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L. Visscher (2011)

Lucas VisscherVU University Amsterdam

An introduction to

Relativistic Quantum Chemistry

Lucas Visscher - VU UniversityAmsterdam


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Outline

Day 1: Introduction and foundations Introduction Dirac equation Relativistic effects

Day 2: All electron methods 4-component methods Perturbation theory of relativistic effects 2-component methods

Day 3: Effective core potentials The frozen core approximation Model potentials Pseudopotentials


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Assumptions in Quantum Chemistry

Born-Oppenheimer approximation Electronic and nuclear motion can be decoupled Electronic energies for motion around clamped nuclei provide

potential energy surfacesfor nuclear motion

Coupling between surfaces can be studied by perturbation theory

Nuclear charge distribution Point nucleus approximation Nuclear deformations are treated in perturbation theory

Relativity The speed of electrons is always far below the speed of light Goal is to find time-independent wave functions (stationary states) Magnetic effects are neglected ortreated in perturbation theory


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Time-independent Dirac equation

Nuclei appear parametrically via their charge,position, and (if applicable) magnetic moment

Separate the time and position variables

H(r, t) = i(r,t)t

(r, t) =(r)(t)

H(r) =E(r)

(t) = eEt/ i

Time dependent Schrdinger or Dirac equation

Time independent Schrdinger or Dirac equation


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Lucas Visscher ACMM - VU University Amsterdam - 5Lucas Visscher ACMM - VU University Amsterdam - 5

Diracs view

Dirac (1929)

The general theory of quantum mechanics is nowalmost complete, the imperfection that still remainbeing in connection with the exact fitting in of thetheory with relativistic ideas. These give rise todifficulties only when high speed particles areinvolved, and are therefore of no importance inthe consideration of atomic and molecularstructure and ordinary chemical reactions in wichit is, indeed, usually sufficiently accurate if oneneglects relativity variation of mass with velocityand assumes only Coulomb forces between thevarious electrons and atomic nuclei.

The fundamental laws necessary for themathematical treatment of large parts of physicsand the whole of chemistry are thus fully known,

and the difficulty lies only in the fact thatapplication of these laws leads to equations thatare too complex to be solved.


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6

Later insights

Pekka Pyykk and Jean-Paul Desclaux (1979) The chemical difference between the fifth row and the

sixth row seems to contain large, if not dominant,relativistic contributions which, however, enter in anindividualistic manner for the various columns andtheir various oxidation states, explaining, for example,both the inertness of Hg and the stability ofHg2

2+.These relativistic effects are particularly strongaround gold. A detailed understanding ofthe interplay

between relativistic and shell-structure effects will formthe impact of relativity on chemistry.

Jan Almlf & Odd Gropen (1996) While the incorporation of these effects sometimes

increases the computation labor, the increase isgenerally reasonable, and certainly much less than in,

e.g. the transition from semiempirical to ab initiomethods for routine quantum chemistry applications.We predict, therefore, that relativistic corrections inone form or anotherwill be included in the majority ofall quantum chemistry calculations before the end ofthis decade.


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Special relativity

Postulate 1: All inertial frames are equivalent

Postulate 2: The laws of physics have the same form inall inertial frames

Lorentz coordinate transformations mix time and spacePostulates hold for electromagnetism (Maxwell relations)Postulates do not hold for Newtonian mechanicsDevelop quantum theory from classical relativistic equations

and make sure electron spin is described7


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Non-relativistic quantization 1

The nonrelativistic Hamilton function

8

Quantization

H=T+V=p

2

2m+ q r( )

H i

t; pi

H(r,t) = i

t

(r, t)

H=

2m

2+ q(r)


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Non-relativistic quantization 2

The nonrelativistic Hamilton function

9

Quantization

H= T+V =

2

2m+ q r( )

= p qA

H i

t; pi

H(r, t) = i

t

(r, t)

H=

2m

2+

iq

2m A+ A ( )+

q2

2m

A2+q(r)

Mechanical () and canonical momentum (p)Principle of minimal electromagnetic coupling

Coulomb gauge: A( ) = 0


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Spin and non-relativistic quantization 1

We can, however, also write the the Hamilton function as

10

Quantization

E= q+ ( )

2

2m

i,

j$% &'

+

= 2ij

H= q+ 12m

i+ qA( ){ }2

= q 2

2m ( )

2

+

q2

2m A( )

2

+

iq

2m ( ), A( )[ ]+

Kronecker delta and Levi-Civita tensor,Summation over repeated indices

i j =ij+ iijk kxyz = zxy = yzx =1

xzy = zyx = yxz = 1


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A r( ) f(r) = f(r)A r( )( )

=f(r)( ) A r( )+ f(r)A r( )

= A f(r)( ) +Bf(r)

