5 Appendix: Detailed Description of the Models
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184
O. Mafra Lopes Jr. et al.
where only two of ϕ2 , ϕ3 , ϕ4 are linearly independent. Please notice that S(Ω ) is
simplified as all products between the functions centered on different atoms are
vanishing.
References
1. Savin A (2002) In: Sen KD (ed) Reviews of modern quantum chemistry: a celebration of the
contributions of Robert G. Parr. World Scientific, Singapore, p 43
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14. Noury S, Krokidis X, Fuster F, Silvi B (1999) Comput Chem 23:597
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Izmaylov AF, Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K, Fukuda R,
Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Montgomery JA Jr,
Peralta JE, Ogliaro F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov VN,
Kobayashi R, Normand J, Raghavachari K, Rendell A, Burant JC, Iyengar SS, Tomasi J,
Cossi M, Rega N, Millam JM, Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo J,
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Chapter 11
Density Scaling for Excited States
´ Nagy
A.
Abstract The theory for a single excited state based on Kato’s theorem is revisited.
Density scaling proposed by Chan and Handy is used to construct a Kohn-Sham
scheme with a scaled density. It is shown that there exists a value of the scaling factor
for which the correlation energy disappears. Generalized OPM and KLI methods
incorporating correlation are proposed. A ζ KLI method as simple as the original
KLI method is presented for excited states.
11.1 Introduction
Density functional theory [1] in its original form is a ground-state theory which
is valid for the lowest-energy state in each symmetry class [2, 3]. It was, however,
applied to excited states as well, starting with the transition state method of Slater
[4]. The first rigorous generalization for excited states was proposed by Theophilou
[5]. The variational principle for excited states was studied by Perdew and Levy [6]
and Lieb [7]. Formalisms for excited states were also provided by Fritsche [8] and
English et al. [9]. Gross, Oliveira and Kohn [10] worked out the theory of unequally
weighted ensembles of excited states. Relativistic generalization of this formalism
was also done [11]. A theory of excited states was presented utilizing GăorlingLevy perturbation theory [12, 13] and a quasi-local-density approximation [14] was
proposed. Coordinate scaling and adiabatic connection were studied [15, 16]. (For
reviews of excited-state theories see [17, 18].) The optimized potential method was
generalized to ensembles [19–24].
The ensemble theory has the disadvantage that one has to calculate all the
ensemble energies lying under the given ensemble energy to obtain the desired
´ Nagy ( )
A.
Department of Theoretical Physics, University of Debrecen, H–4010 Debrecen, Hungary
e-mail: anagy@phys.unideb.hu
P.E. Hoggan et al. (eds.), Advances in the Theory of Quantum Systems in Chemistry
and Physics, Progress in Theoretical Chemistry and Physics 22,
DOI 10.1007/978-94-007-2076-3 11, © Springer Science+Business Media B.V. 2012
185
´ Nagy
A.
186
excitation energy. It is especially inconvenient to use it if one is interested in highly
excited states. That is why it is important to extend density functional theory to a
single excited state. Two theories for a single excited state [25, 26, 28, 29] exist.
A non-variational theory [25–27] based on Kato’s theorem and a variational density
functional theory [28, 29]. In this paper the non-variational theory is extended and
combined with density scaling [30, 31].
Density scaling was proposed by Chan and Handy [32]. In density scaling the
density n(r) is changed to ζ n(r). It is shown that there exist a value of the scaling
factor for which the correlation energy disappears. The optimized potential method
(OPM) [60] and the Krieger-Li-Iafrate (KLI) [61] approach are generalized to
incorporate correlation. In this paper only a non-degenerate excited state is treated.
Section 11.2 presents the non-variational theory. In Sect. 11.3 Kohn-Sham-like
equations are obtained through adiabatic connection. Density scaling is applied
to obtain a generalized Kohn-Sham scheme in Sect. 11.4. The optimized potential
and the KLI methods are generalized in Sect. 11.5. The last section is devoted to
illustrative examples and discussion.
We mention that there are other noteworthy single excited-state theories: the
stationary-principle theory of Găorling [33], the formalism of Sahni [34] or the local
scaling approach of Ludena and Kryachko [35]. Beyond the time-independent theories mentioned above the time-dependent density functional theory [36] provides
an alternative (see e.g. [37]).
11.2 Non-variational Theory for a Single Excited State
The consequence of the Hohenberg-Kohn theorem is that the ground-state electron
density determines all molecular properties. E. Bright Wilson [38] noticed that
Kato’s theorem [39, 40] provides an explicit procedure for constructing the Hamiltonian of a Coulomb system from the electron density:
Zβ = −
1 ∂ n(r)
¯
2n(r)
¯
∂r
.
