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Journal Article•DOI•

Geometry optimizations in the zero order regular approximation for relativistic effects.

Erik van Lenthe, Andreas W. Ehlers, Evert Jan Baerends
29 Apr 1999-Journal of Chemical Physics (American Institute of Physics)-Vol. 110, Iss: 18, pp 8943-8953
TL;DR: In this paper, the energy gradients in the zeroth order regular approximation (ZORA) to the Dirac equation were derived for the transition metal complexes W(CO), Os(CO)5, and Pt (CO)4.
Abstract: Analytical expressions are derived for the evaluation of energy gradients in the zeroth order regular approximation (ZORA) to the Dirac equation. The electrostatic shift approximation is used to avoid gauge dependence problems. Comparison is made to the quasirelativistic Pauli method, the limitations of which are highlighted. The structures and first metal-carbonyl bond dissociation energies for the transition metal complexes W(CO)6, Os(CO)5, and Pt(CO)4 are calculated, and basis set effects are investigated.

Summary (2 min read)

Jump to: [Introduction] – [III. GEOMETRY OPTIMIZATIONS WITH ZORA] – [A. Derivation of energy gradients for the ZORA ESA energy] – [IV. BASIS SET REQUIREMENTS IN THE FROZEN CORE APPROXIMATION] – [V. THE PAULI HAMILTONIAN AND THE QUASIRELATIVISTIC METHOD: FROZEN CORES] – [VI. RESULTS AND DISCUSSION] and [VII. CONCLUSION]

Introduction

  • VU Research Portal Geometry optimizations in the zero order regular approximation for relativistic effects.
  • Bond energies can be calculated accurately with the ZORA method using the electrostatic shift approximation ~ESA!, described in Ref. 6.

III. GEOMETRY OPTIMIZATIONS WITH ZORA

  • In this section expressions are derived for the evaluation of energy gradients in the ~SR!.
  • Next, the implementation in the ADF program system is briefly discussed.

A. Derivation of energy gradients for the ZORA ESA energy

  • The difference in energy between a molecule and its constituting atoms ~fragments!.
  • Note the occurrence of the same operator T@V# , containing the molecular potential V in both the molecular and atomic ‘‘kinetic energy’’ terms.
  • Ense or copyright; see http://jcp.aip.org/about/rights_and_permissions method the model potential of atom A is usually not so large at distances between A and B which are in the order of ~or larger than!.
  • The authors implementation of the analytical gradients for ZORA is based on a modification of the implementation of energy gradients in the nonrelativistic16,17 and quasirelativistic case7 in the ADF program.

IV. BASIS SET REQUIREMENTS IN THE FROZEN CORE APPROXIMATION

  • In the ADF program suite the frozen core approximation is used routinely and can also be applied in the SR ZORA method ~not yet with ZORA including spin–orbit coupling!.
  • The set of basis functions xm is now transformed into a set of coreorthogonal functions x̄m by forming a linear combination of each xm with a set of core orthogonalization functions xk core , one for each core orbital, x̄m5xm1 ( k51 M xk coreCkm .
  • In fact, the exponents of the core orthogonalization functions are optimized, along with those of the valence basis functions, in atomic calculations, in such a way that the ‘‘valence’’ plus ‘‘core’’ sets give an optimal description of the valence atomic orbitals, including their core tails.
  • If the authors add a 1s-type STO with z5115 they see incipient variational collapse of the valence s orbital energies towards ~although not yet anywhere near!.
  • The authors can perform the ZORA calculation with basis functions orthogonalized on SR ZORA core orbitals and investigate how large the error becomes.

V. THE PAULI HAMILTONIAN AND THE QUASIRELATIVISTIC METHOD: FROZEN CORES

  • The Pauli Hamiltonian in general poses no problems for bound electrons if one uses it in a first order perturbation to 130.37.129.78.
  • Snijders and Baerends24 proposed a method for the calculation of relativistic effects in a perturbative procedure, where also first order effects in the change of the density are taken into account.
  • The nonrelativistic basis sets use one core orthogonalization function per core orbital to orthogonalize on the accurately calculated nonrelativistic core orbitals.
  • With good valence basis sets, the nonrelativistic solutions will be accurately described and a good performance of the PAULI FOPT as well as QR Pauli is expected.
  • In the QR calculation, however, the 6s orbital energy now becomes 2485 a.u., showing the drastic effect of variational collapse.

VI. RESULTS AND DISCUSSION

  • In this section the authors test their implementation of the calculation of the analytical energy gradients for SR ZORA on some small molecules using density functional theory.
  • The remaining differences in the results of the two methods can be explained partly by the use of different basis sets, but the authors think that also a part of the deviations is due to the gauge dependence problems of the ZORA ~MP!.
  • In Ref. 9 the molecular model potential is constructed in such a way that distant atoms do not contribute, but in the AuH molecule the H atom is still so close to the Au atom, that a non-negligible contribution remains, which is larger if the atoms are closer.
  • Using the standard ADF basis sets IV for the atoms, very good results were obtained for the optimized bond lengths, if the results are compared with the SR ZORA ESA results ~deviations less than 0.01 Å!. FBDE’s appear to be overestimated by 3–5 kcal/mol ~10– 30 %!.
  • On the other hand, as the authors have seen in Table V, in the SR ZORA ESA larger basis sets are not a problem, and one can study the convergence of the results to the basis set limit, and it is possible to test the frozen core approximation using all electron basis sets.

VII. CONCLUSION

  • Expressions have been derived for the evaluation of energy gradients in the ZORA ESA method and were imple- ense or copyright; see http://jcp.aip.org/about/rights_and_permissions.
  • It was shown that these analytical expressions are easier to evaluate than the expressions following from the recently developed ZORA ~MP!.
  • Method based on the Pauli Hamiltonian previously implemented in the ADF program, in the SR ZORA ESA method it is possible to study the convergence of the optimized bond lengths and bond energies with respect to the basis set limit, and one can test the frozen core approximation using all electron basis sets.

