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7/30/2019 1 Basic Introduction of Computational Chemistry
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Basic Introduction ofComputational Chemistry
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What is Computational Chemistry?
Definition
Some examples
The relation to the real world
Cartoons from "scientist at work; work, scientists, work!"
http://web.archive.org/web/20070125125308/http:/www.nearingzero.net/work/work.htmlhttp://web.archive.org/web/20070125125308/http:/www.nearingzero.net/work/work.html7/30/2019 1 Basic Introduction of Computational Chemistry
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Computational Chemistry is
A branch of chemistry
That uses equations encapsulating the behavior ofmatter on an atomistic scale and
Uses computers to solve these equations
To calculate structures and properties
Of molecules, gases, liquids and solidsTo explain or predict chemical phenomena.
See also:
Wikipedia,
http://www.chem.yorku.ca/profs/renef/whatiscc.html [11/10/2010]
http://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.html[11/10/2010]
3
http://www.chem.yorku.ca/profs/renef/whatiscc.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/compchem.htmlhttp://www.chem.yorku.ca/profs/renef/whatiscc.html7/30/2019 1 Basic Introduction of Computational Chemistry
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Computational Chemistry is
Including:
Electron dynamicsTime independent ab initiocalculations
Semi-empiricalcalculations
Classical moleculardynamics
Embedded models
Coarse grained models
Not including:
Quantum chromo-dynamics
Calculations on Jellium
Continuum models
Computational fluid
dynamicsData mining
Rule based derivations
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Computational Chemistry is
To explain or predict chemical phenomena:
Phenomenon is any observable occurrenceTherefore computational chemistry has to connect withpractical/experimental chemistry
In many cases fruitful projects live at the interface betweencomputational and experimental chemistry because:
Both domains criticize each other leading to improvedapproaches
Both domains are complementary as results that areinaccessible in the one domain might be easilyaccessible in the other
Agreeing on the problem helps focus the invested effort
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Where do you start?
Selection of energy expressionsHartree-Fock / Density Functional Theory
Moller-Plesset Perturbation TheoryCoupled ClusterQuantum Mechanics / Molecular MechanicsMolecular Mechanics
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You start with
Everything starts with an energy expression
Calculations either minimize to obtain:the ground state
equilibrium geometries
Or differentiate to obtain properties:
Infra-red spectraNMR spectra
Polarizabilities
Or add constraints toOptimize reaction pathways (NEB, string method, ParaReal)
The choice of the energy expression determines theachievable accuracy
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Energy expressions NWChem supports
Effective 1-Electron Models
Hartree-Fock and Density Functional TheoryPlane wave formulation
Local basis set formulation
Correlated Models
Mller-Plesset Perturbation TheoryCoupled Cluster
Combined Quantum Mechanical / MolecularMechanics (QM/MM)
Molecular Mechanics
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Hartree-Fock & Density Functional Theory I
Plane wave & Local basis
The energy expression is derived from a singledeterminant wave function approximation
Replace the exchange with a functional to go from
Hartree-Fock to DFTUse different basis sets for different problems
Plane waves for infinite condensed phase systems
Local basis sets for finite systems
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D=iCiCi
F=h+J(D)+K(D)
+(D)Vxc(D)drFC=C
Hartree-Fock & Density Functional Theory II
Plane wave & Local basis
Minimize energy withrespect to Ci and IIterative processcycling until Self-Consistency
GivesThe total energy E
The molecular orbitals Ci
The orbital energies i
10C
C
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Hartree-Fock & Density Functional Theory III
Local Basis Sets
Largest quantities arethe density, Fock,
overlap, 1-electronmatrices
Memory needed O(N2)Replicated data O(N2)per node
Distributed data O(N2) forwhole calculation
Memory requirements Computational Complexity
Main cost is theevaluation of the 2-
electron integralsTakes O(N2)-O(N4) work
O(N4) for small systems
O(N2) in the large N limit
For large N the linearalgebra becomesdominant at O(N3)
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Hartree-Fock & Density Functional Theory IV
Plane waves
Largest quantities arethe density, Fock,
overlap, 1-electronmatrices
Memory needed O(N2)Replicated data O(N2)per node
Distributed data O(N2) forwhole calculation
Memory requirements ComputationalComplexity
Main cost stems fromthe Fourier transforms
For small systems andlarge processor countsdominated by FFTscosting O(N2*ln(N))
For large systems the
non-local operator andorthogonalization areimportant costing O(N3)
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Mller-Plesset Perturbation Theory I
Assumes that electron correlation effects are small
