1. Coordinate Systems for Representing Molecules : 1. 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 # angle dihedral

2. Definitions Potential Energy Surfaces (PES): 1. 1.Assume the energy (E) of a molecule is only a function of the nuclear coordinates.* 2. 2.Mapping the E of the molecule as a function of the coordinates = PES 3. 3.1 st 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: (m e /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. local equilibrium Transition state

3. Definitions Molecular Graphics: 1. 1.Ball and stick 2. 2.Space filling / CPK (Corey-Pauling-Koltun) 3. 3.Useful, but atomic representations are arbitrary 4. 4.Software: a) a)Rasmol (http://www.umass.edu/microbio/rasmol/) b) b)VMD (http://www.ks.uiuc.edu/Research/vmd/) c) c)pov-ray (http://www.povray.org/) Example ‘xyz’ file: 5 Comment O 1.2 3.7 3.5 N 2.3 4.5 5.6 N 4.3 5.7 7.3 C 0.3 3.6 7.7 H 2.2 3.8 8.3 number of atoms in the list atom type, x, y, z

4. Definitions Computer Hardware (software will be discussed with each topic) 1. 1.Parallel vs. Serial 2. 2.Linux/Unix environment 3. 3.Software: fortran, C/C++, commercial packages (see handout) 4. 4.Processors: Intel Pentium/Xeon/Itanium, AMD Opteron, DEC Alpha, Power PC 5. 5.Networking: gigabit ethernet, Myrinet, Infiniband, etc. 6. 6.Beowulf cluster: ~8-64 processors 7. 7.Supercomputers: 1,000+ processors (national labs, universities, government)

5. Definitions Math Concepts, Section 1.10 (Leach) Be familiar with… 1. 1.Vectors 2. 2.Matrices/eigenvectors/eigenvalues 3. 3.Complex #’s 4. 4.Lagrange multipliers 5. 5.Fourier transforms

6. Introduction to Quantum Chemistry Origin of Quantum Mechanics: Early 1900’s – microscopic systems do NOT obey the same rules that macroscopic particles obey – Newtonian mechanics NOT SUFFICIENT!! FSOLUTION: Develop a new set of mechanics to describe these systems. FRESULT: Quantum Mechanics (QM) – energy is quantized vs. previous continuous approximation (Newtonian). Postulate of QM: a wave function, , exists for any chemical system and operators (functions) act on the wave function to return observable properties. Mathematically: H  =E   H is an operator (it operates on  ) and E is a scalar value for some observable property. In matrix form, H could be an N×N square matrix, and  an N-element column vector.  The complex conjugate of  is denoted  * and the product |  *| is known as the probability density of the wave function. This is also abbreviated as |  | 2. Suppose we have a single particle, e, in a box, with the wave function  e, as shown below. What is the value of the following expression?

7. Introduction to Quantum Chemistry

10. n

11. FWhat is the probability of finding e in the left half of the box? FAlthough simple, this is powerful information – calculate dipole moments, approximate electrostatic charges, etc. Restrictions of the Wave Function:  Normalized integral over all space of |  | 2 must be unity   must be quadratically integrable   must be continuous and single-valued  The first derivative of  should be continuousy Wave Function analogous to an oracle – when queried with questions (by an operator), it returns answers (observables). Hamiltonian Operator The Hamiltonian operator (H) is an operator that returns the system energy, E, as an eigenvalue. Mathematically, this is the Schrödinger equation shown earlier: H  =E 

12. Introduction to Quantum Chemistry H takes into account five different contribution with respect to a molecule or atom: 1.The KE of the electrons 2.The KE of the protons 3.Attraction between the electrons and protons 4.Interelectronic repulsions 5.Internuclear repulsions ** In some special cases, it is necessary to account for other peripheral interactions, such as those of applied fields. The full Hamiltonian (including all five of these interactions) is written as: i and j run over electrons, k and l run over nuclei, m is mass, e is the charge on an electron, Z is the atomic number, r is the distance between particles, and the Laplacian operator (“del-squared”) is:

13. Introduction to Quantum Chemistry The Variational Principle Challenge: obtaining the set of orthonormal wave functions.  Pick an arbitrary function , where:   is an eigenfunction and is a linear combination of the orthonormal wavefunctions,  i FNormalization criteria restricts the coefficients: What is the energy of our system using  ?