Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager
|
|
- Reynard Hicks
- 6 years ago
- Views:
Transcription
1 Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France.
2 Notations Molecular orbitals: φ p (r) = µ C µp χ µ (r) φ p φ q = δ pq Non-orthogonal set of atomic orbitals (Gaussian functions): χ µ χ ν = S µν Hamiltonian in second quantization: Ĥ = p,q h pq Ê pq p,q,r,s ) pr qs (Êpq Ê rs δ qr Ê ps where h pq = [ dr φ p (r) r ] + v ne(r) φ q (r) and pr qs = dr 1 dr 2 φ p (r 1 )φ r (r 2 ) 1 r 1 r 2 φ q(r 1 )φ s (r 2 ) = (pq rs) Institut de Chimie, Strasbourg, France Page 2
3 Institut de Chimie, Strasbourg, France Page 3 Variational and non-variational approximations The exact electronic ground state Ψ 0 and its energy E 0 can be obtained two ways: E 0 = min Ψ Ψ Ĥ Ψ Ψ Ψ = Ψ 0 Ĥ Ψ 0 Ψ 0 Ψ 0 Ĥ Ψ 0 = E 0 Ψ 0 Approximate parametrized ground-state wave function: Ψ(λ 0 ) where λ 0 denotes the complete set of optimized parameters. Variational calculation Non-variational calculation Ψ(λ) Ĥ Ψ(λ) = 0 Ĥ Ψ(λ) E(λ) Ψ(λ) = 0 for λ = λ 0 λ Ψ(λ) Ψ(λ) λ=λ0 Hartree-Fock (HF) Configuration Interaction (CI) Multi-Configurational Self-Consistent Field (MCSCF) Many-Body Perturbation Theory (MBPT) Coupled Cluster (CC)
4 Institut de Chimie, Strasbourg, France Page FCI HF MP2 H 2 (1 1 Σ g +, aug cc pvqz) Energy (a.u.) Interatomic distance (a.u.)
5 Institut de Chimie, Strasbourg, France Page 5 Short-range dynamical correlation LUMO LUMO LUMO HOMO HOMO HOMO Ground-state configuration singly-excited conf. doubly-excited conf.
6 Institut de Chimie, Strasbourg, France Page 6 Static correlation LUMO HOMO LUMO HOMO LUMO HOMO
7 Institut de Chimie, Strasbourg, France Page 7 H 2 in the equilibrium geometry: Static correlation Ψ 0 = C 0 1σ α g 1σβ g +... where C 0 2 = 98% no static correlation In the dissociation limit: H A... H B and NOT H A... H+ B or H+ A... H B φ 1σg (r) = 1 ) (φ 1sA (r) + φ 1sB (r) 2 and φ 1σu (r) = 1 ) (φ 1sA (r) φ 1sB (r) 2 ( 1s α A 1sβ B + 1sα B 1sβ A + 1sα A 1sβ A + 1sα B 1sβ B ) 1σ α g 1σ β g = 1 2 ( 1s α A 1sβ B + 1sα B 1sβ A 1sα A 1sβ A 1sα B 1sβ B ) 1σ α u 1σ β u = 1 2 Ψ 0 = 1 2 ( 1σ α g 1σβ g 1σα u 1σβ u ) strong static correlation
8 Institut de Chimie, Strasbourg, France Page 8 EXERCISE: H 2 in a minimal basis (1) Show that the Hamiltonian matrix for H 2 can be written in the basis of the two single-determinant states 1σ α g 1σ β g and 1σ α u 1σ β u as follows, [Ĥ] = E g K K E u, where for i = g, u, E i = 2h ii + 1σ i 1σ i 1σ i 1σ i, h ii = 1σ i ĥ 1σ i, K = 1σ u 1σ u 1σ g 1σ g. (2) In the following, we use the minimal basis consisting of the two 1s atomic orbitals. Explain why, in the dissociation limit, E g = E u and K = 1 2 1s1s 1s1s > 0. (3) Conclude that, in the dissociation limit, the ground state is multiconfigurational and does correspond to two neutral hydrogen atoms with energy (E g K).
9 Institut de Chimie, Strasbourg, France Page FCI HF CISD 2/2 H 2 (1 1 Σ g +, aug cc pvqz) Energy (a.u.) Interatomic distance (a.u.)
