5 The ζOPM and ζ KLI Methods for a Single Excited State
Tải bản đầy đủ - 0trang
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)
´ Nagy
A.
194
The total electron density ni can be expressed with the Hartree-Fock like spinorbitals
ψζ i, j :
ni (r) = ζ ∑ λζ i, j |ψζ i, j (r)|2 ,
(11.43)
j
while the exchange kernel wζ i (r, r ) takes the form
wζ i (r, r ) = ∑ λζ i, j ψζ∗i, j (r )ψζ i, j (r)/|r − r |.
(11.44)
j
After multiplying Eq. 11.41 by ζ λζ i, j ψζ∗i, j and summing for all orbitals we obtain
−
1
λζ i, j ψζ∗i, j ∇2 ψζ i, j + (v + vζ Ji + vζ Si )ni = ζ ∑ λζ i, j εζ i, j |ψζ i, j |2 ; (11.45)
2∑
j
j
where vζ Si is the Slater potential:
vζ Si (r) =
ζ
ψζ∗i, j (r)vζ xi, j (r)ψζ i, j (r)
ni (r) ∑
j
(11.46)
and vζ xi, j is the Hartree-Fock-like exchange potential
vζ xi, j (r)ψζ i, j (r) = −
dr w(r, r )ψζ i, j (r ),
(11.47)
Now the Kohn-Sham equations (11.24) are multiplied by ζ λζ i, j u∗ζ i, j and summed
for all orbitals
−
1
λζ i, j u∗ζ i, j ∇2 uζ i, j + vζ KSi ni = ζ ∑ λζ i, j εζ i, j |uζ i, j |2 .
2∑
j
j
(11.48)
Now we consider the case when both Eqs. 11.43 and 11.25 provide the same excitedstate density ni . Moreover, it is supposed that the Hartree-Fock-like orbitals ψζ i, j
can be approximated by the scaled Kohn-Sham orbitals uζ i, j . Then comparing
Eqs. 11.45 and 11.48 we obtain the generalized ζ KLI approximation for the KohnSham potential:
vζ KSi = v + vζ Ji + vζ xi ,
(11.49)
where
vζ xi = vζ Si +
ζ
uζ i, j |λζ i, j vζ xi − vζ xi, j |uζ i, j |uζ i, j |2 ,
ni ∑
j
(11.50)
11 Density Scaling for Excited States
195
Table 11.1 The values of ζc
and qc for which the ζ KLI
and experimental total
energies are equal for the
ground and some excited
states of the Li atom
Configuration
1s2 2s
1s2 2p
1s2 3s
1s2 3p
ζc
1.00961
1.00939
1.00871
1.00881
qc
0.02856
0.02791
0.02590
0.02621
Table 11.2 The values of ζc
and qc for which the ζ KLI
and experimental total
energies are equal for the
ground and some excited
states of the Na atom
Configuration
1s2 2s2 2p6 3s
1s2 2s2 2p6 3p
1s2 2s2 2p6 4s
1s2 2s2 2p6 4p
ζc
1.00407
1.00398
1.00393
1.00392
qc
0.04454
0.04364
0.04308
0.04295
For ζ = 1 Eq. 11.49 gives the original KLI exchange potential. As we used
Hartree-Fock like expression we obtained only the exchange. The results above are
valid for any value of ζ . We search that ζc for which the correlation energy disappears. For that value of ζ the ζ KLI method provides a very simple approximation
that includes correlation.
11.6 Illustrative Examples and Discussion
As an illustration the ground- and some excited state-energies of Li and Na atoms
have been calculated using the ζ KLI method. The value ζc for which the ζ KLI
and experimental energies [63] are equal has been determined. Tables 11.1 and 11.2
present these values ζc . ζ > 1 means that the scaled number of electrons Nζ is
smaller than the true electron number N. The difference qc = N − Nζ is also shown
in the Tables. The values of ζc and qc for the ground- and excited states are not the
same but their difference is small.
The ζ KLI method is as simple as the original KLI method. But it contains
correlation as well. The ζ KLI method is not exact, because Eζc ,i [ni ] = 0 is valid only
for a single density and the functional derivative, that is, the correlation potential is
not zero.
Acknowledgements This work is supported by the TAMOP 4.2.1/B-09/1/KONV-2010-0007
project. The project is co-financed by the European Union and the European Social Fund. Grant
OTKA No. K67923 is also gratefully acknowledged.
