1 Interpretation of the Kohn--Sham Electron--Interaction Energy Functional EeeKS [ρ] and Its Derivative vee(r)
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188
5 Physical Interpretation of Kohn–Sham Density Functional Theory …
with
F eff (r) = E ee (r) + Z tc (r).
(5.6)
KS
[ρ]/δρ(r) is that it is
Hence, the physical meaning of the functional derivative δ E ee
the work done to bring the model fermion from some reference point at infinity to its
position at r in the force of the conservative field F eff (r). Since ∇ × F eff (r) = 0, this
work done is path independent. Once again, the electron–interaction and Correlation–
Kinetic contributions to the functional derivative vee (r) are explicitly defined via
Q–DFT.
From the above interpretation of the potential energy vee (r) we have
∇vee (r) = ∇
KS
[ρ]
δ E ee
δρ(r)
= −F eff (r),
(5.7)
or, equivalently employing (5.1) and (5.6) that
∇
δTc
δ E ee
+
δρ(r) δρ(r)
= −(E ee (r) + Z tc (r)).
(5.8)
This equation relates the functional derivatives of E ee and Tc to the component fields
E ee (r) and Z tc (r). Note, however, that
∇
δ E ee
δρ(r)
= −E ee (r),
(5.9)
∇
δTc
δρ(r)
= −Z tc (r).
(5.10)
and
These inequalities hold whether or not the fields E ee (r) and Z tc (r) are conservative.
The equality of the functional derivatives to the fields is that given by (5.5) or (5.8).
Since the pair–correlation density may also be written as g(rr ) = ρ(r ) + ρxc (rr ),
where ρxc (rr ) is the Fermi–Coulomb hole charge, the electron–interaction field
E ee (r) of (4.56) may be expressed as the sum of its Hartree E H (r) and Pauli–Coulomb
E xc (r) components:
(5.11)
E ee (r) = E H (r) + E xc (r),
where
E H (r) =
ρ(r )(r − r )
dr and E xc (r) =
|r − r |3
ρxc (rr )(r − r )
dr .
|r − r |3
(5.12)
5.1 Interpretation of the Kohn–Sham Electron–Interaction Energy Functional …
189
As the field E H (r) is due to a static or local charge distribution ρ(r), it may be
expressed as
(5.13)
E H (r) = −∇WH (r),
with the scalar potential energy WH (r) being
WH (r) =
ρ(r )
dr .
|r − r |
(5.14)
Thus, ∇ × E H (r) = 0. Equivalently, the potential energy WH (r) is the work done in
the conservative field E H (r):
WH (r) = −
r
∞
E H (r ) · d .
(5.15)
A comparison of (4.91) and (5.14) shows that
vH (r) = WH (r).
(5.16)
Thus, the physical interpretation of the functional derivative δ E H [ρ]/δρ(r) is that
it is the work done to move a model fermion from its reference point at infinity to its
position at r in the force of the conservative field E H (r). Equivalently
∇
δ E H [ρ]
δρ(r)
= −E H (r).
(5.17)
The Hartree energy functional E H [ρ] of (4.88), which is the energy of self–interaction
of the density, may also be expressed in terms of the Hartree field E H (r) as
EH =
ρ(r)r · E H (r)dr.
(5.18)
Again, employing the partitioning of the pair–correlation density g(rr ) into its
local and nonlocal components, we can write the quantum–mechanical electron–
interaction energy E ee as
E ee = E H + E xc ,
(5.19)
where E xc is the Pauli–Coulomb energy. Thus, the KS electron–interaction energy
functional (5.1) is
KS
= E H + E xc + Tc .
(5.20)
E ee
Comparison with (5.19) then defines the KS ‘exchange–correlation’ energy functional
in terms of the Pauli and Coulomb correlations and Correlation–Kinetic effects as
KS
[ρ] = E xc + Tc ,
E xc
(5.21)
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5 Physical Interpretation of Kohn–Sham Density Functional Theory …
where E xc is expressed in terms of the Pauli–Coulomb field E xc (r) as
E xc =
ρ(r)r · E xc (r)dr,
(5.22)
and with Tc as previously defined by (5.3).
