Electron Correlation Methods
|
|
- Chloe Scott
- 5 years ago
- Views:
Transcription
1 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 < 0 - represents a measure for the error introduced by the HF approximation Dynamical correlation Non-dynamical correlation related to the movements of the individual electrons - short range effect - related to the fact that in certain circumstances the ground state ψ SD wave-function is not a good approximation to the true ground state because there are other Slater determinants with comparable energies multideterminantal wave-function Ψ = a + HF 0Ψ aiψi i usually a 0 1 Frank Jensen, Introduction to Computational Chemistry, John Wiley and Sons, New York, 1999
2 Excited Slater Determinants (ESD) Suppose we have N electrons and K basis functions used to expand the MOs ESD RHF formalism will give N/2 occupied MOs and K-N/2 virtual MOs obtained by replacing MOs which are occupied in the HF determinant by unoccupied MOs - singly, doubly, triply, quadruply, etc. excited relative to the HF determinant Total number of ESD depends on the size of the basis set If all the possible ESD (in a given basis set) are included then all the electron correlation energy is recovered
3 Methods including electron correlation are two-dimensional!!
4 In many cases the interest is only in calculating the correlation energy associated with the valence electrons Frozen Core Approximation (FCA) = limiting the number of ESD to only those which can be generated by exciting the valence electrons - it is not justified in terms of total energy because the correlation of core electrons gives substantial contribution. However, it is essentially a constant factor which drops out when relative energies are calculated Methods for taking the electron correlation into account: Configuration Interaction (CI) Many Body Perturbation Theory (MBPT) Moller-Plesset (MP) Theory Coupled Cluster (CC)
5 Configuration Interaction (CI) -based on the variational principle, the trial wave-function being expressed as a linear combination of Slater determinants The expansion coefficients are determined by imposing that the energy should be a minimum. The MOs used for building the excited determinants are taken from HF calculation and held fixed ΨCI = a0ψscf + asψs + adψd + atψt +... S In the large basis set limit, all electron correlation methods scale at least as K 5 D T Example Molecule: H 2 O Basis set: 6-31G(d) => 19BF => 38 spin MOs (10 occupied, 28 virtual) The total number of excited determinants will be C 38 = Many of them will have different spin multiplicity and can therefore be left out in the calculation. Generating only the singlet Configurational State Functions (CSF) we still obtain determinants Full CI method is only feasible for very small systems!!! 10
6 Configuration State Functions Consider a single excitation from the RHF reference. Both Φ RHF and Φ (1) have S z =0, but Φ (1) is not an eigenfunction of S 2. Φ RHF Φ (1) Linear combination of singly excited determinants is an eigenfunction of S 2. Configuration State Function, CSF (Spin Adapted Configuration, SAC) Singlet CSF Only CSFs that have the same multiplicity as the HF reference Φ( 1,2) = φ 1 α(1)φ 2 β(2) φ 1 α(2)φ 2 β(1)
7 Truncated CI methods Ψ CI = a0 ΨSCF + asψs + adψd + at ΨT +... s Truncating the expansion given above at level one => CIS - CI with only single excited determinants CID - CI with only doubly excited determinants CISD - CI with Singles and Doubles (scales as K 6 ) CISDT - CI with Singles, Doubles and Triples (scales as K 8 ) CISDTQ - CI with Singles, Doubles, Triples and Quadruples (scales as K 10 ) - gives results close to the full CI - can only be applied to small molecules and small basis sets D T CISD - the only CI method which is generally feasible for a large variety of systems - recovers 80-90% of the available correlation energy
