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Electronic Structure Calculations
Introduction and Motivation
a) To calculate any property of a molecule, the molecule’s wavefunction is needed.
b) The wavefunction function is found by solving the Schrödinger equation using the
Hamiltonian of the molecule.
c) The Hamiltonian of the molecule includes
1) Kinetic energy terms for all the electrons
2)
Coulomb attraction terms between each pairing of an electron and a nucleus
3)
Coulomb repulsion terms between each pairing of electrons
4)
Coulomb exchange terms (related to the antisymmetry requirement of the
wavefunction.
5)
Coulomb repulsion terms between nuclei
d) To find the wavefunction
a) Assume that the molecule orbitals that comprise the wavefunction can be written
as an unknown Linear Combination of Atomic Orbitals (LCAO).
b) Use the variation principle to find the best LCAO that minimizes the energy.
Molecular Properties
Electronic Energy
Definition: The electronic energy is the energy lost when all of the particles (electrons
and nuclei) assemble from infinity.
- These energies are considerably greater than the “normal” chemical energies of
several hundred kJ/mol.
- The calculations are done with the nuclei fixed in a single arrangement.
- These calculations are also known as single point energy calculations.
Optimal Molecular Geometry
- Changing the position of the nuclei will change Coulombic energy and therefore
change the arrangement of the electrons.
- To find the optimized molecular geometry, the nuclei are shifted multiple times to
find which nuclear arrangement will yield the lowest electronic energy.
- An optimized geometry should give and bond angles and distance that are close to
experimental values.
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Optimal Conformations
- Sometimes geometry optimization calculations produce molecular geometries that
yield a “local minimum” rather than a “global minimum”.
- Conformer searches move the nuclei into different conformations to find the one
that yields the lowest energy.
- To find an accurate geometry of large molecules and ring molecules, conformer
searching is usually necessary.
Dipole Moments/Magnetic Properties
- Once a wavefunction is found, molecular properties can be found be calculating
expectation values.
- Calculating dipole moments (and other multipole moments) are valuable in the
study of intermolecular forces and certain types of spectroscopy.
- Magnetic properties such as NMR shielding tensors can be calculated (used to
assign peaks in an NMR spectrum).
Population Analyses
- With a population analysis, one can calculate the charges of atoms within the
molecule by calculating the number of electrons within each of the atomic orbitals
used to make the molecule orbital.
- Clever application of population analyses can be used to describe the bonding
within a molecule based on the overlap of atomic orbitals. One application would
be examine the hybridization of an atom involved in a chemical bond.
Electron Densities/Electrostatic Potentials/Molecular Orbitals
- The wavefunction can be used to find the electron density at any point (therefore,
every point) within a molecule. These calculations can be used to visualize the total
electron cloud of the molecule.
- Another useful property is the electrostatic potential, that is, the voltage that
surrounds a molecule.
- The electrostatic potential is useful since it shows the tendency of how an outside
charge will be attracted to the molecule. (Negative charge is attracted to positive
potential and positive charge is attracted to negative potential.)
- The total wavefunction is product of molecular orbitals. One can break apart the
wavefunction and visualize the individual orbitals.
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Electronic Transition Energies
- Most calculations yield two types of molecular orbitals
a) Occupied orbitals – orbitals that are occupied by a pair of electrons in the
molecule’s ground state
b) Virtual orbitals – orbitals higher in energy than the occupied orbitals where
electron may go when the molecule is in an excited state.
- By taking the difference between the energies of the individual orbitals, one
calculate transition energies.
- Also important are transition intensities (how likely is the transition to occur). With
the total electronic wavefunction and the vibrational wavefunctions, one can
calculated electronic transition intensities.
Vibrational Transition Energies
- By moving the nuclei slightly, a vibration’s potential energy curve can be
calculated.
- From the potential energy curve, the vibration’s force constant can be found.
- Thus vibrational energies and frequencies can be found.
Thermochemical Properties
- Quantum mechanics connects to thermodynamics through the study of statistical
thermodynamics.
- By taking a special sum of quantum mechanical energies, one can calculate any
thermodynamic quantities such as the enthalpy or entropy.
- Thus after finding electronic, vibrational, rotational energies, thermodynamic
quantities can be found.
Transition States
- We can find transition states (activated complexes) for reactions by searching for
vibrations with negative force constants. Such a situation corresponds to a potential
energy that is approximately quadratic but is concave down rather than concave up.
- Vibrations with negative force constant yield vibrational imaginary frequencies.
Thus looking at a vibration with imaginary frequencies should yield information
about the transition state.
