Part 2 Electronic Structure Theory

Transcription

1 HS 2014: Vorlesung Part 2 Electronic Structure Theory Dr. Mar'n O. Steinhauser Fraunhofer Ins'tute for High- Speed Dynamics, Ernst- Mach- Ins'tut, EMI, Freiburg, Germany mar(n.steinhauser@unibas.ch or mar(n.steinhauser@emi.fraunhofer.de Web: hdp:// 1 1 CONTENTS Course Outline: 1. Introduc'on 2. The Schrödinger Equa'on 3. The Electronic Schrödinger Equa'on 4. The Hartree- Fock Approxima'on 5. Configura'on Interac'on (CI) 6. Density Func'onal Theory 2 2 1

2 Suggested Literature Chemistry Related: A. Szabo, N. S. Ostlund, Modern Quantum Chemistry, Mineola, NY, Dover Publica(ons, 1996 F. Jensen, Introduc4on to Computa4onal Chemistry 2nd ed., New York, John Wiley & Sons, 2007 N. Ira Levine, Quantum Chemistry 5th ed., Upper Saddle River, NJ, Pren(ce Hall, 1999 W. Koch, M. C. Holthausen, A Chemist s Guide to Density Func4onal Theory, Weinheim, Wiley- VCH, 2001 Quantum Theory: C. Cohen- Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, New York, John Wiley & Sons, 1977 R. G. Parr, W. Yang, Density- Func4onal Theory of Atoms and Molecules, New York, Oxford University Press, 1989 J. J. Sakurai, Modern Quantum Mechanics, Addison- Wesley, 1994 P. A. M. Dirac, Lectures on Quantum Mechanics, Mineola, New York, Dover Publica(ons, 2001 Computer Simula'on and Modeling: M. P. Allen, D. J. Tildesley, Computer Simula4on of Liquids, Clarendon Press, Oxford, 1987 D. C. Rappaport, The Art of Molecular Dynamics Simula4on, University Press, Cambridge, 1995 J. M. Thijssen, Computa4onal Physics, Cambridge New York, Melbourne, Cambridge University Press, 1999 M. O. Steinhauser, Mul4scale Modeling of Fluids and Solids Theory and Applica4ons, Springer, Heidelberg, Berlin, New York, 2008 Kurt Binder, D. W. Heermann, Monte- Carlo Simula4ons in Sta4s4cal Physics, Heidelberg, London, New York, Springer INTRODUCTION 4 4 2

3 Hierarchical Organiza'on of Nature Engineering Models Experiments m s Con(nuum Materials Science Materials Science Crystals Composites Chemistry Molecules Physics Atoms 10 9 m s Dr. M. Steinhauser, Fraunhofer EMI, Hierarchical Organiza'on of Sod and Hard Maeer Nano- and Microtechnology Aggregates Molecules Atoms Subatomic Particles Classical Physics Quantum Theory SUBATOMIC ATOMIC MICRO MESO Particle Accelerators Subatomic Particles Growing Complexity of Hard Matter Probe Microscopy (AFM) Electron Microscopy Atoms Genes Light Microscopy Chromosomes Self-Organization of Soft Matter M. Steinhauser, Mul4scale Modeling of Fluids and Solids, Springer 2008 Materials Science & Engineering MACRO Length (nm) Virus Bacterium Cellular Life Nano and Macromolecular Organisms Biology 6 6 3

4 Time/Length Scales and Associated Theories/Methods time [s] length [m] Makro Meso Mikro Nano (m) 0 10 (mm) 10 3 (µm) 10 6 (nm) 10 9 Large Scale FEM, Plasticity, Elasticity, Heat Transport, etc. Cellular Automata Scale-independent Percolation Models Crystal Plasticity Meso Particle Dynamics Taylor, Sachs, Voigt, Reuss, Hashin-Shtrikman Models c Kinetic Multi Phase Potts-Models t DΔc = 0 Ginzburg-Landau Models, Kinetic Phase Models Dissipative Particle Dynamics (DPD) Bead-Spring-Models Classical Molecular Dynamics i Topological Networks, Dislocation Dynamics Local Electron Density Functional Theory Metropolis Monte-Carlo, Hartree-Fock σ =σ(ε) m x = Ĥ Ψ = E Ψ Dr. M. Steinhauser, Fraunhofer EMI, i f i Paradigm: Lower Scales Determine Behavior on Larger Scales 10-2 m 10-5 m Length Loss of Information Less Transferability Mesoscopic Methods Continuum Methods 10-8 m 10-9 m m Electronic Methods Atomistic Methods s s 10-9 s 10-6 s 10-3 s 10 0 s Time M. Steinhauser, Mul4scale Modeling of Fluids and Solids, Springer

5 Fundamental Par'cles: The Standard Model (Fermions) Spektrum der WissenschaI 12/2013 The only par'cles relevant for Chemistry Whiteboard Notes 9 9 Fundamental Par'cles (Fermions) Specula(ons about the inner structure of quarks, called PREONS DropofRain WaterMolecule H#Atom Proton Quarks m m m m < m Preon(hypthetical) eitherpointparticleor stillinnerstructure Spektrum der WissenschaI 12/

6 Standard Model: 4 Fundamental Interac'ons Interac'on Par'cles Range (m) Rela've Strength Strong Quarks < Weak Quarks, Leptons < Electromagne(c Charged Par(cles 1 Gravita(on Mass Par(cles F. Jensen, Introduc4on to Computa4onal Chemistry, Wiley, 2007 Quantum Electrodynamics (QED): The Coulomb- Poten(al between two electrons is the zero- th order term of an expansion with respect to (1/c 2 ) V Coulomb ij = V(r ij ) = q iq j = q iq j 4πε 0 r ij r ij q j r ij q i This is the only relevant force for chemistry Keep in Mind: The Standard Model of Maeer Atoms are made by massive, point- like nuclei (protons and neutrons) Surrounded by (ghtly bound, rigid shells of core electrons Bound together by a glue of valence electrons

7 So, why Do We Really Need Electronic Structure Theory? For Example: Stability of Different Structures and Bonding Electronic, Op(cal or Magne(c Proper'es of (small) molecules Dynamics of Chemical Reac(ons and Bondings Diels- Alder Reac'on: 1,3- butadiene + ethylene > cyclohexane hdp:// pages/jim/ El. Structure Theory: Insight into Chemical Phenomena First Principles (ab- ini(o) Electronic Structure Theory (EST) Hartree- Fock Approxima'on + Electron Correla'on Molecular Equilibrium Geometries Energy Benchmarks Op(cal (Spectroscopic) Proper(es Transi(on States Support Experiments

8 Material Proper'es from First Principles Energy at our living condi(ons (300 K): 0.04 ev (kine(c energy of an atom in an ideal gas: 3/2 kbt). Differences in bonding energies are within one order of magnitude of 0.29 ev (hydrogen bond). Binding energy of an electron to a proton (hydrogen): ev = 1 Rydberg (Ry) = 0.5 Hartree (Ha) = 0.5 a.u. Thecore(1s)electronsaretightlyboundtonuclei. Chemistryhappenswiththevalenceelectrons Wave Character of Maeer Louis de Broglie ( ): Application of the relativistic equations of motion to matter in the same way as to light (electromagnetism) Wave length of a particle in AngstrØm λ A = h p = h mv = Momentum of the particle Velocity of the particle E= p2 2m h 2mE E ev [ ] Compare with λ = h p = h mc Experimental Verification by Clinton Davisson und Lester Germer: Scattering of Electrons off a Ni-Cristal (1924) Result: No diffuse (isotropic) reflection, but preferred scattering angles L. De Broglie Wave lenght of photons C. Davisson and L. Germer

9 When is a Par'cle like a Wave? Wavelength Momentum = Planck s constant Descrip'on of a Wave: An Amplitude Field Ψ ( r,t ) Ψ ( r,t ) We need to know the amplitude of the excita(on (the field) at every point and every instant