8 Multi-Configuration Self-Consistent Field Method (MCSCF) - is the CI method in which the MOs are also varied, along with the coefficients of the CI expansion MCSCF methods - are mainly used for generating a qualitatively correct wave-function - recover the static part of the correlation (the energy lowering is due to the greater flexibility in the wave-function) dynamic correlation the correlation of the electrons motions In MCSCF methods the necessary configurations must be selected CASSCF (Complete Active Space SCF) - the selection of the configurations is done by partitioning the MOs into active and inactive spaces active MOs - some of the highest occupied and some of the lowest unoccupied MOs Within the active MOs a full CI is performed A more complete notation for this kind of methods is: [n,m]-casscf - n electrons are distributed in all possible ways in m orbitals
9 H 2 O MOs Carry out Full CI and orbital optimization within a small active space. Six-electron in six-orbital MCSCF is shown (written as [6,6]-CASSCF) Complete Active Space Self-consistent Field (CASSCF) Why? 1. To have a better description of the ground or excited state. Some molecules are not welldescribed by a single Slater determinant, e.g. O To describe bond breaking/formation; Transition States. 3. Open-shell system, especially low-spin. 4. Low lying energy level(s); mixing with the ground state produces a better description of the electronic state. Ψ HF
10
11 Alternative to CASSCF Restricted Active Space SCF (RASSCF) RASSCF the active MOs are further divided into three sections: RAS1, RAS2 and RAS3 RAS1 space MOs doubly occupied in the HF reference determinant RAS2 space both occupied and virtual MOs in the HF reference determinant RAS3 space MOs empty in the HF reference determinant Configurations in RAS2 are generated by a full CI Additional configurations are generated by allowing for example a maximum of two electrons to be excited from RAS1 and a maximum of two electrons to be excited to RAS3 RASSCF combines a full CI in a small number of MOs (RAS2) and a CISD in a larger MO space (RAS1 and RAS3)
12 Multi-reference Configuration Interaction - involve the excitations of electrons out of all the determinants which enter the MCSCF
13 MØller-Plesset Perturbation Theory - a perturbational method in which the unperturbed Hamiltonian is chosen as a sum over Fock operators N N N N H 0 = F i hi ( Jij Kij ) = + = hi + 2 i= 1 i= 1 j= 1 i= 1 V ee The sum of Fock operators counts the average electron-electron repulsion twice and the perturbation is chosen the difference: Vee 2 V ee where V ee represents the exact operator for the electron-electron repulsion It can be shown (Jensen, pag.127) that the zero order wave-function is the HF determinant while the zero order energy is just the sum of MO energies. Also, the first order energy is exactly the HF energy so that in this approach the correlation energy is recovered starting with the second order correction (MP2 method) In addition, the first contribution to the correlation energy involves a sum over doubly excited determinants which can be generated by promoting two electrons from occupied MOs i and j to virtual MOs a and b. The explicit formula for the second order Moller-Plesset correction is: E( MP2) = occ vir [ Φ Φ Φ Φ Φ Φ Φ Φ ] i j i< j a< b εi + ε j ε a a b MP2 method - scales as K 5 - accounts for cca % of the correlation energy - is fairly inexpensive (from the computational resources perspective) for systems with reasonable number of basis functions ( ) i j ε b b a 2
14 Coupled Cluster (CC) Methods The idea in CC methods is to include all corrections of a given type to infinite order. The wave-function is written as: Ψ cc = e T Ψ0 e = 1+ T + T +... = where: k = with the cluster operator given by: T = T1 + T2 + T T N T 2 Acting on the HF reference wave function, the T i operator generates all i-th excited Slater determinants: T Ψ T Ψ 0 = = occ i occ vir a vir i< j a< b t a i t Ψ ab ij a i Ψ ab ij The exponential operator may be rewritten as: e T = 1+ T 1 + T T T 3 + T T T T k! k +... First term generates the reference HF wave-function Second term generates all singly excited determinants First parentheses generates all doubly excited states (true doubly excited states by T 2 or product of singly excited states by the product T 1 T 1
