Computational Quantum Chemistry Part I' Obtaining Properties

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Computational Quantum ChemistryPart I.
Obtaining Properties
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Properties are usually the objective.

• May require accurate, precisely known numbers
• Necessary for accurate design, costing, safety
  analysis
• Cost and time for calculation may be secondary
• Often, accurate trends and estimates are at least
  as valuable
• Can be correlated with data to get high-accuracy
  predictions
• Can identify relationships between structure and
  properties
• A quick, sufficiently accurate number or trend
  may be of enormous value in early stages of
  product and process development, for for
  operations, or for troubleshooting
• Great data are best but also theory-based
  predictions

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What properties do we want?

• Phase and reaction equilibria
• Bond and interaction energies
• Ideal-gas thermochemistry
• Thermochemistry and equations of state for real
  gases, liquids, solids, mixtures
• Adsorption and solvation
• Reaction kinetics
• Rate constants, products
• Transport properties
• Interaction energies, dipole
• µ, kthermal, DAB

• Analytical information
• Spectroscopy Frequencies, UV / Vis /IR
  absorptivity
• GC elution times
• Mass spectrometric ionization potentials and
  cross-sections, fragmentation patterns
• NMR shifts
• Mechanical properties of hard and soft condensed
  matter
• Electronic and optical properties of solids

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Restate What kind of properties come directly
from computational quantum chemistry?
Quantum-mechanical energies

• Energies, structures optimized with respect to
  energy, harmonic frequencies, and other
  properties based on zero-kelvin electronic
  structures
• Interpret with theory to get derived properties
  and properties at higher temperatures
• The theoretical basis for most of this
  translation is

Statistical mechanics
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Simplest properties are interaction energies
Here, the van der Waals well for an Ar dimer.
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Simplest chemical bonds are much stronger.
UB3LYP/6-311G(3df,3dp) with basis-set
superposition error correction
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At zero K, define the dissociation energy D0 as
the well depth less zero-point energy.
Alternate view is that D0 E0(dissociated
partners) - E0(molecule) ZPE, where ZPE is
the zero-K energy of the stretching vibration.
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Geometry is then found by optimizing computed
energy with respect to coordinates (here, 1).
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Vibrational frequencies (at 0 K) are calculated
using parabolic approximation to well bottom.

• How many? Need 3Natoms coordinates to define
  molecule
• If free translational motion in 3 dimensions,
  then three translational degrees of freedom
• Likewise for free rotation 3 d.f. if nonlinear,
  2 if linear
• Thus, 3Natoms-5 (nonlinear) or 3Natoms-6
  (nonlinear) vibrations
• For diatomic, ?2E/?r2 force constant k for r
  dimensionless
• F ( ma m?2r/?t2) -kr is a harmonic
  oscillator in Newtonian mechanics (Hookes law)
• Harmonic frequency is (k/m)1/2/2p s-1 or
  (k/m)1/2/2pc cm-1 (wavenumbers)
• For polyatomic, analyze Hessian matrix
  ?2E/?ri?rj instead

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Next, determine ideal-gas thermochemistry.

• Start with ?fH0 and understand how energies are
  given
• We recognize that energies are not absolute, but
  rather must be defined relative to some reference
• We use the elements in their equilibrium states
  at standard pressure, typically 1 atm or 1 bar
  (0.1 MPa)

• From ab initio calculations, energy is typically
  referenced to the constituent atoms, fully
  dissociated. Get ?fH0 from

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To go further, we need statistical mechanics.

• The partition function q(V,T)?exp(-?i /?T)
  arises naturally in the development of
  Maxwell-Boltzmann and Bose-Einstein statistics
• Quantum mechanics gives the quantized values of
  energy and thus the partition functions for
• Translational degrees of freedom
• External rotational degrees of freedom (linear or
  nonlinear rotors)
• Rovibrational degrees of freedom (stretches,
  bends, other harmonic oscillators, and internal
  rotors)
• Electronic d.f. require only ?electronic and
  degeneracy.

