Introduction to Computational Chemistry
|
|
- Phebe Martha Turner
- 6 years ago
- Views:
Transcription
1 Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry Chemicum 4th floor September 10, 2013 Lecture 3. Electron correlation methods September 20 1 / 24
2 Hartree-Fock limit In HF the electronic wave function is approximated by a single Slater determinant Not flexible enough to account for electron correlation therefore the Hartree-Fock limit is always above the exact energy Some electron correlation is already found in the electron exchange term HF wf Better description for the wavefunction needed for more accurate results exact wf Lecture 3. Electron correlation methods September 20 2 / 24
3 Electron correlation Some common nomenclature found in literature Fermi correlation arises from the Pauli antisymmetry of the wave function and is taken into account already at the single-determinant level. Example: The spatial part of wavefunction of two electrons with the same spin can be written as ψ(r 1,r 2 ) = 1 2 [φ 1 (r 1 )φ 2 (r 2 ) φ 1 (r 2 )φ 2 (r 1 )] (1) where r 1 and r 2 are the space coordinates of electrons 1 and 2, respectively. The wavefunction vanishes at the point where the two electron coincide. Around each electron there will be a hole in which there is less electron with a same spin: the Fermi hole. This exchange force is relatively localized. Lecture 3. Electron correlation methods September 20 3 / 24
4 Electron correlation Static correlation is the deviation from the exact energy caused by an attempt to represent a wavefunction by just one determinant when at least two are really needed Bond dissociation Excited states Near-degeneracy of electronic configurations (for example a singlet diradical CH 2 ). Dynamical correlation is associated with the instantaneous correlation among the electrons arising from their mutual Coulombic repulsion. Quite predictable, the major contribution is around 1 ev for each closed shell pair. Well accounted for by DFT functionals, perturbation theory, configuration interaction, and coupled cluster methods. Coulomb hole: The probability of finding two electrons at the same point in space is 0 as the repulsion becomes infinite. For the wavefunction approximated by Hartree-Fock method this requirement is not fulfilled. There is no explicit separation between dynamical and static correlations. Lecture 3. Electron correlation methods September 20 4 / 24
5 Configuration interaction (CI) Configuration interaction (CI) has the following characteristics: A post-hartreefock linear variational method. Solves the nonrelativistic Schrödinger equation within the BO approximation for a multi-electron system. Describes the linear combination of Slater determinants used for the wave function. Orbital occupation (for instance, 1s 2 2s 2 2p 1...) interaction means the mixing of different electronic configurations (states). In contrast to the HartreeFock method, in order to account for electron correlation, CI uses a variational wave function that is a linear combination of configuration state functions (CSFs) built from spin orbitals, Ψ = C i Φ i = C 0 Φ 0 +C 1 Φ 1... (2) i=0 Lecture 3. Electron correlation methods September 20 5 / 24
6 Configuration interaction (CI) Coefficients from the wavefunction expansion are determined by a variational optimization respect to the electronic energy HC = E CI C (3) where H is the Hamiltonian matrix with matrix elements H ij = φ i H φ j (4) The construction of the CI wavefunction may be carried out by diagonalization of the Hamiltonian matrix, but in reality iterative techniques are used to extract eigenvalues and eigenfunctions (Newton s method). Lecture 3. Electron correlation methods September 20 6 / 24
7 Configuration interaction (CI) The first term in the CI-expansion is normally the Hartree Fock determinant The other CSFs can be characterised by the number of spin orbitals that are swapped with virtual orbitals from the Hartree Fock determinant If only one spin orbital differs, we describe this as a single excitation determinant If two spin orbitals differ it is a double excitation determinant and so on This is used to limit the number of determinants in the expansion which is called the CI-space. The eigenvalues are the energies of the ground and some electronically excited states. By this it is possible to calculate energy differences (excitation energies) with CI-methods. Lecture 3. Electron correlation methods September 20 7 / 24
8 Example: CI calculation for Helium atom Lets us begin with two-configuration wavefunction expressed as a linear combination of hydrogenic wavefunctions having the form Ψ 1,2 = c 1 ψ 1 +c 2 ψ 2, (5) where ψ 1 arises from the configuration 1s 2 and ψ 2 arises from the configuration 1s2s. Specifically, the two wavefunctions are ψ 1 = 1 1s(1)α(1) 1s(1)β(1) 2 1s(2)α(2) 1s(2)β(2) = 1s 1s (6) ψ 2 = 1 2 ( 1s 2s + 1s2s ) (7) Lecture 3. Electron correlation methods September 20 8 / 24
