CI/CEPA. Introduction CI Size Consistency Derivation CEPA EPV Results Remarks
|
|
|
- Irene Ray
- 5 years ago
- Views:
Transcription
1 CI/CEPA Introducton CI Sze Consstency Dervaton CEPA EPV Results Remarks 1
2 HY=EY H e Ψ e = E e Ψe Expanson of the Many-Electron Wave Functon Ψ HF Ψ e = Φ o + a a C Φ + ab ab C j Φ j +... Φo = A φ 1 φ 2... φ ι... φ j... j Expanson of the One-Electron Orbtals c m n φ = χ m m 2
3 CI Determnant!= " 1 " 2 " 3! Confguraton State Functon (CSF) #= Confguraton c! Spn Egenfuncton ( )) 2m Full CI - all determnants (for m orbtals and n electrons n $ {# wth same occupaton} Hamltonan s an n-electron operator, so the wavefuncton s a lnear combnaton of of n-electron functons (e.g. determnants) H =! h() +!"# 1" electron! 1 r j < j $ 2"electron a Slater rule jkl H abc = 0 (>2 dfferences) + N.R. +% so double replacements most mportant wth rulng! 0 Sngle and Double CI (SDCI)! 1 2 n ( ) 2.( m " 1 n 2 ) 2 Explan where the formula s come from and calculate # confgs for e.g. water n a DZP bass and e.g. benzene 3
4 Practcal CI Dmensons of CI problem : or more No explct matrx avalable HC=Z Drectly from ntegrals (faster) Only teratve methods possble Lanczos, (Jacob)Davdson* Power Method Largest component projected out * Subspace expansons E. R. Davdson, J. Comput. Phys. 17, 87 (1975). {b} : Bb =! b g = # " b Bg = # g Bb = #"! b B n g = # g B n b = #"! n b H. J. J. van Dam, J. H. van Lenthe, G. L. G. Slejpen & H. A. van der Vorst An mprovement of Davdson's Iteraton method. Applcatons to MRCI and MRCEPA Calculatons Journal of Computatonal Chemstry 17, (1996) Try ths 4
5 Use of CI More accurate (E corr =E HF -E CI ) Near Degeneracy. Excted States; E CI=E exact n MR(D)CI! 0 = # c "! = # c jab C( $ a, j $ b)! 0 MRDCI - select excted states usng PT Sngles CI = TDA! = # c a C( " a)! HF a ab j =>! 0 (=! HF ),! 0,! 1,! 2 for bg bass 5
6 Monomers (SDCI) Dmer (R= ) Sze Consstency 0! A =! A 0 +! A S +! A D! B =! B 0 +! B S +! B D! AB = (! 0 A +! S D A +! ) A " (! 0 B +! S D B +! ) B =! A 0! B 0 +! A 0! B S +!+! A 0! B D +!+! A S! B D +!+! A D! B D 0 =! AB S +! AB +! D AB +! T AB Q +! AB Not n SDCI CI Ne 2 TZVP R= CEPA0 Ne Ne Ne Ne δ = δ =
7 Davdson Correcton for CI! = " 0 + c D " D + ( c Q " Q ) SDCI : # corr = $E SDCI = E SDCI % E 0 say c Q & c D 2 = 1% c 0 2 ; contrbutons quadruples to doubles & # corr Davdson correcton : ( 2 1 % c 0 )# corr Davdson gves a contrbuton for a sngle electron par Pople correcton corrects ths (thus better for few electrons) Better CCSD and CEPA - See n what follows 7
8 Sze Consstency 1 Sngle Determnant Domnant SCF Sze consstency SDCI CEPA CCSD E A +B! E A + E B OK OK! = e S +D! 0 Varatonal Yes No No 8
9 Full CI Intermedate Normalsaton Secular Equatons: SD #! = " 0 + c "!#" # $! SD SD! 0 H " E # =! 0 H " E! 0 + $ c! 0 H! SD = 0 SD TQ + # c j " j +% j SD # E corr = E! E 0 = c SD " 0 H " SD! SD H " E # =! SD H " E # SD + $ TQ c j! SD H! TQ j = 0 TQ j 9
10 Coupled Cluster! = e T! 0 = ( 1 + T T 2 +! )! 0 2 Separated Systems A and B 0! AB = " 1A " 2A " 3A " 4 A " 1B " 1B = A" 1A " 2A " 3A " 4 A " 1B " 1B 0! AB = =! A 0! B 0 T = T A + T B (! AB = e T A +T ) B! 0 A! 0 B = e T A! 0 Ae T B! 0 B =! A #! B CI! AB = ( 1 + T A + T B )! 0 A! 0 B " ( 1 + T A )! 0 A ( 1 + T B )! 0 B = ( 1 + T A + T B + T A T B )! 0 0 A! B Confrm the (lack of) sze consstency of CI and CCD (wthout usng the exponentonal) 10
