Cours-Relativite.pdf

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CNRS Universite de Lille 1

Relativistic Quantum Chemistry


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Outline

Part I: Introduction

1 Three cornerstones of non-relativistic Quantum Chemistry

2 The electronic problem

3 Quantum chemical methods

4 Which are the relativistic elements?

5 The limitations of nonrelativistic quantum chemistry

Part II: Relativistic effects

6 Scalar relativistic effects

7 Illustrations of scalar relativistic effects

8 The spin-orbit interaction

9 Illustration of the spin-orbit interaction

Part III: The Dirac equation and four-component calculations

Valerie Vallet ([email protected]) |CNRS Universite de Lille 1


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Outline (2)

Part IV: Two-component relativistic theory

10 How important are the small components?

11 Unitary transformations

12 The Pauli Hamiltonian

13 Regular approximations

14 The Douglas-Kroll-Hess Hamiltonian

15 Summary of approximate 2c-Hamiltonians

Part V: Core approximations

16 Core approximations - Valence only approaches

17 Pseudopotentials

18 Comments on the use of effective core potentials

Part VI: One-component relativistic methods

19 Which spin-orbit operator?

20 Consequences of spin-orbit coupling

21 1c approaches for the treatment of SO-couplingValerie Vallet ([email protected]) |CNRS Universite de Lille 1


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Outline (3)

Part VII: Illustrations of relativistic effects

Part VIII: Final comments



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Bibliography

Relativistic Quantum Chemistry - The Fundamental Theory of Molecular

Science

M. Reiher, A. Wolf, Wiley (2009)

Introduction to Relativistitc Quantum Chemistry

K. G. Dyall, K. Fgri, Oxford University Press (2007)



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Part I

Introduction

1 Three cornerstones of non-relativistic Quantum Chemistry

2 The electronic problem

3 Quantum chemical methods

4 Which are the relativistic elements?

5 The limitations of nonrelativistic quantum chemistry



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Three cornerstones of non-relativistic Quantum Chemistry

We assume that:

1 molecular systems follow theBorn-Oppenheimerapproximation

use the concept ofPotential Energy Surfaces (PES)2 nuclear charge can be described by afinite-size model

(e.g. use of gaussian type basis functions)

3 electrons move slow enough to be described by a non-relativistic theory



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The electronic problem

The electronic Hamiltonian can be written as

Hel =VNN+i

h(i) + 12i


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Quantum chemical methods

Many quantum chemical methods employ thevariational principlestating

that the expectation value of the Hamiltonian with respect to some trialfunction is always above the exact energy (obtained using the exact wave

function)

trial|Hel|trial el|Hel|el =Eexact

The simplest quantum chemical method,Hartree-Fock, employs a simple

Slater determinant as trial function and finds the orbitals which minimizesthe energy. In practice they are found by solving the Hartree-Fock equation

F(r)i(r) =ii(r);F = f+Vmean

ee {k}

where the Fock operator is an effective one-electron operator containing the

mean field of the other electrons in the molecule. More elaborate methods

such as Configuration Interaction (CI) and Coupled Cluster, employ linearcombination of Slater determinants to capture the full electron correlation.

Still, the majority of todays quantum chemical calculations are based on

density functional theory (DFT)which replaces the complicated electronic

wave functionel(r1, rn)by the much simpler electron density(r).



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The limitations of nonrelativistic quantum chemistry

Fairly early one realized that nonrelativisitc theory was unable to explain

certain trends in observed properties of atoms and molecules

Metal-carbon bond length in the group 12 [Raoet al. 1960]

Non-relativistic QC: bond length should increase from Zn, Cd, to Hg Experimentally: bond length increases from Zn to Cd and then decreases from

Cd to Hg The decrease in bond length is due to relativistic effects!

Ionization potentials of the pblock elements

Need for a relativistic quantum chemical formalism



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Part II

Relativistic effects

6 Scalar relativistic effects

7 Illustrations of scalar relativistic effects

8 The spin-orbit interaction

9 Illustration of the spin-orbit interaction



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Scalar relativistic effects: hydrogen-like atoms

In atomic units the average speed of the 1selectron is equal to the nuclear

charge

v1s=Z a.u. andc=137.0359998a.u.

The relativistic mass increase of the 1selectron is thus determined by the

nuclear chargem=me=

me1 Z2/c2

The Bohr radius is inversely proportional to electron mass

a0 =

402

m

Relativity will contract orbitals of one-electron atoms, e.g.

Au78+: Z/c = 58% 18% relativistic contraction of the 1sorbital



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Scalar relativistic effects: polyelectronic atoms

The effect of the other electrons is to screen the nuclear charge

The relativistic contraction of orbitals will increase screening of nuclear

charge and thus indirectly favor orbital expansion.

In practice we find: s,porbitals : contraction d,forbitals : expansion



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Colour of gold

The colours of silver and gold are related to

the transition between the(n 1)dandnsbands. For silver this transition is in the

ultraviolet, giving the metallic cluster. For

gold it is in the visible, but only when

relativistic effects are included.