10 Dirac Nota'on: Representa'on of Vektors/Operators Ket- Vector in Hilbert space (a complex, linear vector space with with an infinite, countable number of dimensions and a scalar product defined) Ψ = Ψ ( r ) Ψ Orthonormality: Ψ i Bra Vector Ket Vector ( r )Ψ j ( r )d 3 r = Ψ i Ψ j = δ ij Expecta(on Values: ( r )HΨi ( r )d 3 r = Ψ i Ψ i H Ψ i = E i Element of a Product Hilbert Space H 1 H 2 Whiteboard Notes Expansion of Vectors in a Basis: Gram- Schmidt Algorithm Startwithasetofnlinearlyindependentvectors { ϕ,..., ϕ 1 n } Definethefirstvector ϕ 1 = ϕ 1 ϕ 1 ϕ 2 istobeconstructedfrom ϕ 2 andbeorthonormalto ϕ 1 Hence,onehastosubtractthatpartof ϕ 2 whichpointsin thedirectionof ϕ 1 : ϕ 1 η 2 = ϕ 2 ϕ 1 ϕ 2 ϕ 1,i.e. ϕ 1 η 2 = 0 ϕ 2 Normalizethisvector: ϕ = η 2 2 η 2 For ϕ 3 wehavetoremovebothcontributionsin ϕ 2 ϕ 1 ϕ 2 ϕ 1 η 2 thedirections ϕ 1 and ϕ

11 Gram- Schmidt Algorithm: Summary Step1: ϕ = ϕ 1 1 ϕ 1 Stepn 1: fork = 2,3,...,n : k 1 η k = ϕ k ϕ i ϕ k ϕ i,and ϕ k = η k η k i=1 Forageneralorthonormalset,youhave: ϕ i ϕ j = δ ij ϕ i ϕ j = δ ij This simple rela'on between the basis vectors makes calcula'ons much easier Whiteboard Notes Classical States vs. Quantum States ψ x,t π q i, p i PhaseSpaceofClassicalMechanicsisspanned by6n coordinates Timeevolutionofclassicalphasespacetrajectory π ( t) isgivenby: q i = H ;p p i = H ;i = 1,..., N i q i So we have: H ( p i,q i,t) H p i,q i,t 0 π ( t 0 ) π ( t) InQM:q i andp i cannotbedeterminedexactlyatthesametime

12 Classical States Some Remarks: vs. Quantum States ψ x,t π q i, p i 1.Thestate ψ isastatevectorofaqmsystem("pure"state). 2.Thetransition ψ α ψ ( α complex)isthesamestate. 3.Forseveralinteractingsystems, ψ representsthetotalstate.,duetomeasurementsorexternalinfluences. 4. ψ = ψ t 5.ψ ( x,t )isjustaparticularrepresentationofaquantum mechanicalstatewhichstressesitspositiondependency Classical States Further Remarks: vs. Quantum States ψ x,t π q i, p i densityoperator:ρ ψ = p ψ ψ = ψ ψ = P ψ Amoregeneraldescriptionofqmstatesisprovidedbythe Justtheprojection Probabilityoffindingthesysteminstate ψ p = 1(purestate) ontostate ψ Formixedstates,onehas: ˆρ ψ = p i ψ i ψ i i Probabilityoffindingthesystemin onethe(possiblymany)purestates ψ i

13 Prepara'on of a pure quantum State ψ We measure the quan(zed angular momentum j, with a measurement (beam split) apparatus measurement apparatus j z Par(cle beam C Beam is split by the apparatus into two beams, + and - - part of the beam is blinded out, e.g. absorbed by the cover C Prepara'on of a pure quantum State ψ Now we abstract from measuring a par(cular quan(ty: Here, we measure just some quan(ty A which has a discrete spectrum: SpliDer + Absorber = Filter P(a i ) measurementofa andprepofsystemin state a i

14 Prepara'on of a pure quantum State ψ Assume: property B is measurable precisely with A at the same (me: Definitionofapurestatepreparation: ψ = a i,b j,...,z m P( z m ) P b j P a i ϕ The Postulates of Quantum Mechanics I The postulates of QM connect the mathema(cal formalism, i.e. the used objects of Hilbert Space with a physical interpreta(on, i.e. with meaning POSTULATE 1: Measuring apparatus of a par(cular physical quan(ty (observable) POSTULATE 2: Pure quantum mechanical state of a system Linear, hermi(an operator A Hilbert- Vector ψ POSTULATE 3: Measurement Interactionofqm systemwithmeasuringapparatus Applica(on of operator A to the state ψ : A ψ = a i a i a ψ i a i a ψ FilterP a i

15 The Postulates of Quantum Mechanics II The postulates of QM connect the mathema(cal formalism, i.e. the used objects of Hilbert Space with a physical interpreta(on, i.e. with meaning POSTULATE 4: Results of Measurements Eigenvaluesa i oftheoperatora Exactly which eigenvalue is going to be measured, cannot be predicted. Therefore, one needs POSTULATE 5: Probabilityp ( a ψ i )ofmeasuringa i p( a i ψ ) = a i ψ Final Remarks Quantum Mechanics only offers answers to ques(ons of the following kind: a) Which results for a par(cular measurement are at all possible? - actual value pertains to the spectrum of an operator, i.e. its eigenvalues - Each operator represents a measurement apparatus - Thus, this ques(on actually pertains to a property of the measurement apparatus, independent of the inves(gated system b) What is the probability with which a certain result of a measurement is observed? - This is answered by Postulate 5: P p = 1only,if ψ isidenticalto theeigenstate a i of ψ

16 2. The Schrödinger Equa'on The Time- Dependent Schrödinger Equa'on A one- par4cle equa(on H ( r,t )Ψ( r,t ) = "2 2m 2 Ψ ( r,t ) + V ( r,t )Ψ ( r,t ) = i" Ψ ( r,t ) t 1925 onwards: W. Heisenberg (Matrix Formula(on) E. Schrödinger (Wave Equa(on) P. A. M. Dirac (Transforma(on Theory)

17 The Sta'onary Schrödinger Equa'on H ( r,t )Ψ( r,t ) = "2 2m 2 Ψ ( r,t ) + V ( r,t )Ψ ( r,t ) = i" Ψ ( r,t ) t SeparationofVariables r,t Ψ ( r,t ) = ϕ ( r ) f ( t) Whiteboard Notes Solu'on of the One- Electron Schrödinger Equa'on The Free Electron: 2 2m 2 ϕ ( x)+ = Eϕ ( x) = exp i ϕ x 2mE x Wave Func'on: Ψ( x) = ϕ ( x) f ( t) exp i( kx ωt) The probability density of finding the electron at posi(on r at (me t Ψ ( r,t ) 2 = ϕ r does not depend on 'me. exp i " Et 2 = ϕ ( r )

18 Example: Metal Surfaces A more realis(c poten(al V ( x) V 0 x A Central Poten'al: Separa'on of Variables ( r,ϑ,ϕ) H = 2 2m 2 + V ( r) 2 = Δ = 2 x y z 2 Ψ Elm r = RElm r Y lm ϑ,ϕ Separa(on of Variables leads to an algebraic eigenvalue problem. hdp://

19 Electronic Structure Calcula'ons Two Important Concepts for Solving the Schrödinger Equa(on on a Computer: 1. Matrix Formula(on 2. Ritz Varia(onal Principle Matrix Formula'on HΨ ( r ) = EΨ ( r ) H Ψ = E Ψ This is the equa(on we want to solve Ψ = { }completesetoforthogonalfunctions c n ϕ n ϕ n n=1 Orthonormaliza(on is always possible (Gram- Schmidt Algorithm) ϕ m H Ψ = E ϕ m Ψ Basis Set of Func(ons (in computa(ons: a finite set, i.e. n=1,...k) k c n ϕ m H ϕ n = Ec m The coefficients c m are our solu(ons of the linear n=1 algebraic problem

