**Correlation effects and the Møller-Plesset method**

Correlation effects and the Møller-Plesset method

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

MAE715-Atomistic Modeling of Materials

Instructor:Nicholas Zabaras, 101 Rhodes Hall, zabaras@cornell.edu

Teaching Assistant:Adam Shai, ass42@cornell.edu

Lectures:MW, 4:30-5:45, Rhodes 178 Recitation:Tuesdays, 4:30-5:45, Rhodes 163 http://mpdc.mae.cornell.edu/Courses/MAE715/MAE715.html

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Course description

¾The course is aimed at graduate students in engineering, physics and chemistry with interests in understanding the fundamentals behind the methods and software (e.g. Moldy, Gaussian, Gulp, Quantum Espresso, Abinit) for computing electronic structure based properties of materials.

¾Emphasis is given to physical models of interatomicforces from Lennard-Jones models to self-consistent all-electron solution of the quantum mechanical problem.

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Course requirements

¾The course material is self-contained but an earlier exposure to quantum mechanics and solid state physics is desirable.

¾Background expected of is knowledge of basic quantum mechanics and condensed matter physics (at the level of Aschroftand Mermin) e.g. reciprocal lattice space, the theory of bands, phonons, electronic structure property estimation.

¾Some knowledge of more advanced concepts such as the ideas of quasi-particles and the random phase approximation are useful to understand material from the recommended references.

¾Knowledge of programming is only needed at an elementary level. However, you will need to obtain (if you don’t have one) a computer access e.g. an account at the Cornell CAC or CMR clusters. We will help you but you will be responsible for installing the needed programs in these computers.

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Homework and final project

9the use of ab initio energies with MC to address adsorption of H on metallic |

¾The course will include weekly computational assignments and a project. HWs will address between other topics: 9the use of potentials for computing thermodynamic properties 9ab initio simulations for property and band diagram calculations 9molecular dynamics (fracture of nanocrystals) surfaces

¾An individual project needs to be proposed within the first month of the course 9It needs to be directly related to the class but can be related to your research if there are common elements 9Can address a problem from the literature with known solution 9It needs to have a computational component 9A 6 page report is due at the end of the course and an oral presentation is required.

All HWs need to be submitted in an electronic format with attachments of all input output data from the computer codes you used. Also you need to provide links to any papers used from the literature for property estimation or comparison of your results.

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Recommended Textbooks

¾(1) Electronic structure of materials, A. P. Sutton ¾(2) Methods of Electronic Structure Calculations, M. Springborg

¾(3) Electronic structure: Basic theory & practical methods, R. M. Martin

¾(4) Interatomic Forces in Condensed Matter, M. Finnis

¾(5) Atomic and Electronic Structure of Solids, E. Kaxiras

¾(6) Essentials of Computational Chemistry, C.J. Cramer

¾(7) Computational Physics, J.M. Thijsen

Notes and literature material will be provided hopefully 1 hour before each lecture. These notes will be compiled from the above references and up to date journal publications

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

MAE715- SYLLABUS

1.Essential Quantum Mechanics (Schrödinger equation, Born-Oppenheimer approximation, the variational principle and the matrix eigenvalue problem, electronic spin, spin orbitals, molecular orbital theory, valence bond theory, many body problem). 2.Hartree-Fock approximation, mean field theory, the charge density, pseudopotentials, solution of H-F equations, basis sets selection, correlation, perturbation theory. 3.Density functional theory, Kohn-Sham equations, exchange-correlation functionals, the plane-wave pseudopotential method, LDA/GGA approximations. 4.Periodicity and band structures, applications to crystal structure and property prediction in solids, band gap calculations, finite temperature DFT calculations. 5.Energetics & structure from empirical potentials, pair-wise potentials, pseudopotentials, cluster expansions. Molecular dynamics, Car-Parrinello, first-principles MD. 6.Statistical thermodynamics and MC, applications of ab-initio based MC to adsorption. 7.Ab-initio thermodynamics and structure prediction. 8.Spatial coarse graining methods, accelerated MD, kinetic MC, mesoscale models

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

The atomic view point

“If in some cataclysm all scientific knowledge were to be destroyed and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis that all things are made of atoms - little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see there is an enormous amount of information about the world, if just a little imagination and thinking are applied.”

