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An introduction toRelativistic Quantum
Chemistry
Lucas VisscherVrije Universiteit Amsterdam
2
The extra dimension
Method
Hartree-Fock Full CI
Basisset
Minimal
Complete
Hamiltonian
Dirac-Coulomb-Breit
NR
3
Course outline: lecture 1+2 Relativistic Quantum Chemistry
Special Relativity The Dirac equation
• Free particles• Second quantization and QED : a short detour• Hydrogenic atom
Approximate Hamiltonians Breit-Pauli perturbation theory The regular approximation (ZORA) The Douglas-Kroll-Heß method Four-component methods Direct perturbation theory
4
Course outline: lecture 3 Effective Core Potentials
Basic assumptions Ab initio Model Potentials Energy-Consistent Pseudopotentials Shape-Consistent Pseudopotentials
Computational aspects All-electron or valence-only ? Wave Function Theory or Density Functional Theory ? Spin-orbit or scalar relativistic ?
Relativistic effects in chemistry Dissociation energies Bond lengths and bond strengths Dipole moments NMR shieldings Electric Field Gradients
5
Coordinate transformation Galilean transformation
Simple addition of velocities, no speed limit!
w =dx
dt=
d( " x + vt)
dt=
d " x
dt+ v = " w + v
!
x = " x + vt
y = " y
z = " z
6
Maxwell
137
7
Thought experiment 1 Two rotating double stars A and B
Does their light reach earth at different times ? Do we observe one star at two positions ? NO -> The speed of light (c) does not depend on the
motion of the emitting stars Is there some immobile substance (ether) that transmits
the radiation? NO -> Need better theory of mechanics
8
Thought experiment 2 Take two observers inside and outside a moving train The train passes the stationary observer, waiting for the
railroad sign, on its way to a nearby tunnel... They both know the speed of light and wonder when the
light of the railroad sign will illuminate the tunnel
The observer outside has an easy job : t = distance / c The observer inside needs to correct for the fact that the
tunnel is moving towards him (and the light) and gets aslightly smaller t
Their conclusion: with c constant, t needs to be relative
9
Special relativity c constant and t variable gives
Galileo We need a new transformation
!
x = " # x + v # t ( )
y = # y
z = # z
t =$ # t + % # x ( )
!
x = " x + vt
y = " y
z = " z
!
x2
+ y2
+ z2
= c2t2
" x 2
+ " y 2
+ " z 2
= c2 " t
2
Scaling factor
No dependence on y and zsince clocks in the yz planewould disagree (reciprocalrelation between the frames)
10
Lorentz transformation Substitute this ansatz in the unprimed equations and solve
Lorentz transformation
Time and spatial coordinates transform into each other 4-dimensional space-time coordinate system Nonrelativistic limit (c → ∞) gives Galileo transformation!
x = " # x + v # t ( )
y = # y
z = # z
t = " # t +v # x
c2
$
% &
'
( )
!
r = " r + vv # " r ( ) $ %1( )
v2
+ $ " t
&
' (
)
* +
t = $ " t +v # " r ( )c2
&
' (
)
* +
Generalize to 3d
!
" = # = (1$v2
c2)$1/ 2 % =
v
c2
11
Relativistic Quantum Mechanics
1905 : STR Einstein : “E = mc2”
1926 : QM Schrödinger equation
1928 : RQM Dirac equation
1949 : QED Tomonaga, Schwinger &
Feynman
12
Quantization
!
H = T +V =" 2
2m+ q# r( )
" = p$ qA
!
H " ih#
#t ; p"$ih%
ˆ H &(r, t) = ih#
#t&(r,t)
ˆ H = $h
2mˆ % 2 +
iqh
2mˆ % ' ˆ A + ˆ A ' ˆ % ( ) +
q2
2mˆ A 2 + q ˆ ( (r)
Non-relativistic quantizationThe nonrelativistic Hamiltonian function
13
Spin and non-relativistic quantization 1
We can also write the the Hamiltonian function as
Quantization
!
E = q" +# $ %( )
2
2m
# i# j = &ij + i'ijk# k
!
ˆ H = q ˆ " +1
2m# $ %ih ˆ & + q ˆ A ( ){ }
2
= q ˆ " %h
2
2m# $ ˆ & ( )
2
+q
2
2m# $ ˆ A ( )
2
+iqh
2m# $ ˆ & ( ), # $ ˆ A ( )[ ]
+
Kronecker delta and Levi-Civita tensor
!
ˆ " #A r( ) f (r) = ˆ " # f (r)A r( )( )
= ˆ " f (r)( ) #A r( ) + f (r) ˆ " #A r( )
= $ ˆ A # ˆ " f (r)( ) + Bf (r)
!
ˆ H = "h
2mˆ # 2 + q ˆ $ +
q2
2mˆ A
2
+iqh
2mˆ # % ˆ A + ˆ A % ˆ # ( ) "
qh
2m& % ˆ # ' ˆ A + ˆ A ' ˆ # ( )
Spin and non-relativistic quantization 2! "u( ) ! " v( ) = u " v( ) + i! " u # v( )
!
ˆ H = ˆ T + q ˆ " + iq ˆ A # ˆ $ +q
2
2
ˆ A 2 %
q
2& #B
A is a multiplicative operator
chain rule
Use definition of B
in atomic units
15
Spin in NR quantum mechanicsThe Pauli Hamiltonian in two-component form
Second derivatives w.r.t. position, first derivative w.r.t. timeLinear in scalar, quadratic in vector potential→ Can not be Lorentz-invariant
Ad hoc introduction of spin.The anomalous g-factor(ratio magnetic moment to the intrinsic angularmomentum) is not well explained
No spin-orbit coupling
!
"1
2#2 + q$ + iqA % # +
q2
2mA2 "
q
2Bz "
q
2Bx " iBy( )
"q
2Bx + iBy( ) "
1
2#2 + q$ + iqA % # +
q2
2mA2 +
q
2Bz
&
'
( ( (
)
*
+ + +
16
Relativistic quantization 1Take the classical relativistic energy expression
!