15 The second parentheses generates all triply excited states, true (T 3 ) or products triples (T 1 T 2, T 1 T 1 T 1 ) The energy is given by: occ vir ( ab a b b a t + )( Φ Φ Φ Φ Φ Φ Φ Φ ) ij ti t j ti t j i j a b i j b a E = E + cc 0 i< j a< b So, the coupled cluster correlation energy is determined completely by the singles and doubles amplitudes and the two-electron MO integrals Truncated Coupled Cluster Methods If all T N operators are included in T the CC wave-function is equivalent to full CI wavefunction, but this is possible only for the smallest systems. Truncation of T Including only the T 1 operator there will be no improvement over HF, the lowest level of approximation being T=T 2 ( CCD=Coupled Cluster Doubles) If T=T 1 +T 2 CCSD scales as K 6 the only generally applicable model If T=T 1 +T 2 +T 3 CCSDT scales as K 8
16 Basis Set Superposition Error
17
18
19 Algorithm for BSSE 1. Optimize the geometry of the complex => E 2. Optimize the geometry of the fragments => E A and E B 3. Single point calculation on the optimized geometry of the complex with Counter keyword => δ 4. Calculate E CP corrected = E+δ 5. E CP = E CP corrected E A A (A)-E B B (B) OR 1. Optimize the geometry of the complex with the Counter keyword => E CP corrected 2. Optimize the geometry of the fragments => E A and E B 3. E CP = E CP corrected E A A (A)-E B B (B)
20 Quantum chemical calculations are frequently used to estimate strengths of hydrogen bonds. We can distinguish between intermolecular and intra-molecular hydrogen bonds. The first of these are usually much more straightforward to deal with. 1. Intermolecular Hydrogen Bond energies In this case is is normal to define the hydrogen bond energy as the energy of the hydrogen bonded complex minus the energies of the constituent molecules/ions. Let us first consider a simple example with high (C 3v ) symmetry H 3 N...HF Electronic energy (a.u.) NH HF H 3 N HF Counterpoise Correction E HB NH 3 + HF H 3 N HF E HB = x ( )= 38.8 kj/mol E HB = E( H 3 N... HF ) E(HF in basis of H 3 N... HF ) - E(H 3 N in basis of HN 3... HF)
21 Practical aspects The Massage keyword requests that the molecule specification and basis set data be modified after it is generated. The Massage keyword thus makes it possible to add additional uncontracted basis functions to a standard basis set. The following input file performs a portion of a counterpoise calculation, removing the HF molecule but leaving its basis functions. Note that the dummy atom is not included in the numbering of the centers. # HF/6-31G* Massage Test HF + H2O interaction energy: HF removed 0 1 X H F 2 rhf O 2 rho H 4 roh 2 ahoh H 4 roh 2 ahoh rhf rho roh 0.94 ahoh References: 1. Pedro Salvador Sedano, Implementation and Application of BSSE Schemes to the Theoretical Modeling of Weak Intermolecular Interactions, PhD Thesis, Department of Chemistry and Institute of Computational Chemistry, University of Girona; 2. M. L. SENENT, S. WILSON, Intramolecular Basis Set Superposition Errors, International Journal of Quantum Chemistry, Vol. 82, (2001) 3. A. BENDE, Á. VIBÓK, G. J. HALÁSZ, S. SUHAI, BSSE-Free Description of the Formamide Dimers, International Journal of Quantum Chemistry, Vol. 84, (2001) 1 Nuc Nuc 0.0 Gaussian Help
22 Exercise Calculate the interaction energies in the DNA base pairs Adenine-Thymine and Cytosine-Guanine. Consider the BSSE You can look for pdb files of DNA bases at: Adenine-Thymine base pair Guanine-Cytosine base pair
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
More informationElectron Correlation Methods
Electron Correltion Methods HF method: electron-electron interction is replced by n verge interction E HF c E 0 E HF E 0 exct ground stte energy E HF HF energy for given bsis set HF Ec 0 - represents mesure
More informationLecture 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,
More informationElectron 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
More informationAN 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,
More informationPerformance 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
More informationIntroduction 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,
More informationMethods 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