9 Example: CI calculation for Helium atom Since both ψ 1 and ψ 2 describe the 1 S 0 states (S = L = 0), there will be no vanishing matrix elements of Ĥ. If we represent these matrix elements by H ij = ψ i Ĥ ψ j (i,j = 1 or 2), the secular determinant to be solved is H 11 E H 12 H 22 E = 0. (8) H 21 The diagonal matrix elements H 11 and H 22 are just the energies of single configurational calculations for the ground and exited states. Keeping in mind that the spin portions integrate out separately to unity ( α α = 1, α β = 0, etc.), we obtain for the H 11 H 11 = 1s(1)1s(2) Ĥ 1s(1)1s(2) = 2ǫ 1 +J 11 (9) where we represent the 1s orbital by subscript 1. The energy ǫ has the same form as the hydrogen atom energy ǫ n = Z2 2n 2 = 2 n 2 (10) Lecture 3. Electron correlation methods September 20 9 / 24
10 Example: CI calculation for Helium atom Similarly the result for H 22 is derived as H 22 = 1 2 1s 2s 1s2s Ĥ 1s 2s 1s2s (11) Which can be expanded to the form H 22 = 1 ( 1s 2s Ĥ 1s 2s ) 2 1s 2s Ĥ 1s2s + 1s2s Ĥ 1s2s ), 2 (12) which leads to an energy The off-diagonal matrix element becomes H 22 = ǫ 1 +ǫ 2 +J 12 +K 12 (13) H 12 = 1 2 ( 1s 1s Ĥ 1s 2s ) 1s 1s Ĥ 1s2s ) (14) Lecture 3. Electron correlation methods September / 24
11 Example: CI calculation for Helium atom which is simplified because of vanishing spin integrals and thus one can obtain H 12 = 2 1s(1)1s(2) 1 r 12 1s(1)2s(2) (15) Combining the appropriate integrals, the matrix elements become H 11 = 2.75 (16) H 22 = (17) H 12 = (18) The roots of the quadratic formula (produced by secular determinant) are E = and The lower root represents the ground state whose experimental energy is Note that this improves the single configurational result The higher root represents the lowest excited state (experimental energy = 2.15). Lecture 3. Electron correlation methods September / 24
12 Full CI The expansion to the full set of Slater determinants (SD) or CSFs by distributing all electrons among all orbitals is called full CI (FCI) expansion. FCI exactly solves the electronic Schrödinger equation within the space spanned by the one-particle basis set. In FCI, the number SDs increase very rapidly with the number of electrons and number of orbitals. For example, when distributing 10 electrons to 10 orbitals the number of SDs is This illustrates the intractability of the FCI for any but the smallest electronic systems. Practical solution: Truncation of the CI-expansion Truncating the CI-space is important to save computational time. For example, the method CID is limited to double excitations only. The method CISD is limited to single and double excitations. These methods, CID and CISD, are in many standard programs. In the early days of CI-calculations, truncated expansion was constructed by selecting individual configurations. Today, such methods are impractical and even problematic because the relative importance of configurations changes across the potential energy surface. Lecture 3. Electron correlation methods September / 24
13 Static and dynamical correlation CI-expansion truncation is handled differently between static or dynamical correlation. In the treatment of static correlation in addition to the dominant configurations, near degenerate configurations are chosen (referred as reference configurations). Active orbitals (typically valence orbitals) form complete active space (CAS) in which no restrictions are placed on the occupations. The inactive (core) and secondary orbitals are always doubly occupied or unoccupied. Dynamical correlation is subsequently treated by generating excitations from reference space. Unified framework for above treatments is called restricted active space (RAS). Lecture 3. Electron correlation methods September / 24
14 Some problems (and solutions) of CI Excitation energies of truncated CI-methods are generally too high because the excited states are not that well correlated as the ground state is. The Davidson correction can be used to estimate a correction to the CISD energy to account for higher excitations. It allows one to estimate the value of the full configuration interaction energy from a limited configuration interaction expansion result, although more precisely it estimates the energy of configuration interaction up to quadruple excitations (CISDTQ) from the energy of configuration interaction up to double excitations (CISD). It uses the formula δe Q = (1 C 2 0)(E CISD E HF ) (19) where C 0 is the coefficient of the Hartree Fock wavefunction in the CISD-expansion Lecture 3. Electron correlation methods September / 24
15 Size consistency CI-methods are not size-consistent and size-extensive Size-inconsistency means that the energy of two infinitely separated particles is not double the energy of the single particle. This property is of particular importance to obtain correctly behaving dissociation curves. Size-extensivity, on the other hand, refers to the correct (linear) scaling of a method with the number of electrons. The Davidson correction can be used. Quadratic configuration interaction (QCI) is an extension CI that corrects for size-consistency errors in the all singles and double excitation CI methods (CISD). This method is linked to coupled cluster (CC) theory. Accounts for important four-electron correlation effects by including quadruple excitations Lecture 3. Electron correlation methods September / 24