20 Matrix Formula'on k n=1 c n ϕ m H ϕ n = Ec m Matrix elements H mn k n=1 H 11 H 1k " # " H k1 H kk H mn c n = Ec m c 1 " c k = E We have transformed an analysis problem to a (discrete) linear algebra problem Whiteboard Notes c 1 " c k Ritz Varia'onal Principle E[ Ψ] = Ψ H Ψ Ψ Ψ A Func(onal, i.e. a linear operator that has as argument a whole set of func(ons A systema(c trial- and- error Method to find out the minimum energy of a trial wave func(on: E Ψ IfE Ψ wavefunctionandviceversa The Algorithm (What to do): Calculate a Number [ ] E 0 foranytrialfunctionψ [ ] = E 0 thenψisthegroundstate

21 Example: Energy of a Hydrogen Atom E[ Ψ] = Ψ H Ψ Ψ Ψ Ψ a = C exp( αr) Ψ a Ψ a = π C 2 Ψ α 3 a Ψ a = π C 2 2α Ψ a 1 r Ψ a = π C 2 α 2 Whiteboard Notes The Electronic Schrödinger Equa'on

22 The Sta'onary SE of Nelectrons and M nuclei H ( r, R )Ψ( r, R ) = EΨ ( r, R ) Ψ ( r, R ) 2 gives probability density for finding electrons at position r = { r 1, r 2, r 3," r N } and nuclei at R = { R 1, R 2, R 3," R M } Whiteboard Notes Digression into Atomic Units We let every coordinate r j be represented in terms of: r j a 0 rj, with r j being a dimensionless quan(ty and a 0 having units of length. The kine(c and poten(al energies then become: T = 2 2m 1 a 0 2 j V en = Ze 2 a 0 r j V = 1 1 ee e2 a 0 r ji The goal is to factor out e 2 a 0 both in the kine(c and poten(al energies: T = e2 2 1 a 0 2m e 2 a 0 2 j andv = e2 Z r j +1 r ji a 0 Choosing a 0 = 2 e 2 m = 0.5 Å = 1 Bohr with e2 a 0 = 1Hartree = 27.21eV allows to write: T = j andv = Z r j

23 N M The Hamiltonian for electrons and nuclei The Hamiltonian H contains (in atomic units): the electronic kinetic energy: ˆTe = 1 2 the nuclear kinetic energy: ˆTn = 1 2 N 2 j j=1 M j=1 M M j 1 j 2 the electron-nuclear Coulomb attraction: ˆVen = Z j 1 r k R j the nuclear-nuclear Coulomb repulsion: ˆVnn = + Z j Z k 1 R k R j j=1 M j<k=1 N the electron-electron Coulomb repulsion: ˆVee = + 1 r k r j H can contain more terms if external electric or magnetic fields are present. j<k=1 N k=1 The Born- Oppenheimer Approxima'on Max Born Robert Oppenheimer Expansionwithrespecttotheratioofmasses m e m p

24 The Born- Oppenheimer Approxima'on The Concept: Adiaba(c Separability of the fast electronic and slow nuclei movement: H = T n + T e + V ne + V ee + V nn = T n + H e TheelectronicHamiltonianH e containsallofh exceptt n Since the electrons move much faster than the nuclei, the Hamiltonian can be simplified by fixing the posi(ons of the nuclei at specified loca(ons R, i.e. T n = 0. We treat only the electrons as quantum par(cles, in the field of the fixed (slowly moving) nuclei (which are treated as classical par(cles). This gives rise to the concept of poten'al surfaces The Electronic Schrödinger Equa'on The electronic Hamiltonian is: H e = T e + V ne + V ee + V nn And thus we solve the electronic Schrödinger Equa'on (ese) with the electronic wave func(on : r, R H e Ψ k Ψ k electrons coordinates ( r, R ) = E k Ψ k ( r, R ) only a parametric dependence on the coordinates of the nuclei

25 Expansion of the electronic Schrödinger Wave Func'on BecauseH e ishermitian,itselectroniceigenfunctionsψ K formacomplete andorthonormalsetoffunctionsof r. So,Ψcanbeexpandedinthe(electronicwavefunctions)Ψ K : Ψ r, R N = Ψ k r, R k=1 ϕ k ( R ) Expansion of a func(on in a (abstract) linear vector space of func'ons Notetheanalogytoordinaryvectoranalysis: a = TheΨ k ( r, R )andϕ k ( R )dependon R,becauseH e doessothroughv en andv nn. 3 k=1 e i a i Expansion of a vector in a linear vector space of ordinary (Euklidean) vectors A Differen'al Equa'on for the func'ons ϕ k R Thisexpansioncanbeusedintheoriginal,fullSE,includingT n, i.e.thekineticenergyofthenuclei: H ( r, R )Ψ( r, R ) = EΨ ( r, R ) : H e 1 2 M 1 M j j=1 2 j E k Ψ k ( r, R )ϕ k R = 0(3.1) MultiplyingwithΨ l ( r, R ) fromleftandintegratingovertheelectronic degreesoffreedomd 3 ryieldsthecoupledchannelequations. Theseareequationsforthecoefficientsϕ k ( R )

26 The Coupled Channel Equa'ons for the Orbitals Wehad: H e = E l R N + Ψ l k=1 N + Ψ l k=1 0 = E l R M j=1 1 M 2 j=1 M j 1 ( r, R ) j E ϕ l M 1 M j j=1 ( r, R ) M 1 M j j=1 R + 2 j Ψ k ( r, R )d 3 rϕ k R j Ψ k ( r, R ) d 3 r j ϕ k + 1 M j 2 j E ϕ l ( R) Whiteboard Notes R Ifweignoreallnon-adiabaticcouplingterms,weobtainaSEforthe vibrational,rotational,translationalmotionofthenucleionthel-thenergysurfacee l 1 M 2 M j 1 j=1 2 j E k Ψ k ( r, R )ϕ k R = 0. ϕ k R R : The Energy Surface Each electronic state has ist own set of rot./vib. wave func(ons and energies: E l R 1 M 2 j=1 M j 1 2 j E ljmυ ϕ ljmυ R = 0 l = 1 l = 0 This is the electronic- vibra(onal- rota(onal separa(on one sees in textbooks

27 Limita'ons of the Born- Oppenheimer Approxima'on The total wavefunc(on is limited to one electronic surface, i.e. to a par(cular electronic state. Rela(vis(c effects are ignored. These are negligible for lighter elements (Z<36), but not for the 4th period or higher. Electron Spin must be introduced as a ad- hoc hypothesis in non- rela(vis(c quantum mechanics Back to the Electronic Schrödinger Equa'on... There are major difficul'es in solving the electronic SE: N N N 1 1 TheoperatorV ee = = = 1 2 k> j r jk isatwo.electronoperator. j=1 k> j r jk N N 1 j=1 k=1 r jk j k ThisiswhatmakestheelectronicSEnotseparable ThismeansΨisnotrigorouslyaproductoffunctionsof individualelectroncoordinates,calledspatialorbitalsφ ( r ): ForexampleforaBor.Atomwehave: Ψ r 1, r 2, r 3, r 4, ( r 5 ) φ 1 ( r1 ) φ 2 ( r2 ) φ 3 ( r3 ) φ 4 ( r4 ) φ 5 r5 27

28 Spa'al Orbitals and Spin Orbitals Approximate+the+two.electron+operator+V ee +by+a+one.electron+operator,+i.e.+ Approx V ee ( i, j) v ee ( i) = v HF ( i). In+this+approximation,+the+electronic+wave+function+Ψ ( r i )+can+be+written+as+ product+of+single+electron+spatial+orbitals: Ψ r i n = φ i ( ri ) i=1 Spa'al Orbitals You+obtain+spin+orbitals+χ i as+products+of+spatial+orbitals+with+the+ spin+functions+α ( ω )+( )+or+β ( ω )+( )+,+i.+e.+χ i ( xi ) = φ i ( ri )α ++or+χ i xi Ψ x i xi n = χ i ( xi ) i=1 = φ i ri Spin Orbitals where+ x i = ( r i,ω )+is+the+set+of+cartesian+coordinates+ r i +and+spin+coordinates+ω. β More Problems with Solving the ese: Cusp Condi'ons Another Difficulty in Solving the ese: The allowed wave func(ons have cusps Kine(c energy of the electron Electron- nuclear adrac(on Thefactors ( 1 r k r k Z r k )Ψ and ( 1 r k r kl +1 r kl )Ψ willblowupunlessso4calledcuspconditionsareobeyedbyψ asr k 0andr kl 0. Cusp Condi'ons: r k Ψ = Z Ψasr k 0 r kl Ψ = Ψasr kl 0 electron- nucleus electron- electron

