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Martin Head-Gordon Chemical Accuracy from Electronic Structure

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Martin Head-Gordon Chemical Accuracy from Electronic Structure Martin Head-Gordon This presentation will probably Outline involve audience Modern Electronic Structure Methods: discussion, which will create action items. Use PowerPo1in. t tOo kveeerp vtriaecwk o of f standard wavefunction-based methods 2. Wavefunction-based theories. these action items during your presentation• Mean field, perturbation theory, coupled cluster theory • In Slide Show, click on the right mo2us. e Fbuatstotn methods for dynamic correlations MARTIN HEAD-GORDON, • Select “Meeting Minder” • Select the “Action Items” tab Department of Chemistry, University of California, and • Type in action items as they come up3. Fast methods for static correlations (with applications) Chemical Sciences Division, • Click OK to dismiss this box Lawrence Berkeley National Laboratory This will automatically create Berkeley CA 94720, USA an Action Item slide at the end of your presentation with your points entered. Complexity Fundamental approximations • Many-body Schrödinger equation is a partial differential • (1) The one-body problem (“atomic orbital basis”) equation in 3n unknowns– the positions of the electrons. • One-electron functions are expanded in a finite basis. • Exact (brute force) solution will scale approximately • We use atom-centered functions with some maximum exponentially with the number of electrons. angular momentum (L) on atoms of a given period. • Standardized basis sets of increasing L are available. • When done in a given basis of 1-electron expansion • (2) The n-body problem (“electron correlation method”) functions, this is “full configuration interaction” (FCI). • More about this in a minute… mean field is the simplest • Largest FCI’s involve many-body expansions containing possible treatment. billions of terms, for molecules with 2 or 3 of atoms (!). • Well-defined electronic structure models are completely • Approximations are imperative. Accuracy vs feasibility. specified by these two approximations. Branches of the family tree Wave function approaches to electronic structure • Wavefunction-based electronic structure theory: • Hartree-Fock (MO) theory (mean field): A3-A4 cost. • Minimize the energy by varying the wavefunction • HF: 99%! “Correlation energy” is roughly 1 eV per pair. • Tremendously complicated unknown function: • Perturbative treatment of the electron correlations: A5 • Modeling the wavefunction yields “model chemistries” • MP2: 80% of the last 1%. “Pair correlations” (doubles). • Density functional theory • Treat single & double substitutions self-consistently: A6 • The unknown is very simple: • CCSD: 95% of the last 1%. • Hohenberg-Kohn theorem guarantees that: • True functional is unknown and probably unknowable • Correct for the triple substitutions perturbatively: A7 • Modeling the functional gives DFT model chemistries. • CCSD(T): > 99% of the last 1%. Chemical accuracy. Chemical Accuracy from Electronic Structure Calculations 1 Martin Head-Gordon State of the art thermochemistry Mean field (Hartree-Fock) theory • Up to 4 first row atoms: ± 0.3 kcal/mol • Each electron moves in the average field of all others • Martin, Taylor, Dunning, Helgaker, Klopper… – Wavefunction is an antisymmetrized product of “orbitals” • include relativistic effects, core correlation, anharmonicity • CCSD(T) with extrapolation to the complete basis set limit • limited by (?): extrapolation, sometimes CCSD(T) itself. – The energy is minimized with respect to variations of the orbitals, giving the Hartree-Fock equations: • Up to 10 or 15 first row atoms: ± 1 kcal/mol • Curtiss, Raghavachari, Pople: G2/G3 methods – F is the Fock operator, which is a one-particle Hamiltonian • “CBS” methods of Petersson et al with electron-repulsions averaged: • based on methods like CCSD(T), with either additivity corrections for basis set, or extrapolation. Chemistry with the Hartree-Fock model Dodging the correlation bullet • Size-consistent; exact for isolated electrons • Look at energy changes where correlation errors cancel. • Absolute energies: about 99% of exact • The remaining energy is the “correlation energy” • Energy change for isogyric reactions: a bit better – Isogyric reactions conserve the total number of electron pairs • Atomization energies: a disaster… – Example: H2 + Cl ⎯→ HCl + H • The correlation energy of all separated electrons is the error • Thus roughly 1eV error (ie. 100 kJ mol) per separated pair • Energy change for isodesmic reactions: much better! • Optimized molecular geometries: are reasonable… – Isodesmic reactions conserve numbers of each type of electron • Bondlengths are systematically slightly too short pair (different bond pairs, lone pairs, etc) • Vibrational frequencies are systematically