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Entropy, energy, and heat capacity can be
expressed in terms of the partition function(s).
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Simplest treatment is of ideal gas, beginning
with the translation degrees of freedom.

• Quantum mechanics for pure translation in 3-D
  gives

• Note the standard-state pressure in the last
  equation

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Rigid-rotor model for external rotation
introduces the moment of inertia I and rotational
symmetry ?ext.
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Add harmonic oscillators with frequencies ?i and
electronic degeneracy of go.

• For each harmonic oscillator,

• It is convenient to redefine zero for vibrational
  energy as zero rather than 0.5h? this shift
  requires the zero-point energy correction to
  energy. As a result,

• If only the ground electronic state contributes,
  then (Cvo)elec0 and (So)elecRln go. Otherwise,
  need g1 ?1.

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Taken together, they give us ideal-gas Cpo and
So, and integration over T gives ?fH298o.

• Even for gases, there are further complications
  beyond the Rigid-Rotor Harmonic Oscillator model
  (RRHO)
• Low-frequency modes may be fully excited
• Anharmonic behaviors like free and hindered
  internal rotors
• We can generally deal with the statistical
  mechanics that complicate these issues
• Computational chemistry even can calculate
  anharmonicities like shape of the potential well
  or barriers to rotation
• Likewise, we can calculate terms needed to model
  thermochemistry of liquids, solutions, and solids
• Likewise for phase equilibrium and transport
  properties.

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Now examine kinetics from quantum chemistry.

• We have already discussed how to locate
  transition states along the minimum energy
  path
• A stationary point (?E/?? 0) with respect to
  all displacements
• A minimum with respect to all displacements
  except the one corresponding to the reaction
  coordinate
• More precisely, all but one eigenvalue of the
  Hessian matrix of second derivatives are positive
  (real frequencies) or zero (for the overall
  translational and rotational degrees of freedom
• The exception Motion along the reaction
  coordinate
• It corresponds to a frequency ? that is an
  imaginary number
• If ei?t is a sinusoidal oscillation, then ? is
  exponential change

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The entire minimum energy path may not be a
simple motion, but the transition state is still
separable.
Potential energy surface for O-O bond fission in
CH2CHOO B3LYP/6-31G(d) Kinetics analysis based
on O-O reaction-coordinate-driving calculation at
B3LYP/6-311G(d,p)
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Consider transition-state thermochemistry.

• It has a geometrical structure, electronic state,
  and vibrations, so assume we can calculate q,
  H, S, Cp
• For classical transition-state theory, Eyring
  assumed
• At equilibrium, TS would obey equilibrium
  relations with reactant
• The reaction coordinate would be a separable
  degree of freedom
• Thus, with it treated as a 1-D translation or a
  vibration,

• Recognizing the form of a thermochemical Keq,

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Computational quantum chemistry gives very useful
numbers for Eact, also can give good A-factor.

• For gas kinetics, calculate H, S, Cp, ?S(T),
  ?H(T)
• Reaction coordinate contributes zero to S
• Standard-state correction is necessary for
  bimolecular reactions
• Eact, like bond energy, may be adequate for
  comparisons
• Most other factors can be handled
• If reaction coordinate involves H motion and low
  T, quantum-mechanical tunneling may occur (use
  calculated barrier shape)
• High-pressure limit is required (use RRKM, Master
  Equation)
• Low-frequency modes like internal rotors give the
  most uncertainty in ?S, but we can calculate
  barriers
• In principle, the same for anharmonicity of
  vibrations

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Other properties are predicted, too Advances in
methods have been aided by demand.

• Good semi-empirical and ab initio calculations
  for excited states give pigment and dye behaviors
• Solvation models by Tomasi and others make
  liquid-phase behaviors more calculable
• Hybrid methods have proven powerful
• QM/MM for biomolecule structure and ab-initio
  molecular dynamics for ordered condensed phases
  calculate interactions as dynamics calculations
  proceed
• Spatial extrapolation such as embedded-atom
  models of catalysts and Morokumas ONIOM method
  connect or extrapolate domains of different-level
  calculations

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Computational Quantum ChemistryPart II.
Principles and Methods.
In parallel, see the faces behind the names.
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Ab initio is widely but loosely used to mean
from first principles.