11 CC to CEPA! 44 D! 0 22 =! 1! 55 D 33 =! 2! 4455 Q 2233 =! 3 x x x! 45 D 23 =! 4 CCD! c = c " c 55 33! c Q 3 # c D D 1.c 2! H! 2233! "! Q 23 =! 2233 = "( 35 35) 55 =! 0 H! 33 c 3 Q! 1 D H! 3 Q = c 1 D c 2 D! 0 H! 2 D But 11
12 ! 0 H " E # SD = 0 SDCI => CEPA! SD H " E # =! SD H " E # SD + $ TQ c j! SD H! TQ j = 0! SD H " E # =! SD H " E # SD + c SD TQ j SD c SD j $ =! SD H " E # SD + c SD.E corr =! SD H " E + E corr # SD = 0 j! 0 H! j SD Do the dervaton of CEPA0; How can you see t overestmates and how would other CEPA s correct ths; Is t an egenvalue problem? e.g. C. Zrz, R.Ahlrchs n Electron Correlaton. Proceedngs of the Daresbury study weekend, November
13 EPV Excluson Prncple Volaton 44 44! 0! 22! 33 55! 22 x x x 45! 23! "! =! Q! "! =! Q ACPF(Ahlrchs) E corr! n elec " 2 n elec AQCC(Szalay) E corr! n " 2 elec n elec n elec "1 * E corr ( )( n elec " 3) ( ) * E corr CEPA1/2 correcton per orbtal-par 13
14 Sngle Reference Sze consstency Be dmer at Å TZVP (harmonc) bass Method SCF CI CI + Davdson CI + Pople CEPA1 CEPA0 ACPF AQCC MP2 MP3 SC-error -2.5E E E E E E E E E E-10 14
15 .Energes and sze consstency errors (n mh) for (O 2 ) 2. Monomer Dmer SC-Error MRCI Davdson Pople MR-ACPF MR-AQCC MRDCEPA MR-ACEPA MR-CEPA No symmetry restrctons are appled n generatng the CAS reference space DZP bass ; R=1.2 Å Monomer wavefuncton 6-electron π CAS; 15
16 Fnal Remarks CI s not sze consstent and ths can really hurt f not treated e.g. the E corelaton ~ n nstead of n. The major part of the sze consstency can be treated by checkng the separate consttuent parts; (cures the problem at huge dstance) The dfferental part s ntractable CCSD, CEPA and MPn are ntrnscally sze consstent (but not varatonal) No MR-CEPA s really sze consstent 16
Electronic Structure for Excited States (multiconfigurational methods) Spiridoula Matsika
Electronc Structure for Excted States (multconfguratonal methods) Sprdoula Matska Excted Electronc States Theoretcal treatment of excted states s needed for: UV/Vs electronc spectroscopy Photochemstry
Introduction to Density Functional Theory. Jeremie Zaffran 2 nd year-msc. (Nanochemistry)
Introducton to Densty Functonal Theory Jereme Zaffran nd year-msc. (anochemstry) A- Hartree appromatons Born- Oppenhemer appromaton H H H e The goal of computatonal chemstry H e??? Let s remnd H e T e
ψ ij has the eigenvalue
Moller Plesset Perturbaton Theory In Moller-Plesset (MP) perturbaton theory one taes the unperturbed Hamltonan for an atom or molecule as the sum of the one partcle Foc operators H F() where the egenfunctons
5.03, Inorganic Chemistry Prof. Daniel G. Nocera Lecture 2 May 11: Ligand Field Theory
5.03, Inorganc Chemstry Prof. Danel G. Nocera Lecture May : Lgand Feld Theory The lgand feld problem s defned by the followng Hamltonan, h p Η = wth E n = KE = where = m m x y z h m Ze r hydrogen atom
Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
where the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt
Physcs 543 Quantum Mechancs II Fall 998 Hartree-Fock and the Self-consstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.
