Ab Initio and Semiempirical Methods

Ab initio and Semiempirical

Methods

Ab Initio◦ Only have parameters associated with the basis

set of Aos used, in linear combination, to describe MOs.

DFT (Density Functional Theory)◦ Similar to ab initio, attempts to describe how the

energy of a molecule depends on the electron density of the molecule.

Semi-empirical◦ Use many explicit parameters associated with the

specific atomic number (element) of an atom.

Quantum Mechanics Calculations

Comparison of Quantum Mechanics Calculation Methods

Ab Initio (and DFT) Semiempirical

accurate ab initio or DFT calculations require enormous computation and are only suitable for the molecular systems with small or medium size.

Fast method (important for biomolecules)

for specific and well-parameterized molecular systems, these methods can calculate values that are closer to experiment than lower level ab initio and DFT.

Comparison of Quantum Mechanics Calculation Methods

Ab Initio (and DFT) Semiempirical

The ab initio or density functional methods may overcome the problem on having parameters before running a calculation.

However they are slower than any molecular mechanics and semi-empirical methods.

The accuracy of a (molecular mechanics or) semi-empirical quantum mechanics method depends on the database used to parameterize the method.

A disadvantage of these methods is that you must have parameters available before running a calculation.

Only have parameters associated with the basis set of atomic orbitals (contracted basis functions) used, in linear combination, to describe the molecular orbitals.

These basis sets are usually associated with shells, such as an s shell, sp shell, etc.

The basis set is the only parameter in ab initio calculations

Ab Initio Method

Variational Calculation on a system involves the ff steps:1. Write down the Ĥ for the system2. Select mathematical functional form Ψ as

the trial wavefunction. (should have variable parameters)

3. Minimize

With respect to variations in parameters

Ab Initio Method

“from the beginning” - without any previous knowledge of a subject being studied.

Used to describe calculations in which no use is made of experimental data.

All three previous steps listed are explicitly performed.

Self-consistent field (SCF) method◦ Most common type of ab initio calculation for

atoms or molecules. Hartree-Fock

Ab Initio Method

Can be performed at the Hartree-Fock level of approximation, equivalent to a SCF calculation or at a post Hartree-Fock level which includes the effects of correlation — defined to be everything that the Hartree-Fock level of approximation leaves out of a non-relativistic solution to the Schrödinger equation (within the clamped-nuclei Born-Oppenheimer approximation).

Ab Initio Method

All the integrals over AO basis functions are computed and the Fock matrix of the SCF computation is formed from the integrals.◦ Fock Matrix

Divides into two parts One-electron Hamiltonian matrix, H, and the two-

electron matrix, G, with the matrix elements

Ab Initio Method

involves the calculation of the following types of integrals:1. Overlap integrals:

2. Kinetic energy integrals:

3. Nuclear-electron attraction energy integrals:

4. Electron-electron repulsion energy integrals:

Ab Initio Method

The basic approximation is that the total wave function is a single Slater determinant and the resultant expression of MO-LCAO basis functions.

In other words, an ab initio calculation can be initiated once a basis for the LCAO is chosen. Mathematically , any set of functions can be a basis for an ab initio calculation.

Ab Initio Method

Two important criteria1. We want a basis set that is capable of

describing the actual wavefunction well enough to give chemically useful results.

2. We want a basis set that leads to integrals Fij and Sij (two-electron integrals) that we can evaluate reasonably accurately and quickly on a computer.

Gaussian and Slater-type-orbital (STO)◦ Dominate the area of ab initio molecular

calculations

Basis Sets for ab initio calculations

v

vvii cψ

In order to form the Fock matrix of an ab initio calculation, all the core-Hamiltonian matrix elements, Hµν, and two-electron integrals (µν|λσ) have to be computed.

If the total number of basis functions is m, the total number of the core Hamiltonian matrix elements is

Two-Electron Integrals

)1m(m2

1N1

after considering the symmetry of the core Hamiltonian matrix element, Hµν= Hνµ; and the total number of the two-electron integrals is

Two-Electron Integrals

2m3m2mm8

1N 2342

Table 11-2 provides information on energies for a number of atoms in their ground states.

Self-consistent-field energies are presented for three levels of basis set complexity.

In the STO single-ζ level, a minimal basis set of one STO per occupied AO is used, and the energy is minimized with respect to independent variation of every orbital exponent ζ.

The STO double-ζ basis set is similar except that there are two STOs for each AO, the only restriction being that the STOs have the same spherical harmonics as the AOs to which they correspond.

Example on atoms

Example on atoms

The Hartree–Fock energies are estimated by extrapolating from more extensive basis sets, and represent the limit achievable for the SCF approach using a complete basis set.

Example on atoms

Example on atoms

The ff. observations can be made:1. The improvement in energy obtained when one goes

from a single-ζ to a double-ζ STO basis set is substantial, especially for atoms of higher Z.

2. The agreement between the optimized double-ζ data and the HF energies is quite good. Even for neon, the error is only about 10−2 a.u. (0.27 eV). Thus, for atoms, the double-ζ basis is capable of almost exhausting the energy capabilities of a single-configuration wavefunction.

3. The disagreement between HF and “exact” energies (i.e., the correlation energy) grows progressively larger down the list. For neon it is almost 0.4 a.u. (10 eV), which is an unacceptable error in chemical measurements.

Example on atoms

Example on atoms

First, let us compare HF and exact energies for molecules as we did for atoms and see how large the errors due to correlation are.

The results are not too different from those for atoms having the same number of electrons, as shown in Table 11-4; that is, the correlation energies for molecules having ten electrons (CH4,NH3,H2O, HF) are about the same as that for neon, whereas that for the 18-electron molecule H2O2 is more like the correlation energy for argon.

But this is only a very rough rule of thumb. We have already indicated that the correlation energy in a molecule varies with bond length, a factor not present in atomic problems.

Example on molecules


Calculation on the OH Radical (reported by Cade and Hou)◦ A good example of the capabilities of the

extended basis set LCAO-MO-SCF technique on a small molecule

Minimal basis set STOs for OH includes◦ For oxygen

1s, 2s, 2px, 2py, and 2pz STOs.

◦ For hydrogen 1s AO


Near Hartree-Fock Wavefunction for the OH molecule in its Ground-State Configuration (1σ2 2σ2 3σ31π3)at an internuclear separation R = 1.8342 a.u.

Cade and Hou chose a more extensive basis.◦ Oxygen is the site for two 1s, two 2s, one

3s, four 2p, one 4f, eight 2pπ, two 3dπ , and four 4fπ STOs.

◦ On hydrogen, there are two 1s, one 2s, one 2pσ , two 2pπ , and two 3dπ STOs.

The orbital exponents for all of these STOs have been optimized, and the resulting wavefunction is of “near-Hartree–Fock” quality.

Example on molecules (OH Radical)

The Optimized ζ values appear in Table 11-5. The STO labeled σ2p´o is located on oxygen and has the formula


There are threeσ-type MOs and two π-type MOs to accommodate the nine electrons of this radical. One π-type MO is

and the other occupied π-type MO would be the same except with x instead of y. (The z axis is coincident with the internuclear axis.)

It is evident that writing out the complete wavefunction given in Table 11-5 would result in a very cumbersome expression. It is a nontrivial problem to relate an accurate but bulky wavefunction such as this to the kinds of simple conceptual schemes chemists like to use.

One solution is to have a computer produce contour diagrams of the MOs. Such plots for the valence MOs 2σ, 3σ, and 1π of Table 11-5 are presented in Fig. 11-4



Example on small molecules

Listed are dipole moments for ground and some excited states of diatomic molecules.

The dipole moments computed from near-HF wavefunctions contain substantial errors.

It can be seen that CI greatly improves dipole moments. It has been observed that inclusion of singly excited configurations is very important in obtaining an accurate dipole moment.

Example on small molecules

Combine the theoretical form with parameters fitted from experimental data◦ “semi” - partially◦ “empirical” - based on or characterized by observation and

experiment instead of theory use many explicit parameters associated with the

specific atomic number (element) of an atom. The concept of atom types is not used in the

conventional quantum mechanics methods. Semi-empirical quantum mechanics methods use a

rigorous quantum mechanical formulation combined with the use of empirical parameters obtained from comparison with experiment.

Semiempirical Methods

Extended Hückel Method◦ Simplest and fastest but least accurate◦ Provides approx shape and energy ordering of MOs,

also approx form of an electron density map◦ neglects all electron-electron interactions

NDO (Neglect of Differential Overlap ) Methods ◦ neglects some electron-electron interactions.◦ In some part of calculations they neglect the effects

of any overlap density between atomic orbitals.◦ This reduces the number of electron-electron

interaction integrals to calculate, which would otherwise be too time-consuming for all but the smallest molecules.

