Relativity

1
Relativity Electron Correlation

• Lucas Visscher
• Vrije Universiteit Amsterdam

2
The extra dimension
3
Special Relativity

• Essential to describe fast particles.
• Speed of light (300,000,000 m/s) is upper limit
  to velocity.
• Mass of particle depends on its velocity
• From exact solution of Schrödinger equation for
  one electron around a nucleus (H-like) with
  charge 80 (Hg)
• - Mean velocity is 58 of speed of light
• - Electron mass increases by 23

4
Heavy Elements Relativity

• Not the weight but the nuclear charge Z matters
• Heavy Elements have a large Z and many electrons
• Everything beyond Ln is heavy. Superheavy are
  the new elements (Zgt100)
• Electrons move fast in all heavy element molecules

5
Visible Relativistic Effects

• Non-relativistic gold is silver
• The 5d-6s transition is shifted from the UV to
  the visible part of the spectrum by relativistic
  effects
• Phosphorescence
• Singlet-triplet transitions are allowed because
  the non-relativistic quantum numbers are not exact

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Course outline

• Relativistic Quantum Theory
• The Dirac equation
• Quantum Electro Dynamics
• Approximated Hamiltonians
• Molecular relativistic calculations
• Basis set expansions
• Computational aspects
• Electron correlation
• Many-Body Perturbation Theory
• Choice of active space
• Configuration Interaction methods
• Kramers-restricted algorithms
• Coupled Cluster methods
• Applications (two examples)

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Relativististic Quantum Mechanics

• 1905 STR
• Einstein E mc2
• 1926 QM
• Schrödinger equation
• 1928 RQM
• Dirac equation
• 1949 QED
• Tomonaga, Schwinger Feynman

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Non-relativistic quantization

• The classical nonrelativistic energy expression

Quantization
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Non-relativistic quantization

• Write the classical nonrelativistic energy
  expression as

Quantization
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Non-relativistic quantization
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Spin in NR quantum mechanics

• The Pauli Hamiltonian

• Coulomb gauge ? A 0
• Two component wave functions
• Second derivative wrt position, first derivative
  wrt time
• Linear in scalar, quadratic in vector potential
• Not Lorentz-invariant
• Ad hoc introduction of spin
• Origin of anomalous g-factor (ratio magnetic
  moment (m) to intrinsic angular momentum (s) is
  two) is unclear
• No spin-orbit coupling

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Relativistic quantization

• Equivalent classical relativistic energy
  expressions

Quantization 1. (after series exp.)
Relativistically corr. Schrödinger Eq. 2.
Klein-Gordon Equation (spin 0 particles) 3. Dirac
equation (spin 1/2 particles)
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The Dirac Hamiltonian

• No gauge specified
• Four component wave functions
• Linear in derivatives wrt time and position
• Linear in scalar and vector potentials
• Lorentz invariant
• Spin g-factor is exactly 2

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Time-independent Dirac equation

• The nuclei do not move with relativistic speeds.
  Stationary frame of reference (Born-Oppenheimer
  approximation)
• Apply separation of variables

15
Free particle Dirac equation

• Take case V 0
• Use plane wave trial function

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Free particle Dirac equation

• Two doubly degenerate solutions
• Classical energy expression
• Prediction of negative energy solutions !

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Free particle Dirac equation

• Wave function for E E
• Upper components Large components
• Lower components Small components

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Free particle Dirac equation

• Wave function for E E-
• Role of large and small components is reversed
• Charge conjugation symmetry
• Does the variational principle apply? Variational
  Collapse.

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Dirac sea of electrons

• All negative energy solutions are filled
• The Pauli principle forbids double occupancy
• Holes in the filled sea show up as particles with
  positive charge positrons (discovered in 1933)
• Infinite background charge

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Second Quantization

• Introduce a m-dimensional Fock space F(m)
• States are defined by the occupation number
  vector n
• The vacuum has all n0
• We use an orthonormal basis

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Second Quantization

• Second quantized operators
• Creation operator
• Annihilation operator
• Define all operators in terms of these elementary
  operators

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Second Quantization

• Commutation relations
• Number operator
• Hamilton operator

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Fock space Hamiltonian

• Positive and negative energy solutions define a
  Fock space Hamiltonian

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Renormalization

• Subtract energy from negative energy states
• Re-interpretation
• Normal ordered Hamiltonian

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Quantum Electro Dynamics

• Positive energy for positrons
• Total charge

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Dressed particles

• The QED Hamiltonian depends on the positive and
  negative energy solutions of the Dirac equation
• The Dirac equation depends on the external
  potential
• Different realizations possible
• Free particle solutions (Feynman,1948)
• Fixed external potential (Furry,1951)
• External mean-field potential (fuzzy)
• Particles in one representation are
  quasiparticles (dressed with ep-pairs) in another
  representation

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Electron-electron interaction

• Add photon field and interaction term
• Electron-electron interaction is automatically
  retarded by the finite velocity of light
• Corrections to the Dirac equation and the
  instantaneous Coulomb interaction
• Feynman (NP 1965) diagrams
• Breit interaction (1929) (Order c-2)
• Vacuum Polarization Self Energy Lamb shift
  (NP 1955) (c-3)

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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

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Approximate Hamiltonians

• Traditional derivations
• Focus on the positive energy solutions
• Apply unitary transformations to reduce coupling
  between large and small components
• Neglect remaining coupling terms
• Neglect SO-coupling terms
• Via the quaternion Modified Dirac equation
• Focus on the positive energy solutions
• Introduce pseudo large component
• Neglect SO-coupling terms
• Approximate the metric (renormalize)

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The Dirac Hamiltonian
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Foldy-Wouthuysen transformation

• Use a unitary transformation to decouple large
  and small components
• Exact expressions only known for the free
  particle case

Picture change
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Douglas-Kroll-Hess method

• Idea
• Transform bare-nucleus Hamiltonian with
  free-particle tranformation matrix, followed by
  additional transformations to reduce size of
  remaining off-diagonal elements
• Assumptions
• The transformation is based on the Furry picture
  potential can not include mean-field of
  electrons
• The usual implementations neglect the
  transformation of the two-electron interaction
  and often also SO-coupling
• Advantages-Disadvantages
• A Method is variationally stable
• A Slight modification of existing code required
  (replacement of one-electron nuclear attraction
  integrals), fast implementation
• A good results in practice, significant
  improvement over Breit-Pauli
• D Complicated operators, matrix elements can
  not be calculated analytically
• D Two-electron terms are hard to evaluate
• D Interactions with external field need to be
  represented by transformed operators (picture
  change)

33
Four-component methods

• Idea
• Expand Dirac equation in separate basis sets for
  the large and small components
• Use kinetic balance condition to prevent
  variational collapse
• Advantages-Disadvantages
• A No approximations made
• A Matrix elements over the operators are easily
  evaluated
• D Many more two-electron integrals need to be
  handled
• D The Fock matrix is (twice as) large
• D Symmetry need be handled by double group
  theory
• A No picture change problems

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Quaternion algebra

• One real and three imaginary parts
• Commutation relations
• Quaternion gradient

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Quaternion Dirac equation

• Unitary transformation matrix
• Block diagonal Hamiltonian matrix

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Quaternion Dirac equation

• Two independent blocks
• Degenerate solutions Kramers pairs

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Positive energy solutions

• Subtract rest mass energy
• Express small component in terms of large
  component

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Pseudo Large Component

• Define an auxilliary function
• Transform the Dirac equation accordingly

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Quaternion Modified Dirac Equation

• Write in terms of a metric matrix
• No approximation made so far
• Pseudolarge component may be expanded in the same
  basis as the large component
• SO-coupling terms are contained in the imaginary
  part of the Hamilton operator

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Non-relativistic equation

• Take limit c??
• Pseudolarge component may be eliminated
• Large component wave function becomes the
  Schrödinger wave function
• Provides starting point for (Direct) Perturbation
  Theory

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Zeroth Order Regular Approximation

• Take limit c?? only in the metric
• Pseudolarge component may be eliminated
• Large component wave function becomes the ZORA
  wave function
• Good approximation for relativistic effects
• Not invariant for constant shifts in the potential

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Neglect of SO-coupling terms

