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BIOS 203: Lecture 2 - introduction to electronic structure theory

The document serves as an introduction to electronic structure theory, discussing the necessity of quantum mechanics in understanding phenomena like optical and magnetic properties, and the role of potential energy surfaces. It outlines key concepts such as the Schrödinger equation, Hartree-Fock equations, and Density Functional Theory (DFT), highlighting the challenges in calculating electron correlations and approximating energy states. Additionally, it provides references for further reading on quantum mechanics and electronic structure methods.

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Introduction to electronic structure theory! Heather J Kulik! hkulik@stanford.edu! 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 ) from Schrödinger equation! $ 2m ' wikipedia! d What we usually are 2) 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!!
Hydrogen atom! 3 TISE ! 2 $ (n ' ! '1)! ' ! /2 ! 2!+1 Solution:" ! n!m (r, ! , " ) = # & e ! Ln'!'1 ( ! )(Y!m (! , " ) " na0 % 2n(n + !)! Laguerre Spherical polynomials! harmonics! Quantum numbers:" n = 1, 2, 3,… ! = 0,1, 2,…, n !1 m = !!,…, !.
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 " " ? ! > $! " & " B(
A note about notation! !"#$%&'()$*+,"#%)-'./0 1 " .# 0 ! .# 0 1 " ! &$ $ $ '7')$#5)6)-",'2+-3#%)-4 $ 98 % 1 " ! ' ' &$ 1 " !&' ' $ $ 98 % (:;'9<'=>><'''''?@?=>'A#)*%4#%3'!)B:,%-6')2'!"#:$%",4''CC D:$;$"-B'E:B:$'"-B'F%3),"'!"$G"$%
Atoms: Electrons and Nuclei! ˆ = T + V + V +V H 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 is than 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-particle orbitals:! ! ! ! ! ! ! ! ! ! (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 depends on others, can solve together iteratively to reach a solution.! ! Cons: Wave function does not obey Pauli exclusion principle, no explicit correlation.!
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).!
Hartree-Fock Equations! With a Slater determinant wavefunction to represent our many-body wavefunction:! ! ! ! ! (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 " n We 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+1 Pros: 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 basis of atomic-like wavefunctions.! Minimal: one basis function per A.O.! Double/triple/4/5/6 zeta: two/three/four/five/six basis functions per A. O.! Split valence: one A.O. for core, more for valence.! Polarization: From mixing l orbital with l+1! Diffuse functions: letting electrons move away from 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 to 6-311G: 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 particle in a box.! ! ! ! 9 n=3! 9 Pros: Can extrapolate to complete basis set limit, unlike challenges with localized basis set. Can n=2! describe delocalized electrons straightforwardly.! ! 4 4 Cons: Need many functions to describe strong oscillations, e.g. near core of nucleus. Often have 1 n=1! 1 to use pseudopotentials (effective core potentials).! 0 a
What about correlation?! What kind of correlation are we talking about?" A common definition:" Ecorr = Eexact ! EHF In 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 same spin 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 sparse Pros: Introduces dynamic correlation.! s sparse e ! Cons: Expensive calculation – many possible excitations, poor scaling. ! very extremely extremely 0 ! sparse sparse sparse In practice: Truncated excitations to maintain 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 coefficient static and dynamic space, combinatorial explosion of CSFs. Still correlation.! 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 ?! ρ Fundamental 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 " E0 Trial wavefunction from 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 external Interacting 1-to-1
 Non-interacting potential such that their mapping! density is the same as the particles! 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 is the 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 often including gradient of the density! equated with rungs ! on a ladder…" Meta-GGAs: Adding in Laplacian of the density.! ! Hybrids: Add in H-F exchange (% is empirical).! ! DFT+vDW: Add in semi-empirical treatment of van der Waals! ! General approaches: fit against analytical forms in HEG vs. heavily parameterized for test set against experiment. !
DFT in practice! Errors in geometries can be quite small, considerably smaller than Hartree-Fock.! ! Errors in energetics vary widely. Can get errors as little as a few kcal/mol (needed for mechanistic predictions).! ! Key challenges: open shell character, multi-reference character, 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).!