16 Example: CI results for water Compared to CISD-method, the simpler and less computationally expensive MP2-method gives superior results when size of the system increases (MP2 is size extensive). For water monomer, MP2 recovers 94% of correlation energy which remains similar with increasing system (cc-pvtz basis). For stretched water monomer (bond length doubled) CISD recovers only 80.2% of the correlation energy. With the Davidson correction added, the error is reduced to 3%. When the number of monomers increases, the degradation in the performance is even more severe for the equilibrium geometry. For eight monomers, the CISD wavefunction recovers only half of the correlation energy and the Davidson correction remain more or less the constant. Lecture 3. Electron correlation methods September / 24
17 Coupled Cluster method First observations: Coupled cluster (CC) method, especially The CCSD(T), has become the gold-standard of quantum chemistry. CC theory was poised to describe essentially all the quantities of interest in chemistry, and has now been shown numerically to offer the most predictive, widely applicable results in the field. The computatinal cost is very high. So, in practice, it is limited to relatively small systems. Some facts: Coupled cluster (CC) is a numerical technique used for describing many-body systems. The method was initially developed in the 1950s for studying nuclear physics phenomena, but became more frequently used when in 1966 it was formulate for electron correlation in atoms and molecules. It starts from the Hartree-Fock molecular orbital method and adds a correction term to take into account electron correlation. Lecture 3. Electron correlation methods September / 24
18 Coupled Cluster method The wavefunction of the coupled-cluster theory is written in terms of exponential functions: Ψ = eˆt Φ 0 (20) where Φ 0 is a Slater determinant usually constructed from Hartree Fock molecular orbitals. ˆT is an excitation operator which, when acting on Φ0, produces a linear combination of excited Slater determinants. The exponential approach guarantees the size extensivity of the solution. However, it depends on the size consistency of the reference wave function. A drawback of the method is that it is not variational E[Φ] = Φ e ˆTHeˆT Φ Φ Φ (21) which for truncated cluster expansion becomes E[Φ] = Ω H Θ Φ Φ (22) where Ω and Θ are different functions Lecture 3. Electron correlation methods September / 24
19 Coupled Cluster method The cluster operator is written in the form ˆT = ˆT 1 + ˆT 2 + ˆT (23) where ˆT 1 is the operator of all single excitations, ˆT 2 is the operator of all double excitations and so forth. The exponential operator eˆt may be expanded into Taylor series: eˆt = 1+ ˆT + ˆT 2 2! +... = 1+ ˆT 1 + ˆT 2 + ˆT ˆT 1 ˆT2 + ˆT (24) In practice the expansion of ˆT into individual excitation operators is terminated at the second or slightly higher level of excitation. Slater determinants excited more than n times contribute to the wave function because of the non-linear nature of the exponential function. Therefore, coupled cluster terminated at ˆT n usually recovers more correlation energy than CI with maximum n excitations. Lecture 3. Electron correlation methods September / 24
20 Coupled Cluster method The classification of traditional coupled-cluster methods rests on the highest number of excitations allowed in the definition of ˆT. The abbreviations for coupled-cluster methods usually begin with the letters CC (for coupled cluster) followed by S - for single excitations (shortened to singles in coupled-cluster terminology) D - for double excitations (doubles) T - for triple excitations (triples) Q - for quadruple excitations (quadruples) Thus, the operator in CCSDT has the form ˆT = ˆT 1 + ˆT 2 + ˆT 3 (25) Terms in round brackets indicate that these terms are calculated based on perturbation theory. For example, a CCSD(T) approach simply means: It includes singles and doubles fully Triples are calculated with perturbation theory. Lecture 3. Electron correlation methods September / 24
21 Example: Atomization energies Table 1: Calculated and experimental electronic atomization energies (kj/mol) Molecule HF CCSD Exp. F H HF H 2 O O CO C 2 H CH CCSD calculations produce qualitatively correct results Eventhought CCSD is expensive method, it is unfortunately not very accurate Lecture 3. Electron correlation methods September / 24
22 Example: Reaction enthalpies Table 2: Calculated and experimental electronic reaction enthalpies (kj/mol) Reaction HF CCSD Experiment CO + H 2 CH 2 O H 2 O + F 2 HOF + HF N 2 + 3H 2 2NH C 2 H 2 + H 2 C 2 H CO 2 + 4H 2 CH 4 + 2H 2 O CH 2 C 2 H O 3 + 3H 2 3H 2 O CCSD recovers majority of the electron correlation energy CCSD calculations do not achieve chemical accuracy (4 kj/mol) Lecture 3. Electron correlation methods September / 24
