David M. Whisnant
Wofford College

and

Lisa S. Lever
University of South Carolina - Upstate

 

Computational chemistry is a new field of chemistry that has fully developed in the past ten years or so. Computers have become powerful enough to reliably calculate molecular properties with an accuracy that approaches, or sometimes even exceeds, the quality of experimental results. Computer-aided chemistry, if used wisely, can be an important tool in the hands of experimental chemists. These web pages provide the student with an introduction to computational chemistry with information on modern computational methods. You may select a topic from the main menu at right or read consecutively through the pages using the arrow buttons. These reference pages can be printed more concisely using the links below.

 

Tools of Computational Chemistry

Computational Results

 

 


 

Tools of Computational Chemistry

Modern computational chemistry puts a broad set of tools at the disposal of chemists. Although they are closely related, we can separate these tools into two classes - molecular modeling and information management.

 

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Molecular Modeling

Molecular modeling involves the development of mathematical models of molecules that can be used to predict and interpret their properties. There are two types of molecular modeling - molecular mechanics and quantum mechanics.

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Molecular Mechanics

Molecular mechanics is a classical mechanical model that represents a molecule as a group of atoms held together by elastic bonds. In a nutshell, molecular mechanics looks at the bonds as springs which can be stretched, compressed, bent at the bond angles, and twisted in torsional angles. Interactions between nonbonded atoms also are considered. The sum of all these forces is called the force field of the molecule. A molecular mechanics force field is constructed and parameterized by comparison with a number of molecules, for instance a group of alkanes. This force field then can be used for other molecules similar to those for which it was parameterized. To make a molecular mechanics calculation, a force field is chosen and suitable molecular structure values (natural bond lengths, angles, etc.) are set. The structure then is optimized by changing the structure incrementally to minimize the strain energy and spread it over the entire molecule. This minimization is orders of magnitude faster than a quantum mechanical calculation on an equivalent molecule.

Molecular mechanics is a valuable tool for predicting geometries and heats of formation of molecules for which a force field is available. It is a good way to compare different conformations of the same molecule, for instance. Molecular mechanics does have two weaknesses, however. First, force fields are based on the properties of known, similar molecules. If you are interested in the properties of a new type of molecule an appropriate force field probably will not be available for that molecule. Second, because molecular mechanics models look at molecules as sets of springs, they cannot be used to predict electronic properties of molecules, such as dipole moments and spectroscopy. To make predictions about the electronic properties of a molecule, you must use quantum mechanical models.

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Quantum Mechanics

To make a quantum mechanical model of the electronic structure of a molecule, we must solve the Schrödinger equation.


The Hamiltonian operator, H, depends on the kinetic and potential energies of the nuclei and electrons in the atom or molecule. The wavefunction, , will give us information about the probability of finding the electrons in different places in the molecule. The energy, E, is related to the energies of individual electrons, which can be used to help interpret electronic spectroscopy.

Solving the Schrödinger equation is a very difficult problem and cannot be done without making approximations. Two types of approximations are the Born-Oppenheimer approximation and the independent electron approximation.

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The Born-Oppenheimer Approximation


In the Born-Oppenheimer approximation, the positions of the nuclei are taken to be fixed so that the internuclear distances are constant. This is a sensible approximation because the massive nuclei are essentially immobile in comparison with light electrons. We first choose a geometry (with fixed internuclear distances) for a molecule and solve the Schrödinger equation for that geometry. We then change the geometry slightly and solve the equation again. This continues until we find an optimum geometry with the lowest energy.


 

 

 

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Molecular Orbital (MO) Theory

Another approximation (the independent electron approximation) commonly made is that the wavefunction, , can be written as a product of one-electron functions. The one-electron wavefunctions are called molecular orbitals - the molecular equivalent of atomic orbitals. Each molecular orbital then is expressed as a combination of the atomic orbitals from the atoms that make up the molecule. For example, the simplest molecular orbital function for the H2 molecule is written as c11s1 + c21s2, where 1si is a hydrogen 1s atomic orbital function and ci is a parameter. This method is called LCAO-MO theory for Linear Combination of Atomic Orbitals - Molecular Orbital Theory.

