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- 1. Introduction to electronicstructure theory! Heather J Kulik! hkulik@stanford.edu! 02/27/13!
- 2. Why we need quantum!Potential energy surfaces: Bonding and structure: explicit or for force ﬁeld from ﬁrst principles!development! experiment! ! QM! bent linear! Sc! Ti! V! Cr! Mn! Fe! Co! Ni! Cu! Zn! !"
- 3. Why we need quantum!Interesting phenomena depend on what the electrons are doing!!Optical properties! Catalysis! Magnetic properties!
- 4. Quantum mechanics in brief! Time-dependent Schrödinger equation! " !2 2 " " " $#(r, t) ! " #(r, t) +V (r, t)#(r, t) = i! 2m $t Stationary ! ! V (r, t) = V (r ) potential!1) Spatial, time- # !2 2 !& ! !independent %! " +V (r )(! (r ) = E! (r ) fromSchrödinger equation! $ 2m wikipedia! d What we usually are2) Temporal part! i! f (t) = Ef (t) solving in quantum dt chemistry!
- 5. Inﬁnite square well! n 2! 2 !2 # !2 2 "& " " En = ! " +V (r )(! (r ) = E! (r ) ∞! 2ma 2 ∞!TISE:" % $ 2m # 0, % 0!x!a 16 16 for:! V (x) = $ n=4! % ", & x < 0, x > a ! 2 d 2! (x) Inside well:! ! 2 = E! (x) 2m dx 9 n=3! 9 ODE Solution:! ! (x) = Aeikx + Be!ikx , k = 2mE ! n=2! Boundary conditions:! ! (0) = ! (a) = 0 4 4 Simpliﬁed solution:" 1 n=1! 1 2 ! n! $ ! n (x) = sin # x &, n = 1, 2, 3… 0 a a " a % Related to probability of where the electron is!!
- 6. Hydrogen atom! 3TISE ! 2 $ (n ! 1)! ! /2 ! 2!+1Solution:" ! n!m (r, ! , " ) = # & e ! Ln!1 ( ! )(Y!m (! , " ) " na0 % 2n(n + !)! Laguerre Spherical polynomials! harmonics!Quantumnumbers:"n = 1, 2, 3,…! = 0,1, 2,…, n !1m = !!,…, !.
- 7. A note about notation! !"#$%&%()*#+,-./012#(#*%-3()4 ! * ! ! !! = ! (r ) = ! ! !i(6-7*"(285=9:"(;<1#5=ij (r )! j (r )dr !i ! j = " ket" bra" ket" ?! > @ " ?! > # ?! > $! " # "# * ! ! !! ! (r )! j (r )dr = !i ! j = "ij i B @ ?! > ?! > A " " ? ! > $! " & " B(
- 8. A note about notation! !"#$%&()$*+,"#%)-./0 1 " .# 0 ! .# 0 1 " ! &$ $ $ 7)$#5)6)-",2+-3#%)-4 $ 98 % 1 " ! &$ 1 " !& $ $ 98 % (:;9<=>><?@?=>A#)*%4#%3!)B:,%-6)2!"#:$%",4CC D:$;$"-BE:B:$"-BF%3),"!"$G"$%
- 9. Atoms: Electrons and Nuclei!ˆ = T + V + V +VH ˆ ˆ ˆ e e!e e!N N!N 1" 2" 3" 4" 1 # ! ! &1" Te = ! # "i 2 3" ˆ Ve!N ( = "%"V RI ! ri ( ) 2 i i $ I quantum kinetic energy of electrons" electrostatic electron- nucleus attraction"2" ˆ = "" ! 1 ! Ve!e 4" VN!N i j>i r ! rj i electrostatic nucleus- electron-electron interaction" nucleus repulsion"
- 10. Born-Oppenheimer!ˆ ! ! ! ! ! ! ! !H ! (r1,…, rn , R1,…, RN ) = Etot! (r1,…, rn , R1,…, RN ) We decouple the electrons and nuclei. Only the electrons are treated as quantum wavefunctions in the ﬁeld of ﬁxed or slow nuclei.! ! Adiabatic: no coupling between electronic surfaces.! " Born-Oppenheimer: Ionic motion does not inﬂuence electronic surface.!
