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Ab Initio Molecular Orbital Theory

Ab Initio Molecular Orbital Theory. Ab Initio Theory. Means “from first principles;” this implies that no (few) assumptions are made, and that the method is ‘pure’ from a theoretical standpoint, but that is not really true.

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Ab Initio Molecular Orbital Theory

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  1. Ab Initio Molecular Orbital Theory

  2. Ab Initio Theory • Means “from first principles;” this implies that no (few) assumptions are made, and that the method is ‘pure’ from a theoretical standpoint, but that is not really true. • Even ab initio is based on approximate solutions to the Schrödinger equation: e Y = H Y • This can be solved exactly only for H atom!

  3. Ab Initio Theory... • Generally one of two types: • Hartree-Fock • considers each electron to experience effects of all of the other electrons combined • A Correlated method (several) • considers individual electron interactions (to some extent) • CPU time scales as n3 to n4 where n = # basis functions (orbitals)

  4. Hartree-Fock Method • Utilizes three approximations to allow “solution” of many-e- Schrödinger equation • Born-Oppenheimer approximation • electrons act independently of nuclei • Hartree-Fock approximation • electrons experience the ‘field’ of all other electrons as a group, not individually • LCAO • Molecular orbitals can be constructed as linear combinations of atom-centered orbitals

  5. Schrödinger equation: kinetic energy (nuc.) kinetic energy (elect.) 2 kinetic energy terms plus 3 Coulombic energy terms: (one attractive, 2 repulsive)

  6. Schrödinger equation after Born-Oppenheimer Approximation 0 kinetic energy (nuc.) kinetic energy (elect.) 1 kinetic energy term plus 2 Coulombic energy terms: (one attractive, 1 repulsive) plus a constant for nuclei constant

  7. Basis Sets • Combinations of mathematical functions used to represent atomic orbitals • Minimal • H: 1s C, N, O: 1s, 2s, 2px, 2py, 2pz (all three needed to maintain spherical symmetry) • Slater type orbitals (STO) • too difficult to solve analytically when combined • Gaussian type orbitals (GTO) • simpler to manipulate mathematically; combinations of Gaussian (exp) functions can approximate STO’s

  8. Gaussian Type Orbitals • STO-3G • Slater type orbitals approximated by three primitive Gaussian functions • Used as default basis set in semi-empirical MO calculations (AM1, PM3) • Better approximations using combinations of Gaussian functions have been developed and are generally employed in ab initio work

  9. Split Basis Sets • Minimal (small) basis sets such as STO-3G do not adequately describe non-spherical (anisotropic) electron distribution in molecules • ‘Split’ valence basis sets (3-21G; 6-31G, etc.) were developed to overcome this problem • Each split valence atomic orbital is composed of a variable proportion of two (or more) functions of different size or radial extent

  10. Split Basis Sets... (a and b are normalized coefficients; their sum is 1)

  11. Split Basis Sets... • 3-21G • commonly used simple split basis set; OK for HF geometry calculations on 1st row elements, not good for heavier elements or for accurate energies • 3 primitive Gaussian functions for inner core (subvalence) electrons • 2 Gaussians for contracted (small) valence orbitals • 1 Gaussian for extended (large) valence orbitals

  12. More Split Basis Sets...with modifications • 6-31G, 6-311G (the latter has two different sizes of extended Gaussian functions for valence orbitals) • Polarization functions • 6-31G(d), or 6-31G(d,p) [formerly 6-31G* (or **)] • (adds ‘d’ function to ‘heavy’ atoms, ‘p’ function to H, He) (a and b are coefficients whose sum is 1)

  13. More Split Basis Sets…and still more modifications • Diffuse functions • 6-31+G • adds an additional, larger p function to heavy (non-hydrogen or helium) atoms; indicated by + before G • 6-31++G • adds an additional, larger p function to heavy (non-hydrogen or helium) atoms and an additional larger s function to light elements (hydrogen and helium) • Diffuse functions are useful in describing anions, molecules with lone pairs of e-, excited states, TS.

  14. Basis Sets: Common Combinations • 6-31G(d) Common ‘moderate’ basis set • 6-31G(d,p) • 6-31+G(d,p) Good compromise • 6-311++G(d,p) • Many other basis sets are in use, and basis sets can be modified/customized/optimized easily.

  15. Optimization of a Basis Set • To optimize basis set parameters, one relies on the principle that HF theory is variational; that is, it converges toward the “true” value of the energy and will never go below that value. • The constants and exponents that describe the Gaussian functions are varied sequentially until the lowest energy is obtained. • Such a basis set may only apply to that individual molecule, however.

  16. Optimization of a Basis Set... Exponent a Coefficient d g(a, r) = c (xnymzl) e-ar 2 ... s 1 1.00 0.052 1.0 p 3 1.00 1.034919 -0.340155 0.5738330.441884 0.2030630.604344 p 1 1.00 0.0790990.288949 p 1 1.00 0.0291.0 d 1 1.00 0.2411.0 cp = Sdpgp A series of energy calculations is performed using different values of one of the parameters in the basis set; the value is then set for that parameter based on what value gives the lowest energy.

