Definition and Calculation of Electron Densities in Relativistic Quantum

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GRC 2010

Deﬁnition and Calculation of Electron Densities in Relativistic Quantum Theory

Markus Reiher

Laboratorium f¨ur Physikalische Chemie, ETH Zurich, Switzerland

http://www.reiher.ethz.ch

Gordon Research Conference on Electron Distribution and Chemical Bonding

Mount Holyoke College, 15 July 2010

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

Electron density

Powerful quantity for structure and reactivity studies

Topological analyses provide deeper understanding of bonding

In contrast to wave function, it is available in experiment

Fully determines a given system (→ Hohenberg–Kohn theorems, DFT)

Basis for conceptual DFT

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

But ... why relativistic quantum theory?

This is the most fundamental many-electron theory !

(Just an observation: If you make a strong statement about your favourite analysis tool, make sure that it ﬁts in this framework.)

Practical reason: It is mandatory for heavy-element chemistry (i.e., beyond organic chemistry)

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

World of relativistic quantum theory

Quantum Electrodynamics Local QED Model Potentials

first−quantized, N−electron Dirac−Fock−Coulomb (DFC) Hamiltonian

2−component methods 4−comp. Dirac−Fock & post−Dirac−Fock methods

transformation techniques elimination techniques numerical methods basis set methods (atoms) (molecules) DKH transformation (Pauli approximation) regular approximations clumsy, accurate, but less efficient basis sets: X−operator techniques (BSS,IOTC) GRASP (Oxford) ZORA, FORA MOLFDIR (FW transformation) Desclaux/Indelicato DIRAC NESC (Dyall) Johnson BERTHA ADRIEN (M.R.) UTChem DFT (e.g., BDF) Scalar−relativistic 1−component formulations by . . . neglect of spin−dependent terms or use of ECPs

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

For all details on the theory see:

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

How to deﬁne the electron distribution?

Born interpretation: abs. square of wave function (1 electron only)

ρ(r)= |Ψ(r)|2

Many-electron systems: integrate out all but one electronic coordinate

+∞ +∞ ! 3 3 2 ρ(r) = N d r2 · · · d rN |Ψ({ri},t)| Z−∞ Z−∞ ... yields sum of squared occupied (spin) orbitals for Hartree–Fock or DFT wave function occ 2 ρ(r) → |ψi(r)| Xi=1 → Let us look at a simple example ...

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

Squared orbital of Hg79+

1e+08

9e+07 Dirac,fn,-3529.440

8e+07

7e+07

6e+07

5e+07

4e+07

squared radial function 3e+07

2e+07

1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

Squared orbital of Hg79+

1e+08

9e+07 Dirac, fn, -3529.440 a.u. Dirac, pn, -3532.192 a.u. 8e+07

7e+07

6e+07

5e+07

4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07

1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

Squared orbital of Hg79+

1e+08

9e+07 DKH(5,0), fn, -3530.397 a.u. Dirac, fn, -3529.440 a.u. 8e+07 Dirac, pn, -3532.192 a.u.

7e+07

6e+07 finite nucleus DKH(n,0)

5e+07

4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07

1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

Squared orbital of Hg79+

1e+08

9e+07 DKH(5,0), fn, -3530.397 a.u. Dirac, fn, -3529.440 a.u. 8e+07 basis set artifact Dirac, pn, -3532.192 a.u. DKH(5,0), pn, -3532.367 a.u. 7e+07 point charge nucleus DKH(n,0)

6e+07 finite nucleus DKH(n,0)

5e+07

4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07 DKH(n,0)

1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

Squared orbital of Hg79+

1e+08

9e+07 DKH(5,0), fn, -3530.397 a.u. Dirac, fn, -3529.440 a.u. 8e+07 basis set artifact Dirac, pn, -3532.192 a.u. DKH(5,0), pn, -3532.367 a.u. 7e+07 point charge nucleus DKH(n,0) DKH(5,0), pn, lbs, -3532.454 a.u. DKH(5,0), fn, lbs, -3530.397 a.u. 6e+07 finite nucleus DKH(n,0)

