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ELECTRONIC STRUCTURE THEORY

ELECTRONIC STRUCTURE THEORY. Navigating Chemical Compound Space for Materials and Bio Design: Tutorials K. N. Houk Department of Chemistry and Biochemistry UCLA March 16, 2011. Generalities and history Wavefunction electronic structure theory Benchmarking, accuracies

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ELECTRONIC STRUCTURE THEORY

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  1. ELECTRONIC STRUCTURE THEORY Navigating Chemical Compound Space for Materials and Bio Design: Tutorials K. N. Houk Department of Chemistry and Biochemistry UCLA March 16, 2011

  2. Generalities and history Wavefunction electronic structure theory Benchmarking, accuracies General programs for quantum mechanics calculations Some applications from our group Navigating Chemical Compound Space for Materials and Bio Design: Tutorials Electronic Structure Theory Thanks to six great postdocs in my group: Peng Liu Gonzalo Jimenez Silvia Osuna NihanCelebi Steven Wheeler Arik Cohen

  3. Born–Oppenheimer Orbital Approximation Quantum Mechanics Reproduce and PredictChemistry? WFT DFT Thomas–Fermi–Dirac Relativistic Effects (Dirac) Heisenberg–Schrödinger Hohenberg–Kohn SchrödingerEq. Møller–Plesset: MP2, MP3, ... CI, MCSCF, GVB, CCT ? Post-HF Methods KS Methods Non LDA Kohn–Sham Local DensityApproximation(LDA) Hartree–Fock–Slater Hartree–Fock–Slater Hartree–Fock CompleteBasis Set BLYP BP86 BPW91 LCAO LCAO Roothaan–Hall Ab initio KS-LDA Methods LSDA, Xa Approximate Hamiltonian Generalized GradientApproximation (GGA) SVWN Parametrization Semiempirical HybridMethods HMO, PPP EH, CNDO, INDO MNDO, AM1, PM3, PM6 B3LYP B3P86 B3PW91 Half & Half

  4. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. Paul A. M. Dirac Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 123, No. 792 (1929) It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. The Nobel Prize in Physics 1933 Erwin Schrödinger, Paul A.M. Dirac 65 years later…..

  5. John Pople Walter Kohn The Nobel Prize in Chemistry 1998 The Nobel Prize in Chemistry 1998 was divided equally between Walter Kohn "for his development of the density-functional theory" and John A. Pople "for his development of computational methods in quantum chemistry".

  6. Gaussian, Inc. (since 1987)

  7. Introduction to ab initio Molecular Orbital Theory Born-Oppenheimer Approximation Electronic Schrödinger Equation Kineticenergy Coulomb attraction (nuclei-electrons) Electronicrepulsion Ab Initio Molecular Orbital theory consists of a family of methods to solve approximatelythe Electronic Schrödinger Equation without parameterization

  8. = 1 (normalization) Dirac “bra-ket” notation for integrals The Schrödinger equation can be solved analytically (‘exactly’) only for the simplest systems (H, He+). Variational Principle: E'(F’') Approximateenergies, trialwavefunctions E'' (F'') E'''(F''') Exactenergy, realwavefunction Eelec (Yelec)

  9. Hartree-Fock Theory Assume Ye as a single antisymmetric product of one-electron functions (molecular orbitals) For a general N-electron system, we can write this antisymmetric product as a Slater Determinant

  10. Linear Combination of Atomic Orbitals Expansion of orbitals in terms of some basis functions centered on the nuclei: Unoccupied (virt) c coefficients b a basisfunctions k j Linear Combination of Atomic Orbitals (LCAO) i Occupied (occ)

  11. Hartree-Fock equations (eigenvalue equations) for each molecular orbital: molecular orbital orbital energy Fock operator Exchange operator Coulomb operator

  12. Substituting this expansion in the Schrödinger equation solution: Minimum Roothaan-Hall equations solved by an iterative numerical method: self-consistent field (SCF) ci and E are unknown Solution yields N “occupied” orbitals and (M – N) “unoccupied” orbitals

  13. The SCF Procedure Adapted from Cramer, C. J., Essentials of Computational Chemistry, Theories and Models. Second ed.; Wiley: 2004.

  14. The SCF Procedure and Geometry Optimization Density matrix D describes how much each basis function contributes to elec. Adapted from Cramer, C. J., Essentials of Computational Chemistry, Theories and Models. Second ed.; Wiley: 2004.