The KS ‘exchange–correlation’ potential energy vxc (r) is the work done to bring
the model fermion from a reference point at infinity to its position at r in the conservative field F xctc (r):
vxc (r) =
KS
[ρ]
δ E xc
=−
δρ(r)
r
∞
F xctc (r ) · d ,
(5.23)
where
F xctc (r) = E xc (r) + Z tc (r).
(5.24)
This follows from (5.5) using the fact that the Hartree field E H (r) is conservative so
that ∇ × F xctc (r) = 0. Equivalently,
∇vxc (r) = ∇
KS
[ρ]
δ E xc
δρ(r)
= −(E xc (r) + Z tc (r)).
(5.25)
KS
[ρ] and its functional
Thus, the KS ‘exchange–correlation’ energy functional E xc
derivative vxc (r) can be expressed in terms of the Pauli–Coulomb E xc (r) and
KS
[ρ]
Correlation–Kinetic Z tc (r) fields. Hence, the dependence of the functional E xc
and its derivative vxc (r) on the separate electron correlations due to the Pauli principle, Coulomb repulsion, and Correlation–Kinetic effects is explicitly defined within
the framework of Q–DFT.
Substituting (5.21) into (5.25) leads to
∇
δTc
δ E xc
+
δρ(r) δρ(r)
= −(E xc (r) + Z tc (r)),
(5.26)
which relates the functional derivative of the quantum–mechanical exchange–
correlation E xc and Correlation–Kinetic Tc energies to the fields E xc (r) and Z tc (r)
that give rise to them, respectively. Again, irrespective of whether the field E xc (r) is
conservative or not
δ E xc
= −E xc (r).
(5.27)
∇
δρ(r)
Thus, we see that the mathematical entities of KS–DFT, viz. the electron-interaction
KS
KS
[ρ], its Hartree E H [ρ] and ‘exchange-correlation’ E xc
[ρ]
energy functional E ee
components, and their respective functional derivatives vee (r), v H (r), and vxc (r) can
all be afforded a rigorous physical interpretation.
5.1 Interpretation of the Kohn–Sham Electron–Interaction Energy Functional …
191
We next turn to the physical interpretation of the KS ‘exchange’ E xKS [ρ] and
‘correlation’ E KS
c [ρ] energy functionals, and of their respective derivatives vx (r) and
vc (r) in terms of the various electron correlations. This is achieved [15, 16] via
adiabatic coupling constant perturbation theory [17] as applied to both Q–DFT and
KS–DFT. The interpretations then follow on comparison of terms of equal order. We
begin by first describing the adiabatic coupling constant scheme, and Q–DFT and
KS–DFT within this framework.
5.2 Adiabatic Coupling Constant Scheme
In the adiabatic coupling constant (λ) scheme [5–7], the Hamiltonian Hˆ λ is defined
as
(5.28)
Hˆ λ = Tˆ + Vˆλ + λUˆ ; 0 ≤ λ ≤ 1,
where Tˆ and Uˆ are the usual kinetic and electron interaction operators and where the
external potential energy operator Vˆλ = i vλ (ri ). The corresponding Schrödinger
equation is
(5.29)
H λ ψ λ (X) = E λ ψ λ (X),
where ψ λ (X) is the ground state wavefunction for interaction strength λ. The real
interacting system corresponds to λ = 1. In (5.28), the operator Vˆλ is constrained so
that its addition to λUˆ leads to the density for the real system, i.e. the wavefunction
λ
ˆ
= ρλ=1 (r) = ρ(r). Equivalently,
ψ λ (X) is such that the expectation ψ λ |ρ(r)|ψ
the ground state density is independent of λ. For each value of λ, the energy E λ is
Eλ = T λ +
λ
ρ(r)vλ (r)dr + E ee
,
(5.30)
λ
= ψ λ |λUˆ |ψ λ , the electron–
where T λ = ψ λ |Tˆ |ψ λ is the kinetic energy, and E ee
interaction energy. The equivalent constrained search definition of ψ λ (X) is that it
is an antisymmetric wavefunction which yields the density ρ(r) and minimizes the
expectation ψ λ |Tˆ + λUˆ |ψ λ .
