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Introduction to
Relativistic Quantum
Chemistry
Xiaoyan Cao
Institute for Theoretical Chemistry
University of Cologne, Germany
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Contents
1. Special Relativity
2. Relativistic Wave Equation
3. Relativistic Electromagnetic Interactions
4. Relativistic Symmetry
5. All-electron Methods
6. Valence-electron Methods
7. Spin-Orbit Configuration Interaction Methods
8. Relativistic Effects in Chemistry
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1. Special Relativity
1.1 The situation before 1900: two dark clouds on the horizon
1.2 Special relativity: Einstein‟s two postulates about the relativity and the constancy of the c
1.3 Consequences
1.3.1 Lorentz transformation
1.3.2 Velocities transformation
1.3.3 Mass transformation
1.3.4 Relativistic energy
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Two Dark Clouds on the Horizon
Michelson-Morley experiment Special
theory of relativity
Blackbody radiation Quantum
mechanics
“The beauty and clearness of the dynamical theory, which
asserts heat and light to be modes of motion, is at
present obscured by two clouds.” Kelvin in 1900
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Galilean Transformation
O X
Y
Z
O' X'
Y'
Z'
v
'
'
'
' ( )'
y y
z z
x x t
dx d x tw w
dt dt
No speed limit
’
P
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James Clerk Maxwell
(1831-1879)
The first permanent color
photograph (tarton ribbon), taken
by J.C. Maxwell in 1861
The basic laws of electricity and
magnetism (Maxwell‟s equation
,1865):
The electro-magnetic waves travel at
the speed of light c with respect to
the hypothesized ether (reference
system).
The speed of light does not vary with
the speed of the source.
The electro-magnetic waves travel
at different speed with respect to
different reference system in
Galilean transformation.
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Michelson-Morley Experiment
(1887, Nobel Prize in 1907)Purpose: detect a difference in the speed of light in two different directions:
parallel to, and perpendicular to, the motion of the Earth around the Sun.
Results: no measurable difference between the speed of light in the two
directions.
Edward Morley
(1838-1923)
German-born American 7全国理论及量子化学暑期学校,2010,北京
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Einstein’s Two Postulates (1905)
Postulate of relativity
The laws of physics are identical in all inertial
frames.
Postulate of the constancy of the speed of light
In empty space light signals propagate in
straight lines with speed c in all inertial frames.
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1.3-7 Consequences 97/28/2010
Lorentz transformation (1904)
In the coordinate system :
0)()(222
3
1
tcxi
i
In the coordinate system ':
0)()(2'22'
3
1
tcx i
i
A coordinate transformation „ obeying above equations is called a
Lorentz transformation.
Introducing a four-vector r=(x,y,z,ict) with length r·r=x2+y2+z2-c2t2 is
invariant under Lorentz transformation
Hendrik Antoon Lorentz
(1853-1928)
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1.3-8 Consequences 107/28/2010
2 2
2
2 2
2 1 2 12' '
2 12
2
' '
2 1 2 1
relative "simu
'
'
'1 /
/'
1 /
1
==>
ltaneity"
in ', if in
==>
y y
z z
x utx
u c
t ux ct
u c
ut t x x
ct tu
c
t t t t
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Time dilation
Twin paradox
J.C. Hafele, R.E. Keating, Around-the-world Atomic Clocks: Predicted
Relativistic Time Gains, Science 177 (1972)166
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Lorentz contraction
Ladder Paradox12全国理论及量子化学暑期学校,2010,北京
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Composition of velocities
Velocities do not simply „Add‟, for example if a
battle plane is moving at the speed of light
relative to an observer, and the battle plane
fires a missile at the speed of light relative to
the plane, the missile does not exceed the
speed of light relative to the observer.
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Equivalence of mass and energy
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'21 ( / )
oo
mm
u c
2 2 4 2 2 2 4
0
2
0 is the rest energy
E m c p c m c
m c
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2. Relativistic wave equation
22 2 2 2 4
0
2 2 2 4
0
ˆ( )
ˆˆ
2
ˆ
i Ht
pH V E p c m c
m
H p c m c V
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Klein-Gordon Equation for free spin-zero particle
(Fock, Gordon, Klein, Kudar, 1926)
2 2 2 1/ 2
2 2 4 2 2
ˆ( ) [ ( ) ]
ˆ( ) [ ]
i c m c pt
i m c p ct
Oscar
Klein
1894-1977
Walter
Gordon
1893-
1939
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Dirac’s free particle equation
(P. Dirac 1928)
Suppose: the Hamiltonian is linear in first order time
(t) and space (x y z) derivative. The most general free-
particle wave equation is:
2 2 4 2 2
Klein-Gordon Equation:
ˆ( ) [ )]i m c p ct
2
2
( , , , )
ˆˆ ˆ ˆ ( , , , )
ˆˆ ˆ ˆ, , and are unkown constants.
ˆ ˆˆ
x y z
x y z
D
i x y z tt
c i c i c i mc x y z tx y z
i i c mc ht
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18
1 2 3
ˆ0 0ˆ ˆˆ 00
1 0 0 1 0 1 0ˆ ˆ ˆ
0 1 1 0 0 0 1
Relativistic is 4 1 matrix
E corresponds to electron-like and
positron-like (discovered in 1932)
I
I
iI
i
solutions, respectively.
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Dirac (D) one particle Hamiltonian
2ˆˆ ˆ ˆ ( )Dh c p mc V r
V(ri) denotes the electrostatic potential generated
by the -th nucleus at the position of the electron
2
1( )Z e
V rr
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( )
( )
( )
(
ˆˆ ( ) ( , , , )
( )2
( )2
( )2
( )2
iP r Et
p
iP r Et
p
iP r Et
p
i
p
Eigenvalue Eigenfunction
H P h t f x y z t
E P t N ea
E P t N ea
aE P t N e
aE P t N e
)
2 22
3/
2
2
2 2
cos sin1 2 2
, ; ; ;22
sin cos2 2
P r Et
i i
i i
eE mc
acp
eE mc
where NE
e e
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Free-electron solutions of the
time-independent Dirac equation
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22 2 2 2
22 2
+
The ratio of the norms for E>0
1
4
Charge density
=
Current density
a c pR
cE mc
j c
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Hydrogen solutions of the time-
independent Dirac equation
The four-component wave function can be expressed as a pair
of two component spinors:
1
2
3
4
1 1
2 2
2 2
0 0
( )
( )
1 ( ); 1 ( )
; ; 2 1 ;
jmjm
jmk
jmjm
n k n k
F r
r
iG r
r
F e G e
a b mcr k
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1/ 2
22 2 2 2
2 42 6
2 3
1 1 1 1, ,
2 2 2 2
, ,
1 1 1 1, ,
2 2 2 2
1
2 2( 1);
1
2 2( 1)
Four
1 ( ) /
1 31 ( )
2 2 4
qua
j m j m
j m j m
j m j
n
m
j m j mY Y
j j
j m
E mc n k k
mc On n k
jY Y
n
m
j j
ˆntumn numbers for H
1,2,3
1 3 5 1, , , ,
2 2 2 2
1( ) 1, 2, ,
2
, 1, ,
n
j n
k j n
m j j j
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Quantum number and labels for H
n k j l+ symbol j=1/2, 3/2,…, n-1/2
k=±(j+1/2)=±1,±2, …,
+n
l+=j+1/2 for k<0
l+=j-1/2 for k>0
1 1 1/2 0 1s1/2
2 2 3/2 1 2p3/2
-1 1/2 1 2p1/2
1 1/2 0 2s1/2
3 3 5/2 2 3d5/2
2 3/2 1 3p3/2
-2 3/2 2 3d3/2
1 1/2 0 3s1/2
-1 1/2 1 3p1/2
2p1/2 and 2s1/2, 3p3/2 and 3d3/2, 3s1/2 and 3p1/2 are degenerate.
