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5 The ζOPM and ζ KLI Methods for a Single Excited State

5 The ζOPM and ζ KLI Methods for a Single Excited State

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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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Washington, DC



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



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