10 Institut de Chimie, Strasbourg, France Page 10 Multi-Configurational Self-Consistent Field model (MCSCF) The MCSCF model consists in performing a CI calculation with a reoptimization of the orbitals Ψ(κ, C) = e ˆκ ( ξ ) C ξ det ξ where C { C ξ } ξ The MCSCF model is a multiconfigurational extension of HF which aims at describing static correlation: a limited number of determinants should be sufficient. Short-range dynamical correlation is treated afterwards (post-mcscf models) Choice of the determinants: active space H...H 2 electrons in 2 orbitals (1σ g, 1σ u ) 2/2 Be 2 electrons in 4 orbitals (2s, 2p x, 2p y, 2p z ) 2/4
11 Institut de Chimie, Strasbourg, France Page 11 Multi-Configurational Self-Consistent Field model (MCSCF) Complete Active Space (CAS) for Be: 1s 2 2s 2, 1s 2 2p 2 x, 1s2 2p 2 y, 1s2 2p 2 z, if all the determinants are included in the MCSCF calculation CASSCF if a Restricted Active Space (RAS) is used RASSCF The orbital space is now divided in three: doubly occupied molecular orbitals (inactive) φ i, φ j,... 1s active molecular orbitals φ u, φ v,... 2s, 2p x, 2p y, 2p z unoccupied molecular orbitals (virtuals) φ a, φ b,... 3s, 3p, 3d,...
12 Institut de Chimie, Strasbourg, France Page 12 a, b,... u, v,... i, j,...
13 Institut de Chimie, Strasbourg, France Page H 2 (FCI, aug cc pvqz) 1σ g 1σ u 1.6 Ψ 0 σ 2 g 1σ g +1σ u orbital occupation bond distance [units of a 0 ] Ψ ( σ 2 g σ2 u )
14 Institut de Chimie, Strasbourg, France Page 14 Multi-Configurational Self-Consistent Field model (MCSCF) EX6: In order to illustrate with H 2 the fact that active orbitals can be partially occupied, show that the active part of the density matrix A D, defined as A D vw = Ψ Êvw Ψ, where Ψ = 1 ( ) 1σ α 1 + c 2 g 1σg β c 1σu α 1σu β, equals A D = c 2 0 2c c 2 Note: In the particular case of a single determinantal wave function (c = 0) the active density matrix reduces to
15 Institut de Chimie, Strasbourg, France Page 15 Iterative optimization of the orbital rotation vector κ and the CI coefficients C i : Ψ (0) = C (0) i i normalized starting wave function i Ψ(λ) = e ˆκ Ψ(0) + ˆQ δ 1 + δ ˆQ δ convenient parametrization λ = ˆQ = 1 Ψ (0) Ψ (0), δ = i δ i i, Ψ (0) ˆQ δ = 0, Ψ(λ) Ψ(λ) = 1. κ pq. δ i. p > q MCSCF energy expression: E(λ) = Ψ(λ) Ĥ Ψ(λ) Variational optimization: E [1] λ + = and E c[1] λ + = E(λ) δ λ+ Eo[1] λ + E c[1] λ + = 0 where E o[1] λ + = E(λ) κ λ+
16 Institut de Chimie, Strasbourg, France Page 16 Newton method: E(λ) E(0) + λ T E [1] λt E [2] 0 λ E[1] λ E [1] E[2] 0 λ + = 0 E [2] 0 λ + }{{} = E[1] 0 Newton step Convergence reached when E [1] 0 = 0 ( ) EX7: Show that E o[1] 0,pq = Ψ(0) [Êpq Êqp, Ĥ] Ψ(0) and E c[1] 0 = 2 H CAS E(0) C (0) where H CAS ij = i Ĥ j and C(0) =. C (0) i. Note: E o[1] 0 = 0 is known as generalized Brillouin theorem.
17 Institut de Chimie, Strasbourg, France Page FCI HF CISD 2/2 MCSCF 2/2 H 2 (1 1 Σ g +, aug cc pvqz) Energy (a.u.) Interatomic distance (a.u.)
18 Institut de Chimie, Strasbourg, France Page 18 Multi-Reference Perturbation Theory (MRPT) Multiconfigurational extension of MP2: unperturbed wave function Φ 0 Ψ (0) unperturbed energy E (0) = 2 i ε i??? perturbers D??? zeroth-order excited energies E (0) + ε a + ε b ε i ε j??? unperturbed Hamiltonian ˆF E (0) Φ 0 Φ 0 + D D ˆF D D D??? Standard approaches are CASPT2 a and N-electron valence state PT2 (NEVPT2) b a K. Andersson, P. Å. Malmqvist, and B. O. Roos, J.Chem. Phys. 96, 1218 (1992). b C. Angeli, R. Cimiraglia, and J.P. Malrieu, J. Chem. Phys. 117, 9138 (2002).
19 Institut de Chimie, Strasbourg, France Page 19 Energy (a.u.) FCI HF CISD 2/2 MCSCF 2/2 sc NEVPT2 H 2 (1 1 Σ g +, aug cc pvqz) Interatomic distance (a.u.)