The degeneracy is removed by the “Lamb-shift”, a quantum electrodynamical
effects of O(3), e.g., the splitting of 2p1/2 and 2s1/2 is only 0.004meV.
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Qualitative Conclusions
1. States with same n, l but different j are
spin-orbit split.
2. The radial density (F2+G2) has no nodes.
3. The radial electron density suffers a
relativistic contraction.
4. Normalization is no problem.
5. The solutions for K=1 have a singularity
at the origin.
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2 2( ) ( ) ( )r g r f r
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29 7/28/2010
We assume the Born-Oppenheimer approximation to
hold and neglect external fields:
ˆ( , )ˆˆ ( )n n N
D
i i j
Z ZH g i jh i
r
The indices i and j denote electrons, and nuclei. Z is
the charge of the nucleus . Dirac-Coulomb (DC)
Hamiltonian (gc(i,j)=1/rij);DC-Gaunt (DCG) Hamiltonian
(in addition the Gaunt interaction); DC-Breit(DCB)
Hamiltonian (in addition the gauge term)
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3. Relativistic Electromagnetic Interactions
Chemical concepts Molecules ---
aggregates of atoms linked by
electromagnetic interactions
Proper relativistic description of these
interaction
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Interaction energy of two charged particles
41 21 2
12 2 2 4
2
1 2
0
2
1 11
2 2
( )( )1ˆ ( , )
2
4
1/i ij j ij
CB i j
i i
j
j
i
j
u r u ru u uV O
c c r c
r rg i j
q
r
q
r
rr
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4. Relativistic Symmetry
• one of the great unifying principles of
physics and chemistry.
• Provides us with valuable information
about the properties and behavior of the
system.
• Simplify quantum chemical calculations on
the system.
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Double GroupThe character of the representation for state wave function under
a symmetry operation which consists
sin(
of ro
1/
tation by an ang
2
le
is
)( )
sin
given by:
( ), int egral( 2 )
( ), h lf / 2 a i
j j
j
ntegral
Rotation by 2 treated as a symmetry operation but not as an
identity operation.
Any ordinary rotation group is expanded by taking the product
of this new operation, i.e., the relativistic
group must be the direct
product of the nonrelativistic group and the group , .
New group will contain twice as many operations and more
classes a
Double
nd
gr
repres
oup is
entations.
a symmetry gr
E E
oup of the Dirac equation.F. A. Cotton, Chemical Applications of Group Theory, John Wiley & Sons, Inc. 1971
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Character Tables for D4' and D4
D4'E R C4 C4
3 C2 2C2' 2C2"
C43R C4R C2R 2C2
’R 2C2"R
1 A1' 1 1 1 1 1 1 1
2 A2' 1 1 1 1 1 -1 -1
3 B1' 1 1 -1 -1 1 1 -1
4 B2' 1 1 -1 -1 1 -1 1
5 E1' 2 2 0 0 -2 0 0
6 E2' 2 -2 21/2 -21/2 0 0 0
7 E3' 2 -2 -21/2 21/2 0 0 0
D4
E 2C4 C2 2C2' 2C2
"
A1 1 1 1 1 1
A2 1 1 1 -1 -1
B1 1 -1 1 1 -1
B2 1 -1 1 -1 1
E 2 0 -2 0 0
34全国理论及量子化学暑期学校,2010,北京
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Spin and SU(2) Group
22
An unitary transformation for a genaral rotation by an angle around
an axis along the unit vector n in spin space can be written as :
ˆ cos sin2 2
Where,
is Pauli m
in
U e I i n
atrix.
ˆAll have determinant 1 and form a group SU(2)
==> Special unitary group of dimension 2
U
35全国理论及量子化学暑期学校,2010,北京
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Spatial Rotations and the SO(3) Group
ˆ cos
Rotation by an angle about an axis in the direction of unit vector
is given by
0 0 0 0 0 1 0 1
cos ( )s
0
0 0 1 ; 0 0 0 ;
0 1 0 1 0
in
0
,iX n T
n
T
i jij
x y z
n
nn n n
iX iX
R e nn i X n where
iX
1 0 0
0 0 0
Due to the tracelessness of matrices, all transformation matrices
have unit dterminants and form a group SO(3)
==> special orthogonal group in three dimensions
qX
36全国理论及量子化学暑期学校,2010,北京
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Transformation of operators
' 1
1
1
'
ˆThe transformed operator ' under symmetry operation
ˆ is:
ˆ For a general operator effecting a rotation by
ˆan angle around an ax
ˆ ˆˆ ˆ'
is n
ˆ ˆ ˆˆ ˆ ˆ
ˆ ˆˆ ˆ ˆ
=
n
n n n
n n n
Q
U
U U R
r R r
Q
R R
Q
1 r̂
37全国理论及量子化学暑期学校,2010,北京
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Space inversion
Relativistic inversion operator is written as:
ˆˆ ˆ ˆ, where is acting in 3.
==> The upper and lower components of the
spinor have different parity.
==> 4-spinors as a whole transform as only one
ir
R RI I I
rep and can be labeled with the symmetry
of the large components.
38全国理论及量子化学暑期学校,2010,北京
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Reflections and rotation-reflection
2
The double-group reflection operator:
ˆ , where define a twofold axis.
The rotation-reflection:
1 1 1ˆ , , 1.2
==> The large and small components have the
opposite parity u
q R q
n h n R m
I C q
S C I C where nm n
nder the rotation-reflection operations.