E " q# = m2c4 + c 2$ 2[ ]
1/ 2
Quantization recipe gives
After series expansion of the square root this couldprovide relativistic corrections to the Schrödinger Equation
Disadvantage : Difficult to define the square root operatorin terms of a series expansion (A and p do not commute).Not explored much.
!
"E = mc2"
!
ih"#
"t= m
2c4
+ c2$ 2 # % q&#
Without EM-fields
17
Relativistic quantization 2Eliminate the square root prior to quantization
!
E " q#( )2
= m2c4 + c 2$ 2
Quantization
Klein-Gordon Equation
Lorentz invariant No spin
The KG-equation is used for mesons (that have no spin)
!
ih"
"t# q ˆ $
%
& '
(
) *
2
+ = m2c
4 + c 2ˆ , 2( )+
!
"*r( )" r( )# dr = f (t) Charge is conserved, particle number is not
18
Relativistic quantization 3Define a new type of “square root”
Quantization
The Dirac equation
Suitable for relativistic description of electrons
!
ih"#
"t= $mc 2 + c% & ˆ ' + q ˆ ( ( )#!
E " q# = $mc 2 + c% & '
% i,% j[ ]+
= 2(ij ) % i,$[ ]+
= 0 ) $ 2 =1
19
The Dirac equation
!
"mc 2 + c# $ % + q&( )' r,t( ) = ih(' r,t( )(t
First derivatives with respect to time and position Linear in scalar and vector potentials
Can be shown to be Lorentz invariant
Alpha and Beta are conventionally represented bythe following set of 4-component matrices
!
"x =0 # x
# x 0
$
% &
'
( ) "y =
0 # y
# y 0
$
% &
'
( ) " z =
0 # z
# z 0
$
% &
'
( ) * =
I 0
0 +I
$
% &
'
( )
20
The Dirac Hamiltonian
!
ˆ H = "mc2
+ c# $ ˆ % + q&
=
mc2
+ q& 0 c% z c(% x ' i% y )
0 mc2
+ q& c(% x + i% y ) 'c% z
c% z c(% x ' i% y ) 'mc2
+ q& 0
c(% x + i% y ) 'c% z 0 'mc2
+ q&
(
)
* * * *
+
,
- - - -
Four component wave function : why ?
1) Spin doubles the components
2) Negative energy solutions : E < -mc2
21
Densities
!
" r, t( ) = q#†r, t( )# r, t( )
• Charge density
• Current density
• Conservation relation!
j r,t( ) = q"† r,t( ) c# " r,t( )
!
"# r, t( )"t
+$ % j r,t( ) = 0
22
Time-independent Dirac equation The nuclei do not move with relativistic speeds with
respect to each other Take a stationary frame of reference (Born-
Oppenheimer approximation) Separate the time and position variables
!
ˆ H "(r, t) = ih#"(r,t)
#t
"(r, t) = $(r)%(t)
ˆ H $(r) = E $(r)
%(t) = eEt / ih
Time dependent Dirac equation
Time independent Dirac equation
23
Free particle Dirac equation Take simplest case : φ= 0 and A = 0 Use plane wave trial function
!
"(r) = eik#r
a1
a2
a3
a4
$
%
& & & &
'
(
) ) ) )
E *mc2( )a1 * chkza3 * chk*a4 = 0
E *mc2( )a2 * chk+a3 + chkza4 = 0
*chkza1 * chk*a2 + E + mc2( )a3 = 0
*chk+a1 + chkza2 + E + mc2( )a4 = 0
!
k± = kx ± iky
Non-relativistic functional form with constants aithat are to be determined
After insertion into time-independentDirac equation
24
Free particle Dirac equation
Two doubly degenerate solutions
Compare to classical energy expression
Quantization (for particles in a box) and prediction ofnegative energy solutions
E2
!m2
c4
! c2
h2
k2( ) = 0
E+ = + m2
c4
+ c2
h2
k2
E! = ! m2c4 + c2h2k2
E = m2
c4
+ c2
p2
25
Free particle Dirac equation Wave function for E = E+
Upper components are the “Large components” Lower components are the “Small components”
!
a2 = 0 ; a3 = a1
chkz
E+ + mc2
; a4 = a1
chk+
E+ + mc2
h k " p << mc
a3 = a1
cpz
mc2
+ m2c
4+ c
2p
2# a1
pz
2mc
a4 # a1
p+
2mc
For particles moving with “nonrelativistic” velocities
26
Free particle Dirac equation Wave function for E = E-
Role of large and small components is reversed Charge conjugation symmetry Can we apply the variational principle ? Variational Collapse
!
a4
= 0
a1
= a3
chkz
E" "mc2
# a3
pz
"2mc
a2
= a3
chk+
E" "mc2# a
3
p+
"2mc
27
Dirac sea of electrons All negative energy
solutions are filled The Pauli principle
forbids doubleoccupancy
Holes in the filled seashow up as particleswith positive charge :positrons (discoveredin 1933)
Infinite backgroundcharge
Electronlike continuum solutions
Positronlike continuum solutions
Electronlike bound solutions
28
Second Quantization Introduce a m-dimensional Fock space F(m)
States are defined by the occupation number vector n
The vacuum has all n=0
We use an orthonormal basis
n = n1, n2,K,n
m
ni= 0,1
vac = 0, 0,K, 0
n k = !nk
vac vac = 1
29
Second Quantization Second quantized operators
Creation operator
Annihilation operator
Define all operators in terms of these elementary operators
ai
†n
1,K, n
i,K,n
m= 0 (n
i=1)
ai
†n
1,K, n
i,K,n
m= C
in
1,K,1,K, n
m (n
i= 0)
ai
†vac = 0,K,1,K, 0
ain
1,K, n
i,K, n
m= C
in
1,K, 0,K,n
m (n
i= 1)
ain
1,K, n
i,K, n
m= 0 (n
i= 0)
aivac = 0
ˆ ! = !kl
ˆ a k
†ˆ a
l
k ,l =1
m
"
30
Fock space HamiltonianPositive and negative energy solutions define a Fock space Hamiltonian
!
