# Introduction to Computational Chemistry - Jensen Fr.

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be performed less rigorously, trying to illustrate the flow of arguments,

rather than focus on mathematical details.

3.1 The Adiabatic and Born-Oppenheimer Approximations

Let us first review the Born-Oppenheimer approximation in a bit more

detail.1 The total Hamilton operator can be written as the kinetic and

potential energies of the nuclei and electrons.

Htot = Tn + Te + Vne + Vee + Vnn (3-2)

The Hamilton operator is first transformed to the centre of mass system,

where it may be

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ELECTRONIC STRUCTURE METHODS

written as (using atomic units, see Appendix D):

Htot - Tn + He + Hmp He = Te + Vne+Ve(, + Vnn

1 c V (3'3)

Here He is the electronic Hamilton operator and Hmp is called the mass-

polarization (Aftot is the total mass of all the nuclei and the sum is

over all electrons). We note that He depends only on the nuclear

positions (via Vne and Vnn, see eq. (3.23)) and not on their momenta.

Assume for the moment that the full set of solutions to the

electronic Schrodinger

equation is available, where R denotes nuclear positions and r electronic

coordinates.

He(R)*,-(R,r)=?;(R)^(R,r), i - 1,2... oo (3.4)

Since the Hamilton operator is hermitic (J O/IO^a = J Oo-I*O;*n1a), the

solutions can be chosen to be orthogonal and normalized (orthonormal).

4>*(R,r)4>j(R,r)dr =

* , . . (3.5)

fiij = 0; i o j

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 coefficients being functions of the nuclear

coordinates.

r) = J2 Oo(E)O,-(E, r) (3.6)

;=i

Inserting eq. (3.6) into the Schrodinger equation (3.1) gives

CO 00

? (Tn + He + Io?)Oo(AA-(E, a) = Au ? Oo(AA(^ A) (3.7) ;= 1 a=1

The nuclear kinetic energy is essentially a differential operator, and we

may write it as:

T = V__________- V2 = V2

" V 2Ma a

/ a2 a2 d2 \

\dX2J dY2J dZl)

2

where the mass dependence, sign and summation is implicitly included in

the Vn

3.1 THE ADIABATIC AND BORN-OPPENHEIMER APPROXIMATIONS

55

symbol. Expanding (3.7) gives

II ro

53 (V* + Hc + Io?)OoO,- - Etot Oi/Oa 1=1 i-1

00 00

{o2(OoOa) + IaOo-O, + Io?OoO,} = Ao Oo-O/

1=1 i=\

00 II

53 {^[(O^iOi,-) + (OO?IO,)] + OoIaO,: + Oo-Iu?O/} = Ao 53

/=1 i=l

IN

53 {Oa(^Oo) + 2(?IO;)(OIOU) + Oo-(o2o.) + o i.?.o. + o o'Io?O/}

1=1

II

= ?"53 OoOa (3.9)

(=1

where we have used the fact that He and Hmp only act on the electronic

wave function, and O, is an exact solution to the electronic Schrodinger

equation (eq. (3.4)). We will now use the orthonormality of the Oa by

multiplying from the left by a specific electronic wave function O* and

integrating over the electron coordinates.

We will take the opportunity at this point to introduce the bra-ket

notation.

O'^UOIo = (O|I|O)

(3.10)

O*Odv = (O|O)

The bra {n\ denotes a complex conjugate wave function with quantum number

n standing to the left of the operator, while the ket \m), denotes a wave

function with quantum number m standing to the right of the operator, and

the combined bracket denotes that the whole expression should be

integrated over all coordinates. Such a

bracket is often referred to as a matrix element. The orthonormality

condition eq. (3.5)

can then be written as.

(O/ O/) h (3.11)

With this change in notation, eq. (3.9) becomes after integration

V^ni + Ej4>ui + 53 {2{O;|Oi|Oa}(OiOia) + (OO|O*>OA

i= 1

(3.12)

+ <O;-|Io?|O,)Oo} = ?,oOe

The electronic wave function has now been removed from the first two

terms while the curly bracket contains terms which couple different

electronic states. The first two of these are the first- and second-order

iii-adiabatic coupling elements, respectively, while the last is the mass

polarization. The non-adiabatic coupling elements are important for

systems involving more than one electronic surface, such as photochemical

reactions.

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ELECTRONIC STRUCTURE METHODS

In the adiabatic approximation 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 - j survive). Except for

spatially degenerate wave functions, the diagonal first-order non-

adiabatic coupling element is zero.

(\2n + Ej + <O^|Oo) + = Etot*nJ (3.13)

Neglecting the mass polarization and reintroducing the kinetic energy

operator gives (Tn + Ej + (O;|o2|O;-"Oi;- =Etot4>nj (3.14)

or more explicitly

(Tn + Ej(R) + U{E))Ou(E) - ?totWn;(R) (3.15)

The f/(R) term is known as the diagonal correction, and is smaller than

?/(R) by a factor roughly equal to the ratio of the electronic and

nuclear masses (eq. (3.8)). It is usually a slowly varying function of R,

and the shape of the energy surface is therefore determined almost

exclusively by E/R).2 In the Bom-Oppenheimer (BO) approximation the

diagonal correction is neglected, and the resulting equation takes on the

usual Schrodinger form, where the electronic energy plays the role of a

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