39全国理论及量子化学暑期学校,2010,北京
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Time reversal
0 0
0
The relativistic time-reversal operator for one-electron system:
ˆ ˆ ˆ, Where, is complex conjugation operator
ˆ ( ) , , are real
a double time reversal changes
num
the sig
b s
n
r
e .
yK i K K
K b id b id b d
0 1 2
.
For more than one electron:
ˆ ˆ ˆ ˆK=U , Where U= ( ) ( )( ) ( )
of the wave function
y y y nyK i i i i
40全国理论及量子化学暑期学校,2010,北京
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Kramers’ theorem (1930)
In the absence of external vector potentials and the provided potentials
being invariant with respect to time reversal, for a system with half-
integer spin, the energy levels are at least doubly degenerate, and
any degeneracy is even-fold.
1
1 -1
( ) and ( )form a Kramer pair.
ˆ ˆ ˆKT ( ) K ( ) ( )
Examples for Kramer pair:
1s and 1s ; 2p and 2p
t t
t t t
41全国理论及量子化学暑期学校,2010,北京
Kramers symmetry and SU(2)G, GO(3) are used to simplify
methods for quantum chemical calculations on relativistic systems
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5. All-electron Methodology
5.1 Four-Component Methods
5.2 Spin Separations and the Modified Dirac
Equation
5.3 The Foldy-Wouthuysen Transformation
5.4 Douglas-Kroll-Hess Hamiltonian
5.5 Elimination of Small Components (Wood-
Boring Hamiltonian , Pauli Hamiltonian,
ZORA)
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5.2 Spin seperation and the
modified Dirac equation
43
(Exact) Separation of Spin-Free and
Spin-Dependent Terms of the Dirac-
Coulomb-Breit Hamiltonian (Dyall, J. Chem. Phys.
100(1994)2118), Kutzelnigg, Int. J. Quant. Chem. 25(1984)107)
• Scalar Relativistic Hamiltonian, real; spin-orbit term may be treated at different levels of theory.
• No complicated additional integrals compared to a non-relativistic calculation.
• Number of two-electron integrals is only a factor of 3 or 4 higher than in non-relativistic calculations
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7/28/2010 5.2 Spin separation and the
modified Dirac equation
44
2
4
2
(spin-free) (spin-dependen
ˆˆ ˆ, ( ) ( )
ˆ ˆ( ') ( ) 0 (1)
ˆ ˆ( ) ( ' 2 ) 0 (2)
From (2)
ˆ
, we have:
ˆ ˆˆ ˆ ˆˆ ˆ ˆ( )( ) t)
Basic Idea:
L
DS
L S
L S
h c p I mc V
A B A B i A
r
V E c
E c
B
p
c p V m
1
2
2
' ˆ ˆ ˆ ˆ2 1 ( ) ( ) (3)2
substituting (3) to (1) and (2), multiplication of the
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ(2) from the left by , with ( )( ) 2
2
S L LE Vmc p p
mc
pp p p mT
mc
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7/28/2010 5.2 Spin seperation and the
modified Dirac equation
45
2
2
2
Modified Dirac equation:
( ') 0
1 ˆ ˆ ˆ ˆ' 0(2 )
with =1/c and application of the initial identity one gets:
ˆ ˆ (spin-free)/ 4
0 0+
0 / 4
L L
L L
V E T
T p V E pmc
D EG
V TD
T p V p T
i
(spin-dependˆ ˆ )ˆ ent
pV p
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7/28/2010 5.2 Spin seperation and the
modified Dirac equation
46
2
modified metric and modified wave function:
1 0G=
0 / 2
L
L
T
Also possible: corresponding modification of the two-particle terms, i.e.,
the Coulomb-, Gaunt-, Breit- interaction
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7/28/2010 5.3 Foldy-Wouthusen
Transformation
47
1. Foldy-Wouthuysen (FW) Transformation: Decoupling of large and
small components up to a given order of 4
2 2
ˆDirac equation:
ˆˆ ˆˆ ˆ
: block-diagonal and commute with
: off-diagonal and anti
Ev
commute with
Transformed wavefunction
en ( ) terms
Od
Tran
d (O
s
) te
fo
r
rme
ms
d H
iS
i Ht
H mc eV c mc H H
e
ˆ ˆ ˆ ˆ
amiltonian
ˆ ˆ'
Choice of S: elimination of odd terms
is is is isH e He e i e it t
Two traditional ways to derive two-component Hamiltonians
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2 2
4
2 2
3 2 43 2
2
2
(correctiDarwin ter on to thm
The new Dirac equation for positive energy solution:
ˆi ,t
1
1
8 2
2 8 4ˆ
1
,2
LLH where
eH mc eV S
eS B
m c
eE
m
m c
eS
mcB
m m
Oc
w r
c
E
h S
E
e e
e second term) : the 5th term, results from the zitterbewegung of
the electron over a region of a magnit
Interaction energy
ude comparable to
of a moving magne
the co
tic mo
mpton wavelengt
ment and an
h.
exte : the 6th term
(the third term): the 7th term (mass-velocity ter
Corrections to po
Co
te
rnal electric fie
ntial of magnetic
rrections to
moment in e
kinetic term of
xternal magnetic
e
f
lec
iel
l
d
tron m)
(
d
the fourth term): the 8th term.
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Difficulties caused by the FW
transformation
• Higher and higher power of P involved in the Hamiltonian.
• Powers of the P higher than 2 are not bounded.
Only variationally useful for the lowest-order terms
Only used in perturbation theory.
A. Farazdel and V.H. Smith Jr., Int. J. Quantum Chem., 29,311 (1986)
W. Kutzelnigg, Z. Phys. D 15, 27(1990)
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Siegfried Wouthuysen (1916-
1996), Dutch Physicist
L.L. Foldy, (1919-2001), born in
Sabinov, Czechoslovakia with
Hungarian roots, immigrated to
the US in 1921
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7/28/2010 5.4 Douglas-Kroll-Hess
Hamiltonian
51
Starting with free-particle Foldy-Wouthuysen (fpFW
Provide an extension in powers of potential energy instead
ˆof P ==> Produced opera
The Douglas-Kroll transformation:
tors can be used variationally.
