Computational Quantum Chemistry. II. Principles and Methods. 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
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Computational Quantum Chemistry. II. Principles and Methods.
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
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Computational Quantum Chemistry. II. Principles and Methods.
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
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Computational Quantum Chemistry. II. Principles and Methods.
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
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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.
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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
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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
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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:
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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.
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Entropy, energy, and heat capacity can beexpressed in terms of the partition function(s).
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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
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Rigid-rotor model for external rotation introducesthe moment of inertia I and rotational symmetry
ext .
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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.
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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.
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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
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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)
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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,
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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
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Computational Quantum Chemistry. II. Principles and Methods.
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
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Computational Quantum Chemistry. II. Principles and Methods.
Computational Quantum Chemistry
Part II. Principles and Methods.
In parallel, see the faces behind the names.
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Computational Quantum Chemistry. II. Principles and Methods.
“ 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!
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Computational Quantum Chemistry. II. Principles and Methods.
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
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and Methods.
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
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Computational Quantum Chemistry. II. Principles and Methods.
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
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and Methods.
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).
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Computational Quantum Chemistry. II. Principles and Methods.
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]
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Computational Quantum Chemistry. II. Principles and Methods.
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)
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Computational Quantum Chemistry. II. Principles and Methods.
H-atom eigenfunctions correspond tohydrogenic atomic orbitals.
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Computational Quantum Chemistry. II. Principles and Methods.
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:
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Computational Quantum Chemistry. II. Principles and Methods.
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
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and Methods.
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”)
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Computational Quantum Chemistry. II. Principles and Methods.
– 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)
Simulate the real functionality (2).
• Allow size variation.
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Computational Quantum Chemistry. II. Principles and Methods.
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., 6-311++G(3df,2pd)
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Computational Quantum Chemistry. II. Principles and Methods.
Simulate the real functionality (4).
• 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).
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and Methods.
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
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and 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.
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Computational Quantum Chemistry. II. Principles and Methods.
Pople emphasizes matrix extrapolation.
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and Methods.
Results can be seen with ethylene.
Calculations and graphics at HF/3-21G* withMacSpartan Plus (Wavefunction Inc.).
Electron
density HOMO;
LUMO
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and Methods.
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!
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Computational Quantum Chemistry. II. Principles and Methods.
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.
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Computational Quantum Chemistry. II. Principles and Methods.
With these tools, we can move fromoverall formulas... to sketches...
(C33 N3H43)
FeCl2,
a liganded di
(methyl imide
xylenyl) aniline ...
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Computational Quantum Chemistry. II. Principles and Methods.
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/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.