PHYSICAL REVIEW B 81, 245105 共2010兲
Projector augmented wave formulation of Hartree-Fock calculations of electronic structure
Xiao Xu* and N. A. W. Holzwarth†
Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, USA
共Received 19 February 2010; revised manuscript received 5 May 2010; published 4 June 2010兲
The projector augmented wave 共PAW兲 formalism developed by Blöchl 关Phys. Rev. B 50, 17953 共1994兲兴 has
been demonstrated to be an accurate and efficient pseudopotential-like scheme for electronic-structure calculations within density-functional theory. We have extended this formalism to treat the integral-differential
equations of Hartree-Fock 共HF兲 theory, demonstrating that the PAW-HF method is able to calculate valence
energies with the same accuracy as the frozen-core orbital approximation. We show that for some elements,
electrons in core states provide significant contributions to the valence exchange energy and we develop
schemes for incorporating their effects into the PAW-HF formalism.
DOI: 10.1103/PhysRevB.81.245105
PACS number共s兲: 31.15.xr, 71.15.Ap, 71.15.Dx
I. INTRODUCTION
Recently, there has been renewed interest in using the
Hartree-Fock approximation as a component of electronicstructure calculations.1–6 Many of these calculations use
plane-wave representations for the valence wave functions
together with norm-conserving pseudopotentials to represent
the effects of the core electrons. In this paper, we develop a
projected augmented wave 共PAW兲 共Refs. 7–12兲 formulation
of Hartree-Fock theory. Previous work by the VASP group13
has shown that the PAW formalism is able to accurately
evaluate the Fock operator, including all of its multipole moments. In this paper, we explore how the PAW formalism,
developed for Kohn-Sham theory14 can be modified to work
with the integral-differential equations of Hartree-Fock
theory. In particular, we show how the relationship between
the basis and projector functions should be modified, and
also examine the role of the core electrons within the treatment. In this paper, the detailed analysis focuses on the treatment of spherical atoms; application of the Hartree-Fock
PAW formalism to nonspherical, spin polarized, and/or multicomponent systems can be obtained with a straightforward
extension of the basic equations.
The paper is organized as follows. In Sec. II, we review
the all-electron 共AE兲 formalisms, comparing Kohn-Sham and
Hartree-Fock equations. In Sec. III we examine the accuracy
of various frozen-core 共FC兲 approximations within both
Kohn-Sham and Hartree-Fock treatments. The PAW formalism is presented in Sec. IV and example functions are presented in Sec. V. The summary and conclusions are given in
Sec. VI. Some additional details of the formalism are given
in Appendices A and B.
II. ALL-ELECTRON FORMALISMS
In all of the treatments of the electronic structure of atoms, we will use one-electron orbitals which can be written
in the form
⌿ p共r兲 ⬅ ⌿nplpmp共r兲 ⬅
␺ p共r兲
Y lpmp共r̂兲,
r
共1兲
where symbols p 共and q , s , t , . . .兲 denotes a shell index, with
n p denoting the principle quantum number and l p denoting
1098-0121/2010/81共24兲/245105共14兲
the angular momentum. Y lpmp共r̂兲 is the spherical harmonic
function and ␺ p共r兲 denotes the radial portion of the wave
function. The number of electrons in a shell is given by N p
ⱕ 2共2l p + 1兲. For simplicity, we consider only the averaged
electronic configuration so that the electron density ␳共r兲
⬅ ␳共r兲 is spherically symmetric
␳共r兲 ⬅
n共r兲
4␲r2
where
n共r兲 = 兺 N p兩␺ p共r兲兩2 .
共2兲
p
The total electronic energy in all of the treatments can be
written as a sum of four terms
Etot = EK + EN + EH + Exc .
共3兲
Here the kinetic energy EK and the nuclear energy EN can be
evaluated in terms of the one-electron orbitals and densities
in the usual way. The Hartree energy can be expressed in
terms of the electron density using the expression
EH =
e2
2
冕冕
d3rd3r⬘
␳共r兲␳共r⬘兲
.
兩r − r⬘兩
共4兲
We note that this definition of the Hartree energy includes
the so-called electron self-interaction15 which should, in
principle, be corrected within the exchange-correlation contribution. The form of the exchange-correlation energy Exc or
the exchange-only energy Ex depends on the particular approximation scheme we are using and will be discussed in
more detail below.
A. Kohn-Sham theory
Within the local density approximation 共LDA兲 共Ref. 16兲
or generalized gradient approximation 共GGA兲 共Ref. 17兲, the
exchange-correlation functional is assumed to have an explicit dependence on the electron density ␳共r兲 which is usually written in the form
Exc =
冕
d3rf xc共␳共r兲,兩ⵜ␳共r兲兩兲.
共5兲
For the case of GGA, the functional depends explicitly
not only on the density but also on the gradient of the density. The one-electron orbitals are solutions of the KohnSham equations of the form
245105-1
©2010 The American Physical Society
PHYSICAL REVIEW B 81, 245105 共2010兲
XIAO XU AND N. A. W. HOLZWARTH
KS
HKS⌿KS
p 共r兲 = ␧ p⌿ p 共r兲,
共6兲
B. Hartree-Fock formalism
共7兲
Within Hartree-Fock theory, the total energy of a spherical
atom is given by Eq. 共3兲 with the replacement of “exchangecorrelation” energy with the Fock exchange energy
where the Kohn-Sham Hamiltonian takes the form
HKS = K + VKS共r兲.
Here K denotes the kinetic energy operator and VKS共r兲 denotes the Kohn-Sham potential which is determined from the
density derivatives of the last three terms in Eq. 共3兲 with
VKS共r兲 = VN共r兲 + VH共r兲 + Vxc共r兲.
Ze2
,
r
VH共r兲 ⬅ e2
冕
d 3r ⬘
␳共r⬘兲
兩r − r⬘兩
␦Exc
.
␦␳共r兲
共9兲
共10兲
The self-consistent solution of the Kohn-Sham equations is
obtained by minimizing the total energy 关Eq. 共3兲兴 with Exc
defined by Eq. 共5兲 subject to the constraint that Kohn-Sham
orbitals ⌿KS
p 共r兲 are eigenstates of the Kohn-Sham Hamiltonian 共7兲. Since the Kohn-Sham Hamiltonian is Hermitian,
the eigenstates ⌿KS
p 共r兲 are unique up to normalization.
⌰Lpq ⬅
冦
冉
l p L lq
1
N pN q
0 0 0
2
冕冕
d3rd3r⬘
⌿ⴱp共r兲⌿q共r兲⌿ⴱq共r⬘兲⌿ p共r⬘兲
,
兩r − r⬘兩
共11兲
where the notation implies that the summation is taken over
all occupied states of the same spin. For a spin and configuration averaged atom, the Fock exchange energy can be
evaluated by using a moment expansion of the Coulomb
kernel
and the exchange-correlation potential is defined in terms of
the functional derivative with respect to the electron density
␳共r兲
Vxc共r兲 ⬅
e2 ⬘
兺
2 pq
共8兲
For a spherical atom the potential components take the form
VN共r兲 ⬅ −
Ex ⬅ −
冊
Ex = − 兺
l p+lq
兺
pq L=兩l p−lq兩
RLpq;st ⬅ e2
冕冕
dr dr⬘
L
r⬍
ⴱ
ⴱ
L+1 ␺ p共r兲␺q共r兲␺s 共r⬘兲␺t共r⬘兲.
r⬎
共13兲
The Fock weight factor for the moment L for the spherically
averaged atom is given by19–21
for p ⫽ q
冉
1
4l p + 2 l p L lq
N p共N p − 1兲
2
4l p + 1 0 0 0
冊
2
冧
共14兲
for p = q and L ⫽ 0
for p = q and L = 0.
Np
␦FHF共兵␺HF
p 共r兲其,兵␭qp其兲
= 0.
␦␺HFⴱ
p
共兵␺HF
p 共r兲其,兵␭qp其兲
HF
HF
= Etot共兵␺HF
p 其兲 − 兺 N p␭qp共具⌿ p 兩⌿q 典 − ␦qp兲, 共15兲
qp
where ␭qp denotes a Lagrange multiplier. The minimization
is obtained by self-consistently solving the integraldifferential equations resulting from the functional
derivative.
共16兲
The Hartree-Fock equations take the form
HHF共r兲␺HF
p 共r兲 + X p共r兲 −
F
共12兲
This exchange expression uses radial integrals similar to
those introduced by Condon and Shortley18
2
In addition to the Fock exchange, this weight factor includes
the Hartree self-interaction correction.