As an example, the two lowest energy molecular orbitals of the H
2 molecule are shown here.

These can be thought of as combinations of 1s orbitals from the two hydrogen atoms. The molecular orbital on the left is made by two H atom 1s orbitals combining constructively. This is a bonding MO because it helps hold the molecule together. The molecular orbital on the right is due to the destructive interference of the two 1s orbitals and is said to be antibonding.

Although molecular orbitals are written as combinations of atomic orbitals from the atoms in the molecule,
molecular orbitals are not atomic orbitals. They are analogous to atomic orbitals, but instead of being defined for atoms, molecular orbitals are characteristic of the molecule as a whole.

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Ab Initio and Semiempirical Calculations

Because of the large number of particles in a molecule (benzene, for instance, has 12 nuclei and 78 electrons) computer programs are used to do the calculations necessary for the solution of the Schrödinger equation. These calculations involve an enormous number of difficult integrals for large molecules. Ab initio computational methods solve all of these integrals without approximation. Ab initio methods are the most reliable for small and medium-sized molecules, but are prohibitively time-consuming for large molecules (20 atoms or so for PCs; around 100 atoms if workstations are available ).

For larger molecules,
semiempirical methods have been developed which ignore or approximate some of the integrals used in ab initio methods. To compensate for neglecting the integrals, the semiempirical methods introduce parameters based on molecular data. Commercial software packages are available for both ab initio and semiempirical calculations.

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Semiempirical Molecular Orbital Theory

 

Semiempirical molecular orbital theory methods have been developed which ignore or approximate some of the integrals used in the solution of the Schrödinger equation. To compensate for neglecting the integrals, the semiempirical methods introduce parameters based on molecular data. The most commonly used semiempirical methods are included in MOPAC (the Molecular Orbital PACkage) computer programs. The CAChe for Windows software package includes several computational chemistry tools including a version of MOPAC. The CAChe MOPAC offers two of the most reliable semiempirical methods (PM3 and AM1) for predicting heats of formation, ground state geometries, and ionization potentials. It also includes the ZINDO program, which does a good job predicting the visible-UV bands for molecules containing hydrogen and first or second period elements.


 

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Chemical Accuracy

Bond lengths and bond angles

What kind of accuracy do we desire from quantum mechanical calculations? Ideally a “chemically accurate” calculation should give the results below2 .

Bond lengths: calculated values within 0. 01 - 0.02 Å of experiment.

Bond angles: calculated values within 1- 2° of experiment.

Semiempirical methods can only be used for elements for which they have been parameterized, but because these elements are common ones in organic compounds, semiempirical calculations can be quite useful. Semiempirical results do not always satisfy the criteria we set for chemical accuracy. PM3 calculations,3 for instance, generally give bond lengths within ±0.036Å and bond angles within ±3.9°, not always “accurate” but still pretty good.

Electronic energies and heats of formation

Quantum mechanical calculations also can be used to predict electronic energies and heats of formation. Chemically accurate energies should be within 1 kJ/mol (0.2 kcal/mol), a challenging task.4

The average errors for PM3 calculations of heats of formation are tabulated below.5

Type of Compound

Type of Compound

All C, H, N, O

4.4

Organic cations

9.5

Hydrocarbons

3.6

Organics with F, Si, Cl, Br, I

5.7

Cyclic Hydrocarbons

2.4

Compounds with S

12.1

Hydrocarbons, double bonds

2.8

Compounds with P

11.5

Hydrocarbons, triple bonds

5.6

Closed shell anions

8.8

Aromatic hydrocarbons

4.1

Neutral radicals

7.4

Organics with N, O

5.2



Semiempirical methods should only be used to predict the properties of molecules for which reliable parameters are available. Experience also has shown that semiempirical methods do not do well for problems involving hydrogen bonding, transition states, and poorly parameterized molecules6. If you are interested in the highest accuracy or in problems such as the these, ab initio calculations should be used if possible.

When you are comparing semiempirical or ab initio predictions with experiment you should remember that computational models generally are of
gas-phase molecules. There are ways of computationally modeling molecules in solution, either at the ab initio or semiempirical level, but we will not get into this subject here.