More informationComputational 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
More informationIntroduction to multiconfigurational quantum chemistry. 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. Notations
More information0 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
More informationMO 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
More informationNWChem: Coupled Cluster Method (Tensor Contraction Engine)
NWChem: Coupled Cluster Method (Tensor Contraction Engine) Why CC is important?! Correlation effects are important!! CC is size-extensive theory: can be used to describe dissociation processes.! Higher-order
More informationExercise 1: Structure and dipole moment of a small molecule
Introduction to computational chemistry Exercise 1: Structure and dipole moment of a small molecule Vesa Hänninen 1 Introduction In this exercise the equilibrium structure and the dipole moment of a small
More informationOVERVIEW 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
More informationBeyond 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
More informationAb initio calculations for potential energy surfaces. D. Talbi GRAAL- Montpellier
Ab initio calculations for potential energy surfaces D. Talbi GRAAL- Montpellier A theoretical study of a reaction is a two step process I-Electronic calculations : techniques of quantum chemistry potential
More information1 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
More information4 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
More informationComputational 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
More informationJack 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
More informationElectric properties of molecules
Electric properties of molecules For a molecule in a uniform electric fielde the Hamiltonian has the form: Ĥ(E) = Ĥ + E ˆµ x where we assume that the field is directed along the x axis and ˆµ x is the
More informationNWChem: 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
More informationElectron 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
More informationChemistry 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
More informationChemistry 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
More information2~: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._..._.
More informationIntroduction 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
More informationCOPYRIGHTED 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
More informationMulticonfigurational 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,
More informationBuilding 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
More informationIntroduction to Computational Quantum Chemistry: Theory
Introduction to Computational Quantum Chemistry: Theory Dr Andrew Gilbert Rm 118, Craig Building, RSC 3108 Course Lectures 2007 Introduction Hartree Fock Theory Configuration Interaction Lectures 1 Introduction
More informationLec20 Fri 3mar17
564-17 Lec20 Fri 3mar17 [PDF]GAUSSIAN 09W TUTORIAL www.molcalx.com.cn/wp-content/uploads/2015/01/gaussian09w_tutorial.pdf by A Tomberg - Cited by 8 - Related articles GAUSSIAN 09W TUTORIAL. AN INTRODUCTION
More informationAB INITIO METHODS IN COMPUTATIONAL QUANTUM CHEMISTRY
AB INITIO METHODS IN COMPUTATIONAL QUANTUM CHEMISTRY Aneesh. M.H A theoretical study on the regioselectivity of electrophilic reactions of heterosubstituted allyl systems Thesis. Department of Chemistry,
More informationQuantum 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
More informationQUANTUM CHEMISTRY PROJECT 3: ATOMIC AND MOLECULAR STRUCTURE
Chemistry 460 Fall 2017 Dr. Jean M. Standard November 1, 2017 QUANTUM CHEMISTRY PROJECT 3: ATOMIC AND MOLECULAR STRUCTURE OUTLINE In this project, you will carry out quantum mechanical calculations of
More informationPractical 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
More informationJack 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
More informationComputational Chemistry: Molecular Simulations with Chemical and Biological Applications. Prof. M. Meuwly, Dr. PA Cazade, Dr. M.-W.