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Geometry optimizations in the zero order regular approximation for relativistic effects.
van Lenthe, E.; Ehlers, A.W.; Baerends, E.J.
published in
Journal of Chemical Physics
1999
DOI (link to publisher)
10.1063/1.478813
document version
Publisher's PDF, also known as Version of record
Link to publication in VU Research Portal
citation for published version (APA)
van Lenthe, E., Ehlers, A. W., & Baerends, E. J. (1999). Geometry optimizations in the zero order regular
approximation for relativistic effects. Journal of Chemical Physics, 110, 8943-8953.
https://doi.org/10.1063/1.478813
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Download date: 10. Aug. 2022
Geometry optimizations in the zero order regular approximation
for relativistic effects
Erik van Lenthe, Andreas Ehlers, and Evert-Jan Baerends
Afdeling Theoretisch Chemie, Scheikundig Laboratorium der Vrije Universiteit, De Boelelaan 1083,
1081 HV Amsterdam, The Netherlands
~Received 19 October 1998; accepted 2 February 1999!
Analytical expressions are derived for the evaluation of energy gradients in the zeroth order regular
approximation ~ZORA! to the Dirac equation. The electrostatic shift approximation is used to avoid
gauge dependence problems. Comparison is made to the quasirelativistic Pauli method, the
limitations of which are highlighted. The structures and first metal-carbonyl bond dissociation
energies for the transition metal complexes W~CO!
6
,Os~CO!
5
, and Pt~CO!
4
are calculated, and
basis set effects are investigated. © 1999 American Institute of Physics. @S0021-9606~99!30317-2#
I. INTRODUCTION
In the present paper expressions are derived for the
evaluation of energy gradients of the zeroth order regular
approximation ~ZORA!~Refs. 1–3! to the Dirac equation.
The regular expansion, which leads to the ZORA Hamil-
tonian, remains valid even for a Coulombic potential. This is
in contrast to the expansion that leads to the Pauli Hamil-
tonian, which is divergent for a Coulombic potential.
Harriman
4
already used the regular expansion, but called it
the modified partitioning of the Dirac equation. It was shown
in Ref. 5, that the ZORA Hamiltonian is bounded from be-
low for Coulombic potentials. Exact solutions for the hydro-
genic ions were given and in Ref. 6 it was shown that the
scaled ZORA energies in that case are exactly equal to the
Dirac energies.
Bond energies can be calculated accurately with the
ZORA method using the electrostatic shift approximation
~ESA!, described in Ref. 6. With this method geometry op-
timizations can be performed if bond energies for different
geometries are compared. For diatomics this pointwise trac-
ing of the energy surface is still manageable, but for poly-
atomic atoms it will be cumbersome. Therefore it is desirable
to have analytic expressions for the energy gradients. We
present in Sec. III of this paper the derivation of analytic
energy derivatives within the framework of the ZORA ESA
method. Section IV discusses the use of a frozen core and
basis set requirements for ZORA calculations.
In Sec. VI results of geometry optimizations are pre-
sented for a series of small molecules ~diatomics! employing
the scalar relativistic ~SR! ZORA method, i.e., without spin–
orbit coupling. The results are compared with results ob-
tained from a pointwise calculation of bond energies in the
SR ZORA ESA method. The SR ZORA optimized geom-
etries have also been obtained for W~CO!
6
,Os~CO!
5
, and
Pt~CO!
4
and are compared with geometries obtained with a
quasirelativistic method based on the Pauli Hamiltonian for
the same compounds, both calculated in this work with vari-
ous basis sets and published ones.
7
It is well known that the
Pauli Hamiltonian containing the first order relativistic cor-
rection terms ~Darwin, mass–velocity, and spin–orbit cou-
pling! is not bounded from below. One may nevertheless try
to diagonalize the Pauli Hamiltonian in a restricted ~valence!
space. This is usually denoted as the quasirelativistic
method.
8
In order to avoid variational collapse in the QR-
Pauli method, frozen cores have to be employed. Before en-
tering the comparison with the present results and following
up on the discussion of the use of frozen cores in the ZORA
method, we discuss in Sec. V the stability problems of the
quasirelativistic Pauli method in relation to the choice of
both core orthogonalization functions and valence basis sets.
Recently, van Wu
¨
llen
9
proposed a modification of the
ZORA method, which uses a model potential in the ZORA
kinetic energy operator. For this method, called ZORA ~MP!,
he derived analytical expressions for the energy gradients.
The purpose of the ZORA~MP! method was to eliminate the
gauge dependence of the ZORA approach. However, we will
show that a ~small! gauge dependence problem still exists in
this ZORA ~MP! method, which is not present in the ZORA
ESA method. Moreover, we will show that the analytical
expressions for the energy gradients following from the
ZORA ESA method are easier to evaluate than the expres-
sions following from the ZORA ~MP! method.
A different variationally stable relativistic method devel-
oped for atomic and molecular calculations by Hess
10
uses
the Douglas–Kroll transformation.
11
A density-functional
implementation has been provided by Knappe and Ro
¨
sch,
12
with the implementation of analytical energy gradients by
Nasluzov and Ro
¨
sch.
13
These schemes rely on momentum
space evaluation of integrals and require the assumption of
completeness of the finite basis sets employed in practical
calculations. It is an advantage of the ZORA approach that
the required matrix elements can easily be evaluated without
further approximations in schemes that rely on 3D numerical
integration, see, e.g., Refs. 14 and 15, making this method
very straightforwardly applicable to molecules.
Our implementation of the analytical gradients for
ZORA is based on a modification of the implementation of
energy gradients in the nonrelativistic
16,17
and the quasirela-
tivistic case
7
in the Amsterdam density functional ~ADF!
program.
18,15,19
JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 18 8 MAY 1999
89430021-9606/99/110(18)/8943/11/$15.00 © 1999 American Institute of Physics
Downloaded 13 Mar 2011 to 130.37.129.78. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
In our calculations we will use density functional theory
~DFT!, employing the usual ~nonrelativistic! density func-
