Ab Initio Molecular Orbital Theory

Ab Initio Theory n 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. n Even ab initio is based on approximate solutions to the Schrödinger equation:  n This can be solved exactly only for H atom!

Ab Initio Theory... n Generally one of two types: – Hartree-Fock n considers each electron to experience effects of all of the other electrons combined – A Correlated method (several) n considers individual electron interactions (to some extent) n CPU time scales as n 3 to n 4 where n = # basis functions (orbitals)

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

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

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

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

Gaussian Type Orbitals n 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

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

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

Split Basis Sets... n 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

More Split Basis Sets... with modifications n 6-31G, 6-311G (the latter has two different sizes of extended Gaussian functions for valence orbitals) n 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)

More Split Basis Sets… and still more modifications n Diffuse functions – 6-31+G n adds an additional, larger p function to heavy (non- hydrogen or helium) atoms; indicated by + before G – G n 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.

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

Optimization of a Basis Set n 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. n The constants and exponents that describe the Gaussian functions are varied sequentially until the lowest energy is obtained. n Such a basis set may only apply to that individual molecule, however.

Optimization of a Basis Set s p p p d 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. Exponent  Coefficient d g (a, r) = c (x n y m z l ) e -  r 2  p =  d p g p

Descriptions of Some Basis Sets n STO-3G A minimal basis set. The fastest, but the least accurate basis set in common use. n 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. n 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. n 6-31G** Adds polarization functions to hydrogens. This can improve the total energy of the system. n 6-31+G* Adds extra diffuse functions to heavy atoms. This can sometimes improve results for systems with large anions. n 6-311G* Adds more flexibility to the basis set. n 6-311G** Adds polarization functions to hydrogens of the 6-311G* basis set. n G** Add diffuse functions to 6-311G**.

Descriptions of Some Basis Sets... n G(2df,2pd) Improves the polarization of G**. n cc-pCVTZ Similar to 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. n 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. n 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). n aug-cc-pCVQZ Augments cc-pCVQZ with some diffuse (aug: spdf) and some core polarization functions (C: 3s3p2d1f, 9s).

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

Effect of Basis Set Choice on Computation Cost (cpu time) Basis Set # basis Energy (au) SCFRelative functionscyclestime STO-3G G G G* G* G* G** G(2df,2pd) G(3df,3pd) cc-pCVTZ cc-pCVQZ aug-cc-pCVQZ

Effect of Basis Set Size on Energy and CPU Time

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

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

Configuration Interaction n 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 n These additional orbitals require considerable additional cpu time, however n Designation: QCIS(D,T) [singles, doubles, triples]

Moller-Plesset Perturbation Theory MP2 n To the HF wavefunction is added a correction corresponding to exciting 2 electrons to higher energy HF MO’s (called MP2) n 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)

Effect of Level of Theory on Computational Cost (cpu time) Methylcyclcohexane single point geom. opt. MMFF (molec mech) -- 1 AM1 (semi-empirical) HF / 3-21G HF / 6-31G* 75012,650 MP2 / 6-31G*12, Lysergic acid (20 heavy atoms) AM HF / 3-21G 2, ,250 MP2 / 6-31G*19,350 --

Ave. errors in Energy of Eq. geom (kcal/mol) and scaling factor (rel. cpu time req’d) Energy Scaling MethodError(kcal/mol)Factor MP210.4 N 5 MP3 5.0 N 6 CISD 5.8 N 6 CCD 2.4 N 6 QCISD 1.9 N 6 MP4 1.3 N 7 MP5 0.8 N 7 MP6 0.3 N 9 QCISD(T) 0.3 N 9 CCSDTQ 0.01 N 10

Summary of Choices:

Summary of Ab Initio MO Theory n Generally, accuracy of results depends on the degree of electron correlation and the size of the basis set used. n 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. n Most calculations represent a compromise.