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The contrasting neighbors

1064C Mp -39C

12.5 kJ/mol Hfus 2.29 kJ/mol9.29 kJ/mol Sfus 9.81 kJ/mol

19.32g/cm3 13.53g/cm3

426 kS/m Conductivity 10.4 kS/m

dimer gas phase monomer[Xe]4f145d106s1 [Xe]4f145d106s2

pseudo halogen pseudo noble gas

Without relativistic effects mercury would probably not be a liquid at room

temperature !



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Spin-orbit interaction

hSO =

1

2m2

c2

s [V p]

with V = Zr

hSO =

Z

2m2c2r3s s

The spin-orbit interaction isnotthe interaction between spin and angular

momentum of an electron. An electron moving alone in space is subject to no

spin-orbit interaction !

The basic mechanism of the spin-orbit interaction is magnetic induction:

An electron which moves in a molecular field will feel a magnetic field in its rest

frame, in addition to an electric field. The spin-orbit term describes the interaction

of the spin of the electron with this magnetic field due to the relative motion of the

charges.

This operator couples the degrees of freedom associated with spin and space

and therefore makes it impossible to treat spin and spatial symmetry separately.

Spin magnetization : m=

ii i;

collinear magnetization: s=mz=

non-collinear magnetization: s= m



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Heavy open-shell molecule: I+2

[C. Van Wullen,J. Comput. Chem., 2002,23, 779785]

Energy := 0Eh Energy := 0.001469972Eh



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Excited states of heavy closed-shell molecule: UO2+2

Energy = -28180.31982230 Eh

LDA excitation energies

1.87 eV (3g)

1.92 eV (2g)

2.42 eV (2g)

Energy := -28180.31982230 Eh EhLDA excitation energies

1.81 eV (3g)

2.01 eV (2g)

2.34 eV (2g)



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Spin-orbit coupling in atoms

In the absence of spin-orbit coupling atomic electronic states are characterizedby total orbital angular momentumL and total spinS and denoted as 2S+1L. With

spin-orbit interaction only the total angular momentum

J= L S, L+S

is conserved.

The ground state configuration of oxygen is 1s22s22p4 which in a non-relativistic

framework (LS-coupling) gives rise to three states.

Term L S J Level (cm1)3P 2 0.000

1 1 1 158.265

0 226.9771D 2 0 2 15867.8621S 0 0 0 33792.583



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Atomic oxygen emissions in atmospheric aurora

Transition Wavelength(A) Type Lifetime(s)

Green line 1S0 1D2 5577 E2 0.75

Red line 1D2 3P2 6300 M1 110



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Part III

The Dirac equation and four-component calculations



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Construction of the Hamiltonian

Non-relativistic free particle

ENR =

p2

2m

Relativistic free particle

E= m2c4 +c2p2; , mc2]or [+mc2, Connection the relativistic and non-relativistic energy

E=mc2

1+ p2

2m2c2 = mc2

rest mass+

p2

2m

p4

8m3c2 kinetic energy+



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Principle of minimal electromagnetic coupling

[M. Gell-Mann,Nuovo Cimento Suppl., 1956,4, 848]

Electric and magnetic fields can be expressed in terms of scalar and vector

potentials

E= A

t

; B= A

The Hamiltonian of a particle interacting with external fields is obtained from

the free-particle Hamiltonian through the substitutions:

p =p qA Electron:q= e p=p +eA

E E+e



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The Dirac equation in a molecular field

Classically we have

E+enuc= A

t ; B= A

with

hD =E; hD=mc2+c(p)+VeN; VeN= enuc

=

L

S

=

L

L

S

S

; =

I2 0202 I2

; =

02 02

;

where appears the 2 2 Pauli spin matrices

x =

0 1

1 0

, y =

0 ii 0

, z=

1 0

0 1



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Contributions from the large and small component densities

p4c(r) =pL(r)pS(r)

here shown for iodobenzene

The highly local and atomic nature of the small components can be exploited to

significantly reduce the computational cost of 4-component molecular calculations



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The two-electron interaction

The fully relativistic two-electron interaction requires knowledge of the

complete history of the two moving particles and can only be given as aperturbation expansion inc2:

Coulomb termO(c0)(including spin-same orbit (SSO))

gCoulij =

I4 I4rij

; charge-chargeinteraction

Breit termO(c2)g

Breitij =g

Gauntij +g

gauge

ij

Gaunt term (including spin-other orbit (SOO))

gGauntij =

ci ci

c2rij ; current-currentinteraction

Gauge-dependent term:

ggauge

ij =

(ci i)(ci i)rij

2c2



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Physical interpretation of the DCB Hamiltonian

one-body Hamiltonian

c : kinetic energyOne-body spin-orbit coupling

Vext :

mc2 rest mass energy

Two-body Hamiltonian

1

r12: Coulomb interaction

spin-same-orbit interaction

Gaunt:

1

2

1 2r12 : spin-other-orbit interaction

spin-spin interaction

orbit-orbit interaction

Gauge dep.: 1

2

(1 r12)(2 r12)

r312: orbit-orbit interaction



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Part IV

Two-component relativistic theory

10 How important are the small components?

11 Unitary transformations

12 The Pauli Hamiltonian

13 Regular approximations

14 The Douglas-Kroll-Hess Hamiltonian

15 Summary of approximate 2c-Hamiltonians


N f h ll


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Nature of the small components

Reducing computational cost

Use the no-pair approximation

atomic character of the contributions of the S components

one can neglect the multi-center integral block in the S comp. (cost

reduction)

Transforming the 4-component equation to a two-component one

Beware: Do not ignore the non-negligible contributions of the

small-components: L and S are coupled!