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Chapter 12
Finite Element Method in Density Functional
Theory Electronic Structure Calculations
ˇ ık, Robert Cimrman, Maty´asˇ Nov´ak,
Jiˇr´ı Vack´arˇ , Ondˇrej Cert´
ˇ
Ondˇrej Sipr, and Jiˇr´ı Pleˇsek
Abstract We summarize an ab-initio real-space approach to electronic structure
calculations based on the finite-element method. This approach brings a new
quality to solving Kohn Sham equations, calculating electronic states, total energy,
Hellmann–Feynman forces and material properties particularly for non-crystalline,
non-periodic structures. Precise, fully non-local, environment-reflecting real-space
ab-initio pseudopotentials increase the efficiency by treating the core-electrons
separately, without imposing any kind of frozen-core approximation. Contrary to
the variety of well established k-space methods that are based on Bloch’s theorem
ˇ
J. Vack´aˇr ( ) • O. Sipr
Institute of Physics, Academy of Sciences of the Czech Republic
e-mail: vackar@fzu.cz; sipr@fzu.cz
ˇ ık
O. Cert´
Institute of Physics, Academy of Sciences of the Czech Republic,
Na Slovance 2, 182 21 Praha 8, Czech Republic
Charles University in Prague, Faculty of Mathematics and Physics, Ke Karlovu 3,
121 16 Praha 2, Czech Republic
University of Nevada, Reno, USA
e-mail: ondrej@certik.cz
R. Cimrman
University of West Bohemia, New Technologies Research Centre, Univerzitn´ı 8,
306 14 Plzeˇn, Czech Republic
e-mail: cimrman3@students.zcu.cz
M. Nov´ak
Charles University in Prague, Faculty of Mathematics and Physics, Ke Karlovu 3,
121 16 Praha 2, Czech Republic
e-mail: info@czechmodels.cz
J. Pleˇsek
Institute of Thermomechanics, Academy of Sciences of the Czech Republic,
Dolejˇskova 1402/5, 182 00 Praha 8, Czech Republic
e-mail: plesek@it.cas.cz
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 12, © Springer Science+Business Media B.V. 2012
199
200
J. Vack´aˇr et al.
and applicable to periodic structures, we don’t assume periodicity in any respect.
The main asset of the present approach is the efficient combination of an excellent
convergence control of standard, universal basis of industrially proved finiteelement method and high precision of ab-initio pseudopotentials with applicability
not restricted to periodic environment.
12.1 Introduction
For understanding and predicting material properties such as density, elasticity,
magnetization or hardness from first principles quantum mechanical calculations,
a reliable and efficient tool for electronic structure calculations is necessary. The
reciprocal space methods, to which most attention has been dedicated so far, are very
powerful and sophisticated but by their nature are suitable mostly for crystals. For
systems without translational symmetry such as metallic clusters, defects, quantum
dots, adsorbates and nanocrystals, use of real-space methods is more promising.
We introduce new ab-initio real-space method based on (1) density functional
theory, (2) finite element method, and (3) environment-reflecting pseudopotentials.
It opens various ways for further development and applications: restricted periodic
boundary conditions in a desired sub-region or in a requisite direction (e.g. for nonperiodic objects with bonds to periodic surroundings), adaptive finite-element mesh
and basis playing the role of variational parameters (hp-adaptivity) and various
approaches to Hellman-Feynman forces and sensitivity analysis for structural
optimizations and molecular dynamics.
The present method focuses on solving Kohn Sham equations and calculating
electronic states, total energy and material properties of non-crystalline, nonperiodic structures. Contrary to the variety of well established k-space methods that
are based on Bloch’s theorem and applicable to periodic structures, we don’t assume
periodicity in any respect. Precise ab-initio environment-reflecting pseudopotentials
proven within the plane wave approach are connected with real space finite-element
basis in the present approach. The main expected asset of the present approach is
the combination of efficiency and high precision of ab-initio pseudopotentials with
universal applicability, universal basis and excellent convergence control of finiteelement method not restricted to periodic environment.
In the next sections, we give a general overview how the Density Functional
Theory is applied to electronic structure calculations within the framework of
the finite-element method. We show how to incorporate pseudopotentials into the
equations, explaining some technical difficulties that had to be solved and sorting
all the ideas out and presenting them in a fashion applicable to our problem.
12.2 Density Functional Theory and Pseudopotentials
Using the approach described e.g. in [3, 6–8, 10] and [1], i.e. making use of the
Hohenberg-Kohn theorem and applying the ab-initio pseudopotential approach, the
many-particle Schăodinger equation is decomposed into the KohnSham equations
12 Finite Element Method in Density Functional Theory Electronic Structure...
1
− ∇2 + VH (r) + Vxc(r) + Vˆ (r) ψi (r) = εi ψ (r)
2
201
(12.1)
which yield the orbitals ψi that reproduce the density n(r) of the original interacting
system. The core electrons, separated from valence electrons, are represented by a
non-local Hermitian operator Vˆ together with nuclear charge. The density is formed
by the sum over the occupied single-electron states in a system of N electron
N
n(r) = ∑ |ψi (r)|2
(12.2)
i
and VH is the electrostatic potential obtained as a solution to the Poisson equation
VH (r) =
n(r ) 3
d r.
|r − r |
1
2
(12.3)
All the non-electrostatic interactions are represented by the exchange-correlation
potential term Vxc (r) = δ Exc [n]/δ n(r), where Exc is the exchange and correlation
energy.
Kohn–Sham equations are solved within the iterative scheme described e.g. in
[4] and [9].
