Cited from https://faculty.uscupstate.edu/llever/Computational%20Chemistry/ab%20initio.html.
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. We have covered some of these approximations in the Semiempirical MO Theory handout. In this handout we focus on ab initio methods of solving the equation, in which no integrals are neglected in the course of the calculation.
The Born-Oppenheimer Approximation
The first approximation is known as the Born-Oppenheimer approximation, in which we take the positions of the nuclei to be fixed so that the internuclear distances are constant. Because nuclei are very heavy in comparison with electrons, to a good approximation we can think of the electrons moving in the field of fixed nuclei. 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.
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.
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 f_{i} in the Hartree-Fock equations? The most common way of handling this is to define
where v_{i} 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, v_{i}. 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).
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.
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 tonot 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.
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 H_{2} 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 c_{2}, 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 {1s_{A}, 1s_{B}} 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 {1s_{A}, 1s_{B}, 2s_{A}, 2s_{B}}would do a better job approximating the SCF orbital than {1s_{A}, 1s_{B}}.
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 {1s_{H}, 1s_{H}, 1s_{H}, 1s_{H}, 1s_{C}, 2s_{C}, 2p_{xC}, 2p_{yC}, 2p_{zC}}.
Basis Sets^{1} |
Characteristics |
Basis Set Example (CH_{4}) | ||
---|---|---|---|---|
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}, 2p_{y}, 2p_{z} |
||
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}, 2p_{y}, 2p_{z}, 2s’, 2p_{x}‘, 2p_{y}‘, 2p_{z}‘ |
||
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}, 2p_{y}, 2p_{z}, 2s’, 2p_{x}‘, 2p_{y}‘, 2p_{z}‘, 3d_{x}^{2}, 3d_{y}^{2}, 3d_{z}^{2}, 3d_{xy}, 3d_{xz}, 3d_{yz} |
||
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, 2p_{x}, 2p_{y}, 2p_{z}
C: 1s, 2s, 2p_{x}, 2p_{y}, 2p_{z}, 2s’, 2p_{x}‘, 2p_{y}‘, 2p_{z}‘, 3d_{x}^{2}, 3d_{y}^{2}, 3d_{z}^{2}, 3d_{xy}, 3d_{xz}, 3d_{yz} |
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 below. ^{1}
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.
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 states^{2}. 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 = f_{i} + 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 calledMP2, 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.
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.
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. ^{1}
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
References
^{1}J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1995-96, p 102.
^{2}J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1996, p 115.
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