Computational Chemistry: Molecular Simulations with Chemical and Biological Applications Prof. M. Meuwly, Dr. PA Cazade, Dr. M.-W. Lee Overview 1. Electronic Structure of Molecules 1.1 The Electronic Problem
More informationIntroduction to Computational Chemistry: Theory
Introduction to Computational Chemistry: Theory Dr Andrew Gilbert Rm 118, Craig Building, RSC andrew.gilbert@anu.edu.au 3023 Course Lectures Introduction Hartree Fock Theory Basis Sets Lecture 1 1 Introduction
More informationChemistry 334 Part 2: Computational Quantum Chemistry
Chemistry 334 Part 2: Computational Quantum Chemistry 1. Definition Louis Scudiero, Ben Shepler and Kirk Peterson Washington State University January 2006 Computational chemistry is an area of theoretical
More informationChemistry 4560/5560 Molecular Modeling Fall 2014
Final Exam Name:. User s guide: 1. Read questions carefully and make sure you understand them before answering (if not, ask). 2. Answer only the question that is asked, not a different question. 3. Unless
More informationCopyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and
Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere
More informationOther 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
More informationHigh-level Quantum Chemistry Methods and Benchmark Datasets for Molecules
High-level Quantum Chemistry Methods and Benchmark Datasets for Molecules Markus Schneider Fritz Haber Institute of the MPS, Berlin, Germany École Polytechnique Fédérale de Lausanne, Switzerland دانشگاه
More informationQuantum Chemical and Dynamical Tools for Solving Photochemical Problems
2.165430 3.413060 3.889592 9 H 3.413060 2.165430 1.099610 2.165430 3.413060 10 H 3.889592 3.413060 2.165430 1.099610 2.165430 11 H 3.413060 3.889592 3.413060 2.165430 1.099610 12 H 2.165430 3.413060 3.889592
More information( 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
More informationVol. 9 COMPUTATIONAL CHEMISTRY 319
Vol. 9 COMPUTATIONAL CHEMISTRY 319 COMPUTATIONAL QUANTUM CHEMISTRY FOR FREE-RADICAL POLYMERIZATION Introduction Chemistry is traditionally thought of as an experimental science, but recent rapid and continuing
More informationJack 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
More informationJack 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
More informationWave 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
More informationLUMO + 1 LUMO. Tómas Arnar Guðmundsson Report 2 Reikniefnafræði G
Q1: Display all the MOs for N2 in your report and classify each one of them as bonding, antibonding or non-bonding, and say whether the symmetry of the orbital is σ or π. Sketch a molecular orbital diagram
More informationComputational 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
More informationI. CSFs Are Used to Express the Full N-Electron Wavefunction
Chapter 11 One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N- Electron Configuration Functions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon
More informationChemistry 433 Computational Chemistry Fall Semester 2002 Dr. Rainer Glaser
Chemistry 433 Computational Chemistry Fall Semester 2002 Dr. Rainer Glaser Second 1-Hour Examination Electron Correlation Methods Wednesday, November 6, 2002, 11:00-11:50 Name: Answer Key Question 1. Electron
More informationComputational Chemistry
Computational Chemistry Physical Chemistry Course Autumn 2015 Lecturers: Dos. Vesa Hänninen and Dr Garold Murdachaew vesa.hanninen@helsinki.fi Room B407 http://www.helsinki.fi/kemia/fysikaalinen/opetus/
More informationHighly accurate quantum-chemical calculations
1 Highly accurate quantum-chemical calculations T. Helgaker Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, Norway A. C. Hennum and T. Ruden, University
More informationThe 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
More informationuse the backs of pages as needed
CHEMISTRY 4021/8021 Q1) Propose a simple, united-atom molecular mechanics force-field needed to generate a potential energy surface for an isolated molecule of acetone (Me 2 CO). I.e., provide an energy
More informationLecture 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
More informationSession 1. Introduction to Computational Chemistry. Computational (chemistry education) and/or (Computational chemistry) education
Session 1 Introduction to Computational Chemistry 1 Introduction to Computational Chemistry Computational (chemistry education) and/or (Computational chemistry) education First one: Use computational tools
