Numerical Methods

Variational Principle

Say you have a system for which you know what the energy depends on, or in other words, you know the Hamiltonian H. If you cannot solve the Schr��dinger equation to figure out the wavefunction, you can guess any normalized wavefunction whatsoever, say ��, and it turns out that the expectation value of the hamiltonian for your guessed wavefunction will be greater than the actual ground state energy. Or in other words:

<math>E_

Unknown macro: {ground}

\le \left\langle\phi|H|\phi\right\rangle </math>

</center>

This holds for any �� you could have guessed!

For a hamiltonian <math>H</math> that describes the studied system and any normalizable function <math>\Psi</math> with arguments appropriate for the unknown wave function of the system, we define the functional.

<math> \varepsilon\left[\Psi\right] = \frac{\left\langle\Psi|\hat

Unknown macro: {H}

|\Psi\right\rangle}

Unknown macro: {leftlanglePsi|Psirightrangle}

</math>

</center>

The variational principle states that

<math>\varepsilon \geq E_0</math>, where <math>E_0</math> is the lowest energy eigenstate (ground state) of the hamiltonian
<math>\varepsilon = E_0</math> if and only if <math>\Psi</math> is exactly equal to the wave function of the ground state of the studied system.

The variational principle formulated above is the basis of the variational method used in quantum mechanics and quantum chemistry to find approximations to the ground state.

Your guessed wavefunction, <math>\psi</math>, can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal)

<math>\phi = \sum_

Unknown macro: {n}

c_

\psi_

Unknown macro: {n}

\,</math>

</center>

Then, to find the expectation value of the hamiltonian:

<math>\left\langle\phi|H|\phi\right\rangle = \left\langle\sum_

c_

Unknown macro: {n}

\psi_

|H|\sum_

Unknown macro: {m}

c_

\psi_

Unknown macro: {m}

\right\rangle \,</math>

<math>\left\langle\phi|H|\phi\right\rangle = \sum_

Unknown macro: {n}

\sum_

\left\langle c_

Unknown macro: {n}

\psi_

|E_

Unknown macro: {m}

c_

\psi_

Unknown macro: {m}

\right\rangle \,</math>

<math>\left\langle\phi|H|\phi\right\rangle = \sum_

Unknown macro: {n}

\sum_

c_

Unknown macro: {n}

^*c_

Unknown macro: {m}

E_

\left\langle\psi_

|\psi_

Unknown macro: {m}

\right\rangle \,</math>

<math>\left\langle\phi|H|\phi\right\rangle = \sum_

Unknown macro: {n}

c_

|^2 E_

Unknown macro: {n}

\,</math>

</center>

Now, the ground state energy is the lowest energy possible, i.e. <math>E_

\ge E_

Unknown macro: {g}

</math>. Therefore, if the guessed wave function <math>\psi</math> is normalized:

<math>\left\langle\phi|H|\phi\right\rangle \ge E_

\sum_

Unknown macro: {n}

c_

|^2 = E_

Unknown macro: {g}

\,</math>

<math>E[\phi] - E_o = \frac

Unknown macro: {sum_n |a_n|^2(E_n - E_o)}

Unknown macro: {sum_n |a_n|^2}

</math>

</center>

Energy of a Hydrogen Atom

<math>E_

Unknown macro: {alpha}

= \frac{ < \Psi_

| \hat H | \Psi_

Unknown macro: {alpha}

> }{ \Psi_

| \Psi_

Unknown macro: {alpha}

> }</math>

<math>\Psi_

= C e^{(- \alpha r)</math>

<math>< \Psi_

Unknown macro: {alpha}

\Psi_

> = \pi \frac

Unknown macro: {C^2}

Unknown macro: {alpha^3}

</math>

<math>< \Psi_

Unknown macro: {alpha}

\frac
Unknown macro: {1}

Unknown macro: {2}
\nabla^2

\Psi_

> = \pi \frac

Unknown macro: {2 alpha}

</math>

<math>< \Psi_

Unknown macro: {alpha}

\frac
Unknown macro: {1}

Unknown macro: {r}
\nabla^2

\Psi_

> = \pi \frac

Unknown macro: {C^2}

Unknown macro: {alpha^2}

</math>

</center>

Energy of a collection of atoms

<math>\hat H = \hat T_e + \hat V_

Unknown macro: {e-e}

+ \hat V_

Unknown macro: {e-N}

+ V_

Unknown macro: {N-N}

</math>

<math>T_e: \mbox

Unknown macro: {quantum kinetic energy of the electrons}

</math>

<math>\hat T_e = - \frac

Unknown macro: {1}

Unknown macro: {2}

\sum_i \nabla_i^2</math>

<math>V_

: \mbox

Unknown macro: {electron-electron interactions}

</math>

<math>\hat V_

Unknown macro: {e-e}

= \sum_i \sum_

Unknown macro: {j>1}

\frac

Unknown macro: {1}

Unknown macro: {|vec r_i vec r_j|}

</math>

<math>V_

Unknown macro: {e-N}

: \mbox

Unknown macro: {electrostatic electron-nucleus attraction}

</math>

<math>\hat V_

= \sum_i \left [\sum_

Unknown macro: {I}

V ( \vec R_I - \vec r_i ) \right]</math>

<math>V_

Unknown macro: {N-N}

: \mbox

Unknown macro: {electrostatic nucleus-nucleus repulsion}

</math>

</center>

Molecules and Solids: Electrons and Nuclei

<math>\hat H \Psi ( \vec r_1, ..., \vec r_n, \vec R_1, ...., \vec R_N ) = E_

Unknown macro: {tot}

\Psi ( \vec r_1, ..., \vec r_n, \vec R_1, ...., \vec R_N )</math>

</center>

Treat only the electrons as quantum particles in the field of fixed or slowly varying nuclei. This is generally called the adiabatic or Born-Oppenheimer approximation. "Adiabatic" means that there is no coupling between different electronic surfaces; "B-O" implies there is no influence of the ionic motion on one electronic surface.

Linear Combination of Atomic Orbitals

This is the most common approach to find out the ground-state solution. It allows a meaningful definition of "hybridization", "bonding" and "anti-bonding" orbitals. It is also known as LCAO, LCAO-MO (molecular orbitals), or tight-binding (solids). The trial wavefunction is a linear combination of atomic orbitals. The variational parameters are the coefficients.

<math>\Psi_

Unknown macro: {trial}

= \sum_

Unknown macro: {I, (nlm)}

= c_

Unknown macro: {(nlm)}

^I \Psi_

^I (\vec r - \vec R_I)</math>

<math>E_

Unknown macro: {LCAO}

= \mbox

Unknown macro: {min}

\frac{ < \Psi_

| \hat H | \Psi_

Unknown macro: {trial}

> }{ <\Psi_

| \Psi_

Unknown macro: {trial}

></math>

</center>

Huckel approach

Huckel: planar / quasi-planar systems with delocalized <math>\pi</math> bonding. Consider two parameters.

<math>\alpha</math>: matrix element between same orbital
<math>\beta</math>: matrix element between neighboring orbitals
Hamiltonian between further neighbors is zero

Consider benzene

<math>\det \begin

Unknown macro: {bmatrix}

\alpha - E & \beta & 0 & 0 & 0 & \beta
\beta & \alpha - E & \beta & 0 & 0 & 0
0 & \beta & \alpha - E & \beta & 0 & 0
0 & 0 & \beta & \alpha - E & \beta & 0
0 & 0 & 0 & \beta & \alpha - E & \beta
\beta & 0 & 0 & 0 & \beta & \alpha - E \end

= 0</math>

</center>

Reference:

3.23 Fall 2006 Course Notes http://stellar.mit.edu/S/course/3/fa06/3.23/courseMaterial/topics/topic2/lectureNotes/lectures_13_and_14/lectures_13_and_14.pdf

Hartree

Reduce the interacting many-electron system to an individual electron problem in an effective potential. This potential should be determined self-consistently by all other electrons in the system. Neglect the antisymmetric requirement as a first step, and the total wavefunction for a system with <math>N</math> electrons can be written as the product of one-electron wavefunctions.

<math>\Psi ( \vec r_1, ..., \vec r_N ) = \Pi_

Unknown macro: {i=1}

^N \Psi_i (\vec r_1)</math>

</center>

Hartree suggested a variational calculation to minimize the energy.

