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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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Relativistic quantization 2
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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Relativistic quantization 3
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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Free particle Dirac equation
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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Free particle Dirac equation
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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Free particle Dirac equation
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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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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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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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