tionals for the exchange-correlation energy; local density
functionals ~LDA! with gradient correction ~GGC! terms
added, namely, the Becke correction for exchange
20
and the
Perdew correction for correlation.
21
II. THE ZORA EQUATION
The ZORA equation is the zeroth order of the regular
expansion of the Dirac equation. If only a time-independent
electric field is present, the one-electron ~SR! ZORA Kohn–
Sham equations can be written in atomic units (p52i“)as
~
V1T
@
V
#
!
C
i
5
e
i
C
i
, ~1!
with
T
zora
@
V
#
5
s
•p
c
2
2c
2
2 V
s
•p5 p•
c
2
2c
2
2 V
p
1
c
2
~
2c
2
2 V
!
2
s
•
~
“V3 p
!
, ~2a!
T
SR
zora
@
V
#
5 p•
c
2
2c
2
2 V
p. ~2b!
Here use is made of the identity,
~
s
•a
!
~
s
•b
!
5 a• b1 i
s
•
~
a3 b
!
~3!
for the Pauli spin matrices
s
. The effective molecular
Kohn–Sham potential V used in our calculations is the sum
of the nuclear potential, the Coulomb potential due to the
total electron density, and the exchange-correlation potential,
for which we will use nonrelativistic approximations. The
ZORA kinetic energy operator T
zora
, depends on the molecu-
lar Kohn–Sham potential. The scalar relativistic ~SR! ZORA
kinetic energy operator T
SR
zora
, is the ZORA kinetic energy
operator without spin–orbit coupling. This operator can be
used in cases where spin–orbit coupling is not important. For
convenience we will refer to the ~SR! ZORA kinetic energy
with T
@
V
#
.
In Ref. 22 it was observed that replacing the molecular
potential by the sum of the potentials of the neutral spherical
reference atoms V
SA
in the kinetic energy operator is not a
severe approximation, thus
T
@
V
SA
#
'T
@
V
#
. ~4!
This procedure was called the sum of atoms potential ap-
proximation ~SAPA!. This has the advantage that when the
ZORA Kohn–Sham equations are solved self-consistently
~SCF! using a basis set, one only needs to calculate the
ZORA kinetic energy matrix once, instead of in every cycle
in the SCF scheme if the full molecular potential is used.
An improved one-electron energy can be obtained by
using the scaled ZORA energy expression
6
e
i
scaled
5
E
i
zora
11
^
C
i
u
Q
@
V
#
u
C
i
&
, ~5!
with
Q
zora
@
V
#
5
s
•p
c
2
~
2c
2
2 V
!
2
s
•p, ~6a!
Q
SR
zora
@
V
#
5 p•
c
2
~
2c
2
2 V
!
2
p. ~6b!
If, for example, SAPA is used for T
@
V
#
then the same ap-
proximation has to be used for Q
@
V
#
.
III. GEOMETRY OPTIMIZATIONS WITH ZORA
In this section expressions are derived for the evaluation
of energy gradients in the ~SR! ZORA case. Next, the imple-
mentation in the ADF program system is briefly discussed.
A. Derivation of energy gradients for the ZORA ESA
energy
The difference in energy between a molecule and its
constituting atoms ~fragments! A, calculated according to the
~SR! ZORA ESA method,
6
is
DE
ESA
5
1
2
(
A,BÞA
N
Z
A
Z
B
u
R
A
2 R
B
u
1
(
i
occ
^
C
i
u
T
@
V
#
u
C
i
&
2
(
A
N
E
r
~
1
!
Z
A
u
R
A
2 r
1
u
d11
1
2
EE
r
~
1
!
r
~
2
!
r
12
d1d2
1 E
XC
@
r
#
2
(
A
N
S
(
j
occ
^
F
j
A
u
T
@
V
#
u
F
j
A
&
2
E
r
A
~
1
!
Z
A
u
R
A
2 r
1
u
d11
1
2
EE
r
A
~
1
!
r
A
~
2
!
r
12
d1d2
1 E
XC
@
r
A
#
D
, ~7!
with
r
5
(
i
occ
C
i
†
C
i
, ~8a!
r
A
5
(
j
occ
~
F
j
A
!
†
F
j
A
, ~8b!
C
i
is a molecular orbital, and F
j
A
is a fragment orbital. The
energy difference DE
ESA
was derived from the difference in
the scaled ~SR! ZORA total energies. Note the occurrence of
the same operator T
@
V
#
, containing the molecular potential
V in both the molecular and atomic ‘‘kinetic energy’’ terms.
This is a consequence of the combined use of the scaled
ZORA method and the ESA approximation, cf. Ref. 6, and is
crucial for avoiding gauge dependency problems as well as
obtaining numerically stable energy differences. In Ref. 6 the
scaled ZORA total energy was found to be very accurate in
comparison with fully relativistic results.
Suppose the molecular potential V present in the kinetic
energy operator T
@
V
#
does not depend on the molecular or-
bitals C
i
, as it is the case for SAPA, for example. We will
call this potential V
fix
~for SAPA V
fix
5 V
SA
!. Now finding
8944 J. Chem. Phys., Vol. 110, No. 18, 8 May 1999 van Lenthe, Ehlers, and Baerends
Downloaded 13 Mar 2011 to 130.37.129.78. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
the orthonormal orbitals C
i
which minimize the energy dif-
ference DE
ESA
is equivalent to solving the molecular one
electron ZORA equations,
@
V1 T
@
V
fix
##
C
i
5
e
i
C
i
. ~9!
An alternative is to first solve the one-electron ~SR! ZORA
equation ~1! self-consistently with a potential V in the kinetic
energy operator that does depend on the orbitals C
i
. After-
wards one can then fix this potential, and use this fixed po-
tential V
fix
in the kinetic energy operator in Eq. ~7!. One can
then vary the orbitals C
i
in Eq. ~7! to find the minimum, thus
without changing V
fix
in the kinetic energy operator, which is
equivalent to finding the solutions of the ZORA one-electron
equations that were already solved. In this sense the ZORA
ESA energy is stationary with respect to orbital variations.
The potential V
fix
, however, still depends on the geometry of
the molecule, which is important in the case of geometry
optimizations.
In an atomic basis set expansion the ZORA molecular
orbitals C
i
are expressed as a sum over coefficients times
primitive atomic basis functions
x
n
, each centered at one
particular nucleus,
C
i
5
(
n
C
n
i
x
n
. ~10!
If we take the derivative of the energy difference Eq. ~7! with
respect to a nuclear displacement X
A
of nucleus A, we have
to take into account the change in the coefficients C
n
i
~indi-
rect derivative! as well as the change in the atomic basis
functions
x
n
themselves ~direct derivative!, due to the dis-
placement. We will now assume that we have solved the
one-electron ~SR! ZORA equation ~9! with optimal coeffi-
cients C
n
i
. As in the nonrelativistic case the indirect deriva-
tive can be transformed into a direct derivative
23
(
i
occ
(
n
]
DE
ESA
C
n
i
]
C
n
i
]
X
A
52
(
i
occ
2
e
i
K
]
C
i
]
X
A
U
C
i
L
, ~11!
where
]
C
i
/
]
X
A
represents the direct derivative of C
i
with
respect to X
A
, thus
]
C
i
]
X
A
5
(
n
C
n
i
]x
n
]
X
A
. ~12!
The kinetic energy operator in Eq. ~7! is the same for
both the molecule and the constituting atoms ~fragments!,
and contains the molecular potential. This is the only differ-
ence with a similar expression in the nonrelativistic case and
it is important in the case of geometry optimizations, which
we will now consider.
The difference in the kinetic energy between a molecule