about 10% high – Example: H2C=CH−CH3 + CH4 ⎯→ H2C=CH2 + C2H6 Schematic view of the Hamiltonian matrix Double substitutions describe pair correlations • Group determinants by substitution level from HF • Correlated fluctuations of 2 electrons – 0, s(ingles), d(oubles), t(riples), etc… • In general, they are quartic in number. • Look at the coupling of HF (0) to higher substitutions – HF orbitals are optimized to zero coupling to singles – Empty levels – HF wavefunction cannot couple to triples (or higher) • Double substitutions are the leading correction to HF – Occupied levels • Correlations decay rapidly with separation. – Exponentially between functions of the same electron – Algebraically (R-3) between the 2 electrons (→dispersion) Chemical Accuracy from Electronic Structure Calculations 2 Martin Head-Gordon Truncated configuration interaction Møller-Plesset perturbation theory • The exact (full CI) wavefunction is a linear superposition • Apply perturbation theory to electronic structure, with: Cˆ Cˆ Cˆ Cˆ !exact = ( 1 + 2 + 3 +…+ n )" HF – The zero order Hamiltonian is the Fock operator: – As discussed before, this is exponentially expensive! • Truncated configuration for electron correlation? – Since the canonical MO’s are eigenvectors of f: – For instance include all single + double substitutions (CISD) • Not useful because it fails to be size-consistent – Then the zero order problem is: – CISD is exact for 1 H2 molecule (only 2 electrons) – But is not exact for 2 H2’s (that would require CISDTQ) 1st and 2nd order expansions MP2 correlation energy expression • Expansion to first order gives the Hartree-Fock energy • The 2nd order expression in terms of doubles (d) is: • Or in terms of the occupied (i,j) and empty (a,b) orbitals: • Leading estimate of correlation energy is from 2nd order: – MP2 correlation energy is negative definite – Recall from the Hamiltonian that the coupling is just doubles – Computational complexity is 5th order in molecule size. MP2 as a theoretical model chemistry Higher order corrections: MP3 and MP4 • MP2 is size-consistent. Proper treatment of dispersion. • MP3 energy introduces coupling between doubles: • For closed shell molecules, accuracy is much better than HF – Bond lengths are (properly) longer, frequencies lower. – MP3 is A6 in computational complexity – Directly reflects role of antibonding orbitals • MP4 includes singles, doubles, triples & quadruples: • For open shell molecules, accuracy is erratic. – MP2 is sensitive to spin-contamination in UHF. • Efficient computational algorithms are available: – Cubic memory (OVN), (larger) cubic disk space – The MP4 triples are A7 to compute – Cost is asymptotically 5th order, but often lower. – The triples energy contribution is “chemically significant” – MP3 and MP4 energies are erratic in accuracy Chemical Accuracy from Electronic Structure Calculations 3 Martin Head-Gordon Coupled cluster theory Finding the coupled cluster amplitudes • The exact wavefunction may be written as the • Variational CC energy is intractable (can you see why?) exponential of the sum of substitution operators E exp Tˆ Hˆ exp Tˆ exp Tˆ exp Tˆ VCC = ( )! ( )! ( )! ( )! exp Tˆ Tˆ Tˆ Tˆ !exact = ( 1 + 2 + 3 +…+ n ) " • Instead we define CC equations by subspace projection: – Project with reference (0), and all substitutions in T, to give • Truncation of T defines limited coupled cluster models. enough equations to determine the energy, and T amplitudes. • Simplest models are CCD (doubles) and CCSD ! Hˆ " E exp Tˆ ! = 0 ! = exp Tˆ + Tˆ " p ( ) ( ) CCSD ( 1 2 ) • Truncated CC models are clearly size-consistent: e.g. • This gives tractable equations, but the resulting energy does not obey a variational bound. – The quality of the results will be best for cases when the single reference is the dominant part of the wavefunction Energy & amplitude equations for CCSD CCSD as a theoretical model chemistry • Project with the reference (0) to determine the energy • Exact for a pair of electrons ! (Hˆ " E)exp(Tˆ )! = 0 • Size-consistent • Analysis in terms of the Moller-Plesset series: ! E = " Hˆ exp Tˆ " = " Hˆ 1+ Tˆ + Tˆ + 1 Tˆ 2 " – Complete in second order ( ) ( 1 2 2 1 ) – Complete in third order • Project with singles (s) & doubles (d) for amplitudes: – Missing the triples in fourth order – Contains some fifth and higher order terms ˆ ˆ ˆ 1 ˆ 2 ˆ ˆ 1 ˆ 3 !s H 1+ T1 + T2 + 2 T1 + T1T2 + 3! T1 ! = 0 • A significant improvement over MP2 for problem cases (radicals!) ( ) C – Not much improvement for closed shell molecules ˆ ˆ ˆ 1 ˆ 2 ˆ ˆ 1 ˆ 3 1 ˆ 2 1 ˆ 2 ˆ 1 ˆ 4 !d H 1+ T1 + T2 + T1 + T1T2 + T1 + T2 + T1 T2 + T1 ! = 0 – Occasional non-variational pathologies (E.g. stretched bonds) ( 2 3! 2 2 4! ) C • Not chemically accurate for relative energies – Non-linear equations with A6 computational complexity. – Reflects the fact that the triples are
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