2
0
1
1 0 0 0 1 1
2 4 2 2
0
2
1
1
1
2
0 1
2
) transformation:
ˆ ˆˆˆ ˆ ˆU
ˆˆ ˆ ˆ ˆ ˆ ˆU U
ˆ ,
ˆ ˆˆ ˆˆ ( ) ( )
ˆ ˆ ˆˆ , , ( )
ˆ ˆˆ ˆ( ), ( )
2
p p D
D
p
p p p p
p p p
p
p p
p p
A I R H c p mc V
H H O
E m c c p
A V R VR A O c
O A R V A O c
E mc c pA O c R O c
E E mc
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7/28/2010 5.4 Douglas-Kroll-Hess
Hamiltonian
1/ 22
1 1 1 1
' ' '
1 1
' '
1/ 2
2
2
2 1 0 0
ˆ ˆ ˆ ˆDefine : 1 is an integral operator with the kernel:
ˆ ˆ ˆ ˆˆ ˆ, ' , 'ˆˆ , ' , '
ˆ ˆ ˆ ˆ
2 1For an atom, , '
'
ˆ ˆH
p p p p p p
p p p p
D
U W W W
A R V p p A A V p p R AW p p O p p
E E E E
V p p zep p
U U H U
1 1
1 1 1 1 1 1
2 4
1 1 1
5
1 1
1ˆ ˆ ˆ ˆ ˆˆ ˆ, , ,2
1 ˆ ˆ ˆˆEven terms: ( ), , , ( )2
ˆ ˆOdd term : , ( )
p p
p
U E W W W E
O c W W E O c
W O c
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' 1 1
4
DKH DKH
1
For Coulomb interaction ( ) ˆ 1/
ˆ ˆ ˆ ˆˆ ˆ ˆ
The most frequently used spin-averaged one-component DKH operator
1ˆ ˆH ,H
ˆ ˆˆ (
ij ij
ij i j ij i j ij
p eff
i i i j ij
eff p ext p ex
g r
g U U g U U g
E i V i cr
V i i A i V i R i V
2
2 2 2 4
2
ˆˆ )
ˆ ( ) ˆˆ ˆ, , ( )2
t p p
p
p p p
p p
i R i A i
E i mc cp iA i R i E i p i c m c
E i E i mc
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7/28/2010 5.4 Douglas-Kroll-Hess
Hamiltonian
54
Douglas-Kroll-Hess Hamiltonian
•Regular spin-free and/or spin-dependent Hamiltonian;
variational or perturbational treatment possible, also in DFT(J.
Chem. Phys. 96(1992)6322.).
•Spin-free formulation without correction to two-electron
terms requires only a little additional effort compared to non-
relativistic work
•Correct to second order in the external potential .
Douglas, Kroll, Ann. Phys. 82(1974)89; Hess, Phys. Rev.A 32(1985)756; 33 (1986)
3742; A39(1989)6016; J. Chem. Phys., 96(1992)1227; Chem. Phys. Lett. 184
(1991) 491.
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7/28/2010 5.5.1 Wood-Boring Hamiltonian 55
Two traditional ways to derive
two-component Hamiltonians
2. Elimination of Small Components
2
4
2
1
2
Wood-Boring(Cowan-
ˆˆ ˆ, ( ) ( )
ˆ ˆ( ) ( ) 0
ˆ ˆ( ) ( 2 ) 0
1 ˆ ˆ ˆ ˆ( )(1 ) ( )2 2
Energy-dependent non-hermi
Gri
ti
ffin) equation
L
DS
L S
L S
L L L
h c p I c V r
V c p
c p V mc
E Vp p V E
mc
an Hamiltonian
J.H. Wood and A.M. Boring, Phys. Rev. B 18(1978)2701
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7/28/2010 5.5.1 Wood-Boring Hamiltonian 56
2
2 2
22
MV
Within the central field approximation, for one-electron atom
the radial equation:
( ) ( ) ( )
1 ( 1)( ) nonrelativistic Hamiltonian
2 2
H ( ) mass-velocity2
H
S MV D SO nk nk nk
s
nk
H H H H F r F r
d l lH V r
dr r
V r
2
D
2
SO
21
1Darwin
4
1H Spin-orbit (SO) term
4
(1 ( ) )2
nk
nk
nk nk
dV dB
dr dr r
dV kB
dr r
B V r
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Wood Boring approach :
(1) Ignore the contribution of the G to the self-consistent field V
(2) WB equation for F is solved self-consistently.
(3) G is obtained after F is obtained.
They found the obtained G turn out to besurprisingly close to those of the exact method.
Leads to nonorthogonal orbitals and has been mainly used in atomic finite difference calculations.
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7/28/2010 5.5.2 Pauli Hamiltonian 58
1
2 20
1
2
Expansion of Wood-Boring Hamiltonian with
V-E' '1-
2mc 2
1 'ˆ ˆ ˆ ˆ( )(1 ) ( )2 2
zero order : non-relativistic Schrodinger equation
after first two te
k
k
L L L
V E
mc
E Vp p V E
mc
2rms (order ) :
Historically first reduction of Dirac equation to tw
Pauli-Hamiltonia
o-component .
n
form
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2
2
2
22
2
42
2
2
3 2
4
3 2
22
2
2
1 1 ˆ ˆ
2
1
ˆ
2
: Correction to kinetic energy
: spin-orbit correction
: Contact potential, no classical analogue
ˆ
8
1 ˆ ˆ
8
ˆ
8
8
2
dVS L
m c r dr
dVS L
m c r dr
p
m c
p
Vm c
Vm c
m
H
c
pV
m
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Problems of Pauli Hamiltonian
• Singularity at r=0
• Hold for v2/c2<<1
• P4 is not a well-defined operator on the
appropriate Hilbert space.
Often used in perturbation theory
Magnitude of the correction is quite
sensitive to the contraction of the basis
sets.
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Finite nuclear models (Dyall and Knut,Introduction to Relativistic Quantum Chemistry, P115)
2
nuc 0
nuc 0
nuc
A Uniformly charged sphere
B Fermi two-parameter distribution
C Gaussian distribu
3 ,, 2
;0,
,
1 exp
tion
/
nuc
nuc nuc nucnuc
nuc
nuc
nuc
Z rr r
r r r rr V r
r r Zr r
r
rr r s
r
2
0 2
3exp
2
nuc
rmsZ r
r
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7/28/2010 5.5.2 Pauli Hamiltonian 62
Transformation of the Dirac-Coulomb-
Breit Hamiltonian to order 2 yields the
Breit-Pauli Hamiltonian which is useful
for first-order perturbation theory
evaluation of relativistic effects and
yields satisfactory relativistic corrections
to the energy up to the first and second
transition metal row.
(Itoh, Rev. Mod. Phys. 37(1965)159)
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5.5.3 ZORA Hamiltonian
2
1
2
21
2 2
EExpansion of Wood-Boring Hamiltonian in terms of X=
2mc
1 ˆ ˆ ˆ ˆ
Zeroth order r
( )(1 ) ( )2m 2
ˆ ˆ ˆ ˆ( ) (1
egular approximation (ZORA
) ( )2 2
usef) u
L L L
L L L
V
E Vp p V E
mc
c Ep p V E
mc V mc V
2
2
l for DFT
ˆ ˆ ˆ ˆ( ) ( )2
E. Van Lenthe et al. J. Chem. Phys. 99(1993)4597; 101(1994)9783
ZORAcp p V E
mc V
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ZORA Hamiltonian
• Reproduce spin-orbit splittings well but will be deficient in the spin-free relativistic corrections.
• The ZORA equation is bounded from below and variational: the Dirac negative-energy states (<-2mc2) get translated to appear above the positive-energy states: they are mapped from (-, -2mc2) to (2mc2,).
• Widely used in DFT: get accurate results for one-electron energies and densities of the valence orbitals.
2
2 2
ˆ is missing!
4
p E
m c
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Relativistic PseudopotentialsX.Cao and M. Dolg, in book “Relativistic Methods for Chemists”, edited by Barysz and
Ishikawa, Springer UK, 2010
6.1 (Generalized) Phillips-Kleinman Equation
6.2 Valence electron model Hamiltonian
6.3 Analytical form of PPs
6.4 Core-Polarization potentials
6.5 Core-core/nucleus repulsion corrections
6.6 Energy-consistent PPs
6.7 Shape-consistent PPs
6.8 Model potential method
6.9 DFT-Based effective core potentials
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Approximations: core-valence separation .
frozen-core approximation.
Effective core potentials (ECP), i.e., model
potentials (MP) and/or pseudopotentials
(PP),
if needed augmented by core-polarization
potentials (CPP)**corrects for the frozen-core approximation at both the Hartree-Fock (static
core polarization) and the correlated (dynamical core-polarization, i.e core-
valence correlation) level.
7/28/201066
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Advantages
• Reduction of the computational effort.
• Relativistic effects can be included implicitly by means of a suitable parameterization. quasirelativistic (one- or two- component) method
• All elements in the same group of the periodic table can be treated on equal footing. higher accuracy in studies of trends within a group.
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Disadvantages
reduction of accuracy.
critical (especially for f-elements):
choice of the core (small, medium, and
large closed-shell cores, but also open-
shell cores possible).
change on the form of some operators,
e.g., spin-orbit operator.
7/28/201068
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7/28/2010 6.1 Phillips-Kleinman equaiton 69
Generalized Phillips-Kleinman equation(many-electron system, Weeks and Rice, JCP, 49(6):2741, 1968)
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ˆ, 1
ˆ , 0
: a set of orthonormal function, not necessarily the
ˆeigenfunction of
ˆ
.
ˆ
GPK
v v v v
c c v p
c
v
GPK
v v v v c
c
v
v p v p
V H P PH PH P E P
P P
E
E
H
H
H
V
It is formidable to solve PK or GPK, but they provide a formal
theoretical basis for the subsequent development of PPs applied
nowdays in QC calculations.
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1 1( )
2
( )
denotes the charge of the core .
( )v v
c cc cp
n n
v i
i i j ij
v
p
N
Hr
n n Z
V i V V
Q
Q
Valence-only model Hamiltonian
operator:
Core-electron interaction:
Core-core interaction
) )
( )
:
((c
c
N
c i
i
c
N
cc
QV r
r
Q QV
V i
rVr
2
Core-polarization poten
1
2
tial:
cpp fV
7/28/201070
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1 1/ 2
1/ 20
,
Semilocal (ab initio) pseudopotentials:
Spin-orbit-averaged (scalar-relativisti
( ) ( )
(
c)
) ( ) ( )
(
for :
)
m
j
L l
c i i lj ij l
l
j
lj j j
m j
c
lj L i L
la i i
V VV r r P i r
P i ljm i ljm i
V r r
r V
V V
1
0
1
,
1
, 1/ 2 , 1/ 2
( )
( ) ( ) ( )
2(
Spin-orbi
)2 1
(
t operato
) (
:
)
r
l
L
i l i
l
l
l l l
m l
Li
c so i l i i l
l
l i l l i l
L L
l i
l
r P i r
P i lm i lm i
rV r P i
V
Vl s P i
l
V r V r V r
7/28/201071
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7/28/2010 6.6 Energy-consistent PPs 72
2
lj
2
l
2
l
L ljk lk
V exp
V exp
V exp
typically V 0 and n =n 0
Gaussian expansion of radial parts:
'S
tuttgart pseudopotentials'
a
:
r
ljk
lk
lk
n
i ljk i ljk i
k
n
i lk i lk i
k
n
i lk i lk i
k
r A r a r
r A r a r
r A r a r
e chosen.
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7/28/2010 6.4 CPP 73
CPP: Why?
• Core-valence correlation (dynamic core-
polarization) neglected Leading contribution
in an AE CI treatment would be single
excitations from the core orbitals coupled to
single and higher excitations from the occupied
valence orbitals to the virtual ones.
• Frozen core approximation (static core-
polarization missing) the induced error may
become large for LPP and only a few valence
electrons.
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7/28/2010 6.4 CPP 74
2
denotes the dipole polarizability of the core
electric field at core generated by all other cores ,
nuclei and valence electrons.
Problem: Only apply to large distance of
1
2
the
cpp
f
V f
3
(J.C.P 80,3297(1984))
polarizing charge(s)
from the polarized core(s) .
Meyer et. al have suggested a cut-off function F
removing the singularities, the field at core then reads as:
i
i i
rf F
r
, ,3
2
,
2
,
: Cutoff function
1 exp
1 exp
ne
nc
i e c
i e e i
c c
rr Q F r
r
F
F r r
F r r
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7/28/2010 6.4 CPP 75
Difficulties exsiting for CPP
• One- and two-particle contributions arising
from the valence electrons as well as the
cores/nuclei complex of integral
evaluation over Cartesian Gaussian
functions, energy gradients for geometry
optimization are still missing.
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7/28/2010 6.5 Core-core/nucleus repulsion
correction
76
cc
and can be obtained directly by fitting to
the difference between the electrostatic po
To correct point charge repulsion model (Born-
V
tential of
the
Mayer-typ
( ) exp(
e ansa
tz):
)
B b
r B b r
atomic core electron system modelled by the ECP
and the Coulomb potential due to the ECP core charge,
multiplied with the charge of the approaching nucleus.
G. Igel, U. Wedig, M. Dolg, P. Fuentealba, H. Preuss, H. Stoll, R. Frey, JCP,
1984, 81:2737-2740.
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7/28/2010 6.6 Energy-consistent PPs 77
2
A multitude of electronic configurations/states/levels of
the neutral atom and the low-charged i
Energy adjust
ons.
Total valence energy obtained from finite-differe
ment:
: min
:
: n
PP AE
I I
I
PP
I
E E
I
E
ce
valence-only calculations.
All-electron reference data from Wood-Boring quasirelativistic
HF approach, or in the most recent version finite-difference AE MCDHF
calculations based on the DC r D
:
o
AE
IE
CB Hamiltonian.