and its constituting atoms ~fragments! A, calculated accord-
ing to the ~SR! ZORA ESA method is
DT
ESA
@
V
#
5
(
i
occ
^
C
i
u
T
@
V
#
u
C
i
&
2
(
A
N
(
j
occ
^
F
j
A
u
T
@
V
#
u
F
j
A
&
,
~13!
with F
j
A
the fragment orbitals. For deep core states the
(
A
(
j
F
j
A
runs over fragment orbitals F
j
A
, or with suitable
symmetry adaptation, over symmetry combinations of frag-
ment orbitals that each match a corresponding molecular or-
bital C
i
which it very closely resembles. This molecular or-
bital formed by a combination of fragment orbitals we call
f
i
. In the same way we can also make molecular orbitals
f
i
from the valence fragment orbitals, but then it is no longer
guaranteed that there are molecular orbitals C
i
that they
closely resemble. We have to remember that the number of
occupied molecular orbitals may be different from the total
number of occupied fragment orbitals. However, we will as-
sume that the number of occupied deep core levels is the
same.
In a linear combination of atomic orbitals ~LCAO! ex-
pansion the ZORA molecular orbitals C
i
can be expressed as
a sum over single atomic contributions
C
i
5
(
A
N
C
i
A
, ~14a!
C
i
A
5
(
n
P A
C
n
i
x
n
A
. ~14b!
As we did for C
i
we express
f
i
as a sum over single
atomic contributions
f
i
A
. The molecular orbitals
f
i
are con-
structed in such a way that
f
i
A
only has a contribution of one
of the fragment orbitals F
j
A
on fragment A. This means that
we can write
(
A
N
(
j
occ
^
F
j
A
u
T
@
V
#
u
F
j
A
&
5
(
A
N
(
i
occ
^
f
i
A
u
T
@
V
#
u
f
i
A
&
. ~15!
The direct derivative of the kinetic energy difference Eq.
~13! with respect to a nuclear displacement X
A
of nucleus A
is
]
DT
ESA
]
X
A
5
(
i
occ
2
K
]
C
i
A
]
X
A
u
T
@
V
#
u
C
i
L
1
(
i
occ
K
C
i
U
]
T
@
V
#
]
X
A
U
C
i
L
2
(
i
occ
2
K
]
f
i
A
]
X
A
u
T
@
V
#
u
f
i
A
L
2
(
i
occ
(
B
N
K
f
i
B
U
]
T
@
V
#
]
X
A
U
f
i
B
L
. ~16!
The one-center contributions in this equation are
(
i
occ
2
K
]
C
i
A
]
X
A
u
T
@
V
#
u
C
i
A
L
1
(
i
occ
(
B
N
K
C
i
B
U
]
T
@
V
#
]
X
A
U
C
i
B
L
2
(
i
occ
2
K
]
f
i
A
]
X
A
u
T
@
V
#
u
f
i
A
L
2
(
i
occ
(
B
N
K
f
i
B
U
]
T
@
V
#
]
X
A
U
f
i
B
L
5
(
i
occ
(
BÞA
N
S
2
K
C
i
A
U
]
T
@
V
#
]
X
B
U
C
i
A
L
1
K
f
i
A
U
]
T
@
V
#
]
X
B
U
f
i
A
L
1
K
C
i
B
U
]
T
@
V
#
]
X
A
U
C
i
B
L
2
K
f
i
B
U
]
T
@
V
#
]
X
A
U
f
i
B
L
D
. ~17!
For valence orbitals each term in itself is very small, since
]
]
X
B
c
2
2c
2
2 V
5
c
2
~
2c
2
2 V
!
2
]
V
]
X
B
~18!
8945J. Chem. Phys., Vol. 110, No. 18, 8 May 1999 van Lenthe, Ehlers, and Baerends
Downloaded 13 Mar 2011 to 130.37.129.78. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
is of order c
2 2
, which means that each term in Eq. ~17! is of
order c
2 2
. Only for deep core levels these terms can be of
importance ~see also the end of this section!, but for these
deep core levels C
i
A
is very close to
f
i
A
and these terms will
cancel each other. We therefore neglect these one-center
terms altogether and we are left with
]
DT
ESA
]
X
A
'
(
i
occ
(
BÞA
N
S
2
K
]
C
i
A
]
X
A
u
T
@
V
#
u
C
i
B
L
1 2
K
C
i
A
U
]
T
@
V
#
]
X
A
U
C
i
B
L
1 2
K
C
i
B
U
]
T
@
V
#
]
X
A
U
(
CÞA,B
N
C
i
C
L
D
5
(
i
occ
(
BÞA
N
S
K
]
C
i
A
]
X
A
u
T
@
V
#
u
C
i
B
L
2
K
C
i
A
u
T
@
V
#
u
]
C
i
B
]
X
B
L
D
1
(
i
occ
(
BÞA
N
S
K
C
i
A
U
]
T
@
V
#
]
X
A
2
]
T
@
V
#
]
X
B
U
C
i
B
L
1 2
K
C
i
B
U
]
T
@
V
#
]
X
A
U
(
CÞA,B
N
C
i
C
L
D
. ~19!
The matrix elements which include a derivative of the ZORA
kinetic energy with respect to a nuclear displacement will be
very small since they are of order c
2 2
and involve two-
center integrals. We can therefore further approximate this
expression by
]
DT
ESA
]
X
A
'
(
i
occ
(
BÞA
N
S
K
]
C
i
A
]
X
A
u
T
@
V
#
u
C
i
B
L
2
K
C
i
A
u
T
@
V
#
u
]
C
i
B
]
X
B
L
D
. ~20!
This expression is simple to evaluate and obeys the transla-
tional invariance condition, which states that if the whole
molecule is translated, the total energy does not change. We
can compare this with the nonrelativistic expression for the
gradient of the kinetic energy
]
T
NR
]
X
A
5
(
i
occ
2
K
]
C
i
A
]
X
A
U
T
NR
u
C
i
&
5
(
i
occ
(
BÞA
N
S
K
]
C
i
A
]
X
A
U
T
NR
U
C
i
B
L
2
K
C
i
A
U
T
NR
U
]
C
i
B
]
X
B
L
D
.
~21!
The total derivative of the energy difference Eq. ~7! with
respect to a nuclear displacement X
A
of nucleus A is
dDE
ESA
dX
A
5
(
i
occ
2
K
]
C
i
A
]
X
A
u
V1 T
@
V
#
2
e
i
u
C
i
L
1
(
BÞA
N
Z
A
Z
B
~
X
A
2 X
B
!
u
R
A
2 R
B
u
3
2
E
r
~
1
!
Z
A
~
X
A
2 x
1
!
u
R
A
2 r
1
u
3
d1
1
(
i
occ
(
B
N
K
C
i
A
U
]
T
@
V
#
]
X
B
U
C
i
L
, ~22!
since
(
i
occ
(
BÞA
N
S
K
]
C
i
A
]
X
A
u
T
@
V
#
u
C
i
B
L
2
K
C
i
A
u
T
@
V
#
u
]
C
i
B
]
X
B
L
D
5
(
i
occ
2
K
]
C
i
A
]
X
A
u
T
@
V
#
u
C
i
L
1
(
i
occ
(
B
N
K
C
i
A
U
]
T
@
V
#
]
X
B
U
C
i
L
.
~23!
Compared to a similar nonrelativistic expression there is
an extra term
(
i
occ
(
B
N
K
C
i
A
U
]
T
@
V
#
]
X
B
U
C
i
L
. ~24!
This term mimics the gradient of the interaction due to an
effective small component density, which would be present
if the Dirac equation was used.
We may compare Eq. ~20! with a recently derived ana-
lytical expressions for the ZORA kinetic energy gradient in
the ZORA ~MP! method by van Wu
¨
llen,
9
]
T
ZORA~MP!
]
X
A
5
(
i
occ
2
K
]
C
i
A
]
X
A
u
T
@
V
#
u
C
i
L
1
(
i
occ
K
C
i
U
]
T
@
V
#
]
X
A
U
C
i
L
. ~25!
The major difference with the ZORA ESA method @see also
Eq. ~23!# are one-center contributions
(
i
occ
(
BÞA
N
S
K
C
i
B
U
]
T
@
V
#
]
X
A
U
C
i
B
L
2
K
C
i
A
U
]
T
@
V
#
]
X
B
U
C
i
A
L
D
,
~26!
which are present in the ZORA ~MP! method, but which are
not present in the ZORA ESA method. These one-center
contributions can cause problems if the model potential on
atom A used in the ZORA ~MP! method has a finite value in
the core region of atom B, which depends on the distance
between A and B. In this case the ZORA kinetic energy of
the ~deep! core orbitals on atom B will depend on the actual
distance between A and B, even if these ~deep! core orbitals
do not change shape. This is the gauge dependence problem
of ZORA, see also Ref. 6, which is solved if the ZORA ESA
method is used. In the ZORA ~MP! method the model po-
tential of atom A is usually not so large at distances between
A and B which are in the order of ~or larger than! typical
bond lengths between A and B. This means that in general
the errors in the optimized geometries and bond energies will
8946 J. Chem. Phys., Vol. 110, No. 18, 8 May 1999 van Lenthe, Ehlers, and Baerends
Downloaded 13 Mar 2011 to 130.37.129.78. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Citations
More filters
Journal Article•DOI•
Chemistry with ADF

[...]

G. te Velde, F.M. Bickelhaupt1, Evert Jan Baerends1, C. Fonseca Guerra1, S. J. A. van Gisbergen1, J.G. Snijders, T. Ziegler2 •
VU University Amsterdam1, University of Calgary2
15 Jul 2001-Journal of Computational Chemistry
TL;DR: The “Activation‐strain TS interaction” (ATS) model of chemical reactivity is reviewed as a conceptual framework for understanding how activation barriers of various types of reaction mechanisms arise and how they may be controlled, for example, in organic chemistry or homogeneous catalysis.
Abstract: We present the theoretical and technical foundations of the Amsterdam Density Functional (ADF) program with a survey of the characteristics of the code (numerical integration, density fitting for the Coulomb potential, and STO basis functions). Recent developments enhance the efficiency of ADF (e.g., parallelization, near order-N scaling, QM/MM) and its functionality (e.g., NMR chemical shifts, COSMO solvent effects, ZORA relativistic method, excitation energies, frequency-dependent (hyper)polarizabilities, atomic VDD charges). In the Applications section we discuss the physical model of the electronic structure and the chemical bond, i.e., the Kohn–Sham molecular orbital (MO) theory, and illustrate the power of the Kohn–Sham MO model in conjunction with the ADF-typical fragment approach to quantitatively understand and predict chemical phenomena. We review the “Activation-strain TS interaction” (ATS) model of chemical reactivity as a conceptual framework for understanding how activation barriers of various types of (competing) reaction mechanisms arise and how they may be controlled, for example, in organic chemistry or homogeneous catalysis. Finally, we include a brief discussion of exemplary applications in the field of biochemistry (structure and bonding of DNA) and of time-dependent density functional theory (TDDFT) to indicate how this development further reinforces the ADF tools for the analysis of chemical phenomena. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 931–967, 2001

8,490 citations

Journal Article•DOI•
Optimized Slater-type basis sets for the elements 1-118.

[...]

E. van Lenthe1, Evert Jan Baerends1•
VU University Amsterdam1
15 Jul 2003-Journal of Computational Chemistry
TL;DR: Seven different types of Slater type basis sets for the elements H (Z = 1) up to E118, ranging from a double zeta valence quality up to a quadruple zetavalence quality, are tested in their performance in neutral atomic and diatomic oxide calculations.
Abstract: Seven different types of Slater type basis sets for the elements H (Z = 1) up to E118 (Z = 118), ranging from a double zeta valence quality up to a quadruple zeta valence quality, are tested in their performance in neutral atomic and diatomic oxide calculations. The exponents of the Slater type functions are optimized for the use in (scalar relativistic) zeroth-order regular approximated (ZORA) equations. Atomic tests reveal that, on average, the absolute basis set error of 0.03 kcal/mol in the density functional calculation of the valence spinor energies of the neutral atoms with the largest all electron basis set of quadruple zeta quality is lower than the average absolute difference of 0.16 kcal/mol in these valence spinor energies if one compares the results of ZORA equation with those of the fully relativistic Dirac equation. This average absolute basis set error increases to about 1 kcal/mol for the all electron basis sets of triple zeta valence quality, and to approximately 4 kcal/mol for the all electron basis sets of double zeta quality. The molecular tests reveal that, on average, the calculated atomization energies of 118 neutral diatomic oxides MO, where the nuclear charge Z of M ranges from Z = 1-118, with the all electron basis sets of triple zeta quality with two polarization functions added are within 1-2 kcal/mol of the benchmark results with the much larger all electron basis sets, which are of quadruple zeta valence quality with four polarization functions added. The accuracy is reduced to about 4-5 kcal/mol if only one polarization function is used in the triple zeta basis sets, and further reduced to approximately 20 kcal/mol if the all electron basis sets of double zeta quality are used. The inclusion of g-type STOs to the large benchmark basis sets had an effect of less than 1 kcal/mol in the calculation of the atomization energies of the group 2 and group 14 diatomic oxides. The basis sets that are optimized for calculations using the frozen core approximation (frozen core basis sets) have a restricted basis set in the core region compared to the all electron basis sets. On average, the use of these frozen core basis sets give atomic basis set errors that are approximately twice as large as the corresponding all electron basis set errors and molecular atomization energies that are close to the corresponding all electron results. Only if spin-orbit coupling is included in the frozen core calculations larger errors are found, especially for the heavier elements, due to the additional approximation that is made that the basis functions are orthogonalized on scalar relativistic core orbitals.

2,112 citations

Journal Article•DOI•
An atlas of two-dimensional materials

[...]

Pere Miró1, Martha Audiffred1, Thomas Heine1•
Jacobs University Bremen1
18 Aug 2014-Chemical Society Reviews
TL;DR: This Atlas demonstrates the large diversity of electronic properties, including band gaps and electron mobilities of atomically thin materials, as well as rare earth, semimetals, transition metal chalcogenides and halides, and finally synthetic organic 2D materials, exemplified by 2D covalent organic frameworks.
Abstract: The discovery of graphene and other two-dimensional (2D) materials together with recent advances in exfoliation techniques have set the foundations for the manufacturing of single layered sheets from any layered 3D material. The family of 2D materials encompasses a wide selection of compositions including almost all the elements of the periodic table. This derives into a rich variety of electronic properties including metals, semimetals, insulators and semiconductors with direct and indirect band gaps ranging from ultraviolet to infrared throughout the visible range. Thus, they have the potential to play a fundamental role in the future of nanoelectronics, optoelectronics and the assembly of novel ultrathin and flexible devices. We categorize the 2D materials according to their structure, composition and electronic properties. In this review we distinguish atomically thin materials (graphene, silicene, germanene, and their saturated forms; hexagonal boron nitride; silicon carbide), rare earth, semimetals, transition metal chalcogenides and halides, and finally synthetic organic 2D materials, exemplified by 2D covalent organic frameworks. Our exhaustive data collection presented in this Atlas demonstrates the large diversity of electronic properties, including band gaps and electron mobilities. The key points of modern computational approaches applied to 2D materials are presented with special emphasis to cover their range of application, peculiarities and pitfalls.

1,136 citations

Journal Article•DOI•
Understanding the M(NHC) (NHC = N-heterocyclic carbene) bond

[...]

Heiko Jacobsen, Andrea Correa1, Albert Poater1, Chiara Costabile1, Luigi Cavallo1 •
University of Salerno1
01 Mar 2009-Coordination Chemistry Reviews
TL;DR: In this article, the authors provide an overview of the main contributions achieved by the application of advanced computational techniques in the N-heterocyclic carbenes (NHC) bond.

614 citations

Journal Article•DOI•
High-efficiency blue and white organic light-emitting devices incorporating a blue iridium carbene complex.

[...]

Hisahiro Sasabe1, Jun-ichi Takamatsu1, Takao Motoyama1, Soichi Watanabe, Gerhard Wagenblast, Nicolle Langer, Oliver Molt, Evelyn Fuchs, Christian Lennartz, Junji Kido1 •
Yamagata University1
24 Nov 2010-Advanced Materials
TL;DR: High-effi ciency white organic light-emitting devices (OLEDs) have great potential for energy saving solid-state lighting and eco-friendly fl at-display panels and are expected to open new designs in lighting technology, such as transparent lighting panels or luminescent wallpapers because of being able to form paper-like thin fi lms.
Abstract: High-effi ciency white organic light-emitting devices (OLEDs) have great potential for energy saving solid-state lighting and eco-friendly fl at-display panels. [ 1 ] In addition, white OLEDs are expected to open new designs in lighting technology, such as transparent lighting panels or luminescent wallpapers because of being able to form paper-like thin fi lms. Phosphorescent OLED technology is an imperative methodology to realize higheffi ciency white OLEDs because phosphors, such as fac -tris(2phenylpyridine)iridium( III ) [Ir(ppy) 3 ] and iridium( III )bis(4,6(difl uorophenyl)pyridinatoN , C 2 ′ )picolinate (FIrpic) enable an internal effi ciency as high as 100% converting both singlet and triplet excitons into photons. [ 2 ] There are two effective approaches to obtain white OLEDs by using phosphors. One is to combine a blue fl uorophore and phosphors for the other colors, a so-called hybrid white OLED. [ 3 ] A key requirement is the use of a blue fl uorophore with higher triplet energy ( E T1 ) than that of the other phosphors. The blue fl uorophore also needs to have a high photoluminescent quantum yield ( η PL ). Schwartz and coworkers reported hybrid white OLEDs with a power effi ciency at 1000 cd m − 2 ( η p,1000 ) of 22 lm W − 1 (external quantum effi ciency (EQE) of 10.4%) by using N , N ′ -di-1-naphthalenylN , N ′ -diphenyl-[1,1 ′ :4 ′ ,1 ′ ′ :4 ′ ′ ,1 ′ ′ ′ -quaterphenyl]-4,4 ′ ′ ′ -diamine (4P-NPD) as a blue fl uorophore, and Ir(ppy) 3 and iridium( III ) bis(2-methyldibenzo-[ f , h ]quinoxaline)(acetylacetonate) [Ir(MDQ) 2 ( acac)] as green and red phosphors, respectively. [ 4 ]

499 citations

References
More filters
Journal Article•DOI•
Density-functional exchange-energy approximation with correct asymptotic behavior.

[...]

Axel D. Becke1•
Queen's University1
15 Sep 1988-Physical Review A
TL;DR: This work reports a gradient-corrected exchange-energy functional, containing only one parameter, that fits the exact Hartree-Fock exchange energies of a wide variety of atomic systems with remarkable accuracy, surpassing the performance of previous functionals containing two parameters or more.
Abstract: Current gradient-corrected density-functional approximations for the exchange energies of atomic and molecular systems fail to reproduce the correct 1/r asymptotic behavior of the exchange-energy density. Here we report a gradient-corrected exchange-energy functional with the proper asymptotic limit. Our functional, containing only one parameter, fits the exact Hartree-Fock exchange energies of a wide variety of atomic systems with remarkable accuracy, surpassing the performance of previous functionals containing two parameters or more.

45,683 citations

Journal Article•DOI•
Relativistic regular two‐component Hamiltonians

[...]

E. van Lenthe, Evert Jan Baerends, J. G. Snijders
15 Sep 1993-Journal of Chemical Physics
TL;DR: In this article, potential-dependent transformations are used to transform the four-component Dirac Hamiltonian to effective two-component regular Hamiltonians, which already contain the most important relativistic effects, including spin-orbit coupling.
Abstract: In this paper, potential‐dependent transformations are used to transform the four‐component Dirac Hamiltonian to effective two‐component regular Hamiltonians. To zeroth order, the expansions give second order differential equations (just like the Schrodinger equation), which already contain the most important relativistic effects, including spin–orbit coupling. One of the zero order Hamiltonians is identical to the one obtained earlier by Chang, Pelissier, and Durand [Phys. Scr. 34, 394 (1986)]. Self‐consistent all‐electron and frozen‐core calculations are performed as well as first order perturbation calculations for the case of the uranium atom using these Hamiltonians. They give very accurate results, especially for the one‐electron energies and densities of the valence orbitals.

3,585 citations

Journal Article•DOI•
Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation

[...]

