Coordinate Systems for

Representing Molecules:

1. Cartesian (x,y,z) – common in MM

2. Internal coordinates (Z-matrix) – common in QM

** It is easy to convert between the two forms…HOW?

• Cartesian internal coordinates• internal coordinates Cartesian

Definitions

origin

site # type bl site # site # site #angle dihedral

DefinitionsPotential Energy Surfaces (PES):

1. Assume the energy (E) of a molecule is only a function of the nuclear coordinates.*

2. Mapping the E of the molecule as a function of the coordinates = PES

3. 1st derivative = 0, then all forces on atoms = 0. This is often a stationary point (may instead be a transition state).

* The Born-Oppenheimer approximation** is typically used in QM. This assumes that the electronic and nuclear motions of the molecules can be separated and is valid within the range: (me/m)0.25<<1. Mathematically this approximation can be written as:

** J. Robert Oppenheimer (1904-1967) was a graduate student of Born in 1927. Oppenheimer directed the Los Alamos Laboratory that developed the atomic bomb during WWII.

scoordinate nuclearq

scoordinate electronicq

qqqqq

i

Nieli

)();(),(

localequilibrium

Transitionstate

DefinitionsDefinitionsMolecular Graphics:

1. Ball and stick2. Space filling / CPK (Corey-Pauling-Koltun)3. Useful, but atomic representations are arbitrary4. Software:

a) Rasmol (http://www.umass.edu/microbio/rasmol/)b) VMD (http://www.ks.uiuc.edu/Research/vmd/)c) pov-ray (http://www.povray.org/)

Example ‘xyz’ file:5CommentO 1.2 3.7 3.5N 2.3 4.5 5.6N 4.3 5.7 7.3C 0.3 3.6 7.7H 2.2 3.8 8.3

number of atoms in the list

atom type, x, y, z

DefinitionsDefinitionsComputer Hardware

(software will be discussed with each topic)

1. Parallel vs. Serial

2. Linux/Unix environment

3. Software: fortran, C/C++, commercial packages (see handout)

4. Processors: Intel Pentium/Xeon/Itanium, AMD Opteron, DEC Alpha, Power PC

5. Networking: gigabit ethernet, Myrinet, Infiniband, etc.

6. Beowulf cluster: ~8-64 processors

7. Supercomputers: 1,000+ processors (national labs, universities, government)

DefinitionsDefinitionsMath Concepts, Section 1.10 (Leach)

Be familiar with…

1. Vectors

2. Matrices/eigenvectors/eigenvalues

3. Complex #’s

4. Lagrange multipliers

5. Fourier transforms

Introduction to Quantum ChemistryIntroduction to Quantum ChemistryOrigin of Quantum MechanicsOrigin of Quantum Mechanics: : Early 1900’s – microscopic systems do NOT obey Early 1900’s – microscopic systems do NOT obey

the same rules that macroscopic particles obey – the same rules that macroscopic particles obey – Newtonian mechanics NOT Newtonian mechanics NOT SUFFICIENT!!SUFFICIENT!!

SOLUTION:SOLUTION: Develop a new set of mechanics to describe these systems. Develop a new set of mechanics to describe these systems.

RESULT:RESULT: Quantum Mechanics (QM) – energy is Quantum Mechanics (QM) – energy is quantizedquantized vs. previous vs. previous continuouscontinuous approximation (Newtonian). approximation (Newtonian).

Postulate of QMPostulate of QM: a : a wave functionwave function, , , exists for any chemical system and operators , exists for any chemical system and operators (functions) act on the wave function to return observable properties. (functions) act on the wave function to return observable properties. Mathematically:Mathematically:

HH==EE

HH is an is an operatoroperator (it operates on (it operates on ) and ) and EE is a scalar value for some observable is a scalar value for some observable property. In matrix form, property. In matrix form, HH could be an N could be an N××N square matrix, and N square matrix, and an N-element an N-element column vector.column vector.