ˆ H Total
= ˆ H ++
+ ˆ H +"
+ ˆ H "+
+ ˆ H ""
ˆ H ++
= H pqˆ a p
† ˆ a qp,q
E#E+
$ ˆ H ""
= H%&ˆ a %
† ˆ a &% ,&
E#E"
$
ˆ H +":pair creation
= H p%ˆ a p
† ˆ a %%
E#E"
$p
E#E +
$
ˆ H "+:pair annihilation
= H%pˆ a %
† ˆ a pp
E#E +
$%
E#E"
$
31
Renormalization1. Subtract energy from the occupied negative energy
states
2. Re-interpretation
3. Normal ordered Hamiltonian
ˆ a p† = ˆ b p
† ˆ a p =
ˆ b p
ˆ a !† = ˆ b ! ˆ a ! =
ˆ b !†
!
ˆ H QED = H pq
ˆ b p† ˆ b q
p,q
electrons
" + H p#ˆ b p
† ˆ b #† + H#p
ˆ b #ˆ b p( )
#
pos.
"p
el.
" $ H#%ˆ b #
† ˆ b %# ,%
positrons
"
!
ˆ H QED
= ˆ H Total
" E0
= ˆ H Total
" ˆ H Total
Due to the anticommutation relation
32
Quantum Electro Dynamics
Positive energy for positrons
Total charge is also redefined
!
QvacQED
= "e vac ˆ N QED vac
= "e vac bp† bp " b#
†b##
positronstates
$p
electronstates
$ vac = 0
!
E(1p;0e) = K, 1,K;K ˆ H QED
K,1,K;K
= K,1,K;K " H#$b#†b$
#,$
positronstates
% K,1,K;K = "E& ' mc2
Neg. Pos. Neg. Pos.
γ
33
Dressed particles The QED Hamiltonian depends on the positive and
negative energy solutions of the Dirac equation. TheDirac equation depends on the external potential
Common choices Free particle solutions (Feynman,1948) Fixed external potential (Furry,1951) External + some mean-field potential (“fuzzy”)
Particles in one representation are quasiparticles(dressed with ep-pairs) in another representation
Different no-pair approximations possible
34
Electron-electron interaction Add quantized EM-field and interaction term
Electron-electron interaction is automaticallyretarded by the finite velocity of light
Corrections to the Dirac equation and theinstantaneous Coulomb interaction can be derived Feynman (NP 1965) diagrams
• Breit interaction (1929) (Order c-2)• Vacuum Polarization + Self Energy = Lamb shift (NP 1955) (c-3)
= p ! states;e! states; photons
ˆ H QED , full = ˆ H e+ p + ˆ H photons+ ˆ H e+ p, photons
35
Electron-electron interaction Three terms up to order c-2
Coulomb, Gaunt and retardation terms First correction describes the current-current interaction Second correction describes retardation
!
gCoulomb"Breit
1,2( ) =1
r12
"1
c2r12
c#1 $ c#2
"1
2c2c#1 $%1( ) c#2 $% 2( )r12
36
Dirac-Coulomb-Breit Hamiltonian Second quantization is merely convenient for our
purposes, but becomes essential when goingbeyond the No-Pair approximation
Page 68 of Book 1 has everything we need:
Matrix elements are complex and (therefore) haveless permutational symmetry
We want to compute these matrix elements, so weneed to go back to first quantization and basis setexpansion techniques.....
!
ˆ H = hij
D
i, j
" ai
†a j +
1
2gijkl
C + gijkl
B( )i, j ,k,l
" ai
†ak
†ala j
37
MO-integrals in quaternion formL. Visscher, J. Comp. Chem. 23 (2002) 759.
Gµ! ,"#VW ,XY =
$µV †
(r1 )$!W (r1 )$"
X †(r2 )$#
Y (r2 )
r12
dr1dr2%%
Bµ! ,pqXX,&1 2e&1 2
= cµpX,&1 c!q
X,& 2e&1
* e&2
&1 =0
3
'&1 =0
3
'
GpqrsDirac(Coulomb &1 2&3 4 = Bµ! , pq
XX ,&1 2Gµ!"#XXYYB
"#,rs
Y Y,& 3 4
" ,#
NY
'µ ,!
NX
'Y
L ,S
'X
L,S
' &12,&34 = 0,1,2,3( )
GpqrsLévy(Leblond = Bµ!,pq
LL ,0 Gµ!"#LLLLB
"# ,rs
LL,0
" ,#
NL
'µ ,!
NL
'
Gpqrsspinfree = Bµ!,pq
XX ,0 Gµ!"#XXYYB
"#,rs
Y Y,0
" ,#
NY
'µ ,!
NX
'Y
L ,S
'X
L ,S
'
GpqrsTwo( spinor&1 2&3 4 = Bµ! ,pq
LL ,&1 2Gµ!"#LLLLB
"# ,rs
LL,&3 4
" ,#
NL
'µ ,!
NL
' &12,& 34 = 0,1,2, 3( )
38
Discussed in Lecture 1
!
"mc 2 + c# $ ˆ % + q&( )' r( ) = E' r( ) Dirac equation
Electronlike continuum solutions
Positronlike continuum solutions
Electronlike bound solutions
Lorentz invariance
Renormalization (QED)
Choosing the reference DiracHamiltonian in QED: we needorbitals
No-Pair approximation andsecond quantized Hamiltonian
Breit interaction
39
One more exact solution of the Dirac equation
The hydrogenic atom
This equation can be solved exactly byseparating the radial and angular variables
The derivation and energy is found invarious textbooks.
!
mc2 "
Z
rc# $ p
c# $ p "mc 2 "Z
r
%
&
' ' '
(
)
* * *
+ Lr( )
+ Sr( )
%
& '
(
) * = E
+ Lr( )
+ Sr( )
%
& '
(
) *
40
The hydrogenic atom The exact energy expression
Scalar relativistic corrections :
Spin-orbit couping :
!
E = mc2/ 1+
Z /c
n " j "1
2+ ( j +1/2)
2 "Z2
c2
#
$
% %
&
% %
'
(
% %
)
% %
2
j = l ± s
!