- 11. Variational Principle! ˆ | !# "! | H E[!] = "! | !#Energy of a trial wavefunction is greater Unless the trial wavefunction isthan the energy of the exact solution:! the exact solution:! E[!] " E0 E[!] = E0 If we have the ground state energy, we have the ground state wavefunction.!
- 12. Hartree Equations!If we assume many-body wavefunction can be represented as single-particleorbitals:!! ! ! ! ! ! !! ! (r , r ,…, r ) = ! (r )! (r )"! (r ) 1 2 n 1 1 2 2 n n!We obtain Hartree equations directly from variational principle + Time-independent Schrodinger equation:!! & 1 ! ! ! 2 1 !) ! ! (! "i + %V ( RI ! ri ) + % # ! j (rj ) ! ! drj +! i (ri ) = !" i (ri ) 2 ( 2 I j$i rj ! ri + *Pros: Self-consistent equations – each equation for single particle dependson others, can solve together iteratively to reach a solution.!!Cons: Wave function does not obey Pauli exclusion principle, no explicitcorrelation.!
- 13. Slater determinant! ! ! ! !" (r1 ) ! " (r1 ) " !" (r1 ) ! ! ! ! ! ! 1 !" (r2 ) ! " (r2 ) " !" (r2 )! (r1, r2 ,…, rn ) = n! " " # " ! ! ! !" (rn ) ! " (rn ) " !" (rn ) Satisﬁes anti-symmetry for Pauli exclusion principle: Slater determinant changes signs when we exchange two electrons (columns in the matrix).!
- 14. Hartree-Fock Equations!With a Slater determinant wavefunction to represent our many-bodywavefunction:! ! ! ! ! (r ) ! (r ) " ! (r )! ! ! ! ! ! " ! 1 ! (r ) ! (r ) " ! (r ) 1 " 1 " 1 " 2 " 2 " 2! ! (r , r ,…, r ) = n! 1 " 2 n " # " ! ! ! ! (r ) ! (r ) " ! (r )! " n " n " nWe may now obtain analogous Hartree-Fock equations:! Standard notation! Bra-ket notation!$ 1 2 ! ! ! Coulomb integral!&! "i + #V ( RI ! ri ))! " (ri ) + 1-electron integrals! ˆ Jij = ! i! j " ee ! i! j% 2 I ( "i2 Exchange integral!$ hi = ! i ! +Ve!N ! i * ! 1 ! ! ! 2 ˆ K ij = ! i! j " ee ! j! i&# * ! µ (rj ) ! ! ! µ (rj )drj )! " (ri ) !&µ% rj ! ri ) ( $ N N!1 N ! 1 ! ! ! ! ˆ ESlater [! ] = ! Slater H ! Slater = " hi + " " ( Jij ! K ij )#& * ! µ* (rj ) ! ! rj ! ri ! " (rj )drj )! µ (ri ) = !" # (ri ) µ & % ) ( i=1 i=1 j=i+1Pros: Anti-symmetrized, exchange is exact.!Cons: Only correlation comes from anti-symmetrized Slater.!
- 15. Self-consistency! Initial guess Form Diagonalize MOs from operators/ Fock matrix! XYZ! Fock matrix! Are we no! Done! yes! converged?!
