Intro to Quantum Chemistry

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Intro to Quantum Chemistry

• We need to solve the Schrodinger equation for all
  the electrons moving in the external field of the
  nuclei
• Ill assume youve had a pchem course in quantum
  mechanics or equivalent, but will review some
  basics

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Time independentSchrodinger equationin 1D

• is the wave function, which cant be
  directly measured
• E is the energy eigenvalue
• V(x) is the potential energy operator
• The first term is the kinetic energy operator
  acting on the wave function
• This is an eigenvalue differential equation
• Generally well be interested in the ground state
  solution so we can solve this TISE (no time
  evolution)

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Multiple electrons

• Imagine we have 6 electrons, they move in 1D, and
  they dont interact with each other
• Solving the SE for the electrons requires
  something more than just the equation above
• We need to account for the indistinguishability
  of the electrons. Thats the Pauli principle.
• One way of stating the Pauli principle is that,
  if we exchange two electrons, the sign of the
  wave function changes. Another way is that no
  two electrons can have the same set of quantum
  numbers. But remember we need to include spin as
  a quantum number.

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Independent electrons

• If 2 electrons are independent (dont interact)
  then the total wavefunction for the electrons
  breaks down into a product form, eg
• The simple expression for 2 electrons above
  neglects the spin part of the wave function,
  weve written just the spatial part. The
  anti-symmetry comes from the spin part.

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Independent electrons

• For this case of independent electrons, the total
  electron density is simply
• Doubly occupied states assumed also
• This is something that can be directly measured
  from Xray scattering
• Lets look at the solutions for the HO

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Harmonic Oscillator

• Potential
• Eigenvalues
• Wavefunctions

where
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Electrons in harmonic well

• Now we can view 6 non-interacting electrons in a
  harmonic well as occupying these 3 spatial states
• Each one gets 2 electrons, one with spin up and
  one with spin down
• Take a look at the spatial wavefunctions for the
  HO

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Harmonic oscillator wavefunctions
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Orthogonality

• Notice that the wave functions for different
  energy levels are orthogonal, and a given state
  is normalized
• Exercises 1) prove orthogonality for the first 3
  states of the harmonic oscillator listed above 2)
  show that the wave functions are solutions of the
  TISE. We will later see how the formula for the
  electron density is derived.

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Electron density

• So if we add up the squares for these 3 states,
  then we get an electron density with 3 small
  peaks in it
• Those peaks are purely due to quantum mechanics.
  Classical mechanics would predict a simple
  gaussian function for any temperature
• So these kind of oscillations are indicative of
  quantum effects

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More on the Pauli principle

• We need to give more details on the Pauli
  principle, namely how could we construct a
  wavefunction?
• In general we want to solve for the very
  complicated
• The coordinates listed here x1 etc are schematic
  for the 3 spatial coordinates of electron 1 and
  the spin coordinate, that is up or down

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Pauli principle

• But we know from chemistry that atoms have shell
  structures and some degree of independent
  particle behavior is maintained, even in
  multi-electron atoms. That is we can talk about
  1s, 2s, 2p, etc states and that has some meaning
• So we construct a many-electron wavefunction
  built up from products of H-atom type states
• Lets consider the He and Li atoms

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He atom wavefunction

• Spatial part is 1s(1)1s(2). The 1 and 2
  label the electron coordinates. Note this
  function is not antisymmetric wrt interchange of
  1 and 2.
• Spin part ??is up and ? is down possible combos
  are ?(1) ?(2), ?(1) ?(2), ?(1) ?(2)?(1) ?(2),
  ?(1) ?(2)-?(1) ?(2)
• First 3 are symmetric, last one is antisymmetric,
  thus we need that one for ground state
• We wont worry about normalizing here

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Li atom and Slater determinant

• Construct an approximate wavefunction for
  3-electron case? Slater determinant fits the
  bill
• Note that electron number labels are in the
  columns, particle states are in rows
• This expression satisfies PEP

Exercise Write out the Slater determinant for
the He atom and show that you get the
wavefunction discussed above
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Many-el wavefunctions

• Note that this is not the final picture, even
  though it does satisfy the PEP. We have
  approximated the complicated many electron wave
  function as some linear combination of products
  of one-electron states times spin functions.
  Those one-electron states are definitely
  approximate. The amazing thing is that this
  approximation does so well in practice. We will
  see later that the above approximation is the
  Hartree-Fock approximation.