<math>E = \frac

Unknown macro: { <Psi | H | Psi > }

Unknown macro: { <Psi | Psi ></math> </center> If <math>Psi</math> were the exact ground state wavefunction of the system, then <math>E</math> would be the ground state energy. The variational principle states that <math>E</math> is stationary with respect to variation of <math>Psi</math>, and is an upper bound to the ground state energy. The procedure leads to a set of Hartree equations. <center> <math>left [vec r - vec r'|}

\right|- \frac

Unknown macro: {h^2}

Unknown macro: {2m}

\nabla^2 + v(\vec r) + \sum'_j e^2 \int \frac

Unknown macro: {Psi_j^* (vec r') Psi_j (vec r') d vec r'}

{] \Psi_i(\vec r) = \epsilon_i \Psi_i (\vec r)</math>

</center>

The prime is used to rule out the possibility of <math>j=i</math> and <math>\epsilon_i</math> are variational parameters which look like the one-electron energy eigenvalues.

Reference

Introduction to Condensed Matter Physics http://books.google.com/books?vid=ISBN9812387110&id=-iuYN5arHwoC&pg=RA5-PA310&lpg=RA5-PA310&ots=RB5Msh3G_H&dq=Hartree&sig=zfRivZs-fGzfb_srOfn36UjaPnE#PRA5-PA309,M1

Mean-field approach

Independent particle model (Hartree): each electron moves in an effective potential representing the attraction of the nuclei and the average effect of the repulsive interactions of the other electrons
This average repulsion is the electrostatic repulsion of the average charge density of all other electrons.

Hartree Equations

The Hartree equations can be obtained directly from the variational principle, once the search is restricted to the many-body wavefunctions that are written as the product of spin-orbitals. Independent electrons are considered.

<math>\left [| \psi_j (\vec r_j ||^2 \frac

Unknown macro: {1}

{|\vec r_j - \vec r_i| d \vec r_j \right|\frac

Unknown macro: {2}

\nabla_i^2 + \sum_i V (\vec R_I - \vec r_i) + \sum_

Unknown macro: {j ne i}

\int] \psi_i (\vec r_i) = \epsilon \psi_i (\vec r_i)</math>

</center>

The self-consistent field

The single-particle Hartree operator is self-consistent. It depends on itself on the orbitals that are the solution of all other Hartree equations.
There are <math>n</math> simultaneous integro-differential equations for the <math>n</math> orbitals
Solution is achieved iteratively

Reference

3.012 Fundamentals of Materials Science Bonding: OCW Lecture Notes http://ocw.mit.edu/NR/rdonlyres/Materials-Science-and-Engineering/3-012Fall-2005/24F5BF05-5D52-4026-8E08-82C0112E643F/0/lec11b.pdf

Spin Statistics

All elementary particles are either fermions (half-integer spins) or bosons (integer)
A wavefunction that is antisymmetric by exchange is associated with a set of identical (indistinguishable) fermions
- <math>\Psi(\vec r_1, \vec r_2,...,\vec r_j,...,\vec r_k,...,\vec r_n) = \Psi(\vec r_1, \vec r_2,...,\vec r_k,...,\vec r_j,...,\vec r_n)</math>
For bosons, it is symmetric

Reference

3.012 Fundamentals of Materials Science Bonding: OCW Lecture Notes http://ocw.mit.edu/NR/rdonlyres/Materials-Science-and-Engineering/3-012Fall-2005/24F5BF05-5D52-4026-8E08-82C0112E643F/0/lec11b.pdf

Slater determinant

An antisymmetric wavefunction is constructed via a Slater determinant of the individual orbitals instead of just a product, as in the Hartree approach)

<math>\Psi(\vec r_1, \vec r_2, ..., \vec r_n) = \frac

Unknown macro: {1}

{\sqrt{n!}} \begin

Unknown macro: {Vmatrix}

\psi_

Unknown macro: {alpha}

(\vec r_1 ) & \psi_

Unknown macro: {beta}

(\vec r_1 ) & ... & \psi_

Unknown macro: {v}

(\vec r_1 )
\psi_

(\vec r_2 ) & \psi_

Unknown macro: {beta}

(\vec r_2 ) & ... & \psi_

Unknown macro: {v}

(\vec r_2 )
... & ... & ... & ...
\psi_

Unknown macro: {alpha}

(\vec r_n ) & \psi_

(\vec r_n ) & ... & \psi_

Unknown macro: {v}

(\vec r_n )\end

</math>

</center>

Consider an example of two electrons in <math>H_2</math>.