John P. Perdew1, Wang Yue1•
Tulane University1
15 Jun 1986-Physical Review B
TL;DR: Improvements over other simple functionals are also found in the exchange contributions to the valence-shell removal energy of an atom and to the surface energy of jellium within the infinite barrier model.
Abstract: The electronic exchange energy as a functional of the density may be approximated as ${E}_{x}[n]={A}_{x}\ensuremath{\int}{d}^{3}r{n}^{\frac{4}{3}}F(s)$, where $s=\frac{|\ensuremath{ abla}n|}{2{k}_{F}n}$, ${k}_{F}={(3{\ensuremath{\pi}}^{2}n)}^{\frac{1}{3}}$, and $F(s)={(1+1.296{s}^{2}+14{s}^{4}+0.2{s}^{6})}^{\frac{1}{15}}$. The basis for this approximation is the gradient expansion of the exchange hole, with real-space cutoffs chosen to guarantee that the hole is negative everywhere and represents a deficit of one electron. Unlike the previously publsihed version of it, this functional is simple enough to be applied routinely in self-consistent calculations for atoms, molecules, and solids. Calculated exchange energies for atoms fall within 1% of Hartree-Fock values. Significant improvements over other simple functionals are also found in the exchange contributions to the valence-shell removal energy of an atom and to the surface energy of jellium within the infinite barrier model.

3,500 citations

Journal Article•DOI•
Relativistic total energy using regular approximations

[...]

E. van Lenthe, Evert Jan Baerends, J. G. Snijders
01 Dec 1994-Journal of Chemical Physics
TL;DR: In this paper, a simple scaling of the ZORA one-electron Hamiltonian is shown to yield energies for the hydrogenlike atom that are exactly equal to the Dirac energies.
Abstract: In this paper we will discuss relativistic total energies using the zeroth order regular approximation (ZORA). A simple scaling of the ZORA one‐electron Hamiltonian is shown to yield energies for the hydrogenlike atom that are exactly equal to the Dirac energies. The regular approximation is not gauge invariant in each order, but the scaled ZORA energy can be shown to be exactly gauge invariant for hydrogenic ions. It is practically gauge invariant for many‐electron systems and proves superior to the (unscaled) first order regular approximation for atomic ionization energies. The regular approximation, if scaled, can therefore be applied already in zeroth order to molecular bond energies. Scalar relativistic density functional all‐electron and frozen core calculations on diatomics, consisting of copper, silver, and gold and their hydrides are presented. We used exchange‐correlation energy functionals commonly used in nonrelativistic calculations; both in the local‐density approximation (LDA) and including...

2,645 citations

Journal Article•DOI•
Quantum electrodynamical corrections to the fine structure of helium

[...]

Marvin Douglas1, Norman M. Kroll1•
University of California, San Diego1
01 Jan 1974-Annals of Physics
TL;DR: In this article, order α6mc2 corrections to the fine structure splitting of the He4 atom were investigated based on the covariant Bethe-Salpeter equation including external potential to take account of the nuclear Coulomb field.

2,455 citations

Related Papers (5)
Chemistry with ADF

[...]

15 Jul 2001-Journal of Computational Chemistry
G. te Velde, F.M. Bickelhaupt, Evert Jan Baerends, C. Fonseca Guerra, S. J. A. van Gisbergen, J.G. Snijders, T. Ziegler 
Relativistic regular two‐component Hamiltonians

[...]

15 Sep 1993-Journal of Chemical Physics
E. van Lenthe, Evert Jan Baerends, J. G. Snijders
Density-functional exchange-energy approximation with correct asymptotic behavior.

[...]

15 Sep 1988-Physical Review A
Axel D. Becke
Density-functional approximation for the correlation energy of the inhomogeneous electron gas

[...]

15 Jun 1986-Physical Review B
John P. Perdew
Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis

[...]

01 Aug 1980-Canadian Journal of Physics
S. H. Vosko, L. Wilk, Marwan E. Nusair
Frequently Asked Questions (9)
Q1. What are the contributions in "Geometry optimizations in the zero order regular approximation for relativistic effects" ?

In this paper, the authors derived the energy gradients in the ZORA ESA method and showed that these analytical expressions are easier to evaluate than the expressions following from the recently developed MP2 # level of theory using relativistic effective core potentials, with QR Pauli DFT calculations, with DPT density functional calculations, and with Douglas-Kroll-Hess density functional calculation. 

Method based on the Pauli Hamiltonian previously implemented in the ADF program, in the SR ZORA ESA method it is possible to study the convergence of the optimized bond lengths and bond energies with respect to the basis set limit, and one can test the frozen core approximation using all electron basis sets. Comparisons have been made to results of previous ab initio calculations at the @ CCSD~T ! //MP2 # level of theory using relativistic effective core potentials, with QR Pauli DFT calculations, with DPT density functional calculations, and with Douglas– Kroll–Hess density functional calculations. 

The resulting electrostatic shift in the molecular model potential in the region of the Au nucleus will lower the ZORA kinetic energy of the ~deep! 

The optimized bond distances of these pointwise calculations were always within 0.002 Å of the analytically calculated distances. 

For the heavier atoms one also should add corelike basis functions, which are able to describe the core tail of the valence orbitals more accurately. 

This collapse is caused by admixing of core character, due to the orthogonalization on NR core orbitals, whereas one should have orthogonalized on SR ZORA core orbitals. 

For the heavier atoms, these basis sets contain extra 1s and 2p STO functions, in order to describe the core orbitals accurately. 

This is due to the fact that the core wiggles of especially the s-type valence electrons do not behave like Slater-type orbitals near the nucleus, but more like Diractype orbitals which are of the formrh21e2zr, ~33!where h does not have to be an integer. 

The remaining differences in the results of the two methods can be explained partly by the use of different basis sets, but the authors think that also a part of the deviations is due to the gauge dependence problems of the ZORA ~MP!