20 Institut de Chimie, Strasbourg, France Page 20 Multi-state MCSCF approach State-averaged MCSCF model: simultaneous optimization of the ground and the lowest N 1 excited states at the MCSCF level. Iterative procedure: N initial orthonormal states are built from the same set of orbitals. Ψ (0) I = i C (0) I,i i, I = 1,..., N Double-exponential parametrization: Ψ I (κ, S) = e ˆκ e Ŝ Ψ (0) I where Ŝ = N J=1 K>J S KJ ( Ψ (0) K Ψ(0) J ) Ψ(0) J Ψ(0) K and i i i = K Ψ (0) K Ψ(0) K
21 Institut de Chimie, Strasbourg, France Page 21 Multi-state MCSCF approach Gross Oliveira Kohn (GOK) variational principle for an ensemble of ground and excited states: For any set {Ψ I } I=1,N of N orthonormal states, the following inequality holds, N w I Ψ I Ĥ Ψ I I=1 N w I E I I=1 where E 1 E 2... E N are the N lowest exact eigenvalues of ordered as follows, Ĥ, and the weights are w 1 w 2... w N > 0. EXERCISE: Prove the theorem in the particular case of two states by using Theophilou s variational principle: Ψ 1 Ĥ Ψ 1 + Ψ 2 Ĥ Ψ 2 E 1 + E 2. Hint: Show that ] w 1 Ψ 1 Ĥ Ψ 1 + w 2 Ψ 2 Ĥ Ψ 2 = w 2 [ Ψ 1 Ĥ Ψ 1 + Ψ 2 Ĥ Ψ 2 + (w 1 w 2 ) Ψ 1 Ĥ Ψ 1
22 Institut de Chimie, Strasbourg, France Page 22 EXERCISE: Proof of Theophilou s variational principle for two states (1) Let = Ψ { 1 Ĥ Ψ } 1 + Ψ 2 Ĥ Ψ 2 E 1 E 2. We consider the complete basis of the exact eigenvectors ΨI of Ĥ with eigenvalues {E I} I=1,2,... I=1,2,... Both trial wavefunctions can be expanded in that basis as follows, Ψ K = I C KI Ψ I, K = 1, 2. Show that = 2 I=1(p I 1)E I + I>2 p I E I where p I = C 2 1I + C2 2I. (2) Show that = 2 I=1(1 p I )(E 2 E I ) + I>2 p I (E I E 2 ). Hint: prove first that I p I = 2. (3) Let us now decompose the two first eigenvectors (I = 1, 2) in the basis of the trial wavefunctions and the orthogonal complement: Ψ I = C 1I Ψ 1 + C 2I Ψ 2 + ˆQ 12 Ψ I where 2 ˆQ 12 = 1 Ψ K Ψ K. Explain why p I 1 when I = 1, 2 and conclude. K=1
23 Multi-state MCSCF approach State-averaged energy: E(κ, S) = N w I Ψ I (κ, S) Ĥ Ψ I(κ, S) I=1 where w I are arbitrary weights. In the so-called "equal weight" state-averaged MCSCF calculation w I = 1 N. Variational optimization: E(κ, S) κ = E(κ, S) S = 0 Note that, in contrast to the exact theory, converged individual energies (and therefore excitation energies) may vary with the weights. This is due to the orbital optimization. Short-range dynamical correlation is usually recovered within multi-reference perturbation theory (multi-state CASPT2 or NEVPT2 for example) Institut de Chimie, Strasbourg, France Page 23
Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager
Introduction to muticonfigurationa quantum chemistry Emmanue Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS Second schoo in computationa physics,
Second quantization. Emmanuel Fromager
Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Institut
Linear response time-dependent density functional theory
Linear response time-dependent density functional theory Emmanuel Fromager Laboratoire de Chimie Quantique, Université de Strasbourg, France fromagere@unistra.fr Emmanuel Fromager (UdS) Cours RFCT, Strasbourg,
Electron Correlation Methods
Electron Correlation Methods HF method: electron-electron interaction is replaced by an average interaction E HF c = E 0 E HF E 0 exact ground state energy E HF HF energy for a given basis set HF E c
Introduction to Computational Chemistry
Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry Chemicum 4th floor vesa.hanninen@helsinki.fi September 10, 2013 Lecture 3. Electron correlation methods September
Multiconfigurational Quantum Chemistry. Björn O. Roos as told by RL Department of Theoretical Chemistry Chemical Center Lund University Sweden
Multiconfigurational Quantum Chemistry Björn O. Roos as told by RL Department of Theoretical Chemistry Chemical Center Lund University Sweden April 20, 2009 1 The Slater determinant Using the spin-orbitals,
Time-dependent linear response theory: exact and approximate formulations. Emmanuel Fromager
June 215 ISTPC 215 summer school, Aussois, France Page 1 Time-dependent linear response theory: exact and approximate formulations Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie
Electron Correlation - Methods beyond Hartree-Fock
Electron Correlation - Methods beyond Hartree-Fock how to approach chemical accuracy Alexander A. Auer Max-Planck-Institute for Chemical Energy Conversion, Mülheim September 4, 2014 MMER Summerschool 2014
Introduction to density-functional theory. Emmanuel Fromager
Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Institut
Introduction to Electronic Structure Theory
CSC/PRACE Spring School in Computational Chemistry 2017 Introduction to Electronic Structure Theory Mikael Johansson http://www.iki.fi/~mpjohans Objective: To get familiarised with the, subjectively chosen,
Optimization of quantum Monte Carlo wave functions by energy minimization