(11.1)
r=Rβ
Here n¯ denotes the angular average of the density n and the right-hand side is
evaluated at the position of nucleus β . From Eq. 11.1, the cusps of the density show
us where the nuclei are (Rβ ) and what are their atomic numbers Zβ . The integral of
the density provides the number of electrons:
N=
n(r)dr.
(11.2)
Therefore one can readily obtain the Hamiltonian from the density and then
determine every property of the system by solving the Schrăodinger equation. One
11 Density Scaling for Excited States
187
has to emphasize, however, that this argument holds only for Coulomb systems.
By contrast, the density functional theory formulated by Hohenberg and Kohn is
valid for any local external potential.
Kato’s theorem is valid not only for the ground state but also for the excited
states. Consequently, if the density ni of the i-th excited state is known, the
Hamiltonian Hˆ is also in principle known and its eigenvalue problem
Hˆ Ψk = EkΨk
(k = 0, 1, ..., i, ...)
(11.3)
can be solved, where
Hˆ = Tˆ + Vˆ + Vˆee .
Tˆ =
N
1
− ∇2j ,
2
∑
j=1
Vˆee =
N−1
(11.4)
(11.5)
N
k=1
1
j=k+1 |rk − r j |
N
M
∑ ∑
(11.6)
and
Vˆ =
−ZJ
∑ ∑ |rk − RJ |
(11.7)
k=1 J=1
are the kinetic energy, the electron-electron energy and the electron-nuclear energy
operators, respectively.
We emphasize that there are certain special cases, where Eq. 11.1 does not
provide the atomic number. The simplest example is the 2p excited state of the
hydrogen atom, where the density
n2p (r) = cr2 e−Zr
(11.8)
and the derivative of the density are zero at the nucleus. Though Kato’s theorem
(11.1) is valid in this case, too, it does not provide the atomic number. Similar
cases occur in other highly excited atoms, ions or molecules, for which the spherical
average of the derivative of the wave function is zero at a nucleus, that is where we
have no s-electrons.
Pack and Brown [41] derived cusp relations for the wave functions of these
systems. The corresponding cusp relations for the density [42,43] were also derived.
Let us define
ηl (r) =
n(r)
¯
,
r2l
(11.9)
´ Nagy
A.
188
where l is the smallest integer for which ηl is not zero at the nucleus. The
corresponding cusp relations for the density are
∂ η l (r)
∂r
=−
r=0
2Z l
η (0).
l+1
(11.10)
For the example of a one-electron atom in the 2p state, Eq. 11.9 leads to
η2p (r) =
n2p
= ce−Zr
r2l
(11.11)
and the cusp relation has the form:
− 2Z η2p(0) = 2η2p (0).
(11.12)
So we can again readily obtain the atomic number from the electron density. Other
useful cusp relations have also been derived [44, 45]. There are several other works
concerning the cusp of the density [46–55].
11.3 Kohn-Sham-Like Equations
Making use of adiabatic connection [2, 56] Kohn-Sham-like equations can be
derived. We suppose the existence of a continuous path between the interacting and
the non-interacting systems. The density ni of the i-th electron state is the same
along the path.
Hˆ iα Ψkα = Ekα Ψkα ,
(11.13)
Hˆ iα = Tˆ + αVˆee + Vˆiα .
(11.14)
where
The subscript i denotes that the density of the given excited state is supposed to be
the same for any value of the coupling constant α . α = 1 corresponds to the fully
interacting case, while α = 0 gives the non-interacting system:
Hˆ i0Ψk0 = Ek0Ψk0 .
(11.15)
For α = 1 the Hamiltonian Hˆ iα is independent of i. For any other value of α
the ‘adiabatic’ Hamiltonian depends on i and we have different Hamiltonians for
different excited states. Thus the non-interacting Hamiltonian (α = 0) is different for
different excited states. If there are several ‘external’ potentials V α =0 leading to the
11 Density Scaling for Excited States
189
same density ni we select that potential for which the non-interacting kinetic energy
is closest to the interacting one. It is important to emphasize that this procedure only
works for the Coulomb case.
To apply the Kohn-Sham-like equations (11.15) one has to find an approximation
to the potential of the non-interacting system. The optimized potential method [60]
can be generalized for a single excited state, too. It was shown [25] that because
the energy is stationary at the true wave function, the energy is stationary at the
true potential. This is the consequence of the well-known fact that, when the energy
is considered to be a functional of the wave function, the only stationary points of
E[Ψ ] are those associated with the eigenvalues/eigenvectors of the Hamiltonian
δE
=0
δΨk
(k = 1, ..., i, ...).
(11.16)
The density ni of a given excited state determines the Hamiltonian and via
adiabatic connection the non-interacting effective potential Viα =0 . Therefore we can
consider the total energy as a functional of the non-interacting effective potential:
E[Ψi ] = E[Ψi [Vi0 ]].