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Basis Sets
Introduction: A Problem
To find a wavefunction for a molecule, we attempt to find an appropriate Linear
Combination of Atomic Orbitals (LCAO). To do the linear combination, we need
atomic orbitals. The question arises: What atomic orbitals shall we use? The atomic
orbitals of the hydrogenic atom seem like a sensible choice. However, those orbitals
were found without considering any electron repulsion or electron exchange.
Solution: Slater orbitals
A solution can be found by using orbitals that use a couple of adjustable parameters, an
effective nuclear charge, Z , and an effective principal quantum number, n*.
  r, ,    Nr
 n 1

e
 Zr 
   
 n a0 
Yml  ,  
Orbitals of this form are called Slater orbitals. The Z and n* are a way of accounting
for the shielding of the nuclear charge from the other electrons in the atom. For each of
the orbitals in each atom, a set of Z and n* can be found through pain-staking
comparisons of calculated results with experimental results. Thus once thus this painstaking optimization is done; we are able to use the Slater orbitals as a basis set to
construct molecular orbitals.
A New Problem
The problem with using Slater orbitals is that there are no shortcuts when we need to
perform integrals such as  *H d with exponential functions in the atomic orbitals.
The computational resources needed to calculate such integrals are expensive. However
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there are lots of shortcuts to do integrals using Gaussian functions, er .
A New Solution
To take advantage of the ease of doing integrals with Gaussian functions, we need to find
a way to approximate Slater orbitals using Gaussian functions. Modern basis sets use a
sum of Gaussian functions to approximate each of the Slater atomic orbitals.
(Additionally, Cartesian functions replace the spherical harmonics.)
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Nomenclature of Gaussian Basis Sets
STO – nG (STO – 3G)
- The n refers to a fixed sum of n Gaussian functions used to describe each Slater
orbital.
6-31G
- The first number (6) refers to how many Gaussian functions are used in a fixed sum
to describe to core orbitals.
- The second two numbers related the valence orbitals are described by a fixed sum
of 3 Gaussian functions and 1 other Gaussian function. When doing the variational
calculation, both sets are allowed to be varied independent of each other.
6-31G*
- The * refers to the inclusion of d orbitals to describe the p block atoms.
- These d orbitals are named as polarization functions.
6-31G**
- The extra * refers to the inclusion of p orbitals (as polarization functions) to
describe the hydrogen atom.
6-311G
- The second three numbers related the valence orbitals are described by a fixed sum
of 3 Gaussian functions, another 1 Gaussian function and another 1 Gaussian
function. When doing the variational calculation, all three sets are allowed to be
varied independent of each other.
6-31+G
- The + refers to the inclusion Gaussian functions with very small exponents for the
non-hydrogen atoms.
- These diffuse functions allow electron density very far away from the nucleus to be
adequately described.
6-31++G
- The extra + refers to the inclusion of diffuse functions for the hydrogen atoms.
6-31++G(3d, 2p)
- The 3d refers to the 3 extra d functions used to describe the main group elements.
- The 2p refers to the 2 extra p functions used to describe the hydrogen atoms.
Last Word
The bigger the basis set used, the better the calculation will be. However, bigger basis
sets mean more computer time. Practical choice of a basis set means balancing accuracy
with computation time.
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Molecular Mechanics Calculations
Method
1. Treat each atom in a molecule, not as a collection of electrons with a nucleus, but as a
collection of electric fields (force fields).
a) Different atoms have different force fields.
b) Different bonding schemes of the same element have different force fields
- I.e., a doubly bonded oxygen has a different force field than an oxygen with
single bonds.
2. Connect the atoms, using routine rules of chemical bonding (Lewis structures and
such).
3. Calculate an interaction energy with all of the electric fields from the atoms.
4. Move the atoms to different geometries to find a geometry with a minimum
interaction energy.
5. Force fields that are commonly used.
a) SYBYL
b) MMFF
Applications
1. Initial guess of molecular geometry for small molecules
2. Finding molecular geometries for large molecules such as proteins and nucleic acids.
Ab Initio Methods
Method
1. Treat each electron in a molecule independently.
a) A restricted calculation put two electrons into each orbital without consideration
of the electron spin
b) An unrestricted calculation considers each spin orbital to be independent, e.g.,
1s is independent of 1s.
2. Construct Hamiltonian including a method to treat electron repulsion and exchange.
3. All ab initio methods begin with a Hartree-Fock calculation where the electron
repulsion is treated as the interaction between the electron and an average of electron
density of the other electrons in the molecule.
4. The difference between the true energy and the HF energy is called the correlation
energy.