ENR
= "Z2
2
41
42
Orbital stabilisationAlkali metals
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150
Nuclear Charge
nonrelativistic
relativistic
H, Li, Na, K, Rb, Cs, Fr, 119
43
Spin-orbit splittingGroup 13
0.0
0.1
0.2
0.3
0.4
0 50 100 150
Nuclear Charge
nonrelativistic
relativistic
relativistic
B, Al, Ga, In, Tl, 113
44
Orbital destabilisationGroup 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, 112
45
Orbital contraction <r> of the outermost s-orbital
Alkali metals
1
2
3
4
5
6
7
8
0 50 100 150
Nuclear Charge
nonrelativistic
relativistic
46
Orbital contraction <r> of the outermost p-orbital
Group 13
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0 50 100 150
Nuclear charge
nonrelativisticrelativisticrelativistic
47
Orbital expansion <r> of the outermost d-orbital
Group 12
0.8
1.0
1.2
1.4
1.6
1.8
2.0
20 70 120
Nuclear Charge
nonrelativisticrelativisticrelativistic
48
The hydrogenic atom The exact energy expression
Can be expanded to!
E = mc2/ 1+
Z /c
n " j "1
2+ ( j +1/2)
2 "Z2
c2
#
$
% %
&
% %
'
(
% %
)
% %
2
!
E = mc2 "
Z2
2n2
+Z4
2n4c2
3
4"
n
j +1
2
#
$ %
& %
'
( %
) %
+OZ6
c4
*
+ ,
-
. /
!
1+ x( )"1
2 =1"1
2x +
3
8x2"K
49
Approximate Hamiltonians
Find operators that can describe these scalar relativistic and spin-orbit coupling corrections in molecular systems
Start by decoupling the large and small component equations
Rewrite the lower equation as
!
V" L + c# $ p" S = E" L
c# $ p" L + V % 2mc 2( )" S = E" S
!
" Sr( ) = 1#
E #V
2mc2
$
% &
'
( )
#1* + p
2mc" L
r( )
= K E,r( )* + p
2mc" L
r( )
!
K E,r( ) = 1"E "V
2mc2
#
$ %
&
' (
"1
50
Approximate Hamiltonians
Substitute in the upper equation
Unnormalized Elimination of the Small Component
!
1
2m" # p( )K E,r( ) " # p( ) +V
$ % &
' ( ) * L
r( ) = E* Lr( )
!
K E,r( ) = 1"E "V
2mc2
#
$ %
&
' (
"1
51
Pauli Hamiltonian
Crudest approximation :
Take K=1 but keep the magnetic field
!
1
2m" # p( ) " # p( ) +V
$ % &
' ( ) * L
r( ) = E* Lr( )
p2
2m+V
$ % &
' ( ) * L
r( ) = E* Lr( )
!
K E,r( ) =1
!
1
2m" # $( ) " # $( ) +V
% & '
( ) * + L
r( ) = E+ Lr( )
Schrödinger equation
Pauli equation
52
Breit-Pauli Hamiltonian
Find an operator that normalizes the wave function :
Multiply the UESC equation by N-1
!
" = N" L
N = 1+1
4m2c2# $ p( )K 2 # $ p( )
!
N"1 1
2m# $ p( )K E,r( ) # $ p( ) +V
% & '
( ) * N
"1N+ L
r( ) = N"1E+ L
r( )
!
N"1 1
2m# $ p( )K E,r( ) # $ p( ) +V
% & '
( ) * N
"1+ r( ) = EN"2+ r( )
53
Breit-Pauli Hamiltonian
Use series expansions and keep terms up to order c-2
!
ˆ N "1 = 1+
1
4m2c
2# $p( )K 2 # $p( )
%
& ' (
) *
"1/ 2
=1"1
8m2c
2# $p( )K 2 # $p( ) +K
=1"p
2
8m2c
2+ O(c
"4)
!
ˆ N "2 = 1+
1
4m2c
2# $p( )K 2 # $p( )
%
& ' (
) *
"1
=1"p
2
4m2c
2+ O(c
"4)
!
K2 = 1"
E "V
2mc2
#
$ %
&
' (
"2
=1+O c"2( )
54
Breit-Pauli Hamiltonian
Expansion of K
Subsitute and keep only terms to order c-2
!
K E,r( ) = 1+E "V( )2mc
2
#
$ %
&
' (
"1
=1"E "V( )2mc
2+O(c"4 )
!
ˆ N "1 ˆ V +
1
2m# $p( )K # $p( )
%
& ' (
) * ˆ N "1+ = EN
"2+
!
ˆ V + ˆ T +" #p( )V " #p( ) $ Ep
2 $Tp2 $
1
2p
2,V[ ]
+
4m2c
2
%
&
' ' '
(
)
* * * + = E $
Ep2
4m2c
2
,
- .
/
0 1 +
55
The energy dependent term on the lhs was cancelled by the rhs
Further simplify the equation using
Result : The Breit-Pauli equation
!
" #p( )V " #p( ) = pV( ) #p+Vp2 + i" # pV( ) $p
%1
2p2,V[ ]
+= %
1
2p2V( ) % pV( ) #p%Vp2
Breit-Pauli Hamiltonian
!
ˆ H BP = ˆ T + ˆ V +
" #p( )V " #p( ) $Tp2$
1
2p
2,V[ ]
+
4m2c
2
Darwin Mass-Velocity Spin-Orbit
!
ˆ H BP = ˆ T + ˆ V "
p2V
8m2c
2"
p4
8m3c
2+
i# $ pV( ) %p4m
2c
2
56
Expectation values for the hydrogen atom
!
< ˆ H Darwin
>=Z
4
2n3c
2(l = 0)
!
< ˆ H Darwin
>= 0 (l > 0)
!
< ˆ H MV
>=Z
4
2n4c
2
3
4"
n
l +1
2
#
$ %
& %
'
( %
) %
!
< ˆ H SO >=
Z4
2n3c
2
l
l 2l +1( ) l +1( )j = l +1/2( )
!