The complex conjugate of The complex conjugate of is denoted is denoted * and the product |* and the product |*| is known as *| is known as the the probability densityprobability density of the wave function. This is also abbreviated as | of the wave function. This is also abbreviated as |||22..

Suppose we have a single particle, e, in a box, with the wave function Suppose we have a single particle, e, in a box, with the wave function ee, as shown , as shown

below. What is the value of the following expression?below. What is the value of the following expression?

?* dxdydzee

Introduction to Quantum ChemistryIntroduction to Quantum Chemistry

Introduction to Quantum ChemistryIntroduction to Quantum Chemistry

Introduction to Quantum ChemistryIntroduction to Quantum Chemistry

Introduction to Quantum Chemistry

n

Introduction to Quantum ChemistryIntroduction to Quantum Chemistry What is the What is the probabilityprobability of finding e in the left half of the box? of finding e in the left half of the box?

Although simple, this is powerful information – calculate dipole moments, Although simple, this is powerful information – calculate dipole moments, approximate electrostatic charges, etc.approximate electrostatic charges, etc.

Restrictions of the Wave Function:Restrictions of the Wave Function:

Normalized integral over all space of |Normalized integral over all space of |||22 must be unity must be unity

must be quadratically integrablemust be quadratically integrable

must be continuous and single-valuedmust be continuous and single-valued

The first derivative of The first derivative of should be continuousy should be continuousy

Wave FunctionWave Function analogous to an analogous to an oracleoracle – when queried with questions (by an – when queried with questions (by an operator), it returns answers (observables).operator), it returns answers (observables).

Hamiltonian OperatorHamiltonian Operator

The Hamiltonian operator (The Hamiltonian operator (HH) is an operator that returns the system energy, ) is an operator that returns the system energy, EE, , as an eigenvalue. Mathematically, this is the Schras an eigenvalue. Mathematically, this is the Schröödinger equation shown dinger equation shown earlier:earlier:

HH==EE

Introduction to Quantum ChemistryIntroduction to Quantum ChemistryHH takes into account five different contribution with respect to a molecule or atom: takes into account five different contribution with respect to a molecule or atom:

1.1. The KE of the electronsThe KE of the electrons

2.2. The KE of the protonsThe KE of the protons

3.3. Attraction between the electrons and protonsAttraction between the electrons and protons

4.4. Interelectronic repulsionsInterelectronic repulsions

5.5. Internuclear repulsionsInternuclear repulsions

** In some special cases, it is necessary to account for other peripheral interactions, such as ** In some special cases, it is necessary to account for other peripheral interactions, such as those of applied fields.those of applied fields.

The full Hamiltonian (including all five of these interactions) is written as:The full Hamiltonian (including all five of these interactions) is written as:

i k i k ji lk kl

lk

ijik

kk

ki

e r

ZZe

r

e

r

Ze

mmH

2222

22

2

22

ii and and jj run over electrons, run over electrons, kk and and ll run over nuclei, run over nuclei, mm is mass, is mass, ee is the charge on an is the charge on an electron, electron, ZZ is the atomic number, is the atomic number, rr is the distance between particles, and the Laplacian is the distance between particles, and the Laplacian operator (“del-squared”) is: operator (“del-squared”) is:

2 = 2

x2 + 2

y2 + 2

z2

Introduction to Quantum ChemistryIntroduction to Quantum ChemistryThe Variational PrincipleThe Variational Principle

Challenge:Challenge: obtaining the set of orthonormal wave functions.obtaining the set of orthonormal wave functions.

Pick an arbitrary function Pick an arbitrary function , where:, where: i

iic

is an eigenfunction and is a linear combination of the orthonormal wavefunctions, is an eigenfunction and is a linear combination of the orthonormal wavefunctions, ii

Normalization criteria restricts the coefficients:Normalization criteria restricts the coefficients:

1

1

2

2

ii

ijijji

ijjiji

jjj

iii

ccc

dcc

dccd

r

rr

What is the What is the energyenergy of our system using of our system using ??