes nuclear positions and relectronic coordinates.

(3.5) The Hamiltonian operator is Hermitian, eq. (3.6).

(3.6) The Hermitian property means that the solutions can be chosen to be orthogonal and normalized (orthonormal).

(3.7)

Without introducing any approximations, the total (exact) wave function can be written as an expansion in the complete set of electronic functions, with the expansion coeffi- cients being functions of the nuclear coordinates.

(3.8) Inserting eq. (3.8) into the Schrödinger equation (3.1) gives eq. (3.9).

(3.9) The nuclear kinetic energy is a sum of differential operators.

(3.10)

We have here introduced the ∇n2symbol, which implicitly includes the mass depend- ence, sign and summation. Expanding out (3.8) gives eq. (3.11).

Tn

a a a

a

a a a

a

a a a

M

X Y Z

X Y Z

= − ∇ = ∇

∇ =  



∇ = + +

 



∑

₂^{1 2 2}

2 2

2 2

2 2

2 n

, ,

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

T_n+H_e+H_{mp n} R R r, _{tot n} R R r,

( ) ( ) ( )= ( ) ( )

=

∞

=

∑

^{Ψi Ψi}

∑

∞ ^{Ψ Ψ} i

i i

i

E

1 1

Ψtot(R r, )= Ψn( ) (RΨ R r, )

=

∑

∞ ^{i i} i 1

Ψi Ψj ij Ψ Ψi j ij ij

ij

d

i j i j

* , ,

, , R r R r r

( ) ( ) = ↔ =

= =

= ≠

∫

^{d d}

d d

1 0

Ψi*HΨjdr= ΨjH*Ψi*dr ↔ ΨiHΨj = ΨjHΨi *

∫ ∫

H Re( ) (Ψi R r, )=Ei( ) (RΨi R r, ); i=1 2, ,. . .,∞

(3.11)

Here we have used the fact that H_eand H_mponly act on the electronic wave function, and the fact that Ψiis an exact solution to the electronic Schrödinger equation (eq.

(3.5)). We will now use the orthonormality of the Ψiby multiplying from the left by a specificelectronic wave function Ψj* and integrate over the electron coordinates.

(3.12) The electronic wave function has now been removed from the first two terms while the curly bracket contains terms that couple different electronic states. The first two of these are the first- and second-order non-adiabatic coupling elements, respectively, while the last is the mass polarization. The non-adiabatic coupling elements are impor- tant for systems involving more than one electronic surface, such as photochemical reactions.

In the adiabaticapproximation the form of the total wave function is restricted to one electronic surface, i.e. all coupling elements in eq. (3.12) are neglected (only the terms with i=jsurvive). Except for spatially degenerate wave functions, the diagonal first-order non-adiabatic coupling elements are zero.

(3.13) Neglecting the mass-polarization and reintroducing the kinetic energy operator gives eq. (3.14).

(3.14) This can also be written as in eq. (3.15).

(3.15) The U(R) term is known as the diagonal correction, and is smaller than Ej(R) by a factor roughly equal to the ratio of the electronic and nuclear masses. It is usually a slowly varying function of R, and the shape of the energy surface is therefore deter- mined almost exclusively by Ej(R).³ In the Born–Oppenheimer approximation, the diagonal correction is neglected, and the resulting equation takes on the usual Schrödinger form, where the electronic energy plays the role of a potential energy.

(3.16) In the Born–Oppenheimer picture, the nuclei move on a potential energy surface(PES) which is a solution to the electronicSchrödinger equation. The PES is independent of

Tn+ ( )R n R tot n R ( ^Ej ) ( )Ψj =^E Ψj( ) T_n+ ( )R + ( )R _n R _{tot n} R ( ^Ej ^U ) ( )Ψj =^E Ψj( )

Tn+ + ∇n n tot n

(

^{Ej Ψj 2Ψ Ψj}

)

^{j=E Ψj}

∇ + + ∇ +

(

n2 E_j Ψ_j n2Ψ_j Ψ_jHmpΨ Ψ_j

)

n_j=EtotΨn_j

∇ + +  ∇ (∇ )+ ∇ +





=

=

∑

∞

n n n

n n n n n

mp n

tot n 2

2

1

Ψ Ψ 2Ψ Ψ Ψ Ψ Ψ Ψ

Ψ Ψ Ψ Ψ

j j j

j i i j i i

j i i

i

E E j

H

∇ + +

( )

⁼

∇ + +

{ }

=

∇ ( ∇ + ∇ )+ +

{ }

=

∞

=

∞

=

∞

=

∞

=

∑

∑

∑ ∑

n e mp n tot n

n n e n mp n tot n

n n n n n n e n mp

2

1 1

2

1 1

H H

H H

H H

Ψ Ψ Ψ Ψ

Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ

Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ

i i i i

i i

i i i i i i

i

i i i

i i i i i i i i

i

E E

1

1 1

2

1 2 1

2

∞

=

∞

=

∞

=

∞

∑ ∑

∑ ∑

=

(

∇

)