The λ = 0 case corresponds to the S system of noninteracting fermions defined
by the differential equation (4.76). The potential energy (4.75) of these fermions is
vs (r) = vλ=1 (r) + vee (r). Since the density ρ(r) is independent of λ, we may also
write
λ
(r).
(5.31)
vs (r) = vλ (r) + vee
The ground state energy E λ may then be expressed as
E λ = Ts +
λ
ρ(r)vλ (r)dr + E ee
+ Tcλ
(5.32)
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5 Physical Interpretation of Kohn–Sham Density Functional Theory …
or as
Eλ =
i
−
λ
λ
ρ(r)vee
(r)dr + E ee
+ Tcλ ,
(5.33)
i
where Tcλ = T λ − Ts is the Correlation–Kinetic energy for coupling strength λ.
5.2.1 Q–DFT Within Adiabatic Coupling Constant
Framework
The ‘Quantal Newtonian’ first law and integral virial theorem derived for the fully
interacting case are equally valid for the adiabatically coupled system. Thus, the
corresponding Q–DFT equations are the same as described in Chap. 3 but with the
appropriate λ dependence. (The S system components of these equations remain
unchanged.) The Q–DFT equations within the adiabatic coupling constant framework
are summarized below.
The pair–correlation density g λ (rr ) quantal source is
ˆ
)|ψ λ /ρ(r),
g λ (rr ) = ψ λ | P(rr
= ρ(r ) +
ρλxc (rr
(5.34)
),
= ρ(r ) + ρx (rr ) +
(5.35)
ρλc (rr
),
(5.36)
where the Fermi–Coulomb ρλxc (rr ), Fermi ρx (rr ), and Coulomb ρλc (rr ) holes satisfy
the charge conservation sum rules
ρλxc (rr )dr = −1;
ρλc (rr )dr = 0.
ρx (rr )dr = −1;
(5.37)
The Fermi hole ρx (rr ) = −|γs (rr )|2 /2ρ(r), where γs (rr ) is the S system Dirac
density matrix. The spinless single–particle density matrix source of the adiabatically
ˆ
)|ψ λ .
coupled system is γ λ (rr ) = ψ λ |γ(rr
λ
The electron–interaction field E ee (r) is then
g λ (rr )(r − r )
,
|r − r |3
= λE H (r) + λE λxc (r),
E λee (r) = λ
= λE H (r) + λE x (r) +
λE λc (r),
(5.38)
(5.39)
(5.40)
5.2 Adiabatic Coupling Constant Scheme
193
where
E H (r) =
E x (r) =
ρ(r )(r − r )
dr ; E λxc (r) =
|r − r |3
ρx (rr )(r − r )
dr ; E λc (r) =
|r − r |3
ρλxc (rr )(r − r )
dr ;
|r − r |3
ρλc (rr )(r − r )
dr .
|r − r |3
(5.41)
(5.42)
The Correlation–Kinetic field Z λtc (r) is
Z λtc (r) =
1
[z s (r; [γs ]) − z λ (r; [γ λ ])],
ρ(r)
(5.43)
λ
λ
where the component z αλ (r) of the field z λ (r) is z αλ = 2 β ∂tαβ
(r)/∂rβ , and tαβ
(r)
1
λ
2
2
is the kinetic energy density tensor tαβ (r) = 4 (∂ /∂rα ∂rβ + ∂ /∂rβ ∂rα )
γ λ (r r )|r =r =r . The field z s (r) is similarly derived from the idempotent Dirac density matrix γs (rr ) via the S-system tensor ts,αβ (r).