<math>\Psi(\vec r_1, \vec r_2) = \frac

{\sqrt{2}} \begin

Unknown macro: {Vmatrix}

\psi_

Unknown macro: {alpha}

(\vec r_1 ) & \psi_

Unknown macro: {beta}

(\vec r_1 )
\psi_

(\vec r_1 ) & \psi_

Unknown macro: {beta}

(\vec r_2 ) \end

</math>

<math>\Psi(\vec r_1, \vec r_2) = \frac

Unknown macro: {1}

{\sqrt{2}} \left [\psi_

Unknown macro: {alpha}

(\vec r_1 ) \psi_

Unknown macro: {beta}

(\vec r_1 ) - \psi_

(\vec r_2 ) \psi_

Unknown macro: {beta}

(\vec r_2 ) \right]</math>

</center>

Reference

3.012 Fundamentals of Materials Science Bonding: OCW Lecture Notes http://ocw.mit.edu/NR/rdonlyres/Materials-Science-and-Engineering/3-012Fall-2005/24F5BF05-5D52-4026-8E08-82C0112E643F/0/lec11b.pdf

Pauli Principle

If two states are identical, the determinant vanishes. There can't be two electrons in the same quantum state.

Reference

3.012 Fundamentals of Materials Science Bonding: OCW Lecture Notes http://ocw.mit.edu/NR/rdonlyres/Materials-Science-and-Engineering/3-012Fall-2005/24F5BF05-5D52-4026-8E08-82C0112E643F/0/lec11b.pdf

Hartree-Fock

In computational physics and computational chemistry, the Hartree-Fock (HF) method is an approximate method for the determination of the ground-state wavefunction and ground-state energy of a quantum many-body system.

The Hartree-Fock method assumes that the exact, N-body wavefunction of the system can be approximated by a single Slater determinant (in the case where the particles are fermions) or by a single permanent (in the case of bosons) of N spin-orbitals. Invoking the variational principle one can derive a set of N coupled equations for the N spin-orbitals. Solution of these equations yields the Hartree-Fock wavefunction and energy of the system, which are approximations of the exact ones.

The Hartree-Fock method finds its typical application in the solution of the electronic Schr��dinger equation of atoms, molecules and solids but it has also found widespread use in nuclear physics. See Hartree-Fock-Boloyubov for a discussion of its application in nuclear structure theory. The rest of this article will focus on applications in electronic structure theory.

The Hartree-Fock method is also called, especially in the older literature, the self-consistent field method (SCF) because the resulting equations are almost universally solved by means of an iterative, fixed-point type algorithm (see the following section for more details). This solution scheme is not the only one possible and is not specific of the Hartree-Fock method. Therefore "self-consistent field" is a potentially ambiguous denomination.

Algorithm

The Hartree-Fock method is typically used to solve the time-independent Schr��dinger equation for a multi-electron atom or molecule described in the fixed-nuclei approximation by the electronic molecular Hamiltonian. Because of the complexity of the differential equations for any but the smallest systems, the problem is usually impossible to solve analytically, and so the numerical technique of iteration is used. The method makes four major simplifications in order to deal with this task:

The Born-Oppenheimer approximation is inherently assumed. The true wavefunction is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons.
Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely non-relativistic.
The basis set is composed of a finite number of orthogonal functions. The true wavefunction is a linear combination of functions from a complete basis set.
The energy eigenfunctions are assumed to be anti-symmetrized linear combinations of products of one-electron wavefunctions. The effects of electron correlation, beyond that of exchange energy resulting from the anti-symmetrization of the wavefunction, are completely neglected.

The variational theorem states that, for a time-independent Hamiltonian operator, any trial wavefunction will have an energy expectation value that is greater than or equal to the true ground state wavefunction corresponding to the given Hamiltonian. Because of this, the Hartree-Fock energy is an upper bound to the true ground state energy of a given molecule. The limit of the Hartree-Fock energy as the basis set becomes infinite is called the Hartree-Fock limit. It is a unique set of one-electron orbitals, and their eigenvalues.

The starting point for the Hartree-Fock method is a set of approximate one-electron orbitals. For an atomic calculation, these are typically the orbitals for a hydrogenic atom (an atom with only one electron, but the appropriate nuclear charge). For a molecular or crystalline calculation, the initial approximate one-electron wavefunctions are typically a linear combination of atomic orbitals. This gives a collection of one electron orbitals that, due to the fermionic nature of electrons, must be anti-symmetric. This antisymmetry is achieved through the use of a Slater determinant.