2 t l ti i ti H ilt i


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2-component relativistic Hamiltonian

Starting from the Dirac equation in a molecular fieldVext+mc

2 c( p)c( p) Vext mc

2

L

S

=

L

S

E

we look for the formal decoupling of the large and small components.

Two approaches can be distinguished

1 decoupling through theelimination of the small components2 Foldy-Wouthuysen transformationdecoupling by a unitary transformation

It can be shown that the two approaches are equivalent; we thus only

consider the unitary transformation


F ld W th t f ti


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Foldy-Wouthuysen transformation

We look for a unitary transformation Uthat formally decouples the large andsmall components and transforms a 4-spinor to a 2-spinor such that

U hLL hLS

hSL hSS

U= h++ 00 h

The exact transformation can be written as a product of two transformations

U= W1W2 = +

+ , + =

1

1+ =

11+

The first transformation decouples the large and small components

W1 = 1 1

whereas the second one re-establishes normalization

W2 = + 0

0 Valerie Vallet ([email protected]) |CNRS Universite de Lille 1

Foldy Wouthuysen transformation (2)


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Foldy-Wouthuysen transformation (2)

The first transformation W1 gives (W

1 HD W1)hLL+ hLS + hSL+ hSS hLL +hLS hSL + hSShLL hLS +hSL+ hSS hLL

hLS hSL +hSS

For exact decoupling, the off-diagonal elements which are Hermitian

conjugates must be zero

hSL+ hSS= hLL+ hLS

After applying the second transformation W2, we obtain the hamiltonian forthe positive-energy states

h++ =

11+ hLL+ hLS + hSL+ hSS ( hLL+ hLS)

11+ =

1+ [hLL+ hLS]

11+


Foldy Wouthuysen transformation (3)


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Foldy-Wouthuysen transformation (3)

The identification of appears when on considers the application of the

unitary transformation on the wave function

U

L

S

=

+(

L + S)

(S L)

Since we want the 2-components negative energy solutions to be zero:

S = L

The two-component positive energy solutions take the form:

+ = 1

1+

(L + S)

= 11+

(L + L)

=

1+ L

These are the large componentsrenormalized by the operator


Coupling of the large and small components


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Coupling of the large and small components

The exact decoupling implies that

S = L, = (2mc2 V+E)1c( p)

The coupling is energy dependent no closed form of the coupling operator

In the nonrelativistic limit (c ), the coupling is energy independent

= 1

2mc( p)

We can rewrite the coupling operator as

= 12mc2

K(E)c( p), withK(E) = 1 V E2mc2

1


The Pauli Hamiltonian (1)


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Using the definition of we can define a Hamiltonian acting on the largecomponents only:

(V E)L + 1

2m( p)

1

V E

2mc2

1( p)L =0

expansion of the bracket in powers of 1/2mc2. The validity of this expansiondepends on |V E| being smaller than 2mc2

1

V E

2mc2

1=1+

V E

2mc2 +

V E

4m2c4 + . . .

defining the operator H+ as

[H+ E] L = V E+ 12m( p)( p) + 14m2c2 ( p)(V E)( p) + . . . Making use of the Dirac identity( u)( v) = (u v)I2+ i u v

( p)( p) = p2

( p)(V E)( p) = (V E)p2 +[ (V) p i(V) p] fValerie Vallet ([email protected]) |CNRS Universite de Lille 1



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We now need to normalize the wave function+ = OL acting on thepositive energy components:

O = 1+ Taylor expansion yields

O = 1+

1

8mc2 ( p) 1+ 2(V E)2mc2 + . . . ( p) + . . .= 1+

1

8mc2p

2 + O(c4)

O1 = 1 1

8mc2p

2 + O(c4)

The normalization transformation yields

O H+ O1OL =EOL =E+

or by premultiplying by O2

O

1 H+

O

1

+

=E

O

2

+




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T+V+

1

4m2c2

( p)V( p) Ep2 Tp2

1

2(p2V+Vp2)

+

= E 14m2c2 Ep2+ some developments yields:

1

2(p2V+Vp2) = 2

1

2(2V) + (V) +V2

( p)V( p) = 2 (V) +V2+ (V) p




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This is the Pauli Hamiltonian

HPauli =T+V p4

8m3c2 mass-velocity

+

2(2V)

8m2c2 Darwin

+

4m2c2 (V) p

spin-orbit

Mass-velocity term

Relativistic mass correction

E=mc2

1+ p2

m2c2 = mc2

rest mass+

p2

2m

p4

8m3c2

kinetic energy+ . . .

