Computational Quantum Chemistry Part I: Obtaining Properties

8/3/2019 Computational Quantum Chemistry Part I: Obtaining Properties

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Computational Quantum Chemistry. II. Principles and Methods.

Computational Quantum Chemistry

Part I. Obtaining Properties




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 asvaluable – 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 processdevelopment, for for operations, or for troubleshooting

• Great data are best; but also theory-based predictions








Restate: What kind of properties come directlyfrom computational quantum chemistry?

• Energies, structures optimized with respect toenergy, harmonic frequencies, and other propertiesbased 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

Quantum-mechanical energies

Statistical mechanicsStatistical mechanics




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.

-60

-40

-20

0

20

40

60

80

100

0 1 2 3

Br-Br, angstroms

UB3LYP/6-311++G(3df,3dp) withbasis-set superposition error correction



At zero K, define the dissociation energy D0 asthe well depth less zero-point energy.

Alternate view isthat

D

0=

E 0(dissociatedpartners)

- [E 0(molecule) +ZPE],

where ZPE is thezero-K energy of the stretchingvibration.



Geometry is then found by optimizing computedenergy with respect to coordinates (here, 1).

Transitionstate

Ground state - minimumw.r.t. all coordinates

Minimum w.r.t. all butreaction coordinate



Vibrational frequencies (at 0 K) are calculatedusing parabolic approximation to well bottom.

• How many? Need 3N atoms 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, 3N atoms-5 (nonlinear) or 3N atoms-6 (nonlinear) vibrations

• For diatomic, ∂2E /∂r 2 = force constant k [for r dimensionless]

– F (= ma = m∂2r /∂t 2) = -kr is a harmonic oscillator in Newtonianmechanics (Hooke’s law)

– Harmonic frequency is (k/m)1/2/2π s-1 or (k/m)1/2/2πc cm-1 (wavenumbers)

• For polyatomic, analyze Hessian matrix [∂2E /∂r

i∂r

j] instead



Next, determine ideal-gas thermochemistry.

• Start with ∆f H 0° 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 standardpressure, typically 1 atm or 1 bar (0.1 MPa):

– From ab initio calculations, energy is typically referenced tothe constituent atoms, fully dissociated. Get ∆f H 0° from:



To go further, we need statistical mechanics.

• The partition function q(V,T)=∑exp(-i /T ) arisesnaturally in the development of Maxwell-Boltzmannand 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 anddegeneracy.



Entropy, energy, and heat capacity can beexpressed in terms of the partition function(s).



Simplest treatment is of ideal gas, beginningwith the translation degrees of freedom.

• Quantum mechanics for pure translation in 3-D gives:

• Note the standard-state pressure in the last equation



Rigid-rotor model for external rotation introducesthe moment of inertia I and rotational symmetry

ext .



Add harmonic oscillators with frequencies i andelectronic degeneracy of g

o.

• For each harmonic oscillator,

– It is convenient to redefine zero for vibrational energy aszero rather than 0.5h; this shift requires the zero-pointenergy correction to energy. As a result,

• If only the ground electronic state contributes, then(C v

o )elec =0 and (S

o )elec =R·ln g o. Otherwise, need g 1

& 1.



Taken together, they give us ideal-gas C po

andS

o, and integration over T gives ∆

f H

298

o.

• Even for gases, there are further complications beyondthe 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 anharmonicitieslike shape of the potential well or barriers to rotation

• Likewise, we can calculate terms needed to modelthermochemistry of liquids, solutions, and solids

• Likewise for phase equilibrium and transport properties.



Now examine kinetics from quantum chemistry.

• We have already discussed how to locate transitionstates along the “minimum energy path”: – A stationary point (∂E/∂ = 0) with respect to all displacements

– A minimum with respect to all displacements except the onecorresponding 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 e it is a sinusoidal oscillation, then ‡ is exponential change



The entire minimum energy path may not be a simplemotion, but the transition state is still separable.

05

1015202530

3540

45

50

55Potential energysurface for O-O

bond fission inCH2CHOO·B3LYP/6-31G(d);Kinetics analysis

based on

O-O reaction-

coordinate-driving calculation at

B3LYP/6-311+G(d,p)



Consider transition-state thermochemistry.

• It has a geometrical structure, electronic state, andvibrations, so assume we can calculate q‡ , H ‡ , S ‡ , C p

‡

• 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 K eq,



Computational quantum chemistry gives very usefulnumbers for E

act , also can give good A-factor.

• For gas kinetics, calculate H ‡ , S ‡ , C p‡ , ∆S ‡ (T), ∆H ‡ (T)

– Reaction coordinate contributes zero to S ‡

– Standard-state correction is necessary for bimolecular reactions

– E act , 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




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 Morokuma’s ONIOM method; connect or extrapolate domains of different-level calculations




Computational Quantum Chemistry

Part II. Principles and Methods.

In parallel, see the faces behind the names.

http://mooni.fccj.org/~ethall/quantum/quant.htm




“ Ab initio” is widely but loosely used to mean“from first principles.”

• Actually, there is considerable use of assumed formsof functionalities and fitted parameters.

• John Pople noted that this interpretation of the Latinis by adoption rather than intent. In its first use:

• The two groups of Parr, Craig, and Ross [J ChemPhys 18, 1561 (1951)] had carried out some of thefirst calculations separately across the Atlantic - and

thus described each set of calcs as being ab initio!




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, 6-311++G(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




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 bondedmolecule like H2 or CH4 in vacuo.

• Most simply, the goal of electronic structure

calculations is energy.

• However, usually we want

energy of an optimizedstructure and the energy’svariation with structure.




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 3Nmomenta p j and 3N coordinates q j ( j =1,N) such that:

where:

H = Kinetic Energy (T) + Potential Energy (V)

= Total Energy




For quantum mechanics,a Hamiltonian operator is used instead.

• Obtain a Hamiltonian function for a wave using theHamiltonian operator:

to obtain:

where is the “wavefunction,” an eigenfunctionof the equation

• Born recognized that 2 is the probability density function




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

• Separation of variables gives (q) and thus the usualform of the Schroedinger or Schrödinger equation:

• If the electron motions can be separated from thenuclear motions (the Born-Oppenheimer

approximation), then the electronic structure can besolved for any set of nuclear positions.




e-

proton

Easiest to consider H atom first as a prototype.

• 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).




Restate the nonrelativistic electronicHamiltonian in atomic units.

• With distances in bohr (1 bohr = 0.529 Å) andwith energies in hartrees (1 hartree = 627.5 kcal/mol),

(After Hehre et al., 1986)

where

• [Breaks down when electrons approach the speed of light,the case for innermost electrons around heavy atoms]




Set up , the system wavefunction.

• Need functionality (form) and parameters.

• (1) Use one-electron orbital functions (“basisfunctions”) to ...

• (2) Compose the many-electron molecular orbitals by linear combination, then ...

• (3) Compose the system from ’s.

• Wavefunction must be “antisymmetric” – Exchanging identical electrons in should give -

– Characteristic of a “fermion”; vs. “bosons” (symmetric)




H-atom eigenfunctions correspond tohydrogenic atomic orbitals.




Construct each MO i by LCAO.

• Lennard-Jones (1929) proposed treatingmolecular orbitals as linear combinationsof atomic orbitals (LCAO):

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

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




Molecular includes each electron.

• First, include spin (=-1/2,+1/2) of each e-. – Define a one-electron spin orbital, ( x,y,z ,) composed of a

molecular orbital ( x,y,z ) multiplied by a spin wavefunction

or .• Next, compose as a determinant of ’s.

– Interchange row => Change sign; Functionally antisymmetric.

<- Electron 1 in all ’s;<- Electron 2 in all ’s;

<- Electron n in all ’s




However, basis functions i need not be purelyhydrogenic - indeed, they cannot be.

• Form of basis functions must yield accuratedescriptions of orbitals.

• Hydrogenic orbitals are reasonable starting points,

but real orbitals: – Don’t 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.




Simulate the real functionality (1).

• Start with a function that describes hydrogenic orbitalswell.