At this point, a new approximate Hamiltonian operator, called the Fock operator, is constructed. The first terms in this Hamiltonian are a sum of kinetic energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear-electronic coulombic attraction terms. The final set of terms models the electronic coulombic repulsion terms between each electron with a sum. The sum is composed of a net repulsion energy for each electron in the system, which is calculated by treating all of the other electrons within the molecule as a smooth distribution of negative charge. This is the major simplification inherent in the Hartree-Fock method, and is equivalent to the fourth simplification in the above list, (see post-Hartree-Fock).

The newly constructed Fock operator is then used as the Hamiltonian in the time-independent Schr��dinger Equation. Solving the equation yields a new set of approximate one-electron orbitals. This new set of orbitals is then used to construct a new Fock operator, as in the preceding paragraph, beginning the cycle again. The procedure is stopped when the change in total electronic energy is negligible between two iterations. In this way, a set of so-called "self-consistent" one-electron orbitals are calculated. The Hartree-Fock electronic wavefunction is then equal to the Slater determinant of these approximate one-electron wavefunctions. From the Hartree-Fock wavefunction, any chemical property of the system in question can be calculated in an approximate manner.

Mathematical Formulation

Because the electron-electron repulsion term of the electronic molecular Hamiltonian involves the coordinates of two different electrons, it is necessary to reformulate it in an approximate way. Under this approximation, (outlined under Hartree-Fock algorithm), all of the terms of the exact Hamiltonian except the nuclear-nuclear repulsion term are re-expressed as the sum of one-electron operators outlined below. The "(1)" following each operator symbol simply indicates that the operator is 1-electron in nature.

<math>\hat F(1) = \hat H^

Unknown macro: {core}

(1)+\sum_

Unknown macro: {j=1}

^

Unknown macro: {n/2}

[2\hat J_j(1)-\hat K_j(1)]</math>

</center>

where:

</center>

is the one-electron Fock operator,

<math>\hat H^

(1)=-\frac

\nabla^2_1 - \sum_

Unknown macro: {alpha}

\frac

Unknown macro: {Z_alpha}

{r_{1\alpha}}</math>

</center>

is the one-electron core Hamiltonian,

</center>

is the Coulomb operator, defining the electron-electron repulsion energy due to the j-th electron,

</center>

is the exchange operator, defining the electron exchange energy. Finding the Hartree-Fock one-electron wavefunctions is now equivalent to solving the eigenfunction equation:

<math>\hat F(1)\phi_i(1)=\epsilon_i \phi_i(1)</math>

</center>

where <math>\phi_i(1)</math> are a set of one-electron wavefunctions, called the Hartree-Fock Molecular Orbitals.

Example-benzene

Using the Huckel model, consider only the relevant molecular orbitals around the HOMO-LUMO gap. This is given by combinations of <math>p_z</math> orbitals sitting on each carbon atom. The basis consists of six orbitals, one on each of the six carbons of benzene. The diagaonal elements of the Hamiltonian are <math>\alpha</math> and the off-diagonal elements are zero. In the case that two orbitals are considered as next neighbors, the matrix elements is given by <math>-|\beta|</math>.

There are six eigenvalues found when solving the matrix involved with the Hamiltonian. As the number of atoms increases, the bandwidth between the lowest and the highest states tends to a constant (<math>-\beta</math>) and the gap goes to zero. In the rings the Hamiltonian commutes with the rotation operator. The formalism is identical to the case of solids, and the <math>N</math> atoms in the ring are equivalent to the <math>N</math> unit cells of the Born-von Karman conditions.

Hartree-Fock Equations

The Hartree-Fock equations are, again, obtained from the variational principle: we look for the minimum of the many-electron Schrodinger equation in the class of all wavefunctions that are written as a single Slater determinant.