 

Scaling Vibrational Frequencies

In the last part of the job output from a frequency calculation you will find the predicted vibrational frequencies (cm-1) of the normal modes of the molecule. Also supplied are the predicted intensities of the IR and Raman bands corresponding to these normal modes.

 

                  1             2             3   
                  B1            B2            A1 
Frequencies -- 1335.5948     1383.4094    1679.4157 
 
 
                  4             5             6 
                  A1            A1            B2 
Frequencies -- 2027.8231     3160.8817    3232.9970 

 

Computational results usually have systematic errors. In the case of Hartree-Fock level calculations, for instance, it is known that calculated frequency values are almost always too high by 10% - 12%. To compensate for this systematic error, it is usual to multiply frequencies predicted at the HF/6-31G(d) level by an empirical factor of 0.893. Similarly, frequencies calculated at the MP2/6-31G(d) level are scaled by 0.943 8.

 

The predicted frequencies after applying the 0.893 scale factor are listed below.

 

                       1             2             3 
                       B1            B2            A1 
Scaled Frequencies -- 1193          1235          1450 
 
 
                       4             5             6 
                       A1            A1            B2 
Scaled Frequencies -- 1811          2822          2887 

 

 

 

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Molecular Orbital Surfaces

Molecular orbitals are not real physical quantities. Orbitals are a mathematical convenience that help us think about bonding and reactivity, but they are not physical observables. In fact, several different sets of molecular orbitals can lead to the same energy. Nevertheless, they are quite useful. We will use ethylene as an example to illustrate MO concepts.

We often can classify molecular orbitals as sigma or pi orbitals. A sigma ( ) orbital has cylindrical symmetry about the internuclear axis. The hydrogen orbitals below are sigma orbitals. As we saw earlier a bonding orbital has a high electron probability density between the nuclei. An antibonding orbital has a node between adjacent nuclei with lobes of opposite mathematical characteristics (shown as different colors).

 

 

A pi ( )orbital has “up and down” properties like an atomic p-orbital. Bonding and antibonding orbitals for ethylene are shown below.

 

A few acetone molecular orbital surfaces displayed by CAChe following a PM3 calculation help illustrate these concepts.

Acetone molecular orbitals:

LUMO +1

LUMO

HOMO

HOMO -1

HOMO -2

 

 

 

 

 

 

The first orbital is a -orbital because the lobes are pointing at each other along an internuclear axis and have rotational symmetry about that axis. It is antibonding because lobes of a different color are adjacent to each other. The atomic orbitals that contribute to this molecular orbital are not obvious from the picture.


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This is the lowest unoccupied molecular orbital (LUMO). This is a -orbital because the lobes are perpendicular to an internuclear axis. It is antibonding because lobes of a different color are adjacent to each other. This molecular orbital appears to be predominantly made of a p-orbital on the central carbon atom and a p-orbital on the oxygen atom.

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This is the highest occupied molecular orbital (HOMO). It is a nonbonding orbital because the molecular orbital lobes are located on atoms that are not bonded. A nonbonding orbital does not help hold the molecule together. Electrons in a nonbonding orbital are a little like lone pairs of electrons in a Lewis structure. You can see that a p-orbital on the oxygen atom is a major contributor to this molecular orbital.

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This is a -orbital. It is bonding because lobes of the same color are next to each other. As you can see, two p-orbitals (one on the carbon atom and one on the oxygen) overlap to form a molecular orbital that distributes electrons around the central carbon and the oxygen atom.

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This is a bonding -orbital. It has two portions, each of which appears to be formed by the overlap of a p-orbital on a carbon atom and an s-orbital on one of the hydrogen atoms.

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The Electrostatic Potential


You may remember from physics that a distribution of electric charge creates an electric potential in the surrounding space. A positive electric potential means that a positive charge will be repelled in that region of space. A negative electric potential means that a positive charge will be attracted. A molecule is a collection of charges that will have an electric potential - commonly called the “electrostatic potential.” The electrostatic potential is a physical property of a molecule related to how a molecule is first “seen” or “felt” by another approaching species7. A portion of a molecule that has a negative electrostatic potential will be susceptible to electrophilic attack - the more negative the better. It is not as straightforward to use electrostatic potentials to predict nucleophilic attack.

