Introduction to QuantumChemical ElectronicStructure Methods

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Introduction toQuantum-Chemical
Electronic-Structure Methods

• SFB 663 Graduate Course
• Lecture 2

Timo Fleig
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Overview

• What is Quantum Chemistry ?
• General Ideas of Quantum Chemistry
• A Mathematical Primer
• Basis Sets
• Hartree-Fock Theory
• Density-Functional Theory
• Correlated Wave-Function Methods

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What is Quantum Chemistry?

• Quantum Mechanics applied to Atoms and Molecules
• Aim Understanding of Electronic Structure
• Solution of the electronic Schrödinger equation
• Derived Properties of Atoms and Molecules
• Pillar 1 The Methods !
• Pillar 2 Applications

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QC Modern Usage of Methods

• Molecular Applications
• DFT 55
• Wave Function Methods 45
• Efficiency vs. Accuracy
• Criteria for choice?

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General Ideas One and Many Particles

• Many-particle Schrödinger equation

• Much too complicated to solve exactly (even for
  3 particles!)
• Introduction of a one-particle space

• Expand the many-particle wave function in the
• one-particle functions

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A Mathematical Primer

• The linear (matrix) eigenvalue problem

SEQ Introduction of some orthonormal
basis Projection and integration Orthonormal
basis ! Matrix equation for one vector Matrix
equation for all vectors
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General Ideas Mean-field theory

• One-particle idea
• Hartree-Fock picture
• Electrons move in averaged field of other
  electrons
• Effective one-particle theory
• No correlation described

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General ideas Electron Correlation

• Pair density is flat in mean-field theory
• Coulomb repulsion explicit in correlated theory
• Coulomb cusp in correlated picture
• Difficult to describe accurately

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Ideas One-particle functions

• Hydrogen atom radial wave functions
• Cusp at origin
• Increasing number of nodes
• Varying radial amplitudes
• Exact H atom solutions as model

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Ideas One-particle functions

• Hydrogen atom radial distribution functions
• Probability shifts away from the nucleus (with
  increasing principal quantum number)
• Nodal structure

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Angular functions

• Solutions of the angular equations of H atom
• Reflect atomic symmetry!
• Either (complex) spherical harmonics
• or real linear combinations (figure)
• Always the same for all atoms (atomic basis
  functions)

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One-particle basis sets General Concept

• Atoms-in-Molecule approach Atomic electron
  density is only weakly perturbed in formation of
  molecule
• Hydrogen-like functions centered at atoms in
  molecule
• Additional functions for specific purposes
  (polarization, correlation, bonding etc.)

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One-particle basis setsSTOs and GTOs

• Slater-Type Orbitals are H like
• STOs have cusp
• Gaussian Type Orbitals have no cusp
• GTOs decline more strongly
• a exponent

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Basis sets Why GTOs ?
Situation in molecules

• Use of atom-centered GTOs
• Products of two GTOs at centers A, B gives GTO at
  center E
• Hence Ease of integration!
• Much faster than with STOs, despite increased
  number

Atomic centers A,B
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Basis sets Radial shifts

• Higher principal quantum numbers n
• Higher angular momenta p,d,f
• Radial shifts through prefactors
• General expression

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Basis sets Nodal structure

• Linear combination of individual GTOs
• Introduction of signs
• Description of radial nodes through several GTOs
• Analogous for higher n, l

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Basis sets Polarization functions

• Polarization of atomic density upon formation of
  chemical bond
• Additional polarization functions for description
• Molecular field breaks atomic symmetry !
• Use of higher angular-momentum functions
• E.g. pz for polarization of s functions (s.a.)

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Basis sets Correlation functions

• Angular correlation
• Increased probability of finding electrons on
  either side of axis
• Introduction of higher angular momentum functions
• E.g. p functions in H2 molecule

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Basis sets Correlation functions

• Radial correlation
• Probability of finding electrons at different
  distance from nucleus
• Add functions of higher principal quantum number
• E.g. 2s, 3s in H2

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Standard basis sets Overview

• STO-NG N GTO per STO
• Split-valence 6-31G() sets SZ
  (core)/DZ(valence), polarization functions
• Z(zeta) Number of contracted functions
• (aug)-cc-(p)VXZ Correlation-consistent basis
  sets
• XZV(PP) similar to above, less systematic
• ANO (Atomic Natural Orbital) basis sets

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BS Correlation-consistent series

• (aug)-cc-(p)VXZ
• X contracted functions per occupied orbital
• Atomic Hartree-Fock for optimization of occupied
  orbitals
• CISD (s.b.) for optimization of additional
  correlating primitives
• Valence-correlated atomic ground states!