– Slater functions; e.g.,

– Gaussian functions; e.g.,• No s cusp at r =0

• However, all analytical integrals

– Linear combinations of

gaussians; e.g., STO-3G• 3 Gaussian “primitives” tosimulate a STO

• (“Minimal basis set”)






– 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., 6-31+G or 6-311++G)


• Allow size variation.





• 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., 6-311++G(3df,2pd)





• A noteworthy improvement is the set of CompleteBasis Set methods of Petersson. – Better parameterization of finite basis sets.

– Extrapolation method to estimate how result changes due toadding infinitely more s,p,d,f orbitals

• Another basis-set improvement is development of Effective Core Potentials.

– As noted before, for transition metals, innermost electronsare at relativistic velocities

– Capture their energetics with effective core potentials

– For example, LANL2DZ (Los Alamos N.L. #2 Double Zeta).




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 is the true wavefunction, then for any model

antisymmetric wavefunction , E ()>E (). Therefore theproblem becomes a minimization of energy with respect tothe adjustable parameters, the C

µi

’s and ’s.




The Hartree-Fock result omits electron-electroninteraction (“electron correlation”).

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

• i is diagonal matrix of one-electron energies of the i .

• F , the Fock matrix, includes the Hamiltonian for a single electroninteracting 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.

or




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 e’s.

• “Full CI” includes the Hartree-Fock ground-statedeterminant and all possible variations.

– The wavefunction becomeswhere s includes all combinations

of substituting electrons into H-F virtualorbitals.

– The a’s are optimized; not so practical.




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 i elements in the SCF

determinant are substituted into virtual orbitals one and two at a timeand excited-state energies are calculated.

• CASSCF (Complete Active Space SCF) is better: Onlya few excited-state orbitals are considered, but theyare re-optimized rather than the SCF orbitals.

• Other variants: QCISD, Coupled Cluster methods.




Perturbation Theory is an alternative.

• Møller and Plesset (1934) developed an electronicHamiltonian based on an exactly solvable form 0 and a perturbation operator:

• A consequence is that the wavefunction and theenergy are perturbations of the Hartree-Fock results,

including electron-electron correlation effects that H-Fomits.

• Most significant: MP2 and MP4.




Pople emphasizes matrix extrapolation.




Compound methods aim at extrapolation.

• The G1, G2, and G3 methods of Pople and co-workerscalculate energies in cells of their matrix, then projectmore accurately.

– G2 gave ave. error in ∆H f of ±1.59 kcal/mol. – G3 gives ave. error in ∆H f of ±1.02 kcal/mol.

• CBS methods are compound methods that giveimpressive results.

• Melius and Binkley’s BAC-MP4 is based onMP4/6-31G(d,p)//HF/6-31G(d) calculation.




Besides energy, calculations give electrondensity, HOMO, LUMO.

• Electron density (from electron probability densityfunction = 2) is an effective representation of molecular shape.

• Each molecular orbital is calculated, including highest-energy occupied MO (HOMO) and lowest-energyunoccupied MO (LUMO)

• HOMO-LUMO gap is useful for Frontier MO theory and

for band gap analysis.




Results can be seen with ethylene.

Calculations and graphics at HF/3-21G* withMacSpartan Plus (Wavefunction Inc.).

Electron

density HOMO;

LUMO




An increasingly important approachis density-functional theory.

• From Hohenberg and Kohn (1964) – Energy is a functional of electron density: E[]

– Ground-state only, but exact minimizes E[]

• Then Kohn and Sham (1965) – Variational equations for a “local” functional:

where E xc contains electron correlation.




Local density functionals aren’t 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 Becke’s BLYP, B3LYP, BH&HLYP

• Why do it? – Handle bigger molecules! Include correlation!




Other properties can be calculated.

• Frequencies from ∂2 E /∂r 2 (fix HF*0.891).

• Dipole moments.

• NMR shifts.

• Solution behavior.

• Ideal-gas thermochemistry.

• Transition-state-theory rate constants.




With these tools, we can move fromoverall formulas... to sketches...

(C33 N3H43)

FeCl2,

a liganded di

(methyl imide

xylenyl) aniline ...




To quantitative 3-D functionality.



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/compcourse/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.

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Computational Quantum Chemistry Part I: Obtaining Properties