5e+07

4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07 DKH(n,0)

1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction

Squared orbital of Hg79+

1e+08

4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07 DKH(n,0)

1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline

Electron density in quantum theory

Calculation of electron density distributions

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline

Electron density in quantum theory Deﬁnition of electron density

Approximate Hamiltonian operators

Electron density distributions for various Hamiltonian operators

Calculation of electron density distributions Eﬀects from approximate Hamiltonian operators

A diﬃcult case: contact densities

Electron density in conceptual theories

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Deﬁnition of electron density in quantum theory

Electron density is an observable → Operator can be assigned

Density operator deﬁned as

N (3) ρˆr = δ (r − ri) Xi=1 (compare to classical expression for set of point charges) Expectation value yields particle density ! ρ(r) = hΨ({ri},t)|ρˆr|Ψ({ri},t)i

+∞ +∞ +∞ 3 3 3 ∗ = d r1 d r2 · · · d rN Ψ ({ri},t) ρr Ψ({ri},t) Z−∞ Z−∞ Z−∞ Charge density:

ρc(r)= −e ρ(r)

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Deﬁnition of electron density in quantum theory

Continuity equation

The continuity equation is the fundamental equation relating electron and current density

∂ρ + ∇ · j = 0 ∂t → Can be used to deﬁne these quantities !

Deduction of continuity equation from expectation value of density operator uniquely deﬁnes electron distribution of an N-electron system in any QM (model) theory.

Note: the continuity equation also follows from Heisenberg equation of motion for the density operator

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Deﬁnition of electron density in quantum theory

Consider change of electron density with time

Apply the Ehrenfest theorem

dhΨ|Oˆ|Ψi i ∂Oˆ = hΨ|[H,ˆ Oˆ]|Ψi + Ψ Ψ dt ~ ∂t

Insert density operator

d Ψ({ri},t) ρr Ψ({ri},t) i = Ψ({ri},t) [H,ˆ ρˆr] Ψ({ri},t) dt ~

∂ρ(r,t) −∇ · j ∂t | {z } | {z } (2nd term on r.h.s. vanishes as density operator does not depend on t)

Choice of Hamiltonian Hˆ and wave function Ψ determine the current density j

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline

Electron density in quantum theory Deﬁnition of electron density

Approximate Hamiltonian operators

Electron density distributions for various Hamiltonian operators

Calculation of electron density distributions Eﬀects from approximate Hamiltonian operators

A diﬃcult case: contact densities

Electron density in conceptual theories

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Approximate Hamiltonian operators

Dirac–Coulomb–Breit (DCB) operator

Fundamental Hamiltonian for M nuclei and N electrons:

M 2 N M M 2 N N pI ZI ZJ e HˆDCB = + hˆD(i)+ + g(i, j) 2mI |RI − RJ | XI=1 Xi=1 XI=1 J=XI+1 Xi=1 j=Xi+1 ‘Fully relativistic’ single-electron Hamiltonian

M 2 ˆ e 2 ZI e hD(i)= cαi · pˆi + A + (βi − 1)mec − eφ − c |rˆi − RI | XI=1 Dirac matrices αi =(αx, αy, αz) and β 10 0 0 0 0 σ 0 0 σ 0 0 σ 0 0 x 0 0 y 0 0 z 01 0 0 αx = , αy = , αz = , β = − σ 0 0 σ 0 0 σ 0 0 0 0 1 0 „ x 0 0 « „ y 0 0 « „ z 0 0 « „ 0 0 0 −1 «

which are built from the Pauli spin matrices σ =(σx,σy,σz)

0 1 0 i 1 0 σx = 1 0 , σy = −i 0 , σz = 0 −1 ` ´ ` ´ ` ´

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Approximate Hamiltonian operators

