Lecture 2 of BIOS 203 mini-course taught by Heather Kulik at Stanford University. Introduction to electronic structure theory. http://bios203.stanford.edu or email email@example.com for more information.
BIOS 203: Lecture 2 - introduction to electronic structure theory
Introduction to electronicstructure theory! Heather J Kulik! firstname.lastname@example.org! 02/27/13!
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! !"
Why we need quantum!Interesting phenomena depend on what the electrons are doing!!Optical properties! Catalysis! Magnetic properties!
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!
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!!
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"
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.!
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.!
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$i rj ! ri + *Pros: Self-consistent equations – each equation for single particle dependson others, can solve together iteratively to reach a solution.!!Cons: Wave function does not obey Pauli exclusion principle, no explicitcorrelation.!
Hartree-Fock Equations!With a Slater determinant wavefunction to represent our many-bodywavefunction:! ! ! ! ! (r ) ! (r ) " ! (r )! ! ! ! ! ! " ! 1 ! (r ) ! (r ) " ! (r ) 1 " 1 " 1 " 2 " 2 " 2! ! (r , r ,…, r ) = n! 1 " 2 n " # " ! ! ! ! (r ) ! (r ) " ! (r )! " n " n " nWe may now obtain analogous Hartree-Fock equations:! Standard notation! Bra-ket notation!$ 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.!
Self-consistency! Initial guess Form Diagonalize MOs from operators/ Fock matrix! XYZ! Fock matrix! Are we no! Done! yes! converged?!
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.!
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
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.!
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…!
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 %
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
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!!
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
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"
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!
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. !
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…!!!
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).!