23 Example: Water energies Table 3: Deviation of CI and CC energies from non-relativistic exact results (within B-O approximation) for H 2 O (mhartree = kj/mol) Method r e 1.5 r e 2 r e Hartree Fock CID CISD CISDT CISDTQ CCD CCSD CCSDT CCSDTQ CI converges (too) slowly to exact energy CC has superior performance but show fluctuations Lecture 3. Electron correlation methods September / 24
24 Historical facts The CC method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear physics. In 1966 Jiri Cek (and later together with Josef Paldus) reformulated the method for electron correlation in atoms and molecules. Kümmel comments: I always found it quite remarkable that a quantum chemist would open an issue of a nuclear physics journal. I myself at the time had almost gave up the CC method as not tractable and, of course, I never looked into the quantum chemistry journals. The result was that I learnt about Ji ri s work as late as in the early seventies, when he sent me a big parcel with reprints of the many papers he and Joe Paldus had written until then. Lecture 3. Electron correlation methods September / 24
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,
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 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 informationElectron 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
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 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 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 informationIntroduction to Computational Chemistry
Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry room B430, Chemicum 4th floor vesa.hanninen@helsinki.fi September 3, 2013 Introduction and theoretical backround September
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationBasis sets for electron correlation
Basis sets for electron correlation Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 12th Sostrup Summer School Quantum Chemistry
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 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 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 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 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 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 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 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 information3: 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
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 information(1/2) M α 2 α, ˆTe = i. 1 r i r j, ˆV NN = α>β
Chemistry 26 Spectroscopy Week # The Born-Oppenheimer Approximation, H + 2. Born-Oppenheimer approximation As for atoms, all information about a molecule is contained in the wave function Ψ, which is the
More informationLecture 9. Hartree Fock Method and Koopman s Theorem
Lecture 9 Hartree Fock Method and Koopman s Theorem Ψ(N) is approximated as a single slater determinant Φ of N orthogonal One electron spin-orbitals. One electron orbital φ i = φ i (r) χ i (σ) χ i (σ)
More informationElectronic structure theory: Fundamentals to frontiers. 1. Hartree-Fock theory
Electronic structure theory: Fundamentals to frontiers. 1. Hartree-Fock theory MARTIN HEAD-GORDON, Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley National
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 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 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 informationv(r i r j ) = h(r i )+ 1 N
Chapter 1 Hartree-Fock Theory 1.1 Formalism For N electrons in an external potential V ext (r), the many-electron Hamiltonian can be written as follows: N H = [ p i i=1 m +V ext(r i )]+ 1 N N v(r i r j
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 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 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 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 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 informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
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 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 informationYingwei 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
More informationCalculations 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
More informationConvergence properties of the coupled-cluster method: the accurate calculation of molecular properties for light systems
1 Convergence properties of the coupled-cluster method: the accurate calculation of molecular properties for light systems T. Helgaker Centre for Theoretical and Computational Chemistry, Department of
More informationConsequently, the exact eigenfunctions of the Hamiltonian are also eigenfunctions of the two spin operators
VI. SPIN-ADAPTED CONFIGURATIONS A. Preliminary Considerations We have described the spin of a single electron by the two spin functions α(ω) α and β(ω) β. In this Sect. we will discuss spin in more detail
More informationChem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components.