Electron–electron interaction

Is actually the reason why DCB is approximate; ﬁrst-quantized

Coulomb–Breit operator (Gaussian units) e2 α · α (α · (r − r ))(α · (r − r )) g(i, j)= 1 − i j − i i j j i j |r − r | 2 2|r − r |2 i j i j → Includes Coulomb and magnetic interactions → Derives from classical Darwin energy → Considers retardation eﬀects on transmission of interaction

Gaunt interaction is ﬁrst approximation to Coulomb–Breit operator e2 g(i, j)= [1 − αi · αj] |ri − rj| (neglects retardation; includes instantaneous magnetic interaction)

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Approximate Hamiltonian operators

Consequences of the Dirac formalism

Since 4×4 matrices appear in the Dirac Hamiltonian, orbitals have four components (4-spinors)

1 2 3 4 T L S T ψi = (ψi , ψi , ψi , ψi ) = (ψi , ψi )

L S [ψi ≡large component (2-spinor) ψi ≡small component (2-spinor)]

Dirac equation yields positive- (electronic) and negative-energy (positronic) eigenstates

Negative-energy states are source of pathologies: → Hamiltonian is no longer bound from below → calculation of positronic states is an unnecessary burden → Brown–Ravenhall disease (never observed in calculations)

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Approximate Hamiltonian operators

Douglas–Kroll–Hess theory (DKH)

Unitary transformation U block-diagonalizes the Dirac Hamiltonian

0 0 h+ : operator for electronic states † h+ 0 0 hbd = UhDU = 0 0 h− h− : operator for positronic states 0 0 Also, wave function is transformed: ψ˜ = Uψ → ψ˜L Then, any property requires transformation of operator Ψ|Oˆ|Ψ = UΨ|UOUˆ †|UΨ = Ψ˜ L|O˜LL|Ψ˜ L Hence,D squaredE DKHD orbitals do notE yieldD the densityE !

Technical detail: U is constructed as product of inﬁnitely many unitary

transformations, U = U∞ · · · U3U2U1U0 each Taylor-expanded in terms of an anti-hermitian operator

Yields order-by-order approximation for orbitals and operators

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Approximate Hamiltonian operators

Regular approximations

Relate small and large components by energy-dependent X-operator (derivable!)

∞ k S cσ · p L cσ · p εi L ψi = 2 ψi = 2 2 ψi εi − V + 2mec 2mec − V V − 2mec Xk=0 X(εi) Truncation| yields n{z-th oder} regular approximation (nORA)

c2 n ε k ˆ σ p i σ p hKORA = V + ( · ) 2 2 ( · ) 2mec − V " V − 2mec # Xk=0 Zeroth-order regular approximation (ZORA): c2 hˆ σ p σ p V ZORA = · 2 · + 2mec − V

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Approximate Hamiltonian operators

Non-relativistic limit — Schr¨odinger quantum mechanics

Dirac equation requires 4-spinors

DKH and ZORA reduce the spinor from four to two components

... in their scalar-relativistic variant: to one component

For an inﬁnite speed of light one arrives at the well-known Schr¨odinger Hamiltonian: M 2 N 2 M M 2 pI pi ZI ZJ e Hˆnon−rel. = + + 2mI 2me |RI − RJ | XI=1 Xi=1 XI=1 J=XI+1 N N e2 N M Z e2 + − I |ri − rj| |ri − RI | Xi=1 j=Xi+1 Xi=1 XI=1 Standard reference in quantum chemistry

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline

Electron density in quantum theory Deﬁnition of electron density

Approximate Hamiltonian operators

Electron density distributions for various Hamiltonian operators

Calculation of electron density distributions Eﬀects from approximate Hamiltonian operators

A diﬃcult case: contact densities

Electron density in conceptual theories

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Electron density distributions for various Hamilton Operators

Electron density from single Slater determinant of 4-spinors

Insert Slater determinant (SD) Θ into expectation value (+ apply Slater–Condon rules)