< ˆ H SO >=
Z4
2n3c
2
"l "1
l 2l +1( ) l +1( )j = l "1/2( )
57
Approximate relativistic Hamiltonians
Can we improve upon the Breit-Pauli Hamiltonian ?
A short wish list :
1. The Hamiltonian should resemble the SchrödingerHamiltonian as much as possible
2. It should describe the scalar relativistic effects3. It should describe the spin-orbit coupling effect4. It should be variationally stable5. It should be easy to implement6. Errors relative to the Dirac solutions should be small
and systematically improvable
7. It should be well-named....
58
Regular approximations What did we do wrong ? Check the expansion parameter
E should be small relative to 2mc2
Orbital energies vary over a range of -0.1 to 5,000 au Twice the rest mass energy is 37,558 au This difference should be large enough
V should be small relative to 2mc2
The potential is dominated by the nuclear attraction close to the nuclei
Take r = 10-4 au and Z=6 (carbon) : V = 60,000 au Is this inside the nucleus ? No : the RMS radius is 4.7 10-5 au for C.
!
K E,r( ) = 1+E "V( )2mc
2
#
$ %
&
' (
"1
=1"E "V( )2mc
2+O(c"4 )
!
V " #Z
r
59
Regular approximations Can we find a better expansion parameter ? Yes !
E should be small relative to 2mc2 - V V is negative which improves the expansion close to the nuclei
Zeroth order in this expansion
Zeroth order equation does describe SO-coupling and scalarrelativistic corrections
Gauge dependence of the energy Affects ionization energies, structures Gauge independence can be achieved various ways
!
K E,r( ) = 1+E "V( )2mc
2
#
$ %
&
' (
"1
= 1"V
2mc2
)
* +
,
- . "1
1+E
2mc2 "V
)
* +
,
- . "1
!
1
2m" # p( ) 1$
V
2mc2
%
& '
(
) * $1
" # p( ) +V+ , -
. / 0 1 ZORA
r( ) = E1 ZORAr( )
!
V "V + C E " E + C #EC
2mc2
60
Approximations to K(E,r) for the 1s orbital of Fm99+
61
8
6
4
2
0
r!!
(a.u
.)
10-5
10-4
10-3
10-2
10-1
100
r (a.u.)
1s orbital
DIRAC ZORA NR
Uranium atom
62
0.5
0.4
0.3
0.2
0.1
0.0
r!!
(a.u
.)
10-5
10-4
10-3
10-2
10-1
100
101
r (a.u.)
7s orbital
DIRAC ZORA NR
Uranium atom
63
Foldy-Wouthuysen transformations Use an energy-independent unitary transformation to
decouple the large and small component equations
Exact expressions are only known for the free particleproblem
!
UHDU
"1U#
i
D = EU#i
D
HFW = U ˆ H
DU
"1 =H
+0
0 H"
$
% &
'
( )
#i
FW 4,(+) = U#i
D(+) =#
i
FW
0
$
% &
'
( )
ˆ U =1+ X
†X( )
"1
2 1+ X†X( )
"1
2 X†
X 1+ XX†( )"
1
2 1+ XX†( )"
1
2
$
%
& & &
'
(
) ) )
Picture change
!
X =1
2mcK ".p( )
64
Douglas-Kroll-Hess method Idea
Transform “bare-nucleus Hamiltonian” with the known free-particletranformation matrix, followed by additional transformations to reduce size ofremaining off-diagonal elements to some order in the potential
Assumptions The transformation is based on the Furry picture : potential does not include
mean-field of electrons The conventional implementations neglect the transformation of the two-
electron interaction and often also the SO-coupling terms
Advantages-Disadvantages Method is variationally stable Slight modification of existing code required (replacement of one-electron
nuclear attraction integrals), fast implementation Good results in practice, significant improvement over Breit-Pauli Complicated operators, matrix elements can not be calculated analytically Two-electron terms are hard to evaluate Interactions with external field need to be represented by transformed operators
(picture change)
65
The second-order Hamiltonian
The Douglas-Kroll-Hess Hamiltonian
!
HDKH 2 = Ep + Ap V + ".Pp( )V ".Pp( )( )Ap +W
1EpW1
+1
2W1
2Ep +
1
2EpW1
2
!
W1
=ApRpV p, " p ( )A " p # ApV p, " p ( )R " p A " p
E p + E " p
!
H(2)
=H
DKH 2H12
(2)
H21
(2)H22
(2)
"
# $
%
& '
Douglas-Kroll-Hess method
!
Rp =c" #p
Ep + mc2
=" #Pp
!
Ap =Ep + 2mc
2
2Ep!
Ep = m2c4
+ c2p2 Free particle energy operator
Kinematical factor
66
General operator transformations Barysz-Sadlej-Snijders (1997) Reiher (200X) van Wüllen (200X)
Higher order two electron effects Samzow, Hess, Jansen (1992) Park and Almlöf (1994) Hirao (2003-present)
Infinite order via matrix representations Ilias and Jensen (2005)
Higher order approaches
67
Four-component methods Idea
Expand Dirac equation in separate basis sets for the largeand small components
Use kinetic balance condition to prevent “variationalcollapse”
Advantages-Disadvantages No approximations made Matrix elements over the operators are easily evaluated Many more two-electron integrals need to be handled The Fock matrix is twice as large No picture change problems
68
Hartree-Fock Self Consistent Field
1. Construct Fock operator
2. Find eigensolutions
3. Check convergence
4. Compute energy
!
F = " # c 2 + c$ %p+V + J j &K j
j
occupiedorbitals
'
!
F" r1( ) = #" r
1( )
!
" new{ } = " old{ } ?
!
EHF
= EKinetic
+ EPotential
+ EElec . Re p.
!
EHF
= < i |
i
occupiedorbitals
" # $ c 2+ c% &p | i > + < i |
i
occupiedorbitals
" V | i > +1
2< i |
i, j
occupiedorbitals
" J j 'K j | i >
69
!
