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. Beyond the HF
Approximation
Shyue Ping Ong

2. Overview
In this lecture, we will only touch on conceptual
underpinnings of how correlation is included,
without going too much into the details of the
methods. Most of the advanced methods are far
too computationally expensive and limited to
small system sizes, which makes them less
useful for the materials scientist at this time. It
suffices that you understand them at a conceptual
level, and if you are interested (or they become
more accessible in future), there are many
excellent works on the subject.
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3. Limitations of HF
All correlation, other than exchange, is ignored in
HF
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Ecorr = Eexact − EHF

4. So how might one improve on HF?
HF utilizes a single
determinant. An obvious
extension is
Types of correlation
• Dynamic correlation: From
ignoring dynamic electron-electron
interactions. Typically c0 is much
larger than other coefficients.
• Non-dynamical correlation:
Arises from single determinant
nature of HF. Several ci with
similar magnitude as c0.
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ψ = c0ψHF +c1ψ1 +c2ψ2 +…
Degenerate frontier orbitals cannot be
represented with single determinant!

5. Multiconfiguration SCF
Optimize orbitals for a combination of
configurations (orbital occupations)
• Configuration state function (CSF): molecular spin state and occupation
number of orbitals
• Active space: orbitals that are allowed to be partially occupied (based on
chemistry of interest)
Scaling
CAS: Complete active space (CASSCF)
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# of singlet CSFs for m electrons in n orbitals =
n!(n +1)!
m
2
!
"
#
$
%
&!
m
2
+1
!
"
#
$
%
&! n −
m
2
!
"
#
$
%
&! n −
m
2
+1
!
"
#
$
%
&!

6. Full Configuration Interaction (CI)
CASSCF calculation of all orbitals and all electrons
Best possible calculation within limits of basis set
For small systems, can be used to benchmark other
methods
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Full CI
Infinite
Basis Set
Exact
solution to
Schodinger
Equation

7. Limiting excitations in CI
CIS (CI singles)
• Used for excited
states
• No use for ground
states
CID (CI doubles)
CISD (CI singles
doubles)
• N6 scaling
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8. Møller–Plesset perturbation theory
Treats exact Hamiltonian as a perturbation on sum of one-
electron Fock operators
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H = H(0)
+ λV = fi
i
∑ + λV
Expanding the ground-state eigenfunctions and eigenvalues as Taylor series in λ,
ψ =ψ(0)
+ λψ(1)
+ λ2
ψ(2)
+…
a = a(0)
+ λa(1)
+ λ2
a(2)
+…
where ψ(k)
=
1
k!
∂k
ψ
∂λk
and a(k)
=
1
k!
∂k
a
∂λk
H ψ = a ψ
∴(H(0)
+ λV) λk
ψ(k)
∑ = λk
a(k)
λk
ψ(k)
∑∑
By equating powers of λ and imposing normalization, we can derive
a(k)
, which are the kth
order corrections to a(0)
.

9. MPnTheory
MP1 is simply HF
MP2
• Second-order energy correction
• Analytic gradients available
• N5 scaling
MPn > 2
• No analytic gradients available
• > 95% of electron correlation at n=4.
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10. Issues in PerturbationApproach
Perturbation theory works best when perturbation
is small (convergence of Taylor series expansion)
• In MPn, perturbation is full electron-electron repulsion!
MPn is not variational! (possible for correlation to
be larger than exact, but in practice, basis set
limitations cause errors in opposite direction)
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11. Coupled-Cluster
Full-CI wave function can be described as
If we truncate at T2
CCSD(T)
• Includes single/triples coupling term
• Analytic gradients and second derivatives available
• Gold-standard in most quantum chemistry calculations
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ψ = eT
ψHF
where T = T1 + T2 + T3 +…+ Tn is the cluster operator
ψCCSD = (1+(T1 + T2 )+
(T1 + T2 )2
2!
+…)ψHF

12. Practical Considerations
Basis set convergence is a bigger problem for correlated
calculations
Performance vs Accuracy
• HF < MP2 ~ MP3 < CCD < CISD < QCISD ~CCSD < MP4 < QCISD(T) ~ CCSD(T)
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Highly expensive, but accurate!

13. Relative accuracy of variational methods –
Dissociation of HF (hydrogen fluoride)
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Foresman, J. B.; Frisch, Ae. Exploring Chemistry With Electronic Structure Methods: A Guide to Using
Gaussian; Gaussian, 1996.

14. Ionization potentials
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Errors introduced via the
truncation of the space at
different excitation levels and
the effect of this on the IP.
The two systems are oxygen
in an aug-cc-pVQZ basis and
neon in an aug-cc-pVTZ basis
set. The dashed lines indicate
the difference in the total
energy of each species
compared to the FCI limit, and
the solid lines indicate the
error in the IP with each
species truncated at the given
excitation level.
J. Chem. Phys. 132, 174104 (2010); http://dx.doi.org/
10.1063/1.3407895

15. Relative computational cost – C5H12
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Foresman, J. B.; Frisch, Ae. Exploring Chemistry With Electronic Structure Methods: A Guide to Using
Gaussian; Gaussian, 1996.

16. Bond lengths
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Cramer, C. J. Essentials of Computational Chemistry:
Theories and Models; 2004.

17. Parameterized methods
G2/G3 theory for accurate thermochemistry (errors < 4 kcal / mol)
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18. References
Essentials of Computational Chemistry: Theories
and Models by Christopher J. Cramer
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