Lecture 10: Relativistic effects Einstein s great idea as applied to chemistry: why gold is yellow and lead batteries function
Course exam Examination on Monday May 7, 2.00 pm 4.00 pm Content The whole set of lecture notes The review articles explicitly mentioned in the lectures Exercises, especially #2 s (the ones addressed in the exercise sessions) You are allowed to take 4 pages of notes with you to the test
One- and N-electron expansions N-electron exp pansion (#SD s) exact solution in a given one-electron basis Exact solution the basis set limit for a given N-electron model One-electron expansion (#bf s)
Limitations of the molecular electronic Hamiltonian Born-Oppenheimer approximation, point-like nuclei Non-relativistic kinematics & interaction Electrostatic interaction only
Theory of special relativity 1. The laws of physics are the same in all inertial frames of reference 2. The speed of light in vacuum (c) is the same in all inertial frames of reference These are satisfied under the Lorentz transformation where
Klein-Gordon equation The Scrödinger equation is not Lorentz invariant The straightforward introduction of relativity to the SE leads to the Klein-Gordon equation Problematic negative densities Adequate equation for spinless particles
Dirac equation To avoid negative energy densities involved in the Klein-Gordon equation, Dirac introduced a linearized special relativistic wave equation Here a and b are defined such that the relativistic energy-momentum relation is satisfied, yielding where
Dirac equation By multiplying the Dirac equation from the left with b and introducing the gamma matrices we can write the Dirac equation in a compact form Dirac equation is a fully Lorentz invariant equation for a free particle with a spin The solution of the equation is a four-component Dirac spinor Negative energy solutions are positronic solutions
Relativistic many-electron Hamiltonians It would be tempting to just introduce the Coulombic interaction to the Dirac equation Instanteneous interaction => not Lorentz invariant The fully Lorentz invariant two-particle Hamiltonian is known as Bethe-Salpeter equation Difficult to solve Separate time variable for each particle
Relativistic many-electron Hamiltonians The working equation for incorporating relativistic effects to many-electron quantum mechanics is obtained through the Dirac-Coulomb-Breit Hamiltonian where the Breit term
Dirac-Fock method Relativistic quantum chemistry means finding the four-component solution of the Dirac-Coulomb-Breit Hamiltonian Self-consistent field solution has some problems related to non-systematicity in variationality ( finite basis set disease ), very large basis sets needed Continuum dissolution Also correlated four-component methods have been formulated
Quasi-relativistic Hamiltonians It sounds natural to decouple the positive and negative energy solutions and work on the positive energy solutions only Rewrite the Dirac equation in the presence of potential V as where p is the general momentum Hence
Quasi-relativistic Hamiltonians Decouple positive and negative energy solutions via the Foldy-Wouthuysen transformation: When expanding X in the small component equation (ESC) with respect to we get
Breit-Pauli Hamiltonian When applying FW transformation to the DCB Hamiltonian, we obtain the Breit-Pauli Hamiltonian NR Coulomb interaction NR kinematics and Zeeman interaction Relativistic correction to kinematics Spin-orbit interaction Darwin term
Breit-Pauli Hamiltonian Coulombic repulsion Two-electron spin-orbit interaction Two-electron Darwin term Orbit-orbit interaction Spin-other-orbit interaction Spin-spin interaction
Breit-Pauli Hamiltonian The Breit-Pauli Hamiltonian has some inherent caveats The expansion term not always feasible Terms with the Delta-function Unphysical eigenvalue spectrum ( variational collapse ) Numerical studies demonstrate however the practical feasibility of the BP approach
ZORA Hamiltonian Zeroth-order regular approximation (ZORA) Expand the X in ESC with respect to Alleviates some pathologies of the Breit-Pauli Hamiltonian It has its own problems, however It has been revised sevaral times (ERA, MERA, IORA,...)
Douglas-Kroll-Hess Hamiltonian Obtained through a slightly modified FW transformation from DCB Hamiltonian Bounded from below, approaches correct NR limit Terms difficult to interpret
Relativistic corrections Relativistic effect : the difference between NR and DCB Hamiltonians at the same level of theory Relativistic corrections to ground state energy visible already in molecules containing only light elements! E.g. water -52 me h (larger than CC triples contribution)
Manifestations of relativity Orbital contraction P. Pyykkö, CSC Spring School 2012 Radial distribution functions of atomic orbitals of Hg + by NR and relativistic theory V. M. Burke, I. P. Grant, Proc. Phys. Soc. (London) 90, 297 (1967)
Manifestations of relativity Orbital contraction P. Pyykkö, CSC Spring School 2012 P. Pyykkö, J. P. Desclaux, Acc. Chem. Res. 12, 276 (1979).
Manifestations of relativity The seven rules that explain the periodic table Main vertical rule. First shell with every l (1s, 2p, 3d, 4f, 5g) is anomalously small. <r> increases with n for others. Main horisontal trend: <r> decreases with Z. P. Pyykkö, CSC Spring School 2012 Main periodicity: Filled shells stable. NR half-filled ones also. Partial screening effects. Lanthanide contraction due to filling the 4f shell on 6s and 6p shells. Analogous 3d, 2p and 1s effects. Relativistic contraction and stabilization (s, p). Relativistic expansion and destabilization (d, f). Spin-orbit splitting (p, d, f shells).
Manifestation of relativity Colors of metals BiPh 5, violet: LUMO shift down. PbCl 2-6, yellow: LUMO shift down. Metallic gold: 5d band shifts up, 6s Fermi level shifts down. Pb(NO 2 ) 2, yellow. Singlettriplet mixing of the nitrite, due to spin-orbit coupling of the heavy metal. P. Pyykkö, CSC Spring School 2012
Manifestations of relativity About 1.7-1.8 V of the 2.1 V standard cell voltage come from relativity. Cars, indeed, start due to relativity. P. Pyykkö, CSC Spring School 2012 Rajeev Ahuja, Andreas Blomqvist, Peter Larsson, Pekka Pyykkö and Patryk Zaleski-Ejgierd, Physical Review Letters 106, 018301 (2011). Compared to tin, relativistic effects once again scale as Z 2. Largest contribution comes from PbO 2. Next one from PbSO 4.
QED & electroweak effects How about physics beyond the Dirac-Coulomb-Breit Hamiltonian? Quantum electrodynamics (vacuum polarization): 0.1% energy contribution for heavy atoms Weak interactions: O(10-17 ) E h It is however possible that these are the reason for molecular homochirality of biological macromolecules
Summary Non-relativistic Schrödinger equation omits many important phenomena, even those very visible in everyday life Four-component calculations, especially correlated ones technically difficult and computationally expensive Possible to include the most imporant effects via quasi-relativistic approaches Relativity is the last train from physics to chemistry (?)
Homework & practicalities Study the Chapter 7 and a very recent review article on relativistic effects: Autschbach, J. Chem. Phys. 136, 150902 (2012) Exercise session on Friday Deadline for the exercises ( #3 s ) to be returned Course summary, project work digestion and (possible) special topic presentations on Wed May 2 No lecture next Wednesday (Apr 25) Browse through all the lecture slides and the lecture notes for the summary session Deadline for the project works (in writing)