In order to find the Hartree-Fock orbitals 兵␺HF
p 共r兲其, the
total energy in Eq. 共3兲 is optimized as a function of the
orbitals with orthonormalization constraints. The objective
function of this optimization is given by
HF
1 L L
⌰ R
.
2 pq pq;qp
兺
q;Nq⬎0
␭qp␺HF
q 共r兲 = 0.
共17兲
Here
HHF共r兲 ⬅ K + VN共r兲 + VH共r兲.
共18兲
The nuclear and Hartree potentials have the same form as
given in Eq. 共9兲. The exchange kernel function is given by
245105-2
PHYSICAL REVIEW B 81, 245105 共2010兲
PROJECTOR AUGMENTED WAVE FORMULATION OF…
cv
vv
Eval = EKv + ENv + EH
+ EH
+ Exc .
l +l
X p共r兲 ⬅
p q
1 L L
1 ␦Ex
=
−
⌰ pqWqp共r兲␺HF
兺
兺
q 共r兲,
ⴱ
N p ␦␺ p
q L=兩l p−lq兩 N p
共19兲
where
L
共r兲 ⬅ e2
Wqp
冕
dr⬘
L
r⬍
L+1
r⬎
HF
␺HFⴱ
q 共r⬘兲␺ p 共r⬘兲.
共21兲
where
共0兲
共0兲 HF共0兲
HF共0兲
R共0兲
共r兲 − ⑀共0兲
共r兲. 共22兲
p ⬅ − X p 共r兲 + 兺 ␭qp ␺q
p ␺p
q
共0兲
Here the parameters ⑀共0兲
p ⬇ ␭ pp are introduced to further sta22
At convergence 兵␺HF共0兲
共r兲其
bilize the solution.
p
HF共1兲
⬅ 兵␺ p 共r兲其. The results obtained in this way25 are essentially identical to those obtained from the ATSP package developed by Fischer et al.21 That is, the values of ␭qp from the
two programs agree within 10−4 Ry and graphs the radial
wave functions ␺HF
q 共r兲 are superposable.
III. FROZEN-CORE FORMALISMS
The notion of the frozen-core approximation is that the
inner-shell electrons of any atom remain approximately constant and insensitive to a variety of atomic bonding and compositional environments. In practice, after an all-electron calculation for an atom in a reference state, the shell indices
p , q , . . . are divided in to core states 共c兲 and valence states
共v兲. The total electron density is partitioned into core and
valence contributions
␳共r兲 = ␳c共r兲 + ␳v共r兲,
共23兲
where
␳v共r兲 ⬅
nv共r兲
4␲r2
with
nv共r兲 = 兺 Nv兩␺v共r兲兩2 .
and valence nuclear poHere, the valence kinetic energy
tential energy ENv can be evaluated in terms of the oneelectron valence orbitals and the valence density, respectively. The two contributions to the Hartree energy are
divided into a core-valence interaction
共20兲
Self-consistently solving the Hartree-Fock equations is
equivalent to minimizing the total energy Eq. 共3兲 with Exc
given by the Fock exchange energy defined by Eq. 共12兲. In
order to stabilize the numerical algorithm used to solve Eq.
共17兲, it is convenient to use linear combinations of HartreeFock wave functions which diagonalize blocks of the ␭qp
matrix which correspond to closed shells of the same angular
momentum lq = l p. For simplicity, rather than directly solving
the integral-differential equations, we use an iterative
technique.21–24 Starting with an initial guess for the radial
共r兲其 the approximate form of the Hartreecomponents 兵␺HF共0兲
p
Fock equations is a set of inhomogeneous differential equa共r兲其 of the
tions for the updated radial components 兵␺HF共1兲
p
form
HF共1兲
共r兲 = R共0兲
关HHF共0兲共r兲 − ⑀共0兲
p 兴␺ p
p ,
共25兲
EKv
cv
= e2
EH
d3rd3r⬘
␳c共r兲␳v共r⬘兲
⬅
兩r − r⬘兩
冕
c
共r兲nv共r兲
drVH
共26兲
and a valence-valence interaction
vv
=
EH
e2
2
冕冕
d3rd3r⬘
␳v共r兲␳v共r⬘兲 1
⬅
2
兩r − r⬘兩
冕
v
drVH
共r兲nv共r兲.
共27兲
The Kohn-Sham and Hartree-Fock treatments of the valence
contributions to the exchange-correlation energy, Exc, are different as explained below.
A. Frozen core in Kohn-Sham theory
The frozen-core approximation within Kohn-Sham theory
has been well described in the literature.26 The local density
and generalized gradient functionals are nonlinear functions
of the density and therefore, it is not possible to determine a
valence only contribution to Exc in Eq. 共25兲. In practice, Eq.
共5兲 is evaluated using the “frozen-core” density ␳c共r兲 plus the
self-consistent valence density ␳v共r兲. The self-consistent valence radial functions 兵␺v共r兲其 are determined from solving
the Kohn-Sham Eq. 共6兲 while the frozen-core density ␳c共r兲 is
not updated. The frozen-core approximation comes from the
fact that the core radial functions are generally not solutions
of those Kohn-Sham equations as the valence states change
from the reference configuration. Also, the valence orbitals
are generally no longer exactly orthogonal to the core states.
Our past experience27 shows this to be a very good approximation; some of the quantitative results are included in Fig.
2.
B. Frozen core in Hartree-Fock theory
There are at least two different methods of formulating a
frozen-core approximation within Hartree-Fock theory; we
use the terminology of “frozen-core orbital” and “frozencore potential” to distinguish the two schemes. The frozencore orbital approximation within Hartree-Fock theory has
been well described in the literature.24,28 Since the exchange
energy is formed from products of pairs of electron orbitals
and can be divided into core-valence and valence-valence
contributions
Exc → E共val兲
x
共24兲
v
冕冕
where
E共val兲
= Ecxv + Exvv .
x
共28兲
Here
Here we are using the c and v labels to refer to both the
category label and the label for the individual states in each
category. In general, instead of calculating the “total” electron energy, it is convenient to define the valence electron
energy in the form
Ecxv = − 兺
lv+lc
兺
vc L=兩lv−lc兩
and
245105-3
⌰LvcRLvc;cv
共29兲
PHYSICAL REVIEW B 81, 245105 共2010兲
vv
20
10
|∆∆E| (Ry)
|Ex |
cv
|Ex |
10
|Ex| (Ry)
0
Sc
Ti
6
|Ex |
4
|Ex |
V
Cr Mn Fe
Co
Ni
Cu
cv
2
0
10
K
Ca
//
Zn
Ga
Ge
As
Se
Br
FIG. 1. 共Color online兲 Plot of exchange energy magnitudes 兩Ecxv兩
and 兩Evv
x 兩 defined in Eqs. 共29兲 and 共30兲, respectively, for spherically
averaged ground-state atoms in the fourth row of the periodic table.
In these plots, the configuration of Ar defines the core states. The
energy values are given in Ry units.
Exvv
=−兺
lv+lv⬘
兺
vv⬘ L=兩lv−lv⬘兩
1 L L
.
⌰ R
2 vv⬘ vv⬘;v⬘v
共30兲
The magnitude of the valence exchange energy and the relative strength of core-valence and valence-valence contributions is strongly dependent on the atomic species in the calculation. For example, for elements in the second row of the
periodic table, Li-F calculated in their ground states using
the He configuration as core states, we find Ecxv / Exvv ⬇ 0.1.
For elements in the third row of the periodic table, Na-Cl
calculated in their ground states using the Ne configuration
as core states, we find Ecxv / Exvv ⬇ 0.2. For the 4s4p materials
in the fourth row of the periodic table, K-Ca and Zn-Br calculated in their ground states using the Ar configuration as
core states, we find Ecxv / Exvv ⬇ 0.3. On the other hand, for the
4s3d materials in the fourth row of the periodic table, Sc-Cu
calculated in the 4s23dx state using the Ar configuration as
core states, we find Ecxv / Exvv ⬇ 1. In addition to these trends in
the Ecxv / Exvv ratios, the magnitudes of the exchange energies
for the fourth row elements graphed in Fig. 1 show a monotonic increase with increasing numbers of electrons.
The self-consistent solution of the Hartree-Fock equations
in the frozen-core orbital approximation corresponds to solving Eq. 共17兲 only for the valence orbitals 共p ⬅ v兲 while leaving the core orbitals ␺HF
c 共r兲 fixed to the functions found for
the reference state. It is important to require that the valence
orbitals ⌿HF
v 共r兲 remain strictly orthogonal to each other as
well as to the core orbitals
HF
具⌿HF
v 兩⌿c 典 = 0
2
x
1
x+1
4s 3d => 4s 3d
-3
Sc
-2
2
10
V
Ti
x
1
4s 4p => 4s 4p
Cr Mn Fe Co Ni Cu
x+1
LDA
HF
HF(V)
-3
10
10
-4
Zn
Ga
Ge
As
Br
Se
FIG. 2. 共Color online兲 Plots of the magnitudes of ⌬⌬E 关Eq.
共39兲兴 in Ry units for elements in the fourth row of the periodic table
with the Ar core configuration. Three types of results are compared:
“LDA” denotes the Kohn-Sham formalism using the local density
approximation 共Ref. 16兲; “HF” denotes the Hartree-Fock formalism
using the frozen-core orbital approach using Eq. 共17兲; “HF共V兲” denotes the and Hartree-Fock formalism using the frozen-core potential approach using Eq. 共33兲 and 共36兲.
question is whether one can find a fixed potential Vcxv共r兲
which can accurately approximate the core-valence exchange
interaction in the effective single-particle Hamiltonian
c
v
HHF共V兲共r兲 ⬅ K + VN共r兲 + VH
共r兲 + VH
共r兲 + Vcxv共r兲. 共32兲
The corresponding integral-differential equations for the
wave functions in the Hartree-Fock frozen-core potential approximation take the form
␭qv␺HF
兺
q 共r兲,
q;N ⬎0
共33兲
1 L
HF
L
⌰ W 共r兲¯␺v⬘ 共r兲.
Nv vv⬘ v⬘v
共34兲
vv
HHF共V兲共r兲¯␺HF
v 共r兲 + Xv 共r兲 =
q
where
Xvv
v 共r兲
⬅−兺
lv+lv⬘
兺
v⬘ L=兩lv−lc⬘兩
The sum over q in Eq. 共33兲 includes both the frozen-core
wave functions and the modified valence functions ¯␺HF
v 共r兲.
The corresponding valence-core exchange energy is then
given by
共31兲
in order to avoid spurious contributions to the valence-core
exchange energy Ecxv of Eq. 共29兲.
For pseudopotential treatments of electronic structure, it is
often desirable to eliminate the core wave functions and to
represent their effects with contributions to the
pseudopotential.4–6 In principle, pseudopotential formulations are based on corresponding full wave function frozencore potential approximations. The frozen-core potential
c
共r兲 due to core electrons in the Hartree energy has already
VH
been defined in Eq. 共26兲 and requires no additional approximation beyond the frozen-core orbital approximation. The
-1
-2
10
vv
|∆∆E| (Ry)
|Ex| (Ry)
XIAO XU AND N. A. W. HOLZWARTH
Eccv ⬇ Ēcxv ⬅
冕
drVcxv共r兲nv共r兲.
共35兲
We have invested some effort in evaluating the frozen-core
potential approximation. One of the frozen-core potential
forms is chosen to be
Vcxv共r兲
where
245105-4
冉
=
兺v NvXcvv共r兲␺HF
v 共r兲
nv共r兲
,
ref
共36兲
PHYSICAL REVIEW B 81, 245105 共2010兲
PROJECTOR AUGMENTED WAVE FORMULATION OF…
Xcvv共r兲 ⬅ − 兺
lv+lc
兺
c L=兩lv−lc兩
1 L L
⌰ W 共r兲␺HF
c 共r兲.
N v vc cv
共37兲
This form is motivated by the fact that for any configuration,
including the reference configuration, the exact expression
for the core-valence exchange energy can be written
Ecxv =
冕
dr 兺 NvXcvv共r兲␺HF
v 共r兲,
共38兲
v
As a quantitative measure of the frozen-core error, we define
the difference of the excitation energy calculated in the
frozen-core approximation relative to the excitation energy
calculated in an all-electron treatment to be
teresting to compare a frozen-core potential approximation
within the OEP formalism32 to this frozen-core potential approximation of the Hartree-Fock formalism.
Despite the numerical error observed for frozen-core potential treatments of the Fock functional discussed above, the
corresponding norm-conserving pseudopotential treatments
within Hartree-Fock theory4–6 and exchange-only OEP
theory29 have been quite successful. In fact the importance of
core-electron effects within exchange-only OEP theory has
been a topic of debate in the recent literature,33 and several
authors34–36 have shown that it is possible to get quite reasonable results with valence-only pseudopotentials. The
PAW formulation of electronic-structure calculations, provides a well-defined method for examining the core-electron
effects more carefully in both Hartree-Fock and Kohn-Sham
formalisms.37
ground
excited
ground
excited
⌬⌬E ⬅ 共Etot
− Etot
兲 AE − 共Eval
− Eval
兲 FC .
共39兲
Using this measure, results for excitation energies of elements in the fourth row of the periodic table, where the
frozen-core configuration is that of the Ar atom, are illustrated in Fig. 2. For the 4s4p materials, the Hartree-Fock
frozen-core orbital approximation has an error of 2
⫻ 10−3 Ry or less which is similar to the frozen-core density
error obtained using LDA. For the 3d materials, the frozencore errors are 9 ⫻ 10−3 Ry or less for the Hartree-Fock
frozen-core orbital approximation which is similar 共within a
factor of 2兲 to the frozen-core density error obtained using
LDA. For the lighter elements, we have found HF frozencore orbital and LDA frozen-core density errors to be
10−4 Ry or less. Of course, all of these errors can be reduced
by treating the upper core states 共semicore states兲 as valence
states. By contrast, results for the frozen-core error obtained
by using the frozen-core potential of Eq. 共36兲 show the error
in the valence energies of excited states to be larger by a
factor of 5–10 than that of the frozen-core orbital approximation as shown in Fig. 2. We have also examined the second and third row elements of the periodic table, finding the
frozen-core orbital error to be smaller than that of the frozencore potential approximation by a factor 10 or more. While
for most materials that we have studied, the frozen-core orbital error is considerably smaller than that of the frozen-core
potential, it is clear that the errors of both schemes are controllable. It is also quite possible that the frozen-core potential error can be reduced by improving the form29 of the
frozen-core potential Vcxv共r兲 over that given in Eq. 共36兲.
However, the fact remains that the functional forms of Eqs.
共35兲 and 共38兲 are different and it is not surprising that they
should give different results as the valence configuration
changes.
The optimized effective potential 共OEP兲 method30,31 is designed to find a local potential VOEP
x 共r兲 for use in the KohnSham Hamiltonian consistent with the Fock exchange functional. However, in the OEP formulation, the VOEP
x 共r兲
potential is used only to determine the Kohn-Sham wave
functions which are in turn used to determine the exchange
energy using the functional form of Eq. 共12兲. It will be in-
IV. PAW FORMALISM
The PAW formalism was developed by Blöchl7 and
implemented by a number of authors.7,9,11,12,27,38 It is similar
to the ultrasoft pseudopotential approach of Vanderbilt.39 The
formalism needs a set of basis and projector functions for
each atom a which represent the valence states. Using the
atomic shell nomenclature defined in Eq. 共1兲, we will denote
these as ⌽ai 共r兲 for an all-electron basis function, ⌽̃ai 共r兲 for
the corresponding pseudoelectron basis function, and P̃ai 共r兲
for the corresponding projector function. For these functions,
the “shell” indices i that enumerate the basis functions include the valence atomic states and may also include continuum and other states defined in the radial range 0 ⱕ r
ⱕ rac 共the “augmentation” region兲 in order to increase the
“completeness” of the basis set. For each formalism, it is
required that the all-electron basis functions 兵⌽ai 共r兲其 are solutions of the differential or differential integral equations
corresponding to the reference configuration of the atom
within the augmentation region. For the case of Kohn-Sham
theory, they are eigenstates of the Kohn-Sham Hamiltonian
共6兲. For the case of Hartree-Fock theory they are solutions of
the Hartree-Fock Eq. 共17兲 共Ref. 40兲 with special consideration for continuum states.41 The projector functions satisfy
the relationship42
具P̃ai 兩⌽̃aj 典 = ␦ij
共40兲
and the radial pseudofunctions have the property
˜ a共r兲 = ␾a共r兲
␾
i
i
for
r ⬎ rac .
共41兲
Here rac denotes the matching radius for atom a. The radial
function associated with the projector function pai 共r兲 is constructed to be spatially localized within the augmentation
sphere rac .
A key idea of the PAW formalism is the transformation
between a calculated pseudowave function 关⌿̃v共r兲兴 corresponding to a valence state of the system and the corresponding fully nodal wave function 关⌿v共r兲兴 of that state which is
given by7
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PHYSICAL REVIEW B 81, 245105 共2010兲
XIAO XU AND N. A. W. HOLZWARTH
⌿v共r兲 = ⌿̃v共r兲 + 兺 关⌽ai 共r − Ra兲 − ⌽̃ai 共r − Ra兲兴具P̃ai 兩⌿̃v典.
共H̃KS共r兲 − ␧ai 兲⌽̃ai 共r兲 = 兺 P̃aj 共r兲具⌽̃aj 兩H̃KS − ␧ai 兩⌽̃ai 典. 共50兲
j
ai
共42兲
The sum ai is over atom center sites a and basis functions i.
With this transformation and a few additional terms, it is
possible to evaluate the valence electron energy of the system as a combination of smooth pseudopotential-like contributions plus a sum of atom-centered corrections in the form
Evale =
Ẽvale
+兺
共Eavale − Ẽavale兲
atom-centered corrections
共43兲
A. PAW formulation of Kohn-Sham theory
The detailed expression of the PAW valence energy in the
local density approximation has been reported extensively in
the literature and only the main points will be discussed here
and a few additional details are given in Appendix A. Since
the valence energy in Eq. 共43兲 is an explicit functional of the
electron density which in turn is an explicit functional of the
valence pseudowave functions, the Kohn-Sham equations in
the PAW formalism can be evaluated7 in terms of the functional derivative
␦Evale
␦⌿̃KSⴱ
v 共r兲
PAW
PAW KS
= HKS
共r兲⌿̃KS
⌿̃v 共r兲.
v 共r兲 = ␧vO
共44兲
This equation must be solved self-consistently with the orthonormalization constraint
PAW
具⌿̃KS
兩⌿̃v⬘ 典 = ␦vv⬘ .
v 兩O
KS
共45兲
Here the PAW Hamiltonian takes the form
PAW
共r兲 = H̃KS共r兲 + 兺 兩P̃ai 典Daij具P̃aj 兩
HKS
共46兲
OPAW = 1 + 兺 兩P̃ai 典Oaij具P̃aj 兩.
共47兲
aij
and
aij
Here the Kohn-Sham pseudo-Hamiltonian has the form
H̃KS共r兲 ⬅ K + ṼKS共r兲.
Daij ref = 具⌽ai 兩HKS兩⌽aj 典 ref − 具⌽̃ai 兩H̃KS兩⌽̃aj 典 ref .
.
a
pseudoenergy
This relationship is consistent with the requirements that
each pseudowave function basis function ⌽̃ai 共r兲 must be a
solution of the Kohn-Sham PAW Eqs. 共44兲 of the reference
state and with the special form of the one-center Hamiltonian
matrix elements of the reference state
In practice, the projector functions 兵P̃ai 共r兲其 are determined by
solving Eq. 共50兲 from a knowledge of the reference pseudoHamiltonian H̃KS共r兲, the basis functions 兵⌽ai 共r兲 , ⌽̃ai 共r兲其, and
the eigenenergies 兵␧ai 其, following a similar procedure developed by Vanderbilt.39
B. PAW formulation of Hartree-Fock theory
For Hartree-Fock theory, the PAW valence-energy expression can be put in the form of Eq. 共43兲 as in the case of
Kohn-Sham theory. The main differences come in the form
of the treatment of the Fock exchange term and in the representation of the frozen-core orbitals. From our analysis of
the full wave function frozen-core approximation discussed
in Sec. III B, we decided to use the frozen-core orbital approach for our PAW implementation.
For the purpose of this formulation, the valence pseudoorbitals ⌿̃HF
v 共r兲 are treated separately from the core orbitals.
As in the case of the Kohn-Sham formulation, it is assumed
that they are spanned by the basis pseudofunctions 兵⌽̃ai 共r兲其
within the augmentation sphere.
Most of the core wave functions ⌿HF
c 共r兲 are contained
within the augmentation sphere and for those we can define a
trivial core pseudowave function ⌿̃HF
c 共r兲 ⬅ 0. For some materials it is possible that at most one core orbital per l channel
will have a nontrivial amplitude for r ⬎ rac . In such a case, it
is convenient to define a continuous pseudocore orbital ˜␺ac 共r兲
with ˜␺ac 共r兲 ⬅ ␺ac 共r兲 for r ⬎ rac , in a similar spirit to the
pseudocore functions defined for the so-called nonlinear core
corrections in norm-conserving pseudopotentials.43 In our
case, we define
共48兲
˜␺a共r兲 ⬅
c
The overlap matrix element is given by
Oaij = 具⌽ai 兩⌽aj 典 − 具⌽̃ai 兩⌽̃aj 典.
共49兲
The general form of ṼKS共r兲 as well as of the one center
matrix elements Daij has been given in many
references.7,11,27,38 For convenience the contributions from
terms other than exchange and correlation are given in Appendix A.
There are several alternative schemes7,39 to construct the
basis and projector functions for the PAW-Kohn-Sham formalism. In terms of the pseudo Hamiltonian of the reference
state, the projector functions are related to the pseudowave
function basis according to
共51兲
再
rlc+1 Pn共r兲 for r ⱕ rac
␺ac 共r兲
冎
for r ⬎ rac .
共52兲
The nth order polynomial Pn共r兲 is chosen to ensure that ˜␺ac 共r兲
is continuous up to n − 1 derivatives. An example of this
construction is shown for the 3d core state of Ge in Fig. 3.
In order to define localized contributions from Coulombic
interactions, it is necessary to define “compensation” charge7
moments of the form
aL
m̂aL
ij 共r兲 ⬅ mij
冕
r2sL共r兲
du u
L+2
sL共u兲
Here sL共r兲 is a smooth shape function
245105-6
a
⬅ maL
ij gL共r兲.
共53兲
PHYSICAL REVIEW B 81, 245105 共2010兲
PROJECTOR AUGMENTED WAVE FORMULATION OF…
a
aL
M̂ Lvc共r兲 ⬅ 兺 具⌿̃HF
v 兩P̃i 典m̂ic 共r兲.
1.2
ψc(r) (bohr
-1/2
)
1
ψc
Ge 3d
The significance of this moment is that, within the accuracy
of the PAW transformation, the Lth moment of
HF
关␺HFⴱ
v 共r兲␺v⬘ 共r兲兴 is the same as the Lth moment of
关˜␺HFⴱ共r兲˜␺HF共r兲 + M̂ L 共r兲兴. For the moment representing core-
0.8
0.6
~
ψ
c
0.2
0
0
rc
1
2
3
r (bohr)
4
FIG. 3. 共Color online兲 Radial Hartree-Fock wave function ␺c共r兲
for the 3d core state of Ge compared with the constructed pseudowave function ˜␺c共r兲.
sL共r兲 = rLk共r兲
valence wave-function products, the moment coefficient maL
ic
corresponding to valence-core contributions is defined by an
˜ a → ˜␺a.
expression similar to Eq. 共56兲 with ␾aj → ␺ac and ␾
j
c
Because of orthogonality properties of core and valence
states of the reference system and because ˜␺ac ⬅ 0 for many
cases, there are several simplifications of the terms involving
the core functions as discussed in more detail in Appendix B.
By using these compensation charge moments, the pseudoexchange integrals take the form
共54兲
Ẽxvv
or it could be similar smooth function such as one involving
a spherical Bessel function sL共r兲 ⬅ jL共␬r兲. More often we
have chosen
k共r兲 =
冋冦
sin共␲r/rac 兲
共␲r/rac 兲
册
2
for r ⬍
rac
冧
共55兲
where
R̃vv⬘;v⬘v ⬅ e2
The charge moment coefficient is given by
˜ a共r兲␾
˜ a共r兲兴.
dr rL关␾ai 共r兲␾aj 共r兲 − ␾
i
j
=−兺
vv⬘ L=兩lv−lv⬘兩
Exvv = Ẽxvv + 兺 共Eax vv − Ẽax vv兲,
冕冕
=
Ẽcxv
+兺
−
v
Ẽac
x 兲.