 

At right is an electrostatic potential surface of acetone displayed by CAChe using MM/PM3 geometry with PM3 wavefunction. The surface is color coded according to electrostatic potential (blue is negative and red is positive). What part of the acetone molecule appears to be more susceptible to electrophilic attack?

 

 

 

 

 

 

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The Electron Density

The electron density surface depicts locations around the molecule where the electron probability density is equal. This gives an idea of the size of the molecule and its susceptibility to electrophilic attack.

Below is an electron density surface of acetone displayed by CAChe using MM/PM3 geometry with PM3 wavefunction. The surface color reflects the magnitude and polarity of the electrostatic potential. Gray, violet and blue colors correspond to a negative electrostatic potential - regions of the molecule susceptible to electrophilic attack.

 

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Ab Initio Molecular Orbital Theory

 

To make a quantum mechanical model of the electronic structure of a molecule, we must solve the Schrödinger equation.


Solving this equation is a very difficult problem and cannot be done without making approximations. In ab initio methods, no integrals are neglected in the course of the calculation. One approximation is the Born-Oppenheimer approximation in which the positions of the nuclei are fixed so that the internuclear distances are constant. We first choose a geometry (with fixed internuclear distances) for a molecule and solve the Schrödinger equation for that geometry. We then change the geometry slightly and solve the equation again. This continues until we find an optimum geometry with the lowest energy.

 

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The Independent Electron Approximation

 

 

 

When more than one electron is present, the Schrödinger equation is impossible to solve because of the interelectron terms in the Hamiltonian.

Consider, for instance, the Hamiltonian for the hydrogen molecule in the Born-Oppenheimer approximation.

 

 

 


The first two terms are due to the kinetic energy of the electrons. The last six terms express the potential energy of the system of four particles.

The potential energy term due to the repulsion of the electrons makes the Schrödinger equation impossible to solve.

 


To produce a solvable Schrödinger equation we assume that the Hamiltonian is a sum of one-electron functions, f
i, with an approximate potential energy that takes the average interaction of the electrons into account. This leads to a set of one-electron equations, called the Hartree-Fock equations, where is a one-electron wavefunction.


The total wavefunction that is a solution to the total Schrödinger equation, , is approximated as the product of the solutions to the one-electron equations.

This product must be adjusted to satisfy the Pauli Exclusion principle, but we won't get into that here. If you are familiar with determinants, it involves writing the wavefunction as a determinant.

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The Hartree-Fock Self-Consistent Field (SCF) Approximation

The question remains about the approximate potential energy in the one-electron functions that take the average interaction of the electrons into account. What is the form of the functions fi in the Hartree-Fock equations? The most common way of handling this is to define

where vi is an average potential energy due to the interaction of one electron with all the other electrons and nuclei in the molecule. The average potential depends on the orbitals, , of the other electrons, which means we must solve the Hartree-Fock equations iteratively.

The iterative solution of the Hartree-Fock equation is as follows.

1. Guess reasonable one-electron orbitals (wavefunctions), , and calculate the average potential energies, v
i.

2. Using the variation principle, solve the Hartree-Fock equations,

to give new one-electron orbitals, . Use these new orbitals to calculate new and improved average potential energies, vi. Because the solution of the Hartree-Fock equations depends on the variation principle, the Hartree-Fock energy should be higher than the true energy.

3. Repeat the second step until the one-electron orbitals and potential energies don't change (are self-consistent).

 

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Restricted Hartree-Fock Calculations

To take the Pauli Principle into account, we must include electron spin in our wavefunctions. The orbitals that are calculated by the Hartree-Fock method actually are spin orbitals that are a product of a spatial wavefunction and a spin function.

In a spin orbital, is the spatial wavefunction describing the probability of finding the electron in space and or are spin wavefunctions.

For a
closed shell system, in which all of the electrons are paired, during the solution of the self-consistent field equations, we can restrict the solution so that the spatial wavefunctions for paired electrons are the same. This is called a restricted Hartree-Fock (RHF) calculation and generally is used for molecules in which all the electrons are paired. When the spin functions are removed, we are left with a set of spatial orbitals, each occupied by two electrons.