--Richard Feynman

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Introduction to Atomistic Modeling

The ASCI (Advanced Simulation Computing Initiative) program related to | |

reliability of nuclear weapons has a large atomistic component.It includes |

Atomistic modeling is not a new subject. It is widely used in industry & academia. all aspects of multiscale materials modeling:

Ageing of Plutonium, corrosion of alloys, etc. Radiation displacement: MC + MD, Fusion ignition, etc.

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

The earth simulator

The earth simulator used to model environmental processes (climate change, etc.).

¾Enormous computation resources are invested to deal with environmental problems.

¾Abinitio calculation: crust and mantle (large pressures)

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

The earth’s core

People use modeling when experimentation is impossible, difficult or expensive, e.g.

¾Computing properties of the earth’s core: what is the composition, phases and their properties of the mantle in the earth.

¾Experiments cannot be done at 300 GPAs! In addition, temperature cannot be controlled as well.

A lots of researchers work on computational quantum mechanics on phases under high-pressure.

Dario Alfè

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

High pressure phase diagrams

-Composition, phases and their properties of the mantle in the earth. -Experiments cannot be done at high pressures (~300 GPAs) and with controlled temperature. -Thermal property estimation

-Optimize electrical and magnetic properties

Nature401, 462-464 (30 September 1999)

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Predictive nano-chemistry

¾Divide the system into separate regions

–Full quantum treatment for all atoms within active cluster

–Distant interstitial atoms interact through potentials based on MD or other classical representations

–Further out atoms can be treated as a single entity through a renormalized classical interaction

¾Research issues

–Mathematically rigorous separation of length scales –Well defined measures of the accuracy of the model

–Nature of errors

–Interfacing to strongly correlated wavefunctions

–Physical motivated mathematical description to interface quantum and classical regions

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Some issues with multiscale problems

¾Length and time scales: e.g. long-time MD even of small systems is prohibited, long-time simulation implies large cell or improved boundary conditions to avoid finite-size effect.

¾Rare events, strongly correlated phenomena

¾High dimensionality

¾Lack of techniques for quantification of error and systematic improvement

¾Materials-by-design: Adjusting molecular structure to influence macroscopic material or electronic properties

¾ Multi-scale ↔ multi-level/resolution ↔ multi-physics

¾Self-consistent coupling with models at coarser length scales quantum ↔classical ↔effective medium

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

An integrated multiscale approach to molecular electronic devices

¾Develop theory, algorithms & software to understand, and eventually to design, self-assembled molecular electronic devices.

Schematic of the Cui et al. experiment

Geometry of a typical quantum conductance calculation. Electrons traveling from the left lead can be reflected or transmitted on the contact region. Similarly, the electrons can be reflected at the conductor. Schematic of BDT molecule attached to gold electrons

BDT molecules adsorbed on the (1) surface of gold. Top: side view; bottom: view from above surface.

R. Harrison

P. T. Cummings & Y. Leng

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Alloy phase stability calculations r easing for ma tion energies

XY 2 (most stable)

Prototype struct ure s

MgZn 2

SiS 2

PbCl 2

SiO 2

Calculate stable phase structures of new alloys

Search for lowest energy configuration using techniques such as:

Au 3 Cu

AuCu

Au Cu

AuCu

Simple cubic Simple cubic

Known metastable structure

Known high pressure phases, are stabilized by entropy

Low temperature structure identified by DFT cF8 cP46 cI16 tI4 0.0637 eV

0.1660 eV

0 eV Most stable

0.3264 eV Silicon phase structures

AuCu low temperature phase diagram

•Cluster Expansions • Multibody Expansions

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

16 Position

Use multibody expansions to generate ab initio accuracy potential energy surfaces (PES) with great computational savings

Utilize PES to understand absorbate-surface interactions

Towards designing topology and chemical structure of surfaces to enhance adsorption

Generating the n-dimensional PES Utilizing PES to find optimal configurationConstructing the optimal topology

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Visualization of electron densities

See article in the science section of the New York Times (September 7, 1999) on the visualization of electron densities and orbitals in copper oxide.