H=

2m

2+ q+ q2

2mA

2

+

iq

2m A + A ( )

q

2m A + A ( )

Spin and non-relativistic quantization 2

u( ) v( ) = u v( )+ i u v( )

H= T+ q+ iqA + q22

A2q

2B

A is a multiplicative operator

chain rule

Use definition of B

in atomic units


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Spin in NR quantum mechanics

The Pauli Hamiltonian in two-component form

12

Second derivatives w.r.t. position, first derivative w.r.t. timeLinear in scalar, quadratic in vector potential

Is not Lorentz-invariant

LAd hocintroduction of spin. No explanation for theanomalous g-factor (ratio of 2 between magnetic momentand intrinsic angular momentum)

LNo interaction between angular momenta due to theorbital and spin: spin-orbit coupling is relativistic effect

1

22 + q+ iqA +

q2

2mA

2 q

2Bz

q

2Bx iBy( )

q

2Bx + iBy( )

1

22 + q+ iqA +

q2

2mA

2+

q

2Bz

&

'

(((

)

*

+++


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Relativistic quantization 1

Take the classical relativistic energy expression

13

E q= m2c

4+ c

2

2[ ]1/ 2

Quantization recipe gives

After series expansion of the square root this could provide

relativistic corrections to the Schrdinger Equation

Disadvantage : Difficult to define the square root operator

in terms of a series expansion (A and p do not commute).

Not explored much.

"E = mc2"

it = q+ m2c4+ c

2

2

Without EM-fields


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Eliminate the square root before quantization

14

E q( )2

= m2c4+ c

2

2

Quantization

Klein-Gordon Equation

JLorentz invariantLNo spinLThe KG-equation can be used for spinless particles

i

tq%&'

(

)*

2

= m2

c4

+ c2

2( )

*r( ) r( ) dr = f(t) Charge is conserved, particle number is not


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Define a new type ofsquare root

15

Quantization

The Dirac equation

Suitable for description ofone electron- Relativistic kinematics- Charged spin particle

i

t= mc 2 + c + q( )

E q= mc2 + c

i,j[ ]+

= 2ij i,[ ]+ = 0 2=1


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The Dirac equation

JFirst derivatives with respect to time and positionJLinear in scalar and vector potentialsJLorentz invariant and are 4-component matrices

16

mc2

+ c + q( ) r,t( ) = i r,t( )

t

x=

0 x

x 0

$

%&

'

()

y=

0 y

y 0

$

%&

'

()

z=

0 z

z 0

$

%&

'

() =

I 0

0 I

$

%&

'

()


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17

The Dirac Hamiltonian

H= mc2

+ c + q

=

mc2+ q 0 cz c(x iy )

0 mc2+ q c(x + iy ) cz

cz c(x iy ) mc2+ q 0

c(x + iy ) cz 0 mc 2 + q

(

)

***

*

+

,

---

-

Four component wave function

1) Spin doubles the number of components

2) Relativity doubles the number of components again


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Charge and current density

Charge density

Current density

Continuity relation

18

r, t( ) = q r,t( ) r, t( )

j r,t( ) = q r,t( ) c r,t( )

r, t( )t + j r,t( ) = 0

c is the relativistic velocity operator


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19

Free particle Dirac equation

Take simplest case : = 0 and A = 0 Use plane wave trial function

(r) = eikr

a1

a2

a3

a4

$

%

&&

&

&

'

(

))

)

)

Emc2( )a1 ckza3 cka4 = 0

Emc2( )a2 ck+a3 + ckza4 = 0

ckza1 cka2 + E+ mc2( )a3 = 0ck

+a1 + ckza2 + E+ mc

2( )a4 = 0

k= kx iky

Non-relativistic functional form with constants aithat are to be determined

After insertion into time-independentDirac equation


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Two doubly degenerate solutions

Compare to classical energy expression

Quantization (for particles in a box) and prediction ofnegative energy solutions

E2m

2c4 c

22k2( ) = 0

E+=+ m

2c4+ c

22k2

E = m2

c4

+ c2

2

k2

E= m2c4+ c

2p

2


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Wave function for E = E+

Upper components are the Large components Lower components are the Small components

a2 = 0 ; a3 = a1ckz

E++ mc

2; a4 = a1

ck+

E++ mc

2

k p


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Wave function for E = E-

Role of large and small components is reversed Application of variational principle is more difficult

MVariational Collapse Minmax optimization instead of straight minimization

a4= 0

a1= a

3

ckz

Emc

2 a

3

pz

2mc

a2= a

3

ck+

Emc

2 a

3

p+

2mc


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Lucas Visscher ACMM - VU University Amsterdam - 23Lucas Visscher ACMM - VU University Amsterdam - 23