Semiempirical Methods

The neglect of electron-electron interactions in the Extended Hückel model has several consequences.

For example, the atomic orbital binding energies are fixed and do not depend on charge density . With the more accurate NDO semi-empirical treatments, these energies are appropriately sensitive to the surrounding molecular environment.

Limitations of Extended Hückel

HyperChem cannot perform a geometry optimization or molecular dynamics simulation using Extended Hückel.

Stable molecules can collapse, with nuclei piled on top of one another, or they can dissociate into atoms. With the commonly used parameters, the water molecule is predicted to be linear .

Limitations of Extended Hückel

1. CNDO (Complete Neglect of Differential Overlap)

2. INDO (Intermediate NDO)3. MINDO/3 (Modified INDO, version 3)4. MNDO (Modified NDO)5. MNDO/d6. AM17. PM38. ZINDO/1 (Dr . Michael Zerner’s INDO versions)9. ZINDO/S10. TNDO (Typed Neglect of Differential Overlap)

NDO Methods

Use the HF approximation to solve the Schrodinger eqn◦ HF deals w/ several kinds of electron-electron interaction.

Coulumb interaction - the repulsion between an electron in one atomic orbital and an electron in another (or the same, if doubly occupied)

The identical nature of electrons requires a correction for electrons of the same spin, and this is often described as if it were a real interaction. The contributions to matrix elements describing this correction are called atomic exchange integrals.

NDO Methods

CNDO – simplest metod; The electron repulsion between electrons in different orbitals depends only on the nature of the atoms involved, and not on the particular orbital.

◦ Disadvantage: because it neglects almost all exchange integrals, it cannot calculate differences between states of multiplicity arising from the same electronic configuration. CNDO treats singlet-triplet energy gaps poorly .

CNDO, INDO, MINDO/3, ZINDO/1, and ZINDO/S Methods

INDO method corrects some of the worst problems with CNDO.

For example, INDO exchange integrals between electrons on the same atom need not be equal, but can depend on the orbitals involved. Though this introduces more parameters, additional computation time is negligible.

INDO and MINDO/3 (Modified INDO, version 3) methods are different implementations of the same approximation.


ZINDO/1 and ZINDO/S are Dr . Michael Zerner’s INDO versions ◦ used for molecular systems with transition metals. ◦ ZINDO/1 is expected to give geometries of

molecules, and ◦ ZINDO/S is parametrized to give UV spectra


The NDDO (Neglect of Diatomic Differential Overlap) approximation - basis for the MNDO, MNDO/d, AM1, and PM3 methods.

They have an additional class of electron repulsion integrals.

This class includes the overlap density between two orbitals centered on the same atom interacting with the overlap density between two orbitals also centered on a single (but possibly different) atom.

This is a significant step toward calculating the effects of electron-electron interactions on different atoms.

MNDO, MNDO/d, AM1, and PM3 Methods

AM1◦ most accurate computational method included in

HyperChem◦ often the best method for collecting quantitative

information PM3

◦ Functionally similar to AM1 but uses an alternative parameter set

MNDO, MNDO/d, AM1, and PM3 Methods

new semi-empirical method with considerable research needed before its parameters are fully known for the variety of situations in which it might be used.

It is essentially identical to CNDO (TNDO/1) or INDO (TNDO/2).

The only difference is that the semi-empirical parameters are a function not of the atomic number but of the atom type.

It uses Amber atom types and is in some sense a combination of molecular mechanics and semi-empirical quantum mechanics.

TNDO Methods

work best for molecules with electrons that are spread evenly throughout, with no significant charge polarization. ◦ Hydrocarbons are classic examples: all NDO

methods work well on nonpolar hydrocarbons. ◦ Molecular systems with heteroatoms provide a

better test of these methods. ◦ Groups with several electronegative atoms close

together, such as NO2, are the most difficult to treat. ◦ Among inorganic, main group compounds, even the

best semi-empirical methods can fail dramatically with, for example, interhalogen molecules.