  17. Descriptions of Some Basis Sets • STO-3G A minimal basis set. The fastest, but the least accurate basis set in common use. • 3-21G(*) A simple basis set with added flexibility, and polarization functions on atoms heavier than Ne. This is the simplest basis set that gives reasonable results. • 6-31G* A significant improvement on 3-21G(*), 6-31G* adds polarization to all atoms, and improves the modeling of core electrons. 6-31G* is often considered the best compromise of speed and accuracy, and is the most commonly used basis set. • 6-31G** Adds polarization functions to hydrogens. This can improve the total energy of the system. • 6-31+G* Adds extra diffuse functions to heavy atoms. This can sometimes improve results for systems with large anions. • 6-311G* Adds more flexibility to the basis set. • 6-311G** Adds polarization functions to hydrogens of the 6-311G* basis set. • 6-311++G** Add diffuse functions to 6-311G**.

  18. Descriptions of Some Basis Sets... • 6-311++G(2df,2pd) Improves the polarization of 6-311++G**. • cc-pCVTZ Similar to 6-311++G(2df,2pd) but with a more descriptive core (7s) and different S/P splitting; (7-711S, 311P). The letters in the name stand for 'correlation consistent polarized Core and Valence Triple Zeta' and has been designed specifically for post HF methods. • G3Large An extension of 6-311G** with more flexible polarization functions (2df) and polarization of the core electrons. (3d2f on Na-Ar). This basis set is used as the 'limiting HF' basis set in the G3 method. • cc-pCVQZ A systematic extension of cc-pCVTZ with more flexible valence orbital (8-8111S, 3111P), more polarization functions (3d2fg) (= Quadruple Zeta) and a more accurate description of the core (8s). • aug-cc-pCVQZ Augments cc-pCVQZ with some diffuse (aug: spdf) and some core polarization functions (C: 3s3p2d1f, 9s).

  19. Effect of Basis Set Choice on Computation Cost (cpu time) axial-methylcyclohexane on SGI Indigo2 (Spartan cpu time in sec.) Method/Basis Sets.p.opt. AM1/STO-3G ~1 10 HF/STO-3G 72 983 HF/ 3-21G(d) 193 2214 HF/ 6-31G(d,p) 2632 34655 (9.6 h) (approaching “HF limit”; energy [not shown] decreases w/ larger basis set)

  20. Effect of Basis Set Choice on Computation Cost (cpu time)

  21. Effect of Basis Set Size on Energy and CPU Time

  22. Correlated Methods • The limitation of HF theory is the HF approximation itself...the independence of electrons, which is obviously not true! • To better approach the Schrödinger equation, electron correlation is incorporated by one of two methods: • Configuration Interaction • Moller-Plesset Perturbation Theory

  23. Correlated Methods • Include more explicit interaction of electrons • Most begin with HF wavefunction, then incorporate varying amounts of electron-electron interaction by mixing in excited state determinants with ground state determinants • The limit of infinite basis setand complete electron correlation is the Schrödinger equation itself!

  24. Configuration Interaction • Post Hartree-Fock, CI adds to the ground state wavefunction single, double, or triple (or more) substitutions (excitations). Higher level CI corrections invoke an increasing number of virtual (unoccupied) orbitals • These additional orbitals require considerable additional cpu time, however • Designation: QCIS(D,T) [singles, doubles, triples]

  25. Moller-Plesset Perturbation Theory • To the HF wavefunction is added a correction corresponding to exciting 2 electrons to higher energy HF MO’s (called MP2) • Different levels of perturbation are possible: • MP2 (tends to overcorrect the correlation energy) • MP3 (better?, but more costly computationally) • MP4 (better?, but even more costly computationally)

  26. Effect of Level of Theory on Computational Cost (cpu time) Methylcyclcohexanesingle pointgeom. opt. MMFF (molec mech) -- 1 AM1 (semi-empirical) -- 10 HF / 3-21G 120 1750 HF / 6-31G* 750 12,650 MP2 / 6-31G* 12,250 -- Lysergic acid (20 heavy atoms) AM1 -- 430 HF / 3-21G 2,360 118,250 MP2 / 6-31G* 19,350 --

  27. Ave. errors in Energy of Eq. geom (kcal/mol) and scaling factor (rel. cpu time req’d) Energy Scaling Method Error(kcal/mol) Factor MP2 10.4 N5 MP3 5.0 N6 CISD 5.8 N6 CCD 2.4 N6 QCISD 1.9 N6 MP4 1.3 N7 MP5 0.8 N7 MP6 0.3 N9 QCISD(T) 0.3 N9 CCSDTQ 0.01 N10

  28. Summary of Choices:

  29. Summary of Ab Initio MO Theory • Generally, accuracy of results depends on the degree of electron correlation and the size of the basis set used. • The cost of the calculation (cpu time required) increases rapidly as the basis set size is increased and as the amount of electron correlation increases. • Most calculations represent a compromise.

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