A how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
Intermolecular force fields and how they can be determined
Intermolecular force felds and how they can be determned Ad van der Avord Unversty of Njmegen Han-sur-Lesse, December 23 p.1 Equaton of state (Van der Waals) of non-deal gas ( p + a )( ) V 2 V b = kt repulson
Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department
24. Atomic Spectra, Term Symbols and Hund s Rules
Page of 4. Atomc Spectra, Term Symbols and Hund s Rules Date: 5 October 00 Suggested Readng: Chapters 8-8 to 8- of the text. Introducton Electron confguratons, at least n the forms used n general chemstry
Multi-electron atoms (11) 2010 update Extend the H-atom picture to more than 1 electron: H-atom sol'n use for N-elect., assume product wavefct.
Mult-electron atoms (11) 2010 update Extend the H-atom pcture to more than 1 electron: VII 33 H-atom sol'n use for -elect., assume product wavefct. n ψ = φn l m where: ψ mult electron w/fct φ n l m one
Errors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
Probabilistic method to determine electron correlation energy
Probablstc method to determne electron elaton energy T.R.S. Prasanna Department of Metallurgcal Engneerng and Materals Scence Indan Insttute of Technology, Bombay Mumba 400076 Inda A new method to determne
Hongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)
ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of
Canonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
5.04, Principles of Inorganic Chemistry II MIT Department of Chemistry Lecture 32: Vibrational Spectroscopy and the IR
5.0, Prncples of Inorganc Chemstry II MIT Department of Chemstry Lecture 3: Vbratonal Spectroscopy and the IR Vbratonal spectroscopy s confned to the 00-5000 cm - spectral regon. The absorpton of a photon
Electronic Quantum Monte Carlo Calculations of Energies and Atomic Forces for Diatomic and Polyatomic Molecules
RESERVE HIS SPACE Electronc Quantum Monte Carlo Calculatons of Energes and Atomc Forces for Datomc and Polyatomc Molecules Myung Won Lee 1, Massmo Mella 2, and Andrew M. Rappe 1,* 1 he Maknen heoretcal
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
THEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
5. Response properties in ab initio schemes
5. Response propertes n ab nto schemes A number of mportant physcal observables s expressed va dervatves of total energy (or free energy) E. Examples are: E R 2 E R a R b forces on the nucle; crtcal ponts
Lecture 14: Forces and Stresses
The Nuts and Bolts of Frst-Prncples Smulaton Lecture 14: Forces and Stresses Durham, 6th-13th December 2001 CASTEP Developers Group wth support from the ESF ψ k Network Overvew of Lecture Why bother? Theoretcal
International Journal of Pure and Applied Sciences and Technology
Int. J. Pure Appl. Sc. Technol., 4() (03), pp. 5-30 Internatonal Journal of Pure and Appled Scences and Technology ISSN 9-607 Avalable onlne at www.jopaasat.n Research Paper Schrödnger State Space Matrx
Robert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations
Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and x-ray exctatons 9-01 By gong through the procedure ndcated n the text, develop the tme-ndependent Schroednger equaton
Singular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
ψ = i c i u i c i a i b i u i = i b 0 0 b 0 0
Quantum Mechancs, Advanced Course FMFN/FYSN7 Solutons Sheet Soluton. Lets denote the two operators by  and ˆB, the set of egenstates by { u }, and the egenvalues as  u = a u and ˆB u = b u. Snce the
Poisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
Chapter 7. Ab initio Theory
Chapter 7. Ab nto Theory Ab nto: from the frst prnples. I. Roothaan-Hall approah Assumng Born-Oppenhemer approxmaton, the eletron Hamltonan: Hˆ Tˆ e + V en + V ee Z r a a a + > r Wavefunton Slater determnant:
1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:
68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse
Solutions to Problems Fundamentals of Condensed Matter Physics
Solutons to Problems Fundamentals of Condensed Matter Physcs Marvn L. Cohen Unversty of Calforna, Berkeley Steven G. Loue Unversty of Calforna, Berkeley c Cambrdge Unversty Press 016 1 Acknowledgement
2. Hartree-Fock formalism
2. Hartree-Fock formalsm 2.1 Slater determnants One speaks of the Hartree-Fock (HF) approxmaton and of the HF method. The former s also called the self-consstent feld approxmaton, or mean feld approxmaton.