Global shift of the AE reference energies, typically of the order of 1% or
less of the ground state total valence ene .
:
rgy
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7/28/201079
Applications of “Stuttgart-Cologne” PPs
(SDD)WB PPs Year Sum of the times cited
without self-citations
4d and 5d transition elements 1990 1928
heavy main group (13-17) 1993 1126
3d transition elements 1987 870
4f-in-core Ln 1989 364
4f-in-valence Ln 1989 (Basis: 2001) 189
5f-in-valence An 1994(Basis:2003) 248
5f-in-core An (tri-) 2007 16
DHF/DCB PPs
group16-18 atoms 2003 294
Group 11-12 atoms 2005 140
4d 2007 58
U 2009 1
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80 7/28/2010
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7/28/2010 6.7 Shape-consistent PPs 81
Methods of adjustment of ab
initio pseudopotentials:• Orbital adjustment: shape-consistent
Reference data: all-electron valence orbitals and orbital energies (independent particle model)Pitzer, Christiansen, …; Durand, Barthelat, …; Hay, Wadt; Stevens, …)
• Energy adjustment: energy-consistentReference data: all-electron total valence energies (quantum mechanical observables; independent particle model and beyond)Stoll, Dolg, Schwerdtfeger, …
Advantage: independent of the quality of the wavefunction (SCF, MCSCF, CI, CC), e.g., adjustment in an intermediate coupling scheme possible !
Disadvantage: relatively high computational effort; problems with neutral or negative charged cores.
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7/28/2010 6.7 Shape-consistent PPs 82
Requirement for p in the shape-
consistent PP
,
,
,
2
2
for
for
is AE valence orbital
is radially nodeless and smooth in the core region.
The pseudopotential fufill the below radial Fock equation:
1
2
v lj c
p lj
lj c
v lj
lj
PP
lj
r r rr
f r r r
r
f r
V
d l
dr
2
, ,
, ', ' ' , , ,2
2
,
( 1) ˆ2
ˆ ˆlj k lj k
PP
lj p lj p l j p lj v lj p lj
n rPP
i lj k lj
ij k
lV W r r
r
QV r A r e P
r
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7/28/2010 6.7 Shape-consistent PPs 83
Hay and Wadt: avaliable for main group and transition elements
based on scalar-relativistic Cowan-Griffin AE calculations LANL**JCP,1985(82):299- 310, 270-282, 284-298
• remains normalized.
• Fl(r) and its first 3 derivatives match v and
its first 3 derivatives at rc .
2
,
2 3 40 1 2 3 4
2
0, ,0
for
3 in the non-relativistic case
b= +2 for relativistic case
1 1+1= 1 ( 1) (1 )
2 4
For relativistic s orbitals the choice 3 and
a
irlip l
i
bclj
l l
C r e
f r r a a r a r a r a r r r
b l
l l Z
b f
s 6 degree polynomial.th
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7/28/2010 6.7 Shape-consistent PPs 84
A useful criterion for more compact Gaussian expansions for PPs JC Barthelat, P Durand, A Serafini, Mol. Phys. 33, 159-180(1977)
1/ 22
, ,
, , , , , ,
, ,
The minimization of the following operator norm:
ˆ ˆ with
ˆ
, obtained with the exact tabulated on a grid from
the radial Fock equation.
p lj p lj
v lj p lj p lj v lj p lj p lj
pp
p lj v lj ljV
, ,, obtained with the analytical potential pp
p lj v lj ljV
Available for almost all elements, as well as for heavier atoms based on DHF AE
calculations applying the DC Hamiltonian. WJ Stevens et al, Can. J. Chem. 1992,
70:612-630, JCP, 98:5555-5565 1993, JCP, 81, 6026-6033(1984)
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7/28/2010 6.7 Shape-consistent PPs 85
Generalized relativistic ECPTitov, Mosyagin
2
, ', ' ' , , ,2 2
2 1
, , , ', ' ' ,2 2
1 ( 1)
Possible problem exsiti
ˆ2 2
1 ( 1
ng in
) ˆ2 2
In the
shape-consistent PPs:
case
PP
lj p lj p l j p lj v lj p lj
PP
lj p lj v lj p lj p l j p lj
d l lV W r r
dr r
d l lV r W r
dr r
1
,
Soluti
of pseudo-valence orbitals with nodes, singularity appear in
the PPs due to the term .
(1) Most of shape-consistent PPs are derived for positive ions (FC errors may
be large) w
o
hich are
n:
p lj r
Phys
chosen in such a way that this problem can not occur.
(2) Interpolating the potentials in the vicinity of the nodes (GRECP)
additional nonlocal terms added besides the standard
semi-local form.
ics of atomic nuclei, 2003, 66(6):1152-1162.
more parameters, not supported by most of the standard quantum chemistry
codes.
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7/28/2010 6.8 Model potential 86
Huzinaga-Cantu equationS. Huzinaga ,AA Cantu, JCP, 1971,55:5543-5549; S. Huzinaga ,D McWilliams, AA
Cantu, Adv. Quantum. Chem. 7, 187-220, 1973
' '
ˆ ,
, ' ,
basis sets do not need to represent
ˆ 2
Comparing to AE HF equation:
Comp
core orbitals
==> smaller basis sets th
aring to Philli
an
ps-Kleinman eq
AE!
c c c v v
a a a
a a
c
aa
v
F a v c
a a v c
F
here keeps its correct nodal structure
in Phillips-Kleinman does not necessarily have radial nodes
==> larger basis sets
uation:
than PPs!
v
p
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7/28/2010 6.8 Model potential 87
Molecular valence-electron model Hamiltonian
2
,
,,
,
,
,
,
A molecular MP is considered to contain an assembly of
non-overlapping core levels:
1ˆ ˆˆ ,2
ˆ
ˆ2
ˆˆ
ˆ ˆ
ˆ
,
MP
v i MP
i i j i i i
MP X
c
c
c
C
C
Q Q QH g i j
P i
V ir r
V i
nJ
r
V i
V
V i
V
,,
, , ,
c,
ˆ
2
ˆ ˆ and stand for the usual Coulomb and exchange operators
related
AIMP (ab initio model
ˆ
potentials, Huzinaga, Seijo, Barandia
to the core orbital ,
l
ran
ocal sph
):
c
c
c c c
c
X
c c
P
K
J K
2
k
,
erically symmetric model potential to represent the
Coulomb core-valence interaction:
1
constraint for least-squares fitting:
Not costly for the
ˆ
calculations of such
k ir
k
ki
k
i
c
C C er
r
C
V
n
integrals! Any desired
accuracy can be easily achieved.