More informationRelativistic and correlated calculations on the ground, excited, and ionized states of iodine
Relativistic and correlated calculations on the ground, excited, and ionized states of iodine W. A. de Jong, L. Visscher, a) and W. C. Nieuwpoort Laboratory for Chemical Physics and Materials Science Centre,
More informationMolecular 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
More informationMRCI calculations in MOLPRO
1 MRCI calculations in MOLPRO Molpro is a software package written in Fortran and maintained by H.J. Werner and P.J. Knowles. It is often used for performing sophisticated electronic structure calculations,
More informationQUANTUM 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
More informationHandbook 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
More informationHartree-Fock-Roothan Self-Consistent Field Method
Hartree-Fock-Roothan Self-Consistent Field Method 1. Helium Here is a summary of the derivation of the Hartree-Fock equations presented in class. First consider the ground state of He and start with with
More informationFeet on the potential energy surface, head in the π clouds
Graduate Theses and Dissertations Iowa State University Capstones, Theses and Dissertations 2011 Feet on the potential energy surface, head in the π clouds Quentin Anthony Smith Iowa State University Follow
More informationSupporting 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
More informationLecture 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
More informationThe Accurate Calculation of Molecular Energies and Properties: A Tour of High-Accuracy Quantum-Chemical Methods
1 The Accurate Calculation of Molecular Energies and Properties: A Tour of High-Accuracy Quantum-Chemical Methods T. Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry,
More informationPotential Energy Surfaces for Quantum Dynamics Simulations: From ab initio Computations to Vibrational State Determinations
Potential Energy Surfaces for Quantum Dynamics Simulations: From ab initio Computations to Vibrational State Determinations by Ekadashi Pradhan A thesis submitted in partial fulfillment of the requirements
More informationAccurate 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,
More informationComputational Material Science Part II. Ito Chao ( ) Institute of Chemistry Academia Sinica
Computational Material Science Part II Ito Chao ( ) Institute of Chemistry Academia Sinica Ab Initio Implementations of Hartree-Fock Molecular Orbital Theory Fundamental assumption of HF theory: each electron
More informationThe chemical Hamiltonian approach (CHA) [1] Convergence Acceleration Techniques for Non-Hermitian SCF Problems PEDRO SALVADOR.
Convergence Acceleration Techniques for Non-Hermitian SCF Problems PEDRO SALVADOR Institute of Computational Chemistry, University of Girona, Campus Montilivi s/n, 17071 Girona, Spain Received 10 October
More informationCharge-Transfer and Dispersion Energies in Water Clusters
II.26 Charge-Transfer and Dispersion Energies in Water Clusters Suehiro Iwata 1,2, Pradipta Bandyopadhyay 3, Sotiris S. Xantheas 4 1)Toyota Physical and Chemical Research Institute (2008-2012, fellow)
More informationJack 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
More informationCoupled-Cluster Perturbative Triples for Bond Breaking
Coupled-Cluster Perturbative Triples for Bond Breaking Andrew G. Taube and Rodney J. Bartlett Quantum Theory Project University of Florida INT CC Meeting Seattle July 8, 2008 Why does chemistry need triples?
More informationComputational Chemistry. An Introduction to Molecular Dynamic Simulations
Computational Chemistry An Introduction to Molecular Dynamic Simulations Computational chemistry simulates chemical structures and reactions numerically, based in full or in part on the fundamental laws
More informationCOMPUTATIONAL MODELING OF SMALL MOLECULES. Rebecca J. Weber, B.S. Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY
COMPUTATIONAL MODELING OF SMALL MOLECULES Rebecca J. Weber, B.S. Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS December 2015 APPROVED: Angela K. Wilson, Major Professor
More informationCHEMISTRY 4021/8021 MIDTERM EXAM 1 SPRING 2014
CHEMISTRY 4021/8021 Q1) Propose a simple, united-atom molecular mechanics force-field needed to generate a potential energy surface for an isolated molecule of acetone (Me 2 CO). I.e., provide an energy
More informationCHEM3023: 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
