Theor Chem Acc (2015) 134:86
1 3
(this conclusion was drawn not only for ethane molecule,
but for every system considered as yet).
3 The LCAO Hamiltonian
In the followings we shall use, besides the non-orthogonal
set of original basis orbitals {χ μ } , also the respective Löwdin-orthogonalized set {ψ ν } of orbitals:
where S
−
1
2
μν
is an element of the −
1
2 th power of the overlap
matrix, as well as the biorthogonal set {ϕ ρ } :
All the three sets span the same subspace of the one-electron functions.
We introduce creation and annihilation operators corresponding to each set of these orbitals. In order to distinguish to what type of orbitals the given creation or annihilation orbital is pertinent, we shall use Longuet-Higgins’
[ 16 ] notations ˆ
ψ +
ν , ˆ
χ +
μ and ˆ
ϕ +
ρ for the creation operators and
ˆ
ψ −
ν , ˆ
χ −
μ and ˆ
ϕ −
ρ for the annihilation ones. The annihilation
operators are defi ned as the adjoints of the respective creation operators:
The creation operators transform in the same manner as the
respective orbitals do, i.e., according to Eqs. ( 11 ) and ( 12 ).
However, standard Fermion anticommutation rules hold
only for the creation and annihilation operators defi ned for
the orthonormalized set {ψ ν }
while
and
respectively. Owing to the presence of the (inverse) overlap matrix elements in the anticommutators ( 15 ) and
( 16 ), Fermion anticommutation rules hold for the mixed
anticommutators
(11)
ψ ν =
μ
S
−
1
2
μν
χ μ ,
(12)
ϕ ρ =
μ
S
−1
μρ χ μ .
(13)
ˆ
ψ
−
ν = ( ˆ
ψ
+
ν )
†
ˆ
χ
−
μ = ( ˆ
χ
+
μ )
†
ˆ
ϕ
−
ρ = ( ˆ
ϕ
+
ρ )
† .
(14)
{ ˆ
ψ
+
ν ; ˆ
ψ
−
μ } = ˆ
ψ
+
ν
ˆ
ψ
−
μ + ˆ
ψ
+
μ
ˆ
ψ
−
ν = δ μν ,
(15)
{ ˆ
χ
+
ν ; ˆ
χ
−
μ } = ˆ
χ
+
ν ˆ
χ
−
μ + ˆ
χ
+
μ ˆ
χ
−
ν = S μν ,
(16)
{ ˆ
ϕ
+
ν ; ˆ
ϕ
−
μ } = ˆ
ϕ
+
ν ˆ
ϕ
−
μ + ˆ
ϕ
+
μ ˆ
ϕ
−
ν = S
−1
μν ,
(17)
{ ˆ
χ
+
ν ; ˆ
ϕ
−
μ } = δ μν .
and
This means that when acting to right on a string of creation operators ˆ
χ +
μ in a “ket”, operator ˆ
ϕ −
μ behaves as a
conventional annihilation operator does, and analogously,
when acting to left on a string of annihilation operators
ˆ
χ −
μ = ( ˆ
χ +
μ ) † in a “bra”, operator ˆ
ϕ +
μ behaves as a conventional creation operator.
The LCAO version of the Born–Oppenheimer Hamiltonian has a standard form in terms of the creation and annihilation operators referring to the Löwdin-orthogonalized
basis [ 16 , 17 ]:
Here the fi rst sum describes the internuclear repulsion,
h
μν = =ψ μ | ˆ
h|ψ ν is the matrix element of the one-electron
Hamiltonian
in the Löwdin-orthogonalized basis, and [ψ μ ψ ν |ψ ρ ψ τ ]
is a two-electron integral in that basis and the [12|12]
convention.
Using the transformations ( 11 ), ( 12 ) connecting the
different sets of the orbitals (and thus also the respective
creation and annihilation operators,) one can transform the
Hamiltonian ( 19 ) into several equivalent forms. We shall
present here two of them.
Using “biorthogonal” operators
In one version we collect pairs of matrices S
−
1
2 into
matrices S −1 , and express the Hamiltonian in terms of the
one- and two-electron integrals over the original overlapping basis orbitals and of the “biorthogonal” creation and
annihilation operators ˆ
ϕ +
μ , ˆ
ϕ −
ν :
where h μν and [μν|κρ] are the one- and two-electron integrals calculated for the overlapping set of basis orbitals
{χ μ } .
As it is known [ 17 ], the expectation value E of operator ˆ
H can be expressed through the matrix representations
(18)
{ ˆ
ϕ
+
ν ; ˆ
χ
−
μ } = δ μν .
(19)
ˆ
H =
A Z A Z B
R AB
+
μ,ν
h
μν
ˆ
ψ
+
μ
ˆ
ψ
−
ν
+
1
2
μ,ν,ρ,τ
[ψ μ ψ ν |ψ ρ ψ τ ] ˆ
ψ
+
μ
ˆ
ψ
+
ν
ˆ
ψ
−
τ
ˆ
ψ
−
ρ .
(20)
ˆ
h = −
1
2
−
A
Z A
r A
,
(21)
ˆ
H =
A Z A Z B
R AB
+
μ,ν
h μν ˆ
ϕ
+
μ ˆ
ϕ
−
ν
+
1
2
μ,ν,κ,ρ
[μν|κρ] ˆ
ϕ
+
μ ˆ
ϕ
+
ν ˆ
ϕ
−
ρ ˆ
ϕ
−
κ .
34
Reprinted from the journal