An example would be the restricted Hartree-Fock solution to the Schrödinger equation for the hydrogen molecule, H
2. This would lead to two spatial orbitals, one occupied by the pair of electrons and one unoccupied. The orbitals holding electrons are called occupied orbitals and the unoccupied orbitals are called virtual orbitals.

 

 

 

 

 

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Unrestricted Hartree-Fock Calculations

 

For open shell systems that contain unpaired electrons, the assumption made in the restricted Hartree-Fock method obviously won't work. There is more than one way of handling this type of problem. One way is to not constrain pairs of electrons to occupy the same spatial orbital - the unrestricted Hartree-Fock (UHF) method. In this method there are two sets of spatial orbitals - those with spin up () electrons and those with spin down () electrons. This leads to two sets of orbitals as pictured at the right and to a lower energy than if the restricted method were used.

 


 

 

 

 

 

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Basis Sets

 

For molecular calculations, the Hartree-Fock SCF equations

still cannot be solved without one further approximation. To solve the equations, each SCF orbital,, is written as a linear combination of atomic orbitals. For instance, for the H2 molecule, the simplest approximation is to write each spatial SCF orbital as a combination of 1s atomic orbitals, each centered on one of the protons.

 


This reduces the problem to solving for the coefficients, c
1 and c2, since the atomic orbitals do not change.

The set of atomic orbitals that is chosen to represent the SCF orbitals is called a
basis set. The {1sA, 1sB} basis set shown above is a minimal basis set - the smallest set of orbitals possible that describe an SCF orbital. Usually, the quality of a basis set depends on its size. For instance, a larger basis set, such as {1sA, 1sB, 2sA, 2sB}would do a better job approximating the SCF orbital than {1sA, 1sB}.


For many-electron atoms, we don't know the actual mathematical functions for the atomic orbitals, so substitutes are used - usually either Slater-type orbitals (STO) or Gaussian-type orbitals (GTO). We won't concern ourselves with the exact form of STO and GTO. Suffice it to say that they are chosen to behave mathematically like the actual atomic orbitals:
s-type, p-type, d-type, and f-type, for instance. A few commonly used basis sets are listed below. The symbol of the basis set is given in the left column and the characteristics of the basis set in the center. At the right is the basis set that would be used to represent methane. For instance, the STO-3G basis set for methane would be {1sH, 1sH, 1sH, 1sH, 1sC, 2sC, 2pxC, 2pyC, 2pzC}.

Basis Sets8

Characteristics

Basis Set Example (CH4)

STO-3G

A minimal basis set (although not the smallest possible) using three GTOs to approximate each STO. This basis set should only be used for qualitative results on very large systems

Each H: 1s

C: 1s, 2s, 2p
x, 2py, 2pz

3-21G

Inner shell basis functions made of three GTOs. Valence s- and p-orbitals each represented by two basis functions (one made of two GTOs, the other of a single GTO). Use for very large molecules for which 6-31G is too expensive.

Each H: 1s, 1s'

C: 1s, 2s, 2p
x, 2py, 2pz, 2s', 2px', 2py', 2pz'

6-31G(d)
(6-31G*)

Inner shell basis functions made of six GTOs. Valence s- and p- orbitals each represented by two basis functions (one made of three GTOs, the other of a single GTO). Adds six d-type basis functions to non-hydrogen atoms. This is a popular basis set that often is used for medium and large systems.

Each H: 1s, 2s

C: 1s, 2s, 2p
x, 2py, 2pz, 2s', 2px', 2py', 2pz', 3dx2, 3dy2, 3dz2, 3dxy, 3dxz, 3dyz

6-31G(d,p)
(6-31G**)

Like 6-31G(d) except p-type functions also are added for hydrogen atoms. Use when hydrogens are of interest and for final, accurate energy calculations.