Optimization of quantum Monte Carlo wave functions by energy minimization Julien Toulouse, Roland Assaraf, Cyrus J. Umrigar Laboratoire de Chimie Théorique, Université Pierre et Marie Curie and CNRS, Paris,
We also deduced that transformations between Slater determinants are always of the form
.3 Hartree-Fock The Hartree-Fock method assumes that the true N-body ground state wave function can be approximated by a single Slater determinant and minimizes the energy among all possible choices of
QUANTUM CHEMISTRY FOR TRANSITION METALS
QUANTUM CHEMISTRY FOR TRANSITION METALS Outline I Introduction II Correlation Static correlation effects MC methods DFT III Relativity Generalities From 4 to 1 components Effective core potential Outline
OVERVIEW OF QUANTUM CHEMISTRY METHODS
OVERVIEW OF QUANTUM CHEMISTRY METHODS Outline I Generalities Correlation, basis sets Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods III Density
4 Post-Hartree Fock Methods: MPn and Configuration Interaction
4 Post-Hartree Fock Methods: MPn and Configuration Interaction In the limit of a complete basis, the Hartree-Fock (HF) energy in the complete basis set limit (ECBS HF ) yields an upper boundary to the
Lecture 4: Hartree-Fock Theory
Lecture 4: Hartree-Fock Theory One determinant to rule them all, One determinant to find them, One determinant to bring them all and in the darkness bind them Second quantization rehearsal The formalism
Wave function methods for the electronic Schrödinger equation
Wave function methods for the electronic Schrödinger equation Zürich 2008 DFG Reseach Center Matheon: Mathematics in Key Technologies A7: Numerical Discretization Methods in Quantum Chemistry DFG Priority
Multiconfigurational methods in MOLCAS
Multiconfigurational methods in MOLCAS Valera Veryazov Valera.Veryazov@teokem.lu.se Division of Theoretical Chemistry Lund University Sweden CASSCF//CASPT2 p. 1/?? Overview What is correlation? CASPT2
Lecture 3: Quantum Satis*
Lecture 3: Quantum Satis* Last remarks about many-electron quantum mechanics. Everything re-quantized! * As much as needed, enough. Electron correlation Pauli principle Fermi correlation Correlation energy
Electron Correlation
Electron Correlation Levels of QM Theory HΨ=EΨ Born-Oppenheimer approximation Nuclear equation: H n Ψ n =E n Ψ n Electronic equation: H e Ψ e =E e Ψ e Single determinant SCF Semi-empirical methods Correlation
Supplementary Information for: Hydrogen abstraction by photoexcited benzophenone: consequences for DNA photosensitization
Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics. This journal is the Owner Societies 2016 Supplementary Information for: Hydrogen abstraction by photoexcited benzophenone:
Introduction to Second-quantization I
Introduction to Second-quantization I Jeppe Olsen Lundbeck Foundation Center for Theoretical Chemistry Department of Chemistry, University of Aarhus September 19, 2011 Jeppe Olsen (Aarhus) Second quantization
Beyond the Hartree-Fock Approximation: Configuration Interaction
Beyond the Hartree-Fock Approximation: Configuration Interaction The Hartree-Fock (HF) method uses a single determinant (single electronic configuration) description of the electronic wavefunction. For
1 Rayleigh-Schrödinger Perturbation Theory
1 Rayleigh-Schrödinger Perturbation Theory All perturbative techniques depend upon a few simple assumptions. The first of these is that we have a mathematical expression for a physical quantity for which
Quantum Chemistry Methods
1 Quantum Chemistry Methods T. Helgaker, Department of Chemistry, University of Oslo, Norway The electronic Schrödinger equation Hartree Fock theory self-consistent field theory basis functions and basis
Recent advances in quantum Monte Carlo for quantum chemistry: optimization of wave functions and calculation of observables
Recent advances in quantum Monte Carlo for quantum chemistry: optimization of wave functions and calculation of observables Julien Toulouse 1, Cyrus J. Umrigar 2, Roland Assaraf 1 1 Laboratoire de Chimie
Coupled cluster theory: Analytic derivatives, molecular properties, and response theory
Coupled cluster theory: Analytic derivatives, molecular properties, and response theory Wim Klopper and David P. Tew Lehrstuhl für Theoretische Chemie Institut für Physikalische Chemie Universität Karlsruhe
Yingwei Wang Computational Quantum Chemistry 1 Hartree energy 2. 2 Many-body system 2. 3 Born-Oppenheimer approximation 2
Purdue University CHM 67300 Computational Quantum Chemistry REVIEW Yingwei Wang October 10, 2013 Review: Prof Slipchenko s class, Fall 2013 Contents 1 Hartree energy 2 2 Many-body system 2 3 Born-Oppenheimer
( R)Ψ el ( r;r) = E el ( R)Ψ el ( r;r)
Born Oppenheimer Approximation: Ĥ el ( R)Ψ el ( r;r) = E el ( R)Ψ el ( r;r) For a molecule with N electrons and M nuclei: Ĥ el What is E el (R)? s* potential surface Reaction Barrier Unstable intermediate
Methods for Treating Electron Correlation CHEM 430
Methods for Treating Electron Correlation CHEM 430 Electron Correlation Energy in the Hartree-Fock approximation, each electron sees the average density of all of the other electrons two electrons cannot