Eax vv
−
Ẽax vv
兺
vv⬘ L=兩lv−l⬘v兩
HF
aL
aL
⫻具⌿̃v⬘ 兩P̃ak 典具P̃al 兩⌿̃HF
v 典共Rij;kl − R̃ij;kl兲
−
v
Ẽac
x
共57兲
Here
2
RaL
ij;kl ⬅ e
=−兺
lv+lc
兺
vc L=兩lv−lc兩
HF
a
a
⌰Lvc 兺 具⌿̃HF
v 兩P̃i 典具P̃ j 兩⌿̃v 典
ij
冕冕
dr dr⬘
L
r⬍
L+1
r⬎
共62兲
aⴱ
a
a
␾aⴱ
i 共r兲␾ j 共r兲␾k 共r⬘兲␾l 共r⬘兲
共63兲
and
2
R̃aL
ij;kl ⬅ e
冕冕
dr dr⬘
L
r⬍
L+1
r⬎
˜ aⴱ共r兲␾
˜ a共r兲 + m̂aL共r兲兴
关␾
j
ij
i
˜ aⴱ共r⬘兲␾
˜ a共r⬘兲 + m̂aL共r⬘兲兴.
⫻ 关␾
l
kl
k
HF
aij
共61兲
and
The
is to use the compensation charge components to
ensure that the long-range behavior of individual contributions to the Fock integrals are correct while using the grouping of the smooth pseudoterms and the one-center terms to
cancel out the unphysical pseudocontributions near the atom
centers. Below, explicit expressions are given for a spherical
atom in which case there is only one atomic center a. The
ideas are easily generalized to multicenter systems.
HF
For a given product of two valence states, ␺HFⴱ
v 共r兲␺v⬘ 共r兲,
we can define a compensation charge moment of order L
a
a
aL
M̂ vv⬘共r兲 ⬅ 兺 具⌿̃HF
v 兩P̃i 典具P̃ j 兩⌿̃v⬘ 典m̂ij 共r兲,
共60兲
1 L
HF
⌰ 兺 具⌿̃HF兩P̃a典具P̃aj 兩⌿̃v⬘ 典
2 vv⬘ ijkl v i
aL
⫻共RaL
ic;cj − R̃ic;cj 兲.
trick13
共59兲
L
r⬍
L
˜ HFⴱ ˜ HF
L+1 关␺v 共r兲␺v⬘ 共r兲 + M̂ vv⬘共r兲兴
r⬎
lv+l⬘v
=−兺
a
L
⌰LvcR̃Lvc;cv ,
with an identical expression for v⬘ → c. The corresponding
one-center contributions take the form
v
Eac
x
v
共Eac
x
兺
dr dr⬘
a
Ecxv
lv+lc
HFⴱ
L
⫻ 关˜␺v⬘ 共r⬘兲˜␺HF
v 共r⬘兲 + M̂ v⬘v共r⬘兲兴
共56兲
The appropriate values of L are given by 兩li − l j兩 ⱕ L ⱕ li + l j.
These compensation charge terms were used by Blöchl7 to
evaluate the Hartree energy contributions and are embedded
in the PAW-Kohn-Sham formalism discussed above. They
are also important for evaluating the Fock exchange terms13
so that they can be written in the PAW form 共43兲
1 L L
,
⌰ R̃
2 vv⬘ vv⬘;v⬘v
兺
vc L=兩lv−lc兩
L
冕
lv+lv⬘
Ẽcxv = − 兺
for r ⱖ rac .
0
vv⬘
v⬘
v
0.4
maL
ij ⬅
共58兲
ai
共64兲
These expressions for the valence electron contributions to
245105-7
PHYSICAL REVIEW B 81, 245105 共2010兲
XIAO XU AND N. A. W. HOLZWARTH
the exchange energy are equivalent to results given by previous workers.13,44
From the PAW representations of the exchange energies
and of the other energy terms, the PAW Hartree-Fock equations can be derived by taking the functional derivative the
energy with respect to the valence pseudo-orbitals. The selfconsistent Hartree-Fock equations for the PAW formulation
take the form
Xaiv = − 兺
lv+lv⬘
兺
v⬘ L=兩lv−lv⬘兩
1 L
HF
HF
⌰ 关 兺 具P̃a兩⌿̃ 典具⌿̃v⬘ 兩P̃ak 典具P̃al 兩⌿̃HF
v 典
Nv vv⬘ jkl j v⬘
a
aL
⫻共RaL
ij;kl − R̃ij;kl兲兴 + 兺 具P̃ j 兩⌿̃v⬘ 典Zv⬘v;ij
HF
aL
j
−兺
lv+lc
兺
c L=兩lv−lc兩
PAW
PAW
PAW HF
共r兲⌿̃HF
共r兲 − 兺 ␭qvOHF
⌿̃q 共r兲 = 0.
HHF
v 共r兲 + Xv
1 L
aL
aL
aL
⌰ 关 兺 具P̃aj 兩⌿̃HF
v 典共Ric;cj − R̃ic;cj 兲 + Zcv;ic兴,
N v vc j
共73兲
q
共65兲
where
These equations must be solved self-consistently with
orthonormalization constraint
PAW
兩⌿̃HF
具⌿̃HF
q 典 = ␦ vq .
v 兩O
共66兲
This differs slightly from the orthonormalization constraint
in the Kohn-Sham case 关Eq. 共45兲兴 since the index q can refer
to either a valence state v or a 共frozen兲 core state c for which
the relevant matrix elements take the form
ZaL
qv;ij ⬅
ai
H̃HF共r兲⌽̃ai 共r兲 + X̃i共r兲 −
Here the matrix coefficients are given by
⌳aji ⬅ 具⌽̃aj 兩H̃HF兩⌽̃ai 典 + 具⌽̃aj 兩X̃i典 −
In general, these terms are very small if not
where
identically zero. The single particle term of Eq. 共65兲 takes the
form
共68兲
aij
where the pseudo Hamiltonian-type terms depend on the
pseudopotentials due to the nuclear and Hartree interactions
共69兲
The exchange function term takes the form
共70兲
ai
X̃v共r兲 ⬅ − 兺
兺
q L=兩lv−lq兩
1 L L ˜ HF
⌰ W̃ 共r兲␺q 共r兲.
N v vq qv
共76兲
where the differential H̃HF共r兲 and integral X̃i共r兲 operators as
well as the Lagrange multipliers ␭qi are all evaluated for the
reference configuration. Analogous to Eq. 共50兲 for the PAWKohn-Sham formalism, for the PAW-Hartree-Fock formalism
Eq. 共75兲 is used to determine the projector functions 兵P̃aj 共r兲其.
Equation 共75兲 is consistent with the requirement that the
pseudowave function basis functions ⌽̃ai 共r兲 must be a solution of the PAW Hartree-Fock Eq. 共65兲 for the reference
configuration of the atom and the following identities. For
the reference state, the single-particle terms satisfy the relationship
a
a
HF a
HF a
DaHF
ij ref = 具⌽i 兩H 兩⌽ j 典 ref − 具⌽̃i 兩H̃ 兩⌽̃ j 典 ref
共71兲
冕
dr⬘
L
r⬍
L
˜ HFⴱ
˜ HF
L+1 关␺q 共r⬘兲␺v 共r⬘兲 + M qv共r⬘兲兴.
r⬎
共72兲
The one center matrix element for the pseudoexchange kernel function takes the form
共77兲
and the two-particle terms satisfy the relationship45
Xaiv ref = 具⌽ai 兩Xv典 ref − 具⌽̃ai 兩X̃v典 ref.
In this expression, the summation over q includes both valence orbitals which are updated self-consistently and core
orbitals which are frozen. The interaction function W̃Lqv共r兲 is
the pseudoanalog of Eq. 共20兲
W̃Lqv共r兲 ⬅ e2
␭qi具⌽̃aj 兩⌿̃HF
兺
q 典,
q;N ⬎0
q
where the pseudoexchange kernel function takes the form
lv+lq
a
a
␭qi⌿̃HF
兺
q 共r兲 = 兺 P̃ j 共r兲⌳ ji .
q;N ⬎0
j
q
共67兲
共r兲 = X̃v共r兲 + 兺 兩P̃ai 典Xaiv ,
XPAW
v
共74兲
The relationship between the projector and basis functions is
somewhat different from that of the Kohn-Sham formalism,
taking the form
Oaic ⬅ ma0
ic .
H̃HF共r兲 ⬅ K + ṼN共r兲 + ṼH共r兲.
drW̃Lqv共r兲m̂aL
ij 共r兲.
