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AN INTRODUCTION TO QUANTUM CHEMISTRY

1AN INTRODUCTION TOQUANTUM CHEMISTRYMark S. GordonIowa State University2 OUTLINE Theoretical Background in QuantumChemistry Overview of GAMESS Program Applications3 QUANTUM CHEMISTRY In principle, solve Schr dinger Equation Not possible for many-electron atoms ormolecules due to many-body problem Requires two levels of approximation4 FIRST APPROXIMATION Born-Oppenheimer Approximation Assumes we can study behavior ofelectrons in a field of frozen nuclei Correct H: Hexact = Tel + Vel-el + Tnuc + Vnuc-nuc + Vel-nuc Vel-nuc = electron-nucleus cross term: notseparable, so fix nuclear positions Happrox = Tel + Vel-el + Vel-nuc = Hel5 FIRST APPROXIMATION Born-Oppenheimer Approximation Assumes we can study behavior ofelectrons in a field of frozen nuclei Usually OK in ground electronic state:Assumes electronic and nuclear motionsare independent: not really true.

LCAO APPROXIMATION • Increase # AO’s - approach exact HF • Requires complete (infinite) basis χ µ • Computational effort increases ~ N4 – Double # AO’s, effort goes up by factor of 16! – Need to balance accuracy with CPU time, memory ψ ι = ∑ µ χµ Cµ i

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Transcription of AN INTRODUCTION TO QUANTUM CHEMISTRY

1 1AN INTRODUCTION TOQUANTUM CHEMISTRYMark S. GordonIowa State University2 OUTLINE Theoretical Background in QuantumChemistry Overview of GAMESS Program Applications3 QUANTUM CHEMISTRY In principle, solve Schr dinger Equation Not possible for many-electron atoms ormolecules due to many-body problem Requires two levels of approximation4 FIRST APPROXIMATION Born-Oppenheimer Approximation Assumes we can study behavior ofelectrons in a field of frozen nuclei Correct H: Hexact = Tel + Vel-el + Tnuc + Vnuc-nuc + Vel-nuc Vel-nuc = electron-nucleus cross term: notseparable, so fix nuclear positions Happrox = Tel + Vel-el + Vel-nuc = Hel5 FIRST APPROXIMATION Born-Oppenheimer Approximation Assumes we can study behavior ofelectrons in a field of frozen nuclei Usually OK in ground electronic state:Assumes electronic and nuclear motionsare independent: not really true.

2 More problematic in excited states, wheredifferent surfaces may cross: gives rise tonon-adiabatic (vibronic) APPROXIMATION Born-Oppenheimer Approximation Solve electronic Schr dinger equation atsuccessive (frozen) nuclear configurations For a diatomic molecule ( , H2), Born-Oppenhimer Approximation yields potentialenergy (PE) curve: energy as a function ofinter-nuclear distance, R. Bound curve: minimum at finite R Repulsive curve:No stable molecular structure7 ERRepulsive CurveERBound CurveRe8 FIRST APPROXIMATION Born-Oppenheimer Approximation Diatomic Molecules: number of points onPE curve determined by number of valuesof R Polyatomic molecules more complicated: Usually many more coordinates (3N-6) generate Potential Energy Surface (PES) Required number of points increasesexponentially with number of atoms9 SECOND APPROXIMATION Electronic Hamiltonian: H = Tel + Vel-nuc + Vel-el Vel-el not separable: Requires orbital approximation Independent particle model.

3 Assumes eachelectron moves in its own orbital: ignorescorrelation of behavior of an electron withother electrons Can lead to serious problems10 ORBITAL APPROXIMATION Hartree product (hp) expressed as a productof spinorbitals = i i i = space orbital, i = spin function ( , ) Ignoring repulsions and parametrizing leads to H ckel, extended H ckel Theory Tight Binding Approximation Can be very useful for extended systems hp = 1(1) 2(2).. N(N)11 ORBITAL APPROXIMATION Recover electron repulsion by using Orbital wave function (approximation) Correct Hamiltonian Leads to Variational Principle: <E> = < | | > Eexact Using exact Hamiltonian provides an upper bound Can systematically approach the exact energy12 ORBITAL APPROXIMATION Pauli Principle requires antisymmetry: Wavefunction must be antisymmetric to exchangeof any two electrons Accomplished by the antisymmetrizer For closed shell species (all electrons paired)antisymmetric wavefunction can berepresented by a Slater determinant ofspinorbitals: = hp = | 1(1) 2(2).

4 N(N)|13 ORBITAL APPROXIMATION For more complex species (one or more openshells) antisymmetric wavefunction must beexpressed as a linear combination of Slaterdeterminants Optimization of the orbitals (minimization ofthe energy with respect to all orbitals), basedon the Variational Principle) leads to:14 HARTREE-FOCK METHOD Optimization of orbitals leads to F i = i i F = Fock operator = hi + i(2Ji - Ki) for closedshells i = optimized orbital i = orbital energy15 HARTREE-FOCK METHOD Closed Shells: Restricted Hartree-Fock (RHF) Open Shells:Two Approaches Restricted open-shell HF (ROHF)** Wavefunction is proper spin eigenfunction: S(S+1) Most orbitals are doubly occupied =| 1 1 2 2.

