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.

Even ab initio is based on approximate solutions to the Schrödinger equation:

This can be solved exactly only for H atom!

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)

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

Schrödinger equation:

1MA

2A

_ h282H = 2

a_ h2

82m

electrons

A

>

nuclei

a ZA

rAa_ e2

nuclei electrons

aA

ZAZBrAB

e2nuclei

A 1

rabe2

electrons

baB >++

kinetic energy (nuc.) kinetic energy (elect.)

2 kinetic energy terms plus3 Coulombic energy terms:(one attractive, 2 repulsive)

H = E

H = Hamiltonion operator

Schrödinger equation after Born-Oppenheimer Approximation

1MA

2A

_ h282H = 2

a_ h2

82m

electrons

A

>

nuclei

a ZA

rAa_ e2

nuclei electrons

aA

ZAZBrAB

e2nuclei

A 1

rabe2

electrons

baB >++

kinetic energy (nuc.) kinetic energy (elect.)

1 kinetic energy term plus2 Coulombic energy terms:(one attractive, 1 repulsive)plus a constant for nuclei

0

constant

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

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

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

Split Basis Sets...

=ba +for a "tighter" bond: (a > b)

for a "looser" bond: (b > a)

=ba +

(a and b are normalized coefficients; their sum is 1)

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

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 b =

(a and b are coefficients whose sum is 1)

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.

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.

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.

Optimization of a Basis Set...

Energy vs. d basis parameter

2.2

2.25

2.3

2.35

2.4

2.45

2.5

0 0.1 0.2 0.3 0.4a

Ener

gy

...

s 1 1.000.052 1.0p 3 1.001.034919 -0.3401550.573833 0.4418840.203063 0.604344p 1 1.000.079099 0.288949p 1 1.000.029 1.0d 1 1.000.241 1.0

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 a Coefficient d g(a, r) = c (xnymzl) e-ar 2

p = dpgp

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**.

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).

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 10HF/STO-3G 72 983HF/ 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)

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

Basis Set# basis

Energy (au)SCF Relative

functions cycles timeSTO-3G 26 -189.534 688 69 14 0.053-21G 48 -190.886 407 54 14 0.26-31G 48 -191.874 189 82 14 0.36-31G* 72 -191.960 613 31 15 16-311G* 90 -192.001 883 12 15 36-311+G* 106 -192.005 994 08 15 66-311++G** 130 -192.015 295 56 15 256-311++G(2df,2pd) 226 -192.029 578 61 15 886-311++G(3df,3pd) 264 -192.031 627 88 31 235cc-pCVTZ 204 -192.032 898 46 15 82cc-pCVQZ 400 -192.046 642 88 30 3400aug-cc-pCVQZ 712 -192.047 735 33 19 41000

Effect of Basis Set Size on Energy and CPU Time

Energy (Blue) and CPU Time (Red) vs. Basis Set Size

-192.5

-192

-191.5

-191

-190.5

-190

-189.5

-1890 2 4 6 8 10 12

Basis Set size

Ener

gy (a

u)

0

10000

20000

30000

40000

50000

0 2 4 6 8 10 12

Rela

tive

CPU

Tim

e

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

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 set and complete electron correlation is the Schrödinger equation itself!

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]

Moller-Plesset Perturbation Theory

To the HF wavefunction is added a correction corresponding to exciting 2 electrons to higher energy HF MO’s (called MP2MP2)

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) -- 1AM1 (semi-empirical) -- 10HF / 3-21G 120 1750HF / 6-31G* 750 12,650MP2 / 6-31G* 12,250 --

Lysergic acid (20 heavy atoms)AM1 -- 430HF / 3-21G 2,360 118,250MP2 / 6-31G* 19,350 --

Ave. errors in Energy of Eq. geom (kcal/mol) and scaling factor (rel. cpu time req’d)

Energy Scaling

Method Error(kcal/mol) FactorMP2 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

Summary of Choices:

STO-3G3-21G

6-31G

6-311G

6-311G(d,p)

6-311++G(d,p)

MP2 MP3 MP4QCISD MP6 QCISD(T) (full CI)HF

6-311++G(2df,2pd)cc-pVTZ

aug-cc-pCVQZ

(infinite basis set) Exact Solution of the Schrodinger Equation

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.