• Delete imaginary terms in QMD Hamiltonian
• Delete imaginary terms in ZORA Hamiltonian

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Approximate relativistic equations

• Approximation of the full Hamiltonian via
• Approximated metric, elimination of pseudolarge
  (small) component
• One component quaternion equation (ZORA)
• Deletion of quaternion imaginary terms
• Two component real equation (spinfree Dirac)
• AB
• One component real equation (scalar ZORA)

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Hartree-Fock Self Consistent Field

• Construct Fock operator
• Find eigensolutions
• Check convergence
• Compute energy

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Fock operator
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Basis set expansion techniques

• In the modified Dirac equation
• In the original Dirac equation

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Basis set expansion techniques

• Define density matrix
• Matrix representation of Fock operator
• Energy calculation

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Choice of expansion functions

• Large (pseudolarge) component
• Atoms Sturmians, Slaters or Gaussians
• Molecules Spherical or Cartesian Gaussians
• Small component
• Same type as large component
• Should fulfill kinetic balance relation

Restricted KB
Unrestricted KB
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Relativistic basis sets

• Non-relativistic
• Add additional tight functions to get correct
  SO-splittings
• Use kinetic balance for generation of small
  component primitives
• Recontraction is required
• Douglas-Kroll
• Use kinetic balance for generation of small
  component primitives
• Recontraction is required
• Relativistically optimized
• Universal, Even tempered, l-optimized,
  j-optimized
• Family type (i ? g ? d ? s h ? f ? p) for
  kinetic balance
• Usually applied in uncontracted form

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The small component wavefunction

• The large component wave function resembles the
  non-relativistic wave function
• Exact relation between large and small component
  wave functions

• Small component wave function is related to the
  first derivative of large component wave function
• Prefactor damps singularity in the vicinity of
  nuclei
• The small component wave function is an
  embarrassingly local quantity !

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Electron Density of Uranyl
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Towards linear scaling

• Observation Major bottleneck lies in processing
  of (SSLL) and (SSSS) electron repulsion
  integrals
• Simple Coulombic Correction Neglect all
  (SSSS) integrals
• Accurate for most practical purposes
• Method requires an a posteriori correction based
  on neglected electronic charge
• May be inadequate for sensitive properties that
  probe the wave function around the nuclei
• (SSLL) type integrals strongly dominate
  calculation time
• Neglect/approximate multi-center (SSXX)
  integrals
• Balance nuclear attraction and electron repulsion
• No a posteriori corrections necessary
• Work associated with (SSLL) type integrals is
  also reduced
• Simple but approximation is fixed
• Density fitting for Coulomb interaction
• Use experience from DFT and linear schaling in
  large molecules
• Implementation in progress

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Computational scaling
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Computational scaling
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Relativistic electron correlation

• Many-Body Perturbation Theory
• Integral-direct implementation of MP2
• Configuration Interaction
• Full CI to about 10,000 determinants
• Direct CI to about 5,000,000 determinants
• Spinfree CI to 1,000,000,000 determinants
• Coupled Cluster
• CCSD(T) to a few million amplitudes
• Fockspace (MR) CCSD (EA, IE)
• Multi-Configuration Self Consistent Field
• CI to about 100,000 determinants
• Computational bottlenecks
• Transformation of 2-e. integrals to the molecular
  spinor basis
• Memory use in the CI and CC modules

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Second quantization

• Hamilton operator
• Fock operator
• Mean field

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Many-Body Perturbation Theory

• Perturbation
• First order energy

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Many-Body Perturbation Theory

• Second order energy
• Definitions

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Some tricks used in index transformation

• Permutational symmetry. If index ranges are all
  equal
• Form only (pqXX) with p lt q
• Form only (pqrs) with r lt s
• Particle symmetry. If the (SSSS) type integrals
  are to be neglected
• Form only (pqLL) and not (pqSS)
• Symmetrize at the end by adding (pqrs)
  (pqrs) (rspq)
• Used in MOLFDIR, but last step is difficult to
  parallelize
• Direct evaluation of quantity of interest (direct
  MP2)
• Point group symmetry
• Transform only symmetry unique integrals
  (MOLFDIR)
• Use real arithmetic in higher pointgroups (DIRAC)
• Quaternion algebra
• Loop over Kramers paired spinors instead of
  spinors

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MO-integrals in quaternion form L. Visscher, J.
Comp. Chem. 23 (2002) 759.
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The small component wavefunction

• The large component wave function resembles the
  non-relativistic wave function
• Exact relation between large and small component
  wave functions

• Small component wave function is related to the
  first derivative of large component wave function
• Prefactor damps singularity in the vicinity of
  nuclei
• The small component wave function is an
  embarrassingly local quantity !

