Introduction to Computational Chemistry. NSF Computational Nanotechnology and Molecular Engineering Pan-American Advanced Studies Institutes (PASI) Workshop January 5-16, 2004. California Institute of Technology, Pasadena, CA. Andrew S. Ichimura. For the Beginner….
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NSF Computational Nanotechnology and Molecular Engineering Pan-American Advanced Studies Institutes (PASI) Workshop January 5-16, 2004
California Institute of Technology, Pasadena, CA
Andrew S. Ichimura
There are three main problems:
1. Deciphering the language.
2. Technical implementation.
3. Quality assessment.
Calculating molecular structures and relative energies.
Ab initio electronic structure theory
Electron Correlation (MP2, CI, CC, etc.)
Goal: Insight into chemical phenomena.
What is a molecule?
A molecule is “composed” of atoms, or, more generally as a collection of charged particles, positive nuclei and negative electrons.
The interaction between charged particles is described by;
Coulomb interaction between these charged particles is the only important physical force necessary to describe chemical phenomena.
In Classical Mechanics, the dynamics of a system (i.e. how the system evolves in time) is described by Newton’s 2nd Law:
F = force
a = acceleration
r = position vector
m = particle mass
In Quantum Mechanics, particle behavior is described in terms of a wavefunction, Y.
Time-dependent Schrödinger Equation
If H is time-independent, the time-dependence of Y may be separated out as a simple phase factor.
Time-Independent Schrödinger Equation
Describes the particle-wave duality of electrons.
Sum of kinetic (T) and potential (V) energy
When these expressions are used in the time-independent Schrodinger Equation, the dynamics of all electrons and nuclei in a molecule or atom are taken into account.
H. + H.
The electronic Hamiltonian becomes,
B.O. approx.; fixed nuclear coord.
GOAL: Solve the electronic Schrödinger equation, HeY=EY.
PROBLEM: Exact solutions can only be found for one-electron systems, e.g., H2+.
SOLUTION: Use the variational principle to generate approximate solutions.
Variational principle - If an approximate wavefunction is used in HeY=EY, then the energy must be greater than or equal to the exact energy. The equality holds when Y is the exact wavefunction.
In practice: Generate the “best” trial function that has a number of adjustable parameters. The energy is minimized as a function of these parameters.
The energy is calculated as an expectation value of the Hamiltonian operator:
Introduce “bra-ket” notation,
complex conjugate , left
Combined bracket denotes integration over all coordinates.
If the wavefunctions are orthogonal and normalized (orthonormal),
Since electrons are fermions, S=1/2,the total electronic wavefunction must be antisymmetric (change sign) with respect to the interchange of any two electron coordinates. (Pauli principle - no two electrons can have the same set of quantum numbers.)
Antisymmetric wavefunctions can be written as
Consider a two electron system, e.g. He or H2. A suitable antisymmetric wavefunction to describe the ground state is:
(He: f1 =f2 = 1s)
(H2: f1 = f2 = fbonding MO)
Each electron resides in a spin-orbital, a product of spatial and spin functions.
Interchange the coordinates of the two electrons,
A more general way to represent antisymmetric electronic wavefunctions is in the form of a determinant. For the two-electron case,
For an N-electron N-spinorbital wavefunction,
A Slater Determinant (SD) satisfies the antisymmetry requirement.
Columns are one-electron wavefunctions, molecular orbitals.
Rows contain the electron coordinates.
One more approximation: The trial wavefunction will consist of a single SD.
Now the variational principle is used to derive the Hartree-Fock equations...
(1) Reformulate the Slater Determinant as,
Depends on two electrons
One electron terms
One-electron operator - describes electron i, moving in the field of the nuclei.
Two-electron operator - interelectron repulsion.
(3) Calculation of the energy.
Expectation value over Slater Determinant
Examine specific integrals:
Nuclear repulsion does not depend on electron coordinates.
The one-electron operator acts only on electron 1 and yields an energy, h1, that depends only on the kinetic energy and attraction to all nuclei.
Coulomb integral, J12: represents the classical repulsion between two charge distributions f12(1) and f22(2).
Exchange integral, K12: no classical analogue. Responsible for chemical bonds.
Sum of one-electron, Coulomb, and exchange integrals, and Vnn.
To apply the variational principle, the Coulomb and Exchange integrals are written as operators,
The objective now is to find the best orbitals (fi, MOs) that minimize the energy (or at least remain stationary with respect to further changes in fi), while maintaining orthonormality of fi.
Function to optimize.
Rewrite in terms of another function.
Define Lagrange function.
Constrained optimization of L.
Change in L with respect to small changes in fi should be zero.
Effective one-electron operator, associated with the variation in the energy.
Change in energy in terms of the Fock operator.
According to the variational principle, the best orbitals, fi, will make dL=0.
After some algebra, the final expression becomes:
HF equations in terms of Canonical MOs and diagonal Lagrange multipliers.
Lagrange multipliers can be interpreted as MO energies.
The HF equations cast in this way, form a set of pseudo-eigenvalue equations.
A specific Fock orbital can only be determined once all the other occupied orbitals are known.
The HF equations are solved iteratively. Guess, calculate the energy, improve the guess, recalculate, etc.
A set of orbitals that is a solution to the HF equations are called Self-consistent Field (SCF) orbitals.
The Canonical MOs are a convenient set of functions to use in the variational procedure, but they are not unique from the standpoint of calculating the energy.
The ionization energy is well approximated by the orbital energy, ei.
* Calculated according to Koopman’s theorem.
Two criteria for selecting basis functions.
I) They should be physically meaningful.
ii) computation of the integrals should be tractable.
(Molecular orbital = Linear Combination of Atomic Orbitals - LCAO).
LCAO - MO representation
Coefficients are variational parameters
HF equations in the AO basis
Matrix representation of HF eqns.
Roothaan-Hall equations (closed shell)
Fab - element of the Fock matrix
Sab - overlap of two AOs
Total Energy in AO basis
One-electron integrals, M2
Two-electron integrals, M4
Computed at the start; do not change
Products of AO coeff form Density Matrix, D
Obtain initial guess
for coeff., cai,form
the initial Dgd
Form the Fock matrix
Diagonalize the Fock Matrix
Form new Density Matrix
Restricted Hartree-Fock (RHF)
For even electron, closed-shell singlet states, electrons in a given MO with a and b spin are constrained to have the same spatial dependence.
Restricted Open-shell Hartree-Fock (ROHF)
The spatial part of the doubly occupied orbitals are restricted to be the same.
Unrestricted Hartree-fock (UHF)
a and b spinorbitals have different spatial parts.
Consequence of using a single Slater determinant and the Self-consistent Field equations:
Electron-electron repulsion is included as an average effect. The electron repulsion felt by one electron is an average potential field of all the others, assuming that their spatial distribution is represented by orbitals. This is sometimes referred to as the Mean Field Approximation.
Electron correlation has been neglected!!!