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7/28/2010 6.8 Model potential 88
,
,
Spectral representation of nonlocal exchange part in the space
defined by a set of functions centered on core :
Since the exchange part is short-ranged, only a moderate
ˆ
p
X i p pq q
p q
V r i A i
number
of is needed, often is chosen to be identical to the primitive
functions of the valence basis sets. are calculated during the input
processing of each AIMP calculations.
ˆ 2
p
pq
c c
c
P i
A
2
, 2
are represented by sufficiently large AE basis sets
SO operator (fit to the WB SO term) :
ˆ ˆˆ ˆ ˆ
type AIMP also available.
lk i
c
c
rlkcv so l l
l k i
BV e P i l sP i
r
DKH
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7/28/2010 6.9 DFT-based ECPs 89
DFT-based MP combined with LSD-VWN, Andzelm, Radzio, Salahub, JCP, 83, 4573-4580,1985
Starting from the Kohn-Sham(KS) equations for a spin-polarized
system of valence electrons, and assuming orthogonality between
and of spin ( , ), Huzinaga-Cantu equation may be
rewritten as:
v
v c
n
,
2
, , ,
, , ,
, ,
ˆ 2
ˆ ˆ ˆ
' '1ˆ2 '
denote the spin-up ans spin-down densities for the valence
orbitals,
c c c v v v
c
MP
v
v
v xc v v
v v
v v v
F
F F V
r drQF v
r r r
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7/28/2010 6.9 DFT-based ECPs 90
2
,
,MP ,
,MP
,MP ,MP ,
,
( ) , is an occupation number
ˆ ˆ ˆ ˆV 2
ˆThe molecular V is written as a sum over the atomic MPs:
ˆ ˆ ˆ ˆV V
'ˆ
vn
v i i i
i
MP
c c c
c
MP
cMP c
r f r f
V P P
V P
r drnV
r
2
, , ,
,
,
'
'
ˆThe analytical form of may be written as:
ˆ ,
The core orbitals for the projection operator are approximated
by a least square fit procedur
k
xc c c
MP
rMP
k k c
k k
vr r
V
eV A A n
r
e using an expansion of Gaussian
functions.
The reference atomic orbitals were obtained form CG/WB-type
LSD-VWN finite-difference atomic calculations.
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7/28/2010 6.9 DFT-based ECPs 91
Norm-conserving DFT-based shape-consistent PPsH through Pu, Hamann, Schlueter, and Chiang, Phys. Rev Lett. 1979, 43:1494-1497
Comparing to ab initio shape-consistent PPs:
• p,lj is radial KS orbitals at the original AE orbital energies v,lj
• Additional norm-conserving properties of p,lj :
1. The integral from 0 to r of the real and pseudo charge densities agree for r>rc for each valence state.
2. The logarithmic derivatives of the real and pseudo wavefunction and their first energy derivatives agree for r>rc
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7/28/2010 6.9 DFT-based ECPs 92
Density functional semi-core PPs
from H to Am DSPP, Delley, Phys. Rev. B 66, 155125-1-155125-9,2002
• Suggested for use with local orbital
methods.
• Based on a minimization of errors with the
norm conservation conditions for two to
three relevent ionic configurations of the
atom.
• AE reference were defined using PBE
functional.
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Generation of PPs
• Choice of reference data
(AE/DHF/DC, AE/DHF/DCB)
• Choice of the core (energy,
spatial shape)
• Pseudopotential adjustment
(shape- energy- consistent)
• Valence basis sets optimization
(generalized/segmented
contraction)
• Calibration studies (atoms,
molecules)
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Spin-orbit configuration
interaction methods7.1 Breit-Pauli spin-orbit operators
7.2 Mean-field approximations for spin-orbit
interaction
7.3 SO-CI calculations (one-step methods,
two-step methods)
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Douglas-Kroll-Transformed Spin-Orbit Operators(Hess, 1997, Ber. Bunsen Ges. Phys. Chem. 101,1)
1 2
`
1 1 13
2 2
Applying the DK transformation to the DCB Hamiltonian
and seperate spin by using
we obtain:
ˆ ˆ ˆ ,
ˆ
ˆ ,
SO SO SO
i i j
SO ii i i i
i
SO
u v u v i u v
H H i H i j
rH i Z f p p f p
r
H i j f
23
2 23
2
1 22 2
, ,
2 , ,
, , ,2
ij
i j i i i j
ij
ij
i j j i j i
ij
i ji ii i j i
i i i
rp p p f p p
r
rf p p p f p p
r
A A ccA E mcf p f p p A
E mc E mc E
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Breit-Pauli Spin-Orbit Operators
2 2 2 4 2 2 4 3 2
2
2
1 2
1 2
`
1
By keeping only the lowest-order term
/ 2 / 8 ...
12
1( ) ( , )
2
ˆ become so called Breit-Pauli spin-orbit operators:
ˆ ˆ ˆ ,
ˆ
i
p
i
p
i i j
SO
SO SO SO
i i j
E p c m c mc p m p m c
mc
E mcA
E
f p f p pmc
H
H H i H i j
H
2 2 2 23
2
3
2 3
2
1
2
1
2
1
ˆ ,
S
ij
j i
i
O ii i
i
ij
i
ij
O
j
i
S
ri Z S
rS
rS p
m c r
pm c r
H i pm c r
j
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Two-electron SO term• Opposite sign comparing to the one-
electron contribution.
• 20 to 50% of the total spin-orbit splitting.
• The number of two-electron SO integrals is larger than the number of two-electron coulomb integrals by almost an order of magnitude.
Look for an approximation in which only one-electron SO integrals are evaluated.
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Mean-Field Approximations for
Spin-Orbit Interaction• Due to the short-range property of the spin-orbit two-
electron operator, one can neglect integrals that have contributions from more than one atomic center in the molecule.
• Neglect integrals caused by doubly excited state.
• Averaging the two-electron contribution to the spin-orbit matrix element over the valence shell.
The two-electron spin-orbit integrals are replaced with atomic mean-field integrals (AMFI, introduced by Hess et al. (1996)). One-electron operator for the spin-orbit interacton obtained !
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, ,
1
,
, , 1/ 2
0
, 1/ 2
1
ˆ ˆ ˆ ( ) ( ) ( )
Scalar-relativistic PP:
ˆ ˆ( ) ( )
ˆ ( ) ( ) ( )
Spin-orbit P
ˆ ˆ ˆ( ) ( 1)2 1
P:
l
cv i c a i c so i
L
c a i l i L i l L
l
i
l
i
c so i l l l l
l
l
l l l
m l
V rV r
V r V r V r
lP i l P
V r V r V r P i V r
P i l m
i
m i i
l
l
1
1
,
1
2 ˆ ˆˆ ˆ( )2 1
L
Ll i
c so i l i i l
l
V rV r P i l s P i
l
Relativistic PPs
WC Ermler, YS Lee, PA Christiansen, KS Pitzer, CPL, 1981, 81:70-74
RM Pitzer, NW Winter, JPC, 1988, 92:3061-3063
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One-step SO-CI: double group CI
(DGCI)
• CIDBG (Christiansen)
• SICCI (DiLabio, selected intermediate coupling CI)
• LUCIA (Esser, Lund CI Approach)
The breakdown of the nonrelativistic symmetries in the CI process makes high correlated treatment very difficult.