29 Cusps of the Wave Func'on Approxima(ons with zero slopes When we try to approximately solve the ese, we should use trial func(ons that have cusps. Slater- type orbitals e ζ r k have cusps at nuclei, but Gaussians e αr k 2 do not We rarely use func(ons with e- e cusps (but we should). Whiteboard Notes Mean Field Approach Ignorethenon*separabilityinafirstapproachandapproximateV ee byapotentialwhichisactuallyone*electronadditive. This is called MEAN FIELD APPROACH Independent Par'cle Model (Hartree- Products): Each electron moves in an effec(ve poten(al, represen(ng the adrac(on of the nuclei and the average effect of the repulsive interac(ons of the other electrons. This average repulsion is the electrosta(c repulsion of the average charge density of all other electrons

30 The Hartree Equa'ons: Effec'vely One- Par'cle Equa'ons The Hartree Equa4ons can be obtained directly from the varia(onal principle, once the search is restricted to the many- body wave func(ons that are wriden as the product of single orbitals (i.e. we are working with independent electrons). Hartree- Product: Ψ r 1,..., r ( n ) = ϕ 1 ( r1 )ϕ 2 ( r2 )"ϕ n ( rn ) i + V R l r i l + ϕ j ( rj ) 2 1 j i r j r i d r j ϕ i ri = εϕ i ( ri ) The Hartree- Equa'ons: Self- Consistent Field The single par(cle Hartree operator is self- consistent It depends on the orbitals that are the solu(on of all other Hartree equa(ons. We have n simultaneous integro- differen'al equa'ons for the n orbitals. Solu(on is achieved itera(vely, or: self- consistently i + V R l r i l + ϕ j ( rj ) 2 1 j i r j r i d r j ϕ i ri = εϕ i ( ri )

31 Itera'ons to Self- Consistency Ini(al guess at the orbitals Construc(on of all the operators Solu(on of the single par(cle SE With this new set of orbitals, construct the Hartree operators again Iterate procedure un(l it (hopefully) converges The Differen'al Analyzer Vannevar Bush and the Differen(al Analyzer ( MIT Boston, 1931)

32 What is missing in the Hartree Approxima'on? The wave func(on does not include correla(on The wave func(on is not an(symmetric A liele History on the Spin Sta's'cs Theorem All elementary par(cles are either fermions (with half- integer spins) or bosons (integer spin) Observa'on: Spin Symmetry is connected with Sta(s(cs (Fermi- Dirac or Bose- Einstein) First formulated by Markus Fierz (1939) Reformulated in a more systema(c way by Wolfgang Pauli (1940) More conceptual argument provided by Julian Schwinger (1950)

33 A liele History on the Spin Sta's'cs Theorem Feynman Lectures on Physics: [...] An explanation has been worked out by Pauli from complicated arguments of QFT and relativity [...] but we haven t found a way of reproducing his arguments on an elementary level...this probably means that we do not have a complete understanding of the fundamental principle involved [...] Quantum Mechanics by C. Cohen Tanoudjii, Footnote on page 1387: [...] the spin statistics theorem is just an empirical law. These hypotheses [of Pauli] may not all be correct, and discovery of a boson of half-integral spin or a fermion of integral spin remains possible Spin- Sta's'cs All elementary par(cles are either fermions (with half- integer spins) or bosons (integer spin) A set of iden(cal (indis(nguishable) fermions has a wave func(on that is an(symmetric by exchange of par(cles Ψ ( r 1, r 2,..., r j,..., r l,..., r N ) = Ψ ( r 1, r 2,..., r l,..., r j,..., r N ) In a non- rela(vis(c theory, an(symmetry of the wave func(on must be introduced as an addi(onal axiom

34 Slater Determinant and Pauli Principle An an(symmetric wave func(on is constructed via a Slater determinant of the individual orbitals (instead of just a product, as in the Hartree approach). electron coordinates Ψ ( x 1, x 2,..., x N ) = 1 N χ α x1 χ α x2 χ α χ β ( x1 ) " χ ν ( x1 ) χ β ( x2 ) " χ ν ( x2 ) # # $ # xn orbitals χ β ( xn ) " χ ν ( xn ) Levi- Civita Symbol = 1 ε αβγ "ν χ α χ β ( x2 )"χ ν xn N P[ αβγ "ν ] all permuta(ons Pauli principle: If two states are iden(cal, the determinant vanishes, i.e. we can t have two electrons in the same quantum state x1 4. The Hartree- Fock Approxima'on Douglas Hartree ( ) Vladimir Fock ( )

35 The Hartree- Fock Approxima'on: What is the Idea? The Problem: How to find a good mean- field poten(al? Approxima'on: In the electronic Hamiltonian, assume to be a one- par(cle operator, i.e V ee V MF V MF The wave func(on Ψ can then be wriden as a product of one- electron spin orbitals (Slater determinant), and the Hamiltonian is addi(ve. V ee In the HF- Approxima'on, you only use one single determinant trial func'on: = χ 1 χ 2 χ n Ψ HF Whiteboard Notes The Energy Expression (Expecta'on Value) E HF = Ψ HF H e Ψ HF = ( Ψ HF * ) H e ( Ψ HF )d 3 x TheformulafortheenergyofanySlaterdeterminantisthen E HF = i h i + 1 ii jj 2 ij ji, i ij wheretheone5electronintegralisgivenby i h i = d 3 * x 1 χ i ( x1 )h x ( 1 ) χ j ( x1 ), h = T e + V en + V nn andatwo5electronintegralisgivenby ij kl = d 3 x 1 d 3 * x 2 χ i x1 χ j ( x1 ) 1 r 12 χ k χ l x2 * x

36 Varia'onal Principle with Constraints The Hartree- Fock method determines the set of spin orbitals which minimize the energy and give us this best single determinant. { χ i } L = E HF χ i λ ij i j δ ij λ ij { } where are the undetermined Lagrange mul(pliers and i j is the overlap between spin orbitals and j, i.e * i j = χ i χ j = χ i ( x) χ j ( x)d 3 x { } As a result of this varia(on, i.e. δl χ i = 0, one obtains the Hartree- Fock Equa'ons Whiteboard Notes i The Hartree- Fock Equa'ons Ψ ( x 1,..., x n ) = χ 1 χ 2 χ n, i + V R l r i n µ=1 n µ=1 χ µ * χ µ * M l=1 ( x j ) ( x j ) 1 r j r χ µ x j i 1 r j r χ λ x j i χ λ ( x i ) + d 3 r j χ λ d 3 r j χ µ x = r,ω x i x i Posi'on Spin (α,β) = ε λ χ λ ( xi ) Whiteboard Notes

37 Equa'ons with Constraints: Lagrangian Mul'pliers Here, we want to consider the general method of solving equa(ons which have certain constraints by using Lagrangian Mul(pliers λ. For example, we derived the HF- Equa(ons using: { χ i } { } L = E HF χ i λ ij i j δ ij General Func'onal We actually minimize the energy func'onal, i.e. the expecta'on value of the Hamiltonian by varying { } χ i This is our constraint: We demand the func'ons to be orthonormal { } χ i Lagrange Mul'plier(s) (one for each constraint) Whiteboard Notes The Hartree- Fock Equa'ons Succinct Nota(on, introducing appropriate operators: n µ=1 n µ=1 χ µ * χ µ * ( x j ) ( x j ) h x ( i ) χ λ ( xi ) + 1 r j r χ µ x j i 1 r j r χ λ x j i d 3 r j χ λ d 3 r j χ µ x i x i = ε λ χ λ ( xi )