12.2.1 Semilocal and Separable Potentials
The pseudopotential having the form
Vˆ = Vloc (ρ ) + ∑ |lm Vl (ρ ) lm| ,
(12.4)
lm
usually denoted as semilocal l−dependent, is a general hermitian operator in the
spherically symmetric problem (i.e. Vˆ = R−1Vˆ R) and it is radially local. This form
is general, i.e. any such operator can be written in the form (12.4). Equivalently, in
the r representation:
V (r, r ) = r|Vˆ |r = Vloc (ρ )δ (r − r ) +
δ (ρ − ρ )
∑ Ylm (ˆr)Vl (ρ )Ylm∗ (ˆr )
ρ2
lm
The first term doesn’t cause a problem. Let’s denote the second term (which is
semilocal) simply by v:
v = ∑ |lm Vl (ρ ) lm|
lm
Let’s choose a complete but otherwise arbitrary set of functions |φi (they contain
both a radial and an angular dependence) and define a matrix U by the equation
J. Vack´aˇr et al.
202
∑ Ui j
φ j |v|φk = δik
j
then (|ψ = |φk αk ):
v |ψ = ∑ v |φi δik αk = ∑ v |φi Ui j φ j |v|φk αk = ∑ v |φi Ui j φ j |v|ψ
ik
ij
i jk
So any Hermitian operator (including v) can be transformed exactly into the
following form
v = ∑ v |φi Ui j φ j | v
(12.5)
ij
We diagonalize the matrix U by choosing such functions |φ¯i for which the matrix
φ¯ j |v|φ¯k (and hence the corresponding matrix U) is equal to identity. We can find
such functions for example using the Gram–Schmidt orthogonalization procedure
on |φi with a norm f |v|g (for functions f and g), more on that later. Then
v = ∑ v |φ¯i
i
1
φ¯i |v|φ¯i
φ¯i | v = ∑ v |φ¯i φ¯i | v
(12.6)
i
We could take any |φi and orthogonalize them. But because we have v in the form of
(12.4), we will be using |φi in the form |φi = |Rnl |lm , because it turns out we will
only need to orthogonalize the radial parts. The first term in (12.6) then corresponds
to the KB [5] potential. Taking more terms leads to more accurate results without
ghost states.
Let’s look at the orthogonalization. We start with the wavefunctions:
|φi = |Rnl |lm
where Rnl (ρ ) = ρ |Rnl and i goes over all possible triplets (nlm).
We can also relate the i and n, l, m using this formula
inlm =
n−1
l−1
k=1
k=0
∑ k2 + ∑ (2k + 1)
+ (l + m + 1) =
(n − 1)n(2n − 1)
+ l(l + 1) + m + 1
6
The operator v acts on these |φi like this
r|v|φi = r|v|Rnl |lm = rˆ | ρ |Vl (ρ )|Rnl |lm = Vl (ρ )Rnl (ρ )Ylm (ˆr)
Now we need to construct new orthogonal set of functions |φ¯i satisfying
φ¯i |v|φ¯ j = δi j
12 Finite Element Method in Density Functional Theory Electronic Structure...
203
This can be done using several methods, we chose the Gram–Schmidt orthogonalization procedure, which works according to the following scheme:
|φ˜1 = 1
1
φ1 |v|φ1
|φ1 ;
|φ˜2 = 1 − |φ¯1 φ¯1 | v
|φ¯1 =
1
|φ2 ;
φ2 |v|φ2
|φ˜3 = 1 − |φ¯1 φ¯1 | v − |φ¯2 φ¯2 | v
...
1
φ3 |v|φ3
|φ 3 ;
1
φ˜1 |v|φ˜1
|φ˜1
|φ¯2 =
1
|φ˜2
˜
φ2 |v|φ˜2
|φ¯3 =
1
|φ˜3
˜
φ3 |v|φ˜3
We can verify by a direct calculation that this procedure ensures
φ¯i |v|φ¯ j = δi j
It may be useful to compute the normalization factors explicitly:
φ˜1 |v|φ˜1 = 1
φ2 |v|φ¯1 φ¯1 |v|φ2
φ˜2 |v|φ˜2 = 1 −
φ2 |v|φ2
φ3 |v|φ¯1 φ¯1 |v|φ3 + φ3 |v|φ¯2 φ¯2 |v|φ3
φ˜3 |v|φ˜3 = 1 −
φ3 |v|φ3
...
we can also write down a first few orthogonal vectors explicitly:
|φ¯1 =
|φ¯2 =
|φ1
φ1 |v|φ1
|φ2 φ1 |v|φ1 − |φ1 φ1 |v|φ2
( φ1 |v|φ1 φ2 |v|φ2 − φ2 |v|φ1 φ1 |v|φ2 ) φ1 |v|φ1 φ2 |v|φ2
Now the crucial observation is
lm| Rnl |v|Rn l |l m = Rnl |Vl (ρ )|Rn l δll δmm
which means that φi |v|φ j = 0 if |φi and |φ j have different l or m. In other
words |φi and |φ j for different |lm are already orthogonal. Thus the G-S