SD ρDCB(r) = hΘ({rj}) |ρr| Θ({rj})i

N (3) = ψi(ri) δ (r − ri) ψi(ri) ri i=1 D E X

One-electron integrals then yield fully relativistic density N SD † ρDCB(r) = ψi (r) · ψi(r) Xi=1

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Electron density distributions for various Hamilton Operators

Current density for the Dirac–Coulomb–Breit operator

Use continuity equation N ∂ Ψ ρr Ψ i = Ψ cαi · p , ρr Ψ ∂t ~ i * " i=1 # + X

(3) = −∇ · cN Ψ α1δ (r − r1) Ψ (multiplicative operators cancel inD commutator) E

! (3) Hence, DCB current density jDCB = cN Ψ α1δ (r − r1) Ψ now only depends on choice of four-component many-electron wave

function Ψ

For a single Slater determinant, the relativistic current density reads N SD † jDCB(r) = c ψi (r) · α · ψi(r) Xi=1 Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Electron density distributions for various Hamilton Operators

Two-component approaches For DKH, density operator must be transformed !

∂ † (1) (3) (1)† UΨ UρrU UΨ = −∇·cN UΨ U α1δ (r − r1)U UΨ ∂t D ˜ E D E Ψ |{z} DKH electron and current densities for one Slater determinant:

N SD ˜ (i) (3) (i)† ˜ ρDKH(r) = ψi U δ (r − ri)U ψi i=1 D E X N SD ˜ (i) (3) (i)† ˜ jDKH(r) = c ψi U αiδ (r − ri)U ψi i=1 D E X

ZORA electron density can be obtained by back-transformation to four components using then the DCB expression.

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Electron density distributions for various Hamilton Operators

Electron density in the non-relativistic framework

Insert Schr¨odinger Hamiltonian into continuity equation:

∂ Ψ ρr Ψ N~ = −∇ · [hΨ∗∇ Ψi − h(∇ Ψ)Ψ∗i] t m 1 1 d 2 ei

For single Slater determinant, electron and current densities are

N SD ∗ ρNR(r) = ψi (r)ψi(r) Xi=1 ~ N SD ∗ ∗ jNR(r) = [ψi (r)∇ψi(r) − (∇ψi(r))ψi (r)] 2mei Xi=1

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline

Electron density in quantum theory Deﬁnition of electron density

Approximate Hamiltonian operators

Electron density distributions for various Hamiltonian operators

Calculation of electron density distributions Eﬀects from approximate Hamiltonian operators

A diﬃcult case: contact densities

Electron density in conceptual theories

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Eﬀects from approximate Hamilton operators

Comparison of Dirac and non-relativistic electron densities

First pictures of Dirac electron density for Hydrogen by White (1931).

Followed by Burke & Grant (1967) who considered H-like atoms.

Relativistic contraction of s- and p1/2-shells.

Expansion of high-angular-momentum shells.

No nodes in the relativistic radial density (except at the origin).

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Eﬀects from approximate Hamilton operators

DCB, DKH, ZORA-SO for acetylene Ni, Pd, Pt complexes Diﬀerence electron densities

ρ4comp(r) − ρnonrel(r): panels a, b and c

ρZORA(r) − ρnonrel(r): panels d,e and f

ρDKH10(r) − ρnonrel(r): panels g, h and i

Relativistic eﬀects most pronounced for Pt

Scalar-relativistic DKH covers most relativistic eﬀects G. Eickerling, R. Mastalerz, V. Herz, H.-J. Himmel, W. Scherer, M. Reiher, J. Chem. Theory Comput. 3 2007 2182 Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Eﬀects from approximate Hamilton operators

Signiﬁcance of electron correlation eﬀects ?

Quite many studies on small, light molecules.

Early one on H2 by Bader and Chandra: Hartree–Fock (HF) overestimates density in central bonding region. Density around nuclei is underestimated by Hartree–Fock.