^{+ ∇}( )(∇ )+

(

∇

)

^{+ +}







= E

E E

i i i

i i i i

i i i i i i i

i

i i i

tot n

n n n n n

n n n n mp

tot n

Ψ Ψ

Ψ Ψ Ψ Ψ

Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ

H

the nuclear masses (i.e. it is the same for isotopic molecules), but this is not the case when working in the adiabatic approximation since the diagonal correction (and mass- polarization) depends on the nuclear masses. Solving eq. (3.16) for the nuclear wave function leads to energy levels for molecular vibrations and rotations (Section 13.5), which in turn are the fundamentals for many forms of spectroscopy, such as infrared (IR), Raman, microwave, etc.

The Born–Oppenheimer (and adiabatic) approximation is usually a good approxi- mation but breaks down when two (or more) solutions to the electronic Schrödinger equation come close together energetically.⁴Consider for example stretching the bond in the LiF molecule. Near the equilibrium distance the molecule is very polarized, i.e.

described essentially by an ionic wave function, Li⁺F⁻. The molecule, however, dissoci- ates into neutral atoms (all bonds break homolytically in the gas phase), i.e. the wave function at long distance is of a covalent type, Li⋅F⋅. At the equilibrium distance, the covalent wave function is higher in energy than the ionic, but the situation reverses as the bond distance increases. At some point they must “cross”. However, as they have the same symmetry, they do not actually cross, but make an avoided crossing. In the region of the avoided crossing, the wave function changes from being mainly ionic to covalent over a short distance, and the adiabatic, and therefore also the Born–Oppen- heimer, approximation, breaks down. This is illustrated in Figure 3.2, where the two states have been calculated by a state average MCSCF procedure using the aug-cc- pVTZ basis set. The energy of the ionic state is given by the solid line, while the energy of the covalent state is shown by the dashed line. For bond distances near 6 Å, the lowest energy wave function suddenly switches from being almost ionic to being cova- lent, and the two states come within ~15 kJ/mol of each other. In this region the Born–Oppenheimer approximation becomes poor.

0 200 400 600 800 1000

1 2 3 4 5 6 7 8 9

Li–F distance (Å)

Energy (kJ/mol)

Li⁺F^-

Li⁺F^-

Li·F·

Li·F·

Avoided crossing region

Figure 3.2 Avoided crossing of potential energy curves for LiF

For the majority of systems the Born–Oppenheimer approximation introduces only very small errors. The diagonal Born–Oppenheimer correction(DBOC) can be evalu- ated relatively easy, as it is just the second derivative of the electronic wave function with respect to the nuclear coordinates, and is therefore closely related to the nuclear gradient and second derivative of the energy (Section 10.8).

(3.17) The largest effect is expected for hydrogen-containing molecules, since hydrogen has the lightest nucleus. The absolute magnitude of the DBOC for H₂O is ~7 kJ/mol, but the effect for the barrier towards linearity is only ~0.17 kJ/mol.⁵For the BH mol- ecule, the equilibrium bond length elongates by ~0.0007 Å when the DBOC is included, and the harmonic vibrational frequency changes by ~2 cm⁻¹. For systems with heavier nuclei, the effects are expected to be substantially smaller.

When the Born–Oppenheimer approximation is expected to be poor, the non- adiabatic corrections will be large, and a better strategy in such cases may be to take the quantum nature of the nuclei into account directly. Starting from eq. (3.8), both the nuclear and electronic parts may be described by determinantal-based wave func- tions expanded in Gaussian basis sets. Each of the two wave functions (electronic and nuclear) can be described at different levels of approximations, with mean-field methods (i.e. Hartree–Fock) being the first step. The energy spectrum arising from such methods directly gives both nuclear (e.g. vibrations) and electronic states, but there are still some open questions as to how to formulate a consistent theory for actually car- rying out such calculations.⁶

It should be noted that once methods beyond the Born–Oppenheimer approxima- tion are employed, concepts such as molecular geometries become blurred and energy surfaces no longer exist. Nuclei are delocalized in a quantum description, and a “bond length” is no longer a unique quantity, but must be defined according to the experi- ment that it is compared with. An X-ray structure, for example, measures the scatter- ing of electromagnetic radiation by the electron density, neutron diffraction measures the scattering by the nuclei, while a microwave experiment measures the moments of inertia. With nuclei as delocalized wave packages, these quantities must be obtained as averages over the electronic and nuclear wave function components.