Problem: The mass-velocity terms has no lower bound Pauli Hamiltoniancannot be used in variational calculations




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Darwin term

For a point nuclear potential

V = Z

r , V =

Z

r3r, 2V =4Z(r)

Darwin term

hDarwin =

2Z

2m2c2(r)

it is proportional to the charge density at the nucleus and it therefore never

negative. The Darwin term will never lower the energy.

Spin-orbit term

Rewritten ashSO = Z2m2c2r3

s l

The spin-orbit term described the interaction of the spin of the electron with

this magnetic field due to the relative motion of the charges.

This operator couples the degrees of freedom associated with spin and

space, and makes it impossible to treat spin and spatial symmetry separatelyValerie Vallet ([email protected]) |CNRS Universite de Lille 1

Regular approximations


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g pp

Exact coupling

= 1

2mc2K(E)c( p), withK(E) =

1

V E

2mc2

1 Pauli Hamiltonianonly valid if |V E| smaller than 2mc2. But close to the

nucleus |V E|/2mc2 >1

= 1

2mc2K(E)c( p), withK(E) =

1

V E

2mc2

1 Regular approximation:Use instead the 2mc2 Vexpansion, since it is

always positive definite for the nuclear potential and always greater than

2mc2. In addition since the potential becomes infinite at a point nucleus,

havingV in the denominator regularizes the expansion at the nucleus

K(E) = 2mc2

2mc2 V

1+

E

2mc2 V

1


The ZORA Hamiltonian


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Zeroth-Order Regular Approximation (ZORA) (renormalization is ignored):

HZORA =V+ 12m( p) 2mc2

2mc2 V( p)

the second term can be seen as an effective kinetic energy operator that

goes to the nonrelativistic one when V 0.

Taylor expansion of(2mc2 V)1

HZORA = V+ 1

2m( p)

1+

V

2mc2 +

V2

4mc4 . . .

( p)

= V+T+ 1

4m2c2( p)V( p) . . .

ZORA contains nomass-velocity term, only parts of the Darwin term, but all

spin-orbit interactions arising from the nuclei

Important parts of the scalar relativistic effects are missing

The nuclear potentialVin the denominator is best handled on a numerical

grid, so the ZORA Hamiltonian is particularly suited for DFT calculations

(SeeADF, orDIRACpackages)


Foldy-Wouthuysen transformation for a free particle (1)


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y y p ( )

We have seen that only expansions in terms of the potential energy rather

than the momentum yield regular operators

Start with the free-particle Foldy-Wouthuysen transformation (Note that isrenamed R2)

= c( p)

Ep+mc2 = R2, Ep= m

2c4 +c2p2

Normalization factor is:

A= 1

1+

=

Ep+mc2

2Ep O(c0)

Transformed hamiltonian

H =

Ep 0

0 Ep

+ A

V+ R2VR2 [R2, V]

[R2, V] V+ R2VR2

A




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Hamiltonian for the positive-energy levels is

H++

=Ep+ A

V+

c( p)

Ep+mc2V

c( p)

Ep+mc2

A

Expansion in powers of 1/cyields the Pauli Hamiltonian

HPauli =V+T+ 1

4m2c4( p)V( p)

p4

8m3c2

p2V

8m2c2

Vp2

8m2c2

While Pauli is singular (no variational treatment), the presence of the

kinematic factors A and(Ep+mc2)regularizes things




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In large momentum limit (electron close to the nucleus) Ep cp A 1/2 R2 p/p this transformation ensures bound states (variational treatment possible)

Further decoupling by canceling the off-diagonal element

H+ = A[R2, V] A

second transformation with W = 2 such as

H+ = WH++ WH

++ H

++ H

W

= WEp WA[V+ R2VR2]A WA[R2, V] AW

A[R2, V] A EpW+ A[V+ R2VR2] AW =0

exact decoupling imposes:

WEp+EpW = A[R2, V]A + [ A[V+ R2VR2], W] WA[R2, V] AW


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Approximate decoupling: Douglas-Kroll-Hess transf.


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Douglas-Kroll-Hess transformation: decoupling in V

WEp+EpW = A[R2, V] A

O(V1)+ [A[V+ R2VR2], W]

O(V2) WA[R2, V] AW

O(V3) A[R2, V] A Transformation of the one-electron Hamiltonian; transformation of the

two-electron integrals is neglected (classical Coulomb potential)

Possible separation of spin-orbit coupling using the Dirac identity


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Summary of approximate relativistic Hamiltonians


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The exact decoupling of the large and small components is generally not

possible and depends on the external potential

Approximate decoupling may lead to highly singular operators (problem for

variational calculations!)

Decoupling significantly reduces computational cost Relativistic 1-component (scalar) Hamiltonian and its spin-orbit counterpart

can be obtained by elimination of the spin

Any property operator should be subjected to the same transformation!!!

Picture change. If neglected:

small errors on valence properties significant ones on nucleus properties!!!


