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© L. Visscher (2011)
Lucas VisscherVU University Amsterdam
An introduction toRelativistic Quantum Chemistry
Lucas Visscher – Vrije Universiteit Amsterdam
Outline● Why do we care about relativity in chemistry ?
● What are relativistic effects ?● Relativistic effects in atoms● Relativistic effects in molecules
● How can we include relativity in calculations ?● Perturbation theory of relativistic effects● 2-component methods● 4-component methods● Effective Core Potentials
● Some recommendations / best practices● Comparison light / heavy element calculations● Importance of spin-orbit coupling
Assumptions in Quantum Chemistry
● Born-Oppenheimer approximation● Electronic and nuclear motion can be decoupled● Electronic energies for motion around clamped nuclei provide
potential energy surfaces for nuclear motion● Coupling between surfaces is 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 fields can be neglected or treated in perturbation theory
Lucas Visscher – ACMM - VU University Amsterdam - 4Lucas Visscher – ACMM - VU University Amsterdam - 4
Dirac’s view on relativity● Dirac (1929)
● The general theory of quantum mechanics is nowalmost complete, the imperfection that still remainbeing in connection with the exact fitting in of the theory with relativistic ideas. These give rise todifficulties only when high speed particles are involved, and are therefore of no importance in the 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 the various electrons and atomic nuclei.
● The fundamental laws necessary for the mathematical treatment of large parts of physics and the whole of chemistry are thus fully known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.
5
Later insights● Pekka Pyykkö 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 an individualistic manner for the various columns and their various oxidation states, explaining, for example, both the inertness of Hg and the stability of Hg2
2+.These relativistic effects are particularly strong around gold. A detailed understanding of the interplay between relativistic and shell-structure effects will form the impact of relativity on chemistry.
● Jan Almlöf & Odd Gropen (1996)● While the incorporation of these effects sometimes
increases the computation labor, the increase is generally reasonable, and certainly much less than in, e.g. the transition from semiempirical to ab initio methods for routine quantum chemistry applications. We predict, therefore, that relativistic corrections in one form or another will be included in the majority of all quantum chemistry calculations before the end of this decade.
6
The extra dimension
Method
Hartree-Fock Full CI
Basisset
Minimal
Complete
Hamiltonian
Dirac-Coulomb-Breit
NR
Development of relativistic molecular electronic structure theory
The hydrogenic atom: Energies
● The exact non-relativistic energy
● The exact relativistic energy
● Spin-orbit couping :
7
€
E = mc 2 / 1+Z /c
n − j − 12
+ ( j +1/2)2 − Z2
c 2
#
$ % %
& % %
'
( % %
) % %
2
j = l ± s
€
E NR= −Z 2
2n2
Energy depends on orbital and spin variables
8
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
ns -
orb
ital energ
y a
u
nonrelativistic
relativistic
H, Li, Na, K, Rb, Cs, Fr, 119
9
Orbital contraction
● The outermost s-orbital shrinks substantiallyAlkali 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 <
r> in a
u
nonrelativistic
relativistic
10
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 150Nuclear Charge
nonrelativisticrelativisticrelativistic
Group 12
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 50 100 150Nuclear Charge
nonrelativisticrelativisticrelativistic
Zn, Cd, Hg, Cn
11
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 150Nuclear charge
nonrelativisticrelativisticrelativistic
Group 12
0.8
1.0
1.2
1.4
1.6
1.8
2.0
20 70 120Nuclear Charge
nonrelativisticrelativisticrelativistic
12
Ln-An contraction
● Ln-An contraction is partly caused by relativistic effects● Trend expected from the atomic calculations can be
checked by calculations on LnF, AnF, LnH3 and AnH3molecules.
13
Atomization energies
• Example: Halogen molecules• Molecular energy is hardly affected by SO-
coupling (SO quenching) • First order perturbation theory
sg
su*
pg*
pu
sg,1/2
su,1/2*
pg,1/2*
pu,1/2
RelativisticNonrelativistic
pu,3/2
pg,3/2*
14
Atomization energies
• Atomic asymptotes are lowered by SO-coupling• First order perturbation theory
Nonrelativistic
px py pz
Relativistic
p1/2 p3/2 p3/2
Nonrelativistic
Relativistic
SO-splitting
2P
2P3/2
Relativistic effect on atomization energies (kcal/mol)
Spin-Orbit effect on atomization energies is well-reproduced by correcting only the separate atoms limit
-35
-30
-25
-20
-15
-10
-5
0
F2 Cl2 Br2 I2 At2
HF HF+G MP2 CCSD CISD CCSD(T)
Relativistic effect on vibrational frequencies (cm-1)
Bond weakening due to admixture of the antibonding sigma orbital. This is also due to spin-orbit coupling.