- 16. Basis sets: a technical note!Molecular wavefunction from the combination of some basisof atomic-like wavefunctions.!Minimal: one basis function per A.O.!Double/triple/4/5/6 zeta: two/three/four/ﬁve/sixbasis functions per A. O.!Split valence: one A.O. for core, more forvalence.!Polarization: From mixing l orbital with l+1!Diffuse functions: letting electrons move awayfrom the nucleus (key for anions)!Notation examples: 6-31G: Split-valence, double-zeta: Core is 6,valence is 3 in one, 1 in the other.!6-31G(d)/6-31G*: adds d polarization.! Easier to6-311G: split-valence triple-zeta.! work with:!6-31+G: adds diffuse functions.!
- 17. Basis sets: other considerations!Condensed matter simulations often ∞! ∞!use plane wave basis sets.!!We construct our wave function from a summation 16 16 n=4!of standing waves, like the solutions to the particlein a box.!!!! 9 n=3! 9Pros: Can extrapolate to complete basis set limit,unlike challenges with localized basis set. Can n=2!describe delocalized electrons straightforwardly.!! 4 4Cons: Need many functions to describe strongoscillations, e.g. near core of nucleus. Often have 1 n=1! 1to use pseudopotentials (effective core potentials).! 0 a
- 18. What about correlation?!What kind of correlation are we talking about?"A common deﬁnition:" Ecorr = Eexact ! EHFIn Hartree-Fock, each electron experiences repulsion ! ! ! !" (r1 ) ! " (r1 ) " !" (r1 )from an average electron cloud, motion of individual ! ! ! ! ! ! 1 !" (r2 ) ! " (r2 ) " !" (r2 )electrons is not correlated.! ! (r , r ,…, r ) = 1 2 n n! " " # "! ! ! ! !" (rn ) ! " (rn ) " !" (rn )But Slater determiant enforces: two electrons of samespin are in the same space, a kind of correlation.!Dynamic correlation: how one electron repels another when they are nearby.!!Static correlation: multi-determinantal character: i.e. one Slater is not enough!!!Why we need correlation: accurate relative energetics, van der Waals, etc.!
- 19. Conﬁguration Interaction! ! 0 = !1! 2 !! I !! K HF Ehf 0 dense 0 !1 = !1! 2 !! K+1 !! K HF 0 dense sparse very sparse d e extremely n sparse sparsePros: Introduces dynamic correlation.! s sparse e!Cons: Expensive calculation – manypossible excitations, poor scaling. ! very extremely extremely 0! sparse sparse sparseIn practice: Truncated excitations tomaintain accuracy and minimize cost. Diagonalize CI, roots are solutions:(e.g. CCSDT, CISD…)" ground state, ﬁrst excited state…!
- 20. Multi-Conﬁguration SCF! • CSF - Conﬁguration state function.! ! = c0 ! 0 + c1!1 + c2 ! 2 +… • WFN gets multiple CSFs, multiple determinants per CSF.! • Obtain variational optimum for shape of MO s and for weight of CSF.! • Orbital energies are undeﬁned, only occupation matters:!Pros: Introduces both Cons: Difﬁcult to ﬁnd true minimum in coefﬁcientstatic and dynamic space, combinatorial explosion of CSFs. Stillcorrelation.! lacking dynamic correlation (MRCI)!! n!(n +1)! N= CH3OH! !m$ !m $ ! m$ ! m $ m=14,n=12! # &!# +1&!# n &!# n +1&! N=169,884!! "2%"2 %" 2%" 2 %
- 21. Density functional theory! Is there a way to work with the probability # !2 2 "& " " density instead of the wave function?!TISE:" %! " +V (r )(! (r ) = E! (r ) $ 2m Ψ ?! ρFundamental principles of density functional theory:! Vext! 1st Hohenberg-Kohn theorem (1964):" E, Ψ Vext and N,ρ deﬁne the quantum problem! Charge density uniquely determines the potential." N,ρ
- 22. 2nd Hohenberg-Kohn theorem!A variational principle for DFT:" ! ! ! ! !Ev [ ! (r )] = F[ ! (r )]+ ! vext (r )! (r )dr " E0Trial wavefunctionfrom trial density:" Our universal Energy of the functional with an exact solution."Ψ ρ initial guess."BUT we still donʼt know what our functional form is!!
- 23. The Kohn-Sham approach ! A unique mapping!" Non-interacting K-S electrons live in an externalInteracting 1-to-1 Non-interacting potential such that their mapping! density is the same as theparticles! particles! fully-interacting density.!3N variables! 3 spatial variables!Kohn-Sham equations:" ˆ ! (r) = #! 1 " 2 + " (r) + " (r) + " (r)&! (r) = # ! (r) H KS i % $ 2 H xc ext ( i i i N n(r) ! E xc 2! H (r) = " r ! r dr ! xc (r) = n(r) = ! !i (r) ! n(r) i=1
- 24. Exc: Exchange & Correlation!In principle, DFT is exact…"" ! E xc…but we need the dependence ! xc (r) =of Exc with our density." ! n(r)What is Exc?! E xc [ ! (r)] = !T [ ! (r)] + !Vee [ ! (r)] Correction to Non-classical kinetic energy corrections to from interacting electron-electron nature of electrons" repulsion"
- 25. Understanding our Exc!Typically model exchange and correlation separately. !!Exchange-correlation hole is often ﬁt in terms of model system. This isthe space whereby other electrons are excluded around a given electron.!"Local density approximation:"!Exchange modeled on HEG ! ! !Correlation on HEG!! 1/3! LDA 3" 3 % high limit! ! c = A ln(rs ) + B + rs (C ln(rs ) + D) E x [ ! ] = ! $ ( ! (r)4/3 dr! 4#" &! 1 ! g0 g1 $! low limit! ! c = # + 3/2 +... & 2 " rs rs % Correlated quantum chemical techniques are used to interpolate between these limits!
- 26. Many other ﬂavors of Exc!Generalized gradient approximations: Exc variants are oftenincluding gradient of the density! equated with rungs! on a ladder…"Meta-GGAs: Adding in Laplacian of thedensity.!!Hybrids: Add in H-F exchange (% isempirical).!!DFT+vDW: Add in semi-empirical treatment ofvan der Waals!!General approaches: ﬁt against analyticalforms in HEG vs. heavily parameterized for testset against experiment. !
- 27. DFT in practice!Errors in geometries can be quite small, considerablysmaller than Hartree-Fock.!!Errors in energetics vary widely. Can get errors as littleas a few kcal/mol (needed for mechanistic predictions).!!Key challenges: open shell character, multi-referencecharacter, lack of London dispersion forces, non-covalently bonded complexes, self-interaction errors:charge-transfer bound complexes, over-delocalization,etc…!!!
- 28. Follow-up reading!• Quantum mechanics! – C. Cohen-Tannoudji, B. Diu, and F. Laloe Quantum Mechanics: Vol 1 and 2. Wiley-VCH (1992).! – D. A. Mcquarrie and J. D. Simon Physical Chemistry: A molecular approach. University Science books (1997).! – D. J. Grifﬁths Introduction to quantum mechanics. Addison-Wesley (2004). !• Hartree-Fock theory and extensions:! – R. J. Bartlett “Many-body perturbation theory and coupled cluster theory for electron correlation in molecules” Ann. Rev. Phys. Chem. (1981).! – K. B. Lipkowitz, D. B. Boyd, R. J. Bartlett, J. F. Stanton “Applications of Post-Hartree-Fock Methods: A tutorial” Rev. Comput. Chem. (2007).!• Density functional theory:! – K. Burke “Perspective on density functional theory” J. Chem. Phys. (2012).! – K. Burke. The ABC of DFT http://chem.ps.uci.edu/~kieron/dft/book/! – P. Mori-Sanchez, A. J. Cohen, and W. Yang “Localization and delocalization errors in density functional theory and implications for band-gap prediction” Phys. Rev. Lett. (2008).!

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