UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.

1. Molecule properties
from QM modeling
Shyue Ping Ong

2. 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

3. The MaterialsWorld
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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

4. 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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5. What do you get from QM?
Energies
Geometries
Charge densities and spectroscopic properties
And their derivatives…
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6. Energies and eigenvalues
Most direct output from QM calculations
Accuracy have been discussed in previous
lectures
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7. 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 +
1
2
!
"
#
$
%
&hω where ω =
1
2π
k
µ
where k is the force constants

8. 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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9. 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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10. 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

11. 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 ln N + N

12. 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
!
"
#
$
%
&
3
2
V
Utrans =
3
2
RT
Strans
0
= R ln
2πMk BT
h2
!
"
#
$
%
&
3
2 V
NA
'
(
)
)
*
+
,
,
+
5
2
-
.
/
0/
1
2
/
3/

13. 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 /kBT
i=1
3N−6
∏
Svib
0
= R
hωi
kBT(ehωi /kBT
−1)
− ln(1−e−hωi /kBT
)
#
$
%
&
'
(
i=1
3N−6
∑

14. 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

15. 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.

16. Practical reaction calculations
Let’s say we are interested in calculating the
following reaction energies from QM
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Reaction 1
N2 (g)+3H2 (g) → 2NH3(g)
Reaction 2
C(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

17. NANO266
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ΔH0
f ,298(M) = E(M)+ ZPE(M)+[H298(M)− H0 (M)]
− E(Xz )+[H298(Xz )− H0 (Xz )]{ }
z
atoms
∑ + ΔH0
f ,298
z
atoms
∑ (Xz )

18. 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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19. 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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xk
yl
zm
= Zi xi
k
yi
l
zi
m
i
atoms
∑ − ψ(r) xi
k
yi
l
zi
m
i
electrons
∑∫ ψ(r)dr

20. 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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