Computational Quantum Chemistry Part I' Obtaining Properties - PowerPoint PPT Presentation

1 / 56
About This Presentation
Title:

Computational Quantum Chemistry Part I' Obtaining Properties

Description:

... their equilibrium states at standard pressure, typically 1 atm or 1 bar (0.1 MPa) ... CBS methods are compound methods that give impressive results. ... – PowerPoint PPT presentation

Number of Views:228
Avg rating:3.0/5.0
Slides: 57
Provided by: phill111
Category:

less

Transcript and Presenter's Notes

Title: Computational Quantum Chemistry Part I' Obtaining Properties


1
Computational Quantum ChemistryPart I.
Obtaining Properties
2
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

3
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

4
(No Transcript)
5
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
6
Simplest properties are interaction energies
Here, the van der Waals well for an Ar dimer.
7
Simplest chemical bonds are much stronger.
UB3LYP/6-311G(3df,3dp) with basis-set
superposition error correction
8
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.
9
Geometry is then found by optimizing computed
energy with respect to coordinates (here, 1).
10
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

11
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

12
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.

13
Entropy, energy, and heat capacity can be
expressed in terms of the partition function(s).
14
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

15
Rigid-rotor model for external rotation
introduces the moment of inertia I and rotational
symmetry ?ext.
16
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.

17
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.

18
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

19
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)
20
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,

21
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

22
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

23
Computational Quantum ChemistryPart II.
Principles and Methods.
In parallel, see the faces behind the names.
24
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!

25
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

26
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.

27
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
28
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

29
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.

30
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
31
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
32
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)

33
H-atom eigenfunctions y correspond to hydrogenic
atomic orbitals.
34
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

35
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.

36
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.

37
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)

38
(No Transcript)
39
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)

40
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)

41
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).

42
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.

43
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.

44
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.

45
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.

46
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.

47
Pople emphasizes matrix extrapolation.
48
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.

49
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.

50
Results can be seen with ethylene.
Electron density
HOMO LUMO
  • Calculations and graphics at HF/3-21G with
    MacSpartan Plus (Wavefunction Inc.).

51
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.
52
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!

53
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.

54
With these tools, we can move from overall
formulas... to sketches...
(C33N3H43)FeCl2, a liganded di(methyl imide
xylenyl) aniline ...
55
To quantitative 3-D functionality.
56
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.
Write a Comment
User Comments (0)
About PowerShow.com