63
Core electrons choice of active space

• All-electron correlation calculations are not
  feasible
• Approximation of core electrons
• Correlate the core electrons at a lower level of
  theory (e.g. MP2)
• Include core electrons only at HF level of theory
• Use atomic orbitals for core electrons (Frozen
  Core)
• Model frozen core by a Model Potential
• Model frozen core by a Relativistic Effective
  Core Potential
• Error correction and additional features
• Estimate higher order correlation effects in
  another basis set
• Use a core correlation potential
• Use a core polarization potential
• Include valence relativistic effects in RECP

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Valence electrons choice of method

• Method depends on system studied
• 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, Multireference CI or MR-CC
• SOC-inclusive methods are usually required
• Try to identify active center

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Direct Configuration Interaction

• Write wave function as linear combination of
  determinants
• Define sigma and error vector
• Obtain sigma vector directly from MO-integrals

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Coupling Coefficients

• SOC prohibits use of alpha and beta-strings or
  other spin-adapted schemes !
• Insert resolution of identity to work with
  one-electron coupling coefficients
• Use graphical techniques to index determinants
  and evaluate coupling coefficients (/-1 or 0)
• Use Abelian point group symmetry if possible
• Evaluation may be rate-determining step in
  general CI calculations

67
Reverse lexical ordering of determinants

• Example 246 is represented by the thick line
  in the graph
• Vertex weights W(P,k) are number of paths to
  vertex
• Recursion formula
• W(P, k) W(P-1, k-1) W(P-1, k)
• Arc weights Y (P,k)
• Vertical arcs have zero weight.
• Weight of arc Y(P,k) connecting (P-1, k-1) and
  (P,k) is W(P-1, k).
• If a vertex (P-1, k) lies outside the graph, its
  weight and the weight of Y(P, k) are zero.
• Index of determinant is

Ref. W. Duch, Graphical Representation of Model
Spaces, Lecture Notes in Chemistry 42, (Springer
Berlin 1986).
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Evaluation of coupling coefficients

• Example Coupling between 236 and 356 is a
  loop in the graph
• Value of coupling coefficients is
• Example
• nLoop 1, ? 13, ? 19
• This means
• Multiply integral (2,5) with CI coefficient 19
  to contribute to sigma vector element 13
• Split interactions in head, loop tail

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Fast evaluation of coupling coefficients

• ?? 1 YHead  YTail Yupper
• n? 1 YHead  YTail Ylower
• Graphs facilitates analysis, evaluation and
  storage of couplings for different integral
  classes.
• E.g. loops over head will give short loops with
  stride 1
• Scheme may be adapted to include Abelian symmetry
• Division of space into restricted active
  subspaces can be used to make the algorithms more
  efficient at the expensive of increasing code
  length

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Kramers-restricted CI

• Rewrite Hamiltonian in terms of Kramers pairs
• Block CI-vector by counting the number of
  unpaired electrons
• Use modified non-relativistic CI algorithms

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Kramers-restricted CI

• X-operators are defined as linear combinations of
  the original excitation operators, e.g.
• Reduces memory that is needed for the algorithm
• Facilitates approximations and reduction to
  spinfree form

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How to include an SO-operator in CI

• 1. First order quasi-degenerate perturbation
  theory (inclusion after CI step)
• Can also be used with unbound operators (Pauli
  form)
• Is computationally efficient (one step procedure)
• Offers convenient (conventional) interpretation
  scheme
• Important couplings to excited states may be
  missed
• 2. Limited variational theory (inclusion in CI
  step)
• Unbound operators (Pauli form) are acceptable
• Does only increase the CI effort, no influence on
  HF and MO-transformation
• Interpretation is non-conventional
• Accuracy is limited when orbital relaxation
  effects are important
• 3. Variational theory (inclusion in SCF step)
• Can only be used with bound operators
• Is computationally demanding (symmetry breaking
  already in SCF)
• Interpretation is non-conventional
• Should be the most accurate theory