  15. The Hartree-Fock approximation can be applied with or without restrictions on the spins of the MOs.  Restricted (RHF) Closed shell a/b E RHF singlet

  16. The Hartree-Fock approximation can be applied with or without restrictions on the spins of the MOs.  Restricted (RHF, ROHF)and unrestricted (UHF) solutions: Closed shell Open shell a/b a/b a b a b E RHF singlet ROHF doublet UHF doublet UHF singlet

  17. What molecular properties can be calculated? R: Nuclear positions F: Externalelectricfield B: Externalmagneticfield I: Internalmagneticfield Harmonic vibrational frequencies (IR) Electric polarizability Magneticsusceptibility Energygradient (Forces) Spin-spin coupling (J) Electric dipolemoment IR absorptionintensities Magneticdipolemoment Nuclear magneticshielding (d) Hyperfinecouplingconstants (EPR) Circular dichroism … and many others

  18. PotentialEnergy Surface (PES) E r1 r2 Nuclear coordinates

  19. Basis Sets STO Slater type orbitals (STOs) Gaussian type orbitals (GTOs) The analyticalform of the two-electronintegrals is computationallyexpensive. The quadratic dependence on rmakes the analytical form of the two-electron integrals quite easy. Linear combination of GTOs

  20. Classification of Basis Sets Every occupied atomic orbital is represented using a single basis function, which corresponds to the smallest set that one could consider. Minimal Basis Sets First row elements: twos-functions (1s and 2s) and one set of p-functions (2px, 2py, 2pz) A better representation can be obtained combining 2 GTOs in a different proportion to represent every atomic orbital. Double Zeta (DZ) First row elements:four s-functions (1s, 1s’, 2s, 2s’) and two sets of p-functions(2px, 2py, 2pz and 2px’, 2py’, 2pz’) Triple Zeta (DZ) Quadruple Zeta (DZ) Examples of Basis sets split valence STO-3G, 3-21G, 6-31G, 6-311G, cc-pVDZ, cc-pVTZ, … Calculations are usually simplified applying a DZ only for the valence-orbitals, and a single GTO is used to represent the inner-shell orbitals. Coulomb repulsion effects Pauli principle Relativistic effects Valence electrons EXPLICITLY Core electrons POTENTIAL Effective Core Potential (ECP)

  21. Semi-empirical Methods: Overview 1989 INDO/S ZINDO/S Ridley, Zerner MINDO/3 Bingham, Dewar, Lo 1953 1980 1930 1973 2000 2006 1993 SAM1 Dewar, Jie, Yu 1965 RM1 Rocha Stewart PM3 Stewart Methods restricted to π-electrons: AM1/d Voityuk Rosch CNDO Pople HMO Hückel PPP Pople SINDO1 Nanda, Jug 1996 2002 1963 2007 1977 1985 MNDO Dewar, Thiel MNDO/d Thiel, Voityuk EHT Hoffmann PM6 Stewart AM1 Dewar,Stewart PDDG/PM3 PDDG/MNDO Jorgensen Methods restricted to all valence electrons: CNDO (Complete Neglect of Differential Overlap) INDO (Intermediate Neglect of Differential Overlap) NDDO (Neglect of Diatomic Differential Overlap)

  22. Semi-empirical Methods: Overview Methods restricted to π-electrons Methods restricted to all valence electrons: 1930 1980 1965 1963 1977 2007 CNDO (Complete Neglect of Differential Overlap) INDO (Intermediate Neglect of Differential Overlap) NDDO (Neglect of Diatomic Differential Overlap) NDDO INDO Overlap matrix: UNIT matrix + + All 2-center 2e- integrals (not Coulomb) NEGLECTED 1e- integrals involving 3 centers = ZERO All integrals involving different atomic orbitals IGNORED 3- and 4-center 2e- integrals NEGLECTED ELECTRON CORRELATION EFFECTS INCLUDED Use of empirical parameters Remaining integrals: PARAMETERIZED

  23. Semi-empirical Methods: Benchmarks Repasky, M. P.; Chandrasekhar, J.; Jorgensen, W. L. J. Comp. Chem.2002, 23, 1601.

  24. Semi-empirical Methods: Benchmarks Repasky, M. P.; Chandrasekhar, J.; Jorgensen, W. L. J. Comp. Chem.2002, 23, 1601.

  25. Roothaan–Hall Hartree–Fock SchrödingerEq. Electron Correlation monodeterminantalapproximation treatedthe averageCoulombicinteraction of the electrons Ab initio neglected instantaneouselectron-electron interactions (electron correlation) HF limit Correlation Energy (not a physicalentity) Exactsolution overestimated energy Limitations of HF Theory

  26. The Correlation Energy Hartree-Fock calculations recover ~99% of total energy Why is the correlation energy so important? E H · + H· H–H HF underestimated binding energy “exact”at HF level HF energy Exact (correlated) energy • Due to the absence of correlation energy, HF calculations usually lead to: • - too large stretching bond energies   too large activation energies for bond formation reactions.  • too short bonds • too large vibrational frequencies • wavefunctions with a too ionic character.

  27. Electron Correlation Methods • To go beyond HF, • must include electron-electron interaction explicitly (Electron Correlation) • must also move beyond the single-determinant picture Electron Correlation Methods Configuration Interaction (CI) Coupled Cluster (CC) Many Body Perturbation Theory (MBPT) CISD CISD(T) CISDT CISDTQ …… CCSD CCSD(T) CCSDT QCISD QCIST(T) …… MP2 MP3 MP4 ……

  28. Configuration Interaction (CI) CI: wavefunction expansion of Slater determinants in which electrons are “excited” to unoccupied orbitals. c b a k c c Unoccupied (vir) j b b i a a … HF: S-type: D-type: k k Occupied (occ) j j Full CI: include all possible Slater determinants i i

  29. Solving CI Secular Equations Solving the set of CI secular equations == diagonalizing the CI matrix Hij is evaluated by expanding it in a sum product of MO’s MO’s are expanded in AO’s

  30. Combinatorial Issues with CI Calculations Number of electronic configurations grows factorially with the basis-set size a Number of singlet configurations for H2O with 6-31G(d) basis set (19 basis functions) b R. J. Harrison and N. C. Handy, Chem. Phys. Lett.95, 386 (1983). c Ne = number of electrons

  31. Truncated CI Methods In Practice, we truncate the N-particle expansion: a Number of singlet configurations for H2O with 6-31G(d) basis set (19 basis functions) b R. J. Harrison and N. C. Handy, Chem. Phys. Lett.95, 386 (1983). c Ne = number of electrons

  32. Coupled Cluster (CC) Theory Alternatively, the CI wavefunction can be described as The excitation operator having I excitations from the reference generate all possible determinants Truncated Coupled Cluster theory: CCSD: CCSD(T): CCSD with perturbative triples corrections CCSD(T) with large basis-set is the “gold standard” for a single ground state calculation.

  33. CCSD(T) Procedure

  34. Perturbation Theory basic idea: Treat correlation into a series of corrections to an unperturbed starting point • Start with a system with known Hamiltonian, , eigenvalues, , and eigenfunctions, . • Calculate the changes in these eigenvalues and eigenfunctions that result from a small change, or perturbation, in the Hamiltonian for the system. perturbation total Hamiltonian unperturbed system

  35. Møller-Plesset Perturbation Theory Second order Møller-Plesset Perturbation Theory (MP2) Third order Møller-PlessetPerturbation Theory (MP3) additionally includes the third order correction to the energy. Nth order Møller-Plesset Perturbation Theory is called MPn. Calculations up to MP4 are common.

  36. HF MP3 E MP4 MP2 Møller-Plesset Perturbation Theory Advantages of MP methods: • MP2 captures ~ 90% of electron correlation Disadvantages of MP methods: • MP methods are not variational SCS-MP2

  37. Extrapolation Methods Exact Solution to the Schrödinger Equation complete basis set … Basis Set Increase Accuracy TZ2P DZP DZ HF/minimal BS SZ HF MP2 QCISD CCSD CCSD(T) … Full CI Electron Correlation

  38. A Simple Example of Extrapolation Method complete basis set  Ecorr HFlarge BS CCSD(T) large BS … Basis Set TZ2P DZP EBS  EBS DZ Ecorr HFsmall BS CCSD(T)small BS SZ HF MP2 QCISD CCSD CCSD(T) … Full CI Electron Correlation

  39. Extrapolation Methods • Common Extrapolation Methods: • Gaussian-n (G2, G2(MP2), G3, etc.) • Complete basis set (CBS-Q, CBS-QB3, CBS-RAD, etc.) • Weizmann-n (W1, W2, etc.) • HEAT (thermochemistry calculations) • Focal point methods

  40. G2 Theory • Geometry optimized at the HF and MP2/6-31G(d) level. MP4 with a relatively small basis set zero-point vibrational energy at the HF/6-31G(d) level basis set corrections to the 6-311+G(3df,2p) basis set higher-level correction correlation energy corrections to the QCISD(T) level L. A. Curtiss, K. Raghavachari, G. W. Trucks, and J. A. Pople, J. Chem. Phys., 94 (1991) 7221-30.