F =
V + J j "K j
j
# c $.p( ) " K j
j
#
c $.p( ) " K j
j
# V " 2c2 + J j "K j
j
#
%
&
' ' '
(
)
* * *
Fock operator
!
Jj r1( ) =" j
L†
r2( )" j
Lr2( ) +" j
S†
r2( )" j
Sr2( )
r12
dr2
=#$j r2( )r12
dr2#
!
Kj" i
Lr1( ) = Kj
LL" i
Lr1( ) + Kj
SL" i
Lr1( )
=" j
L†
r2( )" i
Lr2( )
r12
dr2" j
Lr1( ) +#
" j
L†
r2( )" i
Lr2( )
r12
dr2" j
Sr1( )#
!
Kj" i
Sr1( ) = Kj
LS" i
Sr1( ) + Kj
SS" i
Sr1( )
=" j
S†
r2( )" i
Sr2( )
r12
dr2" j
Lr1( ) +#
" j
S†
r2( )" i
Sr2( )
r12
dr2" j
Sr1( )#
70
Basis set expansion Use different expansion sets for the large and small
component parts of the wave function
Kinetic balance condition
Recovers the non-relativistic limit
!
"ir( ) =
#µ
Lr( )0
$
% &
'
( )
µ=1
NL
* cµi
L +0
#+Sr( )
$
% &
'
( ) c+i
S
+ =1
NS
*
!
" Sr( ) =
# $ p
2mc" Lr( )
!
"#L*
r( )T$ "%Lr( )dr =
1
2"#L*
r( ) & 'p( )$ "µ
Sr( )dr
µ=1
NS
( ) "µ
S*
r( ) & 'p( )$ "%Lr( )dr
!
TLL =
1
2" #p( )
LS
" #p( )SL Resolution of identity
71
Choice of expansion functions Large component
Atoms : Sturmians, Slaters or Gaussians Molecules : Spherical or Cartesian Gaussians
Small component Same type as large component Should fulfill kinetic balance relation
!
"P
S{ } = # $p( )"P
L{ }
!
"P
S{ } =#"P
L
#x,#"P
L
#y,#"P
L
#z
$ % &
' ( )
Restricted KB Unrestricted KB
72
Condition : (! . p)(! . p) = p2
Kinetic Balance
Schrödinger equation
Dirac equation
c !
Cartesian Gaussian basis
Large Component
Small Component
s p d
s p d f
73
The large component wave function resembles thenon-relativistic wave function
Exact relation between large and small componentwave functions
The small component density
Small component wave function is related to the firstderivative of large component wave function
The small component density is anembarrassingly local quantity !
!
( " S =
#1
2c1+
E #V
2c2
$
% &
'
( )
#1( *
( " L
Electron Density of Uranyl
Large component Small component
75
Spinfree Dirac equation Define an auxilliary function
Transform the Dirac equation accordingly
Separate scalar and spin-dependent part andneglect the spin-dependent terms if desired
!
" S =1
2mc# $p( )% L
!
V T
T" #p( )V " #p( )4mc
2$T
%
&
' '
(
)
* * + L
, L
%
& '
(
) * = E
1 0
0T
2mc2
%
& ' '
(
) * * + L
, L
%
& '
(
) *
Relation holds by definition
76
Direct perturbation theory Consider the modified Dirac equation
Non-relativistic limit is related to the Lévy-Leblond equation
Define a perturbation theory with as first (or second) orderperturbations
!
V T
T" #p( )V " #p( )4mc
2$T
%
&
' '
(
)
* * + L
, L
%
& '
(
) * = E
1 0
0T
2mc2
%
& ' '
(
) * * + L
, L
%
& '
(
) *
!
V T
T "T
#
$ %
&
' ( ) L
* L
#
$ %
&
' ( = E
NR1 0
0 0
#
$ %
&
' ( ) L
* L
#
$ %
&
' (
!
V " #p( )" #p( ) $2m
%
& '
(
) * + u
+ l
%
& '
(
) * = E
LL1 0
0 0
%
& '
(
) * + u
+ l
%
& '
(
) *
!
H(1)
=0 0
0 "Vc"2#
$ %
&
' (
!
S(1)
=0 0
0 c"2
#
$ %
&
' (
77
Discussed in Lecture 2
Two-component methods
Breit-Pauli perturbation theory
Regular approach (ZORA)
Douglas-Kroll-Hess method
4-component methods
Direct Perturbation Theory
!
1
2m" # p( )K E,r( ) " # p( ) +V
$ % &
' ( ) * L
r( ) = E* Lr( )
78
Valence-Only approaches All-electron calculations are not always feasible or necessary
Hierarchy of approximations for “core” electrons1. Correlate the core electrons at a lower level of theory (e.g. MP2)2. Include core electrons only at HF level of theory3. Use atomic orbitals for core electrons (Frozen Core)4. Model frozen core by a Model Potential5. Model frozen core by a Relativistic Effective Core Potential
Error correction and additional features1. Estimate higher order correlation effects in another basis set2. Use a core correlation potential3. Use a core polarization potential4.5. Include valence relativistic effects in RECP
79
Consider the Fock operator
Identify localized (atomic) core orbitals and partition
Coulomb potential goes to zero at large distance, contains correctiondue to imperfect screening of nuclei at short distance
Exchange contribution depends on the overlap : short range Approximation made : atomic core orbitals are not allowed to change
upon molecule formation
!
F = hkinetic "
ZA
rAA
Nuclei
# + Jc
A "Kc
A
c
core
#A
Nuclei
# + Jv"K
v
v
valence
#
F = hkinetic "
ZA
*
rAA
Nuclei
# + Jv"K
v
v
valence
# + "ZA
core
rA
+ Jc
A
c
core
#$
% &
'
( )
A
Nuclei
# " Kc
A
c
core
#A
Nuclei
#
Frozen Core approximation
!
F = hkinetic "ZA
rAA
Nuclei
# + J j "K j( )j
occupiedorbitals
#
!