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Two-component relativistic methods


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2c implementations of most quantum chemical methods (like in 4c)

Dirac Hartree Fock (DHF) methods Post DHF methods: MP2, Coupled-Cluster methods (CC), CISD, CISDTQ,

Full CI

Multi-configuration Self Consistent field

MRCI methods (GAS-CI), IH(FSCC)

Density Functional Theory

Two-component packages

DIRAChttp://wiki.chem.vu.nl/dirac/

UTChem,http://utchem.qcl.t.u-tokyo.ac.jp/

ADFhttp://www.scm.com/

TURBOMOLEhttp://www.cosmologic.de/

NWCHEM http://www.emsl.pnl.gov/capabilities/computing/nwchem/


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http://wiki.chem.vu.nl/dirac/http://utchem.qcl.t.u-tokyo.ac.jp/http://www.scm.com/http://www.cosmologic.de/http://www.emsl.pnl.gov/capabilities/computing/nwchem/http://www.emsl.pnl.gov/capabilities/computing/nwchem/http://www.cosmologic.de/http://www.scm.com/http://utchem.qcl.t.u-tokyo.ac.jp/http://wiki.chem.vu.nl/dirac/


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Part V

Core approximations

16 Core approximations - Valence only approaches

17 Pseudopotentials

18 Comments on the use of effective core potentials


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Need for frozen core approximations


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In the context ofchemical applications

Core orbitalsare not involved

in heavy elements (second, third transition series, lanthanides, actinides,

superheavy elements...) large number of core orbitals

core orbitals are essentially atomic like in a molecule or material core orbitals supply anon-localstatic potential that can be evaluated once in

the calculation

relativistic effects are too a large extent localized in the core region

include relativistic effects in the core potential

treat valence orbitals non-relativistically


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Hierarchy of core approximations


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1 Correlate the core electrons at a lower level of theory (e.g. MP2)

2 Include core electrons only at HF level of theory and freeze them in post-HF

treatments

3 Use atomic orbitals for core electrons (Frozen Core)

4 Model frozen core byEffective Core PotentialsorPseudopotentials

Advantages of core approximations

Stepwise reduction of computational cost

Onemust control the accuracy

choose the right core: core-valence separation either on the basis of energetic(orbital energies) or spatial (shape, radial maxima or expectation values of

orbitals) arguments. check different sizesof the core need for all-electron benchmark calculations

Relativistic effects can be included inEffective Core Potentialsby suitable

parametrization


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Valence-only Hamiltonians (1)


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Search for an Hamiltonian which acts on the valence electrons

Hv =

nvi

hv(i) + i


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Expression ofhv(i)and hv(i)

hv(i) = 12

i+Vcv(i) and gv(i,j) = 1rij

Relativistic effects in the parametrization ofVcv(i),the effective corepotential, which describes the interaction of one valence electron with all

nuclei and cores present.

The molecular pseudopotential is assumed to be a superposition of atomicpseudopotentials, with the Coulomb attraction between point charges as the

leading term

Vcv(i) =N

Q

ri+ Vcv(ri)

+ . . .

For the core-core term, the point-charge interaction is also the first term:

Vcv(i) =N


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Reduce the basis set to describe valence orbitals

Phillips-Kleinman equation (1959) with an effective one-electron Hamiltonian

framework

Fock equation for a valence orbitalv

Fvv=vv+c=v

cvc

Transformation using a projector(1 Pc)onto the valence space(Pc=

c

|cc|)

(1 Pc)Fvv =vv (1)

Reduction of the basis set by mixing v andc, thus removing the radialnodes in the core region

p = Np

v+c=v

cc

v= (Np)

1(1 Pc)p

Valerie Vallet ([email protected])|

CNRS Universite de Lille 1

P.55/89

Pseudopotentials (2)


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By inserting into equation1, we obtain

(1 Pc)Fv(1 Pc)p=v(1 Pc)p

Using the so-called generalized Phillips-Kleinman pseudopotential

VGPK = PcFv FvPc+PcFvPc+ vPc

If one assumes the core orbitalscto be also eigenfunctions of the valenceFock operator, i.e,[Fv,Pc], one gets a simplified pseudo engenvalueproblem:

(Fv+VPK)p=vp

containing the so-called Phillips-Kleinman pseudopotential

VPK =

c=v

(v c) |cc|


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Pseudopotentials (3)


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Fornvvalence electrons, one substitutes

Fv

nvi

Fv(i) +i


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In this form, we have not achieved computational saving because the core

orbitals are still present

Elimination of the core with a suitable model potential

hv(i) =

(1 Pc(k))Fv(i)(1 Pc(k)) +EvPc

1

2i+Vcv(i)

gv(i,j) =

(1 Pc(k))(1 Pc(k))g(i,j)(1 Pc(k)(1 Pc(k))

1

rij


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Analytical form of pseudopotentials


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[L. R. Kahn and W. A. Goddard, J. Chem. Phys., 1972,56, 2685]

Semi-local expression of a pseudopotential

Vcv(ri) =L1l=0

l+1/2j=|l1/2|

V

lj (ri) V

L (ri)

Plj(i) +V

L (ri)

with

Plj(i) =

jmj=j

|ljmjljmj|

Spin-orbit averaged pseudopotential (scalar-relativistic)

Vcv,av(ri) =L1

l=0 V

l (ri) V

L (ri) Pl (i) +V

L (ri)

with

Pl (i) =

lml=l

|lmllml|


P.59/89

Analytical form of pseudopotentials (2)


59/89

Spin-orbit pseudopotential:

Vcv,SO(ri) =

L1l=1

Vl (ri)

2l+1

lP

l,l+1/2 (l+1)P

l,l1/2

=L1

l=12Vl (ri)

2l+1Pll sPl

with

Vl (ri) =V

l,l+1/2(ri) V

l,l1/2(ri)

Gaussian expansion of the radial parts (easy integrals!) - Fitted parameters

in red (cf. pseudopotentials librairies)

V

lj (ri) =

k

Aljkr

nljki exp (a

ljkr

2i)


P.60/89

Analytical form of pseudopotentials (3)


60/89

For large (polarizable cores)...