Chem 44 Review for Exam Hydrogenic atoms: The Coulomb energy between two point charges Ze and e: V r Ze r Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative
More informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules Lecture 4 Molecular orbitals C.-K. Skylaris Learning outcomes Be able to manipulate expressions involving spin orbitals and molecular orbitals Be able to write down
More informationonly two orbitals, and therefore only two combinations to worry about, but things get
131 Lecture 1 It is fairly easy to write down an antisymmetric wavefunction for helium since there are only two orbitals, and therefore only two combinations to worry about, but things get complicated
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 informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation
Lecture 17 Page 1 Lecture 17 L17.P1 Review Schrödinger equation The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent
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 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 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 informationT. Helgaker, Department of Chemistry, University of Oslo, Norway. T. Ruden, University of Oslo, Norway. W. Klopper, University of Karlsruhe, Germany
1 The a priori calculation of molecular properties to chemical accuarcy T. Helgaker, Department of Chemistry, University of Oslo, Norway T. Ruden, University of Oslo, Norway W. Klopper, University of Karlsruhe,
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 informationMulti-reference Density Functional Theory. COLUMBUS Workshop Argonne National Laboratory 15 August 2005
Multi-reference Density Functional Theory COLUMBUS Workshop Argonne National Laboratory 15 August 2005 Capt Eric V. Beck Air Force Institute of Technology Department of Engineering Physics 2950 Hobson
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 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 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 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 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 informationAn 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
More informationIntroduction to numerical projects
Introduction to numerical projects Here follows a brief recipe and recommendation on how to write a report for each project. Give a short description of the nature of the problem and the eventual numerical
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 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 informationThe Rigorous Calculation of Molecular Properties to Chemical Accuracy. T. Helgaker, Department of Chemistry, University of Oslo, Norway
1 The Rigorous Calculation of Molecular Properties to Chemical Accuracy T. Helgaker, Department of Chemistry, University of Oslo, Norway A. C. Hennum and T. Ruden, University of Oslo, Norway S. Coriani,
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 informationDensity Functional Theory
Chemistry 380.37 Fall 2015 Dr. Jean M. Standard October 28, 2015 Density Functional Theory What is a Functional? A functional is a general mathematical quantity that represents a rule to convert a function
More informationChapter 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
More informationIntroduction to Hartree-Fock Molecular Orbital Theory
Introduction to Hartree-Fock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology Origins of Mathematical Modeling in Chemistry Plato (ca. 428-347
More informationElectron States of Diatomic Molecules
IISER Pune March 2018 Hamiltonian for a Diatomic Molecule The hamiltonian for a diatomic molecule can be considered to be made up of three terms Ĥ = ˆT N + ˆT el + ˆV where ˆT N is the kinetic energy operator
More informationMolecular Magnetic Properties
Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 12th Sostrup Summer School Quantum Chemistry and
More informationMolecular Magnetic Properties
Molecular Magnetic 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, Oslo,
More informationJoint ICTP-IAEA Workshop on Fusion Plasma Modelling using Atomic and Molecular Data January 2012
2327-3 Joint ICTP-IAEA Workshop on Fusion Plasma Modelling using Atomic and Molecular Data 23-27 January 2012 Qunatum Methods for Plasma-Facing Materials Alain ALLOUCHE Univ.de Provence, Lab.de la Phys.
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 informationIntroduction 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
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 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 informationChemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron):
April 6th, 24 Chemistry 2A 2nd Midterm. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (-electron): E n = m e Z 2 e 4 /2 2 n 2 = E Z 2 /n 2, n =, 2, 3,... where Ze is
More informationIntroduction 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
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 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 informationIntroduction 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
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 informationThe 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
More informationOptimization 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,
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 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 informationSection 3 Electronic Configurations, Term Symbols, and States
Section 3 Electronic Configurations, Term Symbols, and States Introductory Remarks- The Orbital, Configuration, and State Pictures of Electronic Structure One of the goals of quantum chemistry is to allow
More informationDensity Functional Theory. Martin Lüders Daresbury Laboratory
Density Functional Theory Martin Lüders Daresbury Laboratory Ab initio Calculations Hamiltonian: (without external fields, non-relativistic) impossible to solve exactly!! Electrons Nuclei Electron-Nuclei
More informationCOUPLED-CLUSTER CALCULATIONS OF GROUND AND EXCITED STATES OF NUCLEI
COUPLED-CLUSTER CALCULATIONS OF GROUND AND EXCITED STATES OF NUCLEI Marta Włoch, a Jeffrey R. Gour, a and Piotr Piecuch a,b a Department of Chemistry,Michigan State University, East Lansing, MI 48824 b
More informationDENSITY FUNCTIONAL THEORY FOR NON-THEORISTS JOHN P. PERDEW DEPARTMENTS OF PHYSICS AND CHEMISTRY TEMPLE UNIVERSITY
DENSITY FUNCTIONAL THEORY FOR NON-THEORISTS JOHN P. PERDEW DEPARTMENTS OF PHYSICS AND CHEMISTRY TEMPLE UNIVERSITY A TUTORIAL FOR PHYSICAL SCIENTISTS WHO MAY OR MAY NOT HATE EQUATIONS AND PROOFS REFERENCES
More information