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# Introduction to Computational Chemistry - Jensen Fr.

Jensen Fr. Introduction to Computational Chemistry - John Wiley , 2001. - 444 p. Previous << 1 .. 29 30 31 32 33 34 < 35 > 36 37 38 39 40 41 .. 223 >> Next 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
54
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.
56
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-
(\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 Previous << 1 .. 29 30 31 32 33 34 < 35 > 36 37 38 39 40 41 .. 223 >> Next 