Core-polarization potential

Vcpp= 1

2 f2

Electrostatic field and cut-off factor

f =

i

rir3i

1 exp er

2i

ne+=

Qrr3

(1 exp (cr2))

nc


P.61/89

Adjustment of the parametersAljk,nljk,a

ljk (1)


61/89

Shape-adjustment

Take a valence orbitalv

Make a nodeless pseudo-orbitalv Invert the Schrodinger equation and obtain the parameters

Impossible for large valence space!

Not very precise


P.62/89

Adjustment of the parametersAljk,nljk,a

ljk (2)


62/89

Energy-adjustment (Dolg, Stoll, Stuttgart-Koln) pseudopotentials

Atomic adjustment to total valence energiesEAE,valenceI =E

AEI E

AEcore

multitude of chemically relevant electronic configurations/states/levels of

the neutral atom and low-charged ions !

Optimize the parameters with respect to

S=

I

wI

E

AE,valenceI E

PP,valenceI

2:=min

Adjustment can be made to any method

currently best method: (average- level) multi-configuration

Dirac-Hartree-Fock reference data based on the Dirac-Coulomb-Breit

Hamiltonian


P.63/89

Comments on effective core potentials


63/89

Where to get the parameters

In most quantum chemistry programs (GAUSSIAN, NWCHEM, MOLCAS,

MOLPRO, etc...)

http://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.html

https://bse.pnl.gov/bse/portal

Always prefer the pseudopotentials from the Koln Stuttgart group because

they are far more accurate!


P.64/89

Comments on pseudopotentials (2)
http://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.htmlhttp://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.htmlhttps://bse.pnl.gov/bse/portalhttps://bse.pnl.gov/bse/portalhttp://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.htmlhttp://www.theochem.uni-stuttgart.de/pseudopotentials/index.en.html


64/89

Illustration of the accuracy of the Koln-Stuttgart pseudopotentials


P.65/89

Warnings about DFT and pseudopotentials?


65/89

DFT is assuming a local external potential ....

More seriously (Thomas-Fermi expression of the kinetic energy)

(c+ v)1/3

=1/3c +

1/3v

Nevertheless, a lot of people (including myself!) do it with good results

Dont ask me for an in-depth understanding!


P.66/89


66/89

Part VI

One-component relativistic methods

19 Which spin-orbit operator?

20 Consequences of spin-orbit coupling

21 1c approaches for the treatment of SO-coupling


P.67/89

Philosophy of one-component approaches


67/89

Split scalar relativistic effects and spin-orbit coupling

In two-component or pseudopotentials we can separate scalar relativistic

effects from spin-orbit coupling

First run a scalar relativistic

Treat spin-orbit couplinga posteriori

Which spin-orbit operator? Pseudopotential

Dont use all-electron spin-orbit hamiltonian (you have pseudo-orbitals)

Choose the spin-orbit pseudopotential that is paired with your scalarrelativistic pseudopotential


P.68/89

Which spin-orbit operator? All-electron calculations


68/89

Breit-Pauli Hamiltonian

The one-electron term: HSO1el = e2

4m2c2

k

Z ik

rkr3k

pk

Two-electron term:

HSO2el =

e2

2m2c2

k=l

i( k+2 l) rkl

r3kl pk

Z/r3 divergence whenr 0

Dont use it in variational calculations


P.69/89

Which spin-orbit operator? All-electron calculations


69/89

With DKH Hamiltonian: no-pair spin-orbit Hamiltonian

The one-electron term:

HSO1el =

k

Ak

Ek+mc2

ik

rkr3k

pk

Ak

Ek+mc2

Ak = Ek+mc22Ek The two-electron term has two contributions:

HSO2el =

k=lAkAl

ik

Ek+mc2

rklr3kl

pk

1

Ek+mc2

AkAl

+k=l

AkAl 2ik

Ek+mc2 rkl

r3kl

pl

1El+mc2

AkAl


P.70/89

Mean-field approach to spin-orbit coupling


70/89

Calculate two-electron contribution (shielding) from a fixed configuration

Effective one-electron integrals

Fock-operator technique

Hmean-fieldij = i|H

SO1el |j

+ 1

2

k,fixed{nk}

nk{ik|HSO2el |jkik|H

SO2el |kj ki|H

SO2el |jk}


P.71/89

Mean-field approach: atomic approach


71/89

Spin-orbit coupling is short-ranged : r3 behavior atomic approximation

Compute the Mean-Field integrals for each atom separately

Use atomic orbitals and ground-state average occupations

need for atomic natural basis sets

Atomic Mean-Field SO integral approach (AMFI code, Bernd

Schimmelpfennig)