(11.17)
Utilizing Eq. 11.16 we obtain
δE
=
δ Vi0
δ E δΨi
+ c.c. = 0.
δΨi δ Vi0
(11.18)
So an optimized effective potential can be found for the given excited state.
The KLI approximation to the optimized effective potential can also be derived
straightforwardly [25].
Exchange identities utilizing the principle of adiabatic connection and coordinate
scaling and a generalized ‘Koopmans’ theorem’ were derived and the excited-state
effective potential was constructed [57]. The differential virial theorem was also
derived for a single excited state [58].
11.4 Density Scaling for a Single Excited State
Now, the density scaling is applied to obtain a Kohn-Sham scheme with a scaled
density. Here, we suppose a non-degenerate excited state. Extension to degenerate
excited states will be detailed elsewhere. Consider a non-interacting system with
excited state density nζ i (r) = ni (r)/ζ , where ζ = N/Nζ is a positive number. In the
present theory we suppose that ζ is larger but close to 1. If the original real system
has N-electrons
ni (r)dr = N,
(11.19)
´ Nagy
A.
190
the Kohn-Sham system with the scaled density has Nζ -electrons:
nζ i (r)dr = Nζ .
(11.20)
N is always integer, but Nζ is generally non-integer. Therefore, the Kohn-Sham-like
equations will differ from the ones corresponding to the N-electron Kohn-Sham
system (11.15). To construct another Kohn-Sham system we define the density
nζ i = (1 − q)ni + qnion,
(11.21)
q = N − Nζ = N(1 − 1/ζ ).
(11.22)
where
We consider only that case for which q is a small positive number: q << 1. The
second term in Eq. 11.21 corresponds to the density of the ion that is obtained after
ionization. It is the ground state of the non-interacting N − 1-electron system if we
consider an excitation where the electron is excited from the highest occupied level
to a highest state. It can be an excited state of the non-interacting N − 1-electron
system in other cases. The Kohn-Sham system is a non-interacting system with the
scaled density nζ i . The non-interacting kinetic energy can be constructed from the
non-interacting wave function Ψi0 of the original ‘Kohn-Sham’ system (Eq. 11.15)
0 of the ion:
and the non-interacting wave function Ψion
0 ˆ ion
0
Tζ0 [ni ] = ζ (1 − q) Ψi0 |Tˆ |Ψi0 + q Ψion
|T |Ψion
.
(11.23)
The Kohn-Sham equations with the scaled density have the form
1
− ∇2 + vζ KSi (r) uζ i, j = εζ i, j uζ i, j ,
2
(11.24)
where the scaled Kohn-Sham density has the form
M
nζ i (r) = ∑ λζ i, j |uζ i, j (r)|2 .
(11.25)
j
λζ i, j are the occupation numbers and M is the largest integer for which λζ i, j = 0.
uζ i, j are the scaled Kohn-Sham orbitals. The non-interacting kinetic energy takes
the form
M
1
Tζ0 [ni ] = ζ ∑ λζ i, j uζ i, j | − ∇2 |uζ i, j .
2
j
(11.26)
11 Density Scaling for Excited States
191
In the original theory the exchange-correlation energy Exci [ni ] is defined by the
total energy expression
E[ni ] = T 0 [ni ] + J[ni] + Exci [ni ] +
ni (r)v(r)dr,
(11.27)
where
J[ni ] =
1
2
ni (r1 )ni (r2 )
dr1 dr2
|r1 − r2|
(11.28)
is the classical Coulomb energy and v is the external potential.
Similarly, in this Kohn-Sham theory with the scaled density the exchangecorrelation energy Eζ xc [ni ] is defined by the total energy expression
E[ni ] = Tζ0 [ni ] + J[ni] + Eζ xci [ni ] +
ni (r)v(r)dr.
(11.29)
A comparison of Eqs. 11.27 and 11.29 leads to the important relation
T 0 [ni ] + Exci [ni ] = Tζ0 [ni ] + Eζ xci [ni ].
(11.30)
Obviously, Tζ0=1 = T 0 . The relationship between Tζ0 and T 0 derived by Chan and
Handy [32] is valid for the excited state
Tζ [ni ]0 = ζ T 0 [nζ i ].
(11.31)
Now, we define the correlation energy as follows:
Eζ ci [ni ] = Ψi |Tˆ + Vˆee |Ψi
0
0
−[(1 − q) Ψi0 |ζ Tˆ + ζ 2Vˆee |Ψi0 + q Ψion
|ζ Tˆ ion + ζ 2Vˆeeion |Ψion
]. (11.32)
Theorem 11.1. There exists a parameter ζi for which the correlation energy
disappears: Eζ ci = 0.
Proof. If ζi = 1, Eζ ci is equal to the correlation energy Eci of the original theory.