Multiconfigurational and multireference methods
Multiconfigurational and multireference methods Peter R. Taylor and Jeppe Olsen, Torre Normanna, September 2017 September 13-15, 2017 Contents of the lectures Wednesday, Sept. 13 Molecular orbitals for
Supporting information
Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics. This journal is the Owner Societies 2014 Supporting information Resonant Raman Spectra of Molecules with Diradical Character:
Lecture 5: More about one- Final words about the Hartree-Fock theory. First step above it by the Møller-Plesset perturbation theory.
Lecture 5: More about one- determinant wave functions Final words about the Hartree-Fock theory. First step above it by the Møller-Plesset perturbation theory. Items from Lecture 4 Could the Koopmans theorem
AN INTRODUCTION TO QUANTUM CHEMISTRY. Mark S. Gordon Iowa State University
AN INTRODUCTION TO QUANTUM CHEMISTRY Mark S. Gordon Iowa State University 1 OUTLINE Theoretical Background in Quantum Chemistry Overview of GAMESS Program Applications 2 QUANTUM CHEMISTRY In principle,
Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found.
Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found. In applying quantum mechanics to 'real' chemical problems, one is usually faced with a Schrödinger
Optimization of quantum Monte Carlo (QMC) wave functions by energy minimization
Optimization of quantum Monte Carlo (QMC) wave functions by energy minimization Julien Toulouse, Cyrus Umrigar, Roland Assaraf Cornell Theory Center, Cornell University, Ithaca, New York, USA. Laboratoire
Molecular Simulation I
Molecular Simulation I Quantum Chemistry Classical Mechanics E = Ψ H Ψ ΨΨ U = E bond +E angle +E torsion +E non-bond Jeffry D. Madura Department of Chemistry & Biochemistry Center for Computational Sciences
Performance of Hartree-Fock and Correlated Methods
Chemistry 460 Fall 2017 Dr. Jean M. Standard December 4, 2017 Performance of Hartree-Fock and Correlated Methods Hartree-Fock Methods Hartree-Fock methods generally yield optimized geomtries and molecular
Supporting Information
Supporting Information Computational Evidence of Inversion of 1 L a and 1 L b -Derived Excited States in Naphthalene Excimer Formation from ab Initio Multireference Theory with Large Active Space: DMRG-CASPT2
Practical Issues on the Use of the CASPT2/CASSCF Method in Modeling Photochemistry: the Selection and Protection of an Active Space
Practical Issues on the Use of the CASPT2/CASSCF Method in Modeling Photochemistry: the Selection and Protection of an Active Space Roland Lindh Dept. of Chemistry Ångström The Theoretical Chemistry Programme
The Overhauser Instability
The Overhauser Instability Zoltán Radnai and Richard Needs TCM Group ESDG Talk 14th February 2007 Typeset by FoilTEX Introduction Hartree-Fock theory and Homogeneous Electron Gas Noncollinear spins and
Calculations of band structures
Chemistry and Physics at Albany Planning for the Future Calculations of band structures using wave-function based correlation methods Elke Pahl Centre of Theoretical Chemistry and Physics Institute of
Introduction to DFTB. Marcus Elstner. July 28, 2006
Introduction to DFTB Marcus Elstner July 28, 2006 I. Non-selfconsistent solution of the KS equations DFT can treat up to 100 atoms in routine applications, sometimes even more and about several ps in MD
Computational Chemistry I
Computational Chemistry I Text book Cramer: Essentials of Quantum Chemistry, Wiley (2 ed.) Chapter 3. Post Hartree-Fock methods (Cramer: chapter 7) There are many ways to improve the HF method. Most of
Multiconfigurational methods for the f-elements
Multiconfigurational methods for the f-elements Andy Kerridge Winter School in Theoretical f-element Chemistry Helsinki, Finland, December 5 th -8 th, 204 Overview CASPT2: reference weights and intruder
A Whirlwind Introduction to Coupled Cluster Response Theory. 1 Time-Independent Coupled Cluster Theory
A Whirlwind Introduction to Coupled Cluster Response Theory T. Daniel Crawford, Virginia Tech, Blacksburg, Virginia, U.S.A. 1 Time-Independent Coupled Cluster Theory If the Hamiltonian is independent of
0 + E (1) and the first correction to the ground state energy is given by
1 Problem set 9 Handout: 1/24 Due date: 1/31 Problem 1 Prove that the energy to first order for the lowest-energy state of a perturbed system is an upper bound for the exact energy of the lowest-energy
Multi-configuration time-dependent density-functional theory based on range separation
Multi-configuration time-dependent density-functional theory based on range separation Emmanuel Fromager, Stefan Knecht, and Hans Jørgen Aa. Jensen Citation: J. Chem. Phys. 138, 8411 213); doi: 1.163/1.4792199
Extended Møller-Plesset perturbation theory for dynamical and static correlations
Extended Møller-Plesset perturbation theory for dynamical and static correlations The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.
Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer approx.- energy surfaces 2. Mean-field (Hartree-Fock) theory- orbitals 3. Pros and cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usually does HF-how? 6. Basis sets and notations
Importing ab-initio theory into DFT: Some applications of the Lieb variation principle
Importing ab-initio theory into DFT: Some applications of the Lieb variation principle Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department
Quantum Monte Carlo wave functions and their optimization for quantum chemistry
Quantum Monte Carlo wave functions and their optimization for quantum chemistry Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France CEA Saclay, SPhN Orme des Merisiers April 2015 Outline
MO Calculation for a Diatomic Molecule. /4 0 ) i=1 j>i (1/r ij )
MO Calculation for a Diatomic Molecule Introduction The properties of any molecular system can in principle be found by looking at the solutions to the corresponding time independent Schrodinger equation
Simulating Chemistry using Quantum Computers, Annu. Rev. Phys. Chem. 62, 185 (2011).
I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M-H. Yung, A. Aspuru-Guzik, Simulating Chemistry using Quantum Computers, Annu. Rev. Phys. Chem. 6, 185 (011). Martes Cuántico Universidad de Zaragoza, 19th
Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer approx.- energy surfaces 2. Mean-field (Hartree-Fock) theory- orbitals 3. Pros and cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usually does HF-how? 6. Basis sets and notations
3: Many electrons. Orbital symmetries. l =2 1. m l
3: Many electrons Orbital symmetries Atomic orbitals are labelled according to the principal quantum number, n, and the orbital angular momentum quantum number, l. Electrons in a diatomic molecule experience
Hartree-Fock Theory. ˆf(r)χ i (x) = ɛ i χ i (x) (1) Z A K=1 I. DERIVATION OF THE HARTREE-FOCK EQUATIONS. A. The energy of a Slater determinant
Hartree-Fock Theory The HF approximation plays a crucial role in chemistry and constitutes the starting point for more elaborate treatments of electron correlation. Furthermore, many semi-empirical methods
Hartree-Fock. It can also be viewed as the lowest order term in perturbative expansion (linear in U) with respect to interaction between electrons.
Hartree-Fock It is probably the simplest method to treat the many-particle system. The dynamic many particle problem is replaced by an effective one-electron problem: electron is moving in an effective
Chemistry 3502/4502. Final Exam Part I. May 14, 2005
Chemistry 3502/4502 Final Exam Part I May 14, 2005 1. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle (e) The
Chemistry 3502/4502. Final Exam Part I. May 14, 2005
Advocacy chit Chemistry 350/450 Final Exam Part I May 4, 005. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle
The calculation of the universal density functional by Lieb maximization
The calculation of the universal density functional by Lieb maximization Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,
Building a wavefunction within the Complete-Active. Cluster with Singles and Doubles formalism: straightforward description of quasidegeneracy
Building a wavefunction within the Complete-Active Active-Space Coupled-Cluster Cluster with Singles and Doubles formalism: straightforward description of quasidegeneracy Dmitry I. Lyakh (Karazin Kharkiv
Computational Chemistry. Ab initio methods seek to solve the Schrödinger equation.