38 The Hartree- Fock Equa'ons Succinct Nota(on, introducing appropriate operators: h x ( i ) χ λ ( xi ) + µ µ J µ K µ χ λ χ λ xi xi = ε λ χ λ ( xi ) Coulomb J µ and Exchange K µ Operators Thesetwooperatorsaredefinedbytheiractiononafunction,i.e.as: J µ K µ x1 χ λ ( x1 ) = d 3 * x 2 χ µ ( x2 ) x1 χ λ ( x1 ) = d 3 * x 2 χ µ ( x2 ) 1 r 1 r χ µ x2 2 1 r 1 r χ λ x2 2 χ λ χ µ x1 x

39 The Hartree- Fock Equa'ons and the Fock Operator f The Fock- Operator is: ( x i ) = h x ( i ) + J µ ( xi ) K µ ( xi ) µ one- electron operator Coulomb Operator Exchange Operator The Hartree- Fock Equa'ons are: f ( x i ) χ i ( xi ) = ε i χ i ( xi ) A Physical Picture of Coulomb and Exchange Interac'ons J 1,2 = χ 1 x J 1 x 2 1 r r 2 χ 2 ( x ) d 3 K 1,2 = χ 1 ( x) χ 2 x 2 χ 2 ( x ) d 3 xd 3 x = 2 x = χ 1 x 1 r r χ 2 x J 2 ( x)d 3 x χ 1 x d 3 xd 3 x Generally,J µ > K µ χ 1 x χ2 Overlap region x

40 Hartree s Defini'on of the Mean- Field Poten'al V MF So,wecandefineV MF intermsofthej andk integrals. Ithasthecharacteristics,that Ψ H Ψ = Ψ H e Ψ H e = T e + V en + V nn + J K " # $# Inperturbationtheory: H H e = V ee J K V MF Real Hamiltonian using the true V ee HF Hamiltonian using V MF Thereisnofirst,orderperturbationcorrectiontotheenergy,because: Ψ H Ψ = Ψ H e Ψ ThechoiceofH e formsthebasisofmoller,plessetperturbationtheory(mpn) Back to the Hartree- Fock Equa'ons (HFEs) f The Fock- Operator is: ( x i ) = h x ( i ) + J µ ( xi ) K µ ( xi ) µ one- electron operator Coulomb Operator Exchange Operator The Hartree- Fock Equa'ons are: f ( x i ) χ i ( xi ) = ε i χ i ( xi )

41 How Does One Solve the HF Equa'ons? TheorbitalsareusuallydeterminedbycarryingoutaHFcalculation. Thisisnotdone ( exceptinrarecases )bysolvingthehfsecondorderpartial differentialequationsin3ndimensionsonaspatialgridbutbyexpandingthe spinwavefunctionχ i intermsofso?calledatomicorbital(ao)basisfunctions (becausetheyareusuallycenteredonatoms)usingthelcao?moexpansion: χ i = φ ν f φ µ C µi = ε i φ ν φ µ C µi µ=1 µ=1 k µ=1 C µi φ µ (typically,theφ µ arenotorthogonal) foreachspinorbitali.insertingthisexpansionintof x i reducesthehfcalculationtoamatrixeigenvalueequation: k k χ i ( xi ) = ε i χ i ( xi ) The Roothaan Equa'ons: Matrix Representa'on of HFEs Wehaveobtainedthematrixequations k φ ν f φ µ C µi = ε i φ ν φ µ C µi µ=1 µ=1 IntroducingtheFockmatrixF νµ andtheoverlapmatrixs νµ inaobasis: k F νµ = φ ν f φ µ S νµ = φ ν φ µ µ=1 µ=1 The$HF'Roothan$Equations:$The$matrix$eigenvalue$problem $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ k k $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$F νµ C µi = ε i S νµ C µi or: $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$FC = εsc SCFProcedure

42 SCF Calcula'on of the Fock- Operator Matrix Elements φ ν f φ µ = φ ν φ µ + φ ν Z A φ µ r + A C kη C kγ φ ν ( r )φη r k=occ φ ν ( r )φη r A 1 1 r r φ γ ( r )φ µ r r r φ µ ( r )φ γ r and$the$overlap$integrals$are:$ φ ν φ µ Thenumberoftheseone.andtwo.electronintegralsscaleswith thebasissetsizemasm 2 andm 4. ThecomputationaleffortforsolvingaM M eigenvalueproblemscalesasm SCF Calcula'on of the Fock- Operator Matrix Elements χ ν f χ µ = χ ν χ µ + χ ν Z A χ µ r + A C kη C kγ χ ν r k=occ The Self- Consistent χ ν ( r ) Field χη r Procedure: r r χ γ ( r ) χ µ r 1.TheFockmatrixisformedusingtheinitalC kµ coefficients. 2.TheHF'RoothanEquationsaresolvedtoobtain"new"C coefficients. kµ 3.ThisiterationisrepeateduntiltheC kµ inoneiterationareidentical A 1 1 χη ( r ) bycompletelyignoringallj andk terms. r r χ µ r χγ ( r ) tothoseobtainedinthenextsolutionofthefockmatrix. SomeprogramsguesstheinitalC kµ coefficientsoftheoccupiedspinorbitals 42

43 The Self- Consistent Field (SCF) Procedure as Flow Chart Input 3D coordinates of atomic nuclei START Ini4al Guess Molecular Orbitals (1- electron vectors) Formulate Fock Matrix Diagonalize Fock Matrix iterations END Calculate Proper(es YES SCF Converged? NO What is Actually Missing in the Hartree- Fock Equa'ons? Correla(ons (missing by defini(on ) Dynamical Correla(on (electrons get too close to each other in mean- field approxima(on) Sta(c Correla(on (single determinant approxima(on is not good enough to describe bond breaking) Possible Solu(on to this? ConfigurationInteraction(CI)

44 Koopmans Theorem Some ini(al remarks... Koopmans'Theorem Note: It is o~en misspelled as: Koopman'sTheorem Koopmans Theorem: A Liele Quiz Tjalling Charles Koopmans Koopmans was awarded a Nobel Prize for his contribu(ons to the field of: a) Nuclear Physics b) Quantum Chemistry c) Economics d) None of the above Joint winner, with Leonid Kantorovich,

45 Koopmans Theorem (KT) AgoodthingabouttheHFV MF isthattheorbitalenergiesε K giveapproximate ionizationpotentialsandelectronaffinities.thisiscalledkoopmans'theorem. ( J ij K ij ) + V nn HFEnergyforN electrons:e N = Removeane fromorbitalnumberk: k E N 1 k ΔE = E N E N 1 N = h k + J ik K ik N h i + 1 N N i=1 2 i=1 j=1 N 1 = h i i=1 i=1 = ε k N 1 N 1 i=1 ( J ij K ij ) + V nn The energy of an e - in orbital k is equal to the energy required to remove the electron, to give the corresponding ion j=1 Koopmans Theorem (KT) Addane toorbitalnumberk +1(avirtualorbital): N+1 E k+1 N+1 = h i + 1 i=1 2 N+1 N+1 N+1 ΔE = E k+1 N+1 E N = h k+1 + J i,k+1 K i,k+1 = ε k+1 i=1 i=1 j=1 ( J ij K ij ) + V nn The occupied orbital energies tell us about ioniza(on poten(als (IP). The virtual orbital energies tell us about electron affini(es (EA)

46 Koopmans Theorem (KT): Things to be Cau'ous About KT is a sta(c approach: Orbitals of the ion are the same as in the unionized state (they are frozen ) Bad approxima(on if considerable orbital relaxa(on occurs; (Ioniza(on Poten(al too big). Electron correla(on is neglected Whiteboard Notes Configura'on Interac'on

