John von Neumann Institute for Computing

Effective Core Potentials

Michael Dolg

published in

Modern Methods and Algorithms of Quantum Chemistry,

Proceedings, Second Edition, J. Grotendorst (Ed.),
John von Neumann Institute for Computing, Jülich,
NIC Series, Vol. 3, ISBN 3-00-005834-6, pp. 507-540, 2000.

c 2000 by John von Neumann Institute for Computing

Permission to make digital or hard copies of portions of this work for
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 EFFECTIVE CORE POTENTIALS

MICHAEL DOLG
Institut für Physikalische und Theoretische Chemie,
Wegelerstr. 12, 53115 Bonn
Germany
E-mail: dolg@thch.uni-bonn.de

After a brief review of relativistic effects on the electronic structure of atoms and
molecules the basic ideas of the relativistic ab initio effective core potential method
are outlined. The underlying approximations as well as the differences between the
two commonly used versions of the approach, i.e., model potentials and pseudopo-
tentials, are discussed. The article then focusses on the adjustment of atomic
shape-consistent and energy-consistent pseudopotentials, as well as on correspond-
ing core polarization potentials. Finally, the results of some calibration calculations
for the homonuclear dimers of the halogen atoms are presented.

1 Introduction

The present manuscript discusses the two branches of effective core potential (ECP)
approaches, i.e., the model potential (MP) and the pseudopotential (PP) tech-
niques. The main focus is on those ECP schemes which proved to be successful in
atomic and molecular relativistic electronic structure calculations during the past
decade, and moreover, due to the authors own history, the presentation is some-
what biased towards the discussion of energy-consistent ab initio pseudopotentials.
It is neither intended to give a complete overview over all effective core potential
approaches developed since the pioneering work of Hellmann and Gombas around
1935, nor to cover all schemes currently on the market. In particular techniques
developed especially for density functional theory and/or plane wave based com-
putational approaches have been left out. A number of reviews on effective core
potentials has been published during the last three decades and the reader is referred
to them for more detailed information 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 .

1.1 Relativistic effects

Accurate ab initio electronic structure calculations for systems with heavy ele-
ments require the inclusion of relativistic effects, cf., e.g., the extensive bibli-
ographies of relativistic calculations collected by Pyykkö 21,22,23 . Although this
fact is nowadays generally acknowledged and the discussion of relativistic effects
begins to be included in (quantum) chemical textbooks, a very brief and in-
complete outline of relativistic effects will be given here in order to make more
plausible why even for systems with second row elements a relativistic effective
core potential study may be more accurate than a nonrelativistic all-electron
investigation. Several excellent review articles focussing on relativistic effects
exist 24,25,26,27,28,29,30,31,32,33,34,35,36,37,38 .
For hydrogen and hydrogen-like ions with a point nucleus of charge Z the rela-

1
tivistic Schrödinger equation, i.e., the Dirac equation, is analytically solvable:
!2
2 Z/c
Enκ = ±c [1 + p ]−1/2 − c2 . (1)
n − |κ| + κ2 − (Z/c)2
Here c denotes the velocity of light (c ≈ 137.0359895 a.u.). The relativistic quantum
number κ is defined in terms of the quantum numbers of orbital and total angular
momentum, l and j, as
κ = ∓(j + 1/2) for j = l ± 1/2 . (2)
It is observed that in contrast to the nonrelativistic case two sets of solutions exist,
which are separated by ≈ 2c2 . This is due to the fact that the Dirac equation is
not only a wave equation valid for an electron, but rather for spin-1/2 particles
as electrons and positrons. The solutions near the zero of energy are called elec-
tronic states and essentially correspond to the nonrelativistic solutions, whereas
those near −2c2 are called positronic states. The wavefunction turns out to be a
four-component vector (four-spinor), the two upper components (upper bispinor)
being large for the electronic states, the two lower ones (lower bispinor) being large
for the positronic states (charge degrees of freedom in the wavefunction). Since
the focus in relativistic quantum chemistry is on electrons, it is common to use the
terms large components and small components for the upper and lower components,
respectively. The odd and even components may be related to spin up and down,
respectively, of the particle (spin degrees of freedom). A Taylor expansion of Eq. 1
shows for the electronic states that the nonrelativistic energy increases as Z 2 and
the relativistic corrections to it as Z 4 . However, since the prefactor of the rela-
tivistic energy contributions contains 1/c2 the corrections are expected to become
chemically important only for heavy nuclei. The substitution of the nonrelativistic
Hamiltonian by a relativistic one leads to the so-called direct relativistic effects, i.e.,
a stabilization and a contraction of the hydrogenic functions. It is further observed
that not all states with the same main quantum number n are degenerate as it is
the case for the nonrelativistic solutions. In particular, states with the same nl are
split into two subsets for li0 (spin-orbit splitting).
The total nonrelativistic Hartree-Fock energy of the rare gas atoms He, Ne, Ar,
Kr, Xe and Rn is approximately proportional to Z 2.37 , the correlation corrections
(as estimated from local density functional calculations including a self-interaction
correction) to Z 1.16 and the relativistic corrections (as estimated from quasirela-
tivistic Wood-Boring calculations) to Z 4.34 (Fig. 1). Focussing on the one-electron
functions rather than the total energy one observes a stabilization and contraction
for valence s and p shells, but a destabilization and expansion for valence d and f
shells (Figs. 2, 3). Besides the direct relativistic effects causing the stabilization
and contraction as well as the splitting of the p, d, f, ... shells, so-called indirect
relativistic effects or relativistic self-consistent field effects are present. The con-
traction of the inner shells causes a more efficient screening of the nuclear charge
for the outer shells, thus leading to a decreased effective nuclear charge and an ex-
pansion and destabilization. Direct and indirect effects act on all shells, but direct
effects dominate for s and p valence shells, whereas indirect effects dominate for
d and f valence shells. Relativistic effects on orbitals have direct consequences

2
 5
4
3
log(−energy [a.u.]) 2
1
0
−1
−2
∆E(relativity)
−3
∆E(correlation)
−4 E(Hartree−Fock)
−5
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
log(nuclear charge [a.u.])

Figure 1. Total nonrelativistic Hartree-Fock energy, relativistic corrections (estimated from Wood-
Boring calculations) and correlation contributions (estimated from correlation energy density func-
tional calculations) for rare gas atoms.

−0.1

−0.2
orbital energy (a.u.)

−0.3 DHF ns1/2

DHF (n−1)d3/2
DHF (n−1)d5/2
−0.4 HF ns
HF (n−1)d

−0.5

−0.6
Cu Ag Au 111 E

Figure 2. Nonrelativistic Hartree-Fock (HF) and relativistic Dirac-Hartree-Fock (DHF) orbital

energies for the valence shells of the coinage metals (n = 4, 5, 6, 7 for Cu, Ag, Au and Eka-Au,
respectively).

on quantum mechanical observables, e.g., the ionization potentials of the coinage

metals are enhanced due to the relativistic stabilization of the valence s shell (Fig.
4). Clearly, since the energy and shape of valence orbitals is affected by relativis-
tic effects, these are also important for chemical bonding. Quite often relativity
leads to a bond length contraction, e.g., for the coinage metal hydrides (Fig. 5).
In some rare cases, e.g., for some lanthanide or actinide systems 36 , slight bond

3
 4.5
4.0

radius expectation value (a.u.)

3.5
3.0 DHF ns1/2
DHF (n−1)d3/2
2.5 DHF (n−1)d5/2
HF ns
2.0
HF (n−1)d
1.5
1.0
0.5
Cu Ag Au 111 E

Figure 3. Nonrelativistic Hartree-Fock (HF) and relativistic Dirac-Hartree-Fock (DHF) orbital

radius expectation values hri for the valence shells of the coinage metals (n = 4, 5, 6, 7 for Cu,
Ag, Au and Eka-Au, respectively.

11
HF
10 DHF
Exp.
9
IP(eV)

5
Cu Ag Au E
111

Figure 4. Nonrelativistic Hartree-Fock (HF), relativistic Dirac-Hartree-Fock (DHF) and experi-

mental (Exp.) ionization potentials of the coinage metals. The experimental result for Eka-Au
actually corresponds to the result of a high level correlated relativistic calculation (Eliav et al.,
Phys. Rev. Lett. 73, 3203 (1994)).

length expansions are found. One may relate the relativistic bond length changes
to contractions or expansions of the valence orbitals mainly involved in bonding,
but alternative explanations are also valid 27,33 . Besides bond lengths also binding
energies and vibrational constants are influenced by relativistic effects. In simple
cases bond stabilization or destabilization may be estimated on the basis of atomic
data, e.g., for a mainly ionic A+ B− system the relativistic effects in the ionization

4
potential of A and the electron affinity of B roughly determine the relativistic effect
on the binding energy. Spin-orbit coupling lowers the energy of atoms with open p,
d, and/or f shells. In molecules the lowering of the energy is typically much smaller
due to the usually smaller number of unpaired electrons and the lower symmetry
of the system. This often leads to a net destabilization of the bond by spin-orbit
effects. In special cases, e.g., for the essentially van der Waals bonded dimer Hg2 ,
spin-orbit effects can also increase the binding energy 39 .

2.1
HF
2.0
DHF
Exp.
1.9
bond length (Å)

1.8

1.7

1.6

1.5

1.4
CuH AgH AuH (111E)H

Figure 5. Nonrelativistic Hartree-Fock (HF), relativistic Dirac-Hartree-Fock (DHF) and experi-

mental (Exp.) bond lengths of the coinage metal hydrides. The experimental result for the Eka-Au
hydride actually corresponds to the result of a high level correlated relativistic calculation (Seth
et al., Chem. Phys. Lett. 250, 461 (1996)).

1.2 Computational savings

More familiar than relativistic effects is to the general chemist the idea that only
the valence electrons of an atom determine, at least qualitatively, its chemical be-
havior. The effective core potential approach is based on this experience and tries
to provide a valence-only Hamiltonian which models in actual calculations for va-
lence properties of atoms and molecules as accurately as possible the corresponding
all-electron results. The main motivation to develop such schemes was initially the
reduction of the computational effort, when only the chemically relevant subset of
electrons is treated explicitly. Today, with a far advanced computer technology
at hand and significantly improved algorithms implemented in quantum chemical
program packages, the main advantage of effective core potentials is the ease with
which relativistic effects can be included in the calculations.

5
2 All-electron Hamiltonian

Relativistic all-electron approaches are discussed here in brief for two reasons: on
one hand relativistic ab initio effective core potentials are derived from (atomic)
all-electron relativistic calculations, on the other hand they are often calibrated in
atomic and molecular calculations against the results from all-electron relativistic
calculations.
Starting point of the following considerations is a general configuration space
Hamiltonian for n electrons and N nuclei, where we assume the Born-Oppenheimer
approximation to hold and neglect external fields.
n n N
X X X Zλ Zµ
H= h(i) + g(i, j) + . (3)
i
rλµ
ihj λhµ

The indices i and j denote electrons, λ and µ nuclei. Zλ is the charge of the nucleus
λ. For the one- and two-particle operators h and g various expressions can be
inserted (e.g., relativistic, quasirelativistic or nonrelativistic; all-electron or valence-
only). The basic goal of quantum chemical methods is usually the approximate
solution of the time-independent Schrödinger equation for a specific Hamiltonian,
the system being in the state I, i.e.,
HΨI = EI ΨI . (4)
The most accurate electronic structure calculations nowadays applicable for atoms,
molecules and also solids are based on the Dirac (D) one-particle Hamiltonian
X
hD (i) = c~αi p~i + (β i − I4 )c2 + Vλ (riλ ) , (5)
λ

which is correct to all orders of the fine-structure constant α = 1/c. In these

equations I4 denotes the 4 × 4 unit matrix, and p~i = −i∇ ~ i is the momentum
operator for the i-th electron. α ~ i is a three-component vector whose elements
together with β i are the 4 × 4 Dirac matrices
~0 σ
   
I2 0 ~
β= and ~ =
α , (6)
0 −I2 ~ ~0
σ
which can be expressed in terms of the three-component vector of the 2 × 2 Pauli
~,
matrices σ
     
0 1 0 −i 1 0
σx = , σy = , σz = , (7)
1 0 i 0 0 −1
and the 2 × 2 unit matrix I2 . The rest energy c2 of the electron was subtracted
from Eq. 5 in order to achieve a better compatibility to the nonrelativistic case,
i.e., as in Eq. 1 the zero of energy corresponds to a free electron without kinetic
energy. Vλ (riλ ) denotes the electrostatic potential generated by the λ-th nucleus at
the position of the i-th electron
Zλ
Vλ (riλ ) = − . (8)
riλ

6
In some cases a finite nucleus is used, e.g., a Gaussian-type charge distribution
Z ∞
0 2
ρλ (r) = ρλ exp(−ηλ r ) with 4π dr r2 ρλ (r) = Zλ . (9)
0
The parameter ηλ can be determined from the nuclear radius Rλ , which is itself
derived from the nuclear mass according to
1/3
ηλ = 3/(2Rλ2 ) with Rλ = 2.2677 × 10−5 Mλ a0 . (10)
Other charge distributions, e.g., a finite hard sphere or a Fermi-type nuclear model,
are also used. The coupling of the upper and lower components of the wavefunction
~ ip
via α ~i requires either kinetically balanced basis sets or the imposal of appropriate
boundary conditions in order to avoid the so-called finite basis set disease.
The two-particle terms used in such calculations are either the nonrelativistic
electrostatic Coulomb (C) interaction (yielding the Dirac-Coulomb (DC) Hamilto-
nian correct to O(α0 ))
1
gC (i, j) = , (11)
rij
or in addition the magnetic Gaunt (G) interaction (yielding the Dirac-Coulomb-
Gaunt (DCG) Hamiltonian correct to O(α0 ))
1 ~ iα
α ~j
gCG (i, j) = − , (12)
rij rij
or in addition the retardation of the interaction due to the finite velocity of light,
as it is accounted for in the frequency-independent Breit (B) interaction (yielding
the Dirac-Coulomb-Breit (DCB) Hamiltonian correct to O(α2 ))
1 1 (~
αi~rij )(~αj ~rij )
gCB (i, j) = − [~ ~j +
αi α 2 ]. (13)
rij 2rij rij
For further details the reader is referred to, e.g., a review article by Kutzelnigg 28 .
The Gaunt- and Breit-interaction is often not treated variationally but rather by
first-order perturbation theory after a variational treatment of the Dirac-Coulomb-
Hamiltonian. The contribution of higher-order corrections such as the vaccuum
polarization or self-energy of the electron can be derived from quantum electrody-
namics (QED), but are usually neglected due to their negligible impact on chemical
properties.
In principle problems of relativistic electronic structure calculations arise from
the fact that the Dirac-Hamiltonian is not bounded from below and an energy-
variation without additional precautions could lead to a variational collapse of
the desired electronic solution into the positronic states. In addition, at the many-
electron level an infinite number of unbound states with one electron in the positive
and one in the negative continuuum are degenerate with the desired bound solution.
A mixing-in of these unphysical states is possible without changing the energy and
might lead to the so-called continuum dissolution or Brown-Ravenhall disease. Both
problems are avoided if the Hamiltonian is, at least formally, projected onto the
electronic states by means of suitable operators P+ (no-pair Hamiltonian):
Hnp = P+ HP+ . (14)

7
The Douglas-Kroll transformation 40 of the Dirac-Coulomb Hamiltonian in its im-
plementation by Heß 41,42,43,44,45 leads to one of the currently most successful and
popular forms of a relativistic no-pair Hamiltonian. The one-electron terms of the
Douglas-Kroll-Heß (DKH) Hamiltonian have the form

1
hDKH (i) = Ei − Ai [V (i) + Ri V (i)Ri ]Ai − W1 (i)Ei W1 (i) − {W1 (i)2 , Ei } (15)
2

s
Ei + c 2 c~
σ i p~i
q
with E i = E pi = c p~i 2 + c2 , Ai = Api = , Ri = ,
2Ei Ei + c 2

and {} denoting an anticommutator. W1 (i) is an integral operator with the kernel

V (~ ~ 0)
p, p
W1 (~ ~ 0 ) = Ap (Rp − Rp0 )Ap0
p, p , (16)
E p + E p0

where V (~p, p~ 0 ) is the Fourier transform of the external Potential V (i). The two-
electron terms
1 1 1 1
gDKH (i, j) = Ai Aj [ + Ri Ri + Rj Rj + Ri Rj Ri Rj ]Ai Aj (17)
rij rij rij rij

increase the computational cost significantly, but have only small effects on the
results and are therefore usually neglected, i.e., the unmodified Coulomb interaction
is used.
A straightforward elimination of the small components from the Dirac equation
leads to the two-component Wood-Boring (WB) equation 46 , which exactly yields
the (electronic) eigenvalues of the Dirac Hamiltonian upon iterating the energy-
dependent Hamiltonian

1 Ei − V (i) −1 X
hW B (i) = (~
σi p
~i )(1 + ) (~
σ i p
~ i ) + Vλ (riλ ) . (18)
2 2c2
λ

Due to the energy-dependence of the Hamiltonian the Wood-Boring approach leads

to nonorthogonal orbitals and has been mainly used in atomic finite difference
calculations as an alternative to the more involved Dirac-Hartree-Fock calculations.
The relation

(~
σ j p~j )f (rj )(~
σ j p~j ) = p
~j f (rj )~
pj + i~
σ j [(~
pj f (rj )) × p~j ] (19)

allows the partitioning of spin-independent and spin-dependent parts and there-

fore the derivation of a scalar-relativistic DKH or WB Hamiltonian. This is also
obtained by formally replacing σ ~ i p~i by p
~i in Eqs. 15 and 18.
The WB approach was used to generate both model potentials as well as pseu-
dopotentials. The DKH method was applied together with model potentials and
to provide molecular all-electron results for calibration studies with valence-only
schemes (cf. below).

8
3 Valence-only Hamiltonian

A significant reduction of the computational effort in quantum chemical investiga-

tions can be achieved by restriction of the actual calculations to the valence electron
system and the implicit inclusion of the influence of the chemically inert atomic
cores by means of suitable parametrized effective (core) potentials. This approach
is in line with the chemists view that mainly the valence electrons of an element
determine its chemical behavior, cf., e.g., the periodic table of elements. From
a quantum mechanical point of view the partitioning of a many-electron system
into subsystems is not possible, since electrons as elementary particles are indis-
tinguishable. However, in the framework of effective one-particle approximations
like Hartree-Fock or Dirac-Hartree-Fock theory a definition of core and valence or-
bitals/shells is possible either on the basis of energetic (orbital energies) or spatial
(shape, radial maxima or expectation values of orbitals) arguments. If the core
shells of a system are determined for one bonding situation, e.g., the free atoms,
and then transferred to other bonding situations, e.g., the molecule, one speeks of
the frozen-core or frozen-orbital approximation. This approach is underlying all
valence-only schemes (cf., however, section 6). It is important to realize, however,
that the chemists qualitative view of partitioning core and valence shells is usually
not suitable for quantitative calculations, e.g., treatment of Ti ([18 Ar] 3d2 4s2 3 F2
ground state) or Ce ([54 Xe] 4f1 5d1 6s2 1 G4 ground state) as a four valence electron
systems leads to poor or even disastrous results 36 , whereas it works very well for
C ([2 He] 2s2 2p2 3 P0 ground state). The reason is the presence of partially occu-
pied valence shells which have the same or even lower main quantum number as
the fully occupied core shells. Although based on orbital energies the separation
between core and valence shell may be reasonable, it is poor from a spatial point
of view: the Ti 3d shell has its maximum density close to the one of the 3s and 3p
shells, the Ce 4f shell has its maximum density even closer to the nucleus than the
5s and 5p shells. A change in the valence electron configuration in these compact
valence orbitals, e.g., when looking at an excited atomic state or when forming a
chemical bond, leads to too large changes of the shielding of the nuclear charge for
the most diffuse core orbitals and consequently to a breakdown of the frozen core
approximation. The most reliable effective core potentials have a separation of core
and valence shells according to the main quantum number, e.g., 3s and 3p for Ti
and 4s, 4p, 4d, 5s and 5p for Ce have to be included in the valence shell.
In effective core potential theory an effective model Hamiltonian approximation
for Hnp is seeked, which only acts on the states formed by the valence electrons:

nv
X nv
X
Hv = hv (i) + gv (i, j) + Vcc + Vcpp . (20)
i ihj

The subscripts c and v denote core and valence, respectively. hv and gv stand for
effective one- and two-electron operators, Vcc represents the repulsion between all
cores and nuclei of the system, and Vcpp is a core polarization potential (CPP). nv

9
denotes the number of valence electrons treated explicitly in the calculations
N
X
nv = n − (Zλ − Qλ ) . (21)
λ

Here Qλ denotes the charge of the core λ. Several choices exist for the formula-
tion of such a valence-only model Hamiltonian, i.e., four-, two- or one-component
approaches and explicit or implicit treatment of relativity. Since a reasonable com-
promise between accuracy and efficiency is desired, the standard effective core po-
tential schemes use the implicit treatment of relativity (i.e., a nonrelativistic ki-
netic energy operator and inclusion of relativistic effects via parametrization of the
effective core potential) and a one-component (scalar-quasirelativistic) or a two-
component (quasirelativistic) treatment. Moreover, one may decide to keep the
radial nodal structure of the (atomic) valence orbitals unchanged (model poten-
tials, MP), or formally apply a pseudoorbital transformation to have the energeti-
cally lowest (atomic) valence orbital of each lj or l quantum number without radial
nodes (pseudopotentials, PP).
Scalar-quasirelativistic and quasirelativistic effective core potentials use a for-
mally nonrelativistic model Hamiltonian
1 1
hv (i) = − ∆i + Vcv (i) and gv (i, j) = . (22)
2 rij
Relativistic contributions result only from the parametrization of the effective core
potential Vcv , which describes the interaction of a valence electron with all nuclei
and cores present in the system. The molecular pseudopotential is assumed to be
a superposition of atomic pseudopotentials, with the Coulomb attraction between
point charges as the leading term
N
X Qλ λ
Vcv (i) = (− + ∆Vcv (~rλi )) + ... . (23)
rλi
λ

For the interaction between nuclei and cores the point charge approximation also
is the first term
N
X Qλ Qµ
Vcc = ( + ∆Vccλµ (rλµ )) + ... . (24)
rλµ
λhµ

λ
It is hoped that a suitable parametrization of ∆Vcv and ∆Vccλµ is able to compensate
for all errors resulting from the simplifications of the original valence Hamiltonian.

3.1 Model Potentials

The most straightforward approach to come to an effective core potential is to
use the Fock operator Fv of a valence orbital ϕVa and to simplify the effective one
particle potential
N N
X Zλ X X Qλ λ
− + (2Jc (i) − Kc (i)) = (− + ∆Vcv (~rλi )) , (25)
rλi c
r λi
λ λ

10
where the first sum is over all nuclei λ with charge Zλ and the second over all
core orbitals c. Jc and Kc denote the usual Coulomb and exchange operators. A
first approximation is the assumption of non-overlapping cores, so that the sec-
ond sum on the lhs can also be regarded as a superposition of one-center terms.
A second approximation follows from the goal that relativistic effects should be
treated implicitly. Therefore not only Vcv is approximated but also an additive
relativistic correction term Vrel . In order to obtain the relevant atomic potentials
λ λ
Vrel + Vcv a two- or one-component quasirelativistic atomic all-electron calculation
is performed. The most widely used variant of the method are the ab initio model
potentials (AIMP) of Seijo, Barandiarán and coworkers 48,49,50,51,52,53 , where the
quasirelativistic Hamiltonian proposed by Wood and Boring (WB) 46 for density
functional calculations is used in the framework of Hartree-Fock theory according
to the scheme outlined by Cowan and Griffin (CG) 47 . The WB and CG approaches
correspond essentially to the use of an energy-dependent one-particle Hamiltonian,
which results from the elimination of the small components from the Dirac-equation,
within the Hartree-Fock scheme, disregarding any resulting non-orthogonality be-
tween orbitals of equal lj.
The AIMP method in its present form starts from a quasirelativistic all-electron
Hartree-Fock calculation for the atom under consideration in a suitable electronic
state and approximates the operators on the lhs of Eq. 25 for an atomic core λ as
described in the following.
The long-range local Coulombic (C) part is spherical and is represented by
a linear combination of Gaussians with prefactors 1/r, i.e., a local radial model
potential
Zλ − Q λ X 1 X λ −αλk rλi2
− +2 Jcλ (i) = Ck e = ∆VCλ (i) . (26)
rλi rλi
c∈λ k

The exponents αλkand coefficients Ckλ are adjusted

P to the all-electron potential in a
least-squares sense under the constraint that k Ckλ = Zλ − Qλ in order to enforce
the correct asymptotic behavior of the model potential. Since the evaluation of
integrals over such a local potential is not costly, any desired accuracy can be easily
achieved by using a sufficiently long expansion. The nonlocal exchange (X) part is
substituted by its spectral representation in the space defined by a set of functions
χλp centered on core λ
X X
− Kcλ (i) = |χλp (i)iAλpq hχλq (i)| = ∆VXλ (i) . (27)
c∈λ p,q

It should be noted that this model potential operator yields the same one-center
integrals as the true core exchange operator as long as the basis functions can be
represented by the set of the χλp . Two- and three-center integrals are approximated.
Since, in contrast to the Coulomb part, the exchange part is short ranged, a moder-
ate number of functions χλp is needed and the one-center approximation is expected
to be very good, at least for not too large cores. In practical applications the basis
used in the spectral representation is chosen to be identical to the primitive func-
tions of the valence basis set used for the atom under consideration and the Aλpq
are calculated during the input processing of each AIMP calculation.

11
 With the Coulomb and exchange parts of the model potential discussed so far
the core-like solutions of the valence Fock equation still would fall below the desired
valence-like solutions. In order to prevent the valence-orbitals to collapse into the
core during a variational treatment and to retain a Aufbau principle for the valence
electron system, the core-orbitals are shifted to higher energies by means of a shift
operator
X
P λ (i) = (Dcλ )|ϕλc (i)ihϕλc (i)| . (28)
c∈λ

Here the ϕλc denote the core orbitals localized on core λ. For practical calculations
they are represented by a sufficiently large (all-electron) basis set. In principle only
Dcλ → ∞ would effect a strict orthogonality between core and valence orbitals, how-
ever the more or less arbitrary choice Dcλ = −2λc is usually made due to numerical
reasons. With this choice there is not strict orthogonality between core and valence
orbitals, but the resulting errors are expected to be small.
The approach which has been described so far is the nonrelativistic AIMP
method. It should be noted that for the derivation of the model potential
λ
∆Vcv,av (i) = ∆VCλ (i) + ∆VXλ (i) + P λ (i) (29)
no valence properties, e.g., valence orbitals or valence orbital energies, have
been used in the nonrelativistic AIMP approach. The scalar-quasirelativistic
and quasirelativistic extensions of the AIMP approach are called CG-AIMP (one-
component) and WB-AIMP (two-component), respectively.
For an one-electron atom in the central field approximation one obtains from
Eq. 18 the following radial equation:
(HS + HM V + HD + HSO ) Pnκ (r) = nκ Pnκ (r) . (30)
The nonrelativistic Schrödinger Hamiltonian
1 d2 l(l + 1)
hS (i) = − + + V (r) (31)
2 dr2 2r2
is augmented by three energy-dependent relativistic terms, i.e., a mass-velocity
(MV), a Darwin (D) and a spin-orbit (SO) term
α2 2 α2 dV d 1
HM V = − [nκ − V (r)] , HD = − Bnκ ( − ), (32)
2 4 dr dr r
−1
α2 dV α2

κ+1
HSO =− Bnκ , Bnκ = 1+ [nκ − V (r)] .
4 dr r 2
In the many-electron case the correct nonlocal Hartree-Fock potential is used
in Eq. 30, but a local approximation to it in Eqs. 32. In the CG-AIMP approach
the mass-velocity and Darwin operators are cast together with the exchange terms
into their spectral representation Eq. 27. The valence orbital energies nκ are kept
fixed during the extraction process and are also used for any semi-core orbitals of
the same κ, which are included in the AIMP valence space. A similar strategy is
followed in order to deal with the first derivative of the valence orbital in the Darwin
term. It should be noted, however, that due to the use of relativistic core orbitals

12
and core orbital energies relativistic contributions are also present in the Coulomb
and shift terms of the AIMP. The WB-AIMP method adds to this a representation
of the spin-orbit operator in the form
X X Bλ λ 2
λ
∆Vcv,so (i) = ( lk −βlk
2 e rλi
)Plλ (i)~lλi~si Plλ (i) , (33)
rλi
l k

where ~lλi = ~rλi × p~i and ~si denote the operators of orbital angular momentum
and spin, respectively, and Plλ is the projection operator onto the subspace of
λ
angular quantum number l with respect to core λ. The coefficients Blk and ex-
λ
ponents βlk are determined by means of a least-squares fit to the radial compo-
nents of the Wood-Boring spin-orbit term. We note here in passing, that the
Hamiltonian proposed by Cowan and Griffin is not identical to the spin-orbit
averaged form of the Hamiltonian proposed by Wood and Boring 90 . The one-
component Cowan-Griffin equation is identical to the Wood-Boring equation for
l = 0, but it yields the eigenvalues of the Klein-Gordon equation (valid for a
spin-0 particle) for l i 0. The reason is that in addition to the spin-orbit term
the Darwin term was also neglected for l i 0 by Cowan and Griffin. The CG-
AIMP approach, however, uses in fact the properly spin-averaged Wood-Boring
Hamiltonian and not the Cowan-Griffin Hamiltonian. Ab initio model poten-
tial parameters and corresponding basis sets are available on the internet under
http://www.qui.uam.es/Data/AIMPLibs.html . Since the model potential ap-
proach yields valence orbitals which have the same nodal structure as the all-
electron orbitals, it is possible to combine the approach with an explicit treatment
of relativistic effects in the valence shell, e.g., in the framework of the DKH no-pair
Hamiltonian 54,55 . Corresponding ab initio model potential parameters are available
on the internet under http://www.thch.uni-bonn.de/tc/TCB.download.html .

3.2 Pseudopotentials
The pseudopotential method was first developed by Hellmann 56 and Gombás 57
around 1935. The quantum mechanical foundations of the method have been inves-
tigated later by Fényes and Szépfalusy in the framework of Hartree and Hartree-
Fock theory, respectively. The approach became more popular after the work of
Preuss 58 for molecules and Phillips and Kleinman 59 for solids. Many of the ap-
proximations underlying the method were discussed extensively in the literature,
e.g., cf. papers of Weeks et al.1 and Dixon and Robertson 3 . However, since the
modern pseudopotentials used today have little in common with the formulas one
obtains by a strict derivation of the theory, only a rough derivation in the framework
of nonrelativistic Hartree-Fock theory is presented in the following.
The space of orthonormal orbitals of a system with a single valence electron
outside a closed shell core may be partitioned into a subspace for the doubly oc-
cupied core orbitals ϕc and a subspace for the singly occupied valence orbital ϕv .
For the moment the space of the unoccupied virtual orbitals is not considered. The
Fock equation for the valence orbital ϕv
X
F v ϕv =  v ϕv + vc ϕc (34)
c6=v

13
(Fv denotes the Fock operator) can be transformed by application of 1 − Pc from
the left into a pseudo eigenvalue equation
(1 − Pc )Fv ϕv = v ϕv , (35)
with the projector Pc on the subspace of the core orbitals
X
Pc = |ϕc ihϕc | . (36)
c

Reductions in the basis set used to represent the valence orbital ϕv can be only
achieved if by admixture of core orbitals ϕc the radial nodes are eliminated and the
shape of the resulting pseudo (p) valence orbital ϕp is as smooth as possible in the
core region (pseudoorbital transformation)
X
ϕp = Np (ϕv + ωc ϕc ) . (37)
c6=v