Key Concepts, Methods and Machinery - lecture 2 -
B. Roos V. A. Fok C. C. J. Roothaan W. Hetler Quantum Mechancs W. Paul J. Čížek P.A.M. Drac Key Concepts, Methods and Machnery - lecture - J. A. Pople M. S. Plesset N. Bohr E. Hückel E. Schrödnger and
2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
Uncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
Non-interacting Spin-1/2 Particles in Non-commuting External Magnetic Fields
EJTP 6, No. 0 009) 43 56 Electronc Journal of Theoretcal Physcs Non-nteractng Spn-1/ Partcles n Non-commutng External Magnetc Felds Kunle Adegoke Physcs Department, Obafem Awolowo Unversty, Ile-Ife, Ngera
Programming Project 1: Molecular Geometry and Rotational Constants
Programmng Project 1: Molecular Geometry and Rotatonal Constants Center for Computatonal Chemstry Unversty of Georga Athens, Georga 30602 Summer 2012 1 Introducton Ths programmng project s desgned to provde
Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
GREENBERGER- HORNE- ZEILINGER (GHZ) STATES IN QUANTUM DOT MOLECULE
GREENERGER- ORNE- ZEILINGER (GZ) STATES IN QUANTUM DOT MOLECULE A. SARMA and P. AWRYLAK QUANTUM TEORY GROUP INSTITUTE FOR MICROSTRUCTURAL SCIENCES NATIONAL RESEARC COUNCIL OF CANADA OTTAWA, KAOR,CANADA
The Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
Advanced School on Quantum Monte Carlo Methods in Physics and Chemistry. 21 January - 1 February, Worm algorithm
1929-15 Advanced School on Quantum onte Carlo ethods n Physcs and Chemstry 21 January - 1 February, 2008 Worm algorthm. Prokofev Unversty of assachusetts, Amherst WOR ALGORITH FOR CLASSICAL AD QUATU STATISTICAL
This chapter illustrates the idea that all properties of the homogeneous electron gas (HEG) can be calculated from electron density.
1 Unform Electron Gas Ths chapter llustrates the dea that all propertes of the homogeneous electron gas (HEG) can be calculated from electron densty. Intutve Representaton of Densty Electron densty n s
p 1 c 2 + p 2 c 2 + p 3 c p m c 2
Where to put a faclty? Gven locatons p 1,..., p m n R n of m houses, want to choose a locaton c n R n for the fre staton. Want c to be as close as possble to all the house. We know how to measure dstance
Lecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
Linear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
14 The Postulates of Quantum mechanics
14 The Postulates of Quantum mechancs Postulate 1: The state of a system s descrbed completely n terms of a state vector Ψ(r, t), whch s quadratcally ntegrable. Postulate 2: To every physcally observable
The GW approximation in 90 minutes or so. F. Bruneval Service de Recherches de Métallurgie Physique CEA, DEN
The GW approxmaton n 90 mnutes or so Servce de Recherches de Métallurge Physque CEA, DEN DFT tutoral, Lyon december 2012 Outlne I. Standard DFT suffers from the band gap problem II. Introducton of the
SUPPLEMENTARY INFORMATION
do: 0.08/nature09 I. Resonant absorpton of XUV pulses n Kr + usng the reduced densty matrx approach The quantum beats nvestgated n ths paper are the result of nterference between two exctaton paths of
An Alternative Scheme for Calculating the Unrestricted Hartree-Fock Equation: Application to the Boron and Neon Atoms.
An Alternatve Scheme for Calculatng the Unrestrcted Hartree-Fock Equaton: Applcaton to the Boron and Neon Atoms Mtyasu MIYASITA 1,2, Katsuhko HIGUCHI 1, Masahko HIGUCHI 3 1 Graduate School of Advanced
Planar maps and continued fractions
Batz 2010 p. 1/4 Planar maps and contnued fractons n collaboraton wth Jéréme Boutter 1 3 3 2 0 3 1 maps and dstances: generaltes Batz 2010 p. 2/4 Batz 2010 p. 3/4 degree of a face = number of ncdent edge
Electron-Impact Double Ionization of the H 2
I R A P 6(), Dec. 5, pp. 9- Electron-Impact Double Ionzaton of the H olecule Internatonal Scence Press ISSN: 9-59 Electron-Impact Double Ionzaton of the H olecule. S. PINDZOLA AND J. COLGAN Department
Efficient, General Point Cloud Registration with Kernel Feature Maps
Effcent, General Pont Cloud Regstraton wth Kernel Feature Maps Hanchen Xong, Sandor Szedmak, Justus Pater Insttute of Computer Scence Unversty of Innsbruck 30 May 2013 Hanchen Xong (Un.Innsbruck) 3D Regstraton
APPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
Complex Atoms; The Exclusion Principle and the Periodic System
Complex Atoms; The Excluson Prncple and the Perodc System In order to understand the electron dstrbutons n atoms, another prncple s needed. Ths s the Paul excluson prncple: No two electrons n an atom can
Effective Valence Shell Hamiltonian Calculations on Spin-Orbit Coupling of SiH, SiH +, and SiH 2+
Spn-Orbt Couplng of SH Bull. Korean Chem. Soc. 2003, Vol. 24, No. 6 723 Effecte Valence Shell Hamltonan Calculatons on Spn-Orbt Couplng of SH, SH +, and SH 2+ Ye Won Chang and Hosung Sun * Department of
10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
This column is a continuation of our previous column
Comparson of Goodness of Ft Statstcs for Lnear Regresson, Part II The authors contnue ther dscusson of the correlaton coeffcent n developng a calbraton for quanttatve analyss. Jerome Workman Jr. and Howard
Quantum states of deuterons in palladium
Tsuchda K. Quantum states of deuterons n palladum. n Tenth Internatonal Conference on Cold Fuson. 003. Cambrdge MA: LENR-CANR.org. Ths paper was presented at the 10th Internatonal Conference on Cold Fuson.
LECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
Numerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
Mechanics Physics 151
Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( )
THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
Multiple Sound Source Location in 3D Space with a Synchronized Neural System
Multple Sound Source Locaton n D Space wth a Synchronzed Neural System Yum Takzawa and Atsush Fukasawa Insttute of Statstcal Mathematcs Research Organzaton of Informaton and Systems 0- Mdor-cho, Tachkawa,
Consistency & Convergence
/9/007 CHE 374 Computatonal Methods n Engneerng Ordnary Dfferental Equatons Consstency, Convergence, Stablty, Stffness and Adaptve and Implct Methods ODE s n MATLAB, etc Consstency & Convergence Consstency
Causal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
Bézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
A quote of the week (or camel of the week): There is no expedience to which a man will not go to avoid the labor of thinking. Thomas A.
A quote of the week (or camel of the week): here s no expedence to whch a man wll not go to avod the labor of thnkng. homas A. Edson Hess law. Algorthm S Select a reacton, possbly contanng specfc compounds
Using TranSIESTA (II): Integration contour and tbtrans
Usng TranSIESTA (II): Integraton contour and tbtrans Frederco D. Novaes December 15, 2009 Outlne Usng TranSIESTA The ntegraton contour Electron densty from GF Why go complex? The non-equlbrum stuaton Usng
Lecture 5: Ideal monatomic gas
Lecture 5: Ideal monatomc gas Statstcal mechancs of the perfect gas Ams: Key new concepts and methods: Countng states Waves n a box. Demonstraton that β / kt Heat, work and Entropy n statstcal mechancs
However, since P is a symmetric idempotent matrix, of P are either 0 or 1 [Eigen-values
Fall 007 Soluton to Mdterm Examnaton STAT 7 Dr. Goel. [0 ponts] For the general lnear model = X + ε, wth uncorrelated errors havng mean zero and varance σ, suppose that the desgn matrx X s not necessarly
Linear system of the Schrödinger equation Notes on Quantum Mechanics
Lnear sstem of the Schrödnger equaton Notes on Quantum Mechancs htt://quantum.bu.edu/notes/quantummechancs/lnearsstems.df Last udated Wednesda, October 9, 003 :0:08 Corght 003 Dan Dll (dan@bu.edu) Deartment
Answers Problem Set 2 Chem 314A Williamsen Spring 2000
Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%
Slater-Condon Rules. Antisymmetrization Operator APPENDIX M
APPENDIX M Slater-Condon Rules The Slater determnants represent somethng lke the daly bread of quantum chemsts. Our goal s to learn how to use the Slater determnants when they are nvolved n calculaton
Other methods to consider electron correlation: Coupled-Cluster and Perturbation Theory
Other methods to consider electron correlation: Coupled-Cluster and Perturbation Theory Péter G. Szalay Eötvös Loránd University Institute of Chemistry H-1518 Budapest, P.O.Box 32, Hungary szalay@chem.elte.hu
Chapter Newton s Method
Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve
Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
Georgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
6. Hamilton s Equations
6. Hamlton s Equatons Mchael Fowler A Dynamcal System s Path n Confguraton Sace and n State Sace The story so far: For a mechancal system wth n degrees of freedom, the satal confguraton at some nstant
CHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
Quantum chemistry on parallel computer systems using COLUMBUS, TURBOMOLE, NWCHEM
Quantum chemstry on parallel computer systems usng COLUMBUS, TUBOMOLE, NWCHEM Felx Plasser Unversty of Venna VSC Meetng 27 28/February/202 Quantum chemstry Quantum chemstry solve the electronc Schrödnger
8. Superfluid to Mott-insulator transition