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BS Performance of cc series
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BS Performance of cc series

• He atom wave functions Coulomb cusp
• Full black Correlated, dashed grey HF, full
  grey exact

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Hartree-Fock Theory

• Electron-electron interaction

Partitioning into Hartree-Fock (mean-field)
potential V fluctuation potential Fock Operator
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HF Physics of the Fock operator

• One-particle term h Kinetic energy of electron
  Interaction electron-nuclei
• Hartree-Fock potential
• Coulomb interaction (of electron with
  charge density, classical)
• Exchange interaction (quantum mechanical,
• relates to Spin and antisymmetry)

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Hartree-Fock Theory Ansatz

• Linear expansion of atomic or molecular orbitals
  in (contracted) basis functions

• Variational optimization of expansion
  coefficients c
• Variational Energy expectation value is upper
  bound for exact energy

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HF The wave function

• Ansatz for the many-particle wave function
• Slater determinant

• Singlet ground-state of Helium atom
• Consists of spatial and spin part
• Contains antisymmetry in spin part (singlet)
• Orbital coefficients are optimized for this
  Determinant

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HF Spin and the Fermi hole

• Probability of finding an electron at the same
  position of another electron, equal spin
  projections
• Consequence Coulomb repulsion is diminished
• Included in HF theory

• Fermi correlation

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Hartree-Fock equations

• f depends on solutions
• Iterative solution of equation system
• Upon self-consistency

• Optimized (spin) orbitals
• (Spin) orbital energies

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HF Optimized solutions

• Optimized spin orbitals
• Set of occupied spin orbitals
• Set of unoccupied, virtual spin orbitals (basis
  set dimension typically is larger than problem
  size!)

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HF Total energy

• HF theory delivers gt99 of the full electronic
  energy
• Accuracy also depends on Hamiltonian
• Electron correlation is neglected
• It accounts for the (important) residual
• (spectra, potential curves, etc.)

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Retrospective Many-particle problem

• Increasing basis set HF limit
• Increasing no. of determinants Full-CI limit
• Combined towards the exact electronic solution

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HF Potential energy curves

• Restricted HF does not dissociate correctly
• Unrestricted HF can compensate
• HF bond lengths too short
• Full CI deviation basis set incompleteness

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HF Excitation energies (SCI)

• Singles CI as (near)-equivalent to HF for excited
  states
• FCI/ANO-RCC is very accurate
• SCI excitation energy errors
• 0.2 - gt1.0 eV

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Correlation Methods Overview

• Objective Account for dynamic correlation
• Wave-function based methods
• Configuration Interaction
• Mo/ller-Plesset Perturbation Theory
• Coupled Cluster
• Electron-density based methods
• Density-Functional theory

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Density Functional Theory

• Total energy as functional of the electron
  density

• S Reference system homogeneous,
  non-interacting
• electron gas (Fermi gas)
• v External potential (nuclei)
• J Classical interaction with charge density
• xc Exchange-correlation density functional (!)

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DFT Procedure

• Variation of energy functional w.r.t. density

• yields set of effective one-particle equations
  

• v Kohn-Sham potential (contains ext. pot., J,
  xc)
• u KS orbitals
• e KS orbital energies

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DFT XC functionals

• Local Density Approximation (LDA)
• Assumption No gradient of electron density
  in Exc
• X part Exact Dirac functional from Fermi gas
• C part From Quantum-Monte-Carlo simulations on
  Fermi gas

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DFT XC functionals LDA

• A High density, large kinetic energy, LDA
  approximation unimportant
• B Small density gradient, LDA is good
• C Bonding region, large gradient, LDA fails !

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DFT XC functionals.Gradient corrections

• Generalized Gradient Approximation (GGA)
• Density gradient correction from response
  theory (TD perturbation theory)
• Lee-Yang-Parr (LYP) functionals
• Gradient corrections from He atom
  two-particle density, parameter fitting

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DFT XC functionalsHybrid functionals

• Becke (B) LYP, B3LYP functionals
• Contain a contribution of HF exchange
  interaction
• Reduction of self-energy error
• (Introduced via different modelling of X in
  XC functionals and J functionals)
• Exact XC functional is unknown!

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DFT Range of application

• Closed-shell ground states (single-reference
  picture)
• Excited states with single excitation character
• Geometry optimizations of such states
• Kohn-Sham orbitals for more advanced methods
  (CI,MCSCF)

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DFT Specific failuresCT-excited states (TDDFT)

• Single-excitation dominated !
• Monomer valence-excited states correct
• CT-excited too low in energy
• Errors in the order of 1 eV

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DFT CT-excited states
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DFT CT-excited states

• Long-range behavior of CT state wrong !
• Magenta CIS
• Blue half-and half BLYP
• Green B3LYP
• Red LB94
• Black SVWN

Explanation Essentially self-energy error
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DFT Perspectives and Problems

• Development of new functionals (solving CT
  problem)
• Ab-initio based DFT ?
• Other failures/difficulties
• Rydberg-excited states
• Extended p systems
• Doubly-excited states (and higher)
• Molecular hyperpolarizabilities

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Configuration Interaction (CI)

• Principles
• Based on HF (or MCSCF) wave function
  (orbitals)
• Linear expansion of many-particle wave
  function

c expansion coefficients Y Slater determinants,
ground and excited Variational optimization
Eigenvalue problem
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CI Optimization and properties
Variation of energy-expectation value

• Upper bound for ground- and excited-state
  energies
• Solution wave function is a vector of
  coefficients (and associated determinants) per
  electronic state
• Allows for precise scrutiny and analysis

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CI The expansion (parameterization)
Number of terms (excluding symmetry)

• N occupied spin orbitals
• M-N virtual spin orbitals (basis set!)