Np denotes a normalization factor depending on the coefficients ωc . The original

valence orbital with the full nodal structure in terms of the pseudo valence orbital
with the simplified nodal structure
ϕv = (Np )−1 (1 − Pc )ϕp (38)
may be inserted into the pseudo eigenvalue problem Eq. 35
(1 − Pc )Fv (1 − Pc )ϕp = v (1 − Pc )ϕp . (39)
1
Using the so-called generalized Phillips-Kleinman pseudopotential
V GP K = −Pc Fv − Fv Pc + Pc Fv Pc + v Pc (40)
one recovers again a pseudo eigenvalue problem for the pseudo valence orbital
(Fv + V GP K )ϕp = v ϕp . (41)
If one assumes the core orbitals ϕc to be also eigenfunctions of the Fock operator
Fv , i.e., [Fv , Pc ] = 0, and uses the idempotency of the projection operator Pc = Pcn
(n ≤ 1), one recovers a simplified pseudo eigenvalue problem
(Fv + V P K )ϕp = v ϕp (42)
59
containing the so-called Phillips-Kleinman pseudopotential
X
V PK = (v − c )|ϕc ihϕc | . (43)
c6=v

The transition from a single valence electron to nv valence electron requires formally
in Eq. 39 the following substitutions:
nv
X nv
X
Fv 7−→ Fv (i) + g(i, j) , v 7−→ E v , (44)
i ihj
nv
Y
(1 − Pc ) 7−→ (1 − Pc (i)) , ϕp 7−→ Ψp .
i

14
The sum of effective one-particle operators Fv (i) has to be augmented by the in-
terelectronic interaction terms g(i, j) between the valence electrons. The valence
model Hamiltonian Hv then reads
nv
Y nv
X nv
X nv
Y
Hv = (1 − Pc (k))[ Fv (i) + g(i, j)] (1 − Pc (l)) (45)
k i ihj l
nv
X nv
X
+ Ev [ Pc (i) − Pc (i)Pc (j) − + ...] + Vcc .
i ihj

This form of valence model Hamiltonian is essentially useless for practical calcula-
tions, since it contains complicated many-electron operators due to the introduc-
tion of products of projection operators. In addition the use of such a Hamiltonian
would not bring about any computational savings with respect to an all-electron
treatment, since the derivation given so far essentially consists of a rewriting of the
Fock equation for a valence orbital in a different form. Reductions in the compu-
tational effort can be only achieved by elimination of the core electron system and
simulation of its influence on the valence electrons by introducing a suitable model
Hamiltonian:
1
hv (i) = [(1 − Pc (i))Fv (i)(1 − Pc (i)) + Ev Pc (i)] 7−→ − ∆i + Vcv (i) , (46)
2

1
gv (i, j) = [(1 − Pc (i))(1 − Pc (j))g(i, j)(1 − Pc (i))(1 − Pc (j))] 7−→ .
rij

4 Analytical form of pseudopotentials

λ
The simplest and historically the first choice is the local ansatz for ∆Vcv in Eq. 22,
however, such an ansatz is too inaccurate and therefore has soon been replaced by
a so-called semilocal form. In case of quasirelativistic pseudopotentials, i.e., when
spin-orbit coupling is included, the semilocal ansatz in two-component form may
be written as
L−1 l+1/2
X X
λ
∆Vcv (~rλi ) = (Vljλ (rλi ) − VLλ (rλi ))Pljλ (i) + VLλ (rλi ) . (47)
l=0 j=|l−1/2|

Pljλ denotes a projection operator on spinor spherical harmonics centered at the

core λ
j
X
Pljλ (i) = Pl,l±1/2
λ
(i) = Pκλ (i) = | λljmj (i)ihλljmj (i) | . (48)
mj =−j

For scalar-quasirelativistic calculations, i.e., when spin-orbit coupling is neglected,

a one-component form may be obtained by averaging over the spin
L−1
X
λ
∆Vcv,av (~rλi ) = (Vlλ (rλi ) − VLλ (rλi ))Plλ (i) + VLλ (rλi ) . (49)
l=0

15
The projection operator Plλ refers now to the spherical harmonics centered at the
core λ
l
X
Plλ (i) = | λlml (i)ihλlml (i) | . (50)
ml =−l

A spin-orbit operator may be defined

L−1
λ
X ∆Vlλ (rλi ) λ λ
∆Vcv,so (~rλi ) = [lPl,l+1/2 (i) − (l + 1)Pl,l−1/2 (i)] (51)
2l + 1
l=1

which contains essentially the difference between the two-component pseudopoten-

tials
∆Vlλ (rλi ) = Vl,l+1/2
λ λ
(rλi ) − Vl,l−1/2 (rλi ) . (52)
For practical calculations it is advantageous to separate space and spin
L−1
λ
X 2∆Vlλ (rλi ) λ ~
∆Vcv,so (~rλi ) = Pl (i)lλi~si Plλ (i) . (53)
2l + 1
l=1

The potentials Vljλ and Vlλ (l = 0 to l = L) respectively ∆Vlλ (l = 1 to l = L -1)

are represented as a linear combination of Gaussians multiplied by powers of the
electron-core distance:
X nλ
Vljλ (rλi ) = Aλljk rλiljk exp(−aλljk rλi
2
), (54)
k

nλ
X
[∆]Vlλ (rλi ) = [∆]Aλlk rλilk exp(−aλlk rλi
2
). (55)
k

The necessary one-electron integrals over cartesian Gaussians have been presented,
e.g., by McMurchie and Davidson 60 or by Pitzer and Winter 61,62 . Alternatively,
making use of the operator identity
+l
(l) (l) (l)
X X
|lml iVl (r)hlml | = |χi iAij hχj | . (56)
ml =−l i,j

a nonlocal representation in a (nearly) complete auxiliary basis set can be used

(l)
instead 63 . Once the constants Aij have been determined the integral evaluation is
reduced to overlap integrals (between the auxiliary basis and the actual molecular
basis sets) and therefore the derivatives with respect to the nuclear coordinates
needed in geometry optimizations become much easier to evaluate.
In case of large cores a correction to the point charge repulsion model in Eq. 24
is needed. A Born-Mayer type ansatz proved to be quite successful
∆Vccλµ (rλµ ) = Bλµ exp(−bλµ rλµ ) . (57)
For a core-nucleus repulsion the parameters Bλµ and bλµ can be obtained directly
from the electrostatic potential of the atomic core electron system, for a core-
core repulsion the deviation from the point charge model has to be determined

16
by Hartree-Fock or Dirac-Hartree-Fock calculations for the interaction between the
frozen cores.
Relativistic pseudopotentials to be used in four-component Dirac-Hartree-Fock
calculations can also be successfully generated and used 64 , however, the advantage
of obtaining accurate results at a low computational cost is certainly lost within
this scheme.

5 Adjustment of pseudopotentials

5.1 Shape-consistent pseudopotentials

The origin of shape-consistent pseudopotentials 65,66 lies in the insight that the ad-
mixture of only core orbitals to valence orbitals in order to remove the radial nodes
leads to too contracted pseudo valence orbitals and finally as a consequence to poor
molecular results, e.g., to too short bond distances. It has been recognized about
20 years ago that it is indispensable to have the same shape of the pseudo valence
orbital and the original valence orbital in the spatial valence region, where chemical
bonding occurs. Formally this requires also an admixture of virtual orbitals in Eq.
37. Since these are usually not obtained in finite difference atomic calculations,
another approach was developed. Starting point is an atomic all-electron calcu-
lation at the nonrelativistic, scalar-relativistic or quasirelativistic Hartree-Fock or
the Dirac-Hartree-Fock level. In the latter case the small components are discarded
and the large components of the energetically lowest valence shell of each quantum
number lj are considered as valence orbitals after renormalization. To generate the
pseudo valence orbitals ϕp,lj the original valence orbitals ϕv,lj are kept unchanged
outside a certain matching radius rc separating the spatial core and valence regions
(shape-consistency; exactly achieved only for the reference state), whereas inside
the matching radius the nodal structure is discarded and replaced by a smooth and
in the interval [0,rc ] nodeless polynomial expansion flj (r):