Theory Computational Chemistry Ab initio methods seek to solve the Schrödinger equation. Molecular orbital theory expresses the solution as a linear combination of atomic orbitals. Density functional theory
Extending Kohn-Sham density-functional theory
Extending Kohn-Sham density-functional theory Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Email: julien.toulouse@upmc.fr Web page: www.lct.jussieu.fr/pagesperso/toulouse/ January
Computational Methods. Chem 561
Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational
SCF calculation on HeH +
SCF calculation on HeH + Markus Meuwly Department of Chemistry, University of Basel, Basel, Switzerland Abstract This document describes the main steps involved in carrying out a SCF calculation on the
Introduction to the calculation of molecular properties by response theory
Introduction to the calculation of molecular properties by response theory Julien Toulouse Laboratoire de Chimie Théorique Université Pierre et Marie Curie et CNRS, 75005 Paris, France julien.toulouse@upmc.fr
Chemistry Publications
Chemistry Publications Chemistry 2007 Accurate Ab Initio Potential Energy Curve of F2. I. Nonrelativistic Full Valence Configuration Interaction Energies Using the Correlation Energy Extrapolation by Intrinsic
arxiv: v2 [physics.chem-ph] 14 Jan 2013
Multi-configuration time-dependent density-functional theory based on range separation Emmanuel Fromager a, Stefan Knecht b and Hans Jørgen Aa. Jensen b a Laboratoire de Chimie Quantique, arxiv:1211.4829v2
No. 2 lectronic state and potential energy function for UH where ρ = r r e, r being the interatomic distance and r e its equilibrium value. How
Vol 12 No 2, February 2003 cfl 2003 Chin. Phys. Soc. 1009-1963/2003/12(02)/0154-05 Chinese Physics and IOP Publishing Ltd lectronic state and potential energy function for UH 2+* Wang Hong-Yan( Ψ) a)y,
An Introduction to Quantum Chemistry and Potential Energy Surfaces. Benjamin G. Levine
An Introduction to Quantum Chemistry and Potential Energy Surfaces Benjamin G. Levine This Week s Lecture Potential energy surfaces What are they? What are they good for? How do we use them to solve chemical
Inclusion of Mean Field Effects in the Two-Mode Model of BECs in a Double Well
Inclusion of Mean Field Effects in the Two-Mode Model of BECs in a Double Well David Masiello in collaboration with Bill Reinhardt University of Washington, Seattle August 8th, 2005 masiello@u.washington.edu
The MCSCF Method *, Molecular Orbitals, Reference Spaces and COLUMBUS Input
The MCSCF Method *, Molecular Orbitals, Reference Spaces and COLUMBUS Input Hans Lischka University of Vienna *Excerpt of a course presented by R. Shepard, Argonne National Laboratory, at the Workshop
Special 5: Wavefunction Analysis and Visualization
Special 5: Wavefunction Analysis and Visualization Felix Plasser Institute for Theoretical Chemistry, University of Vienna COLUMBUS in China Tianjin, October 10 14, 2016 F. Plasser Wavefunction Analysis
2~:J~ -ryej- r- 2 Jr. A - f3. sr(djk nv~tor rn~ +~ rvjs (::-CJ) ::;-1-.'--~ -. rhd. ('-.Ji.L.~ )- r'-d)c, -r/~ JJr - 2~d ~2-Jr fn'6.
.~, ~ I, sr(djk nv~tor rn~ +~ rvjs (::-CJ) ::;-1-.'--~ -. rhd. ('-.Ji.L.~ )- r'-d)c, -r/~ JJr - 2~d ~2-Jr fn'6.)1e'" 21t-ol Je C'...-------- lj-vi, J? Jr Jr \Ji 2~:J~ -ryej- r- 2 Jr A - f3 c _,~,= ~,.,w._..._.
Handbook of Computational Quantum Chemistry. DAVID B. COOK The Department of Chemistry, University of Sheffield
Handbook of Computational Quantum Chemistry DAVID B. COOK The Department of Chemistry, University of Sheffield Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1998 CONTENTS 1 Mechanics and molecules 1 1.1
arxiv: v2 [physics.chem-ph] 8 Jun 2018
Version 12 June 2018 Multiconfigurational Short-Range Density-Functional Theory for Open-Shell Systems Erik Donovan Hedegård, 1, a) Julien Toulouse, 2 3, b) and Hans Jørgen Aagaard Jensen 1) Department
Hartree, Hartree-Fock and post-hf methods
Hartree, Hartree-Fock and post-hf methods MSE697 fall 2015 Nicolas Onofrio School of Materials Engineering DLR 428 Purdue University nonofrio@purdue.edu 1 The curse of dimensionality Let s consider a multi
MODERN ELECTRONIC STRUCTURE THEORY: Electron Correlation
5.61 Physical Chemistry Lecture #30 1 MODERN ELECTRONIC STRUCTURE THEORY: Electron Correlation In the previous lecture, we covered all the ingredients necessary to choose a good atomic orbital basis set.