The double-group symmetries multiply rougly 6 times the number of determinants arising from a given spatial configuration.
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Two-step (CILs+ SO) methods
• First step: For a large defined target space T, extensive CI calculations are carried out in a scalar relativistic approximation for all the LS states under interest (Em, m)
reduced representation m spanned on a determinantal intermediate model subspace S (S is smaller than the target space T).
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Second step:
• Add all spatial and spin degenerate
components to m, thus defining the model
pace S'.
• In order to take into account the effects of
the largest CI calculations in the target
space, a Bloch-type effective Hamiltonian
is defined on the model space.
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ˆ ˆ
ˆ 1
'
ˆ,
ˆ is the spin-orbit operator.
LS
m n m n nm
SO
m n nm
m
LS
m n nm m
SO
H H
H
where S
H E
H
Problem: The repolarization of the wave function by spin-orbit
interaction, which becomes important, cannot be taken into
account easily.
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EPCISO: effective and polarized CI-SO
• The spin-orbit repolarization of the wave
function is included by means of singly-
excited configurations on the mode space.
• The full Hamiltonian diagonalized on the
basis of determinants, accounting for
electronic correlation by means of an
effective Hamiltonian.
V. Vallet, L. Maron, C. Teichteil, JP Flament, JCP, 2000, 113:1391-1402
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EPCISO (effective and polarized CI-SO)
• First step: extensive CI calculations (SO free) for a large defined target space T.
• Second step: Choice of S„
Choice of an intermediate reference subspace Minit of determinants belonging to T, e.g., states which have the biggest weights in the wave functions.
all the excited configurations which have a nonzero spin-orbit interaction with the reference configurations belonging to Minit are created. The variational subspace Mvar includes all the reference configurations and the SO dominant singly-excited configurations.
In order to get S2 eigenfunction, all possible determinants arising from the configurations Mvar are generated and the determinantal subspace Md
var are created. On which the full Hamiltonian is represented.
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var
0 0 0
0
0 0
,
0 0* 0
0 , ,
, ,
ˆ
; are determinants
Spin-free effective Bloch-type Hamiltonian
ˆ ˆ
ˆ ˆ ˆ ˆ
: obtained from a sophisticated co
d
LS
m m m
m m i
i M
LS LS
m i m j m m
m i j
LS total LS so
m m m
m
H E
C i i
H H C C i E E j
H E H H H
E
rrelation treatment
and projected on the basis of determinants.
: model wave function is crucial for the
accurate calculations of observables, e.g. transitio
Proble
n mom nt.
m
e
m
V. Vallet, L. Maron, C. Teichteil, JP Flament, JCP, 2000, 113:1391-1402
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全国理论及量子化学暑期学校,2010,北京
114
Energies of the 4fn+16s2/5fn+17s2 and 4fn−15d26s2/5fn−16d27s2 configurations with respect to the
4fn5d16s2/5fn6d17s2 configurations of the lanthanides/actinides from Dirac–Hartree–Fock calculations
M. Dolg,X. Cao, Lanthanides and actinides: Computational methods, in: Computational inorganic and bioinorganic chemistry. Ed.
by Edward I. Solomon, R. Bruce King and Robert A. Scott, John Wiley & Sons, Ltd, 2009, ISBN-13:978-0470699973.
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Third (left bars) and fourth (right bars) ionization potentials of the actinides estimated by PP multi-reference averaged
coupled-pair functional (MR-ACPF) calculations including spin-orbit estimated and extrapolation to the basis set limit
(dotted bars). Relativistic contributions estimated from the difference of MCDHF (DCB Hamiltonian) and Hartree-Fock
calculations (striped bars) and electron correlation contribution estimated from PP MR-ACPF correlation energies
extrapolated to the basis set limit (filled bars). Crosses with error bars denote the experimentally measured values for Th
and the semiempirical estimates for U
X. Cao, M. Dolg, Coord. Chem. Rev., 2006,250(7-8):900-910
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Total relativistic contributions estimated from the difference of MCDHF (DCB Hamiltonian) and
Hartree-Fock calculations (striped bars) and spin-orbit contributions estimated from pseudopotential
calculations with and without spin-orbit operator (filled bars) to the third (left bars) and fourth (right
bars) ionization potentials of the actinides
Note: Calculations of “chemical accuracy” have to be accurate to 0.05 eV !
X. Cao, M. Dolg, Coord. Chem. Rev., 2006,250(7-8):900-910
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全国理论及量子化学暑期学校,2010,北京
117
NR RE
6A 0.0 0.0
6A 658 0.0
6637 0.0
65246 4785
66256 5378
4A6519 7507
4A6958 7507
48407 7507
2A24737 25132
2A26405 25900
229785 29587
8A28446 26208
8A35470 36721
Calculated relative energies (cm-1) for low-lying states of FeOH
obtained with (RE)/without (NR) taking into account relativistic effects
at CASSCF level (X. Cao, Chem. Phys., 2005,311, 203-208)
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Calculated bond lengths (Å) and relative energy (cm-1) to the lowest state
(6A for NR, 6 for RE, cf. Table 1) for some low-lying states of FeOH at
CASSCF level X. Cao, Chem. Phys., 2005,311, 203-208
全国理论及量子化学暑期学校,2010,北京
118
NR RE
6 RFe-O 1.8237 1.7922
RO-H 0.9461 0.9422
E 641 0.0
6 RFe-O 1.8618 1.8272
RO-H 0.9464 0.9426
E 5246 4785
6 RFe-O 1.8480 1.8057
RO-H 0.9493 0.9453
E 6278 5410
4 RFe-O 1.8097 1.7773
RO-H 0.9295 0.9258
E 8429 7529
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Further reading
• Introduction to Relativistic Quantum Chemistry ,
K.G. Dyall and K. Fagri Jr, Oxford University Press, New
York, 2007.
• Relativistic Methods for Chemists Edited by Maria
Barysz and Yasuyuki Ishikawa ,Springer, 2010
全国理论及量子化学暑期学校,2010,北京
119
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120 7/28/2010
Acknowledgements
• Prof. Dr. Michael Dolg , University of Cologne
• Prof. Dr. Weihai Fang, Beijing Normal University