More informationHandbook 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
More informationChem 4502 Introduction to Quantum Mechanics and Spectroscopy 3 Credits Fall Semester 2014 Laura Gagliardi. Lecture 28, December 08, 2014
Chem 4502 Introduction to Quantum Mechanics and Spectroscopy 3 Credits Fall Semester 2014 Laura Gagliardi Lecture 28, December 08, 2014 Solved Homework Water, H 2 O, involves 2 hydrogen atoms and an oxygen
More informationTheoretical methods that help understanding the structure and reactivity of gas phase ions
International Journal of Mass Spectrometry 240 (2005) 37 99 Review Theoretical methods that help understanding the structure and reactivity of gas phase ions J.M. Mercero a, J.M. Matxain a, X. Lopez a,
More informationAnalysis Of Chemical Bonding Using Ab Initio Valence Bond Theory
Analysis Of Chemical Bonding Using Ab Initio Valence Bond Theory Analyse van chemische bindingen met ab initio valentiebindingstheorie (met een samenvatting in het Nederlands) Proefschrift ter verkrijging
More informationCoupled-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
More information2 Electronic structure theory
Electronic structure theory. Generalities.. Born-Oppenheimer approximation revisited In Sec..3 (lecture 3) the Born-Oppenheimer approximation was introduced (see also, for instance, [Tannor.]). We are
More informationIntroduction to Hartree-Fock calculations in Spartan
EE5 in 2008 Hannes Jónsson Introduction to Hartree-Fock calculations in Spartan In this exercise, you will get to use state of the art software for carrying out calculations of wavefunctions for molecues,
More informationSupporting Information: Predicting the Ionic Product of Water
Supporting Information: Predicting the Ionic Product of Water Eva Perlt 1,+, Michael von Domaros 1,+, Barbara Kirchner 1, Ralf Ludwig 2, and Frank Weinhold 3,* 1 Mulliken Center for Theoretical Chemistry,
More information4πε. me 1,2,3,... 1 n. H atom 4. in a.u. atomic units. energy: 1 a.u. = ev distance 1 a.u. = Å
H atom 4 E a me =, n=,,3,... 8ε 0 0 π me e e 0 hn ε h = = 0.59Å E = me (4 πε ) 4 e 0 n n in a.u. atomic units E = r = Z n nao Z = e = me = 4πε = 0 energy: a.u. = 7. ev distance a.u. = 0.59 Å General results
More informationSYSTEMATIC APPROACHES TO PREDICTIVE COMPUTATIONAL CHEMISTRY USING THE CORRELATION CONSISTENT BASIS SETS. Brian P. Prascher, B.S.
SYSTEMATIC APPROACHES TO PREDICTIVE COMPUTATIONAL CHEMISTRY USING THE CORRELATION CONSISTENT BASIS SETS Brian P. Prascher, B.S. Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF
More informationMolecular Magnetism. Magnetic Resonance Parameters. Trygve Helgaker
Molecular Magnetism Magnetic Resonance Parameters Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Laboratoire de Chimie Théorique,
More informationCOUPLED-CLUSTER BASED METHODS FOR EXCITED STATE ENERGIES AND GRADIENTS STEVEN RAY GWALTNEY
COUPLED-CLUSTER BASED METHODS FOR EXCITED STATE ENERGIES AND GRADIENTS By STEVEN RAY GWALTNEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
More informationThis is called a singlet or spin singlet, because the so called multiplicity, or number of possible orientations of the total spin, which is
9. Open shell systems The derivation of Hartree-Fock equations (Chapter 7) was done for a special case of a closed shell systems. Closed shell means that each MO is occupied by two electrons with the opposite
More informationHartree, 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
More informationQuantum Chemistry. NC State University. Lecture 5. The electronic structure of molecules Absorption spectroscopy Fluorescence spectroscopy
Quantum Chemistry Lecture 5 The electronic structure of molecules Absorption spectroscopy Fluorescence spectroscopy NC State University 3.5 Selective absorption and emission by atmospheric gases (source:
More informationCASSCF and NEVPT2 calculations: Ground and excited states of multireference systems. A case study of Ni(CO)4 and the magnetic system NArO
CASSCF and NEVPT2 calculations: Ground and excited states of multireference systems. A case study of Ni(CO)4 and the magnetic system NArO The ground states of many molecules are often well described by
More informationHints on Using the Orca Program
Computational Chemistry Workshops West Ridge Research Building-UAF Campus 9:00am-4:00pm, Room 009 Electronic Structure - July 19-21, 2016 Molecular Dynamics - July 26-28, 2016 Hints on Using the Orca Program
More informationΨ = 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
More information