Introduction to

Relativistic Quantum

Chemistry

Xiaoyan Cao

Institute for Theoretical Chemistry

University of Cologne, Germany

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

2全国理论及量子化学暑期学校，2010，北京

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

3全国理论及量子化学暑期学校，2010，北京

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

4全国理论及量子化学暑期学校，2010，北京

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

5全国理论及量子化学暑期学校，2010，北京

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.

6全国理论及量子化学暑期学校，2010，北京

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

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.

8全国理论及量子化学暑期学校，2010，北京

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)

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

Time dilation

Twin paradox

J.C. Hafele, R.E. Keating, Around-the-world Atomic Clocks: Predicted

Relativistic Time Gains, Science 177 (1972)166

11

全国理论及量子化学暑期学校，2010，北京

Lorentz contraction

Ladder Paradox12全国理论及量子化学暑期学校，2010，北京

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.

13全国理论及量子化学暑期学校，2010，北京

Equivalence of mass and energy

14全国理论及量子化学暑期学校，2010，北京

'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

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

15全国理论及量子化学暑期学校，2010，北京

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

16全国理论及量子化学暑期学校，2010，北京

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

17全国理论及量子化学暑期学校，2010，北京

全国理论及量子化学暑期学校，2010，北京

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.

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

19全国理论及量子化学暑期学校，2010，北京

( )

( )

( )

(

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

20全国理论及量子化学暑期学校，2010，北京

Free-electron solutions of the

time-independent Dirac equation

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

21全国理论及量子化学暑期学校，2010，北京

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

22全国理论及量子化学暑期学校，2010，北京

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

23全国理论及量子化学暑期学校，2010，北京

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.

24全国理论及量子化学暑期学校，2010，北京

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.

25全国理论及量子化学暑期学校，2010，北京

2 2( ) ( ) ( )r g r f r

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28全国理论及量子化学暑期学校，2010，北京

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)

3. Relativistic Electromagnetic Interactions

Chemical concepts Molecules ---

aggregates of atoms linked by

electromagnetic interactions

Proper relativistic description of these

interaction

30

全国理论及量子化学暑期学校，2010，北京

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

31 全国理论及量子化学暑期学校，2010，北京

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.

32全国理论及量子化学暑期学校，2010，北京

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

33全国理论及量子化学暑期学校，2010，北京

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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.

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)

Siegfried Wouthuysen (1916-

1996), Dutch Physicist

L.L. Foldy, (1919-2001), born in

Sabinov, Czechoslovakia with

Hungarian roots, immigrated to

the US in 1921

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

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

' 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

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.

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

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

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.

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

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

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.

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

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)

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

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

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

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

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.

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

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.

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

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

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.

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.

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

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.

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.

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

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

80 7/28/2010

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.

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

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

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)

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.

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

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.

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

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

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.

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

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.

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)

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)

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

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

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.

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 !

, ,

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

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.

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).

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.

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.

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

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.

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

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.

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

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

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)

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

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

120 7/28/2010

Acknowledgements

• Prof. Dr. Michael Dolg , University of Cologne

• Prof. Dr. Weihai Fang, Beijing Normal University