Splitting identical with full SO-operator within a few wave numbers

Available inDALTON,DeMon,Dirac,MOLCAS,ORCA


P.72/89

Consequences of spin-orbit coupling


72/89

H=Hnr+i

Ap4i +B2iV+ 4m2c2 i

sirir3i piHso(i) =f(r)l s=f(r) (lxsx+lysy+lzsz) =f(r) {(l+s+ ls+) /2+lzsz}

non-relativistic term has certain symmetry properties (atom, molecule,

special symmetry group)C2v, D6h. . .-)3P, 2S, 3, 2+, 1B1,

2B2u . . .

Scalar relativistic terms keep these properties (blue)

Commute withL2, Lz etS2, Sz for atoms

with L2zetS2, Sz for linear molecules

invariant with respect to symmetry operation in the general case In atoms and linear molecules, spin-orbit operators dont commute with

L2, Lz etS2, Sz


P.73/89

Consequences of spin-orbit coupling


73/89

In polyatomic molecules, spin functions have special properties IfCz()rotation around thez-axis is a symmetry operation (H2O):

[Cz()]2 = Cz(2)is the identity For spin functionsCz(2)| =?|alpha = | Double group symmetry


P.74/89

1c approaches for the treatment of SO-coupling


74/89

1-component HF/MCSCF

SR all-electron basis SR-PP + basis set

Post-HF/Post-MCSCF treatment to include dynamical correlation and SO

coupling

One-electron SO operator singly-excited configurations

Slow convergence of dynamic correlation (single, double, ..., excited

configurations)

Intermediate coupling scheme: SO relaxation of valence orbitals is important for heavy main group atoms dense spectra in transition metals and actinides


P.75/89

Treat correlation and spin-orbit coupling together (1-step

approach)


75/89



Spin-orbit

configuration interaction

Double group

symmetry

SO couples various space and spin symmetries large SOCI space

Make use of double group symmetry and Direct CI

COLUMBUShttp://www.univie.ac.at/columbus/


P.76/89

Treat correlation and SO in a 2-step fashion
http://www.univie.ac.at/columbus/http://www.univie.ac.at/columbus/


76/89



electron correlation

DFT or WFT correlated method

1) Couple the correlated spin-free states

2) Small SOCI with an effective Ham.MOLCAS, MOLPRO

EPCISO

Since SO converges faster (small CI space)

MOLCAS (RASSI module)http://www.teokem.lu.se/molcas/

MOLPRO (MRCI module)http://www.teokem.lu.se/molcas/

EPCISO (interface with MOLCAS)[email protected]


P.77/89
http://www.teokem.lu.se/molcas/http://www.teokem.lu.se/molcas/http://localhost/var/www/apps/conversion/tmp/scratch_7/[email protected]://localhost/var/www/apps/conversion/tmp/scratch_7/[email protected]://www.teokem.lu.se/molcas/http://www.teokem.lu.se/molcas/


77/89

Part VII

Illustrations of relativistic in chemistry: more common

than you thought


P.78/89

Relativistic effects in chemistry

Direct effects


78/89

Direct effects

contractionof the core-penetrating orbitals

sorbitals,p1/2,p3/2 orbitals in core

Energetic stabilization: higher ionization energy, higher electron affinity,

smaller polarizability

Direct effects indirect effects ond,f, orbitals and valenceporbitals

nuclear charge is shielded to a larger extent because of direct effect on

core-penetrating orbitals (in particular of the semi-core)

relativisticexpansionof core non-penetrating orbitals

energetic destabilization

smaller ionization energy, larger polarizability in turn, stabilization of

core-penetrating orbitals in next shell

gold maximum


P.79/89

U4+ atomic spectrum


79/89

-200

-150

-100

-50

0

Non Relativistic Spin-Free Spin-Orbit

4s

4p

4d


P.80/89

Relativistic Effects on Atomic Shell structures

[S J Rose et al J Phys B: At Mol Phys 1978 11 1171]


80/89

[S. J. Rose et al.,J. Phys. B: At. Mol. Phys., 1978,11, 1171]

Direct (dynamics) and indirect (potential) effects on orbital energies (eV)

Dynamics: Dirac Schrodinger

Potential: Rel Nonrel Rel Nonrel

Au 6s -7.94 -7.97 -6.18 -6.01

Tl 6p1/2 -5.81 -6.79 -4.58 -5.24

Tl 6p3/2 -4.79 -5.63 -4.46 -5.24Lu 5d3/2 -5.25 -7.32 -4.74 -6.63

Lu 5d5/2 -5.01 -6.90 -4.81 -6.63

Direct effects dominate for Au 6sand Tl 6p1/2 Compensation of direct and indirect effects for Tl 6p3/2

Indirect effects dominate for Lu


P.81/89

Relativistic effects on molecular structures

[C. L. Collins et al.,J. Chem. Phys., 1995,102, 2024]