ϕv,lj (r) for r ≥ rc
ϕv,lj (r) → ϕp,lj (r) = (58)
flj (r) for rhrc

The free parameters in flj are determined by normalization and continuity condi-
tions, e.g., matching of flj and ϕv,lj as well as their derivatives at rc . The choice of
rc as well as the choice of flj is in certain limits arbitrary and a matter of experience.
Having a nodeless and smooth pseudo valence orbital ϕp,lj and the correspond-
ing orbital energy v,lj at hand, the corresponding radial Fock equation

1 d2 l(l + 1)
(− + + VljP P (r) + Wp,lj [{ϕp0 ,l0 j 0 }])ϕp,lj (r) = v,lj ϕp,lj (r) (59)
2 dr2 2r2
can be solved pointwise for the unknown pseudopotential VljP P for each combination
lj of interest. In Eq. 59 the term Wp,lj stands for an effective valence Coulomb
and exchange potential for ϕp,lj . Relativistic effects enter the potentials implicitly
via the value of the orbital energy v,lj and the shape of the pseudo valence orbital
outside the matching radius. The resulting potentials VljP P are tabulated on a grid

17
and are usually fitted to a linear combination of Gaussian functions according to
Q XX 2
V PP = − + ( Alj,k rnlj,k −2 e−αlj,k r )Plj . (60)
r
lj k

Shape-consistent pseudopotentials including spin-orbit operators based on Dirac-

Hartree-Fock calculations using the Dirac-Coulomb Hamiltonian have been gen-
erated by Christiansen, Ermler and coworkers 67,68,69,70,71,72,73,74,75,76 . The po-
tentials and corresponding valence basis sets are available on the internet under
http://www.clarkson.edu/˜pac/reps.html . A similar set for main group and
transition elements based on scalar-relativistic Cowan-Griffin all-electron calcula-
tions was published by Hay and Wadt 77,78,79,80,81 . Another almost complete set
of pseudopotentials has been published by Stevens and coworkers 82,83,84 .

5.2 Energy-consistent pseudopotentials

Energy-consistent ab initio pseudopotentials developed from energy-adjusted
semiempirical pseudopotentials, i.e., potentials which were fitted to reproduce the
experimental atomic spectrum. Due to the problems to account accurately for
valence correlation effects, such semiempirical energy-adjustment could only be
performed successfully for one-valence electron systems. The results for alkaline
and alkaline-earth systems were quite good, however, due to the limited validity
of the frozen-core approximation when going from a highly charged one-valence
electron ion to a neutral atom or nearly neutral ion, it essentially failed for other
elements, especially transition metals. However, the idea to fit exclusively to quan-
tum mechanical observables like total valence energies (note that these may be
written as sums of ionization potentials and excitation energies) instead of to rely
on quantities like orbitals and orbital energies only meaningful in an approximate
one-particle picture is very appealing and the approach regained attention in the
ab initio framework 85 .
In the most recent version of the energy-consistent pseudopotential approach
the reference data is derived from finite-difference all-electron multi-configuration
Dirac-Hartree-Fock calculations based on the Dirac-Coulomb or Dirac-Coulomb-
Breit Hamiltonian 86 . These calculations are performed for a multitude of electronic
configurations/states/levels I of the neutral atom and the low-charged ions. The
total valence energies EIAE derived from these calculations define the pseudopoten-
tial parameters for a given ansatz in a least-squares sense. A corresponding set
of finite-difference valence-only calculations (especially the same coupling scheme
and correlation treatment has to be applied) is performed to generate the total
valence energies EIP P , and the parameters are varied in such a way that the sum
of weighted squared errors in the total valence energies becomes a minimum, i.e.,
X 2
(wI [EIP P − EIAE ] ) := min . (61)
I

In principle this formalism can be used to generate one-, two- and also four-
component pseudopotentials at any desired level of relativity (nonrelativistic
Schrödinger, or relativistic Wood-Boring, Douglas-Kroll-Hess, Dirac-Coulomb or

18
Dirac-Coulomb-Breit Hamiltonian; implicit or explicit treatment of relativity in
the valence shell) and electron correlation (single- or multi-configurational wave-
functions, e.g., the use of an intermediate coupling scheme is possible). The pseudo
valence orbitals usually agree very well with the all-electron orbitals in the va-
lence region, cf., e.g., Fig. 6. Parameters of energy-consistent ab initio pseudopo-

0.9
AE 5s1/2
0.7 AE 5p1/2
AE 5p3/2
0.5 PP 5s1/2
PP 5p1/2
PP 5p3/2
P(a.u.)

0.3

0.1

−0.1

−0.3

−0.5
0 1 2 3 4 5 6 7 8
r(a.u.)

Figure 6. Valence spinors of the iodine atom in the [46 Pd] 5s2 5p5 ground state configuration
from average-level all-electron (AE) multiconfiguration Dirac-Hartree-Fock calculations and cor-
responding valence-only calculations using a relativistic energy-consistent 7-valence-electron pseu-
dopotential (PP).

tentials and corresponding valence basis sets have been presented for almost all
elements of the periodic table by Dolg, Preuss, Schwerdtfeger, Stoll and cowork-
ers 85,86,87,88,89,90,91,92,93,94,95,96,97 . They are also available on the internet under
http://www.theochem.uni-stuttgart.de . Since the functional form of energy-
consistent pseudopotentials is identical to the one of shape-consistent pseudopoten-
tials, both types of pseudopotentials can be used in standard quantum chemical pro-
gram packages (COLUMBUS, GAUSSIAN, GAMESS, MOLPRO, TURBOMOLE,
...) as well as polymer or solid state codes using Gaussian basis sets (CRYSTAL,
WANNIER, ...).

5.3 Limitations of accuracy

Effective core potentials are usually derived for atomic systems at the finite dif-
ference level and used in subsequent molecular calculations using finite basis sets.
They are designed to model the more accurate all-electron calculations at low cost,
but without significant loss of accuracy. Unfortunately the correct relativistic all-
electron Hamiltonian for a many-electron system is not known and the various
pseudopotentials merely model the existing approximate formulations. For most
cases of chemical interest, e.g., geometries and binding energies, it usually does not

19
matter which particular Hamiltonian model is used, i.e., typically errors due to the
finite basis set expansion or the limited correlation treatment are much larger than
the small differences between the various all-electron models. For very accurate cal-

Table 1. Relative average energy of a configuration of Hg from all-electron (AE) multi-

configuration Dirac-Hartree-Fock (DHF) average level calculations using the Dirac-Coulomb (DC)
Hamiltonian with a finite nucleus with Fermi charge distribution (fn) or a point nucleus (pn).
Contributions from the frequency-dependent Breit (B) interaction (frequency of the exchanged
photon 103 cm−1 ) and estimated contributions from quantum electrodynamics (QED, i.e., self-
interaction and vacuum polarization) were evaluated in first-order perturbation theory. Errors
of energy-consistent pseudopotentials (PP) with 20 valence electrons and different numbers of
adjustable parameters with respect to the AE DHF(DC,pn)+B+QED data. All values in cm −1 .

configuration AE, DHF contribution error

(DC)+B+QED
fn pn B QED PPa PPb
2
Hg 6s 0 0 0.0 0.0 0.0 0.0
6s1 6p1 35632.3 35674.4 -52.5 -18.7 1.3 0.0
Hg+ 6s1 68842.1 68885.1 -98.6 -11.6 -0.1 0.0
7s1 154127.4 154206.2 -220.6 -42.4 -0.4 0.0
8s1 178127.5 178215.5 -238.4 -41.7 1.1 0.1
9s1 188751.0 188843.2 -244.1 -40.6 1.6 -0.1
6p1 122036.8 122128.9 -154.2 -41.8 0.6 0.0
7p1 167514.3 167609.2 -224.1 -40.3 -3.3 0.0
8p1 183808.0 183903.6 -238.5 -40.0 -0.8 0.0
9p1 191697.2 191793.1 -244.0 -39.6 0.6 0.0
Hg++ 206962.2 207058.4 -249.8 -39.5 2.6 0.0
a energy-consistent pseudopotential with 26 adjustable parameters.
b energy-consistent pseudopotential with 54 adjustable parameters.