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BIOS 203: Lecture 2 - introduction to electronic structure theory

  • 1.
    Introduction to electronic structuretheory! Heather J Kulik! hkulik@stanford.edu! 02/27/13!
  • 2.
    Why we needquantum! 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 needquantum! Interesting phenomena depend on what the electrons are doing!! Optical properties! Catalysis! Magnetic 
 properties!
  • 4.
    Quantum mechanics inbrief! 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 ) from Schrödinger equation! $ 2m ' wikipedia! d What we usually are 2) 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! 3 TISE ! 2 $ (n ' ! '1)! ' ! /2 ! 2!+1 Solution:" ! n!m (r, ! , " ) = # & e ! Ln'!'1 ( ! )(Y!m (! , " ) " na0 % 2n(n + !)! Laguerre Spherical polynomials! harmonics! Quantum numbers:" n = 1, 2, 3,… ! = 0,1, 2,…, n !1 m = !!,…, !.
  • 7.
    A note aboutnotation! !"#$%&%'()*#+,-./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 " " ? ! > $! " & " B(
  • 8.
    A note aboutnotation! !"#$%&'()$*+,"#%)-'./0 1 " .# 0 ! .# 0 1 " ! &$ $ $ '7')$#5)6)-",'2+-3#%)-4 $ 98 % 1 " ! ' ' &$ 1 " !&' ' $ $ 98 % (:;'9<'=>><'''''?@?=>'A#)*%4#%3'!)B:,%-6')2'!"#:$%",4''CC D:$;$"-B'E:B:$'"-B'F%3),"'!"$G"$%
  • 9.
    Atoms: Electrons andNuclei! ˆ = T + V + V +V H 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 is than 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 weassume many-body wavefunction can be represented as single-particle orbitals:! ! ! ! ! ! ! ! ! ! (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 depends on others, can solve together iteratively to reach a solution.! ! Cons: Wave function does not obey Pauli exclusion principle, no explicit correlation.!
  • 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 aSlater determinant wavefunction to represent our many-body wavefunction:! ! ! ! ! (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 " n We 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+1 Pros: 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: atechnical note! Molecular wavefunction from the combination of some basis of atomic-like wavefunctions.! Minimal: one basis function per A.O.! Double/triple/4/5/6 zeta: two/three/four/five/six basis functions per A. O.! Split valence: one A.O. for core, more for valence.! Polarization: From mixing l orbital with l+1! Diffuse functions: letting electrons move away from 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 to 6-311G: 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 particle in a box.! ! ! ! 9 n=3! 9 Pros: Can extrapolate to complete basis set limit, unlike challenges with localized basis set. Can n=2! describe delocalized electrons straightforwardly.! ! 4 4 Cons: Need many functions to describe strong oscillations, e.g. near core of nucleus. Often have 1 n=1! 1 to use pseudopotentials (effective core potentials).! 0 a
  • 18.
    What about correlation?! Whatkind of correlation are we talking about?" A common definition:" Ecorr = Eexact ! EHF In 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 same spin 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 sparse Pros: Introduces dynamic correlation.! s sparse e ! Cons: Expensive calculation – many possible excitations, poor scaling. ! very extremely extremely 0 ! sparse sparse sparse In practice: Truncated excitations to maintain 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 coefficient static and dynamic space, combinatorial explosion of CSFs. Still correlation.! 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 ?! ρ Fundamental 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! Avariational principle for DFT:" ! ! ! ! ! Ev [ ! '(r )] = F[ ! '(r )]+ ! vext (r )! '(r )dr " E0 Trial wavefunction from 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 external Interacting 1-to-1
 Non-interacting potential such that their mapping! density is the same as the particles! 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! Typicallymodel exchange and correlation separately. ! ! Exchange-correlation hole is often fit in terms of model system. This is the 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 flavorsof Exc! Generalized gradient approximations: Exc variants are often including gradient of the density! equated with rungs ! on a ladder…" Meta-GGAs: Adding in Laplacian of the density.! ! Hybrids: Add in H-F exchange (% is empirical).! ! DFT+vDW: Add in semi-empirical treatment of van der Waals! ! General approaches: fit against analytical forms in HEG vs. heavily parameterized for test set against experiment. !
  • 27.
    DFT in practice! Errorsin geometries can be quite small, considerably smaller than Hartree-Fock.! ! Errors in energetics vary widely. Can get errors as little as a few kcal/mol (needed for mechanistic predictions).! ! Key challenges: open shell character, multi-reference character, lack of London dispersion forces, non- covalently bonded complexes, self-interaction errors: charge-transfer bound complexes, over-delocalization, etc…! ! !
  • 28.
    Follow-up reading! •  Quantummechanics! –  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).!