81/89

Bond lengths in pm, dissociation energies in eV, harmonic frequencies in

cm1

Molecule Method rSCFe rMP2e D

SCFe D

MP2e

SCF

MP2

CuHNR 156.9 145.4 1.416 2.585 1642 2024DKH 154.2 142.9 1.476 2.708 1698 2100RECP 154.3 142.9 1.465 2.696 1690 2095DC 154.1 142.8 1.477 2.711 1699 2101Exp 146.3 2.85 1941

AgHNR 177.9 166.3 1.126 1.986 1473 1699DKH 170.1 158.7 1.229 2.190 1602 1870RECP 170.0 158.4 1.224 2.189 1607 1882DC 170.0 158.5 1.233 2.195 1605 1873Exp 161.8 2.39 1760

AuHNR 183.1 171.1 1.084 1.901 1464 1169DKH 157.6 149.8 1.727 3.042 2045 2495RECP 157.1 149.5 1.751 3.075 2076 2512DC 157.0 149.7 1.778 3.114 2067 2496Exp 152.4 3.36 2305


P.82/89

Effects on chemical reactions


82/89

SCF reaction energies in kJ/mol for the reaction XH4 XH2 +H2

Method Si Ge Sn Pb

NR 263 190 129 89

RECP 195 102 -31

DC 261 177 97 -26

Large relativistic effects in Pb

Stabilization of the 6sand destabilization of 6pdecreasessp3 hybridization

inPbH4, thus making the reaction exothermic

In some systems, spin-orbit coupling can enable crossing between states ofdifferent multiplicities (inter-system crossings)


P.83/89

Effects on NMR shieldings


83/89

[P. Hrobarik et al.,J. Phys. Chem. A, 2011,115, 56545659]

Isotropic hydrogen shielding parameters (ppm)


P.84/89

Relativity and the Lead-Acid Battery

[R. Ahuja et al., Phys. Rev. Lett., 2011,106, 018301]


84/89

The lead-acid battery reaction

Pb(s)+ PbO2(s)+ 2 H2SO4(aq) 2 PbSO4(s)+ 2 H2O(l)

The nonrelativistic (NR), scalar relativistic (SR),

and fully relativistic (FR) energy shifts (in eV) for

the solids involved in the lead-acid-battery reaction.

Values for both M = Sn ( green) and M = Pb (black)

are given.

Electromotoric force in eV

experimental: 2.107 V

average fully relativistic DFT value: + 2.13 V


P.85/89

Spin-orbit effects in structural chemistry


85/89

Strong influence of SO coupling on heavy element structures

SO makes Ptn clusters flat (n = 25)

In Cs18Tl8O6 the system exhibit an open-shell degenerate HOMO within a

scalar relativistic approximation.

With SO coupling a closed-shell electronic system is obtained in accordance

with the diamagnetic behavior of this crystal.


P.86/89


86/89

Part VIII

Comments


P.87/89

Some comments


87/89

How to choose between 4c, 2c, 1c + SOCI? Best method depends on the system studied Which property are you interested in? What is the accuracy you are looking for? It depends on whether you look at

chemical reactions, spectroscopy, or molecular properties Which computational capacities do you have access to?

For closed shell systems, one-component methods work well

Dont use non-relativistically contracted basis sets

As usual correlation is important, esp. as there are often a lot of close lying

states with different correlation effects

There are a lot of PPs, ECPs, AIMPs on the market. If you are not sure,

compare to some all-electron method, perhaps even four-component

An approximate method with a good basis set should be preferable to amore accurate method with a too small basis set.


P.88/89

Summary points


88/89

Beware of the importance of relativistic effects

1 The classical examples of relativistic effects in chemistry remain and have

been included in most chemistry textbooks.

2 One of the oldest examples, which deserves more attention, is the

SO-induced NMR heavy-atom shift.

3 Investigators continue to discover new examples, such as the heavy-element

batteries.

4 Catalysis is one of the most important applications of relativistic quantum

chemistry.

5 The SO effects in structural chemistry have been identified only recently

after technical progress.


P.89/89

Further reading

Relativistic Quantum Mechanics


89/89

Q

R. E. Moss,Advanced Molecular Quantum Mechanics - An Introduction to Relativistic Quantum Mechanics andthe Quantum Theory of Radiation, Springer Netherlands, 1973

P. Strange, Relativistic Quantum Mechanics with Applications in Condensed Matter and Atomic Physics,

Cambridge Univ. Press, 1998, p. 594

Relativistic Quantum Chemical methods

P. Schwerdtfeger, Relativistic Electronic Structure Theory: Part 1, Fundamentals, ed. P. Schwerdtfeger, Elsevier,

Amsterdam, 2002

K. G. Dyall and K. Fgri, Introduction to relativistic quantum chemistry, Oxford University Press, New York, 2007

M. Reiher and A. Wolf,Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science,

WILEY-VCH Verlag, 2009

Applications P. Schwerdtfeger, Relativistic Electronic Structure Theory: Part 2, Applications, ed. P. Schwerdtfeger, Elsevier,

Amsterdam, 2004

P. Pyykko,Ann. Phys., 2012,63, 4564


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