126
6 Charge Response Kernel for Electronic Polarization
on the electronic structure. The partial charge Q a is given with the expected value
at an electronic (many-body) wavefunction by
Q a =
| ˆ
Q a |
.
The partial charge Q a is related to the derivative of the total energy E by the
Hellmann-Feynman theorem,
∂E
∂V a
=
∂
∂V a
| ˆ
H |
=
|
∂ ˆ
H
∂V a
|
=
| ˆ
Q a |
= Q a .
(6.6)
This relation (6.6) is valid when the wavefunction satisfies the variational
principle. The CRK is consequently given by the second-order derivative of the
total energy E,
K ab =
∂Q a
∂V b
=
∂ 2 E
∂V a ∂V b
.
(6.7)
The above Eq. (6.7) obviously indicates that K ab is a symmetric matrix, K ab = K ba .
ESP charge Then we define the ESP charge Q a and its operator ˆ
Q a . The partial
charge Q a is determined from electron density and position of nuclei, and the electron density is given as a function of one-electron spatial coordinate. Accordingly, its
operator ˆ
Q a is presented with the one-electron operator for population distribution
ˆ
n a (i) and a function of the nuclear position by
ˆ
Q a = −e
N e
i=1
ˆ
n a (i) + Q
nuc
a .
(6.8)
The first term of Eq. (6.8) is a sum of one-electron operators ˆ
n a (i) for equivalent
N e electrons, and represents the occupation number of electrons at each site a. The
second term Q nuc
a accounts for the contribution of nuclear charge. Q a is represented
as the expectation value of ˆ
Q a ,
Q a =
| ˆ
Q a |
= −e
AO
p,q
D pq
p| ˆ
n a |q
+ Q
nuc
a ,
(6.9)
where p, q denotes the basis sets (atomic orbitals in conventional quantum chemistry), and D pq is the one-electron density matrix. The last expression of Eq. (6.9)
including the density matrix is valid also for DFT, which does not explicitly treat
the wavefunction.
The ESP charge Q a is determined so as to reproduce the surrounding electrostatic
potential, as illustrated in Fig. 6.2. First we introduce a set of monitoring points
6 Charge Response Kernel for Electronic Polarization
on the electronic structure. The partial charge Q a is given with the expected value
at an electronic (many-body) wavefunction by
Q a =
| ˆ
Q a |
.
The partial charge Q a is related to the derivative of the total energy E by the
Hellmann-Feynman theorem,
∂E
∂V a
=
∂
∂V a
| ˆ
H |
=
|
∂ ˆ
H
∂V a
|
=
| ˆ
Q a |
= Q a .
(6.6)
This relation (6.6) is valid when the wavefunction satisfies the variational
principle. The CRK is consequently given by the second-order derivative of the
total energy E,
K ab =
∂Q a
∂V b
=
∂ 2 E
∂V a ∂V b
.
(6.7)
The above Eq. (6.7) obviously indicates that K ab is a symmetric matrix, K ab = K ba .
ESP charge Then we define the ESP charge Q a and its operator ˆ
Q a . The partial
charge Q a is determined from electron density and position of nuclei, and the electron density is given as a function of one-electron spatial coordinate. Accordingly, its
operator ˆ
Q a is presented with the one-electron operator for population distribution
ˆ
n a (i) and a function of the nuclear position by
ˆ
Q a = −e
N e
i=1
ˆ
n a (i) + Q
nuc
a .
(6.8)
The first term of Eq. (6.8) is a sum of one-electron operators ˆ
n a (i) for equivalent
N e electrons, and represents the occupation number of electrons at each site a. The
second term Q nuc
a accounts for the contribution of nuclear charge. Q a is represented
as the expectation value of ˆ
Q a ,
Q a =
| ˆ
Q a |
= −e
AO
p,q
D pq
p| ˆ
n a |q
+ Q
nuc
a ,
(6.9)
where p, q denotes the basis sets (atomic orbitals in conventional quantum chemistry), and D pq is the one-electron density matrix. The last expression of Eq. (6.9)
including the density matrix is valid also for DFT, which does not explicitly treat
the wavefunction.
The ESP charge Q a is determined so as to reproduce the surrounding electrostatic
potential, as illustrated in Fig. 6.2. First we introduce a set of monitoring points