Download Molecular Quantum Chemistry

Molecular Quantum
Chemistry
Dimensionality
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How many dimensions are there, formally? To define
an atom’s location in 3-dimensional space in
principle takes 3 coordinates (e.g., x, y, and z in
some laboratory reference frame)
But, the problem should not depend on the absolute
location of the atoms, only on their location relative
to one another (i.e., the molecular geometry)
So, a typical system has the same dimensionality as
the number of degrees of freedom of the molecule
or system (3N-6 where N>2 is the number of atoms).
Potential energy surface
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Normally used for single molecules - for collections
of molecules, thermodynamics comes into play,
although people still think in terms of PES.
Captures the idea that each structure - that is,
geometry - has associated with it a unique energy
Since geometry changes are smooth, this creates a
smooth energy “landscape”
Chemistry becomes topology...
Potential energy surface
Simplest PES is 1-Dimensional - the diatomic
bond stretch:
Potential energy surface
For more dimensions we take “slices”:
From Cramer, Essentials of Computational Chemistry
Potential energy surface
A reaction pathway is a 1-D slice of a PES.
ABCA+BC pathway:
From Cramer, Essentials of Computational Chemistry
Potential energy surface
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Based on the Born-Oppenheimer approximation
Inherently classical with respect to the nuclei (but
quantum in the electronics)
Does not show kinetic energy! Total energy (PE +
KE) is conserved at a constant value.
At T = 0 K, our molecule will want to be at the lowest
possible potential energy, i.e., at a minimum on the
PES. Finding ways of locate the minima efficiently is
essential in computational chemistry
Between any two minima (valley bottoms) the lowest
energy path will pass through a maximum at a
saddle point. In chemistry, we call that saddle point a
transition-state structure.
Potential energy surface
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From Lewars, Computational Chemistry
Features: minima, saddle
points;
How do we describe these
mathematically?
Reaction coordinates are
usually not one of the
normal coordinates, but
composites.
Potential energy surface
Fundamental points:
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It’s clear the PES is useful, so how can I construct it
for an arbitrary system (defined simply by the
molecular formula)?
It seems that for equilibrium and rate constants, I
don’t actually need to know the whole surface, only
the energies of critical points (minima and saddle
points) - is there a way to find these without
mapping out the entire surface in detail?
If I don’t do the whole surface, can I be sure that I
know about the locations of all the critical points and
how they relate to one another?
What about the KE?
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Remember that we are only considering
nuclei for the PES.
It is intrinsically related to temperature. At 0K,
only vibrational kinetic energy.
In more accurate calculations, ZPE should be
added to the PES.
We need to go to statistical mechanics to
truly incorporate temperature!
Basis sets - LCAO solution
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The form of the Fock matrix elements is:
Density matrix elements (the
total electron density in the
overlap region of θl and θm)
Basis sets - LCAO solution
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SCF procedure:
Initial density
matrix guess
Form Fock
matrix
iterate
Diagonalize
Fock matrix
Form new
density matrix
2-e integrals
HF - SCF solution
Points to note:
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Fock operator depends on occupied MO’s only;
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The procedure produces Nbasis total MO’s;
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There are Nbasis-Nelec (or Nbasis-Nelec/2) unoccupied MO’s;
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These have no direct physical interpretation except as electron
affinities.
There are Nbasis4 two-electron integrals to solve - this is how the
method scales with problem size.
This can be reduced in actual calculations trough a number of tricks usually down to Nbasis2.
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Other basis sets may be used (like Gaussians or multiple zeta STO’s).
More about this in the basis sets section.
As basis set size is increases, a lower energy is obtained (as per the
variational principle). This is never the exact solution of the Sch.
equation.
Hartree-Fock
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For a molecule with n electrons, n spinorbitals are
obtained. Int(n/2) spatial orbitals are occupied
(occupied orbitals), n-Int(n/2) are unoccupied
(virtual orbitals).
If n is even and all the electrons are paired up
(closed shell), one can assume that the spatial
components are the same for a pair of electrons in
the same orbital (a restricted Hartree-Fock wavefn):
In this case, one only needs to find n/2 spatial
orbitals through the HF procedure.
Hartree-Fock
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Open shells occur when there are electrons that are not
paired up: when n is even but there is a triplet state (2S
+1=3) or when n is odd (a doublet: 2S+1=2)
In the restricted open shell formalism, all spatial orbitals for
paired-up electrons are considered to be identical. For Li,
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Often for a chemical problem a better way to go is an
unrestricted calculation, where all spinorbitals are calculated
independently:
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UHF gives a lower energy than RHF, but the problem is that
the UHF wavefunction is not an eigenfunction of the spin
operator S2: spin is undefined for a UHF wavefunction (this
is called the spin contamination problem).