For a general system of nuclei and electrons, the total Hamiltonian (in atomic units) is
: mass of nucleus I
-sum over pairs of electrons i and j separated by distance 
-electron-electron repulsion
-sum over nuclei I of charge 
and electrons i,
where separation between nucleus I and electron i is given by 
-nuclear-electron attraction
-sum over pairs of nuclei I and J separated by distance 
-nuclear-nuclear repulsion
Born-Oppenheimer approximation: decouple electronic and nuclear motion
The Schrödinger equation for the electrons is
where the electronic Hamiltonian 
is
The nuclear Hamiltonian is
Born-Oppenheimer approximation:
nuclei move in the average field of the electrons
A. Hartree-Fock Molecular Orbital Methods
A. Szabo and N.S. Ostlund, Modern Quantum Chemistry, (McGraw Hill, Inc., New York, 1989).
Hartree-Fock theory is a way to solve for 
and
Notation
-an electron is described by spatial coordinates 
and a spin coordinate 
(up or down)
-combine these 4 coordinates in the variable 
-for simplicity, write electronic wave function as 
(i.e. omit argument 
)
Antisymmetry principle (Pauli exclusion principle):
a many-electron wave function is antisymmetric with respect to the interchange of the coordinate of any two electrons
The simplest antisymmetric wave function that describes the ground state of an N-electron
system is a single Slater determinant:
is a spin orbital and is a product of a spatial and a spin orbital
After some algebraic manipulations, we can show that
where
Variational principle: minimize the energy 
subject to orthonormality constraint on spin orbitals
The Hartree-Fock energy is an upper bound to the exact energy.
-Expand the spin orbitals in a basis set 
(typically atom-centered Gaussian functions so that integrals can be calculated easily)
-Vary the coefficients to minimize the energy subject to the orthonormality constraint
This variational procedure leads to the Hartree-Fock equations
where 
is the exchange operator, defined as
This can be written as
where
Eliminate spin
RHF (restricted Hartree-Fock)
-all electrons paired
-each spatial orbital doubly occupied
where
The problem can be formulated as an eigenvalue equation ( Hartree-Fock-Roothaan equations)
F: the Fock matrix
S: overlap matrix
C: the coefficient matrix (each column contains expansion coefficients for a molecular orbital)
: diagonal matrix of orbital energies 
SCF (self-consistent-field procedure)
Since F depends on C, the eigenvalue equation must be solved self-consistently
Start with an initial guess for C
Repeat the following steps until convergence
1. calculate F(C)
2. derive a new C by solving the eigenvalue equation 
B. Correlation
the difference between the exact nonrelativistic energy and the Hartree-Fock
energy in the limit of a complete basis set
Adding correlation to ab initio methods:
1. perturbation theory (i.e., MP2)
-total Hamiltonian is divided into two pieces
, which has known eigenfunctions and eigenvalues
V, which is a perturbation
-the exact energy is expressed as an infinite sum of contributions that will
converge if V is small
2. configuration interaction (CI)
-wave function is represented as a linear combination of N-electron trial functions
(i.e. N-electron Slater determinants)
-linear variational method is used to minimize the energy
3. other methods: coupled cluster, MCSCF
C. Semiempirical methods
approximate the integrals (or matrix elements) with analytical functions containing
parameters that are fit by comparing calculated information such as energy and
geometry to experimental data
Advantages
-the integrals are faster to compute
-correlation can be included by fitting the parameters to experiment
Disadvantage
-the fitting procedure is subjective and not always accurate
Examples: MNDO, AM1
D. Density functional theory
T. Ziegler, Chem. Rev. 91, 651 (1991).
The energy of an electronic system can be expressed in terms of its
density
where the sum is over the N one-electron spatial orbitals 
The electronic energy of an N-electron system is (without approximation) is:
-the exchange-correlation energy
-many different approximations have been used
Local density approximations
(LDA): based on the homogeneous electron gas
Kohn-Sham equations for the 1-electron orbitals:
is the exchange-correlation potential and is the functional derivative of
:
Thus the DFT equations can be formulated similary to the Hartree-Fock equations, and
the resulting eigenvalue equation can also be solved self-consistently.
For calculations on solids the basis functions are typically plane waves instead of Gaussian functions.
E. Valence bond theory
-wave function is represented by a linear combination of pure resonance structures
(instead of individual nuclei and electrons)
-particularly useful for reactions where physical insight 
a small number of significant resonance structures
Examples:
1. Diatomic molecule HF
=HF (covalent)
=H 
F 
(ionic)
total wavefunction is a linear combination of these 2 resonance structures:
2. Simple proton transfer reaction
RXH + Y RX + HY
=RXH Y:
=RX: HY
=RX: H Y:
total wavefunction is a linear combination of these 3 resonance structures:
Energy is 
Variational principle minimize
To obtain the coefficients C and the ground state energy, solve the equation
HC=SCe
H: Hamiltonian matrix with elements
lowest eigenvalue is the ground state energy
Empirical valence bond (EVB) method
Arieh Warshel, Computer Modeling of Chemical Reactions in Enzymes and Solutions,
(John Wiley 
Sons, Inc., New York, 1991)
-approximate matrix elements of H with empirical analytical functions
-assume overlap matrix S is the identity matrix (i.e. assume that effects of non-zero overlap integrals can be absorbed in parametrization of off-diagonal elements of H
: energy of resonance structure i in the gas phase
off-diagonal elements 
: coupling between the resonance structures i and j in gas phase
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