Practical Uses of NDO Methods

Energies of Molecules Geometries of Molecules Energies of Transition States Molecular Orbital Energies and Ionization Pote

ntials Dipole Moments Electrostatic Potential Atomic Charges Chemical Reactivity Vibrational Analysis and Infrared Spectroscopy UV-Visible Spectra

Results of Semi-Empirical Calculations

Lowe, John P. and Kirk A. Peterson. 2006. Quantum Chemistry, Third Edition. Elsevier Inc. pp. 348, 372◦ Chapter 11-The SCF-LACAO-MO Method and

Extensions Section 11-1 Ab Initio Calculations

HyperChem Release 8 Manual Computational Chemistry: Part 2 Theory and Methods. 2002. HyperCube, Inc. pp225, 260, 278-310

References

Energies of Molecules◦ The total energy in an Molecular Orbital

calculation is the net result of electronic kinetic energies and the interactions between all electrons and atomic cores in the system.


The table shows the data on molecules, including hydrocarbons, strained ring systems, molecules with heteroatoms, radicals, and ions comes from a review by Stewarta.

For most organic molecules, AM1 reports heats of formation accurate to within a few kilocalories per mol.

For some molecules (particularly inorganic compounds with several halogens, such as perchloryl fluoride, even the best semi-empirical method fails completely .

Results of Semi-Empirical Calculations (Energies of Molecules)

(a)Stewart, J. J. P . MOPAC: A Semiempirical Molecular Orbital Program J. Computer-Aided Mol. Design4, 1–105:1990.

Geometries of Molecules◦ Shapes of molecules; The calculations have

varying degrees of accuracy and take more time than molecular mechanics methods. The accuracy of the results depends on the molecule.


Energies of Transition States◦ The following data is from Stewart’s review.


Molecular Orbital Energies and Ionization Potentials

Dipole Moments◦ The molecular dipole moment is perhaps the

simplest experimen-tal measure of charge density in a molecule.


Electrostatic Potential◦ Electron distribution governs the electrostatic

potential of molecules. ◦ describes the interaction of energy of the

molecular system with a positive point charge. ◦ useful for finding sites of reaction in a molecule:

positively charged species tend to attack where the electrostatic potential is strongly negative (electrophilic attack).


HyperChem displays the electrostatic potential as a contour plot. (Example is formamide (NH2CHO))


Atomic Charges◦ Calculated atomic charges are a different matter . ◦ There are various ways to define atomic charges.

HyperChem uses Mulliken atomic charges, which are commonly used in Molecular Orbital theory.

◦ These quantities have only an approximate relation to experiment; their values are sensitive to the basis set and to the method of calculation.

Chemical Reactivity◦ Atomic Charges and Reactivity

You can estimate sites of ionic reactivity from the atomic charges in a molecule, particularly for reactions involving “hard” nucleophiles and electrophiles.


◦ Frontier Molecular Orbitals You can interpret the stereochemistry and rates of

many reactions involving “soft” electrophiles and nucleophiles—in particular pericyclic reactions—in terms of the properties of Frontier orbitals.

Example: An example of Frontier Orbital theory is in predicting sites of nitration (electrophilic attack) on aromatic compounds. If you plot the HOMO as a contour map, the region of highest density (regardless of sign) is generally the site of electrophilic attack. Alternatively, you can look in the log file for the atom with the largest molecular orbital coefficient. This is generally the site of reaction.


In this example, the HOMO is plotted one Ångstrom above the plane of the molecule. Since it is of π symmetry, it has a node in the plane of the molecule. It shows the site of electrophilic attack at the carbon adjacent to the oxygen atom.

This is also the experimentally observed site.

The orbital comes from an Extended Hückel calculation of an MM+ optimized geometry .


Vibrational Analysis and Infrared Spectroscopy◦ The quality of the vibrational frequencies varies

widely with the semi-empirical method that is used. Generally , AM1, and PM3 are in closer agreement with experiment than methods based on CNDO or INDO.

◦ The vibrational frequencies are derived from the harmonic approximation, which assumes that the potential surface has a quadratic form.


Vibrational Analysis and Infrared Spectroscopy◦ The following table shows the accuracy of

computed fundamental frequencies for CO2 (cm-1):


UV-visible Spectra The longest wavelength absorption transition for

ethene calcu-lated by HyperChem using PM3 is 207 nm, which compares favor-ably with the experimental value of 190-200 nm.


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Ab Initio and Semiempirical Methods