5 N n n+1 n+2 ..|16 HARTREE-FOCK METHOD Second Approach for Open Shells Unrestricted HF (UHF) Different orbitals for different spins ( , ) Wavefunction is not a proper spin eigenfunction Can often get spin contamination : spin expectationvalue that is significantly different from the correct value Indicator that wavefunction may not be reliable17 HARTREE-FOCK METHOD Closed Shells: Restricted Hartree-Fock (RHF) Open Shells Restricted open-shell HF (ROHF)** Unrestricted HF (UHF) HF assumes molecule can be described by asimple Lewis structure Must be solved iteratively (SCF)18 LCAO APPROXIMATION are AO s: basis functions C i are expansion coefficients Approximation to Hartree-Fock FC = Sc Still solve interatively for C i and i = C i19 LCAO APPROXIMATION Increase # AO s - approach exact HF Requires complete (infinite) basis Computational effort increases ~ N4 Double # AO s, effort goes up by factor of 16!

6 Need to balance accuracy with CPU time, memory = C i20 COMMON BASIS SETS Minimal basis set One AO for each orbital occupied in atom 1s for H, 1s,2s,2p for C, 1s,2s,2p,3s,2p for Si Often reasonable geometries for simple systems Poor energy-related quantities, other properties Double zeta (DZ) basis set Two AO s for each occupied orbital in atom Better geometries, properties, poor energetics21 COMMON BASIS SETS Double zeta plus polarization (DZP) Add polarization functions to each atom 1p for H, 2d for C, 3d for Si, 3f for Ti Smallest reasonable basis for correlated calcs Triple zeta plus polarization (TZV) Diffuse functions for: Anions Weakly bound species (H-bonding, VDW)22 TYPES OF BASIS FUNCTIONS Slater/exponential functions: e- r Closest to H-atom solutions Required integrals don t have closed form Gaussian functions: exp(- r2) Required integrals have closed form Less accurate than exponential functions Solution.

7 Systematic linear combinations ofgaussians23 POPLE BASIS SETS 6-31G Each inner shell AO is combination of 6 gaussians Valence region split into inner & outer regions Inner valence expanded in 3 gaussians Outer valence represented by single gaussian 6-31++G(d,p)adds d functions on each heavy atom p functions on each H Diffuse functions on all atoms (++)24 DUNNING BASIS SETS Correlation consistent basis sets Range from double zeta plus polarization tohextuple zeta plus polarization plus diffuse Best choice for very accurate studies Best choice for weakly bound species25 LIMITATIONS OF HF METHOD Correlation error:motion of electrons notcorrelated due to independent particle model Geometries often reliable Energies generally not reliable Improvements can come from: Perturbation Theory Variational Principle26 IMPROVEMENTS TO HF METHOD Perturbation theory:MP2, MP4.

8 Based on adding successive improvements toboth wavefunction and energy In principle, leads to exact result, but perturbationmust be small Hartree-Fock-based perturbation theory originallydue to Moller & Plesset (MP); popularized byPople and Bartlett (MBPT)27 PERTURBATION THEORY Computationally efficient Often does not converge MP2 often gives better results than MP3, MP4, .. Not appropriate if compound is not welldescribed by a simple Lewis structure Computational effort~N5 (MP2), N6 (MP4)28 PERTURBATION THEORY Alternative is coupled cluster (CC) theory Wavefunction is written as = 0eT = 0 (1+T+T2+T3+..) 0 may be HF T = cluster operator = T1(1e)+T2(2e)+T3(3e)+.

9 Most popular is CCSD(T) Includes singles, doubles, perturbative triples Scales ~ N729 WHAT DO WE DO IF OURSYSTEM CANNOT BE WELLREPRESENTED BY A SINGLESIMPLE LEWIS STRUCTURE?30 , well separated significant diradical characterboth wavefunctions important "pure" diradical: two wavefunctionsmake equal contributions31 IMPROVEMENTS TO HF METHOD Variational Principle Configuration Interaction (~N7) = HF + S + D + T + .. S = all single excitations D = all double excitations, .. L wdin (1955): Complete CI gives exactwavefunction for the given atomic basis Complete CI generally impossible for any butsmallest atoms and diatomic molecules, due tothe number of configurations involved Orbitals not re-optimized in CI32 IMPROVEMENTS TO HF METHOD For near-degeneracies, critical to re-optimize the orbitals Called MC (Multi-configurational) SCF Configurations included in MCSCF definedby active space.

10 Those orbitals and electrons involved in process Include all configurations generated bydistributing active electrons among active orbitals Called complete active space (CAS)SCF33 MCSCF Usually scales ~N5-6, but can be worse Necessary for Diradicals Unsaturated transition metals Excited states Often transition states CASSCF accounts for near-degeneracies Still need to correct for rest of electroncorrelation: dynamic correlation 34 MULTI-REFERENCE METHODS Multi-reference CI: MRCI CI from set of MCSCF configurations Most commonly stops at singles and doubles MR(SD)CI Very demanding ~ impossible to go past 14 electrons in 14 orbitals Multi-reference perturbation theory More efficient than MRCI Not usually as accurate as MRCI35 SUMMARY OF METHODS Perturbation theory Efficient Size-consistent Often ill-behaved ( , non-convergent)


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