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CNRS – Université de Lille 1 Relativistic Quantum Chemistry Outline P. 2 / 80 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 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Outline (2) P. 3 / 80 Part III: The Dirac equation and four-component calculations 10 Special relativity: Galileo vs. Lorentz Transformation 11 From classical mechanics to quantum chemistry Non-relativistic quantization Relativistic quantization 12 The Dirac Equation One-electron part of the Four-Component Hamiltonian Two-electron part of the Four-Component Hamiltonian Part IV: Two-component relativistic theory 13 How important are the small components? 14 Two-component Hamiltonians Breit-Pauli Hamiltonian Regular Approximations Douglas-Kroll-Hess Hamiltonian 15 Summary of 2-component Hamiltonians eXact 2-Component (X2C) Hamiltonians Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Outline (3) P. 4 / 80 Part V: Core approximations 16 Core approximations - Valence only approaches 17 Accuracy of pseudopotentials 18 Where can you get the PP parameters? 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 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Outline (4) P. 5 / 80 Part VII: Illustrations of relativistic effects 22 Direct and indirect effects on orbitals and properties 23 Effects on atomic shell structures 24 Effects on molecular structures 25 Effects on chemical reactions 26 Effects on NMR shieldings 27 Effects on solid-state band structures 28 Effects on structural chemistry Part VIII: Final comments 29 Check-list before you start a relativistic calculation 30 Relativity affects many properties Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 P. 6 / 80 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 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Three Cornerstones of Non-Relativistic QC P. 7 / 80 We assume that: 1 molecular systems follow the Born-Oppenheimer approximation → use the concept of Potential Energy Surfaces (PES) 2 nuclear charge can be described by a finite-size model (e.g. use of gaussian type basis functions) 3 electrons move slow enough to be described by a non-relativistic theory Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 The Electronic Problem P. 8 / 80 The electronic Hamiltonian can be written as Ĥ el = VNN + X i ĥ(i ) + 1X ĝ (i , j ); ĥ(i ) = hˆ0 (i ) + ven (i ) 2 i <j The problematic term is the two-electron interaction which connects all electron coordinates. In the absence of any electron-electron interaction the electronic problem becomes separable, that is, we can solve it for each electron separately ĥ(r)ϕi (r) = εi ϕi (r) The resulting electronic wave function is then written as a Slater determinant of orbitals Ψ(r1 , ·rn ) = |ϕ1 (r1 )ϕ2 (r2 ) · · · ϕn (rn )| Although the electron interaction can not be ignored, Slater determinants and orbitals are key ingredients in most quantum chemical methods. Orbitals are typically expanded in some atom-centered basis ϕ(r) = X µ Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 χµ (r)cµi Quantum Chemical Methods P. 9 / 80 Many quantum chemical methods employ the variational principle stating that the expectation value of the Hamiltonian with respect to some trial function is always above the exact energy (obtained using the exact wave function) hΨtrial |Ĥ el |Ψtrial i ≥ hΨel |Ĥ el |Ψel i = E exact The simplest quantum chemical method, Hartree-Fock, employs a simple Slater determinant as trial function and finds the orbitals which minimizes the energy. In practice they are found by solving the Hartree-Fock equation mean F̂ (r)ϕi (r) = εi ϕi (r); F̂ = f̂ + Vee {ϕk } 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 linear combination of Slater determinants to capture the full electron correlation. Still, the majority of today’s quantum chemical calculations are based on density functional theory (DFT) which replaces the complicated electronic wave function Ψel (r1 , · · · rn ) by the much simpler electron density ρ(r). Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Which Are the Relativistic Elements? P. 10 / 80 Los Alamos National Laboratory Chemistry Division 1A 1 H 1 1s hydrogen 1.008 3 Be lithium beryllium 11 [He]2s1 6.941 Periodic Table of the Elements 2A 4 Li 3A 5 B [Ne]3s1 22.99 19 37 [He]2s22p2 [He]2s22p3 [He]2s22p4 [He]2s22p5 12.01 14.01 16.00 19.00 13 [Ne]3s2 magnesium 24.31 20 [Ar]4s2 40.08 boron 55 [Xe]6s1 132.9 87 Fr [Rn]7s1 francium (223) 3B 21 Sc [Ar]4s23d2 44.96 47.88 38 39 [Kr]5s24d1 88.91 56 Ba [Xe]6s2 barium 137.3 4B 22 Ti [Ar]4s23d1 scandium [Kr]5s2 Sr strontium cesium Al 87.62 Rb rubidium Cs F 10.81 calcium [Kr]5s1 7A 9 [He]2s22p1 potassium 85.47 O 12 Ca [Ar]4s1 1s2 6A 8 9.012 K 39.10 5A 7 N [He]2s2 Na Mg sodium 4A 6 C 8A 2 He Y yttrium titanium 40 Zr [Kr]5s24d2 zirconium 91.22 50.94 41 chromium manganese [Ar]4s13d5 52.00 42 Nb Mo [Kr]5s14d4 niobium 92.91 7B 25 Mn [Kr]5s14d5 molybdenum 95.94 26 Fe 8B 27 Co [Ne]3s23p1 28 Ni [Ar]4s23d5 [Ar]4s23d6 [Ar]4s23d7 [Ar]4s23d8 54.94 55.85 58.93 58.69 43 iron 44 cobalt 45 nickel 46 Tc Ru Rh Pd technetium ruthenium rhodium palladium [Kr]5s24d5 (98) [Kr]5s14d7 101.1 [Kr]5s14d8 102.9 [Kr]4d10 106.4 11B 29 Cu [Ar]4s13d10 copper 63.55 47 Ag 12B 30 aluminum [Ar]4s23d10 [Ar]4s23d104p1 Zn zinc 65.39 48 Cd 26.98 31 Ga gallium 69.72 49 In [Kr]5s14d10 [Kr]5s24d10 [Kr]5s24d105p1 107.9 112.4 114.8 silver cadmium indium [Ne]3s23p2 silicon 28.09 32 Ge [Ar]4s23d104p2 germanium 72.58 50 Sn [Kr]5s24d105p2 tin 118.7 15 P [Ne]3s23p3 phosphorus 30.97 33 As oxygen 16 S [Ne]3s23p4 sulfur 32.07 34 Se [Ar]4s23d104p3 [Ar]4s23d104p4 74.92 78.96 arsenic 51 selenium fluorine 17 Cl [Ne]3s23p5 chlorine 35.45 35 Br [Ar]4s23d104p5 bromine 79.90 Ne [He]2s22p6 neon 20.18 18 Ar [Ne]3s23p6 argon 39.95 36 Kr [Ar]4s23d104p6 krypton 83.80 52 53 [Kr]5s24d105p3 [Kr]5s24d105p4 [Kr]5s24d105p5 [Kr]5s24d105p6 121.8 127.6 126.9 131.3 Sb antimony Te tellurium 54 Xe xenon 72 [Xe]6s24f145d2 [Xe]6s24f145d3 [Xe]6s24f145d4 [Xe]6s24f145d5 [Xe]6s24f145d6 [Xe]6s24f145d7 [Xe]6s14f145d9 [Xe]6s14f145d10 [Xe]6s24f145d10 [Xe]6s24f145d106p1 [Xe]6s24f145d106p2 [Xe]6s24f145d106p3 [Xe]6s24f145d106p4 [Xe]6s24f145d106p5 [Xe]6s24f145d106p6 178.5 180.9 183.9 186.2 190.2 190.2 195.1 197.0 200.5 204.4 207.2 208.9 (209) (210) (222) 110 111 112 (272) (277) 89 [Rn]7s26d1 hafnium actinium (227) 104 58 Ce [Xe]6s24f15d1 cerium 140.1 90 Th [Rn]7s26d2 thorium 232.0 [Rn]7s25f146d2 rutherfordium (257) 59 Pr [Xe]6s24f3 praseodymium 140.9 91 Pa [Rn]7s25f26d1 protactinium (231) 73 Ta tantalum 74 W tungsten 105 106 [Rn]7s25f146d3 [Rn]7s25f146d4 Db dubnium (260) 60 Sg seaborgium (263) 61 75 Re rhenium 107 Bh [Rn]7s25f146d5 bohrium (262) 62 76 Os osmium 108 Hs [Rn]7s25f146d6 hassium (265) 63 Nd Pm Sm Eu [Xe]6s24f4 neodymium 144.2 92 U [Rn]7s25f36d1 uranium (238) [Xe]6s24f5 promethium (147) 93 Np [Rn]7s25f46d1 neptunium (237) [Xe]6s24f6 samarium (150.4) 94 77 Ir iridium 109 Mt [Rn]7s25f146d7 meitnerium (266) 64 80 Hg mercury 65 66 67 Dy Ho dysprosium holmium 95 96 [Rn]7s25f76d1 curium (247) [Xe]6s24f9 158.9 97 Bk [Rn]7s25f9 berkelium (247) [Xe]6s24f10 162.5 98 Cf [Rn]7s25f10 californium (249) 68 99 100 astatine 86 Rn radon 116 Uuo (298) (?) 69 70 Yb 71 [Xe]6s24f14 [Xe]6s24f145d1 168.9 173.0 175.0 ytterbium 101 102 [Rn]7s25f13 [Rn]7s25f14 mendelevium (256) nobelium (254) 118 Lu [Xe]6s24f13 Fm Md No fermium (253) 85 At Uuh thulium [Rn]7s25f12 Po polonium (296) Tm 167.3 Es 83 Bi bismuth Uuq erbium [Xe]6s24f12 164.9 [Rn]7s25f11 lead Er [Xe]6s24f11 einsteinium (254) 82 Pb 114 Ds Uuu Uub darmstadtium (271) terbium [Rn]7s25f7 81 Tl thallium [Rn]7s15f146d9 Tb 157.3 americium (243) gold Gd [Xe]6s24f75d1 152.0 Pu Am Cm [Rn]7s25f6 plutonium (242) 79 Au gadolinium [Xe]6s24f7 europium 78 Pt platinum 84 I iodine 57 88 Actinide Series~ [Ar]4s23d3 vanadium 6B 24 Cr Si nitrogen 10 138.9 La* Hf [Rn]7s2 Lanthanide Series* V 14 4.003 [Xe]6s25d1 lanthanum Ra Ac~ Rf radium (226) 5B 23 carbon helium lutetium 103 Lr [Rn]7s25f146d1 lawrencium (257) element names in blue are liquids at room temperature element names in red are gases at room temperature element names in black are solids at room temperature Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 The Limitations of Nonrelativistic QC P. 11 / 80 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 [Rao et al. 1960] I I I 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 p block elements Need for a relativistic quantum chemical formalism Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 P. 12 / 80 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 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Scalar Relativistic Effects: Hydrogen-Like Atoms P. 13 / 80 In atomic units the average speed of the 1s electron is equal to the nuclear charge v1s = Z a.u. and c = 137.0359998a.u. The relativistic mass increase of the 1s electron is thus determined by the nuclear charge me m = γ me = p 1 − Z 2 /c 2 The Bohr radius is inversely proportional to electron mass a0 = 4πε0 ~2 m Relativity will contract orbitals of one-electron atoms, e.g. Au78+ : Z/c = 58% ⇒ 18% relativistic contraction of the 1s orbital Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Scalar Relativistic Effects: Polyelectronic Atoms P. 14 / 80 • • 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 observe I • I s, p d, f s, p orbitals : contraction d, f orbitals : expansion Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Colour of Gold P. 15 / 80 The colours of silver and gold are related to the transition between the (n − 1)d and ns bands. For silver this transition is in the ultraviolet, giving the metallic cluster. For gold it is in the visible, but only when effects are included. (nrelativistic 1)d ns Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 The Contrasting Neighbors P. 16 / 80 1064◦ C 12.5 kJ/mol 9.29 kJ/mol 19.32 g/cm3 426 kS/m dimer Mp ∆Hfus ∆Sfus ρ Conductivity gas phase -39◦ C 2.29 kJ/mol 9.81 kJ/mol 13.53 g/cm3 10.4 kS/m monomer [Xe]4f 14 5d10 6s1 [Xe]4f 14 5d10 6s2 pseudo halogen pseudo noble gas Without relativistic effects mercury would probably not be a liquid at room temperature! Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Spin-Orbit Interaction P. 17 / 80 h SO = 1 s · [∇V × p] 2m2 c 2 with V = − Zr Z s·l 2m2 c 2 r 3 The spin-orbit interaction is not the 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. h SO = Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Spin-Orbit Coupling in Atoms P. 18 / 80 In the absence of spin-orbit coupling atomic electronic states are characterized by total orbital angular momentum L and total spin S and denoted as 2S +1 L. With spin-orbit interaction only the total angular momentum J = kL − S k, · · · kL + S k is conserved. The ground state configuration of oxygen is 1s2 2s2 2p4 which in a non-relativistic framework (LS-coupling) gives rise to three states. Term 3 P 1 1 D S L S 1 1 2 0 0 0 J 2 1 0 2 0 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Level (cm−1 ) 0.000 158.265 226.977 15867.862 33792.583 Atomic Oxygen Emissions in Atmospheric Aurora P. 19 / 80 Green line Red line Transition S 0 → 1 D2 1 D2 → 3 P2 1 Wavelength(Å) 5577 6300 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Type E2 M1 Lifetime(s) 0.75 110 P. 20 / 80 Part III The Dirac equation and four-component calculations 10 Special relativity: Galileo vs. Lorentz Transformation 11 From classical mechanics to quantum chemistry Non-relativistic quantization Relativistic quantization 12 The Dirac Equation One-electron part of the Four-Component Hamiltonian Two-electron part of the Four-Component Hamiltonian Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Galileo Transformation P. 21 / 80 Motion in the x-direction 0 x = x + vt y = y 0 0 z = z t = t 0 Transforms coordinates between two different reference frames Velocities are additive and time is constant. No “speed limit” Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Lorentz Transformation P. 22 / 80 In the x-direction x = Generalized to 3 dimensions γ(x 0 + vt ) y = y 0 z = z 0 t = γ(t 0 + r = 0 r +v vt 0 ) c2 γ= 1− Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 t = v2 c2 − 12 γ t0 + (v · r0 ) (γ − 1) v2 0 v·r c2 + γt 0 Recall Non-Relativistic Quantization P. 23 / 80 Spin-Free Non-relativistic Hamiltonian H =T +V = π = π2 + q φ(r ) 2m p − qA Non-relativistic Hamiltonian including spin H= (σ · π)2 2m Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 + q φ(r ) Including Spin in the Non-relativistic Hamiltonian P. 24 / 80 Non-relativistic Pauli Hamiltonian. H = = HΨ = (σ · π)2 2m π2 2m i~ − q~ σ · B + qφ 2m ∂ Ψ ∂t Introduction of spin may appear ad hoc No spin-orbit coupling Not Lorentz invariant Linear in φ but quadratic in A Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 + q φ(r ) Relativistic Quantization P. 25 / 80 Classical relativistic energy expression including EM-fiels. 2 4 2 E − qφ = m c +c π π = p − qA 2 Quantization results in the Klein-Gordon Equation (i ~ ∂ − q φ)2 Ψ = (m2 c 4 + c 2 π 2 )Ψ ∂t Lorentz invariant but doesn’t include spin KG-equation is not suitable for electrons Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Relativistic Quantization P. 26 / 80 Generalize the concept of the square root in E − q φ equation 2 E − q φ = β mc + c α · π Require cross-terms drop out by introducing parameters β2 = 1 + [αi , αj ] = 2δij [αi , β]+ = 0 Quantization yields the Dirac equation which can be used to describe electrons! Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 A way to Describe Electrons Relativistically P. 27 / 80 The Dirac equation (β mc 2 + c α · π + q φ)Ψ(r , t ) = i ~ ∂ Ψ(r , t ) ∂t Lorentz invariant First derivative with respect to time AND position Linear in scalar AND vector potentials Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 What are the Parameters? P. 28 / 80 Parametrization includes the Pauli matrices αx = αy = αz = β = 0 σx σx 0 0 σy σy 0 0 σz σz 0 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 I2 0 0 −I2 Requires a Four-Component Wave Function P. 29 / 80 2 H = β mc + c α · π + q φ The Dirac Hamiltonian mc 2 + q φ 0 H= c πz c (πx + i πy ) 0 c πz mc 2 + q φ c (πx − i πy ) −c πz c (πx + i πy ) −mc 2 + q φ 1 Spin doubles the components 2 Has negative energy solutions: E < −mc 2 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 0 c (πx − i πy ) −c πz 0 −mc 2 + q φ How Can we Treat Many-Electron Systems? P. 30 / 80 Exact relativistic many-electron Hamiltonian is unknown Born-Oppenheimer Approximation I I I Theory remains relativistic Decouples nuclear and electronic motion Simplifies relationship between time and space coordinates Add two-electron terms via the Coulomb potential? What relativistic effects should the external potential include? I I Electron feels the motion of other electrons after some time (i.e. a retarded, velocity-dependent potential) Electron generates B as it moves and can interact with another electrons spin (i.e. spin-orbit coupling, SOC) Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Dirac-Coulomb-Breit Hamiltonian