• Actually, there is considerable use of assumed
  forms of functionalities and fitted parameters.
• John Pople noted that this interpretation of the
  Latin is by adoption rather than intent. In its
  first use
• The two groups of Parr, Craig, and Ross J Chem
  Phys 18, 1561 (1951) had carried out some of the
  first calculations separately across the Atlantic
  - and thus described each set of calcs as being
  ab initio!

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Three key features of theory are required for ab
initio calculations.

• Understand how initial specification of nuclear
  positions is used to calculate energy
• Solving the Schrödinger equation
• Understand basis sets and how to choose them
• Functions that represent the atomic orbitals
• e.g., 3-21G, 6-311G(3df,2pd), cc-pVTZ
• Understand levels of theory and how to choose
  them
• Wavefunction methods Hartree-Fock, MP4, CI, CAS
• Density functional methods LYP, B3LYP, etc.
• Compound methods CBS, G3
• Semiempirical methods AM1, PM3

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Initially, restrict our discussion to an
isolated molecule.

• Equivalent to an ideal gas, but may be a cluster
  of atoms, strongly bonded or weakly interacting.
• Easiest to think of a small, covalently bonded
  molecule like H2 or CH4 in vacuo.
• Most simply, the goal of electronic structure
  calculations is energy.

• However, usually we want energy of an optimized
  structure and the energys variation with
  structure.

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Begin with the Hamiltonian function, an
effective, classical way to calculate energy.

• Express energy of a single classical particle or
  an N-particle collection as a Hamiltonian
  function of the 3N momenta pj and 3N coordinates
  qj (j1,N) such that

where H Kinetic Energy (T) Potential
Energy (V) Total Energy
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For quantum mechanics, a Hamiltonian operator is
used instead.

• Obtain a Hamiltonian function for a wave using
  the Hamiltonian operator

to obtain
where Y is the wavefunction, an eigenfunction
of the equation

• Born recognized that Y2 is the probability
  density function

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For quantum molecular dynamics, retain t
Otherwise, t-independent.

• Separation of variables gives Y(q) and thus the
  usual form of the Schroedinger or Schrödinger
  equation

• If the electron motions can be separated from the
  nuclear motions (the Born-Oppenheimer
  approximation), then the electronic structure can
  be solved for any set of nuclear positions.

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Easiest to consider H atom first as a prototype.
e-

• Three energies
• Kinetic energy of the nucleus.
• Kinetic energy of the electron.
• Proton-electron attraction.
• With more atoms, also
• Internuclear repulsion
• Electron-electron repulsion.
• Electrons are in specific quantum states called
  orbitals.
• They can be in excited states (higher-energy
  orbitals).

proton
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Restate the nonrelativistic electronic
Hamiltonian in atomic units.
With distances in bohr (1 bohr 0.529 Å) and
with energies in hartrees (1 hartree 627.5
kcal/mol),
where
(After Hehre et al., 1986)
Breaks down when electrons approach the speed
of light, the case for innermost electrons around
heavy atoms
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Set up Y, the system wavefunction.

• Need functionality (form) and parameters.
• (1) Use one-electron orbital functions (basis
  functions) to ...
• (2) Compose the many-electron molecular orbitals
  y by linear combination, then ...
• (3) Compose the system Y from ys.
• Wavefunction Y must be antisymmetric
• Exchanging identical electrons in Y should give
  -Y
• Characteristic of a fermion vs. bosons
  (symmetric)

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H-atom eigenfunctions y correspond to hydrogenic
atomic orbitals.
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Construct each MO yi by LCAO.