culations of excitation energies, ionization potentials and electron affinities, or for

a detailed investigation of errors inherent in the effective core potential approach,
however, such differences might become important. Tables 1 and 2 demonstrate
that for very special cases like Hg, with a closed 5d10 -shell in all electronic states
considered, a small-core energy-consistent pseudopotential using a semilocal ansatz
reaches an accuracy of 10 cm−1 , which is well below the effects of the nuclear model,
the Breit interaction or higher-order quantum electrodynamical contributions. We
also note that differences between results obtained with a frequency-dependent
Breit term and the corresponding low-frequency limit amount to up to 10 cm−1 .
Moreover, the quantum electrodynamic corrections listed in tables 1 and 2 might
change by up to 20 cm−1 when more recent methods of their estimation are applied
98,99
. Therefore, it is important to state exactly which relativistic all-electron model
the effective core potential simulates and, when comparing effective core potentials
of different origins, to separate differences in the underlying all-electron approach
from errors in the potential itself, e.g., due to the size of the core, the method of
adjustment or the form of the valence model Hamiltonian. In this context we want
to point out that the seemingly large errors for energy-adjusted pseudopotentials

20
reported by Mosyagin et al. 100,101 are mainly due to the invalid comparison of
Wood-Boring-energy-adjusted and Dirac-Fock-orbital-adjusted pseudopotentials to
all-electron Dirac-Fock data, i.e., differences in the all-electron model are considered
to be pseudopotential errors.
Note that in the above example of Hg the average energy of a configuration (table
1) and the fine-structure (table 2) of one-valence electron states is more accurately
represented than the fine-structure of the 6s1 6p1 configuration. The small errors
in the latter case are a consequence of the pseudoorbital transformation and the
overestimation of the 6s-6p exchange integral with pseudo-valence spinors. This
error could be reduced further upon using a smaller core, but the efficiency of
the approach would be sacrificed. It is also obvious from the compiled data that
the accuracy of the valence model Hamiltonian is also a question of the number
of adjustable parameters. Claims that such very high accuracy as demonstrated
here can only be achieved by adding nonlocal terms for outer core orbitals to the
usual semilocal terms 100,101 appear to be invalid, at least for energy-consistent
pseudopotentials. Moreover, additional nonlocal terms obviously do not improve
the performance for atomic states with a 5d9 occupation or in molecular calculations
(cf., e.g., tables III and XVII in Mosyagin et al. 100 ).

Table 2. As table 1, but for fine-structure splittings. All values in cm−1 .

configuration splitting AE,DHF contribution error

(DC)+B+QED
fn pn B QED PPa PPb
1 1 3 3
Hg 6s 6p P1 - P0 1987.7 1988.6 -25.5 0.9 -14.7 3.0
3
P2 -3 P0 6082.6 6084.8 -96.8 2.9 -28.3 -3.5
1
P1 -3 P0 22994.4 22982.3 -72.4 2.2 -12.4 -9.4
Hg+ 6p1 2
P3/2 -2 P1/2 7765.3 7768.8 -132.8 4.8 -14.8 -0.1
7p1 2
P3/2 -2 P1/2 2136.8 2137.9 -29.0 1.1 -1.7 0.2
8p1 2
P3/2 -2 P1/2 939.4 939.9 -12.1 0.4 -4.6 -0.3
9p1 2
P3/2 -2 P1/2 498.7 498.9 -6.2 0.2 -3.5 0.0
a energy-consistent pseudopotential with 26 adjustable parameters.
b energy-consistent pseudopotential with 54 adjustable parameters.

6 Core Polarization Potentials

The frozen-core approximation is underlying the effective core potential schemes.

One may ask if it is possible to account for static (polarization of the core at the
Hartree-Fock level) and dynamic (core-valence correlation) polarization of the cores
in a both efficient and accurate way. The core polarization potential (CPP) ap-
proach originally developed by Meyer and coworkers 102 for all-electron calculations
and adapted by the Stuttgart group 103 for pseudopotential calculations proved to
be quite successful in the past. The core polarization potential is written in the

21
following form
1X ~2 X λ
Vcpp = − αλ f λ + V (i) . (62)
2
λ λ,i

Here αλ denotes the dipole polarizability of the core λ and f~λ is the electric field at
core λ generated by all other cores and nuclei as well as all valence electrons. Since
the validity of the underlying multipole expansion breaks down for small distances
from the core λ, the field has to be multiplied by a cut-off function:
X ~riλ ~rµλ
f~λ = −
X
3 (1 − exp(−δeλ riλ
2
))ne + Qµ 3 (1 − exp(−δcλ rµλ
2
))nc . (63)
i
r iλ r µλ
µ6=λ

The necessary integrals over cartesian Gaussian functions have been presented by
Schwerdtfeger and Silberbach 104 . In those cases where ns and np valence orbitals
are present together with (n-1)d and (n-2)f valence orbitals, e.g., for Cs, it proved
to be more accurate to augment the core polarization potential by a short-range
local potential 64
V λ (i) = C λ exp(−γ λ riλ
2
). (64)
105
A l-dependent cut-off function in Eq. 63 might even be more accurate .

7 Calibration Studies

Calibration studies, especially on molecules, are very important for effective core
potential methods. Excellent results in atomic calculations are a necessary pre-
requisite for successful molecular calculations, but provide no guarantee for them.
Therefore, effective core potentials should be systematically tested on atoms and
small molecules before using them in larger systems. This is especially necessary
for cases where a large core is used for economical reasons.
A number of such molecular calibration studies has been performed in the past
for energy-consistent pseudopotentials 106,107,108,109,110,111 . Comparison is made to
experimental data and/or all-electron results. Some care has to be taken before
drawing final conclusions on the quality of pseudopotentials. Usually all molecular
calculations are performed using finite basis sets, both at the one-electron and the
many-electron level. The truncation of these basis sets leads to errors both at the
all-electron and at the pseudopotential level. Most of the time it is relatively easy to
generate basis sets of nearly the same quality at the one-particle level, e.g., by aug-
menting the standard all-electron and pseudopotential basis sets (which of course
have to be of the same quality for the valence shells) by the same polarization and
correlation functions. It is recommended, however, to approach the basis set limit,
at least up to a given angular quantum number, as closely as possible. At the many-
particle level it is sometimes more difficult to come to directly comparable basis
sets. As an example imagine a large-core pseudopotential augmented by a core po-
larization potential, which both accounts for static and dynamic core polarization.
Static core polarization occurs in the all-electron calculations automatically at the
self-consistent field level and can be accounted for in the frozen-core case by single
excitations out of the spherical core. Comparing the results of such all-electron

22
 2.85

SCF
2.80 CCSD(T)
CCSD(T)+SO
bond length (Å) Exp.
2.75

2.70

2.65

2.60
sp spd spdf spdfg
basis set

Figure 7. Bond length of the iodine dimer I2 depending on the basis set. A subset of a
25s21p14d4f3g basis set and a relativistic energy-consistent 7-valence electron pseudopotential
augmented by a core polarization potential is used. The experimental value is indicated by a
vertical dashed line. The scalar-quasirelativistic pseudopotential calculations at the Hartree-Fock
self-consistent field (SCF) and coupled-cluster with single, double and perturbative triple excita-
tions (CCSD(T)) level of theory use an uncontracted (25s21p14d4f3g) Gaussian type basis set.
Spin-orbit corrections (+SO) were derived from limited two-component configuration interaction
calculations using the quasirelativistic pseudopotential and a contracted [3s3p1d1f] valence basis
set of polarized triple-zeta quality.

calculations to pseudopotential Hartree-Fock calculations including the core polar-

ization potential is not entirely correct, however, since the latter also accounts for
core-valence correlation. This effect can be modelled in all-electron calculations by
single excitations out of the spherical core and simultaneous single excitations in
the valence shell. Thus, comparing the results of an all-electron calculation where
all electrons are correlated to a correlated pseudopotential calculation with a core
polarization potential is also not entirely correct, since the former calculation also
accounts for core-core correlation effects. In addition, attention has to be paid with
respect to the relativistic contributions taken into account in the Hamiltonian, i.e.,
the relativistic scheme used in the all-electron reference calculations should not be
different, e.g., more approximate, from the scheme used to obtain the reference
data in the pseudopotential generation. As an example for a calibration study the
results for the iodine dimer I2 in its ground state are compared to Hartree-Fock
and coupled-cluster calculations using a relativistic energy-consistent 7-valence elec-
tron pseudopotential together with an uncontracted (25s21p14d4f3g) basis set. The
pseudopotential was augmented by a core polarization potential. The results for
the bond length, binding energy and vibrational frequency in dependence on the
highest angular quantum number used in the basis set are given in Figs. 7 to 9. It
is clear from these graphs that the experimental values are only approached for a
large basis set and after inclusion of spin-orbit effects. It should be mentioned here
that the core polarization potential also makes significant contributions, e.g., -0.03

23
 2.25
2.00
1.75
binding energy (eV) 1.50
1.25
1.00
0.75
SCF
0.50 CCSD(T)
CCSD(T)+SO
0.25 Exp.
0.00
sp spd spdf spdfg
basis set

Figure 8. As figure 7, but for the binding energy.