73
Coupled Cluster

• Write wave function in exponential form
• CCSD Restriction to single double excitations
• Energy expression (I,J occupied, A, B virtual)

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The CCSD equations

• Equations for T1 and T2 amplitudes

75
The CCSD intermediates

• Intermediates used in T1 and T2 equations

76
Evaluation of the ltvvvvgt integral contribution

• Contribution to be evaluated
• Use point group symmetry
• Write contraction as BLAS DGEMM or ZGEMM
• Parallelize over integral batches

77
RELCCSD implementation

• Preliminary information consists of two files
  containing symmetry information and the
  transformed one- and two-electron integrals.
• Symmetry-information is only passed on via the
  appropriate multiplication table of an Abelian
  group. This was used to run non-relativistic
  calculations efficiently by defining the table as
  the direct product of a single group with a
  subgroup of SU(2).
• All contractions are performed via calls to
  BLAS-routines (mostly DGEMM).
• Contractions involving the largest integral
  classes are parellized by distributing the
  MO-integrals.
• Prerequisite is that all amplitudes can be held
  in memory simultaneously.

78
Kramers-restricted CC

• For closed shells systems one can define
• Rewrite equations in terms of unique quantities
• Reduction of factor 2 in number of amplitudes
• Reduction by factor 8 in number of operations
  necessary
• Comparison with optimal spinfree algorithm
• KRCCSD is max. 32 times more expensive than NR
  SR-CCSD

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The Fock space CCSD method

• Single reference CCSD Fock space CCSD
• The states in the model space correspond to
• Annihilation of an electron from an active
  occupied orbital (1h,0e) sector Ionization energy
• Creation of an electron in an active virtual
  orbital (0h, 1e) sector Electron Affinity
• Beware of intruder states

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Model spaces

• Example One creation and one annihilation (1h,
  1e)
• Excitation operators

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Implementation of FSCCSD

• 1. Take equations for T1 and T2 amplitudes
• 2. Include the active virtuals in the occupied
  set
• I ? E (0h,1e) sector
• 3. Include the active occupied in the virtual set
  
• A ? V (1h, 0e) sector
• 4. Add the additional folded diagrams that
  involve the effective Hamiltonian (is calculated
  automatically as TEF or TVW parts of the extended
  array of T1 amplitudes)
• 5. Solve equations sector by sector for improved
  convergence. Technically all amplitudes are
  calculated simultaneously.
• Computational scaling (Nv)4 . (No)2 ? (NvNao)4
  . (NoNav)2
• (Nv)2 . (No)4 ? (NvNao)2 . (NoNav)4
• Feasible as long as Nav ltlt No

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Parallelization

• Distribution of work
• Most steps are dominated by AO-integral
  evaluation and are parallelized by distributing
  over the integral calculation tasks (master-slave
  algorithm)
• The CC algorithms are formulated in MO-basis and
  are parallelized by distribution of the
  transformed integrals (fixed distribution, no
  master necessary)
• Difficult aspects
• HF DFT the algorithms scale well but need
  substantial memory (due to the large Fock
  matrices on each node)
• 4-index trade-off between CPU and memory
  efficiency is difficult for high-angular momentum
  function shells
• CC communication of intermediate quantities is
  needed (synchronization steps) and slows down
  calculation on Beowulf-type architectures

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Performance of parallel contractions

• Small testcase
• AlCl, 26 e, 70 orbitals
• SGI Origin R1200 risc

• Larger runs (e.g. CUO, 34e, 200 orbitals)
• Itanium cluster PNNL, up to 128 nodes used, ATLAS
  library
• Peak performance 300 Gflops in 4-virtual
  contraction
• Overall performance is less due to communication
  overhead (standard MPI library) in other
  contractions, use up to 8 processors.