• Lennard-Jones (1929) proposed treating molecular
  orbitals as linear combinations of atomic
  orbitals (LCAO)

• Linear combination of s orbital on one atom with
  s or p orbital on another gives s bond

• Linear combination of p orbital on one atom with
  p orbital on another gives p bond

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Molecular Y includes each electron.

• First, include spin (x-1/2,1/2) of each e-.
• Define a one-electron spin orbital, c(x,y,z,x)
  composed of a molecular orbital y(x,y,z)
  multiplied by a spin wavefunction a(x) or b(x).
• Next, compose Y as a determinant of cs.

lt- Electron 1 in all cs lt- Electron 2 in all
cs lt- Electron n in all cs

• Interchange row gt Change sign \ Functionally
  antisymmetric.

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However, basis functions fi need not be purely
hydrogenic - indeed, they cannot be.

• Form of basis functions must yield accurate
  descriptions of orbitals.
• Hydrogenic orbitals are reasonable starting
  points, but real orbitals
• Dont have fixed sizes,
• Are distorted by polarization, and
• Involve both valence electrons (the outermost,
  bonding shell) and non-valence electrons.
• Hydrogenic s-orbital has a cusp at zero, which
  turns out to cause problems.

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Simulate the real functionality (1).

• Start with a function that describes hydrogenic
  orbitals well.

• Slater functions e.g.,
• Gaussian functions e.g.,
• No s cusp at r0
• However, all analytical integrals
• Linear combinations of
  gaussians e.g., STO-3G
• 3 Gaussian primitives to simulate a STO
• (Minimal basis set)

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Simulate the real functionality (2).

• Allow size variation.

• Alternatively, size adjustment only for outermost
  electrons (split-valence set) to speed calcs
• For example, the 6-31G set
• Inner orbitals of fixed size based on 6
  primitives each
• Valence orbitals with 3 primitives for contracted
  limit, 1 primitive for diffuse limit
• Additional very diffuse limits may be added
  (e.g., 6-31G or 6-311G)

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Simulate the real functionality (3).

• Allow shape distortion (polarization).
• Usually achieved by mixing orbital types

• For example, consider the 6-31G(d,p) or 6-31G
  set
• Add d polarization to p valence orbitals, p
  character to s
• Can get complicated e.g., 6-311G(3df,2pd)

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Simulate the real functionality (4).

• A noteworthy improvement is the set of Complete
  Basis Set methods of Petersson.
• Better parameterization of finite basis sets.
• Extrapolation method to estimate how result
  changes due to adding infinitely more s,p,d,f
  orbitals
• Another basis-set improvement is development of
  Effective Core Potentials.
• As noted before, for transition metals, innermost
  electrons are at relativistic velocities
• Capture their energetics with effective core
  potentials
• For example, LANL2DZ (Los Alamos N.L. 2 Double
  Zeta).

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The third aspect is solution method.

• Hartree-Fock theory is the base level of
  wavefunction-based ab initio calculation.
• First crucial aspect of the theory
  The variational principle.
• If Y is the true wavefunction, then for any model
  antisymmetric wavefunction F, E(F)gtE(Y).
  Therefore the problem becomes a minimization of
  energy with respect to the adjustable parameters,
  the Cµis and ls.

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The Hartree-Fock result omits electron-electron
interaction (electron correlation).

• The variational principle led to the
  Roothaan-Hall equations (1951) for closed-shell
  wavefunctions

or

• ei is diagonal matrix of one-electron energies
  of the yi.
• F, the Fock matrix, includes the Hamiltonian for
  a single electron interacting with nuclei and a
  self-consistent field of other electrons S is an
  atomic-orbital overlap matrix.
• All electrons paired (RHF) there are analogous
  UHF equations.

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One improvement is to use Configuration
Interaction.

• Hartree-Fock theory is limited by its neglect of
  electron-electron correlation.
• Electrons interact with a SCF, not individual
  es.
• Full CI includes the Hartree-Fock ground-state
  determinant and all possible variations.

• The wavefunction becomes
  where s includes all combinations
  of substituting electrons into H-F virtual
  orbitals.
• The as are optimized not so practical.