250
240
vibrational constant (cm )
−1

230
220
210
200 SCF
CCSD(T)
190 CCSD(T)+SO
Exp.
180
170
sp spd spdf spdfg
basis set

Figure 9. As figure 7, but for the vibrational constant.

Å, +0.09 eV and +3 cm−1 for the bond length, binding energy and vibrational con-
stant, respectively, at the Hartree-Fock level. The performance of energy-consistent
quasirelativistic 7-valence electron pseudopotentials for all halogen elements has
been investigated in a study of the monohydrides and homonuclear dimers 112 .
Special attention was also paid to the accuracy of valence correlation energies ob-
tained with pseudo valence orbitals 113,114 . Some of the results for the dimers is
presented in tables 3 to 5. The applied basis sets were uncontracted all-electron
basis sets: (15s9p5d4f3g) for fluorine, (21s13p5d4f3g) for chlorine, (22s17p11d4f3g)

24
Table 3. Bond lengths Re (Å) of the homonuclear halogen dimers from all-electron (AE) Douglas-
Kroll-Heß (DKH) and valence-only pseudopotential (PP) Hartree-Fock self-consistent field (SCF)
calculations. Core-valence correlation and valence correlation are accounted for by a core polar-
ization potential (CPP) and a coupled-cluster treatment with singles, doubles and perturbative
triples (CCSD(T)) including spin-orbit corrections (+SO). For the basis sets cf. the text.

F2 Cl2 Br2 I2 At2

AE DKH,SCF 1.327 1.975 2.273 2.671 2.843
PP,SCF 1.324 1.964 2.266 2.669 2.861
PP+CPP,SCF 1.323 1.958 2.252 2.639 2.822
PP+CPP,CCSD(T)+SO 1.409 1.982 2.281 2.668 2.979
Exp. 1.412 1.988 2.281 2.666

Table 4. As table 3, but for vibrational constants ωe (cm−1 ).

F2 Cl2 Br2 I2 At2

AE DKH,SCF 1267a 615 354 232 169
PP,SCF 1271 619 356 238 168
PP+CPP,SCF 1273 622 359 241 172
PP+CPP,CCSD(T)+SO 927 561 324 215 117
Exp. 917 560 325 215
a
nonrelativistic result.

Table 5. As table 3, but for binding energies De (eV).

F2 Cl2 Br2 I2 At2

AE DKH,SCF -1.07 1.23 1.01 0.92 0.81
PP,SCF -1.03 1.26 1.04 0.95 0.79
PP+CPP,SCF -1.03 1.27 1.08 1.04 0.95
PP+CPP,CCSD(T)+SO 1.66 2.44 1.95 1.57 0.80
Exp. 1.66 2.51 1.99 1.56

for bromine, (25s20p14d4f3g) for iodine, and (27s22p19d13f3g) for astatine. At

the HF level the calibration for the scalar-quasirelativistic pseudopotentials was
against all-electron calculations using the Douglas-Kroll-Heß Hamiltonian, whereas
at the CCSD(T) level including spin-orbit corrections from limited two-component
CI calculations the calibration was with respect to experimental data.
A typical example demonstrating that pseudopotentials account for the major
relativistic effects quite accurately and the largest errors in practical calculations are
actually due to finite basis sets and too limited correlation treatments is provided
by a series of theoretical investigations of gold monofluoride AuF (cf. table 6). All
calculations used the same scalar-relativistic energy-consistent 19-valence-electron
pseudopotential for Au 115 , but the quality of the valence-only calculations was
systematically increased during the years. Originally it was believed that AuF is
not a stable molecule, until its existence was first predicted theoretically 115,116

25
Table 6. Bond length Re (Å), binding energy De (eV) and vibrational constant ωe (cm−1 ) of gold
monofluoride AuF in the 1 Σ+ ground state. All theoretical results have been obtained with a 19-
valence-electron energy-consistent pseudopotential adjusted to multi-configuration Dirac-Hartree-
Fock reference data 115 using different basis sets and valence correlation methods.

method basis year Re De ωe

CISD+SCC(LD)115 A 1989 1.978 2.24 509
CEPA-1115 A 1989 1.991 2.52 488
Exp.117 1992 560
QCISD(T)116 B 1994 1.939 3.09 539
Exp.118 1994 3.2
CCSD(T)119 C 1997 1.909 3.29 573
MRCI+SCC(S)119 C 1997 1.916 3.14 562
MRACPF119 C 1997 1.916 3.20 560
Basis sets: A: Au (8s6p5d1f)/[7s3p4d1f], F (13s8p1d)/[7s3p1d];
B: Au (10s8p7d1f)/[9s5p6d1f], F (15s10p2d1f)/[9s7p2d1f];
C: Au (10s8p7d4f2g)/[9s5p6d4f2g], F (13s7p4d3f2g)/[6s5p4d3f2g] (aug-cc-p-vqz).

and later also proven experimentally 117,118 . The most recent calculations 119
are
in excellent agreement with the available experimental data.

8 A few hints for practical calculations

Some of the simple hints for practical applications of effective core potentials given
in the following may appear to be trivial or superfluous for some of the readers,
but experience during the last years showed that they may be welcome by the more
application-oriented ones who are less familiar with the methods.
Effective core potentials are usually a good and safe choice when properties re-
lated to the valence electron system are to be investigated. It should always be
remembered, however, that the size of the core not only determines the compu-
tational effort, but it also influences the accuracy of the results. Small-core and
medium-core potentials are usually safe to use, whereas the range of large-core po-
tentials is much more limited. In the latter case it might be important to include
a core-core and/or core-nucleus repulsion correction as well as a core polarization
potential. It is not a wise decision to simply neglect these terms, e.g., because the
CPP is not implemented in GAUSSIAN yet.
When using an effective core potential for the first time always do an atomic
test calculation first, e.g., for the ionization potential or electron affinity, in order to
check the correctness of your input and/or the programs library data. Especially in
pseudopotential calculations well-known sources of input errors are the 1/r n prefac-
tors used in some parametrizations or the presence/absence of a local potential. It
is recommended to use the valence basis set coming with the effective core potential,
possibly augmented by additional diffuse and polarizations functions. Especially in
case of pseudopotentials, where the detailed innermost shape of the pseudoorbitals
is essentially arbitrary, it is not recommended to use (contracted) all-electron basis

26
sets or valence basis sets from other potentials, since significant basis set superposi-
tion errors may result. However, the added diffuse and polarization functions may
safely be taken from all-electron or other effective core potential basis sets.
When comparing to other all-electron or valence-only calculations use basis
sets and correlation treatments of the same quality and make sure that relativistic
effects are included at similar levels. Note that in all-electron calculations basis set
superposition errors tend to be larger than in valence-only calculations.

Acknowledgments

The author is grateful to H. Stoll (Stuttgart) for more than 15 years of cooperation
on the field of pseudopotentials. Financial support of the Deutsche Forschungsge-
meinschaft and the Fonds der Chemischen Industrie is also acknowledged.

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