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5. Two common methods exist to attempt to calculate some of the correlation energy,
thus yield a total electronic energy lower than the HF energy.
a) Møller-Plesset Perturbation Theory
1. The electron repulsion is treated as a perturbation of the HF wavefunctions.
2. Perturbation theory is applied from second-order (MP2 method) to fifthorder (MP5).
3. Energies may be lower than true energy. (whereas HF energy can’t be lower.)
b) Configuration Interaction (CI)
1. Recall that the ground state wavefunction is a Slater determinant that
accounts for the indistinguishability and antisymmetry of the electrons.
2. The Configuration Interaction method includes determinants where some of
the ground state occupied orbitals are replaced with virtual orbitals.
a) When one occupied orbital is replaced with a virtual orbital, the
determinant is called a singly excited determinant.
b) When two occupied orbitals are replaced with two virtual orbitals, the
determinant is called a doubly excited determinant, etc…
c) Various CI methods are different in what kind of determinants are
summed together to make the wavefunction.
i) CIS – wavefunction is variationally determined sum of singly excited
determinants.
ii) CISD – wavefunction is a sum of singly and double excited
determinants.
iii) CISDTQ – wavefunction is a sum of singly, doubly, triply,
quadruply excited determinants.
iv) Full CI – wavefunction includes all possible determinants.
d) CI calculations are very expensive; however, they usually do an
excellent job in calculating most of the correlation energy. As the
number of determinants used in the calculation increases, the electronic
energy decreases.
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Density Functional Theory
An alternative method in which all electrons are considered in the calculation is density
functional theory (DFT).
Method
1. Find a Hamiltonian is terms of the total electron density rather than the wavefunction.
a) Many terms of the Hamiltonian will be similar
b) The correlation and exchange terms will be different.
2. Use a HF calculation to find the electron density, then use DFT to find more accurate
electron density.
3. Different DFT methods vary on how the correlation and exchange terms are
constructed.
a) Vosko, Wilks and Nusair (VWN)
b) Becke 3 term correlation with Lee-Yang-Parr exchange (B3LYP)
c) Becke exchange, Perdew correlation (B3P86)
d) Becke exchange, Perdew and Wang correlation (B3PW91)
4. In theory, DFT methods are much simpler to use to calculate; however, the accuracy
of the method is dependent on the quality of the exchange functional used. DFT
methods compare in accuracy to HF methods presently but won’t improve until better
functionals can be found.
Semi-Empirical Methods
1. Calculations can be made much simpler if only the valence electrons can be
considered. Semi-empirical calculations use only valence electrons as well as
experimental based parameters for various integrals needed to greatly simplify to
variational calculations.
2. Methods vary on the parameters used to treat the core electrons as well as parameters
used to treat bonding interactions between neighboring atoms.
- Parameters are often set to attempt to reproduce thermochemical data.
3. The advantage of semi-empirical methods is that bigger molecules and most
chemically interesting systems can be modeled since few electrons are being used in
the calculations.
4. The first semi-empirical method was the Huckel method which will be studied later in
the course.
5. Some modern methods include
a) MNDO
b) ZINDO
c) AM1
d) PM3
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Atoms in Molecules (AIM)
After a molecular wavefunction is found, the electron density can be found from the
square of the wavefunction. The electron density is a three-dimensional scalar function
that can be analyzed in terms of its maxima and minima to understand the structure and
properties of the molecule.
Critical points
A three-dimensional scalar function commonly has four types of critical points.
Nuclear attractors
When a point has maximum density in all three perpendicular directions, the point is
labeled as a (3, -3) critical point.
Such critical points correspond to the position of nuclei.
Bond critical points
When a point has maximum density in two directions and a minimum in one
direction, the point is labeled as a (3, -1) critical point.
Such a point corresponds to the electron density between two atoms that have formed
a chemical bond.
Ring critical points
When a point has maximum density in one direction and a minimum in two
directions, the point is labeled as a (3, +1) critical point.
Such a point corresponds to the electron density in the middle of ring such as
cyclopropane or benzene.
A ring must form when n atoms are connected with n bonds where n>2.
Cage critical points
When a point has a minimum in all three directions, the point is labeled as a (3, +3)
critical point.
Such a point corresponds to the electron density in the middle of cage such as
tetrahedrane, C4H4 or cubane, C8H8.
A cage usually will form where 3 or more rings share 3 atoms.
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Laplacian
Recall that the Laplacian is the scalar second derivative of a scalar function. In one
dimension, the second derivative yield information about the curvature of the function.
In relation to electron density, the second derivative yields information about the
tendency of the electron density to concentrate or deplete.
a) A negative Laplacian shows where electron density is concentrating.
1) “Lone pairs” are found where the Laplacian is negative.
b) A positive Laplacian shows where the electron density is depleting.
c) The Laplacian at a bond critical point yields information about the nature of the
bonding.
1) A negative Laplacian (charge concentration) indicates covalent bonding
2) A positive Laplacian (charge depletion) indicates ionic bonding.