The Hartree-Fock energy is the 1st order correctedenergy
The 2nd and 3rd order corrected energy can becalculated from the 1st order corrected wavefunction (2N+1 rule)
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Mller-Plesset Perturbation Theory II
The zeroth order energy isthe sum of occupiedorbital energiesThe first order energy is theHartree-Fock energy
The energy correctiongives an estimate of the
interaction of the Hartree-Fock determinant with allsingly and doublesubstituted determinants
The total energy scalescorrectly with system size
It is ill defined if the HOMOand LUMO aredegenerate
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2
2
2
,
,
HF MP
MP
i j occ i j r sr s virt
E E E
ij rsE
http://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theory [11/19/2010]
http://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theoryhttp://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theoryhttp://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theoryhttp://en.wikipedia.org/wiki/Moller-Plesset_perturbation_theory7/30/2019 1 Basic Introduction of Computational Chemistry
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Mller-Plesset Perturbation Theory III
The MO basis 2-electron integrals
require the dominantamount of storageThis takes O(N4) storage
Can be reduced toO(N3) by treating theintegral in batches
Memory requirements Computationalcomplexity
Transforming the integralsrequires a summationover all basis functions for
every integralThis takes O(N5) work
If not all transformedintegrals are stored thenthere is an extra cost of
calculating all theintegrals for every batchat O(N4)
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Coupled Cluster I
Sums some corrections to infinite order
Involves singly, doubly, triply-substituted determinantsSimplest form is CCSD
Often necessary to include triply-substituteddeterminants (at least perturbatively), i.e. CCSD(T)
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Coupled Cluster II
The wavefunction is
expressed in anexponential form
The operatorTcontainssingle and doublesubstitution operators
with associatedamplitudes
The vector equation:Solve top two lines for theamplitudes di
r, dijrs
The bottom line gives thetotal energy
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1 2
,
,
00
T HF
r rs
i r i ij r s i j
i occ i j occ
r virt r s virt
rs T T HF
ij
r T T HF
i
HF T T HF
e
T T T
d a a d a a a a
e He
e He
Ee He
http://en.wikipedia.org/wiki/Coupled_cluster[11/19/2010]
http://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_clusterhttp://en.wikipedia.org/wiki/Coupled_cluster7/30/2019 1 Basic Introduction of Computational Chemistry
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Coupled Cluster III
The main objects tostore are thetransformed 2-electron
integrals and theamplitudes
This costs O(N4) storage
Local memory depends
on tile sizes and level oftheoryCCSD O(nt
4)
CCSD(T)O(nt6)
Memory requirements Computational complexity
The main cost are thetensor contractions
For CCSD they can beformulated so that theytake O(N6) work
For CCSD(T) theadditional perturbative
step dominates at O(N7
)
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QM/MM Models I
Describe local chemistry under the influence of an
environmentQuantum region treated with ab-initio method of choice
Surrounded by classical region treated with molecularmechanics
Coupled by electrostatics, constraints, link-atoms, etc.
Crucial part is the coupling of different energyexpressions
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QM/MM Models II
The scheme we are
considering is an hybridscheme (not anadditive scheme likeONIOM)
This scheme is valid for
any situation that theMM and QM methodscan describe, thelimitation lies in theinterface region
20
internal external
external
, , ,
, , ,
( ')d '
'
QM MM
QM QM QM
I iIQM
I MM I MM i QMI I i
vdW
E E r R E r R
E E r R E R
Z ZZ rE r
R r R R
E R
http://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.html [11/20/2010]http://www.salilab.org/~ben/talk.pdf [11/20/2010]http://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.html [11/20/2010]
http://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.htmlhttp://www.salilab.org/~ben/talk.pdfhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.chem.umn.edu/groups/gao/qmmm_notes/LEC_HYB.htmlhttp://www.salilab.org/~ben/talk.pdfhttp://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.htmlhttp://www.ccl.net/cca/documents/dyoung/topics-orig/qmmm.html7/30/2019 1 Basic Introduction of Computational Chemistry
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QM/MM Models III
Dominated by thememory requirementsof the QM method
See chosen QMmethod, N is now thesize of the QM region
Memory requirements Computational complexity
Dominated by thecomplexity of the QMmethod
See chosen QM method,N is now the size of theQM region
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Molecular Mechanics I
Energy of system expressed in terms of relative
positions of atomsThe parameters depend on the atom and theenvironment
A carbon atom is different than a oxygen atom
A carbon atom bound to 3 other atoms is different from one
bound to 4 other atomsA carbon atom bound to hydrogen is different from onebound to fluorine
Etc.