47 Configura'on Interac'on (CI) iinnormalhartree,fock,thehfdeterminantisbuilt fromthelowestenergysingle,electronstates. in basisfunctionsandk electronscanfillupk χ i with lowestenergies:ψ 0 = χ χ χ HF 1 2 K itheideaofciistomixinexcitedstatesfromvirtualorbitals Ψ 1 HF = χ 1 χ 2 χ K+1 χ K iuseannewvariationalwavefunction: Ψ = c 0 Ψ 0 + c 1 Ψ 1 + c 2 Ψ 2 + FULLCI(FCI) Whiteboard Notes But: Full CI (FCI) Fails in Prac'se Full CI is not a prac(cal quantum chemical method to study chemical problems. Estimation)of)Computational)Complexity)of)Full)CI M = p 3N M : )#)of)parameters N : ))#)of)electrons p : )))#)of)parameters)per)coordinate I cannot foresee any technological progress which would make such calcula(ons feasible. Walter Kohn, Nobel lecture 1998 John Pople Nobel Laureate 1998 Today's(Computers: M = 10 12, (p = 3 N 8 Chemically(relevant(molecule: N = 100, p = 3 M = As(comparison: T Universe s atoms N Universe

48 Quantum Chemistry: A Hierarchy of Approxima'ons Additional approxima(ons HΨ = EΨ Ψ HF- Roothan equa'ons single determinant Addi(on of more determinants Semi- empirical methods Convergence to exact solu(on Basis Sets Arrange Four Electrons in Six Orbitals 1 Ground Configura(on HF Virtual Orbitals Occupied Orbitals

49 Basis Sets Arrange Four Electrons in Six Orbitals 1 Ground Configura(on HF 32 Single Excita(ons S Virtual Orbitals Occupied Orbitals Basis Sets Arrange Four Electrons in Six Orbitals 1 Ground Configura(on HF 32 Single Excita(ons S 168 Double Excita(ons D Virtual Orbitals Occupied Orbitals

50 Basis Sets Arrange Four Electrons in Six Orbitals 1 Ground Configura(on HF 32 Single Excita(ons S 168 Double Excita(ons D 224 Triple Excita(ons T Virtual Orbitals Occupied Orbitals Basis Sets Arrange Four Electrons in Six Orbitals 1 Ground Configura(on HF 32 Single Excita(ons S 168 Double Excita(ons D 224 Triple Excita(ons T 70 Quadruple Excita(ons Q Virtual Orbitals Occupied Orbitals

51 Basis Sets Arrange Four Electrons in Six Orbitals 1 Ground Configura(on HF 32 Single Excita(ons S 168 Double Excita(ons D 224 Triple Excita(ons T 70 Quadruple Excita(ons Q 495 Slater Determinants Virtual Orbitals Occupied Orbitals Basis Sets Arrange Four Electrons in Six Orbitals 1 Ground Configura(on HF 32 Single Excita(ons S 168 Double Excita(ons D 224 Triple Excita(ons T 70 Quadruple Excita(ons Q 495 Slater Determinants Virtual Orbitals Occupied Orbitals

52 Basis Sets: Two Independent Aspects to Solving the ese Arrange Four Electrons in Six Orbitals Basis One electron models complete exact HF minimal HF exact SE basis- set limit for a given N- electron model Full CI (FCI) N electron models Quality of a calcula'on depends upon the quality of basis sets Mul(- electron basis: Slater determinants available to expand the wave func(on One- electron basis: atomic orbitals available to expand the molecular orbitals Full mul(- electron basis recovers all correla(on within a given one- electron basis Full one- electron basis recovers the SCF energy without limita(on on the shape of MO What are Basis Sets? Generically, a basis set is a collec(on of vectors which spans a space in which a problem is solved î, r ĵ, ˆkdefineaCartesian,3Dlinearvectorspace In quantum chemistry, the basis set usually refers to the set of (non- orthogonal) one- par(cle func(ons used to build molecular orbitals. LCAO- MO approxima(on: MO s are built from AO s An orbital is a ONE- electron wave func(on AO s are represented by atom- centered Gaussians in most quantum chemistry programs (GTO s) Some older programs used Slater func(ons (STOs) Physicists like plane wave basis sets (Bloch s Theorem)

53 Slater- Type Atomic Orbitals (STOs) φ STO nlm theatomicnuclei. ( r,ϑ,ϕ) = N nlmς Y lm ( ϑ,ϕ)r n 1 e ζ r STOsarecharacterizedbyquantumnumbersn,l,andmandexponents (whichcharacterizetheradialsize)ζ andareusuallylocatedononeof Cartesian STOs: φ STO abc in isanormalizationconstant ( x, y,z) = Nx a y b z c e ζ r ia, b, ccontroltheangularmomentuml = a + b + c iζ controlsthewidthoftheorbital(largeζ givestightfunction, smallζ givesdiffusefunction) itheseareh:atom:like,atleastfor1s;however,theylack radialnodesandarenotpuresphericalharmonics icorrectshort:rangeandlong:rangebehavior radial distribu(on Gaussian- Type Atomic Orbitals (GTOs) φ GTO abc ( x, y,z) = Nx a y b z c e ζ r2 iagain,a,b,ccontroltheangularmomentuml = a + b + c,i.e.theydetail theangularshapeandthedirectionoftheorbital iagain,ζ controlsthewidthoftheorbital(radialsize) inolongerh =atom=like,evenfor1s imucheasiertocompute(gaussianproducttheorem) ialmostuniversallyusedbyquantumchemists tooflatatrnear0 fallsofftooquicklyatlarger radial distribu(on

54 Contracted Gaussian- Type Orbitals (CGTOs) To overcome cusp weakness of GTO func(ons, it is common to combine two, three, or more GTOs into new func(ons called contracted GTOs, or CGTOs. However: This does not really correctly produce a cusp because every Gaussian has zero slope at r = 0 as shown: SzaboandOstlund,ModernQuantumChemistry Contracted Gaussian- Type Orbitals Problem: STOs are more accurate, but it takes longer to compute the integrals using them Solution: Use a linear combina(on of enough GTOs to mimic an STO φ CGTO abc ( x, y,z) = N c i x a y b z c e ζ i r2 Contracted GTO N i=1 Primi(ve GTO s CGTOs are used most commonly for inner orbitals & GTOs for outer shells Unortunate use of nomenclature: A combina(on of n Gaussians to mimic an an STO is o~en called an STO- ng basis, even though it is made of CGTO s

55 Different Categories of Basis Sets Most AO basis sets contain a mixture of different classes (or categories) of func'ons: Fundamental core and valence basis func'ons: Polariza(on func(ons Diffuse func(ons Rydberg func(ons Core and valence: Minimal basis: the number of CGTOs equals the number of core and valence atomic orbitals in the atom E.g. Carbon: one (ght s- type CGTO, one looser s- type CGTO and a set of three looser p- type CGTOs How good are Minimal STOs and GTOs? Comparison of minimal STO (top) and and GTO (bodom) Carbon 1s, 2s, 2p radial basis func(ons, compared to the results obtained using a large AO basis. Jack Simmons, An Introduc4on to Theore4cal Chemistry

56 Polariza'on Func'ons Polariza(on func(ons are func(ons with one higher angular momentum than appears in the atom s valence orbital space For example: d func(ons for C, N, and O, p func(ons for H withexponentsζ orα whichcausetheirradialsizestobesimilar tothesizesofthevalenceorbitals Note: the polariza'on p orbitals of H are similar in size to the valence 1s orbital and the polariza(on dorbitals of C are similar in size to the 2s and 2p orbitals, not like the valence d orbitals of C. Polariza(on func(ons give angular flexibility to the LCAO- MO process in forming molecular orbitals from valence atomic orbitals. Polariza(on func(ons also allow for angular correla(ons in describing the correlated mo(ons of electrons Polariza'on Func'ons s, p, d, and f angular func(ons showing how they span ever more of angle- space as L increases