<math>\left [-\frac

Unknown macro: {1}

Unknown macro: {2}

\nabla_i^2 + \sum_I V (\vec R_I - \vec r_i ) \right] \psi_

Unknown macro: {lambda}

(\vec r_i) + \left [| \psi_

Unknown macro: {mu}

(\vec r_j )||^2 \frac

Unknown macro: {|vec r_j - vec r_i | }

d \vec r_j \right|\sum_

Unknown macro: {mu}

\int] \psi_

Unknown macro: {lambda}

) - \sum_

\left [\vec r_j - \vec r_i |} \psi_

Unknown macro: {mu}

(\vec r_i) \psi_

Unknown macro: {lambda}

(\vec r_j) d \vec r_j \right|\int \psi_

^* \frac

Unknown macro: {1}

{] = \epsilon \psi_

Unknown macro: {lambda}

(\vec r_i)
</math>

<math>\Psi(\vec r_i,...,\vec r_n) = ||Slater||</math>

</center>

Reference

3.012 Fundamentals of Materials Science Bonding: OCW Lecture Notes http://ocw.mit.edu/NR/rdonlyres/Materials-Science-and-Engineering/3-012Fall-2005/24F5BF05-5D52-4026-8E08-82C0112E643F/0/lec11b.pdf

Tight binding

In the tight binding model, it is assumed that the full Hamiltonian <math>H</math> of the system may be approximated by the Hamiltonian of an isolated atom centred at each lattice point. The atomic orbitals <math>\Psi_n</math>, which are eigenfunctions of the single atom Hamiltonian Hat, are assumed to be very small at distances exceeding the lattice constant. This is what is meant by tight-binding. It is further assumed that any corrections to the atomic potential <math>\Delta U</math>, which are required to obtain the full Hamiltonian <math>H</math> of the system, are appreciable only when the atomic orbitals are small. The solution to the time-independent single electron Schr��dinger equation <math>\psi</math> is then assumed to be a linear combination of atomic orbitals

<math>\phi(\vec

Unknown macro: {r}

) = \sum_n b_n \psi_n(\vec

)</math>

</center>

This leads to a matrix equation for the coefficients <math>b_n</math> and Bloch energies <math>\varepsilon</math> of the form below.

<math>\varepsilon(\vec

Unknown macro: {k}

) = E_m - {\beta_m + \sum_{\vec

Unknown macro: {R}

\neq 0} \gamma_m(\vec

) e^{i \vec

\cdot \vec{R}}\over b_m + \sum_{\vec

Unknown macro: {R}

\neq 0} \alpha_m(\vec

) e^{i \vec

Unknown macro: {k}

\cdot \vec

Unknown macro: {R}

}}</math>

</center>

where <math>E_m</math> is the energy of the <math>m</math>th atomic level, and below are the overlap integrals.

<math> \beta_m = -\int \psi_m^*(\vec

Unknown macro: {r}

)\Delta U(\vec

) \phi(\vec

Unknown macro: {r}

) d\vec

</math>

<math> \alpha_m(\vec

) = \int \psi_m^*(\vec

Unknown macro: {r}

) \phi(\vec

-\vec

Unknown macro: {R}

) d\vec

Unknown macro: {r}

</math>

<math> \gamma_m(\vec

) = -\int \psi_m^*(\vec

Unknown macro: {r}

) \Delta U(\vec

) \phi(\vec

Unknown macro: {r}

-\vec

Unknown macro: {R}

) d\vec

</math>

</center>

The tight binding model is typically used for calculations of electronic band structure and energy gaps in the static regime. However, in combination with other methods such as the random phase approximation (RPA) model, the dynamic response of systems may also be studied.

Koopman's Theorems

The total energy is invariant under unitary transformation. It is not the sum of the canonical MO orbital energies. The ionization energy and electron affinity are given by the eigenvalue of the respective molecular orbital in the frozen orbitals approximation.

What is missing

Correlations(by definition)

Dynamical correlations: the electrons get too close to each other in H.-F.
Static correlations: a single determinant variational class is not good enough

Spin contamination

Even if the energy is correct (variational, quadratic) other properties might not (e.g. the UHF spin is an equal mixture of singlet and triplet)

Reference:
*3.23 Fall 2006 Course Notes http://stellar.mit.edu/S/course/3/fa06/3.23/courseMaterial/topics/topic2/lectureNotes/lectures_13_and_14/lectures_13_and_14.pdf

Child pages

Numerical Methods

Numerical Methods

Variational Principle

Energy of a Hydrogen Atom

Energy of a collection of atoms

Molecules and Solids: Electrons and Nuclei

Linear Combination of Atomic Orbitals

Huckel approach

Hartree

Spin Statistics

Slater determinant

Pauli Principle

Hartree-Fock

Algorithm

Mathematical Formulation

Example-benzene

Hartree-Fock Equations

Tight binding

Koopman's Theorems

What is missing