For the system of electrons defined by the Schrödinger equation (5.29), the
λ
(r) of the S system is the work done to
electron–interaction potential energy vee
move the model fermion in the conservative field F eff,λ (r):
λ
(r) = −
vee
where
r
∞
F eff,λ (r ) · d ,
F eff,λ (r) = E λee (r) + Z λtc (r),
(5.44)
(5.45)
and ∇ × F eff,λ (r) = 0. For systems of symmetry such that the fields E λee (r) and
λ
(r) may be written as
Z λtc (r) are separately conservative, the potential energy vee
λ
vee
(r) = Weeλ (r) + Wtλc (r)
= λWH (r) +
(5.46)
λ
λWxc
(r)
= λWH (r) + λWx (r)
+ Wtλc (r)
+ λWcλ (r)
(5.47)
+ Wtλc (r),
(5.48)
λ
where Weeλ (r), WH (r), Wxc
(r), Wx (r), Wcλ (r), Wtλc (r) are, respectively, the work done
λ
in the fields E ee (r), E H (r), E λxc (r), E x (r), E λc (r), and Z λtc (r). The work done WH (r)
in the Hartree field E H (r) may also be expressed as WH (r) = dr ρ(r )/|r − r |.
λ
λ
, Hartree E Hλ , Pauli–Coulomb E xc
, Pauli E xλ , Coulomb
The electron–interaction E ee
λ
λ
E c , and Correlation–Kinetic Tc energies are expressed in integral virial form in terms
of the respective fields as:
λ
=
E ee
drρ(r)r · E λee (r);
(5.49)
194
5 Physical Interpretation of Kohn–Sham Density Functional Theory …
E Hλ = λE H ,
(5.50)
with
EH =
drρ(r)r · E H (r);
drρ(r)r · E λxc (r);
λ
=λ
E xc
E xλ = λE x ,
(5.51)
(5.52)
(5.53)
with
Ex =
drρ(r)r · E x (r);
(5.54)
E cλ = λ
drρ(r)r · E λc (r);
(5.55)
1
2
drρ(r)r · Z λtc (r).
(5.56)
and
Tcλ =
Note that these energy expressions are valid irrespective of whether or not the individual fields are conservative. Observe that the Hartree E Hλ and Pauli E xλ energies
scale linearly with λ.
5.2.2 KS–DFT Within Adiabatic Coupling Constant
Framework
KS–DFT employs the fact that the wavefunction ψ λ (X) of the adiabatically coupled
system is a functional of the density ρ(r). Hence, the ground state energy E λ as
obtained from the model S system is expressed as
E λ [ρ] = Ts +
KS,λ
drρ(r)vλ (r) + E ee
[ρ],
(5.57)
KS,λ
[ρ] is the KS electron–interaction energy functional. The energy funcwhere E ee
KS,λ
[ρ] is further divided into the Hartree E Hλ [ρ] and KS ‘exchange–
tional E ee
KS,λ
[ρ] components, the latter functional being further subdivided into
correlation’ E xc
an ‘exchange’ E xKS,λ [ρ] and ‘correlation’ E cKS,λ [ρ] component. Thus,
KS,λ
KS,λ
[ρ] = E Hλ [ρ] + E xc
[ρ],
E ee
=
E Hλ [ρ]
+
E xKS,λ [ρ]
+
(5.58)
E cKS,λ [ρ].
(5.59)
5.2 Adiabatic Coupling Constant Scheme
195
Here E Hλ [ρ], the Coulomb self–energy, is
E Hλ [ρ] = λE H [ρ],
with
E H [ρ] =
(5.60)
ρ(r)ρ(r )
drdr ,
|r − r |
1
2
(5.61)
and [18]
E xKS,λ [ρ] = λE xKS [ρ],
with
E xKS [ρ] =
1
2
ρ(r)ρx (rr )
drdr .
|r − r |
(5.62)
(5.63)
The Hartree functional E H [ρ] is obviously equivalent to the Q–DFT Hartree energy
E H of (5.51). The KS ‘exchange’ energy functional E xKS [ρ] expression is also equivalent to the Q–DFT Pauli energy E x of (5.54) since the source for these energies—the
Fermi hole or Dirac density matrix—is the same provided the orbitals are the same.
