3.1 Structured Bodies
89
χ
nl
GT O = Cr
2n−l−2 e
−αr
2 Y lm ) and l, and m are not the above mentioned usual quantum
numbers. As apparent from the name in order for the STO-N G to maintain the same
accuracy as the STO a larger number (N ) of Gaussian functions are used (STO-3G
and STO-4G). In that case, one optimizes αs to fit the exponential part of the STOs.
3.1.4 The Electronic Structure of Molecules
When considering the electronic structure of molecules
5 the only practical approach
is the adoption of a numerical procedure that, starting from a limited set of tentative
independent particles wavefunction, generates a succession of solutions possibly
converging to a sufficiently accurate many-electron wavefunction. The method of
election for working such a succession is to use the variation method that minimizes
the electronic energy E e using the variational principle
∗
(r)| ˆ
H |(r)
∗ (r)|(r)
= E e ≥ E 0
(3.12)
with E 0 being the exact (unknown) energy value and (r) being a proper antisymmetric 3K -dimensional wavefunction. In Eq. 3.12 (r) can be expanded in any
resonable approximation to the independent particles wavefunction φ(r) more accurate than the above spinorbitals χ nlms generated by ignoring the electron–electron
interaction.
Let us, for example, assume that the wavefunctions φ are real, calculated using the
one-electron Hamiltonian ˆ
H of Eq. 2.18 including the electron–electron interaction
ˆ
V ee , normalized and mutually orthogonal. This implies that the following relationships hold
S i j =
φ i (r 1 )φ j (r 2 )dτ = δ i j
(3.13)
E e =
K
i=1
h ii +
1
2
K
i=1
(J ii − K ii ).
(3.14)
with
h ii =
φ i (r 1 )
−
1
2
∇
2
i −
N
a=1
Z a
r ai
φ i (r 1 )dτ 1
(3.15)
being the contribution of one-electron terms to energy (where electron 1 has been
arbitrarily assigned to orbital φ i and dτ 1 represents the integration with respect to
5 Although we only considered the equilibrium geometry in the above formulation one can also use
the same equations at any fixed nuclear geometry. In that case it is more convenient to include in
the notation the fixed nuclear geometry
R as a fixed parameter in the notation.
89
χ
nl
GT O = Cr
2n−l−2 e
−αr
2 Y lm ) and l, and m are not the above mentioned usual quantum
numbers. As apparent from the name in order for the STO-N G to maintain the same
accuracy as the STO a larger number (N ) of Gaussian functions are used (STO-3G
and STO-4G). In that case, one optimizes αs to fit the exponential part of the STOs.
3.1.4 The Electronic Structure of Molecules
When considering the electronic structure of molecules
5 the only practical approach
is the adoption of a numerical procedure that, starting from a limited set of tentative
independent particles wavefunction, generates a succession of solutions possibly
converging to a sufficiently accurate many-electron wavefunction. The method of
election for working such a succession is to use the variation method that minimizes
the electronic energy E e using the variational principle
∗
(r)| ˆ
H |(r)
∗ (r)|(r)
= E e ≥ E 0
(3.12)
with E 0 being the exact (unknown) energy value and (r) being a proper antisymmetric 3K -dimensional wavefunction. In Eq. 3.12 (r) can be expanded in any
resonable approximation to the independent particles wavefunction φ(r) more accurate than the above spinorbitals χ nlms generated by ignoring the electron–electron
interaction.
Let us, for example, assume that the wavefunctions φ are real, calculated using the
one-electron Hamiltonian ˆ
H of Eq. 2.18 including the electron–electron interaction
ˆ
V ee , normalized and mutually orthogonal. This implies that the following relationships hold
S i j =
φ i (r 1 )φ j (r 2 )dτ = δ i j
(3.13)
E e =
K
i=1
h ii +
1
2
K
i=1
(J ii − K ii ).
(3.14)
with
h ii =
φ i (r 1 )
−
1
2
∇
2
i −
N
a=1
Z a
r ai
φ i (r 1 )dτ 1
(3.15)
being the contribution of one-electron terms to energy (where electron 1 has been
arbitrarily assigned to orbital φ i and dτ 1 represents the integration with respect to
5 Although we only considered the equilibrium geometry in the above formulation one can also use
the same equations at any fixed nuclear geometry. In that case it is more convenient to include in
the notation the fixed nuclear geometry
R as a fixed parameter in the notation.