Each H: 1s, 2s, 2px, 2py, 2pz

C: 1s, 2s, 2p
x, 2py, 2pz, 2s', 2px', 2py', 2pz', 3dx2, 3dy2, 3dz2, 3dxy, 3dxz, 3dyz

Generally, the larger the basis set the more accurate the calculation (within limits) and the more computer time that is required. As an example, consider the calculation of the bond length of H-F using different basis sets, as shown below8.

 

Basis Set

Bond Length (Å)

| Error (Å) |

6-31G(d)

0.93497

0.017

6-31G(d,p)

0.92099

0.003

6-31+G(d,p)

0.94208

0.025

6-31++G(d,p)

0.92643

0.009

6-311G(d,p)

0.91312

0.004

6-311++G(d,p)

0.91720

0.000

Experimental

0.917


You might notice that although the large basis set, 6-311++G(d,p), predicts the correct answer to within 0.001 Å, several others are correct to within 0.01 Å (well within the criteria of chemical accuracy). Although a larger basis set usually gives better results, you often have diminishing returns as you choose larger sets. A point may be reached beyond which the additional computer time is not worth it.

 

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Post-SCF Calculations

Even with a very large basis set calculation, Hartree-Fock results are not exact because they rely on the independent electron approximation. Hartree-Fock SCF Theory is a good base-level theory that is reasonably good at computing the structures and vibrational frequencies of stable molecules and some transition states9. Electrons are not independent, though. We say that they are correlated with each other and that the Hartree-Fock method neglects electron correlation. This means that Hartree-Fock calculations do not do a good job modeling the energetics of reactions or bond dissociation. There are several ways of correcting SCF results to take electron correlation into account.

One method of taking electron correlation into account is Møller-Plesset many-body perturbation theory, which is used after a RHF or UHF calculation has been made. It is assumed that the relationship between the exact and Hartree-Fock Hamiltonians is expressed by an additional term, H(1), so that H = fi + H(1). Calculations based on this assumption lead to corrections that can improve SCF results. Various levels of perturbation theory can be applied to the problem. They are called MP2, MP3, MP4, etc. MP2 calculations are not time-consuming and usually give quite accurate geometries and about one-half of the correlation energy. Because perturbation theory is not based on the variation principle, the energy predicted by MP calculations can fall below the actual energy.

Another important method of correcting for the correlation energy is
configuration interaction (CI). Conceptually we can think of CI calculations as using the variation principle to combine various SCF excited states with the SCF ground state, which lowers its energy. We won't use CI calculations in our exercises at this level.

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SCF Molecular Orbitals

When calculating molecular orbitals, you should remember that molecular orbitals are not real physical quantities. Orbitals are a mathematical convenience that help us think about bonding and reactivity, but they are not physical observables. In fact, several different sets of molecular orbitals can lead to the same energy. Nevertheless, they are quite useful. We will use ethylene as an example to illustrate MO concepts.

The
basis functions in SCF molecular orbitals are like atomic orbitals. A RHF/6-31G(d) calculation on ethylene uses 38 basis functions (15 for each carbon and 2 for each hydrogen). Since the molecular orbital wavefunction is expanded in terms of the all the basis functions,

it might seem that constructing a picture of the orbital would be difficult. Luckily, most of the coefficients are zero, so the molecular orbitals are easy to picture. Consider, for instance, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of ethylene.


HOMO:


The HOMO is a bonding-orbital.

 

 


LUMO:


The LUMO is an antibonding -orbital.



 

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References

1J. H. Krieger, “Computational Chemistry Impact”, C&E News, 1997, May 12, p 30.; E. K. Wilson, “Computers Customize Combinatorial Libraries”, C&E News, 1998, April 27, p 31.

2J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1995-96, p. 118.

3J. Stewart, J. Comp. Chem., 1989, 10, p. 209-220.

4P. W. Atkins; R. S. Friedman, Molecular Quantum Mechanics, 3rd Ed., Oxford, New York, 1997, p. 276.

5M. C. Zerner, Rev. Comp. Chem., Vol. 2, VCH, New York, 1991, pp 313-365.

6J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1995-96, p. 113.

7P. Politzer and J. S. Murray, Rev. Comp. Chem., Vol. 2, VCH, 1991, p. 273.

8J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1995-96, p 102.

9J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1996, p 115.



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This page was last updated on March, 2004.