P. 31 / 80 Correction to the Coulomb operator obtained as a leading term in the Quantum Electrondynamics (QED) expansion. Coulomb term O (c 0 ) (including spin-”same” orbit (SSO)) Coul ĝij = I4 · I4 ; charge-charge interaction rij Breit term O (c −2 ) Breit ĝij I Gaunt term (including spin-”other” orbit (SOO)) ĝijGaunt = − I = ĝijGaunt + ĝijgauge c αi · c αi c 2 rij ; current-current (magnetic) interaction Gauge-dependent term: gauge ĝij =− (c αi · ∇i )(c αi · ∇i )rij ; retardation term 2c 2 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Dirac-Coulomb-Breit Hamiltonian P. 32 / 80 Dirac-Coulomb-Breit Hamiltonian H DCB = X D hij ai+ aj + ij X CB gijkl ai+ ak+ aj al i <j ;k <l But what about the wavefunction? Still have positive and negative energy solutions!! Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Rewriting the Dirac Eq. in a “Two-Component Form” P. 33 / 80 Shift the diagonal of the Dirac Hamiltonian by −mc 2 This aligns the relativistic and nonrelativistic energy scales In a two-component form: V cσ · π cσ · π −2mc 2 + V ΨL (r) ΨL (r) =E ΨS (r) ΨS (r) ΨL (r) and ΨS (r) are exact eigenfunctions of the Dirac equation For simplification, A = 0 and π = p Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 P. 34 / 80 Part IV Two-component relativistic theory 13 How important are the small components? 14 Two-component Hamiltonians Breit-Pauli Hamiltonian Regular Approximations Douglas-Kroll-Hess Hamiltonian 15 Summary of 2-component Hamiltonians eXact 2-Component (X2C) Hamiltonians Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Nature of the Small Components P. 35 / 80 Contributions from the large and small component densities 4c L S p (r) = p (r)p (r) Reducing computational cost Exploit the highly local and atomic nature of the small components Use the no-pair approximation atomic character of the contributions of the S components ⇒ neglect the multi-center integral block in the S components 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! Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Unnormalized Elimination of the Small Components P. 36 / 80 Find a relationship between the two components ΨS (r) = χΨL (r) χ = K (E , r) σ·p 2mc Eliminate ΨS (r) to obtain a two-component problem 1 (σ · p)K (E , r)(σ · p) + V 2m ΨL (r) = E ΨL (r) K (E , r) = 1− V −E 2mc 2 −1 Exact but not an eigenvalue equation D D The resulting two component hamiltonian is H ++ = H11 + χH12 Starting point for approximations. The simplest approximation, K (E , r), leads to the non-relativistic Pauli Hamiltonian Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Breit-Pauli Hamiltonian P. 37 / 80 Begin with Unnormalized Elimination of the Small Components (UESC) equation UESC equation ( 1 V −E (σ · p) 1 − 2m 2mc 2 (σ · p) + V Expand K (E , r) in terms of small parameter represent relativistic effects ) −1 ΨL (r) = E ΨL (r) V −E to give operators that 2mc 2 Need to renormalize ΨL (r) (expansion occurs in normalization term as well): H BP = H Pauli + H Darwin + H MV + H SO Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Breit-Pauli Hamiltonian P. 38 / 80 The very useful Dirac identity (σ · p)V (σ · p) = pV · p + i σ · (pV × p) Breit-Pauli Operators (so called Scalar Rel. Corrections in blue). H Darwin = MV = SO = H H 1 (∇2 V ) 8m2 c 2 p4 − 3 2 8m c 1 σ · ((∇V ) × p) 4m2 c 2 Good to first order, but variational collapse at higher order Used for light elements where first order is sufficient Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Regular Approximations P. 39 / 80 V −E only valid when this term is small 2mc 2 Not small near the nuclei (|V − E |/2mc 2 > 1) E where expansion is valid for whole region of Expand in terms of 2mc 2 − V space In Breit-Pauli, expanding in terms of K (E , r ) = 1− V −E 2mc 2 −1 = 1− V 2mc 2 −1 Expansion is variationally stable unlike Breit-Pauli Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 1+ E 2mc 2 − V −1 Zero Order Regular Approximation (ZORA) P. 40 / 80 In practice, include only first term in the Regular Approximation K (E , r) = V 2mc 2 −1 leading to the ZORA Hamiltonian: H ZORA = = 1− 2mc 2 1 (σ · p) (σ · p) 2m 2mc 2 − V 1 V +T + (σ · p)V (σ · p) + . . . 4m2 c 2 V+ this formula shows that the ZORA Hamiltonian contains no mass velocity term, only parts of the Darwin term, but all spin-orbit interactions arising from nuclei Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Douglas-Kroll-Hess P. 41 / 80 Ideally: Achieve decoupling of the large and small components of the Dirac Hamiltonian via a unitary transformation H DKH h+ = UH U = 0 D † 0 h− Discard chemically uninteresting negative energy solutions h+ would require a two component wavefunction If completely decoupled, the positive energy eigenvalues are the same as those for the Dirac equation Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 What Can We Use for U in UH D U † ? P. 42 / 80 Foldy-Wouthuysen Transformation (exact for the free particle problem) " U = X = 1 (1 + X † X )− 2 1 X (1 + XX † )− 2 K (E , r) 1 (1 + X † X )− 2 X † 1 (1 + XX † )− 2 # (σ · p) 2mc In DKH, use FW transformation for the bare-nucleus Hamiltonian Douglas and Kroll suggested step-wise decoupling could be done by parameterizing additional transformations Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 DKH Hamiltonian P. 43 / 80 Second Order DKH Hamiltonian " H (2) H DKH2 = (2) H21 (2) H12 (2) H22 # Variationally stable Scalar DKH2 is used most often Good results in practice Matrix elements cannot be computed analytically due to the complicated operators If including an external field, need to transform the operators Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Summary of Approximate Relativistic Hamiltonians P. 44 / 80 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: I I small errors on valence properties significant ones on nucleus properties!!! Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 eXact 2-Component (X2C) Hamiltonians P. 45 / 80 1 Solve the Dirac equation on matrix form 2 Extract the coupling χ from the solutions 3 Construct the transformation matrix U, next hX 2C Advantages of X2C reproduces exactly the positive-energy spectrum of the Dirac Hamiltonian all matrix manipulations; no new operators to program explicit representation of transformation matrix; any property operator can be transformed on the fly, no picture change error Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Two-Component Relativistic Methods P. 46 / 80 2c implementations of most quantum chemical methods (like in 4c) Dirac Hartree Fock (DHF) methods Post DHF methods: MP2, Coupled-Cluster methods (CC), CISD, Full CI Multi-configuration Self Consistent field MRCI methods (GAS-CI), IH(FSCC) Density Functional Theory Two-component packages DIRAC http://wiki.chem.vu.nl/dirac/ UTChem, http://utchem.qcl.t.u-tokyo.ac.jp/ ADF http://www.scm.com/ TURBOMOLE http://www.cosmologic.de/ ReSpect http://rel-qchem.sav.sk/ NWCHEM http://www.emsl.pnl.gov/capabilities/computing/nwchem/ Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 P. 47 / 80 Part V Core approximations 16 Core approximations - Valence only approaches 17 Accuracy of pseudopotentials 18 Where can you get the PP parameters? Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Need for Frozen Core Approximations P. 48 / 80 In the context of chemical applications Core orbitals are 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 a non-local static 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 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Idea: Restriction of the Calculations to Valence Electrons P. 49 / 80 Abbreviations: multiconfiguration Dirac–Hartree–Fock calculations based on the Dirac–Coulomb Hamiltonian perturbatively including the Breit interaction (MCDHF/DC+B); small-core pseudopotential (SPP); density functional theory (DFT) Significant computational savings ⇒ study of large complexes; computational effort concentrated on valence part Implicit inclusion of major relativistic effects Consideration of spin–orbit (SO) effects ⇒ SO operator: perturbative treatment subsequent to scalar-relativistic calculation ⇒ MCDHF/DC+B SPP: variational inclusion from the beginning Application in nonrel. framework of ab initio and DFT calculations ⇒ GAUSSIAN, MOLCAS, TURBOMOLE, CRYSTAL, DIRAC, NWCHEM ... PPs: smaller basis sets ⇒ reduced basis set superposition error LPPs: avoiding of open shells ⇒ no multiconfigurational treatment; good convergence Computational effort ⇔ Accuracy of results Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Approximations P. 50 / 80 Core–valence separation (choice of the core) I I Frozen-core approximation I I neglect of core–valence correlation core-polarization potentials (CPPs) account for dynamic core-polarization consideration of all important configurations in the adjustment CPPs account for static core-polarization Replacement of the core electrons by an ECP I convenient form with adjustable parameters Point charge Coulomb repulsion between nuclei/cores Application of pseudo valence orbitals I I I corrections for penetrating/overlapping cores simplified nodal structure in core region overestimation of valence correlation energies? Don’t apply PPs to investigate core properties as NMR shifts! Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Different Kinds of Effective Core Potentials P. 51 / 80 M. Dolg and X. Cao. “The relativistic energy-consistent ab initio pseudopotential approach and its application to lanthanide and actinide compounds”. In: Recent Advances in Relativistic Molecular Theory. Ed. by K. Hirao and Y. Ishikawa. Vol. 6. New Jersey: World Scientific, 2004, pp. 1–35 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Choice of the Pseudopotential Core P. 52 / 80 M. Dolg and X. Cao. “The relativistic energy-consistent ab initio pseudopotential approach and its application to lanthanide and actinide compounds”. In: Recent Advances in Relativistic Molecular Theory. Ed. by K. Hirao and Y. Ishikawa. Vol. 6. New Jersey: World