ZA
*= Z
A" Z
Core
VCoulomb VExchange
80
Core polarization and overlap Polarizability of the core can modeled by a classical core
polarization potential (see also book II, formula 45.9)
Need a cut-off factor in the field since the multipole expansionis only valid outside the core
Can be extended to model core-correlation and core-valencecorrelation as well
The overlap between cores is assumed to be zero and thusneglects the exchange repulsion and nuclear attractionbetween neighbour cores
For “large core” calculations this requires a correction
!
VCPP
A= "
1
2fA
T#AfA
A
$ Field from the electrons and the other nucleiat the position of core A
Polarizability of core A
81
Ab Initio Model PotentialsReplace the exact, non-local, frozen core potential by a modelpotential plus a projection operator
Density fit of spherical density, can be done toarbitrary precision
!
VFrozen core
A = Jc
A " ZA
core( )c
core
# " Kc
A
c
core
# $VCoul
A +VExch
A + PCore
A
!
VCoul
A=1
rA
ci
Ae"# i
ArA2
i
$
!
VExch
A= " r S rs
"1s Kc
At S tu
"1u
r,s,t ,u
primitivebasison A
#c
core
#
Resolution of identity with non-orthogonal functions
!
PCore
A= c B
c
Ac
c
core
"
Level shift that enforces orthogonality to the core
82
Ab Initio Model Potentials
No freely adjustable parameters Keeps nodal structure of the valence orbitals Core orbitals in the virtual space Relativistic effects can be included in the reference
Fock operator Cowan-Griffin Hamiltonian Wood-Boring Hamiltonian Douglas-Kroll-Hess Hamiltonian
Can also be used to generate “no-valence” MPs Improves description of ions in crystals May require iterative generation scheme See example from the work of Seijo in the green hand-out
83
Effective Core PotentialsReduce the basis set used to describe the valence orbitals
These pseudopotentials are determined via a fitting procedure. Theytake care of Coulomb and Exchange and core-valence orthogonality.
!
VFrozen core
ArA( ) " ML
ArA( ) + lml f l
ArA( )
ml =# l
l
$l= 0
L#1
$ lml
Phillips and Kleinman : shift core orbitals to make themdegenerate with the valence orbitals
!
"v
{ }# ˜ " v{ }
!
Fv" F
v+ #
v$#
c( )c
% c c
Make nodeless pseudo-orbitals
!
VFrozen core
ArA( ) " ML
ArA( ) + ljm j f lj
ArA( ) ljm j
ml =# l
l
$j= l#1/ 2
l+1/ 2
$l= 0
L#1
$
Scalar
Spin-Orbit
84
Shape consistent ECPs “American school” : Christiansen, Ermler, Pitzer “French school” : Barthelat, Durand, Heully, Teichteil Make nodeless pseudo-orbitals that resemble the true valence
orbital in the bonding region
Absolute correlation energy may be overestimated relative tocorrelation calculations done with the real orbitals
Fit is sometimes done to the large component of Dirac wavefunction (picture change error)
Reasonable accuracy for bond lengths and frequencies Available in many program packages
!
"v r( )# ˜ " v r( ) ="v r( ) r $ rC( )f r( ) r < rC( )
% & '
Original orbital in the outer region
Smooth polynomial expansion in the inner region
85
Energy consistent ECPs “German school” : Stoll, Preuss, Dolg Semi-empirical or ab initio approach that tries to reproduce the
low-energy atomic spectrum (using correlated calculations)
Provides good accuracy for many elements and bondingsituations Difference in correlation due to nodeless valence orbitals is
automatically included in the fit Small cores are often necessary to obtain stable results Available in many program packages
!
min wI EI
PP " EI
Reference( )2
I
LowlyingLevels
#$
%
& & &
'
(
) ) )
86
Methods to treat relativity “Best” method depends on system studied See exercise (and answer) 10
Closed shells and simple open shells Use a size-extensive and economical method SOC-inclusive method may be required
Complicated open shells, bond breaking MCSCF, Multi-Reference CI or MR-CC SOC-inclusive methods are usually required Mean-field description of SO (AMFI) is usually sufficient
Use “best practice” and experience from calculations on lightelements
87
Non-relativistic gold is silver The 5d-6s transition is shifted from the UV to the visible part of the
spectrum by scalar relativistic effects
Phosphorescence Singlet-triplet transitions are allowed because the non-relativistic
quantum numbers are not exact
6s
Visible Relativistic Effects
5d
6s
5d
Non-Relativistic Relativistic
88
Mercury: Dipole polarizability Calculation via 4-
component time-dependentHartree-Fock (4c TD-DFTis nowadays also possible)
A. RelativisticB. NonrelativisticC. Breit-Pauli Pert. Theory
6s→6p transitions “Forbidden” 1S0 → 3P1
“Allowed” 1S0 → 1P1
Relativistic
Nonrelativistic
BP PT
Experiment
89
Closed shell molecules Some studies with all-electron single
reference methods
Analyze relativistic effects for diatomicmolecules
90
Atomization energies• Example: Halogen molecules• Molecular energy is hardly affected by SO-
coupling (SO quenching)• First order perturbation theory
σg
σu∗
πg∗
πu
σg,1/2
σu,1/2∗
πg,1/2∗
πu,1/2
RelativisticNonrelativistic
πu,3/2
πg,3/2∗
91
Atomization energies• Atomic asymptotes are lowered by SO-coupling• First order perturbation theory
Nonrelativistic
px py pz
Relativistic
p1/2 p3/2 p3/2
SO-splitting
2P
2P3/2
Relativistic effect on atomization energies (kcal/mol)
Mainly SO-coupling : relativistic effect on atomizationenergies can be estimated by correction to the asymptote
Relativistic effect on harmonic frequencies (cm-1)
Bond weakening due to admixture of the antibonding sigma orbital. This is a second order spin-orbit effect
Relativistic effect on equilibrium distances (Å)
Note the underestimation by Hartree-Fock in At2
95
Atomization energies• Example: Hydrogen halides• SO-coupling is again mostly quenched• First order perturbation theory• Strong sigma-pi mixing in ultra-relativistic H117
σ
σ∗
π
σ1/2
σ1/2∗
π1/2
RelativisticNonrelativistic
π3/2
Dissociation Energiesaug-pVTZ
-25
-20
-15
-10
- 5
0
HF HCl HBr HI HAtR
ela
tiv
istic
shif
t
in
De
(kcal/
mo
l)
HF
HF+G
MP2
CCSD
CISD
CISD+Q
CCSD(T)
Mainly SO-coupling : a good estimate for atomizationenergies can be obtained by correcting only the asymptote
Vibrational Frequenciesaug-pVTZ
-250
-200
-150
-100
-50
0
HF HCl HBr HI HAt
Rela
tiv
isti
c s
hif
t in
!
e (
cm
-1)
HF
HF+G
MP2
CCSD
CISD
CISD+Q
CCSD(T)
Bond weakening due to admixture of the antibonding sigma orbital
Bond Lengths
aug-pVTZ
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
HF HCl HBr HI HAt
Rela
tivis
tic
shif
t in
re
(Å
)
HF
HF+G
MP2
CCSD
CISD
CISD+Q
CCSD(T)
Competition between scalar and spin-orbit effectsTotal effect is small (< 0.01 Å) in this case
99
6
5
4
3
2
1
dis
socia
tion e
nerg
y (
eV
)
AuHHI
TlH
IF
PbO TlF
Au2
PbTe
Bi2
I2
TlI
spin-orbit effect in: fragments (atoms) molecule
SR ZORA ZORA Exp
100
Groundstate of thalliumhydrideK. Faegri Jr.. and L. Visscher, Theor. Chem Acc. 105 (2001) 265.
Goal : Provide benchmark values for this standard testcase Hamiltonian : Dirac-Coulomb-(Gaunt) Correlation space : up to 36 electrons (6s, 6p; 4f, 5s, 5p, 5d)
Method and# electrons corr.
Re(pm)
Ke(N/m)
!
(cm-1)De
(eV)
MP2* 14 186.2 121 1437 1.83
DC-CCSD(T)* 14 188.5 111 1376 2.07
DC-CCSD(T) 14 187.6 113.3 1385 2.00
DC-CCSD(T) 20 187.4 112.1 1378 1.98
DC-CCSD(T) 36 187.4 111.1 1371 1.98
DCG-CCSD(T) 36 187.7 111.9 1376 2.06
experiment 186.8 114.4 1391 2.06
*Seth, Schwerdtfeger and Faegri (1999) calculations with contracted basis sets.
101
Open shell molecules Two studies with all-electron single and
multireference methods
Analyze relativistic effects
Fine structure splitting in radicals
Fine structure splittings XO molecules
150
200
250
300
350
150 200 250 300 350 400
Experimental FSS (cm - 1)
Experiment
DCG-Hartree-FockDCG-CCSD
DCG-CCSD-T
• Valence iso-electronic systems O2–, FO, ClO
• Breit interaction and correlation should be included for accurate results
O2–
FO
ClO
103
Platinumhydride PtH molecule : jj-coupling instead of LS-coupling scheme Pt (5d96s1) + H (1s1) → PtH (5d9σ2)
104
Molecular properties Relativistic effects on some molecular
properties
105
Dipole moment of HI
Relativistic effects
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
-1 -0.5 0 0.5 1
(R-Re)
dip
ole
mom
ent
(au
)
NR CCSD(T)
SF CCSD(T)
DC CCSD(T)
exp.
106
20253035404550
HF HCl HBr HI
NR SOSRRel
NMR: 1H shielding trends
107
NMR: 13C shielding trends
Data from Malkina et al., Chem. Phys. Lett. 1998Mean-field SO method employed.
108
Extracting nuclear structure information fromSpectroscopy & Quantum Chemistry
Nuclear Quadrupole Moments
The coupling between the nuclearquadrupole moment Q and the electricfield gradient (EFG) at the nucleus qgives an energy splitting that dependson the orientation of the nuclear spin.This can be observed with highprecision in microwave (rotational)spectroscopy on diatomic molecules.
Quantum chemistry gives q and canthus be used to obtain accurate valuesof Q or to predict and rationalize NQRor NMR observations.
EQ =e2qzz Q 3mI
2 ! I(I +1)[ ]4I(2I !1)
Molecularrotation
Nuclear spin
109
Iodine
Spread in Nuclear Quadrupole Moments (mbarn)
-500 -400 -300 -200 -100 0 100 200
NR-HF
SF-HF
DC-HF
ZORA4-DFT(BP)
DC-MP2
DC-CCSD
DC-CCSD(T)
TlI
AuI
AgI
CuI
I2
IBr
ICl
IF
HI
110
Further reading Relativistic Quantum Mechanics
R. E. Moss, Advanced molecular quantum mechanics.(Chapman & Hall, London, 1973).
P. Strange, Relativistic Quantum Mechanics. (CambridgeUniversity Press, Cambridge, 1998).
Relativistic Quantum Chemical methods Relativistic Electronic Structure Theory - Part 1 :
Fundamentals, ed. P. Schwerdtfeger (Elsevier, A’dam, 2002). Theoretical chemistry and physics of heavy and superheavy
elements, ed. U. Kaldor and S. Wilson (Kluwer, Dordrecht,2003.
Relativistic Effects in Heavy-Element Chemistry and Physics,edited by B. A. Hess (Wiley, Chichester, 2003).
Applications Relativistic Electronic Structure Theory - Part 2 :
Applications, ed. P. Schwerdtfeger (Elsevier, Amsterdam,2004).
111
Typographical errors
Page 601 & Formula 4 (answer to question 3) : c missing
Page 604 : division symbol missing
Answer to question 2
!
j r,t( ) = q"† r,t( )c#" r, t( )
!
E = mc2/ 1+
Z /c
n " j "1
2+ ( j +1/2)
2 "Z2
c2
#
$
% %
&
% %
'
(
% %
)
% %
2
!
" #X( ) " #Y( ) =" i" j XiY j
=1
2" i," j[ ]
+XiYj +
1
2" i," j[ ]
$XiYj
= %ij XiY j + i&ijk" kXiY j
= X #Y+ i" # X'Y( )