57 Diffuse Func'ons: Designa'ons of Basis Set Size Minimal: one basis func(on (STO, GTO, or CGTO) for each atomic orbital in each atom Double- zeta (DZ): two basis func(ons for each AO Triple- zeta (TZ): three basis func(ons for each AO etc., for quadruple- zeta (QZ), 5Z, 6Z,... The use of different- sized basis func(ons allows the orbital to get bigger or smaller when other atoms approach it. So it adds flexibility to the LCAO process to adequately describe anisotropic electron dis(bu(ons in molecules by allowing molecular orbitals of variable diffuseness as the local electronega(vity of the atom varies. Necessary for anions, Rydberg states, very electronega(ve atoms or dispersion (vdw) Diffuse func(ons have small zeta exponents to hold the electron far away from the nucleus H H - Cl Cl H... H... Br Diffuse Func'ons: Designa'ons of Basis Set Size Minimal: one basis func(on (STO, GTO, or CGTO) for each atomic orbital in each atom Double- zeta (DZ): two basis func(ons for each AO Triple- zeta (TZ): three basis func(ons for each AO etc., for quadruple- zeta (QZ), 5Z, 6Z,... MinimalBasisSet 2s1p 1 2s1p = 5 Virtual Orbitals Occupied Orbitals We keep on adding addi(onal func(ons of varying sizes with valence angular momentum

58 Diffuse Func'ons: Designa'ons of Basis Set Size Minimal: one basis func(on (STO, GTO, or CGTO) for each atomic orbital in each atom Double- zeta (DZ): two basis func(ons for each AO Triple- zeta (TZ): three basis func(ons for each AO etc., for quadruple- zeta (QZ), 5Z, 6Z,... Double(ζ BasisSet 4s2 p 2 2s1p = 10 Virtual Orbitals Occupied Orbitals We keep on adding addi(onal func(ons of varying sizes with valence angular momentum Diffuse Func'ons: Designa'ons of Basis Set Size Minimal: one basis func(on (STO, GTO, or CGTO) for each atomic orbital in each atom Double- zeta (DZ): two basis func(ons for each AO Triple- zeta (TZ): three basis func(ons for each AO etc., for quadruple- zeta (QZ), 5Z, 6Z,... Triple(ζ BasisSet 6s3p 3 2s1p = 15 Virtual Orbitals Occupied Orbitals We keep on adding addi(onal func(ons of varying sizes with valence angular momentum

59 Diffuse Func'ons: Designa'ons of Basis Set Size Minimal: one basis func(on (STO, GTO, or CGTO) for each atomic orbital in each atom Double- zeta (DZ): two basis func(ons for each AO Triple- zeta (TZ): three basis func(ons for each AO etc., for quadruple- zeta (QZ), 5Z, 6Z,... Quadruple"ζ BasisSet 8s4 p 4 2s1p = 20 Virtual Orbitals Occupied Orbitals We keep on adding addi(onal func(ons of varying sizes with valence angular momentum Split- Valence Basis Sets A split- valence basis uses only one basis func(on for each core AO and a large basis for the valence AOs Flexibility more important for valence orbitals, which are chemically important MinimalBasisSet 2s1p 1 2s1p = 5 Occupied Virtual Virtual MinimalBasisSet 2s1p Occupied 1s +1 ( 1s1p) =

60 Split- Valence Basis Sets A split- valence basis uses only one basis func(on for each core AO and a large basis for the valence AOs Flexibility more important for valence orbitals, which are chemically important Double(ζ BasisSet 4s2 p 2 2s1p = 10 Virtual Double(ζ Split(Valence BasisSet ( 3s2 p) 1s + 2 ( 1s1p) = 9 Virtual Occupied Occupied Split- Valence Basis Sets A split- valence basis uses only one basis func(on for each core AO and a large basis for the valence AOs Flexibility more important for valence orbitals, which are chemically important Triple(ζ BasisSet 6s3p 3 2s1p = 15 Virtual Triple(ζ Split(Valence = 13 BasisSet 4s3p 1s + 3 1s1p Virtual Occupied Occupied

61 Split- Valence Basis Sets A split- valence basis uses only one basis func(on for each core AO and a large basis for the valence AOs Flexibility more important for valence orbitals, which are chemically important Virtual Quadruple"ζ BasisSet 8s4 p 4 2s1p = 20 Quadruple"ζ Split"Valence = 17 BasisSet 5s4 p 1s + 4 1s1p Virtual Occupied Occupied Meet the Basis Sets: Many Acronyms This is a par(al lis(ng by a prac((oner irritated by basis set acronyms

62 Meet the Basis Sets: Dunning- Style Bases Developed by Thon Dunning (UT, ORNL) with higher order correla(on methods Basis set nota'on looks like: aug- cc- pvtz, cc- pvqz, pvdz VDZ, VTZ, VQZ or V5Z specifies at what level the valence (V) AOs are described. Nothing is said about the core orbitals because each of them is described by a single contracted Gaussian type basis orbital. cc specifies that the orbital exponents and contrac(on coefficients were determined by requiring the atomic energies computed using a correlated method to agree to within some tolerance with experimental data. If cc is missing, the AO exponents and contrac(on coefficients were determined to make the Hartree- Fock atomic state energies agree with experiment to some precision. p specifies that polariza(on basis orbitals have been included in the basis. In the Dunning nota(on, aug is used to say that diffuse func'ons have been added, but the number and kind depend on how the valence basis is described. At the pvdz level, one s, one p, and one d diffuse func(on appear; at pvtz a diffuse f func(on also is present; at pvqz a diffuse g set is also added; and at pv5z a diffuse h set is present Pople- Style Bases: 6-31+G**, 3-21G*, G*, G Developed by the late Nobel Laureate, John Pople, and popularized by the Gaussian set of programs Basis set nota'on looks like: k-nlm++g** or k-nlm++g(idf,jpd) Core/Valence func'ons: k primi(ve GTO for core electrons n primi(ve GTO for inner valence orbitals e.g. 3-21G, 6-31G, 6-311G l primi(ve GTO for medium valence orbitals m primi(ve GTO for outer valence orbitals Polarizaton func'ons: * indicates one set d polariza(on func(ons added to heavy atoms (non- H), alt. (d) ** indicates one set d polariza(on func(ons added to heavy atoms and one set p polariza(on func(ons added to H atom, alt. (d,p) idf indicates i set d and one set f polariza(on func(ons added to heavy atoms idf,jpd indicates i set d and one set f polariza(on func(ons added to heavy atoms and j set p and one set d polariza(on func(ons added to H atom Diffuse func'ons: e.g. 6-31G*, 6-311G(d,p) + indicates one set p diffuse func(ons added to heavy atoms (non- H) ++ indicates one set p diffuse func(ons added to heavy atoms and 1 s diffuse func(on added to H atom e.g G

63 Meet the Basis Sets: Others Atomic Natural Orbitals (ANO) Developed by J. Amlof and P. R. Taylor at NASA Ames ANO designed to reproduce the natural orbitals for correlated (CISD) calcula(ons on atoms Generally contracted - more complicated scheme to implement Very large basis sets (consequently very expensive) but thorough and numerous levels of trunca(on are available Properties Basis Sets PBS: polarized basis set of Sadlej, double- zeta + diffuse, developed for calcula(on of electrical proper(es, good for dipole moments, polarizabili(es, excited states WMR: generally contracted basis sets by Widmark, Malmqvist, and Roos for atomic and molecular proper(es, rather large Pseudopoten'als / Effec've Core Poten'als Core orbitals generally not affected by changes in chemical bonding but require many basis func(ons to describe accurately Treat core electrons as averaged poten(als rather than actual par(cles pseudopoten(als Advantages of greater efficiency since calcula(on not dominated by unnecessary integrals Advantage of incorpora(ng rela(vis(c effects Developed by removing core- dominated basis func(ons, then reop(mizing the remaining basis func(ons in the presence of the pseudopoten(al Poten(al is linear combina(on of Gaussians chosen to model the core electrons, orthogonal to valence electrons in basis func(ons Very important for heavy atoms, especially transi(on metals

