Title: Relativity
1Relativity Electron Correlation
- Lucas Visscher
- Vrije Universiteit Amsterdam
2The extra dimension
3Special 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
4Heavy 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
5Visible 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
6Course 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)
7Relativististic Quantum Mechanics
- 1905 STR
- Einstein E mc2
- 1926 QM
- Schrödinger equation
- 1928 RQM
- Dirac equation
- 1949 QED
- Tomonaga, Schwinger Feynman
8Non-relativistic quantization
- The classical nonrelativistic energy expression
Quantization
9Non-relativistic quantization
- Write the classical nonrelativistic energy
expression as
Quantization
10Non-relativistic quantization
11Spin in NR quantum mechanics
- 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
12Relativistic 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)
13The 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
14Time-independent Dirac equation
- The nuclei do not move with relativistic speeds.
Stationary frame of reference (Born-Oppenheimer
approximation) - Apply separation of variables
15Free particle Dirac equation
- Take case V 0
- Use plane wave trial function
16Free particle Dirac equation
- Two doubly degenerate solutions
- Classical energy expression
- Prediction of negative energy solutions !
17Free particle Dirac equation
- Wave function for E E
- Upper components Large components
- Lower components Small components
18Free 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.
19Dirac 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
20Second 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
21Second Quantization
- Second quantized operators
- Creation operator
- Annihilation operator
- Define all operators in terms of these elementary
operators
22Second Quantization
- Commutation relations
- Number operator
- Hamilton operator
23Fock space Hamiltonian
- Positive and negative energy solutions define a
Fock space Hamiltonian
24Renormalization
- Subtract energy from negative energy states
- Re-interpretation
- Normal ordered Hamiltonian
25Quantum Electro Dynamics
- Positive energy for positrons
- Total charge
26Dressed 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
27Electron-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)
28Electron-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
29Approximate 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)
30The Dirac Hamiltonian
31Foldy-Wouthuysen transformation
- Use a unitary transformation to decouple large
and small components - Exact expressions only known for the free
particle case
Picture change
32Douglas-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)
33Four-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
34Quaternion algebra
- One real and three imaginary parts
- Commutation relations
- Quaternion gradient
35Quaternion Dirac equation
- Unitary transformation matrix
- Block diagonal Hamiltonian matrix
36Quaternion Dirac equation
- Two independent blocks
- Degenerate solutions Kramers pairs
37Positive energy solutions
- Subtract rest mass energy
- Express small component in terms of large
component
38Pseudo Large Component
- Define an auxilliary function
- Transform the Dirac equation accordingly
39Quaternion 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
40Non-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
41Zeroth 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
42Neglect of SO-coupling terms
- Delete imaginary terms in QMD Hamiltonian
- Delete imaginary terms in ZORA Hamiltonian
43Approximate 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)
44Hartree-Fock Self Consistent Field
- Construct Fock operator
- Find eigensolutions
- Check convergence
- Compute energy
45Fock operator
46Basis set expansion techniques
- In the modified Dirac equation
- In the original Dirac equation
47Basis set expansion techniques
- Define density matrix
- Matrix representation of Fock operator
- Energy calculation
48Choice 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
49(No Transcript)
50Relativistic 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
51The 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 !
52Electron Density of Uranyl
53Towards 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
54Computational scaling
55Computational scaling
56 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
57Second quantization
- Hamilton operator
- Fock operator
- Mean field
58Many-Body Perturbation Theory
- Perturbation
- First order energy
59Many-Body Perturbation Theory
- Second order energy
- Definitions
60Some 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
61MO-integrals in quaternion form L. Visscher, J.
Comp. Chem. 23 (2002) 759.
62The 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
64 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
65Direct Configuration Interaction
- Write wave function as linear combination of
determinants - Define sigma and error vector
- Obtain sigma vector directly from MO-integrals
66Coupling 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
67Reverse 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).
68Evaluation 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
69Fast 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
70Kramers-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
71Kramers-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
72How 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
73Coupled Cluster
- Write wave function in exponential form
- CCSD Restriction to single double excitations
- Energy expression (I,J occupied, A, B virtual)
74The CCSD equations
- Equations for T1 and T2 amplitudes
-
75The CCSD intermediates
- Intermediates used in T1 and T2 equations
-
76Evaluation of the ltvvvvgt integral contribution
- Contribution to be evaluated
- Use point group symmetry
- Write contraction as BLAS DGEMM or ZGEMM
- Parallelize over integral batches
77RELCCSD 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.
78Kramers-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
79The 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
80Model spaces
- Example One creation and one annihilation (1h,
1e) - Excitation operators
81Implementation 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
82Parallelization
- 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
83Performance 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.
84Accuracy of electronic structure calculations
Computational scaling yNx
85Extracting 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
86EFG 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
87Iodine
88NQMs 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
91HI 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.
92Computational 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
93Basis set convergence
94Electron correlation
- CCSD and CCSD(T) work very well (average
deviation 1.2). - MP2 works well at Re but fails in describing the
distance dependence.
95Analysis of relativistic effects
96Comparison 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).
97HI 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.
98Further 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).