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Molecular mechanics II
Energy terms (parameters)
Bond distances (kAB, r
0
AB)Bond angles (kABC, 0ABC)
Dihedral angles(vABCD;n,
0ABCD)
Van der Waals interactions(AB,AB)
Electrostatic interactions (AB)
The parameters are defined inspecial files
For a molecule the parametersare extracted and stored in the
topology fileThis force field is only valid nearequilibrium geometries
23
201
2
,
201
2
, ,
01;2
, , ,
1
2 12 6,
12
,
1 cos
AB AB AB
A B
ABC ABC ABC
A B C
ABCD n ABCD ABCD
A B C D n
AB AB
A B AB AB
AB A B
A B AB
E k r r
k
v n
r r
q q
r
W. Cornell, P. Cieplak, C. Bayly, I. Gould, K. Merz, Jr., D. Ferguson, D. Spellmeyer, T. Fox, J. Caldwell, P.Kollman, J. Am. Chem. Soc., (1996) 118, 2309, http://dx.doi.org/10.1021/ja955032e
http://dx.doi.org/10.1021/ja955032ehttp://dx.doi.org/10.1021/ja955032e7/30/2019 1 Basic Introduction of Computational Chemistry
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Molecular Mechanics III
Main data objects are theatomic positions
Storage O(N)
Memory requirements Computational complexity
Most terms involve localinteractions between atomsconnected by bonds, thesecost O(N) work
Bond terms
Angle terms
Dihedral angle terms
The remaining two termsinvolve non-local interactions,
cost at worst O(N2
), butimplemented using the particlemesh Ewald summation it costsO(N*log(N))
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Summarizing methods
Method Memory Complexity Strengths
MolecularDynamics
O(N) O(N*ln(N)) Conformational sampling/Freeenergy calculations
Hartree-Fock/DFT O(N2) O(N3) Equilibrium geometries, 1-
electron properties, also excited
states
Mller-Plesset O(N4) O(N5) Medium accuracy correlationenergies, dispersive interactions
Coupled-Cluster O(N4) O(N6)-O(N7) High accuracy correlation
energies, reaction barriers
QM/MM * * Efficient calculations on complex
systems, ground state andexcited state properties
25
* Depends on the methods combined in the QM/MM framework.
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What properties might you want to
calculate?
Energies
Equilibrium geometries
Infrared spectra
UV/Vis spectra
NMR chemical shifts
Reaction energies
Thermodynamics
Transition states
Reaction pathways
Polarizabilities
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Energy evaluations
Having chosen an energy expression we cancalculate energies and their differences
Bonding energies
Isomerization energies
Conformational change energies
Identification of the spin state
Electron affinities and ionization potentials
For QM methods the wavefunction and for MMmethods the partial charges are also obtained. Thisallows the calculation of
Molecular potentials (including docking potentials)
Analysis of the charge and/or spin distribution
Natural bond order analysis
Multi-pole moments
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Gradient evaluations
Differentiating the energy with respect to the nuclear
coordinates we get gradients which allows thecalculation ofEquilibrium and transition state geometries
Forces to do dynamics
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Hessian evaluations I
Differentiating the energy twice with respect to the
nuclear coordinates gives the Hessian which allowscalculatingThe molecular vibrational modes and frequencies (if allfrequencies are positive you are at a minimum, if one isnegative you are at a transition state)
Infra-red spectra
Initial search directions for transition state searches
Hessians are implemented for the Hartree-Fock andDensity Functional Theory methods
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Hessian evaluation II
In the effective 1-electron models aperturbed Fock and
density matrix needsto be stored for everyatomic coordinate
The memory requiredis therefore O(N3)
Memory requirements Computational complexity
To compute theperturbed densitymatrices a linear system
of dimension O(N2) hasto be solved for everyatomic coordinate
The number ofoperations needed is
O(N5)
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Magnetic properties, e.g. NMR
The chemical shift is calculated as a mixed second
derivative of the energy with respect to the nuclearmagnetic moment and the external magnetic field.
Often the nuclear magnetic moment is treated as aperturbation
Note thatThe paramagnetic and diamagnetic tensors are notrotationally invariant
The total isotropic and an-isotropic shifts are rotationallyinvariant
Requires the solution of the CPHF equations at O(N4)
31
http://en.wikipedia.org/wiki/Chemical_shift[11/23/2010]
l i bili i
http://en.wikipedia.org/wiki/Chemical_shifthttp://en.wikipedia.org/wiki/Chemical_shift7/30/2019 1 Basic Introduction of Computational Chemistry
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Polarizabilities
Adding an external electric field to the Hamiltonian
and differentiating the energy with respect to thefield strength gives polarizabilityHartree-Fock and DFT
CCSD, CCSDT
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Li I
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Linear response I
Adding a time dependent electric field to the
Hamiltonian, substituting it in the dependentSchrodinger equation, and expanding the time-dependent density in a series an equation for the firstorder correction can be obtained.
This expression is transformed from the time domain
to the frequency domain to obtain an equation forthe excitation energies
Solving this equation for every root of interest has acost of the same order a the corresponding Hartree-Fock or DFT calculation, both in memory
requirements as in the computational complexity.