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CI Excitation Level
The Hamiltonian matrix in determinant basis

• Double excitations couple directly to reference
  state
• All others couple indirectly (coupled linear
  equations)
• Strategy Select important higher excitations !

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CI Modern ApproachesMulti-reference CI

• Selection of an active space (chemical/physical
  criteria)
• Typically correlating (antibonding) virtuals
  included
• Higher excitations within active space
• Singles and Doubles from active reference space
  into rem. virtuals

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CI MRCIGeneral Active Space (GAS) CI

• Advanced technique
• Multiple active spaces
• Arbitrary occupation
• Very general, very flexible
• Hard to implement
• Inner spaces with complete (CAS) exp.

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CI General features
Basis set correlation energy

• Truncated CI is not size-consistent (the
  correlation energy does not scale correctly with
  the number of particles/subsystems)
• The error increases with the number of particles
• General applicability to closed- and open-shell
  systems

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CI Performance Potential curves

• H2O symmetric dissociation curve, cc-pVDZ basis
  set
• CISD good bond length, harmonic freq., poor
  dissociation
• CISDT little improvement
• CISDTQ accurate, but expensive !

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CI performance Potential curves

• MRSDCI/cc-pVDZ curves and error, FCI curve
• Very accurate for complete PEC
• Computationally tractable
• Important Selection of significant active space
  !

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CI Performance excited states

• SDCI fails, MRSDCI requires averaged orbitals to
  be accurate() here ground-state orbitals !

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Mo/ller-Plesset Perturbation Theory (MPPT)

• Principles
• Idea Electron correlation is a perturbation
  (lt1 of total energy)
• Based on Rayleigh-Schrödinger PT
• Not variational
• Closed-shell (ground) reference state
• Single-reference method !
• Size-consistent

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MPPT Procedure

• Assume solution of zeroth-order problem (HF
  problem)

• Definition of a perturbing potential,
  fluctuation potential

• Total electronic Hamiltonian

• Apply Rayleigh-Schrödinger PT

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MPPT 1st-order correction

• By inserting V we obtain

• c are the optimized spin orbitals
• This is a sum of two-particle integrals known
  from HF theory
• The first term is a Coulomb-, the second an
  exchange term
• E0(0) E0(1) are just the HF energy !

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MPPT 2nd-order correction

• n must be a double excitation from reference
  state
• Sum-over-states expression from RS PT
• A little algebra yields

MP2 correction

• Contains known integrals and orbital energies
  from HF
• Easy to implement, cheap to evaluate

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MPPT Higher-order corrections

• Obtained straightforwardly from RS PT expressions
• Contain higher than double excitations from the
  reference state
• E0(3) contains only double excitations
• E0(4) contains up to quadruple excitations
• etc.

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MPPT Convergence of the series

• Even models correct more strongly (excitation
  level)
• Bond stretching hampers convergence

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MPPT Potential energy curves

• H2O, cc-pVDZ basis
• MP2 Good bond lengths (geometry opt.)
• Fails when bond is stretched
• All models Wrong dissociation (single-reference!)
  
• (Dashed UMPn)

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MPPT Curves at higher order

• All MP models H2 molecule, cc-pVQZ basis
• diverge when bond is stretched

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MPPT variant CASPT2

• Multi-reference perturbation theory for ground-
  and excited states
• Complete-Active-Space expansion in active
  reference space
• MP2 correction for all obtained reference
  functions

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Coupled Cluster (CC) Model

• Principles
• Single-determinant reference state
• Projective solution (non-variational!)
• Product (exponential) parameterization

x is an operator generating a doubly excited
determinant Products of coefficients (c) occur
(non-linear)!
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CC Form of the wave function
General parameterization

• Coefficients are called amplitudes (t)
• Single terms and products occur !
• Consequence CC theory is size-consistent

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CC Potential energy curves

• H2O, cc-pVDZ
• Rapid convergence to exact solution (compact wave
  function)
• CCSD dissociates correctly (qual.), higher
  excitations as products
• (T) triple excitations by PT
• bump beginning of bond breaking

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CC Excited states CC response

• (RI) CC response theory (TDPT)
• Approximation hierarchy CCS-CC2-CCSD-CC3
• Errors decrease rapidly
• System dependent (NA bases !)

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CC Excited states CT states

• CC gives correct excitation energies for CT
  states (CT Singles naturally included)
• DFT/SVWN gives spurious results

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Outlook

• Electronic-structure methods for light molecules
  (Zlt30) are established
• Exception Multi-reference CC is still disputed
• Program packages for deriving properties from
  most models are available
• Relativistic method analogues for heavy elements
  trail behind somewhat

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Further reading

• TC2, TC3 manuscripts for download on my web pages
• Atkins, P.W., Molecular Quantum Mechanics
• Szabo, A. and Ostlund, N.S., Modern Quantum
  Chemistry
• Helgaker, T., Jo/rgensen, P., and Olsen, J.,
  Molecular Electronic Structure Theory

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Thanks for your attention and patience !!!