64 Diffuse Basis Func'ons in the Web Website for Diffuse Basis Func(ons tabulated off the shelf: hdps://bse.pnl.gov/bse/portal Website Snapshot taken in Dec Diffuse Basis Func'ons in the Web Locate the website Search by basis set name Search by element Available for many QC programs Download basis set and pseudopoten(als Website Snapshot taken in Dec

65 Example: cc- pvdz for H A basis set of contracted GTOs needs to specify the exponents ζ's These are given here in the format of the program Gaussian (with exponents first).. and the contraction coefficients c i 's Example: cc- pvdz for C

66 6. Density Func'onal Theory What s the Big Deal With DFT? It is fast ( N ) 3 It does not need wave func(ons WOW

67 DFT: Key Breakthrough Publica'ons P. Hohenberg, W. Kohn, Inhomogeneous Electron Gas, Phys. Rev. 136, B864 (1964) W. Kohn, L. J. Sham, Self- Consistent Equa4ons Including Exchange and Correla4on Effects, Phys. Rev. 140, A1133 (1965) Walter Kohn Lu J. Sham Pierre C. Hohenberg The Thomas- Fermi Model (1927) Let s try to find an expression for the energy as a func(onal of the charge density ρ F ρ E ρ = E +V +V kin ext ee The kine(c energy is the tricky term: how do we get the curvature of a wavefunc(on from the charge density? Answer: Local Density Approxima'on (LDA)

68 Local Density Approxima'on Wetakeρateverypointtocorrespondtothe kineticenergydensityofthehomogeneouselectrongas 5 = Aρ T r 3 r non- homogeneous charge density E Th Fe 5 ρ = A ρ 3 ( r )d 3 r + ρ ( r )V ext ( r )d 3 r + 1 ρ ( r 1 )ρ ( r 2 ) 2 r 1 r d 3 r 1 d 3 r Issues with Thomas- Fermi Approach No theore(cal basis the idea of a func(onal is not jus(fied It does not include exchange effects but Dirac proposed to add the LDA exchange energy: E ex 4 3 = C ρ ( r ) E ex Dirac ( Becke,&88) = E ex Dirac ρ To account for the fact that varies strongly in some regions, Becke introduced a gradient- correc(on to Dirac exchange d 3 r 4 γ ρ where%x = ρ 4 3 ρ andγ = ( ) 3 x γ xsinh 1 x d 3 r

69 Density- Func'onal Theory (DFT) TheexternalpotentialV ext andthenumbern ofelectrons completelydefinethequantumproblem The wave func(ons are in principle uniquely determined, via the Schrödinger equa(on All system proper(es follow from the wavefunc(ons Theenergy(andeverythingelse)isthusafunctional ofv ext andn The Hohenberg- Kohn Theorems (1965) The density as the basic variable: the external poten(al r determines uniquely the ground state charge density ρ r, and the opposite is also true: the ground state charge density ρ r r. V ext V ext determines uniquely the external poten(al

70 The First Hohenberg- Kohn Theorem Proof:&Suppose,&one&knows&ρ ( r )&at&all&points& r.& Then&ρ ( r )&can&determine&n &by&n = d 3 rρ ( r ) If&one&knows&N,&one&can&write&T e &and&v ee &in&the&hamiltonian&h. Assume&now,&that&there&are&two&distinct&potentials:&V ext ( r )&and&v ext ( r )&which&form&two& Hamiltonians&H &and& H,&respectively&having&the&same&number&of&electrons&N &but differing&only&in&v ext &and& V ext. Further,&assume&one&uses&H &and& H &to&solve&the&schrödinger&equation&for&their&groundcstate energies&and&wave&functions&e 0,&Ψ( r ),&and&e 0,& Ψ ( r ). Finally,&assume&that&Ψ ( r )&and& Ψ ( r )&have&the&same&onecelectron&density,&i.e. &&&&&&&&&&&&&&&&&&&&&& 2 Ψ d 3 r 2 d 3 r 3 d 3 r N = ρ r = Ψ 2 d 3 r 2 d 3 r 3 d 3 r N The First Hohenberg- Kohn Theorem Proofcontinued:Ifwethinkof Ψ weknowthat E 0 < Ψ H Ψ = Ψ H Ψ + ρ r V ext r = E 0 + ρ ( r ) V ext ( r ) V ext ( r ) d 3 r(1) ( r )astrialvariationalwavefunctionforthehamiltonianh, V ext r d 3 r Similarly,takingΨ ( r )astrialvariationalwavefunctionforthehamiltonian H, E 0 < E 0 + ρ ( r ) V ext ( r ) V ext ( r ) d 3 r(2) Addingthetwoequations(1)and(2)gives E 0 + E 0 < E 0 + E 0, aclearcontradiction. So,therecannotbetwodistinctV ext ( r )thatgivethesamen andthesameρ. Hence,foranygivenρtherecanonlybeoneV ext ( r )

71 The Universal Func'onal F ρ The ground state density determines the poten(al of the Schrödinger equa(on, and thus the wavefunc(ons F ρ r = Ψ T +V Ψ ee ρ r determines its ground state wavefunc(on, that can be taken as a trial wave func(on in this external poten(al Ψ H Ψ = Ψ T +V ee +V ext Ψ = F ρ r + ρv r d 3 r The Second Hohenberg- Kohn Theorem The varia(onal principle: We have a new Schrödinger- like equa(on, expressed in terms of the charge density only. E V ρ r = F ρ r + V ext r ρ ( r )d 3 r E

72 The Non- Interac'ng Unique Mapping The Kohn- Sham System: A reference system is introduced, the Kohn- Sham electrons These electrons do not interact, and exist in an external poten(al, the Kohn- Sham poten(al, such that their ground- state charge density is IDENTICAL to the charge density of the interac(ng system The Non- Interac'ng Unique Mapping For a system of non- interac(ng electrons, the Slater determinant is the EXACT wavefunc(on The kine(c energy of the non- interac(ng system is well defined F n r E H = T n r S + E n r H + E n r exch n ( r ) = 1 n ( r 1 )n( r 2 ) 2 r 1 r d 3 r 1 d 3 r

73 The Kohn- Sham Equa'ons V H ( r ) +V exch ( r ) +V ext r V H r = ψ i n r r r d 3 r,v exch n r r N = ψ i ( r ) i=1 = H KS ψ i ( r ) = ε i ψ i ( r ) r = δ E exch δ n ( r ) Approxima'on of V exch r? How to approximate the exchange correla(on interac(on? Local Density Approxima(on (LDA, the Thomas- Fermi idea) again Ceperley and Alder, Phys. Rev. Lett. (1980) (Quantum MC Numerical data) Perdew & Zunger, PRB 23, 5048 (1981) (Analytical Interpolation) The beginning of DFT as a practical approach Generalized Gradient Approxima(on (GGA) - uses both ρ and ρ - examples are BLYP AM

74 Procedure of Solving the Kohn- Sham (KS) Equa'ons The$KS$procedure$is$as$follows: 1.$An$atomic$orbital$is$chosen 3.$The$electron$density$is$computed$as$ρ r = n i φ i ( r ) 2 4.#This#density#is#used#in#the#KS#equations#to#find#new#EF# { φ i }and$ev ε i i { } 5.#These#new#φ i areusedtocomputeanewdensity,whichisusedtosolve anewsetofksequations.thisprocesscontinuesuntilconvergence. 6.Oncetheconvergedρ ( r )istdetermined,theenergycanbecomputedusing E ρ = n φ i i r i T e φ i ( r ) + V ext ( r ) ρ ( r ρ ( r )ρ( r )d 3 r + e2 ) 2 r r d 3 rd 3 r + E exch ρ Some Pros and Cons Solving the KS equa(ons scales like M 3, so DFT is cheap. Current func(onals seem to be predy good, so results are good. Unlike varia(onal and perturba(ve wavefunc(on methods, there is no agreed upon systema(c scheme for improving func(onals. Excited states: would the same func(onal give you the excited- state s energy if you plugged in a ρ for an excited state? Kieron Burke, J. Chem. Phys.,

75 Perspec'ves on Density Func'onal Theory Kieron Burke, J. Chem. Phys.,