Dirac sea of electrons

Negative energy solutions are alloccupied

Pauli principle applies

J Holes in this sea of electrons areseen as particles with positive

charge: positrons (1933)

L Infinite background charge

QED (Quantum Electrodynamics)to properly account for

contribution of negative energystates

No-pair approximation

mc2

-mc2

2 e 3 e +1 e+2mc2


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More than one electron

General form of a time-independent Hamiltonian

Wave function

Difference between relativistic and non-relativistic calculations isin the calculation of integrals overh and g

Second-quantized form of equations is identical to non-relativistic theory when using the no-pair approximation

H= hii=1

N

+1

2

gij

ji

N

i=1

N

N x 4 components 1,

,N( )

, i,, j,( ) = , j,, i,( ) anti-symmetry


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25

Electron-electron interactions

In molecular calculations:

Coulomb, Gaunt and retardation terms Zeroth order is the instantaneous electrostatic interactions First correction describes the magnetic interactions Second correction describes retardation of the interaction

g12

CoulombBreit=

1112

r12

=

1

r12

1

c

2

r12

c1c

2

1

2c2

c1

1( ) c2 2( )r12

Coulomb: diagonal operator

Gaunt: off-diagonal operator

Retardation


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26

The hydrogenic atom

Starting point for the LCAO approach Can be solved by separating the radial and angular

variables (see Dyall & Faegri or Reiher & Wolf)

The exact solutions help in devising basis setapproaches and in understanding the chemicalbonding in the relativistic regime

mc2

Z

r

c pc p mc2 Z

r

#

$

%

%%%

&

'

(

(((

L

r( )

Sr( )

#

$

%%%%

&

'

((((

= E

L

r( )

Sr( )

#

$

%%%%

&

'

((((


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The hydrogenic atom: Energies

The exact non-relativistic energy

The exact relativistic energy

Spin-orbit couping :

27

E= mc2

/ 1+Z/c

n j1

2+ (j+1/2)

2 Z

2

c2

#

$%%

&%%

'

(%%

)%%

2

j = l s

ENR=

Z2

2n2

Energy depends on orbital and spin variables


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28

The hydrogenic atom: Orbitals

Write orbitals as product of radial and angular (2-spinorfunctions)

Solutions to the radial equation

L r( )S r( )

#

$%

&

'(=

1

r

Pn r( ),m ,( )

iQn r( ),m ,( )

#

$%

&

'(

Pn r( ) = Nn

Per

rF1r( )+ F2 r( )( )

Qn r( ) = Nn

QerrF1r( ) F2 r( )( )

Rnl

r( ) = NnlR

e 2E( )r

rl +1

F r( )

Large component

Small component

Nonrelativistic

=

2E 1+

E

2mc 2

$

%&

'

()

= 2

Z2

c2<

l 0 1 1 2 2 3 3

j 1/2 1/2 3/2 3/2 5/2 5/2 7/2 -1 1 -2 2 -3 3 -4

s1/2 p1/2 p3/2 d3/2 d5/2 f5/2 f7/2


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Orbital stabilization: increase in ionization energy

Alkali metals

0.10

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.20

0 20 40 60 80 100 120 140

Nuclear Charge

nsorbitaenergya nonrelativistic

relativistic

H, Li, Na, K, Rb, Cs, Fr, 119


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30

Orbital destabilization and spin-orbit splitting

B, Al, Ga, In, Tl, 113

Group 13

0.0

0.1

0.2

0.3

0.4

0 50 100 150

Nuclear Charge

nonrelativistic

relativistic

relativistic

Group 12

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 50 100 150

Nuclear Charge

nonrelativistic

relativistic

relativistic

Zn, Cd, Hg, Cn


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Orbital contraction

The outermost s-orbital becomes more compactAlkali metals

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

0 20 40 60 80 100 120 140

Nuclear Charge

ns

in

au

nonrelativistic

relativistic


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Orbital expansion

The outermost p- and d-orbitals expand

Group 13

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 50 100 150

Nuclear charge

nonrelativisticrelativisticrelativistic

Group 12

0.8

1.0

1.2

1.4

1.6

1.8

2.0

20 70 120Nuclear Charge

nonrelativisticrelativisticrelativistic


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Ln-An contraction

Ln-An contraction is partly caused by relativisticeffects

Trend expected from the atomic calculations isconfirmed by calculations on LnF, AnF, LnH3 and

AnH3 molecules.


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Summary

Relativity

Dirac equation for electron in non-quantized electromagnetic potential

Mean-field electronic potential non-relativistic 2-electron operators before/after finding bound state solutions

(a.k. atomic or molecular orbitals)

Higher-order corrections from QED

Chemistry Scalar relativistic effects

stabilization/destabilization contraction/expansion

Spin-orbit coupling Important for heavy elements

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