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Elementary QM (Parr/Yang)

• Schrodinger equation for many electrons

where
and
(interaction of el with nuclei)
We are now using atomic units
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Variational Principle

• Expectation value of total energy
• Imagine approx wave function is linear
  combination of exact eigenstates
• We know
• So using orthogonality
• Thus the approx energy is always above the true
  ground state

Exercise prove for yourself the variational
principle
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Variational Principle

• So what we try to do is to vary parameters in our
  wave function to minimize the total energy. Then
  we have the best solution possible under our
  assumptions.
• Hartree-Fock theory applies the variational
  principle after assuming that we have a single
  Slater determinant for our wave function.

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Hartree-Fock theory

• HF wave function
• We plug this into the expression for the total
  energy, and get (assuming normalization)

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HF energies

• One particle
• Coulomb
• Exchange
• Notice JiiKii

Exercise show that the above expressions are
obtained from the HF wavefunction
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Minimize wrt Psi variation with orthonormality
constraint

• HF equations where

Notice this last operation is nonlocal, that is
the exchange operator is nonlocal, while Coulomb
operator is local
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HF energies

• We can express the total energy in terms of the
  Lagrange multipliers

Exercise prove the above expression for the
energy
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Total molecular energies

• Energy including n-n interactions
• What is missing? We assumed a Slater determinant
  wave function, and in reality the wave function
  is more complex. This leads to the concept of
  correlation energy. HF theory is a mean-field
  theory

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Electron density

• The electron density is defined as

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Hellmann-Feynman Theorem

• Forces on nuclei are purely electrostatic, due to
  electrons and other nuclei (only need electron
  density)
• This formula is used in the ab initio simulations
  discussed earlier

Exercise try to prove the H-F theorem above.
What conditions are necessary?
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Variational method

• Levine has a good description of both variational
  and perturbation methods
• Variational method express the wavefunction as
  function of some parameters
• Minimize the total energy wrt to variations of
  those parameters

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Perturbation theory

• Start with a known solution (could be hydrogenic
  states, or solution to HF equations)
• Then

Exercise evaluate the 1st order correction to
the energy of the harmonic oscillator with a
perturbation
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Moller-Plesset (MP) Theory

• This is a commonly used PT to go beyond HF theory
• It uses the HF solution as the zeroth-order
  state, then obtains corrections to energies and
  wave functions
• There are MP2, MP3, etc approximations
• Electron correlation is included approximately.
  MP2 does pretty well, and is computationally
  manageable.

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Density matrices

• Density matrices yield a compression of
  information, giving a physical quantity that can
  be used to represent the total energy in a more
  compact form
• Well see that when we can represent the wave
  function as a Slater determinant, the DM is easy
  to calculate, and looks like a simple extension
  of the electron density
• Oscillations of the DM in space are a
  manifestation of QM, and the decay properties of
  the DM allow for efficient calculations

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Density Matrices

• Projection operator
• Density operator
• This includes possibility of populations of
  multiple states at finite temperature
• Were mainly interested in ground state
  electronic structure, a pure state
• Then idempotency

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Reduced density matrices

• We can obtain low dimensional DMs by integrating
  out over other coordinates
• For example
• We can express the exact total energy in terms of
  only

and
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DM for HF theory

• We can express the HF total energy in terms of
• And for a single determinant the 1-DM is very
  simple
• Exercise show this using the orthogonality of
  the states

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Properties of the 1-DM

• The value of the 1-DM for
  is the electron density
• So the function starts at with the
  electron density and then eventually decays
  towards zero at large distances
• The decay is exponential if there is a nonzero
  HOMO/LUMO gap, that is if the material is not a
  metal