Lecture 2 Part A Intro to ab initio methods Recommended reading: Leach, Chapters 2 & 3 for QM methods For more QM methods: Essentials of Computational Chemistry by C.J. Cramer, Wiley (2002) 1
ab initio ab initio - from the beginning The concise Oxford Dictionary, Oxford University Press, 2001 ab initio calculation - A method of calculating atomic and molecular structure directly from the first principles of quantum mechanics, without using quantities derived from experiment (such as ionization energies found by spectroscopy) as parameters. A Dictionary of Chemistry, Oxford University Press, 2000 The most chemically accurate, physically precise computation possible. The holy grail of computational chemists. 2
Fundamentals The postulates and theorems of quantum mechanics form the rigorous foundation for the prediction of observable chemical and physical properties of matter from first principles. Any model of a material and its behavior, regardless of its source, must ultimately find its basis in quantum mechanics. All models of materials include QM either explicitly or implicitly. 3
Fundamentals Matter is made of atoms, which are composed of electrons, protons, and neutrons. Over 75 years ago the laws of QM as formulated by Schrodinger, Dirac and others made it theoretically possible to understand and calculate how electrons and atomic nuclei interact to form matter. However, the solution of the governing equations of QM is too difficult to solve exactly for anything but the simplest of systems (like the hydrogen atom). 4
QM Methods Over the past 10 years, ground-breaking advances in the development of QM techniques now allow QM calculations on molecular systems of real, practical interest. Strength of carbon nanotubes Optical spectra of quantum dots Interaction of biological molecules (docking) with surfaces, other bio-molecules, etc. Energy of nanostructures on surfaces Molecular structure and reactivity of complex molecules (e.g. buckyballs and related carbonaceous molecules) Bond strengths, angles for macromolecules. Fracture of inorganic matter. 5
QM Methods Doing a QM simulation or calculation means including the electrons explicitly. With QM methods, we can calculate properties that depend upon the electronic distribution, and to study processes like chemical reactions in which bonds are formed and broken. The explicit consideration of electrons distinguishes QM models and methods from classical force field models and methods. 6
Different QM Methods Several approaches exist. The two main ones are: Molecular orbital theory Came from chemistry, since primarily developed for individual molecules, gases and now liquids. Two flavors Ab initio: all electrons included (considered exact) Semi-empirical: only valence electrons included Density functional theory Came from physics and materials science community, since originally conceived for solids. All electrons included via electronic density (considered exact). 7
Fundamentals The fundamental postulates of QM assert that microscopic systems are describable by wave functions that completely characterize all of the physical properties of the system. A wavefunction squared is a probability density. There are QM operators corresponding to every physical observable that, when applied to a wave function, allow the prediction of the probability of finding the system to exhibit a particular value or range of values for that observable. x Ψ = xψ Eigenvalue eqn for position. 8
Fundamentals The operator that returns the system energy is called the Hamiltonian operator H. Eigenvalue equation for system energy HΨ = EΨ Time-independent Schrodinger Equation H = " $ i h 2 # 2 i " 2m e Kinetic energy of electrons $ k h 2 # 2 k " 2m k Kinetic energy of nuclei $ i $ k Potential energy of electrons & nuclei e 2 Z k r ik + $ i< j e 2 r ij + Potential energy of electrons $ k<l e 2 Z k Z l r kl Potential energy of nuclei 9
Born-0ppenheimer Approximation Need to simplify! Neutrons & protons are >1800 times more massive than electrons, and therefore move much more slowly. Thus, electronic relaxation is for all practical purposes instantaneous with respect to nuclear motion. We can decouple the motion, and consider the electronelectron interactions independently of the nuclear interactions. This is the Born-Oppenheimer approximation. For nearly all situations relevant to soft matter, this assumption is entirely justified. (H el +V n )Ψ el (q i ;q k ) = E el Ψ el (q i ;q k ) The Electronic Schrodinger Equation 10
QM Methods Goal of all QM methods in use today: Solve the electronic Schrodinger equation for the ground state energy of a system and the wavefunction that describes the positions of all the electrons. The energy is calculated for a given trial wavefunction, and the best wavefunction is found as that wavefunction that minimizes the energy. 11
QM Methods Solving SE is not so easy! Anything containing more than two elementary particles (i.e. one e- and one nucleon) can t be solved exactly: the many-body problem. Even after invoking Born-Oppenheimer, still can t solve exactly for anything containing more than two electrons. So -- all QM methods used today are APPROXIMATE after all, even if considered exact! That is, they provide approximate solutions to the Schrodinger equation. Some are more approximate than others. 12
Molecular Orbital Theory MOT is expressed in terms of molecular wave functions called molecular orbitals. Most popular implementation: write molecular orbital as a linear combination of atomic orbitals φ (LCAO): Eq. 2.68 in Leach K $ µ=1 " i = a µi # µ K = # atomic orbitals Many different ways of writing basis set, which leads to many different methods and implementations of MOT. 13