Scientific, 2004, pp. 1–35 Energetic separation: Spatial separation: valence space 5f 0 valence space valence space 6d 2 5f 0 5spdf Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 7s2 6spd 6spd 7s 7s (4 electrons) (12 electrons) (30 electrons) ⇒ not useful ⇒ LPP (large-core PP) ⇒ SPP Are f-in-core Pseudopotential Useful? P. 53 / 80 Comparison between Ce (lanthanide) and Th (actinide): Stronger indirect relativistic destabilization and expansion of Th 5f Th 5f is more diffuse than Ce 4f Th 5f amplitude is small, but not negligible in the valence space Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Valence-only Model Hamiltonian P. 54 / 80 The valence-only model Hamiltonian for a molecule (in a.u.) is given as Ĥv = − nv nv nv N N X X X X QI QJ 1 1X I + Vcv (i ) + + V̂CPP ∆i + 2 rij RIJ i <j i I I <J I The effective core potentials Vcv describe all interactions with the core system. scalar-relativistic, one-component I V̂cv (i ) = − i QI riI + lmax X X l =0 I 2 AIlk exp (−alk riI )P̂lI (i ) with P̂lI (i ) = l X |lm, I ihlm, I | m= −l k quasi-relativistic, two-component I V̂cv (i ) = − QI riI + lmax X j =l +1/2 X X l =0 j =|l −1/2| I 2 AIljk exp (−aljk riI )P̂ljI (i ) with P̂ljI (i ) = k j X |ljm, I ihljm, I | m= −j I I I V̂PP (i ) = V̂PP ,SA (i ) + V̂PP ,SO (i ) Relativistic effects result only from the PP parametrization to relativistic reference data! Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Adjustment of Energy-Consistent Pseudopotentials P. 55 / 80 1 2 Selection of reference configurations ⇒ consideration of all chemical important states AE AE computation of the total energies EIAE and the core energy Ecore AE ⇒ calculation of the total valence energies EIAE ,V = EIAE − Ecore 3 Check how many Gaussians are needed 4 Guess for the free parameters AImk and aImk (m = l , lj) of the Gaussians 5 Adjustment of the PP parameters by a least-squares fit to EIAE ,V S= X AE ,valence wI E I − EIPP ,valence 2 := min I , Adjustment can be made to any method currently best method: (average- level) multi-configuration Dirac-Hartree-Fock (MCDHF) reference data based on the Dirac-Coulomb-Breit Hamiltonian 6 Test if parameters deviate smoothly with increasing nuclear charge 7 Comparison of the PP pseudo valence and the AE valence orbitals 8 Optimization of a corresponding basis set Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Example of Calibration for Atoms P. 56 / 80 Illustration of the accuracy of the Köln-Stuttgart pseudopotentials Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Example of Calibration for Molecules P. 57 / 80 Test systems: AcF, ThF3+ , PaF4+ (closed-shell molecules) Method: MCDHF/DC+B SPPs HF + MP21 in DIRAC Reference: AE DHF + MP2 using DC as well as X2C/DKH2 + AMFI Hamiltonians Active space: all electrons in the PP calculation; virtual orbitals ≤ 100 a.u. Basis sets: F aug-cc-pVQZ; Ac: PP (16s15p12d10f6g3h), AE (37s34p27d21f6g3h); Th: PP (16s15p12d10f9g5h1i), AE (37s34p26d23f9g5h1i); Pa: PP (16s15p12d10f10g5h2i), AE (37s34p26d23f10g5h2i) Bond distances (Å2 ) and force constants (N/m) [Weigand et al. manuscript] ThF3+ AcF Re DC X2C DKH2 PP ∆DC /PP % DC/PP HF 2.130 2.129 2.129 2.124 0.006 0.3 ke MP2 2.105 2.105 2.104 2.099 0.006 0.3 HF 287.4 287.6 287.8 293.2 5.8 2.0 Re MP2 301.7 301.8 302.1 309.8 8.1 2.7 HF 1.878 1.877 1.877 1.876 0.002 0.1 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 PaF4+ ke MP2 1.878 1.878 1.877 1.876 0.002 0.1 HF 712.3 712.3 712.2 724.4 12.1 1.7 Re MP2 692.3 691.5 691.6 704.3 12.0 1.7 HF 1.772 1.771 1.771 1.772 0.0001 0.01 ke MP2 1.802 1.802 1.802 1.803 0.001 0.06 HF 986.1 985.7 985.3 993.0 6.9 0.7 MP2 804.7 800.3 798.5 795.2 9.5 1.2 Accessing the PP Parameters? P. 58 / 80 Program or web databases 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 Köln Stuttgart group because they are far more accurate! Important review for heavy elements: M Dolg and X. Cao. “Relativistic Pseudopotentials: Their Development and Scope of Applications”. In: Chem. Rev. 112 (2012), pp. 403–480. DOI: 10.1021/cr2001383 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 P. 59 / 80 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 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Philosophy of One-Component Approaches P. 60 / 80 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 (DFT, HF, post-HF, multiconfigurational, etc...) Treat spin-orbit coupling a posteriori Choose the proper SO Hamiltonian if you use relativistic pseudopotentials: Don’t use all-electron spin-orbit hamiltonian: you have pseudo-orbitals!!! Must use the spin-orbit pseudopotential that is paired with your scalar relativistic pseudopotential Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Choose the Proper SO Hamiltonian for All-Electron Calculations (1) P. 61 / 80 Breit-Pauli Hamiltonian SO The one-electron term: Ĥ1el = Two-electron term: e2 ~ X r~k ~ Z i σ ~ · × pk k 4m2 c 2 rk3 k SO Ĥ2el = − r~kl e2 ~ X ~ i ( σ ~ + 2 σ ~ ) · × pk k l 2m2 c 2 rkl3 k 6=l 3 Z /r divergence when r → 0 Don’t use