Malcolm W. Browne, "Glue of Molecular Existence is Finally Unveiled," New York Times, 7 September 1999.

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Electron density in a peroxide

Electron density visualization in a peroxide (Lead Titanate, PbTiO3 , a ferroelectric) -work of N. Marzari at MIT.

One can obtain computationally orbitals with fine detail.

MAE 715 –Atomistic Modeling of Materials

N. Zabaras (1/20/2009) 19

Cu-O bonding in copper oxide superconductors

Cu-O bond (experiments) Ti-O bond (theory)

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Diels-Alder Reaction *

20 http://www.wag.caltech .edu/hom e-pages/jim /

¾The Diels-Alder reaction is an organic chemical reaction (a cycloaddition) between a conjugated diene and a substituted alkene (the dienophile), to form a substituted cyclohexene system.

¾The reaction can proceed even if some of the atoms in the newly-formed ring are not carbon.

¾Requires little energy to form the very useful cyclohexene ring

¾Nobel price in Chemistry (1950)

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Predicting crystal structure

15000 DFT calculations to find the ground states structures in 80 binary metallic systems completed in about 6 months (Prof. Ceder’s work, MIT )

¾Pattern generation and recognition using statistical techniques (machine learning)

¾Making predictions about other materials

¾Computing trends

¾Predictive discovery, micro alloying, materials-by-design

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Phase diagrams: Metastability of Al-Li

An example of using ab initio calculations to investigate the metastability of Al-Li:

¾Al-Li was a key alloy for air-frames in the 1980’s (Al-Cu is currently used)

¾Li is the lightest solid element so substituting the Cu and using Al-Li alloys was supposed to make light alloys –however, Al-Li alloys are not very weldable.

¾From the Al-Li phase diagram (stable phases), we can see that there is an fcc solid solution in equilibrium with a compound. ¾Metastable precipitate strengthens the materialbut is too short lived to analyze experimentally

¾Using computation, a phase that is metastable can easily be made to become stable by not letting the system evolve (to a stable phase).

Sluiter et al. (1990) PRB 42 (16) 1990, p. 10460

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Fracture initiation, failure and crack propagation

¾Brittle vs ductile fracture ¾Fracture in thin films

¾ Dislocation dynamics

¾Material failure

Using modified material-specific potential formulation: Molecular dynamics, Hyper dynamics, Quantum MD

IBM-LLNL Research

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Spanning the length scales in fracture simulations

24 Europhys. Lett., 4 (6), p. 783-787 (1998)

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Spanning the length scales in fracture simulations

¾The coupling-of-length-scalesmethod (CLS) integrates the MD simulation with finite element (FE) and tight-binding (TB) simulations operative in adjoining spatial regions.

¾TB Hamiltonians are used where the smallest length scales are relevant and a quantum mechanical description of the forces on atoms is required, e.g.at a crack tip.

¾On intermediate scales, MD describes the motion of each atom which interacts with other atoms through materials-specific atomic potentials.

¾On the longest length scales a FE simulation includes the coupling to long-wavelength elastic modes. The hand-shaking' between these different descriptions is accomplished seamlessly .

J. Q. Broughton, F. Abraham, R. E. Rudd, N. Bernstein, E. Kaxiras

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

Surface calculations: energies, affinities, absorption

¾Effect of faceting ¾Melting and wetting on metal surfaces

¾Computing affinities for other materials

¾Effect of shape and structure on properties

F. Di Tolla, E. Tosatti and F. Ercolessi Phy Rev Let 74(16) 1995, p. 3201

MAE 715 –Atomistic Modeling of Materials N. Zabaras (1/20/2009)

The lowest energy defect in Silicon

¾Si is an open structureleading to different defects vs. those in a closed-packed metal. Computation has made considerable impact in understanding defects in Si.

¾In a closed-pack metal, a vacancy is just taking an atom out and an interstitial is putting an atom in the interstitial position. In Si, when you take an atom out you have a significant rearrangement of the other atoms.

¾If you have theories for how the vacancy or interstitial atom arrangements should look like, you can test them and compare them computationally, calculate their energy, etc.

S. Goedecker, T. Deutsch, and L. Billard (2002) PRL 8 (23) 2002, p. 235501