Other studies comparing CI and HF come to similar conclusions: → density shifted from bonding region towards nuclei when considering electron-correlation eﬀects

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Eﬀects from approximate Hamilton operators

Correlation eﬀects for TM complexes — Fe(NO)2+ example

ρ (r) −ρ (r) a) CASSCF HF 2 Diﬀerence densities 1 (HF≡ no correlation) 0 Fe NO

-1

-2 Around Fe atom, correlation

-2 0 2 4 6 8 ρ b) (r) −ρ (r) density exhibits four minima 2 DFT,BP86 HF

0 Minima are surrounded by four

-1 maxima

-2

-2 0 2 4 6 8 → electron density is shifted from ρ (r) − ρ (r) c) CASSCF DFT,BP86 2 bonding region to atomic nuclei

0 S. Fux, M. Reiher, Structure & Bonding (2010), submitted. -1

-2

-2 0 2 4 6 8

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline

Electron density in quantum theory Deﬁnition of electron density

Approximate Hamiltonian operators

Electron density distributions for various Hamiltonian operators

Calculation of electron density distributions Eﬀects from approximate Hamiltonian operators

A diﬃcult case: contact densities

Electron density in conceptual theories

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 A diﬃcult case: contact density

DKH density at the atomic nucleus 1e+08

4e+07 finite nucleus DKH(9,9) squared radial function 3e+07 point charge nucleus Dirac 2e+07 DKH(n,0)

1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr R. Mastalerz, R. Lindh, M. Reiher, Chem. Phys. Lett. 465 2008 157

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline

Electron density in quantum theory Deﬁnition of electron density

Approximate Hamiltonian operators

Electron density distributions for various Hamiltonian operators

Calculation of electron density distributions Eﬀects from approximate Hamiltonian operators

A diﬃcult case: contact densities

Electron density in conceptual theories

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Electron density in conceptual theories Reactivity descriptors in conceptual DFT

symbol descriptor energy derivative ∂E µ chemical potential „ ∂N «v(r) δE ρ(r) electron density „ δv(r) «N 2 ′ δ E δρ(r) χ(r, r ) linear response function ′ = „ δv(r)δv(r ) «N „ δv(r) «N ∂2E ∂µ η chemical hardness 2 = „ ∂N «v(r) „ ∂N «v(r) 1 ∂N S chemical softness = η „ ∂µ «v(r) δ∂E ∂ρ(r) f(r) Fukui function = „ δv(r)∂N « „ ∂N «v(r)

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Electron density in conceptual theories

Fukui function f −(r) for 3 heavy-element molecules

relativistic non−relativistic Relativistic eﬀects most a) b) pronounced for PbCl2 and Bi2H4 PbCl 2 → changes chemical interpretation

c) d)

(CH3)2SAuCl: Bi2 H2 comparatively small relativistic eﬀects

e) f)

(CH 3 ) 2 SAuCl N. Sablon, R. Mastalerz, F. De Proft, P. Geerlings, M. Reiher, Theor. Chem. Acc. (2010), in press.

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Summary and Conclusion

Summary and Conclusion foundations of electron density in relativistic theory

numerical eﬀects on the calculated density peculiarities of approximate relativistic methods ‘relativistic eﬀects’ on density and descriptors of conceptual DFT

take home message: choose any relativistic all-electron theory for density calculations

Review of the material just presented: S. Fux, M. Reiher, Structure & Bonding (2010), submitted. General presentation: M. Reiher, A. Wolf, Relativistic Quantum Chemistry, Wiley-VCH, 2009.

Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Acknowledgments

Acknowledgments

Samuel Fux Remigius Mastalerz

... and the other members of my group

Collaborations: G. Eickerling, W. Scherer (Augsburg); R. Lindh (Uppsala); H.-J. Himmel (Heidelberg); N. Sablon, F. De Proft, P. Geerlings (Brussels) Financial support: DFG priority programme, ETH Zurich, SNF

Electron Density in (Relativistic) Quantum Theory Markus Reiher