it in variational calculations and for heavy elements Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Choose the Proper SO Hamiltonian for All-Electron Calculations (2) P. 62 / 80 One and two electron spin-orbit contributions derived at the first order DKH level The one-electron term: SO Ĥ1el = X k r Ak = Ak Ek + mc 2 i σ~k · r~k × p~k rk3 Ak Ek + mc 2 Ek + mc 2 2Ek The two-electron term has two contributions: SO Ĥ2el = X Ak Al k 6=l + X k 6=l Ak Al i σ~k · Ek + mc 2 2i σ~k · Ek + mc 2 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 r~kl rkl3 r~kl rkl3 × p~k × p~l 1 Ek + mc 2 1 El + mc 2 Ak Al Ak Al Choose the Proper SO Hamiltonian for All-Electron Calculations (3) P. 63 / 80 Mean-field approach to spin-orbit coupling Calculate two-electron contribution (shielding) from a fixed configuration Effective one-electron integrals Fock-operator technique mean-field Ĥij = + SO hi |Ĥ1el |j i 1 X 2 SO SO SO nk {hik |Ĥ2el |jk ihik |Ĥ2el |kj i − hki |Ĥ2el |jk i} k ,fixed {nk } Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Choose the Proper SO Hamiltonian for All-Electron Calculations (4) P. 64 / 80 Atomic Mean-field approach to spin-orbit coupling Spin-orbit coupling is short-ranged : r −3 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 in DALTON, DeMon, Dirac, MOLCAS, ORCA Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Consequences of Spin-orbit Coupling P. 65 / 80 = H Hnr + X ~iV Api + B ∇ 4 2 + i Hso (i ) = ~ 4m2 c 2 X i ~si · ~ri ri3 ∧ ~pi f (r )~l · ~s = f (r ) (lx sx + ly sy + lz sz ) = f (r ) {(l+ s− + l− s+ ) /2 + lz sz } non-relativistic term has certain symmetry properties (atom, molecule, special symmetry group: C2v , D6h . . .-); 3 P, 2 S, 3 Π, 2 Σ+ , 1 B1 , 2 B2u . . . Scalar relativistic terms keep these properties (blue) Commute with L2 , Lz et S 2 , Sz for atoms, with L2z et S 2 , Sz for linear molecules. It is invariant with respect to symmetry operation in the general case In atoms and linear molecules, spin-orbit operators don’t commute with L2 , Lz et S 2 , Sz In polyatomic molecules, spin functions have special properties I I I If Cz (π) rotation around the z-axis is a symmetry operation (H2 O): [Cz (π)]2 = Cz (2π) is the identity For spin functions Cz (2π)|αi =?|alphai 6= |αi Double group symmetry Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 1c Approaches for the Treatment of SO-Coupling P. 66 / 80 SR all-electron basis SR-PP + basis set 1-component HF/MCSCF 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: I I SO relaxation of valence orbitals is important for heavy main group atoms dense spectra in transition metals and actinides Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Treat Correlation and SO in a 2-Step Approach P. 67 / 80 SR all-electron basis SR-PP + basis set 1-component HF/MCSCF 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] Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 P. 68 / 80 Part VII Illustrations of relativistic in chemistry: more common than you thought 22 Direct and indirect effects on orbitals and properties 23 Effects on atomic shell structures 24 Effects on molecular structures 25 Effects on chemical reactions 26 Effects on NMR shieldings 27 Effects on solid-state band structures 28 Effects on structural chemistry Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Relativistic effects in chemistry P. 69 / 80 Direct effects contraction of the core-penetrating orbitals s orbitals, p1/2 , p3/2 orbitals in core Energetic stabilization: higher ionization energy, higher electron affinity, smaller polarizability Indirect effects indirect effects on d, f , orbitals and valence p orbitals nuclear charge is shielded to a larger extent because of direct effect on core-penetrating orbitals (in particular of the semi-core) relativistic expansion of core non-penetrating orbitals energetic destabilization smaller ionization energy, larger polarizability in turn, stabilization of core-penetrating orbitals in next shell gold maximum Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 U4+ atomic spectrum P. 70 / 80 Non Relativistic Spin-Free 0 -50 -100 4d -150 4p 4s -200 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Spin-Orbit Relativistic Effects on Atomic Shell structures P. 71 / 80 S. J. Rose, I. P. Grant, and N. C. Pyper. “The direct and indirect effects in the relativistic modification of atomic valence orbitals”. In: J. Phys. B: At. Mol. Phys. 11.7 (1978), p. 1171. DOI: 10.1088/0022- 3700/11/7/016 Direct (dynamics) and indirect (potential) effects on orbital energies (eV) Dynamics: Potential: Au 6s Tl 6p1/2 Tl 6p3/2 Lu 5d3/2 Lu 5d5/2 Dirac Rel Nonrel -7.94 -7.97 -5.81 -6.79 -4.79 -5.63 -5.25 -7.32 -5.01 -6.90 Schrödinger Rel Nonrel -6.18 -6.01 -4.58 -5.24 -4.46 -5.24 -4.74 -6.63 -4.81 -6.63 Direct effects dominate for Au 6s and Tl 6p1/2 Compensation of direct and indirect effects for Tl 6p3/2 Indirect effects dominate for Lu Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Relativistic Effects on Molecular Structures P. 72 / 80 C. L. Collins, K. G. Dyall, and H. F. Schaefer III. “Relativistic and correlation effects in CuH, AgH, and AuH: Comparison of various relativistic methods”. In: J. Chem. Phys. 102 (1995), p. 2024. DOI: 10.1063/1.468724 Bond lengths in pm, dissociation energies in eV, harmonic frequencies in cm−1 Molecule CuH Method reSCF reMP2 DeSCF DeMP2 ω SCF ω MP2 NR DKH RECP DC Exp 156.9 154.2 154.3 154.1 145.4 142.9 142.9 142.8 146.3 1.416 1.476 1.465 1.477 2.585 2.708 2.696 2.711 2.85 1642 1698 1690 1699 2024 2100 2095 2101 1941 NR DKH RECP DC Exp 177.9 170.1 170.0 170.0 166.3 158.7 158.4 158.5 161.8 1.126 1.229 1.224 1.233 1.986 2.190 2.189 2.195 2.39 1473 1602 1607 1605 1699 1870 1882 1873 1760 NR DKH RECP DC Exp 183.1 157.6 157.1 157.0 171.1 149.8 149.5 149.7 152.4 1.084 1.727 1.751 1.778 1.901 3.042 3.075 3.114 3.36 1464 2045 2076 2067 1169 2495 2512 2496 2305 AgH AuH Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Effects on Chemical Reactions P. 73 / 80 K. G. Dyall. “All-electron molecular Dirac-Hartree-Fock calculations: Properties of the XH4 and XH2 molecules and the reaction energy XH4 − → XH2 + H2 , X=Si, Ge, Sn, Pb”. In: J. Chem. Phys. 96 (1992), p. 1210. DOI: 10.1063/1.462208 SCF reaction energies in kJ/mol for the reaction XH4 −−→ XH2 + H2 Method NR RECP DHF Si 263 261 Ge 190 195 177 Sn 129 102 97 Pb 89 -31 -26 Large relativistic effects in Pb Stabilization of the 6s and destabilization of 6p decreases sp3 hybridization in PbH4 , thus making the reaction exothermic In some systems, spin-orbit coupling can enable crossing between states of different multiplicities (inter-system crossings) Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Effects on NMR Shieldings P. 74 / 80 Peter Hrobárik et al. “Relativistic Four-Component DFT Calculations of 1H NMR Chemical Shifts in Transition-Metal Hydride Complexes: Unusual High-Field Shifts Beyond the Buckingham–Stephens Model”. In: J. Phys. Chem. A 115.22 (2011), pp. 5654–5659. DOI: 10.1021/jp202327z Isotropic hydrogen shielding parameters (ppm) Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Relativity and the Lead-Acid Battery P. 75 / 80 Rajeev Ahuja et al. “Relativity and the Lead-Acid Battery”. In: Phys. Rev. Lett. 106 (1 2011), p. 018301. DOI: 10.1103/PhysRevLett.106.018301 The lead-acid battery reaction Pb(s) + PbO2(s) + 2 H2 SO4(aq) −−→ 2 PbSO4(s) + 2 H2 O(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 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Spin-orbit effects in structural chemistry P. 76 / 80 Ulrich Wedig et al. “Homoatomic Stella Quadrangula [Tl8 ]6 – in Cs18 Tl8 O6 , Interplay of Spin-Orbit Coupling, and Jahn-Teller Distortion”. In: J. Am. Chem. Soc. 132.35 (2010), pp. 12458–12463. DOI: 10.1021/ja1051022 Strong influence of SO coupling on heavy element structures SO makes Ptn clusters flat (n = 2–5) In Cs18 Tl8 O6 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. Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 P. 77 / 80 Part VIII Comments 29 Check-list before you start a relativistic calculation 30 Relativity affects many properties Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 How to Choose Between 4c, 2c, 1c + SOCI Approaches? P. 78 / 80 Check-list to choose a relativistic approach How to choose the relativistic description? I I I I 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 Don’t 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 a more accurate method with a too small basis set. Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Summary Points P. 79 / 80 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. Valérie Vallet ([email protected]) | CNRS – Université de Lille 1 Further Reading P. 80 / 80 Relativistic Quantum Mechanics R. E. Moss. Advanced Molecular Quantum Mechanics - An Introduction to Relativistic Quantum Mechanics and the Quantum Theory of Radiation. Springer Netherlands, 1973. ISBN : 978-94-009-5690-2. DOI : 10.1007/978-94-009-5688-9 P. Strange. Relativistic Quantum Mechanics with Applications in Condensed Matter and Atomic Physics. Cambridge Univ. Press, 1998, p. 594. ISBN : 9780521565837 Relativistic Quantum Chemical methods P. Schwerdtfeger. Relativistic Electronic Structure Theory: Part 1, Fundamentals. Ed. by P. Schwerdtfeger. Amsterdam: Elsevier, 2002. ISBN : 9780444512499 K. G. Dyall and K. Fægri. Introduction to relativistic quantum chemistry. New York: Oxford University Press, 2007 M. Reiher and A. Wolf. Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science, 2nd Edition. WILEY-VCH Verlag, 2014. ISBN : 978-3-527-33415-5 Applications P. Schwerdtfeger. Relativistic Electronic Structure Theory: Part 2, Applications. Ed. by P. Schwerdtfeger. Amsterdam: Elsevier, 2004. ISBN : 978-0-444-51299-4 Pekka Pyykkö. “Relativistic Effects in Chemistry: More Common Than You Thought”. In: Ann. Phys. 63.1 (2012), pp. 45–64. DOI: 10.1146/annurev-physchem-032511-143755 Valérie Vallet ([email protected]) | CNRS – Université de Lille 1
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