共75兲
HF
HF a
HF
HF
a
PAW
具⌿̃HF
v 兩OHF 兩⌿̃c 典 ⬅ 具⌿̃v 兩⌿̃c 典 + 兺 具⌿̃v 兩P̃i 典Oic ,
PAW
a
共r兲 = H̃HF + 兺 兩P̃ai 典DaHF
HHF
ij 具P̃ j 兩,
冕
共78兲
Once the basis and projector functions are determined, the
solution of self-consistent PAW-Hartree-Fock Eqs. 共65兲 can
be obtained using an iterative method similar to that of the
all-electron or frozen core Eqs. 共17兲. For any guess of the
共r兲其, the Lagrange mulvalence pseudowave functions 兵˜␺HF共0兲
v
tipliers ␭共0兲
can
be
estimated
by
multiplying
both sides of
v⬘v
共r兲 and integrating over all space. The
Eq. 共65兲 by ⌿̃HF共0兲
v⬘
Lagrange multipliers ␭共0兲
cv corresponding to the interaction between core states and valence states can be estimated from
the form
245105-8
PHYSICAL REVIEW B 81, 245105 共2010兲
PROJECTOR AUGMENTED WAVE FORMULATION OF…
other numerical considerations of the Hartree-Fock equations
must be taken into account as well.兲
Ge
ψv(r) (bohr
-1/2
)
0.5
V. EXAMPLE PAW FUNCTIONS
0
4s HF
4s LDA
4p HF
4p LDA
-0.5
0
2
4
6
r (bohr)
8
10
FIG. 4. 共Color online兲 Radial valence wave functions for Ge,
comparing Kohn-Sham 共LDA兲 and HF.
HF共0兲
HF共0兲
HF
a
HF
␭共0兲
典 + 兺 DaHF
典
cv = 具⌿̃c 兩H̃ 兩⌿̃v
cj 具P̃ j 兩⌿̃v
j
共0兲
a共0兲
+ 具⌿̃HF
c 兩X̃v 典 + Xcv .
共79兲
Having derived the form of the PAW-HF equations, we are
now in a position to examine the role of the core and
pseudocore orbitals. It is clear that from the way that the
pseudocore orbitals have been defined, ˜␺ac 共r兲 ⬇ 0, so that the
main contributions of the core orbitals are expressed in the
atom centered radial Coulomb integrals RaL
ic;cj similar to the
expression defined in Eq. 共63兲. When ˜␺ac 共r兲 ⬅ 0, many of the
matrix elements and Hamiltonian terms simplify as discussed
in Appendix B. In practice, the main contributions can be
expressed in terms of the RaL
ic;cj integrals which contribute to
the core-valence exchange energy Ecxv as defined in Eqs. 共57兲
and 共62兲. They also contribute to the pseudoexchange kernel
function Xaiv defined in Eq. 共73兲. Since all of these corevalence contributions depend on constant matrix elements or
functions that can be precalculated and stored, they should
not substantially increase the computational requirements of
PAW-Hartree-Fock over that of PAW-Kohn-Sham. 共Of
course the plane-wave treatment of the Fock operator and
There are many adjustable parameters in the construction
of the PAW basis and projector functions which may be used
to find a set of functions that “span” the space of pseudowave functions in one-to-one correspondence with the fully
nodal frozen-core wave functions for the range of electronic
configurations of interest. In fact, the shapes of the HartreeFock valence wave functions are, in general, similar to the
shapes of the corresponding Kohn-Sham valence wave functions. This is illustrated in Fig. 4 for the example of Ge.
Therefore, it should be possible to construct PAW-HF basis
and projector sets with the help of the large literature describing the process for Kohn-Sham calculations.7,11,39,46–49
In constructing the pseudobasis functions, for each atom
we vary only the augmentation radius rac and construct the
˜ a共r兲 from all-electron basis functions
pseudobasis functions ␾
i
a
␾i 共r兲 using the polynomial form introduced by Vanderbilt in
the construction of the ultrasoft pseudopotential scheme.39
关This also is the scheme for constructing the nontrivial
pseudocore functions ˜␺c共r兲 of Eq. 共52兲.兴 The other adjustable
a
共r兲
function in this construction is the localized potential Ṽloc
defined in Eq. 共A4兲. It is often efficient to choose this
potential11 as an unscreened norm-conserving pseudopotential derived from the Troullier-Martins50 construction for a
high angular-momentum scattering state. Since the integral
form of the Hartree-Fock equations slightly complicates the
Troullier-Martins construction, we chose another method of
a
共r兲. A simple, but reasonable choice is
constructing Ṽloc
a
共r兲 = V0k共r兲
Ṽloc
using the shape function defined in Eq. 共55兲 and an adjustable amplitude V0. Table I lists the Hartree-Fock valence
energies of several atoms comparing the PAW energies with
the corresponding frozen-core results and also comparing the
TABLE I. Hartree-Fock valence energies for several configuration averaged atoms calculated with the
frozen core orbital approximation 共FC兲 and the PAW formalism. Results including 共兩˜␺c兩 ⬎ 0兲 and excluding
共兩˜␺c兩 ⬅ 0兲 pseudocore orbitals are compared. The augmentation radii 共rac 兲 are given in bohr units, the local
potential amplitudes 共V0兲 are given in Ry units, and the valence energies for the ground and excited states are
given in Ry units. The core configurations are He for C, Ne for Si, and Ar for Ge.
Atom
Type
C
C
C
Si
Si
Si
Ge
Ge
Ge
FC
PAW
PAW
FC
PAW
PAW
FC
PAW
PAW
共80兲
兩˜␺c兩
rac
V0
⬎0
⬅0
1.3
1.3
2.0
2.0
⬎0
⬅0
2.0
2.0
3.0
3.0
⬎0
⬅0
2.2
2.2
3.0
3.0
245105-9
Eval共ns2np2兲
Eval共ns1np3兲
−10.5990
−10.5990
−10.5990
−7.3147
−7.3147
−7.3147
−7.2257
−7.2258
−7.2258
−9.9542
−9.9541
−9.9541
−6.8070
−6.8066
−6.8070
−6.6800
−6.6796
−6.6800
PHYSICAL REVIEW B 81, 245105 共2010兲
XIAO XU AND N. A. W. HOLZWARTH
HF
HF
LDA
-1/2
4
)
~
ψc(r) = 0
(bohr
-1/2
~a
pi (r) (bohr )
3
Ge 4p
6
~
|ψc(r)| > 0
2
rc
1
2
3
r (bohr)
4
1 ~I
pi
0
3
5
effects of including or excluding the pseudocore orbitals. In
these cases, we have been able to achieve excellent agreement between the valence energies calculated with PAW and
the frozen-core orbital schemes. The convergence of the
PAW-HF equations are sensitive to the choices of rac and V0
but the energies obtained including or excluding the smooth
core functions are very close.
Our results provide information on the role of the
pseudocore orbitals of the upper core states. Of the materials
presented in Table I, the core effects are most prominent for
Ge. In Fig. 5 two examples of projector functions for the 4p
states of Ge are presented, comparing the effects of including
or excluding the pseudocore functions and also comparing
with the analogous LDA projector function. We see that the
projector obtained by including the pseudocore functions is
quite similar in shape to that the LDA formulation. On the
other hand, by setting the pseudocore functions to zero, the
projector function extends considerably beyond rac . The reason for this behavior can be explained from the defining Eq.
共75兲. By construction, when the smooth core wave functions
are included in Eq. 共75兲, for r ⱖ rac the left-hand side becomes
identical to the left-hand side of the all-electron Hartree-Fock
Eq. 共17兲 and therefore vanishes. When the smooth core functions are omitted, the two equations become equal only when
core wave function amplitudes have become negligible so
that the left-hand side of Eq. 共75兲 and therefore p̃ai 共r兲, remains nonzero beyond r = rac . The example shown in Fig. 5
illustrates one of the more extreme cases of this effect, which
is related to the extended shape of the Ge 3d core state
shown in Fig. 3. However, the results in Table I show that the
corresponding results on the valence energy of the atom are
negligible. How these extended projectors behave when used
to describe multiatomic systems will be an interesting question to study in future work.
A more strenuous test of the PAW-HF formalism is the
ionization of Fe based on an Ar core configuration and valence configurations 4s23d6 → 4s13d6. The augmentation radius was chosen to be rac = 2.0 bohr and the valence orbital
matching radii were chosen to be 2.0 and 1.6 bohr for the 4s
and 3d states, respectively.51 The reference state was constructed from the configuration averaged ground state of
(bohr
-1/2
)
FIG. 5. 共Color online兲 Radial PAW projector functions for 4p
states of Ge, comparing effects of including and excluding
pseudocore orbital functions ˜␺c共r兲 within the HF formulation and
also comparing the corresponding LDA projector. These projectors
where used to obtain the results presented in Table I.