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Accuracy of electronic structure calculations
Computational scaling yNx
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Extracting nuclear structure information
fromSpectroscopy Quantum Chemistry

• Nuclear Quadrupole Moments
• The coupling between the nuclear quadrupole
  moment Q and the electric field gradient (EFG) at
  the nucleus q gives an energy splitting that
  depends on the orientation of the nuclear spin.
  This can be observed with high precision in
  microwave (rotational) spectroscopy on diatomic
  molecules.
• Quantum chemistry gives q and can thus be used to
  obtain accurate values of Q or to predict and
  rationalize NQR or NMR observations.

Molecular rotation
Nuclear spin
86
EFG calculations Computational details

• DC-CCSD(T)
• Large uncontracted GTO type basis sets
• Freeze (deep) core orbitals in correlation
  calculation
• Leave out core-like virtuals in the correlation
  calculation
• HF analytical derivative calculation
• Add on correlation effect via CCSD(T) finite
  field calculations
• Picture change corrections (PCNQM model) are not
  necessary
• DCSF-CCSD(T) (only scalar relativistic effects)
• Check the basis set truncation error
• Effect of core correlation (also via DC-MP2)
• DC-HF
• Check the basis set truncation error

87
Iodine
88
NQMs Conclusions

• ZORA-DFT gives smaller NQMs
• Mean absolute deviations
• MAD(HF) gtgt MAD(DFT) gt MAD(MP2)
• Computational Efficiency
• Quantitative accuracy requires large basis sets.
• Current bottleneck is the 4-index transformation.
• Spin-orbit effects are small, spinfree algorithms
  can well be used to test convergence with basis
  set.
• Standard value for Al was confirmed, the values
  for Ga, In and I were to be revised.

89
Spin-Orbit CI

• Obtain orbitals with the spinfree DC-HF method.
• Add the SOC operator in the CI step
• PT The active space consists of the valence np
  orbitals
• SOCI-X CI with single excitations to virtuals
  below X au.
• Full Use the full DC-HF method to generate
  orbitals

90
Thallium Fine Structure Splitting

• Extend the active space with p-core orbitals
• Allow single excitations from core to valence
• Approximates proper orthogonalization condition
• Full relaxation is necessary for quantitative
  accuracy

91
HI dipole moment motivation

• Discrepancy between recent the DK SR calculations
  of Ilias et al. (TCA, 2003) and experiment.
• Anomalous distance dependence of the dipole
  moment in contrast to the other hydrogen
  halides, the dipole moment of HI decreases around
  the equilibrium geometry, the maximum value of
  the dipole moment lies before Re.

92
Computational details

• Basis Set
• For H a slightly modified (and decontracted)
  basis of Sadlej.
• For I Basis set optimization with respect to
  the dipole moment. We increase the basis until
  this property converges. First at the
  Dirac-Coulomb Hartree-Fock level and continue
  with Spin-Free CCSD.
• Note we use fully uncontracted basis sets.
• Correlation method
• CCSD(T)
• 26 electrons correlated
• virtuals up to 50 au are taken into account
• Hamiltonian
• Dirac-Coulomb Hamiltonian
• (SSSS) integrals neglected

93
Basis set convergence
94
Electron correlation

• CCSD and CCSD(T) work very well (average
  deviation 1.2).
• MP2 works well at Re but fails in describing the
  distance dependence.

95
Analysis of relativistic effects
96
Comparison with experiment

• Experimental data H. Riris, C. B. Carlisle, D.
  E. Cooper, L.-G. Wang, T. F. Gallagher, and R. H.
  Tipping, J. Mol. Spectrosc. 146, 381 (1991).

97
HI conclusions

• To obtain the right slope of the dipole moment
  function of HI both scalar relativistic and
  spin-orbit effects are important.
• The right sign for the slope of the dipole moment
  function at Re in previous studies seems to be
  due to a fortituous cancellation of effects.
• With the Dirac-Coulomb Coupled-Cluster method
  using a large basis set, we obtain excellent
  agreement between theory and experiment.
• ? The good agreement between our state of the art
  calculations and experiment indicates that the
  experiment is accurate.

98
Further reading

• Relativistic Quantum Mechanics
• 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,
  Adam, 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).