As a consequence, the scaling of these functionals with λ is also linear.
λ
(r) of the S system is
In KS–DFT, the electron–interaction potential energy vee
KS,λ
defined as the functional derivative of E ee (r), so that
KS,λ
[ρ]
δ E ee
δρ(r)
λ
= vHλ (r) + vxc
(r)
λ
(r) =
vee
=
vHλ (r)
+
where
vxλ (r)
+
(5.64)
(5.65)
vcλ (r),
(5.66)
δ E Hλ [ρ]
,
δρ(r)
(5.67)
vxλ (r) =
δ E xKS,λ [ρ]
.
δρ(r)
(5.68)
vcλ (r) =
δ E cKS,λ [ρ]
.
δρ(r)
(5.69)
vHλ (r) =
Here vxλ (r) and vcλ (r) are the KS ‘exchange’ and ‘correlation’ potential energies,
respectively. From the scaling relationships for E Hλ [ρ] and E xKS,λ [ρ], we see that the
corresponding functional derivatives also scale linearly [18]:
vHλ (r) = λvH (r),
(5.70)
196
5 Physical Interpretation of Kohn–Sham Density Functional Theory …
with
vH (r) =
and
δ E H [ρ]
=
δρ(r)
ρ(r )
dr ,
|r − r |
vxλ (r) = λvx (r),
with
vx (r) =
δ E xKS [ρ]
.
δρ(r)
(5.71)
(5.72)
(5.73)
As noted previously, the functional derivative vH (r) = WH (r).
5.2.3 Q–DFT and KS–DFT in Terms of the Adiabatic
Coupling Constant Perturbation Expansion
The relationship between the KS ‘exchange’ E xKS [ρ] and ‘correlation’ E cKS [ρ] energy
functionals, their functional derivatives vx (r) and vc (r), and the fields of Q–DFT
is achieved [15, 16] by expressing these fields in terms of the coupling constant
perturbation expansion. Thus, the wavefunction ψ λ (X) is expanded as
ψ λ (X) =
{φi }(X) + λψ1 (X) + λ2 ψ2 (X) + . . .
(5.74)
where {φi } is the Slater determinant of the S system. The resulting pair–correlation
density g λ (rr ) and single–particle density matrix γ λ (rr ) are
and
g λ (rr ) = gs (rr ) + λg1c (rr ) + λ2 g2c (rr ) + . . . ,
(5.75)
γ λ (rr) = γs (rr ) + λγ1c (rr ) + λ2 γ2c (rr ) + . . . ,
(5.76)
gs (rr ) = ρ(r ) + ρx (rr ),
(5.77)
where
ˆ
ˆ 1 , etc. The fields E λee (r) of (5.38) and Z λt (r) of
+ ψ| O|ψ
and O1c (rr ) = ψ1 | O|ψ
c
(5.43), which arise from these sources, are then
E λee (r) = λE H (r) + λE x (r) + λ2 E c,1 (r) + λ3 E c,2 (r) + . . . ,
and
Z λtc (r) = −λZ tc ,1 (r) − λ2 Z tc ,2 (r) − λ3 Z tc ,3 (r) − . . . ,
(5.78)
(5.79)
5.2 Adiabatic Coupling Constant Scheme
197
where E c,1 (r) = dr g1c (rr )(r − r )/|r − r |3 and Z tc ,1 (r) = z(r; [γ1c ])/ρ(r), etc.
The expansions of these fields can then be employed in the expressions for the
electron–interaction and correlation–kinetic energy and potential energy.
For systems with symmetry such that the individual fields are conservative, the
work done Weeλ (r) and Wtλc (r) in the fields E λee (r) and Z tλc (r), respectively, may then
be expressed as
Weeλ (r) = λWH (r) + λWx (r) + λ2 Wc,1 (r) + λ3 Wc,2 (r) + . . . ,
and
Wtλc (r) = −λWtc ,1 (r) − λ2 Wtc ,2 − . . . ,
(5.80)
(5.81)
where Wc,1 (r), Wtc ,1 (r), etc. are the work done in the fields E c,1 (r), Z tc ,1 (r), respectively, etc.