33
http://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdf
Li II
http://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdfhttp://www.physik.fu-berlin.de/~ag-gross/articles/pdf/MG03.pdf7/30/2019 1 Basic Introduction of Computational Chemistry
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Linear response II
The equations haveN
occ*N
virtsolutions
Note that the vectorsare normalized butdifferently so than yourusual wavefunction
The orbital energydifference is a mainterm in the excitationenergy
In the case of pureDFT with large
molecules most of theintegrals involving Fxcvanish as this is a localkernel
34
http://www.tddft.org/TDDFT2008/lectures/IT2.pdf
* *
,
,
2
1 2
1 2
1 0
0 1
1
,
ia jb ij ab a i H xc
ia jb H xc
xc
A B X X
B A Y Y
X X Y Y
A ia F F jb
B ia F F jb
fF r r
r r
EOM CCSD/CCSD(T)
http://www.tddft.org/TDDFT2008/lectures/IT2.pdfhttp://www.tddft.org/TDDFT2008/lectures/IT2.pdf7/30/2019 1 Basic Introduction of Computational Chemistry
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EOM CCSD/CCSD(T)
A Coupled Clustermethod for excitedstates
Depends on the groundstate cluster amplitudes
Memory andcomputationalcomplexity similar tocorresponding CoupledCluster method
35
,
,
,
, , ,
T HF
k k
k k
s
k i s i
i s
st
k ij s t i j
i j s t
R e
R r
r a a
r a a a a
T T HF HF
k k ke He R E R
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Which methods do you pick?
Factors in decision making
Available functionality
Accuracy aspects
Th d i i ki f t
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The decision making factors
The method of choice for a particular problem
depends on a number of factors:The availability of the functionalityThe accuracy or appropriateness of the method
The memory requirements and computational cost of themethod
37
A il bl f ti lit
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Available functionality
38
MM HF/DFT MP2 CC QM/MMEnergy
Gradients
Hessians
Polarizabilities
Excited states
NMR
Gradients and Hessians can always be obtained by
numerical differentiation but this is slow.
http://www.nwchem-sw.org/
A i t f th d
http://www.nwchem-sw.org/http://www.nwchem-sw.org/http://www.nwchem-sw.org/http://www.nwchem-sw.org/7/30/2019 1 Basic Introduction of Computational Chemistry
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Appropriateness of method
Too complex a question to answer here in generalFor example consider bond breaking
Molecular Mechanics cannot be used as this is explicitly notpart of the energy expressionHF/DFT can be used but accuracy limited near transition states(unrestricted formulation yields better energies, but often spin-contaminated wavefunctions)Moller-Plesset cannot be used as near degeneracies causesingularities
CCSD or CCSD(T) can be used with good accuracyQM/MM designed for these kinds of calculations of course withthe right choice of QM region
So check your methods before you decide, if necessaryperform some test calculations on a small problem.Often methods that are not a natural fit have been
extended, e.g. dispersion corrections in DFTBottom line: Know your methods!
39
W i
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Wrapping up
Further readingAcknowledgements
Questions
Further reading
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Further reading
D. Young, Computational Chemistry: A Practical
Guide for Applying Techniques to Real WorldProblems, Wiley-Interscience, 2001, ISBN:0471333689.
C.J. Cramer, Essentials of Computational Chemistry:Theories and Models, Wiley, 2004, ISBN:0470091827.
R.A.L. Jones Soft Condensed Matter, Oxford
University Press, 2002, ISBN:0198505892.
41
Further reading (Books)
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Further reading (Books)
GeneralD. Young, Computational Chemistry: A Practical Guide for Applying Techniques to RealWorld ProblemsC.J. Cramer, Essentials of Computational Chemistry: Theories and Models
F. Jensen, Introduction to Computational Chemistry
Molecular dynamicsFrenkel & Smit, Understanding Molecular Simulation
Allen & Tilldesley, Computer Simulation of Liquids
Leach, MolecularModelling: Principles & Applications
Condensed phaseR.M. Martin, Electronic Structure: basic theory and practical methods
J. Kohanoff, Electronic Structure Calculations for Solids and Molecules
D. Marx, J. Hutter, Ab Initio Molecular Dynamics
Quantum chemistryOstlund & Szabo, Modern Quantum Chemistry
Helgaker, Jorgensen, Olsen, Molecular Electronic Structure Theory
McWeeny, Methods of Molecular Quantum Chemistry
Parr & Yang, Density Functional Theory of Atoms & MoleculesMarques et al, Time-Dependent Density Functional Theory
OtherJanssen & Nielsen, Parallel Computing in Quantum Chemistry
Shavitt & Bartlett, Many-Body Methods in Chemistry and Physics
42
Acknowledgement
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Acknowledgement
This work was produced using EMSL, a national
scientific user facility sponsored by the Departmentof Energy's Office of Biological and EnvironmentalResearch and located at Pacific Northwest NationalLaboratory.
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Questions?
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Questions?