Fe 4s
x 1/3
2
0
-2
0
x 1/3
2
~ II
pi
~
φi
φi
Fe 3d
I
p~i ~ II
pi
1 φ
i
0
0
φ~i
0.5
1
1.5
r (bohr)
2
2.5
FIG. 6. 共Color online兲 Plots of the radial basis and projector
functions for the 4s and 3d states of Fe. The parameters of the
construction are described in the text. Label “I” refers to 兩˜␺c兩 ⬎ 0
and V0 = −2 Ry and label “II” refers to 兩˜␺c兩 ⬅ 0 and V0 = −6 Ry,
affecting only the shapes of the projector functions.
4s23d6. The shapes of the corresponding PAW basis and projector functions are shown in Fig. 6. Compared to the example of Ge discussed above, the effects of the pseudocore
orbitals on the shapes of the projector functions are small, as
shown in these plots. For Fe and the other transition metal
atoms, we find that the local potential amplitude V0 plays an
important role in stabilizing the PAW-HF integral-differential
equations. Table II lists the computed ionization energies,
comparing the all-electron, frozen-core orbital, and PAW results for various choices of the local potential amplitude V0
and the inclusion or exclusion of the smooth core orbital
functions. For some of these choices, the calculation diverges
or converges to a result with significant deviation from the
frozen-core ionization energy. For the two best choices of
parameters shown in Table II—PAW 共I兲 corresponding to
TABLE II. Hartree-Fock ionization energies 共in Ry units兲 for
configuration averaged Fe关E共4s13d6兲 − E共4s23d6兲兴 calculated using
AE, FC orbital, and PAW methods. The FC and PAW results are
based on the Ar core configuration as described in the text. The
amplitude of local potential V0 is specified in Ry. Results including
共兩˜␺c兩 ⬎ 0兲 and excluding 共兩˜␺c兩 ⬅ 0兲 the pseudocore wave functions
are compared. “x” indicates that the calculation diverged.
V0
−2
−4
−6
245105-10
AE
FC
0.4991
0.4992
PAW 共兩˜␺c兩 ⬎ 0兲
PAW 共兩˜␺c兩 ⬅ 0兲
0.5002
0.4904
x
x
0.6288
0.4992
PHYSICAL REVIEW B 81, 245105 共2010兲
PROJECTOR AUGMENTED WAVE FORMULATION OF…
1
ψv(r) (bohr
-1/2
)
0.6
0.4
0.2
0
-0.2
-1/2
)
0.6
ψv(r) (bohr
6
Fe 4s 3d
2
Fe 4s 3d
6
0.4
ψv FC
~ PAW(I)
ψ
v
ψv PAW(I)
~ PAW(II)
ψ
v
ψv PAW(II)
0.2
0
-0.2
0
2
4
r (bohr)
6
FIG. 7. 共Color online兲 Plots of the Hartree-Fock radial 4s orbitals for Fe, comparing results for the ground-state configuration
共lower panel兲 and ionized configuration 共upper panel兲 and the FC
orbital results. The PAW results are based on the calculation of the
pseudofunctions ˜␺v共r兲 and using the PAW transformation Eq. 共42兲.
The results for the fully nodal 4s wave functions are virtually indistinguishable on this scale.
兩˜␺c兩 ⬎ 0 and V0 = −2 Ry and PAW 共II兲 corresponding to 兩˜␺c兩
⬅ 0 and V0 = −6 Ry, the calculated 4s orbital for configuration averaged Fe in its ground and ionized states are very
close to the frozen-core orbital results as shown in Fig. 7.
In this Fe example, the 3d radial wave functions for Fe in
its ionized configuration changes very little compared with
that of the reference configuration. In order to extend this
study of transition-metal atoms to include configurations
which alter the occupancy of the 3d shell, we find that it will
be necessary to start with the Ne core configuration and use
twice as many basis and projector functions.
VI. SUMMARY AND CONCLUSIONS
In this manuscript, we have derived the necessary equations and demonstrated examples of the PAW-HF formalism
for atoms across the periodic table. For these examples, we
have shown that with the proper choice of augmentation radii, local pseudopotentials, etc., atomic PAW calculations
achieve the same numerical accuracy as the frozen-core orbital approximation within Hartree-Fock theory. More generally, we have demonstrated that exchange contributions from
interactions of core and valence electron states are significant
for some elements and that they can be treated accurately
within the integral-differential equations of the PAW-HF formalism 关Eq. 共65兲兴.
In addition, we have numerically compared the frozencore orbital and the frozen-core potential approximations for
atoms across the periodic table. For our choice of potential
form, the frozen-core orbital approximation gave results
closer to the all-electron Hartree-Fock treatment. Further
work may give additional insight on this issue.
At the end of Sec. III we posed the question of how core
electrons are folded into norm-conserving pseudopotential
treatments of Hartree-Fock and related formalisms. Within
the PAW-HF formulation based on the frozen-core orbital
approximation, we see that most of the core-electron effects
are treated with stored matrix elements. However, we have
seen for some atoms such as Ge, the upper core states have
significant extension beyond the augmentation sphere. For
these cases, we introduced the smooth pseudocore functions
˜␺a共r兲. We have explored how these extended core orbital
c
functions affect the extent of the PAW-HF projector functions as defined by Eq. 共75兲. In particular, we have shown
several examples which show that by including 兩˜␺ac 共r兲兩 ⬎ 0,
we can ensure that the projector functions are strictly contained within the augmentation sphere as suggested by the
original formulation of the PAW theory.7 However if we set
兩˜␺ac 共r兲兩 ⬅ 0, the effects of the extended core functions are incorporated into extended parts of the projector functions
p̃ai 共r兲. For cases we studied, the excitation energies of the
two approaches were very similar. It remains to be determined whether these extended projector functions can be
successfully used to calculated the electronic structure of
multicenter systems.
In addition to further study of the question of extended
core states, further work needs to be done to optimize the
choice of PAW parameters such as the augmentation radii rac
a
and the optimal form of the local potential Ṽloc
共r兲. In addition, the use of continuum functions within Hartree-Fock
theory for the purpose of augmenting the span of the
pseudobasis functions is somewhat more complicated than
within Kohn-Sham theory. In the latter case, the continuum
states are solutions of a Hermitian differential operator. In
the case of the Hartree-Fock equations of spherically averaged atoms, the unoccupied continuum states are not as welldefined. In addition to these issues, the next step will be to
extend the analysis to multicenter and spin-polarized systems. Another interesting extension of this work will be to
treat some of the many recent hybrid Hartree-Fock and generalized gradient exchange-correlation functionals that have
reported in the recent literature.52 In addition, there have
been several groups1,2 who are developing methods to evaluate Hartree-Fock energies augmented with various types of
correlation energy formalisms, using orbitals derived from
Kohn-Sham calculations. At the Hartree-Fock level, since a
self-consistent Hartree-Fock calculation determines an unconstrained minimum of the energy, the energy determined
from Kohn-Sham orbitals is an overestimate. Conceivably,
self-consistent PAW-Hartree-Fock approach along the lines
developed in the present manuscript could improve these energy estimates.
ACKNOWLEDGMENTS
This work was supported by NSF under Grants No. NSF
DMR-0405456, No. DMR-0427055, and No. DMR-
245105-11
PHYSICAL REVIEW B 81, 245105 共2010兲
XIAO XU AND N. A. W. HOLZWARTH
冕
冕
0705239, and benefited from discussions with Yaojun A. Du
at Wake Forest University, Leeor Kronig of the Weizmann
Institute of Science, Weitao Yang of Duke University, John
Perdew of Tulane University, Jianmin Tao of Los Alamos
National Laboratory, and Alan Wright and Normand Modine
of Sandia National Laboratory. Computations were performed on the Wake Forest University DEAC cluster, a centrally managed resource with support provided in part by the
University.
The Coulomb interaction between valence electron pseudoorbitals can be expressed in terms of radial integrals similar
to those of the exchange contributions in Eq. 共60兲
APPENDIX A: ADDITIONAL PAW EXPRESSIONS
The nonexchange 共correlation兲 contributions to the atomcentered contributions to the valence energy can be written
in the form
The nonexchange 共correlation兲 contributions to the valence energy and Hamiltonian equations take the same form
in both PAW-Kohn-Sham and PAW-Hartree-Fock formulations. There are various ways of evaluating the terms as described in several publications.7,9,27,49 For completeness we
give the formulations that we have used in the present work
on spherical atoms. The nonexchange 共correlation兲 contributions to the pseudoenergy of Eq. 共43兲 takes the form
cv
vv
+ ẼH
.