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Partial CI calculations are feasible.

• CIS (CI with Singles substitutions), CISD,
  CISD(T) (CI with Singles, Doubles, and
  approximate Triples)
• CI calculations where the occupied ci elements in
  the SCF determinant are substituted into virtual
  orbitals one and two at a time and excited-state
  energies are calculated.
• CASSCF (Complete Active Space SCF) is better
  Only a few excited-state orbitals are considered,
  but they are re-optimized rather than the SCF
  orbitals.
• Other variants QCISD, Coupled Cluster methods.

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Perturbation Theory is an alternative.

• Møller and Plesset (1934) developed an electronic
  Hamiltonian based on an exactly solvable form H0
  and a perturbation operator

• A consequence is that the wavefunction Y and the
  energy are perturbations of the Hartree-Fock
  results, including electron-electron correlation
  effects that H-F omits.
• Most significant MP2 and MP4.

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Pople emphasizes matrix extrapolation.
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Compound methods aim at extrapolation.

• The G1, G2, and G3 methods of Pople and
  co-workers calculate energies in cells of their
  matrix, then project more accurately.
• G2 gave ave. error in ?Hf of 1.59 kcal/mol.
• G3 gives ave. error in ?Hf of 1.02 kcal/mol.
• CBS methods are compound methods that give
  impressive results.
• Melius and Binkleys BAC-MP4 is based on
  MP4/6-31G(d,p)//HF/6-31G(d) calculation.

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Besides energy, calculations give electron
density, HOMO, LUMO.

• Electron density (from electron probability
  density function Y2) is an effective
  representation of molecular shape.
• Each molecular orbital is calculated, including
  highest-energy occupied MO (HOMO) and
  lowest-energy unoccupied MO (LUMO)
• HOMO-LUMO gap is useful for Frontier MO theory
  and for band gap analysis.

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Results can be seen with ethylene.
Electron density
HOMO LUMO

• Calculations and graphics at HF/3-21G with
  MacSpartan Plus (Wavefunction Inc.).

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An increasingly important approach is
density-functional theory.

• From Hohenberg and Kohn (1964)
• Energy is a functional of electron density Er
• Ground-state only, but exact r minimizes Er
• Then Kohn and Sham (1965)
• Variational equations for a local functional

where Exc contains electron correlation.
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Local density functionals arent very useful for
molecules, but...

• Kohn and Sham had

• Need nonlocal effects of gradient,
• Even more interesting Hybrid functionals
• Combine Hartree-Fock and DFT contributions
• Axel Beckes BLYP, B3LYP, BHHLYP
• Why do it?
• Handle bigger molecules! Include correlation!

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Other properties can be calculated.

• Frequencies from ?2E/?r2 (fix wHF0.891).
• Dipole moments.
• NMR shifts.
• Solution behavior.
• Ideal-gas thermochemistry.
• Transition-state-theory rate constants.

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With these tools, we can move from overall
formulas... to sketches...
(C33N3H43)FeCl2, a liganded di(methyl imide
xylenyl) aniline ...
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To quantitative 3-D functionality.
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Close References for further study.

• J.B. Foresman and Æ. Frisch, Exploring Chemistry
  with Electronic Structure Methods, 2nd Ed.,
  Gaussian Inc., 1996.
• W. J. Hehre, L. Radom, P. v. R. Schleyer, and J.
  A. Pople, Ab Initio Molecular Orbital Theory
  (Wiley, New York, 1986).
• T. H. Dunning, J. Chem. Phys. 90, 1007-1023
  (1989).
• H. Borkent, "Computational Chemistry and Org.
  Synthesis," http//www.caos.kun.nl/7Eborkent/comp
  course/comp.html
• J. P. Simons, Theoretical Chemistry,
  http//simons.hec.utah.edu/TheoryPage/
• D.A. McQuarrie, Statistical Mechanics, Harper
  Row, 1976.
• S.W. Benson, Thermochemical Kinetics, 2nd Ed,
  Wiley, 1976.