Consider a small change in ζi and notice that from the definition (11.32) follows
that Eζ ci is almost quadratic in ζ . Eζ ci = 0 means that
0 ˆ ion
0
ζ 2 [(1 − q) Ψi0 |Vˆee |Ψi0 + q Ψion
|Vee |Ψion
] + ζ [(1 − q) Ψi0 |Tˆ |Ψi0
0 ˆ ion
0
|T |Ψion
] − Ψi |Tˆ + Vˆee |Ψi = 0.
+q Ψion
(11.33)
´ Nagy
A.
192
This equation has solutions as
0 ˆ ion
0 2
|T |Ψion
] + 4[(1 − q) Ψi0 |Vˆee |Ψi0
[(1 − q) Ψi0 |Tˆ |Ψi0 + q Ψion
0 ˆ ion
0
+q Ψion
|Vee |Ψion
] Ψi |Tˆ + Vˆee|Ψi > 0.
(11.34)
Consequently, there exist a value of ζ for which Eζ ci = 0. Note that Eq. 11.33 has
two solutions, however, the other solution is not close to 1. Moreover it can even be
negative and thus physically not acceptable.
11.5 The ζ OPM and ζ KLI Methods for a Single Excited State
In order to perform calculations one needs explicit expressions for the functionals.
In the ground-state theory, exchange can be treated exactly (or very accurately) via
the optimized potential method [60] (or KLI method [61]). Now, these methods are
combined with density scaling for a single excited state.
In the optimized potential method the following problem is solved: find the
potential such that when it is given a small variation, the energy of the system
remains stationary:
δE
= 0.
δV
(11.35)
From the fact the energy is stationary at the true wave function follows that the
energy is stationary at the true potential. It is well-known that considering the energy
as a functional of the wave function E[Ψ ], the eigenvalues of the Hamiltonian are
stationary points of E
δE
=0
δΨk
(k = 1, ..., i, ...),
(11.36)
and only the eigenvalues are stationary points.
As we emphasized above from the density of a given excited state ni one can
obtain the Hamiltonian, the eigenvalues and eigenfunctions and through adiabatic
connection the Kohn-Sham potential Viα =0 and certainly the solution of the KohnSham equations leads to the density ni :
ni → Hˆ → Ek , Ψk
(k = 1, ..., i, ...) → viKS → ni .
(11.37)
(k = 1, ..., i, ...) → vζ iKS → nζ i .
(11.38)
For the scaled density we also have
nζ i → Hˆ → Ek , Ψk
11 Density Scaling for Excited States
193
Thus, we can consider the total energy as a functional of the Kohn-Sham potential:
E[Ψi ] = E[Ψi [vζ iKS ]].
(11.39)
Making use of Eq. 11.16 we obtain
δE
=
δ Vζ0i
δ E δΨi
+ c.c. = 0.
δΨi δ Vζ0i
(11.40)
Thus, from the fact that the energy is stationary at the true wave function follows
that the energy is stationary at the true potential. (We mention in passing that there
is a condition of this sort in the Levy-Nagy theory [28]. It is also one of the key
results in the potential functional theory of Yang et al. [59]).
However, as the energy is only stationary and not minimum at the true density it
is difficult to find adequate approximations. The Kohn-Sham wave function should
be orthogonal to the exact Kohn-Sham wave function(s) of the lower state(s). Since
the exact Kohn-Sham wave functions are not known, one is satisfied if approximate
orthogonality with respect to the approximate lower Kohn-Sham wave function(s)
is assured.
In the ground-state theory exchange can be treated exactly via the optimized
potential method [60]. This method has been generalized for excited states [19, 25]
and extension for the scaled density is straightforward. To find the optimized
potential is very tedious even in the ground-state. However, Krieger, Li and
Iafrate [61] introduced a very accurate approximation. This method can be readily
generalized to excited states [19,25]. An extension to the scaled density is presented
here using an alternative derivation of the KLI approximation [62].
Both the OPM and KLI methods can be applied when the total energy is known
as a functional of the one-electron orbitals. Let us consider the exchange-only case.
The exchange energy is known. The expression is the same as the Hartree-Fock
one, but contains the Kohn-Sham orbitals. As we have just shown above for a
certain critical value of scaling factor ζ the correlation energy disappears. Using
the stationary principle the first variation of the total energy with respect to the oneelectron orbitals leads to Hartree-Fock-like equations:
1
− ∇2 ψζ i, j (r) + v(r) + vζ Ji (r) ψζ i, j (r)− dr wζ i (r, r )ψζ i, j (r ) = εζ i, j ψζ i, j (r),
2
(11.41)
where v is the external potential and vζ Ji is the classical Coulomb potential:
vζ Ji (r) =
dr ni (r )/|r − r |.
(11.42)