Molecular Orbital Theory Dozens of approaches for writing basis sets (e.g. in terms of Gaussian wavefunctions, or as linear combos of Gaussians). Different implementations retain different numbers of terms. Semi-empirical MOT methods consider only valence electrons. Some methods include electron exchange. Some methods include electron correlation. 14
Density Functional Theory A different approach for solving Schrodinger s equation for the ground state energies of matter. Based on theory of Hohenberg and Kohn (1964) which states that it is not necessary to consider the motion of each individual electron in the system. Instead, it suffices to know the average number of electrons at any one point in space. The HK theorem enables us to write E el as a functional of the electron density ρ. To perform a DFT calculation, one optimizes the energy with respect to the electron probability density, rather than with respect to the electronic wave function. For a given density, the lowest energy is the best one. 15
Density Functional Theory In the commonly used Kohn-Sham implementation, the density is written in terms of one-electron molecular orbitals called Kohn-Sham orbitals. This allows the energy to be optimized by solving a set of one-electron Schrodinger equations (the KS equations), but with electron correlation included. This is a key advantage of the DFT method - it s easier to include electron correlation. 16
Density Functional Theory In DFT calculations, the MO s are written as linear combinations of atomic orbitals (LCAO) or basis functions which can be represented using Gaussian functions, plane waves, etc. N $ i=1 " = a i # i 17
Density Functional Theory Different choices of basis sets, how many terms to use, of what type, contribute to difficulty of calculation. More than a few hundred light atoms is still too time-consuming, even on big computers. For molecules or systems with large numbers of electrons, pseudopotentials are used to represent the wavefunctions of valence electrons, and the core is treated in a simplified way. This is a basic introductory summary of the DFT method. 18
Implementing MOT & DFT in Computer Code John Pople at Northwestern is a pioneer in computational quantum chemistry. In 1970 Pople developed Gaussian, a quantum chemistry code that solves approximations of the SE for molecules. In 1990, he included DFT in Gaussian. This brought state-of-the-art QM computational methods to the masses. Ab initio and semi-empirical methods (especially MOT methods) have revolutionized the Pharmaceuticals Industry, and are now playing a major role in materials R&D. 19
1998 Nobel Prize in Chemistry Walter Kohn "for his development of the density-functional theory" John Pople "for his development of computational methods in quantum chemistry" 20
QM Codes for Materials Research Commonly used codes: Gaussian: ab initio MOT and DFT Available in Cerius 2 from Accelrys NWChem: ab initio MOT and DFT www.emsl.pnl.gov/pub/docs/nwchem 21
DFT Codes for Materials Research DFT codes used in materials research: VASP - Vienna ab initio simulation package Siesta - from Spain Abinit Gaussian Castep - Cambridge sequential total energy package DMol 3 22
Applications of ab initio computations using DMol 3 23
Electronic band structure of POSS cubes functionalized with n benzene molecules (n = 0-8) -10-8 -6-4 -2 0 2 4 6 8 10 Energy (ev) 24
Electronic band structure of POSS cubes functionalized with acene molecules (benzene, naphtalene, anthracene, tetracene, and pentacene) -10-8 -6-4 -2 0 2 4 6 8 10 Energy (ev) Pure poss 25
Electron densities of acene-functionalized POSS Molecule Band Gap (ev) Pure Acene (ev) P-POSS T-POSS A-POSS N-POSS B-POSS POSS 0.999 1.486 2.295 3.325 4.842 8.4 1.335 1.597 3.268 3.348 13.776 - HOMO LUMO 26
Electron densities of silica nanotubes HOMO LUMO 27
Future developments of ab initio methods Used for calculating quantities like reaction rates, bond strengths and angles, heats of formation, solubility, etc.. Any properties that depend critically on electron distribution. Lots of activity in MOT and DFT methods. Order N methods: The Holy Grail Ab initio - computational effort now scales with the number of electrons to a power n<4. Semi-empirical - with only valence electrons, can get order N scaling? 28
Too slow! Even with all of the advances of the past 10 years in ab initio methods ignoring some of the electrons (pseudopotentials) implementation of ab initio codes on parallel machines it is still not possible to use solely QM methods to simulate systems that contain more than a few thousand atoms (a chunk of matter containing less than 1 nm on a side), or for more than a picosecond for a very small number of atoms. 29
Still too slow! Ten years from now we may gain an order of magnitude in what can be simulated, but this is still not sufficient for many problems in soft matter. Assembly (especially if hierarchical) Mechanical properties of composites Rheology Development of structure on length scales of 10 to 100 s of nanometers. Enter classical force fields. 30
Classical vs. ab initio methods Classical No electronic properties ab initio Electronic details included Phenomenological potential energy surface (typically 2-body contributions) Difficult to describe bond breaking/formation Can do up to a billion particles Potential energy surface calculated directly from Schrodinger equation (many body terms included automatically) Describes bond breaking/formation Limited to several hundred atoms with significant dynamics 31