8. Superflud to Mott-nsulator transton Overvew Optcal lattce potentals Soluton of the Schrödnger equaton for perodc potentals Band structure Bloch oscllaton of bosonc and fermonc atoms n optcal lattces
ISQS 6348 Final Open notes, no books. Points out of 100 in parentheses. Y 1 ε 2
ISQS 6348 Fnal Open notes, no books. Ponts out of 100 n parentheses. 1. The followng path dagram s gven: ε 1 Y 1 ε F Y 1.A. (10) Wrte down the usual model and assumptons that are mpled by ths dagram. Soluton:
Research Article Green s Theorem for Sign Data
Internatonal Scholarly Research Network ISRN Appled Mathematcs Volume 2012, Artcle ID 539359, 10 pages do:10.5402/2012/539359 Research Artcle Green s Theorem for Sgn Data Lous M. Houston The Unversty of
Calculation of Converged Rovibrational Energies and Partition Function for Methane using Vibrational-Rotational Configuration Interaction
February 2, 2004 Prepared for publcaton n J. Chem. Phys. Calculaton of Converged Rovbratonal Energes and Partton Functon for Methane usng Vbratonal-Rotatonal Confguraton Interacton Arndam Chakraborty,
12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
Physics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
SIO 224. m(r) =(ρ(r),k s (r),µ(r))
SIO 224 1. A bref look at resoluton analyss Here s some background for the Masters and Gubbns resoluton paper. Global Earth models are usually found teratvely by assumng a startng model and fndng small
Lecture notes for FYS KJM 4480
Lecture notes for FYS KJM 4480 Quantum mechancs for many-partcle systems Smen Kvaal October 9, 2015 Contents 1 Fundamental formalsm 3 1.1 Many-partcle systems........................................ 3
8.323 Relativistic Quantum Field Theory I
MI OpenCourseWare http://ocw.mt.edu 8.323 Relatvstc Quantum Feld heory I Sprng 2008 For nformaton about ctng these materals or our erms of Use, vst: http://ocw.mt.edu/terms. MASSACHUSES INSIUE OF ECHNOLOGY
Vector Norms. Chapter 7 Iterative Techniques in Matrix Algebra. Cauchy-Bunyakovsky-Schwarz Inequality for Sums. Distances. Convergence.
Vector Norms Chapter 7 Iteratve Technques n Matrx Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematcs Unversty of Calforna, Berkeley Math 128B Numercal Analyss Defnton A vector norm
Bezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
Joint Statistical Meetings - Biopharmaceutical Section
Iteratve Ch-Square Test for Equvalence of Multple Treatment Groups Te-Hua Ng*, U.S. Food and Drug Admnstraton 1401 Rockvlle Pke, #200S, HFM-217, Rockvlle, MD 20852-1448 Key Words: Equvalence Testng; Actve
LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
Detecting Attribute Dependencies from Query Feedback
Detectng Attrbute Dependences from Query Feedback Peter J. Haas 1, Faban Hueske 2, Volker Markl 1 1 IBM Almaden Research Center 2 Unverstät Ulm VLDB 2007 Peter J. Haas The Problem: Detectng (Parwse) Dependent
5.76 Lecture #5 2/07/94 Page 1 of 10 pages. Lecture #5: Atoms: 1e and Alkali. centrifugal term ( +1)
5.76 Lecture #5 /07/94 Page 1 of 10 pages 1e Atoms: H, H + e, L +, etc. coupled and uncoupled bass sets Lecture #5: Atoms: 1e and Alkal centrfugal term (+1) r radal Schrödnger Equaton spn-orbt l s r 3
The exponential map of GL(N)
The exponental map of GLN arxv:hep-th/9604049v 9 Apr 996 Alexander Laufer Department of physcs Unversty of Konstanz P.O. 5560 M 678 78434 KONSTANZ Aprl 9, 996 Abstract A fnte expanson of the exponental
Density matrix. c α (t)φ α (q)
Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n
) is the unite step-function, which signifies that the second term of the right-hand side of the
Casmr nteracton of excted meda n electromagnetc felds Yury Sherkunov Introducton The long-range electrc dpole nteracton between an excted atom and a ground-state atom s consdered n ref. [1,] wth the help
CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,
Inductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors