NANO266 - Lecture 6- Molecule Properties from Quantum Mechanical Modeling
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Transcript of NANO266 - Lecture 6- Molecule Properties from Quantum Mechanical Modeling
Molecule properties from QM modeling
Shyue Ping Ong
Our journey so far
Schrodinger Equation
Variational Approaches
Hartree Fock
Including Correlation with
Hartree Fock
Density Functional
Theory
Local density approximation
Generalized gradient
approximation
Hybrids
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It’s time to see what we can do with these
The Materials World
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Molecules
Isolated gas phase
Typically use localized basis functions, e.g.,
Gaussians
Everything else (liquids,
amorphous solids, etc.)
Too complex for direct QM!
(at the moment)
But can work reasonable
models sometimes
Crystalline solids
Periodic infinite solid
Plane-wave approaches
Overview
In this lecture, we will • Survey the study of properties of isolated
molecules using quantum mechanical approaches.
• Connect calculations with real world properties • Discuss performance and accuracy
Lab 1: Study of ammonia formation using QM
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What do you get from QM?
Energies Geometries Charge densities and spectroscopic properties
And their derivatives…
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Energies and eigenvalues
Most direct output from QM calculations Accuracy have been discussed in previous lectures
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Vibrational frequencies and energies
Harmonic oscillator assumption To obtain the force constants, one simply needs to calculate the 2nd derivative of the energy with respect to bond stretching at equilibrium bond geometry Can be done analytically for HF, MP2, DFT, CISD, CCSD
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E = n+ 12
!
"#
$
%&hω where ω =
12π
kµ
where k is the force constants
Scaling factors for vibrational frequencies
To account for systematic errors in predicted vibrational frequencies
E.g., HF overemphasizes bonding and all force constants (and frequencies) are too large
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Ensemble thermodynamic ensembles
QM gives the single molecule energies Question: How do we get ensemble thermodynamic variables from single-molecule calculations?
Answer: Statistical mechanics
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A brief recap of statistical mechanics
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Z(N,V,T ) = e−Ei (N ,V )kBT
i∑
U = kBT2 ∂ lnZ
∂T$
%&
'
()N ,V
H =U +PV
s = kB lnZ + kB∂ lnZ∂T
$
%&
'
()N ,V
G = H −TS
Assumption: Ideal gas molecules
Since ideal gas molecules do not interact,
The molecular partition function can be further broken down into separable components
Combining the results, we have
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Z(N,V,T ) = z(V,T )N
N! where z(V,T ) is the molecular partition function.
z(V,T ) = zelec (T )ztrans (V,T )zrot (T )zvib(T )
ln Z(N,V,T )( ) = N zelec (T )+ ztrans (V,T )+ zrot (T )+ zvib(T )[ ]− N lnN + N
Components of the partition function
Electronic • Typically, excited states are much higher in energy and make no significant
contribution to partition function below a few 1000K. => Just the electronic energy from QM.
• If there is a non-singlet ground state, there are contributions to the electronic entropy.
Translation (Particle in box)
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ztrans (V,T ) =2πMk BT
h2!
"#
$
%&
32V
Utrans =32RT
Strans0 = R ln 2πMk BT
h2!
"#
$
%&
32 VNA
'
(
))
*
+
,,+52
-
./
0/
1
2/
3/
Components of the partition function
Vibrational • Based on quantum mechanical harmonic oscillator assumption
(3N – 6 degrees of freedom)
Rotational
• Linear and non-linear molecules to be treated separately • Refer to statistical mechanics textbook
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zvib(T ) =1
1− e−hω /kBT
Uvib =1
1− e−hωi /kBTi=1
3N−6
∏
Svib0 = R hωi
kBT (ehωi /kBT −1)
− ln(1− e−hωi /kBT )#
$%
&
'(
i=1
3N−6
∑
Typical calculation procedure for enthalpies
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Geometry optimization
(GO)
• Typically at a lower level of theory and smaller basis set
Frequency calculation
• Same level of theory as GO
• Obtain vibrational and other contributions to free energy
SCF energy calculation
• Higher level of theory and basis set
Selecting model chemistries
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Foresman, J. B.; Frisch, Ae. Exploring Chemistry With Electronic Structure Methods: A Guide to Using Gaussian; Gaussian, 1996.
Practical reaction calculations
Let’s say we are interested in calculating the following reaction energies from QM
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Reaction 1N2 (g)+3H2 (g)→ 2NH3(g)
Reaction 2C(s)+O2 (g)→ 2CO2 (g)
This one is easy. I just calculate the energies in the gas state for each of the molecules with the statistical corrections! -> Subject of Lab 1
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ΔH 0f ,298(M ) = E(M )+ ZPE(M )+[H298(M )−H0 (M )]
− E(Xz )+[H298(Xz )−H0 (Xz )]{ }z
atoms
∑ + ΔH 0f ,298
z
atoms
∑ (Xz )
Ionization energies and electron affinities
Koopman’s Theorem • HOMO energy as estimate of vertical IE fairly reasonable due to
canceling of basis set incompleteness and correlation errors in Hartree-Fock
• Though a corollary of Koopman exists for DFT for the exact xc functional, in practice eigenvalues from inexact DFT are poor estimates.
ΔSCF
• Calculate energy of molecule in neutral and positively / negatively charged
• Generally works well if diffuse functions are used to model ions with diffuse electron clouds.
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Charge distribution properties
Multipole moments Partial atomic charges
• Class II charges – determined by partitioning of wave functions (a somewhat arbitrary process)
• Mulliken approach – partition according to degree atomic orbitals contribute to wave function
• Lowdin – Transform AO basis functions to orthonormal set • Natural population analysis (NPA) – Orthogonalization in four-
step process to render electron density as compact as possible before Mulliken analysis
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xkylzm = Zixik yi
lzim
i
atoms
∑ − ψ(r) xik yi
lzim
i
electrons
∑∫ ψ(r)dr
NMR spectral properties
General recommendation is very large basis sets (at least triple-ζ) and lots of diffuse and polarization functions Not possible to predict chemical shift for nuclei of heavy atoms with effective core potentials For molecules comprising first row atoms, heavy-atom chemical shifts can be obtained with a fair degree of accuracy, even with HF (though DFT and MP2 fares much better).
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