Time-independent molecular properties
Time-independent molecular properties Trygve Helgaker Hylleraas Centre, Department of Chemistry, University of Oslo, Norway and Centre for Advanced Study at the Norwegian Academy of Science and Letters,
Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule. Vesa Hänninen
Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule Vesa Hänninen 1 Introduction In this exercise the equilibrium structure and the electronic energy
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
Handbook of Computational Quantum Chemistry
Handbook of Computational Quantum Chemistry David B. Cook Dept. of Chemistry University of Sheffield DOVER PUBLICATIONS, INC. Mineola, New York F Contents 1 Mechanics and molecules 1 1.1 1.2 1.3 1.4 1.5
0 belonging to the unperturbed Hamiltonian H 0 are known
Time Independent Perturbation Theory D Perturbation theory is used in two qualitatively different contexts in quantum chemistry. It allows one to estimate (because perturbation theory is usually employed
NWChem: Coupled Cluster Method (Tensor Contraction Engine)
NWChem: Coupled Cluster Method (ensor Contraction Engine) What we want to solve H Ψ = E Ψ Many Particle Systems Molecular/Atomic Physics, Quantum Chemistry (electronic Schrödinger equations) Solid State
Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer approx.- energy surfaces 2. Mean-field (Hartree-Fock) theory- orbitals 3. Pros and cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usually does HF-how? 6. Basis sets and notations
Lecture 4: methods and terminology, part II
So theory guys have got it made in rooms free of pollution. Instead of problems with the reflux, they have only solutions... In other words, experimentalists will likely die of cancer From working hard,
Chapter 2 Quantum chemistry using auxiliary field Monte Carlo
Chapter 2 Quantum chemistry using auxiliary field Monte Carlo 1. The Hubbard-Stratonovich Transformation 2. Neuhauser s shifted contour 3. Calculation of forces and PESs 4. Multireference AFMC 5. Examples
Accurate description of potential energy surfaces by ab initio methods : a review and application to ozone
Accurate description of potential energy surfaces by ab initio methods : a review and application to ozone Péter G. Szalay Laboratory of Theoretical Chemistry Institute of Chemistry Eötvös Loránd University,
COPYRIGHTED MATERIAL. Quantum Mechanics for Organic Chemistry &CHAPTER 1
&CHAPTER 1 Quantum Mechanics for Organic Chemistry Computational chemistry, as explored in this book, will be restricted to quantum mechanical descriptions of the molecules of interest. This should not
arxiv: v2 [physics.chem-ph] 11 Mar 2011
Quantum Monte Carlo with Jastrow-Valence-Bond wave functions Benoît Braïda 1, Julien Toulouse 1, Michel Caffarel 2, and C. J. Umrigar 3 1 Laboratoire de Chimie Théorique, Université Pierre et Marie Curie
Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation.
Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation. Bastien Mussard, Julien Toulouse Laboratoire de Chimie Théorique Institut du Calcul
Pseudo-Hermitian eigenvalue equations in linear-response electronic-structure theory
1/11 Pseudo-Hermitian eigenvalue equations in linear-response electronic-structure theory Julien Toulouse Université Pierre & Marie Curie and CNRS, 4 place Jussieu, Paris, France Web page: www.lct.jussieu.fr/pagesperso/toulouse/
Coupled-cluster and perturbation methods for macromolecules
Coupled-cluster and perturbation methods for macromolecules So Hirata Quantum Theory Project and MacroCenter Departments of Chemistry & Physics, University of Florida Contents Accurate electronic structure
Ψ = Σ m C i,m φ 1...φ m...φ N + φ 1...φ i...φ N.
Chapter 19 Corrections to the mean-field model are needed to describe the instantaneous Coulombic interactions among the electrons. This is achieved by including more than one Slater determinant in the
Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer approx.- energy surfaces 2. Mean-field (Hartree-Fock) theory- orbitals 3. Pros and cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usually does HF-how? 6. Basis sets and notations
CHEM6085: Density Functional Theory Lecture 10
CHEM6085: Density Functional Theory Lecture 10 1) Spin-polarised calculations 2) Geometry optimisation C.-K. Skylaris 1 Unpaired electrons So far we have developed Kohn-Sham DFT for the case of paired
CHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules Lecture 5 The Hartree-Fock method C.-K. Skylaris Learning outcomes Be able to use the variational principle in quantum calculations Be able to construct Fock operators
In this lecture we will understand how the molecular orbitals are formed from the interaction of atomic orbitals.
Lecture 7 Title: Understanding of Molecular Orbital Page-1 In this lecture we will understand how the molecular orbitals are formed from the interaction of atomic orbitals. We will see how the electrons
Valence bond theory accounts, at least qualitatively, for the stability of the covalent bond in terms of overlapping atomic orbitals.
Molecular Orbital Theory Valence bond theory accounts, at least qualitatively, for the stability of the covalent bond in terms of overlapping atomic orbitals. Using the concept of hybridization, valence