-50
-40
-30
-20
-10
0
10
F2 Cl2 Br2 I2 At2
HF HF+G MP2 CCSD CISD CCSD(T)
Relativistic effect on equilibrium distances (Å)
Important and slightly method dependent for 6p elements
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
F2 Cl2 Br2 I2 At2
HF HF+G MP2 CCSD CISD CCSD(T)
18
6
5
4
3
2
1
diss
ocia
tion
ener
gy (
eV)
AuHHI
TlH
IF
PbO TlF
Au2
PbTe
Bi2
I2
TlI
spin-orbit effect in: fragments (atoms) molecule
SR ZORA ZORA Exp
Make orbital diagram and identify possibleSOC effects. Always include scalar effects.
Relativistic effect on atomization energies (kcal/mol)
Slide: courtesy Erik van Lenthe, SCM
Outline● Why do we care about relativity in chemistry ?
● What are relativistic effects ?● Relativistic effects in atoms● Relativistic effects in molecules
● How can we include relativity in calculations ?● Perturbation theory of relativistic effects● 2-component methods● 4-component methods● Effective Core Potentials
● Some recommendations / best practices● Comparison light / heavy element calculations● Importance of spin-orbit coupling
Non-relativistic quantum mechanics
The nonrelativistic Hamiltonian
20
Usually only electric field via scalar potential f
Magnetic fields via vector potential A:
p→𝜋 = p− 𝑞𝑨(minimal coupling substitution)
Zeeman interaction, but no spin-orbit coupling.
H= T + 𝑉 = +,
-.+ 𝑞𝜙
H = +,
-.+ 𝑞𝜙 − 0
1𝑨 2 p+ 0,
1𝐴-
NR quantum mechanics with spin
Pauli Hamiltonian in two-component form
21
Second derivatives w.r.t. position, first derivative w.r.t. timeLinear in scalar (electric), quadratic in vector (magnetic) potential
® Is not Lorentz-invariant
L Ad hoc introduction of electron spin
L Ratio spin angular momentum and magnetic moment incorrect
L No spin-orbit coupling
€
−12∇2 + qφ + iqA ⋅ ∇ +
q2
2mA2 − q
2Bz −
q2Bx − iBy( )
−q2Bx + iBy( ) −
12∇2 + qφ + iqA ⋅ ∇ +
q2
2mA2 +
q2Bz
&
'
( ( (
)
*
+ + +
Dirac equation
J First derivatives with respect to time and positionJ Linear in scalar and vector potentials
J Lorentz invariant
a and b are 4-component matrices
22
€
βmc 2 + cα ⋅ π + qφ( )ψ r,t( ) = i∂ψ r,t( )∂t
€
αx =0 σ x
σ x 0$
% &
'
( ) αy =
0 σ y
σ y 0$
% &
'
( ) α z =
0 σ z
σ z 0$
% &
'
( ) β =
I 00 −I$
% &
'
( )
23
The Dirac Hamiltonian
€
ˆ H = βmc 2 + cα ⋅ ˆ π + qφ
=
mc 2 + qφ 0 cπ z c(π x − iπ y )0 mc 2 + qφ c(π x + iπ y ) −cπ z
cπ z c(π x − iπ y ) −mc 2 + qφ 0c(π x + iπ y ) −cπ z 0 −mc 2 + qφ
(
)
* * * *
+
,
- - - -
Four component wave function
1) Spin doubles the number of components
2) Relativity doubles the number of components again
24
Free particle Dirac equation
● Take simplest case : f= 0 and A = 0● Use plane wave trial function
€
Ψ(r) = eik⋅r
a1a2a3a4
$
%
& & & &
'
(
) ) ) )
E −mc2( )a1 − ckza3 − ck−a4 = 0
E −mc2( )a2 − ck+a3 + ckza4 = 0
−ckza1 − ck−a2 + E + mc2( )a3 = 0
−ck+a1 + ckza2 + E + mc2( )a4 = 0
€
k± = kx ± iky
Non-relativistic functional form with constants aithat are to be determined
After insertion into time-independentDirac equation
Free particle Dirac equation
● Two doubly degenerate solutions
● Compare to classical energy expression
● Quantization (for particles in a box) and prediction of negative energy solutions
€
E2 −m2c4 − c22k2( ) = 0
E+ = + m2c4 + c22k2
E− = − m2c4 + c22k2
E = m2c4 + c2 p2
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+ + mc2 ; a4 = a1ck+
E+ + mc2
k ≡ p << mc
a3 = a1cpz
mc2 + m 2c4 + c2p2≈ a1
pz2mc
a4 ≈ a1p+
2mc
For particles moving with “nonrelativistic” velocities
Lucas Visscher – ACMM - VU University Amsterdam - 27Lucas Visscher – ACMM - VU University Amsterdam - 27
Dirac sea of electrons● Negative energy solutions are all
occupied
● Pauli principle applies
J Holes in this sea of electrons are seen as particles with positive charge: positrons (1933)
L Infinite background charge
¤ QED (Quantum Electrodynamics) to properly account for contribution of negative energy states
¤ No-pair approximation
mc2
-mc2
2 e– 3 e– + 1 e+2mc2
28
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 set approaches and in understanding the chemical bonding in the relativistic regime
mc2 − Zr
cσ ⋅p
cσ ⋅p −mc2 − Zr
#
$
%%%%
&
'
((((
ψ L r( )
ψ S r( )
#
$
%%%%
&
'
((((
= Eψ L r( )
ψ S r( )
#
$
%%%%
&
'
((((
29
The hydrogenic atom: orbitals
● Write orbitals as product of radial and angular (2-spinor functions)
● Solutions to the radial equation€
ψ L r( )ψ S r( )
#
$ %
&
' ( =1r
Pnκ r( )ξκ ,m ϑ ,ϕ( )iQnκ r( )ξ−κ ,m ϑ ,ϕ( )#
$ %
&
' (
€
Pnκ r( ) = NnκP e−λrrγ F1 r( ) + F2 r( )( )
€
Qnκ r( ) = NnκQ e−λrrγ F1 r( ) − F2 r( )( )
€
Rnl r( ) = NnlRe− −2E( )rrl+1F r( )
Large component
Small component
Nonrelativistic €
λ = − −2E 1+E
2mc 2$
% &
'
( )
€
γ = κ 2 −Z 2
c 2< κ
l 0 1 1 2 2 3 3j 1/2 1/2 3/2 3/2 5/2 5/2 7/2k -1 1 -2 2 -3 3 -4
s1/2 p1/2 p3/2 d3/2 d5/2 f5/2 f7/2
More than one electron
● General form of a time-independent Hamiltonian
● Wave function
● Difference between relativistic and non-relativistic calculations is in the calculation of integrals over h and g
● Second-quantized form of equations is identical to non-relativistic theory when using the no-pair approximation
H = hii=1
N
∑ +12
gijj≠i
N
∑i=1
N
∑
N x 4 componentsΨ 1,…,N( )Ψ …, i,…, j,…( ) = −Ψ …, j,…, i,…( ) anti-symmetry
31
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
g12Coulomb−Breit =
1112r12
=1r12
−1c2r12
cα1 ⋅cα2
−12c2
cα1 ⋅∇1( ) cα2 ⋅∇2( )r12
Coulomb: diagonal operator
Gaunt: off-diagonal operator
Retardation
32
Expansion of the energy expression
● The exact Hydrogenic energy expression
● Can be expanded to€
E = mc 2 / 1+Z /c
n − j − 12
+ ( j +1/2)2 − Z2
c 2
#
$ % %
& % %
'
( % %
) % %
2
€
E = mc 2 − Z 2
2n2+
Z 4
2n4c 234−
n
j +12
#
$ %
& %
'
( %
) %
+O Z 6
c 4*
+ ,
-
. /
€
1+ x( )−12 =1− 1
2x +
38x 2 −…
40
Breit-Pauli Hamiltonian applied to hydrogenic atom
€
< ˆ H Darwin >=Z 4
2n3c 2 (l = 0)
€
< ˆ H Darwin >= 0 (l > 0)
€
< ˆ H MV >=Z 4
2n4c 234−
n
l +12
#
$ %
& %
'
( %
) %
€
< ˆ H SO >=Z 4
2n3c 2l
l 2l +1( ) l +1( )j = l +1/2( )
€
< ˆ H SO >=Z 4
2n3c 2−l −1
l 2l +1( ) l +1( )j = l −1/2( )
Positive: reduces nuclear attraction
Always negative: decreases kinetic energy
Operator is delta-function for V = -Z/r
Splitting larger for small n and/or l and large Z
Darwin Mass-Velocity Spin-Orbit
€
ˆ H BP = ˆ T + ˆ V − p2V8m2c 2 −
p4
8m3c 2 +iσ ⋅ pV( ) ×p
4m2c 2
41
Modern relativistic Hamiltonians
The wish list for a relativistic Hamiltonian:
1. It should describe the scalar relativistic effects
2. It should describe the spin-orbit coupling effect
3. There should be a lowest eigenvalue (variational stability)
4. Interpretation: comparison with Schrödinger picture
5. Implementation: easy integrals, efficient coding
6. Errors should be small and systematically improvable
42
Regular Approximation● What is wrong with the BP approach ? 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 nuclear 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 ≈ −Zr
43
0th order regular approximation: ZORA● Can we use 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
J Zeroth order equation does describe SO-coupling and scalar relativistic corrections
L Gauge dependence of the energy● Affects ionization energies, structures● Can be avoided by keeping potential in the denominator fixed
€
K E,r( ) = 1+E −V( )2mc 2
#
$ %
&
' (
−1
= 1− V2mc 2
)
* +
,
- . −1
1+E
2mc 2 −V)
* +
,
- . −1
€
12m
σ ⋅ p( ) 1− V2mc 2
%
& '
(
) * −1
σ ⋅ p( ) +V+ , -
. / 0 ψ ZORA r( ) = Eψ ZORA r( )
€
V →V + C E → E + C − EC2mc 2
44
8
6
4
2
0
r√ρ
(a.u
.)
10-5 10-4 10-3 10-2 10-1 100
r (a.u.)
1s orbital
DIRAC ZORA NR
Uranium atom
Slide: courtesy Erik van Lenthe, SCM
45
0.5
0.4
0.3
0.2
0.1
0.0
r√ρ
(a.u
.)
10-5 10-4 10-3 10-2 10-1 100 101
r (a.u.)
7s orbital
DIRAC ZORA NR
Uranium atom
Picture change: nodes absent in Dirac density
Slide: courtesy Erik van Lenthe, SCM
46
Four-component methods
● Idea● Expand Dirac equation in basis set● Use kinetic balance condition to prevent “variational
collapse”
● Advantages-Disadvantages
J No approximations madeJ Matrix elements over the operators are easily evaluated
L Many more two-electron integralsL The Fock matrix is twice as large
J No picture change
56
Foldy-Wouthuysen transformations
● Define energy-independent unitary transformation to decouple the large and small component equations
● Exact operator expressions are only known for the free particle problem
€
UH DU−1UψiD = EUψ i
D
H FW = U ˆ H DU−1 =H + 00 H−
$
% &
'
( )
ψiFW 4(+) = Uψ i
D(+) =ψ i
FW
0
$
% &
'
( )
ˆ U =1+ X †X( )
−12 1+ X †X( )
−12 X †
− 1+ XX †( )−
12 X 1+ XX †( )
−12
$
%
& & &
'
(
) ) )
Picture change
€
X =12mc
K σ.p( )
59
eXact 2-Component (X2C) theory
1. Define a 4-component basis and compute matrix elements over the one-electron operators
2. Find exact solution to the Dirac equation in this matrix representation
3. Use the eigenvectors to construct an exact decoupling operator in matrix form
4. Transform all other one-electron operators to this decoupled representation
5. Add two-electron Coulomb operator in unmodified form (accept picture change error for this operator)
Idea: Decouple a matrix representation of the Dirac equation
61
Matrix-based X2C approaches
K Fully equivalent to the matrix Dirac equation in the no-pair approximation
J 2-component picture is easily compared to the non-relativistic Schrödinger picture
J Errors made by neglecting corrections to the 2-electron operators are small
K The necessary diagonalization and other matrix manipulations are done before the SCF procedure
K Decoupling from molecular Hartree-Fock solutions: molecular mean-field (X2C-MMF) approach
K 4-component property matrices can be readily transformed to the 2-c picture
62
Valence-Only approaches
● All-electron calculations are not always feasible or necessary
● Hierarchy of approximations for “core” electrons1. Correlate core electrons at a lower level of theory (e.g. MP2)2. Do not correlate core electrons at all (HF-only)3. Use atomic orbitals for core electrons (Frozen Core)4. Model frozen core by a Model Potential (AIMP)5. Model frozen core by a Effective Core Potential (ECP)6. Model frozen core by a Local Pseudopotential (LPP)
● Error correction and additional features1. Estimate higher order correlation effects in another basis set2.3. Use a core polarization potential4.5. Include valence relativistic effects in RECP6. Suitable for orbital-free DFT calculations
63
● Consider the Fock operator
● Identify localized (atomic) core orbitals and partition
● Coulomb potential goes to zero at large distance, contains correction due to imperfect screening of nuclei at short distance
● Exchange potential depends on the overlap with the frozen atomic orbitals: short range
● Approximation made: atomic core orbitals are not allowed to change upon molecule formation, other orbitals stay orthogonal to these AOs
€
F = hkinetic − ZA
rAA
Nuclei
∑ + JcA −Kc
A
c
core
∑A
Nuclei
∑ + Jv −Kvv
valence
∑
F = hkinetic − ZA*
rAA
Nuclei
∑ + Jv −Kvv
valence
∑ + −ZAcore
rA+ Jc
A
c
core
∑$
% &
'
( )
A
Nuclei
∑ − KcA
c
core
∑A
Nuclei
∑
Frozen Core approximation
F = hkinetic − ZA
rAA
Nuclei
∑ + J j −K j( )j
occupiedorbitals
∑
€
ZA* = ZA − ZCore
VCoulomb VExchange
65
Ab Initio Model Potentials
Replace the exact, non-local, frozen core potential by a model potential plus a projection operator
Density fit of spherical density, can be done toarbitrary precision
€
VFrozen coreA = Jc
A − ZAcore( )
c
core
∑ − KcA
c
core
∑ ≈VCoulA +VExch
A + PCoreA
€
VCoulA =
1rA
ciAe−α i
A rA2
i∑
€
VExchA = − r S rs
−1 s KcA t S tu
−1 ur,s,t ,u
primitivebasison A
∑c
core
∑
Resolution of identity with non-orthogonal functions
€
PCoreA = c Bc
A cc
core
∑
Level shift that shifts the core solutions to high energies
66
Nodal structure
Radon ZORA-LDA TZP
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
r (Å)
6s orbital
1s orbital
2s orbital
3s orbital
4s orbital
5s orbital
ψ † r( )ψ r( )
67
Valence density
Radon ZORA-LDA TZP
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
0.00 0.50 1.00 1.50 2.00 2.50 3.00
r (Å)
6s density
6s integrated density
4π !r 2ψ† !r( )ψ !r( )0
r
∫ d !r
4πr2ψ † r( )ψ r( )
68
Valence orbitals
Radon ZORA-LDA TZP
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
r (Å)
6s orbital
1s orbital
2s orbital
3s orbital
4s orbital
5s orbital
ψ † r( )ψ r( )
69
Pseudo orbitals
Radon ZORA-LDA TZP
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
r (Å)
6s orbital
6s pseudo orbital
Matching pointMatching point
70
Effective Core Potentials
Effective Core Potentials allow reduction of the basis set used to describe the valence orbitals by creating smoother orbitals
These nonlocal pseudopotentials are determined via a fitting procedure that optimizes the potential for each l-value. Takes care of Coulomb and Exchange and core-valence orthogonality.
€
VFrozen coreA rA( ) ≈ ML
A rA( ) + lml f lA rA( )
ml =− l
l
∑l= 0
L−1
∑ lml
Phillips and Kleinman : shift core orbitals to make themdegenerate with the valence orbitals
€
ψv{ }→ ˜ ψ v{ }€
Fv → Fv + εv −εc( )c∑ c c
Make nodeless pseudo-orbital by mixing core and valence spinors
€
VFrozen coreA rA( ) ≈ ML
A rA( ) + ljm j f ljA rA( ) ljm j
ml =− l
l
∑j= l−1/ 2
l+1/ 2
∑l= 0
L−1
∑
Scalar
Spin-Orbit
71
Shape consistent ECPs
● “American school” : Christiansen, Ermler, Pitzer● “French school” : Barthelat, Durand, Heully, Teichteil● Make nodeless pseudo-orbitals that resemble the
true valence orbitals in the bonding region
● Fit is sometimes done to the large component of Dirac wave function (picture change error)
● Creating a normalized shape consistent orbital necessarily mixes in virtual orbitals
● Intermolecular overlap integrals are well reproduced● Gives rather accurate bond lengths and structures
€
ψv r( )→ ˜ ψ v r( ) =ψv r( ) r ≥ R( )fv r( ) r < R( )
% & '
Original orbital in the outer region
Smooth polynomial expansion in the inner region
72
ECPs and electron correlation
● Integrals are calculated over pseudospinors● Consider the MP2 valence energy expression
● Orbital energy spectrum is compressed and in particular the intra-atomic 2-electron integrals will be different from the reference all-electron calculation
● Absolute correlation energy may be overestimated relative to correlation calculations done with the unmodified orbitals
● Example : for Pt the radial maximum of the 5d is very close to a node of the 6s. Pseudoizing the 6s will remove this node and overestimate the correlation energy. Remedy : takes also the 5s in the valence
€
EMP 2 =ij ab
2
εi + εi −εa −εba,b
virtual
∑i, j
occupied
∑
€
ij ab pseudo− ij ab original
≠ 0
€
εioriginal −εa
original > εipseudo −εa
pseudo
73
Energy consistent ECPs
● “German school” : Stoll, Preuss, Dolg● Initially semi-empirical, later ab initio approach that tries to
reproduce the low-energy atomic spectrum (using correlated calculations)
● Provides good accuracy for many elements and bonding situations● Difference in correlation energy due to the nodeless valence
orbitals is automatically included in the fit● Small cores may still be necessary to obtain stable results● Cheap core description allows for good valence basis sets● Available in many program packages (a.k.a. “SDD”)
€
min wI EIPP − EI
Reference( )2
I
LowlyingLevels
∑$
%
& & &
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(
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Outline● Why do we care about relativity in chemistry ?
● What are relativistic effects ?● Relativistic effects in atoms● Relativistic effects in molecules
● How can we include relativity in calculations ?● Perturbation theory of relativistic effects● 2-component methods● 4-component methods● Effective Core Potentials
● Some recommendations / best practices● Comparison light / heavy element calculations● Importance of spin-orbit coupling
75
Some recommendations
● “Best” method depends on system and property that is calculated !
● Closed shells and simple open shells ● Use a size-extensive and economical method● SOC-inclusive method may be required
● Complicated open shells, bond breaking● CASSCF/PT2, 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 light elements to decide on the electronic structure method
● Two comparison between heavy/light element calculations● Two examples in which spin-orbit coupling is important
76
Chemistry of heavy elements
● A different world….
● Structure and valence properties: frozen core / ECPs● XPS, NMR, EFG, Mössbauer: all-electron methods
Au2 C20N20H12
electrons 178 160
total energy -36,870 Hartree -987 Hartree
basis functions 48s38p24d18f2g426 functions
240s108p24d684 functions
chemical bonds 1 40
Bond energy 0.1 Hartree2.3 eV
10 Hartree272 eV
77
Spectroscopy of f-elements
● Low-lying electronic states
● Lanthanides: 4f is shielded from environment, possible to work with f-in-core ECPs to get structures
● Actinides: 5f can participate in chemical bonding, small-core ECP or all-electron approach may be necessary
Eu3+ C6H6
electrons 6 f-electrons 6 π-electrons
orbitals 7 6
energies < 0.125 Hartree 3 (with SOC: 8) 1
states < 0.125 Hartree 159 (with SOC:58) 1
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 includedfor accurate results
O2–
FO
ClO
L. Visscher, T.J. Lee, K.G. Dyall, J Chem Phys. 105 (1996) 8769.
79
NMR: 1H shielding trends
20
25
30
35
40
45
50
55
HF HCl HBr HI
sH
(ppm
)
NR NR+SR+SOSRRel
[1] L. Visscher, T. Enevoldsen, T. Saue, H.J.Aa. Jensen, J. Oddershede, J Comput Chem. 20 (1999) 1262–1273.
80
Further reading
Relativistic Quantum Mechanics● M. Reiher and A. Wolf, Relativistic Quantum Chemistry, (Wiley, 2009) ● K. G. Dyall and K. Faegri Jr, Relativistic Quantum Chemistry, (Oxford
University Press, 2007)● R. E. Moss, Advanced molecular quantum mechanics. (Chapman &
Hall, London, 1973).● P. Strange, Relativistic Quantum Mechanics. (Cambridge University
Press, Cambridge, 1998).
Relativistic Quantum Chemical methods● Relativistic Electronic Structure Theory - Part 1 : Fundamentals, ed.
P. Schwerdtfeger (Elsevier, Amsterdam, 2002).● Theoretical chemistry and physics of heavy and superheavy
elements, ed. U. Kaldor and S. Wilson (Kluwer, Dordrecht, 2003.
Applications● Relativistic Electronic Structure Theory - Part 2 : Applications, ed. P.
Schwerdtfeger (Elsevier, Amsterdam, 2004).