  41. Performance of G2 and DFT for Enthalpies of Formation MAE / kcal mol-1 Test Set: G2/97(148 Hf) Curtiss, L. A. et al., J. Chem. Phys. 1997, 106, 1063.

  42. G3 and G4 Theories • G3 • new basis sets for single point energies • spin–orbit correction and correction for core correlation • MAD for G2/97 set: 1.01 kcal/mol • requires less computational time than G2 • G3(MP2) • use MP2 instead of MP4 in single point energy calculations • increase MAD to 1.30 kcal/mol • G3B3: G3 using B3LYP geometries • MAD for G2/97 set: 0.99 kcal/mol • G4 • for molecules with 1st, 2nd, and 3rd row main group atoms • use CCSD(T) instead of QCISD(T) for correlation corrections • B3LYP geometries • Larger basis sets for single point energy calculations G3: a) Curtiss, L. A. et al., J. Chem. Phys.1998, 109, 7764. b) Curtiss, L. A. et al., J. Chem. Phys.1999, 110, 4703. c) Baboul, A. G. et al., J. Chem. Phys. 1999, 110, 7650. G4: Curtiss, L. A. et al., J. Chem. Phys.2007, 126, 084108.

  43. Performance of G3, G4, versus DFT MAE / kcal mol-1 Test Set: G3/05(454 energies) a) Curtiss, L. A. et al., J. Chem. Phys.2005, 123, 124107. b) Curtiss, L. A. et al., J. Chem. Phys.2007, 126, 084108.

  44. Complete Basis Set (CBS) Methods • J. W. Ochterski, G. A. Petersson, and J. A. Montgomery Jr., J. Chem. Phys., 104 (1996) 2598. • J. A. Montgomery Jr., M. J. Frisch, J. W. Ochterski, and G. A. Petersson, J. Chem. Phys., 110 (1999) 2822.

  45. Weizmann-n Theory: W1, W2, W3, W4 • Compute energies of small molecules to within 1 kJ/mol (0.3 kcal/mol) accuracy. • More accurate and computationally demanding than G2, G3, and CBS-QB3. Martin, J. M. L.; de Oliveira, G., J. Chem. Phys.1999, 111, 1843.

  46. HEAT: High accuracy extrapolated ab initio thermochemistry Tajti, A.; Szalay, P. G.; Csaszar, A. G.; Kallay, M.; Gauss, J.; Valeev, E. F.; Flowers, B. A.; Vazquez, J.; Stanton, J. F., J. Chem. Phys. 2004, 121, 11599-11613.

  47. Mean Absolute Error with the G2/97 Data Set Exp. Uncertainty W2 W1 CBS-Q G3 G2 CCSD(T)/aug-cc-pV5Z G2(MP2) CBS-4 CCSD(T)/aug-cc-pVQZ B3LYP/6-311+G(3df,2df,2p)//B3LYP/6-31G(d) B3LYP/6-31+G(d,p)//B3LYP/6-31G(d) B3LYP/6-31G(d)//B3LYP/6-31G(d) MP2/6-311+G(2d,p)//MP2/6-311+G(2d,p) MP2/6-311+G(2d,p)//HF/6-31G(d) PM3 SWVN5/6-311+G(2d,p)//SWVN5/6-311+G(2d,p) AM1 HF/6-311+G(2d,p)//HF/6-31G(d) HF/6-31G(d)//HF/6-31G(d) MAE / kcal mol-1 a) Curtiss, L. A. et al., J. Chem. Phys. 1997, 106, 1063. b) Curtiss, L. A. et al., J. Chem. Phys.1998, 109, 7764. c) Martin, J. M. L. et al., J. Chem. Phys.2001, 114, 6014.

  48. TIMING ISSUES

  49. Timings of QM Calculations Single-pointenergycalculation HF/6-31G(d,p) Gaussian Mainframes Supercomputers Workstations PC 7 10 200 years 6 10 1 week 5 10 1 day 1 hour 4 10 1000 100 1 minute < 30 seconds 10 1965 1970 1975 1980 1985 1990 1995 2000 2005-present

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