The scaling relationship for the KS–DFT functionals E Hλ [ρ] and E xKS,λ [ρ] are
given in the previous section. It has further been shown [17] that the KS ‘correlation’
energy functional E cKS,λ [ρ] commences in second order:
KS
KS
[ρ] + λ3 E c,3
[ρ] + . . . ,
E cKS,λ [ρ] = λ2 E c,2
(5.82)
so that the KS correlation potential too commences in second order:
vcλ (r) = λ2 vc,2 (r) + λ3 vc,3 (r) + . . . ,
(5.83)
KS
KS
[ρ]/δρ(r), etc., and E c,2
[ρ] is the O(λ2 ) KS correlation energy.
where vc,2 (r) = δ E c,2
From (5.44) we have
λ
(r) = −F eff,λ (r),
(5.84)
∇vee
so that on substituting for the field F eff,λ (r) from (5.45) and the KS definition for
the potential vee (r) from (5.66), we have
∇[vHλ (r) + vxλ (r) + vcλ (r)] = −[E λee (r) + Z λtc (r)].
(5.85)
On substitution of the expansions for the various terms as given in the previous
section, and on equating terms of equal order, we obtain the components of the KS
potential in terms of the fields as:
∇v H (r) = −E H (r),
(5.86)
∇vx (r) = −[E x (r) − Z tc ,1 (r)],
∇vc,2 (r) = −[E c,1 (r) − Z tc ,2 (r)],
(5.87)
(5.88)
∇vc,3 (r) = −[E c,2 (r) − Z tc ,3 (r)], etc.
(5.89)
198
5 Physical Interpretation of Kohn–Sham Density Functional Theory …
We are now in the position to provide a rigorous interpretation of the KS ‘exchange’
and ‘correlation’ energy functionals and their functional derivatives in terms of the
electron correlations that contribute to them. However, prior to explaining the interpretation, note that (5.86) is equivalent to (5.17). The physical reason for this equivalence between the functional derivative v H (r) and the field E H (r) is that the latter
is due to the density ρ(r) which is a static charge distribution.
5.3 Interpretation of the Kohn–Sham ‘Exchange’ Energy
Functional ExKS [ρ] and Its Derivative vx (r)
The physical interpretation of the KS ‘exchange’ potential energy vx (r) follows from
(5.87). It is the work done to move the model fermion in a conservative field R(r):
vx (r) =
δ E xKS [ρ]
=−
δρ(r)
r
∞
R(r ) · d ,
(5.90)
where R(r) = E x (r) − Z tc ,1 (r). Since ∇ × R(r) = 0, this work done is path independent. The field R(r), and hence the potential energy vx (r), is therefore representative both of Pauli correlations via the component field E x (r), as well as those due
to part of the Correlation–Kinetic effects through the field Z tc ,1 (r).
For systems with symmetry such that the fields E x (r) and Z tc ,1 (r) are separately
conservative, we may write
vx (r) = Wx (r) + Wtc ,1 (r),
(5.91)
where Wx (r) is the work done in the field E x (r) due to the Fermi hole charge, and
Wtc ,1 (r) the work done in the field Z tc ,1 (r).
The KS ‘exchange’ energy functional E xKS [ρ] is related to its functional derivative
vx (r) by the virial theorem of (4.99). Substituting (5.87) into this equation leads to
E xKS [ρ] −
ρ(r)r · [E x (r) − Z tc ,1 (r)]dr = 0.
(5.92)
Now as noted in Sect. 5.2.2, E xKS [ρ] is equivalent to the Q–DFT Pauli energy E x
provided the same orbitals are employed in their determination. Thus, using the
relationship [19] between E x and E x (r) of (5.54) in the above equation, it follows
that
(5.93)
ρ(r)r · Z tc ,1 (r)dr = 0.
Therefore, although the Correlation–Kinetic field Z tc ,1 (r) contributes explicitly to
the potential energy vx (r), it does not contribute directly to the KS ‘exchange’ energy