3 Local Dielectric Constant Density Analysis of High-k Dielectric Nanomaterial
61
homogeneous constant vector D. Any definite environmental medium is not set.
Here and hereafter, in the expression of expectation values, bra and ket are dropped,
and we can discriminate an operator and its expectation value by the existence of
the hat symbol “ ˆ .”
Local polarizability density tensor is calculated by using a finite difference
method or coupled perturbed Hartree-Fock (CPHF) method. In both computations
of polarization, only electronic contributions are taken into account, and nuclear
contributions are not included. First computation of a finite difference method is
explained. With six results with different electric displacements, D i
1,2 = 0 ±
D i /2 = ±0.0001 [a.u.] (i = x, y, z), that is, ±D x , ±D y and ±D z , the
polarization difference can be calculated as
P
i (x) = P
i (x)| +D j − P
i (x)| −D j .
(3.19)
With these three polarization difference vectors (j = x, y, z), local polarizability
density tensor is calculated with the expression,
P
i (x) = α
ij (x))D
j .
(3.20)
By using CPHF method, linear response to external electric field is strictly calculated, though contamination from nonlinear effect was negligible in the above finite
difference method. Local polarizability is calculated by CPHF method as follows
[18]. Hamiltonian h and density matrix R are expanded with the perturbation
parameter, λ, as
h(λ) = h
(0)
+ λh
(1) ,
(3.21)
R(λ) = R
(0)
+ λR
(1)
+ · · · .
(3.22)
The first order Fock matrix h
F (1) is given as
h
F (1)
= h
(1)
+ G(R
(1) ).
(3.23)
The matrix G is two-electron interaction,
G pq (R) =
r,s
R rs
[ψ p ψ q |ψ r ψ s ] − [ψ p ψ s |ψ r ψ q ]
,
(3.24)
where subscripts p, q, r, s are used for all molecular orbitals (MOs). Then, the firstorder CPHF equation is given by
(( a − i )R
(1)
ai = −h
F (1)
ai ,
(3.25)
where is unperturbed orbital energy, and subscripts a and i are used for virtual and
occupied MOs, respectively. In the computation of local polarizability, perturbation
parameter is taken for external electric field,
61
homogeneous constant vector D. Any definite environmental medium is not set.
Here and hereafter, in the expression of expectation values, bra and ket are dropped,
and we can discriminate an operator and its expectation value by the existence of
the hat symbol “ ˆ .”
Local polarizability density tensor is calculated by using a finite difference
method or coupled perturbed Hartree-Fock (CPHF) method. In both computations
of polarization, only electronic contributions are taken into account, and nuclear
contributions are not included. First computation of a finite difference method is
explained. With six results with different electric displacements, D i
1,2 = 0 ±
D i /2 = ±0.0001 [a.u.] (i = x, y, z), that is, ±D x , ±D y and ±D z , the
polarization difference can be calculated as
P
i (x) = P
i (x)| +D j − P
i (x)| −D j .
(3.19)
With these three polarization difference vectors (j = x, y, z), local polarizability
density tensor is calculated with the expression,
P
i (x) = α
ij (x))D
j .
(3.20)
By using CPHF method, linear response to external electric field is strictly calculated, though contamination from nonlinear effect was negligible in the above finite
difference method. Local polarizability is calculated by CPHF method as follows
[18]. Hamiltonian h and density matrix R are expanded with the perturbation
parameter, λ, as
h(λ) = h
(0)
+ λh
(1) ,
(3.21)
R(λ) = R
(0)
+ λR
(1)
+ · · · .
(3.22)
The first order Fock matrix h
F (1) is given as
h
F (1)
= h
(1)
+ G(R
(1) ).
(3.23)
The matrix G is two-electron interaction,
G pq (R) =
r,s
R rs
[ψ p ψ q |ψ r ψ s ] − [ψ p ψ s |ψ r ψ q ]
,
(3.24)
where subscripts p, q, r, s are used for all molecular orbitals (MOs). Then, the firstorder CPHF equation is given by
(( a − i )R
(1)
ai = −h
F (1)
ai ,
(3.25)
where is unperturbed orbital energy, and subscripts a and i are used for virtual and
occupied MOs, respectively. In the computation of local polarizability, perturbation
parameter is taken for external electric field,