Ẽvale nX = ẼKv + ẼNv + ẼH
共A1兲
a
Qcore
=
vv
=
ẼH
共A2兲
cv
ẼNv + ẼH
=
冕
drṼaf 共r兲ñv共r兲,
共A3兲
⬅
+
a
共r兲
ṽcore
+ 共− Z +
a
a
Qcore
−
ñv共r兲 ⬅ 兺 Nv兩˜␺v共r兲兩2 .
共A5兲
a
共r兲 is a localized potential defined in the range 0
Here Ṽloc
a
ⱕ r ⱕ rc . The potential due to extended core states is given
by
ñac 共r⬘兲
r⬎
The kinetic energy contribution is
Kaij ⬅ 具⌽i兩K兩⌽ j典 − 具⌽̃i兩K兩⌽̃ j典.
共A11兲
The nuclear- and core-electron contribution is given by
关Vaf 兴ij ⬅ 具⌽i兩Vaf 兩⌽ j典 − 具⌽̃i兩Ṽaf 兩⌽̃ j典,
共A12兲
where
Vaf 共r兲 ⬅ −
Z ae 2
c
+ VH
共r兲.
r
共A13兲
The valence-valence Hartree interactions may be evaluated
v⬘
where the interaction integrals were defined in Eqs. 共63兲 and
共64兲.
For evaluating the nonexchange 共correlation兲 terms in the
smooth Hamiltonian, the pseudopotential contributions for
the nuclear and Hartree interactions in both the Kohn-Sham
关Eq. 共48兲兴 and Hartree-Fock 关Eq. 共69兲兴 Hamiltonians can be
written in the form
ṼN共r兲 + ṼH共r兲 = Ṽaf 共r兲 + 兺 NvW̃0vv共r兲,
共A15兲
v
with
ñac 共r兲 ⬅ 兺 Nc兩˜␺ac 共r兲兩2 .
c
共A6兲
The long-range contribution of the nuclear and core-electron
charges is given by the last term of Eq. 共A4兲, where the
compensation charge potential is defined in terms of the
functional form defined in Eq. 共53兲
v̂a0共r兲 ⬅ e2
冊
共A14兲
v
dr⬘
冉
v
a
Q̃core
兲v̂a0共r兲
and
冕
共A9兲
⬘
a0
avv
兴ij ⬅ 兺 兺 Nv⬘具⌿̃v⬘兩P̃ak 典具P̃al 兩⌿̃v⬘典共Ra0
关VH
ij;kl − R̃ij;kl兲,
共A4兲
a
共r兲 ⬅ e2
ṽcore
共A8兲
共A10兲
kl
a
Ṽloc
共r兲
dr ñac 共r兲.
1 avv
兴ij .
⫻ Kaij + 关Vaf 兴ij + 关VH
2
where
Ṽaf 共r兲
a
Q̃core
=
1
兺 N N R̃0 .
2 vv v v⬘ vv;v⬘v⬘
ij
v
The interaction of the valence electron pseudo-orbitals with
the pseudonuclear and core-electron potentials can be combined into a term of the form
and
a
a
− Ẽvale
兲 nX = 兺 兺 Nv具⌿̃v兩P̃ai 典具P̃aj 兩⌿̃v典
共Evale
Here the kinetic energy contribution is given by
ẼKv = 兺 Nv具⌿̃v兩K兩⌿̃v典.
dr nac 共r兲
冕
ga共r⬘兲
dr⬘ 0
r⬎
共A7兲
where the fixed pseudopotential was defined in Eq. 共A4兲 and
the interaction function W̃0vv共r兲 was defined in Eq. 共72兲.
The nonexchange 共correlation兲 contributions to the
Hamiltonian matrix elements appropriate for both the PAWKohn-Sham and PAW-Hartree-Fock formalism is
avv
Daij nX = Kaij + 关Vaf 兴ij + 关VH
兴ij + 关Va0兴ij ,
共A16兲
where the last term comes from the compensation charge
contributions in Eq. 共A9兲
关Va0兴ij ⬅ ma0
ij
with
245105-12
冕
dr v̂a0共r兲共ñv共r兲 + 兺 NvM̂ 0vv共r兲兲. 共A17兲
v
PHYSICAL REVIEW B 81, 245105 共2010兲
PROJECTOR AUGMENTED WAVE FORMULATION OF…
APPENDIX B: SIMPLIFIED EXPRESSIONS FOR CORE
MATRIX ELEMENTS
When the core orbital is localized so that the corresponding pseudocore orbital can be set to zero: ˜␺ac 共r兲 ⬅ 0, many of
the expressions presented in Sec. IV simplify. For example,
the charge moment coefficient in Eq. 共56兲 takes the form
maL
ic =
冕
dr rL␾ai 共r兲␺ac 共r兲.
共B1兲
exchange-energy contributions evaluated with the allelectron core- and valence-basis functions. Of course, in the
complete extension of this formalism to multicenter systems,
the occupancy-angular factor ⌰Lvq defined in Eq. 共14兲 will
also be modified.
The self-consistent PAW-Hartree-Fock Eqs. 共65兲 can, in
principle, contain a contribution from localized core states if
the Lagrange multiplier 兩␭cv兩 ⬎ 0 since the extended overlap
operator reduces to
PAW HF
⌿̃c → 兺 兩P̃ai 典Oaic ,
OHF
The pseudocore-valence contributions to exchange energy
defined after Eq. 共60兲 can be written as
HF
a
a
aL
R̃Lvc;cv = 兺 具⌿̃HF
v 兩P̃i 典具P̃ j 兩⌿̃v 典Mic;cj ,
共B2兲
aij
where
2
MaL
ic;cj ⬅ e
冕冕
dr dr⬘
L
r⬍
L+1
r⬎
aL
共r兲m̂aL
m̂ic
cj 共r⬘兲
共B3兲
represents the interaction of two compensation charge contributions on site a.
The analogous one-center contribution which appears in
the one-center valence-core energy of Eq. 共62兲 as defined by
extending Eq. 共64兲 to core states becomes
aL
R̃aL
ic;cj = Mic;cj
acv
v
Ẽcxv + 兺 共Eac
x − Ẽx 兲 = − 兺
a
lv+lc
兺
vc L=兩lv−lc兩
a
where Oaic ⬅ ma0
ic . If the PAW basis function 兩⌽̃i 典 corresponds
to an occupied valence state, it is constructed to be orthogonal to the core orbitals so that Oaic = 0, otherwise Oaic may be
nonzero.
While the pseudoexchange kernel function X̃v共r兲 defined
in Eq. 共71兲 has no contributions from localized core states,
the one-center matrix element for the pseudoexchange kernel
function Xaiv defined in Eq. 共73兲 does have localized corestate contributions. While the valence-valence terms of Eq.
共73兲 remain as stated, if all of the core states were localized,
their contributions can be expressed in terms of the atomcentered all-electron integrals.
共B4兲
ensuring that the unphysical compensation charge contributions on a single site cancel out of the calculation. If all the
core states were localized, the complete expression for the
valence-core exchange energy would be given by
a
⌰Lvc 兺 具⌿̃HF
v 兩P̃i 典
aL
⫻具P̃aj 兩⌿̃HF
v 典Ric;cj
aij
共B5兲
representing a sum of single-site valence-core-electron
Xaiv vc
=−兺
lv+lc
兺
c L=兩lv−lc兩
1 L
aL
⌰ 兺 具P̃aj 兩⌿̃HF
v 典Ric;cj .
N v vc j
共B7兲
One final point regarding pseudocore orbitals needs clarification. That is, when 兩˜␺c共r兲兩 ⬎ 0, Gram-Schmidt orthogonalization with the generalized overlap operator defined in Eq.
共67兲, is used to orthogonalize the valence states to the core
states. This process violates the notion that core and valence
orbital functions reside in separate function spaces but it can
be consistently implemented provided that Eq. 共67兲 is used
HF
PAW
also to calculate the denominator term 具⌿̃HF
c 兩OHF 兩⌿̃c 典.
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In order to augment the Hartree-Fock basis set with continuum
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In terms of the radial integrals, the notation is meant to imply
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Basis and projector functions for l = 1 channel were constructed
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245105-14